Molecular Spectroscopy: A Quantum Chemistry Approach 3527344616, 9783527344611

Table of contents :
Cover
Molecular Spectroscopy:

A Quantum Chemistry Approach
© 2019
Contents to Volume 1
Contents to Volume 2
Preface
Volume 1
1 Interpretability Meets Accuracy in Computational
Spectroscopy: The Virtual Multifrequency Spectrometer
2 Excited State Dynamics in NTChem
3 Quantum Chemistry for Studying Electronic Spectroscopy
and Dynamics of Complex Molecular Systems
4 Theoretical and Experimental Molecular Spectroscopy
of the Far-Ultraviolet Region
5 Weight Averaged Anharmonic Vibrational Calculations:
Applications to Polypeptide, Lipid Bilayers, and Polymer
Materials
6 Chiral Recognition by Molecular Spectroscopy
7 Quantum Approach of IR Line Shapes of Carboxylic Acids
Using the Linear Response Theory
8 Theoretical Calculations Are a Strong Tool in the
Investigation of Strong Intramolecular Hydrogen Bonds
9 Spectral Simulation for Flexible Molecules in Solution with
Quantum Chemical Calculations
10 Combination Analysis ofMatrix-Isolation Spectroscopy and
DFT Calculation
Volume 2
11 Role of Quantum Chemical Calculations in Elucidating
Chemical Bond Orientation in Surface Spectroscopy
12 Dynamic and Static Quantum Mechanical Studies of
Vibrational Spectra of Hydrogen-Bonded Crystals
13 Quantum Mechanical Simulation of Near-Infrared Spectra:
Applications in Physical and Analytical Chemistry
14 Local Modes of Vibration: The Effect of Low-Frequency
Vibrations
15 Intra- and Intermolecular Vibrations of Organic
Semiconductors and Their Role in Charge Transport
16 Effects of Non-covalent Interactions on Molecular and
Polymer Individuality in Crystals Studied by THz
Spectroscopy and Solid-State Density Functional Theory
17 Calculation of Vibrational Resonance Raman Spectra
of Molecules Using Quantum Chemistry Methods
18 Density Functional Theoretical Study on Surface-Enhanced
Raman Spectroscopy of CH2/NH2 Wagging Modes in p–?
ConjugatedMolecules on Noble Metal Surfaces
19 Modeling Plasmonic Optical Properties Using
Semiempirical Electronic Structure Calculations
Index

Citation preview

Molecular Spectroscopy

Molecular Spectroscopy A Quantum Chemistry Approach

Edited by Yukihiro Ozaki Marek Janusz Wójcik Jürgen Popp

Volume 1

Molecular Spectroscopy A Quantum Chemistry Approach

Edited by Yukihiro Ozaki Marek Janusz Wójcik Jürgen Popp

Volume 2

Editors

Kwansei Gakuin University Department of Chemistry 2-1 Gakuen Kobe Sanda Campus 669-1337 Sanda, Hyogo Japan

All books published by Wiley-VCH are carefully produced. Nevertheless, authors, editors, and publisher do not warrant the information contained in these books, including this book, to be free of errors. Readers are advised to keep in mind that statements, data, illustrations, procedural details or other items may inadvertently be inaccurate.

Marek Janusz Wójcik

Library of Congress Card No.:

Jagiellonian University Department of Chemistry Ingardena 3 30-060 Kraków Poland

applied for

Yukihiro Ozaki

British Library Cataloguing-in-Publication Data

A catalogue record for this book is available from the British Library.

Jürgen Popp

Leibniz-Institut für Photonische Technol Albert-Einstein-Str. 9 07745 Jena Germany Cover Image: Kindly provided by Yukihiro Ozaki, Kwansei Gakuin University, Japan; © Sirinarth Mekvorawuth/EyeEm/Getty Images (Background)

Bibliographic information published by the Deutsche Nationalbibliothek

The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available on the Internet at . © 2019 Wiley-VCH Verlag GmbH & Co. KGaA, Boschstr. 12, 69469 Weinheim, Germany All rights reserved (including those of translation into other languages). No part of this book may be reproduced in any form – by photoprinting, microfilm, or any other means – nor transmitted or translated into a machine language without written permission from the publishers. Registered names, trademarks, etc. used in this book, even when not specifically marked as such, are not to be considered unprotected by law. Print ISBN: 978-3-527-34461-1 ePDF ISBN: 978-3-527-81462-6 ePub ISBN: 978-3-527-81460-2 oBook ISBN: 978-3-527-81459-6 Cover Design: Wiley Typesetting SPi Global, Chennai, India Printing and Binding

Printed on acid-free paper 10 9 8 7 6 5 4 3 2 1

v

Contents to Volume 1 Preface xiii 1

Interpretability Meets Accuracy in Computational Spectroscopy: The Virtual Multifrequency Spectrometer 1 Vincenzo Barone and Cristina Puzzarini

1.1 1.2 1.2.1 1.2.2 1.2.2.1 1.2.2.2 1.2.2.3 1.2.2.4 1.2.3 1.3 1.3.1 1.3.2 1.3.3 1.3.4 1.4 1.4.1

Introduction 2 The Virtual Multifrequency Spectrometer 3 The VMS Framework 3 The VMS Framework: Spectroscopies and Theoretical Background 4 Rotational Spectroscopy 4 Vibrational Spectroscopy 5 Vibronic Spectroscopy 8 Magnetic Spectroscopy 10 The VMS Framework: Quantum Chemical Methods 12 The VMS Framework at Work 14 Rotational Spectroscopy 14 Vibrational Spectroscopy 16 Vibronic Spectroscopy 19 Magnetic Spectroscopy 22 The VMS Framework: Applications 25 A Complete Spectroscopic Characterization: The Glycine Case Study 26 Vibrational Spectroscopy of a Chiral Molecule: The Methyloxirane Case Study 28 From Molecular Structure to Electronic Spectrum: The Pyrimidine Case Study 30 EPR Spectrum in Different Solvents: The TOAC Case Study 35 Conclusions 37 Acknowledgments 38 References 38

1.4.2 1.4.3 1.4.4 1.5

2

Excited State Dynamics in NTChem 43 Takehiro Yonehara, Noriyuki Minezawa, and Takahito Nakajima

2.1 2.2

NTChem 43 Electron Dynamics in a Molecular Aggregate Under a Light Field 44

vi

Contents

2.2.1 2.2.1.1 2.2.1.2 2.2.1.3 2.2.2 2.2.2.1 2.2.2.2 2.2.2.3 2.2.2.4 2.2.3 2.2.3.1 2.2.3.2 2.3 2.3.1 2.3.1.1 2.3.1.2 2.3.1.3 2.3.2 2.3.2.1 2.3.2.2 2.3.2.3 2.3.3 2.3.3.1 2.3.3.2 2.3.4 2.3.4.1 2.3.4.2 2.3.4.3 2.4

Theoretical Background 45 Overview of Electron Dynamics Calculation 45 Schemes Based on Multistate Multi-electron Wavefunctions 46 Schemes Based on the Electron Density Matrix 48 Group Diabatic Fock Scheme 49 Group Diabatic Fock Matrix 49 Transformation of Observable, Fock, and Density Matrices 50 Time Propagation of Density Matrix in GD Representation 51 Initial Density Matrix: Local Excitation and Electron Filling 51 Numerical Demonstration 52 Charge Migration Dynamics: NPTL–TCNE Dimer 53 Unpaired Electron Dynamics: 20-mer Ethylene 54 Trajectory Surface Hopping Molecular Dynamics Simulation 59 Theoretical Background 60 Conical Intersection 60 Nonadiabatic Molecular Dynamics Simulation 61 Electronic Structure Method 62 Theoretical Method 63 TSH Approach 63 LR-TDDFT 63 Nonadiabatic Coupling 64 Example: Photodynamics of Coumarin 65 Computational Details 66 Results and Discussion 66 Future Direction 69 Effects of Environment 69 Accurate Description of S0 /S1 Crossing 69 Spin-Forbidden Transition 70 Summary 70 Acknowledgment 72 References 72

3

Quantum Chemistry for Studying Electronic Spectroscopy and Dynamics of Complex Molecular Systems 79 Hyun Woo Kim, Kyungmin Kim, Soo Wan Park, and Young Min Rhee

3.1

Overview of Quantum Chemical Tools for Studying Electronic Spectroscopy 80 Examples: Quantum Chemical Calculations of Simple Systems 88 Spectral Line Shape 93 Examples: Complex Systems 95 Single Chromophore Case: Retinal in Rhodopsin 95 Isolated Retinal: Early Studies 96 Retinal in Protein: Absorption Predictions 97 MD Simulations with QM/MM 99 Rhodopsin Variants 100 2D Electronic Spectra 100 Multiple Chromophore Case: Bacteriochlorophylls in LH2 102

3.2 3.3 3.4 3.4.1 3.4.1.1 3.4.1.2 3.4.1.3 3.4.1.4 3.4.1.5 3.4.2

Contents

3.4.2.1 3.4.2.2 3.4.2.3

Excited State Energies 103 Coupling Between Excited States 104 Dynamics Simulations 105 References 107

4

Theoretical and Experimental Molecular Spectroscopy of the Far-Ultraviolet Region 119 Masahiro Ehara and Yusuke Morisawa

4.1 4.2 4.2.1 4.2.2 4.3 4.3.1 4.3.2 4.3.3 4.4

Introduction 119 Method 120 Theory and Computational Details 120 ATR-FUV Spectroscopy 123 Results and Discussion 124 FUV Spectra of n-Alkanes in the Liquid and Solid Phases 124 Amides 130 Nylons 136 Summary 141 References 142

5

Weight Averaged Anharmonic Vibrational Calculations: Applications to Polypeptide, Lipid Bilayers, and Polymer Materials 147 Kiyoshi Yagi, Hiroki Otaki, Pai-Chi Li, Bo Thomsen, and Yuji Sugita

5.1 5.2 5.2.1 5.2.2 5.2.3 5.2.4 5.3 5.3.1 5.3.2 5.3.3 5.4

Introduction 147 Method 149 Weight Average Method 149 Electronic Structure Calculations for a Cluster and Local Modes Vibrational Quasi-degenerate Perturbation Theory 151 Computational Procedure 153 Applications 153 Pentapeptide: SIVSF 153 Sphingomyelin Bilayer 157 Hydration of Nylon 6 160 Concluding Remarks and Outlook 163 Acknowledgments 164 References 165

6

Chiral Recognition by Molecular Spectroscopy 171 Magdalena Pecul and Joanna Sadlej

6.1 6.2

Introduction 171 The Physical Manifestation of Optical Activity in Chiroptical Spectroscopic Methods: Theory of the Chiroptical Properties 172 General Background 173 Calculations of VCD Spectra 173 Practical Simulation of VCD Spectra 173 Calculations of ROA Spectra 175

6.2.1 6.2.2 6.2.2.1 6.2.3

150

vii

viii

Contents

6.2.4 6.2.5 6.2.5.1 6.2.6 6.2.6.1 6.2.6.2 6.2.6.3 6.2.6.4 6.3 6.3.1

6.3.1.1 6.3.1.2 6.3.1.3 6.3.1.4 6.3.1.5 6.3.2 6.3.2.1 6.3.2.2 6.3.3 6.4

Calculations of ECD and CPL Spectra 176 Calculations of CPL Spectra 177 Spin-Forbidden Circular Dichroism and Circularly Polarized Phosphorescence (CPP) 177 Electronic Structure Methods, Basis Set Requirements, and Program Packages 179 Vibrational Optical Activity 180 Electronic Structure Methods: ECD, CPL, and CPP 181 The Basis Set Requirements: VCD, ROA, ECD, and CPL 181 Software 182 Selected Case Studies 182 Taking into Account Chemical Environment, Conformational Flexibility, and Vibrational Corrections in the Calculation of Chiroptical Spectra 182 Matrix-Isolation CD Spectra 183 Modeling of Solvent Effects in the Chiroptical Spectra 183 Environment-Induced CD and CPL Activity 184 Conformational Averaging of the Chiroptical Spectra 185 The Concept of Robustness 185 Modeling of Anharmonicity in VCD and ROA Spectra 186 Computational Approaches to Anharmonicity 186 The Role of Anharmonicity in VCD and ROA Spectra: Examples 186 The Induced VCD Intensity Monitored by Experimental or Calculated VCD Spectra 187 Perspective 189 References 191

7

Quantum Approach of IR Line Shapes of Carboxylic Acids Using the Linear Response Theory 199 Paul Blaise, Olivier Henri-Rousseau, and Adina Velcescu

7.1 7.2

Introduction 199 The Characteristics of the Infrared Spectra of Hydrogen-Bonded Species 199 Recall on the General Features of the IR Spectra of Systems Containing Hydrogen Bonds 199 The Linear Response Theory 201 Conditions for a Quantitative Theory 202 The Strong Coupling Theory of Anharmonicity 203 Spectral Density 203 The Model for IR Spectra of Centrosymmetric Dimers 207 The Line Shape 208 Limit Situations 210 Examples 211 Acetic Acid 211 Formic Acid 211 Conclusion 212 References 212

7.2.1 7.2.2 7.2.3 7.3 7.3.1 7.3.2 7.3.3 7.3.4 7.3.5 7.3.5.1 7.3.5.2 7.4

Contents

8

Theoretical Calculations Are a Strong Tool in the Investigation of Strong Intramolecular Hydrogen Bonds 215 Poul Erik Hansen, Aneta Jezierska, Jarosław J. Panek, and Jens Spanget-Larsen

8.1

Introduction and Definition of Types of Intramolecular Hydrogen Bonds 215 Definitions of Strong Intramolecular Hydrogen Bonds 215 Calculation of Structural Parameters 217 Hydrogen Bond Strength 218 Calculation of Energies 220 Tautomerism 223 2D Approaches to Hydrogen Bond Potentials 223 Potential Energy and Free Energy Surfaces 223 Calculation of IR Spectra of Strongly Hydrogen-Bonded Systems 226 The Harmonic Approximation 226 Going Beyond the Harmonic Approximation 228 Simplified, Static Procedures 228 Advanced Calculations on Small Systems: Malonaldehyde and Acetylacetone 229 Larger Systems 232 Car–Parrinello Molecular Dynamics Simulations 232 NMR 234 Introduction 234 Calculation of OH Chemical Shifts 235 Estimation of Ring Current and Anisotropy Effects on OH Chemical Shifts 236 Estimation of Other Effects on OH Chemical Shifts 237 Calculation of Chemical Shifts in Charged Systems 237 Calculation of Deuterium Isotope Effects on Nuclear Shieldings 238 Jameson Approach 238 Car–Parrinello and Two-Dimensional Sampling of Chemical Shifts 239 Principal Component Analysis 240 Solvent Effects 240 Conclusions 241 References 242

8.2 8.3 8.4 8.5 8.6 8.6.1 8.6.2 8.7 8.7.1 8.7.2 8.7.2.1 8.7.2.2 8.7.3 8.7.3.1 8.8 8.8.1 8.8.2 8.8.3 8.8.4 8.8.5 8.8.6 8.8.6.1 8.8.6.2 8.9 8.10 8.11

9

Spectral Simulation for Flexible Molecules in Solution with Quantum Chemical Calculations 253 Yukiteru Katsumoto

9.1 9.2

Introduction 253 Selection of the Calculation Level for Spectral Simulations of Flexible Molecules 254 Simulation of IR Spectra Observed in Solution Phase 257 Competition Between Intramolecular and Intermolecular Interactions 261 Conformational Diversity and Solvation in the Vibrational Spectrum 265

9.3 9.4 9.5

ix

x

Contents

9.6 9.7

Conformational Diversity in the Vibrational Circular Dichroism Spectrum 269 Conformational Diversity in the Electronic Circular Dichroism 272 References 275

10

Combination Analysis of Matrix-Isolation Spectroscopy and DFT Calculation 279 Nobuyuki Akai

10.1 10.2 10.3 10.4 10.4.1 10.4.2 10.4.3 10.4.4 10.5 10.5.1 10.5.2

Introduction 279 Matrix-Isolation Method 280 Adoption of Theory and Basis Set 282 Conformational Analysis 283 1,2-Dichloroethane 283 Chlorobenzaldehyde 285 Vanillin 286 Excitation Light 289 Identification for Unknown Species 290 Rare Tautomer of Cytosine 290 Reversible Isomerization Between Triplet and Singlet Species from 1,8-Diaminonaphthalene 291 Spectrum and Structure of Molecular Complex or Cluster 293 Photoinduced Transient Species 294 Hydroquinone 294 Lowest Electronic Excited Triplet State 296 Conclusion 297 References 298

10.6 10.7 10.7.1 10.7.2 10.8

Contents to Volume 2 11

Role of Quantum Chemical Calculations in Elucidating Chemical Bond Orientation in Surface Spectroscopy 303 Dennis K. Hore

12

Dynamic and Static Quantum Mechanical Studies of Vibrational Spectra of Hydrogen-Bonded Crystals 327 Mateusz Z. Brela, Marek Boczar, Łukasz Boda, and Marek Janusz Wójcik

13

Quantum Mechanical Simulation of Near-Infrared Spectra: Applications in Physical and Analytical Chemistry 353 Krzysztof B. Be´c, Justyna Grabska, Christian W. Huck, and Yukihiro Ozaki

14

Local Modes of Vibration: The Effect of Low-Frequency Vibrations 389 Emil Vogt, Anne S. Hansen, and Henrik G. Kjaergaard

Contents

15

Intra- and Intermolecular Vibrations of Organic Semiconductors and Their Role in Charge Transport 425 Andrey Yu. Sosorev, Ivan Yu. Chernyshov, Dmitry Yu. Paraschuk, and Mikhail V. Vener

16

Effects of Non-covalent Interactions on Molecular and Polymer Individuality in Crystals Studied by THz Spectroscopy and Solid-State Density Functional Theory 459 Feng Zhang, Keisuke Tominaga, Michitoshi Hayashi, and Takashi Nishino

17

Calculation of Vibrational Resonance Raman Spectra of Molecules Using Quantum Chemistry Methods 497 Julien Guthmuller

18

Density Functional Theoretical Study on Surface-Enhanced Raman Spectroscopy of CH2 /NH2 Wagging Modes in p–𝛑 Conjugated Molecules on Noble Metal Surfaces 537 De-Yin Wu, Yan-Li Chen, Yuan-Fei Wu, and Zhong-Qun Tian

19

Modeling Plasmonic Optical Properties Using Semiempirical Electronic Structure Calculations 575 Chelsea M. Mueller, Rebecca L.M. Gieseking, and George C. Schatz Index 597

xi

v

Contents to Volume 1 Preface xiii 1

Interpretability Meets Accuracy in Computational Spectroscopy: The Virtual Multifrequency Spectrometer 1 Vincenzo Barone and Cristina Puzzarini

2

Excited State Dynamics in NTChem 43 Takehiro Yonehara, Noriyuki Minezawa, and Takahito Nakajima

3

Quantum Chemistry for Studying Electronic Spectroscopy and Dynamics of Complex Molecular Systems 79 Hyun Woo Kim, Kyungmin Kim, Soo Wan Park, and Young Min Rhee

4

Theoretical and Experimental Molecular Spectroscopy of the Far-Ultraviolet Region 119 Masahiro Ehara and Yusuke Morisawa

5

Weight Averaged Anharmonic Vibrational Calculations: Applications to Polypeptide, Lipid Bilayers, and Polymer Materials 147 Kiyoshi Yagi, Hiroki Otaki, Pai-Chi Li, Bo Thomsen, and Yuji Sugita

6

Chiral Recognition by Molecular Spectroscopy 171 Magdalena Pecul and Joanna Sadlej

7

Quantum Approach of IR Line Shapes of Carboxylic Acids Using the Linear Response Theory 199 Paul Blaise, Olivier Henri-Rousseau, and Adina Velcescu

8

Theoretical Calculations Are a Strong Tool in the Investigation of Strong Intramolecular Hydrogen Bonds 215 Poul Erik Hansen, Aneta Jezierska, Jarosław J. Panek, and Jens Spanget-Larsen

vi

Contents

9

Spectral Simulation for Flexible Molecules in Solution with Quantum Chemical Calculations 253 Yukiteru Katsumoto

10

Combination Analysis of Matrix-Isolation Spectroscopy and DFT Calculation 279 Nobuyuki Akai

Contents to Volume 2 11

Role of Quantum Chemical Calculations in Elucidating Chemical Bond Orientation in Surface Spectroscopy 303 Dennis K. Hore

11.1 11.2 11.2.1 11.2.2 11.2.3 11.3 11.4

Introduction 303 Vibrational Sum-Frequency Generation Spectroscopy 304 Basic Experimental Details 304 Molecular View 308 SFG Phase Measurement 310 Determination of Bond Polarity 311 Quantum Chemical Calculations for Modeling the Molecular Hyperpolarizability 313 Example 318 Summary 320 Acknowledgments 321 References 321

11.5 11.6

12

Dynamic and Static Quantum Mechanical Studies of Vibrational Spectra of Hydrogen-Bonded Crystals 327 Mateusz Z. Brela, Marek Boczar, Łukasz Boda, and Marek Janusz Wójcik

12.1 12.2 12.3 12.3.1

Introduction 327 Historical and Theoretical Background 329 Applications 332 Vibrational Spectra of Strong Hydrogen Bonds in Oxalic Acid Dihydrate Crystal with Isotopic Substitution Effects 333 Simulations of Infrared Spectra of Crystalline Vitamin C 336 Study on Proton Dynamics of Strong Hydrogen Bonds in Aspirin Crystals 338 The Hydrogen Bond Dynamics in Crystalline Tropolone 343 Summary and Perspectives 345 Acknowledgment 346 References 346

12.3.2 12.3.3 12.3.4 12.4

Contents

13

Quantum Mechanical Simulation of Near-Infrared Spectra: Applications in Physical and Analytical Chemistry 353 Krzysztof B. Be´c, Justyna Grabska, Christian W. Huck, and Yukihiro Ozaki

13.1 13.2

Introduction 353 Overview of the Current Progress in Computational NIR Spectroscopy 355 Basic Molecules 355 Investigations of Intermolecular Interactions and Biomolecules 363 Connecting the Link Between Theoretical and Analytical NIR Spectroscopy 368 Miscellaneous Applications of NIR Spectra Simulation 373 Accurate NIR Studies of Single Mode Anharmonicity by Solving 1D Schrödinger Equation 375 Conclusions 383 References 383

13.2.1 13.2.2 13.2.3 13.2.4 13.2.5 13.3

14

Local Modes of Vibration: The Effect of Low-Frequency Vibrations 389 Emil Vogt, Anne S. Hansen, and Henrik G. Kjaergaard

14.1 14.2 14.2.1 14.2.1.1 14.2.2 14.2.2.1 14.2.3 14.2.3.1 14.3 14.3.1 14.3.2 14.4 14.4.1 14.4.2 14.4.3 14.4.3.1 14.4.3.2 14.4.3.3 14.4.3.4 14.4.3.5

Introduction 389 Local Mode (LM) Models 391 1D LM Model 391 OH Stretching in Sulfuric Acid 393 2D LM Model 394 CH Stretching in Butadiene 397 3D LM Model 400 CH Stretching in Propane 401 Effect of Low-Frequency Modes 402 Methyl Torsional Mode 403 Intermolecular Modes in Bimolecular Complexes 406 Local Mode Intensities 408 Wavefunctions 409 Dipole Moment Function 411 Absolute Intensities 411 Sulfuric Acid: Higher Overtones 411 Dimethylamine: An Intense First Overtone 412 Water Dimer: A Weak First Overtone 413 Effect of Methyl Torsion 414 Effect of Intermolecular Modes in Bimolecular Complexes 416 Summary 417 Appendix 14.A Deriving the LM Hamiltonian 418 References 420

14.5

vii

viii

Contents

15

Intra- and Intermolecular Vibrations of Organic Semiconductors and Their Role in Charge Transport 425 Andrey Yu. Sosorev, Ivan Yu. Chernyshov, Dmitry Yu. Paraschuk, and Mikhail V. Vener

15.1 15.2

Introduction 425 Theoretical Treatment of Coupling Between Intra- and Intermolecular Vibrations in Low-Frequency Region 426 Computations of IR and Raman Spectra by Solid-State DFT 427 Low-Frequency Vibrations of Crystals Formed by Structurally Close Molecules 429 The Role of Inter- and Intramolecular Vibrations in Charge Transport 434 Local and Nonlocal Electron–Phonon Coupling 434 Three Mechanisms Underlying Impact of Electron–Phonon Interaction on Charge Transport in Organic Semiconductors 436 Computational and Experimental Approaches to Electron–Phonon Coupling in Organic Semiconductors 439 Electron–Phonon Coupling in Various Organic Semiconductors 440 Impact of Low-Frequency Vibrations on Charge Transport in Fn -TCNQ Crystal Family 443 Conclusions 449 Acknowledgments 450 References 450

15.2.1 15.2.2 15.3 15.3.1 15.3.2 15.3.3 15.3.4 15.4 15.5

16

Effects of Non-covalent Interactions on Molecular and Polymer Individuality in Crystals Studied by THz Spectroscopy and Solid-State Density Functional Theory 459 Feng Zhang, Keisuke Tominaga, Michitoshi Hayashi, and Takashi Nishino

16.1 16.2 16.3 16.4

A Historical Review of Phonon Modes 459 Theoretical Representation of Non-covalent Interactions 462 A Mode Decomposition Method 465 Interpretation of the Nature of Optical Phonon Modes Controlled by Prototypical Non-covalent Interactions 470 C60 471 Anthracene 473 Adenine 475 𝛼-Glycine and l-Alanine 476 Frequency Sequences of Intermolecular Translations and Librations and Intramolecular Vibrations 478 Mixing Between Intermolecular and Intramolecular Vibrations 479 Application of the DFT-D Method in a Material System scPLA 481 Experimental Evidence Supporting the Mode Assignments 484 Conclusion 486 References 487

16.4.1 16.4.2 16.4.3 16.4.4 16.4.5 16.4.6 16.5 16.6 16.7

Contents

17

Calculation of Vibrational Resonance Raman Spectra of Molecules Using Quantum Chemistry Methods 497 Julien Guthmuller

17.1 17.2

Introduction 497 Theory of Resonance Raman Scattering, Approximations, and Quantum Chemistry Methods 499 Sum-Over-State Formulation of the Vibrational Raman Intensities 499 Normal Raman Scattering in the Double Harmonic Approximation 501 Resonance Raman Intensities 503 Time-Dependent Formulation of Resonance Raman Intensities 506 Resonance Polarizability Derivatives 508 Transform Theory and Simplified Φe Approximation 509 Calculation of the Franck–Condon Overlap Integrals 511 Quantum Chemistry Methods to Calculate Resonance Raman Spectra 515 Illustrative Applications 518 Using the Short-Time Approximation 518 Including Franck–Condon Vibronic Couplings 518 Considering Several Electronic Excited States in Resonance 521 Including Herzberg–Teller Vibronic Couplings 523 Conclusions 523 References 525

17.2.1 17.2.2 17.2.3 17.2.4 17.2.5 17.2.6 17.2.7 17.2.8 17.3 17.3.1 17.3.2 17.3.3 17.3.4 17.4

18

Density Functional Theoretical Study on Surface-Enhanced Raman Spectroscopy of CH2 /NH2 Wagging Modes in p–𝛑 Conjugated Molecules on Noble Metal Surfaces 537 De-Yin Wu, Yan-Li Chen, Yuan-Fei Wu, and Zhong-Qun Tian

18.1 18.2 18.3 18.4 18.5 18.6 18.6.1 18.6.2 18.6.3 18.6.4 18.7 18.8 18.9

Introduction 537 Brief Review of Wagging Vibrational Raman Spectra Normal Mode Analysis 541 Density Functional Theoretical Calculations 546 Raman Intensity 548 Modeling Molecules 549 Aniline 549 Para-substituted Anilines 554 Benzyl Radicals and Its Anion 557 Terminal Olefins 558 Chemical Enhancement Effect 562 The Reason of Broadbands 565 Conclusions 567 Acknowledgments 568 References 568

538

ix

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Contents

19

Modeling Plasmonic Optical Properties Using Semiempirical Electronic Structure Calculations 575 Chelsea M. Mueller, Rebecca L.M. Gieseking, and George C. Schatz

19.1 19.2 19.3

Introduction 575 INDO/CI vs. TD-DFT: Absorption Spectra of Ag Nanoclusters 577 Higher-Order Excitations: The Role of Double Excitations in Absorption 579 Identification of Quadrupolar Plasmonic Excited States 580 Electrochemical Charge Transfer 583 Voltage Effects and the Chemical Mechanism of Surface-Enhanced Raman Scattering 584 Conclusions 590 Acknowledgment 591 References 591

19.4 19.5 19.6 19.7

Index 597

xiii

Preface The purpose of this book is to outline the state-of-the-art quantum chemical approach to molecular spectroscopy. Over the last two decades or so, molecular spectroscopy has made remarkable progress; several novel spectroscopies such as terahertz spectroscopy, tip-enhanced Raman scattering (TERS), and far-ultraviolet (FUV) spectroscopy in condensed phase have emerged. Moreover, existing spectroscopies have shown prominent advances in this period. The advances in spectroscopies lie in the development of theory, instruments, spectral analysis, and applications. In spectral analysis quantum chemical approach is particularly important. It is useful not only for spectral analysis such as band assignments but also for studies of structure, reactions, and physical and chemical properties of molecules. This book aims at making a strong bridge between molecular spectroscopy and quantum chemistry. For the last quarter of a century quantum chemistry has been extensively used for various spectroscopies such as vibrational spectroscopy, electronic spectroscopy, and nuclear magnetic resonance spectroscopy. However, one cannot find a good book that connects spectroscopy and quantum chemistry. This book may be the first one that explains comprehensively how quantum chemical approach can be applied to molecular spectroscopy. It covers FUV spectroscopy, UV–visible spectroscopy, near-infrared (NIR) spectroscopy, IR spectroscopy, far-IR spectroscopy/terahertz spectroscopy, Raman spectroscopy, and NMR spectroscopy. Almost all kinds of molecular spectroscopies are presented in this book. For quantum chemical approaches various new calculation methods are introduced. The recent rapid progress in supercomputers has made it possible to utilize these new methods. For example, anharmonic quantum chemical calculations are becoming popular due to advances in supercomputers. In applications many chapters deal with studies of hydrogen bonding and inter- and intramolecular interactions. In this book, we invited front runners from many countries who are currently very active in the molecular spectroscopy–quantum chemistry field. This book is very useful not only for chemistry but also for applied physics, material sciences, biosciences, and industrial applications. It is suitable for molecular spectroscopists who are interested in quantum chemistry and quantum chemists who are interested in molecular spectroscopy. We hope this book will find many readers among students at graduate level as well as researchers and engineers in academia and industry.

xiv

Preface

Last but not the least, we would be most grateful if the book can inspire readers to use novel quantum chemistry approaches for molecular spectroscopy studies and/or to attempt to develop new approaches by themselves. In closing, we would like to thank Dr. Lifen Yang, Ms. Shirly Samuel, and Mr. Jayakumar Ramprasad of Wiley for their continuous efforts in publishing this book. April 2019

Yukihiro Ozaki, Sanda, Japan Marek Janusz Wójcik, Krakow, Poland Jürgen Popp, Jena, Germany

1

1 Interpretability Meets Accuracy in Computational Spectroscopy: The Virtual Multifrequency Spectrometer Vincenzo Barone 1 and Cristina Puzzarini 2 1 Scuola Normale Superiore, Piazza dei Cavalieri 7, 56126 Pisa, Italy 2

Università di Bologna, Dipartimento di Chimica “Giacomo Ciamician”, Via Selmi 2, 40126 Bologna, Italy

The virtual multifrequency spectrometer (VMS), under active development in our laboratories over the last few years, is shortly described in this chapter by means of selected spectroscopic techniques and a few representative case studies. The VMS project aims to offer an answer to the following question: is it possible to turn strongly specialized research in the field of computational spectroscopy into robust and user-friendly aids to experiments and industrial applications? VMS contains a number of tools devised to increase the interaction between researchers with different background and to push toward new frontiers in computational chemistry. As a matter of fact, the terrific advancements in computational spectroscopy and the wide availability of computational and analytic tools are paving the route toward the study of problems that were previously too difficult or impossible to be solved and let imagine even more ambitious targets for fundamental and applied research. Under such circumstances, a robust, flexible, and user-friendly tool can allow for moving data analysis toward a proactive process of strategic decisions and actions. This chapter starts from these premises, and it proposes a perspective for a new virtual platform aimed at integrating past developments in theory, algorithms, and software with new workflow management and visualization tools. After a short review of the underlying theoretical framework, the features of the principal tools available in the current version of VMS for a selection of spectroscopic techniques are addressed in some details. Next, four case studies are presented, thus aiming to illustrate possible applications of VMS to systems of current interest for both fundamental and applied research. These applications convincingly show that even if several extensions of the software are planned or already under development, VMS represents a powerful and user-friendly tool for both computational and experimentally oriented spectroscopists.

Molecular Spectroscopy: A Quantum Chemistry Approach, First Edition. Edited by Yukihiro Ozaki, Marek Janusz Wójcik, and Jürgen Popp. © 2019 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2019 by Wiley-VCH Verlag GmbH & Co. KGaA.

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1 Interpretability Meets Accuracy in Computational Spectroscopy

1.1 Introduction Spectroscopic techniques provide a wealth of qualitative and quantitative information on the chemical and physical–chemical properties of molecular systems in a variety of environments. Nowadays, sophisticated experimental techniques, mainly based on vibrational, electronic, and resonance spectroscopies, allow studies under various environmental conditions and in a noninvasive fashion [1, 2]. Particularly effective strategies are obtained when different spectroscopic techniques are combined together and further supported and/or integrated by computational approaches. Indeed, not only the spectral analysis is seldom straightforward, but also molecular spectra do not provide direct information on molecular structures, properties, and dynamics [3, 4]. The challenges can be posed by the intrinsic properties and complexity of the system and/or caused by thermal or environmental effects, whose specific roles are not easy to separate and evaluate. In such a context, computational spectroscopy is undoubtedly a powerful and reliable tool to unravel the different contributions to the spectroscopic signal and understand the underlying physical phenomena [5, 6]. However, direct vis-à-vis comparisons between experimental and computed spectroscopic data are still far from being standard. To fill this lack, a virtual multifrequency spectrometer (VMS) (http://dreamslab.sns.it/vms/) has been implemented with the aim of providing a user-friendly access to the latest developments of computational spectroscopy, also to nonspecialists [7–11]. As it will be better explained in the following section, VMS integrates state-of-the-art computational implementations of different spectroscopies with a powerful graphical user interface (GUI) [12], which offers an invaluable aid in preorganizing and displaying the computed spectroscopic information. For the sake of clarity, it should be noted that several codes incorporate implementation of spectroscopic properties at different levels of theory together with graphic engines. However, none of these tools offer the characteristics that should be considered mandatory for state-of-the-art computational spectroscopy (e.g. rigorous treatment of anharmonicity, vibronic contributions, etc.) and/or for flexible user-friendly graphical tools. In particular, it should emphasize the uniqueness of VMS in incorporating both general utilities needed by experimentally oriented scientists (e.g. conversion of theoretical quantities to experimental observables, manipulation of several spectra at the same time, etc.) and advanced tools for theoreticians and developers (e.g. resonance Raman [RR] spectra). The aim of the present chapter is to provide an overview of the VMS software, thus focusing on its peculiarities and unique features. The chapter is organized as follows. In the following section, a brief summary of the general machinery of the VMS program and of the main technical aspects will be provided. This will be followed by a short introduction of the theoretical background for the selected spectroscopies (e.g. rotational, vibrational, vibronic, and magnetic) and of the corresponding quantum chemical (QC) requirements. Then, the current status of VMS will be presented in some detail with specific reference to rotational, vibrational, vibronic, and magnetic spectroscopy. Finally, applications will be illustrated with the help of four case studies, which will allow the capabilities of VMS to be demonstrated. Some general considerations will conclude the chapter.

1.2 The Virtual Multifrequency Spectrometer

1.2 The Virtual Multifrequency Spectrometer VMS is a tool that integrates a wide range of computational and experimental spectroscopic techniques and aims at predicting and analyzing different types of molecular spectra as well as disclosing the static and dynamic physical–chemical information they contain [7]. VMS is mainly composed of two parts, namely, VMS-Comp, which provides access to the latest developments in the field of computational spectroscopy, and VMS-Draw, which provides a powerful GUI for an intuitive interpretation of theoretical outcomes and a direct prediction or comparison to experiment (http://dreamslab.sns.it/vms/) [7]. The spectroscopies supported by VMS are electron spin resonance (ESR), nuclear magnetic resonance (NMR), rotational (microwave [MW]), infrared (IR), vibrational circular dichroism (VCD), nonresonant Raman (nRR), resonance Raman, Raman optical activity (ROA), resonance Raman optical activity (RROA), electronic one-photon absorption (OPA) (i.e. UV–vis) and one-photon emission (OPE) (i.e. fluorescence), electronic circular dichroism (ECD), and circularly polarized luminescence (CPL). 1.2.1

The VMS Framework

Data importer

Raw data

Computations

Anharmonic vibrational spectroscopy

Jobs management

Cw-ESR Vibrationally spectroscopy resolved electronic Rotational spectroscopy spectroscopy

Spectra processing Analysis tools

Comparison tools

Figure 1.1 The framework of the virtual multifrequency spectrometer.

Big data tools

Preprocessing

Inputs creation

Postprocessing

Virtual multi-frequency spectrometer

The framework of the VMS program is graphically shown in Figure 1.1 [7]. The key feature of VMS is to provide a user-friendly access to computational spectroscopy tools also to nonspecialists. VMS integrates a powerful GUI, VMS-Draw, which offers an invaluable aid in the pre- and post-processing stages [12]. This permits a direct way to present the information produced by in vitro and in silico experiments, thus allowing the user to focus the attention on the underlying physical–chemical features without being concerned with technical details. VMS-Draw is interfaced with VMS-Comp [8, 9, 13], which

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1 Interpretability Meets Accuracy in Computational Spectroscopy

takes care of QC computations of the required spectroscopic parameters and all high-performance computing (HPC) aspects [7, 12]. Both VMS-Draw and VMS-Comp modules are either fully embedded with the Gaussian package [14] or loosely bound to other suites of QC programs, such as CFOUR [15]. In the last case, general input–output facilities as well as ad hoc scripts that permit effective interactions with other electronic structure codes than Gaussian have been developed or are still under development (see, for example, Ref. [10]). Overall, VMS has access to almost all computational models and to properties that are not yet available in the reference QC Gaussian suite. In addition to the large availability of QC methods and properties, VMS has the unique feature of allowing state-of-the-art computational spectroscopy studies driven by a flexible user-friendly graphical tool that furthermore includes those general utilities needed by experimentally oriented scientists (e.g. manipulations of several spectra at the same time, spectral normalization, etc.) and advanced tools for theoreticians and developers (e.g. resonance Raman spectroscopy). In the following sections, the theoretical background and the QC requirements for quantitative spectral prediction/analysis of selected spectroscopies are presented together with a description of the spectral simulation facilities and of the corresponding results. 1.2.2 The VMS Framework: Spectroscopies and Theoretical Background The complete list of the spectroscopies available within the VMS software has been given above. In this chapter, we limit ourselves to the discussion of a selection of spectroscopies, namely, the rotational, vibrational, vibronic, and magnetic spectroscopies, for which we provide a short description of the theoretical background. 1.2.2.1

Rotational Spectroscopy

The terms of the effective rotational Hamiltonian are the pure rotational and centrifugal distortion contributions, which describe the rotational energy levels for a given vibrational state, with the ground state usually being the one of interest. While a complete treatment can be found in the literature (see, for example, Ref. [16]), here, we recall just the key aspects of interest. The basic rotational Hamiltonian, within the semirigid rotor approximation, can be written as Hrot = HR + Hqcd + Hscd + · · ·

(1.1)

where Hqcd and Hscd are the quartic and sextic centrifugal terms, respectively. The dots refer to the possibility of including higher-order centrifugal contributions. HR is the rigid rotor Hamiltonian: ∑ eq B𝜏 J2𝜏 (1.2) HR = 𝜏

where

eq B𝜏 eq

has been defined as follows:

B𝜏 =

ℏ2 eq 2hcI𝜏𝜏

(1.3)

1.2 The Virtual Multifrequency Spectrometer

where 𝜏 refers to the inertial axis. From a computational point of view, the equilibrium rotational constants are straightforwardly obtained from the geometry optimization. Even if the equilibrium contribution to rotational constants is the most important, the effect of molecular vibrations cannot be neglected when aiming at a quantitative description of rotational spectra. Therefore, the term describing the dependence of the rotational constants on the vibrational quantum numbers should be incorporated in Eq. (1.3), and equilibrium rotational constants should be replaced by the effective rotational constants that contain the contributions beyond the rigid rotor harmonic oscillator (RRHO) approximation. Their effects on rotational motion can be conveniently described by means of vibrational perturbation theory (VPT), and we refer the reader to, for example, Refs. [16, 17] for a detailed treatment. While there are no corrections at the first order in VPT, at the second order (VPT2), the expression becomes [18]: ( ) N ∑ di eq v 𝛼i,𝜏 vi + (1.4) B𝜏 = B𝜏 − 2 i=1 where the superscript v denotes a specific vibrational state and the sum runs on all fundamental vibrational modes i, with vi being the corresponding quantum number and di its degeneracy order. The 𝛼i,𝜏 values are the so-called vibration–rotation interaction constants and contain three contributions: the first one is a corrective term related to the moment of inertia, the second one is due to the Coriolis interactions, and the last is an anharmonic correction. Therefore, from a computational point of view, anharmonic force field (FF) calculations are required to correct the equilibrium rotational constants for vibrational effects. The quartic centrifugal distortion Hamiltonian is defined as 1∑ 𝜏 JJJJ (1.5) Hqcd = 4 𝜏𝜂𝜍𝜚 𝜏𝜂𝜍𝜚 𝜏 𝜂 𝜍 𝜚 where the tensor 𝜏𝜏𝜂𝜍𝜚 depends only on the harmonic part of the potential energy surface (PES). To obtain the quartic centrifugal distortion parameters actually employed, further contact transformations with purely rotational operators (thus diagonal in the vibrational quantum numbers) are then required. An analogous expression can be written for the sextic centrifugal distortion term Hscd , and the computation of the corresponding sextic centrifugal distortion constants involves harmonic, anharmonic, and Coriolis perturbation terms. Therefore, from a computational point of view, anharmonic force field computations are needed for their determination. To relate the experimental parameters to combinations of 𝜏𝜏𝜂𝜍𝜚 (𝜏𝜏𝜂𝜍𝜚𝜖𝜄 in the case of sextics), it is necessary to further completely reduce the Hamiltonian. Different results are then obtained depending on the reduction chosen; see, for example, Refs. [16, 17, 19]. 1.2.2.2

Vibrational Spectroscopy

For the simulation of vibrational spectra, a purely vibrational Hamiltonian (Hvib ) is commonly used. In the framework of VPT2, which is based on Taylor expansions of the harmonic potential (V ), vibrational (Ev ) energies, and vibrational

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wavefunction, up to the second order [20], the vibrational Hamiltonian is defined as follows: N N N 1∑ 1 ∑ 1 ∑ 2 2 Hvib = 𝜔 (p + qi ) + k qqq + k qqq q 2 i=1 i i 6 i,j,k=1 ijk i j k 24 i,j,k,l=1 ijkl i j k l √ N ∑ eq ∑ 𝜔i 𝜔k + B𝜏 𝜁ij,𝜏 𝜁kl,𝜏 qpq p +U (1.6) 𝜔j 𝜔l i j k l 𝜏 i,j,k,l=1 For asymmetric tops, at the VPT2 level, the energy (Em , in cm−1 ) of a given vibrational state m is given by Em = E0 +

N ∑

vm i 𝜔i +

i=1

[ ] 1 m m m 𝜒ij vm v + + v ) (v i j j 2 i i,j=1 N ∑

(1.7)

where vm is the number of quanta associated with mode i in state m and 𝜔i the i corresponding harmonic wavenumber. E0 is the zero-point vibrational energy, which is defined as follows: [ ] N N N ∑ ∑ kiik kjjk kijk 2 𝜔i ∑ kiijj + + − E0 = 2 i,j=1 32 i,j,k=1 32𝜔k 48(𝜔i + 𝜔j + 𝜔k ) i=1 ] [ N−1 N ∑∑ ∑ Beq (𝜔i − 𝜔j )2 𝜏 2 (1.8) {𝜁ij,𝜏 } 1− − 4 𝜔i 𝜔j 𝜏 i=1 j=i+1 In Eq. (1.7), 𝜒 is the anharmonicity contributions matrix, with its elements given by 5k 2 ∑ (8𝜔i 2 − 3𝜔j 2 )kiij 16𝜒ii = kiiii − iii − 3𝜔i 𝜔j (4𝜔i 2 − 𝜔j 2 ) j=1 N

2

(1.9)

j≠i

4𝜒ij = kiijj − +

2𝜔i kiij

2

−

(4𝜔i 2 − 𝜔j 2 ) (4𝜔j 2 − 𝜔i 2 ) [ N ∑ 2𝜔k (𝜔i 2 + 𝜔j 2 − 𝜔k 2 )kijk 2 Δijk

k=1 k≠i,j

+

2𝜔j kijj 2

4(𝜔i 2 + 𝜔j 2 ) ∑ 𝜔i 𝜔j

𝜏

− −

kiii kijj 𝜔i kiik kjjk

−

kjjj kiij

]

𝜔j

𝜔k

eq

B𝜏 {𝜁ij,𝜏 }2

(1.10)

where Δijk = 𝜔i 4 + 𝜔j 4 + 𝜔k 4 − 2(𝜔i 2 𝜔j 2 + 𝜔i 2 𝜔k 2 + 𝜔j 2 𝜔k 2 )

(1.11)

Transition energies from the ground state 𝜈m are therefore straightforwardly obtained from Eqs. (1.7) and (1.8) as Em − E0 difference. The intensities for a broad range of spectroscopies at the VPT2 level can be obtained by referring to a generic property P, which can depend on either the normal coordinates (q) or their conjugate momenta (p): P = P(0) + P(1) + P(2)

(1.12)

1.2 The Virtual Multifrequency Spectrometer

where P(0) = Peq + s0

N ∑

Pi (a†i + Sai )

(1.13)

Pij qj (a†i + Sai )

(1.14)

i=1

P(1) = s1

N N ∑ ∑ i=1 j=1

P(2) = s2

N N N ∑ ∑∑

Pijk qj qk (a†i + Sai )

(1.15)

i=1 k=1 j=1

In equations above, a†i and ai are the creation and annihilation operators, respectively; s0 , s1 , and s2 are constant factors; and S corresponds to a sign (i.e. it represents the multiplication by +1 or −1). The function of Eq. (1.12) is then used to obtain analytic formulas for the transition moments up to three quanta [21–25] and can be simply related to the property of interest by identifying the variables in Eqs. (1.12)–(1.15) with the actual quantities, as exemplified in Figure 1.2. The electric (𝛍) and magnetic (m) dipoles and the polarizability (𝛂) are used in IR, VCD, and Raman intensities, respectively, whereas the electric dipole–magnetic dipole optical activity (G′ ) and the electric dipole–electric quadrupole (A) tensors also enter the ROA intensities [13]. From a quick inspection of Eqs. (1.9) and (1.10), it is evident that for the VPT2 energies, the denominator might become exceedingly small. This situation leads to the so-called Fermi resonances (FRs), which can be distinguished in type I (𝜔i ≈ 2𝜔j ) and type II (𝜔i ≈ 𝜔j + 𝜔k ). Indeed, a near resonance can be sufficient to obtain unphysical results due to an excessive contribution from anharmonicity. This is a well-known issue of VPT2, which has been extensively studied in the literature [16, 26–39] and needs to be correctly addressed for a successful application of this method. A major difficulty lies in the definition of the resonance conditions. In the literature, several efficient identification processes have been P

P0

Pi

Pji

Pjki

s0

μ

μeq

∂μ ∂qi

∂ 2μ ∂qi qj

∂ 3μ ∂qi qj qk

1

m

0

Mi

∂M ∂qi

∂ 2M ∂qi qj

ih

ih

ih

2

2

2 2

α

αeq

∂α ∂qi

∂ 2α ∂qi qj

∂ 3α ∂qi qj qk

1

G′′

G′eq

∂G′ ∂qi

∂ 2 G′ ∂qi qj

∂ 3 G′ ∂qi qj qk

1

A

Aeq

∂A ∂qi

∂ 2A ∂qi qj

∂ 3A ∂qi qj qk

1

2

2 2 2

s1

s2

1

1

2 2

6 2

1

1

2 2

6 2

1

1

2 2

6 2

1

1

2 2

6 2

S

+1

–1 +1 +1

+1

Figure 1.2 Equivalence relations between the model property P and actual properties.

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presented [26, 28–30, 33, 36–40]. Then, those terms that have been identified as resonant should be removed from the perturbative treatment for the calculation of the energy. This approach is named deperturbed VPT2 (DVPT2). To take into account the missing terms, an ad hoc variational step, which reintroduces the previously discarded terms, can be performed using the DVPT2 vibrational energies as references. We refer to the overall resulting procedure as generalized VPT2 (GVPT2). An alternative approach has been proposed by Kuhler, Truhlar, and Isaacson and denoted degeneracy-corrected PT2 (DCPT2); this is based on replacing the potentially resonant terms with nonresonant forms derived from a model system considering only the two states involved [29]. A shortcoming of this approach is a potential inaccuracy for each replaced terms far from resonance; this can be partially corrected by introducing a switch function that will mix the DCPT2 and VPT2 results for each potentially resonant term, thus leading to the so-called hybrid DCPT2-VPT2 (HDCPT2) [37]. Finally, other types of resonance should be mentioned; these are collectively denoted as Darling–Dennison resonances (DDRs) [26, 30, 32, 39–45] and are commonly treated through a variational procedure, analogous to that used for FRs. In the following, we always refer to GVPT2, and this includes corrections to both Fermi and DDRs. The problem of resonances in intensity calculations has been more scarcely addressed in the literature, and limited is the number of programs supporting them [13, 23, 27, 39, 44]. Since they are related to the mechanical anharmonicity (wavefunction), it is possible to use the analysis for the energy shortly addressed above also for the transition moments. However, an important difference is the impact of DDRs, which can lead to incorrect intensities. Depending on the protocol applied for the definition of DDRs, it may be necessary to complement it with an ad hoc test targeted to handle the most critical cases (for instance, of near-equal energies) [13, 46]. The eigenvectors (LE ) of the matrix diagonalized to introduce the variational contribution of resonances to energies are used to project the deperturbed transition moments on the variationally corrected states following the procedure described in Ref. [23]: T VPT2 ⟨P⟩var I,F = LE ⟨P⟩I,F

(1.16)

In the VMS framework, all the required strategies for a correct derivation of the intensities are implemented. 1.2.2.3

Vibronic Spectroscopy

A reliable description of molecular vibrations in ground and excited electronic states is at the heart of an accurate simulation of vibrational modulation (hereafter vibronic in a broad sense) effects in UV–vis spectra and their chiroptical (e.g. ECD) counterparts. Indeed, experimental spectra originate from the convolution of vibronic transitions, thus usually leading to highly asymmetric band-shaped spectra at both low and high resolution. From a theoretical point of view, a rigorous inclusion of rovibrational effects beyond the standard rigid rotor harmonic oscillator approximation can be performed for small molecules, whereas for larger systems, feasible approaches are currently based on neglecting the rovibrational coupling and on the Franck–Condon (FC) principle at the harmonic level. Within this framework, a general sum-over-state expression

1.2 The Virtual Multifrequency Spectrometer

for the vibronic contributions to the transition between two electronic states has been derived for OPA, OPE, ECD, and CPL [47, 48], and it has been recently extended to resonance Raman, its chiroptical counterpart RROA, and also spin-forbidden transitions [49], with the corresponding intensity being expressed by the following equation: ∑∑ I = 𝛼𝜔𝛽 𝜌𝛾 [⟨𝜓m |dA |𝜓n ⟩ ⋅ {⟨𝜓m |dB |𝜓n ⟩}∗ ]𝛿(𝜔mn − 𝜔) (1.17) m

n

where the sums run over all possible initial m and final n vibronic states, with 𝜌𝛾 being the Boltzmann population; 𝛿 is the Dirac function, and the asterisk is used to denote the conjugate of the dB,mn matrix element. In the equation above, I is a general experimental observable related to intensity (e.g. for OPA, I is the molar absorption coefficient 𝜖(𝜔), and for OPE, I is the energy emitted by one mole per second Iem ∕Nn ), and 𝜔 is the incident frequency. For OPA, OPE, ECD, or CPL, I is obtained by replacing 𝛼, 𝛽, 𝛾, dA,mn , and dB,mn according to 10πNA , 𝛽 = 1, 𝛾 = m, dA,mn = dB,mn = 𝜇mn 3𝜀0 ln(10)ℏc 2 OPE ∶ 𝛼 = , 𝛽 = 4, 𝛾 = n, dA,mn = dB,mn = 𝜇mn 3𝜀0 c3 40πNA ECD ∶ 𝛼 = , 𝛽 = 1, 𝛾 = m, dA,mn = 𝜇mn , dB,mn = ℑ(mmn ) 3𝜀0 ln(10)ℏc 8 , 𝛽 = 4, 𝛾 = n, dA,mn = 𝛍mn , dB,mn = ℑ(mmn ) CPL ∶ 𝛼 = 3𝜀0 c4 In the formulas above, NA is the Avogadro constant, 𝜀0 is the vacuum permittivity, 𝜇mn is the electric transition dipole moment between the vibronic states m and n, and ℑ(mmn ) is the imaginary part of the magnetic transition dipole moment between the vibronic states m and n. For a more detailed description of the theoretical background for calculating I for OPA, OPE, ECD, and CPL, the reader is referred to Refs. [47, 48]. In practical terms, in order to apply Eq. (1.17), additional approximations need to be introduced. Since analytical forms of transition moments are not known, they are usually expanded in power series with respect to the mass-weighted Cartesian normal coordinates Q about the equilibrium geometry of one of the electronic states: ) N ( ∑ 𝜕dX,mn e dX,mn (Q) = dX,mn (Qeq ) + +··· (1.18) 𝜕Qi eq i=1 OPA ∶ 𝛼 =

where X can be either A or B. The expansion of Eq. (1.18) is usually truncated to the first two terms: the zeroth- and first-order first terms, which correspond to the FC approximation [50, 51] and Herzberg–Teller (HT) contribution [52], respectively. The VMS implementation includes both FC and HT terms, thus allowing the proper treatment of the leading contributions in both strongly and weakly allowed transitions. Furthermore, the normal modes of initial and final electronic states are usually different and are related by the so-called Dushinsky transformation [53]: Q = JQ′ + K

(1.19)

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where Q and Q′ represent the mass-weighted normal coordinates of the initial and final electronic states, respectively, and J is the Dushinsky matrix, which describes the projection of the normal coordinate basis vectors of the initial state on those of the final state and represents the rotation of the normal modes upon the transition. Finally, the vector K represents the displacements of the normal modes between the initial and the final state structures. Two general strategies are usually employed. These are the adiabatic model (adiabatic Hessian [AH]), which requires each PES being expanded around its corresponding energy minimum, and the vertical model (vertical Hessian [VH]), which needs the knowledge of both PESs expanded around the energy minimum of the reference electronic state. Further approximations can be obtained by neglecting mode-mixing and frequency change effects, thus leading the adiabatic shift (AS) and vertical gradient (VG) models, respectively. Furthermore, together with the time-independent (TI) sum-over-state approach, in the VMS software a time-dependent (TD) strategy, which employs the exact analytic form (at the harmonic level) of the time evolution, has been implemented [48]. The TI approach is the method of choice when high-resolution spectra are sought; however, it can suffer from convergence issues. These can be overcome with a TD approach, which takes into account all vibrational states, thus leading to fully converged spectra, possibly accounting for temperature effects, without any increase of the computational cost. Being based on the FC principle, the so far described framework is well suited to describe transitions accompanied by small structural changes, and it may fail when the flexibility of the system increases. A first improvement is obtained using internal coordinates instead of the Cartesian ones, because this choice minimizes the coupling between different modes even at an anharmonic level. A general implementation has been developed, which generates automatically a complete set of nonredundant internal coordinates starting from redundant generalized internal coordinates (GICs) [54]. Whenever a complete anharmonic treatment of the system is not feasible, an appealing approach is to divide normal modes in classes that are treated at different levels. This strategy can be efficiently applied, provided that large-amplitude motions (LAMs) can be decoupled from the remaining normal modes. In this respect, it has been shown that GICs provide an efficient reduction of mode couplings also for nontrivial LAMs [55]. 1.2.2.4

Magnetic Spectroscopy

Spin relaxation techniques, such as NMR and ESR spectroscopies, represent powerful and sensitive tools for studying structural and dynamic properties of macromolecular systems. Among magnetic spectroscopies, we focus on ESR, often called electron paramagnetic resonance (EPR) spectroscopy, which is the spectroscopic technique of choice to investigate open-shell species. In particular, ESR is extensively applied to investigate complex biological systems, either directly or with the help of site-directed labeling techniques. In the following, we briefly summarize the theoretical aspects of ESR spectroscopy (a more detailed account can be found, for instance, in Refs. [56, 57]). When dynamical effects are not taken into account and only transition energies are considered, line positions

1.2 The Virtual Multifrequency Spectrometer

and the corresponding line intensities and amplitudes can be analyzed in terms of an effective spin Hamiltonian [58, 59]. The interaction of the electron spin (S) of a radical containing a nucleus of spin I with an external magnetic field (B) can be approximated by the spin Hamiltonian Hs : H s = 𝜇B S ⋅ g ⋅ B + S ⋅ A ⋅ I + · · ·

(1.20)

where the first term is the Zeeman interaction between the electron spin and the external magnetic field in terms of the Bohr magneton, 𝜇B , and the electronic g-tensor. Additional terms that might appear in the equation above are the spin coupling (J) and spin–spin dipolar tensor (T), which are present only when the system under consideration contains more than one unpaired electron (e.g. in the case of biradicals). The second term of Eq. (1.20) is the hyperfine interaction between S and I, described through the hyperfine coupling tensor A, which in turn can be decomposed into two terms: A = aK 𝟏 + A(K) dip

(1.21)

where the first contribution is the isotropic hyperfine coupling constant (hfcc), while the second term is the anisotropic hyperfine coupling tensor. The former is the so-called Fermi contact term, which is related to the spin density at the Kth nucleus under consideration [60] ∑ 𝛼−𝛽 8π ge gK 𝛽 K P𝜇𝜈 ⟨𝜙𝜇 |𝛿(rnK )|𝜙𝜈 ⟩ (1.22) aK = 3 g0 𝜇𝜈 while the anisotropic contribution, also denoted as dipolar hyperfine coupling term, can be derived from the classical expression of interacting dipoles [61]: ∑ 𝛼−𝛽 ge −5 2 = g 𝛽 P𝜇𝜈 ⟨𝜙𝜇 |rnK (rnK 𝛿ij − 3rnK,i rnK,j )|𝜙𝜈 (1.23) A(K) K K ij g0 𝜇𝜈 𝛼−𝛽 is the difference between the density matrices for elecIn both equations, P𝜇𝜈 trons with 𝛼 and 𝛽 spins, i.e. the spin density matrix, g0 is the g-value of the electrons in the radical, and the 𝜙′ s are the basis functions. Therefore, the essential quantities to be calculated are the spin density at the Kth nucleus and the dipole–dipole coupling terms. Consequently, isotropic and anisotropic hyperfine contributions can be easily evaluated as expectation values of the corresponding one-electron operators. The electronic g-tensor can be expressed in terms of second derivatives of the energy with respect to the external field (B) and electron spin (S). The g-tensor itself can be decomposed in its various contributions:

g = ge 𝟏 + Δg ≡ ge 𝟏 + ΔgRMC + ΔgDC + ΔgOZ/SOC

(1.24)

where ge is the free electron g factor (= 2.002319304386(20)), ΔgRMC is the relativistic mass correction, and ΔgDC is the gauge correction. These two terms are usually small and have opposite signs; as a consequence, their contributions tend to cancel out. The last term is a second-order contribution arising from the coupling of the orbital Zeeman (OZ) and the spin–orbit coupling (SOC) operators. Only the SOC term involves a true two-electron operator, but it is

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usually approximated by a one-electron operator involving adjusted effective nuclear charges. An effective evaluation of the g-tensor for quite large systems by QC approaches is relatively recent and has been revolutionized by hybrid density functionals that, when coupled to proper basis sets, offer a very good compromise between accuracy and computational efficiency (see, for example, Refs. [62, 63]). 1.2.3

The VMS Framework: Quantum Chemical Methods

This section is focused on accuracy and interpretability. It is thus devoted to address the computational requirements for obtaining quantitative spectral prediction/analysis. The key point to reach this goal is to greatly reduce the errors associated with computations. This implies to reduce as much as possible the errors due to the truncation of both basis set and wavefunction, the so-called one- and N-electron errors, respectively. To fulfill this task, the best option is to rely on composite schemes, which are approaches that evaluate the contributions important to reach high accuracy at the best possible level and combine them through the additivity approximation [64–68]. For all spectroscopies, the starting point is an accurate and reliable description of the equilibrium structure, which is defined as a minimum on the Born–Oppenheimer (BO) PES. In this context, the best option is offered by the so-called semi-experimental (SE) equilibrium geometry (reSE ), which is obtained by a least squares fit of the experimental vibrational ground state rotational constants of different isotopic species corrected for computed vibrational corrections. In this respect, a new tool, the Molecular Structure Refinement (MSR) program [69], has been recently integrated in the VMS software. From a pure computational point of view, composite schemes based on coupled-cluster techniques including single and double excitations and a perturbative treatment of triples, CCSD(T) [70], and that account for extrapolation to the complete basis set (CBS) and core–valence (CV) correlation effects (i.e. the so-called CCSD(T)/CBS+CV scheme) are able to provide an accuracy similar to that offered by the SE approach [66, 67, 71]. To summarize, nowadays it is possible to determine equilibrium structures with an accuracy of 0.001–0.002 Å for bond lengths and 0.05–0.1∘ for angles. However, composite approaches are computationally expensive. It is therefore important to identify levels of theory that are affordable for medium- to large-sized systems. In this respect, the double-hybrid B2PLYP functional [72] in conjunction with triple-zeta basis sets is known to provide an accuracy of 0.002–0.003 Å for bond distances [73, 74], with maximum errors below 0.01 Å. Such an accuracy is well suited for subsequent spectroscopic applications. Composite schemes can also be applied to the accurate computation of harmonic force fields. According to the literature on this topic (see, for example, Refs. [67, 75, 76]), composite approaches based on CCSD(T) are able to provide harmonic frequencies with an accuracy of 5–10 cm−1 . However, their applicability is limited to small- to medium-sized molecules. As in the case of equilibrium structures, B2PLYP in conjunction with triple-zeta quality basis sets provides an alternative for larger systems, thus showing an accuracy of 8–15 cm−1 . Due to the

1.2 The Virtual Multifrequency Spectrometer

computational requirements, the anharmonic part of force field calculations, i.e. the computation of cubic and quartic semi-diagonal force constants, is usually evaluated at a lower level of theory. While for small molecules the CCSD(T) method can be employed, when increasing the molecular size, global-hybrid (like B3LYP [77, 78]) or double-hybrid functionals are mostly used. Improvements in the accuracy can be achieved by means of hybrid approaches, where the harmonic part of the potential is determined at the CCSD(T) level or even employing a composite scheme, while for the anharmonic part Møller–Plesset theory to second order [79] and density functional theory (DFT) are used. When harmonic frequencies evaluated by means of composite schemes are combined with anharmonic corrections obtained using either B3LYP in conjunction with double-zeta quality basis sets or B2PLYP with triple-zeta quality sets, fundamentals as well as overtone and combination bands are predicted with mean absolute errors (MAEs) of 6–8 and 5–7 cm−1 (see, for example, Ref. [7]), respectively. Analogously, hybrid approaches can also be applied to molecular and magnetic properties. In this case, CCSD(T) and composite schemes are used for highly accurate determinations of equilibrium values, with vibrational corrections computed using DFT approaches. A special note is required for magnetic properties. While it is well demonstrated that CCSD(T) is able to provide accurate results, the basis set issue is a delicate one in the view of obtaining quantitative predictions of the isotropic hfcc’s. Indeed, to correctly describe the spin density at a nucleus, core correlation is necessary for atoms heavier than helium [80]. This is related to the general fact that very tight s primitives are needed to describe the spin density at the nucleus of interest as well as diffuse functions on surrounding atoms are required for a proper description of the spin density [80]. For these reasons, specific basis sets have been set up [81, 82]. Except for high-resolution gas-phase experiments, the recorded spectra are tuned by the surrounding environment (e.g. solvent, matrix, solid). A full QC treatment of a significant portion of the environment is in most cases not feasible, but more importantly, it is not even required. The experimental outcome corresponds to a statistical averaging of environment librations and small-amplitude motions, which, in turn, would require a huge number of barely feasible simulations. In addition, a full QC picture of the environment is not required to gain an accurate physical description of physical–chemical phenomena strongly localized in the probe. An effective strategy is then offered by focused models, where the system is partitioned into a chemically interesting core (e.g. the solute or a part of it) and the environment, which tunes the core’s properties. A suitably high level of theory is retained for the core, whereas the environment is treated in a more approximate way. A popular focused approach is the so-called QM/MM model, where the core is treated by means of QC methods based on the coupled-cluster ansatz (or its equation of motion extension for excited states), whenever possible, or rooted in DFT (or TD-DFT for excited states) for larger systems; the environment retains its atomistic resolution, but it is described classically by a suitable molecular mechanics (MM) force field. In the last few years, a fully polarizable QM/MM/continuum approach, which employs a polarizable, fluctuating charge (FQ) FF, has been

13

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1 Interpretability Meets Accuracy in Computational Spectroscopy

developed to treat aqueous solutions [83–85]. This has been recently extended to linear response equations, first, second, and third analytical derivatives with respect to geometric and electric perturbations, which are the ingredients needed for a proper simulation of experimental spectra [86, 87].

1.3 The VMS Framework at Work As mentioned in Section 1.1, VMS aims at going beyond the current approaches for the analysis and prediction of molecular spectra, i.e. VMS aims at moving from the standard practice of extracting numerical data from experiment to be compared with QC results to a direct comparison between the recorded and computed spectra. This strongly reduces any arbitrariness and allows for a proper account of all chemical–physical information hidden in the molecular spectra. The VMS software has therefore been devised to develop and implement efficiently spectral models able to account for all important vibrational and environmental effects in conjunction with the spectroscopic properties relevant to the technique under consideration as well as to include state-of-the-art QC computations of the latter. As mentioned above, in the following, we focus on rotational, vibrational, vibronic, and magnetic spectroscopies, thus providing an overview of the corresponding VMS modules. 1.3.1

Rotational Spectroscopy

Unlike other softwares purposely developed to support the analysis of MW spectra, the rotational spectroscopy module of the virtual multifrequency spectrometer, VMS-ROT, incorporates QC predictions to be used as starting points for guiding experiments and spectral interpretations. Indeed, in addition to provide a user-friendly access to the latest developments of computational spectroscopy, VMS-ROT is directly interconnected to the Gaussian package [14] as the reference source for the QC calculation of spectroscopic data. However, the extension to other QC packages, e.g. CFOUR [15], is in progress and will be available to external users in the next release. According to the user’s expertise in performing QC calculations, the VMS-ROT allows for computations of any kind, ranging from standard geometry optimizations and frequency calculations to highly accurate composite schemes also including anharmonicity effects. In the first step, i.e. the QC calculation of the relevant spectroscopic data, the user is supported by a “built-in tool” to automatically generate the required input files. In addition to this automatized procedure, the user can remove or add spectroscopic parameters as well as adjust manually their values. Once the results of the QC calculations are ready, the second step, i.e. prediction and/or fitting steps, relies on the SPFIT/SPCAT program developed by Pickett [88]. It should be noted that, while Pickett’s program is the de facto standard for rotational and rovibrational analysis, sometimes the non-straightforward encoding of the spectroscopic parameters may prevent its immediate use. For this reason, VMS-ROT also provides an intuitive and user-friendly GUI for SPFIT and SPCAT, which

1.3 The VMS Framework at Work

allows creating the required input files for the two programs, running them, and analyzing their output at the end of the execution. Once all the required data have been provided, the SPFIT/SPCAT program is executed, with the output details being displayed in a dedicated output textbox. At the end of this second step, the output files are automatically loaded in VMS-ROT. In the case of a fitting procedure, the new values of the spectroscopic parameters are displayed, whereas in the case of spectra prediction, the synthetic spectrum is shown in the plot area. If the first case applies, different tools are available for a fast and efficient analysis of the fitting process: (i) at each iteration, the trend of the root-mean-square (RMS) deviation of the fit is displayed in a dedicated plot; (ii) the correlation matrix among the spectroscopic parameters is graphically visualized as a gray-scale plot, where color saturation provides an intuitive indication of the absolute value, thus allowing one to quantify strong correlations between parameters that can lead to an ill-conditioned fit procedure; and (iii) a panel is devoted to the analysis of the residuals between observed and calculated transition frequencies, thus allowing the user to pick up, at runtime, those that deviate more than a specified threshold, thus permitting the quick identification of wrongly assigned lines. Once loaded in the plot area, the spectrum obtained in the third step (simulation or fit) can be edited by the user. To give an example, the stick spectrum can be convoluted with a suitable line shape function. The so-called VMS-Comparison tool can then be employed to compare computed and experimental spectra. This tool offers advanced manipulation utilities. It is worthwhile noting that if the experimental spectrum is the overall result of the concomitant presence of different species, e.g. conformers and isomers, then different computed spectra, one for each possibly present species, can be combined and weighted according to their relative population. Finally, VMS-ROT also incorporates all the required tools for the assignment of an experimental spectrum based on the predicted spectrum. A sketch of the so-called VMS-Assignment tool is graphically displayed in Figure 1.3, with the test case provided by the rotational spectrum of fenchone [89]. It is noted that the working area shows two horizontal panels: the calculated spectrum is depicted in the top one, whereas the experimental trace can be loaded in the lower panel. When both spectra are loaded into the corresponding panels, the assignment feature can be accessed by clicking on the proper item of the toolbar: for each assigned transition, the list of the upper- and lower-level quantum numbers, the measured and calculated frequency together with their difference, the estimated experimental uncertainty, and the weight of a transition within a blend of overlapped lines are annotated. Noted is that the assignment of a given experimental line, based on the comparison with the computed spectrum, can be performed in a manual or an automatic mode (this option is highlighted in Figure 1.3). Once a set of transitions has been assigned, it is then possible to proceed with the spectral fitting. A peculiar feature of VMS-ROT is that when the fitting process is completed, SPCAT is run as a background process with the newly determined parameters, thus allowing the update of the corresponding calculated spectrum. The spectral assignment procedure is iterated until a satisfactory assignment of the experimental spectrum is obtained.

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1 Interpretability Meets Accuracy in Computational Spectroscopy

(a)

Simulated spectrum window

(b)

Experimental spectrum window

Figure 1.3 Assignment tool of VMS-ROT. (a) Simulated spectrum window: the selection of a transition in the calculated spectrum adds a new line in the assignment table, thus reporting the corresponding quantum numbers and calculated frequency of the selected peak. (b) Experimental spectrum window: the assignment is completed by selecting a transition of the experimental spectrum, which is added to the assignment table box.

As pointed out in the theoretical background, accurate equilibrium structures are the leading terms in the field of rotational spectroscopy. For this reason, the VMS-ROT module has also been equipped with the MSR code [69] for the accurate determination of molecular structures by means of the so-called SE approach [90]. While we refer the reader to the specific literature for a detailed description of the methodology (see, for example, Refs. [91, 92]), as already mentioned above, SE equilibrium geometries are obtained by a least squares fit of experimental vibrational ground state rotational constants of different isotopologues corrected by computed vibrational (and possibly electronic) contributions. The high accuracy of SE equilibrium structures (e.g. 0.001 Å for bond lengths) is well recognized in the literature [93]. However, this approach requires a sufficiently large number of isotopic substitutions in order to guarantee all structural parameters to be determined. When this condition may not be fulfilled, two strategies have been implemented in the MSR package. The first one is based on reducing the dimensionality of the problem by fixing a subset of the parameters to the corresponding QC values. In this respect, the availability of highly accurate geometrical parameters obtained from CCSD(T)-based composite models is a fundamental requisite. An alternative approach is provided by the so-called method of predicate observations [94], which implies to augment the input data set for the fit by estimates of structural parameters weighted appropriately. 1.3.2

Vibrational Spectroscopy

As described in the section dedicated to the theoretical background, the effects of mechanical, electrical, and, possibly, magnetic anharmonicity cannot be

1.3 The VMS Framework at Work

neglected in order to obtain accurate vibrational frequencies and intensities, with VPT2 providing a cost-effective approach for fulfilling this task. As in the case of VMS-ROT, a vibrational spectroscopy module of VMS, VMS-VIB, is directly interconnected to the Gaussian package [14] as the reference source for both QC and VPT2 calculations. Since any meaningful comparison of computed spectra to experimental data requires the experimental conditions to be reproduced, the stick spectrum (a collection of peak positions and intensities) is convoluted with a line shape function (either Gaussian or Lorentzian) with a half width at half maximum (HWHM) suitably chosen. The convolution procedure is usually iterative, and it is performed by VMS-Draw, which is able to make adjustments in the HWHM, to plot the resulting computed spectrum and to compare it with the experimental one in real time. By performing the comparison between computed and experimental spectra, the selection of each peak in the stick spectrum directly provides its assignment to a specific transition. An example is shown in Figure 1.4, which compares experiment and theory for the IR spectrum of uracil recorded in Ar matrix in the 500–2000 cm−1 range [95], with the portion 1600–1800 cm−1 highlighted in the inset. It is first of all noted how well the experimental line shapes and their width are reproduced by the virtual spectrometer. Figure 1.4 also illustrates the inadequacy of the harmonic approximation to even qualitatively reproduce the experimental spectrum: not only the line positions are badly determined, but also – by definition – non-fundamental bands cannot be predicted. The level of theory employed is B3LYP-D3/SNSD, with D3 denoting Grimme’s dispersion correction [96], which provides at the anharmonic level (i.e. VPT2 computations) a maximum absolute error of about 10 cm−1 for line positions. In addition to spectra visualization, in VMS-VIB, other tools are available to analyze the outcome of anharmonic computations. The contribution of the higher derivatives to the anharmonic frequencies are collected in the so-called X matrix, which can be further split into different sub-matrices containing Coriolis, cubic, and quartic terms. Furthermore, the anharmonic correction for each mode can be split into intrinsic anharmonicity, direct coupling with each normal mode, and indirect coupling with more than one additional mode. The first two contributions are collected in the diagonal and off-diagonal elements of the Y matrix, respectively. All these matrices are visualized as gray-color scale plot, where the larger the matrix element is, the darker the color is. Such a visualization allows a quick quantification of all possible correlations between different groups of parameters, thus permitting to point out ill-conditioned optimizations. The analysis of anharmonic couplings provides important information on the system under study, and it can be used, for example, for defining cost-effective reduced dimensionality anharmonic models. To give an example, Figure 1.5 shows the heat diagram for the semi-diagonal cubic force constant Kiij matrix of glycine. Such diagram allows to describe the coupling between the normal modes, in particular the strong coupling between the symmetric NH2 and the asymmetric CH2 stretchings. Molecular systems characterized by some flexibility show nearly isoenergetic isomers and/or conformers. Since these can be concomitantly present in experimental mixtures, the analysis of corresponding experimental spectra is

17

C4

C2

1 0.8

0.6

0.6

C5 0.4

Normalized IR intensity

C6

0.8

0.2

0.4 0.2 Anharm convoluted Anharm stick Harm convoluted Harm stick

0.8 0.8

0.6

0.6

0.4

0.4

0.2

0.2

Experimental IR spectrum of uracil in argon matrix

Convoluted Stick

0

0 3500

1800

1750

3000

1700

1650

2500

Anharmonic IR spectrum of uracil monomer Harmonic IR spectrum of uracil monomer

1600

2000

1500

1000

Wavenumber (cm–1) Fundamentals: ∗ v(C2=0)

∗ v(C4=0)

∗ v(C5=C6)

Non-fundamentals : ∗

Figure 1.4 Comparison of the computed harmonic and anharmonic IR spectra of uracil to the experimental counterpart recorded in Ar matrix.

500

1.3 The VMS Framework at Work

tr (A)s υNH 2 υCH (A)str 2

υNH2(S)str υCH2(S)str

Figure 1.5 The heat diagram for the semi-diagonal cubic force constant, Kiij , matrix of glycine.

further complicated by the fact that all of them should be taken into account based on their relative abundances. While we refer readers to the specialistic literature for the accurate computation of free energies (required for deriving the relative abundance) by means of composite schemes (see, for example, Refs. [67, 68, 97]) and for the approach to compute resonance-free thermodynamic properties beyond the RRHO model [37], we point out here the capability of VMS-VIB in dealing with vibrational spectra of complex molecular mixtures simulated based on the computation of the contributions for all conformers/isomers, weighted by theoretical Boltzmann populations. A specific example, the IR spectrum of glycine, will be discussed in Section 1.4. VMS-VIB furthermore allows IR spectra to be converted from absorbance (which is proportional to the molar extinction coefficient, directly provided by most of the QC programs) to transmittance, which is the de facto standard for experimental recordings. However, since the conversion requires the knowledge of sample concentration and optical path, to avoid any inconsistency, the transmittance spectrum is provided in arbitrary units. In the case of Raman spectra, QC programs usually provide the Raman activity, whereas the principal experimental observable is the Raman cross section, with VMS-VIB allowing the plot of both quantities. In particular, the comparison of harmonic or anharmonic simulated spectra with their experimental counterparts can be performed by plotting the Raman cross section and adjusting the HWHM and line shape function of the peaks. 1.3.3

Vibronic Spectroscopy

As mentioned in the brief discussion of the theoretical background, a reliable description of molecular vibrations is of fundamental importance in both high- and low-resolution electronic spectra. In particular, the simulation of

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1 Interpretability Meets Accuracy in Computational Spectroscopy

vibrationally resolved electronic spectra is mandatory for correctly interpreting high-resolution experimental data in the gas phase that are characterized by intricate band shapes, which originate from the progressions of different vibrational motions. The situation is especially involved when investigating short-life molecules (e.g. free radicals), where different species can contribute to the overall recorded spectrum. In such circumstances the vibronic module of VMS can represent a valuable aid to experimentalists because the setup of the whole simulation is essentially completely automatized: starting from the QC computation of equilibrium geometries and harmonic force fields for all involved electronic states to the simulation of the whole spectrum and its comparison with its experimental counterpart, which can be imported in VMS by different means. Indeed, different tools for the analysis and visualization of results are implemented in VMS. An overall picture is provided by Figure 1.6: depending on the specific data set to be analyzed, different graphical representations, like two-dimensional (2D) or three-dimensional (3D) plots, bar charts, and heat maps, can be produced. More in detail, Figure 1.6 depicts the comparison between the vibrationally resolved computed (red) and experimental (blue) absorption electronic spectra of dideprotonated alizarin, with the scheme of the plot digitization procedure being also shown. The present version of the software does not take into account all the effects tuning the spectral outcome (e.g. non-adiabatic couplings), but a pilot implementation of those contributions in the TD general framework has been recently completed [49] and will be available in the next release of VMS. Nevertheless, the current implementation has already allowed the simulation of qualitatively correct spectral line shapes, even when experimental data encompass large energy intervals and several excited electronic states. This is particularly true if vertical

Experimental spetrum Normalized absorbance

20

Spectra plots

3D visualization

Wavelength (nm)

Computed spectrum

Bar charts

Analysis Heat maps

Structure, properties

Figure 1.6 Comparison between the vibrationally resolved computed (red) and experimental (blue) absorption electronic spectra of dideprotonated alizarin, with the scheme of the plot digitization procedure being shown in the upper inset. In the right panel, a sketch of the different features available for analysis and visualization is depicted.

1.3 The VMS Framework at Work

Exp. Exp. shifted FCHT|AH

Intensity (a.u.)

Stick

–1000

0

1000

2000

3000

4000

5000

6000

7000

Energy (cm–1) relative to the 0–0 origin

Figure 1.7 The A 2 B1 ← X 2 A1 electronic transition of the phenyl radical in the gas phase.

excitations computed by refined post-Hartree–Fock approaches are evaluated at DFT/TD-DFT geometries, the latter approaches being employed also to obtain harmonic force fields. As an example, Figure 1.7 compares the simulated and experimental high-resolution spectra for the A 2 B1 ← X 2 A1 electronic transition of the phenyl radical in the gas phase. It is apparent that the complete AH model including both Franck–Condon and Herzberg–Teller (FCHT) contributions provides results in remarkable agreement with experiment when the whole spectrum is slightly shifted in order to correct for some inadequacy of the computed vertical excitation energy (VE). Vibronic effects play also a role in tuning the band shapes of less resolved electronic spectra, like those recorded in solution. In such circumstances, even when vibronic signatures are not apparent in the spectra, only inclusion of these effects allows for a correct evaluation of relative band intensities. Then, VMS provides an invaluable aid to experimentalists in setting up a proper computational strategy and in the comparison between computed and experimental spectra. To give an example, the UV–vis spectrum of the indigo dye (see Figure 1.8) is dominated by a single electronic transition, but the spectral shape is strongly nonsymmetric, and vibrational modulation plays a significant role both in this connection and in determining the color perceived by the human eye (see Figure 1.8). Vibronic effects can be even more significant for chiral spectroscopies, in both absorption (ECD) and emission (CPL). An example is provided by the CPL spectrum of dimethyloxirane, which does not show any sign alternation at the FC level, but HT effects strongly modify the overall band shape and introduce a sign alternation (see Figure 1.9). In this case (and in a number of other circumstances), the robustness, flexibility, and ease of use of VMS allow for a detailed analysis of the different factors playing a role in determining the experimental outcome.

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1 Interpretability Meets Accuracy in Computational Spectroscopy

Virtual spectroscopic laboratory Comp. B3LYP

Intensity (arbitrary units)

Exp.

200 225 250 275 300 325 350 375 400 425 450 475 500 525 550 575 600 625 650 675 700 725 750 775 800

Wavelength

Figure 1.8 Comparison between the computed and experimental spectra of the indigo dye. Molecular structure is shown together with the color perceived by the human eye. 5

Difference of emitted intensity (μJ/mol)

22

AH|FC AH|FCHT

4 3 2 1 0 –1 –2 –3

47000

48000

49000

50000

51 000

52 000

53 000

54 000

Energy (cm–1)

Figure 1.9 S3 → S0 CPL spectrum of dimethyloxirane simulated by different models.

1.3.4

Magnetic Spectroscopy

As already mentioned, the VMS package contains two different tools for magnetic spectroscopy: one devoted to EPR and the other to NMR spectroscopy. In the context of magnetic spectroscopy, we focus on EPR, whose theoretical background has been briefly summarized in Section 1.2.2.4. The main features

1.3 The VMS Framework at Work

CH3

O

H3C N CH3 CH3 O

Figure 1.10 The 2,2,6,6-tetramethyl-4-piperidone-1-oxyl radical, also known as TEMPONE.

of VMS in the framework of EPR spectroscopy will be sketched with reference to the simulation of the EPR spectrum of the TEMPONE radical (see Figure 1.10), whereas we refer readers to Ref. [11] for a detailed account on the VMS-NMR tools. As with all other spectroscopies, the starting point is the input of the target molecule set up by means of the general graphical utilities of VMS-Draw (which are shared by all the other modules) and the subsequent submission, in background, of the specific QC code, which is, for the purposes of this section, the Gaussian16 (G16) suite of programs for quantum chemistry [14]. Since TEMPONE is a molecule that at room temperature rapidly interconverts between two twisted-crossover structures, the electronic structure computations should include the averaging of the magnetic tensors over the effective LAM connecting these two geometries. The G16 software is able to perform automatically this task together with the perturbative treatment needed to take into account the vibrational averaging for all the other small-amplitude vibrations. The detailed procedure can be summarized by the following steps and a representative screenshot of the whole process is provided in Figure 1.11: Step 1: A 3D graphical representation: The molecule is shown together with the inertial laboratory frame. Step 2: By clicking on the “Set Dynamics” button, a new window is opened where the user can choose the form of the diffusive operator Γ, which describes the overall rotational motion of the probe (possibly coupled to large-amplitude internal rotations) at a coarse-grained level based on the model describing diffusive motions of regular bodies in viscous media [98]. In this case the one rigid body model can be chosen for the rotational averaging. Step 3: By clicking on the “Spin Probes” button, a new window opens, thus allowing a user-friendly setting of the spin Hamiltonian parameters. The setting also includes the choice of the number of unpaired electrons and of the corresponding reference atoms (in the present case, the selection is one electron and the N–O moiety, which defines the reference frame

23

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1 Interpretability Meets Accuracy in Computational Spectroscopy

(a)

(b)

Figure 1.11 Two representative screenshots of the VMS-EPR software: setup of the modeling parameters (a) and of the spectra viewer (b).

for the g-tensor), and it also permits the indication whether spin active nuclei are possibly present in the probe(s) (in the present case, the N atom with nuclear spin 1), which leads to the drawing of the reference frame for the A-tensor. Step 4: In the “Physical Data” tag of the “Parameter Selector,” a number of relevant parameter are set, also including the magnetic field (B = 3197.3

1.4 The VMS Framework: Applications

Step 5:

Step 6:

Step 7:

Step 8: Step 9:

Step 10:

G in the present case), the field sweep (75.7 G), the viscosity of the solvent (water in the present case: 𝜂 = 0.89 cP), and the temperature (T = 298.15 K). The “Additional Data” tag is used to set the intrinsic linewidth to 2.4 G, which is a typical value since the unresolved super-hyperfine coupling of the electron with 12 surrounding hydrogen atoms has to be taken into account. Clicking on the “Diffusion” button in the “Main Control” panel activates the automatic computation of the diffusion tensor and the appearance of the corresponding reference frame in the 3D space, with atoms assuming different colors if they belong to different fragments. In the present case (i.e. a single fragment), all atoms have the same color. In the “Diffusive Environment,” the magnetic tensors can be provided or modified from the values already available to the program by previous quantum mechanical computations. Next, their values can be adjusted by fitting within the “Refine Environment.” It should be noted that in general small corrections, at most, are necessary and, in the present case, the traces of g and A have been slightly adjusted together with the intrinsic linewidth. By clicking the “Load Spectrum” button, an experimental spectrum is loaded, and it can be manipulated by using the general VMS utilities. Now the user is ready to enter the “ESR Environment,” where spectra can be calculated (by setting up and solving the stochastic Liouville equation (SLE) [98]) with or without fitting the experimental counterpart and then plotted. Note that it is possible to run the calculation interactively or via Portable Batch System (PBS). In the present case, very small corrections (j

The coupling terms, V i , V ij , …, can be efficiently generated by a multiresolution method [56] combining different accuracy of electronic structure levels and potential energy functions (a quartic force field (QFF) [57] and a grid potential [58]).

151

152

5 Weight Averaged Anharmonic Vibrational Calculations

In the VSCF method [13, 14], the vibrational wavefunction is approximated to be a Hartree product of one-mode functions: ΨVSCF = q

f ∏

𝜓q(i)i (Qi )

(5.13)

i=1

The VSCF equation is derived from the variational principle, which determines the one-mode functions: [ ] 1 𝜕2 − + V i (Qi ) 𝜓q(i)i = 𝜀q(i)i 𝜓q(i)i (5.14) 2 𝜕Q2i ⟩ ⟨ f f ∏ (j) ∏ (j) (5.15) V i (Qi ) = 𝜓qj ∣ V (Q) ∣ 𝜓qj j≠i

j≠i

Here V i (Qi ) is a mean-field potential for the ith mode averaged over one-mode functions of other modes. Thus, the nonlinear Eq. (5.14) is solved iteratively. Starting from an initial wavefunction (e.g. the harmonic oscillator function), V i (Qi ) is constructed by a current set of one-mode functions and plugged into Eq. (5.14) to obtain a new set of one-mode functions. Then, the procedure is repeated using the new set of functions until the change in the total energy is below a threshold value. VQDPT [28, 29] is an extension to VSCF by the quasi-degenerate perturbation theory [59, 60]. In this method, we consider a state space spanned by VSCF configuration functions and divide the space into P space composed of quasi-degenerate states and its complementary Q space: f ∏

|p⟩ =

𝜓p(i)i (Qi )

(5.16)

i=1

P=

∑

|p⟩⟨p|,

(5.17)

p

Q = 1 − P.

(5.18)

In VQDPT2, the effective Hamiltonian is given as ( ) ∑ ⟨p ∣ Ĥ ∣ q⟩⟨q ∣ Ĥ ∣ p′ ⟩ 1 1 (2) ′ + (0) (Ĥ eff )pp′ = ⟨p ∣ Ĥ ∣ p ⟩ + (0) (0) (0) 2 Ep − Eq Ep′ − Eq q∈Q (5.19) Ep(0)

=

f ∑

𝜀(i) pi

(5.20)

i=1

VQDPT2 energy and wavefunction are given by diagonalization of the effective Hamiltonian. The P and Q spaces are constructed by specifying target vibrational states and a parameter that specifies the number of excitation and de-excitation with respect to the target state. For further details, see Refs. [28, 29].

5.3 Applications

5.2.4

Computational Procedure

To summarize this section, the procedure of the weight average method is given below: (1) Run MD simulations to sample the structure. (2) Analyze the resulting trajectory to find representative clusters and their weight (wt ). (3) For each cluster, (3)-1. Optimize the geometry and calculate the PES using the electronic structure calculations. (3)-2. Compute the vibrational spectrum, i.e. the vibrational energy levels (𝜈q(t) ) and intensities (𝜎q(t) ), using VSCF and VQDPT2. (4) Finally, calculate the total spectrum by a weight average of the spectrum of each cluster (Eq. (5.6)). In this procedure, the computationally intensive steps are (1) and (3)-1. It is crucial to make use of high-performance computers to carry out these steps. MD simulation (step 1) can be accelerated with a modern software that have good parallel efficiency and the support of GPU. In step (3)-1, the PES generation requires electronic structure calculations in many grid points (i.e. different geometries) for each type of clusters. Thus, the calculation is scalable by distributing the grid points to different nodes. Furthermore, the electronic structure calculation itself may be carried out in parallel though the scalability strongly depends on the method and the program of choice. Note that one may use a precomputed spectrum in step (4) bypassing step (3). This is often the case when computing the spectrum in different conditions (e.g. temperature, pressure, etc.), because MD simulations in different conditions affect the weight of clusters but not necessarily the types of clusters. In other words, we may take an alternative route: (i) compute the spectra of all possible clusters, (ii) perform MD simulations with required condition and analyze the trajectory to obtain the weight, and (iii) calculate the total spectrum. In practice, however, “all possible clusters” are not known in advance, so some (preliminary) MD simulations are needed. Nonetheless, constructing a dictionary of cluster/spectrum enables to compute the spectrum of different conditions efficiently.

5.3 Applications 5.3.1

Pentapeptide: SIVSF

Recent advances in the gas-phase laser spectroscopy have made feasible to measure the spectrum of biomolecules in high resolution [61, 62]. In particular, the development of the electrospray ionization and cryogenic ion trap, which enables a nondestructive vaporization of biomolecules in isolated, cold environment, has opened a way to obtain an accurate spectrum of biomolecules. Recently, Ishiuchi, Fujii, and coworkers [63, 64] have measured the IR spectrum of a pentapeptide, SIVSF. SIVSF is a partial sequence of the β2 -adrenoceptor protein near a binding

153

154

5 Weight Averaged Anharmonic Vibrational Calculations

site, and revealing its structure is of great importance to understand the mechanism of molecular recognition. The IR spectrum of SIVSF-NH2 was obtained in a conformer-specific way by the UV/IR double resonance technique in a region of NH and OH stretching frequencies. However, it was difficult to determine the structure of SIVSF-NH2 from the IR spectrum due to many possible conformations the peptide can take and the strong anharmonicity of NH/OH stretching vibrational modes in the presence of HBs. Therefore, we carried out MD simulations and anharmonic vibrational calculations of SIVSF-NH2 [37]. First, the conformation of SIVSF was searched using the replica-exchange molecular dynamics (REMD) method [65–67]. The REMD simulation was carried out for SIVSF-NH2 in vacuum with 12 replicas in a temperature range of 300–1300 K using the NAMD program [68] with the CHARMM36 force field [69]. The system was first equilibrated for 1 ns, followed by a 60 ns production run per replica. The replica exchange was applied every 2 ps. The resulting 30 000 snapshots were used for the k-means clustering analysis. One-hundred and forty four representative conformers were found by setting the clustering radius so as to cover major free energy minima in Ramachandran plots of each residue. Then, the geometry of the conformers were energy minimized using the B3LYP functional with mixed 6-31G(d,p) and 6-31++G(d,p) basis sets using Gaussian 09 [70]. The diffuse functions were added only to N, O, and H atoms involved in the HBs. The five lowest energy conformers were selected for the anharmonic vibrational calculations. VQDPT2 calculations were performed in 80-dimensional space, i.e. 10 target NH/OH stretching modes and 70 other modes that are strongly coupled to the target modes. The anharmonic PES was constructed up to the three-mode coupling level by combining QFF (requiring 160 Hessian calculations) and grid potentials (requiring 153 000 energy calculations) [56]. The structure and spectrum of the five selected conformers are shown in Figure 5.3. The structure of the conformers can be characterized by the HB connectivity of the backbone. Conformers 1, 2, and 3 have a similar structure forming γ and β turns at Val→Ser(1) and Phe→Ile, respectively, and NH of Ser(4) remaining free from HBs. Conformer 4 has a rather different structure with a sequence of γ turns. Although the HB connectivity of conformer 5 looks similar to that of 1, 2, and 3, conformer 5 is more stretched due to the conformation of Ile. These structural features are sensitively probed in the IR spectrum. NHS4 (free), NHF (β turn), and NHV (γ turn) are found in a similar position in conformers 1, 2, and 3. Conformer 4 yields peaks of γ turns below 3250 cm−1 , whereas the stretched structure of conformer 5 yields peaks above 3350 cm−1 . IR spectrum of conformer 1 is found to be in best agreement with the experiment among those of five conformers. The NH/OH stretching frequencies are calculated with a mean absolute deviation of 11.2 cm−1 and a maximum deviation of 16.7 cm for NHV . It is also notable that conformer 1 is more stable than conformer 2 by more than 8 kJ/mol. Therefore, SIVSF-NH2 observed in the experiment is assigned to conformer 1 from both energetic and spectral point of view. In order to further assess the quality of calculation, we compared the calculated NH stretching frequencies with those of the amide backbone of short peptides previously reported by Chin et al. [71] As shown in Figure 5.4, the NH bond free from HB was measured around 3480 cm−1 , whereas β and γ

5.3 Applications

γL turn

γL turn

β turn

γL turn

β turn

OH-π γL turn Conformer 1

Ser(1)

Ile γD turn

γD turn

Ser(4)

Val

Phe

β turn

β turn

OH-π Conformer 2

Ser(1)

γL turn

Ile γL turn

Val

Ser(4)

β turn

Phe

γL turn

β turn

OH-π

γL turn

Ser(1)

Ile

Val

Ser(4)

Phe

Conformer 3

γL turn

γD turn γL turn

γL turn

γD turn

γL turn

γD turn

Ile

Val

Ser(4)

Phe

γD turn Conformer 4

Ser(1)

Figure 5.3 (a) The structure and the HB pattern of five lowest energy conformers. (b) Theoretical IR spectrum of the five conformers compared with the experimental one. The spectrum is constructed using a Lorentzian function with the full width at half maximum of 5 cm−1 . The hydrogen-bonded and free NH stretching modes of the terminal NH2 group, in which one of the NH bonds is hydrogen-bonded, are labeled “hb-NH” and “f-NH,” respectively. The OH and NH stretching modes of residue R are labeled as OHR and NHR , respectively. The symmetric and asymmetric NH2 stretching modes of Phe are denoted as (sym-NH2 )F and (asym-NH2 )F , respectively. The labels colored in green and blue are the NH stretching modes involved in the γ and β turns, respectively. The red labels are the NH stretching modes of Ser(4) free from HB. Source: Otaki et al. [37]. Copyright 2016, Adapted with permission of American Chemical Society.

155

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5 Weight Averaged Anharmonic Vibrational Calculations

γL turn

β turn

β turn

γL turn

β turn

OH-π β turn

(a)

Ser(1)

Conformer 5

Ile

Val

Ser(4)

Phe

Exp.

NHI

NHV NHF (asym-NH2)F (sym-NH2)F NHS4

(hb-NH)S1 NHV

NHF

(hb-NH)F

Conformer 2 (8.4 kJ/mol)

(f -NH)S1 OHS1 OHS4 NHS4 (asym-NH2)F

NHI

NHV

OHS4

(sym-NH2)F

(hb-NH)S1 NHF

Conformer 3 (17.8 kJ/mol)

(f-NH)S1

NHI

NHV NHS4

OHS1

Conformer 1 (0.0 kJ/mol)

OHS1 NHS4

(f-NH)F

OHS1 NHI (hb-NH)F (sym-NH2)S1 NHF (asym-NH2)S1 (f-NH)F

OHS4 Conformer 4 (17.9 kJ/mol) OHS4

Conformer 5 NH I (sym-NH2)F (18.1 kJ/mol) NHF NH V OHS1 (asym-NH2)F NHS4 3100

3200

(b)

Figure 5.3 (Continued)

3300

3400

Wavenumber

3500 (cm–1)

OHS4 3600

3700

5.3 Applications

Figure 5.4 Plots of the NH stretching frequency of the amide group free from HB (crosses), in β turns (triangles), and in γ turns (circles). Filled triangles and circles have a neighboring CO free from HB (left in the inset), whereas open ones have the CO accepting HB (right in the inset). The experimental data are taken from Ref. [71]. Source: Otaki et al. [37]. Copyright 2016, Adapted with permission of American Chemical Society.

HB donor

C O

Exp.

HB donor

H N

C O

H N

β Turns

HB acceptor

free NH

γ Turns

This work free NH

β Turns γ Turns

3200

3250

3300

3350

3400

3450

3500

–1

NH stretching frequency (cm )

turns were redshifted to a range of 3410–3430 cm−1 and 3340–3410 cm−1 , respectively, due to the formation of HBs. The band position showed a further redshift when the neighboring CO of the amide group accepted a HB. The NH frequencies calculated for conformers 1–5 are consistent with these data. In Figure 5.4, the NH stretching frequencies free from HB are calculated in a range of 3450–3470 cm−1 (NHS4 in Figure 5.3), and the frequency becomes lower in the order of β turn, and β and γ turns with a neighboring CO accepting HB. However, some notable differences are also found between theory and experiment. Firstly, the band positions of β turns (NHF of conformers 1–3 and 5) are calculated to be lower than the experimental one by about 50 cm−1 . Note that an amide group that accepts a β turn donates to a γ turn in these conformers. Secondly, two bands of consecutive γ turns in conformer 4 (at 3230 and 3204 cm−1 ) lie beyond the low-frequency limit of the experimental range. These results indicate that a longer sequence of HBs makes the amide groups more polarized, thereby inducing stronger HBs and further redshifts of the NH stretching frequency. Since these structural motifs were not included in the experimental data set for short peptides, systematic theoretical and experimental studies are still needed for characterizing the amide A band of polypeptides. 5.3.2

Sphingomyelin Bilayer

SM is one of the major lipids in biological membrane. The structure and dynamics of SM in a membrane has drawn great attention due to its ability to form a self-assembled domain, which is considered to be a platform of the lipid raft [72–74], i.e. a nano-domain where membrane proteins and functional molecules are integrated to perform biological function efficiently. Unlike other glycerol-based lipids, SM has an amide group (Figure 5.5a) that can both donate and accept HBs and thus can form hydrogen-bonded clusters. Recently, Shirota, Kobayashi, and coworkers [75] have found a strong band at 1643 cm−1 in the Raman spectrum of an SM bilayer. Interestingly, they also observed the same

157

158

5 Weight Averaged Anharmonic Vibrational Calculations

γ-Chain 5

7

H3C(CH2)10 H3C(CH2)n (a)

β-Chain

2′ 3′

H OH 4

6

1′

1 3

2

NH H

O O P O O

N

O

(b)

Figure 5.5 (a) The chemical structure of N-acyl sphingomyelin (SM). (b) The final structure of an SM bilayer obtained from MD simulations at 50 ∘ C (left) and 23 ∘ C (right). Nitrogen atoms of the choline group are drawn in blue. The highlighted waters are those inside the membrane, i.e. beyond the average position of the choline nitrogen atoms along the membrane normal. Source: Yagi et al. 2015 [38]. Reproduced with permission of Royal Society of Chemistry.

band in a mixture of SM and dioleoylphosphatidylcholine (DOPC) but not in that of SM and dipalmitoylphosphatidylcholine (DPPC). Since SM is known to be miscible with DPPC but not with DOPC, the finding suggests that the band is a signal of SM clusters in a bilayer. However, the assignment of the Raman band has been controversial. Although Levin et al. [76] originally assigned the band to the amide I vibration of SM, Lamba et al. [77] later suggested the band to be the bending mode of water molecules. To reveal the origin of the vibrational band, we have carried out vibrational calculations of an SM bilayer. We first constructed an all-atom model of an SM bilayer consisting of 128 and 5120 molecules of SM and water, respectively, using CHARMM-GUI [78] and then carried out two MD simulations at 23 and 50 ∘ C. The system was equilibrated for 10 ns, followed by a 150 ns production run using NAMD [68] simulation package with CHARMM force field for SM [79] and TIP3P [80] for water. The final structures of the two simulations are shown in Figure 5.5b. It is clearly seen that the lipid tails are disordered at 50 ∘ C, whereas they are well ordered at 23 ∘ C. The result is consistent with the observation that the SM bilayer undergoes a phase transition from liquid crystal (LC) to gel phase at 38 ∘ C. Thus, in the following, we denote the simulation at 23 and 50 ∘ C as that of gel and LC phases, respectively. It is also notable in Figure 5.5b that the number of water molecules inside the membrane (highlighted water molecules in Figure 5.5b) is reduced in the gel phase compared with the LC phase. The number of water

5.3 Applications

(A) H N (A)

C O H H O H (H) SM O

H N

C O

H N

(W)

H O

C O

(W) H H O (H) SM

(a) WAAAWW (1.0) WAAAW (1.2) A (1.3) HAAWW (1.3) WAWAWW (1.3) AWAW (1.4) HA (1.6) HAWAW (1.7) AAWW (2.1) WA (2.2) HAAW (2.4) WAWAW (3.0)

MISC (11.8)

HAW (9.0) LC

AAW (3.3) AWW (4.8)

(b)

WAW (15.0)

WAAWW (5.0) HAWW (5.9)

AW (8.6)

WAWW (8.1) WAAW (7.9)

(A)

HAAWAW (1.1) HAWAWW (1.2) WAA (1.3) WAAWW (1.4) HAWW (1.4) A (1.6) AWW (1.8) AAWW (2.0) AA (2.0) HAAWW (2.1) WAWAW (2.5) WA (2.5) AWA (3.1)

MISC WAW (10.0) (10.3) HAWA (9.9) Gel

HAWAW (7.5) HAAW (7.2)

AAW HAA (3.7) (5.3) AWAW (3.9) WAAW (5.1) AW (4.2) WAWW (4.5) HAW (4.5)

Figure 5.6 (a) The HB connectivity of the amide group (A), hydroxyl group (H), and water molecules (W). (b) The type and the statistical weight (in percentage) of hydrogen-bonded SM clusters obtained from the HB analysis on the 100 ns trajectory of an SM bilayer in the LC (left) and gel (right) phases. The cluster types that increase or decrease more than twice upon the change from the LC phase to the gel phase are colored in red or blue, respectively. Source: Yagi et al. 2015 [38]. Reproduced with permission of Royal Society of Chemistry.

molecules per lipid is decreased from 3.9 to 2.1 on average throughout the whole trajectory. Furthermore, the area per lipid shrinks from 55.8 and 47.5 Å2 . Thus, the SM bilayer in the gel phase is more tightly packed and less hydrated than in the LC phase. The HB connectivity of the system was analyzed in the last 100 ns of the trajectory (10 000 frames). The HB clusters were classified based on the connectivity of amide (A), water (W), and hydroxyl (H) groups as illustrated in Figure 5.6a. The dominant cluster types as well as their statistical weight are shown in Figure 5.6b. The most dominant cluster type in both LC and gel phases is WAW, i.e. an amide monomer donating and accepting one water molecule each. However, the major clusters are markedly different after WAW in the LC and gel phase. The amide monomers, which are abundant in the LC phase (e.g. HAW, AW), decrease by half in the gel phase, while amide dimers (e.g. HAWA, HAWAW, HAAW) increase drastically in the gel phase. Therefore, the SM molecules have longer hydrogen-bonded amide groups in the gel phase than in the LC phase. Then, DFT calculations were performed for each type of clusters at the level of B3LYP/6-31(++)G(d,p) (the diffuse functions were added only to N, H, C, and O atoms involved in the HBs) using Gaussian 09 [70]. The amide and hydroxyl groups of SM were represented by a fragment of SM shown in Figure 5.7. After

159

160

5 Weight Averaged Anharmonic Vibrational Calculations

NH H

OH

O A (a)

H

WAW

HAW

(b)

Figure 5.7 (a) The chemical formulas of model molecules used for representing the amide (A) and hydroxyl (H) groups of SM. (b) The structure of WAW and HAW clusters.

the geometry optimization, the harmonic frequency and Raman activity were obtained. The resulting frequency was corrected for each vibrational mode (𝜐) by the following formula: 𝜈𝜐t = f𝜐 × (𝜔t𝜐 + 𝛿𝜔𝜐 ) where f 𝜐 and 𝛿𝜔𝜐 accounted for the anharmonicity and other errors (i.e. due to the level of electronic structure theory and the size of fragment), respectively. These parameters were obtained from harmonic and anharmonic calculations with different electronic structure levels and fragment size for an amide monomer [38]. Using the frequencies thus obtained, the Raman spectrum was obtained by a weight average over clusters. The results for LC and gel phases are shown in Figure 5.8a, b, respectively, together with the experimental spectrum measured at 23 ∘ C in Figure 5.8c. The calculated spectrum shows a prominent band of CC stretching mode around 1685 cm−1 and amide I bands of monomers and dimers in a lower frequency range. In the LC phase, the amide I band of monomers has larger contribution than that of dimers and shows a clear peak around 1665 cm−1 . However, the band overlaps with the CC stretching band, and consequently the amide I band appears only as a vague shoulder in the total spectrum. In contrast, the total spectrum in the gel phase exhibits a clear trace of the amide I band due to an increase in the contribution of amide dimers. The growth of dimers yields a plateau-like shape in a region of 1640–1670 cm−1 , which is in good agreement with the experimental spectrum measured in wet condition. As shown in Figure 5.8c, the Raman measurement is sensitive to the water content of the membrane, where a prominent band grows at 1643 cm−1 as the membrane is dried. Although the spectrum in dry condition is not directly calculated, it is plausible that the reduction of water induces an increase in the number and size of SM cluster and a growth of the amide I band of dimers. Therefore, we assign the Raman band at 1643 cm−1 to the amide I band of hydrogen-bonded SM clusters. 5.3.3

Hydration of Nylon 6

Understanding the effect of hydration on polymer structure is of great importance for polymer materials that function in the presence of water [81–83]. For example, reverse osmosis (RO) membranes filter out ions from solution and are used for desalination of seawater to produce drinking water. Proton conductive membranes, which are used for electrolyte in fuel cell, uptake water molecules

5.3 Applications

(a)

Calc. (LC)

(b)

Calc. (gel)

(c) Exp.

1720

1700

1680

1660

1640

1620

1600

–1

Wavenumber (cm )

Figure 5.8 (a, b) Raman spectrum of an SM bilayer in the LC and gel phases. The blue (broken) and red (solid thick) lines are the amide I bands of the monomers and dimers of the amide group, respectively. The green (solid thin) line is the CC stretching band. The black line is the total intensity. The spectrum is constructed using a Lorentzian function with the full width at half maximum of 15 cm−1 . (c) Raman spectrum measured for SM bilayers at 23 ∘ C in a wet (black) and dry (gray) condition. Source:Yagi et al. 2015 [38]. Reproduced with permission of Royal Society of Chemistry.

from (humid) air and construct internal HB network through which protons can diffuse efficiently. Vibrational spectroscopy, unlike other structural analysis methods, can measure the sample in operating condition and provide the information of hydrated polymer structure [5, 84–87]. However, the interpretation of the spectrum is often difficult due to overlaps of many vibrational bands. Therefore, computations that help assign the spectrum are greatly required. We have calculated the vibrational spectrum of nylon 6 as an example to assess the accuracy of a weight average method [39]. Five systems were prepared consisting of 4 chains of nylon 6 shown in Figure 5.9a including water molecules corresponding to 0.0, 2.3, 4.5, 8.6, and 15.9 wt%. The MD simulation was performed using NAMD [68] with the force field parameters for nylon 6 adapted from CHARMM [69, 79, 88], and TIP3P [80] for water. After initial equilibration,

161

162

5 Weight Averaged Anharmonic Vibrational Calculations

O

O

N

HO

NH2

H (a)

52

2.3 wt% water

(b)

8.6 wt% water

4.5 wt% water

15.9 wt% water

Figure 5.9 (a) The chemical structure of nylon 6. (b) Snapshots of a unit cell of nylon 6–water systems at different water concentrations. Each system contains four 52-mers of nylon 6 as well as a set amount of water molecules. Source: Thomsen et al. [39]. Copyright 2017, Adapted with permission of American Chemical Society.

the production runs were performed for 1 μs in the NPT ensemble at 1 atm and 300 K for each system. Snapshots of hydrated systems are shown in Figure 5.9b. Although water molecules appear alone or in small clusters in low water content, the increase in water content causes a formation of longer water chains and larger clusters. The diffusion coefficient of water molecules was calculated as 2.4, 2.5, 13, and 31 × 10−10 cm2 /s for 2.3, 4.5, 8.6, and 15.9 wt% systems, respectively. These results are in good agreement with the experimental study [89] that reported the diffusion coefficient as 1.0–10 × 10−10 cm2 /s and an exponential rise of the diffusion coefficients after 5 wt%. A total of 100 000 structural snapshots obtained from MD simulations were used to find HB clusters of amide groups and water molecules. For each type of clusters, electronic structure calculations were carried out at the B3LYP level with mixed 6-31++G(d,p) and 6-31G(d,p) basis sets using Gaussian 09 [70]. The amide group of nylon 6 was capped by methyl groups, and the smaller basis sets were

5.4 Concluding Remarks and Outlook

used for the methyl groups. Vibrational calculations were performed in a space of amide I, II, III, and A modes for amide groups and the two OH stretching modes and HOH bending mode for water molecules. Anharmonic PESs were constructed up to the two-mode coupling level (see Eq. (5.12)) using the grid method [58]. VQDPT2 calculations were carried out to obtain the IR spectrum of each cluster. Finally, the total IR spectrum was obtained by a weight average of IR spectra of all clusters for each hydrated system. The difference IR spectrum between 8.6 and 0.0 wt% in a OH/NH stretching region is shown in Figure 5.10a. The comparison between harmonic and anharmonic calculations shows a drastic effect of anharmonicity, not only shifting the band position but also varying the band shape. Note that the overtone bands (7a and 7b) have no intensity in the harmonic approximation and appear only in the anharmonic result. The anharmonic spectrum is found to be in good agreement with the experimental one providing the assignment of observed peaks. A characteristic dip (4) is caused by a loss of amide chain (1A-A-1A shown in Figure 5.10c), which is replaced by those involving water molecules either as an acceptor (1A-A-1W) or a donor (1A1W-A-1A). In the former, the NH stretching band makes a blueshift (4 → 3a), because the HB between A-1W is weaker than A-1A. On the other hand, the donation of a water molecule to the amide group makes the HB stronger, thereby causing a redshift of NH stretching band (4 → 5a). Other bands in this region originate from OH stretching bands of water. The OH bond free from HB has the highest frequency (1), while three- and fourfold coordinated water molecules arise in the intermediate (2, 3b, and 5b) and the lowest frequency (6) range, respectively. These results attest to the quality of the weight average method to calculate spectrum of hydrated polymer material.

5.4 Concluding Remarks and Outlook A weight averaged method for computing the vibrational spectrum of complex systems is reviewed. The method combines MD simulations for structural sampling and anharmonic vibrational calculations based on DFT for computing the spectrum. The idea of local vibration is introduced, and the total spectrum is represented as a weighted sum of local vibrational spectra. Applications to pentapeptide (SIVSF-NH2 ), SM bilayers, and hydrated nylon 6 have shown that vibrational bands of amide group and water molecules are computed with sufficient accuracy comparable to experiment. Further development and extension are now ongoing. Although the applications so far involved only a small number of molecular groups (i.e. amide groups and water molecules), many hydrophilic groups may come into play in realistic systems. Thus, an automated procedure is needed, which divides the system into an ensemble of clusters requiring minimum or no human intuitive inputs. In this regard, how to recognize a cluster of groups (via HBs) is the central issue. Another concern is the validity of extracting a cluster in vacuum neglecting the effect of environment. Although such a simplicity has computational benefit, it also faces some obvious limitations; for example, the current approach is not able to discriminate amide groups in different secondary structures. Accounting for

163

5 Weight Averaged Anharmonic Vibrational Calculations

Harmonic

(a)

3a 2

3800

3600

3400

Wavenumber (b)

Experiment 3187.3 3093.1

3200

3000

(cm–1)

5b

3b

2

1A–W–1A

3a

7a 7b

6

Hydrogen stretches of water 3750 – 3200 cm–1

1

(c)

3296.9

3465.5

4000

3426.1

3b 4 5b

1

Anharmonic

5a

3248.1

ΔAbsorbance (a.u.)

164

1A–W–1A1W

Amide A 3450 – 3250 cm 4

1A–A–1W

1A–A–1A

1W–W–1A1W

1A–W–2A

(d)

–1

6

5a

1A1W–A–1A

1A1W–W–2A

Overtones ~ 3100 cm–1 7b 7a

Water bending 1W–W–1A

Amide II 1A1W–A–1A

Figure 5.10 (a) The harmonic and anharmonic difference IR spectrum of wet (8.6 wt%) and dry (0 wt%) nylon 6 in the OH/NH stretching region compared with the experimental one [84]. The calculated spectra are broadened by a Lorentzian function with a width of 80 cm−1 . The thick dashed lines indicate the contribution of water. (b–d) The representative structures found to absorb most and closest to the peak locations found in the calculated spectra. Source: Thomsen et al. [39]. Copyright 2017, Adapted with permission of American Chemical Society.

the environmental effect with reasonable computational cost is the challenge. Nonetheless, once these problems are overcome, it becomes feasible to compute the spectra of biomolecules and polymers in different conditions of temperature, pressure, and humidity that reveals a signal from functionally important structures. Experimental and computational vibrational analyses will help elucidate the molecular mechanism of biomolecular functions and/or materials performance and will be a powerful tool to lead novel molecular design.

Acknowledgments We thank Prof. M. Fujii and Prof. S. Ishiuchi and Dr. K. Shirota and Dr. T. Kobayashi for providing the experimental data and many stimulating

References

discussions. This research is partially supported by the “Molecular Systems,” “iTHES,” “Integrated Lipidology,” and “Dynamic Structural Biology” projects in RIKEN (to Y. S.), the Center of Innovation Program from Japan Science and Technology Agency (JST), JSPS KAKENHI Grant No. JP26220807 and JP26119006 (to Y. S.), and JSPS KAKENHI Grant No. JP16H00857 (to K. Y.).

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hydrogen bonds in the condensed phase. Chem. Rev. 104: 1887–1914. 2 Barth, A. (2007). Infrared spectroscopy of proteins. Biochim. Biophys. Acta

1767: 1073–1101. 3 Ganim, Z., Chung, H.S., Smith, A.W. et al. (2008). Amide I two-dimensional

infrared spectroscopy of proteins. Acc. Chem. Res. 41: 432–441. 4 Davis, E.M. and Elabd, Y.A. (2013). Water clustering in glassy polymers. J.

Phys. Chem. B 117: 10629–10640. 5 Sammon, C., Deng, C., and Yarwood, J. (2003). Polymer–water interac-

6

7 8

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6 Chiral Recognition by Molecular Spectroscopy Magdalena Pecul 1 and Joanna Sadlej 2 1 University of Warsaw,Department of Chemistry, 1/3 Wójcickiego Street, 02-093 Warsaw, Poland 2 Cardinal Stefan Wyszy´nski University,Faculty of Mathematics and Natural Sciences, 1/3 Wóycickiego Straße, 01-938 Warsaw, Poland

6.1 Introduction Chiral molecules, constituting the basis of all life forms, play a fundamental role in biology, chemistry, and medicine. Most biological molecules consist of only left-handed amino acids and right-handed sugars. The question of the origin of the homochirality of life remains far from answered [2, 3] and constitutes one of the impulses for the studies of chiroptical phenomena. On the other hand, determination of the absolute configuration of chiral molecules and the enantiomeric excess of a given stereoisomer in solution is a field of research of increasing importance. Indeed, the absolute configuration of a chiral system is critical in understanding structure–(biological) activity relationships. The two enantiomers of a chiral compound can exhibit significantly different pharmacological and toxicological effects [4]. The absolute configuration can be determined by X-ray diffraction or by chiral chromatography (GC/HPLC) or by stereo-controlled organic synthesis [5]. However, the former requires high-quality single crystals, whereas the latter is usually laborious, time consuming, and expensive. Therefore, chiroptical spectroscopic methods have experienced significant development in the last decades. Chiral molecules exhibit distinct responses to left and right circularly polarized light in absorption, emission, scattering, and refraction. We are not going to discuss the last topic, focusing instead on circular dichroism (CD) and circularly polarized luminescence (CPL) (absorption and emission, respectively) and Raman optical activity (ROA) (scattering). Electronic circular dichroism (ECD) has been measured at the same time as optical rotation (works of Biot, Arago, and Fresnel [6–8] at the beginning of XIX century) and theory allowing for prediction of its intensities has been published by Rosenfeld [9]. Later on, first measurements of CPL, the emission analog of ECD, have been accomplished (see Ref. [10] for an early review.) ECD has wide applications in many areas of chemistry and biochemistry, and CPL is recognized as one of the very few methods allowing for investigation of structures of electronically excited states. Molecular Spectroscopy: A Quantum Chemistry Approach, First Edition. Edited by Yukihiro Ozaki, Marek Janusz Wójcik, and Jürgen Popp. © 2019 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2019 by Wiley-VCH Verlag GmbH & Co. KGaA.

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Chiroptical methods based on vibrational spectroscopy, under the general name of vibrational optical activity (VOA), have been developed since 1970s [11–14]. The first of them was vibrational circular dichroism (VCD) [11], and the second was vibrational Raman optical activity (VROA/ROA) [15]. The disadvantage of these methods is that the differential response of a chiral molecule to circularly polarized light, the molar extinction coefficients, is proportional to the ratio of internuclear distance and light wavelength and thus in infrared region is ca. 10−2 to 10−3 times weaker than in the UV-VIS. The ratio of ROA to Raman intensity is even smaller. In spite of very low intrinsic intensities, the VCD and VROA methods are expected to provide information concerning molecular conformation, inaccessible by other spectroscopic methods such as NMR or X-ray. Due to progress in instrumentation and experiment, theory, and computations, chiroptical spectroscopy has evolved rapidly and is now very powerful for the reliable determination of absolute configuration, conformational details, and relative conformer populations of chiral molecules and biochemical species. The development of quantum mechanical methods reliably predicting VCD, VROA, ECD, and CPL spectra has increased the confidence in the use of these methods. Several groups have used experimental chiroptical spectroscopic methods together with the versatile numerical quantum chemical calculations, and very interesting results are demonstrated in recent review articles and books [16–25].

6.2 The Physical Manifestation of Optical Activity in Chiroptical Spectroscopic Methods: Theory of the Chiroptical Properties Theoretical models allow to understand better the relationship between the spectroscopic parameters and the structure of the samples, and they are essential for realistic modeling of these parameters as a function of structure. Only in the last 20–25 years, the first principles calculations of the chiroptic spectroscopic parameters have become possible. Intensity parameters of all chiroptical methods under discussion are “linear response properties” of a molecule, i.e. they arise from the terms linear in the field amplitudes when the interaction of a molecule with an electromagnetic wave is present. The four chiroptical spectroscopic methods under discussion are complementary in nature. The chiroptic parameters have been previously reviewed, discussing as well the impact of the environment on these parameters [16–19, 21–25]. Recently, the availability of apparatus and software for the calculations of chiroptical spectra improved significantly [19, 22]. Thus, the main purpose of this review is to document the advance to the present date. We have, therefore, restricted this review to a comprehensive summary of recent results, reported in the last five years on the VCD/ROA and CPL observables for small molecules, along with a brief review of the basic theoretical methods applied to calculate the spectroscopic parameters.

6.2 The Physical Manifestation of Optical Activity in Chiroptical Spectroscopic Methods

6.2.1

General Background

The majority of calculations of chiroptical properties use response theory. The molecule’s electron density responds to the external electromagnetic field, and this response is evaluated as a perturbation of the stationary ground state in powers of the field amplitude in the frequency domain. The parameters of the chiroptical spectra can be related to the response functions, their residues, and poles. 6.2.2

Calculations of VCD Spectra

The intensity of VCD spectra for two enantiomers L and R is of equal magnitude but of opposite sign. That is why the spectra contain information about the absolute configuration of a chiral molecule. VCD is defined as a difference in absorbance A = log10 (I∕I0 ) of a molecule for left vs. right circular polarized radiation in the infrared region of the electromagnetic spectrum ΔA = AL − AR . Under the assumption that the Beer–Lambert’s law is satisfied A = 𝜀(𝜆)Cl (where 𝜀(𝜆) is the molar extinction coefficient, C is the solute molarity, and l is the sample path-length), the difference Δ𝜀(𝜆) = 𝜀L (𝜆) − 𝜀R (𝜆) determines the VCD spectrum. Similarly as in the vibrational absorption spectra (VA), the VCD band frequency contains information on energy of a particular vibrational mode, but unlike in the VA spectra, the VCD band intensity exhibits either a positive or negative sign. There are two main groups of methods used for calculations of VCD spectrum: 1. Most popular are static calculations based on quantum chemical methods: (a) Their implementations rely on magnetic field perturbation theory (MFPT) [20, 26–30]. (b) They also rely on nuclear velocity perturbation theory (NVPT) to compute atomic axial tensors (AAT) and rotational strengths within Kohn–Sham method using plane wave basis set and Carr–Parrinello molecular dynamics (CPMD) [31–33]. This approach contains the anharmonic effects incorporated by molecular dynamics (MD) approach and simulate bulk phase dynamic VCD spectra. (c) The VCD spectra of a special group of molecules (which have weakly interacting coupled units) can be described in framework of coupled dipole excitation model (VCDEC) [34, 35]. 2. Recently, a dynamic description of VCD was proposed, in which the VCD spectra are given by the Fourier transform of the time correlation function (TCF) of magnetic and electric dipole moment in frame of ab initio molecular dynamic (AIMD) approach [36]. Molecular theory of VCD has been derived in different ways by several authors [12, 20]. The most popular and used are the perturbative approaches by Stephens [26–28] and Buckingham [13]. In the following subsection, we shortly describe the formulations based on Stephens’ expression [26–28]. 6.2.2.1

Practical Simulation of VCD Spectra

The intensity of the VA band at the molecular levels is directly related to the square of the electric dipole transition moment (EDTM) of the molecule, called

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dipole strength Dg1,g0 (i), and therefore is always positive. This dipole strength Dg1,g0 (i) for the ith mode is connected with the transition from the ground electronic (g) ground vibrational state (0) described by the Ψg0 wavefunction to the ground electronic (g) excited vibrational state (1) described as Ψg1 . The dipole strength expression also contains an absorption operator proportional to the dipole moment change in the ith mode multiplied by geometrical change of molecule in the mode, i.e. the normal mode coordinate. The corresponding intensity of the VCD band of the ith parent mode is connected to the transition moment called rotational strength Rg1,g0 (i). It represents the imaginary part of the scalar product of the electric dipole transition moment and the magnetic dipole transition moment (MDTM). Thus, the vibronic coupling needs to be considered in order to obtain reasonable VCD spectra. The nuclear terms are calculated from the vibrational modes and frequencies. The sign of the rotational strength for the ith mode is thus determined by the angle 𝛼(i) between the vector of EDTM Dg1,g0 (i) and the vector of the MDTM Mg1,g0 (i). The Eqs. (6.1) and (6.2) define the dipole strength and the rotational strength quantities at the molecular level for the ith vibrational mode: 2 Dg1,g0 (i) = | < Ψg0 |𝜇|Ψg1 > |2 = Eg1,g0 (i)

(6.1)

Rg1,g0 (i) = Im[< Ψg0 |𝜇|Ψg1 >< Ψg1 |m|Ψg0 >] = Eg1,g0 (i)Im[Mg1,g0 (i)] (6.2) It can be seen from Eq. (6.2) that rotational strength may be positive or negative depending on whether the projection of the EDTM onto the MDTM is positive or negative. If two vectors are orthogonal with respect to each other, as in achiral molecules, i.e. the angle 𝛼(i) between them is close to 90∘ , the rotational strength is zero. The vibrational MDTM in Eq. (6.2) is equal to zero for closed-shell molecules in Born–Oppenheimer approximation. To obtain the expressions for VCD intensity, it is thus necessary to go beyond this approximation. The procedure for doing this using perturbation approach is shown in Refs. [26, 27]. Ultimately, the prediction of VA and VCD spectra within the harmonic approximation requires: 1. Frequency calculations at harmonic force field (HFF), i.e. the Hessian. It requires calculations of the energy derivatives with respect to the Cartesian displacement of the 𝛾th atom from equilibrium. 2. The intensity of VA spectra, i.e. the atomic polar ) tensor (APT) defined per ( 𝜕𝜓 g 𝛾 atom 𝛾, P𝛼𝛽 . It requires only the elements 𝜕X 𝛾 . The APT tensor represents 𝛼 the derivative of molecular EDTM of the entire molecule with respect to the Cartesian displacement from equilibrium of the 𝛾th atom. 3. In addition to APT, the intensity of VCD spectra requires the so-called the 𝜕𝜓 atomic axial tensor (AAT) defined per atom 𝛾 with the elements ( 𝜕Hg )0 . The 𝛽 electronic part of AAT tensor is the overlap integral containing the derivatives of the electronic wavefunction of the ground state g with respect to the external magnetic field H𝛽 . Moreover, there arises the question of the origin

6.2 The Physical Manifestation of Optical Activity in Chiroptical Spectroscopic Methods

dependence of the AAT tensors, as it frequently happens in the case of calculations of properties involving the magnetic dipole operators [37]. The described above Stephens’s expression of wavefunction parameters perturbed with respect to nuclear and magnetic perturbation required the solution of (3N + 3) response equations (N is the number of atoms in the molecule). It is indispensable to mention different solution formulae for calculating some elements: to solve the magnetically perturbed self-consistent field (SCF) equations by Bouˇr et al. [38] and alternatively to calculate the AAT by Coriani et al. [39], which yield some savings in computational time compared with Stephens’ one.

6.2.3

Calculations of ROA Spectra

The ROA effect is described by means of the absolute difference between intensities of the inelastically scattered light by a chiral sample, with the incident light circularly polarized left and right, IkR − IkL , where IkL,R are the scattered intensities with linear k polarization for right (R) and left (L) circularly polarized incident light and k denotes the Cartesian component (incident circular polarization Raman optical activity [ICP-ROA]). Alternatively, ROA can be measured as a small circularly polarized component in the scattered light using incident light of fixed polarization, including unpolarized light (scattered circular polarization Raman optical activity [SCP-ROA]). These two approaches are equivalent in the far-from-resonance limit. Barron and Buckingham [40] gave the theoretical background for this effect. The first experimental ROA spectra were published by Barron et al. [41]; latter were confirmed by Hug et al. [42]. The calculations of Raman intensities in harmonic approximation require the knowledge of the electric dipole polarizability derivatives. ROA computations are quite similar, but more involved, since they require also the knowledge of geometry derivatives of two other optical tensors. The essentials of ROA computations are as follow. The differential scattering intensity between right and left circularly polarized light for the z-polarized backward scattering is given by [12, 40] IzR − IzL (180) = 24𝛽(G′ )2 + 8𝛽(A)2

(6.3)

where 𝛽(G ) = ′ 2

𝛽(A)2 =

v 3𝛼kiv G′ vki − 𝛼kk G′ vii

2 1 𝜔 𝛼 v 𝜖 Av 2 rad ki kjl jli

(6.4) (6.5)

𝜔rad is the radiation angular frequency, 𝜖kjl is the unit third rank antisymmetric tensor, and the other quantities are defined below. Here and in the following, the Einstein summation convention is used. Most of the computations of the ROA spectra is carried out within the double harmonic approximation, where the quantities in Eqs. (6.4)–(6.5), defined as

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products of vibrational transition moments, can be described by means of geometric derivatives of optical tensors: ( )( ′ ) 𝜕Gki 1 𝜕𝛼ki v ′v ′ (6.6) 𝛼ki G ki =< 0 ∣ 𝛼ki ∣ 1 >< 1 ∣ Gki ∣ 0 >= 2𝜔 𝜕Q 0 𝜕Q 0 𝛼kiv 𝜖kjl Avjli =< 0 ∣ 𝛼ki ∣ 1 >< 1 ∣ 𝜖kjl Ajli ∣ 0 >=

( ( ) ) 𝜕Ajli 1 𝜕𝛼ki 𝜖kjl 2𝜔 𝜕Q 0 𝜕Q 0 (6.7)

The tensors in Eqs. (6.6)–(6.7) are the electric dipole–electric dipole polarizability 𝜶, the imaginary part of the electric dipole–magnetic dipole polarizability G′ , and the real part of the electric dipole–electric quadrupole polarizability A [12]. Q is the normal coordinate of the vibration under study. The subscript 0 indicates that the quantities are calculated at the equilibrium geometry. The second equality in Eqs. (6.6)–(6.7) is valid only within the harmonic approximation [12]. The 𝜶, G′ , and A tensors can be defined in terms of sum-over-states expressions (see Ref. [43] or in the notation of modern response theory [44]: 𝛼𝛼𝛽 (−𝜔; 𝜔) = −⟨⟨𝜇𝛼 ; 𝜇𝛽 ⟩⟩𝜔

(6.8)

′ G𝛼𝛽 (−𝜔; 𝜔) = −i⟨⟨𝜇𝛼 ; m𝛽 ⟩⟩𝜔

(6.9)

A𝛼,𝛽𝛾 (−𝜔; 𝜔) = ⟨⟨𝜇̂ 𝛼 ; Θ𝛽𝛾 ⟩⟩𝜔

(6.10)

Expressions ⟨⟨A; B⟩⟩𝜔 in the above equations denote linear response functions. Greek letter in subscripts denote Cartesian coordinates. In the limit of the static field, the G′ tensor will vanish. The harmonic approach to ROA has been implemented in a number of program packages at various levels of theory, including time-dependent density functional theory (TD-DFT). It is also possible to go beyond the harmonic approximation and simulate the ROA spectra directly from MD trajectories, in the case of other vibrational spectroscopies. Recently the first method for computing ROA spectra of liquid systems from bulk phase AIMD approach was published [45, 46]. The TCF are considered, and the Fourier transform of these TCF yields the ROA spectrum [45, 46]. A more extensive analysis of the theory of ROA can be found in several reviews on the subject [24, 43, 47]. 6.2.4

Calculations of ECD and CPL Spectra

CD, the differential absorption of left and right circularly polarized light by a sample of one enantiomer, is usually expressed as the difference between the molar extinction coefficients for left and right circularly polarized light 𝜀L (𝜆) − 𝜀R (𝜆). This quantity is related to the rotatory strength n R of the transition between the ground state 0 and the nth excited state (electronic states for electronic CD, vibrational states for VCD). The rotatory strength n R was derived from quantum mechanical theory by Rosenfeld [9] and was shown for isotropic samples to be

6.2 The Physical Manifestation of Optical Activity in Chiroptical Spectroscopic Methods

given as the product of the electric dipole and MDTM, which in atomic units can be written as n

R = ⟨0|𝜇|n⟩ ⋅ ⟨n|m|0⟩

(6.11)

For oriented samples, there is also a contribution from interactions with the electronic quadrupole moment: 3 ̂ 𝛿𝛽 |0⟩ (6.12) 𝜔 𝜖 ⟨0|𝜇̂ 𝛾 |n⟩⟨n|Q 4 n0 𝛼𝛾𝛿 This contribution is purely anisotropic and thus vanishes upon orientational averaging and does not contribute in the case of isotropic samples such as a liquid. In linear response theory, the scalar rotatory strength is calculated as a residue of the linear response function [48, 49]. For a transition from the ground state |0⟩ to an excited state |n⟩, it is in the velocity and length gauges given by, respectively, n Q R𝛼𝛽

n v

=

R =

1 1 ⟨0|p|n⟩ ⋅ ⟨n|L|0⟩ = Tr{ lim (𝜔 − 𝜔n )⟨⟨p; L⟩⟩𝜔 } 𝜔→𝜔n 2𝜔n 2𝜔n

i R = − ⟨0|r|n⟩ ⋅ ⟨n|L|0⟩ = Tr{ lim (𝜔 − 𝜔n )⟨⟨r; L⟩⟩𝜔 } 𝜔→𝜔n 2

n r

(6.13) (6.14)

In these expressions, r, p, and L are the electronic position, momentum, and orbital angular momentum operators, respectively; ℏ𝜔n is the excitation energy of the nth electronic transition. The expression with the momentum operator (velocity gauge) is gauge invariant, while ensuring gauge invariance of the expression with the position operator (length gauge) requires the use of specially constructed atomic or molecular orbitals. The use of London atomic orbitals [50] (also called gauge-invariant atomic orbitals [GIAO]) is the most common choice. Using complex response functions, CD and optical rotation are interrelated, corresponding to the imaginary and real parts of the linear response functions, respectively. 6.2.5

Calculations of CPL Spectra

Intensity of CPL is also evaluated from the rotatory strength, calculated for the equilibrium geometry of the electronic excited state, instead of the ground state as in ECD. Otherwise, the procedure is the same, so we are not going to discuss it again. Instead, some attention is going to be paid to the rotatory strength for spin-forbidden transition, since its implementation is relatively novel. 6.2.5.1 Spin-Forbidden Circular Dichroism and Circularly Polarized Phosphorescence (CPP)

To obtain the rotatory strength for CPP and spin-forbidden CD corresponding to a singlet–triplet transition, one needs to derive an expression for the rotatory strength where the excited state is a triplet state |3 f ⟩ (assuming that the molecule is closed shell and that the ground state, denoted now as |1 0⟩, is a singlet): Rl1 03 f = −

ie2 1 ⟨ 0|r𝛼 |3 f ⟩⟨3 f |L𝛼 |1 0⟩ 2me

(6.15)

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e2 ⟨1 0|p𝛼 |3 f ⟩⟨3 f |L𝛼 |1 0⟩ 2m2e 𝜔3 f

Rv1 03 f =

(6.16)

with ℏ𝜔3 f = E3 f − E1 0 . If evaluated at the ground state geometry, the above rotational strengths give the ECD intensity originating from the singlet to triplet transition in absorption. If evaluated at the equilibrium geometry of the triplet excited state, on the other hand, they yield the difference of intensity of left and right circularly polarized light originating from the triplet to singlet emission process, that is, the intensity of CPP. The transition moments between a singlet state and a triplet state are obtained directly from the residue of the linear response function when relativistic two- or four-component wave functions are used [51, 52], whereas this is not the case for nonrelativistic theories, since the transition moments are forbidden. However, in the nonrelativistic case and within a perturbation theory framework, these dipole transitions become allowed when spin-orbit-perturbed ground and excited state wavefunctions are introduced [53]. In this fashion, as shown in Ref. [54], the phosphorescence dipole strength (and transition rate) from excited state |3 f ⟩ can be obtained as the square of the phosphorescence transition matrix element: Dl1 03 f = e2 ⟨1 0|r𝛼 |3 f ⟩⟨3 f |r𝛼 |1 0⟩ = e2 ℏ2 [ lim (𝜔 − 𝜔3 f )⟨⟨r𝛼 ; Hso , V 𝜔 ⟩⟩0,𝜔 ]2

(6.17)

𝜔→𝜔3 f

e2

Dv1 03 f =

m2e 𝜔23 f

⟨1 0|p𝛼 |3 f ⟩⟨3 f |p𝛼 |1 0⟩

ℏ2 e 2 [ lim (𝜔 − 𝜔3 f )⟨⟨p𝛼 ; Hso , V 𝜔 ⟩⟩0,𝜔 ]2 m2e 𝜔23 f 𝜔→𝜔3 f

=

(6.18)

and the rotational strength of CPP can be computed as [ ] ∑ ⟨1 0|r𝛼 |1 k⟩⟨1 k|Hso |3 f ⟩ ∑ ⟨1 0|Hso |3 k⟩⟨3 k|r𝛼 |3 f ⟩ ie2 l + R1 03 f =− 2 2ℏ me 1 k 𝜔1 k − 𝜔3 f 𝜔3 k 3k (6.19) [ ⋅

∑ ⟨1 0|L𝛼 |1 k⟩⟨1 k|Hso |3 f ⟩ 1k

𝜔1 k − 𝜔3 f

+

∑ ⟨1 0|Hso |3 k⟩⟨3 k|L𝛼 |3 f ⟩ 3k

𝜔1 k

iℏ2 e2 ( lim (𝜔 − 𝜔3 f )⟨⟨r𝛼 ; Hso , V 𝜔 ⟩⟩0,𝜔 ) 2me 𝜔→𝜔3 f ( lim (𝜔 − 𝜔3 f )⟨⟨L𝛼 ; Hso , V 𝜔 ⟩⟩0,𝜔 )†

]† (6.20)

=−

𝜔→𝜔3 f

(6.21)

in the length gauge and as Rv1 03 f =

ℏ2 e 2 ( lim (𝜔 − 𝜔3 f )⟨⟨L𝛼 ; Hso , V 𝜔 ⟩⟩0,𝜔 ) 2m2e 𝜔3 f 𝜔→𝜔3 f ( lim (𝜔 − 𝜔3 f )⟨⟨p𝛼 ; Hso , V 𝜔 ⟩⟩0,𝜔 ) 𝜔→𝜔3 f

in the velocity gauge.

(6.22)

6.2 The Physical Manifestation of Optical Activity in Chiroptical Spectroscopic Methods

In these equations, Hso is the spin–orbit coupling operator; the indexes i and j refer to the electrons; A refers to the nuclei; r ij is the position of particle i relative to particle j; Lij = r ij × pi is the orbital angular momentum of particle i, of spin si , with respect to the position of particle j; ZA is the charge of nucleus A; ge is the electron g factor; and 𝛼 is the fine-structure constant.

6.2.6 Electronic Structure Methods, Basis Set Requirements, and Program Packages Chiroptical properties are sensitive to the approximations made in the electronic structure calculations. Predominantly, the chiroptical linear response methods are based on: (i) density functional theory (DFT) at various levels, despite its shortcomings, (ii) TD-DFT, and (iii) perturbed coupled-cluster single-double-triple (CCSD(T)) wavefunction-based theory (WFT-CC) with large basis sets, which are needed for the calculations of the effect of electron correlation. Hartree–Fock (HF) method is generally not recommended as this approach does not take into consideration the electron correlation effect. What follows is that it became evident early that electron correlation must be included in order to obtain reliable results. Post-HF wavefunction-based methods, like CCSD(T), scale unfavorably with the size of the system. Only for smaller molecules it is possible to perform correlated wavefunction computations, like with coupled cluster (CC) in the approximate singles-doublet (CC2) [16, 18]. As the multiconfiguration self-consistent field (MC SCF) model is well suited to treat the static electron correlation effect, it was implemented for calculations in Refs. [18, 23, 55]. The most popular methods for computational studies of chiroptical properties currently are DFT and TD-DFT method [18, 24]. TD-DFT method refers to static and to linear order response calculations, instead of an explicit time-dependent formalism. Computations of the ECD and OR spectra need TD-DFT methodology, as electronic and optical rotation parameters are functions of frequency of the applied field, while simulations of the VCD spectra does not require the calculations of frequency-dependent dynamic response. Unfortunately, the exact form of the exchange–correlation functional remains unknown. Therefore, many approximate functionals have been developed and tested. Functionals used for calculations of different molecular properties can be grouped into three classes [24, 56]: (i) local-density approximation (LDA), (ii) nonlocal gradient approximation (GGA), and (iii) hybrid functionals. The accuracy of DFT calculations varies greatly with the choice of the functional. Currently, in the semiempirical, Becke-3-Lee–Yang–Parr (B3LYP) hybrid functional and/or BPW91, B3PW91 seems the most popular for rotatory strength calculations [18]. Coulomb-attenuated CAM-B3LYP [57] functional has been shown to improve on B3LYP. As far as the basis sets are concerned, the often utilized aug-cc-pVDZ and aug-cc-pVTZ basis sets led to good results. It has been suggested that in the case of organic molecules, ROA tensors can be calculated with small valence basis sets augmented at the periphery of molecules [58].

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6.2.6.1

Vibrational Optical Activity

VCD Spectra To obtain the rotatory strength for VCD, the knowledge of the electric and magnetic transition dipole moments for the vibrational transitions is needed. VCD calculations are sensitive to the DFT functional and basis sets. Many studies have been show that the hybrid B3LYP or B3PW91 functionals give good agreement with the respective results. As VCD has been recognized as a spectroscopic tool to investigate the intra- and intermolecular interactions in solutions, such systems require additional diffuse/polarization functions, require to include the basis set superposition errors (BSSE) too, and zero-point energy correction (ZPE) in evaluating the Boltzmann factors and the relative binding energies. There are two complementary approaches to the calculation of VCD and ROA spectra from electronic structure theory – the rotatory strength can be calculated either from an approximate electronic wavefunction (WFT) or from an approximate electronic density, using DFT (Kohn–Sham DFT [KST]) [16, 24]. It became evident early that electron correlation must be included in order to obtain reliable results applicable to rotatory strength. As the MC SCF model is well suited to treat the static correlation, it was implemented for calculations in Ref. [55]. However, for larger systems it becomes increasingly difficult to perform these calculations. Recently, DFT, the cost of which is similar to that of a RHF model, plays a dominant role in theoretical investigations of the vibrational rotatory strength. It can be applied to large systems, where the use of MC SCF or CC methods is not feasible. Now, DFT is routinely used to analyze the results of experimental measurements [18, 25]. ROA Spectra ROA computations are quite similar but more involved than VCD.

The ab initio computation of ROA requires calculation of the derivatives of 𝜶, G′ , and A tensors with respect to the nuclear coordinates. The first ab initio spectra were computed in 1990 using the HF approach by Polavarapu et al. [59], who employed the numerical procedure for the nuclear displacement derivatives of polarizabilities. Helgaker et al. [55] presented the GIAO calculations of ROA at the HF and MC SCF levels. An important step in the development of ROA calculations was made by using analytical geometry derivatives and DFT. It has been shown that when the same force field is used, DFT yields the results of similar quality as the CC method. The molecule’s electron density responds to the external field, and this response is evaluated as a perturbation of the stationary ground state in powers of the field amplitude in the frequency domain. Recently, TD-DFT has been developed for the calculations of ROA spectra too. The method has been implemented in a number of program packages at various levels of theory, including TD-DFT. The majority of applications of TD-DFT use response theory. Several ROA calculations have been performed using DFT; however a comparison with CC is available, for instance, for propylene oxide (PO) [22, 58]. When the same force field has been applied at DFT/B3LYP and CCSD approaches, the ROA spectra revealed similarity.

6.2 The Physical Manifestation of Optical Activity in Chiroptical Spectroscopic Methods

In resonance Raman optical activity (RROA) and simulations of surface enhanced resonance Raman optical activity (SERROA), we need to add the damping factor to account for finite lifetime of the electronic excited state [21]. 6.2.6.2

Electronic Structure Methods: ECD, CPL, and CPP

Description of excited states by means of quantum chemical methods is challenging in general, and electronic chiroptical spectra are no exception. Most of the calculations nowadays are carried out using time-dependent (or linear response) DFT, with excitation energies computed as location of the poles and transition moments obtained as residues of the response functions. This approach usually works reliably enough, but when it fails, it is very difficult to improve the calculations. Calculations of CPL intensities require the same transition moments as ECD intensities but are evaluated at the geometry minimum on the excited electronic state potential energy surface (PES). Consequently, in order to calculate CPL spectra, one needs to have tools to optimize molecular geometry in the excited electronic state, which is not trivial. These days, it is usually done by implementing gradients of excitation energy calculated by response method, as in TD-DFT/linear response DFT [60, 61]. The calculations of rotatory strength for singlet–triplet transitions (for the spectra CPP and spin-forbidden CD) as double residue of a quadratic response function have been described in Ref. [54]. 6.2.6.3

The Basis Set Requirements: VCD, ROA, ECD, and CPL

Selection of a proper basis set poses an additional problem in ab initio calculations. The basis set choice for rotatory strength calculations is cumbersome, and only a few suggestions can be provided. It appears from the computational experience that the standard basis sets (for instance, 6-31G* and 6-31G**), usually used for the optimization of geometry, are not adequate to predict the spectra correctly. Satisfactory results can be obtained with the cc-pVXZ correlation-consistent basis set [58]. Raman intensities are known to be sensitive to diffuse augmentation of the basis set. Thus ROA tensors require basis sets with diffuse functions too. Moreover, Raman intensity and ROA intensity differences are also dependent on the quality of the force field via the transformation to normal coordinates [58]. It should also be kept in mind that a description of the interacting systems (by H-bonds or van der Waals interactions), such as those discussed in this review, requires an addition of diffuse functions. In the case of electronic chiroptical spectra, the basis set requirements depend on the nature of the transition and are basically similar as for the EDTMs. It means that the standard basis sets, enriched with diffuse functions, are usually sufficient for transition to valence states, while description of the Rydberg states is problematic. Unlike optical rotation, rotatory strengths converge pretty fast with the basis set size in velocity gauge formulation (which may be preferable than length gauge since it does not require the use of GIAO) [62].

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6.2.6.4

Software

These days, calculations of chiroptical spectra are frequently performed to interpret and supplement experimental results, even by nonspecialists. Software packages that are capable of VCD and ROA computations are available commercially and under noncommercial licenses. We present here, in alphabetical order, the most popular programs implementing analytical derivatives in the calculations of all tensors of DFT/TD-DFT chiroptical response computations: (i) Amsterdam Density Functional (ADF) [63, 64]; (ii) CADPAC [65]; (iii) Dalton (VCD, ROA, GIAO) [66, 67] with its linear scaling version LSDalton (Linear-Scaling Dalton) [68]; (iv) Gaussian (G09), also includes the necessary subroutines for analytical derivatives of all the required ROA tensors [69]; (v) MOVIPAC (VCD) [70]; (vi) NWChem [71]; (vii) ORCA [72]; (viii) Psi4 [73]; and (ix) Turbomole (VCD, ROA, ECD) commercial [74]. Considerable progress has been reported in the CC description, the calculations of the electronic structure, and the molecular properties of molecules at higher levels of approximation of the electron correlation effect. For the CC method, the expression for magnetic dipole vibrational transition moment must be evaluated with consideration of the differing left- and right-hand CC functions. There are now two available quantum chemistry program packages, Dalton [66, 67] and Psi4 [73], capable of computing the chiroptical response functions and CD spectra using the CC linear response method.

6.3 Selected Case Studies This section reviews some important aspects of the computational models along with the recent developments. They are: (i) The modeling of chemical environment, (ii) the role of anharmonicity of CD spectra, and (iii) the induced VCD activity as an access to solute–solvent interactions. These points are illustrated by literature. 6.3.1 Taking into Account Chemical Environment, Conformational Flexibility, and Vibrational Corrections in the Calculation of Chiroptical Spectra Experimental CD spectra can be registered for gas phase, and for the condense phase-solid state, or, mostly, in solution. Water is the favorable solvent for CD spectra ranging from small organic molecules to biomolecules, but not for VCD, for which D2 O is usually required, since the bending vibration mode of H2 O overlaps with the amide mode. In contrast to VCD, ROA spectra of protein and peptide may be observed in water solution; however one has to remember about the poor sensitivity of ROA and reduced signal-to-noise ratio. Environment influences the electronic density and, in specific cases, also molecular conformation. Especially when hydrogen bonds or other specific interactions between solvent and solute are present (as in aqueous solution), it is expected that the geometry of the solute could be modified. Therefore modeling of the solvent effects is essential for rendering CD spectra of given molecules in solution.

6.3 Selected Case Studies

6.3.1.1

Matrix-Isolation CD Spectra

The analysis of VCD spectra provides access to molecular structure. More insights into the conformations can be obtained when a matrix isolation (MI) with the low-temperature matrices spectroscopy is used. The pioneering works by Schlosser et al. [75] and Henderson and Polavarapu [76] have opened a possibility for experimental investigations using the matrix-isolation vibrational circular dichroism spectra (MI-VCD). Later, Tarczay et al. [77–79] published the MI-VCD experimental evidence that the absolute conformation strongly depends on the geometry of the chiral–achiral molecular complex. Based on the analysis of Ar and Kr matrix-isolation VCD spectra of simple models of peptides and protein, Ac-Gly-NHMe and Ac-L-Ala-NHMe, the authors found two hydrogen-bonded conformers of Ac-L-Ala-NHMe, a typical chiral building unit of peptides. The MI-VCD spectra could distinguish between different conformers and could be interpreted more precisely than the MI-IR because the rotatory strength of some vibrational transitions changes as a function of the backbone torsional coordinates. Recently, the MI-VCD spectra of propylene oxide (methyl oxirane, PO) was published and was compared with the anharmonic spectra calculated at the B3LYP/6-311++G(3df,3pd) level [80, 81]. These results are presented in Section 6.3.2.2. 6.3.1.2

Modeling of Solvent Effects in the Chiroptical Spectra

Solvation effects significantly influence the CD and ROA spectra. These interactions can influence: (i) The population of conformational states of the solute molecule. (ii) The single conformer parameters of the solute molecule, when there are few different conformers of solute molecule in the solvent. (iii) The solvent that can determinate the dimension of the aggregates of solute molecules. There are three general approaches to the problem of solvent effects, with some variations extending and mixing these solutions (such as one by Perera et al.) [82]: 1. The supermolecular approach (explicit, discrete), in which solvent molecules and solvated molecule are treated by quantum methods. In the following model, the specific intermolecular interactions are probed explicitly, while bulk solvent effects are neglected. The protocol of this approach involves averaging of the computed parameters for solvated structures along a computation (quantum mechanics/molecular dynamics). 2. A family of continuum models (implicit), in which the solvent is treated as a macroscopic dielectric and isotropic, polarizable continuum model (PCM), while the solute is described by quantum methods. This model is computationally efficient, far less computationally demanding than the explicit one and easy to apply. 3. As a third model one can use the so-called hybrid model, which merges the other two: the first solvation shell is treated by quantum model, as the solvated molecule, while the remaining solvent shells are described by PCM. An extension of hybrid model is the micro-solvation model, called “the cluster-in-a-liquid model,” proposed by Xu group [82]. It is the most promising, yet time-consuming, multiscale approach. The authors advocated a protocol,

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which combines both methods (1) with (2); however, the supermolecular part here is the solute molecule–(solvent)n clusters, and then such a system is placed in a continuum solvent model (for example, PCM, COSMO, or ONIOM models) [82]. The CD spectra are sensitive to environment; therefore this field of research is growing. Solvation is very important also for simulations of VCD and ROA spectra. One can use continuum (implicit) or discrete (explicit) model. Popular examples are the various flavors of PCMs. The hybrid model seems to be the best approach currently available. As a continuum model the integral equation formalism (IEF-PCM) method is the most widespread. In this model a molecule-shaped cavity is used to define the boundary between the solvent and solute molecule [83–86]. Klamt et al. [87, 88] proposed conductor-like screening model (COSMO), in which the solvent is modeled as a conductor instead of a dielectric medium. The description of solvent effect often needs a great number of snapshots from molecular dynamics simulation. Therefore, the cluster-in-a-liquid model merges the chiral solute clusters with explicit model and, when combined with implicit model, yielded very good agreement with the experimental data for water and other protic solvents [82]. It opens up a new generation of methods that can be used for simulations of VCD and ROA spectra and shows better agreement between calculations and experiment than the previously used hybrid methods. The most demanding approach, from a computational viewpoint, would be to perform dynamic simulation of the solute embedded in a large number of solvent molecules, all treated with quantum chemical methods, and it is clearly desirable. Several published studies have examined in careful detail different models of solvation effects and how they work in ROA spectra simulations [89–92]. Summarizing this point let us mention the paper by Hopmann et al. [89] who studied the effect of solvation model on the Raman and ROA spectra of lactamide and 2-aminopropanal. The authors used PCM, supermolecular model, classical MD, and Car–Parrinello MD. Of course, Car–Parrinello dynamic simulations gave the best agreement with the experimental Raman and ROA spectra, better than classical MD simulation, while PCM gave only the basic ROA intensity pattern. However, for flexible many-atom molecules, the required simulation time dramatically rises, and the ab initio MD calculations are impractical. It has been shown, for example, for l-cysteine [90] that in some case a model of hydration by only a few water molecules can reproduce the essential features of the experimental VCD and ROA spectra (see Figure 6.1). 6.3.1.3

Environment-Induced CD and CPL Activity

When a normally achiral molecule is placed in a chiral environment, it can exhibit induced optical activity. This phenomenon has been employed, for example, in using CD of achiral thioflavin-T, an optical probe of protein misfolding in formation of amyloid fibrils [93]. Later, it has been shown that thioflavin-T intercalated in amyloid fibrils exhibits also CPL. Quantum chemical calculations suggest that this optical activity in absorption and emission spectra is most probably mainly due to proximity of aromatic side chains of amino acids: proximity of an aromatic

6.3 Selected Case Studies

Raman (IR+IL)

Experimental 4H2O clusters

ROA (IR–IL)

Experimental 4H2O clusters

2000

1800

1600

1400

1200 ν

1000

800

600

400

[cm–1]

Figure 6.1 Comparison of the Raman (upper graph) and ROA (lower graph) obtained from ´ et al. 2012 [90]. optimized L-cysteine–water clusters with experiment. Source: Kaminski Reproduced with permission from American Chemical Society.

ring has larger effect on the rotatory strength than proximity of any other group that may be present in such environment [94]. 6.3.1.4

Conformational Averaging of the Chiroptical Spectra

Before we compute a chiroptical spectra for a molecule, it is necessary to determine its PES and to perform conformational averaging, because a single static structure at 0 K is not adequate to describe chiroptic properties. Averaging of static gas-phase structures, zero-point energy, temperature vibrational averaging, and inclusion of solvent effects on static structure are needed. If more than one conformer of a molecule is significantly populated in solution at given temperature, the observed spectrum is a linear composition of the population-weighted spectra of each of the conformers present. Vibrational CD can resolve conformers that persist for psec or longer, as vibrational periods are typically ca. 1 ps. Therefore, stereo-information obtained by IR and VCD spectra cannot be obtained by any other physical methods. Moreover, different conformers of a molecule can have different signs of the chiroptical properties. Consequently, when there are several conformers similar in energy, it is essential to take into account all of them. An average over vibrational structures is obtained automatically, if the molecular dynamics simulations or Monte Carlo are performed. 6.3.1.5

The Concept of Robustness

Before we present the examples of the experimental and associated theoretical analysis, let us discuss “the concept of robustness.” The sign of a VCD band

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intensity depends on the angle between the EDTM and MDTM vectors. This means that small conformational changes can have a strong effect on the VCD intensity. Therefore, Nicu and Baerends [95, 96] advocate that the predictive modes for VCD spectra are only these ones, which fulfill “the robustness criteria”: the angle between the EDTM and MDTM should be significantly different from 90∘ . Gobi and Magyarfavi proposed to introduce the so-called dissymmetry factor [97] as a criterion for this. The problem was studied in many papers, as reviewed, for example, by Polavarapu [25]. In addition, to determine the absolute configuration, one can use only the intensity sign of these normal modes, what are sufficiently close to the chiral center, so-called “chiral part” of the molecule [98]. 6.3.2

Modeling of Anharmonicity in VCD and ROA Spectra

Majority of the chiroptical spectra calculations are carried out in frame of double harmonic approximation (both mechanical and electric/magnetic). Similarly as in IR spectra, the anharmonicity effect has been shown to be crucial in achieving the agreement for VCD spectra interpretation. Without anharmonic force fields it is not possible to obtain a close match between computed and experimental vibrational frequencies, as anharmonic effects might modulate the observed overall VCD band shape. Moreover, harmonicity cannot provide any information about the intensities of overtones and combination bands. 6.3.2.1

Computational Approaches to Anharmonicity

Two main schemes to develop the anharmonic approaches were formulated in the framework of (i) perturbative [99, 100] and (ii) variational schemes [101]. The first approach there is a formulation of the method based on the Rayleigh–Schrödinger perturbation method including anharmonic corrections at the second-order level of vibrational perturbation theory referred to as VPT2 by Barone and coworkers [99, 100]. Such calculations have been performed to PO molecule isolated in an argon matrix by MI-VCD method, and the anharmonic spectra assign all bands observed even in the fingerprint region, including the combination modes and overtones [80, 119]. Recently, a virtual multi-frequency spectrometer (VMS) system was developed by Barone group [99, 102]. This system is now extended to a model including both mechanical and electric/magnetic anharmonicity of OR, Raman, VCD, and far-from-resonance ROA spectra [102]. Usually, the measurements are done far from resonance. However, when VPT2 approach (energies and transition moments) could be plagued by resonance, e.g. the presence of Fermi resonances, the VPT2 could remove the resonance, as (i) deperturbed VPT2 denoted (DVPT2) or as (ii) the generalized second-order vibrational perturbation approach (GVPT2) and degeneracy-corrected method DCPT2 [102]. The quality of the anharmonic frequencies depends on the parent harmonic ones. 6.3.2.2

The Role of Anharmonicity in VCD and ROA Spectra: Examples

At this point it is worthwhile to investigate the anharmonic contributions to the simulated spectra. Anharmonic effects have been considered in VCD and ROA spectra calculations (see Refs. [102–105] and references therein). The first

6.3 Selected Case Studies

calculation of the anharmonic VCD for PCM-solvated systems was published by Cappelli et al. [106, 107]. In addition to the anharmonic correction, averaging over the nuclear vibrational motions is desirable for solvated systems. The average of a calculated CD parameter contains a term from the anharmonicity and a term from the curvature of a given CD parameter surface that causes vibrational corrections. This is usually a significant correction (on the order of 20%), although at present such calculations are not routinely performed. As an example a small molecule propylene oxide (PO) has been chosen – PO, the well-known benchmark as a prototype to study anharmonic effects on VCD spectra calculations by Merten et al. [104]. The matrix-isolation MI-IR and matrix-isolation MI-VCD spectra at 10 K and the spectra in solution CCl4 at fingerprint region of both R,S enantiomers are known for this molecule. Figure 6.2a displays the experimental IR, VCD measured in an Ar matrix and in solution, and the calculated harmonic spectra. Figure 6.2b presents the experimental MI-IR and MI-VCD spectra, which are compared with the anharmonic calculated spectra. The results are summarized below: (i) Spectra of both enantiomers of PO show good mirror-image symmetry in both experiment and computational simulations. (ii) All fundamental frequencies in the experimental IR data in solution can be identified in the IR calculated harmonic spectra. (iii) The calculated intensities of VCD spectra at the harmonic approximation correspond well to the experimental ones. (iv) With the help of the anharmonic calculated VCD spectra, one could assign new bands (marked in the figures by asterisks): the overtones of the 𝜈3 and 𝜈2 as very weak bands at 818 and 749 cm−1 and the combination modes at 1362 cm−1 . (v) However, the failure to describe the region in the region 1525–1480 cm−1 is noticed. This means that the description of the not-resolved combination or overtone bands needs further developments of the anharmonic theory. 6.3.3 The Induced VCD Intensity Monitored by Experimental or Calculated VCD Spectra This review aims to address also the question of how to relate the characteristic pattern of the VCD spectra to the intermolecular hydrogen bonding in the studied complexes. We will focus on representative studies, in order to illustrate the importance of the VCD spectra of hydrogen-bonded systems and the possibility to use this method for investigation of the interacting systems and the spectral fingerprint of the induction of VCD intensity1 upon complexation. This review is complementary to the few articles published earlier by Zehnacker and Suhm who discuss, among other problems, the interaction-induced conformational changes [109], by Crassous about the chiral “transfer” in coordination complexes [110], on “induced VCD intensity” in supramolecular systems by Hembury et al. [111], Dobrowolski and Sadlej group [112–116], Xu group [117–121], and Merten [80, 104, 122]. 1 We use the “induced VCD intensity” terminology after Merten, who maintains that: “It is important to note that the induced VCD is not related with ‘transfer of chirality’ as it is often referred to in literature, nor to an ‘induction’ of chirality” [108].

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Figure 6.2 (a) Experimental IR and VCD spectra of PO in the fingerprint region measured in an argon matrix and in CCl4 and the corresponding scaled harmonic spectra at the B3LYP/6-311++G(3df,3pd) level. (b) Fingerprint region of the experimental MI-IR and MI-VCD spectra of PO compared with the calculated anharmonic spectra (B3LYP/6-311++G(3df,3pd)). The asterisks mark previously unassigned bands. Source: Merten et al. 2013 [104]. Reproduced with permission from John Wiley and Sons.

6.4 Perspective

In frame of the fixed partial charge model, “the induced chirality,” according to Merten, “can be explained as a borrowing of magnetic dipole moment intensity from other part of the complex” [108]. The concept of robustness is likely to be useful for both VCD and ROA [123]. State-of-the-art calculations have been essential for clarification of the conformational preferences of the interacting systems. It was found in several mentioned papers that the VCD spectra of different conformers often exhibit not only different shapes but also different signs, making it easier to get conclusive interpretations. Experimentally, the induced VCD intensity effect was observed by Gobi et al. [124], Debie et al. [125], and Xu groups [121, 122, 126]. Very special example of the induced VCD intensity is the observation on the water (solute molecule) bending mode (HOH) (1550–1650 cm−1 ) upon interaction with such chiral solute, as lactic acid, propylene oxide, glycidol, cysteine, and others [126]. This means the water bending mode gained VCD intensity due to the interaction of the chiral solute molecule (i.e. lactic acid) with non-chiral solvent (water). Similar to water, the bending mode of non-chiral ammonia (HNH) (1600 cm−1 ) interacting with chiral methyl lactate in a solid rare gas matrix gains the VCD signal. The experimental VCD spectra of S-propylene oxide (S-PO) are known in a few solvents: benzene, CCl4 , and H2 O [82, 119]. Figure 6.3 presents three series of spectra. Figure 6.3a illustrates the VCD experimental spectra in four solvents. The spectra in water, contrary to others, have an additional band at ca. 1600 cm−1 of different signs for both enantiomers S-PO and R-PO, while for other solvent does not. Achiral water molecule forms a hydrogen bond with chiral PO molecule, and the result of this interaction is the induced VCD intensity at the water binding vibration. To answer the question on the structure of the complexes water–PO, the authors presented in Figure 6.3b the experimental data and compared them with the simulated spectra of two forms of complexes. Noticeably, water can form a complex from the same side as the methyl group (syn form) or from the opposite side (anti form). The syn-PO–water complex has the positive intensity, while the anti-PO–water has negative bending mode of water. The conclusion is that anti-PO–water seems to be the dominating form. Figure 6.3c shown the simulated VCD spectra of the complexes PO–(water)2 and PO–(water)3 , forms syn and anti. This part of Figure 6.3 shows similar VCD intensities in syn and anti forms. Therefore, the conclusion is as follows: aqueous 1 : 1 water–PO complex is the dominant form, and moreover, the anti-form is preferred and contributes to the experimental spectra [82, 119]. Recently, Merten proposed to use the induction of the VCD intensity due to intermolecular interaction between chiral and non-chiral species as a “diagnostic marker” of interactions and proposed to use the introduction of chirality into given complex system to study the interaction between catalyst and reactant [108].

6.4 Perspective The past decade has seen a great progress in experimental and theoretical methodologies for prediction of the vibrational and ECD. In this chapter, we

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Figure 6.3 (a) Experimental VCD spectra of S-PO in various solvents. Dotted line indicates measurement with R-PO. (b) Comparison of the experimental VCD of S-PO in water with the simulated VCD spectra of the anti- and syn-PO–water complex. (c) Simulated VCD spectra of the two most stable conformers of PO–(water)2,3 . Source: Losada et al. 2008 [119]. Reproduced with permission from American Chemical Society.

References

briefly reviewed the application of theoretical studies to understand the spectra of chiral small molecules, especially the vibrational (VCD) and ROA, as well as CPL spectra. We have discussed some recently published aspects of the computational models and methodological developments as: (i) The role of the anharmonicity in VCD and ROA spectra, (ii) the approach to calculate the spin-forbidden CD and circular polarized phosphorescence, (iii) the application in unveiling solvent effects as “clusters-in-a-liquid” approach, and, finally, (iv) the influence of the intermolecular interaction on the CD spectra. However, several challenges still exist in computational chiroptical spectroscopy in the years to come. We would like to mention some major future directions: (i) state-of the-art choice of the method of calculation, e.g. to use CC response methods and to add in DFT approach a long-range corrected functionals for larger systems; (ii) development of new basis sets for the CC and DFT levels of theory, especially because the requirements of the basis set for ROA intensity calculations are more demanding than for VCD calculations; and (iii) effective methods to design molecular dynamics protocols for description of dynamic systems. The development of the NVPT approach within AIMD based in the TCF formalism for the calculations of dynamical spectra in liquid phase [33, 45, 46] is very promising and is highly likely to open a new chapter of chiroptical spectroscopy.

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function methods for calculation of anharmonic vibrational energies and vibrational contributions to molecular properties. Phys. Chem. Chem. Phys. 9: 2942–2953. Bloino, J., Biczysko, J., and Barone, V. (2015). Anharmonic effects on vibrational spectra intensities: infrared, Raman, vibrational circular dichroism, and Raman optical activity. J. Phys. Chem. A 119: 11862–11874. Danˇecˇ ek, P., Kapitán, J., Baumruk, V. et al. (2007). Anharmonic effects in IR, Raman, and Raman optical activity spectra of alanine and proline zwitterions. J. Chem. Phys. 126: 224513. Merten, Ch., Bloino, J., Barone, V., and Xu, Y. (2013). Anharmonicity effects in the vibrational CD spectra of propylene oxide. J. Phys. Chem. Lett. 4: 3424–3428. Bloino, J. (2015). A VPT2 route to near-infrared spectroscopy: the role of mechanical and electrical anharmonicity. J. Phys. Chem. A 119: 5269–5287. Cappelli, C., Bloino, J., Lipparini, F., and Barone, V. (2012). Toward ab initio anharmonic vibrational circular spectra in the condensed phase. Phys. Chem. Lett. 3: 1766–1773. Cappelli, C., Monti, S., Scalmani, G., and Barone, V. (2010). On the calculations of vibrational frequencies for molecules in solution beyond the harmonic approximation. J. Chem. Theory Comput. 6: 1660–1669. Merten, Ch. (2017). Vibrational optical activity as probe for intermolecular interactions. Phys. Chem. Chem. Phys. 19: 18803–18812. Zehnacker, A. and Suhm, M.A. (2008). Chirality recognition between neutral molecules in the gas phase. Angew. Chem. Int. Ed. 47: 6970–6992. Crassous, J. (2009). Chiral transfer in coordination complexes: towards molecular materials. Angew. Chem. Int. Ed. 38: 830–845. Hembury, G.A., Borovkov, W., and Inoue, Y. (2008). Chiral transfer in coordination complexes: towards molecular materials. Chem. Rev. 108: 1–69. Rode, J.E. and Dobrowolski, J.Cz. (2003). VCD technique in determining intermolecular H-bond geometry: a DFT study. J. Mol. Struct. THEOCHEM 637: 81–89. Cz. Dobrowolski, J., Jamróz, M., Kolos, R. et al. (2007). Theoretical prediction and the first IR-matrix observation of several l-cysteine molecule conformers. Chem. Phys. Chem. 8: 1085–1094. Cz. Dobrowolski, J., Jamróz, M.H., Kolos, R. et al. (2013). IR low-temperature matrix, X-ray and ab initio study on l-isoserine conformations. Phys. Chem. Chem. Phys. 12: 10818–10830. Rode, J.E., Cz Dobrowolski, J., and Sadlej, J. (2010). VCD spectroscopy as a novel probe for chirality transfer in molecular interactions. Chem. Soc. Rev. 39: 1478–1488. Rode, J.E., Dobrowolski, J.Cz., and Sadlej, J. (2005). Phenylisoserine in the gas-phase and water: ab initio studies on neutral and zwitterion conformers. J. Mol. Mod. 17: 961–970. Losada, M. and Xu, Y. (2007). Chirality transfer through hydrogen-bonding: Experimental and ab initio analyses of vibrational circular dichroism spectra of methyl lactate in water. Phys. Chem. Chem. Phys. 9: 3127–3135.

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hydrogen-bonding: experimental and ab initio analyses of vibrational circular dichroism spectra of methyl lactate in water. J. Chem. Phys. 128: 014508. Losada, M., Nguyen, P., and Xu, Y. (2008). Vibrational absorption, vibrational circular dichroism, and theoretical studies of methyl lactate self-aggregation and methyl lactate-methanol intermolecular interactions. J. Phys. Chem. A 112: 5621–5627. Yang, G. and Xu, Y. (2009). Probing chiral solute-water hydrogen bonding networks by chirality transfer effects: a vibrational circular dichroism study of glycidol in water. J. Chem. Phys. 130: 164506–164509. Liu, Y., Yang, G., Losada, L., and Xu, Y. (2010). Vibrational absorption, vibrational circular dichroism, and theoretical studies of methyl lactate self-aggregation and methyl lactate-methanol intermolecular interactions. J. Chem. Phys. 132: 234513. Merten, Ch., McDonald, R., and Xu, Y. (2014). Evidence of dihydrogen bonding of a chiral amine-borane complex in solution by VCD spectroscopy. Angew. Chem. Int. Ed. 53: 9940–9943. Tommasini, M., Longhi, G., Mazzeo, G. et al. (2014). Mode robustness in Raman optical activity. J. Chem. Theory Comput. 10: 5520–5527. Gobi, S., Vass, E., Magyarfalvi, G., and Tarczay, G. (2011). Phys. Chem. Chem. Phys. 13: 13972–13984. Debie, E., Bultinck, P., Herrebout, W., and van der Veken, B. (2008). Solvent effects on IR and VCD spectra of natural products: an experimental and theoretical VCD study of pulegone. Phys. Chem. Chem. Phys. 10: 3498–3508. Yang, G. and Xu, Y. (2011). Vibrational circular dichroism spectroscopy of chiral molecules. Top. Curr. Chem. 298: 189–236.

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7 Quantum Approach of IR Line Shapes of Carboxylic Acids Using the Linear Response Theory Paul Blaise, Olivier Henri-Rousseau, and Adina Velcescu Laboratoire de Mathématiques et de Physique (LAMPS EA 4217), 52 Avenue Paul Alduy,, 66860, Perpignan cedex, France

7.1 Introduction In this chapter we will attempt to show how the spectroscopic study of chemical compounds is inextricably linked to quantum theory. We shall take as an introductory example the study of the hydrogen-bonded compounds that have been the subject of nearly a century of sustained efforts. More particularly, we will see how the infrared (IR) spectra of these compounds can theoretically be reproduced with some success. We will thus review the different contributions to the knowledge of these spectra in order to show how the latest theories using quantum concepts are at work to explain the details of these spectra.

7.2 The Characteristics of the Infrared Spectra of Hydrogen-Bonded Species 7.2.1 Recall on the General Features of the IR Spectra of Systems Containing Hydrogen Bonds A single hydrogen-bonded system (H-bonded system) is the interaction between a hydrogen atom H and two electronegative groups X and Y as it is shown in Figure 7.1. While the X—H bond is of a covalent nature, the H· · ·Y H X Y bond is essentially electrostatic in nature. A large number of chemical compounds contain hydroHydrogen bond gen bonds (H-bonds). It goes from water to more complex molecules like DNA. The importance of this type of link has Figure 7.1 Hydrogengiven rise over the years to many experimental and theoret- bonded system. ical works. Among the many theories that explain the particular behavior of hydrogen bonded compounds, some recent ones are ab initio quantum methods, used for instance for the acetic acid dimers in Ref. [37]. Molecular Spectroscopy: A Quantum Chemistry Approach, First Edition. Edited by Yukihiro Ozaki, Marek Janusz Wójcik, and Jürgen Popp. © 2019 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2019 by Wiley-VCH Verlag GmbH & Co. KGaA.

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For our part, we would like to show how an approach involving the fundamental properties of the normal modes of vibration at work in this type of molecule and using the tools of quantum mechanics can be considered for a more intuitive understanding of the processes leading to an explanation of the IR spectra of hydrogen bonded species. This approach concerns the main vibrational modes of a hydrogen-bonded system, which are: – The 𝜈s (X—H) high-frequency stretching mode. – The 𝜈s (X—H· · ·Y) stretching low-frequency mode. The hydrogen bonding has considerable effects on the spectral density of the 𝜈s (X—H) stretching mode by comparison with a free hydrogen stretching band. These effects are: – – – –

A strong shift or the frequencies toward lower values. An extensive broadening of the line shapes with peculiar sub-bands. A strong enhancement of the band intensities. The existence of changes in frequencies and intensities correlative to an isotopic change in the H-bonded system. – The band shapes are often remaining in the same structure in the gas and condensed phases. It is also known that the other IR transitions do not show any important change when a hydrogen bond is formed. For a quantitative information on the properties of an H-bond, the redshifts of the 𝜈s (X—H) stretching mode may be as large as several hundred of cm−1 , and the intensity increases up to thousandfold. The changes of the various properties are interrelated, and it is the interaction energy, often termed strength, that is usually taken as the reference parameter for correlations of the most conspicuous spectroscopic properties involving the bandwidth and more generally the band shape. In the following table are given the order of magnitude of the bandwidths for different kinds of H-bonds with respect to their strength. Very weak Weak Medium Strong ≃15 − 300 cm−1 ≃300 − 1000 cm−1 ≃1000 − 1500 cm−1 ≃2000 cm−1 Moreover there is a reverse relationship between the frequency shift and the length R(X· · ·Y) as well as between this length and the equilibrium distance R(X—H) and between the later and this same shift [1], Novak [2]. Besides, the substitution of deuterium for hydrogen leads to a considerable lowering of the angular frequency. Moreover, the isotopic frequency ratio 𝜈s (X—H)/𝜈s (X—D) may be considerably lower than that of the free X—H group [2]. On the other hand, the large increase of the high-frequency bandwidth is but one challenge for the nuclear dynamics theories of hydrogen bonding. Others are the band asymmetry; the appearance of subsidiary absorption maxima and minima, such as windows; and the peculiar isotope effects on frequency shifts. There are several mechanisms that influence the evolution of the highfrequency band shape. They may be separated into intrinsic and medium-related

7.2 The Characteristics of the Infrared Spectra of Hydrogen-Bonded Species

or extrinsic ones. Band shapes in the low pressure gas phase are obviously owing exclusively to mechanism determined by the nuclear dynamics of the isolated hydrogen-bonded complex; this also is true to a good approximation for dilute noble-gas matrices. The only complementary broadening effect under these conditions is due to intermolecular collisions. In dilute solution, the medium (inert solvent, for instance) takes the important role of the thermal bath. In the more concentrated solution and pure liquid state, several other mechanisms arise such as differences in the metrics of association. These are absent in crystalline solids, but instead the correlation field splitting and phonon coupling effects have to be considered. A theory that is susceptible to take into account these dramatic changes in the IR spectrum may be considered to give some understanding of the dynamics of the hydrogen bond. One may classify theories dealing with the shape of IR spectrum as first and second generation, the first one being qualitative and the last one quantitative. Here we shall limit us to describe the quantitative theories that are lying in the framework of the so-called linear response theory. The physical idea at the basis of all the quantitative theories is the assumption of an anharmonic coupling between the high-frequency mode and the hydrogen bond bridge. Related to this main mechanism must be quoted Fermi resonances, Davydov splitting [3], and tunneling effect. This anharmonic theory will be exposed later. Among the quantitative theories, there are a lot that are involving the linear response theory. Thus, it is suitable to give some information on the linear response theory and its application to absorption spectroscopy. 7.2.2

The Linear Response Theory

The linear response theory describes the way in which a molecular system responds to a small perturbation. Under this assumption the change in the expectation value of any operator is linear in the perturbing source. The response function is then independent of the perturbation and depends only on the system. The main results of this theory were formulated by Kubo in 1957 in an important paper [4]. Consider a group of molecules, the Hamiltonian of which is H, that are exposed to a monochromatic electromagnetic field E(t, 𝜔): E(t) = E0 cos(𝜔t)

(7.1)

If the pulsation 𝜔 of this field is near that of a characteristic transition of the molecule, it is well known that the electromagnetic field will induce transition between the corresponding energy levels of the molecule. If the interaction between the radiation and matter is approximated by the electric dipole interaction, which is valid when the wavelength is large compared to the molecular dimensions, the interaction Hamiltonian describing the coupling V (t, 𝜔) between the field and the dipole moment of the molecule is given by V (t) = 𝜇 ⋅ E(t)

(7.2)

where 𝝁 is the dipole moment operator of the molecule. If the perturbation of the molecule by the electromagnetic field is small, the time evolution operator

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7 Quantum Approach of IR Line Shapes of Carboxylic Acids

U(t) of the system may be expanded up to first order. That leads to the following equation that is linear in the electromagnetic field: U(t) = U∘ (t) +

t

1 ∘ U∘ (t ′ )† {E(t ′ ) ⋅ 𝝁}U∘ (t ′ )dt ′ U (t) ∫0 iℏ

(7.3)

Here U∘ (t) is the time evolution operator of the molecule in the absence of electromagnetic field. Following an approach of Gordon [5], it is shown how, in the framework of this linear response theory, the spectral density I(𝜔) characterizing the molecular transition may be related to the autocorrelation function (ACF) G(t) of the to eq. (7.4) +∞

I(𝜔) =

1 e−i𝜔t G(t)dt 2π ∫−∞

(7.4)

with G(t) = ⟨𝝁† (0) ⋅ 𝝁(t)⟩0

(7.5)

and where 𝝁(0) is the dipole moment operator at initial time and the same operator at time t given in the Heisenberg picture by 𝝁(t) = eHt∕ℏ 𝝁(0)e−Ht∕ℏ

(7.6)

In the linear response theory, attention is focused on the time dependence of the ACF. An important point is that no assumption is made in the obtainment of the ACF of the dipole moment operator about random motion of the dipoles since these dipoles are moving in time according to the Hamiltonian H of the system whatever it is. Thus, it is erroneous to think that a correlation function can be used only when a system is undergoing some kind of stochastic motion. Of course, when there are random motion of the dipoles due to the influence of the medium, the correlation function G(t) describes the decay of our knowledge about the system as it approaches equilibrium. If, for instance, we know the direction of the dipole moment at a given time, then, after a long time and many collisions, it is equally likely to be pointing in any direction. The dipole correlation function is a quantitative statement of just how this loss of memory comes about. 7.2.3

Conditions for a Quantitative Theory

A complete theory of band shapes of hydrogen-bonded systems has to cover the above phenomenology exposed in the precedent section. Obviously the main task is the explanation of the inherent band broadening, but the theory has also to provide for inclusion of the band shaping mechanism operating in the condensed phases and the effects of tunneling that enter the scene with low barrier proton potential functions. Clearly, the theory has to be quantitative and allow the reconstruction of spectra with the use of physically accessible parameters.

7.3 The Strong Coupling Theory of Anharmonicity

In the following we shall be dealing with theoretical studies of the shape of the high stretching vibration of hydrogen bonds that are all performed in the framework of the linear response theory. Instead of dealing with the fitness with experiment, of the diverse most fundamental theoretical approaches involving different mechanisms and formalisms, for which we refer to Maréchal and Witkowski [6], Hofacker et al. [7], Hadzi and Bratos [8] and Sandorfi [9], Maréchal [10, 11], and Sandorfi [12], we have preferred here to emphasize the connection between these diverse treatments, in such a way as appears a matter of doctrine susceptible to give a consistent and unified theoretical tool built up on clear physical basis; thus we have tried to perform unification in the notations and to clarify the links between the demonstrations. For an overview dealing with this question, see Henri-Rousseau and Blaise [13–16]. In the present study, we shall omit the very important question of the IR spectroscopy of water because of its specificity in the subject of hydrogen bond, and thus we shall not treat the corresponding theoretical studies of Rice et al. [17, 18] whose works remain the basic reference. In the following, to avoid a too extensive development, we shall limit to the situations where it is possible to neglect Fermi resonance and tunneling effect. Of course, that leads to restrict to weak or intermediate hydrogen bonds. Then, the basic physical model is the strong anharmonic coupling between the high-frequency and low-frequency stretching mode that we shall now expose.

7.3 The Strong Coupling Theory of Anharmonicity The concept of anharmonic coupling in H-bonded systems is at the basis of the explanation of IR band profiles for that sort of compounds. Already in 1937, Badger and Bauer [19] have stipulated that the spectral properties of hydrogen-bonded compounds are due to the interaction between the X—H mode and the temperature excited intermolecular vibrations. This was precised later by Stepanov [20] who imagined a coupling between the high-frequency X—H stretching mode and a low-frequency X—H· · ·Y one. This sort of coupling was responsible according to Sheppard [21] of Franck–Condon-like substructure of the response to an electromagnetic excitation of the system. Two sorts of anharmonic coupling may be distinguished: (i) the X—H/X—H· · ·Y coupling and (ii) the possible Fermi-type coupling between the X—H stretching mode and some overtones or combinations of intra-molecular bending modes. 7.3.1

Spectral Density

Consider the interaction operator V(t) of each molecule with the monochromatic electric field of angular frequency 𝜔 and of strength E0 . It may be written as V(t) = 𝝁̂ ⋅ E0 (e−i𝜔t + ei𝜔t )

(7.7)

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7 Quantum Approach of IR Line Shapes of Carboxylic Acids

In the following, it will be assumed that the strength E0 of the electric field is weak. Let H∘ be the Hamiltonian of each molecular oscillator, the eigenvalue of which is (7.8) H∘ |Ψ ⟩ = E |Ψ ⟩ with ⟨Ψ |Ψ ⟩ = 𝛿 k

k

k

f

k

fk

where |Ψk ⟩ is an eigenstate of the Hamiltonian H∘ with ∑ |Ψk ⟩⟨Ψk | = 1

(7.9)

and Ek the corresponding eigenvalue. Of course, the full Hamiltonian H of the system formed by a single molecule interacting with the electromagnetic field is the sum of those given by Eqs. (7.7) and (7.8): H = H∘ + V(t) The transition probability at time t for the system described by H to pass from an initial eigenstate |Ψk (0)⟩ to another eigenstate |Ψl (t)⟩ at time t because of the presence of V(t) may be written as |C(l, t | k, 0)|2 = |⟨Ψk (0)|Ψl (t)⟩|2

(7.10)

Now, according to this last equation, resulting from the first-order time-dependent perturbation method, the probability for a given molecular oscillator being at initial time in the ket |Ψk ⟩ to pass at time t to another ket |Ψl ⟩ is given by 2π ̂ l ⟩|2 {𝛿(Ek − El − ℏ𝜔)}t (7.11) (E(t))2 |⟨Ψk |𝝁|Ψ ℏ where E(t, 𝜔) is the electromagnetic field. Besides, the whole rate W of the total energy transfer between the molecular oscillator and the electromagnetic field is the sum of the time derivative of all these transition probabilities multiplied by the corresponding transferred energies ℏ𝜔kl times the probabilities 𝜌kk for the initial states |Ψk ⟩ to be occupied, i.e. ( ) ∑∑ 𝜕|C(l, t | k, 0)|2 𝜌kk (7.12) W= ℏ𝜔kl 𝜕t k l |C(l, t | k, 0)|2 =

where 𝜔kl = (Ek − El )∕ℏ

(7.13)

The probabilities 𝜌kk are the matrix elements of the Boltzmann density operator 𝜌B given by 1 −𝛽H∘ e Z and performed over the states |Ψk ⟩ defined by Eq. (7.8), that is, 𝜌B =

∘ 1 1 (7.14) ⟨Ψ |e−𝛽H |Ψk ⟩ = e−𝛽Ek Z k Z where Z is the partition function and 𝛽 the usual thermal Lagrange parameter equal to the inverse of kB T. Then, in view of Eqs. (7.11) and (7.14), the rate (7.12) 𝜌kk = ⟨Ψk |𝜌B |Ψk ⟩ =

7.3 The Strong Coupling Theory of Anharmonicity

becomes W = W (𝜔) =

∑∑ 2π ⟨Ψk |𝜌B |Ψk ⟩|⟨Ψk |𝝁|Ψl ⟩|2 {𝛿(𝜔kl − 𝜔)}ℏ𝜔kl (E(𝜔))2 ℏ k l (7.15)

On the other hand, since the spectral density I(𝜔) is the intensity, at angular frequency 𝜔, of the energy transfer between the field and the molecular oscillator, it follows that it is simply related to W through W (𝜔) = I(𝜔)ℏ𝜔 so that, due to Eq. (7.15), the spectral density reads ∑∑ 2π ̂ l ⟩|2 {𝛿(𝜔kl − 𝜔)} I(𝜔) = ⟨Ψk |𝜌B |Ψk ⟩|⟨Ψk |𝝁|Ψ (E(𝜔))2 ℏ k l Again, passing from the Dirac delta function to its corresponding integral representation, and writing explicitly the squared modulus of the matrix element of the dipole moment operator, ∑ 2π ̂ l⟩ c⟨Ψk |𝜌B |Ψk ⟩⟨Ψk |𝝁|Ψ (E(𝜔))2 I(𝜔) = ℏ k ∞

⟨Ψl |𝝁|Ψk ⟩

1 e−i(𝜔kl −𝜔)t dt 2π ∫−∞

or, due to Eq. (7.13), I(𝜔) =

∞ ∑∑ 2π 1 ̂ l⟩ ⟨Ψk |𝜌|Ψk ⟩⟨Ψk |𝝁|Ψ (E(𝜔))2 ℏ 2π ∫−∞ k l

̂ −iEk t∕ℏ )|Ψk ⟩ e−i𝜔t dt ⟨Ψl |(eiEl t∕ℏ )𝝁(e and thus I(𝜔) =

∞ ∑∑ 2π 1 ̂ l⟩ ⟨Ψk |𝜌|Ψk ⟩⟨Ψk |𝝁|Ψ (E(𝜔))2 ℏ 2π ∫−∞ k l ∘ ∘ ̂ −iH t∕ℏ )|Ψk ⟩ e−i𝜔t dt ⟨Ψl |(eiH t∕ℏ )𝝁(e

This result reads either ∑∑ 1 2π ̂ l ⟩⟨Ψl (t)|𝝁|Ψ ̂ k (t)⟩ e−i𝜔t dt ⟨Ψk |𝜌|Ψk ⟩⟨Ψk |𝝁|Ψ (E(𝜔))2 ∫ ℏ 2π −∞ k l ∞

I(𝜔)=

(7.16) or I(𝜔) =

∞∑ 2π 1 ⟨Ψk |𝜌B (E(𝜔))2 ℏ 2π ∫−∞ k { } ∑ ⟩ ∘ ∘ |Ψ ⟨Ψ |𝝁̂ ̂ −iH t∕ℏ )|Ψk ⟩ e−i𝜔t dt |Ψl ⟩⟨Ψl | (eiH t∕ℏ )𝝁(e k | k l

205

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7 Quantum Approach of IR Line Shapes of Carboxylic Acids

or, after using the closure relation (7.9) and expressing the Boltzmann density operator 𝜌B via (7.14), we have ∞∑ ∘ 1 2π 1 ⟨Ψk | (e−𝛽H )|Ψk ⟩ I(𝜔) = (E(𝜔))2 ℏ 2π ∫−∞ k Z ∘ ∘ ̂ iH t∕ℏ )𝝁(e ̂ −iH t∕ℏ )|Ψk ⟩ e−i𝜔t dt ⟨Ψk |𝝁(e Then, using the eigenvalues of the density operator, one obtains ∞∑ 2π 1 1 I(𝜔) = ⟨Ψk | (e−𝛽Ek )|Ψk ⟩ (E(𝜔))2 ℏ 2π ∫−∞ k Z ∘ ∘ ̂ iH t∕ℏ )𝝁(e ̂ −iH t∕ℏ )|Ψk ⟩ e−i𝜔t dt ⟨Ψk |𝝁(e or, after commuting the scalar exponential with the bra and using the normalization condition appearing in (7.8) and returning to the Boltzmann density operator, we have I(𝜔) =

∞∑ ∘ ∘ 1 2π ̂ iH t∕ℏ )𝝁(e ̂ −iH t∕ℏ )|Ψk ⟩ ⟨Ψk |𝜌B 𝝁(e (E(𝜔))2 ∫ ℏ 2π −∞ k

or

e−i𝜔t dt

∞∑ 1 2π −i𝜔t ̂ 𝝁(t)|Ψ ̂ ⟨Ψk |𝜌B 𝝁(0) dt (E(𝜔))2 k ⟩e ℏ 2π ∫−∞ k

I(𝜔) =

(7.17)

(7.18)

̂ ̂ is the Heisenberg picture dipole moment operator with 𝝁̂ ≡ 𝝁(0) and where 𝝁(t) at time t: ∘ ∘ ̂ = (eiH t∕ℏ )𝝁(e−iH t∕ℏ ) 𝝁(t) (7.19) Finally, expressing the sum over k as a trace operation, Eq. (7.18) yields ∞

2π 1 (E(𝜔))2 ℏ 2π ∫−∞

I(𝜔) =

−i𝜔t ̂ 𝝁(t)}e ̂ tr{𝜌B 𝝁(0) dt

(7.20)

Thus, this SD (7.20) may be viewed as proportional to the following Fourier transform: ∞

I(𝜔) ∝

∫−∞

G(t)e−i𝜔t dt

(7.21)

where G(t) is the ACF of the dipole moment operator defined by ̂ G(t) = tr{𝜌B 𝝁(0)

̂ 𝝁(t)}

(7.22)

Note that according to the properties of the Fourier transform of the complex ACF, the following equalities hold: ∞

I(𝜔) =

∫−∞

∞

G(t)e−i𝜔t dt = 2Re

∫0

∞

G(t)e−i𝜔t dt = 2Re

∫0

G(t)∗ ei𝜔t dt (7.23)

Equations (7.21) and (7.22) are the two most important results of the linear response theory.

7.3 The Strong Coupling Theory of Anharmonicity

7.3.2

The Model for IR Spectra of Centrosymmetric Dimers

Consider now a cyclic dimer of carboxylic acid involving two H-bond bridges, as depicted in Figure 7.2. The two parts of the dimer are labeled r = a, b. For such dimers, there are two degenerate high-frequency modes and also two degenerate low-frequency H-bond vibrations. The adiabatic approximation leads to description of each moiety by effective Hamiltonians of the H-bond bridge: for a single H-bond bridge, this effective Hamiltonian is either that of an harmonic oscillator, if the fast mode is in its ground state, or that of a driven harmonic oscillator, if the fast mode is excited. When one of the two identical fast modes is excited, then, because of the symmetry of the cyclic dimer and of a possible coupling ℏV ∘ between the two degenerate excited states of the fast mode, an interaction may occur (Davydov coupling), leading to an exchange between the two identical excited parts of the dimer. Of course, this interaction between degenerate excited states is of a nonadiabatic nature, although the adiabatic approximation has been performed to separate the high- and low-frequency motions of each moiety. The details of our calculations may be found in Ref. [22]. ̂2 It may be observed that because of the symmetry of the dimer, there is a C 2 ̂ ̂ operator (with C2 = 𝟏) that exchanges the coordinates Qi of the two H-bond bridges of the cyclic dimer according to ̂ 2 Qb = Qa C

̂ 2 Qa = Qb C

(7.24)

Of course, the same symmetry properties hold for the conjugate momenta, i.e. ̂ 2 Pa = Pb C

̂ 2 Pb = Pa C

(7.25)

| [ℍ{0,0} ] 0 0 || | II | | {1,0} [ℍDav ] = | 0 [ℍII ] V∘ | | | {0,1} | 0 V∘ [ℍII ] || |

(7.26)

with, respectively, ] [ℍ{0,0} II

=

∑ i

(

P2i 2M

+

MΩ2 Qi2 2

) + ℍ{𝜃}

Figure 7.2 Cyclic H-bonded dimers. The action of the parity operator Cˆ 2 on the high- and low-frequency coordinates of the centrosymmetric cyclic dimer exchanges the coordinates. Cˆ 2 Qa = Qb ; Cˆ 2 qa = qb Cˆ 2 Qb = Qa ; Cˆ 2 qb = qa .

with i = a, b.

qa O R

Qa H

O

C

C O

H Qb

O qb

R

207

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7 Quantum Approach of IR Line Shapes of Carboxylic Acids

and ℍ{𝜃} is the Hamiltonian of the thermal bath that is as usually done in the quantum theory of damping [23], figured by an infinite set of harmonic oscillators: ( ) ∑ p̃ 2i 1 {𝜃} 2 2 ̃ 𝜔̃ q̃ (7.27) ℍ = + m ̃i 2m 2 i i i i ) ( 2 ) ( 2 MΩ2 Qb2 Pb Pa MΩ2 Qa2 {1,0} [ℍII ] = + + + 2M 2 2M 2 + bQa + [ℍ{Int} ]a + ℍ{𝜃} + (ℏ𝜔∘ − 𝛼 ∘ ℏΩ) II 2

(7.28)

Let us write [ℍ{k,l} ] = [ℍ{k} ] + [ℍII{l} ]b II II a

(7.29)

Then, owing to the symmetry of the system, it appears that the parity operator exchanges the two Hamiltonians: ̂ 2 [ℍ{1,0} ] = [ℍ{0,1} ] C II II

(7.30)

The physics corresponding to the situation described by the Hamiltonian (7.26) is depicted in Figure 7.3. 7.3.3

The Line Shape

We now give the results of our theory dealing with the IR line shape of the centrosymmetric cyclic dimer under the irreversible influence of the medium. Formally, the line shape is obtained within the linear response theory by the following

V°

|(m)b〉

|{1}b〉 |{0}a〉

|(m)a〉

V°

|(m)b〉

|{0}b〉

|{1}a〉

|(m)a〉

Figure 7.3 Structure of the “Davydov” Hamiltonian. The Davydov coupling parameter V ∘ is coupling two symmetric configurations.

7.3 The Strong Coupling Theory of Anharmonicity

equations: ∞

I(𝜔) ∝ Re

{Gg (t)}({G (+) (t)} + {G

∫0

(−)

∘ (t)})(e−𝛾 t )e−i𝜔t dt

(7.31)

Development of the theory allowed us to write the spectral density on the form I(𝜔) = (𝜇∘ u )2 with I± (𝜔) ∝

∑∑ mg

ng

(I+ (𝜔) + I− (𝜔))

Pmg ng

(7.32)

∑∑ nu

𝜇

2 ± e−𝜆nu Ωt {(1 ± (−1)nu +1 ) + 𝜂 ∘ (1 ∓ (−1)nu +1 )}2 |B{±} nu 𝜇 | (Img ng nu 𝜇 (𝜔))

where 𝜂 ∘ is lying between 0 and 1, reflecting the amount of forbidden transition with {±} }𝜇 ⟩ {B{±} nu 𝜇 } = ⟨(nu )|{𝜉 {±} {H{±} }𝜇 ⟩ = {ℏ𝜔{±} }𝜇 |{𝜉 {±} }𝜇 ⟩ u }|{𝜉

with

(7.33)

) ( √ 1 † ̂2 + 𝛼 ∘ ℏΩ(a† + a)∕ 2 ± V∘ C {H{±} } = ℏΩ a a + u 2

and using the equations √ a|{n}⟩ = n|{n − 1}⟩ a† |{n}⟩ =

√ n + 1|{n + 1}⟩

ˆ 2 |(nu )⟩ = (−1)n |(nu )⟩ C

(7.34)

and where √ (1 + ⟨n⟩mg )⟨n⟩ng 𝛼̃ ∘2(mg +ng ) 𝛼̃ ∘ = 𝛼 ∘ 2 mg !ng ! 1 ℏΩ ⟨n⟩ = 𝜆 with 𝜆 = kB T e −1 𝛾mg ng I±mg ng nu 𝜇 (𝜔) ∝ (𝜔 − Ω±mg ng nu 𝜇 )2 + (𝛾mg ng )2 Pmg ng =

Ω±mg ng nu 𝜇 = 𝜔∘ − {(mg − ng + nu )Ω − 𝜔±𝜇 } − 2𝛼 ∘2 Ω 𝛾mg ng = (mg + ng )̃𝛾 + 𝛾∘ 𝛾̃ = 𝛾

√ 2

Study of cyclic dimers of carboxylic acids has produced recently a large amount of contributions in relation with the present model [24–27].

209

210

7 Quantum Approach of IR Line Shapes of Carboxylic Acids

7.3.4

Limit Situations

Figure 7.4 recapitulates how the SD (7.32) reduces to all the special situations given previously in the literature. In like manner, it would be possible to show [14, 22] that the spectral density (7.31) reduces satisfactorily to many special situations available in the literature specially to those obtained by:

Blaise et al. (2005) [28]

γ =γ° = 0 V =0 γ°=0

V= 0 Boulil et al. (1988) [29] γ° =0

V =0 γ° =0

Maréchal and Witkowski with Davydov coupling (1968)[6]

γ= 0

V=0 Maréchal and Witkowski without Davydov coupling (1968) [6] γ°= 0 γ =0

V =0

γ° =0

Rösch-Ratner (1974) [30] γ =0

Quantum model of direct and indirect dampings (1998) [15] Classical approximation [Q,P] = 0 Semiclassical model (2005) [32]

Sakun (1985) [31]

Commutation of stochastic average and integral on time

Abramczyk (1985) [33]

Memory neglect

Robertson and Yarwood (1978) [34]

Rotation neglect

Slow modulation limit Bratos (1957) [35]

Figure 7.4 Connections between the different theories of IR spectra of H-bonded systems.

7.3 The Strong Coupling Theory of Anharmonicity

(i) Maréchal and Witkowski in the absence of damping. (ii) Rösch and Ratner [30] in the absence of Davydov coupling and indirect damping. (iii) Boulil et al. [29] in the absence of Davydov coupling and direct damping. (iv) Robertson and Yarwood [34] in the semiclassical limit, without Davydov coupling and direct damping. 7.3.5

Examples

7.3.5.1

Acetic Acid

Figure 7.5 gives spectra for CD3 CO2 H dimer in the gas phase at room temperature. The gray spectrum represents the experimental line shapes taken from Novak and coworkers [36], whereas the solid line is the theoretical line shape [28] computed by the aid of Eq. (7.32). The parameters used for the calculations are given in the caption of this figure. 7.3.5.2

Formic Acid

Spectral density

Recently, this model has been applied to obtain the theoretical IR line shape of liquid formic acid. Postulating the coexistence of chain-like and dimer formic acid, Fathi et al. [27] found the best fitting of the theoretical line shape with the experimental spectra. A full quantum theoretical approach has been used to study the 𝜈(O—H) experimental IR line shapes of liquid formic acid. For this purpose, a model that accounts for the proportion of cyclic dimers has been successfully adapted. The present model thus incorporates the strong anharmonic coupling between the high-frequency mode and the H-bond bridge, the Davydov coupling between the excited states of the two moieties, and multiple Fermi resonances between the 𝜈(O—H) (Bu) mode and combinations of some bending modes, together with the quantum direct and indirect dampings. This model reproduces satisfactorily

2000

3500 Angular frequency (cm–1)

Figure 7.5 IR spectrum of the CD3 COOH dimer in the gas phase at room temperature. Grayed: experimental line shape. Solid line: our theoretical approach. Parameters: T = 300 K, Ω = 88 cm−1 , 𝛼 ∘ = 1.19, 𝜔∘ = 3100 cm−1 , V∘ = −1.55Ω, 𝜂 = 0.25, 𝛾 = 0.24Ω, 𝛾 ∘ = 0.10Ω. Source: Blaise et al. 2005 [28]. Reproduced with permission of American Institute of Physics.

211

7 Quantum Approach of IR Line Shapes of Carboxylic Acids

Spectral density

HCOOD

Spectral density

HCOOH

2200

2900 Angular frequency (cm–1)

3600

1850

2350 –1 Angular frequency (cm )

2850

2350 –1 Angular frequency (cm )

2850

DCOOD Spectral density

DCOOH Spectral density

212

2200

2900 Angular frequency (cm–1)

3600

1850

Figure 7.6 Application of the theory to several isotopic species of formic acid. Grayed: experimental spectra. Solid line: our theoretical approach. Source: Details of the calculations are given in Ref. [27].

the main features of the experimental line shapes of liquid hydrogenated and deuterated formic acid by using a minimum set of independent parameters (Figure 7.6).

7.4 Conclusion In this contribution we have discussed the foundations on which our quantum theory of vibrational spectra of hydrogen-bonded compounds is based. This theory, built in the context of linear response and strong an harmonic coupling, appears to encompass all other theories dealing with the same subject, whether it be semiclassical or quantum. The IR spectra of the hydrogen-bonded dimers of carboxylic acids were interpreted taking into account the Davydov coupling between the two halves of the dimer. As an example we show some simulations carried out with our theory on acetic acid and formic acid, which give a satisfactory agreement with the experimental spectra.

References 1 Pimentel, G. and McClellan, A. (1960). The Hydrogen Bond. San Francisco,

CA: Freeman. 2 Novak, A. (1974). Struct. Bond. (Berlin) 18: 177. 3 Davydov, A. (1962). Theory of Molecular Excitons. New York: McGraw Hill. 4 Kubo, R. (1957). J. Phys. Soc. Jpn. 12 (6): 570.

References

5 Gordon, R. (1968). Adv. Magn. Res. 3: 1. 6 Maréchal, Y. and Witkowski, A. (1968). J. Chem. Phys. 48: 3637. 7 Hofacker, G.L., Marechal, Y., and Ratner, M.A. (1976). The dynamical aspects

8

9 10 11 12 13 14

15

16

17 18 19 20 21 22

23 24

25

of hydrogen bonds. In: The Hydrogen Bond, Recent Developments in Theory and Experiment, 1 (ed. W.P. Schuster, G. Zundel, and C. Sandorfy), 295. Amsterdam: North-Holland. Had˜zi, D. and Bratos, S. (1976). In: The Hydrogen Bond Theory (ed. P. Schuster, G. Zundel, and C. Sandorfy), 567. Amsterdam: North Holland Publishing. Sandorfy, C. (1976). In: The Hydrogen Bond Theory (ed. P. Schuster, G. Zundel, and C. Sandorfy), 616. Amsterdam: North Holland Publishing. Maréchal, Y. (1980). In: Molecular Interactions, Chapter 8 (ed. H. Ratajczak and W. Orville-Thomas), 230. Chichester: Wiley. Maréchal, Y. (1987). In: Vibrational Spectra and Structure, Vol. 16 (ed. J. Durig), 312. Amsterdam: Elsevier. Sandorfy, C. (1984). Vibrational Spectra of Hydrogen Bonded Systems in the Gas Phase, Topics in Current Chemistry, vol. 120, 41. Springer-Verlag. Henri-Rousseau, O. and Blaise, P. (1997). In: Theoretical Treatment of Hydrogen Bonding (ed. D. Hadzi), 165. New York: Wiley. Henri-Rousseau, O. and Blaise, P. (1998). The infrared spectral density of weak hydrogen bonds within the linear response theory. In: Advances in Chemical Physics, vol. 103 (ed. I. Prigogine and S.A. Rice), 1–186. New York: Wiley. Henri-Rousseau, O. and Blaise, P. (1998). Theory of weak damped H-bonds: influence of direct and indirect damping on the 𝜈X−H IR spectra. In: Recent Research Developments in Chemical Physics, vol. 2 (ed. Pandalai, 181. Trivandrum. Henri-Rousseau, O., Blaise, P., and Chamma, D. (2002). Infrared lineshapes of weak hydrogen bonds: recent quantum developments. In: Advances in Chemical Physics, vol. 121 (ed. I. Prigogine and S.A. Rice), 241. New York: Wiley. Rice, S. and Sceats, M. (1981). J. Phys. Chem. 85: 1108. Rice, S., Bergren, M., Belch, A. et al. (1983). J. Phys. Chem. 87: 4295. Badger, R. and Bauer, S. (1937). J. Chem. Phys. 5: 839. Stepanov, B. (1946). Nature 157: 808. Sheppard, N. (1959). In: Hydrogen Bonding (ed. D. Hadzi), 85–105. London: Pergamon. Henri-Rousseau, O. and Blaise, P. (2008). Infrared lineshapes of weak hydrogen bonds centrosymmetric cyclic dimers of carboxylic acids. In: Advances in Chemical Physics, vol. 139 (ed. I. Prigogine and S.A. Rice), 245–496. New York: Wiley. Louisell, W.H. (1973). Quantum Statistical Properties of Radiation, 1e. Wiley. Brela, M.Z., Boczar, M., Boda, L., and Wójcik, M.J. (2018). Molecular dynamics simulations of vibrational spectra of hydrogen-bonded systems. In: Frontiers of Quantum Chemistry, (ed. M.J. Wójcik, H. Nakatsuji, B. Kirtman and Y. Ozaki, 353–376. ¿. Singapore: Springer. Grabska, J., Be´c, K.B., Ishigaki, M. et al. (2017). Spectrochim. Acta, Part A 185: 35–44.

213

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26 Wójcik, M.J. (2016). Theoretical modeling of vibrational spectra and proton

27 28 29 30 31 32 33 34 35 36 37

tunneling in hydrogen-bonded systems. In: Advances in Chemical Physics, vol. 160 (ed. I. Prigogine and S.A. Rice), 307–342. New York: Wiley. Fathi, S., Blaise, P., Velcescu, A., and Nasr, S. (2017). Chem. Phys. 492: 12–22. Blaise, P., Wojcik, M.J., and Henri-Rousseau, O. (2005). J. Chem. Phys. 122: 064306. Boulil, B., Henri-Rousseau, O., and Blaise, P. (1988). Chem. Phys. 126: 263. Rösch, N. and Ratner, M. (1974). J. Chem. Phys. 61: 3344. Sakun, V. (1985). Chem. Phys. 99: 457. Blaise, P., Déjardin, P.-M., and Henri-Rousseau, O. (2005). Chem. Phys. 313: 177. Abramczyk, H. (1985). Chem. Phys. 94: 91. Robertson, G. and Yarwood, J. (1978). Chem. Phys. 32: 267. Bratos, S. and Hadzi, D. (1957). J. Chem. Phys. 27: 991. Haurie, M. and Novak, A. (1965). J. Chim. Phys. 62: 146. Be´c, K.B., Futami, Y., Wójcik, M.J. et al. (2016). J. Phys. Chem. A 120 (31): 6170–6183.

215

8 Theoretical Calculations Are a Strong Tool in the Investigation of Strong Intramolecular Hydrogen Bonds Poul Erik Hansen 1 , Aneta Jezierska 2 , Jarosław J. Panek 2 , and Jens Spanget-Larsen 1 1 2

Roskilde University, Department of Science and Environment, DK-4000 Roskilde, Denmark University of Wrocław, Faculty of Chemistry, ul. F. Joliot-Curie 14, 50-383 Wrocław, Poland

8.1 Introduction and Definition of Types of Intramolecular Hydrogen Bonds The present chapter is dealing with strong intramolecular hydrogen bonds. Also, a number of intermolecular strong hydrogen bonds have been investigated [1], but these are not included in the present review. These are often of ionic character and are only relevant in a few cases as most of the strong intramolecular hydrogen bonds are “resonance assisted;” see below. Intramolecular hydrogen bonds can be of many kinds as shown in Figure 8.1. However, not all of them will be strong hydrogen bonds. One way of forming a strong hydrogen bond is to have the hydrogen bond donor and acceptor linked by a double bond or another type of conjugation as shown in Figure 8.2, often referred to as resonance-assisted hydrogen bonds (RAHBs). Charge is another typical way of creating a strong intramolecular hydrogen bond (see Figure 8.1, bottom).

8.2 Definitions of Strong Intramolecular Hydrogen Bonds The definition should be based on the IUPAC recommendation [2] for hydrogen bonds: (i) “The length of the X—H bond usually increases on hydrogen bond formation leading to a red shift of the infrared X—H stretching frequency and an increase in the infrared absorption cross-section for the X—H stretching vibration. The greater the lengthening of the X—H bond the stronger is the H· · ·X bond.” (ii) “The X—H· · ·Y—Z hydrogen bond … typically include[s] pronounced proton deshielding for H in X—H.”

Molecular Spectroscopy: A Quantum Chemistry Approach, First Edition. Edited by Yukihiro Ozaki, Marek Janusz Wójcik, and Jürgen Popp. © 2019 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2019 by Wiley-VCH Verlag GmbH & Co. KGaA.

216

8 Investigation of Strong Intramolecular Hydrogen Bonds

O

N

O

N

N

H

H

H

H

H

O

O

O

O

N

C

C

C

C N

CH N

H

H

O O O

H

N H

N

N

R

Figure 8.1 H-bond motifs without indication of inter- or intramolecular type. H

O

H

H

O

H

O

a

R

O

R

b

H

H3C

O H X

H

O

c

CH3

R

H3C

X

H

d

R

CH3

Figure 8.2 Intramolecular H-bonds with a “double” bond linking the hydrogen bond donor and acceptor.

“Low barrier” hydrogen bonds have been the subject of recent reviews [3, 4]. Hansen and Spanget-Larsen [5] have proposed the following definitions for strong hydrogen bonds in O· · ·O systems based on 1 HO chemical shifts and OH stretching frequencies: XH=OH: 2800 cm–1 > 𝜈 OH > 1800 cm–1 and 19 ppm > 𝜈 OH > 15 ppm. However, in case of OH chemical shifts, these should be corrected for possible ring current effects and anisotropy contributions (Section 8.8.3) [6–8]. Other criteria could be a large 2 ΔC(XD) (two-bond deuterium isotope effects on 13 C chemical shifts) or 4 ΔC(XD) (four-bond deuterium isotope effects on 13 C chemical shifts) or large XH primary isotope effect [9]. Isotope effects have the advantage that they are independent of solvent effects and that ring current effects are eliminated. Some of the parameters mentioned above seem to be correlated. Examples are XH chemical shifts and primary isotope

8.3 Calculation of Structural Parameters

effects [10–12]. OH chemical shifts and the OH stretching wavenumber 𝜈 OH [13] and 𝜈 OH and two-bond isotope effects [14]. This is very useful in the sense that if one parameter cannot be measured, another can be used in describing the intramolecularly hydrogen-bonded system.

8.3 Calculation of Structural Parameters Strong intramolecular hydrogen bonds are complicated to describe on the basis of quantum chemistry methods. They are short and easily modified by external factors, e.g. the substituent effects in the parent molecule, presence of neighboring crystal molecules, or solvent effects. Their inherent feature is very flat potential energy surface (PES) for proton motion; therefore even small external perturbations can strongly modulate the hydrogen bond parameters, including donor–acceptor distance and proton position. Nowadays one can choose from a rich spectrum of quantum chemistry methods, differing by theoretical background (density functional theory (DFT), correlated schemes such as coupled cluster or Møller–Plesset calculus), approach to time dependency (static calculations, molecular dynamics (MD)), environment (explicit and implicit solvent models – polarizable continuum models [15] are of the latter type), and electronic state (TD-DFT and adiabatic diagrammatic connection methods being currently the most used for the excited state properties). One of the examples of difficulty in reproduction of structural parameters by theoretical calculations is picolinic acid N-oxide (PANO) with its strongly anharmonic intramolecular hydrogen bond. Panek et al. [16] demonstrated that inclusion of neighboring molecules and taking into account the state of matter are mandatory. The experimental donor–acceptor O· · ·O distance is 2.426 Å, as measured by neutron diffraction. The gas-phase DFT and MP2 structural optimizations of PANO monomer led to very elongated D· · ·A distance, e.g. 2.519 Å for B3LYP/6-311+G(d,p) and 2.513 Å for MP2 with the same basis set. An attempt to improve the description of the structure by the use of cluster model was only moderately successful, yielding for the monomer, dimer, and trimer at the B3LYP/6-31+G(d,p) the respective D⋅⋅⋅A distances of 2.509, 2.489, and 2.469 Å. On the other hand, transition to fully periodic solid-state plane wave approach yielded the value of 2.451 Å (with BLYP functional and Γ-point approximation), which is a significant improvement over the gas-phase cluster model. An interesting structural question regarding the shape of the proton as a quantum object was studied by Benoit and Marx [17]. Path integral quantization of nuclear motions coupled with Car–Parrinello MD scheme allowed the authors to register the proton distribution along the bond in water ice at 25 K with varying density, leading to a range of donor–acceptor distances (from 2.85 to 2.17 Å). The quantized proton was reproduced for large bond lengths as a disclike cloud, which upon O· · ·O compression to 2.44 –2.36 Å turned into a cigar-like object and upon further compression the disclike appearance was restored. These results were interpreted as a sign of translational tunneling

217

218

8 Investigation of Strong Intramolecular Hydrogen Bonds

between two minima in the “cigar-like” region; the tunneling vanishes, and the disclike shape is seen again, when the bond is compressed, so that a symmetric single well potential is achieved. This result is important for understanding of the role of quantum nuclear effects in strong hydrogen bonds.

8.4 Hydrogen Bond Strength When talking about strong hydrogen bonds, short X· · ·Y bond length, X and Y being typically O, N, or S, has for a long time been considered a measure of hydrogen bond strength [18], so this parameter is available. O· · ·O distances correlate with O· · ·H distances (see Figure 8.3). Sanz et al. [19] claimed that the O· · ·O distance was the only important parameter and supported this by calculation in systems with single bonds instead of double bonds but with the same O· · ·O distance. An example is given in Figure 8.4. It is obvious that data for these compounds fall widely off the correlation (see Figure 8.5) for most of the compounds. Such a procedure is questionable also because of the much higher total energy of such compounds. 1.8 1.75

Series 1

1.7

Series 2

1.65

Series 3

1.6

Series 4

1.55 Series 5

1.5 1.45 2.40

2.45

2.50

2.55

2.60

2.65

2.70

Figure 8.3 Plot of calculated H· · ·O distance vs. calculated O· · ·O distance, both in Angstroms. Series 1: aromatic compounds. Series 2: aromatic compounds with steric strain. Series 3: donor and acceptor connected by double bond. Series 4: donor and acceptor connected by double bond with steric strain. Series 5: aromatic compounds with electron-attracting substituents. O

O H

H

H O H 1

2 H O

H H O

O

4

O

O H

H

H

O

O

H

H 3

Figure 8.4 Compounds with single bond corresponding to salicylaldehyde.

RO...O

8.4 Hydrogen Bond Strength

2.65 2.6 2.55 2.5 2.45 2.4 0.96

Series 1 Series 2 Series 3 Series 4 0.98

1 ROH

1.02

1.04

Series 5 Series 6

Figure 8.5 A plot of the calculated O· · ·O distances vs. the calculated OH bond lengths, both in Angstroms.

Series 1 are compounds with a fixed O· · ·O distance and a single bond of the same length as in a corresponding compound with double bond (see Figure 8.2). Series 2 are aromatic compounds. Series 3 are aromatic compounds with steric strain. Series 4 are donor and acceptor connected by double bond. Series 5 are donor and acceptor connected by double bond with steric strain. Series 6 are aromatic compounds with electron-attracting substituents. Perrin [4] has criticized the O· · ·O distance as a good parameter for assessing hydrogen bond strength. As seen later this is true on a fine scale, but as a general trend there seems to be a correlation between the O· · ·O distance and the hydrogen bond energy. Atoms in molecules [20] can be used to calculate the electron density at the bond critical point of the hydrogen bond. This has been introduced as a parameter to estimate hydrogen bond strength [21–26]. In the first three cases, the hydrogen bonds were intermolecular, and the technique was explored as such, and in the two latter cases the principle was used in intramolecularly hydrogen-bonded systems. Afonin et al. [27] have investigated a series of not so strong intramolecular hydrogen-bonded compounds but including acetylacetone. They estimated the hydrogen bond energies using the Schaefer [28] equation: EHB = Δd + (0.4 ± 0.2). Δd is the difference between the OH chemical shift in a hydrogen-bonded situation and that of a reference non-hydrogen-bonded one, in this case phenol at high dilution at a δOH of 4.69 ppm. It is obvious from that study that acetyl acetone as expected is falling off a plot of EHB vs. the electron density at the bond critical point. The concept of quasi-aromaticity of the six-membered hydrogen-bonded system, marked in bold in Figure 8.6, has been studied with a number of methods. In systems like those in Figure 8.6 and others, H O O quasi-aromaticity has been investigated based on the HOMA index [29–31]. For the corresponding compounds with X=NR, Martyniak et al. [32] also found quasi-aromaticity. In contrast to these R studies, Kleinpeter and Koch [33] found no sign of quasi-aromaticity neither in the beta-diketones nor Figure 8.6 Hydrogenin the equally strongly hydrogen-bonded system bonded system. 1,3,5-triacetyl-2,4,6-trihydroxybenzene (Figure 8.7).

219

220

8 Investigation of Strong Intramolecular Hydrogen Bonds

H

O

O

O

OH

Figure 8.7 1,3,5-Triacetyl-2,4,6-trihydroxybenzene.

HO

O H O O

O H

O

O OH

Strong hydrogen bonding in this compound was ensured both by corrected OH chemical shifts [34] and deuterium isotope effects on 13 C chemical shifts [26]. Also in the o-hydroxy Schiff bases, no quasi-aromaticity was found [35]. The Kleinpeter–Koch approach was in all cases to calculate TSNMRS (see below).

8.5 Calculation of Energies Calculation of energies is clearly very important in assessing the intramolecular hydrogen bond. Several schemes have been suggested. One way is to calculate the energy difference between the hydrogen-bonded and the open form (OH group turned 180∘ ) (later called HB-out [34]), also called “closed and open” [36], and use this as a theoretical measure of the hydrogen bond energy. This has been used in many intermolecular contexts. Cuma et al. [37] used this approach for salicylaldehyde. Grabowski used this on intramolecularly hydrogen-bonded systems such as malonaldehyde [25], and more recently, a large collection of o-hydroxyarylaldehydes have been investigated (see below). Furthermore, this method requires that the OH group of the open form is not involved in steric or other interactions. One important question is of course which theoretical models may lead to reliable energies. In case of o-hydroxyarylaldehydes gas-phase equilibrium geometries and model hydrogen bond energies of the investigated molecules were computed with either B3LYP [38, 39] DFT or second-order Møller–Plesset (MP2) perturbation theory [40, 41] using the 6-311++G(d,p) basis set [42–44]. For a number of species, the geometry optimization failed to converge with MP2/6-311++G(d,p) in spite of several attempts, a problem apparently associated with the inclusion of diffuse functions in the basis set (no problems were observed with MP2/6-311G(d,p)). Grabowski [25] studied malonaldehydes and correlated the calculated hydrogen bond energy with the Q-parameter [45]. However, this approach is not generally applicable [24]. An example of a correlation between O· · ·O distance and hydrogen bond energy (MP2 6-311+G(d,p) is given in Figure 8.8 including a number of malonaldehyde derivatives (marked with diamonds): acetylacetone, malonaldehyde, enol form of 1,3-butanedione (CH3 at C=O and COH), 2-fluoromalonaldehyde, 2-chloromalonaldehyde, 3-acetyl-4-hydroxy-pyran-2-one, and 5-acetyl-4-hydroxy-pyran-2-one (the latter two are similar to the dehydroacetic acid derivatives [46]). The data points for these compounds are close to the correlation line. The data points falling above the diamond data points are salicylate, salicylaldehyde, 6-methylsalicylaldehyde, 5-nitrosalicylaldehyde, methyl acrylate, and (Z)-methyl 3-hydroxybut-2-enoate (the latter two are triangles). It looks like there is a

RO...O

8.5 Calculation of Energies

–20

–15

–10

–5

2.8 2.75 2.7 2.65 2.6 2.55 2.5 2.45

0

5

HB energy

Figure 8.8 Calculated O· · ·O distance (Å) vs. calculated HB-out MP2 energies (kcal/mol). Correlation line is for the diamond points. 18

H

Hydrogen bond energy

17

O OH

16 15 14

OH O H

13 12

H

11

O

y = –59.578x + 168.46 R2 = 0.8541

H O

H

10 9 8 2.5

2.52

y = –42.14x + 120.94 R2 = 0.5902 2.54 2.56 2.58 2.6

O OH

2.62

2.64

2.66

2.68

2.7

R(O···O)

Figure 8.9 Hydrogen bond energies (MP2) in kcal/mol vs. calculated O· · ·O distances in Angstroms. The data are divided into non-sterically hindered (upper) compounds like 4 and 8, as well as salicylaldehyde, 4-hydroxy-, 5-hydroxy-, 4-methoxy-, and 5-nitrosalicylaldehyde and 1-hydroxy-2-naphthaldehyde, and sterically hindered (lower).

difference between compounds with a double bond between the hydrogen bond donor and acceptor and those being aromatic, but why the esters are falling off the line is unclear. The plot of Figure 8.9 shows that a shorter O· · ·O distance is related to a higher hydrogen bond energy. Jablonski et al. [47] suggested a scheme related to HB-out but including all four possibilities in enaminones – ZZ, ZE, EZ, and EE. This approach has not been tested on strong hydrogen bonds and suffers from the fact that it is probably only useful for 1,2-disubstituted double bonds. Rusinska-Roszak [48] has used the molecular tailoring approach (MTA) [49] for a large series (186 compounds) of β-hydroxy-α,β-unsaturated carbonyl compounds to estimate hydrogen bond energies, based on MP2(full)/6311++G(2d,2p) calculations. An average value of 15.2 kcal/mol was obtained, indicating a strong hydrogen bond according to Steiner [50]. The MTA method is illustrated in Figure 8.10. The energy is calculated as EHB = EM1 + EM2 − EM3 − EMH .

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8 Investigation of Strong Intramolecular Hydrogen Bonds

MH

M1

M2

M3

Figure 8.10 Illustration of the MTA fragmentation method. Source: Rusinska-Roszak 2015 [48]. Reproduced with permission of The American Chemical Society. Series 1

Series 2

0.8 0.75 TBDIE

222

0.7 0.65 0.6 15

15.5

16

16.5

17

17.5

HB energy

Figure 8.11 Observed two-bond deuterium isotope effects (TBDIEs) on 13 C chemical shifts in ppm vs. calculated hydrogen bond energies according to the molecular tailoring approach (MTA) in kcal/mol.

Rusinska-Roszak [48] compared the calculated energies to structural parameters and electron density not only at the bond critical point but also at the OH chemical shifts. Unfortunately, all these parameters were calculated, so to some extent it becomes a test of how good a large basis set is to calculate many different parameters. In Figure 8.11 some of the calculated energies are compared with experimental two-bond deuterium isotope effects on 13 C chemical shifts. Series 2 are for unsymmetrical arrangements of the substituents in the symmetrically substituted compounds with either t-butyl or isopropyl groups. For t-butyl the less favored (17.3, 0.745) situation falls wide off the correlation line, whereas this is opposite for the isopropyl group. Here the outlier is for the favored form at (16.0, 0.745). The other outlier is values for acetylacetone (15.2, 0.639). Rusinska-Roszak [51] has also treated a series of o-hydroxy aromatic aldehydes in a narrow range of hydrogen bond energies and only with a few strongly hydrogen-bonded systems like 1,3,5-triacetyl-2,4,6-trihydroxybenzene. The

8.6 Tautomerism

MTA gave mostly smaller hydrogen bond energies than the HB-out method, but the trends are clearly the same for all three types of calculations. Energies of two tautomers have also been calculated for 1,3,5-triacetyl-2,4,6trihydroxybenzene (see Figure 8.7) using a small basis set; the authors concluded that two tautomers existed [52]. This was then proven not to be the case [53].

8.6 Tautomerism 8.6.1

2D Approaches to Hydrogen Bond Potentials

Martyniak et al. [32] have calculated hydrogen bond potential curves for a series of o-hydroxy aromatic Schiff bases (Figure 8.12) as seen in Figure 8.13. These are marked S1–S4. In Figure 8.13, the double well potentials belong to S2 and S3. S1 belongs to the corresponding benzene derivative, and S4 is with the substituents at the 2 and 3 positions of the naphthalene ring. The potential energy diagrams are calculated using B3LYP/6-311++G(d,p) functional and basis set. NH bond lengths are increased gradually and the structure is optimized after each increase. For S2 and S3 tautomerism is clearly present as shown in Figure 8.13. 8.6.2

Potential Energy and Free Energy Surfaces

Contemporary quantum chemistry provides us with numerous methods for accurate calculation of energetic parameters, e.g. coupled cluster with singles, doubles, and perturbative triples (CCSD(T)) or quantum Monte Carlo techniques. The most frequently applied approach for the electronic ground state simulations is DFT. The method was applied to a set of o-hydroxyaryl Schiff bases with strong intramolecular hydrogen bonds [54]. The PES were constructed by displacing the proton on a circular arc defined by the optimal positions of the donor, hydrogen, and acceptor atoms. The use of such uniquely defined path provides a balanced description of the proton migration from the donor to the acceptor. The variety of substituents introduced the inductive and steric effects influence on the intramolecular hydrogen bond PES. All the investigated o-hydroxyaryl Schiff bases exhibit potential proton transfer phenomena in the electronic ground state, which is evidenced by the presence of two minima of the proton PES at the donor and acceptor sides. The presence of nitro substituent in the phenyl ring and methyl group in the Schiff moiety (R = CH3 , X = NO2 , Figure 8.12 Naphthalene Schiff bases. The shown structure refers to S3 in Figure 8.13.

R

R N

A

N+ H

H

O

O–

B

223

224

8 Investigation of Strong Intramolecular Hydrogen Bonds

ΔE (kcal/mol)

A

25

M4 M1

20

15

Figure 8.13 Potential energy curves for S1 [2-(E)-(methylimino)methy]phenol; S2 [2-(E)-(methylimino)methy]-1-naphthol, S3 [1-(E)-(methylimino)methy]-2-naphthol, whereas S4 [3-(E)-(methylimino)methy]-2-naphthol. Source: Martyniak et al. 2012 [32]. Reproduced with permission of the Royal Society of Chemistry.

M3 M2

10 S4 S1 5

S3 S2

0 0.8

1.2

1.6

2.0

O—H (Å)

Y = CH3 or Cl; see Figure 8.14) results in the strongest lowering of the energy barrier and the secondary minimum at the acceptor side compared to the unsubstituted reference structure (R = X = Y = H). DFT also enables calculations of the excited state propH CH3 N O erties via TD-DFT theory. An interesting application for a set of small molecules is presented by Aquino and X R Lischka [55] who studied excited state intramolecular proton transfer (ESIPT) in, for example, malonaldehyde, o-hydroxybenzaldehyde, and salicylic acid using TD-DFT Y and CC2 approaches. These schemes were compared with experimental and accurate CASPT2/MR-AQCC data. It Figure 8.14 Molecular was found that, even with their well-known shortcom- structures of selected ings, both TD-DFT and CC2 are reasonable schemes for investigated the description of ESIPT energy surfaces. Another study o-hydroxyaryl Schiff of intramolecular excited state tautomerism is provided bases (see the text by Błaziak et al. [56]. Twelve derivatives of quinoline for specific R, X, Y). N-oxide with intramolecular hydrogen bonds of various The dashed line indicates the strengths were studied in the ground and excited states. intramolecular While the ground state does not allow for spontaneous hydrogen bond. proton transfer phenomena, for chosen molecules the S1 state was investigated, and in all these cases the ESIPT was predicted to occur, as the minimum at the donor side either vanished or was higher in energy than the minimum at the acceptor side. However, the easy modulation of the bridge parameters enabled us to describe the bridge proton behavior using the free energy rather than PES. 1D potential of mean force (Pmf ) was computed on the basis of Car–Parrinello MD [57] trajectory snapshots for Mannich base-type compound [58]. Further extension of this technique to 2D case (in the gas phase) was presented by Jezierska and Panek [59]. Calculation of 2D free energy surface for a strong intramolecular hydrogen bond was shown for 2-(N-diethylamino-N-oxymethyl)-4,6-dichlorophenol.

8.6 Tautomerism

–160

Bridge O–H ... O angle (°)

–170 180 170 O

160

H O

C2H5 N C2H5

CI

150 140

CI

0.8

1

1.2

1.4

1.6

1.8

2

Donor–proton O–H distance (Å) 0

5

10

15

Free energy (kcal/mol)

Figure 8.15 A two-dimensional free energy map reconstructed from the CPMD simulation with a posteriori inclusion of quantum nuclear effects. Negative values of the angle correspond to the acute O—H· · ·O angle. Source: Jezierska et al. 2008 [61]. Adapted with permission of Wiley.

This compound contains two slightly different molecules in the crystal cell with very short intramolecular hydrogen bonds (of 2.400 and 2.423 Å), as determined previously by neutron diffraction [60]. The bridge proton is strongly delocalized; therefore the application of the 2D Pmf provided a good insight into its topology. The solid-state free energy 1D profile is an almost symmetrical well, with proton centered in the middle, while the gas-phase 1D profile shows a shallow but asymmetrical well centered closer to the donor atom. The 2D free energy surface with D· · ·A distance and D—H· · ·A angle as the independent variables (Figure 8.15) illustrates that the well is also quite broad with respect to angular distortion. Simple but very interesting example of tautomerism within RAHB is provided by malonaldehyde. The formation of a quasi-ring closed by the hydrogen bond and possessing two conjugated double bonds makes the proton transfer relatively easy. Such proton transfer requires, however, internal reorganization of the double bonds, which raises questions on dynamical time dependence of the proton potential function. Wolf et al. [62] carried out ab initio MD simulations within DFT-based projector augmented wave MD scheme, related to the Car–Parrinello formulation. They have shown kinetic aspect of the proton dynamics, where various instantaneous forms of the proton potential (double well or symmetric single well) can occur on a short time scale. Another related example, 3-cyano-2,4-pentanedione, was studied by Durlak et al. [63]. They developed two types of free energy curves for the proton motion on the basis of Car–Parrinello scheme. One is based on classical description of nuclear motions, while the other includes quantum effects (path integral MD). In this strong intramolecular hydrogen bond, the proton is so strongly delocalized in a symmetric single well potential; that inclusion of quantum effects leads only to a small broadening of the well. This important result is in line with the observation

225

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8 Investigation of Strong Intramolecular Hydrogen Bonds

that nuclear quantum effects are not strongly modifying the system behavior when the barrier is either very high (small possibility of tunneling) or very low (freedom of motion already within classical limit). Another method of investigation of the free energy landscapes is metadynamics [64], which uses time-dependent potentials to move the particle out of the local minima and force it to explore the phase space. The history of the added potential terms is then used to reconstruct the free energy map defined by collective variables. The application of the method for strong intramolecular hydrogen bonds is shown in a study on quinoline N-oxide derivatives [65]. The free energy surfaces for 8-hydroxyquinoline N-oxide and quinaldic acid N-oxide have shown that spontaneous proton transfer in the electronic ground state is not favorable. These results are within the limit of classical treatment of nuclei, while some of our previous studies [61] were carried out with inclusion of nuclear quantum effects, indicating broadening of the potential well for strongly anharmonic systems. Car–Parrinello MD is an efficient tool to investigate correlations between the proton motions in interacting hydrogen bonds. The study of free energy surfaces for a zwitterionic proton sponge, 1,8-bis(dimethylamino)-4,5-dihydroxynaphthalene, has shown that indeed the motions of the protons in the N· · ·H· · ·N and O· · ·H· · ·O bridges are coupled [66]. The trans position of the protons (see Figure 8.16) is preferred because it maximizes the interaction of two formal opposite charges. However, the barriers for the proton transfer in both bridges are very small, and the bridge protons are strongly delocalized, which is visible in their strongly anharmonic 1D free energy profiles.

8.7 Calculation of IR Spectra of Strongly Hydrogen-Bonded Systems 8.7.1

The Harmonic Approximation

Historically, IR spectroscopy has been the most important spectroscopic method in the study of hydrogen bonding [67–69], and the possibility to predict the vibrational transitions of hydrogen-bonded systems by theoretical calculations has been of great interest for decades [70]. Most calculations of molecular vibrational frequencies are based on the harmonic approximation. In fact, the basic concepts in vibrational spectroscopy (normal modes; fundamental, overtone, and combination levels; selection rules, etc.) are defined within this approximation. In the harmonic approximation, it is assumed that the potential energy in the vibrational eigenvalue equation can be written as 1 (8.1) 𝜆 Q2 2 in which Q is the displacement along the particular independent normal mode. The constant 𝜆 is related to the harmonic force constant k, the reduced mass 𝜇, the harmonic frequency 𝜈, and the harmonic wavenumber 𝜈̃ through 𝜆 = k∕𝜇 = (2π𝜈)2 = (2πc̃ 𝜈 )2 . The normal modes Qr of the molecule and the corresponding V (Q) =

8.7 Calculation of IR Spectra of Strongly Hydrogen-Bonded Systems

0.5

2

0

1.5 1

–0.5 0

0.5

1

0.5

2

0

1.5 1

–0.5 –1 –1 –0.5

OHO PT coordinate (Å)

2

0

1.5 1

–0.5 1

0.5 0

NHN PT coordinate (Å)

0.5

Free energy (kcal/mol)

NHN PT coordinate (Å)

3 2.5

0.5

1

Solid state CPMD DFT-D2 dispersion

1

0

0.5

OHO PT coordinate (Å)

Solid state CPMD no dispersion

–1 –1 –0.5

0

0.5 0

2

0.5

1.5 0

1

–0.5

0.5

–1 –1 –0.5

OHO PT coordinate (Å)

0

0.5

1

0

OHO PT coordinate (Å)

+ H N

N

N

O

– O

O

H

2.5

1

Free energy (kcal/mol)

–1 –1 –0.5

0.5 0

3 2.5

1

Free energy (kcal/mol)

3 2.5

1

NHN PT coordinate (Å)

Gas-phase CPMD DFT-D2 dispersion Free energy (kcal/mol)

NHN PT coordinate (Å)

Gas-phase CPMD No dispersion

H

H

+ N

O–

Figure 8.16 Correlation of proton motions in double bridges of the zwitterionic 1,8-bis(dimethylamino)-4,5-dihydroxynaphthalene – free energy surfaces reconstructed from CPMD simulations. Qualitative explanation of the correlation is presented below: the bridge protons prefer trans arrangement, because it maximizes attraction between the centers of formal charges of the opposite sign. Source: Jezierska and Panek [66]. Copyright 2015, Adapted with permission of American Chemical Society.

𝜆r values are obtained as the eigenvectors and eigenvalues of the mass-weighted force constant matrix (see, for example, [71]). Hartree–Fock (HF) calculations, i.e. calculations within the molecular orbital approximation, generally predict harmonic vibrational wavenumbers that are about 10% too large relative to the observed ones. Considering a set of 1066 experimental wavenumbers, Scott and Radom [72] established empirical scaling relations and overall root-mean-square (RMS) errors for a variety of quantum chemical procedures. The RMS error for HF calculations was 50–60 cm–1 , depending on the basis set, but inclusion of electron correlation effects in the calculation model provided a significant improvement. In particular, the RMS error for DFT procedures like B3LYP [38, 39] and B3PW91 [38, 73] was found

227

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8 Investigation of Strong Intramolecular Hydrogen Bonds

to be 34 cm–1 , even with a modest basis set like 6-31G*. Increasing the size of the basis set did not improve the correlation [72, 74]. It is noteworthy that MP2 perturbation theory [41, 75] calculations performed relatively poorly, leading to RMS errors around 60 cm–1 , not better than HF calculations. Hence, the efficient DFT procedures are frequently the methods of choice, with B3LYP being the most popular. 8.7.2

Going Beyond the Harmonic Approximation

It is well known that the harmonic approximation tends to fail for strongly hydrogen-bonded systems. The potential energy associated with the stretching motions of strongly hydrogen-bonded OH and NH groups generally deviates from the harmonic approximation, leading to significant anharmonic effects. In strongly anharmonic potentials, coupling with other vibrational modes may lead to distribution of the IR intensity associated with the stretching motion over several transitions, frequently resulting in broad and complex absorption bands. In some cases, a specific OH or NH stretching band may be difficult to identify in the experimental IR spectrum [76–80]. Unfortunately, accurate multidimensional calculations going beyond the harmonic approximation are generally very demanding in terms of theoretical know-how and computational effort (see, for example, [81, 82]). The development of simplified, generally applicable procedures is an area of ongoing research. 8.7.2.1

Simplified, Static Procedures

Szczepaniak et al. [83] presented a simplified but remarkably efficient approach. They studied the IR spectrum of PANO, a compound with a short and strong intramolecular OH· · ·O hydrogen bond. They based their analysis on the harmonic approximation and on the relaxed anharmonic potential energy computed for the stretching motion of the hydrogen-bonded proton. The force constant matrix obtained from the harmonic calculation was modified by replacing pertinent elements with values derived from the computed anharmonic OH stretching potential. The results indicated that the OH stretching coordinate contributes to several normal modes, mixing strongly with other vibrational motions such as OH bending and C=O stretching. The authors obtained an excellent description of the entire IR spectrum of PANO, but general application of their procedure does not seem straightforward. In the vibrational self-consistent field (VSCF) approximation, the total vibrational wave function is written as a product of single mode functions, like in the harmonic approximation, but each vibrational mode is considered to move in the mean field of the rest of the vibrational motions. Each individual mode is thus allowed to couple with the other modes, amounting to a breakdown of the harmonic approximation. The procedure is analogous to the Hartree molecular orbital theory of electronic structure. The VSCF approximation and its developments have been reviewed by Bowman [84], Gerber and Ratner [85], Bowman et al. [81], Roy and Gerber [86], and most recently by Ravichandran and Subrata Banik [87]. The VSCF procedure is implemented in commercially available software packages like GAMESS [88, 89] and MOLPRO [90]. Alparone and

8.7 Calculation of IR Spectra of Strongly Hydrogen-Bonded Systems

Millefiori [91] applied the VSCF and the correlation-corrected VSCF techniques in a study of (Z)-3-hydroxypropenal (malondialdehyde enol), a system with a medium strong intramolecular hydrogen bond. The OH stretching mode was found to couple substantially with other modes, such as OH in-plane bending, leading to reduction of the harmonic value by more than 500 cm–1 . An attractive possibility is application of the second-order vibrational perturbation theory (VPT2) developed by Barone and coworkers [92–94]. This procedure is generally applicable, fully automated, and implemented in commercial software packages like Gaussian [95, 96]. Unfortunately, it is fairly time consuming, requiring a large number of force constant matrix calculations. The VPT2 procedure has been applied to several systems with intramolecular hydrogen bonding, but it tends to overestimate the anharmonic shift of the OH stretching band in OH⋅⋅⋅O systems. Depending on the basis set [13] (Spanget-Larsen, unpublished results), B3LYP calculations on malonaldehyde enol using the VPT2 procedure typically predict OH stretching wavenumbers around 200 cm–1 below the experimental value of 2856 cm–1 [97–101]. Application of the VPT2 procedure to dibenzoylmethane enol with a stronger hydrogen bond yielded OH stretching wavenumbers equal to 2208 cm–1 with B3LYP/6-31G* and 1550 cm–1 with B3LYP/cc-pVDZ [13, 77]. These values indicate a remarkable dependence on the basis set, and both values are significantly lower than the experimental wavenumber close to 2600 cm–1 ; see Figure 8.17 [13, 103]. Finally, Buemi and Zuccarello [104] studied nitromalonamide enol, which has a short and strong intramolecular OH⋅⋅⋅H hydrogen bond. Their B3LYP/6-311++G(d,p) anharmonic VPT2 calculation yielded an OH stretching wavenumber equal to 1428 cm–1 , a value much lower than the observed band center close to 1900 cm–1 [13, 105]. Including the influence of an aqueous solvent by the PCM-SCRF model, Buemi and Zuccarello [104] obtained a wavenumber equal to 856 cm–1 , but the authors considered this prediction to be unreliable. Spanget-Larsen et al. [13] and Hansen and Spanget-Larsen [14] applied the VPT2 procedure to a large number of intramolecularly OH· · ·O hydrogen-bonded systems, ranging from weak to strong hydrogen bonding. As shown in Figure 8.18, they observed an excellent linear correlation between the OH stretching wavenumbers computed with the VPT2 procedure and the corresponding harmonic values. This is an interesting result since the VPT2 calculation is orders of magnitude more time consuming than the harmonic analysis. Moreover, these authors established a linear correlation between wavenumbers of observed band centers and calculated harmonic wavenumbers, the so-called P(Harm) procedure. This procedure enables the prediction of effective OH stretching wavenumbers for this class of compounds from a simple harmonic calculation to an accuracy of about 70 cm–1 (see, for example, Figure 8.17). These results are discussed in recent literature [5, 13, 14]. 8.7.2.2 Advanced Calculations on Small Systems: Malonaldehyde and Acetylacetone

The simplest system with an intramolecular OH· · ·O hydrogen bond is malonaldehyde enol ((Z)-3-hydroxypropenal), with only nine atoms and 21 vibrational

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8 Investigation of Strong Intramolecular Hydrogen Bonds

3500 0.5

3000

2500

2000

1500

O

1000 IR CCI4 solution

Absorbance

O

H

P(Harm) 2643 cm–1 0.0

IR intensity (km/mol)

νOH

B3LYP/cc-pVDZ Anharmonic, VPT2

500

0 B3LYP/cc-pVDZ Harmonic

500

0 3500

νOH

3000

2500

2000

1500

1000

Wavenumber (cm–1)

Figure 8.17 The IR spectrum of dibenzoylmethane enol has puzzled spectroscopists for more than half a century [102]. A B3LYP/cc-pVDZ anharmonic VPT2 frequency calculation predicts a very large anharmonic shift of the OH stretching transition, leading to a wavenumber of 1550 cm–1 [77]. But the P(Harm) procedure predicts a value close to 2600 cm–1 [13], which is consistent with the assignment by Tayyari et al. [103]. The very strong transition predicted close to 1600 cm–1 is associated with the OH in-plane bending vibration. Source: Hansen et al. [77]. Copyright 2006, Top panel adapted with permission of Elsevier B.V.

Figure 8.18 Linear regression of OH stretching wavenumbers computed with the anharmonic VPT2 procedure on the corresponding harmonic values. The calculations were performed with B3LYP/6-31G* for a series of compounds with intramolecular OH· · ·O hydrogen bonding, including also a few species with free OH groups [13, 14]. Source: Spanget-Larsen et al. [13]. Copyright 2011, Adapted with permission of Elsevier B.V.

4000

VPT2 anharmonic (cm–1)

230

3000

2000

1000 2000

3000 Harmonic

(cm–1)

4000

8.7 Calculation of IR Spectra of Strongly Hydrogen-Bonded Systems

degrees of freedom. The observed OH stretching band has a wavenumber of 2856 cm–1 [97–101] corresponding to a medium strong hydrogen bond. The hydrogen-bonded proton is assumed to be situated in a symmetrical double minimum potential. The tunneling splitting has been determined by microwave spectroscopy at 22.58 cm–1 [106–108]. Malonaldehyde has been the test case for several advanced theoretical investigations, focusing in particular on the anharmonic effects on the IR spectrum, the proton transfer rate, and the tunneling splitting. The study of malonaldehyde by Alparone and Millefiori [91] using VSCF procedures was mentioned above. Tayyari et al. [101] constructed a 2D potential energy function, comprising the OH stretching and in-plane bending motions. On the basis of MP2 and B3LYP calculations, they obtained satisfactory agreement with the observed tunneling splitting. The authors predicted a strong coupling between OH stretching and bending modes and concluded that this coupling was responsible for the anomalous vibrational behavior of bent intramolecularly hydrogen-bonded systems. Yagi et al. [109] generated a full-dimensional (21D) potential energy hypersurface for malonaldehyde and studied the proton transfer and the tunneling dynamics by ab initio procedures. On the basis of this potential, Manthe and coworkers [110] applied accurate full-dimensional quantum dynamics methods, using the multi-configurational time-dependent Hartree approach and a diffusion Monte Carlo-based spectral evolution method. Excellent agreement with observed data was obtained with both methods. On the basis of high-level CCSD(T) electronic energies, Bowman and coworkers [111, 112] computed a highly accurate full-dimensional potential energy hypersurface for malonaldehyde enol, obtaining a barrier for intramolecular proton transfer equal to 4.1 kcal/mol. They performed calculations in a 1D truncated model, as well as using a full-dimensional diffusion Monte Carlo-based procedure. The latter resulted in excellent agreement with experimental data. Using the full-dimensional PES previously published by Wang et al. [112], Schröder and Meyer [113] studied the tunneling splitting in low-lying excited vibrational states of malonaldehyde with the multi-configuration time-dependent Hartree method. They obtained good agreement with the spectroscopically determined values for several states. Application of the abovementioned advanced procedures to larger systems is not straightforward. For example, replacing the terminal hydrogen atoms in malonaldehyde enol with methyl groups leads to acetylacetone enol, the simplest member of the series of β-hydroxyketone enols. Acetylacetone comprises 15 atoms, with 39 vibrational degrees of freedom. To our knowledge, calculation of a full-dimensional 39D potential energy hypersurface for this system has so far not been attempted. In general, approximate and truncated procedures must be applied. Mavri and Grdadolnik [114, 115] studied the OH· · ·O proton dynamics in acetylacetone enol by applying a mixed quantum–classical approach. The hydrogen-bonded proton was treated as a quantum particle, while the remaining frame of the molecule was treated according to classical mechanics. The proton

231

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8 Investigation of Strong Intramolecular Hydrogen Bonds

dynamics calculation lead to qualitative prediction of the observed, very broad and complex OH stretching band with a maximum close to 2800 cm–1 . Matanovi´c and Dosli´c [116–118] considered the vibrational spectrum of acetylacetone enol by using reduced-dimensional potential models and VPT2 (see above) calculation of anharmonic effects on the vibrational wavenumbers. They concluded that the torsional dynamics of the methyl groups had a strong impact on the considerable broadening of the observed OH stretching band. Replacing the methyl groups in acetylacetone enol with phenyl groups leads to dibenzoylmethane enol (Figure 8.17). As discussed above, the OH stretching band is believed to be situated around 2600 cm−1 [13, 103]. This assignment was recently supported by the results of an MD simulation [119]. According to the theoretical results of these authors, the very broad, very intense, and irregularly shaped “monster band” between 1700 and 1400 cm−1 (with strong Evans windows) is primarily associated with the OH in-plane bending motion. 8.7.3 8.7.3.1

Larger Systems Car–Parrinello Molecular Dynamics Simulations

The use of methods reproducing time evolution of the molecular properties enables an analysis of the vibrational features directly from the classical nuclei molecular trajectory. One approach is done by Fourier transform of the dipole moments. As a result one obtains not only positions but also intensities of the characteristic bands, which can be corrected for quantum behavior; however, the use of the total dipole moment makes it impossible to separate the contributions of specific groups of atoms. Another possibility is to use the Fourier transform of the atomic velocity autocorrelation function. This yields correct band positions, but intensities are proportional to motion amplitudes and thus arbitrary; however, the spectrum can be decomposed into atomic contributions. One has to keep in mind that the Car–Parrinello or Born–Oppenheimer MD works in the limit of classical nuclei; therefore phenomena such as overtones, Evans holes, etc. will not be reproduced in the simulated IR spectrum. The impact of modulated proton dynamics on the shape of IR spectra was presented in a study on two differently substituted o-hydroxyaryl Schiff bases [120]. An influence of substituent (steric, inductive, and resonance) effects and condensation state (gas phase vs. solid state) on vibrational features was clearly demonstrated. Gas-phase results for 2-(N-methyliminomethyl)-4,6-dichlorophenol indicate that the proton is located at the donor side with only a few events of proton transfer on 8 ps time scale, but when the crystalline form is studied, the proton is transferred to the acceptor side. On the other hand, 2-(N-ethyl-α-iminoethyl)-4-chloro-5-methylphenol exhibits very frequent and longer-living proton transfers to the acceptor side, while the crystal phase also shows proton transfer to the acceptor side. These aspects of the proton dynamics are reflected in the power spectra of atomic velocity (see Figure 8.19), where the first compound in the gas phase possesses a bridge proton stretching signature from ca. 2200 to 3000 cm−1 ; transition to the solid state shifts this stretching region toward higher wavenumbers (2400–3100 cm−1 ), in agreement with the more localized character of the

Gas-phase CPMD 2-(N-methyliminomethyl)-4,6-dichlorophenol H

N

Arbitrary intensity

Arbitrary intensity

O

Solid state (crystal) CPMD 2-(N-methyliminomethyl)-4,6-dichlorophenol

CI

CI

All atoms

All atoms Bridge proton 500

1000

1500

2000

2500

Bridge proton

3000 –1 3500 500 cm

1000

Gas-phase CPMD 2-(N-ethyl-α-iminoethyl)-4-chloro-5methylphenol H

2000

2500

3000 –1 3500 cm

Solid state (crystal) CPMD 2-(N-ethyl-α-iminoethyl)-4-chloro-5methylphenol

N

Arbitrary intensity

Arbitrary intensity

O

1500

H3C CI

All atoms

All atoms Bridge proton 500

1000

1500

2000

2500

3000 –1 3500 500 cm

Bridge proton 1000

1500

2000

2500

3000 –1 3500 cm

Figure 8.19 Atomic velocity power spectra for 2-(N-methyliminomethyl)-4,6-dichlorophenol and 2-(N-ethyl-α-iminoethyl)-4-chloro-5-methylphenol. Bottom lines in each panel correspond to the bridge proton contribution. Source: Jezierska-Mazzarello et al. 2011 [120]. Adapted with the permission of American Institute of Physics.

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8 Investigation of Strong Intramolecular Hydrogen Bonds

proton (albeit at the acceptor side). The mobility of the bridge proton in the gas-phase 2-(N-ethyl-α-iminoethyl)-4-chloro-5-methylphenol is underlined by a very broad signature overlapping with lower wavenumbers (corresponding to heavy atom motions), yielding a continuous region (900–2800 cm−1 ). When the crystal is studied, the proton is localized mostly at the acceptor side, but the anharmonicity of the potential locates the stretching signature from 2200 to 3050 cm−1 . In this study the power spectra of dipole moment trajectory proved informative, because they have shown increased absorption in the stretching regions, corresponding to an increased polarizability of the delocalized proton. A combination of experimental and theoretical approaches was used to gain deeper insight into the bridge proton dynamics of PANO. While the previously mentioned study demonstrated the importance of environmental effects (crystal field), it was necessary to resort to the first-principles MD to obtain quantitative agreement between experimental and theoretical vibrational features [121]. To achieve this, a Car–Parrinello MD of crystalline PANO was performed; it has shown the proton localized at the donor side, with only four events of short-lived intramolecular proton transfer on the time scale of 5 ps. However, when 1D proton potential function was calculated for selected snapshots of CPMD, according to the snapshot–envelope methodology [122], it turned out that large variations of the potential function shape occurred as a result of modulation by heavy atom positions. Each potential function served as a potential term for the vibrational Schrödinger equation. Solutions of individual equations (one for each snapshot) yielded a set of vibrational transition wavenumbers, which were then broadened by a Gaussian term and provided a final envelope (shape) of the stretching mode. It was correctly predicted to have a maximum close to 1400 cm−1 , in agreement with experimental IR spectrum. This methodology allows for introducing quantum nature of nuclear vibrations for a selected mode, in this case – the bridge proton stretching. The snapshot–envelope methodology was also successfully applied to demonstrate differences in vibrational signatures of model o-hydroxy Schiff and Mannich bases, which were, respectively, N-methyl-2-hydroxybenzylidene amine (HBZA) and o-dimethylaminomethylphenol (DMAP) [61]. The power spectra of atomic velocity were able to reproduce fundamental differences in the proton dynamics derived from diverse chemical constitution of these two species; HBZA exhibits a broad region of proton stretching (2000–3000 cm−1 ), while DMAP has a narrower band at higher wavenumbers, 2550–3300 cm−1 . The inclusion of quantum effects via a posteriori snapshot–envelope technique resulted in a broad stretching region for HBZA, from ca. 1000 to 2800 cm−1 , in agreement with experimental gas-phase IR measurement [123]. This shows that the description of intramolecular hydrogen bond in this class of aromatic Schiff bases benefits from inclusion of quantum nuclear effects.

8.8 NMR 8.8.1

Introduction

Calculation of NMR nuclear shieldings is usually done by the gage-invariant atomic orbital (GIAO) method [124]. However, in the past also the local origin

13C

nuclear shieldings

8.8 NMR

90 80 70 60 50 40 30 20 10 0 100

y = –0.8933x + 182.63 R2 = 0.9964 120

140 13C

160

180

200

measured chemical shifts

Figure 8.20 Plot of 13 C nuclear shieldings vs. 13 C measured chemical shifts all in ppm.

gauge (LORG) method [125] has been used with success in calculations of ring current effects [126]. The calculations give nuclear shieldings. These can then be converted to chemical shifts in two ways, either calculating the nuclear shielding of tetramethylsilane (TMS) or another suitable reference compound or plotting the nuclear shielding vs. the observed chemical shifts as demonstrated in Figure 8.20. The present authors prefer the latter as a number of parameters are obtained to judge the quality of the calculated data. These parameters are the R2 , the slope, and the average deviation. This will be discussed later. 8.8.2

Calculation of OH Chemical Shifts

OH chemical shifts are mentioned as an example as the OH chemical shifts often have been used to estimate the hydrogen bond strength of hydrogen bonds in which the OH groups are involved. OH chemical shifts depend on the OH bond length as demonstrated in Figure 8.21. An overview is given by Siskos et al. [127].

Calc. OH bond lengths

1 y = 0.0039x + 0.9356 R2 = 0.9349

0.995 0.99 0.985 0.98 0.975 0.97 10

11

12

13 14 OH chemical shifts

15

16

Figure 8.21 Calculated OH bond lengths in Angstroms vs. OH chemical shifts in ppm for a series of o-hydroxy aromatic aldehydes.

235

236

8 Investigation of Strong Intramolecular Hydrogen Bonds

The OH chemical shifts may also depend on anisotropy effects and in aromatic compounds on ring current effects. The latter has to be estimated and subtracted out before using the OH chemical shifts as a measure of hydrogen bond strength [6].

8.8.3 Estimation of Ring Current and Anisotropy Effects on OH Chemical Shifts Calculation of ring current effects goes very far back. A classic case is the Johnson–Bovey map [128]. Later, Haig and Mallion [129] developed this further. Scheiner has also calculated the shielding at various positions around the benzene ring [130]. More recently, Kleinpeter and Koch [33] has used through-space nuclear magnetic resonance shieldings (TSNMRSs) to calculate ring current effects in a number of cases. The idea is the same as for NICS values [131]. The latter were computed on the basis of MP2/6-311G(d,p) geometries using the GIAO method [124] at the B3LYP/6-311G(d,p) [38, 39] theory level. To calculate the spatial NICS, ghost atoms were placed on a lattice of −10 to +10 Å with a step size of 0.5 Å in the three directions of the Cartesian coordinate system. The resulting 68 921 NICS values, thus obtained, were analyzed and visualized by the SYBYL 7.3 [132] molecular modeling software. The isochemical shielding surfaces are explained in the legend of Figure 8.22. Quantitative indication of the diatropic ring current effect is the closer the distance (in Angstroms) of a certain shielding (deshielding) ICSS from the center of the molecule, the stronger the corresponding ring current effect in 1 H NMR spectroscopy. This type of calculations can then be used to calculate the ring current contribution to OH chemical shifts. The OH proton is positioned at the green spot and the oxygen is left out. This technique has been used on a series of o-hydroxy acyl aromatics and some o-hydroxythioacyl aromatics [6]. An example is shown in Figure 8.22.

(a)

(b)

Figure 8.22 Visualization of the spatial magnetic properties (TSNMRS) of formaldehyde (a) and thioformaldehyde (b) as ICSS (blue represents 5 ppm shielding; cyan, 2 ppm shielding; blue green, 1 ppm shielding; green, 0.5 ppm shielding; yellow, 0.1 ppm shielding; and red, −0.1 ppm deshielding). Source: Hansen et al. 2018 [6]. Reprinted with permission of Elsevier.

8.8 NMR

Figure 8.23 Calculated ring current effects for salicylaldehyde and thiosalicylaldehyde. Source: Hansen et al. 2018 [6]. Reprinted with permission of Elsevier.

O

O

H

R

O

N O HO

(a)

(b)

O

–

H S

0.03 total

0.67 ppm H total

O

O

H

H

R

N+

O

O

H O

H

(c)

Figure 8.24 (a) 10-Hydroxybenzo[h]quinoline; (b) 4′ -substituted 2′ -hydroxyacylaromatic; (c) 5′ -substituted-2′ -hydroxyacylaromatic.

Two examples are salicylaldehyde and thiosalicylaldehyde as seen in Figure 8.23. 8.8.4

Estimation of Other Effects on OH Chemical Shifts

Scheiner [7] has taken a more general approach that also in relevant cases involves ring current effects. He divided the OH chemical shift into two contributions: one that is independent of the hydrogen bonding as such and one dependent on the hydrogen bond. This is done by first calculating the effect for the complete system, and then subtract the effect calculated when the acceptor is absent. From this is subtracted what is termed the positional effect, the effect when the donor group is absent and the chemical shift is calculated at the position of the OH proton. Calculation of chemical shift of the OH proton when the acceptor is removed is clearly problematic in cases in which the acceptor is integrated in the molecule. An example is 10-hydroxybenzo[h]quinolines (see Figure 8.24a). Besides this technical problem, it is not obvious if it is really neutral in systems like Figure 8.24b, c, in which the substituent influences the resonance assistance to remove the acceptor. 8.8.5

Calculation of Chemical Shifts in Charged Systems

Strong intramolecular hydrogen bonds are found in thiophenoxyketenimines (see Figure 8.25). This type of compound has, as seen in the Figure 8.25, probably a strong zwitterionic character. Furthermore, the presence of a “C=S” bond made it difficult to calculate the nuclear shieldings. It was obvious that both calculations of the structure and the calculation of nuclear shieldings were of importance. A large number of functionals, also some that were made to deal especially with aromatic compounds, like the [133, 134] ones, were less than optimal. In the end, calculations at the MP2 level were used with 6-311+G(2d,p) basis set for the NMR calculations and the B3LYP/6-311++G(d,p) functional/basis set for the structure calculations [135].

237

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8 Investigation of Strong Intramolecular Hydrogen Bonds

R

R N

´R

H

N

´R

S

S

X

H

Figure 8.25 Resonance structures of thiophenoxyketenimines.

X A

8.8.6 8.8.6.1

B

Calculation of Deuterium Isotope Effects on Nuclear Shieldings Jameson Approach

Deuterium isotope effects on especially 13 C chemical shifts have been shown to be good measures of hydrogen bond strength [9, 11, 136]. Furthermore, deuterium isotope effects are very useful in establishing tautomeric equilibria in compounds with strong hydrogen bonds [9, 11, 12]. According to Jameson [137], the deuterium isotope effect observed at neighboring atoms can be expressed as follows: ( C) ( C) 𝛿𝜎 𝛿𝜎 ∗ [⟨ΔrAH ⟩ − ⟨ΔrAD ⟩] + [⟨ΔrCA ⟩ − ⟨ΔrCA ⟩∗ ] + · · · 𝜎−𝜎 = 𝛿rAH e 𝛿rCA e (8.2) The largest effect occurs in the bond in which the substituted atom is directly involved. For effects over more bonds, the isotope effect observed at the 13 C nucleus being at a distance of two or more bonds (13 C—A—H, A = C, O, N) from the site of substitution depends on the change in the shielding of the 13 C nucleus occurring in response to the shortening of the CA bond (primary electronic factor) as well as to the shortening of the AH bond (secondary electronic factor) and also on the changed averaged distances AH (primary dynamic factor) and CA (secondary dynamic factor). It has been shown that the product of the secondary electronic factor (δ𝜎 C /δrAH )e and the primary dynamic factor ⟨ΔrAH ⟩ − ⟨ΔrAD ⟩ is more important than the second term in Eq. (8.2) leading to the simple equation ( ) ∑ 𝛿𝜎 ∗ [⟨ΔrCH ⟩ − ⟨ΔrCD ⟩] (8.3) 𝜎−𝜎 = 𝛿rCH e In aromatic molecules and other molecules in which the π-electron delocalization takes place, the slight changes in the AH bond distance due to isotopic substitution may lead to changes in the charge distribution in the molecule and as a consequence to the isotope effect on the chemical shielding. Calculations were made for a series of o-hydroxy acyl aromatics deuterated at the OH position(s) with widely different deuterium isotope effects on 13 C shifts as seen in Figure 8.26 [136, 138, 139]. One checkpoint for these calculations is the comparison of the various long range isotope effects within the same molecule as illustrated in Figure 8.26. The calculations are based on Eq. (8.3). The value of ⟨ΔrOH ⟩ − ⟨ΔrOD ⟩ is easily calculated by varying the OH distance. The (δ𝜎 C /δrOH )e is obtained from a PES calculated by gradually extending the OH bond, allowing the other bonds

8.8 NMR

Figure 8.26 Deuterium isotope effect transmission coefficients. Source: Hansen 1993 [136]. Reproduced with permission of John Wiley & Sons.

O

X

H (D) O

(20)

X=H, R′ or OR

60 10

–30 25

–80 –10

O

200

80 O

–50

N –130

(a)

O –120 20 650

H O

–400 (b)

O

N

–140

–60

–30

H

470

130 O

N

–60

O

O 70

–230

H O

(c)

Figure 8.27 Deuterium isotope effects on 13 C chemical shifts for picolinic N-oxide. (a) Calculated assuming that the OH bond is shortened 0.01 Å. (b) Experimental values. (c) Values calculated using a 2D potential. Vales are in ppb. Source: Hansen 2007 [11]. Reproduced with permission of John Wiley & Sons.

and angles involved to be optimized. Zero point energies are calculated for the H and D species, and the point of the maximum probability is found [138]. In a simpler form just to get a comparison between the isotope effects, the OH bond length is shortened to a fixed amount. A comparison between the experimental and calculated values for PANO is given in Figure 8.27. In this case the isotope effects were calculated as just described above (a) but also using a 2D potential (b) (from [140]). The experimental values are given in (c). A different approach has been taken by Limbach et al. [141] but not used it for strong intramolecular hydrogen bonds. However, this approach in the end is similar to the simpler approach described above. 8.8.6.2

Car–Parrinello and Two-Dimensional Sampling of Chemical Shifts

Calculation of spectroscopic properties based on a Car–Parrinello trajectory is not limited to the vibrational features only. The NMR properties can also be computed, however, at a significant computational cost due to delocalized nature of the plane wave basis set, which is not atom centered [142]. An application of the method to strong intramolecular hydrogen bonds was presented for 2-(N-diethylamino-N-oxymethyl)-4,6-dichlorophenol, a Mannich-based N-oxide with short hydrogen bond, d(O· · ·O) = 2.407 Å. The NMR simulations were performed for two phases: gas form and crystalline form. Snapshots extracted from the CPMD trajectory served as a set of data to calculate shielding parameters. Then, the values for respective protons were averaged over the CPMD run. The experimental 1 H NMR chemical shift of the bridge proton is 17.8 ppm (CD2 Cl2 as a solvent), while the averaged CPMD-based values are 13.5 ppm for the gas phase and 15.1 ppm in the crystal. The agreement is then rather semiquantitative, but large deshielding of the proton in the strong intramolecular hydrogen bond is reproduced.

239

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8 Investigation of Strong Intramolecular Hydrogen Bonds

Treatment of isotope effects in PANO is presented with the use of construction of DFT-based 2D free energy surfaces [140]. At each point of a PES (with O—H and O· · ·O distances as independent variables), 1 H and 13 C NMR chemical shifts were calculated for selected atoms. The solution of 2D vibrational Schrödinger equation combined with the 2D map of chemical shifts yielded expectation values of the chemical shift functions of the chosen nuclei. Primary isotope effects for the bridge proton, 𝛿 H − 𝛿 D , yielded a value of +0.631 ppm in chloroform (experimental value is +0.476 ppm); the secondary effects (13 C) are in worse agreement with the experiment. For the equilibrium systems like Schiff bases (an example is shown in Figure 8.12), the total isotope effect can be formulated as ΔCtot = ΔCint + ΔCeq

(8.4)

ΔCint = XM × ΔCint (M) + (1 − XM )ΔCint (PT)

(8.5)

ΔCeq = ΔX × (δCM − δCPT )

(8.6)

Ratio = (δC2M − δC2PT )∕(δC1′M − δC1′PT )

(8.7)

An important feature is the ratio between two isotope effects as this can be used to estimate if the equilibrium isotope part is dominant. If one assumes as in Eq. (8.5) that the intrinsic contribution for the two carbons C-2 and C-1′ are small compared to the equilibrium part, then plotting data of Eq. (8.7) should lead to a straight line as seen in [143].

8.9 Principal Component Analysis PCA has been used to analyze deuterium 13 C isotope effects and 13 C NMR chemical shifts in Schiff bases [144]. It was shown that the factors obtained could be related to the XH proton chemical shifts involved in the intramolecularly hydrogen bonds. This also makes possible to classify groups of compounds as either tautomeric or “static.” PCA has also been used to analyze bond distances and bond length in the six-membered intramolecularly hydrogen-bonded systems of o-hydroxy nitro, and acyl aromatics in relation to deuterium isotope effects (deuterium at the hydroxyl group) is transmitted via the hydrogen bond [145].

8.10 Solvent Effects Intramolecular hydrogen bond can be modulated by the presence of solvent, to the point of being disintegrated in favor of two separate hydrogen bonds formed with solvent molecules. Such phenomena were studied experimentally [146, 147] indicating that even low-polarity solvents can affect self-association of sterically modified Schiff bases. The subject of solvent effects is challenging for computational approaches due to the large scale of involved systems (bulk

8.11 Conclusions

solvents) and necessity to take into account entropic and dynamical effects. However, instead of using explicit solvation models, where the solvent molecules are included directly in the simulation, one can use continuum models, where the solvent is represented by dielectric continuum, and the solute is placed within a cavity (PCM, COSMO models [15]). Peter Nagy in his review [148] described in detail competition between intra- and intermolecular hydrogen bonds in solution. He pointed out the role of formation of quasi-rings; distortion of dihedral angles was found to be a sensitive parameter of interactions with solvent. Prediction of the intramolecular vs. intermolecular hydrogen bonding pattern in polar (aqueous, some protic organic) solutions is however challenging. The review points out that the gas-phase structure is at least partially maintained in aqueous solution, but the important – and difficult to predict – issue is a shift in the populations of conformers. Generally there is increase in population of the species with disrupted intramolecular hydrogen bonds. Computational studies using first-principles MD are usually concerned with intermolecular hydrogen bonding and solvation shells; a study of trimethylamine N-oxide in water [149] is a very recent example; intramolecular bridges have not received as much attention. One of the best small models for intramolecular hydrogen bonds, hydrogen maleate, was studied by Dopieralski et al. [150], and the presence of the counterions (Na+ , K+ ) was discussed. It was also found that the inclusion of quantum effects with path integral MD scheme led to increased symmetry of the potential well and larger delocalization of the bridge proton.

8.11 Conclusions A key question related to strong intramolecular hydrogen bonds is of course to be able to say that it is indeed strong. The hydrogen bond energy should be a good measure. However, the determination of the hydrogen bond energy is still difficult. The hydrogen-bonded/open approach (HB-out) suffers from the fact that interactions in the open form often make it impossible to use the method. The MTA also seems to have its problems with steric interactions (see Figure 8.11). One possibility suggested recently for o-hydroxy aromatic aldehydes is to use two-bond deuterium isotope effects on 13 C chemical shifts as a measure of hydrogen bond strength (Figure 8.9). However, it is still to be seen if this procedure is generally applicable. Vibrational frequencies can to a large extent be calculated rather well in the harmonic approximation. An exception is the OH stretching frequencies for systems with strong intramolecular hydrogen bonding. Moreover, the IR absorption bands for such systems are frequently broad and diffuse, making it difficult to identify these transitions. Barone and coworkers have developed an anharmonic VPT2 procedure for the calculation of vibrational transitions, providing useful results for a variety of compounds. But this procedure tends to overestimate the anharmonic shifts in hydrogen-bonded systems, as observed for the OH stretching frequencies in a large series of compounds with intramolecular OH· · ·O hydrogen bonds (see,

241

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8 Investigation of Strong Intramolecular Hydrogen Bonds

for example, Figure 8.17). Moreover, it was found that for those compounds, the OH stretching wavenumbers predicted with the VPT2 procedure were essentially linearly related to the corresponding harmonic wavenumbers (Figure 8.18), and an efficient correlation between observed band centers and calculated harmonic OH stretching wavenumbers was established. Car–Parrinello MD has become an established method to investigate intermolecular and intramolecular interactions. As it is shown in the current overview, proton reaction pathways and their consequences for the molecular structure, conformation, and reactivity were successfully studied within the CPMD framework. However, its applications require massive computing power, especially within path integral scheme of quantization of nuclear motions. Other factors that limit the accuracy of CPMD are divided into DFT-related and specific CPMD-related issues. The former ones are too “soft” potential energy functions leading to proton transfer barriers lower than for, e.g. MP2 method, lack of consistent description of dispersion forces, and quality of pseudopotentials. The latter ones are short time step required to reproduce the fictitious orbital dynamics (restricting the routinely available time scale to hundreds of picoseconds only), the “drag effect” (coupling of electronic and ionic degrees of freedom leading to lower wavenumbers as compared with the limit of zero fictitious orbital mass), and the classical treatment of nuclei, which prevents an easy description of tunneling, overtones, etc. The path integral scheme is very costly, and it is limited to ensemble averages, while centroid dynamics – giving access to quantum dynamics – is even more costly. Even taking into account all of the above issues, Car–Parrinello MD enables numerous scientists to study details of dynamics of molecular processes that are not accessible via static approaches (location of minima) or classical force fields.

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9 Spectral Simulation for Flexible Molecules in Solution with Quantum Chemical Calculations Yukiteru Katsumoto Fukuoka University, Faculty of Science, 8-19-1 Nanakuma, Jonan-ku, Fukuoka 814-0180, Japan

9.1 Introduction Computer simulations for molecular structures and properties have moved from a specialized branch of theoretical chemistry to a practical tool used by common chemists. This drastic change has been owing to great advances in user-friendly software packages and computational resources. Moreover, the access to common computer systems, in which useful calculation tools are ready to use, becomes easy time by time. Nowadays, therefore, only a modest background in computational chemistry is required for running a quantum chemical calculation (QCC). The simulation of molecular spectra is one of the most important techniques that enable us to connect an experimental result with the knowledge in quantum chemistry. In general, spectral simulations with QCCs are utilized for investigating the molecular structures including configuration and conformation, the chemical reactions, the molecular interactions, the physical properties such as electromagnetics and optics, and so on. In this chapter, we will focus on the identification of conformation and configuration for flexible molecules in solution. In this context, we will also deal with the interaction between the target and solvents. For flexible molecules in solution, the spectral simulation with QCCs plays a key role, because these molecules may have various conformers and may interact with solvents. It has been known that the density functional theory (DFT) calculation can reproduce the vibrational spectra of organic molecules in terms not only of the wavenumber (after an appropriate scaling) but also the relative intensity. The classical calculations of vibrational bands, the normal mode analysis, are carried out based upon the harmonic oscillator approximation (HOA). Therefore, we need to apply an appropriate scaling factor to the simulated vibrational wavenumbers in order to establish consistency with the experimental wavenumbers containing anharmonicity in nature. Yoshida et al. have performed the normal mode analysis for 205 compounds at the B3LYP/6-311G(d,p) level and have proposed the scaling method for shifting the calculated wavenumbers adjusting to the observed ones [1]. This report has demonstrated that an appropriate scaling

Molecular Spectroscopy: A Quantum Chemistry Approach, First Edition. Edited by Yukihiro Ozaki, Marek Janusz Wójcik, and Jürgen Popp. © 2019 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2019 by Wiley-VCH Verlag GmbH & Co. KGaA.

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of the calculated vibrational wavenumbers with HOA enables us to predict the vibrational wavenumbers of unknown compounds with high accuracy. If target molecules are polyatomic and flexible, the study on the structural isomers, the conformational isomers, and the stereoisomers becomes very important for understanding their vibrational spectra. In liquid, solution, and solid phases, a target molecule may interact with the other molecules. Thus, we have to consider not only the conformation and configuration but also intra- and intermolecular interaction of the targets. In order to study the conformations of a compound by molecular spectroscopies, the key is to find the marker band(s) with the aid of QCCs. In the following sections, we will show several examples for the spectral simulation used in the study on the conformation, configuration, and solvation of flexible molecules. The first step of the simulation is to select an appropriate calculation level in QCCs by comparing the experimental and simulated spectra of the target. If the target molecule is too large to be treated in QCCs (such as a polymer), we should also find an appropriate model compound. When our target is a polymer containing the amide group, for example, a ternary amide compound would be the good model. The second step is to know the solvation effect on the molecular spectrum of the model compounds. In this procedure, the readers will know characteristic features of the solvation in vibrational spectroscopies. The third step is to search which conformer(s) should be observed in a real system. In general, the molecular spectrum of a flexible compound is more complicated than the expectation from its chemical structure, because of the existence of various conformers. The spectrum of a flexible molecule is possibly the sum of spectra arising from individual conformers. In this case, the relative energies of optimized conformers obtained from QCCs are very helpful to evaluate their observability under the experiment [2]. If a target contains an asymmetric carbon, each stereoisomer shows the vibrational circular dichroism (VCD). The VCD plays an important role to determine the absolute configuration of a chiral compound without chromophore [3]. QCCs make a significant contribution when the chiral target possesses various conformers [4]. In such a case, the relative energies and the vibrational intensities of characteristic bands of the conformers become crucial to identify the absolute configuration. This is the same for the simulation of electronic circular dichoism (ECD) spectra, which can be carried out by the time-dependent DFT (TD-DFT) method [5].

9.2 Selection of the Calculation Level for Spectral Simulations of Flexible Molecules The first step of spectral simulations is to select the theoretical method of the calculation. Nowadays, various types of calculation methods are available in QCC packages such as GAMESS and Gaussian. Hartree–Fock (HF) theory is often chosen for the first trial in the optimization process of large and flexible molecules using a simple basis set such as STO-3G and 3-21G. In general, it is considered that HF cannot give an accurate solution in any system, because

9.2 Selection of the Calculation Level for Spectral Simulations of Flexible Molecules

Figure 9.1 Chemical structures of NdMAm and NMAm. N

O

N

O

H NdMAm

NMAm

the mean-field approximation is implied. The advantage of the HF method is that the calculation results can be systematically improved via post-HF methods such as Møller–Plesset (MP) perturbation theory [6], coupled-cluster single-double (CCSD) theory [7, 8], and so on. Because the calculation cost of these methods becomes very high, DFT calculations have often been chosen. Computational costs of DFT calculations are significantly low compared with the post-HF methods. After the approximations concerned with the exchange and correlation interactions, the accuracy of DFT results is improved enough to discuss the electron structure of molecules. In this section, we will show the process for selecting the theoretical method and the basis set, taking the example of the simulation for the infrared (IR) spectrum of N,N-dimethylacetamide (NdMAm, Figure 9.1). First of all, we have to prepare an experimental IR spectrum of NdMAm under less perturbed condition. A matrix-isolation IR spectrum [9] is very helpful for this purpose. If it is difficult, you can also refer the IR spectrum of the target in a dilute solution with a polar solvent. Figure 9.2 shows IR spectra of NdMAm in Ar matrix at 12 K and in cyclohexane solution at 298 K. The Ar matrix-isolation IR spectrum of NdMAm shows a fine structure of vibration bands, because the molecule is trapped and fixed under inert circumstance. Although the IR bands of NdMAm in cyclohexane solution are much broadened, the band positions are similar to those in Ar

IR intensity (a.u.)

(a)

(b)

(c)

(d) 1800 1700 1600 1500 1400 1300 1200 1100 1000 Wavenumber (cm−1)

Figure 9.2 IR spectra of NdMAm (a) in Ar matrix at 12 K, (b) in cyclohexane at 298 K, (c) in neat liquid at 298 K, and (d) in water at 298 K.

255

1009

1181

1271

1407 1395 1354

1504

H2O

1676

9 Spectral Simulation for Flexible Molecules in Solution with Quantum Chemical

IR intensity (a.u.)

256

MP2/6-31G(d)

B3LYP/6-31+G(d)

G96LYP/6-31+G(d)

1800 1700 1600 1500 1400 1300 1200 1100 1000 Wavenumber (cm–1)

Figure 9.3 Comparison between the IR spectrum of NdMAm measured in Ar matrix at 12 K (the upper panel) and the simulated spectra at several theory levels. The scaling factors for MP2/6-31G(d), B3LYP/6-31+G(d), and G96LYP is 0.954, 0.978, and 1.014, respectively.

matrix. The broadening of IR bands observed in solution is owing to the rotation and libration of molecules. Figure 9.3 represents the simulated IR spectra of NdMAm at MP2/6-31G(d), B3LYP/6-31+G(d), and G96LYP/6-31+G(d) levels. These IR spectra are obtained after the geometrical optimization, followed by the calculation of force constants, and then the normal mode analysis that gives rise to the vibrational frequencies under HOA. The IR spectrum obtained at MP2/6-31G(d) is in excellent agreement with the experimental IR spectrum of NdMAm in Ar matrix. For B3LYP/6-31+G(d), the relative intensity of the two bands near 1400 cm−1 is opposite to the experimental result. However, the other bands are similar to the experimental ones in terms of the relative locations in wavenumber. On the other hand, the spectrum obtained from G96LYP/6-31+G(d) region is clearly different from the experimental one, especially in 1350–1550 cm−1 . It should be emphasized that we have to check the relative locations and intensities of the calculated bands throughout the fingerprint region, because the conformational changes are often characterized by the shift in minor bands. In general, the vibrational frequencies of the bands calculated under HOA deviate from the experimental ones containing a contribution from the anharmonicity. This can be readily understood, if we remember that Morse potential is a better approximation for the vibrational structure of a diatomic molecule

9.3 Simulation of IR Spectra Observed in Solution Phase

than the harmonic oscillator potential. Therefore, the vibrational frequencies produced by QCCs with HOA are usually multiplied by a scaling factor (in the range of 0.8–1.0) to better match experimental vibrational frequencies. It is possible to consider that the scaling factor is used to compensate for the fact that the potential energy surface is not harmonic. The wavenumber linear scaling method, which employs a scaling “function” with quadratic form, has also been proposed instead of the scaling “factor” [1]. It is of note that the vibrational mode of the groups containing hydrogen such as CH and OH is significantly influenced from the anharmonicity, whereas a simple scaling factor is often enough for the bands in the fingerprint region. Recently, some of the QCC packages provide the option to calculate anharmonic vibration frequencies, but their calculation cost is basically very high [10]. If your purpose is to simulate a more complicated system based upon the result of a smaller molecular model, the calculation cost becomes the center of attention. Without regarding the computational resource, we could choose the MP2 method, because the simulated IR spectrum faithfully reproduces the experimental result for NdMAm. On the other hand, the cost of the B3LYP method for the geometrical optimization and frequency calculation of NdMAm is less by about five times than that of the MP2 method. For the simulation of the larger system containing NdMAm, therefore, we have selected the B3LYP level of calculations as shown in the following sections.

9.3 Simulation of IR Spectra Observed in Solution Phase In many cases, flexible molecules that we are interested are embedded in condensed phase such as liquid, solid, and solution. Thus, it is very important to know how the intermolecular interaction shifts IR bands. In this section, we will show an example of the spectral simulation of NdMAm in solution. The C=O stretching vibration (𝜈 C=O ) band of NdMAm measured in several solvents is shown in Figure 9.4. The 𝜈 C=O wavenumbers identified by the second derivatives, 𝜈 C=O obs , are summarized in Table 9.1. It is worth notifying that there are two or three peaks in several solvents. The first moment of the 𝜈 C=O band, ⟨𝜈 C=O ⟩, is calculated as ⟨𝜈 C = O ⟩ = ∫ 𝜈 I(𝜈)d𝜈/ ∫ I(𝜈)d𝜈, which corresponds to the mean-center wavenumber of the band. To investigate the solvation effects on the 𝜈 C=O band of NdMAm, one may choose the self-consistent reaction field (SCRF) method as the first approach, in which the solvent is represented by a continuum medium having the macroscopic dielectric property [13]. Here, we have employed the polarizable continuum model using the integral equation formalism variant (IEFPCM) [14–16]. In Figure 9.5, the 𝜈 C=O wavenumbers calculated at B3LYP/6-31+G(d) by IEFPCM with each solvent parameter, 𝜈 C=O calc , are plotted against the dielectric constant, 𝜀r , together with ⟨𝜈 C=O ⟩. When 𝜀r of a solvent is lower than 6, the 𝜈 C=O calc value after scaling is in excellent agreement with ⟨𝜈 C=O ⟩ in each solvent as can be seen in Figure 9.5. On the other hand, if 𝜀r is higher than 6, the ⟨𝜈 C=O ⟩ values deviate from 𝜈 C=O calc that monotonically decreases as 𝜀r increases. For THF, acetone, and the neat liquid,

257

9 Spectral Simulation for Flexible Molecules in Solution with Quantum Chemical

1-Hexanol

THF

1632

1660

1614

1664

Methanol

n-Hexane

1650

1675

1649

Acetone

1700

1614 1635

1603

1666

Water

IR intensity (a.u.)

258

1600 1550 1700 1650 Wavenumber (cm–1)

1600

1550

Figure 9.4 𝜈 C=O band of NdMAm measured in various solvents at 298 K. Table 9.1 𝜈 C=O obs wavenumber observed under various conditions. 𝝂 C=O wavenumber (cm−1 ) Solvent

𝜺r

Vapor

1

1690c)

1690c)

Ar

1.43

1676

1676

n-Hexane

1.89

1675

1666

Cyclohexane

2.22

1673

1667

a)(298 K)

𝝂 C=O

⟨𝝂 C=O ⟩

obs b)

CCl4

2.228

1661

1655

Benzene

2.27

1659

1651

CHCl3

4.81c)

1632

1633

THF

7.6c)

1660

1651

1-Hexanol

13.3

1614

1635

1666

1630

1-Pentanol

13.9

1614

1635

1666

1629

1-Butanol

17.1

1614

1635

1666

1629

1-Propanol

20.1

1614

1635

1666

1627

Acetone

20.7

1649

1645

Ethanol

24.3

1614

1633

1664

1626

Methanol

32.7

1614

1633

1664

1622

Neat (NdMAm)

37.8

1643

1660

1647

Water

78.4

1603

1600

a) Reference [11]. b) Estimated by the second derivative. IR spectra were measured at 293 K (20 ∘ C) and 0.023 M. c) Reference [12].

9.3 Simulation of IR Spectra Observed in Solution Phase

νC = Ocalc × 0.978

1700

Wavenumber (cm–1)

Figure 9.5 Correlation between 𝜈 C=O calc and ⟨𝜈 C=O ⟩ experimentally observed. 𝜈 C=O calc was obtained at the B3LYP/6-31+G(d) level with SCRF models.

1675

THF

1650

1625

Acetone

CHCl3

Neat

Alcohols

1600

Water

0

20

40 εr

60

80

𝜈 C=O calc is much lower than ⟨𝜈 C=O ⟩, indicating that IEFPCM is not appropriate to estimate ⟨𝜈 C=O ⟩ in these solvents. The 𝜈 C=O calc values calculated for alcohols and water seem better, because the lower wavenumber shift is qualitatively reproduced, even though the calculation tends to underestimate 𝜈 C=O for the alcohol solution and to overestimate 𝜈 C=O for the aqueous solution. The most notable point is that the multiple 𝜈 C=O peaks are observed in the experimental IR spectrum of NdMAm in alcohols and the neat liquid, which are impossible to be reproduced by IEFPCM calculations (see Figure 9.4). These results imply that many aspects of the solvation effects on the 𝜈 C=O band are lost in the QCC results with a continuum medium model. Acetone, THF, alcohols, and water molecules interact with NdMAm through the dipole interaction and hydrogen bond. To consider these effects, we have to employ a more realistic model of the solvation. The simplest approach is to calculate an explicit complex of NdMAm and a solvent molecule. Figure 9.6 shows the optimized structures for the complex of NdMAm with various solvent molecules. These calculations were carried out in vacuum at the B3LYP/6-31+G(d) level. Figure 9.7 shows that 𝜈 C=O calc with the scaling factor of 0.971 are plotted against 𝜈 C=O obs , which are listed in Table 9.1. It is of note that the scaling factor is determined so that 𝜈 C=O calc of the NdMAm/n-hexane complex corresponds to 𝜈 C=O obs of NdMAm in the n-hexane solutions. As can be seen in Figure 9.7, the 𝜈 C=O calc values obtained for the complexes are correlated well with 𝜈 C=O obs , although 𝜈 C=O calc tends to be higher by 10 cm−1 than 𝜈 C=O obs . There are two major factors causing the difference between 𝜈 C=O calc and 𝜈 C=O obs : (i) the influences from the surrounding solvents other than those directly interacted and (ii) the insufficiency of the calculation level. For the latter, it has been pointed out that the B3LYP method is not good for calculating the non-covalent (especially dispersion) interactions [17]. However, the result shown here has suggested that an explicit solvation model systematically gives

259

9 Spectral Simulation for Flexible Molecules in Solution with Quantum Chemical

NdMAm + acetone

NdMAm + THF

NdMAm + benzene

NdMAm + ethanol

NdMAm + NdMAm

NdMAm + CHCl3

NdMAm + 2 CH3OH

NdMAm + 3 H2O

Figure 9.6 Optimized structures for the complex of NdMAm and the solvent molecule(s).

Benzene

1680

n-Hexane

NdMAm(a) NdMAm(s)

νC = Ocalc (cm–1)

260

1660

CHCl3

THF

Cyclohexane CCl4 Acetone

CH3OH(m) 1-Propanol(m) 1-Butanol(m) Ethanol(m)

1640

1620

1600 1600

2CH3OH 3H2O

1620

1640 1660 νC = Oobs (cm–1)

1680

Figure 9.7 Correlation between 𝜈 C=O calc for NdMAm/solvent complex (Figure 9.6) and 𝜈 C=O obs listed in Table 9.1. The scaling factor for 𝜈 C=O calc is 0.971, which is determined by referring to 𝜈 C=O obs for the n-hexane solution. NdMAm(a) and NdMAm(s) indicate the antisymmetric and the symmetric modes arising from the dipole coupling. The solid line is drawn as guides for the eye.

9.4 Competition Between Intramolecular and Intermolecular Interactions

a better result than IEFPCM. It is therefore presumed that the solvation at the molecular spectroscopic level should be interpreted in terms of a direct molecular interaction between a solute and a solvent. In Figure 9.7, 𝜈 C=O calc obtained for the complex of NdMAm with the single methanol molecule (NdMAm/CH3 OH) is plotted against 𝜈 C=O obs of 1633–1635 cm−1 observed in the alcohol solutions, whereas 𝜈 C=O calc of the NdMAm/2CH3 OH complex is paired with 𝜈 C=O obs of 1614 cm−1 . It is highly possible that the 𝜈 C=O bands at 1614 and 1633–1635 cm−1 of NdMAm observed in alcohols are owing to mono- and di-hydrogen-bonded C=O groups, respectively. The 𝜈 C=O band at 1664–1666 cm−1 may arise from the interaction between the C=O group of NdMAm and the alkyl group of alcohols. These molecular pictures of the solvation effects on 𝜈 C=O rationalizes the shift in 𝜈 C=O obs of NdMAm. For example, the 𝜈 C=O bands of NdMAm observed for alcohol solutions does not shift depending upon 𝜀r of alcohols that varies from 32.7 (methanol) to 13.3 (1-hexanol). This cannot be explained by a continuum medium model, but be readily understood by a direct molecular interaction. The comparison between 𝜈 C=O obs for the 1-propanol and acetone solutions is also interesting. Although these solvents have similar 𝜀r , the 𝜈 C=O obs values are completely different. This is also not understandable by assuming a continuum medium model. The calculation result for the NdMAm dimer tells us that the two 𝜈 C=O bands in the neat liquid arise from the vibrational coupling: the symmetric (s) and antisymmetric (a) 𝜈 C=O modes. The intermolecular vibrational coupling is often very important for understanding the vibrational bands in condensed phases [18, 19]. An IR active vibration can couple with the same vibration mode of a neighboring molecule via the transition dipole, if some degree of short-range order exists due to the large permanent dipole moments of the molecules. In this section, we described how the solvation affects the 𝜈 C=O wavenumber of NdMAm. Because the shift in the 𝜈 C=O band can be understood based upon an explicit and direct interaction between the C=O group and the solvent, the knowledge obtained here can be applied to the other molecules that contains not only amide groups but also carboxyl, ester, ketone, and so on.

9.4 Competition Between Intramolecular and Intermolecular Interactions When the target molecule contains two or more groups that interact each other, the competition between intramolecular and intermolecular interactions may occur. This situation is often found in the solvation of polymers. Although the framework of simulation for this case is basically the same with the method shown in the previous section, we have to consider the effect of the interaction among the neighbor chains on target bands in addition to the interaction with solvents. For example, the solvation of poly(N,N-dimethylacrylamide) (PNdMAm), which is a vinyl polymer having NdMAm as the side chain, can be interpreted based upon the solvation of NdMAm [20]. When C=O groups are incorporated in polymer chains, however, the access of the solvent molecules

261

9 Spectral Simulation for Flexible Molecules in Solution with Quantum Chemical

n

O

O

HN

O

N H

N H

PNiPAm

dNiPAm-c1

Figure 9.8 Chemical structures of PNiPAm and its dimer model (dNiPAm-c1).

to the C=O groups may be hampered by neighboring chains. Indeed, the 𝜈 C=O envelope of PNdMAm in solvents contains a characteristic band that is not observed for NdMAm. That is to say, 𝜈 C=O of PNdMAm in solution is influenced not only from solvation but also from another kind of effects such as a steric hindrance and an interaction between neighboring amide groups. In this section, we have presented an example of molecular systems, in which the competition between the intramolecular and intermolecular interactions occurs. For this purpose, the solvation of a dimer model of poly(N-isopropylacrylamide) (PNiPAm) has been examined [21]. The side chain of PNiPAm is the secondary amide that can form an intramolecular C=O· · ·H—N hydrogen bond among the side chains (see Figure 9.8 for the chemical structure). In order to consider the effect of the neighboring chains on the solvation of C=O groups in PNiPAm, we can use a simple dimer model such as N,N′ -(2-propyl)-2-methylpentanediamide (dNiPAm-c1; Figure 9.8). Figure 9.9 shows IR spectra in the amide I and II regions of PNiPAm and dNiPAm-c1 in THF and methanol solutions. Because both the IR spectra are very similar, it is assumed that dNiPAm-c1 can be used as a good model for the solvation of PNiPAm. The amide I vibration mode (𝜈 amI ) of PNiPAm arises mainly from C=O stretching (∼80%) and N—H in-plane bending (∼10%). On

1558

1629

PNiPAm

1555

1627

1651

Figure 9.9 IR spectra in the amide I and II region for PNiPAm and dNiPAm in methanol at 298 K. The concentration of each solution was 2 wt%.

1651

1673

IR intensity (a.u.)

262

dNiPAm

1750

1700

1650 1600 1550 Wavenumber (cm–1)

1500

9.4 Competition Between Intramolecular and Intermolecular Interactions

the other hand, the amide II vibration mode is concerned with N—H bending and C—N stretching. Because the contribution of the C=O stretching mode is predominant in 𝜈 amI , the solvation effect on the 𝜈 amI wavenumber is almost the same with that on 𝜈 C=O for NdMAm. For the methanol solution of PNiPAm and dNiPAm-c1, three 𝜈 amI bands were found at 1673, 1651, and 1627–1629 cm−1 . The origin of the band at 1673 cm−1 may be similar to the 𝜈 C=O band of NdMAm at 1664–1666 cm−1 arising from the interaction between the C=O group and the alkyl group of alcohols. It is worth noting that N-methylacetamide (NMAm; Figure 9.1), which is a representative of the secondary amide compound, shows a unimodal peak at 1683 cm−1 in a dilute THF solution [22]. The 𝜈 amI wavenumber of N-isopropylacetamide in vacuum or low-temperature matrix is lower by about 10 cm−1 than those of NMAm [23]. Thus, the band at 1673 cm−1 of dNiPAm-c1 is assignable to C=O groups without hydrogen bonding with methanol molecules. In order to assign the bands at 1651 and 1627–1629 cm−1 of dNiPAm-c1, we have to take into account the influence of the intramolecular interaction in addition to the solvation effects. Because dNiPAm-c1 has two amide groups, these 𝜈 amI bands are possibly involved in both the inter- and intramolecular hydrogen bond. This situation is similar to PNiPAm in methanol, because the polymer also possesses amide groups as the side chain. Note that the interchain C=O⋅⋅⋅H—N hydrogen bond under this experimental condition is negligible, because methanol is a good solvent for PNiPAm and the polymer chain behaves as a swollen coil in methanol [24]. The 𝜈 amI bands at 1651 and 1627–1629 cm−1 are assignable to di-hydrogenbonded and mono-hydrogen-bonded C=O groups, respectively [22]. For dNiPAm-c1 in methanol, there are two possible proton donors: the OH group of solvents and the NH group of the neighboring amide group. In general, it is very difficult to identify the proton donors by referring only to the shift in the 𝜈 amI wavenumber. To demonstrate the competitive feature of the solvation and the intramolecular interaction for dNiPAm in methanol and water, the spectral simulations have been performed. Two conformers of dNiPAm forming the intramolecular C=O· · ·H—N hydrogen bond have been optimized as shown in Figure 9.10; the conformations of the pentyl backbone are trans (t) and gauche (g). The complexes of dNiPAm with water or methanol molecules have also been calculated. In the complex, the solvent molecules form hydrogen bond with the C=O and N—H groups that are not involved in the intramolecular C=O· · ·H—N hydrogen bond. The simulated IR spectra obtained for these optimized geometries are shown in Figure 9.11. The top panel of Figure 9.11 represents IR spectra of dNiPAm in vacuum with t and g conformers. The amide I band around 1660 cm−1 arises from the C=O group involved in the intramolecular C=O· · ·H—N hydrogen bond, whereas the band at ca. 1690 cm−1 originates from the free C=O group. Two amide II bands appear in response to the amide I bands; the band at around 1530 cm−1 is due to the N—H group involved in the intramolecular C=O· · ·H—N hydrogen bond. The middle and bottom panels of Figure 9.11 show IR spectra for the complexes of dNiPAm and solvent molecules. The results are very interesting and suggestive, because both the amide I bands shift to a lower wavenumber without changing their relative

263

264

9 Spectral Simulation for Flexible Molecules in Solution with Quantum Chemical

C N O H Main chain

Trans (t) conformer

Gauche (g) conformer

t + 2CH3OH

g + 2CH3OH

t + 2H2O

g + 2H2O

Figure 9.10 Optimized geometries of trans and gauche conformers of dNiPAm at the B3LYP/6-31G(d) level and those of dNiPAm solvated by water and methanol molecules.

peak positions obtained for the dNiPAm in vacuum. That is, the band around 1645 cm−1 is due to the intramolecular C=O· · ·H—N hydrogen bond, while the bands around 1670 cm−1 arise from the intermolecular C=O· · ·H—O hydrogen bonding with solvent. This kind of behavior has been often reported and known as cooperativity of hydrogen bonds [25]. The result mentioned above is not a deciding factor for the assignment of 𝜈 amI , but just one possibility. At the same time, the simulation result tells us the difficulty of the assignment of IR bands observed for a complex system. To confirm the existence of the intramolecular C=O· · ·H—N hydrogen bond in real systems, we need to carry out an additional experiment such as the H/D exchange. The H/D exchange rate depends on the solvent accessibility and the stability of the intramolecular C=O· · ·H—N hydrogen bond. Actually, the H/D exchange experiment has demonstrated that dNiPAm forms the intramolecular

Amide II C=O···N−H Amide II Free N−H

2

In vacuum

1

IR intensity (103 dm3) (mol cm)

Figure 9.11 (a) Simulated IR spectra for the trans (solid line) and gauche (dotted line) conformers of dNiPAm displayed in Figure 9.10. (b) Simulated IR spectra of dNiPAm hydrogen bonding with methanol molecules shown in Figure 9.6. (c) Simulated IR spectra of dNiPAm hydrogen bonding with water molecules. IR bands marked by asterisk indicate the bands due to the OH bending vibration of water.

Amide I Free C=O Amide I C=O···N−H

9.5 Conformational Diversity and Solvation in the Vibrational Spectrum

0 2

2 CH3OH

t conformer g conformer

1

0 2

2 H2O

1

0 1800

1700

1600

1500

1400

Wavenumber (cm–1)

C=O· · ·H—N hydrogen bond [21]. Thus, it can be concluded that the 𝜈 amI band at 1627–1629 cm−1 contains a contribution not only from solvation of methanol but also from the intramolecular C=O· · ·H—N hydrogen bond.

9.5 Conformational Diversity and Solvation in the Vibrational Spectrum The diversity in conformation often makes it difficult to analysis an experimental IR spectrum of flexible molecules. In this section, the conformation analysis of N,N′ -1,2-ethanediylbis (N,2-dimethyl-propionamide) (EDMPAm; Figure 9.12), which is a dimer model of poly(2-isopropyl-2-oxazoline) [26], has been described. Figure 9.13 represents the IR spectra of EDMPAm in the neat liquid and in the CCl4 solution. In order to search the conformers of EDMPAm, we have performed QCCs of EDMPAm at the B3LYP/6-31+G(d) level. The representative conformers obtained from the geometrical optimization are summarized in Table 9.2 and Figure 9.14. There are six conformers concerned with the torsion of the NCCN dihedral angle. The conformers having a different NCCN torsion angle are identified by the lowercase letters (a, b, etc.), whereas Arabic numerals (1, 2, etc.) indicate the difference in the relative orientations of the isopropyl group. Judging from the relative energy ΔE listed in Table 9.2, it

265

9 Spectral Simulation for Flexible Molecules in Solution with Quantum Chemical

Figure 9.12 Chemical structure of EDMPAm. O N N O

538

837

(a) in CCl4 ×5

751

1088

1126

1477

EDMPAm

IR intensity (a.u.)

266

(b) Neat liquid ×10

(c) Boltzmann weighted ×10

(d) a1 ×10

(e) b1 ×10

(f) a2 1700

1600

1500

1400

1300

1200

1100

1000

900

800

700

600

500

Wavenumber (cm–1)

Figure 9.13 Experimental IR spectra of EDMPAm (a) in CCl4 and (b) in the neat liquid and the simulated IR spectra at the B3LYP/6-31+G(d) level for (d) a1, (e) b1, and (f ) a2. Trace (c) is the Boltzmann-averaged spectrum of a1, b1, and a2.

can be assumed that the conformers a1, a2, and b1 are equally observable in a nonpolar environment such as in the gas phase and in nonpolar solvents. As shown in Figure 9.13, IR spectra calculated for a1 and a2 are very similar to the experimental one. However, there are several bands in the experimental spectra that are conspicuously absent in the simulated spectra for the a-type conformers; these bands appear at 1126, 837, and 538 cm−1 . The shape of the peak mass if near 750 cm−1 calculated for the a-type conformers is also different from those in the experimental one. Those observations indicate the existence of the b1 conformer. Thus, we performed the Boltzmann-weighted average of the simulated spectra of a1, a2, and b1. The excellent agreement between the Boltzmann-weighted spectrum and the experimental IR spectra of EDMPAm is found. That is, the three conformers a1, a2, and b1 coexist for EDMPAm in the CCl4 solution and in the neat liquid. The result can also be confirmed by Raman

9.5 Conformational Diversity and Solvation in the Vibrational Spectrum

Table 9.2 Relative energy ΔE (kJ/mol) from the energy of the most stable conformer of EDMPAm (a1), 𝜙, and 𝜓 for each conformer optimized in vacuum. ID

𝚫E a)(kJ/mol)

𝝓 (∘ )

𝝍 (∘ )

a1

0.000

180.00

11.30

a2

0.446

177.85

14.12

b1

0.0867

62.60

182.7

c1

7.10

53.69

158.6

d3

16.2

170.0

175.6

e3

11.7

78.10

16.88

f4

11.7

−78.09

−2.87

a) Zero point energy was corrected.

C3 N5

C4

N2

a1

b1

c1

d3

e3

f4

Figure 9.14 Optimized geometries of the representative conformers of EDMPAm in vacuum.

spectroscopy, which is more sensitive to the changes in the skeletal conformation than IR spectroscopy. The Raman spectrum of the neat liquid of EDMPAm has been shown in Figure 9.15. The simulated Raman spectra for the conformers a1, a2, and b1 are represented in this figure, together with the Boltzmann-weighted average of the simulated spectra. The characteristic bands for the b1 conformer appear at 1266, 1245, 1199, 1056, 908, and 850 cm−1 . The Boltzmann-weighted Raman spectrum is also in excellent agreement with the experimental one. After refining the conformers of EDMPAm, we can move to the spectral simulation of solvation of the target in the same manner described in the previous

267

908 850

1056

1199

1266 1245

9 Spectral Simulation for Flexible Molecules in Solution with Quantum Chemical

Raman intensity (a.u.)

268

(a) Neat liquid

(b) Boltzmann weighted

(c) a1 (d) b1 (e) a2 1700

1600

1500

1400

1300

1200

1100

1000

900

800

Wavenumber (cm–1)

Figure 9.15 Experimental Raman spectrum of EDMPAm in the neat liquid (a) and the simulated Raman spectra at the B3LYP/6-31+G(d) level for a1 (c), b1 (d), and a2 (e). Trace (b) is the Boltzmann-averaged spectrum of a1, b1, and a2.

section. The upper panel of Figure 9.16 shows the IR spectrum of EDMPAm in D2 O, which significantly differs from that in the CCl4 solution. The 𝜈 C=O band of EDMPAm located at 1648 cm−1 in the CCl4 solution shifts to a lower wavenumber by 48 cm−1 in the D2 O solution. The bands at 1475 and 1409 cm−1 in CCl4 , which are associated with the vibration modes of the —NCH2 CH2 N— backbone, move to a higher wavenumber by 15 ∼ 18 cm−1 in D2 O. It is worth emphasizing that the direction of the shifts in the former and later bands is opposite. To facilitate the interpretation of the IR spectral changes, QCCs for the hydrogen-bonded complex between EDMPAm and D2 O were carried out. The lower panel of Figure 9.16 represents the simulated IR spectrum for the hydrogen-bonded complex, together with the optimized geometry. The input geometry of EDMPAm was the conformer a1, and four water molecules were placed around the C=O groups, forming the C=O· · ·H—O hydrogen bond. Although each carbonyl group is hydrogen-bonded to two water molecules, the optimized geometry of EDMPAm in the complex is not modified significantly from the original a1 conformer in vacuum. The simulation reproduces accurately the band shifts caused by hydration; the 𝜈 C=O band shows a redshift, whereas the two bands around 1500 and 1400 cm−1 undergo a blueshift. The spectral simulation of EDMPAm for the conformation and solvation can be applied to decipher IR spectra of poly(2-isopropyl-2-oxazoline) (PiPrOx) [26]. The analysis of the vibrational spectra of EDMPAm makes it possible to evaluate the conformation and the molecular interaction of PiPrOx in solution.

in CCl4

1600

in D2O

Without D2O

1612

IR intensity (a.u.)

1496 1475 1423 1408

1648

9.6 Conformational Diversity in the Vibrational Circular Dichroism Spectrum

1700

1600

1500

1428 1413

1512 1494

1651

With D2O

1400

a1 + 2H2O

1300

1200

1100

1000

Wavenumber (cm–1)

Figure 9.16 (a) Experimental IR spectra of EDMPAm in CCl4 and D2 O solutions at 2 wt %. (b) Calculated IR spectra of EDMPAm with or without D2 O molecules, together with the optimized structure. The conformation of EDMPAm is trans (a1).

9.6 Conformational Diversity in the Vibrational Circular Dichroism Spectrum The VCD spectroscopy has been known as a powerful tool for determining the absolute configurations of chiral molecules as well as their conformations [3]. In this section, we demonstrate the conformation and configuration analysis on 1-phenylethanol with the aid of QCCs. For the first step, the conformation of dl-1-phenylethanol (racemic) has been examined by the use of the Ar matrix-isolation IR spectroscopy with QCCs. The optimization and the spectral simulation were carried out at the B3LYP/6-311++G(d,p) level. The optimized geometries of the conformers I, II, and III with the configuration of S are shown in Figure 9.17. The conformer I was obtained as the most stable conformer. The relative energy of the conformers II and III are 4.73 and 7.25 kJ/mol, respectively. The upper panel of Figure 9.18 shows the Ar matrix-isolation IR spectrum of dl1-phenylethanol in the 1600–500 cm−1 region measured at 12 K, along with the calculated spectra of conformers I, II, and III. The calculated spectrum of the conformer I is very similar to that for the experimental matrix IR spectrum. By closely inspecting, a tiny amount of the conformer II and III seems to coexist in the Ar matrix.

269

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9 Spectral Simulation for Flexible Molecules in Solution with Quantum Chemical

H

C1

C

O

Figure 9.17 Optimized conformers of (S)-1-phenylethanol obtained at the B3LYP/6-311++G(d,p) level.

C2

Conformer I

Conformer II

Conformer III

The VCD spectrum of (−)-1-phenylethanol has been measured in the CS2 solution at 0.1 M. The upper panel of Figure 9.19 represents the VCD spectrum of (−)-1-phenylethanol. The calculated VCD spectra for the conformers I, II, and III (Figure 9.17) are represented in the lower panel of Figure 9.19. The spectrum of the conformer I captures the feature of the experimental VCD spectrum, suggesting that the bands at 1350, 1252, 1097, 1076, 1047, 994, and 899 cm−1 in the CS2 solution are due to the conformer I. By referring to the optimized geometry, we have concluded that the configuration of (−)-1-phenylethanol is S. There are differences between the experimental VCD spectrum and the calculated one for the conformer I: the existence of the bands at 1369 and 1008 cm−1 , the relative intensity of the bands at 1200 and 1252 cm−1 , and so on. Because the VCD measurements are generally difficult in a very dilute solution, it is highly possible that the experimental spectrum contains a contribution from the intermolecular interactions. Indeed, the OH stretching vibration envelope of (−)-1-phenylethanol under the same experimental condition indicates the coexistence of the monomer and at least dimers [4]. The C—O—H bending, C—H deformation, C—O stretching, and C—C (adjacent to the OH group) stretching modes, which appear in the 1120–1400 cm−1 region, should be largely perturbed by the formation of hydrogen bonds. Figure 9.20 shows the optimized geometries for several possible dimers of (S)-1-phenylethanol with the conformers I and II. The formation of hydrogen bonds among (−)-1-phenylethanol is likely further

9.6 Conformational Diversity in the Vibrational Circular Dichroism Spectrum

997 987 897

704

907

1074 1077

1248

(b) 1445

IR intensity (a.u.)

1454

1254

701

(a)

Conformer I

Conformer II

Conformer III

1600

1400

1200

1000

800

600

Wavenumber (cm−1)

Figure 9.18 (a) IR spectrum of DL-1-phenylethanol (racemic mixture) in the Ar matrix at 12 K. An asterisk symbol (*) indicates the band due to CO2 . (b) Calculated IR spectra of the conformers I, II, and III.

promoted in the neat liquid. As can be seen in Figure 9.21b, the VCD spectra of (−)-1-phenylethanol in the neat liquid are considerably different from that in CS2 solution at 0.1 M. Figure 9.21c represents the calculated VCD spectra of dimers, whose optimized geometries are shown in Figure 9.20. In order to reproduce the experimental VCD spectrum of (−)-1-phenylethanol in the 0.1 M CS2 solution, we calculated a linear combination of the spectra as follows: the conformer I + the dimer II/I + 0.5 × the dimer I/II. As can be seen in Figure 9.21a, the agreement between the sum of calculated VCD spectra and the observed one is fairly good. As a result, it can be interpreted that the bands at 1200 and 1075 cm−1 are originated from the conformer II, forming the intermolecular hydrogen bond. In a similar manner, we can estimate the components in the neat liquid of (−)-1-phenylethanol. The VCD spectrum in the neat liquid can be reproduced by the linear combination of VCD spectra of the dimers: (II/I) + (I/II) + 0.5 × {(II/II) + (I/I)} as shown in Figure 9.21b. The results clearly indicate that VCD spectrum is largely influenced not only by the conformational changes but also by the intermolecular interaction.

271

Figure 9.19 (a) VCD spectrum of (−)-1-phenylethanol observed in the CS2 solution at 0.1 mol dm−3 . (b) Calculated VCD spectra for conformers I, II, and III.

1008 994

1097

1200

1252

1369

899

1350

(a)

1047

1097

9 Spectral Simulation for Flexible Molecules in Solution with Quantum Chemical

VCD intensity (a.u.)

272

(b)

Conformer I

Conformer II

Conformer III

1400

1300

1200

1100

1000

900

800

Wavenumber (cm−1)

l/l

l/ll

ll/l

ll/ll

Figure 9.20 Structures of the dimers of (S)-1-phenylethanol optimized at the B3LYP/6-31+G(d,p) level.

9.7 Conformational Diversity in the Electronic Circular Dichroism TD-DFT calculations enable us to predict not only the ultraviolet and visible (UV–vis) spectrum but also electric circular dichroism (ECD) spectrum of a

1008

1200

(a)

ex. calc.

(b) VCD intensity (a.u.)

Figure 9.21 (a) Observed VCD spectrum of (−)-1-phenylethanol in the CS2 solution at 0.1 mol dm−3 . The dotted line represents a linear combination of the VCD spectra for the monomer and dimers, conformer I + dimer II/I + 0.5 × dimer I/II. (b) Observed VCD spectrum of (−)-1-phenylethanol in the liquid state. The dotted line represents a linear combination of the VCD spectra for the dimers, dimer II/I + dimer I/II + 0.5 × (dimer II/II + dimer I/I). (c) Calculated VCD spectra of the dimers I/II, II/I, I/I, and II/II.

1075

9.7 Conformational Diversity in the Electronic Circular Dichroism

ex. calc.

(c) dimer II/I dimer I/II dimer I/I dimer II/II

1500 1400 1300 1200 1100 1000 Wavenumber (cm−1)

900

800

target molecule. In this section, we will explain the analysis of the ECD spectrum of an indomethacin derivative, 2-[1-(2-chloro-6-iodo-benzoyl)-5-methoxy-2methylindol-3-yl] acetic acid (Cl/I-Im; Figure 9.22). The top panel of Figure 9.23 shows the experimental ECD spectra of (−)- and (+)-Cl/I-Im in ethanol solution. Each enantiomer was obtained by the use of a preparative chiral HPLC [27]. There are various possible conformers for Cl/I-Im according to the rotation of C1′ –C7′ , C7′ –N1, C3–C2′′ , and so on. The rotation of C1′ –C7′ causes the atropisomers named aS or aR, whereas the rotation of C7′ –N1 gives rise to the conformers named trans and cis. The conformers concerned with the C3–C2′′ rotation are distinguished by the Arabic number. The first step is the comprehensive search on the possible conformers of Cl/I-Im. For the mechanical search on the conformers, we can employ a commercial software based on the molecular mechanical calculation (MMC) such as CONFLEX. Thirty-two conformers were found by the CONFLEX calculation with the force field of MMFF94s. In some cases, the stability of conformers estimated by MMC is totally different from that obtained from QCCs. Thus, the optimization of all conformers was carried out using DFT at the B3LYP/6-31G(d) level with the LANL2DZ basis sets for iodine. The most stable conformers are found to be cis-aS-2 and cis-aR-2, which is illustrated

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9 Spectral Simulation for Flexible Molecules in Solution with Quantum Chemical

Figure 9.22 Chemical structure of Cl/I-Im. O

O

2′′ 3

HO N1

Cl 2′

7′

O

1′

l

6′

Cl/l-lm

Figure 9.23 (Top) ECD spectra of (−)- and (+)-Cl/I-Im in methanol. (Bottom) Simulated ECD spectra for (aR)- and (aS)-Cl/I-Im after the Boltzmann-weighted sum of the spectra obtained for several conformers shown in Figure 9.24.

40 (−)-Cl/I-Im (+)-Cl/I-Im

Δε

20

0

–20 Boltzmann weighted

Rotatory strength (a.u.)

274

(aR)-Cl/I-Im (aS)-Cl/I-Im

200

250 300 Wavelength (nm)

350

400

in the top left of Figure 9.24. There are eight conformers, which are shown in Figure 9.24, lying in a range of 2 kJ/mol from the most stable one: cis-aS-1, cis-aS-2, cis-aR-1, cis-aR-2, trans-aS-1, trans-aS-2, trans-aR-1, and trans-aR-2. Note that the energy values were obtained after the zero-point energy correction. It is very interesting that the ECD spectra calculated for cis-aS-2 and cis-aR-2, which are shown in the top right panel of Figure 9.24, were not similar to the experimental ECD spectra observed for (−)- and (+)-Cl/I-Im. Moreover, we could not find a similar ECD spectra calculated (see Figure 9.24). Therefore, we calculated ECD spectra of aR and aS by considering the Boltzmann factor, which can be derived from the relative energy of the conformers. The

References

(cis, aR)-2 (cis, aS)-2

(cis, aS)-2 ΔE = 0.00 kJ/mol (BF = 1.0)

(cis, aR)-2 ΔE = 0.00 kJ/mol (BF = 1.0)

(cis, aS)-1 ΔE = 0.08 kJ/mol (BF = 0.97)

(trans, aS)-1 ΔE = 1.40 kJ/mol (BF = 0.58)

(cis, aR)-1 ΔE = 0.08 kJ/mol (BF = 0.97)

Rotatory strength (a.u.)

(cis, aR)-1 (cis, aS)-1

(trans, aR)-1 (trans, aS)-1

(trans, aR)-1 ΔE = 1.37 kJ/mol (BF = 0.57) (trans, aR)-2 (trans, aS)-2

(trans, aS)-2 ΔE = 1.61 kJ/mol (BF = 0.54)

(trans, aR)-2 ΔE =1.55 kJ/mol (BF = 0.52)

200

250

300

350

400

Wavelength (nm)

Figure 9.24 Simulated ECD spectra for several conformers of (aR)- and (aS)-Cl/I-Im. The relative energy (ΔE) and Boltzmann factor (BF) are denoted under each optimized structure.

Boltzmann-weighted ECD spectra of both aS and aR enantiomers are shown in the lower panel of Figure 9.23. The result is in excellent agreement with the experimental CD of (−)- and (+)-Cl/I-Im in ethanol. This indicates that (−)-Cl/I-Im can be assigned to aS-Cl/I-Im and (+)-Cl/I-Im to aR-Cl/I-Im. Thus, we conclude that various conformers of aS- and aR-Cl/I-Im coexist in the ethanol solution. That is, we must take into account the diversity of the conformers to reproduce an ECD spectrum of flexible molecules in solution.

References 1 Yoshida, H., Ehara, A., and Matsuura, H. (2000). Density functional vibra-

tional analysis using wavenumber-linear scale factors. Chem. Phys. Lett. 325: 477–483. 2 Shin-ya, K., Takahashi, O., Katsumoto, Y., and Ohno, K. (2007). Intramolecular CH· · ·π and CH· · ·O interactions in the conformational stability of

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3 4

5 6 7

8 9

10

11 12

13 14

15

16

17

18

benzyl methyl ether studied by matrix-isolation infrared spectroscopy and theoretical calculations. J. Mol. Struct. 827 (1–3): 155–164. Nafie, L.A., Keiderling, T.A., and Stephens, P.J. (1976). Vibrational circular dichroism. J. Am. Chem. Soc. 98 (10): 2715–2723. Shin-ya, K., Sugeta, H., Shin, S. et al. (2007). Absolute configuration and conformation analysis of 1-phenylethanol by matrix-isolation infrared and vibrational circular dichroism spectroscopy combined with density functional theory calculation. J. Phys. Chem. A 111 (35): 8598–8605. Wakamatsu, S., Takahashi, Y., Tabata, H. et al. (2013). Conformation and atropisomeric properties of indometacin derivatives. Chem. A Eur. J. 19 (22). Møller, C. and Plesset, M.S. (1934). Note on an approximation treatment for many-electron systems. Phys. Rev. 46 (7): 618–622. ˇ Cížek, J. (1966). On the correlation problem in atomic and molecular systems. calculation of wave function components in ursell-type expansion using quantum-field theoretical methods. J. Chem. Phys. 45 (11): 4256–4266. Sinanoˇglu, O. (1962). Many-electron theory of atoms and molecules. I. Shells, electron pairs vs many-electron correlations. J. Chem. Phys. 36 (3): 706–717. Willson, S.P., Andrews, L., Willson, S.P., and Andrews, L. (2006). Matrix isolation infrared spectroscopy. In: Handbook of Vibrational Spectroscopy (ed. P.R. Griffiths). Chichester, UK: Wiley. Adamo, C., Cossi, M., Rega, N., and Barone, V. (2001). New computational strategies for the quantum mechanical study of biological systems in condensed phases. In: Theoretical and Computational Chemistry, 467–538. Elsevier. Lide, D.R. (1993). CRC Handbook of Chemistry and Physics: A Ready-Reference Book of Chemical and Physical Data, 73e. CRC Press. Coleman, W.M. and Gordon, B.M. (1988). Examinations of the matrix isolation Fourier transform infrared spectra of organic compounds: Part XI. Appl. Spectrosc. 42 (6): 1049–1056. Tapia, O. and Goscinski, O. (1975). Self-consistent reaction field theory of solvent effects. Mol. Phys. 29 (6): 1653–1661. Miertuš, S., Scrocco, E., and Tomasi, J. (1981). Electrostatic interaction of a solute with a continuum. A direct utilization of ab initio molecular potentials for the prevision of solvent effects. Chem. Phys. 55 (1): 117–129. Miertũs, S. and Tomasi, J. (1982). Approximate evaluations of the electrostatic free energy and internal energy changes in solution processes. Chem. Phys. 65 (2): 239–245. Pascual-ahuir, J.L., Silla, E., and Tuñon, I. (1994). GEPOL: an improved description of molecular surfaces. III. A new algorithm for the computation of a solvent-excluding surface. J. Comput. Chem. 15 (10): 1127–1138. Grimme, S., Antony, J., Schwabe, T., and Mück-Lichtenfeld, C. (2007). Density functional theory with dispersion corrections for supramolecular structures, aggregates, and complexes of (bio)organic molecules. Org. Biomol. Chem. 5 (5): 741–758. Fini, G. and Mirone, P. (1974). Evidence for short-range orientation effects in dipolar aprotic liquids from vibrational spectroscopy. J. Chem. Soc. Faraday Trans. 2 70 (0): 1776.

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19 Torii, H., Musso, M., Giorgini, G., and Doge, G. (1998). The non-coincidence

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21

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24 25

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effect in highly diluted acetone-CCl4 binary mixtures II. Experimental, theoretical and monte carlo simulation results. Mol. Phys. 94 (5): 821–828. Katsumoto, Y., Tanaka, T., and Ozaki, Y. (2005). Molecular interpretation for the solvation of poly(acrylamide)s. I. Solvent-dependent changes in the C=O stretching band region of poly(N,N-dialkylacrylamide)s. J. Phys. Chem. B 109 (44). Katsumoto, Y., Tanaka, T., Ihara, K. et al. (2007). Contribution of intramolecular C=O· · ·H–N hydrogen bonding to the solvent-induced reentrant phase separation of poly(N-isopropylacrylamide). J. Phys. Chem. B 111 (44): 12730–12737. Eaton, G., Symons, M.C.R., and Rastogi, P.P. (1989). Spectroscopic studies of the solvation of amides with N—H groups. Part 1—The carbonyl group. J. Chem. Soc. Faraday Trans. 1 Phys. Chem. Condens. Phases 85 (10): 3257. Coleman, W.M. and Gordon, B.M. (1989). Examinations of the matrix isolation fourier transform infrared spectra of organic compounds. Part XVI. Appl. Spectrosc. 43 (6): 1008–1016. Kubota, K., Fujishige, S., and Ando, I. (1990). Solution properties of poly(N-isopropylacrylamide) in water. Polym. J. 22 (1): 15–20. Torii, H. (2004). Vibrational interactions in the amide I subspace of the oligomers and hydration clusters of N-methylacetamide. J. Phys. Chem. A 108 (35): 7272–7280. Katsumoto, Y., Tsuchiizu, A., Qiu, X., and Winnik, F.M. (2012). Dissecting the mechanism of the heat-induced phase separation and crystallization of poly(2-isopropyl-2-oxazoline) in water through vibrational spectroscopy and molecular orbital calculations. Macromolecules 45 (8). Takahashi, H., Wakamatsu, S., Tabata, H. et al. (2011). Atropisomerism observed in indometacin derivatives. Org. Lett. 13 (4): 760–763.

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10 Combination Analysis of Matrix-Isolation Spectroscopy and DFT Calculation Nobuyuki Akai Tokyo University of Agriculture and Technology, Graduate School of Bio-Applications and Systems Engineering (BASE), 2-24-16, Naka-cho, Koganei, Tokyo 184-8588, Japan

10.1 Introduction Vibrational spectral analysis had been carried out by experts who practice for at least several years and have to obtain a lot of experience and knowledge for molecular symmetric property, normal vibration, functional group frequencies, etc. Molecular identification of a measured spectrum of unknown chemical species, one of which is intermediate in chemical reactions, has been much difficult for them. Nowadays, calculation chemistry including computer abilities and useful softwares like Gaussian series rapidly develops, leading to that anyone can accomplish molecular identification and band assignments of observed vibrational spectra by comparing with the corresponding simulated ones. Meanwhile, such easy comparisons cause often serious mistakes for band assignment and molecular identification due to the misunderstanding condition of the target molecule between the experiment and theory. A simulated infrared (IR) spectrum depends on theories and basis set and is ordinarily estimated by the harmonic vibration analysis at a geometry that is optimized for single molecule in vacuum. On the other hand, band shapes and peak wavenumbers in experimental spectra are largely changed by many factors on phase, solvent, temperature, concentration, etc.; for example, it is well known that IR spectrum of acetone in CCl4 is different from that in methanol or water. Although such the solvation effect is often estimated by continuum models of solvation like polarizable continuum model (PCM), in which one solute is placed on an empty cavity in a solvent with dielectric field, reproducing an experimental spectrum is not so easy because a target molecule largely interacts with the neighbor solvent through direct interaction force including hydrogen bond, dispersion force, and so on. Gas-phase spectroscopy can obtain a fine spectrum that is close to a line spectrum because the sample molecule is not affected by surrounding molecules. However, even if the samples are cooled to a few Kelvin by a supersonic jet technique, a lot of vibrational rotational lines appear in the gas-phase spectrum, which disturb a direct comparison with simulated vibrational spectrum. If an experimental method to obtain a pure Molecular Spectroscopy: A Quantum Chemistry Approach, First Edition. Edited by Yukihiro Ozaki, Marek Janusz Wójcik, and Jürgen Popp. © 2019 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2019 by Wiley-VCH Verlag GmbH & Co. KGaA.

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10 Combination Analysis of Matrix-Isolation Spectroscopy and DFT Calculation

vibrational spectrum of a single molecule located in a non-interacting solid is possible, the experimental spectrum is easily compared with the corresponding simulated one. The matrix-isolation method as described in this chapter is close to the ideal condition, and some results obtained by the combination method of the matrix-isolation IR spectroscopy with density functional theory (DFT) calculation are introduced below.

10.2 Matrix-Isolation Method The matrix-isolation method [1, 2] is a classical experimental method developed by George C. Pimentel in 1950s [3]. “Matrix isolation” means that the sample molecule is embedded in a chemically inert solid like N2 or noble atoms as shown in Figure 10.1. The sample molecule isolated in the matrix medium is usually made by depositing the sample diluted with excess noble gas on a cooled transparent plate like a CsI crystal for IR measurement at low temperature in a vacuum chamber. The noble gas such as neon or argon is well known to be a chemically inert material that not only reacts with the sample molecules but also interacts with those much less than other materials, and the solid condition inhibits free rotation of the sample molecules in the rigid matrix, resulting in that a pure vibrational spectrum with sharp vibrational bands without rotational lines is obtained. Therefore, IR spectroscopy equipped with the matrix isolation has been frequently used to confirm accuracy of theoretical results, and then matrix-isolation IR spectrum of unknown chemical species has been analyzed with aids of the confirmed theories like the B3LYP functional, which is one of hybrid functionals of DFT method. An example of matrix-isolation IR spectrum is shown in Figure 10.2, where experimental spectra of acetic acid are compared with the theoretical ones estimated at the B3LYP/6-31++G(d,p) level. Seeing two experimental spectra, it is clear that the bandwidths of the matrix-isolation IR spectrum are narrower

Figure 10.1 Schematic diagram of matrix-isolation method. Sample molecules (gray ellipse) are separately embedded in noble-gas (white circle) solid, which are transparent in a wide light region between UV–vis and IR rays.

IR intensity

Absorbance

10.2 Matrix-Isolation Method

(a)

in CCl4 in Ne matrix

(b)

Without scaling factor (c) Scaling factor of 0.98 (d)

2000

1500

1000

Wavenumber (cm–1)

Figure 10.2 Comparison of experimental spectra of acetic acid in CCl4 solvent (a) and isolated in an Ar matrix (b) with a simulated spectrum estimated at the B3LYP/6-31++G(d,p) level without scaling factor (c) and 0.98 used (d).

than those of the IR spectrum measured in CCl4 solvent although CCl4 is a typical nonpolar solvent that interacts with acetic acid much weaker than polar solvents. Wavenumber shifts of absorption bands are also certainly affected by intermolecular interaction; the intense band assigned to the characteristic C=O stretching mode appears at 1788 and 1715 cm−1 in the spectra of the matrix and in CCl4 solvent, respectively, meaning that molecular interaction of noble-gas matrix condition is much weaker than nonpolar solvent CCl4 . It is another advantage that the matrix spectrum can be measured in all the IR region, while record of the region below 830 cm−1 is hard in CCl4 because the asymmetric C—Cl stretching mode appearing around 790 cm−1 with strong IR absorption disturbs. It is noticed that the matrix-isolation IR spectrum as a quasi-ideal solid condition (Figure 10.2b) is satisfactorily reproduced by the B3LYP/6-31++G(d,p) level calculation (Figure 10.2c). Comparing both spectra in detail, the intense matrix bands appear at 989, 1183, 1385, and 1788 cm−1 that are consistent with the theoretical values at 1001, 1206, 1413, and 1822 cm−1 estimated by harmonic oscillator approximation. Using a scaling factor of 0.98 for them, the adjusted theoretical spectrum of Figure 10.2d, where intense bands appear at 981, 1182, 1385, and 1786 cm−1 , is more close to the observed spectrum. As shown in this example, a scaling factor is usually used to correct theoretical wavenumbers estimated by the harmonic vibrational analysis, where the value of scaling factor depends on both theory and basis set; usually a scaling factor of 0.99–0.86 or a linear scaling factor is empirically used [4, 5]. It is noted that the observed bands in the matrix spectra are not completely reproduced by this theory; for example, overtones and combination bands cannot be predicted by the harmonic oscillator approximation. Although such little disconnection between both is inevitable, their spectral patterns in the whole region are satisfactorily consistent with each other for molecular identification. Now anharmonic oscillator calculation can predict better theoretical values, but is

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10 Combination Analysis of Matrix-Isolation Spectroscopy and DFT Calculation

not so easily performed for relatively large molecules because its calculation cost is much higher than the harmonic vibrational analysis. Since a matrix-isolation IR spectrum being quasi-ideal vibrational spectrum is accurately reproduced by simulated one when a suitable theory is selected, unknown species, whose spectrum has been unreported so far, will be identified with an aid of the theoretical calculation. This is the most important point for what numerous combination analyses of matrix-isolation IR spectroscopy with quantum chemical calculation have been carried out recently. Using matrix-isolation method to study molecular reaction such as photolysis, unknown spectra are often obtained because the matrix medium is at cryogenic temperature; ca. 6 K for Ne and 15 K for Ar, resulting in that highly reactive chemical species like radicals or intermediates that are immediately changing into other stable compounds at room temperature, are stably kept in the matrix for long time. Vibrational analysis and identification for such unknown species have been almost impossible before the development of calculation chemistry, but now lots of new chemical species have been identified by the combination method, for example, noble-atom compounds like HArF, HKrCCH, HXeSH, and so on [6–9]. Then, numerous researches using the combination method of matrix-isolation IR spectroscopy and quantum chemical calculation have been reported in these decades [10–12], and here our research topics are introduced as typical examples for matrix-isolation IR spectroscopy with aids of DFT calculation in the following sections.

10.3 Adoption of Theory and Basis Set Since the matrix-isolation IR spectroscopy provides a quasi-ideal vibrational spectrum as described above, the combination method needs more accurate theory than other experimental spectroscopies. If a deficient calculation is used in the combination analysis, a wrong identification or assignment would be concluded. Therefore, one has to choose theory or functional with especially careful consideration for components of the sample molecule. The B3LYP hybrid functional, which is one of the DFT functionals, is well known to be one of the best functionals and is popularly used in the calculation for many molecules, complexes, and clusters. However, the universal method for all molecules does not exist; for example, since dispersion force is not estimated by the B3LYP functional, M06 series functionals or high-level ab initio methods must be used for van der Waals clusters. In addition, a few molecules including H2 O2 and O3 are known to be difficult to be theoretically predicted by DFT functionals, and then high-level coupled-cluster theory like CCSD(T) is needed for these molecules [13–15]. Selection of a basis set is also an important factor to reproduce an observed spectrum. It is known that IR spectra of molecules including sulfur atom are not satisfactorily reproduced by the B3LYP functional with the 6-31++G(d,p) basis set that is one of the most popular basis set. Figure 10.3 shows a comparison of the matrix-isolation spectrum of bis(trifluoromethanesulfonyl)imide (H-TFSI)

10.4 Conformational Analysis

Observed

6-31++G(d,p)

6-311++G(d,p)

6-31++G(df,pd)

6-31++G(3 df,pd)

6-311++G(3 df,3 pd)

1800

1600

1400

1200

1000

800

600

Wavenumber (cm–1)

Figure 10.3 Comparison of the matrix IR spectrum of H-TFSI in a Ne matrix with simulated spectra estimated at B3LYP functional with several basis sets.

with theoretical ones predicted by the B3LYP functional with several basis sets, where no scaling factors are used. It is clear that the B3LYP/6-31++G(d,p) level cannot reproduce the matrix-isolation spectrum, as well as 6-311++G(d,p) and 6-31++G(df,pd) basis sets. The S=O stretching modes around 1450 cm−1 , S—N—S asymmetric stretching mode around 850 cm−1 , and S—N—S bending mode around 600 cm−1 are underestimated, when the simulated spectra are not scaled. On the other hand, the 6-31++G(3df,3pd) and 6-311++G(3df,3pd) basis sets are satisfactorily consistent with the observed spectrum, implying that spectral analysis for molecules including sulfur atom needs at least the B3LYP/6-31++G(3df,3pd) level.

10.4 Conformational Analysis 10.4.1

1,2-Dichloroethane

The gauche and trans forms for 1,2-dichloroethane is known to be a typical example of rotational conformers. The conformational study is one of the successful examples for the combination analysis of matrix-isolation IR spectroscopy with quantum chemical calculation [16, 17]. The population ratio of

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10 Combination Analysis of Matrix-Isolation Spectroscopy and DFT Calculation

two conformers depends on their energy difference and temperature, which is estimated by the Boltzmann distribution law as below: Nl ∕Nm = Al ∕Am × exp(−ΔE∕kT)

(10.1)

where N i , Ai , ΔE, k, and T represent the number of particles in each state, degeneracy, energy difference between both states, the Boltzmann constant, and temperature, respectively. Since the law simply shows that the ratio of the less stable conformer increases with increasing temperature, band assignments for each conformer are possible by measuring IR spectra in the gas phase at several temperatures. The gas-phase spectra include not only vibrational bands but also numerous rotational lines as described above, which makes difficult to perform vibrational analysis. Using the matrix-isolation method, sample molecules diffused from a heated deposition nozzle are immediately frozen on the CsI plate at cryogenic temperature, meaning that the population ratio of the conformers in the matrix solid is equal to that of those at the deposition nozzle. Then, comparing a matrix-isolation IR spectrum composed of the conformers with the corresponding simulated spectra, conformational analysis is easily accomplished. Figure 10.4 shows Ar matrix-isolation IR spectra of 1,2-dichloroethane as an example of deposition temperature dependence. 1,2-Dichloroethane has two conformers; trans is estimated to be more stable than gauche by 7.1 kJ/mol at the B3LYP/6-31++G(d,p) level. The 1232 and 1290 cm−1 bands are assigned to trans CI H

H

H

H

Trans

CI CI

H

CI

H

H

Gauche

284

H

298 K

400 K

1350

1300

1250

1200

Wavenumber (cm–1)

Figure 10.4 Matrix-isolation spectra of 1,2-dichloroethane isolated in Ar solid.

1150

10.4 Conformational Analysis

and gauche, respectively, and the relative absorbance of the 1290 cm−1 band assigned to the less stable conformer is recognized to increase with deposition temperature increasing from 298 to 400 K. If a target molecule has only two conformers as this example of 1,2-dichloroethane, conformational analysis may be not so difficult without matrix-isolation technique by conventional IR spectroscopy for the gas phase. However, a lot of molecules such as amino acids have multiple conformers, and their vibrational spectral patterns are usually very similar to each other, causing difficulty of conformational analysis for the IR spectroscopy. 10.4.2

Chlorobenzaldehyde

Absorbance

Even if a matrix-isolation IR spectrum is composed of ideal sharp bands, conformational analysis for multiple conformers by the deposition temperature dependence experiment is not so easy. In order to distinguish the observed bands to multiple conformers clearly, photoinduced reaction among the conformers is often attempted in the matrix-isolation method because a conformer produced upon light irradiation is stable in the low-temperature condition and unchanged for enough long time to be measured. A simple example of photoinduced conformational changes is shown in Figure 10.5b, where a different spectrum of 2-chlorobenzaldehyde between measured before and after light irradiation is shown; increasing and decreasing

(a)

(b)

IR intensity

× 10

O-cis

H

O CI

O-cis

(c) UV irr. O-trans

O

H CI

1800

1600

1400

1200

Wavenumber

1000

800

O-trans

(cm–1

)

Figure 10.5 Matrix-isolation IR spectra of 2-chlorobenzaldehyde measured before UV irradiation (a), difference spectrum measured between before and after UV irradiation (b), and simulated spectrum composed of the O-cis and O-trans conformers in upside and downside, respectively (c).

285

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10 Combination Analysis of Matrix-Isolation Spectroscopy and DFT Calculation

bands are assigned to be the O-cis conformer as the product and the O-trans conformer as the reactant (Figure 10.5a), respectively [18]. Although the increasing and decreasing bands of both conformers are very similar in this case, almost all bands in the different spectrum are classified to each conformer and compared with the corresponding simulated spectra as shown in Figure 10.5c. By the comparison of the experimental different spectrum with the simulated spectrum, the correspondence between observed and theoretical bands is very obvious; for example, the observed intense bands at 1730, 1474, 1074, 846, and 755 cm−1 in the upper side are consistent with the theoretical 1751 (C=O stretching), 1474 (ring stretching with C—H in-plane bending), 1055 (ring stretching), 839 (ring deformation), and 755 (C—H out-of-plane bending) cm−1 of the O-cis conformer, and the observed bands at 1706, 1447, 1269, and 760 cm−1 in the lower side agree well with 1729 cm−1 (C=O stretching), 1444 cm−1 (ring stretching with C—H in-plane bending), 1265 cm−1 (C—CHO stretching), and 759 cm−1 (C—H out-of-plane bending) of the O-trans conformer. 10.4.3

Vanillin

Here, an actual method of the conformational analysis is explained using the case of a slightly more complicated molecule; vanillin has three rotatable single bonds, and then the number of the conformation is considered to be maximum of 8 (=2 × 2 × 2). First, we have to know the number of stable conformations and their energies using quantum chemical calculation, which leads us to their optimized geometries with energies as shown in Figure 10.6. In the case of vanillin, six conformers are stable structures, and conformers G and H are not found to be stable geometries. If some conformers are estimated to be located on close energy levels, less H

O

O

O Me O

Conformer A (most stable) O

H

H

O

O Me

Conformer E (+30.0)

H

H

Conformer B (+5.1) O

H

O

O

Conformer C (+20.9) O

H

H

Me

Conformer F (+30.6)

O Me H

O O

H

O Me

O Me O

H

O

H

H

O

Conformer D (+24.7) O

H

O O

H

Me

Conformer G (unstable)

O O

H

Me

Conformer H (unstable)

Figure 10.6 Conformers of vanillin with their related energies in kJ/mol estimated at the B3LYP/6-31++G(d,p) level.

10.4 Conformational Analysis

–730 –778

(a)

–819

–872

–1033

–1040/1043

Absorbance

than a few kJ/mol, they exist as a ratio according to the Boltzmann distribution law at a deposition temperature, and then the ratio is expected to be maintained in a low-temperature matrix. Conformer A of vanillin is estimated to be the most stable conformer and 5.1 kJ/mol and more than 20 kJ/mol lower energy than the second stable conformer B and the third stable conformer C, respectively, suggesting that the abundance ratio of conformers A:B:C is predicted to be 89 : 11 : 0 at the deposition temperature of 298 K. Second, the observed matrix IR spectrum is visually compared with the simulated spectral patterns of the candidate conformers in Figure 10.7, showing the observed matrix-isolation IR spectrum of vanillin with the simulated spectra of the conformers A and B, where the observed spectrum seems to agree well with the sum of both simulated spectra. In detail, the observed 1033, 872, 819, and 730 cm−1 bands seem to be consistent with the simulated bands at 1035, 874, 812, and 727 cm−1 for conformer A, and the 778 cm−1 band is consistent with that at 770 cm−1 for conformer B. Although the observed broaden doublet bands appearing at 1040 and 1043 cm−1 seem be assigned to the 1038 cm−1 band for conformer B, it cannot be confirmed by this figure. In order to accurately assign the observed bands to both conformers, one typical method is to perform similar experiments with various deposition temperatures by heating. However, broaden or sprit bands like the observed band around 1040 cm−1 are frequently difficult to assign to each other by the heating method. Third, UV irradiation on the matrix sample is examined to expect conformational changing. Figure 10.8 shows a difference spectrum of the sample

IR intensity

(b) Conformer A

(b) Conformer B

1100

1000

900

800

700

Wavenumber (cm–1)

Figure 10.7 Matrix-isolation IR spectrum of vanillin (a) and simulated spectra of conformers A (b) and B (c).

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10 Combination Analysis of Matrix-Isolation Spectroscopy and DFT Calculation

Absorbance

(a)

(b) Conformer B IR intensity

288

Conformer A 1100

1000

900

800

700

–1

Wavenumber (cm )

Figure 10.8 A difference IR spectrum of vanillin measured between after and before UV irradiation (a) and simulated spectra composed of conformers A and B in downside and upside (b), respectively.

between measured before and after UV light irradiation. It is visually clear that the spectrum has positive and negative bands due to increasing and decreasing species upon the light irradiation. In this case, the spectral patterns in the upper side and lower side are satisfactorily consistent with the simulated spectrum composed of conformers B and A, respectively, meaning that the conformational change from conformers A to B occurs upon the light irradiation. This comparison clearly suggests that the bands at 1040 and 1043 cm−1 are assigned to conformer A and B, respectively, and both bands at 1033 and 1040 cm−1 should be assigned to the predicted 1035 cm−1 band for conformer A. Such a splitting band may be explained by any vibrational coupling due to intense bands in lower wavenumber region; in this case intense bands are expected to appear at 493 and 622 cm−1 for conformer A. The different spectrum includes one more important thing. Carefully comparing the difference spectrum with the deposition one, some band shapes in Figure 10.8a are noticed to be different from the corresponding deposition bands in Figure 10.7a; for example, the intense 1033 cm−1 band in Figure 10.7 is decreasing with weak intensity in Figure 10.8. The discrepancy is caused by third species, bands of which are overlapped with the original bands, produced by the irradiation. Finally, each observed band can be consistent with the calculated ones, judging from the correspondences between the observed and calculated bands in not only their wavenumbers but also IR intensities. These results are usually summarized in a table with their vibrational assignments (Table 10.1).

10.4 Conformational Analysis

Table 10.1 The observed wavenumbers (in cm−1 ) of vanillin compared with the simulated ones estimated at the B3LYP/6-31++G(d,p) level with a scaling factor of 0.98. Conformer A Observed

1040

Conformer B

Assignment

Calculated

Observed

Calculated

1035

1043

1038

O—Me stretch + C—H bend

1033 900

926

922

C—O—C stretch + ring bend

874

850

851

Out-of-plane C—H bend

822/819

812

830

825

813

807

730

727

872

10.4.4

Out-of-plane C—H bend Out-of-plane C—H bend

778

794

In-plane ring bend

770

In-plane CCCCHO bend

Excitation Light

Since such conformational change and/or photolysis depends on wavelength of irradiation light, several conformers are in many case possible to be spectroscopically distinguished by different wavelength irradiation even if a target molecule has more complicated conformers or tautomers. Especially, near-IR or IR excitation has been recently used for study on conformational isomerization via vibrational excited states [19–28] because radiating UV light on sample molecule may cause not only conformational change but also photolysis. The photoinduced conformational change was used in conformational analysis for many molecular systems; i.e. acetylacetone [29–32], which has 8 (2 × 2 × 2) possible conformers as shown in Figure 10.9, is one of the best examples for using this method, where the ccc, ctc, and tct conformers and one tautomer H

O

O

Me

O Me

O

Me

H ccc

H Me Me

H cct

H Me O

Me Me

O

O

O

H

H

Me

O

O

H

Me

Me

H ttc

H

O

H

H ctt

Me O

O

Me

O

H H ctc

Me

O H tct

H tcc

Me

O

Me

Me O

O H ttt

Me

Figure 10.9 Eight enol conformers and keto tautomer of acetylacetone.

H

289

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10 Combination Analysis of Matrix-Isolation Spectroscopy and DFT Calculation

yielded upon UV irradiation are experimentally identified. It is noted that radiation on a matrix sample always causes not only conformational change but also unexpected photoreaction including photolysis, where unknown compounds are often produced. In the next section, some results by the combination analysis for photoreaction products or intermediates, spectra of which has not been known so far, are described.

10.5 Identification for Unknown Species 10.5.1

Rare Tautomer of Cytosine

As described above, the matrix-isolation IR spectroscopy with aids of quantum chemical calculations is one of the most powerful methods for molecular identification of newly produced chemical species. It is not rare that unexpected photoreactions to yield unknown products occur in matrix-isolated samples, and then molecular identifications of the products have to be say almost impossible by empirically analysis from their measured IR spectra. Actually, several bands assigned to “unknown” had been frequently concluded at least before the development of quantum chemical calculation so far. With the development of calculation chemistry, cyclopedic calculations for all candidate species produced from a reactant were carried out at several theory levels, and then existences of unexpected products were found, resulting in that a number of reports for molecular identification by matrix-isolation IR spectroscopy have been drastically increasing during these decades. This section relates with numerous reports using the combination method, and a photoinduced hydrogen-atom migration as a typical and important chemical reaction is introduced firstly. Photoinduced isomerization of cytosine is considered to be a good example for hydrogen migration reaction [33–35]. Here, only two simplified tautomerizations are focused although conformational changes also occur in this molecular system. Cytosine, which is one of the nucleic acid bases, has multiple tautomers as shown in Figure 10.10. The amino-oxo (AO) tautomer is estimated to be the most stable, and amino-hydroxy (AH) is the second stable tautomer with 1.7 kJ/mol higher energy at the B3LYP/6-31++G(d,p) level. The third stable imino-oxo (IO) is 7.7 kJ/mol higher energy than the AO tautomer. (Although the AH, IO, and IH tautomers have multiple conformers in the direction of OH and NH bonds, each most stable conformer is considered in this figure.) A Ne matrix-isolated IR spectrum of cytosine evaporated at ca. 200 ∘ C is obtained as shown in Figure 10.11a, where the AO and AH types are expected NH2

NH2

N

N

N H AO (0)

O

N AH (1.7)

NH

NH

NH OH

N H IO (7.7)

O

NH N IH (56.9)

OH

Figure 10.10 Isomers of cytosine with their relative energies (in kJ/mol) estimated at the B3LYP/6-31++G(d,p) level.

10.5 Identification for Unknown Species

1440 1430

1625

1684

1734 1724

1755

Absorbance

Obsd.

(a) (b)

IR intensity

Calcd. IO

(c)

AO AH

(d)

1800

1700

1600

1500

1400

Wavenumber (cm–1)

Figure 10.11 Matrix-isolation IR spectrum of cytosine (a), photoinduced tautomerization spectrum (b), and simulated spectra composed of IO and AO in upside and downside, respectively (c), and AH (d).

to coexist since their energies are closed to be 1.7 kJ/mol. In fact, the observed spectrum seems to be reproduced by both simulated spectra; i.e. the intense 1734 and 1440 cm−1 bands are surely consistent with 1739 cm−1 (AO) and 1444 cm−1 (AH), respectively. One may have a big concern that assignments of all observed bands to both tautomers can be judged by the way one looks. Wherein, trying UV light (𝜆 ≥ 275 nm) irradiating on the matrix sample for a short time, a little changing of the matrix IR spectrum may be found, which is distinguishable by comparing with the difference spectrum between measured before and after the UV irradiation. The difference spectrum is shown in Figure 10.11b, where (i) the 1734 cm−1 band for AO decreases, (ii) the 1440 cm−1 band for AH slightly increases (the 1430 cm−1 band is assigned to the other conformation of AH), and (iii) the 1755 cm−1 band newly increases, meaning that the observed bands are assigned to three species. Comparing the difference spectrum with the simulated spectra of the expected tautomers, the first, second, and third spectral components are satisfactorily reproduced by the theoretical spectra of AO, AH, and IO, respectively. In this manner, production of the IO type is confirmed, and then its band assignments are possible to be accomplished; i.e. the observed 1755 and 1685 cm−1 bands are consistent with the calculated 1766 cm−1 (C=O stretching) and 1684 cm−1 (C=N stretching), respectively. 10.5.2 Reversible Isomerization Between Triplet and Singlet Species from 1,8-Diaminonaphthalene The matrix-isolation methods, where thermal reaction is inhibited by the cryogenic condition, are generally known to be a powerful technique to detect high-reactive reaction intermediates like radicals as described above, and then many studies focusing on them have been reported so far [36–39]. Especially,

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10 Combination Analysis of Matrix-Isolation Spectroscopy and DFT Calculation

NH

NH2 N

3ANN

NH

Figure 10.12 Photoinduced isomerization between 1,8-aminonaphthylnitrene (3 ANN), 1,8-dihydro-1,8-naphthalenediimine (3 DND), and 1,2-dihydrobenz[cd]indazole (DBI).

3DND

H

H N

N

DBI

molecular identification and structural analysis for carbenes and nitrenes as interesting biradical species need a large contribution of not only experimental spectroscopies but also various theories. Existence of such radical species is in general evidenced by EPR spectroscopy, but the combination method of IR spectroscopy and quantum chemical calculation also accomplishes the purpose although their spin states are not directly detected. An example of them would deal with our recent report [40], which was accidentally found to be an interesting isomerization among three chemical species including a nitrene and biradical as shown in Figure 10.12. The nitrene, 8-amino-1-naphthylnitrene (3 ANN), yielded by UV photolysis of 1,8-diaminonaphthalene changes into the triplet biradical species, 1,8-dihydro-1,8-naphthalenediimine (3 DND), by 𝜆 ≥ 700 nm light irradiation. The related difference spectrum is shown in Figure 10.13a with the corresponding simulated spectrum. The B3LYP/6-31++G(d,p) calculation predicts that the lowest triplet states of both are more stable than their lowest singlet states, and then the simulated spectrum composed of 3 ANN in downside and 3 DND in upside seems to be in good agreement with the observed difference spectrum. In other words, the popular B3LYP calculation can accurately reproduce observed IR spectrum not only in the singlet ground state but also in the triplet ground state, suggesting that geometries in the triplet ground state optimized at the calculation level may be true. Direct determination of a molecular structure in the triplet ground state is not so easy, and the combination method would be useful in the aspect of spin chemistry. In addition, there is one more important thing that the combination method is also able to analyze an isomerization between different spin states as triplet and singlet states. In this molecular system, isomerization between the triplet nitrene and 1,2-dihydrobenz[cd]indazole (DBI) as a molecule in the singlet ground state also occurs by another light irradiation. A difference spectrum related with the isomerization is shown in Figure 10.13c with the corresponding simulation composed of DBI in the upside and 3 ANN in the downside, which reproduces the observed spectrum satisfactorily. In both experimental difference spectra (Figure 10.13a, c), the common decreasing 1604 cm−1 band is assigned

10.6 Spectrum and Structure of Molecular Complex or Cluster

(a)

Absorbance

(b)

3DND 3

ANN

(c)

DBI

(d) 3

ANN

2000

1800

1600

1400

1200

1000

800

600

Wavenumber (cm–1)

Figure 10.13 Matrix IR spectra related with photoinduced isomerization between 3 ANN and 3 DND (a) and 3 ANN and DBI (c) to compare with the corresponding simulated spectra (b) and (d), respectively.

the characteristic C—NH2 stretching mode of 3 ANN by the aid of the DFT calculation. The bond length of C—NH2 is estimated to be 1.373 Å, which is shorter than that in 1,8-diaminonaphthalene, 1.410 Å. Using time-dependent DFT calculation that can estimate absorption wavelength of the three isomers, the photoinduced reversible isomerization mechanism is proposed as shown in Figure 10.12.

10.6 Spectrum and Structure of Molecular Complex or Cluster Molecular complex and small cluster are also studied by the matrix-isolation IR spectroscopy, although several laser spectroscopies in the gas phase, which can separate each complex structure using multiple lasers with a mass analyzer, are usually used for this subject. The advantage of the matrix-isolation IR spectroscopy for studying molecular complex is measurable in wide IR region at one measurement, where experimentally detected multiple bands can be compared with simulated values obtained by quantum chemical calculation in the same manner of single molecule analysis. Two experimental techniques to investigate molecular complex by the matrix-isolation method are used; one is dependence of concentration of the sample in a premixed gas reservoir, and the other is photolysis of single molecule in a matrix. The former is well known to be the standard method for this subject and used in not only almost typical experiments

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10 Combination Analysis of Matrix-Isolation Spectroscopy and DFT Calculation

N

H N

H N

H

N

C

H N

N H

C

H

2.52 Å

Figure 10.14 Photolysis of s-triazine to make cyclic (HCN)3 in a matrix cage.

C

but also the matrix-isolation IR spectroscopy with aids of DFT calculations [41–47]. Instead of diffusional depositing a sample on the CsI plate, a supersonic jet depositing is often used to effectively make more amount of molecular complex [48–51]. Studies on metal complex have investigated by laser ablation of metal solid in the vacuum [52–57]. UV irradiation on a matrix sample sometimes causes dissociation of covalent bonds, resulting in that fragmentations to produce several kinds of small chemical species occur and that a molecular complex composed of the fragments is yielded in a matrix cage [58–63]. The fragments yielded from a single molecule are kept in the same matrix cage; thus, elemental composition of the complex absolutely corresponds to that of the reactant molecule. An example of molecular complex would be introduced using photolysis of s-triazine to produce cyclic (HCN)3 as shown in Figure 10.14 [63]. The fundamental bands of HCN isolated in an Ar matrix appear at 3306, 2098, and 721 cm−1 , which are consistent with the calculated values of 3306 (C—H stretch), 2098 (C≡N stretching), and 721 cm−1 (bending), respectively. In photolysis of s-triazine, two characteristic bands at 764 and 736 cm−1 appeared in the bending region and are assigned to the in-plane (744 cm−1 ) and out-of-plane (736 cm−1 ) bending modes of cyclic (HCN)3 by the aids of DFT calculation, which estimates that the hydrogen bond length between HCN· · ·HCN is 2.52 Å.

10.7 Photoinduced Transient Species 10.7.1

Hydroquinone

The last example for unique IR spectra that are possible to be measurable only by the matrix-isolation IR spectroscopy would be introduced. The spectra are recorded with a conventional FTIR spectrophotometer during light irradiation on matrix samples not after. They may be assigned to be two kinds of photoinduced transient species. One is less stable isomer, which temporarily produced on photoexcitation immediately changes into more stable isomer after stopping the light. Usually, isomers produced in the matrix-isolation method are kept for enough long time to be measured because the low-temperature condition inhibits any thermal reaction overcoming reaction barriers; it is considered that thermal isomerization occurs over barrier height of less than ca. 10 kJ/mol at matrix temperature. However, IR spectra of less stable isomers measured only during light irradiation were sometimes reported, and the isomerization from less stable to other isomers was explained by hydrogen-atom tunneling effect. Molecular identifications for less stable isomers existing during light irradiation were impossible to do without the combination method of matrix-isolation IR spectroscopy and

10.7 Photoinduced Transient Species

Absorbance

quantum chemical calculation. Tunneling isomerization has been reported for several kinds of molecules: formic acids, hydroquinones, thioureas, etc. [64–73]. Here, matrix IR spectrum of hydroquinone is shown in Figure 10.15, where an IR spectrum of hydroquinone immediately is measured after the deposition procedure; a difference spectrum between measured during and after UV light irradiation and the simulated spectrum composed of cis and trans hydroquinone in up- and downsides, respectively, are compared [71]. The energy difference and barrier height between both conformers is calculated to be 0.6 and 10.8 kJ/mol from the energy of trans hydroquinone, which is the stable conformer. It is interesting that the bands in the spectrum observed before UV irradiation (Figure 10.15a) were assigned to only trans hydroquinone but not cis, although an IR spectrum composed of both had been initially expected to be recorded before UV irradiation by the Boltzmann distribution law at the deposition temperature of ca. 30 ∘ C, which has been estimated to 1 : 0.8 for trans/cis. The bands assigned to cis were detected only during UV irradiation as shown in Figure 10.15b, where the increasing and decreasing bands were surely consistent with the spectral patterns of cis and trans, respectively. After UV irradiation, the isomerization from trans to cis occurred in darkness and at cryogenic temperature, although the reaction barrier was estimated to be 10 kJ/mol, suggesting that the isomerization occurred by a hydrogen-atom tunneling effect.

(a) O

H

× 30

(b)

O

H

IR intensity

cis

O

cis

(c)

H

trans H

O

trans 1600

1400

1200

1000

800

Wavenumber (cm–1)

Figure 10.15 IR spectra of hydroquinone (a), UV-induced transient species (b), and simulated spectrum composed of cis and trans conformers of hydroquinone in upside and downside (c), respectively.

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10 Combination Analysis of Matrix-Isolation Spectroscopy and DFT Calculation

Since IR spectra of both isomers are very similar as Figure 10.15, the band assignments and identification are impossible without the combination method of matrix-isolation IR spectroscopy and quantum chemical calculation. 10.7.2

Lowest Electronic Excited Triplet State

Absorbance

The other possibility of photoinduced transient species is the molecule in an electronic excited state, which is usually measured by pump–probe laser spectroscopies. Indeed, short-lifetime species that relax to the ground state by emitting fluorescence or internal conversion are not able to be detected by the matrix-isolation IR spectroscopy. However, IR spectra of a kind of lowest electronically excited triplet (T1 ) states, lifetime of which is prolonged to a few seconds under the low-temperature matrix condition, are often obtained and identified with the aids of DFT calculation. Such T1 species continuously emit phosphorescence for a few or ten second after stopping excited light irradiation on the matrix sample, and their IR spectra can be also measured by a conventional IR spectrophotometer during light excitation [74–82]. An example of measured IR spectra for T1 species is shown in Figure 10.16, where increasing and decreasing species are identified to the T1 and the ground (S0 ) state of 1,4-dicyanobenzene, respectively [79]. It notes that simulated spectral pattern of the T1 state satisfactorily reproduces the observed one with the same accuracy as that of the S0 state, meaning that the B3LYP calculation can predict not only the electronic ground states including the S0 state and triplet ground state for carbenes and nitrenes but also the T1 state. The observed

IR intensity

296

(a)

Transient bands

(b)

T1 state S0 state 2000

1500

1000 –1

Wavenumber (cm )

Figure 10.16 A difference IR spectrum of 1,4-dicyanobenzene measured between during and after UV irradiation (a) and simulated IR spectrum composed of its T1 and S0 states in upside and downside (b), respectively.

10.8 Conclusion

Figure 10.17 Bond length changing of 1,4-dicyanobenzene in the S0 to the T1 states.

1.164 → 1.180 1.435 → 1.388

1.390 → 1.349

1.406 → 1.472

Bond length (in Å)

bands appear at 1993, 1399, 1208, 1101, and 748 cm−1 , which are consistent with the calculated values, 1998 cm−1 (C≡N stretching), 1339 cm−1 (CCC and C—H in-plane bending), 1280 cm−1 (C—C stretching), 1100 cm−1 (C—H in-plane bending), and 758 cm−1 (C—H out-of-plane bending), respectively. Judging from the good consistency between experimental and theoretical wavenumbers estimated by the B3LYP calculation, the geometrical structure in the T1 state is determined as shown in Figure 10.17. Comparing the optimized geometries in both states, the observed large redshift of the C≡N stretching from 2244 cm−1 in the S0 state to 1993 cm−1 in the T1 state is explained by the structural change that the C—C≡N bond includes a part of conjugated double bond (C=C=N) character in the T1 state; the C—C single bond length gets shorter from 1.435 to 1.388 Å, while the C≡N triple bond length gets longer from 1.164 to 1.180 Å. In addition, the lower wavenumber shift of the C—H out-of-plane bending from 841 to 748 cm−1 is also understood by the benzene ring having a part of quinonoid character in the T1 state. Since some calculation softwares like Gaussian series have recently acquired ability to optimize molecular geometry in the electronically excited states, vibrational analysis in some excited states will be performed with the same as that in the ground state.

10.8 Conclusion As described above, the development of calculation chemistry has largely affected all researchers using matrix-isolation IR spectroscopy, and now quantum calculations are indispensable to analysis of a matrix-isolation IR spectrum. Especially, molecular identification and band assignment of photoproducts absolutely need not only expert skills for matrix-isolation IR spectroscopy but also exact knowledge of calculation chemistry. Although numerous “unknown” species could be identified by the combination analysis of matrix-isolation IR spectroscopy and quantum chemical calculation and then lots of photoreaction mechanism was interpreted, the number of undetected chemical species is not small even now. Then, the combination method is expected to continue to be a major part of molecular science and photochemistry.

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Oxford: Oxford University Press. 2 Clark, R.J.H. and Hester, R.E. (1989). Spectroscopy of Matrix Isolated Species.

Wiley. 3 Whittle, E., Dows, D.A., and Pimentel, G.C. (1954). Matrix isolation method

for the experimental study of unstable species. J. Chem. Phys. 22: 1943. 4 Yoshida, H., Ehara, A., and Matsuura, H. (2000). Chem. Phys. Lett. 325:

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11 Role of Quantum Chemical Calculations in Elucidating Chemical Bond Orientation in Surface Spectroscopy Dennis K. Hore University of Victoria, Department of Chemistry, Canada

11.1 Introduction Characterizing the structure of molecules at surfaces is a fundamental step in the pursuit of understanding molecular interactions that are the basis of improved catalyst design [1, 2], biocompatibility of medical implant materials [3–5], enzyme immobilization strategies for biosensors [6–8], and modeling environmental pollutants at ocean surfaces [9] – among a huge range of other applications. In all cases, it is not only the composition of the surface that is of interest but also the orientation and conformation of species at the surface. Consider, for example, the case of protein adsorption at the polymer–solution interface. Knowing the bulk composition of the polymer has little to do with the interfacial structure, as the polymer chains may adopt a conformation, orientation, and composition (in the case of polymer blends) that are drastically different from the bulk on account of their interaction with the aqueous phase. Similarly, the solvent hydrogen bonding environment that is so critical in determining the protein properties in solution is unique at the polymer surface and therefore an important contributor to the protein surface structure. Finally, the protein residue side chain functional groups that preferentially interact with the polymer surface may differ from those that are solvent-accessible in the bulk solution phase. For example, at a hydrophobic polymer surface, proteins may denature as a result of maximizing their exposure of hydrophobic residues, normally buried in the core of their tertiary structure. In summary, interactions between the substrate, solvent, and adsorbed species are all critical in accounting for the interfacial structure. This chapter will focus on vibrational spectroscopy as a probe of such interactions, as it is generally a label-free method that can be used in a wide variety of in situ environments including buried polymer–aqueous interfaces [10, 11], so long as the surface of interest is accessible to light. Vibrational techniques also offer submolecular structural details, as specific chemical functional groups may be probed. This is an advantage over other methods such as the use of fluorophores as there is no perturbation of the system, and one can elucidate the structure of the molecules of interest directly, rather than that of Molecular Spectroscopy: A Quantum Chemistry Approach, First Edition. Edited by Yukihiro Ozaki, Marek Janusz Wójcik, and Jürgen Popp. © 2019 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2019 by Wiley-VCH Verlag GmbH & Co. KGaA.

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an external probe. The challenge, however, is achieving sufficient selectivity of interfacial species – how to probe the polymer surface and interfacial water molecules without being overwhelmed by the response of the same molecules in the bulk polymer or aqueous phase and, similarly, how to selectively probe proteins at the surface without incorporating signals from molecules in solution. As will be described in the following section, one approach is to utilize nonlinear optical techniques based on even orders of the electric susceptibility tensor. By the end of this chapter, our goal is to illustrate that a unique and important aspect of the interfacial structure can be determined with such techniques: the absolute orientation or polarity of chemical bonds. That is to say, we can determine whether a surface-adsorbed methyl group is directed “down” toward the bulk polymer phase or “up” toward the bulk aqueous phase. However, the general scheme by which spectroscopic data results in molecular structure, including this unique polarity information, relies on quantum chemical calculations. It will be our ultimate goal to illustrate this connection.

11.2 Vibrational Sum-Frequency Generation Spectroscopy 11.2.1

Basic Experimental Details

There are several more introductory [12], more advanced [13–15], and more general [16, 17] descriptions of nonlinear vibrational spectroscopy. Our goal here is to provide only the background necessary to illustrate the topic of interest – how quantum mechanics provides a link between the spectroscopy and underlying surface structure. In the most basic version of the experiment, a fixed-frequency visible laser and tunable infrared laser are spatially and temporally overlapped at the interface of interest. As shown in Figure 11.1, this can occur in a transmission or reflection geometry. Reflection geometries are particularly useful at the interface between two condensed phases when one of the materials is not transparent to the incoming infrared beam. Under the electric dipole approximation, materials that lack an inversion center (i.e. there are no points where (x, y, z) → (−x, −y, −z)) have a non-zero value of the second-order (and in fact all even orders) electric susceptibility 𝝌 (2) . Outside of the realm of chiral bulk phases, a E(ωIR)

E(ωvis)

c

b E(ωIR)

E(ωvis)

E(ωSFG) H2O E(ωvis) E(ωIR)

E(ωSFG)

E(ωSFG)

Figure 11.1 Some basic co-propagating geometries for a visible–infrared sum-frequency generation experiment. For exposed solid surfaces, the SFG signal can be collected in a (a) transmission or (b) external reflection configuration. For solid–liquid interfaces, (c) internal reflection configurations can be used.

11.2 Vibrational Sum-Frequency Generation Spectroscopy

or crystals belonging to non-centrosymmetric classes [18, 19], such symmetry is found at all interfaces. Even in the general case where there is no orientation of any molecule in the plane of the surface (at z = 0), the surface normal properties along +z and −z are unique. This situation describes the chemical functional groups of the polymer surface, the water molecules within the interfacial layer, and adsorbed proteins – all have 𝝌 (2) ≠ 0. In this environment, one photon of the visible beam at frequency 𝜔vis and one photon of the infrared beam at 𝜔IR are annihilated. Conservation of energy dictates that a new photon is created with frequency 𝜔vis + 𝜔IR , hence the term sum-frequency generation (SFG). The intensity of the signal, I, may be described as ⋅ Evis (𝜔vis ) ⋅ EIR (𝜔IR )|2 I(𝜔vis + 𝜔IR ) ∝ |𝝌 (2) eff

(11.1)

where E are the electric components of the electromagnetic fields of the incident lasers. Local field correction factors L can be used to relate these surface fields to the incident fields from the laser. The effective susceptibility is defined according to = (e ⋅ L)(e ⋅ L) ∶ 𝝌 (2) (e ⋅ L) 𝝌 (2) eff

(11.2)

where e is a unit polarization vector that considers the projection of E onto the laboratory frame [12, 20] ⎡ ± cos 𝛾 ⎤ e=⎢ 1 ⎥ ⎢ ⎥ ⎣ sin 𝛾 ⎦

(11.3)

and 𝛾 are the beam angles with respect to the surface normal. The correspondence between the incoming/outgoing and surface fields is obtained using ⎡Lxx L=⎢ 0 ⎢ ⎣0

0 Lyy 0

0⎤ 0⎥ ⎥ Lzz ⎦

(11.4)

A detailed discussion of the L factors may be found in the literature [21] and include models such as 2n1 cos 𝛾2 (11.5a) Lxx = n1 cos 𝛾2 + n2 cos 𝛾1 2n1 cos 𝛾1 (11.5b) Lyy = n1 cos 𝛾1 + n2 cos 𝛾2 ( 2) n1 2n2 cos 𝛾1 Lzz = (11.5c) n1 cos 𝛾2 + n2 cos 𝛾1 n′2 where the subscript 1 refers to the incident medium, 2 refers to the refracted medium, and n′ is the ratio of microscopic local field corrections [22]. As shown in Figure 11.2, if we take the square root of the intensity and plot this as a function of the IR beam frequency, we see the resonances in |𝝌 (2) |(𝜔IR ) (Figure 11.2c) that bear some resemblance to what is observed in an IR absorption (Figure 11.2a) or Raman scattering spectrum (Figure 11.2b). This is not surprising, as the energy level diagram shown in Figure 11.3 reveals that the SFG in this case may be thought of as a coincident IR absorption and anti-Stokes Raman scattering event.

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11 Elucidating bond orientation in surface spectroscopy

Im[χ(1)] (a.u.)

1.00 a 0.75 0.50 0.25 0.00 2800

2850

2900 2950 3000 IR wavenumber / cm–1

3050

2850 2900 2950 3000 Stokes Raman shift / cm–1

3050

2850

2900 2950 3000 IR wavenumber / cm–1

3050

2850

2950 3000 2900 IR wavenumber / cm–1

3050

Im[χ(3)] (a.u.)

1.00 b 0.75 0.50 0.25 0.00 2800

|χ(2)| (a.u.)

1.0 0.8

c

0.6 0.4 0.2 2800 1.0

Im[χ(2)] (a.u.)

306

d

0.5 0.0 –0.5 2800

Figure 11.2 (a) 𝕀m{𝝌 (1) } spectra of ethanol obtained from an ATR-IR absorption experiment, (b) 𝕀m{𝝌 (3) } spectra from a spontaneous Raman scattering experiment, and (c) |𝝌 (2) | obtained from a homodyne SFG experiment. If the phase of 𝝌 (2) can be determined, then it is additionally possible to isolate 𝕀m{𝝌 (2) } as shown in (d). Note that the vibrational bands in (d) are pointing in both directions, and this information, together with quantum chemical calculations, is directly connected to the polar orientation of the chemical functional group. Source: Data taken from Ref. [23].

It is important to keep in mind, however, that 𝝌 (2) = 0 in the bulk, so the signal collected here comes exclusively from the interface. This is one reason that SFG spectra may be dramatically different from both IR and Raman spectra in terms of the relative band intensities. This raises the often asked question of to what extent the signal probes the interface. The answer is simply that, for any 𝝌 (2)

11.2 Vibrational Sum-Frequency Generation Spectroscopy

a Electronic excited state

E(ωvis)

Electronic ground state

c

b |ψvitual

E(ωSFG = ωvis + ωIR)

E(ωIR)

|ψ1

|ψ1

|ψ1

|ψ0

|ψ0

|ψ0

Figure 11.3 (a) Energy level diagram illustrating a vibrationally resonant but electronically nonresonant SFG process where 𝜔vis is too low in energy to reach the electronic excited state. Vibrationally and electronically nonresonant processes where 𝜔IR is too (b) high or (c) low in energy to match the vibrational 1 ← 0 transition. Note that panels (b) and (c) describe the situation for 𝛼-quartz in our example, where nonresonant SFG can be produced for any 𝜔IR .

process, signal originates from all regions in which the two incident laser beams are sufficiently overlapped in time and space and the centrosymmetry is broken. The vagueness of this answer presents both challenges and opportunities for SFG spectroscopy [24–26]. For the continuation of this discussion, we can picture a situation where 𝝌 (2) ≠ 0 over a distance of approximately 1 nm on either side of the interface. Such values are in agreement with results of molecular dynamics simulations that can accumulate histograms of the orientation distribution as a function of z [27]. From this point, the connection between the SFG spectra and sought molecular structure involves a more detailed discussion of 𝝌 (2) [28–31] that we will attempt to present in a succinct manner here. As the incident visible and IR fields, and emitted SFG field, are vectors (in general, tensors of rank 1), the second-order susceptibility that connects the three quantities is a tensor of rank 3. In three-dimensional Cartesian coordinates, it is a 3 × 3 × 3 = 27-element quantity. Fortunately, for most of the systems of interest, isotropy in the (x, y)-plane results in only 7 of these 27 elements being non-zero, and choosing a visible frequency far from any electronic resonance makes only 3 of these 7 elements unique. Finally, the values of all unique non-zero elements are determined in the canonical way, by judicious choice of the polarization of the incident laser beams and detected SFG beam. For simplicity, we consider an experiment where the visible beam is s-polarized, the IR beam is p-polarized, and the s-component of the SFG intensity is detected. In this scheme,

(11.6) (2) 𝜒yyz .

so we are able to obtain While additional insight is obtained by varying the polarization combinations [32–34] and measuring additional elements of 𝝌 (2) , this single experiment will suffice to illustrate our point.

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11.2.2

Molecular View

So far we have been describing collections of molecules, where the net response is represented by 𝝌 (2) . Each molecule contributes to this response through the molecular hyperpolarizability. Chemists historically prefer to refer to this quantity as 𝜷, though we shall adopt the notation 𝜶 (2) as it more clearly highlights the connection between the second-order macroscopic and microscopic responses: 𝝌 (2) =

1 𝜀0

N ∑

𝜶 (2)

(11.7)

molecules

where 𝜀0 is the permittivity of vacuum and N is the number of molecules. Equation (11.7) describes the general relationship between two tensors of rank 3. However, it is the details of this expression that will be of interest for the remainder of this chapter. We begin by defining a coordinate system where l, m, n each represents any of the molecular Cartesian coordinates a, b, c. We will then use i, j, k for any of the laboratory (surface) coordinates x, y, z. A 3 × 3 matrix D of nine direction cosines can be used to project elements of 𝜶 (2) into the laboratory frame [35]. In general, this projection is parameterized by three Euler angles: D(𝜃, 𝜙, 𝜓) = Rz (𝜙) ⋅ Ry (𝜃) ⋅ Rz (𝜓)

(11.8)

which can be derived from three elementary rotation matrices R, one for each of the constituent Euler angles [35]. For simplicity, we will consider an isotropic distribution of all angles except for the tilt angle, 𝜃, the angle between the molecular c axis and the surface normal z. We then have 2π

2π

1 D(𝜃, 𝜙, 𝜓) d𝜙 d𝜓 4π2 ∫0 ∫0 0 sin 𝜃 ⎤ ⎡ cos 𝜃 1 0 ⎥ =⎢ 0 ⎥ ⎢ 0 cos 𝜃 ⎦ ⎣− sin 𝜃

D(𝜃) =

(11.9)

and (2) (𝜃) = 𝛼ijk

abc abc abc ∑ ∑∑ l

m

(2) Dil (𝜃) ⋅ Djm (𝜃) ⋅ Dkn (𝜃) ⋅ 𝛼lmn

(11.10)

n

(2) (2) where our notation 𝛼ijk (𝜃) emphasizes the fact that once 𝛼lmn is projected into the laboratory ijk frame by means of D(𝜃), it itself becomes a function of 𝜃. We now return to our example of the single yyz element and add an explicit 𝜃-dependence: (2) (𝜃) = 𝛼yyz

abc abc abc ∑ ∑∑ l

m

(2) Dyl (𝜃) ⋅ Dym (𝜃) ⋅ Dzn (𝜃) ⋅ 𝛼lmn .

(11.11)

n

(2) (2) In other words, each of the 27 elements of 𝛼 (2) in the molecular frame (𝛼aaa , 𝛼aab , (2) (2) etc., all the way through to 𝛼ccc ), has some contribution to 𝛼yyz . The manner in which they contribute depends on the symmetry of the local vibrational mode,

11.2 Vibrational Sum-Frequency Generation Spectroscopy

and the degree to which they contribute depends on the tilt angle 𝜃. We assume that molecules whose structure at the surface differs only in terms of 𝜃 can be represented by the same 𝜶 (2) values in the molecular frame. This is an important consideration, since even when there is no strong evidence for this, it is a common approximation that is used to make the structure elucidation tractable. This precludes the possibility that the interfacial environment results in a perturbation of the electronic structure, such that 𝜶 (2) in the molecular frame is distinct between surface and bulk environments. To proceed, we revisit Eq. (11.7) with the realization that experiments do not provide an opportunity to build up the ensemble average one molecule at a time. Instead, we are interested in relating the measured 𝝌 (2) response to the average structure of the participating molecules. We can express this as 𝝌 (2) =

N (2) ⟨𝜶 ⟩ 𝜀0

(11.12)

where the sum over N molecules in Eq. (11.7) has been replaced by the ensemble average represented by the angular brackets. From this point, we can consider the symmetry of the specific vibrational mode of interest. For example, the methyl symmetric stretch has C3v symmetry that (2) results in only three unique non-zero elements of the hyperpolarizability: 𝛼aac , (2) (2) 𝛼bbc , and 𝛼ccc . After performing the projection into the laboratory frame, we arrive at the expression N (2) ⟨𝛼 (𝜃)⟩ 𝜀0 yyz N (2) (2) (2) (2) (2) (2) + 𝛼aac + 𝛼bbc )⟨cos 𝜃⟩ − (2𝛼ccc − 𝛼aac − 𝛼bbc )⟨cos3 𝜃⟩. (11.13) = (2𝛼ccc 𝜀0

(2) = 𝜒yyz

Here we can see that the trigonometric functions of 𝜃 introduced by elements of the direction cosine matrix have resulted in an expression that depends on the average values of cos 𝜃 and cos3 𝜃. This is another important result of SFG spectroscopy that the dependence of the signal on the molecular tilt angle follows only odd powers of cos 𝜃. For any whole number n, cos2n+1 (π − 𝜃) = −cos2n+1 𝜃. In other words, the SFG field emitted by two molecules with “opposite” orientation, 𝜃 and π − 𝜃, will be out of phase and therefore destructively interfere. This is the origin of the surface specificity of this technique. Each molecule in the bulk polymer and liquid phase generates an SFG field, but no net response is measured since the field cancels with that from other molecules in an isotropic environment where all orientations (including 𝜃 and π − 𝜃) are equally represented. At the surface, molecules must not only be oriented, but they must be oriented in a polar manner in order to generate an SFG signal. A polymer film that is stretched to a high draw ratio results in a very large degree of orientation of the polymer chains along the draw axis, but there is an equal possibility that molecules are oriented in either direction along this axis. Such an alignment will result in large order parameters in polarized IR spectroscopy probing cos2 𝜃 and Raman spectroscopy additionally probing cos4 𝜃. Since cos2n (π − 𝜃) = cos2n 𝜃, IR or Raman signals would add constructively regardless of the polarity of the

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orientation distribution. However, the orientation distribution must be polar, i.e. biased toward one quadrant of the tilt angle, to yield any SFG response. 11.2.3

SFG Phase Measurement

The ability of SFG to selectively probe interfaces affords an additional unique structural parameter that may be understood based on the above discussion. We have seen that a polar orientation of chemical bonds is required to generate SFG signal. The corollary of this is that it should be possible to approach this problem from the other end and determine the bond polarity from the SFG response. The expressions above illustrate that this is indeed possible if 𝝌 (2) is measured, but the required polarity information is lost if only the magnitude of the response |𝝌 (2) | is (2) 2 obtained (Figure 11.2c) in an intensity measurement probing |𝜒yyz | (Eq. (11.1)). The solution is to measure the magnitude and phase of the emitted SFG field and (2) therefore obtain the magnitude and phase of 𝜒yyz . This naturally requires some sort of interferometric detection scheme, generally referred to as heterodyne SFG spectroscopy. There have been several approaches proposed for this [36–52]. The basic premise is that SFG from a material with known 𝝌 (2) phase, referred to as the local oscillator (LO), is combined with SFG produced by the sample under investigation. If the interference between the two SFG sources is constructive, the sample SFG field and LO have the same phase; if they cancel, they are phase-shifted by π rad. Figure 11.4 illustrates the case of the sample of interest and the reference sample having the same phase; the interference will be constructive resulting in a maximum in the detected SFG intensity. Either way, information on the bond polarity may be accessed. In a variation of this experiment, an LO of unknown phase can be used. The sample is later replaced with a reference sample of known phase, and the sample–LO response is compared with the reference–LO response to determine the sample 𝝌 (2) magnitude and phase. However the result is achieved, there (2) (2) are then four options for presenting the data, |𝜒ijk |(𝜔IR ), 𝜙ijk (𝜔IR ), ℝe{𝜒ijk }(𝜔IR ), (2) (2) (2) i𝜙ijk }(𝜔IR ) where 𝜒ijk = |𝜒ijk |e , and the real and imaginary parts are and 𝕀m{𝜒ijk (2) (2) (2) (2) } = |𝜒ijk | cos 𝜙ijk and 𝕀m{𝜒ijk } = |𝜒ijk | sin 𝜙ijk . The imaginary given by ℝe{𝜒ijk spectrum (Figure 11.2d) is particularly valuable. First, it removes the dispersive (2) contributions to the 𝜒ijk line shape that are contributed by the real part. Second, the absorptive bands may be more readily compared with those in the bulk IR and Raman spectra on account of the linear additivity of the response of each (2) normal mode. It is the cross terms present in |𝜒ijk | that give rise to phenomena such as apparent peak frequency shifts. For our discussion, however, the most (2) important attribute of 𝕀m{𝜒ijk }(𝜔IR ) is that it is a bisignate spectrum. As shown in Figure 11.2d, bands may be upward or downward pointing with respect to the baseline, and this is intimately linked to the sought polarity information. What is remaining is to determine whether an upward pointing vibrational band indicates an upward or downward pointing chemical moiety; this will be the final point of our discussion.

11.3 Determination of Bond Polarity Surface χ(2) ≠ 0 Unknown sample

a Visible laser

CH3

EIR Infrared laser

Dark or bright signal

Evis

Evis

EIR

+

=

Evis EIR

Detector

Reference sample Bulk χ(2) ≠ 0

3000

c ESFG

Phase shifting unit

2θ θ Sample (unknown or reference)

IR wavenumber (cm–1)

b

EIR Lens

ELO source

Evis

2950

2900

2850

2800 –45 –35 –15 0 15 30 45 Phase shifting unit tilt angle / °

Figure 11.4 (a) Schematic of a heterodyne SFG experiment, where the visible and infrared laser pulses are split in order to generate SFG simultaneously from the sample of interest and a reference sample. The two sources of SFG are then combined to monitor their interference to yield the magnitude and phase of 𝜒 (2) . (b) One scheme for the practical realization of this experiment, where the visible and infrared beams are made collinear and travel through a bulk material with 𝜒 (2) ≠ 0 in order to generate the LO. The residual visible and IR beams, along with the LO, now reach the sample to produce the SFG interference. In such a scheme, a phase shifting unit is introduced in order to modulate the interference, producing (c) a pattern of fringes as a function of the phase shift and the IR frequency.

11.3 Determination of Bond Polarity We have already established the framework necessary to put together the final piece of this puzzle. We now consider that the resonance line shape of the vibrational hyperpolarizability can be written as a sum of vibrationally nonresonant and resonant contributions, and the resonant portion can in turn be expressed as a sum over all of the 3N − 6 normal modes. In the molecular frame, ∑

3N−6 (2) (2) (𝜔IR ) = 𝛼lmn,NR + 𝛼lmn

(2) 𝛼lmn,q (𝜔IR )

q

∑

blmn,q

q

𝜔q − 𝜔IR − iΓq

3N−6

=

(2) 𝛼lmn,NR

+

(11.14)

√ where i = −1, bq is the quantity responsible for the strength of the interaction (to be discussed in further detail in the following section), 𝜔q is the normal mode

311

312

11 Elucidating bond orientation in surface spectroscopy

frequency, and Γq is the homogeneous line width of the qth normal mode. In the laboratory frame, ∑

3N−6 (2) (2) (𝜔IR , 𝜃) = 𝛼ijk,NR (𝜃) + 𝛼ijk

(2) 𝛼lmn,q (𝜔IR , 𝜃)

q

∑

bijk,q (𝜃)

q

𝜔q − 𝜔IR − iΓq

3N−6 (2) = 𝛼lmn,NR (𝜃) +

(11.15)

where the numerator of our Lorentzian line-shape function contains the important tilt angle dependence of the resonant response. For simplicity, we consider that the vibrationally nonresonant response is purely real, as in the case when all materials are dielectrics. In exceptional cases such as metallic substrates, a more general version of the theory is based on the same concepts [53–55]. If an SFG phase measurement was performed, we then have access to ∑

Byyz,q (𝜃) ⋅ Γq

q

(𝜔q − 𝜔IR )2 + Γ2q

3N−6 (2) }(𝜔IR , 𝜃) = 𝕀m{𝜒yyz

(11.16)

where the macroscopic quantity B(𝜃) is the sum of all contributions of the individual molecular b(𝜃). Note that b (and therefore B) is a real quantity. So the workflow then goes like this: for a given vibrational mode of interest (for example, methyl symmetric stretch), first determine the sign of b(𝜃). In other words, we need to know whether b is positive or negative in the quadrant 0 < 𝜃 < π∕2. Based on the transformation of a tensor of rank 3, we know that it will then have the opposite sign in the quadrant π∕2 < 𝜃 < π (Figure 11.5a and b). Before we discuss the details of the underlying quantum mechanics of b(𝜃), it is useful to point out that it may be possible, in practice, to come up with an experimental scenario that can be used to solve the polarity problem with only this information. That is, imagine that we seek to characterize the methyl groups at the surface of a polymer or the methyl groups of an adsorbed protein. We would prepare a reference sample with methyl groups in similar environments, such as an alkyl chain with a sufficiently short length that we know the methyl groups are oriented with their C-to-H axis directed away from the substrate. We emphasize that at this point, we know nothing about byyz (𝜃). We now either interfere the methyl symmetric stretch SFG field at 2875 cm−1 directly with LO generated from the methyl reference substrate at the same frequency or do the double comparison experiment and compare methyl sample–LO with methyl reference–LO. Either way, if the phase of the two methyl samples is the same (constructive interference observed), we know the polymer methyl groups are oriented in the same way, C-to-H axes on average directed away from the bulk polymer phase. This is true if both samples are installed in the same direction in the experiment. We are, after all, measuring the polarity of the bonds! Although this approach works and is robust, it is not practical for general use, as it assumes that one can always prepare a suitable LO or reference material with the exact same chemical functionality and chemical environment as that of the unknown sample. It would therefore be more useful to have a universal phase reference and then make the polarity assignment based on a known sign of b(𝜃). The phase reference is straightforward, as it can be obtained from the nonresonant response of any material that has 𝝌 (2) ≠ 0 in the bulk, a non-centrosymmetric material such as crystalline quartz.

11.4 Quantum Chemical Calculations for Modeling the Molecular Hyperpolarizability SFG field generated from surface methyl group

π/2 < θ < π rad E(ωvis) Visible field

+E(ωSFG = ωvis = ωIR)

c

H

3C

a

z Valley

π rad phase shift

IR field E(ωIR) 0 < θ < π/2 rad Polymer film

Peak

Solid support

z

CH

–E(ωSFG)

b

+E(ωSFG)

c

–E(ωSFG)

d

3 c π rad phase shift

0 < θ < π/2 rad

OH

z Valley c π rad phase shift

π/2 < θ < π rad c

Peak

HO

z

Figure 11.5 Relationship between the polar orientation of the surface chemical functional group, its quantum chemical properties, and the phase of the generated sum-frequency field. In the case of a methyl group, if the C-to-H direction (defining the molecular c-axis) is oriented in the same quadrant as the surface normal (case b), the emitted SFG field has a phase that is shifted by π rad with respect to a hydroxyl group with its O to H direction oriented in the same manner, away from the bulk solid phase.

11.4 Quantum Chemical Calculations for Modeling the Molecular Hyperpolarizability Referring to the energy level diagram in Figure 11.3, we denote the IR transition dipole moment vector in the molecular (a, b, c) frame using Dirac notation: ∞

̂ 0⟩ = ⟨Ψ1 |𝝁|Ψ

∞

∞

∫−∞ ∫−∞ ∫−∞

̂ 0 da db dc Ψ∗1 𝝁Ψ

(11.17)

where Ψ0 is the vibrational ground state, Ψ1 is the vibrational excited state, 𝝁̂ is the dipole moment operator, and the asterisk denotes the complex conjugate. Since we are operating far from electronic resonance, the anti-Stokes

313

314

11 Elucidating bond orientation in surface spectroscopy

transition induced by the visible laser can be approximated by a rank 2 transition polarizability: ∞

̂ 1⟩ = ⟨Ψ0 |𝜶|Ψ

∞

∫−∞ ∫−∞ ∫−∞ ∞

=

∞

∞

̂ virtual ⊗ Ψ∗virtual 𝝁Ψ ̂ 1 da db dc Ψ∗0 𝝁Ψ

∞

∫−∞ ∫−∞ ∫−∞

̂ 1 da db dc Ψ∗0 𝜶Ψ

(11.18)

where 𝜶̂ is the linear polarizability operator constructed from the outer product of the two (rank 1) transition dipole operators connecting the ground and excited vibrational states through the virtual electronic state. Using this formalism, we can construct the rank 3 molecular hyperpolarizability from the outer product of the two tensors ̂ 1 ⟩ ⊗ ⟨Ψ1 |𝝁|Ψ ̂ 0 ⟩. 𝜶 (2) = ⟨Ψ0 |𝜶|Ψ

(11.19)

Taking an individual (l, m, n) element of the tensor as an example, the result may be obtained by scalar multiplication: (2) = ⟨Ψ0 |𝛼̂ lm |Ψ1 ⟩⟨Ψ1 |𝜇̂ n |Ψ0 ⟩. 𝛼lmn

(11.20)

We therefore require a method for evaluating the two constituent quantities ̂ 0 ⟩ and ⟨Ψ0 |𝜶|Ψ ̂ 1 ⟩. Using a harmonic approximation, we can write the ⟨Ψ1 |𝝁|Ψ transition dipole moment as the derivative of the dipole moment with respect to the normal mode coordinate Q evaluated at the equilibrium geometry Q = 0. For each of the q normal modes, we therefore have [ ] 𝜕𝜇n 1 (11.21) ⟨Ψ1 |𝜇̂ n |Ψ0 ⟩q ≈ √ 2mq 𝜔q 𝜕Qq Q=0 with an example shown in Figure 11.6. For each of the three Cartesian components a, b, and c that describe the molecular axes, we calculate elements of the dipole moment vector for seven geometries representing nuclear displacements along Q. The structure corresponding to Q = 0 has equilibrium bond angles and bond lengths. The three molecules with Q = −1, −2, and −3 are in successive “compressed” states in the vibrational period; Q = +1, +2, and +3 are successively “stretched.” In reality these may not be simple compressions and extensions, as the nuclear coordinates are in general coupled. This is more often encountered for low-frequency modes, as vibrations above 2800 cm−1 such as alkyl C—H stretches are relatively decoupled. Nevertheless, when using quantum mechan̂ 0 ⟩, it is not necessary to make approxiical approaches to determine ⟨Ψ1 |𝜇|Ψ mations regarding the degree to which the modes are coupled. One can instead consider the displacement of all atoms, information contained in the eigenvectors of the Hessian matrix. Since we consider the displacement from the equilibrium geometry to be small, we can model 𝜇l (Q) with a second-order polynomial (solid lines in Figure 11.6). This allows for facile determination of the derivative about Q = 0, as indicated with the dashed lines in Figure 11.6. We can write ⟨Ψ0 |𝛼̂ lm |Ψ1 ⟩ (1) in a similar way: as a derivative of the linear polarizability 𝛼lm [ (1) ] 𝜕𝛼lm 1 ⟨Ψ0 |𝛼̂ lm |Ψ1 ⟩q ≈ √ (11.22) 2mq 𝜔q 𝜕Qq Q=0

11.4 Quantum Chemical Calculations for Modeling the Molecular Hyperpolarizability

0.26 0.24 –3 –2 –1

0

1

2

0.20

–0.25

0.19

–0.30

0.18

–0.35

μc

μb

μa

0.28

0.17

–0.40

0.16

–0.45

3

–3 –2 –1

Q

0

1

2

3

Q

–3 –2 –1

0

1

2

3

Q

Figure 11.6 Values of the dipole moment vector elements 𝜇n for the CH3 symmetric stretch of methanol. The molecule-fixed (l, m, n) coordinates are chosen so that c coincides with the methyl group C3 axis. The calculation was performed using B3LYP/6-31G(d,p) to yield the seven points for each of the nine tensor elements, with Q = 0 corresponding to the equilibrium geometry for this particular normal mode. The points were then fit to a second-order polynomial (solid line) in order to evaluate the tangent at Q = 0 (dashed line).

for the anti-Stokes transition. An example of such a calculation is shown in Figure 11.7. Since we are far from an electronic resonance, the polarizability (1) (1) tensor is symmetric with 𝛼lm = 𝛼ml . As a result, only the six lower triangular elements are shown. Combining the polarizability and dipole moment derivatives via Eq. (11.20) results in [ (1) ] [ ] 𝜕𝛼lm 𝜕𝜇n 1 1 (2) . (11.23) 𝛼lmn ≈ 2mq 𝜔q 𝜕Qq 𝜕Qq Q=0 𝜔q − 𝜔IR − iΓq Q=0

If we now compare with Eq. (11.14), we see that [ (1) ] [ ] 𝜕𝛼lm 𝜕𝜇n 1 blmn = . 2mq 𝜔q 𝜕Qq 𝜕Qq Q=0

(11.24)

Q=0

and bijk (𝜃) =

abc abc abc ∑ ∑∑ l

m

Dil (𝜃) ⋅ Djm (𝜃) ⋅ Dkn (𝜃) ⋅ blmn

(11.25)

n

The above expression has a large applicability for structure elucidation efforts in SFG spectroscopy. Since 𝜒 (2) = N⟨𝛼 (2) ⟩, it follows that B = N ⋅ b(𝜃). If one fits the peaks in a vibrational SFG spectrum according to the normal mode frequencies, then the amplitude of the peak yields B directly. The ratio of amplitudes obtained with different beam polarization yields different elements of the 𝜒 (2) tensor. Such a ratio, for example, Byyz ∕Byzy , is then equal to the ratio of laboratory frame hyperpolarizabilities, byyz (𝜃)∕byzy (𝜃). This ratio is independent of the number of molecules N and is purely a function of the structure, so it can be used to access the tilt angle 𝜃 [14, 33, 56, 57]. However, for our primary interest of bond polarity elucidation, we are concerned only about the sign of b(𝜃). As can be seen in the above expressions, this (1) ∕𝜕Q]Q=0 and [𝜕𝜇n ∕𝜕Q]Q=0 . If both comes down to the sign of the derivatives [𝜕𝛼lm quantities are positive or negative, blmn will be positive; if one quantity is positive and the other is negative, blmn will be negative. Many electronic structure calculation packages determine the value of these derivatives since they are directly

315

11 Elucidating bond orientation in surface spectroscopy 4.8

α(1) aa

4.6 4.4

(1) α(1) ab = α ba

4.2

(1) (1) α ac = α ca

4.0 1

2

3

4.8 4.6 (1) α(1) bc = α cb

4.4 4.2

–3 –2 –1

0

1

2

4.0

3

–3 –2 –1

0

1

2

3

1.0 0.5 0.0

α(1) cc

0.4 0.2 0.0 –0.2 –0.4 –0.6 –0.8 –1.0

0

α(1) bb

0.2 0.1 0.0 –0.1 –0.2 –0.3 –0.4 –0.5 –0.6 –0.7

–3 –2 –1

α(1) cb

α(1) ba

3.8

α(1) ca

316

–0.5 –1.0

–3 –2 –1 0 1 2 3 Q

–1.5

–3 –2 –1 0 1 2 3 Q

21 20 19 18 17 16 15 14

–3 –2 –1 0 1 2 3 Q

(1) Figure 11.7 Values of the linear polarizability tensor elements 𝛼lm for the CH3 symmetric stretch of methanol. The molecule-fixed (l, m, n) coordinates are chosen so that c coincides with the methyl group C3 axis. The calculation was performed using B3LYP/6-31G(d,p) to yield the seven points for each of the nine tensor elements, with Q = 0 corresponding to the equilibrium geometry for this particular normal mode. The points were then fit to a second-order polynomial (solid line) in order to evaluate the tangent at Q = 0 (dashed line). Far from electronic resonance, the Raman tensor is symmetric, so only the lower triangular half is shown.

related to the Raman scattering and IR absorption cross section. However, the Raman cross section is proportional to the square of the transition polarizabil(1) ∕𝜕Q]2Q=0 , and the absorption cross section is related to the square of the ity [𝜕𝛼lm transition dipole moment [𝜕𝜇n ∕𝜕Q]2Q=0 . These last two quantities are useless to us, since we lose the sought sign information if only the squares are reported. If we now continue with our example of the methyl symmetric stretch, we can provide an alternative to the methyl–methyl direct comparison for polarity determination presented in the previous section. Using a calibrated piece of z-cut quartz as the reference sample in the SFG experiment, interference between quartz and the LO can be obtained for every frequency 𝜔IR of interest, since the quartz generates an electronically nonresonant and vibrationally nonresonant SFG response. Assume that we have calibrated our z-cut quartz so that we know the phase of its effective susceptibility element of interest. We are still using the same ssp beam polarizations (s-polarized SFG detected, s-polarized visible, and

11.4 Quantum Chemical Calculations for Modeling the Molecular Hyperpolarizability

p-polarized IR incident). As quartz belongs to symmetry point group 32, we have

·

(11.26)

We then measure our unknown sample at the frequency corresponding to the resonance of its methyl symmetric stretch, 2875 cm−1 . This measurement provides the difference in phase between the effective nonlinear susceptibilities of (2) for our quartz our sample and the quartz reference. Knowing the phase of 𝜒eff in its particular orientation is −π∕2 rad, we can determine the absolute phase of (2) for the sample from 𝜒eff 𝜙sample,eff = Δ𝜙(sample−quartz),eff + 𝜙quartz,eff .

(11.27)

(2) (2) i𝜙sample = |𝜒yyz |e , we next need to consider the In order to arrive at the phase of 𝜒yyz phase of the e and L factors appearing in Eq. (11.2). In this case, the six prefactors are real and positive for both materials, so 𝜙sample = 𝜙sample,eff . However, care must be taken when the x-component of the SFG field is probed, if any of the refractive indices are complex (as in the case of metals) [53, 54, 58], or above the critical angle in an internal reflection geometry (Figure 11.1c) [23], as any of these three situations may result in 𝜙sample ≠ 𝜙sample,eff , so an additional step is required. The (2) values and in particular the signs of the 𝛼lmn elements appearing in Eq. (11.13) tell us that byyz (𝜃) < 0 when 0 < 𝜃 < π∕2 rad and byyz (𝜃) > 0 when π∕2 < 𝜃 < π rad. Since the heterodyne SFG measurement yields the imaginary component of 𝜒 (2) , we can use Eq. (11.16) to see that the quadrant–sign correspondence we have just (2) made enables us to conclude that 𝕀m{𝜒yyz }(𝜔IR = 2875 cm−1 ) = −π∕2 rad when 0 < 𝜃 < π∕2 rad and displays a value of +π∕2 rad when 𝜃 is in the other quadrant. Therefore, by comparing the phase of the methyl symmetric stretch with that of a calibrated quartz sample, we have determined the polarity of the functional group without the use of a reference sample containing methyl groups of known polarity. The real power of this approach is in the flexibility it offers for studying different chemical function groups. Consider, for example, a hydroxyl group with its local c-axis pointing from the oxygen to the hydrogen atom. We carry out the same (1) ∕𝜕Q]Q=0 and [𝜕𝜇n ∕𝜕Q]Q=0 in procedure, calculating all required elements of [𝜕𝛼lm order to arrive at our expression for byyz (𝜃). As shown in Figure 11.5, we now reach the opposite conclusion from the methyl case, finding that byyz (𝜃) > 0 when 0 < 𝜃 < π∕2 rad and byyz (𝜃) < 0 when π∕2 < 𝜃 < π rad. This means that, if the heterodyne SFG experiment reveals that O—H stretch on resonance has the same phase as the z-cut quartz sample, c is antiparallel to z. In other words, the O to H axis is directed down toward the substrate. Figure 11.8 summarizes these principles, illustrating the relationship between the experimental observable (sign of the signal in the heterodyne SFG experiment), sign of the relevant quantum mechanical property b, and the connection to the sought bond orientation including quadrant (polarity) information.

317

11 Elucidating bond orientation in surface spectroscopy

Heterodyne SFG band profile

b@ θ = 0 rad

Polarity

Im{χ(2)}

– Methyl C→H

θ > π/2 rad, C → H down

+ Hydroxyl O→H

Figure 11.8 Relationship between the direction of the observed band in a homodyne and heterodyne SFG experiment (corresponding to the measured amplitude B) and the angle dependence of the quantum mechanical property b(𝜃) that determines the corresponding sign of the vibrational hyperpolarizability.

θ < π/2 rad, O → H up

– Im{χ(2)}

318

Methyl C→H

θ < π/2 rad, C → H up

+ Hydroxyl O→H

θ > π/2 rad, O → H down

11.5 Example As a demonstration of this method, we consider the problem of characterizing the functional groups at the surface of poly(methyl methacrylate). This material, widely known as PMMA or acrylic, is one of the most important industrial polymers with a wide range of uses including the production of biomedical implants. Even though the bulk material structure may be well controlled during the synthesis, the surface structure may be substantially different in terms of the population and orientation of chemical groups [59]. Since PMMA is so ubiquitous, its surface has been characterized using many techniques [60–64]. We have applied heterodyne SFG spectroscopy to characterize the surface ester methyl groups on the PMMA side chain [65]. As described above, we have used the SFG response from a monolayer of trichloro(octadecyl)silane (OTS) on glass as a reference sample. Prior to any quantum chemical calculations, we have determined that, on resonance with the methyl symmetric stretch, the response of the PMMA and OTS samples were in phase as shown in Figure 11.9a. As both materials are dielectric with negligible nonresonant response, interpreting the results of the heterodyne measurement was straightforward. However, the environments of the two methyl groups is radically different, as revealed by the ≈ 80 cm−1 shift in their CH3 symmetric stretch resonances. In the case of OTS, this was observed at 2873 cm−1 , typical for methyl groups at the end of an alkyl chain. But for PMMA, the methyl ester symmetric stretch occurred at 2956 cm−1 . Since this points to an obvious difference in the environment of the functional groups, we need to use quantum chemical calculations to determine the sign of byyz (𝜃). Note that, as we are comparing these two functional groups directly (instead of comparing against the nonresonant response of quartz), it is technically only the difference in sign of byyz (𝜃) between the two methyl environments that is critical here. This point is especially interesting from a chemical standpoint, as the

11.5 Example

c

χ(2) phase (rad)

0 – 1–4 π

CH3 H

CH2 C C

– 1–2 π

OCH3

b

a

a

–20 –10 0 10 20 30 Methyl symmetric stretch frequency shift/cm–1

(1) αaa >0 ∂α(1) aa /∂Q > 0

(1) αbb >0

Q=0

d

+

–

Q=0

Alkyl methyl b e

(1)/∂Q > 0 ∂αbb

c –

b

n

– –34 π –π –30

Ester methyl a

H

+

(1) > 0 αcc (1)/∂Q > 0 ∂αcc

–

Q=0

μc > 0 ∂uc/∂Q < 0

c

f

μc < 0 ∂uc/∂Q < 0

+

–

Q=0

+

Normal mode coordinate c c

g

z

0 < θ < π/2 rad c c

c

Air Bulk polymer

Figure 11.9 (a) SFG 𝜒 (2) phase response of PMMA methyl ester symmetric stretch (blue) and OTS methyl symmetric stretch (red). Since there is an 80 cm−1 difference in resonance frequency, the region ±30 cm−1 of this resonance is shown. (b) Methyl hexanoate molecule, (1) (1) (1) illustrating the local methyl coordinate systems. Values of (c) 𝛼aa , (d) 𝛼bb , (e) 𝛼cc , and (f ) 𝜇c as a function of the vibrational normal mode coordinate Q. From this data it was determined that (g) the PMMA ester methyl groups are pointing, on average, away from the bulk polymer film.

permanent dipole moments of an alkyl and ester methyl group are known to point in opposite directions [66]. The calculations were performed on the small molecule methyl hexanoate (Figure 11.9b), as it contained a representative alkyl and ester methyl group. We determined that byyz (𝜃) did indeed have the same sign (i.e. they were both positive in the same quadrant of 𝜃) for the two methyl environments [65]. Considering the local c axis as the methyl C3 axis for each functional group (Figure 11.9b), it was interesting to note that the dipole moments for each of the seven geometries created along Q had opposite signs of 𝜇c as shown in Figure 11.9f, in agreement with the chemical intuition. However, as we have demonstrated here, it is not the sign of the components of 𝜇 but [𝜕𝜇∕𝜕Q]Q=0 that is relevant to the polarity

319

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11 Elucidating bond orientation in surface spectroscopy

discussion. After determining that byyz (𝜃) of both samples had the same sign in the same quadrant, only then it is possible to conclude that the polarity of the PMMA ester methyl group was the same (C-to-H axis directed away from the bulk polymer film) as that of the OTS methyl group (C-to-H axis directed away from the glass), as illustrated in Figure 11.9G. This result provides a molecular rationalization for the hydrophobicity of the PMMA surface, with a water contact angle in the range 60–70∘ [67, 68]. Technically, materials are considered to be hydrophilic if their contact angle is less than 90∘ , or else they are classified as hydrophobic. However, typical hydrophilic materials have contact angles less than 30∘ . In other words, PMMA is considered hydrophobic for practical applications since it is not as hydrophilic as materials such as poly(ethylene glycol). However, the relatively low contact angle of PMMA is one of the features that makes it favorable for applications involving biomolecular adhesion. Our determination of the ester methyl polarity reveals the torsional angle about the ester O=C—O—CH3 dihedral, with a value close to 180∘ enabling the ester methyl group to point away from the bulk polymer phase. Rotation of this dihedral by 180∘ (if the methyl group were directed toward the bulk polymer film) would further expose the ester oxygen atoms, resulting in an even lower water contact angle. This is an example of how SFG phase measurements, together with quantum chemical calculations, can be used to better understand material properties.

11.6 Summary Nonlinear optical techniques based on even orders of the susceptibility tensor are exclusive probes of broken centrosymmetry. As a member of this family, visible–infrared SFG spectroscopy produces vibrational signatures of species in environments where 𝝌 (2) ≠ 0. In the special cases of interfaces between two achiral isotropic materials, such as an aqueous solution adjacent to an amorphous polymer, such centrosymmetry exists everywhere except at the surface, at z = 0. As a result of the imbalance of forces due to the abrupt change in chemical environment for z > 0 and z < 0, molecules in this narrow interfacial region align themselves in a manner that results in a preference for the polarity of their orientation. In other words, certain chemical groups are directed to a specific bulk phase. When the intensity of the SFG is collected as a function of the infrared frequency, this nonlinear vibrational technique reveals the structure of the interfacial environment. However, further information may be determined by measuring the phase of 𝝌 (2) , often expressed as the imaginary component of the nonlinear susceptibility. In a 𝕀m{𝝌 (2) } spectrum, the sign of each vibrational band is connected to the quadrant of the tilt angle. Quantum chemical calculations can be used to determine the sign of the amplitude of the molecular hyperpolarizability. This sign information, when connected with the sign of 𝕀m{𝝌 (2) }, can ultimately be used to determine the direction in which the functional groups are pointing.

References

Acknowledgments This work was supported by grants from the Natural Sciences and Engineering Research Council of Canada. Equipment was purchased with support from the Canadian Foundation for Innovation and the British Columbia Knowledge Development Fund. Calculations were performed using resources that included Compute Canada and WestGrid clusters. Wei-Chen Yang provided valuable discussion and feedback on the manuscript.

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12 Dynamic and Static Quantum Mechanical Studies of Vibrational Spectra of Hydrogen-Bonded Crystals Mateusz Z. Brela, Marek Boczar, Łukasz Boda, and Marek Janusz Wójcik Jagiellonian University, Faculty of Chemistry, Gronostajowa 2, 30-387 Kraków, Poland

12.1 Introduction In this chapter we would like to introduce the current knowledge about two approaches for simulation of vibration spectra of hydrogen-bonded systems. The first one is the static analysis of vibrational spectra based on calculation of analytical or numerical frequencies in stationary points on the potential energy surface. The second, dynamic approach, focuses on the analysis of the time course of the atom positions and/or dipole moments and, further, the Fourier transform of the autocorrelation function of the dipole moment operator involved in the IR transitions. The main difference between these two approaches is the way of describing and understanding the interaction in studied systems. Both of them, static and dynamic approach, played an important role in understanding the hydrogen bond (HB) phenomena in the last decades [1–9]. It should be emphasized that HB is the important interaction that plays a crucial role in many areas of physics, chemistry, and biology [10–19]. Current trends in molecular modeling, as well as the need of designing new structures and materials, show the great importance of understanding and correct description of interactions in molecular systems. The great theoretician John Pople in his Nobel lecture [20] has described the “features of theoretical models” as predictions: “if the model has been properly validated according to some such criteria, it may be applied to chemical problems to which the answer is unknown or in dispute.” This elucidates the necessity of cooperation between experiment and theory. Recently this concept is realized by modern spectroscopy where comparison between experimental data and calculated results gives possibility to describe and characterize interactions in the studied systems. In the last decades vibrational spectroscopy has been applied in many fields, such as chemistry, biology, and molecular physics [21–30]. In addition to classical approaches, Raman scattering (RS) and infrared (IR) absorption, new optical technologies gave rise to Raman optical activity (ROA), vibrational circular dichroism (VCD), THz time domain (THz-TD), and multiphoton vibrational spectroscopy. However, interpretation of the spectroscopic data

Molecular Spectroscopy: A Quantum Chemistry Approach, First Edition. Edited by Yukihiro Ozaki, Marek Janusz Wójcik, and Jürgen Popp. © 2019 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2019 by Wiley-VCH Verlag GmbH & Co. KGaA.

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obtained from advanced and modern techniques is not easy and straightforward. It should be pointed out that spectroscopic data are very sensitive to inter- and intramolecular interactions. Therefore, analysis of weak interactions is very difficult. Much more reliable quantum mechanical simulations are desirable, allowing one to interpret experimental observations more deeply and model molecular properties in a consistent way. The twentieth century was the age of the characterization of materials, while the twenty-first century is the age of modeling and designing new materials. Modeling desirable molecular properties is not an easy and straight process. Interactions in model systems often involve hydrogen bonds that should be described correctly. For such cases, vibrational spectroscopy is a very powerful and informative method. This was demonstrated during studies of the IR spectra of hydrogen-bonded systems [10–19]. Quantitative reconstruction and interpretation of the complex spectra of hydrogen-bonded systems require reasonable modeling and complete description of the interactions within and between hydrogen bonds and the molecular environment. Molecular dynamics (MD) and static methods give precise information on the nature of hydrogen bonds, which is useful for understanding the molecular properties. Hydrogen bond is one of the most important directional intermolecular interactions. Hydrogen bonding interactions are strong, short-ranged, and highly directional. Their energies lie between a dipole–dipole interaction and a covalent bond. In the last 50 years, many authors classified hydrogen bonds by many factors [31–33]. The most common is classifying hydrogen bonds in three classes: strong, moderate, and weak. Unfortunately, this classification is subjective. However, spectroscopy together with quantum chemistry gave us methods for characterization of the hydrogen bond interactions [34–37]. The main factors used for description of hydrogen bonds are stretching frequency of X—H vibration as well as geometric parameters, such as distance between donor and acceptor and the H—X· · ·Y angle. Theoretical chemistry gave us possibility to calculate these factors. In hydrogen-bonded fluids there exist some properties of hydrogen bonds, which complicate their theoretical description and give rise to a number of macroscopic physical properties that are unique to such fluids. It should be stressed that hydrogen bonds are responsible for the remarkable properties of water, folding of proteins, and self-assembly of many advanced materials. The analysis of hydrogen bond strength is possible using spectroscopy. Shifts of X—H stretching bands involved in the hydrogen bond are connected with the strength of this interaction. Also, their widths and total intensities increase by an order of magnitude. The analysis of vibrational spectra is not so easy because of their complexity. Many bands overlap each other, and their assignments are impossible without theoretical analysis. The main problem for the computational investigation of hydrogen bond phenomena is quantum effects of proton movement. The hydrogen is the lightest atom that can easily tunnel. Taking into account all quantum effects is very complicated. The dynamics of proton motion in hydrogen bonds is determined by a complex interplay of vibrational interactions [38–41]. These interactions are responsible for the complicated structure of the IR and Raman spectra of

12.2 Historical and Theoretical Background

hydrogen-bonded systems [39–41] and for the dynamics of proton tunneling [42, 43]. In this chapter we compare two complementary methods for simulation of vibration spectra: static and dynamic. The first one has been used for a long time. It is based on theoretical quantum mechanical model to analyze proton dynamics in hydrogen-bonded systems. In quantum approach researchers calculate IR frequency not only to compare their results with experimental data but also to find the stationary points on the potential energy surface. The analysis of stationary points as well as transition states in the potential energy surface is valuable especially for understanding catalytic effects. In the second, dynamic approach, the calculations of IR spectra of hydrogen-bonded complexes are based on linear response theory, in which the spectral density is the Fourier transform of the autocorrelation function of the dipole moment operator involved in the IR transitions [44, 45]. Recently Born–Oppenheimer molecular dynamics (BOMD), Car–Parrinello molecular dynamics (CPMD), path integral molecular dynamics (PIMD), hybrid molecular dynamics (QM/MM), and other dynamics methods became very popular [46–48]. They are used to generate trajectories and simulate IR spectra of hydrogen-bonded systems. Each of these methods has some advantages and disadvantages that will be discussed later. In this chapter we also present and compare selected applications of different methods. They are used for studies of the vibrational spectra of different systems, from single molecule to complicated bio-systems. We present several applications for crystals. First, we will present a simulation of IR spectra of the oxalic acid dihydrate crystal, discussed in literature for decades. In this case we will show the deuterium substitution effects described by static as well as dynamic methods. Then, temperature effect in vibrational spectra of vitamin C will be investigated. Afterward, two forms of aspirin crystals will be compared through analysis of their vibration spectra. The last but not least discussed example will be tropolone with unusual dimeric interaction in the crystal structure. The last part of this chapter contains a short summary and indicates possible future directions of applications of various methods in static and dynamic approaches. We discuss also the perspectives of modeling the hydrogen bond interactions by theoretical methods.

12.2 Historical and Theoretical Background Theoretical calculations have been more and more popular as a support for experimental analysis. During the past several years, density functional theory (DFT) has become a promising method in computational chemistry. This method, developed by Kohn and coworkers [49, 50], gave possibility to perform relatively accurate calculations for big systems [51–55]. After that, the expansion of theoretical methods was enormous. Quantum chemical methods are able to predict many important properties, such as electronic structures [56–59], bonding characteristics [60–63], interaction mechanisms [64, 65], optics

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[66–73], magnetic [74–77], and thermal properties [78, 79]. DFT calculations of vibrational spectra, which have been primarily restricted to organic compounds and small polyatomic systems [80–84], were recently used for transition metal complexes [85–101]. Extensively complicated systems, such as metal halides, have been subject of growing interest due to their academic importance and various industrial applications [102–105]. In this point some statement should be done. It is impossible to perform catalytic mechanism analysis without correct and accurate vibrational analysis. Therefore, thermodynamic functions are based on calculations of IR vibrations and intensities. The main problem with the static methods is influence of the environment effects. The vibrational analysis performed for molecules in solvents is extremely complicated. The explicit solvent model in which solvents molecules are included in calculations is exceptionally expensive. The addition of new molecules results in complication of the analysis of spectra. However the vibrational analysis is possible in the implicit solvent model (PCM, COSMO) [106–108]. The interpretation of the stationary points in the potential energy surface changed by some external potentials is still unclear. The molecular dynamics using potentials, based on independent electronic structure calculations, is well established as a powerful tool, especially to investigate many-body condensed matter. The wide and broad description of molecular dynamics technique was an aim of several monographs and reviews [46–48]. The main assumption of adiabatic molecular dynamics is to describe the movement of nuclei in terms of potential energy surfaces designed by electronic energy. The Born–Oppenheimer approach gives us possibility to treat the nuclei and electrons independently. It should be noted that taking into account the quantization of nuclear motion should be done during MD simulation, which requires full quantum treatment. However, full quantum method that includes quantum effects for all atoms is still computationally too expensive to be a “standard” treatment. The calculation of the IR spectra of hydrogen-bonded complexes by MD calculations is based on linear response theory, in which the spectral density is the Fourier transform of the autocorrelation function of the dipole moment operator involved in the IR transition. Recently molecular dynamics was performed by using several approaches that have been used to simulate IR spectra of hydrogen-bonded systems [109, 110]. One of the most popular approach is the BOMD. This method is based on the integration of the classical equation of motion on ab initio molecular potential surface [109–111]. Such approach provides an information about electronic structure from the first principles by using quantum chemistry methods, the same as in static calculations. Ab initio methods may be used to calculate potential energy surface on the fly. The potential surface calculated by ab initio methods is vital for systems with high possibility of breaking and formation of chemical bonds. Such systems are hydrogen-bonded structures, especially with strong hydrogen bonds. It should be stressed that the dynamics of proton motion in hydrogen bonds is determined by a complex interplay of vibrational interactions. These interactions are responsible for the complicated structure of the IR and Raman (R) spectra of hydrogen-bonded systems and for the dynamics of proton

12.2 Historical and Theoretical Background

tunneling. For many such systems ab initio molecular dynamics is a matter of choice. The cost of MD calculations is a cost of ab initio calculations multiplied by number of trajectory steps. The long trajectories are beneficial for spectroscopic investigations, especially in the low-frequency region that represents slow motions. It should be pointed out that time step is an important parameter for an MD simulation. The decreasing time step in numerically calculated trajectory leads to the more accurate simulations. Nevertheless computational cost increases with number of steps. In practice the limit of the maximum time steps is determined by the vibrational period of the mode with the highest energy. The extension of BOMD methods is the CPMD [56] that couples the electronic degrees of freedom with the classical coordinates system by assigning to electrons a fictitious mass. This approximation excludes minimization of the electron wavefunctions at every step in the trajectory. CPMD uses fictitious dynamics to keep the electrons close to the ground state, preventing the need for the costly self-consistent iterative minimization at each time step. In order to stay on the Born–Oppenheimer surface, the time step should be comparatively small. In consequence the computational cost of simulation is relatively small. The costly self-consistent iterative minimization is done only at the first step of the MD simulation. The hybrid methods have been recently developed to analyze the bigger systems, such as proteins. The most common QM/MM method was introduced by Warshel and Levitt in 1976 [112]. The enormous increase of system sizes was a trigger to develop the new method that merges accurate and exact molecular dynamics based on BO approach with the fast and efficient molecular mechanics. These methods are based on the selection of two or more regions: first, treated by quantum mechanics; second, treated by classic molecular mechanics; and others treated in lower level of theory. The main advantage of such approach is the possibility of treatment of chemical reaction in such big systems as proteins and biomolecules. However, an open question is always the selection of size of QM region. Recent study [113] shows that there is no advantage to investigate a big QM region. It should be pointed out that there is small possibility of combining QM and MM regions, which may be done by one of the most popular ONIOM models [114–116]. The hybrid molecular dynamics is more useful for the chemical process analysis than for spectroscopic investigation. However, this method is still expanding. The new QM/MM approaches are developed recently, and new investigations use this method [115]. In the last decade the QM/MM method was successfully applied to understand the selected vibrations of biomolecules. The MD simulations produce trajectories with information about time dependence of some system descriptors. Post-molecular dynamics analysis gave a possibility to analyze the descriptors describing, e.g. electron densities, atomic charges, interaction energies, or nuclei characterization, such as proton potentials. One of the examples is quantization of nuclear motion [117] by snapshot methodology, developed by Mavri, Stare, and coworkers [118–120]. This technique allows us to explore the changes in electronic structure during MD simulations. An approach based on Wannier localization function is

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especially popular [121]. Information about the distribution of Wannier centers that represent pairs of electrons (in close shell calculations) or one electron (in open shell calculations) give us the possibility to discuss changes in interaction character and its direction along MD. The Wannier localization function may be used also for the assignment of the dipole moment, which is useful for crystal structures treated by periodic boundary conditions. This methodology is still pioneering with wide perspectives.

12.3 Applications Hydrogen bonding is an ubiquitous interaction that is important in a diverse range of applications, including inter- and intramolecular interactions, solvation, self-assembly of macromolecules and crystals, and protein folding. However, despite being discovered more than a century ago, the qualifying features of hydrogen bonds require refinement and enhancement. As an example, there have been some new investigations for weak hydrogen bonds between carbonyl methyl groups in polymer structures using molecular dynamics [122]. A variety of different experimental and theoretical evidences are considered characteristic for hydrogen bonding. The analysis of the medium strong intermolecular hydrogen bonds in the 2-hydroxy-5-nitrobenzamide has shown the stepwise difference between static and dynamic approaches [123]. The authors compared the static calculation of dimers present in the crystal structure and performed harmonic analysis of the crystal cell. Subsequently, they run CPMD simulation, and afterward they used obtained trajectories for post-MD analysis. One- and two-dimensional quantization of the proton motion have been done in the instantaneous potentials influenced by the fluctuating environment. At the end, the authors compared calculated spectra using different approaches with the experimental ATR IR spectra. In this work authors focused on the high-frequency region associated with the O—H and N—H stretching modes. That region is very sensitive to the strength of hydrogen bonding. The results of static approximation (frequencies calculated for an isolated dimer) in the harmonic approximation gave poor agreement with the experiment that could not be accepted. The consideration of the crystal field within harmonic approximation slightly improved the results. The significant improvement was obtained by the Fourier transform in the time course of the dipole moment function obtained from the Car–Parrinello trajectories. Further post-MD analysis gave the best agreement with the experiment. Solutions of Schrödinger equations for the snapshots of 1D and 2D proton potentials were used not only for spectroscopic analysis but also for the comparison of the 1D proton potential constructions; see Figure 12.1. The authors have shown that the plain elongation of the O—H bond length gave the best agreement with the 2D treatment and the experimental spectrum. The IR spectra of crystal structures were reconstructed by static calculations as well as ab initio molecular dynamics simulations for the imidazole crystal [124], ascorbic acid [125], and aspirin [126]. These recent ab initio calculations

12.3 Applications

H

H

O

O

O

H 0

x

O

O

φ

O

x 0 Roo/2

Roo/2

S

10.0 1988 cm–1 5.0 0.0

(b)

15.0

15.0 E (kcal/mol)

E (kcal/mol)

15.0

10.0 2286 cm–1

0.0

10.0 2062 cm–1 5.0

5.0

–0.2 0.0 0.2 0.4 0.6 0.8 Internal coordinate x (Å)

E (kcal/mol)

(a)

–0.2 0.0 0.2 0.4 0.6 0.8 Internal coordinate x (Å)

0.0 –0.4 –0.2 0.0 0.2 0.4 0.6 Internal coordinate x (rad)

Figure 12.1 Definition of pathways for one-dimensional proton potentials (a). (b) The corresponding proton potentials with the first two eigenvalues and eigenfunctions. All three proton potentials were obtained from the same snapshot structure. Source: Reprinted (adapted) with permission from Brela et al. [123]. Copyright 2012, American Chemical Society.

show a great potential of this method for spectroscopic investigations of complex systems with hydrogen bonds. The main advantage of this method is that it includes most of factors that play a crucial role in the mechanisms of hydrogen bond dynamics (anharmonicity, couplings between vibrational modes, and intermolecular interactions in crystals). The results reproduce very well the positions of bands and their relative intensities. The biggest deficiency is connected with the half-width reconstruction. Before considering some applications in details, we would like to present the results that show the strength of MD calculations for analysis of complex vibrations. One of the best examples is crystals of water, where water molecules constantly wag, rock, and twist; see Figure 12.2. Gług et al. performed CPMD for analyzing IR spectra of ice Ih and ferroelectric ice XI [127]. The results have shown that librational region exhibits especially large differences in the simulated spectra for two considered forms of ice. They clearly present that theoretical IR spectra of ice forms can be used with success for analyzing experimental data obtained by IR telescopes. They allow to distinguish forms of ice that exist in many phases with different hydrogen bond networks in the universe. In this case IR studies might give us information on the role of molecules in the formation of stars and planets. IR spectroscopy has shown that water, vital molecule for life, is present in space, mostly in its crystalline form – ice. Ice XI, the structural variant of the well-known ice Ih, may exist in conditions present in the interstellar space [128]. 12.3.1 Vibrational Spectra of Strong Hydrogen Bonds in Oxalic Acid Dihydrate Crystal with Isotopic Substitution Effects We discuss the selected applications starting from the oxalic acid dihydrate crystal [129]. Oxalic acid is the substance produced naturally from plants. It has been

333

334

12 Dynamic and Static Quantum Mechanical Studies of Vibrational Spectra a

a

o

b

A′

b

Rock A

Wag

A

B′ B

B

C′ C

Wag

Rock

Twist

C

D′ c

D c

(a)

(b)

D

(c)

Figure 12.2 (a) Subsystems ABCD and A′ B′ C′ D′ in crystal structure of ice XI, (b) three librational modes of water molecule, (c) positive directions of wagging and rocking modes of water molecules for subsystem ABCD. Source: Reprinted (adapted) with permission from Gług et al. [127]. Copyright 2015, Elsevier.

demonstrated that it plays an important role in the prevention of chronic diseases (osteoporosis, obesity). The biological function of oxalic acid in human body is almost unknown, and full understanding of the oxalic acid role has to be investigated [130]. Oxalic acid dihydrate forms white crystals at room temperature with melting point at ∼394 K. The crystal structure of oxalic acid dihydrate was determined many times. The most valuable are neutron diffraction experiments done by Sabine et al. [131] and X-ray measurements by Leiserowitz [132] and Coppens [133]. The unit cell of oxalic acid dihydrate contains two molecules of oxalic acid and four molecules of water. IR spectra of oxalic acid dihydrate have been studied in many experimental and theoretical works. The most interesting is the role of hydrogen bond strengths, which effect is clearly reflected in the hydrogen bond stretching band positions [134–139]. Two types of hydrogen bonds are present in crystal structure; see Figure 12.3. One type connects water molecules as donors and the acid carbonyl groups as acceptors. The second type connects the acid’s hydroxyl groups as the donors to water oxygens as acceptors. This bond is relatively short; at room temperature the O· · ·O distance is 2.499 Å. In this section we present results of ab initio CPMD calculations obtained by Brela et al. [129]. The analysis has been performed for the unit cell of the crystal of oxalic acid dihydrate. The authors calculated vibrational spectra of the oxalic

(a)

(b)

Figure 12.3 Structure (a) and atom labeling (b) in crystal unit cell of oxalic acid dihydrate. The atoms are color coded: carbon, gray; oxygen, red; hydrogen, white. Source: Reprinted (adapted) with permission from Brela et al. [129]. Copyright 2013, Elsevier.

12.3 Applications

III-HB

IV-HB

II-HB

I

I-HB

0

500

1000

1500

2000

2500

3000

3500

4000

ν (cm–1)

Figure 12.4 Infrared spectrum of oxalic acid dihydrate crystal calculated in harmonic approximation. Insert presents fragment of crystal structure of the unit cell. The atoms are color coded: carbon, gray; oxygen, red; hydrogen, white. Source: Reprinted (adapted) with permission from Brela et al. [129]. Copyright 2013, Elsevier.

acid dihydrate using static and dynamic approaches and compared them with the experimental spectra. The great emphasis has been put on the reconstruction of the stretching bands of hydrogen-bonded O—H groups. For this reason the O—H band has been thoroughly examined by DFT methods, CPMD calculations, and the post-MD quantization of nuclear motions. At the beginning of optimization, the geometry of the unit cell of oxalic acid dihydrate was done. After that the frequency calculation for oxalic acid dihydrate in external crystal field has been performed. The calculated spectra are presented in Figure 12.4. It should be emphasized that in these calculations the periodicity of the crystal and thermal fluctuations were taken into account. The vibrational spectrum of the oxalic acid dihydrate crystal has been calculated, analyzed, and compared with experimental data and also with the results of the DFT calculations performed for a crystal cell of the oxalic acid dihydrate. The O—H bands, obtained from CPMD calculations and instantaneous one-dimensional snapshot potentials, extracted from the trajectories, calculated using the Wannier functions [121], were compared with the experimental bands. In this work the authors discussed also the isotopic effects on the spectra. The calculated deuterium shifts of vibrational bands were in good agreement with the experimental data. The results reproduced well the stretching bands of the experimental spectrum. The authors concluded that couplings of similar modes are present in the crystal structure. This investigation proved that CPMD method can be used for analysis of hydrogen-bonded systems. The structure

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12 Dynamic and Static Quantum Mechanical Studies of Vibrational Spectra

1

X H O

O

0

I* (a.u.)

336

1500

1750

2000

2250

2500

2750

3000

ν (cm–1)

Figure 12.5 O—H stretching band contours of oxalic acid dihydrate: black line contour was calculated from individual fundamental vibrational transitions as superposition of Gaussian functions with a half-width of 10 cm−1 . Gray 𝛿 functions represent fundamental vibrational transitions. The red contour represents the spectrum calculated by Fourier transform of the autocorrelation function of the atom’s position obtained from Car–Parrinello trajectory. The bands were normalized. Source: Reprinted (adapted) with permission from Brela et al. [129]. Copyright 2013, Elsevier.

of the O—H stretching band was reproduced well, especially when using the snapshot methodology, as seen in Figure 12.5.

12.3.2

Simulations of Infrared Spectra of Crystalline Vitamin C

One of the most important biochemical compounds is vitamins. Like oxalic acid, they are the natural substances that play a crucial role for the functioning of the human body. The vitamins strongly affect the enzymatic reactions in organisms. There are 13 most essential vitamins, and they can be classified into two kinds: water soluble and fat soluble. Vitamin C (l-ascorbic acid) is water soluble. l-Ascorbic acid has an important role in collagen synthesis reactions. These reactions are important for animals in wound healing and in preventing bleeding from the capillaries. Vitamin C may also act as an antioxidant [140]. It should be emphasized that the function of l-ascorbic acid as vitamin relies not on its antioxidant properties against oxidative stress but upon enzymatic reactions that are stereospecific. Unfortunately the human body cannot synthesize vitamin C. Deficiency in this vitamin causes disease scurvy in humans. Ascorbic acid is widely used as a food additive to prevent oxidation. Pharmacological use of vitamins justifies interest in studying properties of hydrogen-bonded systems present in their structures in order to deepen understanding of their intermolecular interactions [125].

12.3 Applications

c H1 O1

C2 C3

(a)

O3

C4 H3 O4

a I

H4 C5

C1

O2 H2

o

H5 O5

II

H7 C6

II

H8 O6 H6

(b)

I

b

Figure 12.6 Ascorbic acid molecules − labeling of atoms and crystal structure of the ascorbic acid. Source: Reprinted (adapted) with permission from Brela et al. [125]. Copyright 2015, American Chemical Society.

l-Ascorbic acid is a very interesting example of hydrogen-bonded crystal. The unit cell has been built from four molecules, which form hydrogen-bonded network; see Figure 12.6. Molecules of vitamin C form four kinds of hydrogen bonds, from medium strong to weak [125]. This system of hydrogen bonds allows to study a whole range of these important intermolecular interactions. Vitamin C has been widely studied in many theoretical and experimental papers [141–144]. The assignment of vibrational bands of vitamin C was successfully done by static calculations [141]. However, the major problem was to analyze and understand the temperature effects. Experimental study of temperature dependence of band positions in the l-ascorbic acid shows unusual effect: blueshifts with increasing temperature [142]. It should be pointed out that static methods do not describe this temperature effect correctly. To understand this Brela et al. [125] calculated IR spectra associated with the O—H stretching modes that are very sensitive to the strength of hydrogen bonds. The CPMD calculations have been done to generate time course of atom positions as well as dipole moment functions. The anharmonic frequencies were calculated by Fourier transform of these functions. This approach gave reasonably good agreement with the experimental data after inclusion of thermal motion of nuclei; see Figure 12.7. The structure of the experimental spectra shows six peaks: first one corresponds to the broad band with maximum at about 2750 cm−1 , and the second one has the biggest intensity, and it is broad with maximum localized at 3000 cm−1 . The next four peaks are relatively narrow (with the half-widths about 25 cm−1 ), and their maxima are located at 3250, 3330, 3420, and 3500 cm−1 . Differences between spectra calculated at different temperatures were difficult to analyze. Afterward, the authors have assigned each peak in the IR spectrum to hydrogen-bonded O—H groups. Fourier transform of the autocorrelation functions describing atom positions (power spectra), calculated using CPMD for atoms involved in different hydrogen bonds, is shown in Figure 12.8. These methods enabled us to analyze individual hydrogen bonds and their contributions to the spectrum of vitamin C. Bands corresponding to the each hydrogen bond were assigned. Further study has been done through the post-MD analysis, by solving Schrödinger equations for the snapshots of the proton potentials. In this case, the best agreement with the experimental spectra was achieved. The results have shown also double proton minima for the strongest hydrogen bond that

337

Absorbance

12 Dynamic and Static Quantum Mechanical Studies of Vibrational Spectra

0

500

1000

1500

2000

2500

3000

3500

4000

2500

3000

3500

4000

ν (cm–1)

(a) 0.04

0.03 I* (a.u.)

338

0.02

0.01

0.00 0 (b)

500

1000

1500

2000 ν (cm–1)

Figure 12.7 Infrared spectra of L-ascorbic acid. (a) Experimental spectrum and (b) spectrum calculated by the Fourier transform of the dipole moment from the Car–Parrinello trajectory. Source: Reprinted (adapted) with permission from Brela et al. [125]. Copyright 2015, American Chemical Society.

confirms the strength of this interaction. The results of the post-MD analysis have shown the blueshift of band position for each analyzed hydrogen bonds, as seen in Figure 12.9. Brela et al. [125] also discussed the difference between two types of the molecules of l-ascorbic acid molecules present in the crystal. The analysis demonstrated almost negligible impact of different types of the molecules on the positions of bands for strong hydrogen bonds. However, the influence is seen for weaker hydrogen bonds. 12.3.3 Study on Proton Dynamics of Strong Hydrogen Bonds in Aspirin Crystals The next very interesting application is aspirin (acetylsalicylic acid) crystal. Almost all people around the world use aspirin to combat headaches. Aspirin

12.3 Applications

I* (a.u.)

1

2000

2250

2500

2750

3000

3250

3500

3750

4000

ν (cm–1)

Figure 12.8 The power spectra of the atoms in the O—H bonds obtained from Car–Parrinello trajectory. The black color corresponds to hydrogen bond: HB1; red, blue, and green correspond to HB2, HB3, and HB4, respectively. Please see Figure 12.7. Source: Reprinted (adapted) with permission from Brela et al. [125]. Copyright 2015, American Chemical Society.

was discovered as a drug in the eighteenth century. First, it was isolated from plants – the bark of a willow tree. Acetylsalicylic acid was subsequently synthesized in the laboratory. The nineteenth century brought numerous studies on aspirin. Firstly researchers studied the structure and physical properties of aspirin. Later, the scientists concentrate on the design of new drugs [145]. The polymorph forms of crystalline aspirin have been studied widely by the experimental as well as theoretical groups [146–159]. In this work special attention is paid to changes in the interaction network between different aspirin forms I and II. The intramolecular hydrogen bond network is different in form I and form II; see Figure 12.10 [126]. Brela et al. [126] presented full ab initio molecular dynamics of the two types of aspirin crystals. Further, quantization of O—H motions has been done to distinguish the characteristics of hydrogen bond interactions. Proton potentials have been constructed from previously calculated MD trajectories (using BOMD) for supercell, shown in Figure 12.10. In this work the authors attempted analysis of nuclear quantum effects by the time-independent formalism. It was already shown that this methodology is useful and powerful to study the effects of hydrogen bonds on spectroscopic properties [160–163]. The snapshot methodology for quantization of nuclear motion was developed by Mavri, Stare, and coworker [118] and successfully applied to understand hydrogen bond interactions in different phases [6, 122–126, 164]. The posterior quantization has not only a lot of advantages but also many disadvantages. First, this is only an approximation that tries to retrieve lost assumptions. The quantization of nuclear motion should be done during MD simulations, and that would require full quantum treatment. It is not possible. Using wave packets is horribly expensive and tricky especially when we would like to analyze slow motions. Someone may say that a posteriori procedure is

339

12 Dynamic and Static Quantum Mechanical Studies of Vibrational Spectra 1.0

I* (a.u.)

300 K

750 1000 1250 1500 1750 2000 2250 2500 2750 3000 3250 3500 3750 ν (cm–1)

(c) 1.0

I* (a.u.)

150 K

750 1000 1250 1500 1750 2000 2250 2500 2750 3000 3250 3500 3750 ν (cm–1)

(b) 1.5 5K

I* (a.u.)

340

750 1000 1250 1500 1750 2000 2250 2500 2750 3000 3250 3500 3750

(a)

ν (cm–1)

Figure 12.9 The band contours calculated from individual fundamental vibrational transitions obtained from 150 snapshots extracted from trajectory in (a) 5 K, (b) 150 K, and (c) 300 K. Each band is a superposition of Gaussian functions with a half-width of 50 cm−1 . The colors correspond to four kinds of O—H groups: black line corresponds to HB1, red to HB2, green to HB3, and blue to HB4. Source: Reprinted (adapted) with permission from Brela et al. [125]. Copyright 2015, American Chemical Society.

12.3 Applications

Aspirin cyclic dimer

Aspirin form ll

(a)

Aspirin form l

(b)

(c)

Figure 12.10 (a) Structure of cyclic dimer of aspirin. (b,c) Form I and form II of aspirin in the supercell (the double-sized crystal unit cell). The atoms are color coded: carbon, gray; oxygen, red; hydrogen, white. Source: Reprinted with permission from Brela et al. [126]. Copyright 2016, American Chemical Society.

also expensive. Creation for each proton’s potential energy requires to calculate n single points. The considered crystal structure has m hydrogen bonds, which should be analyzed in k steps of trajectory. Thus, quantization methodology involves in total m × k × l single point calculations, which may be more than hundred thousand. However, after MD calculations all single points can be calculated fully independently. The post-MD quantization methodology is fully parallel, and that is its big advantage. Brela et al. clearly demonstrated that the strength of hydrogen bonds is strongly connected with the crystal forms. Moreover, the analysis of the IR spectra and the fluctuation of geometrical parameters along molecular dynamics have been done. The authors connected the interaction in the cyclic dimmers, present in the crystal structures of both forms, with the existence of polymorphs. The results revealed that spontaneous and simultaneous proton transfer occurs in hydrogen bonds in the cyclic dimers. However, only in the form I, two proton minima in the potential function for the hydrogen bond have been observed; see Figure 12.11. The spectroscopic analysis, especially modeling of band contours, shows significant differences in the hydrogen bond strengths.

341

Form l

Form ll

(a)

(a) x

x

y

(c) ΔE (kcal/mol)

200 150 100 50 0 1.9

1.6 1.25

1.25 0.9 0.9 y (Å) 0.6 0.6 x (Å)

1.6

1.9

200 180 160 140 120 100 80 60 40 20

0.8

1

t = 5000 (pm) ν = 2541 (cm–1) 0.8

(c)

1.9 1.6 1.25 0.9 0.6 0.6 0.9 1.25 1.6 1.9 x (Å)

1

1.2 1.4 1.6 1.8 x (Å)

ΔE (kcal/mol)

t = 5000 (pm) ν = 2149 (cm–1) 1.2 1.4 1.6 1.8 y (Å)

ΔE (kcal/mol)

t = 5000 (pm) ν = 2601 (cm–1) 1.2 1.4 1.6 1.8 x (Å)

210 200 190 180 170 160 150 140 130 120 110

210 200 190 180 170 160 150 140 130 120 110

t = 5000 (pm) ν = 2594 (cm–1) 0.8

1

200 200

195

190

190

180

185

170 160 1.9 1.6 1.9 1.6 1.25 1.25 0.9 y (Å) 0.6 0.6 0.9 x (Å)

180 175 170

1.2 1.4 1.6 y (Å)

1.8

1.9 1.6 y (Å)

1

(b)

ΔE (kcal/mol)

0.8

210 200 190 180 170 160 150 140 130 120 110

y (Å)

200 190 180 170 160 150 140 130 120 110

ΔE (kcal/mol)

ΔE (kcal/mol)

(b) 210

y

1.25 0.9 0.6 0.6 0.9 1.25 1.6 1.9 x (Å)

Figure 12.11 (a) Position of second cyclic dimers in form I and form II supercell and direction of proton displacing. (b) One-dimensional proton potentials with the first two eigenvalues and eigenfunctions. (c) The surface plot and contour map of 2D proton potential corresponding to the same snapshots from trajectory as 1D potentials. Source: Reprinted with permission from Brela et al. [126]. Copyright 2016, American Chemical Society.

12.3 Applications

12.3.4

The Hydrogen Bond Dynamics in Crystalline Tropolone

The last discussed application reveals the influence of slow motion on the spectroscopic features and other properties of crystals. The example is tropolone crystal. This molecule is an aromatic non-benzoic compound [165–167] and forms unusual cyclic dimers. The tropolone molecule contains carbonyl and hydroxyl groups attached to aromatic seven-membered ring. This structure is especially interesting because of the high influence of the out-of-plane motions of tropolone ring on the intermolecular hydrogen bonds. Further, this compound has been a very popular model compound for studying tunneling in the ground and excited electronic states [42, 43, 168–180]. In study [181] the authors applied several quantum mechanical methods to understand the relationship between conjugation of 𝜋-electrons with the nuclear motions and its effects in IR spectra. It should be noticed that the intermolecular hydrogen bonds are formed between two aromatic rings, and the influence of 𝜋-electrons on hydrogen bonds must be significant. However, only theoretical analysis of molecular orbitals might give an information on such interaction. Fortunately such analysis may be narrowed to the frontier orbitals (highest occupied molecular orbital [HOMO] and lowest unoccupied molecular orbital [LUMO]) [182]. The main problem solved in this study was the calculation of changes in the frontier orbitals during slow motions. This was solved by the analysis of HOMO and LUMO along BOMD trajectories. This was especially important because of the dynamic character of hydrogen bonds as well as the conjugation between 𝜋-electrons from aromatic ring with the oxygen atoms forming hydrogen bonds. Frontier orbital analysis of one selected structure or average structure obtained from BOMD calculation may not yield all important information. As was already said in previous section, the post-MD quantization methodology, as well as other analysis, e.g. frontier orbital analysis performed along MD trajectories, may be computed by fully parallel treatment. It is a significant advantage while performing calculations in the high-performance clusters. The authors considered supercell containing 18 unit cells. Such system allows to analyze three independent pseudo-dimeric structures present in the crystal of tropolone; see Figure 12.12. The C=O and O—H groups form intermolecular hydrogen bonds, forming pseudo-dimeric structures and coexist with the intramolecular hydrogen bonds. The authors investigated the HOMO and LUMO orbitals of the studied tropolone dimers. Part of the results is shown in Figure 12.13. The analysis of the points from trajectories shows the big fluctuation of the frontier orbitals. The conjunction is observed, when the tropolone dimer structure is planar. Though, the dimers are deformed, the HOMO and LUMO orbitals are localized only in one molecule. The aromatic carbon ring influences hydrogen bonds in dimeric structures only when the structure is planar. The presence of conjunction between these orbitals affects the intermolecular hydrogen bonds more than intramolecular ones. The quantum effects in proton dynamics were also considered for tropolone crystal. The results were compared with the experimental IR spectrum. The

343

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12 Dynamic and Static Quantum Mechanical Studies of Vibrational Spectra

(a)

(b)

(c)

(d)

Figure 12.12 Panel (a) presents the chemical structure of tropolone. Three panels (b–d) show the view along z, y, and x axes, respectively, for tropolone supercell (the tripled size of crystal unit cell along x and y directions and doubled size along z direction). The atoms are color coded: carbon, black; oxygen, red; hydrogen, white. The magnificent oxygen and hydrogen atoms (red balls) contain the analyzed C=O· · ·H hydrogen bonds in three “cyclic dimers.” Source: Reprinted with permission from Brela et al. [181]. Copyright 2018, Elsevier.

Figure 12.13 Results of the analysis of the electronic structure for a selected tropolone dimer obtained from the BOMD simulation of the tropolone supercell, the contour of the highest occupied molecular orbital (HOMO). Source: Reprinted with permission from Brela et al. [181]. Copyright 2018, Elsevier.

calculated band shapes were in reasonable good agreement with the experimental data. It should be pointed out that author’s focus was mainly on the conjunction. A spontaneous delocalization of 𝜋-electrons occurred over oxygen atoms in the cyclic dimers. The conjunction between two aromatic tropolone molecules is strongly connected with the “butterfly” motion of the dimers. The comparison with the experimental spectroscopic data validates the obtained results. Only in the crystal field that pseudo-dimeric weak interaction is stable.

12.4 Summary and Perspectives

12.4 Summary and Perspectives This chapter is aimed to present new applications of ab initio molecular dynamics and static calculations as powerful and complementary tools for simulations and analysis of vibrational spectra of hydrogen-bonded systems. The main advantage of the static methods is the speed and simplicity of its analysis. The simple assignment of band positions may be done after designing stationary points on the potential energy surface. Further analysis may be done because of enormous development of static approaches that include anharmonicity and couplings between motions. Molecular dynamics takes to account the anharmonicity that plays an important role in hydrogen-bonded systems. However, the main advantage of ab initio molecular dynamics is to perform calculations in predefined statistical ensemble, i.e. taking into account thermal effects. Of course, the quality of results depends on proper operations of thermostat. The molecular dynamics simulations give also a possibility to include quantization of motions as well as post-MD analyses. The proton transfer is one of the most important processes in enzymological and biological reactions. It was shown by Warshel and coworker [113] that preorganization and electrostatic field of the neighborhood has a pivotal role for such reactions. In very large systems static methods fail. Correct assignment of the stationary points in multidimensional hyperspace is a difficult task for systems made up of thousands of atoms. There are too many local minima located close to each other. For this reason the choice is molecular dynamics that takes into account many states of the system. In years the popularity of molecular dynamics for simulation of vibronic spectra is increasing. In this chapter we discussed examples concerning several hydrogen bond systems in crystals. However, it should be pointed out that dynamic approaches are naturally more expensive than static. In our opinion, the MD should be applied when it is required for untypical systems, e.g. hydrogen-bonded systems. Recently many studies give deeper interpretations based on MD calculations of spectroscopic properties for complex systems, such as polymers or macromolecules. One of the best examples is the analysis of weak C—H· · ·O=C hydrogen bonds in polyhydroxybutyrate (PHB) [122]. This study has shown the dynamic changes of interactions between acetylic groups and methyl groups. The C—H· · ·O=C interactions have been unstable along MD simulations; the forming process and the braking process occurred at the same time. This effect is not influenced by methyl group rotations and polymer chain movements. It should be pointed out that the consideration of the interaction energy between polymer chains in selected structures is not sufficient to discuss the strength of interchain interactions. Similar conclusions have been drawn from the analysis for two polymorphs of nylon 6 [183]. One of the most promising perspectives is interpretation of new spectral regions. The low-frequency motions, overtone region, and terahertz frequencies are new areas with many open questions that might be answered by MD simulations [184]. Those complex problems include couplings, resonances, and quantum effects that are naturally included in the ab initio MD simulations.

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New powerful supercomputers allow to perform longer simulations. The longer trajectory means lower-frequency regions in our analysis range. However, it seems that the static methods in this case should be more efficient than dynamic approaches. In our opinion the high-frequency region, with analysis of overtones, is a future challenge for analyses using molecular dynamics. At the end we want to stress that spectroscopy is still a developing science. The new challenges are still occurring. Large chemical and biochemical compounds and more multifaceted and complex systems, such as proteins, require new techniques for analysis. The open question is the analysis of the two-dimensional spectroscopies, e.g. 2D IR. We are sure the MD simulations, as well as static approaches, are useful, valuable, and powerful tools that help and strengthen interpretations of vibrational spectra. Combination of these two independent approaches might give some light in the interpretation of new spectroscopic techniques.

Acknowledgment This work was financially supported by the National Science Centre, Poland, grant 2016/21/B/ST4/02102.

References 1 Wójcik, M.J. (2016). Adv. Chem. Phys. 160: 311–346. 2 Brela, M.Z., Boczar, M., Boda, Ł., and Wójcik, M.J. (2018). Molecular

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Dynamics Simulations of Vibrational Spectra of Hydrogen-Bonded Systems, 353–376. Springer. Wójcik, M.J., Gług, M., Boczar, M., and Boda, Ł. (2014). Chem. Phys. Lett. 612: 162–166. Gług, M., Brela, M.Z., Boczar, M. et al. (2017). J. Phys. Chem. B 121: 479–489. Mavri, J., Pirc, G., and Stare, J. (2010). J. Chem. Phys. 132: 224506. Stare, J., Panek, J., Eckert, J. et al. (2008). J. Phys. Chem. A 112 (7): 1576–1586. Flakus, H.T., Hachula, B., and Garbacz, A. (2012). J. Phys. Chem. A 116: 11553–11567. Durlak, P., Latajka, Z., and Berski, S.A. (2009). J. Chem. Phys. 131: 024308–024316. Durlak, P. and Latajka, Z. (2009). Chem. Phys. Lett. 477: 249–254. Hadži, D., Thompson, H.W., and International Union of Pure and Applied Chemistry (1959). Hydrogen Bonding. papers presented at the Symposium on Hydrogen Bonding held at Ljubljana (29 July–3 August 1957). London: Pergamon Press. Pimentel, G.C. and McClellan, A.L. (1960). The Hydrogen Bond. San Francisco: W.H. Freeman; trade distributor: Reinhold Pub. Corp., New York.

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353

13 Quantum Mechanical Simulation of Near-Infrared Spectra: Applications in Physical and Analytical Chemistry Krzysztof B. Be´c 1 , Justyna Grabska 1 , Christian W. Huck 2 , and Yukihiro Ozaki 1 1 Kwansei Gakuin University, Department of Chemistry, School of Science and Technology, 2-1 Gakuen, Sanda, Hyogo 669-1337, Japan 2 Leopold-Franzens University, Institute of Analytical Chemistry and Radiochemistry, CCB-Center for Chemistry and Biomedicine, Innrain 80/82, 6020 Innsbruck, Austria

13.1 Introduction Interpretation of near-infrared (NIR) spectra has often appeared problematic in any case stepping beyond the simplest molecules [1]. The molecular mechanisms standing behind the absorption of radiation in NIR region involve excitations of non-fundamental vibrations, overtones, and combination modes [2, 3]. The primary parameters of the resulting bands, wavenumbers and intensities are ruled by anharmonic effects with inter-mode anharmonicity playing a significant role. The mode–mode couplings and vibrational resonances impose non-straightforward peak shifts and intensity variations. Moreover, the number of non-fundamental vibrations significantly exceeds the number of fundamental ones; this proportion becomes significant for larger molecules [4]. The resulting individual bands overlay strongly, and the observed spectral outline is intrinsically complex as it results from conglomerated contributions. Consequently, broadness and non-homogeneity of experimental bands increase the difficulty of associating the spectral changes with the molecular background, often making the analysis challenging. In the case of infrared (IR or mid-IR; 4000–400 cm−1 ) or Raman spectroscopy, an application of quantum harmonic oscillator, a simplistic model of molecular vibration, most of the time yields a description of molecular vibration adequately accurate to provide a useful picture [5]. Simulation of harmonic spectra (IR, Raman) often in hyphenation with empirical wavenumber scaling can be described as routine approach nowadays. Contrarily, NIR spectroscopy finds no use in harmonic approximation, instead requiring anharmonic calculation schemes. Due to a considerable increase in the demand for computational resources, the examples of NIR spectral simulations have remained rather rare in literature until the theoretical and technological advance has recently made such studies feasible for the molecules extending beyond few atoms in complexity. The theoretical background and practical aspects of that topic have been reviewed by us previously [6, 7]; thus, in the present chapter, these aspects Molecular Spectroscopy: A Quantum Chemistry Approach, First Edition. Edited by Yukihiro Ozaki, Marek Janusz Wójcik, and Jürgen Popp. © 2019 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2019 by Wiley-VCH Verlag GmbH & Co. KGaA.

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will be covered only briefly, while the applications will be focused on the most. As mentioned above, the approximation of molecular vibration as a harmonic oscillator brings significant practical advantages from the computational point of view. The Newton equations of motion for vibration lead to a matrix eigenvalue equation in which the harmonic vibrational frequencies are calculated by diagonalization of the matrix of mass-weighted second derivatives of the potential energy (mass-weighted Hessian). Accordingly, the transition intensities result from the square of the derivatives of the dipole moment with respect to the normal modes. Since Hessian comes as a straightforward output of the geometry optimization step, harmonic frequencies are obtained at a relatively minor cost. The typical overestimation of these values (blueshift) is being commonly addressed with an empirical scaling, thus yielding cost-effective theoretical IR or Raman spectra [8]. While diatomic anharmonicity remains relatively easy to account for, the problem rises in complexity for polyatomic molecules. A number of anharmonic approaches exist; from the point of view of applied spectroscopy, these may be categorized by their accuracy vs. cost balance. Variational calculations are extremely expensive but yield an exact solution, limited only by the accuracy of potential energy evaluation; these are practically applicable to simplest molecules [9]. Applied vibrational spectroscopy requires reasonably efficient anharmonic methods that achieve the ability to compute complex molecules with a controlled penalty of lower quality of description of certain less meaningful factors. Vibrational self-consistent field (VSCF) scheme, which assumes full vibrational wavefunction being factorizable into a set of normal mode wavefunctions, is one of the tools often used for anharmonic simulation of IR spectra [10]. The key feature of the approach is the approximation of intermodal anharmonicity, which accounts for the averaged mode–mode coupling effects; effectively any given mode feels an averaged effect of all other modes. Improved variants have been proposed, i.e. perturbation-corrected VSCF (PT2-VSCF) that uses second-order perturbation theory to correct the VSCF level computations yielding higher accuracy [11]. Interestingly, the loss in accuracy of basic VSCF method decreases with an increasing size of the molecule. The reason lies in an increased averaging following the increasing numbers of modes in the mean-field approximation [12]. Computationally cost effective are the anharmonic methods based on second-order vibrational perturbation theory (VPT2) [13]. In principle, it includes an anharmonic correction in the form of cubic and quartic force constants found by numerical differentiation of the harmonic Hessian at molecular geometries slightly displaced from the equilibrium. VPT2 involves a relatively modest additional computing effort if only bimodal correlations are included. Typically, the applicability of the method has been hindered by its susceptibility to produce meaningless results in case of tightly coupled modes (close degeneracies, i.e. vibrational resonances); this has often invoked customized solutions depending on the molecular system. Efforts have been made to design and implement automated treatment for close degeneracies [14]. Deperturbed (DVPT2) and generalized (GVPT2) evolutions of this approach have appeared recently in which the close degeneracies are identified and removed from the calculations (DVPT2) and reintroduced by means of variational approach (GVPT2) [14]. In effect a robust and general tool

13.2 Overview of the Current Progress in Computational NIR Spectroscopy

has been obtained for anharmonic treatment of even fairly complex molecules. Other anharmonic approaches may be mentioned, e.g. vibrational configuration interaction (VCI) [15] and vibrational coupled-cluster (VCC) [16] methods. Both are computationally highly expensive, although may be found in the literature to be used for simulation of IR spectra of selected simple molecules that feature certain vibrational intricacies [17]. With constant advance in high-power computing, one should see an increasing interest in these methods in the future. A brief note should be made regarding anharmonic calculations of macromolecules and biomolecules. It is known that anharmonic effect play a significant role in some of those cases as some biomolecules, e.g. proteins or nucleic acids, feature strong anharmonicity in the form of low-barrier bond torsions, ring modes in large ring systems, low-energy vibrations in THz region, or hydrogen bond complexes [18, 19]. An even basic and highly approximate account for anharmonicity may grant significant improvements in those cases. As such, these applications are exclusively efficiency oriented to enable feasible studies of large systems [20]. This topic is particularly challenging and has long remained an active area of focused study [12, 20–22]. The NIR spectra simulations so far has mostly been based on either VSCF or VPT2 routes, as particularly the latter one offers very profitable balance between accuracy and demand for computing resources; the examples of these studies will be overviewed beneath.

13.2 Overview of the Current Progress in Computational NIR Spectroscopy 13.2.1

Basic Molecules

Recent years have seen an increasing number of quantum mechanical (QM) simulations of NIR spectra. A number of QM studies on NIR spectra of alcohols have been reported over the last three years [23–26]. These molecules can be considered relatively important for our understanding of their molecular structure [27–31] and conformational isomerism [31, 32]; structure and dynamics of hydrogen bonding; self-association mechanisms and intermolecular interactions [27, 28, 30, 33–36], with highlight of the interactions with solvent molecules (e.g. water, nonpolar solvents); [37, 38] or chiral discrimination [34]; the influences of temperature on the above effects have often been considered [24, 31, 33, 39–42]. Consequently, alcohols have remained among the molecules investigated most frequently in physicochemical NIR studies, although those have often been hindered by the lack of practical availability of spectra simulations. Be´c et al. have recently investigated three basic alcohols (methanol, ethanol and 1-propanol) by employing experimental and theoretical NIR spectroscopy, demonstrating the feasibility of gaining deep insights into the origins of the examined spectra [23]. They have achieved good agreement between the experimental (solution; 3 × 10−5 M in CCl4 ) and calculated spectra including the reproduction of minor bands (Figure 13.1) and the influence of conformational isomerism. This study has well demonstrated the potential of

355

1st ovt. of O—H str.

Experimental

Calculated

Exp.

7200

Calc.

7500 6700

CH3 as. str. + O—H str. CH3 as.′ str. + O—H str. CH3 symm. str. + O-H str.

6200

CH3 as.′ str. + CH3 as. str. 1st ovt. of CH3 as. str. CH3 symm. str. + CH3 as. str. 1st ovt. of CH3 as.’ str. (1) CH3 symm. str. + CH3 as.’ str. and (2) CH3 symm. str. 1st ovt.

5700 5200 Wavenumber (cm–1)

CH3 asym.′def. + O—H str. (1) CH asym.def. + O—H str. 3 (2) and CH3 sym.def. + O—H str. O—H bend (ip) + O—H str. CH3 rock. + O—H str.

4700 4200 3700

Figure 13.1 NIR spectra of diluted methanol; experimental (5 × 10−3 M CCl4 ) and simulated by the use of anharmonic calculations (GVPT2 scheme on DFT-B2PLYP/SNST level of electronic theory and CPCM solvation model of CCl4 ). Source: Be´c et al. 2016 [23]. Reproduced with permission of the PCCP Owner Societies.

Absorbance (a.u.)

Main contributions: 1. CH3 asym.′def. + CH3 as. str. 2. CH3 asym.def. + CH3 as. str. 3. CH3 symm.def. + CH3 as. str. 4. CH3 asym.def. + CH3 symm. str. 5. O—H bend (ip) + CH3 as. str. and more O—H bend (ip) + CH3 symm. str. (?) CH3 rock.′ + CH3 as. str. Main contributions: 1. CH3 rock.′ + CH3 symm. str. or CH3 rock.′ + CH3 as. str. 2. CH3 rock. + CH3 as.’ str. 3. C—O str. + CH3 as.’ str. 4. C—O str. + CH3 symm. str. and more

C—O str. + O-H str.

or

13.2 Overview of the Current Progress in Computational NIR Spectroscopy Wavelength (cm–1) 8000

Absorbance Methanol, dilute in CCl4

9000

7000

6000

5000 νOH + δOH

2νOH (nonbonded)

2νaCH3

2νsCH3

νOH + δCH

4000 νaCH3 + δCH3

νOH + νCO

νsCH3 + δCH3

νOH + diffuse δOH

1100 1200 1300 1400 1500 1600 1700 1800 1900 2000 2100 2200 2300 2400 2500 Wavelength (nm)

Figure 13.2 Band assignments of methanol diluted in CCl4 determined through the usage of classical spectroscopic methods of analysis. Source: Weyer and Lo 2002 [1]. Reproduced with permission of John Wiley & Sons.

(t) 15+1

(g,t) 16+1

(g) 15+1

trans gauche (g) 14+1

(g,t) 13+1

(g) 12+1

(g) 10+1, (g) 11+1

(g) 9+1

(g) 7+1

Absorbance (a.u.)

(t) 10+1

NIR spectra simulation by QM calculations in explaining the spectra forming factors (Figure 13.1) far surpassing those that have been available through the use of classical spectroscopic methods even for a relatively uncomplicated molecule such as methanol (Figure 13.2). Ethanol and 1-propanol on top of featuring more vibrational modes additionally have conformational isomers that give distinct spectral signatures (Figure 13.3). Be´c et al. have showed that with QM simulations it is possible to unambiguously ascribe the band contributions in NIR region, which arise from different conformational isomers [23]. Their simulation accurately reproduced the band-shape details of the 2𝜈OH band as

Exp.

Calc.

5200

5100

5000

4900 4800 –1 Wavenumber (cm )

4700

4600

Figure 13.3 Band assignments in the experimental and calculated NIR spectra of low concentration (5 × 10−3 M CCl4 ) ethanol. The calculated NIR spectrum is based on the CPCM-B2PLYP-D/SNST level of theory. Details of the 5200–4600 cm−1 region. Source: Be´c et al. 2016 [23]. Reproduced with permission of the PCCP Owner Societies.

357

358

13 Quantum Mechanical Simulation of Near-Infrared Spectra

well, where homogeneous peak of methanol is clearly set apart from those of ethanol and 1-propanol due to their conformational flexibility. The simulated spectra reflected well an increasing complexity of the 2𝜈OH band shape due to separation of the band contributions stemming from different conformers. Therefore, the potential of comprehensive explanation of the observed spectra with aid of theoretical calculations has been shown [23]. Be´c et al. in their study of simple alcohol molecules have used deperturbed/generalized second-order vibrational perturbation theory (DVPT2/ GVPT2) in connection with several levels of theory for approximation of the electronic structure of the investigated molecules [23]. A number of basis sets have been considered as well as a selection of three different solvation models within self-consistent reaction field (SCRF) formalism. This addition allowed evaluating a number of approaches according to their feasibility in accurate reflection of NIR transitions; similar comparisons assembled for fundamental (IR) modes are frequent in literature yet remain very rare for non-fundamentals. The general conclusion has been that density functional theory (DFT) methods should be prioritized in applied spectroscopic studies. DFT offers good balance between the final accuracy and computational cost, the latter being a significant factor since anharmonic vibrational analysis involves a substantial resource expense on itself (Table 13.1). Post-Hartree–Fock methods, i.e. second-order Møller–Plesset perturbation (MP2) scheme, have been found to be of limited feasibility for this purpose, both due to the final accuracy and computing time. Selection of DFT functional should follow the specificity of the molecular system and its complexity. For studying molecules in solution, B3LYP and B2PLYP functionals can be recommended, especially with the addition of empirical correction for dispersion, which largely improves the description of non-covalent and long-range interactions. It has been reported that double-hybrid functionals, such as B2PLYP, offer remarkable consistency in the derived NIR frequencies albeit when the time effectiveness of the simulation is prioritized; single-hybrid B3LYP is able to deliver good results as well. A more detailed overview on the dependence of the computational time on the selected level of theory used for VPT2 calculation has been reported by the same authors as well [7]. Calculations based on B2PLYP functional proved to be roughly twice as expensive as those with B3LYP; selecting triple-𝜁 SNST basis set over a small double-𝜁 N07D basis set has been over two times as expensive, at least in the case of methanol molecule. The addition of DFT empirical dispersion correction and implicit solvation model (CPCM, IEF-PCM, or SMD) introduces only a meager overhead (Table 13.1). Appending these two kinds of supplementary calculation steps should be recommended therefore; the former is applicable in general, and the latter to the studies of molecules in solution [23]. To explain the role that temperature plays in affecting the structure and hydrogen bonding of alcohols and similar molecules has been a major topic of physicochemical NIR studies for many years. Very early it has been observed that 2𝜈OH band undergoes a temperature-induced spectral shift in the NIR spectra of diluted alcohols. Furthermore, the band-shape changes have been monitored by the use of second derivative and two-dimensional correlation analysis (2D-COS); the latter powerful technique has often been used in

13.2 Overview of the Current Progress in Computational NIR Spectroscopy

Table 13.1 An exemplary comparison of total computational time for methanol molecule (including geometry optimization, harmonic calculations, and VPT2 treatment). Method

Solvent model

CPU timea)(s)

Wall timea)(s)

Wall time relative ratio

B3LYP/6-31G(d,p)

—

117

353

1

B3LYP/6-31G(d,p)

CPCM

135

375

1.1

B3LYP/6-31G(d,p)

IEF-PCM

147

409

1.2

B3LYP/6-31G(d,p)

SMD

151

422

1.2

B3LYP/N07D

—

157

466

1.3

B3LYP/SNST

—

246

1 345

3.8

B3LYP-D3/SNST

—

246

1 351

3.8

MP2/SNST

—

305

878

2.5

B2PLYP/N07D

—

370

1 060

3.0

B2PLYP/SNSD

—

660

1 860

5.3

B2PLYP/SNST

—

886

2 534

7.2

B2PLYP-FC/SNST

—

887

2 448

6.9

B2PLYP-D/SNST

—

891

2 520

7.1

B2PLYP/SNST

CPCM

928

2 515

7.1

B2PLYP-D/SNST

CPCM

938

2 644

7.5

B2PLYP/SNST

SMD

957

2 580

7.3

B2PLYP/SNST

IEF-PCM

971

2 623

7.4

MP2/aVTZ

—

2 512

6 553

18.6

MP2/aVQZ

—

46 083

119 288

337.9

a) The CPU time and wall time depend on the hardware platform. The presented values are for 24 core Intel Haswell architecture computing node. Source: Be´c et al. 2016 [7]. Reprinted with permission of SAGE Publications.

conjunction with NIR spectroscopy. The studies of butyl alcohols have been helpful in yielding insights into the molecular background standing behind the observed spectral variations. The 2𝜈OH band-shape change accompanies the peak shift in the case of 1-butanol, 2-butanol, and iso-butanol; however, tert-butanol spectrum features only the peak shift. Accordingly, the four kinds of butyl alcohol differ clearly in their structure; while 1-butanol, 2-butanol, and iso-butanol feature different levels of conformational flexibility, tert-butyl alcohol remains effectively inflexible. These observations clearly indicated that the conformational isomerism should be expected to manifest itself in the temperature-dependent NIR spectra of alcohols; yet no decisive explanations have been available at that time. Consequently, the investigation of NIR spectra of aliphatic alcohols supported by QM simulations has been continued by Grabska et al. with their focus on butanols [24]. They have attempted a theoretical reproduction of the temperature-induced spectral variations in NIR region of these molecules. To balance the final accuracy and computational cost in the case of these larger molecules, a hybrid approach to the vibrational analysis has been applied; the harmonic and anharmonic parts have been performed

359

360

13 Quantum Mechanical Simulation of Near-Infrared Spectra

on different levels of electronic theory (B2PLYP/def2-TZVP and B3LYP/SNST, respectively) to maximize the overall efficiency of the calculation. The resulting simulated NIR line shapes have remarkably resembled the experimental ones (Figure 13.4) including the reproduction of fine effects, i.e. shapes of minor bands in the 5200–4500 cm−1 region for all examined molecules (Figure 13.4). This has yielded detailed band assignment and full comprehension of the NIR spectra of diluted butanols [24]. In the following part of their study, Grabska et al. have considered the impact of the temperature on the conformational population of butyl alcohols [24]. By calculating Boltzmann coefficients corresponding to the abundances of all conformational isomers of 1-butanol, 2-butanol, and iso-butanol and subsequent incorporation of the obtained values in the spectra simulation process, they have succeeded in reproducing the temperature-dependent spectral shift and band-shape changes observed experimentally, origins of which have not been decisively explained before. These variations have been monitored by the use of 2D-COS analysis and good agreement between the simulated and experimental 2D plots. This comparison has allowed confirming that the relative changes in the conformational populations at least contribute partially to the observed spectral variability. This work has well demonstrated the usefulness of QM simulations in bringing definite answers to the problems that have often been difficult to classical NIR spectroscopy [24]. A separate thread in the context of NIR spectroscopy of alcohols has been explored by Be´c et al. in their examination of 1-hexanol, cyclohexanol, and phenol [26]. The study has focused on the spectra–structure correlations resulting from the NIR vibrations of an OH group attached to a molecular skeleton of three different kinds: linear and cyclic aliphatic, and aromatic ring. It is well known that the bands due to X–H vibrations, and OH ones in particular, are significantly augmented in NIR spectra. Therefore, OH vibrations are one of the major spectra forming factors, and examination of how the molecular structure affects the NIR spectrum in this context brings important knowledge in the field [26]. The above molecules feature noticeable dissimilarities in their NIR spectra: that of 1-hexanol remains coherent with shorter chain linear alcohols; however, those of cyclohexanol and especially phenol reveal substantial differences (Figure 13.5). On top of detailed band assignments, an elucidation of clear trends in spectra forming factors corresponding to the appearance of combination bands of 𝜈OH mode could have been established. This mode couples strongly to a number of other modes and gives decisive spectral signature in the 5500–4000 cm−1 region (Figure 13.5). The specificity of phenol ring modes, a relatively good separation of the fundamental bands in IR, also protrudes onto NIR region in which well-resolved sharp peaks appear throughout lower frequencies (5500–4000 cm−1 ), which can be considered an uncommon feature in NIR spectra, even among uncomplicated molecules such as methanol (Figure 13.1). The observed signature allows easily discriminating different kinds of alcohols (aliphatic, aromatic) and identifying an OH group attached to an aromatic ring [26]. One can notice that the robustness of VPT2 simulation of spectra appears lower in the case of 2𝜈OH peak, apparent, e.g. in the case of cyclohexanol (Figure 13.1a); the reasons of this are known and will be discussed in detail later in this chapter.

νCH + δCH

2νOH

νOH + δCH νCH + δCOH νCO + νCH

2νCH2(s) 2νCH3(w) νCH + νCH

Absorbance (a.u.)

Absorbance (a.u.)

Exp. Final calc. TGg TGt TTg TGg′ TTt GGg GGt GGg′ GTg′ GTg GTt G′Gg GG′t GG′g

νOH + δCH νOH + δCOH νOH + νCO

Exp. Final calc. G-t G-g+ G-gTg+ TT TgG+g+ G+t G+g-

νOH + δCH 2νCH 2ν δipH(CCCO) νOH + δ H(CCCO) oop νCH + νCH δipH(CCCO) + νCH νOH + νCH

νOH + νCH

8000 (a)

7500

7000

6500

νCH + δCH δH(CCCO) + νCH νCH + τCC δH(CCCO) + δH(CCCO) δCCC + νCH νOH + τCC

2νOH

6000

5500

Wavenumber (cm–1)

5000

4500

4000

8000 (b)

7500

7000

6500

6000

5500

5000

4500

4000

Wavenumber (cm–1)

Figure 13.4 Experimental and simulated (harmonic: B2PLYP/def2-TZVP; VPT2: B3LYP/SNST; CPCM) NIR spectra of butyl alcohols. (a) 1-Butanol; (b) 2-butanol; (c) iso-butanol; (d) tert-butyl alcohol. The contributions of the spectral line shapes corresponding to conformational isomers presented as well (colored lines). Source: Reprinted with permission from Grabska et al. [24]. Copyright 2017, American Chemical Society.

Absorbance (a.u.)

2νOH

δCCH + δCCH νCH + τCC νCH + τCO νCO + τCC

νCH + δCH δCCH + νCH δCCH + τCC δCCH + δCH δCH + τCC

Exp. Final calc.

νCH3 + νCO νCH3 + δCH3 δCOH + νCH3

νCH + δCH 2νCH νCH + νCH νCH + τCC νCH + δCCH νCH + νCH τCC + δCCH δCCH + δCCH

νCH3 + νCC

2νOH

Absorbance (a.u.)

Exp. Final calc. Gg′ Gg Gt Tg Tt

νOH + δCH νCH + δCCH νCO + τCO

νOH + νCH νOH + τCC

νOH + δCH3 νOH + δCOH νOH + νCO 2νCH3 νCH3 + νCH3 νOH + νCH3

8000 (c)

7500

Figure 13.4 (Continued)

7000

6500

6000

5500

Wavenumber (cm–1)

5000

4500

4000

8000 (d)

7500

7000

6500

6000

5500

Wavenumber (cm–1)

5000

4500

4000

13.2 Overview of the Current Progress in Computational NIR Spectroscopy

15 16 14 1719 18 13

1 12 8 7 9 11

Absorbance (a.u.)

3 2 4

10 5 6

Exp.

Calc.

7500

7000

6500

6000

5500

5000

4500

4000

–1

(a)

Wavenumber (cm ) 1

16 17

15

Absorbance (a.u.)

7 8 10 2 6

3

9 13

4 5

11 12 14

Exp.

Calc.

7500

(b)

7000

6500

6000

5500

5000

4500

4000

–1

Wavenumber (cm )

Figure 13.5 Experimental (0.2 M; CCl4 ) and simulated (VPT2//B3LYP/SNST+CPCM) NIR spectra of (a) cyclohexanol and (b) phenol. Source: Be´c et al. 2018 [26]. Reprinted with permission of Elsevier.

13.2.2

Investigations of Intermolecular Interactions and Biomolecules

NIR spectroscopy offers unique advantages for investigation of intermolecular interactions, e.g. properties of hydrogen bonding; accordingly, this has been one of the most focused fields therein [27, 28, 33–38]. Again, the added value that QM simulations have brought into IR and Raman spectroscopies in this context

363

13 Quantum Mechanical Simulation of Near-Infrared Spectra

has remained underdeveloped in the case of NIR spectroscopy due to practical difficulties discussed earlier above. One of the key systems for studies of hydrogen bonding is the cyclic dimer of carboxylic acids, i.e. formic acid or acetic acid. Be´c et al. have recently presented a combined experimental and computational study of acetic acid (CCl4 solution) in NIR region [43]. They focused on spectroscopic properties of the cyclic dimer, which can be observed throughout wide region of concentration. The simulation has reproduced the majority of experimental NIR bands; however, a prominent exception has been noticed. The calculated binary combinations of the stretching and bending OH modes have appeared as strong transitions, leading to the appearance of two sharp and well-resolved peaks in the simulated spectrum; yet, these have been missing in the experimental line shape. Instead, a prominent baseline elevation can be noticed throughout the majority of NIR region (6500–4000 cm−1 and below). The conclusions drawn in the study has been that the aforementioned combination bands undergo a spectral shift and broadening as the result of hydrogen bonding; similar effects are well researched in IR region. To reflect the baseline contribution the simulated bands have been fitted to the experimental spectrum, giving much improved agreement (Figure 13.6). Basing on that, a possible effect of hydrogen bonding on an NIR spectrum has been suggested [43]. A good understanding of NIR spectral properties of simple carboxylic acids has been helpful in the exploration of fatty acids. These key biological molecules can be categorized into short-chain fatty acids (SCFAs), medium-chain fatty acids (MCFAs), and long-chain fatty acids (LCFAs). Fatty acids feature a number of interesting properties from the point of view of physical chemistry, e.g. association mechanisms and hydrogen bonding properties [44–46]; these molecules are Exp. Mod. spectrum Comb. band 1 Comb. band 2 Absorbance (a.u.)

364

7000

6500

6000

5500 5000 Wavenumber (cm–1)

4500

4000

Figure 13.6 Experimental (solution; CCl4 ) and modeled spectrum of acetic acid. Band fitting results for the two combination bands involving OH stretching modes of acetic acid cyclic dimer. Source: Reprinted with permission from Be´c et al. [43]. Copyright 2016, American Chemical Society.

13.2 Overview of the Current Progress in Computational NIR Spectroscopy

H

H C O C H

H

C O

C H H

H

(a)

C C H

(c)

H C

O

H C

H

C O

H

(d)

O C

H

H

C O

H

(e)

H O C O H

(b)

H H

H H C C H

H

H

C

C H

C H

O C O H

Figure 13.7 Molecular structures of the major conformational isomers of the studied SCFAs: (a) propionic acid, (b) butyric acid, (c) acrylic acid, (d) crotonic acid, and (e) vinylacetic acid. Source: Reprinted with permission from Grabska et al. [50]. Copyright 2017, American Chemical Society.

also highly relevant in applied NIR studies, e.g. hyperspectral imaging [47–49]. Selected SCFAs and MCFAs have been investigated by Grabska et al. in their two subsequent studies [50, 51]. In the first one, five kinds of SCFAs (saturated: propionic and butyric acid; unsaturated: acrylic, crotonic, and vinylacetic acid) have been examined [50]. These carboxylic acids are only moderately more complex than acetic acid; thus they have served as good subjects for verifying how much do they retain the specificity of the simple carboxylic acids and what is the impact on NIR spectra of the aliphatic chain structure and the existence of C=C bonds [50]. Grabska et al. have selected the objects of their study in such a way that a set of principle structural features could be investigated (Figure 13.7): the difference between saturated and unsaturated SCFAs, impact of the location of C=C bond (medium-chain, crotonic acid; terminal, acrylic and vinylacetic acid), and exclusive existence of either of the three following structural features: methyl (crotonic acid), sp3 (vinylacetic acid) or sp2 (terminal; acrylic acid and vinylacetic acid), and methylene group [50]. The spectra simulation based on VPT2 anharmonic vibrational analysis and B3LYP/SNST+CPCM level of electronic theory involved full conformational analysis for each of SCFAs [50]. The achieved agreement with the experimental spectra measured in solution phase (0.05 M; CCl4 ) has been very good (Figure 13.8). The baseline elevation phenomenon reported in the case of acetic acid has been observed again in the case of SCFAs [43]. The calculated intensities of the combination bands of two kinds have been orders of magnitude higher than the other ones (Figure 13.8; left column). These combination bands (a + b and a + c) involve the following modes of the hydrogen-bonded OH groups: (a) out-of-phase (or opposite phase) stretching and (b) in-plane bending modes, each time combined with (c) in-phase stretching mode. The band fitting procedure attempted first in the study of acetic acid has been applied in the investigation of SCFAs as well. It has brought much higher agreement with the experimental spectra (Figure 13.8; right column), reflecting the absence of

365

13 Quantum Mechanical Simulation of Near-Infrared Spectra

Absorbance (a.u.)

Absorbance (a.u.)

Exp. Calc. I II

Exp. Calc. 7500

7000

(a)

6500

6000

5500

5000

4500

4000

7500

7000

6500

Wavenumber (cm–1)

6000

5500

5000

4500

4000

4500

4000

4500

4000

Wavenumber (cm–1)

Absorbance (a.u.)

Absorbance (a.u.)

Exp. Calc. I II

Exp. Calc. 7500

7000

(b)

6500

6000

5500

5000

4500

4000

7500

7000

6500

Wavenumber (cm–1)

6000

5500

5000

Wavenumber (cm–1)

Absorbance (a.u.)

Exp. Calc. Absorbance (a.u.)

366

I II

Exp. Calc. 7500

(c)

7000

6500

6000

5500

5000

Wavenumber (cm–1)

4500

4000

7500

7000

6500

6000

5500

5000

Wavenumber (cm–1)

Figure 13.8 Experimental and simulated NIR spectra of MCFAs: (a) propionic acid, (b) butyric acid, (c) acrylic acid, (d) crotonic acid, and (e) vinylacetic acid. Left column: simulated spectra assembled from the raw modeled bands. Right column: the final simulated spectra after band fitting procedure applied to the two combination bands (details in the text). Source: Reprinted with permission from Grabska et al. [50]. Copyright 2017, American Chemical Society.

several intensive sharp peaks (a + b, a + c arising from various conformational isomers in the raw calculated spectra) (Figure 13.8; left column) and instead accurately reproducing the baseline elevation (Figure 13.8; right column). Grabska et al. have succeeded in elucidating consecutive trends throughout several spectral subregions in the NIR spectra of SCFAs. The characteristic discrepancies in overtone and combination contributions have been pointed out; the observed spectral contribution of the CH3 group have remained in full agreement with previous assumptions made in the literature on the basis of experimental data. For the unsaturated compounds the location of the C=C bond has been concluded to impose a clear impact on the corresponding NIR

13.2 Overview of the Current Progress in Computational NIR Spectroscopy

Absorbance (a.u.)

Absorbance (a.u.)

Exp. Calc. I II

Exp. Calc. 7500

7000

(d)

6500

6000

5500

5000

4500

4000

7500

7000

6500

6000

5500

5000

4500

4000

4500

4000

Wavenumber (cm–1)

Wavenumber (cm–1)

Absorbance (a.u.)

Absorbance (a.u.)

Exp. Calc. I II

Exp. Calc. 7500

(e)

7000

6500

6000

5500

5000

Wavenumber (cm–1)

4500

4000

7500

7000

6500

6000

5500

5000

Wavenumber (cm–1)

Figure 13.8 (Continued)

spectrum; in example, appearance of sp2 CH2 group leads to the rise of very specific, well-defined, and intense NIR bands, at 6172/6131, 4746/4734, and 4483/4489 cm−1 for acrylic/vinylacetic acids, respectively (Figure 13.8). It has been suggested that these bands are excellent structural markers due to their high intensity and location in wavenumber regions where no overlapping with the bands of other structures is probable [50]. The above study has been continued with focus on two MCFAs, saturated hexanoic and unsaturated sorbic acid [51]. These molecules feature distinct differences in their NIR spectra, apparent both in very high dilution (where the self-association is insignificant) and also in more concentrated solution (where the spectral features of the cyclic dimers dominate; Figure 13.9). In both these cases, the spectra simulation has given accurate results, and good explanation of the observed features could have been provided. It has been evidenced that the principal shape of the NIR spectra is preserved over wide range of concentrations and even in neat liquid (hexanoic acid) and powder (sorbic acid) [51]. The reported findings have been consistent with those established in the earlier studies of acetic acid and SCFAs [43, 50]. These three investigations [43, 50, 51] have opened a new lane of research in NIR biospectroscopy as accurate NIR simulation of small- and medium-sized biomolecules has become feasible and may develop into the area of more complex ones (e.g. LCFAs, lipids, proteins, nucleic acids) in the foreseeable future. The achievements of theoretical NIR spectroscopy so far could have been used successfully in the interpretation of the data derived from NIR imaging of biosamples [49].

367

13 Quantum Mechanical Simulation of Near-Infrared Spectra

11

Absorbance (a.u.)

10

3 5 4 6 2

1

7

8

12 13 14 18 1517 16

9

Exp.

Calc. 7500

7000

6500

(a)

5500

6000

5000

4500

4000

Wavenumber (cm–1)

16 15 17 18 21 19 20

Absorbance (a.u.)

368

14 12 4

6

8

3 7 5 2

1

1113 10 9

Exp.

Calc. 7500

(b)

7000

6500

6000

5500

Wavenumber

5000

4500

4000

(cm–1)

Figure 13.9 Band assignments proposed for NIR spectra of MCFAs in medium to high concentration (CCl4 ). (a) Hexanoic acid. (b) Sorbic acid. Source: Grabska et al. 2017 [51]. Reprinted with permission of Elsevier.

13.2.3 Connecting the Link Between Theoretical and Analytical NIR Spectroscopy The context of bio-relevant molecules also appears in analytical NIR spectroscopy that is being widely used in quantitative analysis of various types of samples, e.g. raw materials, intermediate, and final products including those of natural

13.2 Overview of the Current Progress in Computational NIR Spectroscopy

origin [52, 53]. The analysis based on NIR spectroscopy in hyphenation with multivariate analytical (MVA) tools (or chemometrics) allows taking advantage of the rich spectral information carried out in an NIR signal in order to correlate it with the sample content [54]. However, the entire procedure is effectively done in a black box, in which the molecular background of the analyzed spectrum is not considered whatsoever. Out of necessity the analytical NIR spectroscopy has been forced to function upon such hindrance, although ideas of incorporating chemical band assignments into chemometric procedures in order to improve the analytical performance for detection/quantification have been reported in literature [55]. In recent years analytical NIR investigations on few occasions have attempted to adopt QM simulation of NIR spectra in order to gain benefits from the physicochemical insights that these provide, effectively supplementing the analytical studies [56–59]. One of the first examples of such joint study has been published by Schmutzler et al. in 2014 in their report of the analytical pathway based on NIR spectroscopy in combination with multivariate data analyses for nondestructive quality control of apples [56]. The article has brought a significant analytical impact in the field of food quality control, as it has reported on a fully automatable, nondestructive device analyzing apples utilizing fiber probe and a surface scanning procedure. The present review will focus on highlighting the novelty introduced in this work in the department of combining the QM spectra simulation and analytical NIR spectroscopy resulting in computer-aided analytical spectroscopy. Malic acid, a dicarboxylic acid that can be found on the outward of apples, is highly relevant for the analytical vibrational spectroscopy of these fruits [56]. Schmutzler et al. have employed anharmonic vibrational analysis of malic acid by means of VSCF method; in specific, a second-order perturbation-corrected scheme (PT2-VSCF) has been employed for improved description of intermodal anharmonicities [56]. Møller–Plesset MP2 method coupled with 6-31G(d,p) basis set has been used; both l- and d-malic acid molecules have been considered, and an explicit approximation of aqueous solution has been involved in calculation in the form of CPCM solvation model of water. The resulting calculated wavenumbers and intensities of NIR transitions of up to three quanta (second overtones and ternary combinations) in comparison with the experimental spectrum are depicted in Figure 13.10; the experimental spectral patterns have been reproduced in a qualitatively correct manner in the simulation. Accordingly, a number of band assignments have been provided. Nevertheless, the authors have concluded that the simplifications they have assumed (quartic force field approximation, relatively simple basis set, and explicit solvent cavity model) may have impacted the final accuracy of the simulation. Moreover, an excessive computational cost of PT2-VSCF approach and its unfavorable scaling with the molecular complexity have been pointed out as considerable hindrances in the studies of larger molecules [56]. Nevertheless, in the case of simpler molecules, PT2-VSCF calculations may successfully be used in supporting role of analytical NIR spectroscopy. Lutz et al. have employed this computational scheme in the development of content quantification procedure of gasoline by means of miniaturized NIR spectroscopy [57]. The study has required a consideration of few types of molecules, representing the chemicals resident in gasoline. The

369

13 Quantum Mechanical Simulation of Near-Infrared Spectra

FT-NIR spectrum of aqueous L-malic acid

50 40 30 20 Absorbance (a.u.)

370

10 0 PT2-VSCF spectrum of aqueous L-malic acid

50 40 30 20 10 10000

9000

8000

6000 7000 Wavenumber (cm–1)

5000

4000

Figure 13.10 The experimental FT-NIR spectrum of aqueous malic acid in comparison with the PT2-VSCF-derived line spectrum. Source: Schmutzler et al. 2013 [56]. Reprinted with permission of Nova Science Publishers.

selection involved n-octane (representing straight chain alkanes), toluene (representing aromatic and branched/alkyl-substituted aromatic hydrocarbons), ethyl tert-butyl ether (representing ethers and branched/alkyl-substituted aliphatic hydrocarbons), and ethanol. These molecules remain sufficiently uncomplicated to avoid the necessity of extensive simplifications in anharmonic analysis based on PT2-VSCF scheme. Accordingly, the authors have employed MP2 method together with TZVP basis set, considerably larger one than that which has been used in the study of malic acid. The QM simulation supplemented the analytical study with complex band assignments and allowed qualitative discrimination of the various contents of gasoline [57]. Analytical NIR spectroscopy is becoming one of the key tools of quality control in the field of phytopharmaceutical industry [52, 53]. Natural samples require highly robust methods of analysis due to complex and varying content. Therefore, the additional and independent insight provided by spectra simulation represents a considerable value therein, i.e. enabling qualitative assessment of the analyzed data or better understanding of the chemometric regression vectors. Kirchler et al. have demonstrated the potential of combining spectra simulation into analytical NIR spectroscopy in their articles reporting on feasibility of miniaturized NIR spectroscopy in the quantification of rosmarinic acid (RA) in Rosmarini folium [58]. RA is the key active compound in a number of traditional plant medicines and features therapeutic and antioxidant properties. It is a relatively large molecule counting 42 atoms, which imposes difficulties in the process of spectra simulations. Vibrational analysis requires a considerable computational resource and time in order to yield anharmonic force field; e.g. intermodal

13.2 Overview of the Current Progress in Computational NIR Spectroscopy

Calculated

1

Experimental

Absorbance (a.u.)

Calc.

2

16 13 15 12 3 11 6

20

17

18 19

10 9 8

5 4

14

7

Exp.

8000

7500

7000

6500

6000

5500

5000

4500

4000

Wavenumber (cm–1)

Figure 13.11 The experimental (powder) and theoretical NIR spectrum of rosmarinic acid obtained in anharmonic GVPT2//DFT-B3LYP/N07D simulation. Source: Kirchler et al. 2017 [58]. Reproduced with permission of The Royal Society of Chemistry.

couplings needed to be evaluated at least for the number of 7140 binary combinations to yield an approximation of the NIR spectrum of RA. For this task VPT2 route at B3LYP/N07D level of theory has proved to deliver good result (Figure 13.11). The chosen functional/basis set offers a very good cost efficiency, and as demonstrated in the case of RA, it may be used in anharmonic simulations of large molecules yielding qualitatively correct result. Even with the simplified treatment of RA molecule in vacuum, the impact of intermolecular interactions could be identified in the experimental spectrum, mostly leading to 2𝜈OH band broadenings; the previously researched effects that lead to the baseline elevation in an NIR spectrum have been helpful in verifying the agreement with the experimental band of RA [58]. The obtained results have allowed to comprehend fully the NIR features of RA, and a set of detailed band assignments have been proposed (Figure 13.11, Table 13.2). Accordingly, Kirchler et al. have used these findings for the qualitative assessment of the partial least squares (PLS) regression coefficient vectors derived for the spectral series measured on different NIR spectrometers including portable devices [58]. The calibration models resulting from the usage of different instruments lead to different coefficient vectors emphasizing discrepant wavenumbers as the most influential in the regression. Kirchler et al. have employed a combined approach involving advanced data analytical methods (i.e. heterocorrelation 2D spectroscopy) and QM simulations in order to better explain this variability and find which RA bands impact the most the analytical NIR spectroscopy on different instruments [58]. As theoretical computational NIR spectroscopy is being under strong development currently, the linkage between it and analytical NIR spectroscopy is also being strengthened. Applied spectroscopy faces complex samples, in which

371

372

13 Quantum Mechanical Simulation of Near-Infrared Spectra

Table 13.2 Band assignments in NIR spectrum of rosmarinic acid, based on GVPT2//DFT-B3LYP/N07D calculation. Wavenumber (cm−1 ) Experimental

Calculated

Major contributions

1

6854.9

6853

2𝜈OH (ar)

2

6767.2

6741

2𝜈OH (ar)

3

∼6680

6645

2𝜈OH (carboxyl)

4

∼6044

6056

2𝜈CH (ar, aliph, in phase)

5

5986.5

6001

2𝜈CH (ar, aliph, opp. phase)

6

5929.7

5930

2𝜈CH (ar) 2𝜈CH (ar) 2𝜈CH (ar) 7

5752.5

5780

[𝜈 as CH2 , 𝜈CH (ar)] + [𝜈 as CH2 , 𝜈CH (ar)]

8

5128.0

5126

[𝜈C=O, 𝛿 ip OH (carboxyl)] + [𝜈OH (carboxyl)]

9

5075.8

5027

[𝛿 ring , 𝛿 ip OH (ar)] + [𝜈OH (ar, para-)] [𝛿 ring , 𝛿 ip OH (ar)] + [𝜈OH (ar, para-)] [𝛿 ring ] + [𝜈OH (ar, para-)]

10

4994.9

4980

[𝛿 ring , 𝛿 ip OH (ar)] + [𝜈OH (ar, meta-)] [𝛿 ring , 𝛿 ip OH (ar)] + [𝜈OH (ar, meta-)]

11

4923.8

4906

[𝛿 ring , 𝛿 ip OH (ar)] + [𝜈OH (ar, para-)] [𝛿 ring , 𝛿 ip OH (ar)] + [𝜈OH (ar, para-)]

12

4860.0

4847

[𝛿 ring , 𝛿 ip OH (ar)] + [𝜈OH (ar, meta-)] [𝛿 ring , 𝛿 ip OH (ar)] + [𝜈OH (ar, meta-)]

13

4788.3

4798

[𝜈CC] + [𝜈OH (ar, para-)] [𝜈CC] + [𝜈OH (ar, para-)] [𝜈CC] + [𝜈OH (ar, meta-)] [𝛿CCH (carboxyl)] + [𝜈OH (carboxyl)] [𝛿CH (ar), 𝛿 ip OH (ar)] + [𝜈OH (ar, para-)] [𝛿CH (ar), 𝛿 ip OH (ar)] + [𝜈OH (ar, para-)]

14

4701.0

4701

15

4629.4

4632

[𝛿CH (aliph)] + [𝜈OH (ar, meta-)] [𝛿CH (ar), 𝛿 ip OH (ar)] + [𝜈OH (ar, meta-)] [𝛿CH (ar), 𝛿 ring , 𝛿 ip OH (ar)] + [𝜈OH (ar, meta-)] [𝛿CH (ar), 𝛿 ring , 𝛿 ip OH (ar)] + [𝜈OH (ar, para-)] [𝛿 ip OH (ar), 𝛿CH (ar), 𝛿 ring ] + [𝜈OH (ar, para-)]

16

4575.7

4757

[𝛿CH (ar), 𝛿 ring , 𝛿 ip OH (ar)] + [𝜈OH (ar, meta-)] [𝛿 ip OH (ar), 𝛿CH (ar), 𝛿 ring ] + [𝜈OH (ar, meta-)]

17

∼4508

4465

[𝛿 ring , 𝛿 ip OH (ar)] + [𝜈CH (ar)] [𝛿 ring ] + [𝜈CH (ar)] (Continued)

13.2 Overview of the Current Progress in Computational NIR Spectroscopy

Table 13.2 (Continued) Wavenumber (cm−1 ) Experimental

Calculated

Major contributions

[𝛿 ring , 𝛿 ip OH (ar)] + [𝜈CH (ar)] [𝜈C—O (carboxyl), 𝛿 ip OH (carboxyl)] + [𝜈OH (carboxyl)] [𝛿 ring ] + [𝜈CH (ar)] [𝛿 ring , 𝛿 ip OH (ar)] + [𝜈CH (ar)] 18

4372.3

4360

[𝛿 ring ] + [𝜈CH (ar)] [𝛿 ring ] + [𝜈CH (ar)]

19

4233.3

4237

[𝛿 sciss CH2 ] + [𝜈 as CH2 , 𝜈CH (ar)] [𝛿 sciss CH2 ] + [𝜈 as CH2 , 𝜈CH (ar)]

20

4179.4

4194

[𝛿CH (aliph)] + [𝜈CH (ar, aliph, opp. phase)] [𝛿 sciss CH2 ] + [𝜈 s CH2 ] [𝛿CH (aliph)] + [𝜈CH (ar, aliph, in phase)]

aliph, moiety connected to aliphatic chain; ar, moiety connected to aromatic ring. Source: Kirchler et al. 2017 [58]. Reproduced with permission of The Royal Society of Chemistry.

the quantified content is often entangled with intermolecular interactions, e.g. as it was presented in the case of malic acid–water complex, which Schmutzler et al. have investigated in their development of quantum chemical approach [56]. Sometimes it is possible to avoid the context of the chemical environment and obtain insightful information from QM simulations, e.g. as demonstrated by Kirchler et al. upon the development of quantification method of rosmarinic acid in medicinal herbs by means of miniaturized NIR spectroscopy [58]. It is believed that the foreseeable future will see more efforts oriented in this direction, which is the exploration of complex systems and direct applications of theoretical NIR spectroscopy in analytical routines. 13.2.4

Miscellaneous Applications of NIR Spectra Simulation

Isotopic substitution has been one of the key phenomena studied by vibrational spectroscopy as it produces a distinct signature in the form of spectral shifts [60–62]. Therefore, it has been used, e.g. for the purpose of band assignments. It is relatively straightforward to follow the spectral changes due to conveniently selected substitutions, e.g. replacement of OH group by OD group or deuteration of methyl group in simple molecules; the resulting spectral features may be comprehended relatively easy in NIR region. Contrarily, an uneven or partial substitution (e.g. deuteration of only part of CH3 group) occurring due to a spontaneous isotope equilibration or imperfect synthesis is far more difficult to follow in an NIR spectrum, as it is difficult to isolate experimentally a given form in order to unequivocally identify its existence in the sample. QM simulations offer unique and straightforward support in such case, as Grabska et al. have presented on the basis of their in-depth investigation of methanol and its deuterated derivatives [25]. They have considered the entirety of isotopomers of methanol

373

13 Quantum Mechanical Simulation of Near-Infrared Spectra

Figure 13.12 Simulated NIR spectra of CXXXOX (X = H, D) molecules. Source: Reprinted with permission from Grabska et al. [25]. Copyright 2017, American Chemical Society.

CH3OD CH3OH CD3OH CD3OD Absorbance (a.u.)

374

CDHHOD CHDDOD CDHDOD CHDHOD CHDDOH CDDHOH CHDHOH CDHHOH

9000

8000

7000

6000

5000

4000

Wavenumber (cm–1)

molecule and simulated the NIR spectra of them (Figure 13.12) [25]. This has allowed identifying the randomly substituted species in the two samples of CH3 OD, which uncovered different levels of contamination by the isotopomers. This accomplishment would not be possible with the use of classical spectroscopy methods. On top of that, they have provided complete band assignments in the spectra of the four major methanol isotopomers, CH3 OH, CH3 OD, CD3 OH, and CD3 OD, which are used routinely in physicochemical spectroscopic studies. In that case they have calculated up to two quanta vibrational transitions, and the simulated spectra involved first and second overtones and binary and ternary combinations [25]. On this occasion it has been feasible to verify the assumption based on previous studies that an acceptable loss of spectral information results from the simplification of NIR spectra modeling to first overtones and binary combinations; e.g. no significant inaccuracy has been observed for the simulated spectra of butanols despite such simplification (Figure 13.4). The examination of the spectra of methanol and its derivatives has confirmed that one should expect to miss about 20% of the spectral information in such case, as estimated from the relationship between the summed integral intensity of the relevant calculated bands. However, this information is rather spread over multiple minor bands and thus becomes “diffused” over the wavelength axis. The second overtones

13.2 Overview of the Current Progress in Computational NIR Spectroscopy

Absorbance (a.u.)

Experimental Calculated Contributions:

9000

2ν 3ν ν+ν ν+ν+ν 2ν + ν

8500

8000

7500

7000

6500

6000

5500

5000

4500

4000

Wavenumber (cm–1)

Figure 13.13 Contributions due to first and second overtones, as well as binary and ternary combination bands into the NIR spectrum of CH3 OH. Source: Reprinted with permission from Grabska et al. [25]. Copyright 2017, American Chemical Society.

and ternary combinations may be expected not to be important for NIR spectra comprehension, although this is likely molecule dependent and exceptions may be found. Higher-order modes play the major role as the spectra shaping factor in the high-frequency region, i.e. over 7200 cm−1 . This matches the working region of some of the newest miniaturized NIR instruments [58]. Therefore, the ability to accurately model the second overtones and ternary combinations may become increasingly important in the nearest future (Figure 13.13). QM calculations also clearly demonstrate the inherent complexity of NIR spectra; this fact has been well known, but computational studies allow reproducing it accurately and visualizing in a straightforward way. In example, Grabska et al. have presented the band overlay in the NIR spectrum of vinylacetic acid (Figure 13.14) [50]. Note that the intensity scale is common for all bands in the figure. Extensive overlapping does not prevent the simulation to reproduce the spectral line shape even throughout the most complex region of 5000–4000 cm−1 , in which the majority of binary combinations appear [50]. 13.2.5 Accurate NIR Studies of Single Mode Anharmonicity by Solving 1D Schrödinger Equation The generally applicable methods of anharmonic vibrational analysis, such as those based on VPT2 or VSCF formalism, fulfill well the demand for simulation of the entire NIR spectra [6]. The band overlapping common in this region imposes the need to model all the spectral contributions in order to derive information useful for the needs of applied spectroscopy. Exceptions exist, e.g. in the form of 2𝜈OH band that most often appears isolated and well resolved in the spectra of nonassociated molecules [3, 23]. Coincidently, this highly anharmonic mode often proves to be troublesome in general VSCF or VPT2

375

7500

Absorbance (a.u.)

Absorbance (a.u.)

Absorbance (a.u.)

7000

6500 6000 5500 Wavenumber (cm–1)

5000 4800 4700 4600 4500 4400 4300 4200 4100 4000 Wavenumber (cm–1)

Exp. Calc. 1st over. bands Binary comb. bands Baseline contrib.

7000

6500

6000

5500

5000

4500

5000

Wavenumber (cm–1)

Figure 13.14 Convolution of NIR bands on the example of spectra simulation for vinylacetic acid. All bands are presented in common intensity; note an extensive band overlay. Source: Reprinted with permission from Grabska et al. [50]. Copyright 2017, American Chemical Society.

13.2 Overview of the Current Progress in Computational NIR Spectroscopy

schemes. These methods are relatively efficient by evaluating the anharmonic potential of molecular vibration based on minimal number of energy evaluation points; consequently, the amount of anharmonicity that is being “captured” by the method is not always adequate for the modes that deviate strongly from the harmonic oscillator [6]. Again, this is intentional simplification in order to make the computational cost of anharmonic calculations reasonable. Ways of improving the efficiency/accuracy balance of anharmonic calculations (e.g. by transformation of the vibrational coordinates) are presently strongly focused in theoretical research [63, 64]. Nevertheless, for highly anharmonic modes, a general VSCF or VPT2 calculations may yield erroneous or misleading results. In such cases, it is advisable to develop a detailed study of the anharmonic potential (based on multipoint energy evaluations) and by solving the corresponding time-independent Schrödinger equation effectively determine the vibrational levels corresponding to a given mode with very high accuracy [38, 65, 66]. There exist a few approaches to the vibrational problem outlined above; main differences lie in the numerical method used for solving the matrix differential equation in eigenvalue problem [65]. In the present chapter, exemplary applications of these calculations to spectroscopy will be reviewed. In example, Be´c et al. have demonstrated the potential of improving this way the quality of simulated NIR spectrum in the case of cyclohexanol (Figure 13.15) [26]. The VPT2 simulation has incorrectly reproduced the 2𝜈OH wavenumber of cyclohexanol, with unreliable result derived for the major conformer (equatorial-trans) of the molecule; effectively, the simulation indicated a splitting of the 2𝜈OH band not observable in the experimental (Figure 13.5a) [26]. 14

×104 10000

Energy (cm–1)

10 8 6

Energy (cm–1)

12

8000 6000 4000 2000 0 –0.2

–0.1

0

0.1 Q (Å)

0.2

0.3

0.4

4 2 0 0

0.5

1

1.5

2

Q (Å)

Figure 13.15 Vibrational potential and vibrational states [B3LYP/6-311G(d,p)] of the OH stretching mode of the main (equatorial-gauche) conformer of cyclohexanol. Source: Be´c et al. 2018 [26]. Reprinted with permission of Elsevier.

377

378

13 Quantum Mechanical Simulation of Near-Infrared Spectra

By probing the vibrational potential along OH stretching mode for the two leading rotamers of cyclohexanol over a dense grid point for energy evaluations [B3LYP/6-311G(d,p)+CPCM(CCl4 )], the majority of anharmonicity has been captured [26]. Subsequently, solving time-independent Schrödinger equation by means of generalized matrix Numerov method allowed resolving the vibrational levels with great accuracy (better than 1 cm−1 ; consequently the only meaningful inaccuracy results from the precision of the energy evaluations) [65]. This approach effectively yields an anharmonic vibrational analysis that is absolutely accurate in practical sense, as only the inaccuracy of the calculation of electronic energy impacts the final agreement with the experiment [65]. It may be argued that this source of error remains nearly constant for similar molecules, i.e. two rotamers of cyclohexanol. In relative sense, this makes comparative studies highly accurate as the wavenumbers derived for such molecules should be exact in relation to each other. Indeed, Be´c et al. have reported that the splitting between 2𝜈OH bands of the two conformers of cyclohexanol, incorrectly predicted by VPT2 calculations to be equal to 260 cm−1 , has been reduced to 30 cm−1 ; this remains in good agreement with the value obtained experimentally (27 cm−1 ) [26]. The exceptional accuracy comes at a cost of largely increased computing time; however, with a reduction in the grid density, examination of larger molecules may become feasible. The ability of resolving vibrational energies and transition intensities with high accuracy has been employed in a series of studies of fine effects induced by the substituents [67] or the solvent and monitored by NIR and also IR spectroscopy [38, 68, 69]. Gonjo et al. have examined the solvent effects on vibrational spectra of phenol and its 2,6-dihalogenated derivatives; the OH stretching mode has been focused as it is sensitive to these effects [38]. They have considered very broad spectral region, covering visible, near-infrared, and infrared regions (15 600–2500 cm−1 ); fundamental, first-, second-, and third-overtone bands have been analyzed. The wide scope of the study required a very reliable solution for the vibrational problem for the needs of data interpretation; thus they have scanned the potential energy curve along the OH stretching coordinate q0 within the range of −0.7q0 to 1.0q0 around the equilibrium with a dense step 0.02q0 by using B3LYP/6-311++G(3df, 3pd) method with isodensity surface polarized continuum model (IPCM) solvation model of the corresponding solvent (see below). Rich data allowed avoiding approximations (such as fitting of Morse function; final accuracy of the vibrational levels has been maintained at the level exceeding 0.001 cm−1 ) and has been used for solving the Schrodinger equation by means of Johnson’s approach [70]. The intensities of the examined transitions have also been derived from the calculation of integrated absorption coefficient (km/mol, base e); thus, a complete and highly reliable calculated wavenumbers and intensities of 𝜈OH, 2𝜈OH, and 3𝜈OH bands have been available for direct interpretation of the experimental spectra. They have focused on studying the hydrogen bonding effects and solvent dependences of phenol and 2,6-difluorophenol, 2,6-dichlorophenol, and 2,6-dibromophenol in the selection of solvents (n-hexane, CCl4 , CHCl3 , and CH2 Cl2 ). Gonjo et al. have plotted the 𝜈OH (𝜈 = 0, 1, 2, and 3) shift from a gas state to a solution state (solvent shift) and have found a linear relationship between them (Figure 13.16). The slope of solvent shift decreases in the order of phenol, 2,6-difluorophenol,

13.2 Overview of the Current Progress in Computational NIR Spectroscopy

The slope of solvent shift (cm–1)

80 CH2Cl2 CHCl3 CCl4 n-hexane

60

40

20

0

Phenol

(a)

2,6-Difluoro 2,6-Dichloro 2,6-Dibromo phenol phenol phenol

The slope of solvent shift (cm–1)

80 CH2Cl2 CHCl3 CCl4 n-hexane

60

40

20

0

Phenol

(b)

2,6-Difluoro 2,6-Dichloro 2,6-Dibromo phenol phenol phenol

The slope of solvent shift (cm–1)

80

60

40

20

0 (c)

CH2Cl2 CHCl3 CCl4 n-hexane

Phenol

2,6-Difluoro 2,6-Dichloro phenol phenol

2,6-Dibromo phenol

Figure 13.16 The slopes of solvent shifts of phenol, 2,6-difluorophenol, 2,6-dichlorophenol, and 2,6-dibromophenol in n-hexane, CCl4 , CHCl3 , and CH2 Cl2 . (a) Observed and (b) calculated with the basis set: 6-311++G(3df,3pd). (c) Calculated with basis set: cc-pVTZ. Source: Reprinted with permission from Gonjo et al. [38]. Copyright 2011, American Chemical Society.

379

380

13 Quantum Mechanical Simulation of Near-Infrared Spectra

and 2,6-dichlorophenol while becoming larger with the increase in the dielectric constant of the solvent. A new physical insight has been obtained in their study; the relative intensities of the OH stretching vibrations of phenol in CCl4 , CHCl3 , and CH2 Cl2 against the intensity of the corresponding OH vibration in n-hexane increase in the fundamental and second overtone but decrease in the first and third overtones (Figure 13.16). Thus, Gonjo et al. have observed a so-called parity in the relative intensities of consecutive transitions. The parity has been concluded to be more prominent for phenol (due to its intermolecular hydrogen bonding) than for 2,6-dihalogenated phenols that feature an intramolecular hydrogen bond. It has been suggested that the intermolecular hydrogen bond between the OH group and the Cl atom plays a key role for the parity and that the intermolecular interaction between the solutes and the solvents (solvent effects) does not have a significant role in the parity [38]. The solvent effects on the OH stretching fundamental and first overtone bands have been further explored for pyrrole molecule by Futami et al. using similar methods [68]. The same selection of solvents has been used (n-hexane, CCl4 , CHCl3 , and CH2 Cl2 ), as these feature gradually changing properties (i.e. dielectric constant, polarity, acidity) and are convenient for spectroscopic usage in NIR and upper IR regions. They have reported that the wavenumbers of the 𝜈NH and 2𝜈NH bands decrease in the order of CCl4 , CHCl3 , and CH2 Cl2 , which is the increasing order of the dielectric constant of the solvents. However, the corresponding absorption intensities increase in the same order, and the intensity increase is more significant for the fundamental than the overtone. The solvent dependence of NH stretching bands of pyrrole is quite different from those due to the formation of hydrogen bonds. The QM calculations have suggested that the decreases in the wavenumbers of both the fundamental and the overtone of the NH stretching mode with the increase in the dielectric constant of the solvents arise from the anharmonicity of vibrational potential and their intensity increases come from the gradual increase in the slope of the dipole moment function (Figure 13.17) [68]. The impact of the dielectric constant of the solvent on X–H stretching mode of solvated molecule has been further explored by Futami et al. in their examination of hydrogen fluoride (HF) [69], which has been selected as the simplest polar molecule. The calculations have revealed that the vibrational potential and dipole moment function of HF molecule vary continuously with a change in the dielectric constant of the solvent. The calculations were carried out at B3LYP/6-311++G(3df,3pd) and CCSD/aug-cc-pVQZ levels, both with SCRF/IPCM solvent cavity model. It has also been found that the absorption intensities of the fundamental increase with the increase of the dielectric constant smoothly, but those of the first, second, and third overtones do not increase continuously. The study has drawn attention to the problem of selection of the electronic theory method, as the B3LYP and CCSD levels yielded significantly different results in the dependence of absorption intensities on the dielectric constant. Not all of the raised questions could have been answered in the study of HF molecule; thus, the investigation of the solvent effects impacting IR/NIR spectra has been continued by Chen et al. who have focused on C=O stretching vibrations of acetone and 2-hexanone [71].

13.2 Overview of the Current Progress in Computational NIR Spectroscopy ε = 10

1.2 0.8 0.4

(a)

300 200 100 0 –100 –200 50 000 45 000 40 000 35 000 30 000 25 000 20 000 15 000 10 000 5000 0 –0.4

ε=1

ε = 10

4 3 2 1 0

Dipole moment (D)

Energy (cm–1)

(b)

0.0

ε=1

(c)

–1 –0.2

0.2 0.4 0.0 Normal coordinate q/q0

0.6

0.8

Figure 13.17 (a) Dependences on the dielectric constant of the potential energy curve, dipole moment function (𝜀 = 1–10), and wavefunction (𝜀 = 1) of NH stretching mode. (b) Difference of the potential energy curve between the calculation result for dielectric constant of 1 and a variety of dielectric constants. (c) Difference of the dipole moment function between the calculation result for dielectric constant of 1 and a variety of dielectric constants. Source: Futami et al. 2012 [68]. Copyright 2011, American Chemical Society.

A similar computational approach has been used for detailed studies of the change in the anharmonicity of vibrational potential and transition dipole moment upon the formation of hydrogen bonding. Futami et al. have investigated NIR and IR spectra of pyrrole, pyridine, and pyrrole–pyridine complex in solution phase (CCl4 ) [72]. The first overtone of the NH stretching vibration of nonbonded pyrrole molecule has clearly been observed at 6856 cm−1 ; however, pyrrole–pyridine complex has not revealed a detectable 𝜈NH band. The theoretical calculations and solving for the one-dimensional Schrödinger equation has allowed obtaining highly reliable data on the vibrational levels and the dipole moment functions of the 𝜈NH mode of nonbonded pyrrole and pyrrole–pyridine complex (Figure 13.18). These results have suggested that the 2𝜈NH transition is weakened upon formation of the NH· · ·N hydrogen bonding. The simulation has reproduced the experimental peak shift and intensity variations induced by the formation of hydrogen-bonded complex [72]. Further insights have resulted from the analysis of one-dimensional wave equation as it has been concluded that upon the formation of hydrogen bonding, the overlap integral of the wavefunction is enhanced but the transition dipole moment diminishes,

381

13 Quantum Mechanical Simulation of Near-Infrared Spectra

40 000

0

x, y

30 000

–5

20 000

–10

10 000

–15

0 –0.4

–20 –0.2

(a)

0.0

0.2

0.4

0.6

0.8

Normal coordinate qM /qMO 50 000 40 000 30 000

x, y

0

z

–5

20 000

–10

10 000

–15

0 –0.4

(b)

5

Dipole moment (D)

Energy (cm–1)

5

z

Dipole moment (D)

50 000

Energy (cm–1)

382

–20 –0.2

0.0

0.2

0.4

0.6

0.8

Normal coordinate qC/qCO

Figure 13.18 Vibrational wavefunctions and dipole moment functions along potential energy curves of the NH stretching mode of (a) pyrrole monomer and (b) pyrrole–pyridine complex calculated at the DFT//B3LYP/6-311++G(3df,3pd) level. Symbols qM0 and qC0 denote units of the normal coordinates for the NH stretching mode in the monomer and the complex, respectively. They are represented by the displacement vectors of atoms in Å units as follows: qM0 = {N1 (0, 0, −0.07), H1 (0, 0, 1.00), H2 (0.01, 0, 0), H3 (−0.01, 0, 0)}; qC0 = {N1 (0, 0, −0.08), H1 (0, 0, 1.00), H2 (0.01, 0, 0), H3 (−0.01, 0, 0), H4 (0, −0.01, −0.01), H5 (0, 0.01, −0.01)}. Source: Futami et al. 2014 [72]. Reprinted with permission of Elsevier.

leading to a profound decrease in the intensity of the 2𝜈NH band to a level at which it is hardly detectable in the experiment. The study has revealed that the transition dipole moment is significantly decreased upon formation of the complex, resulting in the remarkably weak intensity of the overtone mode of the hydrogen-bonded NH group [72]. The above study has further been extended by Futami et al. onto several other complexes, including pyrol–ethylene and pyrrole–acetylene systems featuring NH· · ·π hydrogen bonding [73]. They have found stabilization energy of the NH· · ·π hydrogen bonding being almost one

References

third that of NH· · ·N kind. The NH· · ·π hydrogen bonding formation induces a smaller redshift of the fundamental and first overtone of 𝜈NH bands; the energy shift has been concluded to be depending on the intermolecular force. Despite these findings, it has been also observed that the increasing trend in the fundamental absorption intensity and decreasing trend in the first overtone one remain similar to that of pyrrole–pyridine case, which features NH· · ·N hydrogen bonding [73].

13.3 Conclusions Due to peculiarity of NIR spectroscopy, it receives considerable gains from QM calculations. An extensive computational cost of anharmonic approaches has long been forming the major hindrance in wide-spreading theoretical NIR spectroscopy of reasonably complex molecules. Recent years have witnessed advances in anharmonic theories, which coupled to the ever-growing computer technology have enabled feasible theoretical NIR spectroscopy in connection with applied spectroscopy. A trend in the evolution of NIR spectroscopy can be noticed; interdisciplinary investigations linking basic, theoretical, and analytical NIR spectroscopy increasing complexity of the molecular systems, inclusion of intermolecular interactions, and considerable improvement in the comprehension of fine spectral features should be accounted for the primary accomplishments that were reviewed in this chapter. At the present stage, theoretical NIR spectroscopy still stands at its relatively early stage of development, in opposition to other kinds of vibrational spectroscopy. One should expect further development in the field, with major challenges still remaining, i.e. large biological systems. Even more coherent growth of theoretical NIR spectroscopy in close connection to analytical applications may be envisioned.

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14 Local Modes of Vibration: The Effect of Low-Frequency Vibrations Emil Vogt, Anne S. Hansen, and Henrik G. Kjaergaard University of Copenhagen, Department of Chemistry, Universitetsparken 5, 2100 Copenhagen Ø, Denmark

In this chapter, we consider the analyses of vibrational spectra based on localized vibrational modes (local modes). Section 14.1 of this chapter emphasizes when and why local modes may be preferred over normal modes. In Section 14.2 we construct one-, two-, and three-dimensional (1D, 2D, 3D) effective local mode Hamiltonians and compare calculated and experimental transition frequencies. We will see how parameters in the different local mode Hamiltonians can be obtained from either electronic structure calculations or experimentally observed transition frequencies. Section 14.3 involves examples where the local mode vibrations are significantly affected by low-frequency (large-amplitude) vibrations. We show that modifications of the simple local mode models are necessary to explain experimental spectra. We provide two examples of such low-frequency vibrations: methyl group torsion and intermolecular modes in hydrogen-bonded bimolecular complexes. In Section 14.4 we focus on vibrational transition intensities and show how a combined theoretical and experimental approach can be used to obtain the thermodynamic stability of a bimolecular complex.

14.1 Introduction The conventional approach used to analyze vibrational spectra of polyatomic molecules is based on the normal mode (NM) picture [1]. The NM picture may be derived by Taylor expanding the potential energy surface (PES) in the Cartesian displacements of the atoms from their equilibrium positions. The kinetic energy operator and the PES of the vibrational Hamiltonian are expressed in mass-weighted Cartesian coordinates, and the Taylor expansion of the PES is truncated at second order. The Hamiltonian is then transformed to a basis for which there are no cross-terms, i.e. the Hamiltonian in the new basis describes a set of independent harmonic oscillators. The linear combinations of mass-weighted displacement coordinates that accomplish this separation of modes are called normal coordinates. Because the Taylor expansion of the PES is truncated at second order, the NM picture is only valid for small vibrational Molecular Spectroscopy: A Quantum Chemistry Approach, First Edition. Edited by Yukihiro Ozaki, Marek Janusz Wójcik, and Jürgen Popp. © 2019 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2019 by Wiley-VCH Verlag GmbH & Co. KGaA.

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14 Local Modes of Vibration: The Effect of Low-Frequency Vibrations

displacements from the equilibrium geometry. The NM approximation provides a simple picture of the fundamental transitions in most (isolated) molecules, but significantly decreases in quality when higher energy regions are explored, due to the inherent lack of anharmonicity and localization of NMs. From a mathematical point of view, the logical approach to extend NMs into the higher energy regime is to include higher-order terms in the Taylor expansion of the PES. This can be done with either perturbation theory or through variational methods. In both cases, the highly excited vibrational states will be described by linear combinations of wavefunctions that depend on the normal coordinates. From a chemist point of view, this is not a satisfactory picture because investigating vibrational overtones immediately demands increasing complexity. On the other hand, increased complexity with increasing internal energy seems inevitable as the density of vibrational states significantly increases with increasing internal energy. However, experimental work, starting from the 1930s, suggested that at least for certain modes, the overtone spectrum is fairly simple and shows unexpected regularities [2, 3]. Such regularities are difficult to understand when investigated in terms of a zeroth-order NM Hamiltonian. It turns out that the observed regularities can be understood if the vibrations in question are assumed to be localized [2, 4–7]. This assumption is the foundation of the so-called local mode (LM) picture [8–10]. The fact that such a model is beneficial in the very high energy regime is maybe not surprising, as an analogy for photochemically induced bond dissociation is the breaking of one local bond and not all bonds associated with an NM. The surprising element is rather that for certain vibrations, the transition between normal and local vibrations seems to be convenient much earlier than one would initially expect. There are many crucial benefits to be gained from such an observation as well as in general expressing the Hamiltonian in local coordinates [8]. In the LM model, the natural mathematical progression in terms of a Taylor expansion of the PES is somewhat dismissed, i.e. coupling (off-diagonal) and non-coupling (diagonal) terms are not treated equally. In fact, all diagonal anharmonic contributions are often included in the zeroth-order LM Hamiltonian, and only certain off-diagonal contributions are included as the perturbation, if at all. For vibrational modes where diagonal anharmonicity is more important than coupling, the zeroth-order LM Hamiltonian provides a better starting point than the zeroth-order NM Hamiltonian [5]. The LM picture shows less coupling in the overtone region of high-frequency modes, compared with the NM picture, and as such provides a simplified description of the associated transitions where only few modes carry significant amplitude. The Hamiltonian describing these transitions may therefore be approximated by a reduced dimensionality LM Hamiltonian, where only the few modes of interest are included. In addition to the reduced dimensionality, the LM model is appealing as vibrations are described with internal coordinates, which fits well with the chemists’ style of thinking. There have been a number of reviews on LMs primarily focusing on the energy levels, some of which are referenced here [8–14]. Like most other tasks in computational chemistry, calculating vibrational properties is a trade-off between accuracy and computational cost. If the vibrational model requires a large sampling of the PES, the corresponding

14.2 Local Mode (LM) Models

electronic structure method, used to calculate the electronic energy at each nuclear displacement, cannot be too computationally demanding. LM theory provides a way to only include a few dimensions, with limited compromise on the vibrational model. As such, the PES sampling can be significantly reduced, and more accurate electronic structure methods can be used, compared with full-dimensional vibrational models based on NMs.

14.2 Local Mode (LM) Models In the LM picture individual modes are selected, simplifying the vibrational picture, compared with the NM picture. Expanding the potential energy in a Taylor ̂ is [15] series of the LM coordinates, the vibrational Hamiltonian (H) 1∑ 1∑ Ĥ = Gii p̂ 2i + G p̂ p̂ 2 i 2 i≠j ij i j ∑ 1∑ + Fii q̂ i2 + Fij q̂ i q̂ j 2 i i≠j ∑ ∑ + Vidia + Vijnondia (14.1) i

i≠j

where the G’s are the Wilson G-matrix elements for a fixed nuclear configu2 V |qi =qj =0 ), p̂ is the momenration [1], the F’s are the force constants (Fij = 𝜕q𝜕 𝜕q i

j

tum operator, q̂ is the internal displacement coordinate, and Vidia and Vinondia are higher-order diagonal and off-diagonal elements of the PES. In LM models, the first column in Eq. (14.1) is treated as the unperturbed Hamiltonian (Ĥ (0) ), which include all the contributions from the isolated anharmonic oscillators (AO) of a molecule. The isolated AO are often approximated as Morse oscillators, due to its relatively simple analytical solutions. The second column in Eq. (14.1) represents coupling between oscillators and is treated as a perturbation (Ĥ (1) ). In this section of the chapter, we show how Eq. (14.1) may be truncated to a 1D, 2D, or 3D LM Hamiltonian, depending on the system of interest. The formulation of the G-matrix elements can be complicated for some systems and will not be discussed here. Nevertheless, we provide some G-matrix elements [16] and the derivation of Eq. (14.1) in Appendix 14.A. 14.2.1

1D LM Model

The simplest approximate form of Eq. (14.1) is the 1D case, where only a single anharmonic LM is included in the LM Hamiltonian, or a number of identical decoupled LMs. The 1D LM model can be used to describe XH stretching vibrations, such as the OH stretch in alcohols or the six identical CH stretches in benzene, as these modes often only couple weakly to the other vibrational modes, i.e. they are localized [3, 17]. After introducing the theoretical framework for the 1D LM model, we will apply the 1D LM model to the two identical OH oscillators in sulfuric acid (H2 SO4 ) [18–20].

391

392

14 Local Modes of Vibration: The Effect of Low-Frequency Vibrations

The 1D LM Hamiltonian describes a single mode and the second column in Eq. (14.1) is therefore neglected. The 1D LM Hamiltonian is 1 1 Ĥ 1DLM = Gii p̂ 2i + Fii q̂ i2 + Vidia (14.2) 2 2 1 = Gii p̂ 2i + VAO (qi ) 2 The eigenvalues of Ĥ 1DLM , i.e. the energy expression for a single oscillator, may be expressed as an expansion in (vi + 1∕2): ) ) ( ( Evi 1 1 2 𝜔̃ i − vi + 𝜔̃ i xi + · · · (14.3) = Ẽ vi = vi + hc 2 2 where 𝜔̃ i (harmonic frequency) and 𝜔̃ i xi (anharmonicity) are the LM parameters and vi is the vibrational quantum number. Vibrational transition frequencies (in wavenumbers) can be obtained by subtracting the ground state energy (Ẽ 0 ) from Eq. (14.3): 𝜈̃vi ←0 = Ẽ vi − Ẽ 0 = vi 𝜔̃ i − (v2i + vi )𝜔̃ i xi + · · ·

(14.4)

For most oscillators, including only 𝜔̃ i and 𝜔̃ i xi is sufficient to get accurate transition frequencies and is the exact solutions if VAO is chosen as a Morse potential. Equation (14.3) is a general expansion of the energy and does not specify how the LM parameters should be obtained. One way to obtain the LM parameters is to use a Birge–Sponer-type fit, where Eq. (14.4) is fitted to experimentally observed transition frequencies [21]. Typical LM parameters for XH stretching oscillators obtained by fitting Eq. (14.4) to experimentally observed transition frequencies are given in Table 14.1. The LM parameters can alternatively be obtained from ab initio calculations of the PES. One can show that the LM parameters can be expressed as [22, 26] ( ) 2 5Fiii Gii h 1 √ (14.5) Fii Gii and 𝜔̃ i xi = − Fiiii 𝜔̃ i = 2πc 64π2 cFii 3Fii where Fii , Fiii , and Fiiii are the second-, third-, and fourth-order derivatives of the 1D PES, with respect to the internal displacement coordinates, evaluated at the equilibrium geometry. The 1D PES is generated from a number of equally spaced single-point energy calculations for different XH bond displacements around the equilibrium geometry. Equation (14.5) is derived by using second-order Table 14.1 Typical LM parameters for different bond types. Bond

𝝎̃ (cm−1 )

̃ (cm−1 ) 𝝎x

OH

3803

85

NH

3529

78

CH

3037

62

SH

2686

49

OH, ethylene glycol [22]; NH, dimethylamine [23]; CH, cyclohexane [24]; SH, tert-butylthiol [25].

14.2 Local Mode (LM) Models

Rayleigh–Schrödinger perturbation theory, for which the energy expression is [27] Ev = ⟨v(0) |Ĥ (0) |v(0) ⟩ + ⟨v(0) |Ĥ (1) |v(0) ⟩ + ⟨v(0) |Ĥ (2) |v(0) ⟩ −

∑ |⟨v(0) |Ĥ (1) |v′ (0) ⟩|2 v′ ≠v

(0) Ev(0) ′ − Ev

(14.6)

where |v(0) ⟩ is the vth eigenfunction of the unperturbed Hamiltonian Ĥ (0) . The PES is Taylor expanded to fourth order in qi , and the unperturbed system is chosen as a harmonic oscillator: 1 1 1 1 Ĥ (0) = Fii q̂ i2 + Gii p̂ 2i and Ĥ (1) = Fiii q̂ i3 and Ĥ (2) = F q̂ 4 2 2 6 24 iiii i (14.7) In the harmonic oscillator approximation, the coordinate operator can be written as √ qc,i † ℏGii (14.8) q̂ i = √ (â i + â i ) and qc,i = 𝜔i 2 where â †i is the step-up (creation) operator, â i is the step-down (annihilation) operator, 𝜔i = 2πc𝜔̃ i and qc,i is the classical turning point. The effect of the step-up and step-down operator on the eigenfunctions of the harmonic oscillator is √ √ â †i |vi ⟩ = vi + 1|vi + 1⟩ and â i |vi ⟩ = vi |vi − 1⟩ (14.9) Inserting Eq. (14.7) into Eq. (14.6), and using Eq. (14.9), gives an expansion of the energy in (vi + 1∕2) for the vi th vibrational state. Equation (14.5) is then obtained by comparing the expansion in (vi + 1∕2) with Eq. (14.3). 14.2.1.1

OH Stretching in Sulfuric Acid

H2 SO4 has two equivalent OH bonds. These OH stretching modes are weakly coupled, and we can to a good approximation treat them as two isolated OH stretching oscillators. This is similar to the six identical CH oscillators in benzene [17]. In Figure 14.1, spectra of the fundamental and first overtone OH stretching transitions of H2 SO4 are shown, and the calculated and observed frequencies are compared [18–20]. The simplest approach to determine 𝜔̃ and 𝜔x ̃ is to use a Birge–Sponer plot with the observed ΔvOH = 1–3 transitions. This gives not surprisingly very good agreement with the experimental transition frequencies for these three transitions and reasonable agreement for the higher transitions. Alternatively, the 1D LM PES can be obtained at the CCSD(T)/aug-cc-pV (T+d)Z level of theory by displacing the OH bond from −0.2 to +0.2 Å in 0.05 Å steps around the equilibrium bond [20]. These single-point energy calculations are then used to calculate the LM parameters using Eq. (14.5). With this approach, the agreement with experiments is similar across the range of transitions considered here. The 1D LM models are denoted Morse in Figure 14.1 as the analytical expression for the energy levels of the Morse oscillator and the 1D LM model are similar, a fact we will exploit in later sections of this chapter.

393

14 Local Modes of Vibration: The Effect of Low-Frequency Vibrations 0.30

Absorbance

0.25

O

0.20 0.15

O S

0.10

H

0.05

H O

O

0.00 3500

3550

3600

3650

ν~ (cm−1)

3700

ν (cm−1)

Absorbance

394

v

Expt.

0.020

1

3609

3609

3619

0.015

2

7061

7059

7079

3

10351

10351

10379

4

13490

13483

13520

5

16494

16456

16501

0.010 0.005 0.000 7000 7050

7100

7150

−1

ν (cm )

Morsea Morseb

a: Using experimental LM parameters from observed ΔvOH = 1 – 3 transitions. b: Using ab initio LM parameters.

Figure 14.1 The observed OH stretching transitions of H2 SO4 for ΔvOH = 1 and 2. Calculated and experimental values of the OH stretching transitions (Δ v = 1–5) are also given. Source: Observed and calculated data are taken from Refs. [18–20, 28].

14.2.2

2D LM Model

The description of molecular vibrations improves when additional modes are introduced in the LM model. One example is the two OH stretching vibrations in H2 O, which are attached to the same oxygen atom (as opposed to those in H2 SO4 ) and hence couple strongly. For simplicity, we only include the harmonic coupling between the two oscillators. The 2D LM Hamiltonian is 1 1 Ĥ 2DLM = Gii p̂ 2i + VAO (qi ) + Gjj p̂ 2j + VAO (qj ) 2 2 (14.10) + Gij p̂ i p̂ j + Fij q̂ i q̂ j This LM model is commonly referred to as the “harmonically coupled anharmonic oscillator” (HCAO) model [29, 30]. The unperturbed system may now be defined from the two decoupled anharmonic oscillators. The zeroth-order expression for the unperturbed transition frequencies (in wavenumbers) then becomes (0) Ẽ v(0) − Ẽ 00 = vi 𝜔̃ i − (v2i + vi )𝜔̃ i xi + vj 𝜔̃ j − (v2j + vj )𝜔̃ j xj + · · · i vj

(14.11)

The two oscillators are coupled by treating the last two terms in Eq. (14.10) as a first-order perturbation, and we get Ĥ (1) = Gij p̂ i p̂ j + Fij q̂ i q̂ j

(14.12)

14.2 Local Mode (LM) Models

or in wavenumbers Ĥ (1) = −𝛾ij′ (â †i â j + â i â †j ) + 𝛾ij∗ (â †i â †j + â i â j ) hc where 𝛾ij′

⎞ ⎛ √ ⎜ 1 Gij 1 Fij ⎟ = ⎜− √ + √ 𝜔̃ i 𝜔̃ j ⎟ ⎜ 2 Gii Gjj 2 Fii Fjj ⎟ ⎠ ⎝

𝛾ij∗

⎞ ⎛ √ ⎜ 1 Gij 1 Fij ⎟ = ⎜− √ − √ 𝜔̃ i 𝜔̃ j ⎟ ⎜ 2 Gii Gjj 2 Fii Fjj ⎟ ⎠ ⎝

(14.13)

and

(14.14)

assuming that the coordinate and momentum operators have the step-up and step-down properties, known for the harmonic oscillator. This assumption has numerically been shown to be a reasonable approximation [29]. In the harmonic oscillator approximation, the momentum operator is defined by √ ipc,i † ℏ𝜔i ̂ and pc,i = (14.15) p̂ i = √ (â − a) Gii 2 where 𝜔 = 2πc𝜔̃ and the position operator is defined by Eq. (14.8). The terms of the form â i â j and â †i â †j couple vibrational states that differ by two in the total quantum number, i.e. they couple states that are well separated in energy. The contribution to the vibrational coupling from these terms is therefore expected to be small and may be neglected. With this approximation, Eq. (14.12) reduces to Ĥ (1) = −𝛾ij′ (â †i â j + â i â †j ), and the coupling is described by only one parameter. This parameter can be found from the splitting of the two fundamental transitions. The effect of Ĥ (1) may either be treated through perturbation theory or with variational methods. Ĥ (1) couples states with the same total quantum number, i.e. the quantum number of one oscillator is stepped down, while another is stepped up. If the two oscillators are similar, states with the same total quantum number may be nearly degenerate in the unperturbed system, and Rayleigh–Schrödinger perturbation theory can therefore not be used to treat the effect of Ĥ (1) . Hence, variational methods are often preferred, for which the matrix formalism of quantum mechanics is convenient. The LM parameters can be obtained from ab initio calculations using Eqs. (14.5) and (14.14). Here, one has to construct a new PES to obtain Fij . This PES is calculated by displacing the two modes simultaneously, typically in steps of 0.05 Å starting from the equilibrium geometry. In H2 O, 𝛾ij′ couples the two equivalent OH stretching modes and leads to the well-known symmetric and asymmetric vibrations. The observed splitting between the symmetric and asymmetric fundamental transition is 99 cm−1 [31]. In H2 O2 the two equivalent OH oscillators are located on adjacent oxygen atoms, and the symmetric and asymmetric fundamental splitting is only ∼ 0.8 cm−1 [32]. In H2 SO4 , a sulfur atom bridges the two oxygen atoms, and only the asymmetric fundamental transition is observed, indicating very weak

395

396

14 Local Modes of Vibration: The Effect of Low-Frequency Vibrations

v =0

v =1

v =2 0 0

0 ~ –2 ωx ~ –γ′ ω ~ ~ –γ′ ω–2ωx

1 0

Figure 14.2 2D HCAO Hamiltonian. The same basis states as shown on the right are implied for the bra states.

0 1 ~ ~ –√2γ′ 2 0 2ω–6ωx 0 ~ ~ –√2γ′ 1 1 –√2γ′ 2ω–4ωx ~ ~ –√2γ′ 2ω–6ωx 0 2 0

coupling [33]. The HCAO Hamiltonian for two coupled OH oscillators is seen in Figure 14.2. As seen in Figure 14.2, 𝛾 ′ couples the |1⟩|0⟩ and |0⟩|1⟩ states directly, and the spectral splitting is 2 𝛾 ′ if the two oscillators are equivalent (Ẽ 10 = Ẽ 01 ). The |2⟩|0⟩ and |0⟩|2⟩ states are indirectly coupled, i.e. through the |1⟩|1⟩ state, and based on the HCAO model, the splitting between the symmetric and antisymmetric first overtone (denoted |20⟩+ and |20⟩− ) is predicted to be less than the corresponding splitting for the fundamental transitions. This prediction is in agreement with the experimental results, where the observed |20⟩± splitting in H2 O is 48 cm−1 . The splitting generally decreases with increasing vibrational quanta and the |40⟩± splitting in H2 O is only 3 cm−1 [31]. For nonequivalent oscillators, it is not straightforward to obtain the 2D LM parameters of the unperturbed system directly from the experimental transition frequencies, as the spectral splitting depends on the energy levels of the decoupled oscillators. However, with the butadiene example, Section 14.2.2.1, we illustrate how deuteration can be used to decouple the individual oscillators and determine the LM parameters of the unperturbed system. Another example of a 2D LM Hamiltonian is the inclusion of a lower-frequency bending mode. This could be the SOH bending mode in sulfuric acid (H2 SO4 ). Due to the frequency mix–match of the OH stretching and SOH bending mode, there is little 1:1 coupling between these modes. However, exciting multiple quanta in the SOH bending mode could lead to resonances. The simplest of such resonances is the Fermi resonance that describes the 1:2 coupling. The Fermi resonance coupling is included in the 2D LM model through the first-order Hamiltonian [16, 31]: Ĥ (1) = f r′ (â †i â j â j + â i â †j â †j )

(14.16)

The Fermi resonance coupling is derived in a similar manner as Eq. (14.14), i.e. by assuming that the coordinate and momentum operators have step-up and step-down properties for the anharmonic zeroth-order LM wavefunctions. The Fermi resonance coupling constant, f r′ , therefore contains both kinetic and potential energy coupling and can be calculated with the following expression: [ ( ] )| q p2 )| ( 2 qc,s qc,b p q p 𝜕G 𝜕G c,s 1 c,b c,s c,b c,b | | bb sb − f r′ = + Fsbb √ | | √ + √ hc 𝜕qs ||eq 4 2 𝜕qb ||eq 2 2 4 2 (14.17)

14.2 Local Mode (LM) Models

where eq stands for equilibrium geometry and qc,i and pc,i are defined from Eqs. (14.8) and (14.15), respectively. 14.2.2.1

CH Stretching in Butadiene

1,3-Butadiene has two equivalent vinyl groups, each with three nonequivalent CH bonds. Two of the CH bonds have a common center atom and are expected to couple more strongly than to the third CH bond. To include the coupling, a 2D LM model is used to calculate the CH stretching frequencies. Figure 14.3 shows a spectrum of the sixth overtone CH stretching region of 1,3-butadiene [34]. Three bands are observed, corresponding to transitions to the pure LM states of the three nonequivalent CH stretching vibrations. The 2D LM calculated CH stretching transition frequencies are within 20 cm−1 of the observed values. The CH stretching frequencies are calculated using the HCAO model, where the zeroth-order Hamiltonian describes the individual uncoupled anharmonic oscillators (CH bonds): Ĥ (0) =

3 ( ∑ 1 i=1

2

Gii p̂ 2i + VAO (qi )

) (14.18)

The first-order Hamiltonian describes the coupling between states within a given vibrational manifold, i.e. with the same total number of CH stretching quanta (v = vt + vc + vn ): ′ ′ (â †n â t + â n â †t ) − 𝛾nc (â †n â c + â n â †c ) Ĥ (1) = −𝛾ct′ (â †c â t + â c â †t ) − 𝛾nt

(14.19)

In principle, this first-order Hamiltonian makes the vibrational model of 1,3-butadiene 3D as the pairwise coupling between all three oscillators is ′ ′ included. However, 𝛾ct′ is much larger than 𝛾nt and 𝛾nc , and the terminal CH oscillators may essentially be treated as being decoupled from the nonterminal CH oscillator. To obtain the LM parameters of the unperturbed system (the three 1D anharmonic oscillators), a spectrum of fully deuterated 1,3-butadiene Hn CHt Relative absorbance

Ht

H Hc H

H

CHc ν~ (cm−1)

CHn

18200 18400 18 600 18 800 19 000 19200 –1

Wavenumbers (cm )

t c n

Expt.

2D LMa

0 0 7

18 583

18 586

0 7 0

18 802

18 783

7 0 0

19 015

18 996

a: Using LM parameters obtained from ΔvCH = 4 – 6

Figure 14.3 Spectrum of CH stretching transitions of 1,3-butadiene in the region ΔvCH = 7. The CH stretching frequencies are calculated using the HCAO model with LM parameters from a combined experimental and theoretical approach. Source: Observed and calculated data are taken from Ref. [34].

397

398

14 Local Modes of Vibration: The Effect of Low-Frequency Vibrations

Table 14.2 LM parameters (cm−1 ) for 1,3-butadiene. CHn

CHc

CHt

𝜔̃

3107 ± 4

3137 ± 10

3172 ± 4

𝜔x ̃

55.4 ± 0.7

54.5 ± 1.7

55.4 ± 0.6

𝛾ct′

′ 𝛾nt

′ 𝛾nc

45

−3.5

0.5

Source: Kjaergaard et al. 1993 [34]. Reproduced with permission of American Institute of Physics.

(1,3-butadiene-d6 , 98.99 % D) is measured. The 1,3-butadiene-d6 sample contains small traces of 1,3-butadiene-d5 for which one CH bond and five CD bonds are present. CD and CH stretching frequencies are very different, due to the difference in the corresponding G-matrix elements (the inverse of the reduced √ mass), 𝜔̃ CD ∼ 𝜔̃ CH ∕ 2. The CH stretching mode in 1,3-butadiene-d5 can therefore be seen as a decoupled CH oscillator of 1,3-butadiene. The zeroth-order LM parameters (𝜔̃ and 𝜔x) ̃ are determined from the observed 1,3-butadiene-d5 ΔvCH = 4 − 6 transitions. The coupling between the CH stretches is determined from ab initio calculations using Eq. (14.14). The LM parameters are given in Table 14.2. As seen from Table 14.2, the anharmonicities of the different CH oscillators are very similar, but the harmonic frequencies are different. The harmonic frequency increases in the following order – CHn , CHc , CHt – in agreement with the observed band positions in the spectrum of the Δ v = 7 region (Figure 14.3). The LM parameters are used to set up a Hamiltonian matrix as shown in Figure 14.4, ~ E004 γ′nc γ′nt ~ γ′nc E013 γ′ct ~ γ′nt γ′ct E103

=0 ≠0

v=0 v=1

0 0 4 0 1 3 1 0 3 ~ E040 γ′ct γ′nc ~ γ′ct E130 γ′nt ~ γ′nc γ′nt E031

v=4

0 4 0 1 3 0 0 3 1 ~ E400 γ′ct γ′nt ~ γ′ct E310 γ′nc ~ γ′nt γ′nc E 301

4 0 0 3 1 0 3 0 1

Figure 14.4 HCAO Hamiltonian for 1,3-butadiene. The same basis states (|t⟩|c⟩|n⟩) as shown on the right are implied for the bra states. Only pure LM states and LM combinations states with three quanta in one mode are included. Each off-diagonal element should be multiplied with the appropriate constant from Eq. (14.9).

14.2 Local Mode (LM) Models

Hn

Ht

H Hc Relative absorbance

H

H ν~ (cm−1)

CHt CHc CHn 13 0

31 0

t c n

Expt.

2D LMa

0 0 4

11 326

11 320

0 4 0

11 445

11 419

4 0 0

11 546

11 522

1 3 0

11 799

11 747

3 1 0

11 909

11 864

a: Using LM parameters obtained from ΔvCH = 4 – 6

11 050

11 600

12 150

Figure 14.5 Δv = 4 region of 1,3-butadiene-d5 (top trace) and 1,3-butadiene (bottom trace). The bands at 11 799 and 11 909 cm−1 are assigned to combination bands based on the HCAO calculations. Source: Kjaergaard et al. 1993 [34]. Reproduced with permission of American Institute of Physics.

where we for simplicity have left out LM combination states with excitations in all three modes and the states with two quanta in two modes. As seen in Figure 14.4, the HCAO Hamiltonian for butadiene takes a block-diagonal form with a block for each vibrational manifold. Eigenvalues of the HCAO Hamiltonian (the vibrational energy levels) are found variationally, in this case by diagonalizing the blocks of interest. There are three separate blocks within the v = 4 manifold. Each of these blocks may be treated as a 2 × 2 ′ ′ matrix and a 1 × 1 matrix because 𝛾nt ∼ 0 cm−1 and 𝛾nc ∼ 0 cm−1 . The calculated and experimental transition frequencies for the Δv = 4 manifold are seen in Figure 14.5. Also shown in Figure 14.5 are the experimental spectra of the Δv = 4 region for 1,3-butadiene-d5 and 1,3-butadiene. As seen from Figure 14.5, only two combination bands are observed in the Δv = 4 region of 1,3-butadiene. The observed combination bands in the 1,3-butadiene spectrum correspond to the simultaneous excitation of the two coupled nonequivalent CH bonds. The pure LM states with four quanta in one CH oscillator are located at lower energies in 1,3-butadiene, compared with 1,3-butadiene-d5 , due to the coupling to the combination states. The combination bands between the nonterminal and terminal CH bonds are predicted to have a small intensity and are not observed experimentally [34]. In Section 14.4, we will discuss how intensities of vibrational transition are calculated within the LM model. Another interesting system where the 2D LM model is useful is the description of the OH oscillator and CH oscillators in HCOOH. The CH oscillator has a smaller harmonic frequency and anharmonicity than the OH oscillator. This means that the frequency of one quanta in the CH stretching mode might match the energy difference between states with v and v−1 quanta (𝜔̃ − 2v𝜔x) ̃ in the higher-frequency (and higher anharmonicity) OH stretching mode. It is clear that

399

400

14 Local Modes of Vibration: The Effect of Low-Frequency Vibrations

this energy difference decreases as v increases and there is a potential for tuning in and out of resonance as v increases. This is observed in the overtone spectra in the ΔvOH = 5 region of HCOOH, with substantial mixing of the |5⟩OH |0⟩CH and the |4⟩OH |1⟩CH states [35]. The frequency shifts and intensity mixing disappear when the HCOOH is deuterated to DCOOH, and only the ΔvOH = 5 transition is observed. 14.2.3

3D LM Model

In the previous subsection the 2D LM model was used to treat three oscillators, as only two of the three oscillators were strongly coupled. The 3D LM model is used when all three oscillators couple. In Section 14.2.3.1, we will see that this is the case for the CH stretches in methyl groups. As for the 2D HCAO LM model, only harmonic coupling is included, i.e. the three oscillators in the 3D LM model are coupled pairwise. With this approximation, the 3D LM model is still commonly referred to as the HCAO model. The 3D LM Hamiltonian is 1 1 1 Ĥ 3DLM = Gii p̂ 2i + Gjj p̂ 2j + Gkk p̂ 2k 2 2 2 + Gij p̂ i p̂ j + Gik p̂ i p̂ k + Gjk p̂ j p̂ k + VAO (qi ) + VAO (qj ) + VAO (qk ) +Fij q̂ i q̂ j + Fik q̂ i q̂ k + Fjk q̂ j q̂ k

(14.20)

This Hamiltonian is divided into a zeroth- and first-order Hamiltonian. The zeroth-order Hamiltonian describes the three 1D (decoupled) anharmonic oscillators: 1 1 1 Ĥ (0) = Gii p̂ 2i + Gjj p̂ 2j + Gkk p̂ 2k 2 2 2 +VAO (qi ) + VAO (qj ) + VAO (qk ) (14.21) and the corresponding unperturbed transition frequencies are (0) − Ẽ 000 = vi 𝜔̃ i − (v2i + vi )𝜔̃ i xi Ẽ v(0) i vj vk

+ vj 𝜔̃ j − (v2j + vj )𝜔̃ j xj + vk 𝜔̃ k − (v2k + vk )𝜔̃ k xk + · · ·

(14.22)

The first-order Hamiltonian describes the coupling between the three oscillators: Ĥ (1) = Gij p̂ i p̂ j + Gik p̂ i p̂ k + Gjk p̂ j p̂ k + Fij q̂ i q̂ j + Fik q̂ i q̂ k + Fjk q̂ j q̂ k

(14.23)

which may be written in terms of the LM coupling parameters: Ĥ (1) (14.24) = −𝛾ij′ (â †i â j + â i â †j ) − 𝛾ik′ (â †i â k + â i â †k ) − 𝛾jk′ (â †j â k + â j â †k ) hc As for the 2D LM model, the LM parameters can be obtained from ab initio calculations using Eqs. (14.5) and (14.14). Another example of a 3D Hamiltonian is the inclusion of a bending mode in the 2D XH stretching Hamiltonian. One example is H2 O, in which case we

14.2 Local Mode (LM) Models

end up with two identical OH stretching modes and one HOH bending mode [16, 31]. Since the frequency of the HOH bending mode is approximately half the frequency of the OH stretching modes, the 1:1 coupling term between stretching and bending is not going to contribute significantly; however the Fermi resonance term (1: 2) should be included in the Hamiltonian (Eq. (14.16)). The unperturbed Hamiltonian is Eq. (14.21), and the first-order LM Hamiltonian for H2 O is Ĥ (1) ∑ ′ † ′ [f r (â i â 3 â 3 + â i â †3 â †3 )] − 𝛾12 (â †1 â 2 + â 1 â †2 ) = hc i=1 2

(14.25)

where the 𝛾ij′ term clearly couples the two identical OH stretching modes and leads to the well-known symmetric and asymmetric vibrations that are observed in the spectra. The f r′ terms describes the stretch–bend Fermi resonance (1:2) coupling [31]. 14.2.3.1

CH Stretching in Propane

In its lowest energy structure, propane has a methyl group (CH3 ) with two out-of-plane CH bonds (CHop ) and an in-plane CH bond (CHip ). The 3D LM model is used to describe the CH stretching vibrations of this methyl group, where the two CHop oscillators are identical and hence have equivalent LM parameters and the CHip oscillator is different from the other two. The zeroth-order Hamiltonian describes the individual anharmonic oscillators, and the first-order Hamiltonian describes the coupling between the two CHop oscillators and between a CHop and a CHip oscillator. The vibrational states related to the methyl group are denoted |v1 v2 ⟩|v3 ⟩, where v1 and v2 are the quantum numbers of the two CHop oscillators and v3 is the quantum number of the CHip oscillator. As the two CHop oscillators are identical, many transitions will be degenerate in the unperturbed system, and these split into a symmetric (denoted with a + subscript) and antisymmetric (denoted with a − subscript) linear combination when coupled. Figure 14.6 shows a spectrum of the fifth overtone CH stretching region of propane [36]. Three bands are observed in the Δv = 6 region of propane (Figure 14.6). Two of these bands correspond to the CH stretching vibrations of the CHop and CHip bonds. The interaction between the CH oscillators in the methyl group and the CH oscillators on the methylene group is expected to be weak and is neglected in the vibrational LM model. The 2D LM model is used to describe the CH stretching vibrations of the methylene group. The 3D LM parameters are obtained from a combined experimental (𝜔, ̃ 𝜔x) ̃ and theoretical (𝛾 ′ ) approach and are shown in Table 14.3. As seen from Table 14.3, the (different) CH oscillators have different harmonic frequencies but similar anharmonicities. The LM calculated CH stretching frequencies are within 2 cm−1 of the observed values, as shown in Figure 14.6. The splitting of the |60⟩+ |0⟩ and |60⟩− |0⟩ energy levels in the CH3 group and of the |60⟩+ and |60⟩− of the CH2 group is negligible. In essence, even the CH oscillators that have a common center atom become localized in the high overtone region. However, at lower overtones, the splitting is significant as shown in the 2D example.

401

14 Local Modes of Vibration: The Effect of Low-Frequency Vibrations

CHop Relative absorbance

402

Hop H op CHip

H3C

CHm

Hip

Hm Hm ν~ (cm−1)

15400

15600

15800

16000

Wavenumbers (cm–1)

vopvop vip

Expt.

3D LM

60

15 740

15 738

15 830

15 829

±

0

00 6

Figure 14.6 Absorbance from the CH stretching vibrations of propane in the ΔvCH = 6 region. The calculated |60⟩+ |0⟩ and |60⟩− |0⟩ energy levels are identical and are therefore denoted by |60⟩± |0⟩, labeled CHop in the spectrum. Source: Observed and calculated data are taken from Ref. [36]. Table 14.3 LM parameters (cm−1 ) for propane. CHm

CHop

CHip

𝜔̃ a)

3036 ± 10

3052 ± 7

3060 ± 9

𝜔x ̃ a)

62.9 ± 1.5

61.0 ± 1.1

59.9 ± 1.4

′ 𝛾m,m

′ 𝛾op,op

′ 𝛾op,ip

23.4

25.9

27.4

a) Determined from a Birge–Sponer plot of the Δv = 3–7 transitions. Source: Kjaergaard et al. 1990 [36]. Reproduced with permission of American Journal of Physics.

14.3 Effect of Low-Frequency Modes We have seen that coupling between the XH stretching LMs can be important and that coupling to lower-frequency modes often can be neglected. Exceptions are examples of accidental resonances with bending modes or between OH and CH stretches. However, if low-frequency vibrations in molecules have a significant effect on the LMs, these might also have to be included. In this chapter we give examples of two such cases. One of the earlier much studied examples of coupling between the XH stretching LMs and lower-frequency vibrations is the torsion of a methyl group, which affects the CH stretching frequency and hence the overtone spectra [37–42]. Similarly, some of the intermolecular vibrational modes in bimolecular complexes, with, for example, methanol as a donor molecule, have a significant effect on the OH stretching frequency, which lead to significant changes in calculated vibrational OH stretching frequencies, and the observed band shape [43].

14.3 Effect of Low-Frequency Modes

14.3.1

Methyl Torsional Mode

The time scale of the CH stretching vibrations, relative to the torsion motion, is such that conformationally different CH bonds are observed in CH stretching overtones. In propane, despite the “free rotation” or torsion around the C—C single bonds, two peaks (CHip and CHop ) are observed from the methyl groups in the CH stretching overtone region [44]. The small changes in chemical environment around the CH bonds lead to different vibrational frequencies, and subtle differences are observed due to the correlation between bond length and frequency [45]. For example, in naphthalene, two CH stretching transitions corresponding to the 𝛼 and 𝛽 hydrogens are observed, and in 1,3-butadiene, three CH stretching transitions are observed [34, 46]. However, in a molecule like toluene where the barrier to torsion is very small, the torsional wavefunction is delocalized, and many conformers (different values of the torsional angle) are populated, and the effect of this on the LM overtone spectrum is the appearance of an apparent additional band [45, 47]. To treat the coupling between the high-frequency CH stretching vibration and the low-frequency CH3 torsion, we define the coordinates as shown in Figure 14.7. The barrier to rotation of the methyl group (torsion) is either threefold (o-xylene) or sixfold (p-xylene) depending on the symmetry as shown in Figure 14.8. The potential energy (in wavenumbers) of this torsional mode can be expressed as a Fourier expansion with the leading terms Vtorsion = V0 + V3 cos(3𝜃) + V6 cos(6𝜃)

(14.26)

In p-xylene the barrier is small, and the potential is sixfold (V6 ∼ 10 cm−1 ), whereas o-xylene has a larger threefold component (V3 ∼ 425 cm−1 ) with a small sixfold component (18 cm−1 ). This methyl rotation can be treated as a rigid rotor with the Hamiltonian (in wavenumbers) Htorsion ̃ 2 + Vtorsion ̃ torsion = Bm =H hc H2(H3)

z

y

H1

q2(q3)

q1 C

x

(14.27)

H1 H2 2π/3

θ C

x

C H3

Figure 14.7 The coordinate system used to describe a rotating methyl group. (Left panel) Shows a side view illustrating the CH stretching vibration (displacement qi ). The carbon of the methyl group is in origo. (Right panel) Shows a top view in which the methyl group, illustrated by the dashed lines, has been rotated 𝜃 degrees from the minimum energy conformer shown with solid lines. Source: Rong and Kjaergaard 2002 [48]. Adapted with permission of American Chemical Society.

403

14 Local Modes of Vibration: The Effect of Low-Frequency Vibrations O-Xylene

ω~ (cm–1)

V (cm–1)

600

~ (cm–1) ωx

404

400

5

200

0

0

–5

3350

3280

3300

3260

3250

3240

3200

3220

66

66

65

65.5

64

65

63

64.5 0

60

120 Θ (°)

p-Xylene

10

180

CH3 Torsion, Θ H H

0

60

120

180

H

p-Xylene

Θ (°)

Figure 14.8 Potential (V), frequency (𝜔), ̃ and anharmonicity (𝜔x) ̃ of o- and p-xylene as a function of the torsional angle, Θ. The asterisks represent the HF/6-31G(d) calculated results, and the solid lines the Fourier series fit.

where B̃ is the rotational constant of the methyl group and m is the torsional quantum number. The value of B̃ depends on the bond lengths and angles but is on the order of 5 cm−1 . Torsion of the methyl group leads to a variation in the CH bond length and with it a variation of the frequency and anharmonicity of the associated CH oscillator. The variation can also be expressed in a Fourier series. As seen in Figure 14.8 for o- and p-xylene, the angular variation of these parameters will be dominated by a twofold symmetry in both molecules. The variation in, for example, 𝜔̃ is on the order of 50–100 cm−1 , and depending on the size of the torsional potential itself, this can be a small or a large contribution. We use an adiabatic separation of the slower (lower energy) torsion and the faster (higher energy) CH stretching vibrations. The variation in frequency and anharmonicity contribute to the effective potential of the methyl torsion depending on what vibrational state the molecule is in. Let us consider the simplest example of an isolated CH stretch in the deuterated methyl group CHD2 . Within the adiabatic separation, we would first solve the anharmonic Hamiltonian associated with a 1D CH stretching LM oscillator and subsequently solve the methyl torsional Hamiltonian for a given CH stretching vibrational solution with quantum number v. For the deuterated methyl group CHD2 , the torsional Hamiltonian can be written as [39, 49] ) ( )2 ( ⃗ F⃗ v + 1 − X ̃ 2+𝜴 ⃗ F⃗ v + 1 + V⃗ torsion F⃗ ̃ torsion = Bm H (14.28) 2 2 For ease of notation we use a vector notation. F⃗ is a column vector representing a Fourier series expansion [1, cos(𝜃), ..., cos(6𝜃), ..., sin(𝜃), ..., sin(6𝜃)]T limited to ⃗ and X ⃗ are row vectors of dimension 13 represent13 terms. The coefficients 𝜴 ing the frequency and anharmonicity of the CH stretching oscillators and their

14.3 Effect of Low-Frequency Modes

variation with the torsional angle. V⃗ torsion is a row vector representing the angular dependence of the torsion potential. For a given vibrational level, the sum of the last three terms represents the effective torsional potential, V effective,v , and the torsion effective torsional Hamiltonian for that vibrational level becomes effective,v

⃗ ̃ 2 ⃗ ̃v H torsion = Bm + V torsion F

(14.29)

For the methyl group in propane, the Vtorsion term dominates, whereas in toluene Vtorsion is a minor contribution. For a methyl group, CH3 , we describe the CH stretching oscillators with the HCAO LM model to include some coupling between the CH stretching oscillators and the methyl torsion in a rigid rotor basis. We approximate the total Hamiltonian for a rotating methyl group by [37, 38, 41, 42] [ ] 3 3 ∑ ∑ 1 1 † † 2 ′ ⃗ F⃗ i (vi + ) − X ⃗ F⃗ i (vi + ) − ̃ methyl = H 𝛾ij (â i â j + â i â j ) 𝜴 2 2 i=1 j>i ̃ 2 + V⃗ torsion F⃗ 1 + Bm

(14.30)

where the subscript i refers to the three different CH stretching oscillators (shown ⃗ and X ⃗ as well as 𝛾 ′ terms defines the CH in Figure 14.7). The leading term in the 𝜴 ij stretching part of the Hamiltonian. The HCAO coupling between the CH stretching oscillators is included via the 𝛾ij′ term; however the small angular variation in 𝛾ij′ is ignored [48]. One can use an adiabatic separation of the CH stretching vibration from the methyl torsion and solve the CH stretching part first as described in Sections 14.2.2.1 and 14.2.3.1. For each of the CH stretching eigenfunctions in a given manifold, we then set up the associated methyl torsional Hamiltonian with an effective potential (Eq. (14.28)), which is defined by the coefficients in the eigen⃗ and X. ⃗ Alternatively, one can solve the vectors combined with the coefficients 𝜴 Hamiltonian associated with Eq. (14.30), implying a nonadiabatic separation of the CH stretching and methyl torsion. The penalty for the nonadiabatic approach is that a much larger matrix has to be diagonalized. The vibrational torsional basis functions are a product of three anharmonic oscillator (typically Morse) wavefunctions |v1 ⟩|v2 ⟩|v3 ⟩ multiplied with a real rigid rotor function |m⟩. The real rigid rotor basis functions are essentially cos(m𝜃) and sin(m𝜃) functions, where m is the torsional quantum number. If we limit m to 12, there is a total of 25 torsional states associated with each vibrational basis function. In a given vibrational manifold, v = v1 + v2 + v3 , there are (v + 1)(v + 2)∕2 CH stretching basis states. For the v = 6 manifold, there are 28 vibrational states with 25 torsional states, a total of 700 eigenstates. Thus, from the 25 torsional states associated with the |0⟩|0⟩|0⟩ state, this leads to 17 500 transitions, which contribute to the CH stretching spectrum in the ΔvCH = 6 region. In Section 14.4.3.4 we discuss how the CH stretching transition intensities are calculated for o-xylene and p-xylene and see that the relatively few features in the spectra are modeled well, despite the large number of torsional transition.

405

406

14 Local Modes of Vibration: The Effect of Low-Frequency Vibrations

14.3.2

Intermolecular Modes in Bimolecular Complexes

Formation of a bimolecular hydrogen bound complex introduces additional vibrational modes, intermolecular modes, not present in the isolated monomers [50]. These intermolecular modes arise as the two molecules forming the complex (collectively) lose rotational and translational degrees of freedom. In this section of the chapter, the importance of these low-frequency intermolecular modes and their effect on the bound XH stretching (XHb stretching) vibration in hydrogen bound complexes are illustrated. The adiabatic separation used for the methyl torsion can also be considered in the bimolecular complexes. It was used in water dimer, H2 O⋅H2 O, to separate the O—O stretch (donor–acceptor stretch) and the OHb stretch and showed that the effect of the O—O stretch is a widening of the OHb stretching band [43]. The XH stretching vibration typically redshifts upon hydrogen bonding, and the size of the frequency redshift depends on the hydrogen bond strength. A large redshift corresponds to a strong hydrogen bond [51, 52]. For nearly linear hydrogen bonds, the intermolecular modes in the vibrational model reduces the calculated XH stretching redshift by ∼ 30% [53]. It therefore becomes increasingly important to include the intermolecular modes in the LM model, as the strength of the hydrogen bond increases. The local mode perturbation theory (LMPT) model was developed to include the effect of the intermolecular modes [50, 54, 55]. The LMPT model was originally applied to hydrated complexes: H2 O⋅N2 , H2 O⋅H2 O, and H2 O⋅NH3 . Here, the hydrogen bond donor molecule, H2 O, is described with a 3D LM model, and the effects of six intermolecular modes on the donor vibrational modes are included using Rayleigh–Schrödinger perturbation theory. The formulation of the Hamiltonians included in the LMPT model can be found in Refs. [50, 53–55] and will not be discussed here. The LMPT model is abbreviated using two indices, e.g. 3D+6D LMPT, where the first index (3D) denotes the number of vibrational modes included for the hydrogen bond donor and the second index (6D) denotes the number of intermolecular modes included [50]. For the hydrated complexes, only two of the six intermolecular modes were found to significantly perturb the OHb oscillator. Both of these intermolecular modes are associated with breaking the hydrogen bond [50]. For a series of HF complexes, the LMPT model was simplified to a 1D+2D LMPT model [53]. The two intermolecular modes included are shown in Figure 14.9 for the HF⋅Ar complex. These intermolecular modes strongly affect the bound XH stretching vibration, as they partially break the hydrogen bond, as illustrated by the arrows in Figure 14.9. This effect reduces the overall hydrogen bond strength and hence reduces the overall frequency redshift of the OHb oscillator.

Figure 14.9 Illustration of the two most significant intermolecular modes for HF⋅Ar. Source: Mackeprang et al. 2018 [53]. Reproduced with permission of Elsevier.

14.3 Effect of Low-Frequency Modes

4000

Fundamental FH stretching transition frequency (cm−1)

LM

LMPT

VPT2

Expt.

3950

3900

3850

3800

3750

FH

FH...Ar

FH...FHf

FH...N2

FH...CO2 FHb...FH FH...CO

Figure 14.10 Observed and calculated HF stretching frequencies for the HF monomer and a series of HF complexes. The PES is calculated with the CCSD(T)-F12a/VDZ-F12 ab initio method. Source: Mackeprang et al. 2018 [53]. Reproduced with permission of Elsevier.

Calculated and experimental FH stretching frequencies are shown in Figure 14.10. As seen in Figure 14.10, the discrepancy between the 1D LM calculated and the experimental HF stretching frequencies is 5–50 cm−1 , with the largest discrepancies observed for the strongest HF hydrogen bound complexes. For the stronger bound complexes, accurate FH stretching transition frequencies require descriptions beyond the simple 1D LM model. The 1D+2D LMPT model improves the accuracy of the calculated FH stretching frequencies to within 1–16 cm−1 of the experimental values, similar to the accuracy of the full-dimensional VPT2 model [53]. The increasing importance of intermolecular modes for stronger hydrogen bound complexes is illustrated by comparing FH stretching redshifts for the FH⋅Ar (weak) and FH⋅CO (strong) complexes. For FH⋅Ar, the 1D LM model predicts a FH stretching redshift of 22 cm−1 , which is 6 cm−1 larger than the corresponding LMPT value (16 cm−1 ). For FH⋅CO, the difference between the 1D LM model (171 cm−1 ) and experimental (117 cm−1 ) FH stretching redshifts is 54 cm−1 , whereas the 1D+2D LMPT calculated FH stretching redshift (118 cm−1 ) is in very good agreement with the experimental value. The LMPT model was modified for larger complexes, e.g. alcohol⋅amine complexes [55]. Here, a 2D LM model including the OHb stretching and the COH bending oscillators is employed for the donor, and only two intermolecular modes are included, i.e. a 2D+2D LMPT model. Figure 14.11 compares the observed and calculated fundamental OHb stretching frequencies of MeOH complexes with four different acceptors. In each acceptor, the accepting atom is different, illustrating the similar hydrogen bond strength of O, S, and P, with N being a much stronger acceptor atom. The 2D LM calculated OHb stretching frequencies are within 100 cm−1 of the observed values, with the largest discrepancy observed for the MeOH⋅TMA complex, the complex with the strongest acceptor. In comparison, the 2D+2D LMPT OHb stretching frequencies are less than 40 cm−1 from

407

14 Local Modes of Vibration: The Effect of Low-Frequency Vibrations OH.N Absorbance

408

O

OH.P

Y

v~OH,1←0 (cm–1) Acceptor (Y) TMA TMP DMS DME

OH.S OH.O

3300

H

3400 3500 Wavenumber (cm–1)

Expt. 3355 3541 3569 3579

2D LM 2D+2D LMPT 3337 3260 3522 3487 3529 3490 3553 3500

3600

Figure 14.11 OHb stretching fundamental transition in MeOH complexes. Experimental infrared spectra of the fundamental OHb stretching region in different MeOH complexes (left). Observed and CCSD(T)-F12a/VDZ-F12 calculated frequencies of the OHb stretching fundamental transition (table). Acceptor molecules: trimethylamine (TMA, Y = N), trimethylphosphine (TMP, Y = P), dimethyl sulfide (DMS, Y = S) and dimethyl ether (DME, Y = O).

the observed values, clearly illustrating the importance of including the intermolecular modes in the vibrational model. So far, we have seen that the simple LM models can be used to describe the vibrational XH stretching overtone spectra of different molecules. For isolated XH stretching vibrations (e.g. OH stretch in H2 SO4 , the CH stretch in benzene or naphthalene, and the NH stretch in dimethylamine [DMA]), a simple 1D LM model is sufficient to obtain accurate frequencies. However, for XH stretching vibrations that share a common center atom (e.g. CH stretches in propane and OH stretches in H2 O), the 1D LM model is insufficient, and a 2D or 3D LM Hamiltonian that includes coupling between the LMs is necessary to get good agreement with experiments. We have also seen that it can be necessary to include low-frequency modes (e.g. methyl torsion and intermolecular modes) in the vibrational model to obtain accurate predictions, when these modes affect the XH stretching modes.

14.4 Local Mode Intensities In Sections 14.2 and 14.3, we saw how to construct effective LM Hamiltonians, with LM parameters that could be obtained from experiments, quantum mechanical calculations, or a combination of both. We used these effective Hamiltonians to assign and explain vibrational bands observed in the XH stretching overtone region of different molecules. However, as we saw for 1,3-butadiene (Figure 14.5), the calculated transition frequencies alone were not sufficient to assign the two observed combination bands in the ΔvCH = 4 region. Nevertheless, we argued that only two of the six combination bands were predicted to have significant intensity and could therefore distinguish between vibrational bands of similar energy. Here, we introduce the theory necessary to calculate vibrational transition intensities and apply it to some of the examples given in Sections 14.2 and 14.3. It is essentially these intensities that make

14.4 Local Mode Intensities

overtone spectra relatively simple, as it is primarily the pure LMs that carry intensity. We calculate intensities in terms of dimensionless oscillator strengths. The oscillator strength from the ground state to an excited vibrational state is [36, 56] 2 ⃗ fv←0 = 4.702 × 10−7 (cm × D−2 )𝜈̃v←0 |⟨v|𝝁|0⟩|

(14.31)

⃗ is the dipole moment function (DMF) and ⟨v|⃗ where 𝝁 𝝁|0⟩ is the transition dipole moment. The transition dipole moment is expressed in Debye (D), and the transition frequency, 𝜈̃v←0 = Ẽ v − Ẽ 0 , in cm−1 . Thus, to obtain calculated oscillator strengths, we need vibrational wavefunctions and the DMF. The vibrational wavefunctions are the solutions to the time-independent (vibrational) Schrödinger equation ̂ 1 v2 ...⟩ = E|v1 v2 ...⟩ H|v

(14.32)

for the different reduced dimensionality LM Hamiltonians. The DMF can be found from ab initio calculations. 14.4.1

Wavefunctions

To calculate LM wavefunctions, we need to specify the LM coordinates and obtain the corresponding potentials. In Section 14.2.1.1 we saw that a simple 1D LM model could be used to predict the OH stretching transition frequencies in H2 SO4 . The LM coordinate for an OH stretch is simply the OH bond length (internuclear distance between the oxygen and the hydrogen atoms). This coordinate is linear in the Cartesian coordinates, and the G-matrix element is the inverse of the effective mass (G = 1∕𝜇 = (mO + mH )∕(mO mH )). The potential for the OH stretch can either be represented analytically or numerically (by calculating the electronic energy and nuclear repulsion of H2 SO4 for a large range of OH displacements). One of the simplest analytical potentials that allow for dissociation is the Morse potential [57]: VMorse = D(1 − exp(−aq))2

(14.33)

where q is the bond displacement from equilibrium, D is the dissociation energy (in Joule), and a (in m−1 ) is an inherent Morse parameter. The 1D LM Schrödinger equation is solvable for the Morse potential [57], and the corresponding energy levels (in wavenumbers) are ) ( ) ( 1 2 1 𝜔̃ + v + 𝜔x ̃ (14.34) Ẽ v = v + 2 2 where 𝜔̃ and 𝜔x ̃ are related to the Morse parameters, a and D, in the following way: ( )1∕2 4π𝜔xc ̃ ℏπc𝜔̃ 2 a= and D = (14.35) ℏGii 2𝜔x ̃ If we assume that 𝜔̃ and 𝜔x ̃ are the LM parameters, we obtain a 1D (Morse) potential simply by obtaining the LM parameters with the methods discussed in ̃ = D∕hc = 𝜔̃ 2 ∕(4𝜔x). ̃ Section 14.2.1. If the potential is expressed in cm−1 , then D

409

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14 Local Modes of Vibration: The Effect of Low-Frequency Vibrations

As the 1D LM Schrödinger equation is analytically solvable for the Morse potential, the corresponding wavefunctions are known once the LM parameters are specified. The Morse wavefunctions are [57] 𝛽

|v⟩ = Nv exp(−y∕2)y𝛽v ∕2 Lvv (y)

(14.36) 𝛽 Lvv (y)

where 𝛽v = 1∕x − 2v − 1, y = (1∕x) exp(−aq), is a generalized Laguerre polynomial, and Nv is a normalization constant. The normalization constant is ( ) a𝛽v Γ(v + 1) 1∕2 (14.37) Nv = Γ(𝛽v + v + 1) where Γ is the gamma function. The analytical form of these wavefunctions facilitates evaluation of the transition dipole moment in Eq. (14.31). With the numerical approach, the potential is represented by a discrete set of points, and the 1D nuclear Schrödinger equation is solved numerically. Each point on the PES is calculated by solving the electronic Schrödinger equation and adding the nuclear repulsion for different positions of the nuclei. For the OH stretch in H2 SO4 , the different nuclear positions are generated by displacing the OH bond from −0.30 to 0.70 Å, in small steps (typically 0.025 Å) around the equilibrium geometry. Common for the numerical methods, which can be used to solve the 1D nuclear Schrödinger equation, is that the accuracy generally increases with the number of points in the PES. Therefore, interpolation schemes are often employed to increase the number of points, without having to solve the electronic Schrödinger equation for additional nuclear configurations. Within the 1D approximation, the advantage of the numerical approach is that the performance relies solely on the level of electronic structure method and not on a predetermined shape of the potential. Nevertheless, the drawback is that the potential is not directly linked to the LM parameters. As we saw in Section 14.2, the LM parameters are useful when analyzing an experimental spectrum. Hence, the numerical approach is (often) preferred when the need for quantitative predictions outweighs the need for qualitative information. In the 2D and 3D LM models, the zeroth-order Hamiltonian describes the decoupled anharmonic LM oscillators. The eigenfunctions of these Hamiltonians are hence product states of the individual oscillators, e.g. the Morse oscillator wavefunctions. The first-order Hamiltonian describes the coupling between the individual oscillators. In the HCAO model, the kinetic and potential energy coupling terms are written in terms of step-up and step-down operators (Eq. (14.12)), making it possible to construct a block-diagonal Hamiltonian where only states within the same vibrational manifold are coupled (as we saw in Section 14.2.2.1). The 2D or 3D LM eigenfunctions are then found by diagonalizing each block separately. The resulting eigenfunctions will be linear combinations of the product states within the same vibrational manifold. If the kinetic and potential energy coupling terms are not approximated, all states will be coupled, and the full Hamiltonian needs to be diagonalized by either exact (variational) or approximate (perturbation theory-based) methods. In either case, the size of the Hamiltonian, i.e. the number of basis states included, influences the resulting eigenvalues and eigenfunctions. Therefore, the amount

14.4 Local Mode Intensities

of basis states included is increased until convergence of both eigenvalues and eigenfunctions is reached. 14.4.2

Dipole Moment Function

In all the LM models, the DMF may be represented by a Taylor expansion in the respective internal coordinates. This is reasonable as the dipole is a smoothly varying function with respect to the displacement coordinates. For the 1D LM model, we write the DMF as ⃗ ⃗0 + 𝝁 ⃗ 1q + 𝝁 ⃗ 2 q2 + · · · + 𝝁 ⃗ 6 q6 𝝁(q) =𝝁

(14.38)

⃗ i is the ith expansion coefficient vector, essentially the ith-order derivative where 𝝁 of the DMF with respect to q, evaluated at the equilibrium geometry. We limit the expansion to sixth order. This leads to oscillator strengths of CH stretching transitions that are converged up to the sixth overtone [24]. In general, the DMF is expressed as a Taylor expansion in all internal coordinates. For three internal coordinates we have | ∑ 𝜕 i+j+k 𝜇⃗ || 1 j ⃗ ijk q1i q2 q3k where 𝝁 ⃗ ijk = ⃗ 1 , q2 , q3 ) = 𝝁 (14.39) 𝝁(q i!j!k! 𝜕qi 𝜕qj 𝜕qk || ijk 1 2 3 |eq ⃗ ijk are determined from grids of the dipole moment as a funcThe coefficients 𝝁 tion of bond displacements and are vectors with Cartesian (x, y, or z) components. The dipole moment grids are found by varying the LM coordinates in suitable steps (typically 0.05 Å for XH bond displacements) around the equilibrium geometry and calculating the Cartesian components of the dipole moment vector for each nuclear configuration. Many electronic structure programs orientate molecules automatically, and special care must be taken to prevent reorientation of the molecule when displacing along the respective LMs. The reorientation will not influence the PES but may lead to different oscillator strengths due to changes in the DMF, arising from it being a vector. To limit the effect of vibration–rotation coupling on the vibrational intensities an Eckart frame can be used [31, 58, 59]. However, the effect of the Eckart conditions will decrease with molecular size. 14.4.3

Absolute Intensities

In the following sections we illustrate how oscillator strengths are calculated with the different LM models in order to understand the observed features in the absorption spectra of different molecules. The intensities are calculated with Eq. (14.31) with wavefunctions and DMF described in Sections 14.4.1 and 14.4.2. 14.4.3.1

Sulfuric Acid: Higher Overtones

The intensities of the OH stretching overtone transitions in H2 SO4 are important as these overtone transitions have been shown to drive the photolysis of H2 SO4 to sulfur trioxide (SO3 ) and water in the upper atmosphere [60–62]. The subsequent photolysis of SO3 could explain stratospheric sulfur dioxide (SO2 ) observations.

411

412

14 Local Modes of Vibration: The Effect of Low-Frequency Vibrations

Table 14.4 Calculated OH stretching intensities (oscillator strengths) in sulfuric acid.a) State

Morse (1D LM)

Numeric (1D)

Experiments

1

3.7 × 10−5

3.7 × 10−5

3.4 × 10−5

2

1.3 × 10

−6

−6

1.2 × 10−6

3.7 × 10

−8

−8

4.4 × 10−8

2.0 × 10

−9

−9

3.3 × 10−9

1.9 × 10

−10

−10

2.7 × 10−10

3 4 5

1.4 × 10 5.2 × 10 3.7 × 10 4.3 × 10

a) CCSD(T)/aug-cc-pV(T+d)Z ab initio method used for PES and DMF grids [20].

The more conventional UV photolysis is not likely in H2 SO4 due to the high energy of the lowest-lying electronic states requiring high energy photons that are not present in the atmosphere until at very high altitudes [18]. The coupling between the two OH stretching modes in sulfuric acid is very small. It leads to symmetric and asymmetric transitions that overlap even in the fundamental region. The intensity of the OH stretching transitions can be calculated with the 1D OH stretching LM model described in Section 14.2.1 with the 1D expansion of the DMF (Eq. (14.38)). We can also improve this 1D model by including the SOH bending mode. Due to the frequency mix–match of the OH stretching and SOH bending modes, there is little coupling between these modes, and intensity changes will mainly happen due to accidental resonances, like the Fermi resonance. In Table 14.4 we compare the calculated and observed OH stretching intensities. The difference in using a Morse or full numeric potential to describe the OH stretching oscillator increases as we excite higher overtones, illustrating the need for numeric potentials when accurate absolute intensities of higher overtones are required. 14.4.3.2

Dimethylamine: An Intense First Overtone

The absorbance spectra of the fundamental and first overtone NH stretching vibration of DMA is shown in Figure 14.12 [23]. The intensity of the fundamental transition is extremely weak, about five times weaker than the first NH stretching overtone. This abnormal intensity pattern has been observed for a few amines and for chloroform [23, 63–65]. To explain this intensity pattern, the DMF must be considered. We use a 1D LM model to treat the NH stretching oscillator and the DMF is described by Eq. (14.38). The DMF for the displacement of the NH bond in DMA is also shown in Figure 14.12. The intensity of a transition is proportional to the norm square of the transition dipole moment (Eq. (14.31)). With the DMF expressed as in Eq. (14.38), the transition dipole moment is ̂ +𝝁 ⃗ ⃗ 0 ⟨0|v⟩ + 𝝁 ⃗ 1 ⟨0|q|v⟩ ⃗ 2 ⟨0|q̂ 2 |v⟩ + · · · + 𝝁 ⃗ 6 ⟨0|q̂ 6 |v⟩ (14.40) ⟨0|𝝁(q)|v⟩ =𝝁 ⃗ 0 ⟨0|v⟩ is zero due to orthonormality of the LM wavefunctions. For where 𝝁 v = 1, the value of the integrals, ⟨0|̂qn |1⟩, decreases rapidly when n increases. The ⟨0|̂q|1⟩ integral is the dominant contribution to the fundamental transition,

14.4 Local Mode Intensities

1.0

Δv = 2

Absorbance

0.8 0.6 Δv = 1

0.4 0.2 0.0 3200

3300

3400

6500

6600

6700

Wavenumber (cm–1) (dx2 + dy2 + dz2)1/2 (D)

1.10 DMA 1.09

f v

1.08 1.07

Expt.

1D LM

1

1.01 ×

10–7

1.10 × 10–7

2

5.49 × 10–7

3.78 × 10–7

1.06 1.05 –0.3 –0.2 –0.1 0.0 0.1 0.2 0.3 0.4 q (Å)

Figure 14.12 Observed NH stretching transitions of DMA and the DMF for the displacement of the NH bond in DMA. Source: Data are taken from Ref. [23]. 2

as ⟨0|̂q|1⟩∕Å ∼ 20 × ⟨0|̂q2 |1⟩∕Å . Thus, the intensity of the Δv = 1 transition ̂ ⃗ 1 ⟨0|q|1⟩ term of Eq. (14.40). However, is usually determined primarily by the 𝝁 the first derivative of the DMF for the displacement of the NH bond in DMA (Figure 14.12) is close to zero, which leads to a very small contribution to the transition intensity from this term. For the Δv = 2 transition, the ⟨0|̂q|2⟩∕Å 2 and ⟨0|̂q2 |2⟩∕Å integrals are similar in size, and the intensity of the Δv = 2 ̂ ⃗ 2 ⟨0|q̂ 2 |2⟩ terms of the ⃗ 1 ⟨0|q|2⟩ and 𝝁 transition therefore depends on both the 𝝁 transition dipole moment expansion. Again, the first derivative of the DMF for the displacement of the NH bond in DMA is close to zero, and in DMA the second derivative of the DMF therefore dominates the transition intensity for the Δv = 2 transition. This explains the observation that the fundamental NH stretching vibration is weaker than the first overtone NH stretching vibration. 14.4.3.3

Water Dimer: A Weak First Overtone

The OH oscillator involved in the hydrogen bonding (OHb ) in the water dimer also shows an interesting intensity pattern. The fundamental OHb stretching transition shows the intensity enhancement and frequency redshift typically associated with hydrogen bonding [51, 52]. However, early matrix-isolation experiments did not identify a transition associated with the OHb stretching

413

414

14 Local Modes of Vibration: The Effect of Low-Frequency Vibrations

mode in first overtone region, despite clear observation of the three other OH stretching transitions. These three OH stretching transitions were in the fundamental region weaker than the OHb stretching transition. Again the expansion of the transition dipole moment offers a simple explanation of this observation. In the DMA example the fundamental transition became very weak due to a near zero derivative of the DMF. In the water dimer, the dipole moment derivative with respect to the OHb mode is increased approximately by a factor of 5, compared with that of the OH stretching modes in the water monomer, and this leads to the enhancement of the fundamental transition in the water dimer. The transition intensities in the first overtone are dominated by the ⟨0|̂q|2⟩∕Å 2 and ⟨0|̂q2 |2⟩∕Å matrix elements, which are of the same size and with opposite ⃗ 2 are ⃗ 1 and 𝝁 sign [66]. In water dimer it happens that the main components of 𝝁 very similar and of same sign and this leads to a near cancellation of terms in the expansion of the transition dipole moment. This causes the first OHb stretching overtone to be significantly weaker than the other OH stretching transitions in this region, despite it being stronger than the other three OH stretching transitions in the fundamental region [67, 68]. Just like the enhancement of the fundamental transition, the weaker first overtone transition seems characteristic for hydrogen-bonded stretching transitions [51, 52]. 14.4.3.4

Effect of Methyl Torsion

The DMF of the CH stretching oscillator also varies with torsional angle similar to what we found for the CH stretching frequency and anharmonicity. Thus, to calculate a CH stretching spectrum of a methyl group, this needs to be included in the usual expansion of the DMF (Eq. (14.39)). For a single isolated oscillator as in the CH stretch of a CHD2 group (or the OH stretch in HOONO) [40, 69], one of the dipole moment vector components, e.g. the x-component, becomes ⃗ x F⃗ 𝜇x = QC

(14.41)

⃗ = [1, q1 , q2 , q3 , q4 , q5 , q6 ], the usual expansion of the XH stretching DMF where Q in internal coordinates, F⃗ is the Fourier series expansion expressed earlier, and Cx is the 7 ×13 matrix that contain the dipole moment expansion coefficients found from a set of ab initio calculations similar to what is used for the torsional dependence of 𝜔. ̃ The y- and z-components of the DMF are defined in a similar manner with the expansion coefficients in the corresponding Cy and Cz . To generalize this approach to a methyl group, CH3 , with three HCAO coupled CH stretching oscil⃗ to allow for the two coordinate mixed terms in the dipole lators, we increase Q moment expansion, similar to the mixed terms in the HCAO Hamiltonian. The x-component of the DMF is 𝜇x =

3 ∑

⃗ i Cx F⃗ Q

(14.42)

i=1

⃗ 2 and Q ⃗ 3 defined by permuwhere Q1 = [1, q1 , q12 , ..., q16 , q2 q3 , q22 q3 , q2 q32 ] with Q tations of the appropriate indices. The Cx matrix becomes a 10 ×13 matrix of the dipole moment expansion coefficients and similar for the y- and z-component.

14.4 Local Mode Intensities

10900 (a)

11000

11100

Wavenumber (cm–1)

11200

10 900 (b)

11000

11100

Wavenumber (cm–1)

Figure 14.13 Simulated (top) and observed (bottom) spectra of the methyl region in the ΔvCH = 4 region for o-xylene (a) and toluene (b). HF/6-31G(d) was used to obtain all LM model parameters, with scaling of frequencies and anharmonicities. Source: Rong et al. 2003 [40]. Reproduced with permission of American Chemical Society.

In Figure 14.13, we show the methyl region of ΔvCH = 4, for o-xylene in the left panel, with the observed spectrum in the lower trace. The barrier for methyl torsion is about 425 cm−1 and sufficiently high such that only the two peaks associated with the in-plane and out-of-plane CH stretching oscillators are seen, similar to what has been seen in propane (V3 = 1150 cm−1 ) and dimethyl ether (V3 = 900 cm−1 ) [36, 40, 70]. Thus, in principle, there is no need to include the coupling to the torsion of the methyl group, and the spectrum could be modeled with the HCAO model as shown for propane [36]. However, we also see in Figure 14.13 that including the torsion does give a good agreement with the observed spectrum (bottom traces). In toluene the torsional barrier is very small (V6 = 5 cm−1 ), and the effect of methyl torsion is clear in the spectrum, with a band that almost looks like a new peak arising between the in-plane and out-of-plane CH stretching transitions. This “third” peak was earlier assigned to the “freely rotating methyl group.” Despite the simple-looking structure of the simulated spectra in Figure 14.13, they both arise from 9375 transitions, with the dipole moment expansion clearly selecting transitions in certain regions. The OH oscillator in HOONO is also an interesting example where including a low-frequency torsional mode in the vibrational model is essential [69, 71]. Here, two isomers, cis–cis HOONO and trans–perp HOONO, with vastly different harmonic frequency and anharmonicity are observed. The cis–cis isomer has an intramolecular hydrogen bond that reduces the harmonic frequency by ∼ 250 cm−1 and increases the anharmonicity by ∼ 20 cm−1 , respectively, relative to the trans–perp isomer where there is no hydrogen bond. OH stretching transitions for both HOONO isomers are observed experimentally [72] and are easily distinguished due to the difference in the LM parameters. Although the torsional barrier in cis–cis HOONO is much larger than for toluene, the vibrational spectrum is also composed of many transitions due to the thermal population of the many torsional eigenstates. A similar vibrational model as used for toluene can be used to describe the OH stretching spectrum of HOONO [69].

415

416

14 Local Modes of Vibration: The Effect of Low-Frequency Vibrations

14.4.3.5

Effect of Intermolecular Modes in Bimolecular Complexes

For fundamental XHb stretching transitions, an intensity enhancement is observed upon hydrogen bonding [51, 52]. This enhancement can facilitate detection of hydrogen bound bimolecular complexes. In DMA, the fundamental NH stretching transition is very weak (Section 14.4.3.2), and the enhancement upon complexation can be as high as a factor of 500. In MeOH (and other alcohols), the enhancement is typically around a factor of 20. At room temperature the amount of complex formed is small, typically less than 1 Torr when monomers at pressures (PA and PB ) of around 100 Torr are mixed. The small complex pressures, PA⋅B , cannot be measured directly but can be determined indirectly by [36, 56] PA⋅B = 2.6935 × 10−9 (K−1 Torr m cm)

T ∫ A(𝜈) ̃ d𝜈̃ fcalc l

(14.43)

where ∫ A(𝜈)d ̃ 𝜈̃ is the integrated absorbance and fcalc is the calculated oscillator strength of the observed vibrational transition of the complex. The accuracy of the calculated oscillator strengths is therefore crucial to obtain the right complex pressures. From the complex and monomer pressures, an equilibrium constant (K) of complex formation is obtained: PA⋅B ∕P∘ K= (14.44) PA ∕P∘ × PB ∕P∘ where P∘ is the standard pressure of 1 bar. The Gibbs energy of complex formation is determined from the equilibrium constant: ΔG∘ = −RT ln(K)

(14.45)

For the MeOH⋅DMA complex, detection of both the fundamental OHb stretching vibration and the weaker second overtone NHf stretching vibration is possible. Thus, for the MeOH⋅DMA complex, it is possible to obtain an equilibrium constant from either of these two transitions, combining the observed and calculated intensities [73, 74]. Obviously, the K value obtained from the NHf stretching second overtone should match that obtained from the OHb stretching fundamental transition. Therefore, the dual determination of K indirectly gives an estimate of the accuracy of the method used by comparing K values obtained for the two transitions. The spectra in Figure 14.14 show the absorbance corresponding to the OHb stretching fundamental transition and the NHf stretching second overtone in the MeOH⋅DMA complex. The OH and NH stretching transition intensities were calculated with both the 1D LM model and the 2D+2D LMPT model [55]. Clearly the LMPT model has little effect on the intensity of the NH transition, as the NH bond is not involved in the hydrogen bonding. With the LMPT oscillator strengths, excellent agreement is found between the two independent equilibrium constants. The difference in K between the two determinations corresponds to a difference of ∼ 1 kJ/mol in the Gibbs energy. For comparison, a 7–9 kJ/mol variation is found for a series of purely ab initio calculated Gibbs energies of the MeOH⋅DMA complex [73].

0.30

MeOH.DMA

OH

NHf

0.004

0.15 0.00 3000

0.002 0.000 3200

3400

3600

9400

9600

Absorbance

Absorbance

14.5 Summary

9800

Wavenumber (cm–1)

NHf stretch

OH stretch K

Method

f

K

1D LM

0.11

1.75 ×

2D + 2D LMPT

0.16

1.27 × 10–4

f

10–4

0.17

2.06 × 10–8

0.16

2.07 × 10–8

Figure 14.14 Absorption from the fundamental OH stretching and second overtone NHf stretching vibrations of MeOH⋅DMA complexes. Source: Spectra and oscillators strengths are taken from Ref. [55, 73, 74].

14.5 Summary In this chapter we have seen how reduced dimensionality LM Hamiltonians can be used to explain and predict spectral features associated with high-frequency modes. We have obtained LM parameters either from experimentally observed transition frequencies, ab initio calculations, or a combination of both. We have also seen that it can be necessary to include low-frequency vibrations in the vibrational model in order to explain spectral features. Intensities are inherently more difficult to calculate accurately compared with transition frequencies. Calculated intensities are useful to understand observed spectra and can be essential in facilitating assignment of spectra. For the NH stretching vibration of DMA, the simple 1D LM model predicts the intensity pattern of the fundamental and first overtone transitions, despite it being atypical, f1←0 < f2←0 . Similar, the simple LM model predicts the very weak first OHb stretching overtone of the water dimer despite this mode being the most intense fundamental transition. In 1,3-butadiene, relative intensities of the three nonequivalent CH stretching overtone transitions are correctly predicted by a 3D LM model supporting the assignment. In these examples it is clear that deviation from linearity of the DMF is essential to model the spectra. A linear DMF would be completely useless. In the ΔvCH = 4 region of toluene, a peak arises between the CH stretching vibrations of the in-plane and out-of-plane transitions, which can be explained and modeled by including methyl torsion in the 3D CH stretching LM model. For bimolecular complexes we illustrate that by combining calculated and experimental intensities, the partial pressure of the complex can be determined and that including intermolecular modes in the LM model improves agreement with

417

418

Appendix 14 Local Modes of Vibration: The Effect of Low-Frequency Vibrations

experiments. This enables determination of the equilibrium constant and from that the Gibbs energy of complexation, a quantity that is difficult to calculate solely with ab initio methods.

Appendix 14.A Deriving the LM Hamiltonian We have seen many different LM Hamiltonians in Section 14.2, but have not yet had to discuss how the actual LM Schrödinger equation is constructed. We have also seen that G-matrix elements enter into the different LM Hamiltonians, but neither have we discussed the origin nor formulation of such quantities. Here, we derive the starting point of Section 14.2, i.e. Eq. (14.1), and list some of the G-matrix elements used. The vibrational Hamiltonian in Cartesian coordinates is −ℏ2 ∑ 1 𝜕 2 Ĥ c = + Vc 2 𝛼 m𝛼 𝜕x2𝛼

(14.A.1)

where 𝛼 expresses a summation over the Cartesian coordinates of each atom and Vc is the PES associated solely with the Cartesian coordinates. We introduce a new set of coordinates, {q1 , q2 , ..., q3N−6 } (the LM coordinates), which in general depend on the Cartesian coordinates in a nonlinear way. For linear molecules 3N − 5 coordinates are needed. The vibrational Hamiltonian may be expressed in the LM coordinates (internal displacement coordinates) by transforming both the kinetic and the potential energy. Vc is a function of 3N Cartesian coordinates, and it may not be immediately obvious how a transformation to 3N − 6(5) coordinates is possible. However, the 3N coordinates account for all degrees of freedom, that is, the translational, the rotational, and the vibrational degrees of freedom. The vibrational degrees of freedom are separated from the 3(2) rotational and 3 translational degrees of freedom; hence only 3N − 6(5) degrees of freedom are needed to express the potential. As Vc is unique for each molecule, we will not explicitly transform the Cartesian potential to LM coordinates, but simply replace Vc with V when needed. The transformation of the kinetic energy can be achieved by applying the chain rule [75]: ( )( ) ∑ 𝜕qj 𝜕 −ℏ2 ∑ 1 𝜕 𝜕 −ℏ2 ∑ 1 ∑ 𝜕qi 𝜕 T= = 2 𝛼 m𝛼 𝜕x𝛼 𝜕x𝛼 2 𝛼 m𝛼 𝜕x𝛼 𝜕qi 𝜕x𝛼 𝜕qj i j −ℏ2 ∑ ∑ 1 𝜕qi 𝜕 𝜕qj 𝜕 (14.A.2) 2 ij 𝛼 m𝛼 𝜕x𝛼 𝜕qi 𝜕x𝛼 𝜕qj ( [ ] ) 1 𝜕 𝜕qi 𝜕qj 𝜕 −ℏ2 ∑ ∑ 1 𝜕 𝜕qi 𝜕qj 𝜕 − = 2 ij 𝛼 m𝛼 𝜕qi 𝜕x𝛼 𝜕x𝛼 𝜕qj m𝛼 𝜕qi 𝜕x𝛼 𝜕x𝛼 𝜕qj ( ) [ ] ∑ 1 −ℏ2 ∑ 𝜕 𝜕 𝜕qi 𝜕qj 𝜕 𝜕 = G − 2 ij 𝜕qi ij 𝜕qj m𝛼 𝜕qi 𝜕x𝛼 𝜕x𝛼 𝜕qj 𝛼 =

Appendix 14.A Deriving the LM Hamiltonian

Here, 1∕m𝛼 and the sum over 𝛼 was absorbed into Gij in the last step, such that Gij =

∑ 1 𝜕qi 𝜕qj m𝛼 𝜕x𝛼 𝜕x𝛼 𝛼

(14.A.3)

which is more conveniently written as Gij =

N ∑ 1 ⃗̂ ⃗̂ 𝛽 qj ) (∇𝛽 qi ) ⋅ (∇ m 𝛽 𝛽

(14.A.4)

where 𝛽 indicates a specific atom. In Eq. (14.A.2), a square parenthesis indicates that the derivative is only to be taken within the parenthesis. If the new coordinates (the q’s) are linear in the Cartesian coordinates, the G-matrix elements are constants, and the very last term in Eq. (14.A.2) is zero. This is the case for normal coordinates, as normal coordinates are linear combinations of the Cartesian (displacement) coordinates. However, as the LM coordinates are not linear in the Cartesian coordinates, the G-matrix elements are generally not constants, but rather scalar functions of the LM coordinates. Therefore, each G-matrix element may be expressed as a Taylor expansion in the respective LM coordinate(s) around the equilibrium geometry (eq): Gij = Gij(0) +

∑ nm

n+m 1 𝜕 Gij || | n!m! 𝜕qin 𝜕qjm ||

(14.A.5)

eq

Here, the summation is over all values of n and m where n + m ≥ 1. Although not strictly necessary, it is a fairly good approximation to calculate Gij for a fixed nuclear configuration, i.e. only including the first term (Gij(0) ) in Eq. (14.A.5). With this approximation, the G-matrix elements in Eq. (14.A.2) commute with the 𝜕∕𝜕qi operator, and the LM kinetic energy can be written as T=

1∑ −ℏ2 ∑ 𝜕 𝜕 Gij = G p̂ p̂ 2 ij 𝜕qi 𝜕qj 2 ij ij i j

(14.A.6)

where it is implied that Gij is evaluated at the equilibrium geometry and the last term in Eq. (14.A.2) is omitted. The last term in Eq. (14.A.2) is often called the LM pseudopotential, as it takes the form of a potential when the volume element of integration is Πi dqi [75]. The LM pseudopotential is ( ) 𝜕|G| ℏ2 1 1∕4 ∑ 𝜕 V′ = − |G|−5∕4 Gij (14.A.7) |G| 2 4 𝜕qi 𝜕qj ij where |G| is the determinant of the G-matrix. V ′ is often ignored in LM theory, and the (modified) LM Hamiltonian is 1∑ G p̂ p̂ + V (14.A.8) Ĥ = 2 ij ij i j which may be expressed as Eq. (14.1) by Taylor expanding the potential energy in the LM coordinates.

419

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14 Local Modes of Vibration: The Effect of Low-Frequency Vibrations

The G-matrix elements (ZXH) associated with examples in this chapter are listed below [1, 16, 20]: Gss =

1 1 + mH mX

1 1 1 Gbb = 2 + 2 + rZX mZ rXH mH mX Gsb = − )| | | = | |eq ( )| 𝜕Gsb | | = 𝜕qb ||eq

(

𝜕Gbb 𝜕qs

( 1 2 rZX

sin(𝜙) mX rZX

+

1 2 rXH

2cos(𝜙) − rZX rXH

)

(14.A.9)

2 cos(𝜙) 2 2 − − 2 3 3 mX rZX rXH mH rXH mX rXH − cos(𝜙) mX rZX

where s denotes a stretch and b a bend, 𝜙 is the angle associated with the bending motion, and rij denotes the distance between the ith and jth atoms.

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tions of OH-stretching overtone induced photodissociation of fluorosulfonic and chlorosulfonic acid. Phys. Chem. Chem. Phys. 12 (29): 8277–8284. Thomsen, D.L., Axson, J.L., Schrøder, S.D. et al. (2013). Intramolecular interactions in 2-aminoethanol and 3-aminopropanol. J. Phys. Chem. A 117 (40): 10260–10273. Niefer, B.I., Kjaergaard, H.G., and Henry, B.R. (1993). Intensity of CH-and NH-stretching transitions in the overtone spectra of cyclopropylamine. J. Chem. Phys. 99 (8): 5682–5700. Tarr, A.W. and Zerbetto, F. (1989). Absolute intensities of CH-stretching overtones in chloroform and deuterochloroform. Chem. Phys. Lett. 154 (3): 273–279. Kjaergaard, H.G., Low, G.R., Robinson, T.W., and Howard, D.L. (2002). Calculated OH-stretching vibrational transitions in the water- nitrogen and wateroxygen complexes. J. Phys. Chem. A 106 (38): 8955–8962. Schofield, D.P. and Kjaergaard, H.G. (2003). Calculated OH-stretching and HOH-bending vibrational transitions in the water dimer. Phys. Chem. Chem. Phys. 5 (15): 3100–3105. Kjaergaard, H.G., Garden, A.L., Chaban, G.M. et al. (2008). Calculation of vibrational transition frequencies and intensities in water dimer: comparison of different vibrational approaches. J. Phys. Chem. A 112 (18): 4324–4335. Schofield, D.P., Kjaergaard, H.G., Matthews, J., and Sinha, A. (2005). The OH-stretching and OOH-bending overtone spectrum of HOONO. J. Chem. Phys. 123 (13): 134318. Kjaergaard, H.G., Henry, B.R., and Tarr, A.W. (1991). Intensities in local mode overtone spectra of dimethyl ether and acetone. J. Chem. Phys. 94 (9): 5844–5854. Schofield, D.P. and Kjaergaard, H.G. (2005). Effect of OH internal torsion on the OH-stretching spectrum of cis,cis-HOONO. J. Phys. Chem. A 109 (9): 1810–1814. Fry, J.L., Nizkorodov, S.A., Okumura, M. et al. (2004). cis-cis and trans-perp HOONO: action spectroscopy and isomerization kinetics. J. Chem. Phys. 121 (3): 1432–1448. Du, L., Mackeprang, K., and Kjaergaard, H.G. (2013). Fundamental and overtone vibrational spectroscopy, enthalpy of hydrogen bond formation and equilibrium constant determination of the methanol-dimethylamine complex. Phys. Chem. Chem. Phys. 15 (25): 10194–10206. Hansen, A.S., Maroun, Z., Mackeprang, K. et al. (2016). Accurate thermodynamic properties of gas phase hydrogen bonded complexes. Phys. Chem. Chem. Phys. 18 (34): 23831–23839. Kjaergaard, H.G. and Mortensen, O.S. (1990). The quantum mechanical hamiltonian in curvilinear coordinates: a simple derivation. Am. J. Phys. 58 (4): 344–347.

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15 Intra- and Intermolecular Vibrations of Organic Semiconductors and Their Role in Charge Transport Andrey Yu. Sosorev 1,2 , Ivan Yu. Chernyshov 3 , Dmitry Yu. Paraschuk 1 , and Mikhail V. Vener 3 1 M.V. Lomonosov Moscow State University, Faculty of Physics and International Laser Center, Leninskie Gory, 1/62, Moscow 119991, Russia 2 Institute of Spectroscopy of the Russian Academy of Sciences, Fizicheskaya Street, 5, Troitsk, Moscow 108840, Russia 3 Mendeleev University of Chemical Technology, Department of Quantum Chemistry, Miusskaya Square 9, Moscow 125047, Russia

15.1 Introduction Organic electronics is a rapidly developing high-tech area aimed at production of a new generation of electronic devices with a number of advantages over traditional inorganic ones (mainly based on silicon): flexibility, stretchability, ease of production, shock resistance, transparency, and the ability to create materials for a specific task. Organic light-emitting diodes (OLEDs) have been successfully commercialized; organic solar cells and field-effect transistors are expected to enter the market soon. However, organic semiconductors (OSs), which constitute the active layers of organic electronic devices, generally show charge mobility, 𝜇, much below that of crystalline inorganic semiconductors. Only a few OSs, mainly organic semiconductor crystals (OSCs), exhibit reproducible 𝜇 exceeding that of amorphous silicon (𝜇 ∼ 1 cm2 /(V s)), which is commonly used in modern thin-film electronics [1, 2]. Low charge mobility in OSs hinders efficient operation of electronic devices. In contrast to inorganic semiconductors, organic ones are “soft” materials – they consist of molecules bound by non-covalent forces. This “softness” affects the charge transport properties of OSCs: although charge carriers are highly delocalized within the molecules, their transfer between them, and therefore the charge mobility, is hindered by weak intermolecular electronic coupling [3, 4]. If the electronic coupling between adjacent molecules is considerable, charge delocalization can occur, providing efficient band-like charge transport with high 𝜇. Otherwise, interaction of the charge carriers with vibrations (phonons) – electron–phonon interaction – leads to charge localization and inefficient hopping transport mechanism. Accordingly, charge transport in OSs is controlled by the interplay between intermolecular charge delocalization and localization. Thus, intra- and intermolecular vibrations play a detrimental role Molecular Spectroscopy: A Quantum Chemistry Approach, First Edition. Edited by Yukihiro Ozaki, Marek Janusz Wójcik, and Jürgen Popp. © 2019 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2019 by Wiley-VCH Verlag GmbH & Co. KGaA.

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15 Intra- and Intermolecular Vibrations of Organic Semiconductors

in charge transport in OSCs, and adequate description of charge transport in them requires detailed understanding of the electron–phonon interaction. Recent theoretical studies [3, 4] suggest that in high-mobility OSCs, thermally populated low-frequency (LF) vibrations in the THz frequency range 0–200 cm−1 limit charge delocalization and mobility because they induce strong electron–phonon interaction. However, LF vibrations in organic crystals are complex and include intra- and intermolecular (translational and librational) contributions; therefore, LF vibrational structure of OSCs and its role in charge transport require rigorous theoretical investigation [5]. OSCs with similar molecular structures but different crystal packing, e.g. single crystals of fluorinated tetracyanoquinodimethanes Fn -TCNQ (n = 0, 2, 4), present convenient objects for such study. As recently shown [6], Fn -TCNQ crystals are characterized by significant differences in their LF vibrational spectra, and this was explained by the different crystal packing. In this chapter, we review computational and experimental approaches for description of the vibrational structure of OSCs and its impact on charge transport in these materials. High-frequency modes (associated with intramolecular vibrations) and LF vibrations (associated mainly with intermolecular motion) are considered separately. We focus on the solid-state density functional theory (DFT) study of the LF vibrational modes in OSCs with similar molecular structures but significantly different packing motifs (Fn -TCNQ family), followed by estimation of the contributions of these modes to electron–phonon coupling. Experimental techniques for investigation of the OSCs vibrational structure, as well as the obtained data, are briefly described. Dramatic impact of the number of molecules in the unit cell on the vibrational spectrum and electron–phonon coupling is highlighted. This chapter is organized as follows: we begin with a brief review of the methods for theoretical and experimental descriptions of OSC vibrations in Section 15.2 and provide an example of solid-state DFT computations of LF spectra for the Fn -TCNQ crystals. In Section 15.3, we review computational and experimental approaches for estimation of electron–phonon coupling and the corresponding results reported for various OSCs. Theoretical impacts of various vibrational modes on charge transport in Fn -TCNQ crystal family are discussed in Section 15.4. Finally, we formulate the conclusions on the relationship between the crystal structure, vibrations, and electron–phonon interaction in OSCs in Section 15.5.

15.2 Theoretical Treatment of Coupling Between Intraand Intermolecular Vibrations in Low-Frequency Region LF vibrations of organic crystals are complex due to strong coupling between the intra- and intermolecular vibrations [7]. Rigorous theoretical description of LF vibrations requires the use of periodical ab initio or DFT computations [5]. Reliability of the calculations is typically checked by comparison of the calculated vibrational spectrum, e.g. infrared (IR) absorption or Raman, with

15.2 Theoretical Treatment of Coupling

the experimental one. In this section, we review state-of-the-art theoretical and experimental approaches to the OSC vibrational structure including the LF range and discuss the impact of crystal structure on the LF vibrations using as an example the Fn -TCNQ crystal family. There are two types of phonons (vibrations): optical and acoustic [8]. Optical phonons have a nonzero frequency at the Γ-point (center) of the Brillouin zone and show no dispersion near this long wavelength limit, while acoustic phonons have zero frequency at the Γ-point [9]. In contrast to acoustic ones, optical phonons are directly detectable by vibrational spectroscopy. Only optical phonons are discussed in this chapter. The high-frequency region of vibrational spectrum has been thoroughly investigated, since isolated molecule approximation gives an adequate description of intramolecular vibrations. We are interested in an accurate description of coupling between intra- and intermolecular vibrations in crystals; therefore, the LF region from 0 to 6 THz (0–200 cm−1 , or above 50 μm) is considered below. Different theoretical approaches have been applied in an attempt to interpret the LF spectra of crystalline compounds, e.g. isolated-molecule calculations [10], rigid-molecule approximation [11], empirical force field methods (molecular dynamics) [12], and solid-state (periodic) DFT computations [5, 13–15]. Isolated-molecule calculations cannot reproduce LF spectra in principle due to the neglect of intermolecular interactions [16]. Rigid-molecule approaches are unable to predict energies of vibrations containing mixed inter- and intramolecular vibrations in the LF region [5]. Force field methods have many downfalls due to the overgeneralization and limited transferability of computational parameters [17]. Therefore, to correctly calculate the LF spectrum for a crystalline material, methods that consider the periodic boundary must be used. Solid-state DFT is most commonly applied to perform this type of calculation, and a number of techniques and programs exist that differ in how they define the basis sets defining the electronic density. These may either be localized basis sets (such as the atomic-like orbitals used in DMol [18] and SIESTA [19] and the Gaussian wave functions used in the CRYSTAL [20]) or plane-wave basis sets (such as those implemented in CASTEP [21], VASP [22, 23], CPMD [24], CP2K [25], and Quantum Expresso [26]). 15.2.1

Computations of IR and Raman Spectra by Solid-State DFT

The codes mentioned at the end of the previous section are widely used for calculations of the structural and spectroscopic properties of molecular crystals, e.g. see [9, 27–34]. At the first step, the positions of all atoms in the cell are optimized. The cell parameters are usually borrowed from experiment. In some cases, cell parameters are optimized [9, 30, 34] and in some cases not [29, 33]. It should be noted that the change in the volume of a simulated cell of molecular crystals appears to be insignificant after the full optimization [9, 34–36]. Harmonic frequency calculations verify that the computed structure corresponds to the global/local potential energy surface minima. At the second step, the IR intensities and Raman activities are evaluated. For simplicity, we consider computations of IR/Raman spectra in harmonic and anharmonic approximations separately.

427

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15 Intra- and Intermolecular Vibrations of Organic Semiconductors

IR and Raman spectra are often computed in the so-called double harmonic approximation [37]. In this approximation, the potential energy surface is truncated at the second order, and the property surface (dipole moment and polarizability) at the first order [38]. In the CRYSTAL package, the IR intensities are computed from the dipole moment derivatives [39], while the Raman activities are computed through a coupled perturbed Hartree–Fock/Kohn–Sham method [40]. IR and Raman intensities are evaluated by CASTEP [21] using either density functional perturbation theory (DFPT) [41] or supercells and finite displacements [42]. The vibrational frequencies and mode eigenvectors at the zone center (Γ point) are computed using the DFPT routines implemented VASP [22, 23]. Raman intensities can be evaluated by the Python program VASP_RAMAN.PY [43], which uses the VASP package as back end. IR intensities of the normal modes can be also evaluated [44]. Computation of the anharmonic IR spectrum requires performing of the MD simulations. It is obtained as the Fourier transform of the autocorrelation function of the classical dipole moment M [45], calculated at each point of the MD trajectory: ) ( ∞ 4π𝜔 ei𝜔t ⟨M(t)M(0)⟩ dt (15.1) F(𝜔)Re I(𝜔) = ∫0 𝜀0 chn where I(𝜔) is the relative IR absorption at frequency 𝜔, T is the temperature, k B is the Boltzmann constant, c is the speed of light in vacuum, 𝜀0 is the vacuum permittivity, n is the refractive index (which is treated as constant), and F(𝜔) is a quantum correction factor. Different suggestions are proposed on the particular shape of F(𝜔) [46, 47]. The dipole moment function is obtained by the Berry-phase approach of Resta [48], as implemented in CPMD [24], or using condensed-phase optimized molecular potentials for atomistic simulation studies (COMPASS) force field [49]. The calculated IR spectra are in good agreement with the experimental spectrum found at a nonzero temperature, e.g. see [50, 51]. The evaluation of the Raman intensities using CPMD is not straightforward [52]. Therefore, in interpretation of the LF IR and Raman spectra of molecular crystals, the power spectrum is sometimes used, e.g. see [53, 54]. Nowadays CRYSTAL [13, 14, 16, 55–58], CASTEP [59–61], and VASP [52, 54, 62] are the most commonly used codes for evaluation of IR and Raman spectra of crystalline materials in the THz region. Combined use of IR/Raman spectroscopic and solid-state DFT computations makes it possible to identify and quantify non-covalent interactions [14] and polymorphism [13, 54] in molecular crystals, as well as to perform the molecular characterization in solid state [57, 58, 63]. This approach is particularly effective in the study of molecular crystals with similar chemical structures [56, 61, 64]. Vibrational spectroscopy (IR absorption and Raman scattering) and neutron scattering methods provide experimental verification of the results of calculations. The former detects only vibrational modes at the Γ-point of the Brillouin zone because of zero momentum of phonons. Their intensities are subject to symmetry-dependent selection rules and depend, respectively, on the dipole and polarizability derivatives, whose calculation requires the knowledge of the electronic response to the nuclear motion. On the other hand, incoherent inelastic

15.2 Theoretical Treatment of Coupling

neutron scattering (INS) intensities depend only on the vibration frequencies and eigenvectors and the known scattering cross sections of atoms. Since the neutron wavelength is comparable to the interatomic spacing, the INS spectrum is complicated by the wave vector dependence of the vibrational modes. Experimental studies of the LF vibrations in OSCs are rare. To the best of our knowledge, IR spectra of OSCs below 100 cm−1 have not been reported yet. INS spectra were reported for naphthalene and anthracene [65]. Raman spectra below 100 cm−1 were reported for a few OSCs: naphthalene [66], anthracene [66], tetracene [66, 67], rubrene [67, 68], pentacene [69], oligothiophenes [70], picene [71], BTBT derivatives [54], and TCNQ [72] and its fluorinated derivatives – F2 -TCNQ and F4 -TCNQ [73]. These data were used for polymorph identification [54, 70] and verification of the results of the calculations [73]. 15.2.2 Low-Frequency Vibrations of Crystals Formed by Structurally Close Molecules In isolated-molecule calculations both translation and rotation components are projected out of the Hessian matrix. In solid-state calculations, only translations are projected out. This results in 3N a – 6 normal modes in isolated-molecule calculations [74] and 3N a – 3 normal modes in solid-state calculations [75], where N a is the number of atoms in a nonlinear molecule or a primitive cell, respectively. It is important to distinguish between a primitive (reduced) cell and crystallographic unit cell: the first is a unit cell of minimum volume; the second one is chosen with the symmetry of the crystal in mind [75–77]. The primitive cell is the only unit cell that is relevant for spectroscopy [74]. Three types of vibrations are usually considered in molecular crystals: intramolecular vibrations, translations, and librations [75, 78, 79]. It should be noted that vibrational modes often have mixed character, and the contribution of the particular type of vibration to the considered mode could be found [80]. In crystals formed by molecules with similar structure, intramolecular vibrational modes are rather similar as they are determined mainly by the molecular structure. Their frequencies are usually higher than 200 cm−1 . On the contrary, the intermolecular modes are determined by crystal packing, and their frequencies are typically below 200 cm−1 . Crystal packing can vary greatly in the series of crystals formed by similar molecules. This can be illustrated on crystal family of fluorinated tetracyanoquinodimethanes Fn -TCNQ, n = 0, 2, 4. Crystal packing of Fn -TCNQ, n = 0, 2, 4 [81] are shown in Figure 15.1. TCNQ crystallizes in a brickwork motif [82], in which TCNQ molecule interacts with four neighbors face to face, forming π-stacked 2D layers. These layers abut in a herringbone-like manner producing wavelike CH· · ·N bonded layers (S7 in Figure 7.3). F2 -TCNQ crystal consists of flat layers lying on symmetry planes. The molecules of one layer are located opposite the voids of the adjacent layer; thus each molecule interacts with eight neighbors face to face forming unusual 3D brickwork packing motif. F4 -TCNQ crystallizes in a classic herringbone motif. For more details on the crystal packing of these crystals, see Refs. [73, 83]. Comparative investigation of TCNQ and F2 -TCNQ crystals is of particular interest because their crystal packing differs slightly. The main difference is that

429

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15 Intra- and Intermolecular Vibrations of Organic Semiconductors

(a)

(b)

(c)

Figure 15.1 Crystal packing of crystalline TCNQ (a), F2-TCNQ (b), and F4-TCNQ (c).

the layers in the TCNQ stack in ribbon-like structures, whereas in F2 -TCNQ they stack in flat layers. Raman spectroscopy of F2n -TCNQ crystals shows significant differences in the LF region (Figure 15.2). Remarkably, the experimental frequency of the lowest Raman-active band in F2 -TCNQ (ca. 77 cm−1 ) is about twice higher than that in TCNQ (ca. 39 cm−1 ) and about 1.5 times higher than that in F4 -TCNQ (ca. 55 cm−1 ). Theoretical frequencies of the Raman-active vibrations for Fn -TCNQ crystals are in reasonable agreement with the experimental Raman data; therefore, the experiment and the solid-state DFT computations are in good agreement. Theoretical analysis of relative contributions of intra- and intermolecular vibrations to LF modes reveals 10 modes with at least 20% intermolecular character [80] in TCNQ and only 4 in F2 -TCNQ, c.f. Tables 15.1 and 15.2. In Table 15.2, there are no modes with translational character in the F2 -TCNQ crystal. The different numbers of modes with significant intermolecular character in TCNQ and F2 -TCNQ are associated with the different numbers of molecules per primitive cell Zred . In the TCNQ family, only F2 -TCNQ is characterized by Zred = 1. In the rigid-molecule approximation, the total number of intermolecular vibrations depends on Zred as follows: 3Zred − 3 are translations, and 3Zred are librations [78, 79]. This leads to the absence of translations in crystals with Zred = 1. Translations in molecular crystals are usually characterized by the

Relative intensity (a.u.)

15.2 Theoretical Treatment of Coupling

1.0 0.8 0.6 0.4 0.2 0.0

TCNQ

1.0 0.8 0.6 0.4 0.2 0.0

F2-TCNQ

1.0 0.8 0.6 0.4 0.2 0.0

F4-TCNQ

0

50

100

150

200

Wavenumber (cm–1)

Figure 15.2 Low-frequency Raman spectra for polycrystalline TCNQ (upper panel), F2 -TCNQ (middle panel), and F4 -TCNQ (lower panel). Blue lines are the experimental data. Frequencies of the Raman-active vibrations computed by solid-state DFT (B3LYP/6-31G**) are shown by red bars; their height is proportional to the relative orientationally averaged Raman intensity of the corresponding vibration. The features below 20 cm−1 are artifacts. Source: Chernyshov et al. 2017 [73]. Reproduced with permission of American Chemical Society.

lowest-frequency values [73, 78, 79]. The absence of translations is the main reason of twofold difference in the minimum frequency between F2 -TCNQ and TCNQ. This is supported by the fact that change of Zred from 2 to 4 (TCNQ as compared to F4 -TCNQ) results in only 1.4-fold impact on the minimal frequency [73]. The left panel of Figure 15.3 collates frequencies of the vibrations for the TCNQ family obtained with solid-state and isolated-molecule calculations. Two conclusions follow from this figure: (i) the lowest-frequency value changes monotonically in the considered molecules, whereas this value changes non-monotonically in the crystals; (ii) the number of LF modes is similar in isolated-molecule calculations, whereas this number is proportional to the Zred value in the considered crystals, in line with the findings reported in the previous paragraph. To show the generality of these findings, we consider the crystal family of ortho-, meta-, and para-dicyanobenzenes (1,2-bCN2 [84], 1,3-bCN2 [85], 1,4-bCN2 [86]). Note that both 1,2-bCN2 and 1,4-bCN2 exist in two different crystalline forms (polymorphs) [87, 88]. Only one polymorph was considered for 1,2-bCN2 and 1,4-bCN2 crystals. As a result, Zred equals to 1, 2, and 4 for the dicyanobenzene family (Figure 15.3). Note that crystal packing of 1,4-bCN2 (Z red = 1) is very similar to that in F2 -TCNQ except that layers formed

431

432

15 Intra- and Intermolecular Vibrations of Organic Semiconductors

Table 15.1 Relative contributions (%) of intra- and intermolecular vibrations into LF modes of crystalline TCNQ according to the solid-state B3LYP/6-31G**a) computations. Intermolecular contribution Frequency

Intramolecular contribution (%)

Translation (%)

Libration (%)

IR

Raman

37.98

9.5

0.0

90.5

I

A

40.77

50.6

49.4

0.0

A

I

45.88

24.3

75.7

0.0

A

I

69.29

71.7

28.3

0.0

A

I

72.53

23.6

0.0

76.4

I

A

83.03

37.8

0.0

62.2

I

A

86.74

16.5

0.0

83.5

I

A

88.48

70.5

29.5

0.0

A

I

102.25

43.0

0.0

57.0

I

A

108.34

99.0

1.0

0.0

A

I

109.58

46.4

0.0

53.6

I

A

115.24

91.0

9.0

0.0

A

I

117.40

97.8

2.2

0.0

A

I

120.99

98.9

1.1

0.0

A

I

126.74

98.8

1.2

0.0

A

I

139.95

90.9

0.0

9.1

I

A

143.68

99.6

0.0

0.4

I

A

153.53

99.4

0.6

0.0

A

I

155.86

96.7

0.0

3.3

I

A

159.79

99.1

0.0

0.9

I

A

164.67

98.2

0.0

1.8

I

A

167.52

96.7

0.0

3.3

I

A

170.80

98.6

1.4

0.0

A

I

180.10

92.8

0.0

7.2

I

A

180.32

92.8

0.0

7.2

I

A

IR/Raman-active (inactive) modes are indicated by “A”(“I”). a) Grimme dispersion correction does not change “form” and wavenumber of low-frequency modes significantly.

by 1,4-bCN2 molecules do not lie on the symmetry plane, resulting in a nonzero angle between the layer and the molecule’s plane. The details of the crystal packing of the 1,2-bCN2 and 1,3-bCN2 crystals are given in Ref. [56]. The right panel of Figure 15.3 collates frequencies of the vibrations for the dicyanobenzenes family obtained with solid-state and isolated-molecule calculations. The lowest vibrational frequency in isolated-molecule calculations increases in the 1,4-bCN2 , 1,2-bCN2 , 1,3-bCN2 sequence. In solid state, the lowest vibrational frequency changes in the opposite fashion. It is ca. two times higher in 1,4-bCN2 (Z red = 1) crystal as compared with 1,2-bCN2 (Z red = 2) and

15.2 Theoretical Treatment of Coupling

Table 15.2 Relative contributions (%) of intra- and intermolecular vibrations into low-frequency modes of crystalline F2 -TCNQ according to the solid-state B3LYP/6-31G**a) computations. Intermolecular contribution Frequency (cm−1 )

Intramolecular contribution (%)

75.69

Translation (%)

100

Libration (%)

IR

Raman

0

0

A

I

82.15

56.7

0

43.3

I

A

88.22

24.3

0

75.7

I

A

108.99

79.3

0

20.7

I

A

115.85

100

0

0

A

I

116.12

100

0

0

A

I

0

59.1

I

A

0

0

A

I

97.1

0

2.9

I

A

95.1

137.6

40.9

152.07

100

152.35 171.11

0

4.9

I

A

172.2

100

0

0

A

I

174.78

100

0

0

I

A

IR and Raman activities of the considered vibrations indicated by “A” in the last two columns, respectively. a) Grimme dispersion correction does not change “form” and wavenumber of low-frequency modes significantly.

150

100

50

0 (a)

200 Wavenumbers (cm–1)

Wavenumbers (cm–1)

200

Zred = 1

Zred = 2

Zred = 4

F2-TCNQ

TCNQ

F4-TCNQ

150

100

50

0 (b)

Zred = 1 1,4-bCN2

Zred = 2

Zred = 4

1,2-bCN2 1,3-bCN2

Figure 15.3 Calculated vibrational frequencies in crystals (blue) and the corresponding isolated molecules (red). The lines are guides to the eye showing the behavior of the lowest vibrational frequency. (a) The TCNQ family (b) the dicyanobenzenes family.

433

434

15 Intra- and Intermolecular Vibrations of Organic Semiconductors

1,3-bCN2 (Z red = 4). These results are in agreement with those obtained for TCNQ family and indicate the universality of the relation between the lowest vibrational frequency and Zred . To summarize, crystals with one molecule per primitive cell (Zred = 1) are characterized by significantly higher values of the lowest vibrational frequency with respect to crystals with Zred > 1. LF vibrations in crystals significantly differ from those ones in gas phase; thus the rigid-molecule approximation is not applicable for calculation of IR and Raman spectra, and more rigorous treatment, e.g. solid-state DFT, is required. These findings are important for research of OSCs as intermolecular vibrations have significant impact on charge transport as mentioned above. In Sections 15.3 and 15.4, we discuss this impact in detail.

15.3 The Role of Inter- and Intramolecular Vibrations in Charge Transport 15.3.1

Local and Nonlocal Electron–Phonon Coupling

Intra- and intermolecular vibrations affect the charge transport in OSC due to the electron–phonon interaction (coupling). To introduce the concept of electron–phonon interaction, it is instructive to consider the tight-binding approximation and represent an OSC as a number of lattice sites (molecules) [1, 7]. The electron wave function is decomposed over a basis set of the molecular orbitals for isolated sites – eigenstates of the molecular electronic Hamiltonian, H mol [89]. Typically, only one molecular orbital 𝜑m per site – HOMO for hole transport or lowest unoccupied molecular orbital (LUMO) for electron transport – is considered. The electronic Hamiltonian of the entire system is written as [7, 90] (second quantization formalism is used) ∑ ∑ He = 𝜀m a+m am + tmn a+m an (15.2) m

mn

where am + and am are the creation and annihilation operators for charge carrier on site m, correspondingly; 𝜀m is the energy of charge carrier on this site 𝜀m = ⟨𝜑m (Rm )|Hmol |𝜑m (Rm )⟩

(15.3)

and t mn (denoted sometimes J mn or V mn ) is the charge transfer integral that describes the electronic coupling (interaction) between sites m and n: tmn = ⟨𝜑m (Rm )|Hmol |𝜑n (Rn )⟩

(15.4)

In Eqs. (15.3)–(15.4), Rn are the positions of the nuclei. Thus, in the tightbinding approach, the system description splits into the two tasks: atomic-level computation of 𝜑i , 𝜀n , and t nm and then construction of the wave function for the entire system from these quantities using Eq. (15.2). The 𝜀m and t mn values in Eqs. (15.3)–(15.4) depend on the positions of the nuclei (R) and hence are coupled to the vibrational modes of the system. Similarly, changes in the charge density (e.g. in the process of charge transport) should induce the geometry relaxation (changes in R), which can be regarded

15.3 The Role of Inter- and Intramolecular Vibrations in Charge Transport

as generation of vibrations according to the Franck–Condon principle. The modulation of the electronic structure (e.g. 𝜀m and t mn ) by vibrations and the geometry change following the charge density change is in essence the electron–phonon coupling. The vibrational modulation of 𝜀m is called local (Holstein-type) electron–phonon coupling, while the modulation of t mn is called nonlocal (Peierls-type) coupling. To describe electron–phonon coupling, nuclear motion must be treated, and this is commonly done in the harmonic approximation (see Section 15.2.1). In principle, anharmonicity can be considered; however, such calculations are very cumbersome [91]. Electron–phonon coupling is typically considered as a perturbation of the system Hamiltonian, and only the linear terms are retained. These approximations yield the so-called Holstein–Peierls Hamiltonian [1]: ∑ (0) ∑ (0) 𝜀m a+m am + tmn a+m an H = He + Hph + He−ph = +N

−1∕2

∑∑

+N

−1∕2

∑∑

m

m

ℏ𝜔i (gmi ∗ b+ik

m,n

+ gmi bik )a+m am

i,k

m,n i,k

∗

ℏ𝜔i (g mni b+ik + g mni bik )a+m an +

∑ i

) ( 1 ℏ𝜔i b+i bi + 2 (15.5)

+

where bik and bik are creation and annihilation operators for the phonon of ith vibrational mode with (dimensionless) wave vector k, 1 𝜕𝜀m (15.6) gmi = ℏ𝜔i 𝜕qi is the local electron–phonon coupling constant describing modulation of 𝜀m by this mode (qi is the dimensionless atomic displacement vector for mode i), and g mni =

1 𝜕tmn ℏ𝜔i 𝜕qi

(15.7)

is the nonlocal electron–phonon coupling constant describing the modulation of t mn by this mode. Since translation symmetry requires that gmi = eikRm gi [1], index m in Eq. (15.6) will be omitted below. In Eq. (15.7), m and n indices will be retained to distinguish g mni values for various charge transfer directions in the crystal. Note that g i and g i defined by Eqs. (15.6)–(15.7) are dimensionless; there are other definitions for these quantities (e.g. in Refs. [68, 69], g i are expressed in the energy units), and hence comparing the estimates for electron–phonon coupling from different reports should be done with care. The g i values are usually significant for intramolecular (high-frequency) vibrations, while significant g mni are associated with intermolecular (LF) vibrations. Holstein–Peierls Hamiltonian is a workhorse of state-of-the-art studies of charge transport in OSCs, although the applicability of the perturbation approach to electron–phonon coupling was questioned [4] because of significant modulation of t mn by thermal vibrations. Cumulative characteristic of the local electron–phonon coupling is the polaron-binding energy, 𝜀p , which describes the change in the site energy when the charge is placed at it [92]. Alternatively, the local electron–phonon coupling can be quantified by the intramolecular reorganization energy, 𝜆, which

435

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15 Intra- and Intermolecular Vibrations of Organic Semiconductors

quantifies the total energy cost due to the geometry relaxation when the charge transfers from one molecule to another. In archetypical OS such as oligoacene crystals, the intramolecular reorganization energy constitutes the main part of the total reorganization energy, 𝜆 [93, 94]; we therefore consider 𝜆 ≈ 𝜆intra below. Assuming that the vibrational modes are similar in the neutral and ionized states, the polaron-binding energy is directly proportional to the reorganization energy: 𝜆 ≈ 2𝜀p [1]. The 𝜀p and 𝜆 values are related to g i in the following way ([1]): 𝜀p =

𝜆 ∑ 𝜆i ∑ ℏ𝜔i gi2 = = 2 2 i i

(15.8)

Therefore, the 𝜆i values describe the contributions of the vibrational modes to the local electron–phonon coupling in the units of energy. It is worth noting that in molecular spectroscopy and electron transfer theory, this contribution is frequently expressed in dimensionless Huang–Rhys factors: Si = g i 2 = 𝜆i /ℏ𝜔i [95]. A cumulative characteristic of the nonlocal electron–phonon coupling is the lattice distortion energy, L ([1]): L=

∑ i,mn

Li mn =

∑ g 2mni i,n

ℏ𝜔i

(15.9a)

The larger the 𝜆 and L values, the stronger the coupling (interaction) between the electron and phonon subsystems, i.e. the higher the impact of vibrations on charge transport.

15.3.2 Three Mechanisms Underlying Impact of Electron–Phonon Interaction on Charge Transport in Organic Semiconductors As mentioned in Section 15.3.1, direct solution of the charge transport problem is not practical because of its extreme complexity – there are too many factors of the same order that affect the charge mobility. Various approximations resulted in multiple charge transport models, the most rigorous being the polaron (Holstein–Peierls, based on Eq. (15.5)) [96] and transient localization (dynamic disorder; see below) [4] approaches. In both, charge mobility 𝜇 is determined by the interplay of intermolecular charge delocalization (favorable for charge transport) and charge localization at single or few sites (hindering charge transport). The former is provided by strong electronic coupling, i.e. large t, while the latter is facilitated by strong electron–phonon coupling, i.e. large 𝜆 and/or L. If the electronic coupling overwhelms the electron–phonon coupling, significant intermolecular charge delocalization enables efficient coherent charge transport with high 𝜇 and band-like temperature dependence (d𝜇/dT < 0) observed in several OS [1, 2]. Otherwise, electron–phonon coupling induces charge localization resulting in hopping charge transport with low 𝜇 and thermally activated behavior (d𝜇/dT > 0), which is observed in most OS [1, 2]. Simple reasoning provided, e.g. in Ref. [97], suggests that charge localization occurs in the case 𝜆 > 2t for 1D, 𝜆 > 4t for 2D, and 𝜆 > 6t for 3D charge transport [97]. Therefore, the higher dimensionality of charge transport is favorable for 𝜇.

15.3 The Role of Inter- and Intramolecular Vibrations in Charge Transport Dynamic disorder

E

E E0

(a)

Self-localization

σ = √2LkT

E0

Phonon-assisted charge transfer

E εp

(b)

(c)

Figure 15.4 The three mechanisms underlying the impact of electron–phonon coupling on charge transport. For simplicity, the rigid-body approximation is used. Top panels describe the spatial representation, with sites depicted by blue ellipsoids, their positions in absence of nuclear motion illustrated by dashed lines, and charge carrier shown by circles. Bottom panels correspond to the energetic representation, with the blue line describing the carrier energy at these sites. (a) Thermal intermolecular vibrations introduce dynamic disorder – variance of t and hence in the charge carrier energy at different sites; this disorder is determined by T and L. (b) Charge carrier induces geometry relaxation that forms the potential well and can facilitate self-localization of the carrier. The depth of the well is determined by 𝜀p. (c) Thermal vibrations temporally increase tmn , enhancing the probability of charge hopping between the sites.

We distinguish three mechanisms underlying the impact of electron–phonon coupling on charge transport in OS: dynamic disorder, charge self-localization, and phonon-assisted charge transport. These mechanisms of electron–phonon coupling are illustrated in Figure 15.4 and discussed in the following paragraphs. (i) Dynamic disorder: First, thermally populated vibrations introduce the dynamic (transient) disorder in 𝜀 and t values (Figure 15.4a) and thus determine the energy landscape for charge transport. At ambient temperature, mainly LF vibrations (𝜔 < 200 cm−1 for 300 K) associated primarily with intermolecular motions are populated; they also have large amplitudes due to low 𝜔. Accordingly, these vibrations introduce significant dynamic (i.e. varying in time) disorder in t and can induce charge localization in the energetic “valleys” (Figure 15.4). This is the basic idea of the transient localization scenario suggested by S. Fratini et al. [4] and dynamic disorder model by A. Troisi et al. [3]. The variance of t, 𝜎 t 2 , and hence the impact of dynamic disorder on charge transport increases with L and temperature [69]: 𝜎t 2 = ⟨(Δtmn )2 ⟩ = 2LkT

(15.9b)

where k is the Boltzmann constant. The discussed mechanism of the electron–phonon coupling can be also regarded as scattering of the charge carriers by equilibrium phonons [1].Dynamic disorder was suggested to set the limit for 𝜇 in high-mobility OS, where large t and/or low 𝜆 enable significant intermolecular charge delocalization and coherent charge transport [3, 4, 98]. In this case, 𝜇 should decrease with increasing T since the latter enhances the dynamic disorder according to Eq. (15.9b), therefore, limiting the charge delocalization [99]. The decrease of 𝜇 with T, (band-like 𝜇(T) dependence) was indeed observed in several high-mobility OS [2]. Importantly, within the dynamic disorder (transient localization) model, the lowest vibrational frequency 𝜔0 has dramatically impacted the charge

437

438

15 Intra- and Intermolecular Vibrations of Organic Semiconductors

mobility: the higher 𝜔0 , the higher 𝜇. Strong 𝜇(𝜔0 ) dependence is because lower 𝜔0 results in larger displacements, i.e. larger 𝜎 t . (ii) Self-localization: The second aspect of the electron–phonon interaction is that a moving charge carrier induces changes (“relaxation”) of the molecular geometries and intermolecular distances (Figure 15.4b) and, therefore, alters the 𝜀 and t values. This can facilitate charge localization at one or several sites resulting in a polaron – charge carrier “dressed” by vibrations – or, in other words, by lattice distortion [7, 95]. The geometry relaxation can be interpreted as generation of extra (nonequilibrium) phonons. Self-localization is usually associated with local electron–phonon interaction [1]; however, nonlocal electron–phonon coupling can be also important since intermolecular delocalization alters 𝜆 significantly [97]. Self-localization is a dominating electron–phonon coupling mechanism for low-𝜇 OS and is responsible for hopping charge transport therein [95, 98]. (iii) Phonon-assisted charge transport: Thermal intramolecular vibrations provide the very possibility of charge transport in the hopping models, e.g. in widely used Marcus [100] or small-polaron [95] models. They are responsible for fluctuations of the energies of the charged (occupied) site and adjacent neutral (vacant) ones and making charge transfer possible when the energy of the initial state becomes equal or higher to that of the final state. Since the thermal fluctuations increase with T, thermally activated 𝜇 behavior (d𝜇/dT > 0) is observed in this case. However, exactly these vibrations are responsible for charge localization (see above), and the stronger the local electron–phonon coupling (higher 𝜆), the deeper the energy well that charge needs to overcome and hence the lower 𝜇. The situation is somewhat different for LF vibrations. For a long time, it has been debated whether they can assist charge transport. The physics behind the hypothetical phonon-assisted charge transport is that if the charge is localized at a single site, its hopping rate is proportional to the average t 2 , which increases due to the thermal motion [101]: ⟨tmn 2 ⟩ = tmn0 2 + 𝜎t = t0 2 + 2LkT

(15.9)

The higher the L and T, the higher the t variance 𝜎 t (see Eq. (15.8)), and thus the higher the charge transfer (hopping) rate – in contrast with the case of delocalized charges. However, this hypothesis does not consider the dynamical disorder in 𝜀 introduced by the intermolecular vibrations, which should decrease the hopping rate. Which factor, namely, temporally increased t mn favorable for charge transport or detrimental dynamical disorder that has larger impact, deserves further investigation? To summarize, strong electronic coupling between the molecules (large t) can provide charge delocalization resulting in high 𝜇 (typically above 1 cm2 /(V s) [2]). However, if the electron–phonon coupling overwhelms the electronic coupling (𝜆 and/or L are much larger than t), charge carrier undergoes self-localization at a single or few sites resulting in hopping charge transport with low 𝜇 (typically below 0.1 cm2 /(V s) [102]). Local electron–phonon coupling associated with intramolecular vibrations plays a key role in limiting the charge mobility in this

15.3 The Role of Inter- and Intramolecular Vibrations in Charge Transport

case. Intermolecular vibrations can hypothetically improve the charge transport since they temporally increase t mn and hence the hopping rate; however, the probable positive impact of phonons on charge transport is still under debate. On the contrary, if a charge is delocalized, the central role in limiting 𝜇 plays by the nonlocal electron–phonon coupling associated mainly with intermolecular vibrations, which introduce strong dynamical disorder in t. If LF vibrations are too strong (𝜎 2 = 2LkT > t mn 2 ), the coherent transport is destroyed, resulting in the abovementioned hopping transport [95]. Therefore, to achieve high band-like charge mobility, both local and nonlocal electron–phonon couplings should be reduced. 15.3.3 Computational and Experimental Approaches to Electron–Phonon Coupling in Organic Semiconductors Computation is the most widely used method for obtaining 𝜆 and L values and their decomposition over normal vibrational modes. Quantum chemical approaches (e.g. DFT), molecular mechanics, and QM/MM are widely used for this purpose. To obtain 𝜆 and 𝜆i , calculations for isolated molecules are usually considered to be sufficient [1, 95]. Probably the simplest approach for estimation of the total reorganization energy, 𝜆, is the four-point scheme [1], which is based on calculation of the net energy relaxation for the two molecules, one of which loses and the other acquires an excess charge. A more complicated method, normal mode decomposition (NMD), is employed to reveal the contributions from various vibrational modes, 𝜆i [1, 95]. In this approach, equilibrium geometries and vibrational modes of the neutral and charged (i.e. doublet) states of the molecule are calculated first. Then the difference between the atomic coordinates of the neutral (RN ) and charged (RC ) states, ΔR = RN − RC , is projected on the displacement vectors of the mentioned vibrational modes, Qi , yielding the displacements in units of length and in the normal coordinates, ΔQi . Note that Qi are expressed√ are related to the dimensionless coordinates qi via Qi = (ℏ∕2Mi 𝜔)qi , where Mi is the reduced mass of the ith vibrational mode. These displacements determine 𝜆i according to the textbook formula for the potential energy of the classical harmonic oscillator: m (ΔQi )2 𝜆i = i (15.10) 2𝜔i A slight modification of the NMD method exploits the calculation of the HOMO (LUMO) energy derivatives along the normal modes, 𝜕𝜀H(L) /𝜕Qi , and using Eqs. (15.6) and (15.8) [1]. It is worth noting that in this version of the NMD method, Qi for vibrational modes of either an isolated molecule or crystal can be used (see Ref. [68] for 𝜆i in rubrene obtained using the two variants of the NMD method). However, if only local electron–phonon coupling is under study, it seems more practical to use the modes for isolated molecules, which are calculated much easier. The nonlocal electron–phonon coupling constants, Li , can be determined in the following way. First, vibrational modes of the crystal, Qi , are obtained. Then, the derivatives of the transfer integrals along these modes, dt/dQ, are calculated

439

440

15 Intra- and Intermolecular Vibrations of Organic Semiconductors

from the change of the t values between the equilibrium and displaced states. The t values are usually obtained in the dimer approximation [1]. Calculation of t considering the periodic boundary conditions, e.g. by fitting the periodical electron density with the Wannier functions [103], seems to be more rigorous; however, it is much more complicated and rarely used. The inaccuracy from neglect of the PBC has not been estimated yet. An alternative approach for estimation of the electron–phonon coupling exploits molecular dynamics or QM/MM calculations with successive calculation of ε and t over simulation trajectories [104, 105]. The Fourier transformation of the autocorrelation function yields the spectrum of contributions of various vibrations to the electron–phonon coupling. The advantage of this method is that it allows estimation of the impact of the acoustic modes, while the NMD described above typically considers only Γ-point (optical) phonons unless supercells are regarded [104]. The acoustic phonons were concluded to play a comparable role with the optical ones in charge transport in Refs. [104, 105], while in Ref. [73] their impact was suggested to be insignificant. Thus, the role of acoustic phonons in charge transport in OS requires further investigation. No experimental approaches have been suggested for direct measurement of L and 𝜆 values in OS. Nevertheless, indirect estimation of these values from the experimental data using semiempirical schemes is possible [106]. The technique suitable for estimation of the local electron–phonon interaction is ultraviolet photoelectron spectroscopy (UPS) [107], while information on the nonlocal interaction can be obtained from angle-resolved ultraviolet photoelectron spectroscopy (ARUPS) [108]. To reveal the contributions of various modes to local and nonlocal electron–phonon coupling, Raman spectroscopy was used, although only for a narrow class of OSCs – charge transfer complexes [109, 110]. In Ref. [111], transmission electron microscopy (TEM) was applied to extract the degree of dynamical disorder in a number of OS, which is directly related to L. INS – a powerful method for investigation of vibrations in molecular crystals [112, 113] – can be also used for this purpose. This approach gives a reasonable description for the LF vibrations of molecular crystals, e.g. see [50, 114]. However, INS requires extremely high-quality single crystals, which contain a lot of hydrogen atoms (the INS cross section of the hydrogen atom is at least one order of magnitude larger than that of the others) – this is the reason of rare application of this method for studying the electron–phonon interaction in OSCs. 15.3.4

Electron–Phonon Coupling in Various Organic Semiconductors

A number of studies on electron–phonon coupling in various OS have shown that both local and nonlocal contributions weaken with the increase of the size of the conjugated core of the OS molecule. Figure 15.5a illustrates this trend for the local electron–phonon coupling. The 𝜆 value decreases with the molecular size (number of aromatic rings) for several OS series: oligoacenes, oligothiophenes, oligophenyls, and thienoacenes [115]. The reason of the decrease is that for molecules with larger conjugated cores, acquiring or losing the charge after the charge transfer induces weaker changes in the molecular geometry because

15.3 The Role of Inter- and Intramolecular Vibrations in Charge Transport

Reorganization enery (meV)

400 350 300 250 200 150 100 50 2

3

(a) 20 Nonlocal electron–phonon coupling (meV)

4

5

6

7

Number of aromatic rings Holes Electrons

15

10

5

0 2 (b)

3 4 Number of aromatic rings

5

Figure 15.5 The effect of molecular size on local (a) and nonlocal (b) electron–phonon couplings in model OSCs. (a) Reorganization energies as a function of the number of aromatic rings is plotted for (from top to bottom) fused thiophenes, oligophenyls, oligothiophenes, and oligoacenes. Source: Atahan-Evrenk and Aspuru-Guzik 2014 [115]. Reproduced with permission of Springer. (b) Lattice distortion energy – the measure of nonlocal electron–phonon coupling – is plotted as a function of the number of aromatic rings for oligoacenes. Source: Sanchez-Carrera et al. 2010 [101]. Reproduced with permission of American Chemical Society.

of the lower excess charge density. However, the size of the conjugated core is not the only factor determining 𝜆. For instance, the 𝜆 values for thienoacenes, oligothiophenes, and oligophenyls are considerably larger than those for oligoacenes with the same number of the aromatic rings (Figure 15.5a). The significantly lower 𝜆 values for oligoacenes are commonly attributed to the enhanced rigidity of the fused molecular cores, resulting in efficient π-conjugation [116, 117]. Figure 15.5b illustrates weakening of the nonlocal electron–phonon coupling along the main charge transfer direction(s) (decrease of L) with the increase of the conjugated core size. This plot shows that with the increase of the oligoacene length, L gradually decreases for both hole and electron transport [101]. This

441

15 Intra- and Intermolecular Vibrations of Organic Semiconductors

can be rationalized by strengthening of the intermolecular interactions with the increase of the molecular size and hence suppression of intermolecular motion. Indeed, enthalpy of sublimation that can be considered as a quantitative estimate of the intermolecular interaction gradually increases with the molecular size of oligoacene: it is ∼73 kJ/mol for naphthalene [118, 119], ∼100 kJ/mol for anthracene [119, 120], ∼130 kJ/mol for tetracene [119], and ∼155 kJ/mol for pentacene [119]. To the best of our knowledge, the impact of the molecular size on L was not investigated for other OS series. Since low 𝜆 and L are favorable for charge transport, charge mobility 𝜇 is expected to increase with the size of the molecular conjugated core within a series of analogous compounds with similar packings. This was indeed observed for oligoacenes [2, 121–123] and oligothiophenes [124]. For example, 𝜇 is ca. 1 cm2 /(V s) for naphthalene [121], 2 cm2 /(V s) for anthracene [122], 2.4 cm2 /(V s) for tetracene [123], and up to 20 cm2 /(V s) for pentacene [2]. The search for efficient OSCs for the last decade was thus focused mainly on molecules with large and rigid conjugated cores, e.g. condensed aromatics [82]. Nevertheless, some small-molecule OS crystals have also shown high 𝜇 > 5 cm2 /(V s), e.g. F2 -TCNQ [81] and HM-TTF [125]. The origin of the high charge mobility in crystalline F2 -TCNQ and the impact of various vibrations on charge transport in this OS will be discussed in Section 15.4. Figure 15.6 collates the contributions of various vibrational modes to the local (𝜆i = 2𝜀sp ) and nonlocal (Li = 𝜀d ) electron–phonon coupling in two crystalline 15

Holstein, Ag/Ag Peierls, Ag/Au

10

a c

5

b

εsp, εd (meV)

Pentacene LT

15

Holstein, Ag/Ag Peierls, Ag/Au

Pentacene HT

c

10 b

εsp, εd (meV)

442

5 a

0

0

200

400

600

800

1000

1200

1400

1600

Wavenumber (cm–1)

Figure 15.6 Local and nonlocal electron–phonon couplings in pentacene crystal. Contributions to the small polaron-binding energy (red and green bars) and lattice distortion energy (blue) for two polymorphs of pentacene, low-temperature (LT, top) and high-temperature (HT, bottom); insets show the corresponding crystal structures. Source: Girlando et al. 2011 [69]. Adapted with permission of American Institute of Physics.

15.4 Impact of Low-Frequency Vibrations on Charge Transport in Fn -TCNQ Crystal Family

polymorphs of pentacene, so-called low-temperature (LT) and high-temperature (HT) phases, calculated using solid state DFT [69]. The figure clearly illustrates that nonlocal electron–phonon coupling (shown with blue bars) stems mostly from the LF vibrations (𝜔 < 200 cm−1 ), while local coupling (shown with red and green bars) is associated with high-frequency modes (𝜔 > 200 cm−1 ). The former vibrations are strongly related to the intermolecular interaction determined by crystal packing, while the latter are associated with intramolecular vibrations and are less sensitive to the crystal structure. Accordingly, 𝜆i are similar for the two pentacene polymorphs due to the same molecular structure, while Li differ significantly because of different crystal packings (see insets in Figure 15.6). The largest 𝜆i are for the modes in the range between 1000 and 1700 cm−1 . This range corresponds to the collective vibrations of the aromatic rings [106]. Strong modulation of the C—C and/or C=C bond lengths by these vibrations result in significant modulation of the HOMO and LUMO energies, which provide high 𝜆i according to Eq. (15.6). In other OSCs, e.g. oligothiophenes, significant 𝜆i are observed for torsional modes as well [124]. In the LF range, L can originate from different types of vibrations: librations, translations, and intramolecular vibrations [69]; no general correlation between the percentages of these contributions and Li was established. Experimental data for 𝜆 and L, even indirect, are extremely scarce. Nevertheless, UPS and X-ray photoelectron spectroscopy (XPS) data confirm that the reorganization energy decreases with the molecular size [126, 127]. One of the lowest experimental 𝜆 values, 𝜆 = 40 meV, was observed for phthalocyanines [128]. Angle-resolved photoemission spectroscopy (ARPES) confirms [129] that the effect of temperature (thermally populated LF vibrations) on transport bandwidths, i.e. t values, is less pronounced in larger acenes (pentacene) than in smaller (naphthalene and anthracene) ones. Strong dependence of the charge mobility, 𝜇, on the lowest vibrational frequency, 𝜔0 , which is expected for high-mobility OSCs within the dynamical disorder model, is partially confirmed by the experimental data on the OSCs with close molecular structures. Specifically, 𝜔0 decrease after 13 C isotope substitution in rubrene leads to a decrease in 𝜇 by ∼15%, although the band-like character of charge transport is retained, in complete correspondence with the calculations [130]. In addition, within Fn -TCNQ crystal family discussed in Section 15.2.2, the largest 𝜇 was observed in F2 -TCNQ [81], where 𝜔0 is twice larger than in the two other crystals [73]. Therefore, we conclude that the results of the computations are generally reproduced in the experiment.

15.4 Impact of Low-Frequency Vibrations on Charge Transport in Fn -TCNQ Crystal Family Investigation of crystalline OSCs with similar chemical structures but significantly different packing motifs can through light on the role of intermolecular interactions and nonlocal electron–phonon coupling in OSs. Since the latter is more pronounced for OSCs consisting of very small molecules, the family of

443

444

15 Intra- and Intermolecular Vibrations of Organic Semiconductors

Fn -TCNQ (n = 0, 2, 4) single crystals discussed in Section 15.2.2 is a perfect playground for these investigations. Within this series, F2 -TCNQ shows an outstanding electron mobility of 𝜇 = 7 cm2 /(V s) at room temperature [81], while the other two crystals show significantly lower 𝜇 (1.5 cm2 /(V s) [131] and 0.2 cm2 /(V s) [81], correspondingly). Moreover, crystalline F2 -TCNQ exhibits band-like charge transport with d𝜇/dT < 0 indicating efficient charge delocalization, whereas the other two crystals show temperature-activated transport indicating localized charges [81]. Charge transport in F2 -TCNQ crystal family was addressed theoretically in several studies [73, 83, 97]. Lower 𝜇 in F4 -TCNQ was ascribed to weak electronic coupling between adjacent molecules: calculated maximal t mn in this crystal are as low as ∼30 meV, while in F2 -TCNQ and TCNQ they are ∼70 meV [73, 83, 97]. In other terms, the calculated width of conduction band, W , is much lower in F4 -TCNQ (250 meV) than in TCNQ (500 meV) and F2 -TCNQ (800 meV) [81]. However, significantly different 𝜇 values and qualitatively different 𝜇(T) dependence in F2 -TCNQ and TCNQ cannot be explained by the difference in maximal t values, which are comparable for the two crystals. This cannot be explained by the difference in the local electron–phonon coupling as well: the reorganization energies are similar (𝜆 ∼ 250 meV) for F2 -TCNQ and TCNQ molecules [73, 83, 97]; moreover, 𝜆i spectra are similar [83]. In Ref. [97], outstanding band-like 𝜇 in F2 -TCNQ was attributed to 3D character of the charge transport, while in TCNQ it is only 1D; the difference in dimensionality of charge transport is in accordance with Ref. [81]. The other reason for high 𝜇 in F2 -TCNQ is the suppressed nonlocal electron–phonon interaction [73, 81, 83]. As discussed in Section 15.2.2, the unique crystal structure of F2 -TCNQ with Zred = 1 results in lower number of LF vibrational modes and approximately twice larger value of the lowest vibrational frequency (𝜔0 = 88 cm−1 ) as compared with that for TCNQ (𝜔0 = 38 cm−1 ). According to Ref. [4], L and hence 𝜇 strongly depend on the frequency of a characteristic intermolecular mode (e.g. the lowest vibrational mode with frequency 𝜔0 ): 𝜇 ∝ 𝜔0 3.6 . Recent results of multiscale modeling reported in Ref. [83] corroborate that nonlocal electron–phonon coupling is weaker in the F2 -TCNQ crystal: the relative variance of t, 𝜎 t /⟨t⟩, does not exceed 0.33 in F2 -TCNQ but reaches 0.58 in TCNQ. To gain a more detailed understanding of the impact of vibrations on charge transport in F2 -TCNQ and TCNQ, in the next section we compare the contributions of various vibrational modes (see Section 2.2) to nonlocal electron–phonon interaction along different charge transfer directions, in accordance with our results reported in Ref. [132]. Main charge transfer directions of TCNQ and F2 -TCNQ (t mn > kT) are shown in Figure 15.7; the corresponding t mn values are listed in Table 15.3. Two dimers with the largest t mn in each of the crystals are referred below as Dimers 1 and 2. The former includes molecules 1 and 2 from Figure 15.7; the latter consists of molecules 1 and 3. Figure 15.8 collates the contributions of various vibrations to the nonlocal electron–phonon interaction within Dimer 1 (Li 12 ) and Dimer 2 (Li 13 ). We assume that the larger the Li mn bar, the stronger the impact of the mode on charge transport in direction mn. The bars are divided in accordance with the

15.4 Impact of Low-Frequency Vibrations on Charge Transport in Fn -TCNQ Crystal Family

2

3

1

b a

c (a) 3 2

1 b

a c

(b)

Figure 15.7 Main charge transfer directions of TCNQ (a) and F2 -TCNQ (b) crystals. Directions with tmn > 2kT are shown with green arrows, those with kT < tmn < 2kT are shown with yellow arrows, and those with tmn < kT are omitted (for T = 300 K). Black arrows depict crystal axes. Table 15.3 Main transfer integrals for TCNQ and F2 -TCNQ crystals. Dimer #

Molecules (m–n)

Transfer integral tmn (meV) TCNQ

F2 -TCNQ

1

1–2

60

75

2

1–3

34

54

contribution of librational, translational, and intramolecular motions to the corresponding vibration (see Tables 15.1 and 15.2). Figure 15.8 shows that charge transport in both Dimers 1 and 2 of the crystals under study is most affected by the modes with substantial intermolecular (librational or translational) character. Librations dominate in L for symmetrical dimers: Dimer 1 of TCNQ and Dimers 1 and 2 of F2 -TCNQ, while translations contribute significantly only for asymmetrical Dimer 2 of TCNQ. It is instructive to discuss the modes with the largest Li in a more detailed way. According to Eqs. (15.7) and (15.9a), the Li mn value for a given vibrational mode is determined by its ability to modulate t mn . For n-type transport observed in TCNQ and F2 -TCNQ, t mn in one-electron approximation is determined by the overlapping of the LUMOs of the monomers. LUMO patterns for TCNQ and

445

15 Intra- and Intermolecular Vibrations of Organic Semiconductors

TCNQ

F2-TCNQ 14

Intramolecular Translation Libration

Dimer 1

2

1

10

2 2

0

1

8 6 4

Dimer 2

2

4

0

3

2

Li13 (meV)

1

Dimer 2 1

4

3

2 0

0 40 60 80 100 120 140 160 (a)

Dimer 1

12

Li12 (meV)

Li12 (meV)

4

Li13 (meV)

446

Frequency (cm–1)

80 (b)

100 120 140 160 Frequency (cm–1)

Figure 15.8 Contribution of various vibrational modes to the nonlocal electron–phonon interaction, Li mn , for Dimers 1 and 2 of TCNQ (a) and F2 -TCNQ (b) crystals. The bars are divided according to the percentage of intramolecular (red), translational (green), and librational (blue) motions. Insets show relative orientations of the molecules in the corresponding dimers. Source: Sosorev et al. 2018 [132]. Reproduced with permission of Royal Society of Chemistry.

F2 -TCNQ, as well as charge delocalization within the dimers are illustrated in Figure 15.9. As follows, close contacts between CN groups of adjacent molecules or that between the CN group of one molecule and the phenyl ring of the other are essential for considerable intermolecular electronic coupling. Accordingly, the modes that modulate significantly relative position of these moieties contribute significantly to L. Figure 15.9 shows vibrational patterns (atomic displacements) of the vibrations with the largest Li 12 and Li 13 for TCNQ and F2 -TCNQ. In TCNQ, charge transfer along the Dimer 1 axis is most sensitive to the vibration at 38 cm−1 (Figure 15.8a, top). The vibrational pattern of this mode is shown in Figure 15.10a, b within Dimers 1 and 2, respectively. Substantial impact of this vibration on charge transfer in Dimer 1 (Li 12 = 3.5 meV) stems from the considerable modulation of the contact between the CN1 group of one monomer and phenyl ring of the other: the CN group moves significantly, while the phenyl ring position remains nearly unaffected. This contact is responsible for charge transfer within Dimer 1 (see Figure 15.9). On the contrary, this mode does not alternate the contact between the N atom from CN2 group of one monomer and the π-system of CN2 group of the other monomer, which is responsible for charge transfer within Dimer 2, because of synchronous motion of these groups (Figure 15.10b). This explains negligible Li 13 for this vibration (Figure 15.8a, bottom).

15.4 Impact of Low-Frequency Vibrations on Charge Transport in Fn -TCNQ Crystal Family

F2-TCNQ

TCNQ

(a)

(b)

TCNQ, Dimer 1

F2-TCNQ, Dimer 1

F2-TCNQ, Dimer 2

(c)

(d)

(e)

Figure 15.9 Illustration of intermolecular charge delocalization responsible for charge transport in TCNQ (a) and F2 -TCNQ (b) crystals from the dimer LUMO delocalization. Larger density of dimer LUMO between the molecules corresponds to larger t. (c) LUMO of Dimer 1 of TCNQ. LUMO of Dimer 1 (d) and Dimer 2 (e) of F2 -TCNQ.

In F2 -TCNQ crystal, charge transport along the Dimer 1 and 2 axes is most affected by the modes at 138 and 82 cm−1 , respectively (Figure 15.8b, top). These modes have similar vibrational patterns (Figure 15.10e) consisting of intramolecular vibration (mainly motion of the CN groups) and libration (see Table 15.2). However, in the former mode, CN2 group moves, while CN1 is immobile, and vice versa in the latter mode. Accordingly, the mode at 138 cm−1 shows a dominant impact on charge transfer in Dimer 1 (L12 = 12 meV) due to significant relative ′ displacements of CN1 and CN1 groups (Figure 15.10c) and hence deep modulation of t 12 (Figure 15.10c). On the other hand, the mode at 138 cm−1 does not affect significantly charge transfer along the Dimer 2 axis (Li 13 ∼ 1 meV) because of negligible displacement of the CN2 moiety with respect to the phenyl ring of the second monomer and hence weak modulation of t 13 (Figure 15.10d). On the contrary, the mode at ∼82 cm−1 , which involves considerable motion of the CN2 group, has large Li . Interestingly, the vibrational mode of TCNQ crystal at 83 cm−1 and that of F2 -TCNQ at 82 cm−1 have nearly identical vibrational patterns: out-of-plane motion of CN2 group is mixed with libration. Moreover, they possess one of the largest Li within Dimer 1 (TCNQ) and Dimer 2 (F2 -TCNQ). As follows from Figure 15.7, these dimers have similar structure – the molecules are parallel to each other, and their π-orbitals overlap significantly. This similarity of the dimers structure and molecular structure of TCNQ and F2 -TCNQ explains similar vibrational patterns of the two modes. In-plane librations are also present in both crystals and possess close frequency, 72 and 86 cm−1 for TCNQ and 88 cm−1 for F2 -TCNQ. Two librational modes for TCNQ crystal can be attributed to two molecules in the unit cell.

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15 Intra- and Intermolecular Vibrations of Organic Semiconductors

b

Dimer 1 (J = 60 meV) a 2

TCNQ ω = 38 cm–1

CN1′

Dimer 2 (J = 33 meV) b

CN1

(b)

Dimer 1 (J = 75 meV) b

F2-TCNQ ω = 138 cm–1

CN1′ CN

a CN1

c 1

(d) F2-TCNQ, ω = 82 cm–1

F2-TCNQ, ω = 138 cm

–1

CN1

CN1′

2

2

1

(c)

CN2

Dimer 2 (J = 54 meV) b 3

CN2′

CN1

CN2

CN1

CN2

CN1

(e)

CN1

(f)

c

TCNQ, ω = 83 cm–1

CN1 CN2

c

CN2

c

(a)

1

CN2′

1

a

a

3

CN2

CN1

CN2

(g)

Figure 15.10 The impact of representative LF vibrations on Dimers 1 and 2 of the TCNQ (a, b) and F2 -TCNQ (c, d) crystals and vibrational patterns for similar modes in F2 -TCNQ (e, f ) and TCNQ (g). Atomic displacements are shown by red arrows. Atomic contacts providing intermolecular electronic coupling are shown with blue dotted lines. Green arrows depict the crystal axes. Molecules are numbered according to Figure 15.7. Symmetrically independent CN groups of the molecules are labeled as CN1 and CN2 , the corresponding groups in the second monomer are labeled as CN1′ and CN2′ .

As described here and in Section 15.2, F2 -TCNQ crystal does not show any vibrational modes below 80 cm−1 , while in TCNQ the lowest vibrational frequency is as low as 38 cm−1 , and several modes in the range 38–80 cm−1 play significant role in nonlocal electron–phonon coupling. Since the modes with considerable Li mn have significantly lower frequencies for TCNQ crystal than for F2 -TCNQ, they are more thermally populated in the former, which is in line with stronger electron–phonon interaction in TCNQ reported in Ref. [83]. As was shown in Section 15.2, the considerably higher frequency of the lowest-frequency mode in F2 -TCNQ crystal than in TCNQ one, as well as the lower number of vibrational modes in the former crystal, stem from its unique molecular packing with Zred = 1. Therefore, crystal structure with Zred = 1 is expected to be beneficial for charge transport. Indeed, the authors of Ref. [73] noticed that all the three discovered non-fullerene OSC with electron mobility above 5 cm2 /(V s) known to date, namely, F2 -TCNQ, NDI-CHEX [136], and PDIF-CN2 [133], possess exactly Zred = 1. Some of p-type OSCs with the electron

15.5 Conclusions

mobility above 5 cm2 /(V s), namely, TIPS-Pn [134] and diF-TES-ADT [135], also have Zred = 1. Therefore, proper design of the OS crystal structure with small Zred can result in significant improvement of the charge transport properties.

15.5 Conclusions We have reviewed computational and experimental approaches for description of the vibrational structure of OSCs and its impact on charge transport in these materials (electron–phonon interaction). Particular attention was paid to the LF vibrations, which were recently found to limit charge transport in high-mobility OSCs. These vibrations are complex and generally include both intramolecular and intermolecular contributions; therefore, rigorous theoretical treatment requires the use of periodic ab initio or DFT methods, which were discussed in the beginning of the chapter. Using the Fn -TCNQ single crystal family as an example of OSCs with similar chemical structure but different crystal packing, we have demonstrated how a combination of experimental (X-ray and Raman spectroscopy) and theoretical (solid-state DFT, calculations of transfer integrals, and reorganization energies) methods provides a detailed understanding of the relationship between the OSC molecular and crystal structure, its LF vibrations and electron–phonon coupling (impact of vibrations on charge transport). The clear correlation between the lowest vibrational frequency of the crystal and the charge mobility was found within Fn -TCNQ series: the highest frequency is found for F2 -TCNQ crystal, which possesses the highest mobility. The abovementioned lowest vibrational frequency is tightly related to the crystal structure. The lowest-frequency vibration of TCNQ (libration at ∼40 cm−1 ) is absent in F2 -TCNQ, which was attributed to Zred = 1. On the contrary, mixed intra- and intermolecular vibrations with similar vibrational pattern are present in both TCNQ and F2 -TCNQ at ∼80 cm−1 ; in the latter crystal, it is the lowest-frequency vibration. All the three mentioned vibrational modes modulate substantially charge transfer integrals and hence contribute significantly to the electron–phonon interaction. However, under ambient conditions, the vibration at ∼40 cm−1 in TCNQ is much more populated than the vibrations in F2 -TCNQ. This leads to a significantly larger dynamic disorder in TCNQ, which explains the lower charge mobility in this crystal as compared with F2 -TCNQ. We thus conclude that elimination of several optical LF vibrations and increase of the lowest vibrational frequency can suppress the electron–phonon interaction, resulting in enhanced charge mobility. Thus, crystalline OSs possessing packing motif with one molecule per primitive cell are expected to show high carrier mobility, which is corroborated by charge mobility data: for instance, best n-type OSCs have Zred = 1. We suggest that the frequency of the lowest spectroscopically detectable vibration be used for the search of high-mobility OSCs in the series of similar compounds. We anticipate that the joint experimental and theoretical studies of the interconnection between molecular structure, crystal packing, vibrations, and charge

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transport will provide efficient guidelines for focused search of OSCs with high charge mobility.

Acknowledgments The authors acknowledge funding from Russian Foundation for Basic Research (project #16-32-60204 mol_a_dk) for the work on Sections 15.1 and 15.2 and from Russian Science Foundation (project #18-72-10165) for the work on Section 15.3.

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O· · ·H· · ·O fragment of H5 O2 + : Ab initio classical molecular dynamics and quantum 4D model calculations. J. Chem. Phys. 114 (1): 240–249. Emin, D. (2013). Polarons. Cambridge University Press. Bredas, J.-L., Beljonne, D., Coropceanu, V. et al. (2004). Charge-transfer and energy-transfer processes in π-conjugated oligomers and polymers: a molecular picture. Chem. Rev. 104 (11): 4971–5004. McMahon, D.P. and Troisi, A. (2010). Evaluation of the external reorganization energy of polyacenes. J. Phys. Chem. Lett. 1 (6): 941–946. Li, Y., Coropceanu, V., and Brédas, J.-L. (2016). Charge transport in crystalline organic semiconductors. In: The WSPC Reference on Organic Electronics: Organic Semiconductors (ed. J.-L. Brédas and S.R. Marder), 193–230. Singapore: World Scientific. Ortmann, F., Bechstedt, F., and Hannewald, K. (2011). Charge transport in organic crystals: theory and modelling. Phys. Status Solidi B 248 (3): 511–525. Sosorev, A.Y. (2017). Role of intermolecular charge delocalization and its dimensionality in efficient band-like electron transport in crystalline 2,5-difluoro-7,7,8,8-tetracyanoquinodimethane (F2 -TCNQ). Phys. Chem. Chem. Phys. 19 (37): 25478–25486. Schweicher, G., Olivier, Y., Lemaur, V. et al. (2014). What currently limits charge carrier mobility in crystals of molecular semiconductors? Isr. J. Chem. 54 (5–6): 595–620. Picon, J.-D., Bussac, M.N., and Zuppiroli, L. Quantum coherence and carriers mobility in organic semiconductors. Phys. Rev. B 75 (23): 235106. Marcus, R.A. and Sutin, N. (1985). Electron transfers in chemistry and biology. Biochim. Biophys. Acta 811 (3): 265–322. Sanchez-Carrera, R.S., Paramonov, P., Day, G.M. et al. (2010). Interaction of charge carriers with lattice vibrations in oligoacene crystals from naphthalene to pentacene. J. Am. Chem. Soc. 132 (41): 14437–14446. Troisi, A. (2011). The speed limit for sequential charge hopping in molecular materials. Org. Electron. 12 (12): 1988–1991. Marzari, N., Mostofi, A.A., Yates, J.R. et al. (2012). Maximally localized Wannier functions: theory and applications. Rev. Mod. Phys. 84 (4): 1419–1475. Yi, Y., Coropceanu, V., and Brédas, J.-L. (2012). Nonlocal electron–phonon coupling in the pentacene crystal: beyond the Γ-point approximation. J. Chem. Phys. 137 (16): 164303. Li, Y., Coropceanu, V., and Brédas, J.-L. (2013). Nonlocal electron–phonon coupling in organic semiconductor crystals: the role of acoustic lattice vibrations. J. Chem. Phys. 138 (20): 204713. Closs, G.L. and Miller, J.R. (1988). Intramolecular long-distance electron transfer in organic molecules. Science 240 (4851): 440–447. Coropceanu, V., Malagoli, M., Silva Filho, D.A. et al. (2002). Hole- and electron-vibrational couplings in oligoacene crystals: intramolecular contributions. Phys. Rev. Lett. 89 (27): 275503.

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effect on the band dispersion of organic molecular semiconductors. Nat. Commun. 8 (1): 173. Myers, A.B. (1996). Resonance Raman intensities and charge-transfer reorganization energies. Chem. Rev. 96 (3): 911–926. Pedron, D., Speghini, A., Mulloni, V. et al. (1995). Coupling of electrons to intermolecular phonons in molecular charge transfer dimers: a resonance Raman study. J. Chem. Phys. 103 (8): 2795–2809. Illig, S., Eggeman, A.S., and Troisi, A. (2016). Reducing dynamic disorder in small-molecule organic semiconductors by suppressing large-amplitude thermal motions. Nat. Commun. 7: 10736. Fillaux, F. (2000). Hydrogen bonding and quantum dynamics in the solid state. Int. Rev. Phys. Chem. 19 (4): 553–564. Vener, M.V. (2007). Proton dynamics in hydrogen-bonded crystals. In: Hydrogen-Transfer Reactions. Handbook/Reference Book (ed. J.T. Hynes, J.P. Klinman, H.-H. Limbach, et al.), 273–299. Weinheim: Wiley-VCH. Vener, M.V., Manaev, A.V., Hadži, D. et al. (2008). DFT study of proton dynamics in the potassium hydrogen maleate crystal: the infrared versus the inelastic neutron scattering spectra. Z. Phys. Chem. 222 (8–9): 1349–1358. Atahan-Evrenk, S. and Aspuru-Guzik, A. (2014). Prediction and theoretical characterization of p-type organic semiconductor crystals for field-effect transistor applications. In: Prediction and Calculation of Crystal Structures (ed. S. Atahan-Evrenk and A. Aspuru-Guzik), 95–138. Cham: Springer. Mei, J., Diao, Y., and Appleton, A.L. (2013). Integrated materials design of organic semiconductors for field-effect transistors. J. Am. Chem. Soc. 135 (18): 6724–6746. Silva Filho, D.A., Coropceanu, V., Fichou, D. et al. (2007). Hole-vibronic coupling in oligothiophenes: impact of backbone torsional flexibility on relaxation energies. Philos. Trans. R. Soc. London, Ser. A 365 (1855): 1435–1452. Irving, R.J. (1972). The standard enthalpy of sublimation of naphthalene. J. Chem. Thermodyn. 4 (5): 793–794. Kruif, C.G. (1980). Enthalpies of sublimation and vapour pressures of 11 polycyclic hydrocarbons. J. Chem. Thermodyn. 12 (3): 243–248. Siddiqi, M.A., Siddiqui, R.A., and Atakan, B. (2009). Thermal stability, sublimation pressures, and diffusion coefficients of anthracene, pyrene, and some metal β-diketonates. Chem. Eng. Data 54 (10): 2795–2802. Warta, W., Stehle, R., and Karl, N. (1985). Ultrapure, high mobility organic photoconductors. Appl. Phys. A 36 (3): 163–170. Karl, N. and Marktanner, J. (2001). Electron and hole mobilities in high purity anthracene single crystals. Mol. Cryst. Liq. Cryst. 355 (1): 149–173. Reese, C., Chung, W.-J., Ling, M.-M. et al. (2006). High-performance microscale single-crystal transistors by lithography on an elastomer dielectric. Appl. Phys. Lett. 89 (20): 202108. Yang, X., Wang, L., Wang, C. et al. (2008). Influences of crystal structures and molecular sizes on the charge mobility of organic semiconductors: oligothiophenes. Chem. Mater. 20 (9): 3205–3211.

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125 Pfattner, R., Bromley, S.T., Rovira, C. et al. (2016). Tuning crystal ordering,

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16 Effects of Non-covalent Interactions on Molecular and Polymer Individuality in Crystals Studied by THz Spectroscopy and Solid-State Density Functional Theory Feng Zhang 1 , Keisuke Tominaga 1 , Michitoshi Hayashi 2 , and Takashi Nishino 3 1 Kobe University, Molecular Photoscience Research Center, Chemistry Department, Rokkodai-Cho 1-1, Nada, Kobe 657-8501, Japan 2 National Taiwan University, Center for Condensed Matter Sciences, 1 Roosevelt Road, Section 4, Taipei 10617, Taiwan 3 Kobe University, Graduate School of Engineering, Departments of Chemical Science and Engineering, Rokkodai-Cho 1-1, Nada, Kobe 657-8501, Japan

Advances in terahertz (THz) spectroscopy techniques and solid-state density functional theory (DFT) simulation methods have given a full access to the optical phonon modes of molecular crystals and crystalline polymers. This work has systematically examined the nature of optical phonon modes arising from a variety of prototypical intermolecular interactions acting on molecules or polymers in crystal phases. We particularly focus on interpreting the vibrational properties of molecular individuals under the influence of intermolecular interactions. First, a criterion has been proposed for the determination of the frequency sequences of intermolecular translations and librations. Second, two mixing forms between intermolecular and intramolecular vibrations have been identified. One is strong mixing within similar characteristic frequency regions, while the other is weak mixing of distinct characteristic frequency regions separated by a large gap. The former is predictable from classical mechanics and appears in molecular systems having structural flexibility. The latter is nonclassical and has been illustrated in molecular systems where hydrogen bonds and dipole–dipole interactions are present. An application of THz spectroscopy to determine polymer packing conformations have been elucidated. Firm evidence of the breaking of the helical symmetry has been found. These findings provide new insight into the nature of molecular optical phonon modes and also illuminate the promising potential of THz spectroscopy toward the practical applications in characterization of materials science.

16.1 A Historical Review of Phonon Modes Since Born and coworkers in the early nineteenth century formulated the general theory of lattice dynamics in the ionic crystals [1, 2], phonon modes have been used as the fundamental concept of atomic vibrations in solid matter. Atoms Molecular Spectroscopy: A Quantum Chemistry Approach, First Edition. Edited by Yukihiro Ozaki, Marek Janusz Wójcik, and Jürgen Popp. © 2019 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2019 by Wiley-VCH Verlag GmbH & Co. KGaA.

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in the ionic crystals are connected through uniformly the Coulomb force, and phonon modes are described as relative motions of individual atoms in the Cartesian coordinate systems with specific momentum properties. It then saw the straightforward application of this motion picture to other unique form interaction systems such as metallic crystals controlled by metallic bonds and semiconductor crystals controlled by covalent bonds [3]. Davydov in 1940s and 1950s made a primary contribution to generalizing the phonon mode concept to molecular crystals where two classes of interactions in different orders of magnitude exist [4]. Forces interacting on different molecules are significantly smaller than that on individual atoms composing molecules themselves. Molecules may preserve their individuality in the solid phase. Therefore as the first approximation molecular phonon modes are described, apart from the momentum property, by two classes of motions. The first is intermolecular vibrations associated with the translational and rotational degrees of freedom of each individual molecule. The second class is intramolecular vibrations of the atoms of a molecule relative to one another. In these motions, the center of mass (COM) of each molecule is not displaced and there are no rotations for one molecule as a whole. It is reasonable to claim that the essence of understanding THz spectroscopy of molecular crystals is to examine the effects of non-covalent interactions on molecular individuality from the vibration point of view. Precisely, we need to give answers to the following two questions: (1) What is the sequence of characteristic frequencies of intermolecular translations and librations and intramolecular motions? (2) What are the extents to which the three types of motions mix, and to which do their frequencies distribute? In the early study using inelastic neutron scattering [5–7], Raman [8–10], and far-infrared [11–13] spectroscopies, molecular phonon modes are interpreted under the rigid-body approximation by ignoring the mixing of intermolecular and intramolecular vibrational dynamics. In this approximation, molecular conformations and, thus, intramolecular normal mode eigenvectors are not affected by intermolecular interactions [14, 15]. Davydov [4] developed the oriented gas model to effectively describe intramolecular vibrations in the solid state [4, 16], and Hornig [17] developed several theoretical methods to determine spectroscopic selection rules for molecular phonons, more precisely, for the intermolecular vibrations [17–20]. This rigid-body assumption is, however, not universally true. For example, as illustrated by Civalleri et al. [21, 22], the most stable structure of urea in the gas phase adopts a C 2 -anti conformation, whereas it possesses in bulk C 2v symmetry that corresponds to a transition-state conformation in the gas phase (Figure 16.1). Another well-known example is ammonia [23]. Ammonia in the gas phase has a perfect triangular pyramid structure with C 3v point group symmetry. When forming crystals via largely hydrogen bonding, its symmetry reduces to C 3 , and the mirror reflection symmetric elements are lost. Other examples include naphthalene [24] and anthracene [25], which experience symmetry degeneracies from the gas phase to solid phase, owing to distortions induced by intermolecular interactions. Accompanying the conformational

16.1 A Historical Review of Phonon Modes

Figure 16.1 Molecular structures of urea: (a) C 2v conformation in the crystalline phase; (b) C 2 -anti conformation in the gas phase. Source: Reprinted with permission from Civalleri et al. [21]. Copyright 2007, American Chemical Society.

(a)

(b)

variations are changes in normal mode eigenvectors in which intermolecular eigenvectors are expected to take on intramolecular characteristics and vice versa. This issue was discussed by Pawley and coworkers in the 1970s and 1980s using the classical molecular force field methods [26–29]. Under the assumption that the crystal potential can be expressed as a sum of intermolecular and intramolecular potentials (ignoring the coupling terms between the two), they observed the mixing of intermolecular and intramolecular vibrations in naphthalene, anthracene, tetracyanoethylene, and hexamethylenetetramine. These results yielded the first qualitative explanation of frequency distribution and correlation field splitting (factor group splitting or Davydov splitting) [4] of intramolecular modes from the aspect of normal mode eigenvectors. Mixed intermolecular and intramolecular vibrations were then used to analyze anisotropic displacement parameters in crystallography [30–32]. From the energy perspective, mixing of intermolecular and intramolecular vibrations in the solid phase can be explained as follows. In general, we assume that vibrations with well-defined origins have characteristic frequencies and vibrations with similar characteristic frequencies, as long as their symmetric properties are identical (i.e. belonging to an identical space group representation), may mix [32–34]. Intramolecular vibrations featuring collective backbone vibrations of molecules with multi-structural segments may have vibrational energies similar to intermolecular vibrations. Therefore, the collective intramolecular modes may mix with intermolecular vibrations. In contrast, high-frequency intramolecular vibrations of localized functional groups are not able to mix with intermolecular vibrations. This is our basic viewpoint on molecular phonon modes in terms of classical potential surfaces of molecular crystals. Advances in freely propagating terahertz (THz) generation and detection have enabled a routine access to molecular vibrations in the low-frequency region, approximately 0.1–10 THz or 3–333 cm−1 . A great interest in molecular phonon modes has been stimulated in the past two decades [35–37]. Comprehensive information concerning optical phonon modes, such as frequency, polarizability, and isotopic and thermal shifts, has been reported for various molecular crystals [38–42]. In parallel, the steady development of ab initio solid-state density functional theory (DFT) has enabled the characterization of the electronic structures (i.e. potential energy surfaces) of molecular crystals under the Born–Oppenheimer approximation. In particular, functionals that take account of dispersion forces have allowed normal mode simulations with unprecedented accuracy over the classical force field methods [43–46]. Frequency calculations

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16 Studied by THz Spectroscopy and Solid-State Density Functional Theory

have been reported in a broad range of systems, including explosives [47], polycyclic aromatics [23, 41, 48], carbohydrates [49], pharmaceuticals [50–53], and biomolecules [42, 45, 46, 54–61]. Given the impact of THz spectroscopy and ab initio DFT simulations, we have systematically investigated the vibrational characteristics of molecular phonon modes. We are primarily concerned with the extent to which the conventional knowledge of molecular phonon modes in the THz region is valid and applicable and as to whether there are any new properties in such molecular phonon modes. This work reviews our progress on understanding THz spectroscopy of molecular crystals through an interplay of experiment and theory. The content is divided into four parts. We will first introduce one DFT model that has been used to simulate THz modes of all the crystalline systems of interest and also introduce an independently developed mode decomposition method to characterize the nature of any simulated mode of interest. We will second test the DFT model against a broad spectrum of small molecules with prototypical intermolecular interactions and interpret the nature of the associated optical phonon modes. The bench of testing crystals includes C60 , anthracene, adenine, 𝛼-glycine, and l-alanine crystals [48, 59–61]. As shown in Figure 16.2, in the order from C60 to the amino acids, molecules become increasingly flexible (e.g. in terms of the collective torsional freedom in the backbone single bonds in amino acids), internal covalent bonds become increasingly saturated, and intermolecular interactions increase in complexity and strength (Figure 16.2). Structural information for these systems is provided in Table 16.1. Following the works in the small molecular systems, we will introduce a practical application of THz spectroscopy and DFT simulations in the structural determination of a polymer material. Lastly, we will introduce an experimental method that provides qualitative verification of the mode assignments to an extent as whether a given THz mode is dominated by intermolecular or intramolecular vibrations.

16.2 Theoretical Representation of Non-covalent Interactions For a reliable representation of non-covalent interactions, there is a wide variety of computational methods available. The most accurate class of the methods employ wave functions and utilize coupled-cluster expansions or many-body perturbations. These methods are nevertheless formidable for solid-state calculations due to their steep scaling of the cost with system size. DFT, utilizing electron densities and occupied orbitals, is the only feasible approach at present for ab initio predictions of molecular crystals. According to the definition in symmetry-adapted perturbation theory (SAPT), the non-covalent interactions can involve contributions of four physical origins: electrostatic, exchange, induction, and dispersion [62]. While DFT was able to describe relatively accurately the first three types of interactions [63, 64] in spite of disputes existing [65], DFT was failing badly in representing the potential energy curves of dispersion energies [64, 66].

16.2 Theoretical Representation of Non-covalent Interactions

y

z x

z

y

x

X

Y

(b)

(a) z

#1

#2

Y

y

x

X

z x

Z

y

#2 Z

#1

(c)

y

y z

z

x Y

(d)

X

Z

x

(e)

Figure 16.2 Structural information of the C60 (a), anthracene (b), adenine (c), 𝛼-glycine (d), and L-alanine (e) crystal systems. The left side in each panel shows the three principal axes x, y, and z of the irreducible molecular unit(s); the right side shows the spatial packing structures of all the irreducible molecules in the unit cell. The red, green, and blue lines represent the three unit cell vectors a, b, and c, respectively. For the C60 and L-alanine systems with the cubic and orthorhombic unit cells, respectively, the three crystallographic coordinate axes X, Y, and Z are coincident with the directions of the three unit cell vectors a, b, and c, respectively. For the other three monoclinic systems, the directions of X and Y are coincident with that of a and b, while Z points away from c. Their relationships have been shown explicitly. CH/π and hydrogen bond intermolecular connections are shown in their owner molecular systems. Molecules in different layers in the anthracene and adenine crystals are shown in different colors for clarity. The dipole moments of C60 and anthracene are zero owing to their high symmetry. The dipole moments are 0.17 D for both adenine #1 and #2 and 1.05 and 1.01 for glycine and L-alanine, respectively. Source: Adapted with permission from Zhang et al. [71]. Copyright 2016, John Wiley and Sons.

The current efforts of recovering dispersion energies generally fall into two categories. The first category takes the ab initio spirit. The applied strategies include either the optimization of parameters in existing functionals, or the development of nonlocal correlation density functionals, or the application of “post-DFT” approaches, i.e. employing virtual Kohn–Sham orbitals. The other category takes an empirical strategy, in which the London-type force fields are included as an empirical correction to DFT energy. The most prevalent method is the DFT-D2 model parameterized on benchmark interaction energies by Grimme in 2004 [67] and updated in 2006 [68]. This empirical correction is based on the assumption that total dispersion interaction between larger molecules or solids can be described as a sum of contributions from all pairs of

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16 Studied by THz Spectroscopy and Solid-State Density Functional Theory

atoms. Each pair contributes a terms proportional to the inverse sixth power of its interatomic distances, R, Edisp = −s6

∑ C6paris pairs

R6

fdmp (R)

(16.1)

where a global scaling factor, s6 , was used to adjust the correction strength paris is the dispersion coefficient for each exchange–correlation functional, C6 for a certain atomic pair, and f dmp (R) is a damping function used to prevent Eq. (16.1) from diverging at small R. Civalleri re-parameterized the D2 correction in small-molecule crystals with respect to cohesive energy [69]. The re-parameterized correction term was referred as D* . In terms of the application of quite a number of DFT methods in THz spectroscopy of molecular crystals, the accuracy of energy prediction may not be the major concern. Two most relevant concerns are (i) accuracy in describing potential energy surfaces of molecular crystals, precisely, the shape of the potential surface at the equilibrium, and (ii) computational efficiency. Obviously, efficiency is an adherent advantage of the DFT-D approach over all the ab initio methods. In the past few years, we have tested the DFT-D* approach using a broad spectrum of molecular crystals including fullerene, polycyclic aromatics, carbohydrates, strong H-bond molecules, pharmaceuticals, nucleobases, nucleosides, amino acids, and short-chain peptides. In spite of its lower accuracy in terms of cohesive energy prediction (being general believed to be reliable to ∼10 kJ/mol [68, 69]), the DFT-D* methods have yield consistently satisfactory reproduction of THz normal modes of molecular crystals [70]. An explanation of this observation is that the R−1 variation tendency of the dispersion energy with atomic dis6 tance has been firmly verified in both experimentation and theory and thus have unambiguous physical meaning. A straightforward augment of DFT functionals energy, although inconsistent with the ab initio spirit, represents a with the R−1 6 part of indisputable truth for forming the proper shape of potential surface. In our previous review article [70], we have summarized the calculation results of five benchmark molecules, C60 , anthracene, adenine, 𝛼-glycine, and l-alanine, using a variety of DFT models. In this article, we have revisited all the five systems with a single approach, that is, the PBE [71] functional augmented with the D* dispersion terms, thus referred as the PBE-D* method. The 6-31G(d,p) basis set was employed for C60 owing to the large size of this system, while the 6-311G(d,p) basis set for all other systems. All solid-state DFT simulations with periodic boundary conditions were performed with the CRYSTAL14 software package [44, 72]. The geometries of all the crystals were optimized with a full relaxation strategy, i.e. all the atomic coordinates and the unit cell parameters were allowed to relax. The crystal structures deposited in the Cambridge Structure Database (CSD) were used as the starting points. Refer to Table 16.1 for the detailed structural information. The frequency calculations were conducted under the harmonic approximation. The mass-weighted Hessian matrices in Cartesian coordinates were diagonalized at the gamma point. The IR intensities were calculated using a periodic coupled perturbed Kohn–Sham analytical approach [73, 74]. A shrinking factor (5,5) was used to define the

16.3 A Mode Decomposition Method

Table 16.1 Structural information of molecular crystals. The lattice parameters are the values determined in experimentation; a, b, c and 𝛼, 𝛽, 𝛾 are in units of Å and degree, respectively. C60

Anthracene

Adenine

Glycine

L-Alanine

Space group

Pa3 (205)

P21 /a (14)

P21 /c (14)

P21 /n (14)

P21 21 21 (19)

a

14.040 78

8.37

7.891

5.087

5.927 9

b

14.040 78

6.00

22.242

11.773

12.259 7

c

14.040 78

11.12

7.448

5.460

5.793 9

𝛼

90

90

90

90.00

90

𝛽

90

125.4

113.193

111.99

90

𝛾

90

90

90

90.00

90

Vol.

2 768.048 55

455.2

1201.6

303.2

421.1

Z

4.00

2.0

8

4

4

Z′

0.17

0.5

2

1

1

Temp.

5K

16 K

RT

23 K

23 K

Method

NPD

NPD

XRPD

XRPD

XRPD

Entry in CSD

—

ANTCEN16

KOBFUD

GLYCIN85

LALNIN24

Ref.

[82]

[25]

[102]

[85]

[87]

DFT model

PBE-D* / 6-31G(d,p), s6_0.594

PBE-D* / 6-311G(d,p), s6_0.675

PBE-D* / 6-311G(d,p), s6_0.675

PBE-D* / 6-311G(d,p), s6_0.675

PBE-D* / 6-311G(d,p), s6_0.675

SP, space group; Vol., unit cell volume; Temp., the experimental temperature; NPD, neutron powder diffraction; XRPD, X-ray powder diffraction; Ref., reference (the source of the experimental data). Source: Reprinted with permission from Zhang et al. [71]. Copyright 2016, John Wiley and Sons.

commensurate grid and sampling rate of k points in the reciprocal space for solid-state calculations. All other parameters were used as the default values of CRYSTAL14.

16.3 A Mode Decomposition Method As one-electron wave functions in a crystal obey the Bloch theorem, the direct-space Kohn–Sham equation is effectively resolved in k space through Fourier transformation. Cartesian coordinates in this regard provide a better representation for atomic displacements than external and internal coordinates do. As a result, the physical meaning of a simulated normal mode is not immediately clear until an additional analysis of the underlying intermolecular and intramolecular vibrations is performed. For small-amplitude normal modes, the separation of intermolecular and intramolecular vibrations depends on the Eckart conditions [75, 76]. The Eckart conditions use displacements of the COM to represent intermolecular translations and use libration angles about the three principal molecular axes to represent intermolecular librations [48]. Any pure

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16 Studied by THz Spectroscopy and Solid-State Density Functional Theory

intramolecular vibration should satisfy the conditions that displacements of the COM and the principal libration angles are equal to zero. A particularly convenient means of mode analysis is the potential energy distribution (PED) method that was originally developed for gas-phase molecules [77–79]. It involves a unitary transformation of the potential energy function from a normal mode coordinate system to a user-defined external and internal coordinate system. Thus, PED requires a complete predefined set of internal modes. This task is feasible for small molecules, but becomes formidable for larger ones. If the concern is limited to low-frequency phonon modes associated with a limited number of internal modes, it is appropriate to ignore this problem. Furthermore, we can first separate the intermolecular and intramolecular vibrations by exploiting the Eckart conditions and gain insight into the overall distribution of intramolecular vibrations in the phonon modes for any molecular system. We then determine a minimum basis of internal modes needed for the characterization of the intramolecular vibrations mixed with intermolecular vibrations. In this section, we will introduce an independently developed mode decomposition method that is applicable to small-amplitude molecular motions in all forms of matter in gas, liquid, or solid (crystals or amorphous) phases. In the Born–Oppenheimer approximation, the electronic energy together with the nuclei repulsion energy provides the potential energy of the nuclear motions in a specific crystal lattice. The sum of these two energies as a function of the variations of the crystal structure becomes the potential energy surface V (x) of the nuclear motions. Here x represents symbolically the coordinates of all nuclear positions and the lattice vectors. V (x) has the translational symmetry of the lattice. In the harmonic approximation to the potential energy surface, the Hessian matrix element relating to the variations in terms of the small displacement ui of the ith atom, in any direction in an arbitrary zero cell, with the small displacement uj of the jth atom, and in any direction in the G-cell, is given by [80] ) ( 2 V (x) 𝜕 (16.2) Hij0G = 𝜕u0i 𝜕uGj eq

where the subscript eq stands for the equilibrium geometry of the lattice. The translational invariance of V (x) generates the momentum representation. It therefore allows us to represent the Hessian matrix in momentum space (k or reciprocal space) via Fourier transformation: ∑ exp(ik ⋅ G) Hij0G (16.3) Hij (k) = G

Thus, we see that the Hessian matrix in k space has been block factorized with respect to momentum k. Each block has a dimension of 3N, where N is the number of nuclei per unit cell. This corresponds to a problem of collective atomic vibrations in all unit cells with a specific momentum, (i.e. phonon modes). Because THz spectroscopy examines optical transitions near the Γ-point where k is effectively zero, we will discuss Γ-point phonon modes. Consequently, Eq. (16.3) is reduced to ∑ Hij0G (16.4) Hij (0) = G

16.3 A Mode Decomposition Method

Equation (16.4) implies that we are allowed to consider only atomic vibrations of the zero cell, because all unit cells at the Γ-point vibrate in phase. Harmonic frequencies are then obtained by diagonalizing the mass-weighted Hessian matrix (force constant matrix) F FL = ΛL

(16.5)

where an element of F is given by Hij (0) Fij (0) = √ mi mj

(16.6)

and mi and mj are the relative masses of the ith and jth atoms, respectively, and L is the eigenvector matrix. Λ is a diagonal eigenvalue matrix with element 𝜆n = 4π2 c2 𝜈̃n2 , where 𝜈̃n (in cm−1 ) indicates the frequency of the nth normal mode Qn (n = 1, 2, … , 3N). The normal mode coordinates Q are related to the mass-weighted Cartesian coordinates q through the linear transformation [76]: Q = L−1 q

(16.7)

The complete form of the nth normal mode Qn is expressed by the row vector (√ √ √ √ √ √ n n n n n n Qn = m1 𝛿1X , m1 𝛿1Y , m1 𝛿1Z , … mi 𝛿iX , mi 𝛿iY , mi 𝛿iZ , ) √ √ √ n n n (16.8) … mN 𝛿NZ , mN 𝛿NY , mN 𝛿NZ n n n , 𝛿iY , 𝛿iZ ) represents the displacement vector (atomic units) of the where (𝛿iX ith atom participating in Qn in the crystallographic Cartesian coordinate system. CRYSTAL14 normalizes the atomic displacements with respect to classical amplitudes according to

𝜈n )2 (Qn )2 u(a0 )2 1∕2 hc̃ 𝜈n = 1∕2(2πc̃

(16.9)

where h is the Planck’s constant, u is the unified atomic mass unit, and √ a0 is the Bohr radius. The amplitude of Qn has a reciprocal relationship with 𝜈̃n . In the output, the first three normal modes with negative or zero frequencies represent acoustical motions of the unit cell and will be excluded from further discussion. From a symmetry perspective, the Γ-point corresponds to the long wave limit and does not involve the translation operator. Thus, the Γ-point symmetry is determined by the factor group, a subgroup of the space group (SP) formed by separating out translations [20]. If Qn is a nondegenerate mode, symmetrically equivalent molecules would vibrate in the same fashion, except for phase differences that are determined by the relevant irreducible representation. If Qn is degenerate, symmetrically equivalent molecules would vibrate similarly to the extent that the percentage contributions of intermolecular and intramolecular vibrations are identical; however, the phase relations cannot be determined. If our concern is to only quantify contributions of intermolecular and intramolecular vibrations to the Γ-point phonon modes, then it is acceptable to focus the analysis on vibrations of any irreducible molecular unit(s) in the unit cell. This is because the characteristics of all other molecular units are essentially identical. Henceforth, we introduce a convenient mode analysis for a single molecular basis, which allows the instantaneous separation of intermolecular translations

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16 Studied by THz Spectroscopy and Solid-State Density Functional Theory

and librations from intramolecular vibrations, when the atomic displacement n n n vectors (𝛿iX , 𝛿iY , 𝛿iZ ) of the molecule are available. Intermolecular translations are represented by COM movement along the three axes X, Y , Z of the crystallographic Cartesian coordinate system. The atomic translation vector of an irreducible molecule in the nth normal mode, e.g. (𝛿 n, trans, X , 0, 0) along X, can be obtained by specifying 𝛿 n, trans, X according to 𝛿 n,trans,X =

O 1 ∑ m 𝛿n M i=1 i iX

(16.10)

where M is the relative molecular mass and O is the number of atoms in the molecule. The other two translation vectors (0, 𝛿 n, trans, Y , 0) and (0, 0, 𝛿 n, trans, Z ) along Y and Z, respectively, are obtained similarly. The intermolecular libration, featuring a hindered rotation of the whole molecule as a rigid body, can be decomposed into three independent librations about the three principal molecular axes. Thus we transform the crystallographic Cartesian coordinate system (X, Y , Z) to the principal molecular coordinate system (x, y, z) through an orthogonal transformation [81]. First, the origin of the crystallographic Cartesian coordinate system must be moved to the COM to ensure a common origin. The axes x, y, z are determined by three eigenvectors of the molecular inertia tensor I: ∑ ∑ ∑ − mi Xi Zi ⎤ ⎡ mi (Yi 2 + Zi 2 ) − mi Xi Yi ∑ ∑ ∑ I = ⎢ − mi Yi Xi (16.11) mi (Xi 2 + Zi 2 ) − mi Yi Zi ⎥ ∑ ∑ ∑ ⎢ ⎥ mi (Xi 2 + Yi 2 )⎦ − mi Z i Y i ⎣ − mi Zi Xi where (X i , Y i , Z i ) is the position vector of the ith atom in the crystallographic Cartesian coordinate system. The three principal librations can be extracted in the following way. For libration about x, the atoms move in the yz plane. The libration angle 𝜃 n, lib, x of the molecule in the nth normal mode is given by (O ) ∑ 1 n ′ 𝜃 n,lib,x = m r′ × 𝜹 i (16.12) Ix i=1 i i0 x

n 𝜹′ i

where r′i0 and are the equilibrium position vector (xi , yi , zi ) and the disn placement vector (𝛿ix , 𝛿iyn , 𝛿izn ) of the ith atom in the new coordinate system, respectively. The subscript “x” for the parenthesis signifies a projection of the vector along the x direction. I x is the eigenvalue (or principal moment) of I corresponding to the principal axis x. Consequently, the displacement vector (0, 𝛿iyn,lib,x , 𝛿izn,lib,x ) of the ith atom following the principal libration about x is given by ⎛ 0 ⎞ ⎛xi ⎞ ⎜𝛿 n,lib,x ⎟ = (Rnx − E) ⎜yi ⎟ ⎜ iyn,lib,x ⎟ ⎜ ⎟ ⎝𝛿iz ⎠ ⎝ zi ⎠

(16.13)

16.3 A Mode Decomposition Method

where E is an identity matrix and Rnx is a rotation matrix about x, being given by 0 ⎞ ⎛1 0 Rnx = ⎜0 cos 𝜃xn − sin 𝜃xn ⎟ ⎜ n n ⎟ ⎝0 sin 𝜃x cos 𝜃x ⎠

(16.14)

In a similar manner, we separate the principal librations about y and z and n,lib,y n,lib,y obtain the respective atomic displacement vectors (𝛿ix , 0, 𝛿iz ) and n,lib,z n,lib,z (𝛿ix , 𝛿iy , 0). The rotation matrices about y and z are Rny

⎛ cos 𝜃yn 0 sin 𝜃yn ⎞ ⎛cos 𝜃zn − sin 𝜃zn n ⎟ ⎜ 0 1 0 and Rz = ⎜ sin 𝜃zn cos 𝜃zn = ⎜ ⎜ n n⎟ 0 ⎝− sin 𝜃y 0 cos 𝜃y ⎠ ⎝ 0

0⎞ 0⎟ ⎟ 1⎠

(16.15)

At this point, we transform the principal axes coordinate system back to the crystallographic Cartesian coordinate system to obtain the atomic n,lib,y n,lib,y n,lib,y n,lib,x n,lib,x n,lib,x displacement vectors (𝛿iX , 𝛿iY , 𝛿iZ ), (𝛿iX , 𝛿iY , 𝛿iZ ), and n,lib,z n,lib,z n,lib,z , 𝛿iY , 𝛿iZ ) for the principal librations about x, y, and z, respectively. (𝛿iX We relied on Eqs. (16.10) and (16.12) to separate translations and librations. Derivations are found in Ref. [48]. The displacement vector n,intra n,intra n,intra , 𝛿iY , 𝛿iZ ) of the ith atom involved in the intramolecular vibration (𝛿iX of the nth mode is obtained by subtracting contributions from all translations and the principal librations from its total motion. We define the intramolecular vibration vector Vnintra of a certain molecule in question in Qn as (√ √ n,intra √ n,intra √ n,intra n,intra √ n,intra m1 𝛿1X , m1 𝛿1Y , m1 𝛿1Z , … mi 𝛿iX , mi 𝛿iY , Vnintra = ) √ √ √ √ n,intra n,intra n,intra n,intra mi 𝛿iZ , … mO 𝛿OZ , mO 𝛿OY , mO 𝛿OZ (16.16) The nature of Vnintra will be fully characterized when we define a proper basis of l }, as given by internal modes {S#1 Vnintra =

L ∑

𝛼ln Sl

(16.17)

l=1 l where S#1 satisfies the normalization condition and has the form (√ √ S,l √ S,l √ S,l S,l √ S,l √ S,l l = , m1 𝛿1Y , m1 𝛿1Z , … mi 𝛿iX , mi 𝛿iY , mi 𝛿iZ , S#1 m1 𝛿1X ) √ S,l √ S,l √ S,l (16.18) … mO 𝛿OZ , mO 𝛿OY , mO 𝛿OZ S,l S,l S,l where (𝛿iX , 𝛿iY , 𝛿iZ ) is the displacement vector of the ith atom. The expansion coefficient 𝛼ln is given by 𝛼ln = Vnintra ⋅ Sl . Two situations were specifically considered for constructing a reliable internal mode basis on which intramolecular vibrations in the low-frequency phonon modes could be expanded and characterized. In one situation, the symmetry of molecular configuration in the solid phase is identical (e.g. adenine) or reduced relative to that in the gas phase (e.g. C60 [82–84] and anthracene [25, 48]). More precisely, the site group symmetry of molecules in a crystal is the same as or a subgroup of the point group symmetry in the gas phase (more details are found

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16 Studied by THz Spectroscopy and Solid-State Density Functional Theory

in Halford [19] and Hornig [17]). A firm correlation is thus established between the internal modes of the gas and solid phases, where the gas-phase modes provide a good basis for the solid phase. In the second situation, molecules undergo dramatic conformational changes in the condensation process (e.g. amino acids [59, 85, 86]); thus, the molecule site group in the solid phase is no longer a subgroup of the point group of the gas-phase molecule. Because the correlation between the gas-phase and solid-phase internal modes is lost, the gas-phase internal modes are no longer suitable for the solid phase; a particular approach was therefore developed for constructing the intended solid-phase internal mode basis. A detailed description of this approach has been introduced in our previous works [59, 70]. For ease of readability, we will not include it in this review. Finally, we calculate the amplitudes of the three translations, the three principal librations, and the decomposed internal modes for the nth normal mode by the root-mean-square mass-weighted atomic displacement (RMSMWAD) of all atoms in the molecule according to √ √ O √ ∑ √1 n,l 2 n mi (𝛿iV ) (16.19) Dl = √ √O √ i = 1 √ V = X, Y , Z where l refers to the three translations, the three principal librations, and elementary internal modes. In the previous review, we evaluated the relative strengths of all the elementary vibrational components with respect to their RMSMWAD amplitudes [70]. In this work, we will examine their contributions Pln to the total vibrational energy according to following formula, (Dn )2 Pln = ∑ l n 2 × 100% (Dl )

(16.20)

l

16.4 Interpretation of the Nature of Optical Phonon Modes Controlled by Prototypical Non-covalent Interactions This section will discuss the nature of molecular phonon modes arising from distinct types of non-covalent interactions and its implications to the interpretation of molecular phonon modes. We in particular focus on giving answers to the two questions raised in the introduction, i.e. understanding the effects of various types of non-covalent interactions on the vibrational behaviors of molecular individuality in crystal phase. The credibility of all the frequency calculations was evaluated against the experimental criteria of infrared frequencies and intensities. For C60 [87–91], 𝛼-glycine, and l-alanine [13, 46, 92, 93], the far-IR data reported in literature at temperature lower than 10 K was used. For adenine and anthracene, THz spectra over 30–180 cm−1 were reported in our previous work

16.4 Interpretation of the Nature of Optical Phonon Modes

at 10 K, although high-quality far-IR data are not available [48, 61]. For simplicity, we will not show here all the experimental results. Predictions of the rigid-body approximation will be frequently recalled to compare with the predictions of ab initio calculations. In the rigid-body approximation, intermolecular and intramolecular vibrations are completely separated. In a unit cell with N molecules, each containing O atoms, there are 6N − 3 optical intermolecular modes (excluding the three acoustical translations) and N(3O − 6) optical intramolecular modes classified by irreducible representations of the factor group of the unit cell. Different representations correspond to different phases (symmetries) of the motions – translations, librations, or intramolecular vibrations – of the N molecules. Three group theory analysis techniques allow the determination of the selection rules: factor groups developed by Bhagavantam and Venkatarayudu [18], molecular site groups developed by Halford [19] and Hornig [17], and nuclear site groups originally developed by Mathieu [94] and extended to crystals by Rousseau et al. [20] Although the three methods are, in principle, equivalent, the latter method, which tabulates a comprehensive listing of the irreducible representations for intermolecular and intramolecular modes, will be adopted in this work for the ease of indexing. Group theory analysis informs us of the symmetry properties of the translations, librations, and certain intramolecular vibrations. Only vibrations belonging to identical irreducible representations are allowed to mix. Furthermore, comparison between the rigid-body approximation predictions and the ab initio simulations will yield important insights concerning the applicability and limitation of the rigid-body picture for each prototype molecule. 16.4.1

C60

C60 represents a nonplanar conjugated system composed solely of sp2 -hybridized carbon and has a closed-shell singlet ground electronic state. Each carbon atom bonds with other three atoms through one shorter bond and two longer bonds. The shorter bonds have a slightly higher order than the longer ones, although the difference is not exactly the same as that between double and single bonds. The shorter bonds fuse two hexagons, designated as 6 : 6 bonds. The longer bonds fuse a pentagon to a hexagon, designated as 6 : 5 bonds. Thus, C60 has anisotropic electron distributions: the 6 : 6 bonds compose the most electron-sufficient regions, while the pentagons the most electron-deficient regions. In the solid state, the intermolecular interactions are regarded as an interplay of dispersion forces and electrostatic repulsion. Its low-temperature crystal (below 90 K) possesses Pa3 space group symmetry. Each simple cubic unit cell contains four C60 , siting at four fcc sites with S6 site group symmetry (Figure 16.2a). A small amount of static disorder of two symmetry equivalent configurations is suggested existing in the low-temperature crystal [82, 84, 95]. The majority molecules adopt an orientation with the 6 : 6 bonds of one C60 unit facing the pentagons of adjacent C60 units (P configuration). The minority molecules adopt an orientation with 6 : 6 bonds of one C60 unit facing the hexagons of adjacent C60 units (H configuration). The energy difference between the two configurations are estimated to be 11 meV [84]. Their relative

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16 Studied by THz Spectroscopy and Solid-State Density Functional Theory

Intensity (km/mol)

proportion is suggested as 5.1 : 1 and persists from 5 to 90 K [82]. Above 90 K, C60 molecules hop between these two configurations, and the population of H configuration steadily increases at the expense of P configuration. Above 260 K, C60 molecules undergo rapid reorientation, and a phase transition to a fcc unit cell structure with Fm3m space group symmetry taking place at 260 K [82–84]. We in this work consider only the phonon mode vibrations of C60 in its lowest-temperature phase, i.e. the crystal structure with Pa3 space group symmetry. C60 in the gas phase possesses the highest order – Ih point group symmetry. C60 therefore experiences a symmetry reduction as condensed from gas to solid. According to the report in literature, the lowest frequency of normal modes of gas-phase C60 is around 270 cm−1 . In the crystalline state, only two IR active bands at 40 and 60 cm−1 have been adequately resolved at 10 K below 300 cm−1 [87–91]. The solid-state frequency calculation was performed with the P configuration. As shown in Figure 16.3a, the simulated IR normal modes satisfactorily reproduce the two IR bands. Figure 16.3b shows the percentage contributions of the intermolecular translations and librations and the intramolecular vibrations to the vibrational energy of all the simulated normal modes. We can clearly observe that in the low-frequency region, intramolecular vibrations do not appear at all. The intermolecular librations and

1.0 Crystal 0.5 0.0

(a) 100 Trans.

50 Percentage (%)

472

100 50

Lib.

100 50

Intra. Vib.

0 0 (b)

50

100 Wavenumber

150

200

(cm–1)

Figure 16.3 The simulation results for the C60 crystal. (a) The simulated IR spectrum in the crystalline state. Lorentzian line shapes with an arbitrary full width at half maximum (FWHM) are convolved into all modes to provide a visual guide. The frequency distributions of IR (black) and Raman (red) active modes are shown below the simulated spectrum. (b) The percentage contributions of the intermolecular translations, librations, and intramolecular vibration to the potential energy of each mode.

16.4 Interpretation of the Nature of Optical Phonon Modes

translations successively dominated normal modes in order of frequency and sharply separate. The above observation is completely consistent with the prediction of the rigid-body approximation. The nuclear site group analysis [20] predicts that a C60 crystal with Pa3 space group symmetry has 12 translation modes represented by Au + Eu + 3T u (with degenerate dimensions of 1, 2, and 3, respectively) and twelve libration modes represented by Ag + Eg + 3T g (with degenerate dimensions of 1, 2, and 3, respectively) [96]. Three T u modes are acoustical. Mixing between intermolecular translations and librations are not allowed because they belong to different irreducible representations. In addition, the irreducible representations of three classes of normal modes are consistent with the predictions of the rigid-body approximation (refer to the details in the previous review [70]). Finally, we conclude that in the C60 crystal intermolecular translations and librations and intramolecular vibrations do not mix at all and, thus, the rigid-body approximation is reliable. In fact, it has been repeatedly applied in the past to interpret the vibrational dynamics of C60 in IR [87–91], Raman [91, 97, 98], and inelastic neutron scattering spectroscopies [99, 100].

16.4.2

Anthracene

Anthracene is a planar conjugated hydrocarbon. It possesses D2h point group symmetry in the gas phase. As condensed into the solid phase, anthracene reduces to C i site group symmetry, and the crystal adopts P21 /c space group symmetry. Each base-centered monoclinic unit cell contains two anthracene molecules [25]. The DFT simulation indicates that a single anthracene does not have a permanent dipole moment. In contrast to C60 , CH/π bonds form between adjacent anthracene molecules and determine the directionality of intermolecular interactions (Figure 16.2b). Through the CH/π bonding, each anthracene connects to four surrounding molecules, forming a layered structure. Layers interact via dispersion forces. As shown in Figure 16.4, the gas-phase anthracene has two internal modes, denoted as IM-a and IM-b, in the 0–200 cm−1 frequency region. IM-a belongs to B3u representation and is IR active and Raman inactive, while IM-b belongs to Au representation and is both IR and Raman inactive. As anthracene condenses in crystal phase, on the one hand, both the two internal modes shift toward higher-frequency region and undergo frequency splitting (or correlation field splitting, the split modes belong to Au and Bu representation) due to the intermolecular interactions; on the other hand, IM-b becomes IR active owing to the symmetry reduction. In addition to the frequency split of the gas-phase modes, a set of intermolecular modes appear in crystal phase. As shown in Figure 16.4b, there are three modes dominated by intermolecular translations and six by intermolecular librations. Assuming anthracene is a rigid body, the nuclear site group analysis [20] predicts that there are six translation modes represented by 3Au + 3Bu (three Au + 2Bu are acoustical) and six libration modes represented by 3Ag + 3Bg . This prediction is totally consistent with the result of the ab initio calculations as

473

Intensity (km/mol)

16 Studied by THz Spectroscopy and Solid-State Density Functional Theory

(a)

Percentage (%)

474

b

0

a1

5 Crystal

a2

0 100 50 100 50 100 50 100 50 100 50 0

b1b2

IM-a

Trans. Lib.

IM-b

IM-a IM-b Residue 0

(b)

a

Gas phase

1

50

100 150 Wavenumber (cm–1)

200

Figure 16.4 The simulation results for the anthracene crystal. (a) Compares the simulated spectra in the gas and crystalline states. Lorentzian line shapes with an arbitrary full width at half maximum (FWHM) are convolved into all modes to provide a visual guide. The frequency distributions of IR (black) and Raman (red) active modes are shown below each simulated spectrum. (b) The percentage contributions of the intermolecular translations, librations, and two internal modes, IM-a and IM-b, to the potential energy of each mode. The residue is shown in the lower part as an evaluation of the completeness of the mode decomposition for each mode.

shown in Figure 16.4b. Therefore, the framework of the rigid-body approximation largely works for anthracene. Compared with C60 , a new phenomenon is observed in the anthracene system – mixing between intermolecular and intramolecular vibrations takes place in spite of the weak strength. As highlighted by the dot line frame in Figure 16.4b, the pair of split modes a1 and a2 of the intramolecular IM-a motion contains unbalanced contributions from intermolecular translations. Although being weak in the mixing strength, the mixing with intermolecular translations plays a crucial role in the interpretation of the isotope shifts (IS) of the pair modes. Because if the pair modes are a pure result of the frequency split, they should have equivalent IS. However, the experimental results indicate that the IS (5.5%) of mode a1 is larger than that (4.7%) of mode a2 [41]. This difference arises from a slightly stronger presence of translations in mode a2 (9.0%) relative to that in mode a1 (2.1%), as shown in Figure 16.4b, concerning the IS (2.8%) of pure translation modes is much smaller than that (5.7%) of the IM-a [48]. Finally, we emphasize that the vibrational mixing is completely consistent with the restriction of symmetry. According to the prediction of nuclear site group analysis, the librational motions belong to distinct representations from translations and the IM-a and IM-b motions. As a result, librations do not mix with the other two types of motions at all.

16.4 Interpretation of the Nature of Optical Phonon Modes

16.4.3

Adenine

Adenine is one of the four bases of nucleic acids and contains four nitrogen atoms in the main rings and a NH2 side group. As shown in Figure 16.2c, the configuration of adenine has the lowest-order of symmetry – the C 1 point group. Adenine crystals have P21 /c space group symmetry, with four pairs of adenine dimers at four C 1 sites in one monoclinic unit cell [101]. Therefore, each cell contains eight molecules. The asymmetric configuration produces weak permanent dipole moments. The adenine molecules in each dimer pair are bound by two hydrogen bonds, and each adenine is bound to two other molecules via two hydrogen bonds. The hydrogen bonding network forms layered structures, which stack via π/π and CH/π interactions [102, 103]. Overall, compared with anthracene, the new form contributing to the intermolecular interactions is the hydrogen bonding. The gas-phase adenine has three internal modes, denoted as IM-a, IM-b, and IM-c, respectively, in the 0–260 cm−1 frequency region as shown in Figure 16.5a. When adenine condenses in crystal phase, similarly as the anthracene case, we can still observe a clear trend that the three gas-phase modes shift toward higher frequency and split mainly into eight modes (represented by

Intensity (km/mol)

200

200

b

100

100

Gas phase c

a

0

0

40 Crystal

20

a1–8

c1–8

b1–8

0

Percentage (%)

(a)

0 0

100 50 100 50 100 50 100 50 100 50 100 50 0

50

100

150

200

250 740

760

IM-a 100 50 100

Trans.

50

Lib.

100 50 100

IM-a

0

(b)

1000

IM-b

50 100

IM-c

50

Residue

100 50 100 50

100

150

200

250 740

IM-b

IM-c

760

Wavenumber (cm–1)

Figure 16.5 The simulation results for the adenine crystal. (a) Compares the simulated spectra in the gas and crystalline states. Lorentzian line shapes with an arbitrary full width at half maximum (FWHM) are convolved into all modes to provide a visual guide. The frequency distributions of IR (black) and Raman (red) active modes are shown below each simulated spectrum. (b) The percentage contributions of the intermolecular translations, librations, and three internal modes, IM-a, IM-b, and IM-c, to the potential energy of each mode. Since each irreducible unit in the unit cell is composed of two adenines, the percentage contributions of the decomposed motions are calculated as an average over the two molecules. The residue is shown in the lower part as an evaluation of the completeness of the mode decomposition for each mode.

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2Ag + 2Au + 2Bg + 2Bu ) as there are eight molecules in one unit cell. It is worth noting that the frequency shift, induced by the intermolecular interactions, could be dramatically large for some modes. For example, the frequency of IM-b shifts from 216 cm−1 in the gas phase to around 760 cm−1 in the solid phase. In addition to the above behaviors of the internal modes, a set of intermolecular modes appears in the lower-frequency side of the crystalline spectrum. Under the rigid-body approximation, the nuclear site group analysis [20] predicts 24 translation modes represented by 3Ag + 3Au + 3Bg + 3Bu (three modes Au + 2Bu are acoustical) and 24 libration modes represented by 3Ag + 3Au + 3Bg + 3Bu . The intermolecular translations and librations are consequently allowed by symmetry to mix. As with anthracene, the framework of the rigid-body approximation is largely preserved in adenine. Inspecting Figure 16.5b, we can clearly define two frequency regions with respect to the vibrational characteristics. The first region includes the first 45 modes in which the translations and librations dominate, while the other includes all higher-frequency modes where intramolecular vibrations dominate. Normal modes in the first region have an irreducible representation of 6Ag + 5Au + 6Bg + 4Bu , which is the same as the rigid-body prediction. As with anthracene, the framework of the rigid-body approximation is largely preserved in adenine. Different from anthracene, the mixing between intermolecular and intramolecular vibrations and that between the three types of intramolecular vibrations tend to become more apparent. Such a stronger mixing results in nontrivial distributions of the internal modes away from their characteristic regions. For example, as shown in Figure 16.5a, the characteristic frequency region of IM-a in crystal phase can be identified within 181–202 cm−1 . Owing to the mixing with intermolecular vibrations, IM-a makes remarkable contributions of 9.3% and 18.7% to the normal modes at 124 and 157 cm−1 , respectively. Furthermore, the characteristic frequency of IM-b can be identified in the 748–769 cm−1 frequency region. Through mixing with IM-a and IM-c, IM-b also shows non-ignorable contributions in the frequency regions of 181–202 cm−1 and 235–259 cm−1 . We conclude that the hydrogen bond interaction leads to the mixing of vibrations in completely distinct frequency regions and results in the distributions of certain internal modes in regions remote from their characteristic frequencies. 16.4.4

𝜶-Glycine and L-Alanine

Glycine and l-alanine are amino acids with probably the most complex intermolecular interactions in nature. These two molecules have a neutral structure in the gas phase but undergo dramatic conformational change and adopt zwitterionic configurations in the solid phase as shown in Figure 16.2d,e. Therefore, the comparison between frequency calculation results in the gas and solid phases becomes pointless. The zwitterionic structure induces large molecular dipole moments; as a result, compared with adenine, strong dipole–dipole and dipole-induced dipole interactions occur in the amino acid crystals. In both molecules, robust head-to-tail amino acid chains form via NH· · ·O hydrogen bonds. Glycine forms three different structures, depending on the crystallization conditions [104]. In the 𝛼-polymorph, the chains are linked by hydrogen bonds

16.4 Interpretation of the Nature of Optical Phonon Modes

Intensity (km/mol)

to form double antiparallel layers that interact via mainly dispersion forces [86, 105–107]. The crystal structure has P21 /n space group symmetry, where each monoclinic unit cell contains four glycine molecules with C 1 site group symmetry. The l-alanine crystal has a unique phase where the chains are linked via hydrogen bonds to form columnar structures in three dimensions. The crystal structure has P21 21 21 space group symmetry, with each orthorhombic unit cell containing four l-alanine molecules with C 1 site symmetry [85]. The simulated IR active modes for the 𝛼-glycine and l-alanine crystals are shown in Figures 16.6a and 16.7a, respectively. Figures 16.6b and 16.7b show the percentage contributions of intermolecular translations and librations and two internal modes, IM-a and IM-b, to each mode for the 𝛼-glycine and l-alanine crystals, respectively. Refer to the previous works [59, 70] as to how IM-a and IM-b were constructed. In contrast to the previous three molecules, the backbones of amino acids adopt the linear configurations, so that they possess a skeletal torsional freedom which features the torsion of the carboxylate group with respect to all the remaining atoms. Owing to this structural flexibility, the lowest-lying internal modes IM-a fall into the similar frequency region as the intermolecular vibrations and strongly mix with them. As a result, it is difficult to find a clear boundary between dominant intermolecular and intramolecular vibrations. The characteristic frequency regions of the intramolecular vibrations completely overlap those of intermolecular vibrations. Strong mixing between the two classes of vibrations takes place in the overlapped regions. The rigid-body approximation is no more suitable. In addition, similar to the adenine system, we observe the nontrivial mixing between vibrations with

50

10

Crystal 0

0

Percentage (%)

(a) 100 50 100 50 100 50 10 5 10 5 0

Lib. IM-a IM-b Residue 0

(b)

100 50 100 50 100 50 100 50 100 50 0

Trans.

50

100

150

200 600

IM-a

IM-b

650

Wavenumber (cm-1)

Figure 16.6 The simulation results for the 𝛼-glycine crystal. (a) The simulated spectrum in the crystalline state. Lorentzian line shapes with an arbitrary full width at half maximum (FWHM) are convolved into all modes to provide a visual guide. The frequency distributions of IR (black) and Raman (red) active modes are shown below each simulated spectrum. (b) The percentage contributions of the intermolecular translations, librations, and three internal modes, IM-a, and IM-b, to the potential energy of each mode. The residue is shown in the lower part as an evaluation of the completeness of the mode decomposition for each mode.

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Intensity (km/mol)

16 Studied by THz Spectroscopy and Solid-State Density Functional Theory

10

Crystal

200

0

0

(a)

Percentage (%)

478

100 50 100 50 100 50 10 5 10 5 0

Trans. Lib. IM-a IM-b Residue 0

(b)

50

100

150

200 500

550

100 50 100 50 100 50 100 50 100 50 0 600

IM-a

IM-b

Wavenumber (cm–1)

Figure 16.7 The simulation results for the L-alanine crystal. (a) The simulated spectrum in the crystalline state. Lorentzian line shapes with an arbitrary full width at half maximum (FWHM) are convolved into all modes to provide a visual guide. The frequency distributions of IR (black) and Raman (red) active modes are shown below each simulated spectrum. (b) The percentage contributions of the intermolecular translations, librations, and three internal modes, IM-a, and IM-b, to the potential energy of each mode. The residue is shown in the lower part as an evaluation of the completeness of the mode decomposition for each mode.

distinct characteristic frequencies. Specifically, we examine the mixing between the intimal modes IM-b, which features the NH3 + torsion, and intermolecular vibrations. The characteristic frequencies of NH3 + torsions normally appear in high-frequency regions (640 cm−1 for 𝛼-glycine and 540 cm−1 for l-alanine). As shown in Figures 16.6b and 16.7b, NH3 + torsions have weak but non-ignorable contributions (around 5%) to the normal modes dominated by intermolecular vibrations in the lower-frequency regions about 100 cm−1 . 16.4.5 Frequency Sequences of Intermolecular Translations and Librations and Intramolecular Vibrations In light of a quantitative examination of the origins of optical phonon modes in the five prototypical intermolecular interaction systems, we will in this and the next sections give answers to the two questions raised in the introduction concerning the general issue about interpreting THz spectroscopy. We will first consider the frequency sequence of intermolecular translations and librations and intramolecular vibrations. Overall, the intermolecular vibrations would lie in the lower-frequency end, and intramolecular vibrations appear in the relatively higher-frequency region. The frequency separation between the two forms of vibrations can be as large as 200 cm−1 in the case of C60 . Proceeding from C60 to amino acids, we have observed a clear tendency of the gradual decrease of the lowest-lying intramolecular vibrations with the increase of the skeletal flexibility. We will also observe in the latter section that in molecules composed of multi-segments, e.g. adenosine, a skeletal torsional mode has even lower energy

16.4 Interpretation of the Nature of Optical Phonon Modes

6000

Average moment of inertia (Å2)

Figure 16.8 Comparison of the molecule weights and the average of the moments of inertia about the three principle axes for C60 , anthracene, adenine, L-alanine, and 𝛼-glycine.

C60 anthracene adenine α-glycine L-alanine

4000

2000

0 0

200

400

600

Molecular weight

than intermolecular vibrations. Therefore, the frequency sequence of intramolecular vibrations with respect to intermolecular vibrations depends on the skeletal flexibility of a molecule in question. In terms of intermolecular translations and librations, since both of them arise from intermolecular interactions, we argue that both forms of vibrations would have similar force constant and that their frequency sequence is associated with the ratio of the moment of inertia to molecular weight (MW). We would anticipate that the ratio becomes larger for the case in which the librations have lower frequencies than the translations. To see this, in Figure 16.8 we compare the average moment of inertia about the three principle axes and MW for C60 , anthracene, adenine, glycine, and l-alanine. C60 has the highest ratio of 8.4 Å, corresponding to that all of the most low-lying optical modes are dominated by librations. Anthracene has a ratio of 5.1 Å. Consequently, the libration about mainly the principal axis – x – with the largest moment of inertia dominates the first optical mode (Figure 16.2b). For the other three molecules, adenine, 𝛼-glycine, and l-alanine, the first several optical phonon modes are constantly dominated by translations. As shown in Figure 16.8, the three molecules appear having similar ratios. Fitting the three data points using a linear equation constrained to pass the origin, we get an optimized ratio of 2.3 Å2 . We thus tentatively deduce that any molecules with a ratio of the average moment of inertia to MW around 2.3 Å2 would have a feature wherein translations have the lowest frequencies over librations. The more deviating beyond this criterion, the most likely that librations dominate the lowest-lying optical phonon modes. 16.4.6

Mixing Between Intermolecular and Intramolecular Vibrations

Starting with C60 and proceeding to the amino acids, we observed a gradual overlap of the characteristic frequency regions of intermolecular and intramolecular vibrations with the increase of backbone flexibility, as well as strong mixing of the two classes of vibrations in the overlapping regions in the amino acids. Except for C60 , we also observed mixing of intermolecular and intramolecular vibrations

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16 Studied by THz Spectroscopy and Solid-State Density Functional Theory

in distinct frequency regions; the mixing strength increases from anthracene to adenine and to the amino acids. This phenomenon represents a deviation from the conventional notion that considers mixing unlikely between vibrations with significant energy separation. We have discussed two forms of mixing between intermolecular and intramolecular vibrations. One is strong mixing between vibrations with similar characteristic frequencies, such as intermolecular vibrations and skeletal torsions in the amino acids. The other is weak mixing between vibrations with different characteristic frequencies, such as intermolecular vibrations and the three internal modes of adenine as well as NH+3 torsions of the amino acids; these forms of mixing occur in all the molecular crystals. Note that C60 represents the unique exception in which neither of the two forms of mixing do not take place. The first form is controlled by the molecular structural factor. When molecules become sufficiently flexible to induce intramolecular vibrations having similar energies as those of intermolecular vibrations, mixing will occur. An example is torsional motion around the single bonds of molecular backbones. The second mixing form is different. In the example of NH+3 torsion that originates from rotational motion of a localized group, the potential surface would be quite stiff. The NH+3 torsion is therefore expected to have solely high energy. However, as shown in Figures 16.6 and 16.7, the NH+3 torsions have low-frequency characteristics via mixing with intermolecular vibrations. This phenomenon should be originated deeply from the subtle electronic structures of the intermolecular interactions that can only be treated properly, to some extent, by the modern ab initio approach but be hardly predicted by the empirical force field method. The second form may deepen our understanding of the nature and complexity of THz normal modes of molecular systems in the presence of hydrogen bonds and therefore a mechanism of vibrational energy transfer among normal modes in different frequency regions. Vibrational energy can be transferred among normal modes when the corresponding anharmonic coupling terms are nonzero. For example, strong anharmonic coupling between high-frequency O—H stretching modes and low-frequency hydrogen bonds has been confirmed in liquid-state hydrogen bonding systems [85, 108]. We regard low-frequency vibrations as a thermal bath for dissipation of high-energy vibrations [109–112]. If high- and low-frequency normal modes have completely different regions, it is difficult to understand the atomic pictures underlying such energy transfer mechanism(s). However, if the high- and low-frequency normal modes are allowed to share common characteristics, as revealed in this work, then the shared vibrations will provide energy transfer pathways via anharmonic couplings [113–119]. Given the significance of weak mixing between intermolecular and intramolecular vibrations at distinct characteristic frequencies, the fundamental question becomes mechanistic. In terms of the presence of this form of mixing, the five benchmark molecules divide into two groups: C60 and anthracene where the mixing does not exist and adenine, glycine, and l-alanine where the mixing exists. Compared with first group, the second group has three types of additional intermolecular interactions: hydrogen bond, dipole–dipole, and dipole-induced dipole interactions. We would argue that the three types of interactions play

16.5 Application of the DFT-D Method in a Material System scPLA

a crucial role in enabling the second form of mixing. At the current stage, the detail mechanism is still unclear.

16.5 Application of the DFT-D Method in a Material System scPLA In the above section, we have verified the credibility of the DFT-D* approach in terms of reproducing THz modes arising from a variety of non-covalent interactions. We will in this section elucidate an application of this theoretical tool to a practical system [70]. Symmetry conservation has been used as a principle of determining polymer conformations in the crystalline state [120–122]. The most famous symmetry–conservation law in polymer science is perhaps the equivalence postulate, formulated by the early pioneers of polymer crystallography, Bunn [123], Huggins [124], and Pauling [125]. This postulate claims that the constitutional repeating units – “the smallest constitutional unit, the repetition of which constitutes a regular macromolecule” [120] – of single polymer chains should be related by symmetry. We will illustrate that an interplay of THz spectroscopy and solid-state DFT will provide unambiguous evidence of the violation of this principle with using a prototype polymer crystal – poly(lactic acid) stereocomplex (scPLA) [126]. In scPLA, enantiomeric poly(l-lactic acid) (PLLA) and poly(d-lactic acid) (PDLA) coexist in equal amounts. According to the X-ray diffraction (XRD) measurements [127–129], the crystal possesses R3c space group symmetry. In each primitive cell, a left-handed 31 PLLA helix and a right-handed 31 PDLA helix adopt a parallel packing conformation and conserve a perfect chiral relation via glide plane operations (Figure 16.9). scPLA samples without preferred orientations were prepared by annealing the casted films of an equimolar mixture of PLLA and PDLA. scPLA samples with polymer chains uniaxially aligned were prepared for the polarization THz measurements. Four bands were resolved as shown in the upper panel of Figure 16.10a; bands a and d possess polarizations parallel to the chain axes, while bands b and c possess polarizations perpendicular to the chain axes as shown in the middle and bottom panels of Figure 16.10a, respectively. Concerning the gamma point phonon modes, the R3c space group has three irreducible representations, A1 , A2 , and E [20]. A1 , general to all space groups, represents IR active modes with polarizations parallel to the chain axes. E arises from the 31 screw symmetry and represents double-generate IR active modes with polarizations perpendicular to the chain axes. A2 originates from the glide plane symmetry, representing IR inactive but Raman active modes. In summary, the polarization THz measurements make possible a specific examination of the reproduction of normal modes with A1 and E irreducible representations and consequently a verification as to whether the 31 screw symmetry is a good element. The B3LYP-D* functional [130, 131] and Gaussian’s 6-311G(d,p) basis set [132] were employed for the solid-state DFT calculations. If the R3c space group symmetry is conserved, three bands are predicted as shown in the upper panel

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4

6 3

z

5 y

2

z

4 3 1

x x

2

6

y

5 1 (a)

(b)

Figure 16.9 Crystal structure of scPLA with R3c space group symmetry. The two enantiomorphous helices in one primitive cell are projected onto a plane normal to (a) and parallel to (b) the chain axes (z direction). The y-axis lies on a plane determined by the two chain axes of PLLA and PDLA. The red, green, and blue lines represent the three primitive cell vectors a, b, and c, respectively. Source: Reprinted with permission from Zhang et al. [127]. Copyright 2016, American Chemical Society.

of Figure 16.10b. Two modes (middle panel of Figure 16.10b), bearing the A1 irreducible representation, are assigned to THz bands with polarizations parallel to the chain axes; a pair of degenerate modes (bottom panel of Figure 16.10b), bearing the E irreducible representation, is assigned to the THz bands with polarization perpendicular to the chain axes. One can readily find that THz bands a and d, with polarizations parallel to the chain axes, are rather satisfactory against both the frequency and intensity criteria, yet the reproduction of THz bands b and c, with polarizations perpendicular to the chain axes, is not satisfactory. This observation indicates that E is not a good irreducible representation for crystalline scPLA, and the underpinning 31 screw symmetry should not be preserved. If we remove the 31 screw symmetry and optimize the R3c-geometry in the P1 space group, i.e. imposing no symmetric restriction on the relaxation of atomic coordinates, all the four experimental bands have been qualitatively reproduced as shown in the upper panel of Figure 16.10c. Compared with the R3c-simulation, the symmetry relegation barely affects the reproduction of normal modes with a polarization parallel to the chain axes (middle panel of Figure 16.10c) but remarkably improves the reproduction of normal modes with a polarization perpendicular to the chain axes (bottom panel of Figure 16.10c). The characteristics of the optical phonon modes simulated by breaking the 31 screw symmetry are shown in Figure 16.10d.

16.5 Application of the DFT-D Method in a Material System scPLA

Absorbance

0 1

b

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10

d

0 25 50 75 Wavenumber (cm–1)

(a)

a

25 50 75 Wavenumber (cm–1)

100 (d)

25 50 75 100 Wavenumber (cm–1)

Figure 16.10 Simulation results for scPLA. Panel (a) shows a THz spectrum of scPLA without preferred orientation (top) and THz spectra of scPLA with polymer chains uniaxially parallel (middle) and perpendicular (bottom) to the polarization direction of THz pulses; the frequency region with a reduced signal/noise ratio is shadowed. The XRD and THz spectra were recorded at 293 and 78 K, respectively. Panels (b,c) show the normal modes simulated with R3c and P1 space groups, respectively. In each panel, the top, middle, and bottom parts display the simulated normal modes with the corresponding polarization properties with those of panel (a); Lorentzian line shapes with an arbitrary half width at half maximum (HWHM) are convolved to normal modes for visualization; a frequency distribution pattern of normal modes is shown in the top part (Raman active modes of R3c space group are shown in red). Panel (d) shows the percentage contributions of the intermolecular translations, librations, and intramolecular vibration the potential energy of each mode simulated with the P1 space group symmetry. Since the PLLA and PDLA in the unit cell are irreducible to each other, the percentages of all the decomposed motions are calculated as an average over the two polymers. Source: Adapted with permission from Zhang et al. [127]. Copyright 2016, American Chemical Society.

We next examine how far the 31 screw symmetry is broken. In the example of PLLA, we fixed the atomic positions of the constitutional repeating unit #4 and evaluated the RMSD of the other two repeating units, #5 and #6, from the artificially created ones with perfect 31 screw symmetry with respect to #4 as shown in Figure 16.11a. PDLA was analyzed in the same way as shown in Figure 16.11b. It clearly shows the hydrogen atoms of the CH3 groups, and the oxygen atoms of the C=O groups undergo the strongest deviations in the

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16 Studied by THz Spectroscopy and Solid-State Density Functional Theory

Figure 16.11 Examination of the symmetry breaking extents for PLLA and PDLA simulated with P1 space group. Panels (a,b) show the RMSDs from the 31 screw symmetry for PLLA and PDLA, respectively. RMSDs are examined with respect to all the chemically irreducible atoms. Source: The data was adapted with permission from Zhang et al. [127]. Copyright 2016, American Chemical Society.

(a) 10 Deviation (×10–3 Å)

0 (b) 10

) (C 3

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(C )

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H

) (H

O )

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( C

(

O

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)

C)

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O

484

symmetry breaking process. This result is consistent with the IR experimental results that the interactions between the CH3 and the C=O groups play an important role in the formation of scPLA [133, 134]. The relaxation of these functional groups from their symmetric sites restricted by the R3c space group may result in a stronger interaction between PLLA and PDLA and lower the electronic energy by 0.098 kJ/mol. Note that this work adopts Grimme’s empirical approach to take into account the dispersion forces, whose accuracy in terms of a lattice-energy prediction is in several kJ/mol [135, 136]. The energy calculation result is therefore sound at a qualitative level.

16.6 Experimental Evidence Supporting the Mode Assignments The major part of this work relies on the experimental criteria of frequency and intensity (in most cases) and IS (only for anthracene) to evaluate the credibility of mode assignments. We introduce in this section another experimental dimension on which qualitative evidences of the mode assignments are provided. As we know, intermolecular vibrations arise from intermolecular interactions that are strongly affected by the environments in which molecules are sited. A variation of the packing structures of molecular solids would significantly alter the intermolecular interactions and in consequence lead to a notable change of the THz absorption feature. On the contrary, intramolecular vibrations, relatively less associated with intermolecular interactions, are not expected to be affected by the change of packing structures. By taking the scPLA as an example, the annealed sample shows very high crystallinity and exhibits four clear THz peaks as shown in the lower panels of

16.6 Experimental Evidence Supporting the Mode Assignments 0.6 Non-annealed 0.4 Absorbance

Intensity (cps)

500

0 Annealed

10 × 10

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Nonannealed

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d

0.5 0.0

0 10

(a)

20

30 2θ (°)

40

25

50

(b)

50

75

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Wavenumber (cm–1)

Figure 16.12 Examination of the effect of crystallinity of scPLA on the THz observation. Panel (a) compares the XRD patterns of annealed and non-annealed scPLA. Panel (b) compares the THz spectra of annealed and non-annealed scPLA. Source: The annealed sample data was adapted with permission from Zhang et al. [127]. Copyright 2016, American Chemical Society.

Figures 16.12a,b, respectively. As characterized in Figure 16.10d, mode a features a primary intramolecular vibration and a secondary intermolecular libration. Obviously the strong IR intensity would arise from the dominant intramolecular vibrations. In contrast, the other three modes, b, c, and d, feature a balanced mixing between intermolecular and intramolecular vibrations and show medium IR intensities. If we do not anneal the scPLA sample, the crystallinity dramatically reduces as shown in the upper panel of Figure 16.12a. In this case, most regions inside the crystal become amorphous and the intermolecular interaction patterns are remarkably changed. Consequently, the three modes, b, c, and d, in which intermolecular librations and translations have notable contributions, disappear in the THz spectrum, while mode a, dominated by the intramolecular vibration, remains. A similar verification of the mode assignment is performed to adenosine [61]. The commercial sample has good crystallinity and displays two sharp THz peaks a and b as shown in the lower panels of Figures 16.13a,b, respectively. If we lyophilize the commercial sample to reduce the crystallinity, we observe that mode a remains, while mode b disappears, indicating the former has more likely an intermolecular nature and the latter an intramolecular nature. This implication is completely consistent with the theoretical predictions. As shown in Figure 16.13c, both the two THz modes have been satisfactorily reproduced by the solid-state DFT calculations. The mode characterization in Figure 16.13d shows that mode a features indeed a mixing of a primary intramolecular vibration and a secondary intramolecular libration, while mode b the mixing of a primary intermolecular translations and a secondary intermolecular libration and intramolecular vibration.

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16 Studied by THz Spectroscopy and Solid-State Density Functional Theory

30 Lyophilized

0 Crystalline

10 × 103

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Lyophilized Absorption coefficient (cm–1)

Intensity (km/mol)

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a

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486

a

Lib. 50 100 Intra. vib.

0.0

50 20

(c)

100

40

60

Wavenumber (cm–1)

0 20 (d)

40

60

Wavenumber (cm–1)

Figure 16.13 Examination of the effect of crystallinity of adenosine on the THz observation. Panel (a) compares the XRD patterns of lyophilized and commercial adenosine. Panel (b) compares the THz spectra of lyophilized and commercial adenosine. Panel (c) shows the frequency calculation results for the crystalline adenosine. Lorentzian line shapes with an arbitrary full width at half maximum (FWHM) are convolved into all modes to provide a visual guide. Panel (d) shows the percentage contributions of the intermolecular translations, librations, and intramolecular vibration to the potential energy of each mode.

16.7 Conclusion Using solid-state DFT, we have analyzed optical phonon modes probed by THz spectroscopy. Through a systematic characterization of crystals of C60 , anthracene, adenine, 𝛼-glycine, and l-alanine, we specifically focus on two basic issues concerning mode assignments of molecular phonon modes. One is the frequency distributions of intermolecular translations and librations; the other is the frequency distribution of intramolecular vibrations of molecular individuals when siting in a crystalline field. As for the first issue, we proposed a criterion using the ratio of the average moment of inertia of molecular individuals to MW. If the ratio is around

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41 Kambara, O., Ponseca, C.S., Tominaga, K. et al. (2013). Vibrational mode

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17 Calculation of Vibrational Resonance Raman Spectra of Molecules Using Quantum Chemistry Methods Julien Guthmuller Gda´nsk University of Technology, Faculty of Applied Physics and Mathematics, Department of Theoretical Physics and Quantum Information, Narutowicza 11/12, 80-233 Gda´nsk, Poland

This chapter focuses on the spectroscopy of vibrational resonance Raman scattering. First, the results of time-dependent perturbation theory are summarized in order to define the Raman scattering cross sections. Then, the most commonly employed approximations to calculate Raman intensities are described in both cases of normal (i.e. non-resonant) Raman and resonance Raman scattering. Additionally, the relation between the resonance Raman intensities and the absorption spectrum is presented. Recursive methods to calculate the Franck–Condon overlap integrals, which are required to compute resonance Raman and absorption spectra, are described. Moreover, several quantum chemistry methods that can be employed to calculate the structural, vibrational, and electronic properties of the molecular ground and excited states are reviewed. Finally, few key applications on molecular systems are provided to illustrate the performance of the different approximations and methods. These consist of results obtained by using the short-time approximation, by including Franck–Condon and Herzberg–Teller vibronic couplings, and by considering the situation of several electronic excited states in resonance.

17.1 Introduction The Raman effect was discovered in 1928 by Raman and Krishnan [1], while independent theoretical predictions of this phenomenon had been made by Smekal [2] in 1923 and by Kramers and Heisenberg [3] in 1925 using quantum theory. A description of the early history of the Raman effect can be found, for example, in the detailed review paper of Long [4]. In general, the Raman process consists of an inelastic scattering of light by matter. Because the scattered light carries detailed information about the properties of the scattering materials or molecules, numerous spectroscopies were developed, making use of the Raman effect. When considering molecular systems, the most commonly employed spectroscopy is vibrational Raman scattering (Figure 17.1), in which a transition occurs between two vibronic levels of the molecule. Typically, the vibrational Molecular Spectroscopy: A Quantum Chemistry Approach, First Edition. Edited by Yukihiro Ozaki, Marek Janusz Wójcik, and Jürgen Popp. © 2019 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2019 by Wiley-VCH Verlag GmbH & Co. KGaA.

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17 Calculation of Vibrational Resonance Raman Spectra of Molecules

(a)

V

(b)

(c)

V e

e

ћ

vn = 1 vn = 0

∂Ωeg ∂Qn

eq

eq ћΩeg

ћωL

ћωs

ћωL

u′

ћωeg

ћωs

g u

e

g

g u

un = 1 un = 0

u′

ΔeQn Q

Qn

Figure 17.1 (a) Normal (non-resonant) Raman scattering, (b) resonance Raman scattering, and (c) displaced harmonic oscillator model.

Raman process involves a transition from an initial vibrational state (characterized by the quantum number u) to a final vibrational state (characterized by the quantum number u′ ) of the electronic ground state (g). In the case of Stokes Raman scattering, the energy of the final state is higher than the energy of the initial state, so that the scattered photon energy (ℏ𝜔S ) is smaller than the incident photon energy (ℏ𝜔L ), whereas for anti-Stokes scattering, it is the reverse. In experimental apparatuses, the incident radiation is usually in the visible or ultraviolet region. Depending on the incident light frequency (𝜔L ) and of the investigated compound properties, two types of scattering can be distinguished: (i) normal Raman scattering, in which the incident photon energy is far below the electronic excited state (e) transition energies (ℏ𝜔L ≪ ℏ𝜔eg ) (Figure 17.1a), and (ii) resonance Raman (RR) scattering, in which the incident photon energy is near an electronic excited state transition energy (ℏ𝜔L ≈ ℏ𝜔eg ) (Figure 17.1b). When resonance occurs with the discrete vibronic levels of the state e, the process is referred as discrete RR scattering [5]. In this case, the RR scattering intensities are enhanced by up to 6 orders of magnitude compared with normal Raman scattering. This results in the high sensitivity of RR spectroscopy, allowing detection of diluted species in solution, of complex sample mixtures, and of chromophores present in large molecular systems. Furthermore, the RR intensities depend on the properties of the electronic excited state(s) in resonance and therefore carry specific information about the structures and the dynamics of the resonant electronic excited state(s) (see, e.g. Refs. [6, 7]). The accurate calculation of RR intensities can help to decipher the information carried by the scattered light by establishing a direct connection between the RR intensities and the molecular properties. Therefore, the understanding

17.2 Theory of Resonance Raman Scattering

and interpretation of experimental RR spectra can strongly benefit from theoretical simulations. These can be achieved by combining quantum chemistry (QC) methods to calculate the electronic and vibrational molecular properties, together with appropriate models and approximations to compute the Raman intensities. It is the goal of the present chapter to present the main and most commonly employed approaches to calculate RR spectra using QC methods. This presentation is inspired by previous reports on the topic (see, e.g. Refs. [5, 7–11]) and is organized as follows: in Section 17.2 the theory of Raman scattering is summarized, and several approximations are introduced in order to obtain practical expressions from which the Raman intensities can be calculated using QC methods. Then, in Section 17.3 different applications on molecular systems are described in order to assess the accuracy of the methods and to present the impact of different vibronic effects on the RR intensities. Finally, Section 17.4 provides a conclusion.

17.2 Theory of Resonance Raman Scattering, Approximations, and Quantum Chemistry Methods 17.2.1 Sum-Over-State Formulation of the Vibrational Raman Intensities The quantum mechanical theory of Raman scattering is well known (see, e.g. Refs. [5, 12–14]) and provides a formal expression from which Raman intensities (or Raman cross sections) can be calculated. This expression is derived from time-dependent perturbation theory, in which the interaction between the incident monochromatic radiation (at frequency 𝜔L ) and the molecule is described by the electric dipole approximation. Thus, the first-order response to the incident electric field is characterized by an induced transition electric dipole moment: (𝜇𝜌(1) )i→f = (𝛼𝜌𝜎 )i→f E𝜎

(17.1)

where E𝜌 is a component of the electric field amplitude, (𝜇𝜌(1) )i→f is a component of the induced transition electric dipole moment amplitude for a transition between the initial (i) and final (f) molecular states, and (𝛼 𝜌𝜎 )i → f is the Raman polarizability tensor describing the linear response to the electric field. The knowledge of the Raman polarizability tensor enables the calculation of the Raman scattering intensities. Application of perturbation theory provides a sum-over-state (SOS) expression for (𝛼 𝜌𝜎 )i → f that was originally derived by Kramers, Heisenberg, and Dirac [3, 15] and that takes the following form: { } 1 ∑ ⟨f|𝜇𝜌 |k⟩⟨k|𝜇𝜎 |i⟩ ⟨f|𝜇𝜎 |k⟩⟨k|𝜇𝜌 |i⟩ + (17.2) (𝛼𝜌𝜎 )i→f = ℏ k 𝜔ki − 𝜔L − iΓk 𝜔kf + 𝜔L + iΓk where 𝜇𝜌 is a component of the dipole moment operator, 𝜔ki ≡ (Ek − Ei )/ℏ is the Bohr frequency between the initial (i) and intermediate (k) molecular states, and Γk is a damping factor describing a homogeneous broadening. The inverse of the damping factor Γ−1 is related to the lifetime of the excited state (k). For simplicity, k

499

500

17 Calculation of Vibrational Resonance Raman Spectra of Molecules

the damping factors will be assumed equal (i.e. Γk ≡ Γ). The indices 𝜌 and 𝜎 refer to Cartesian components in a molecule-fixed axis system. In the following, the Born–Oppenheimer (BO) approximation (i.e. adiabatic approximation) is employed, in which the molecular states can be written as a product of the electronic |𝜑⟩, vibrational |𝜃⟩, and rotational |R⟩ wavefunctions. Moreover, in the BO approximation the energy of a given molecular state is obtained as a sum of the electronic, vibrational, and rotational energies. Because most Raman scattering is observed under experimental conditions in which the rotational structure is not resolved, it is usually assumed that the Raman process involves a transition between pure initial and final vibronic states [5, 12]. In that case, the rotational energies appearing in the Bohr frequencies, i.e. 𝜔ki and 𝜔kf , of Eq. (17.2) are neglected. Then, the closure relation over the rotational ∑ states (i.e. |Rr ⟩⟨Rr | = 1) can be applied on the numerators of Eq. (17.2) for the r

intermediate states (k). Additionally, the relation between the molecule-fixed axis system and the laboratory-fixed axis system is obtained by performing classical isotropic averaging, in which all molecular orientations are assumed to be equally possible [5]. In the case of pure vibrational Raman scattering, the initial, intermediate, and final vibronic states take the form |i⟩ = |𝜑g ⟩|𝜃gu ⟩, |k⟩ = |𝜑e ⟩|𝜃ev ⟩, |f⟩ = |𝜑g ⟩|𝜃gu′ ⟩

(17.3)

where g and e represent the electronic ground and excited states, respectively, and the indices u, u′ , and v indicate the associated vibrational quantum numbers. For inelastic scattering, u and u′ are assumed to be different. Next, the transition dipole moment can be written as ⟨i|𝜇𝜌 |k⟩ = ⟨𝜃gu |⟨𝜑g |𝜇𝜌 |𝜑e ⟩|𝜃ev ⟩ = ⟨𝜃gu |(𝜇𝜌 )ge |𝜃ev ⟩

(17.4)

where (𝜇𝜌 )ge is a component of the electronic transition dipole moment between the electronic ground state and an electronic excited state. Using these notations, the Raman polarizability tensor for pure vibrational scattering becomes { 1 ∑ ∑ ⟨𝜃gu′ |(𝜇𝜌 )ge |𝜃ev ⟩⟨𝜃ev |(𝜇𝜎 )eg |𝜃gu ⟩ (𝛼𝜌𝜎 )gu→gu′ = ℏ e≠g v 𝜔eg + 𝜔vu − 𝜔L − iΓ } ⟨𝜃gu′ |(𝜇𝜎 )ge |𝜃ev ⟩⟨𝜃ev |(𝜇𝜌 )eg |𝜃gu ⟩ + (17.5) 𝜔eg + 𝜔vu′ + 𝜔L + iΓ where 𝜔eg and 𝜔vu are the Bohr frequencies associated to the electronic and vibrational energies, respectively. In Eq. (17.5), the term e = g is neglected from the summation over the electronic states. This term is usually small for incident frequencies larger than the vibrational Bohr frequencies, i.e. 𝜔L ≫ 𝜔vu , 𝜔vu′ . By performing isotropic averages of Eq. (17.5), it is possible to define the cross section for a given experimental setup [5, 12, 16]. In particular, for the commonly employed 90∘ scattering geometry consisting of an incident linearly polarized radiation perpendicular to the scattering plane and a detection of all scattered polarizations at an angle of 90∘ with respect to the incident radiation direction,

17.2 Theory of Resonance Raman Scattering

the Raman differential cross section is given by d𝜎gu→gu′ dΩ

=

𝜔L 𝜔3S

1 (45a2 + 5𝛿 2 + 7𝛾 2 ) 16π2 𝜀20 c4 45

(17.6)

where 𝜔S is the frequency of the scattered light and a2 , 𝛿 2 , and 𝛾 2 are the three Raman rotational invariants for randomly oriented molecules: 1 |𝛼 + 𝛼yy + 𝛼zz |2 9 xx 3 𝛿 2 = {|𝛼xy − 𝛼yx |2 + |𝛼xz − 𝛼zx |2 + |𝛼yz − 𝛼zy |2 } 4 3 2 𝛾 = {|𝛼xy + 𝛼yx |2 + |𝛼xz + 𝛼zx |2 + |𝛼yz + 𝛼zy |2 } 4 1 + {|𝛼xx − 𝛼yy |2 + |𝛼xx − 𝛼zz |2 + |𝛼yy − 𝛼zz |2 } 2

a2 =

(17.7)

where in Eq. (17.7) 𝛼 𝜌𝜎 denotes the components of the Raman polarizability tensor of Eq. (17.5). 17.2.2 Normal Raman Scattering in the Double Harmonic Approximation Henceforth, it is assumed that the vibrational modes and frequencies are obtained within the harmonic approximation [17]. In this case, u and v are multi-indices representing the harmonic quantum numbers of the vibrational normal modes of the ground and excited states, respectively, i.e. u ≡ u1 , u2 , … , uM , where M is the number of modes. Additionally, the vibrational states can be written as a product of one-dimensional harmonic oscillator states |𝜒⟩: |𝜃gu ⟩ =

M ∏

|𝜒gul ⟩; |𝜃ev ⟩ =

l=1

M ∏

|𝜒evl ⟩

(17.8)

l=1

The situation of non-resonant Raman scattering is first described (Figure 17.1a), in which the incident photon energy ℏ𝜔L is much smaller than any electronic transition energy (i.e. adiabatic energy ℏ𝜔eg ) but is assumed to be much larger than the vibrational transition energies, i.e. ℏ𝜔eg ≫ ℏ𝜔L ≫ ℏ𝜔vu , ℏ𝜔vu′ . Then, the Placzek [18] approximation consists of neglecting the vibrational Bohr frequencies 𝜔vu and 𝜔vu′ in the denominators of Eq. (17.5). This allows the application of the closure relation over the intermediate vibrational states (i.e. ∑ |𝜃ev ⟩⟨𝜃ev | = 1) in the numerators of Eq. (17.5), which leads to the Raman v

polarizability tensor (𝛼𝜌𝜎 )gu→gu′ = ⟨𝜃gu′ |(𝛼𝜌𝜎 )gg |𝜃gu ⟩

(17.9)

where (𝛼 𝜌𝜎 )gg defines the ground state electronic adiabatic polarizability [5]: } { (𝜇𝜌 )ge (𝜇𝜎 )eg (𝜇𝜎 )ge (𝜇𝜌 )eg 1∑ (𝛼𝜌𝜎 )gg = + (17.10) ℏ e≠g 𝜔eg − 𝜔L − iΓ 𝜔eg + 𝜔L + iΓ

501

502

17 Calculation of Vibrational Resonance Raman Spectra of Molecules

where 𝜔eg is the adiabatic Bohr frequency (Figure 17.1). The tensor (𝛼 𝜌𝜎 )gg depends on the nuclear coordinates and can be developed as a Taylor series: ( ) ∑ 𝜕(𝛼𝜌𝜎 )gg eq Ql + · · · (17.11) (𝛼𝜌𝜎 )gg = (𝛼𝜌𝜎 )gg + 𝜕Ql eq l eq

where (𝛼𝜌𝜎 )gg and (𝜕(𝛼 𝜌𝜎 )gg /𝜕Ql )eq are, respectively, the ground state electronic polarizability and the derivative of the ground state electronic polarizability evaluated at the ground state equilibrium geometry (denoted by eq). The index l indicates a summation over all the mass-weighted normal coordinates Ql of the electronic ground state. Equation (17.11) is truncated after the linear term with respect to Ql . This leads to the so-called double harmonic approximation [17, 19], i.e. the potential energy surface (PES) is restricted to the harmonic quadratic term in Ql , while the polarizability is restricted to the first order in Ql . Reporting Eq. (17.11) into Eq. (17.9) leads to ( ) ∑ 𝜕(𝛼𝜌𝜎 )gg eq (𝛼𝜌𝜎 )gu→gu′ = (𝛼𝜌𝜎 )gg 𝛿u′ u + ⟨𝜃gu′ |Ql |𝜃gu ⟩ (17.12) 𝜕Ql eq l The first term on the right-hand side of Eq. (17.12) is non-zero only when u = u′ and consequently describes an elastic scattering of light (i.e. Rayleigh scattering). The second term on the right-hand side of Eq. (17.12) describes Raman scattering, in which the initial vibrational quantum numbers u are different than the final vibrational quantum numbers u′ . Introducing the one-dimensional harmonic oscillator states |𝜒⟩ from Eq. (17.8) and using the fact that they are orthonormal leads to the Raman polarizability tensor ( ) ∑ 𝜕(𝛼𝜌𝜎 )gg ∏ (𝛼𝜌𝜎 )gu→gu′ = ⟨𝜒gu′ |Ql |𝜒gul ⟩ 𝛿u′′ ul′ (17.13) l l 𝜕Ql eq l l′ ≠l This expression shows that only transitions involving a modification of only one vibrational quantum number, i.e. u1 , … , ul , … , uM → u1 , … , u′l , … , uM , are non-zero. Additionally, from the well-known identity for harmonic oscillators ⎧0 ; u′l ≠ ul ± 1 ⎪√ ⎪ ℏul ; u′l = ul − 1 ⎪ ⟨𝜒gu′ |Ql |𝜒gul ⟩ = ⎨ 2𝜔l l ⎪√ ⎪ ℏ(ul + 1) ; u′l = ul + 1 ⎪ 2𝜔 l ⎩

(17.14)

it is seen that only transitions involving a modification of ±1 in the vibrational quantum number ul are non-zero. Therefore, in the double harmonic approximation, the overtone and combination transitions are forbidden. The cases u′l = ul + 1 and u′l = ul − 1 describe the Stokes and anti-Stokes scattering processes for fundamental transitions, respectively. In particular, by reporting Eq. (17.14) into Eq. (17.13), one obtains the Raman polarizability tensor for a fundamental

17.2 Theory of Resonance Raman Scattering

transition of the type 0 → 1n : √ ( ) 𝜕(𝛼𝜌𝜎 )gg ℏ (𝛼𝜌𝜎 )g0→g1n = 2𝜔n 𝜕Qn eq

(17.15)

where 𝜔n is the vibrational frequency associated to the electronic ground state normal coordinate Qn . The notation g0 means ul = 0 ∀ l and the notation g1n means that u′n = 1 and u′l = 0 ∀l ≠ n. The electronic transition dipole moment can be taken as a real quantity, i.e. (𝜇𝜌 )ge = (𝜇𝜌 )eg ; consequently the ground state electronic polarizability (Eq. (17.10)) and the Raman polarizability tensor (Eq. (17.9)) are symmetric with respect to the permutation of the indices 𝜌 and 𝜎, i.e. 𝛼 𝜌𝜎 = 𝛼 𝜎𝜌 . As a consequence, the rotational invariant 𝛿 2 (Eq. (17.7)) is equal to zero. Next, reporting Eq. (17.15) into Eq. (17.6) provides the Raman differential cross section for a fundamental transition: d𝜎g0→g1n dΩ

=

ℏ𝜔L 𝜔3S 32π2 𝜀20 c4 𝜔n

Sn 45

(17.16)

where Sn is called the Raman activity of the vibrational mode Qn and is defined by 2

2

Sn = 45a′ n + 7𝛾 ′ n

(17.17)

with the Raman rotational invariants a′ 2n and 𝛾 ′ 2n given by ( ( ) ) ) |2 ( 𝜕(𝛼yy )gg 𝜕(𝛼zz )gg 1 || 𝜕(𝛼xx )gg | 2 + + a′ n = | | | 9 || 𝜕Qn 𝜕Q 𝜕Q n n eq eq eq | } {( ) ) ( ( 2 2 | | | 𝜕(𝛼xz )gg | 𝜕(𝛼yz )gg ) |2 | 𝜕(𝛼xy )gg | | | | | | ′2 𝛾n=3 | | +| | +| | | | | | | | 𝜕Qn 𝜕Qn 𝜕Qn eq | eq | eq | | | | {( ( ( ) ) ) ) ( |2 | 𝜕(𝛼xx )gg 𝜕(𝛼yy )gg 𝜕(𝛼zz )gg ||2 1 || 𝜕(𝛼xx )gg | | + − − | +| | | | | 2 || 𝜕Qn eq 𝜕Qn 𝜕Qn eq 𝜕Qn eq || eq | | ( ) |2 } |( 𝜕(𝛼yy )gg ) 𝜕(𝛼zz )gg | | (17.18) +| − | | | 𝜕Q 𝜕Q n n eq eq | | 17.2.3

Resonance Raman Intensities

The situation of RR scattering is now considered (Figure 17.1b), in which the incident photon energy ℏ𝜔L is close (in resonance) to a set of vibronic transition energies. The formalism presented here (see, e.g. Refs. [7–11]) involves an expansion of the electronic transition dipole moment with respect to the normal modes and leads to the definition of the Franck–Condon (FC) and Herzberg–Teller (HT) contributions to the scattering. Assuming that the incident frequency is in resonance with a set of vibronic Bohr frequencies, i.e. 𝜔L ≈ 𝜔eg + 𝜔vu , the first “resonant” term in the right-hand

503

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17 Calculation of Vibrational Resonance Raman Spectra of Molecules

side of Eq. (17.5) will dominate, while the second “non-resonant” term can be neglected. This defines the RR polarizability tensor { } 1 ∑ ∑ ⟨𝜃gu′ |(𝜇𝜌 )ge |𝜃ev ⟩⟨𝜃ev |(𝜇𝜎 )ge |𝜃gu ⟩ RR (𝛼𝜌𝜎 )gu→gu′ = (17.19) ℏ e≠g v 𝜔eg + 𝜔vu − 𝜔L − iΓ The electronic transition dipole moment (𝜇𝜌 )ge depends on the nuclear coordinates and can be developed as a Taylor series: ( ) ∑ 𝜕(𝜇𝜌 )ge eq Ql + · · · (17.20) (𝜇𝜌 )ge = (𝜇𝜌 )ge + 𝜕Ql eq l eq

where (𝜇𝜌 )ge and (𝜕(𝜇𝜌 )ge /𝜕Ql )eq are, respectively, the electronic transition dipole moment and the derivative of the electronic transition dipole moment evaluated at the ground state equilibrium geometry. Herein, the expansion is performed with respect to the ground state (g) normal coordinates Ql . An alternative choice is to expand the transition dipole moment with respect to the excited state (e) normal coordinates. Then, Eq. (17.20) is truncated after the linear term with respect to Ql and is reported in Eq. (17.19). Under this approximation the RR polarizability tensor can be written in the form 1 ∑ RR RR {(FC)RR (17.21) (𝛼𝜌𝜎 )RR e + (FC∕HT)e + (HT)e } gu→gu′ = √ ℏ 2 e≠g RR RR where (FC)RR e , (FC∕HT)e , and (HT)e represent the contributions of the electronic excited state (e) to the scattering. These terms take the following form: √ ∑ ⟨𝜃gu′ ∣ 𝜃ev ⟩⟨𝜃ev ∣ 𝜃gu ⟩ eq eq 2(𝜇𝜌 )ge (𝜇𝜎 )ge (FC)RR e = 𝜔eg + 𝜔vu − 𝜔L − iΓ v { ( ) √ ∑ ∑ ⟨𝜃gu′ ∣ 𝜃ev ⟩⟨𝜃ev |Ql |𝜃gu ⟩ 𝜕(𝜇𝜎 )ge eq RR (𝜇𝜌 )ge (FC∕HT)e = 2 𝜕Ql 𝜔eg + 𝜔vu − 𝜔L − iΓ eq v l } ) ( ∑ ⟨𝜃gu′ |Ql |𝜃ev ⟩⟨𝜃ev ∣ 𝜃gu ⟩ 𝜕(𝜇𝜌 )ge eq (𝜇𝜎 )ge + 𝜕Ql 𝜔eg + 𝜔vu − 𝜔L − iΓ eq v ( ) ( ) √ ∑ 𝜕(𝜇𝜌 )ge ∑ ⟨𝜃gu′ |Ql |𝜃ev ⟩⟨𝜃ev |Ql′ |𝜃gu ⟩ 𝜕(𝜇𝜎 )ge RR (HT)e = 2 𝜕Ql 𝜕Ql′ 𝜔eg + 𝜔vu − 𝜔L − iΓ eq eq v l,l′

(17.22) where ⟨𝜃 gu ∣ 𝜃 ev ⟩ denotes an FC integral describing the overlap between the vibrational wavefunctions of the ground state (g) and excited state (e). Within the harmonic oscillator approximation, the FC overlap integrals are real quantities, i.e. ⟨𝜃 gu ∣ 𝜃 ev ⟩ = ⟨𝜃 ev ∣ 𝜃 gu ⟩. The FC contribution depends on the product eq eq (𝜇𝜌 )ge (𝜇𝜎 )ge , the FC/HT term involves the products between the transition eq dipole moment (𝜇𝜌 )ge and the derivatives (𝜕(𝜇𝜎 )ge /𝜕Ql )eq with respect to all normal coordinates, and the HT contribution involves all products of the type (𝜕(𝜇𝜌 )ge ∕𝜕Ql )eq (𝜕(𝜇𝜎 )ge ∕𝜕Ql′ )eq . Making use of the well-known identity for harmonic oscillators √ √ ℏ √ { ul ⟨𝜃g…ul −1… ∣ 𝜃ev ⟩ + ul + 1⟨𝜃g…ul +1… ∣ 𝜃ev ⟩} ⟨𝜃gu |Ql |𝜃ev ⟩ = 2𝜔l (17.23)

17.2 Theory of Resonance Raman Scattering

the FC/HT and HT terms of Eq. (17.22) can be expressed in function of FC overlap integrals ⟨𝜃 gu ∣ 𝜃 ev ⟩ (see, e.g. Refs. [8, 9, 11]). In the particular case of fundamental transitions 0 → 1n , the evaluation of the FC contribution requires the determination of FC overlap integrals of the type ⟨𝜃 g0 ∣ 𝜃 ev ⟩ and ⟨𝜃g1n ∣ 𝜃ev ⟩ ∀n, ∀v. In addition, the evaluation of the FC/HT and HT contributions requires the calculation of FC overlap integrals of the type ⟨𝜃g2n ∣ 𝜃ev ⟩ and ⟨𝜃g1n 1l ∣ 𝜃ev ⟩. The notation |𝜃g1n 1l ⟩ means that un = 1, ul = 1, and ui = 0 ∀ i ≠ n, l. Within the harmonic approximation, Eqs. (17.21) and (17.22) provide general relations from which the RR intensities can be calculated. This demands the computation of the electronic and vibrational transition frequency, 𝜔eg and eq 𝜔vu , components of the transition dipole moment (𝜇𝜌 )ge and their derivatives (𝜕(𝜇𝜌 )ge /𝜕Ql )eq . These quantities can be obtained from QC calculations (see Section 17.2.8). Additionally, it is necessary to evaluate the FC overlap integrals ⟨𝜃 gu ∣ 𝜃 ev ⟩ between the ground state (g) and the excited states (e). In the general case, in which the ground and excited states have different equilibrium geometries, different vibrational frequencies, and different normal coordinates (i.e. inclusion of Duschinsky rotation effects [20]), the FC overlap integrals can be computed using recursive relations (see Section 17.2.7). However, additional simplifications are often introduced. The most widely employed and simplest approximation to describe the PESs of the ground and excited states is known as the independent mode displaced harmonic oscillator (IMDHO) model, in which it is assumed that the ground and excited states share the same normal coordinates (neglect of Duschinsky rotation) and the same vibrational frequencies. Within the IMDHO model, the difference between the ground and excited state (e) PESs is only determined by the geometrical displacements Δe Ql along the normal coordinates Ql (Figure 17.1c). In this case, the displacements Δe, l (in dimensionless units) can be obtained from the derivatives of the vertical transition frequency Ωeg with respect to the normal coordinates Ql , evaluated at the ground state (g) geometry √ ( √ ) 𝜔l ℏ 𝜕Ωeg (17.24) Δe,l = Δe Ql = − 3∕2 ℏ 𝜕Ql eq 𝜔l Within the IMDHO model, the multidimensional FC overlap integrals ⟨𝜃 gu ∣ 𝜃 ev ⟩ can be written as a product of one-dimensional FC overlap integrals, and simple recursive relations can be derived (see Section 17.2.7). Making use of these relations, the FC contribution for fundamental transitions 0 → 1n can be expressed in the form (see, e.g. Refs. [8, 21]) eq

eq

(FC)RR e = (𝜇𝜌 )ge (𝜇𝜎 )ge Δe,n {Φe (𝜔L ) − Φe (𝜔L − 𝜔n )}

(17.25)

where the function Φe (𝜔L ) is defined as [22, 23] Φe (𝜔L ) =

∑

⟨𝜃g0 ∣ 𝜃ev ⟩2

v

𝜔eg + 𝜔v0 − 𝜔L − iΓ

(17.26)

From Eq. (17.25) it is seen that the RR intensity of the fundamental transitions is related to the geometrical displacements Δe,n , connecting the minima of the ground and excited state PESs, whereas the Φe function describes the dependency of the RR intensity with respect to the incident frequency 𝜔L (i.e. the

505

506

17 Calculation of Vibrational Resonance Raman Spectra of Molecules

so-called RR excitation profile (RREP)). It can also be noted here that the evaluation of the Φe (𝜔L ) function only requires the calculation of FC overlap integrals of the type ⟨𝜃 g0 ∣ 𝜃 ev ⟩. Further simplifications can be obtained by assuming small values of the displacements (i.e. Δe, n ≪ 1). In that case, the overlap between the ground and excited state vibrational wavefunctions is dominated by the overlap integral ⟨𝜃 g0 ∣ 𝜃 e0 ⟩. Therefore, it is approximated that ⟨𝜃 g0 ∣ 𝜃 e0 ⟩ ≈ 1 and that ⟨𝜃 g0 ∣ 𝜃 ev ⟩ ≈ 0 for v ≠ 0. Using the fact that the vibrational Bohr frequency 𝜔00 is zero within the IMDHO model (i.e. no zero-point vibrational energy correction), the Φe (𝜔L ) function reads Φe (𝜔L ) =

1 𝜔eg − 𝜔L − iΓ

(17.27)

where 𝜔eg is the adiabatic Bohr frequency. Thus, by reporting Eq. (17.27) into Eq. (17.25), the FC contribution for fundamental transitions 0 → 1n becomes eq

𝜔n Δe,n

eq

(FC)RR e = (𝜇𝜌 )ge (𝜇𝜎 )ge

(𝜔eg − 𝜔L − iΓ)(𝜔eg − 𝜔L + 𝜔n − iΓ)

(17.28)

Equation (17.28) is known as the small-shift approximation [5, 9]. Assuming a pre-resonance situation with a single electronic excited state (e), i.e. 𝜔eg − 𝜔L ≫ 𝜔n , the vibrational frequency 𝜔n can be neglected in the denominator of Eq. (17.28). Therefore, by reporting Eq. (17.28) into Eq. (17.21) and into Eq. (17.6) (and using 𝜔S = 𝜔L − 𝜔n ≈ 𝜔L ), it follows that the relative RR intensities for fundamental transitions can be approximated by Ig0→g1n ∝ 𝜔2n Δ2e,n

(17.29)

This expression is known as the Savin formula [24]. By making use of Eq. (17.24), it is found that Ig0→g1n ∝

ℏ 𝜔n

(

𝜕Ωeg 𝜕Qn

)2 (17.30) eq

which is generally known as the gradient method or the short-time approximation (STA) [25] (see Section 17.2.4).

17.2.4

Time-Dependent Formulation of Resonance Raman Intensities

An alternative method for the calculation of RR intensities is the time-dependent approach originally developed by Heller and coworkers [25–28], which is based on wave packet dynamics. Considering the FC contribution of the RR polarizability tensor (Eqs. (17.21) and (17.22)) and making use of the mathematical identity 1 =i ∫0 𝜔 − iΓ

+∞

e−i(𝜔−iΓ)t dt

(17.31)

17.2 Theory of Resonance Raman Scattering

the RR polarizability tensor can be written as i ∑ eq eq (𝛼𝜌𝜎 )RR (𝜇 ) (𝜇 ) gu→gu′ = ℏ e≠g 𝜌 ge 𝜎 ge +∞

×

∫0

∑ ⟨𝜃gu′ ∣ 𝜃ev ⟩⟨𝜃ev ∣ 𝜃gu ⟩e−i(𝜔eg +𝜔vu −𝜔L −iΓ)t dt

(17.32)

v

By introducing (i) the vibrational energies Eu and Ev of the ground (g) and excited (e) states, respectively, defining the vibrational Bohr frequency 𝜔vu = (Ev − Eu )/ℏ; (ii) the vibrational Hamiltonian HVib e of the electronic excited |𝜃 state (e), verifying the eigenvalue equation HVib ev ⟩ = Ev |𝜃ev ⟩; and (iii) the e ∑ i − ℏi HVib t e = e− ℏ Ev t |𝜃ev ⟩⟨𝜃ev |, the RR time evolution operator (i.e. propagator) e v

polarizability tensor takes the form (𝛼𝜌𝜎 )RR gu→gu′ =

i ∑ eq eq (𝜇 ) (𝜇 ) ℏ e≠g 𝜌 ge 𝜎 ge ∫0

+∞

⟨f ∣ ie (t)⟩e−i(𝜔eg −Eu ∕ℏ−𝜔L −iΓ)t dt

(17.33)

where |i⟩ ≡ |𝜃 gu ⟩ and |f⟩ ≡ |𝜃gu′ ⟩ are the initial and final vibrational states, i

Vib

respectively, and |ie (t)⟩ ≡ e− ℏ He t |i⟩ defines the time-dependent vibrational wavefunction of the excited state (e). In this formulation, the RR polarizability tensor is expressed as a half-Fourier transform of a time-dependent overlap between the final vibrational state |f⟩ and the initial vibrational wave packet propagated by the excited state vibrational Hamiltonian HVib e . Exact solutions were found for |ie (t)⟩ in the general situation of displaced harmonic oscillators including frequency changes and Duschinsky rotations [28]. The formalism has also been generalized in order to include HT effects (see, e.g. Refs. [29–31]). While the time-independent method of Section 17.2.3 requires the evaluation of infinite summations over vibronic states, the time-dependent formulation necessitates a numerical integration of an unbounded integral. This feature can become attractive for systems with many degrees of freedom. Moreover, the approach also allows the incorporation of inhomogeneous broadening effects at no additional computational cost. A detailed description of the time-dependent method, of its implementations, and of its applications to different molecular systems in association with QC calculations can be found, e.g. in Refs. [29–38]. The time-dependent approach has been also largely employed for the analysis and interpretation of experimental RR data (see, e.g. Refs. [6, 39, 40]). If only short-time dynamics is important, the relative RR intensities for fundamental transitions 0 → 1n can be approximated from the excited state gradients according to Eq. (17.30). This result was derived by Heller et al. [25] and corresponds to the so-called STA. It is valid under one of the following conditions: (i) the incident frequency 𝜔L is in pre-resonance with the electronic excited state; (ii) the homogeneous damping factor Γ is large, so that the e−Γt factor in Eq. (17.33) truncates the integral at short times; and (iii) the system has many displaced vibrational modes with different frequencies. The STA is only applicable when a single electronic excited state is in resonance with the incident frequency, but does not require a harmonic PES for the excited state. However, within the IMDHO model the STA formula is equivalent to the Savin formula (Eq. (17.29)),

507

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17 Calculation of Vibrational Resonance Raman Spectra of Molecules

because in this case the excited state gradients are directly connected to the displacements (Eq. (17.24)).

17.2.5

Resonance Polarizability Derivatives

Another type of methods for the calculation of RR intensities relies on the derivatives of the frequency-dependent complex polarizability with respect to the normal coordinates. This type of methods is based on an extension of the Placzek polarizability theory, which is valid for both non-resonant and RR scattering, and includes a finite damping factor for the electronic excited states (see, e.g. Refs. [41–44]). Applications of these methodologies were performed on several molecular systems either using the linear response formalism [45, 46] or the real-time approach [47, 48] of the time-dependent density functional theory (TD-DFT). The advantages of this approach are that it includes naturally the contributions of all electronic excited states and that it provides a unique formalism for the description of both the non-resonant and resonant cases. However, as seen below it neglects some vibronic coupling effects, and because it is based on the double harmonic approximation, it provides only RR intensities for fundamental transitions. In particular, in the case in which the FC contribution dominates and when only one electronic excited state is in resonance, the method is similar to the STA (Eq. (17.30)) [41]. To get an insight into the simplifications involved in the resonance polarizability derivative methods, the Raman polarizability tensor (Eq. (17.5)) is reconsidered. Instead of neglecting the vibrational Bohr frequencies 𝜔vu and 𝜔vu′ in the denominators (as done in Section 17.2.2 for normal Raman scattering), the clamped nucleus (CN) approximation [19] is introduced, in which the vibronic energies ℏ(𝜔eg + 𝜔vu ) and ℏ(𝜔eg + 𝜔vu′ ) are approximated by the difference of electronic energy ℏΩeg at a fixed arbitrary nuclear configuration Q: eq

eq

ℏ𝜔eg + ℏ𝜔vu = Ee (Qe ) − Eg (Qg ) + Ev − Eu ≈ Ee (Q) − Eg (Q) = ℏΩeg

(17.34)

Similarly to Eq. (17.9), this leads to the Raman polarizability tensor in the CN approximation elec (𝛼𝜌𝜎 )CN gu→gu′ = ⟨𝜃gu′ |(𝛼𝜌𝜎 )gg |𝜃gu ⟩

where (𝛼𝜌𝜎 )elec gg defines the ground state electronic polarizability { } (𝜇𝜌 )ge (𝜇𝜎 )eg (𝜇𝜎 )ge (𝜇𝜌 )eg 1∑ (𝛼𝜌𝜎 )elec = + gg ℏ e≠g Ωeg − 𝜔L − iΓ Ωeg + 𝜔L + iΓ

(17.35)

(17.36)

In Eq. (17.36) the electronic polarizability does not involve the adiabatic Bohr frequencies 𝜔eg , but instead involves the vertical Bohr frequencies Ωeg that have a dependency on the nuclear coordinates. Therefore, in Eq. (17.35) the integration over the nuclear coordinates also operates on the denominators of Eq. (17.36). Next, adopting the double harmonic approximation (see Section 17.2.2) leads to

17.2 Theory of Resonance Raman Scattering

the Raman polarizability tensor for fundamental transitions: ( ) √ 𝜕(𝛼𝜌𝜎 )elec gg ℏ CN (𝛼𝜌𝜎 )g0→g1n = 2𝜔n 𝜕Qn

(17.37)

eq

An expression for the RR polarizability tensor can then be obtained by neglecting the “non-resonant” term on the right-hand side of Eq. (17.36) and by differentiation of (𝛼𝜌𝜎 )elec gg with respect to the normal coordinate Qn [49, 50]: { ( ) ∑ 𝜕Ωeg 1 eq eq CN (𝛼𝜌𝜎 )g0→g1n = √ −(𝜇𝜌 )ge (𝜇𝜎 )ge 𝜕Qn eq 2ℏ𝜔n e≠g ] [ eq eq (Ωeg − 𝜔L )2 − Γ2 + 2iΓ(Ωeg − 𝜔L ) × eq [(Ωeg − 𝜔L )2 + Γ2 ]2 [ ] [ eq ]} ( ( ) ) 𝜕(𝜇𝜌 )ge 𝜕(𝜇𝜎 )ge Ωeg − 𝜔L + iΓ eq eq + (𝜇𝜌 )ge + (𝜇 ) eq 𝜕Qn eq 𝜕Qn eq 𝜎 ge (Ωeg − 𝜔L )2 + Γ2 (17.38) eq

where Ωeg is the vertical Bohr frequency evaluated at the ground state geometry (Figure 17.1c). The first term on the right-hand side of Eq. (17.38) describes the FC contribution, whereas the second term is associated to the FC/HT contribution. Therefore, (𝛼𝜌𝜎 )CN g0→g1n includes both FC and FC/HT effects. However, the term depending on the incident frequency 𝜔L and describing the RREP takes a simple form. Indeed, it presents no dependency on the vibrational frequency 𝜔n and eq involves a single resonance centered at Ωeg . Thus, for a dominant FC contribution and when only one electronic excited state is in resonance, the method reduces to the STA or gradient method (Eq. (17.30)). A more detailed comparison between this approximation and the vibronic formulation of Section 17.2.3 can be found in Ref. [8]. 17.2.6

Transform Theory and Simplified 𝚽e Approximation

The simulation of RR spectra with QC methods is usually performed together with the calculation of the absorption spectrum in order to identify the bands and vibronic transitions that are in resonance with the incident photon energy ℏ𝜔L . The absorption spectrum can be obtained using similar methodologies and approximations as the RR intensities, and as presented below there exists a connection between the absorption band shape and the RREPs. The absorption cross section for transitions from an initial state (i) to an ensemble of final states (f) is given by 4π2 ∑ ∑ 1 Γ |⟨i|𝜇𝜌 |f⟩|2 (17.39) 𝜔L 𝜎Abs (𝜔L ) = 3cℏ π (𝜔fi − 𝜔L )2 + Γ2 f 𝜌={x,y,z} where 𝜔L is the frequency of the incident light, ⟨i|𝜇𝜌 |f⟩ is a component of the transition dipole moment, and 𝜔fi is the Bohr frequency between the initial (i) and final (f) states. A homogeneous broadening is described by a Lorentzian function with a damping factor Γ, i.e. the full width at half maximum (FWHM) is equal

509

510

17 Calculation of Vibrational Resonance Raman Spectra of Molecules

to 2Γ. The BO and harmonic approximations are employed, and the transitions are assumed to occur between the electronic ground state (g) and the electronic excited states (e) of the molecule. Additionally, it is assumed that the initial state is in the vibrational ground state |𝜃 g0 ⟩. Thus, the initial and final vibronic states read |i⟩ = |𝜑g ⟩|𝜃g0 ⟩; |f⟩ = |𝜑e ⟩|𝜃ev ⟩

(17.40)

Then, by keeping only the linear term in the Taylor series of (𝜇𝜌 )ge (Eq. (17.20)) and by adopting the IMDHO model, the absorption cross section can be written as [8] ∑ 4π 𝜎Abs (𝜔L ) = {(FC)Abs + (FC∕HT)Abs + (HT)Abs (17.41) 𝜔L e e e } 3cℏ e≠g where the FC contribution takes the form ∑ eq eq (FC)Abs = (𝜇𝜌 )ge (𝜇𝜌 )ge Im{Φe (𝜔L )} e

(17.42)

𝜌={x,y,z}

and the function Φe is defined by Eq. (17.26). The expressions for the FC/HT and HT contributions are given in Ref. [8] and involve the imaginary parts of a sum of functions Φe evaluated at different frequencies. Equation (17.42) shows that within the FC approximation, the absorption spectrum associated to a single electronic excited state (e) is related to the imaginary part of the function Φe . This fact is exploited in the so-called transform theory of RR scattering [21, 22, 51–54] that was originally derived by Hizhnyakov and Tehver [51]. As seen in Eq. (17.25), the FC contribution to the RR intensities depends on the function Φe . Hence, in most applications [55–58] of the transform theory, the explicit calculation of the sum over vibrational states in the definition of Φe (Eq. (17.26)) is avoided, but instead Φe is extracted from the experimental absorption spectrum. Indeed, Eqs. (17.41) and (17.42) relate the experimental absorption band shape to the imaginary part of Φe . Then, the real part of Φe can be obtained from Kramers–Kronig relations [59], which connects the real and imaginary parts of Φe . Finally, the deduced function Φe is reported in Eq. (17.25), from which RR intensities and RREPs can be simulated. However, in this approach the determination of the RR intensities still requires the evaluation of the displacements Δe, n , which can be obtained from QC calculations. In more recent applications [23, 60, 61], Φe is directly calculated by an explicit summation over the vibrational states in Eq. (17.26). Moreover, within the IMDHO model the FC, FC/HT, and HT contributions for both absorption and RR (fundamental transitions) cross sections can be expressed in terms of Φe [8]. The first advantage of this formulation is that it simplifies the computation of the FC overlap integrals. Indeed, evaluation of the function Φe only requires the calculation of FC overlap integrals of the type ⟨𝜃 g0 ∣ 𝜃 ev ⟩, whereas FC overlap integrals of the general form ⟨𝜃 gu ∣ 𝜃 ev ⟩ are required (see Section 17.2.3) in the general vibronic theory (Eq. (17.22)). The second advantage is that a straightforward simplification of Φe can be introduced. Indeed, it can be computationally advantageous to avoid calculating the FC factors ⟨𝜃 g0 ∣ 𝜃 ev ⟩2 and to replace the determination of the adiabatic Bohr frequency 𝜔eg by the calculation of the

17.2 Theory of Resonance Raman Scattering eq

vertical Bohr frequency Ωeg at the ground state geometry. This can be performed by virtue of the FC principle, which states that the most probable transition (given by the FC factor) occurs vertically from the ground state geometry. Therefore, the value of 𝜔eg + 𝜔v0 in Eq. (17.26) can be approximated by the eq vertical Bohr frequency Ωeg . Application of this simplification along with the ∑ property of the FC factors that v ⟨𝜃 g0 ∣ 𝜃 ev ⟩2 = 1 leads to the so-called simplified Φe approximation [8]: Φe (𝜔L ) ≈

1 eq Ωeg − 𝜔L − iΓ

(17.43)

The small-shift approximation (Eqs. (17.27) and (17.28)) is a particular case of the simplified Φe approximation, in which the largest FC factor is given by ⟨𝜃 g0 ∣ 𝜃 e0 ⟩2 , meaning that the vertical and adiabatic Bohr frequencies almost coincide. The connection between the Φe function and the absorption spectrum in the FC approximation gives an indication that the simplified Φe approximation should be valid in situations when the incident frequency 𝜔L is in resonance with absorption bands, displaying a large broadening and non-resolved vibronic structure. Inserting Eq. (17.43) in Eq. (17.25) and making use of Eq. (17.24) leads to the FC contribution of RR scattering written in the simplified Φe approximation √ ( ) 𝜕Ωeg ℏ eq eq RR (𝜇 ) (𝜇 ) (FC)e = − 𝜔n 𝜌 ge 𝜎 ge 𝜕Qn eq eq

×

eq

(Ωeg − 𝜔L + iΓ)(Ωeg − 𝜔L + 𝜔n + iΓ) eq

eq

[(Ωeg − 𝜔L )2 + Γ2 ][(Ωeg − 𝜔L + 𝜔n )2 + Γ2 ]

(17.44)

Comparing Eq. (17.44) with the expression obtained from the resonance polarizability derivative (Eq. (17.38)) shows that the simplified Φe approximation provides RREP having a dependence on the vibrational frequency 𝜔n and having two eq eq resonances centered at Ωeg and Ωeg + 𝜔n , whereas Eq. (17.38) only provides a eq resonance at Ωeg . In general, the RR intensities and RREPs obtained with the simplified Φe approximation show a better agreement with the general vibronic theory (Section 17.2.3) than the results calculated using the resonance polarizability derivative expression [8] (Eq. (17.38)). 17.2.7

Calculation of the Franck–Condon Overlap Integrals

The calculation of the RR intensities within the SOS formulation (Section 17.2.3) necessitates the computation of FC overlap integrals between the harmonic PESs of the ground and excited states. Several methods and algorithms were proposed in the literature to calculate these integrals [62–80]. Analytical formulas can be employed, but for large molecular systems it is often computationally more attractive to use recursive relations. Such relations can be derived following the approaches of Sharp and Rosenstock [63], Doktorov et al. [65], and Ruhoff [68]. First, the method is illustrated on the simple case of a one-dimensional displaced harmonic oscillator. Then, it is shown how the approach can be generalized to multidimensional harmonic PESs including Duschinsky rotation effects [20].

511

512

17 Calculation of Vibrational Resonance Raman Spectra of Molecules

As indicated by Eq. (17.8), the vibrational states of the ground (g) and excited (e) states are written as a product of one-dimensional harmonic oscillators |𝜒⟩: M M ∏ ∏ |𝜃gu ⟩ = |𝜒gul ⟩; |𝜃ev ⟩ = |𝜒evl ⟩ (17.45) l=1

l=1

To √simplify the notations, √ the dimensionless normal coordinates, i.e. qn = 𝜔n ∕ℏQn and qe,n = 𝜔e,n ∕ℏQe,n , are employed in the following, in which qn and qe, n indicate a ground and an excited state dimensionless normal coordinate, respectively, while 𝜔n and 𝜔e, n are the corresponding vibrational frequencies. The one-dimensional harmonic oscillator wavefunctions of the ground and excited states are [17] 1 2 1 𝜒gun (qn ) = 1 √ Hun (qn )e− 2 qn ; π ∕4 2un u ! n

1 2 1 Hvn (qe,n )e− 2 qe,n 𝜒evn (qe,n ) = 1 √ v ∕ π 4 2 n vn !

(17.46)

where Hun is a Hermite polynomial, which can be obtained from a generating function [81]: ] ] [ un [ vn 𝜕 𝜕 −s2 +2sqe,n −t 2 +2tqn Hun (qn ) = e ; H (q ) = e (17.47) vn e,n 𝜕t un 𝜕svn t=0 s=0 The situation of a one-dimensional displaced harmonic oscillator is now considered (Figure 17.1c), in which the ground and excited state vibrational frequencies are assumed to be equal, i.e. 𝜔n = 𝜔e, n . In this case, the ground and excited state normal coordinates are related by the simple relation qn = qe,n + Δe,n (17.48) Introducing the notation ⟨un ∣ vn ⟩ ≡ ⟨𝜒gun ∣ 𝜒evn ⟩, the FC overlap integral between the vibrational state |𝜒gun ⟩ and |𝜒evn ⟩ (Figure 17.1c) reads +∞

1 2 2 1 Hun (qn )Hvn (qe,n )e− 2 (qn +qe,n ) dqe,n ⟨un ∣ vn ⟩ = √ π2un un !2vn vn ! ∫−∞

(17.49)

Reporting Eqs. (17.47) and (17.48) into Eq. (17.49), and performing the integra√ +∞ −q2 tion over qe, n by making use of the identity ∫−∞ e dq = π, leads to [ un vn ] Δ2 e,n 1 𝜕 𝜕 tΔe,n −sΔe,n +2ts (17.50) e− 4 e ⟨un ∣ vn ⟩ = √ t=0 𝜕t un 𝜕svn 2un un !2vn vn ! s=0 Equation (17.50) provides directly the origin overlap integral Δ2 e,n

(17.51) ⟨0 ∣ 0⟩ = e− 4 Then, recursive relations [62, 68, 82] are obtained by using the properties of the derivatives with respect to the parameters t and s in Eq. (17.50): √ ⎧⟨un ∣ vn ⟩ = √1 [−Δe,n ⟨un ∣ vn − 1⟩ + 2un ⟨un − 1 ∣ vn − 1⟩] ; vn > 0 ⎪ 2vn √ ⎨ 1 [Δe,n ⟨un − 1 ∣ vn ⟩ + 2vn ⟨un − 1 ∣ vn − 1⟩] ; un > 0 ⎪⟨un ∣ vn ⟩ = √ ⎩ 2un (17.52)

17.2 Theory of Resonance Raman Scattering

Thus, any FC overlap integral ⟨un ∣ vn ⟩ can be calculated by recursion from the Eq. (17.52), starting with the origin overlap integral ⟨0 ∣ 0⟩ (Eq. (17.51)). For example, in the case of ⟨0 ∣ vn ⟩ (with vn > 0), the recursive relation is easily obtained from Eq. (17.50) by performing one derivative with respect to the parameter s: [ vn ] Δ2 e,n 1 𝜕 −sΔe,n e− 4 e ⟨0 ∣ vn ⟩ = √ 𝜕svn s=0 2vn vn ! [ v −1 ] Δ2 Δe,n 𝜕n 1 − e,n −sΔe,n 4 e = −√ e √ 𝜕svn −1 2vn 2vn −1 (v − 1)! s=0 n

Δe,n = − √ ⟨0 ∣ vn − 1⟩ 2vn

(17.53)

which is a particular case of the first relation in Eq. (17.52). Applying Eq. (17.53) recursively leads to the analytical expression (−1)vn vn − Δ2e,n ⟨0 ∣ vn ⟩ = √ Δe,n e 4 2vn vn !

(17.54)

Equations (17.53) or (17.54) can be used to calculate the multidimensional FC factors ⟨𝜃 g0 ∣ 𝜃 ev ⟩2 appearing in the definition of the function Φe (𝜔L ) (Eq. (17.26)). Indeed, within the IMDHO model the FC overlap integral ⟨𝜃 g0 ∣ 𝜃 ev ⟩ can be written as a product of one-dimensional FC overlap integrals: ⟨𝜃g0 ∣ 𝜃ev ⟩ = 2

M ∏ l=1

2v

⟨𝜒g0 ∣ 𝜒evl ⟩ = 2

M Δ l ∏ e,l l=1

2vl vl !

e−

Δ2 e,l 2

(17.55)

Thus, the calculation of the FC factors ⟨𝜃 g0 ∣ 𝜃 ev ⟩2 only requires the knowledge of the displacements Δe, l and of the quantum numbers vl . The recursive approach to calculate FC overlap integrals can be generalized [63, 65, 68] to describe the situation of multidimensional harmonic PESs having different vibrational frequencies and different normal coordinates in the ground and excited states. If the molecule does not undergo too large distortions [83, 84] between the equilibrium geometries of both states, the relation between the normal coordinates of the ground and excited states can be approximated by a linear transformation proposed by Duschinsky [20]: Qg = JQe + ke

(17.56)

where Qg and Qe are column matrices containing the mass-weighted normal coordinates of the ground (Ql ) and excited (Qe, l ) states, respectively. ke is the √ column matrix of the displacements, i.e. (ke )l = ℏ∕𝜔l Δe,l , and J is the rotation or Duschinsky matrix defined by J = (Lg )−1 Le

(17.57)

where Lg and Le are the transformation matrices connecting the mass-weighted normal coordinates to the mass-weighted Cartesian coordinates for the ground and excited states, respectively. These matrices are obtained by solving the ground and excited state vibrational normal mode eigenvalue problem in the

513

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17 Calculation of Vibrational Resonance Raman Spectra of Molecules

harmonic approximation [17] (see Section 17.2.8). Using Eqs. (17.56) and (17.45), the origin overlap integral ⟨𝜃 g0 ∣ 𝜃 e0 ⟩ can be evaluated, employing the method of Sharp and Rosenstock [63], which leads to 1 √ ⎛ 2M det(𝚪 𝚪 ) ⎞ 2 g e ⎟ 1 T −1 T ⎜ e 2 ke (𝚪g JS J −1)𝚪g ke ⟨0 ∣ 0⟩ ≡ ⟨𝜃g0 ∣ 𝜃e0 ⟩ = ⎜ ⎟ det(JS) ⎜ ⎟ ⎝ ⎠

(17.58)

where 𝚪g and 𝚪e are the diagonal matrices of the reduced frequencies, i.e. (𝚪g )ll = 𝜔l /ℏ and (𝚪e )ll = 𝜔e, l /ℏ. The matrix S is defined by S = JT 𝚪g J + 𝚪e . Then, the general recursive relations for ⟨u ∣ v⟩ ≡ ⟨𝜃 gu ∣ 𝜃 ev ⟩ can be derived according to the method of Ruhoff [68]: ⎡ √ ⎢ ⟨u ∣ v⟩ = √ ⎢bn ⟨u − 1n ∣ v⟩ + 2(un − 1)Ann ⟨u − 2n ∣ v⟩ 2un ⎢ ⎣ 1

+

M ∑

√

l=1 l≠n

∑ ul (Aln + Anl )⟨u − 1l − 1n ∣ v⟩ + 2 l=1 M

√

un > 0

⎤ vl ⎥ E ⟨u − 1n ∣ v − 1l ⟩⎥ ; 2 ln ⎥ ⎦ (17.59)

and ⎡ √ 1 ⎢ ⟨u ∣ v⟩ = √ ⎢dn ⟨u ∣ v − 1n ⟩ + 2(vn − 1)Cnn ⟨u ∣ v − 2n ⟩ 2vn ⎢ ⎣ +

M ∑

√

l=1 l≠n

∑ vl (Cln + Cnl )⟨u ∣ v − 1l − 1n ⟩ + 2 l=1 M

vn > 0

√

⎤ ul ⎥ Enl ⟨u − 1l ∣ v − 1n ⟩⎥ ; 2 ⎥ ⎦ (17.60)

where the notation ⟨u ∣ v − 1n ⟩ ≡ ⟨u ∣ v1 , … , vn − 1, … , vM ⟩ is employed. The matrices A, b, C, d, and E are defined by 1∕2

1∕2

A = 2𝚪g JS−1 JT 𝚪g − I 1∕2

b = 2𝚪g [I − JS−1 JT 𝚪g ]ke 1∕2

1∕2

C = 2𝚪e S−1 𝚪e − I 1∕2

d = −2𝚪e S−1 JT 𝚪g ke 1∕2

1∕2

E = 4𝚪e S−1 JT 𝚪g

(17.61)

In summary, the calculation of FC overlap integrals requires as input data, (i) the geometries of the ground and excited states (or equivalently the displacements Δe, l ), (ii) the vibrational frequencies of the ground (𝜔l ) and excited (𝜔e, l ) states, and (iii) the orthogonal matrices Lg and Le , connecting the mass-weighted

17.2 Theory of Resonance Raman Scattering

Cartesian coordinates to the mass-weighted normal coordinates. Then, any FC overlap integral ⟨𝜃 gu ∣ 𝜃 ev ⟩ can be obtained by recursion using the Eqs. (17.59) and (17.60) starting with the origin overlap integral ⟨𝜃 g0 ∣ 𝜃 e0 ⟩ (Eq. (17.58)). For example, this methodology was applied to simulate absorption [85–92], emission [93–96], photoelectron [97, 98], electronic circular dichroism [99, 100], and RR [8, 11, 23, 60, 61, 101–108] spectra. 17.2.8 Quantum Chemistry Methods to Calculate Resonance Raman Spectra The presentation of the vibrational Raman scattering described in the previous sections made use of the harmonic approximation for the BO PES of an electronic state. Therefore, the method to calculate the vibrational frequencies and modes is briefly presented. In the harmonic approximation the PES energy E(x) is expanded as a Taylor series up to the quadratic term [17] 1 (17.62) E(x) = Eeq + rT M−1∕2 FM−1∕2 r 2 where Eeq is the electronic energy at the equilibrium geometry, x is the column matrix of the Cartesian coordinates (i.e. displacements with respect to the equilibrium geometry), and r represents the mass-weighted Cartesian coordinates r = M1/2 x, where M is a diagonal matrix containing the atomic masses. The force constant matrix F (i.e. Hessian matrix) is defined by ( 2 ) 𝜕 E (F)ij = (17.63) 𝜕xi 𝜕xj eq The normal coordinates are obtained by diagonalization of the matrix M−1/2 FM−1/2 , which provides the diagonal matrix 𝚲: 𝚲 = LT M−1∕2 FM−1∕2 L

(17.64)

where L is the orthogonal matrix connecting the mass-weighted Cartesian coordinates to the mass-weighted normal coordinates Q = LT r. For example, the matrix L enters the definition of the Duschinsky matrix (Eq. (17.57)), which is used in the calculation of FC overlap integrals (Section 17.2.7). For a nonlinear molecule six eigenvalues in 𝚲 are zero, which corresponds to the translational and rotational degrees of freedom. Thus, the vibrational frequencies are given √ by the remaining eigenvalues according to 𝜔l = (𝚲)ll . Next, by reporting Eq. (17.64) into Eq. (17.62), the potential energy can be expressed in function of normal coordinates: 1 1∑ 2 2 𝜔Q E(Q) = Eeq + QT 𝚲Q = Eeq + 2 2 l=1 l l M

(17.65)

Then, the vibrational Hamiltonian H Vib takes the form of a sum of onedimensional decoupled harmonic oscillators [17]: Vib

H

M M ∑ ∑ ℏ2 𝜕 2 1 2 2 = − + 𝜔Q, 2 2 2 l l 𝜕Ql l=1 l=1

(17.66)

515

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17 Calculation of Vibrational Resonance Raman Spectra of Molecules

where the constant electronic energy Eeq has been excluded in the definition of H Vib . The eigenvectors of H Vib are given by Eq. (17.45), and the eigenvalues are ( ) M ∑ Eu = ℏ𝜔l ul + 12 . l=1

This approach can be applied to both the ground and excited electronic states. It first requires an optimization of the geometry to find the minimum of the PES. Then, the Hessian matrix (Eq. (17.63)) has to be computed at the obtained equilibrium geometry. This can always be realized by numerical differentiation of the electronic energy, although more efficient analytical expressions have been devised for the most common electronic structure methods (see, e.g. Refs. [109, 110]). For the electronic ground state, the most widely applied method is density functional theory (DFT) [111–114], for which hybrid functionals (as, e.g. the well-known B3LYP functional [115, 116]) were shown to give an accurate description of equilibrium geometries, dipole moments, and vibrational frequencies (see, e.g. Refs. [117, 118]). Alternatively, among the wavefunction-based approaches, the Hartree–Fock (HF) method [119] has a comparable computational cost as DFT but a much lower accuracy. Therefore, electron correlation effects need to be included via so-called post-HF methods. This can be performed through perturbation theory, as, e.g. at the second-order Møller-Plesset (MP2) level [120], which provides vibrational frequencies usually of similar accuracy as DFT [121, 122]. Higher accuracy can be obtained from highly correlated methods, e.g. coupled cluster [123, 124] including single and double excitations (CCSD), plus triple excitations at a perturbative level (CCSD(T)). However, these methods have a larger computational cost, and analytical expressions for the derivatives are usually not available in QC programs. Therefore, their application is limited to small molecular systems. It should be mentioned that in most applications using the harmonic approximation (see Section 17.3), the vibrational frequencies are multiplied by a scale factor to correct for anharmonic effects. Scale factors were provided for the most commonly employed QC methods [121, 122, 125]. Additionally to the calculation of harmonic vibrational frequencies and normal modes, most QC programs provide efficient methods to simulate normal Raman spectra within the double harmonic approximation [126–130]. In this case, the reported quantities are usually the Raman activities Sn (Eq. (17.17)) associated to the fundamental transitions. These values are calculated from the derivatives of the electronic polarizability (Eq. (17.37)), in which the damping factor Γ is neglected in the non-resonant regime. In particular, analytical methods [131, 132] were proposed to compute these quantities in the framework of the time-dependent HF and TD-DFT. Concerning the resonant regime, the resonance polarizability derivative methods (Section 17.2.5) were applied, e.g. on the molecules of uracil, pyrene [41], rhodamine 6G [45], pyridine [46], and tyrosine [50] and on several other systems [44, 48, 49, 133–135]. These calculations were performed using the TD-DFT formalism [136–138], assuming a fixed value of the parameter ℏΓ (typically around 0.1 eV).

17.2 Theory of Resonance Raman Scattering

The calculation of the RR intensities with the SOS (Section 17.2.3) or time-dependent (Section 17.2.4) approaches requires the choice of a model for the harmonic PES of the electronic excited state. Within the IMDHO model, the vibrational frequencies and normal modes of the excited state are assumed to be similar to the ground state. Therefore, only the values of the displacements Δe, l need to be determined in this case. These quantities can be obtained using a vertical approach, in which the displacements are calculated from the excited state gradients (Eq. (17.24)) evaluated at the ground state equilibrium geometry. This approach is known as the vertical gradient (VG) model. Alternatively, the displacements can be obtained using an adiabatic approach, in which the Δe, l ’s are calculated from the difference of geometry between the equilibrium structures of the ground and excited states. This approach is known as the adiabatic shift (AS) model and is computationally more expensive because it requires an optimization of the excited state geometry. Obviously, if the IMDHO model gives a genuine description of the excited state PES, then both VG and AS models are equivalent. An improvement of the AS model consists in computing the Hessian matrix of the excited state at its equilibrium geometry. This approach is known as the adiabatic Hessian (AH) model and takes into account the changes in vibrational frequencies and normal modes between both electronic states, therefore, allowing for an inclusion of Duschinsky effects. Finally, one can mention the vertical Hessian (VH) model, in which the excited state Hessian matrix is calculated at the ground state geometry. A more detailed description of these models as well as a comparison of their performances in the simulation of absorption and RR spectra can be found, e.g. in the Refs. [11, 139, 140]. The properties of the excited states, i.e. energies, geometries, vibrational frequencies, transition dipole moments, gradients, etc., can be calculated with QC methods. Among them, the most popular approach is TD-DFT, due to its good compromise between accuracy and computational cost. Nevertheless, several wavefunction-based methods are also widely applied and aim at providing higher accuracy than TD-DFT at the prize of a reasonably increased computational cost. In particular, multiconfigurational methods, such as the RASPT2 [141]/RASSCF [142, 143] (restricted active space perturbation theory of second-order/restricted active space self-consistent field (SCF)) or the well-established CASPT2 [144]/CASSCF [145] (complete active space perturbation theory of second-order/complete active space SCF) approaches, are able to deliver a reliable description of the excited states, irrespective of the wavefunction character. Moreover, different methods derived from the coupled-cluster theory are available and have shown good accuracies in the determination of excited state energies [146, 147], as, e.g. the equation of motion coupled-cluster method restricted to single and double excitations [148] (EOM-CCSD). One can also mention the density matrix renormalization group (DMRG) method [149, 150], which presents great potentialities for the calculation of systems, requiring a large number of active orbitals [151, 152].

517

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17 Calculation of Vibrational Resonance Raman Spectra of Molecules

17.3 Illustrative Applications This part summarizes several applications of the theoretical methods to calculate RR spectra. The results are solely taken from the author’s bibliography but are intended to illustrate some general features of the approximations described in Section 17.2. 17.3.1

Using the Short-Time Approximation

The STA is the simplest approach to calculate a RR spectrum. Therefore, it was widely applied in the literature and was often shown to provide a suitable prediction of the RR intensities (see, e.g. Refs. [60, 153–158]) for an excitation in pre-resonance or in absorption bands, displaying a large broadening and a non-resolved vibronic structure. As mentioned in Section 17.2, this approximation can be only employed if a single electronic excited state is in resonance and it neglects HT effects. As an illustration, Figure 17.2 presents the RR spectra of the julolidinemalononitrile (JM) chromophore, assuming resonance with the S1 state [23, 40, 159]. Figure 17.2a shows that the choice of the exchange–correlation (XC) functional in the TD-DFT method has a large impact on the calculated RR intensities. In general, the accuracy of the obtained RR spectra not only depends on the approximations employed to determine the RR polarizability tensor (see Section 17.2 and applications below) but also depends on the accuracy of the QC methods used to calculate the ground and excited states properties. This has motivated the use of RR intensities for the benchmarking of QC methods, in particular for the evaluation of excited state gradients within the STA [160–162]. Concerning the JM chromophore, a consistent description of both absorption and RR spectra (see Ref. [23] and Figure 17.2b) is obtained if an amount of exact exchange close to 35% is included in the XC functional (so-called B3LYP-35). Additionally, the effect of increasing the polarity of the solvent is well described by the polarizable continuum model (PCM) [163]. In particular, the simulations reproduce the observed variations of the relative intensities of the bands around 1540 and 1600 cm−1 when the solvent is changed. This shows that the use of the STA is relevant for this molecule. 17.3.2

Including Franck–Condon Vibronic Couplings

This part illustrates the effects of different approximations on the RR spectrum of rhodamine 6G (R6G) (Figure 17.3a) for an excitation in the absorption maximum of the S1 state [8, 101, 102] and considering only the FC scattering contribution. The simulation of the RR intensities requires the use of a damping factor Γ, which can be estimated from the broadening of the experimental absorption spectrum. For R6G, a Γ equal to 400 cm−1 reproduces adequately the absorption band shape [101]. Figure 17.3b compares the experimental RR spectrum [164] with simulated RR spectra obtained with the STA (Eq. (17.30)), the simplified Φe method (Eq. (17.43)), the IMDHO model (Eqs. (17.25) and (17.26)), and the general scheme that includes Duschinsky effects (Eq. (17.22)).

17.3 Illustrative Applications

N

Raman intensity (a.u.)

N

C C N

BHandHLYP

B3LYP-35

B3LYP

400

600

1000

800

1200

1400

1600

(a)

Raman intensity (a.u.)

Acetonitrile

Cal. Exp.

*

* Dichloromethane

Cal.

*

Exp.

Cyclohexane

Cal.

* * 400

(b)

1800

Wavenumber (cm–1)

600

800

* 1000

* 1200

* 1400

Exp. 1600

1800

Wavenumber (cm–1)

Figure 17.2 (a) RR spectra of JM calculated with different XC functionals (STA, acetonitrile), (b) comparison between theoretical (STA, B3LYP-35) and experimental [159] RR spectra in several solvents. The RR intensities were broadened by a Lorentzian function with an FWHM of 20 cm−1 . Source: Guthmuller and Champagne 2007 [23]. Reproduced with permission of AIP Publishing.

The STA is in overall agreement with the experimental spectrum for the bands above 1000 cm−1 but significantly underestimates the RR intensities of the two bands below 800 cm−1 . Similar results are obtained when using the resonance polarizability derivative methods [8, 45] described in Section 17.2.5. However, the inclusion of vibronic coupling effects within the IMDHO model or using the simplified Φe approach leads to a notable improvement of the RR spectrum in the low wavenumber region. The inclusion of Duschinsky effects and using displacements Δe, l calculated from the optimized geometries instead of from

519

0.06

CH3

Raman cross section (10–23 cm2)

H 2C O O

H 3C HN H3C

CH3 NH+

O

H2C

CH2

CH3

(a)

(c)

IMDHO model ν1 ν2 ν3 ν4

0.05 0.04 0.03 0.02 0.01

0.00 17 000 18 000 19 000 20000 21 000 22 000 23 000 Wavenumber (cm–1)

ν2

Raman intensity (a.u.)

Duschinsky

ν4 ν3

IMDHO model Simplified Φe STA Exp. 600

(b)

800

1000

1200

1400

Wavenumber (cm–1)

0.12 Raman cross section (10–23 cm2)

ν1

Simplified Φe ν1 ν2 ν3 ν4

0.10 0.08 0.06 0.04 0.02

0.00 17 000 18 000 19 000 20000 21 000 22 000 23 000

1600

(d)

Wavenumber (cm–1)

Figure 17.3 (a) Molecule of R6G, (b) theoretical (B3LYP) and experimental [164] RR spectra. (c,d) Calculated RR excitation profiles. Source: Guthmuller 2016 [8]. Reproduced with permission of AIP Publishing.

17.3 Illustrative Applications

the excited state gradients leads to negligible changes to the RR spectrum (green line) in comparison to the IMDHO model (gray line). This indicates that the PES of the S1 state of R6G is adequately described by a displaced harmonic PES. Other applications investigating the impact of Duschinsky effects can be found, e.g. in Refs. [29, 31, 106, 165]. The RREP of a Raman transition represents the dependency of its intensity with respect to the excitation frequency. Figure 17.3c,d displays the RREPs for four exemplary modes. The shape of the RREP is related to the vibronic couplings present in the IMDHO model. In particular, it is seen that the RREPs of the modes with large vibrational frequencies (v2, v3, and v4) have a different shape as the RREP of the low-frequency mode v1. This different behavior leads to modification of the relative RR intensities when the excitation frequency is varied, i.e. excitation close to 19 000 cm−1 will enhance the low-frequency modes and excitation above 20 000 cm−1 will enhance the high-frequency modes. Such effects are not described by the STA [8], whereas the simplified Φe approach reproduces the main features of the IMDHO model. 17.3.3

Considering Several Electronic Excited States in Resonance

This part presents an example of system in which the contribution of several electronic excited states has to be included in order to simulate the RR spectrum. The investigated compound is a ruthenium complex, containing a so-called 4H-imidazole ligand and two bipyridine ligands (Figure 17.4). It presents a remarkably broad absorption in the visible region, which according to the calculations originates mainly from four excited states with metal-to-ligand or intraligand charge-transfer characters [166]. The RR spectra were calculated for several excitation wavelengths, covering the visible region using the simplified Φe approach and considering only FC scattering. The contributions of the four excited states are included in the Raman polarizability tensor (Eq. (17.21)), which allows the description of possible constructive or destructive interference effects between the states. First, the RR spectra were obtained by using the calculated (TD-DFT/B3LYP) vertical excitation energies (referred as unshifted in Figure 17.4). These results show a good agreement with the experimental spectra at excitation wavelengths of 568 and 413 nm, in which the RR spectrum is dominated by the contribution of a single excited state. However, due to the deviations of the TD-DFT predictions with respect to the experimental excitation energies, larger inaccuracies are obtained in the RR spectra simulated at excitation wavelengths of 482 and 458 nm, in which several excited states are contributing to the RR intensities. Therefore, the RR spectra were also calculated, employing excitation energies that are shifted (referred as shifted on Figure 17.4) so as to better reproduce the experimental intensity pattern in the absorption spectrum. In that case, the calculated RR spectra are in excellent agreement with the experimental data, which shows that the simulations are able to reproduce the wavelength dependence of the RR spectra in this complex situation. Furthermore, these simulations allowed an assignment of the experimental bands to vibrational modes localized either on the bipyridine or 4H-imidazole ligands. Similar calculations for the protonated complex [166] showed that protonation

521

126

185

180 175, 174 171

200

198 196, 195

230 229, 228

246 244 240 234, 233

118

126, 195

Cal. shifted λRR = 458 nm

*

*

Exp. λRR = 458 nm

N N

Ru2+

N

Cal. unshifted λRR = 482 nm

1130 cm–1

1700 1600 1500 1400 1300 1200 ν~ (cm–1)

1100

1038 cm–1 1029 cm–1

Cal. shifted λRR = 482 nm

*

*

N N

N

N N

Cal. unshifted λRR = 413 nm

Cal. shifted λRR = 413 nm

*

*

Exp. λRR = 482 nm

1000

126

200

198

230 229, 228

240 234

126

180 175, 174 171

198 196, 195 186 185

200

230 229, 228

(d) 246 244 240 234, 233

(b)

1130 cm–1

1038 cm–1

*

Exp. λRR = 568 nm

1038 cm–1 1029 cm–1

Cal. shifted λRR = 568 nm

1261 cm–1

198 196, 195

171

Cal. unshifted λRR = 458 nm

1130 cm–1

*

(c)

Cal. unshifted λRR = 568 nm

1278 cm–1

200

230 228

246 240 234, 233

(a)

Exp. λRR = 413 nm

900 1700 1600 1500 1400 1300 1200

1100

1000

900

800

ν~ (cm–1)

Figure 17.4 Calculated (B3LYP) and experimental RR spectra of the [(tbbpy)2 Ru(4H-imidazole)]2+ complex at four different excitation wavelengths. Source: Reprinted with permission from Kupfer et al. [166]. Copyright 2012, American Chemical Society.

17.4 Conclusions

favors a photoinduced charge transfer toward the 4H-imidazole ligand. Other applications considering several electronic excited states in resonance and describing interference effects can be found, e.g. in Refs. [50, 103, 105, 167–170]. 17.3.4

Including Herzberg–Teller Vibronic Couplings

As a last illustration, the effects of HT vibronic couplings are presented for two systems. This concerns the molecules of R6G (see Section 17.3.2) and of trans-porphycene (Figure 17.5). This latter compound corresponds to a constitutional isomer of porphyrin [171]. For both molecules the excitation frequency is assumed to be in resonance with the S1 state. Hence, Figure 17.5 presents a comparison between the experimental RR spectra [164, 172] (black line) and the theoretical results [8, 108] obtained within the IMDHO model, including only the FC contribution (gray line) and including the FC, FC/HT, and HT contributions (green line). In the case of R6G, the HT effects are small. Indeed, they lead mainly to a modification of the relative intensities between the bands v3 and v4, which nevertheless improves the agreement with the experimental spectrum. However, in the case of trans-porphycene, the HT effects are much larger. The RR intensities in the 900–1650 cm−1 range are strongly underestimated when using the FC approximation, whereas the incorporation of the HT contributions leads to an enhancement of the bands, which significantly improves the agreement with the experimental spectrum. Therefore, the HT effects have an important role on the RR spectrum of trans-porphycene, and their inclusion in the computational scheme is mandatory to accurately predict the RR intensities. The different magnitudes of HT effects between R6G and trans-porphycene can be interpreted from the different values of the S1 oscillator strengths (calculated at 0.744 and 0.128 for R6G and trans-porphycene, respectively), which indicates that FC scattering is predominant in R6G, whereas both FC and FC/HT contributions are significant in trans-porphycene [108]. Other applications describing the impact of HT effects can be found, e.g. in Refs. [11, 29, 31, 49].

17.4 Conclusions In this chapter an overview of the theory of vibrational Raman scattering, of the main approximations, and of QC methods has been reported for the calculation of RR intensities in molecular systems. Applications on four illustrative molecules have been described to assess the performance of the different approaches. In particular, the presentation of the RR theory has been made within the harmonic approximation, and the applications have concerned the calculation of RR intensities for fundamental transitions. However, RR intensities associated to overtone and combination transitions can be directly calculated from the RR polarizability tensor given by Eqs. (17.21) and (17.22). Similarly, temperature effects can be described from Eqs. (17.21) and (17.22) by including a Boltzmann distribution of initial states (see, e.g. Refs. [33, 165]).

523

17 Calculation of Vibrational Resonance Raman Spectra of Molecules

Raman intensity (a.u.)

Exp. FC + FC/HT + HT FC

ν3

600

800

1000

1200

1400

ν4

1600

Wavenumber (cm–1)

(a)

Exp. FC + FC/HT + HT FC Raman intensity (a.u.)

524

200 (b)

400

600

800

1000

1200

1400

1600

Wavenumber (cm–1)

Figure 17.5 Calculated (B3LYP) and experimental RR spectra of R6G (a) and trans-porphycene (b). Source: Guthmuller 2016 [8] and Guthmuller 2018 [108]. Reproduced with permission of AIP Publishing.

As illustrated in Section 17.3, the presented methodology provides reliable RR spectra. However, additional improvements can be obtained by inclusion of other effects, for example, by considering (i) the anharmonicity of the ground and excited state PESs (see, e.g. Refs. [100, 165, 173]); (ii) the nonadiabatic effects between coupled electronic states, when the BO approximation is not reliable [139]; and (iii) the interaction with an environment as, e.g. a solvent, a metal surface, a nanoparticle, or a biological surrounding (see, e.g. Refs. [174, 175]).

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In the latter case, the description of the environment provides simulations closer to the real systems investigated in experiment, which facilitates the interpretation and understanding of the measured data. Finally, the accuracy of the RR intensities relies on the accuracy of the QC methods employed to calculate the ground and excited state properties. The constant development of these methods is leading to a constant improvement of the predicted RR spectra, allowing their applications to systems of increasing complexity.

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46 Mohammed, A., Ågren, H., and Norman, P. (2009). Time-dependent density

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18 Density Functional Theoretical Study on Surface-Enhanced Raman Spectroscopy of CH2 /NH2 Wagging Modes in p–𝛑 Conjugated Molecules on Noble Metal Surfaces De-Yin Wu 1 , Yan-Li Chen 1,2 , Yuan-Fei Wu 1 , and Zhong-Qun Tian 1 1 Xiamen University, State Key Laboratory of Physical Chemistry of Solid Surfaces, Collaborative Innovation Center of Chemistry for Energy Materials, and Department of Chemistry, College of Chemistry and Chemical Engineering, Siming South Road 422, Xiamen 361005, PR China 2 Changzhou University, Jiangsu Key Laboratory of Advanced Catalytic Materials and Technology, School of Petrochemical Engineering, Wujin Gehu Road 1, Changzhou 213164, PR China

The wagging mode is a characteristic out-of-plane bending vibration for a series of organic compounds containing —CH2 /—NH2 groups, such as terminal olefins, p-substituted aniline derivatives, and benzyl radicals. The SERS signal of the wagging mode is always sensitive to the interfacial interaction, displaying significant frequency shift and Raman enhancement. To understand the origin of the special SERS signal, density functional theory (DFT) calculation is performed to obtain harmonic vibrational frequencies and Raman intensities of equilibrium structures for the p–π conjugated molecules adsorbed on silver surfaces on the basis of the molecule–metallic cluster model. Our results showed that the frequency shift of the wagging mode is strongly dependent on the hybridization effect – the transformation from sp2 changing to sp3 hybridization causes a dramatic frequency shift and the Raman intensity enhancement. Furthermore, our results also revealed the causes of the remarkable enhancement of this mode in SERS intensity. From the point of view of the frontier molecular orbital interaction and the change of electronic structures, the derivatives of polarizability tensor for the wagging coordinate are quite large appearing at the significant extent of the geometry deformation, closely associated with the p–π conjugation effect and the hybridization property as well as the energy exchange of the frontier molecular orbital. The above factors result in a significant increase in the derivatives of polarizability tensor along with the direction of the wagging vibrations.

18.1 Introduction Surface vibrational optical spectroscopic techniques, including surface-enhanced Raman spectroscopy (SERS) [1–3], surface-enhanced infrared spectroscopy [4, 5], and surface sum-frequency generation, provide information for interfacial processes at the molecular level. More specifically, these spectroscopic signals provide fingerprint information on surface adsorption configurations [6, 7], Molecular Spectroscopy: A Quantum Chemistry Approach, First Edition. Edited by Yukihiro Ozaki, Marek Janusz Wójcik, and Jürgen Popp. © 2019 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2019 by Wiley-VCH Verlag GmbH & Co. KGaA.

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molecule–metal interactions [3], and chemical reactions [8, 9]. In particular, the SERS is especially proper to extract low-frequency peaks and avoid the interference from the solvent molecules, such as water [7]. For some cases, significant changes including relative intensity and peak shifts are observed in the surface vibrational spectra [10–23]. These usually come from the interactions between probe molecules and surfaces or new unknown species produced by surface-induced chemical reactions [13, 21–23]. However, the experimental studies often analyze the spectral peaks by comparing the SERS signals with normal Raman spectra of these molecules in the gas phase or aqueous solution. In this case, the theoretical calculation on the analysis of surface vibrational spectroscopy plays a very important role on analysis of vibrational spectra of molecules adsorbed on metal surfaces. A series of studies showed that quantum chemistry calculations could clarify not only the assignment of vibrational fundamentals but also the enhancement mechanism of the SERS signals [8, 13, 24–26]. Combining SERS with density functional theory (DFT) calculations, the SERS signals can be an indicator of sensitive surface probes to inspect the electronic structure of the adsorption configurations and understand the complex SERS phenomena. The abnormal SERS spectra of the p–π conjugated molecules adsorbed on a metal surface were reported in previous studies. When the molecules containing CH2 /NH2 groups such as aniline, p-aminobenzoic acid, p-aminobenzonitrile, benzyl, and propylene radical adsorbed on silver films or electrodes, an intense and broad Raman band can be observed in their SERS spectra [13, 21–23]. The interpretation of the abnormal SERS bands was often neglected there. In the cases, the adsorption interaction takes place through the lone pair orbital of the amino group or the p orbital of the CH2 group. This causes the change in the hybridization property of the adsorbed groups companied with the p–π conjugation effect. To better understand the complex changes of SERS signals after the probe molecules adsorbed on the metal surface, it is necessary to investigate the relationship of the binding interaction and the Raman signal of the wagging mode by DFT calculations.

18.2 Brief Review of Wagging Vibrational Raman Spectra Figure 18.1 shows the sketch of the NH2 wagging (ωNH2 ) mode of aniline. Similar cases can meet in the CH2 wagging (ωCH2 ) mode of terminal olefins, benzyl, and p-substituted aniline derivatives. To present a whole understanding on the SERS of the wagging mode for olefins and aniline derivatives adsorbed on metal surfaces, we addressed a brief review as follows. Different vibrational spectroscopy techniques have been used to characterize the adsorption state by concerning the ωCH2 mode. Compared with the wagging frequency in free terminal olefins, the ωCH2 SERS signals of adsorbed molecules on silver displayed different shifts dependent on surface coverage and surface adsorption sites [14–20, 27–33]. In previous studies, the blueshift is attributed to the surface adsorption sites with

18.2 Brief Review of Wagging Vibrational Raman Spectra

Figure 18.1 Character vibrational modes related to the NH2 group in free aniline. Twisting

Wagging

Rocking

Scissoring

partially positively charged silver clusters [27, 30–32]. A redshift shoulder peak was also observed in the SERS spectra of ethylene adsorbed on silver films, while its assignment was often neglected though it generally displays a constant frequency. The SERS spectra of terminal olefin were firstly reported by Moskovits and coworkers for ethylene and propylene adsorbed on a Ag substrate [11, 12]. Three peaks were observed at 977, 955, and 917 cm−1 in the range of the wagging mode for ethylene adsorbed on the cold-deposited Ag film. Similar peaks at 971, 944, and 920 cm−1 were observed for ethylene matrices with a very low concentration of silver on a Ag substrate by Brings et al. [10]. The blueshifts for the ωCH2 mode were also observed for ethylene adsorbed on Ag surfaces pre-covered by chlorine or oxygen [27–29]. In the case, the adsorption took place selectively at silver atoms on which a positive charge had been induced by the oxygen atom [34]. In contrast to the blueshift on silver, SERS of ethylene on copper and gold films measured by Otto et al. showed a large redshift of the ωCH2 mode [10, 35–41]. Similar changes of the wagging mode were observed for other terminal olefins adsorbed on Ag surfaces. The Raman spectrum of 1,3-butadiene in solution containing silver ions showed that the wagging vibration shifted to a higher frequency by 25–44 cm−1 [30–32]. An intense and broadband was observed around 800 cm−1 in the SERS of benzyl chloride reduction at a silver electrode in organic electrolyte solution [13]. The signal was assigned to the methylene wagging vibration of adsorbed benzyl radical and its anionic species [13]. Besides, abnormal intense IR signals of the wagging vibration were also observed in propylene [14–17], acrolein, and acrylonitrile [18–20] adsorbed on low-index single crystal surfaces by surface-enhanced infrared spectroscopy. For aromatic amine derivatives adsorbed on silver surfaces, an intense and broad SERS band can be observed at ∼950 cm−1 . The band was not attributed to the NH2 wagging vibration in previous studies, but to the —NH2 rocking vibration in p-aminobenzoic acid (PABA) and p-aminobenzonitrile (PABN) adsorbed on silver electrodes [21–23]. In their adsorption states, the p-position substituted group of aniline should display strong adsorption binding to silver surfaces. For aniline, the NH2 wagging vibration is at around 660 cm−1 , corresponding to the 0 → 3 transition from the double well potential [42, 43],

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18 Density Functional Theoretical Study on Surface-Enhanced Raman Spectroscopy

and its harmonic fundamental was proposed to be 540 cm−1 [44]. In contrast to the free molecule, when the aniline derivatives adsorb on metal surfaces through the amino nitrogen atom, the adsorption interaction can lead the ωNH2 wagging vibrations exhibiting very significant changes in the frequency and intensity. For example, the frequency significantly shifts to around 900 cm−1 [8, 24]. The SERS spectra of o-aminobenzonitrile, m-aminobenzonitrile, and p-aminobenzonitrile displayed a strong and broadband at 900–1000 cm−1 , which was assigned to a mixed vibration of the in-plane CH bending and the NH2 wagging [45]. A recent study showed that a broadband near 850–1050 cm−1 was observed in SERS spectra of some substituted aromatic amine pollutants, like aniline, p-chloroaniline, p-methylaniline, and p-aminobenzoic acid adsorbed on silver sols [46]. A further theoretical analysis for these interesting phenomena is required for unambiguous assignment. Although theoretical studies reported for the adsorption interaction of these molecules on different metal surfaces, the vibrational spectral analysis was neglected. Such reports like a DFT calculation proposed a redshift for the CH2 wagging mode on the top site and a blueshift on the short bridge site for ethylene adsorbed on Ag(110) [47]. Similarly, for 1,3-butadiene adsorbed on silver films, the blueshift of the wagging frequency was explained due to the molecule interacting with interfacial Ag ions or positively charged active sites through the π-bonded configuration [32]. To our best knowledge, there is no theoretical study focusing on the Raman enhancement of the wagging vibrations until our recent studies reported [8, 24]. For p-aminothiophenol interacting with two silver clusters, DFT calculation showed that a relatively strong Raman peak at about 900 cm−1 was attributed to the NH2 wagging vibration coupled with the CH out-of-plane bending vibration of the benzene ring [8]. This was further demonstrated in different aromatic amines adsorbed on noble metal clusters by theoretical investigation [24]. For SERS of benzyl chloride reduced in a rough silver electrode, an abnormal intense Raman peak observed at the reduction potential was assigned to the wagging vibration of the adsorbed CH2 group based on DFT calculations [13]. As mentioned above, it is necessary to clarify why the special NH2 and CH2 wagging vibrations show such significant frequency shifts and special SERS bands. This was interpreted partially due to the change of the hybridization effect from sp3 to sp2 during periodic vibrations of the wagging mode; however, the nonresonantly selective enhancement on the vibration seems to be a general feature for p–π conjugated molecules. All these promote us to do further systematical theoretical analysis on the wagging modes. In this chapter, the SERS of the wagging modes for aromatic amines and terminal olefins adsorbed on silver surfaces are investigated by using DFT calculations. Comparing with the experimental observations, we calculated the titled molecules interacting with neutral silver clusters and positively charged silver clusters. We further summarize the trend of the frequency shift and the Raman enhancement of the wagging vibration. The origin of the phenomena is analyzed in detail through inspecting the binding interaction, geometry deformation, polarizability tensor derivatives, and molecular orbital properties.

18.3 Normal Mode Analysis

18.3 Normal Mode Analysis To determine the adsorbed structure from the observed vibrational spectra, it is necessary to clearly understand the relationship of the vibrational modes and local molecular geometries. This is very important in doing well vibrational assignments of adsorbed molecules. In general, this requires that our theoretical model can describe well the adsorbate–substrate interaction and the reliable geometric structures. Then we can carry out scaled quantum mechanical force field (SQMFF) calculations on the basis of the hessian matrix, which is the so-called matrix of force constants to describe the potential energy surface of molecules adsorbed on metal surfaces. In our calculations, we adopted Pulay’s definition of internal coordinates for different molecular systems containing metal atoms [48]. In the cases, we consider the metal–molecule as a new complete molecule to do the assignment of vibrational fundamentals. This was expected to describe the vibrational fundamentals of adsorbates more proper than an isolated molecule. Thus we first transfer the force constant matrix in the Cartesian coordinate to that in internal coordinate. If there is a symmetric point group, we will consider our system in independent symmetric internal coordinates. We can separate the nonlinear molecule–metal cluster as 3N freedoms of normal modes under the harmonic approximation. Their motions follow Newton’s law of motion as given [49]: d 𝜕T 𝜕V + = 0, dt 𝜕 q̇ i 𝜕qj

j = 1, 2, … , 3N

(18.1)

where T is kinetic energy of studied systems related to the function of the generalized velocity q̇ and V is the potential energy related to the generalized coordinate q. It can be expressed as ) ) 3N ( 3N ( ∑ 1 ∑ 𝜕2V 𝜕V V = V0 + qi + q q + anharmonic terms 𝜕qi 0 2 i,j=1 𝜕qi 𝜕qj 0 i j i=1 = V0 +

3N ∑

fi qi +

i=1

3N 1∑ f q q + anharmonic terms 2 i,j=1 ij i j

(18.2)

where V 0 is the minimum potential energy at equilibrium geometric structure, independent on the nuclear coordinates. ( ) Meanwhile, the force is zero at the equilibrium geometric structure, 𝜕V = 0. When the anharmonic terms are 𝜕qi 0 neglected, the potential energy can be rewritten as 2V =

3N ∑

fij qi qj

(18.3)

i,j=1

where f ij is an element of harmonic Hessian matrix: ( 2 ) 𝜕 V fij = 𝜕qi 𝜕qj 0

(18.4)

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18 Density Functional Theoretical Study on Surface-Enhanced Raman Spectroscopy

This is so-called as the force constant, which has a property f ij = f ji . According to the harmonic vibrational approximation, the kinetic energy and potential energy can be expressed in terms of the velocity and position of nucleic motions: q̈ j +

3N ∑

fij qi = 0,

j = 1, 2, … , 3N

(18.5)

i=1

This is a set of linear differential equations consisted of 3N. Their solutions are 3N harmonic functions. In fact, the 3N equations can be further reduced to 3N − 5 for a linear molecule or 3N − 6 vibrations for a nonlinear molecule according to the Eckart transformation. This may eliminate the translational and rotational degrees of freedom from the total nucleic motions. For a molecule containing N atoms, their displacements (Δx𝛼 , Δy𝛼 , and Δz𝛼 ) of the given configuration (x𝛼 , y𝛼 , and z𝛼 ) with respect to their equilibrium positions (a𝛼 , b𝛼 , and c𝛼 ) are Δx𝛼 = x𝛼 − a𝛼 , Δy𝛼 = y𝛼 − b𝛼 , and Δz𝛼 = z𝛼 − c𝛼 . The degrees of freedom of translation motions can be eliminated according to N ∑ 𝛼=1 N ∑ 𝛼=1 N ∑ 𝛼=1

m𝛼 Δx𝛼 = 0 m𝛼 Δy𝛼 = 0 m𝛼 Δz𝛼 = 0

(18.6)

where m𝛼 is the mass of the 𝛼th atom. Meanwhile the degrees of freedom of rotation motions can be eliminated as N ∑ 𝛼=1 N ∑ 𝛼=1 N ∑ 𝛼=1

m𝛼 (b𝛼 Δz𝛼 − c𝛼 Δy𝛼 ) = 0 m𝛼 (c𝛼 Δx𝛼 − a𝛼 Δz𝛼 ) = 0 m𝛼 (a𝛼 Δy𝛼 − b𝛼 Δx𝛼 ) = 0

(18.7)

If the mass weight coordinate frame was used, the kinetic energy of the studied system can be expressed in a classical form: 2T =

3N ∑

q̇ i2

i=1

where q̇ denotes the velocities of atoms. The relative positions of atoms are √ √ √ q1 = m1 Δx1 , q2 = m1 Δy1 , q3 = m1 Δz1 , √ √ √ q4 = m2 Δx2 , q5 = m2 Δy2 , q6 = m2 Δx2 , …

(18.8)

18.3 Normal Mode Analysis

The potential energy of the molecular systems is given in Taylor expansion: ) ) 3N ( 3N ( ∑ ∑ 𝜕V 𝜕2 V q + q q + anharmonic terms 2V = 2V0 + 2 𝜕qi 0 i i,j=1 𝜕qi 𝜕qj 0 i j i=1 = 2V0 + 2

3N ∑

fi qi +

i=1

3N ∑

fij qi qj + anharmonic terms

(18.9)

i,j=1

( When ) the equilibrium position was chosen as an energy reference, V 0 = 0 and 𝜕V = 0 at the minimum. The harmonic potential energy function is 𝜕q i

0

2V =

3N ∑

(18.10)

fij qi qj

i,j=1

where f ij is the force constant, ( 2 ) 𝜕 V fij = 𝜕qi 𝜕qj 0

(18.11)

and f ij = f ji . The equation of Newton motion under harmonic approximation in the mass-weighted coordinate frame d 𝜕T 𝜕V + = 0, dt 𝜕 q̇ i 𝜕qj

j = 1, 2, … , 3N

(18.12)

can be transformed according to the expression of the kinetic energy and potential energy as q̈ j +

3N ∑

j = 1, 2, … , 3N

fij qi = 0,

(18.13)

i=1

This is a set of 3N linear differential equations. Their general solutions are ( 1 ) qi = Ai cos 𝜆 2 t + 𝜀 (18.14) where Ai , 𝜆, and 𝜀 are amplitude, frequency, and phase of the harmonic vibrations of a studied system. Till now we can obtain the equation of molecular vibrations 3N ∑ (fij − 𝛿ij 𝜆)Ai = 0

j = 1, 2, … , 3N

(18.15)

i=1

where 𝛿 ij is Kronecker 𝛿 symbol. When i = j, 𝛿 ij = 1, rather than it is zero. To obtain non-zero solutions, the secular equation must be satisfied: |f11 − 𝜆 f12 | | | f21 f22 − 𝜆 | | ··· ··· | | | f | 3N,1 f3N,2

| | | f23 · · · f2,3N | |=0 ··· ··· · · · || | f3N,3 · · · f3N,3N − 𝜆|| f13 · · ·

f1,3N

(18.16)

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18 Density Functional Theoretical Study on Surface-Enhanced Raman Spectroscopy

where the eigenvalues of 𝜆 are some special values for obtaining 3N amplitudes of Ai . Each of 𝜆 can be used to determine the coefficients Ai . This can be considered 1

as the amplitude Aik of a given normal mode with vibrational frequency 𝜆k2 ∕2π from 𝜆k . In the case of the normalization of amplitude of the normal mode, we have ∑ lik2 = 1 i

where lik = [ ∑ i

A′ik (A′ik )2

(18.17)

]1 2

Similar to the mass-weighted coordinate frame, we can use the internal coordinate, the degrees of freedom that consisted of linear combination of bond lengths, bond angles, out-of-plane bending angles, and torsions of interest. On the basis of the internal coordinate representation, the normal mode coordinate can be denoted as [49, 50] ∑ lik qi Qk = i

where lik is a transformation coefficient from mass-weighted Cartesian coordinate qi (i = 1, 2, 3, …, 3N) to the normal mode in the internal coordinate. To enhance the physical meaning of vibrational analysis on the basis of bond stretching angle bending, rotation, and torsion of atomic nucleic motions, it is to construct a transformation matrix from the Cartesian coordinate to the internal coordinate. Here we denote the number of the internal coordinate 3N − 6 after canceling the redundant internal coordinate through considering the molecular symmetry: Rt =

3N ∑

Bti Xi ,

t = 1, 2, … , 3N − 6

(or 3N − 5)

(18.18)

i=1

The kinetic energy is written on the basis of the internal coordinates as ∑ 2T = (G−1 )tt′ Ṙ t Ṙ t (18.19) tt ′ −1

where (G )tt′ is closely associated with atomic masses and geometric parameters, such as bond lengths, bond angles, and so on. The potential energy is written as ∑ Ftt′ Rt Rt (18.20) 2V = tt ′

where Ftt′ is the harmonic force constant. The secular equation is |F11 − (G−1 )11 𝜆 F12 − (G−1 )12 𝜆 | | |F21 − (G−1 )21 𝜆 F22 − (G−1 )22 𝜆 | | ··· ··· | | |F − (G−1 ) 𝜆 F − (G−1 ) 𝜆 | n1 n1 n2 n2

· · · F1n − (G−1 )1n 𝜆| | | · · · F2n − (G−1 )2n 𝜆| |=0 | ··· ··· | | −1 · · · Fnn − (G )nn 𝜆||

(18.21)

18.3 Normal Mode Analysis

where 𝜆 = 4π2 v2 , n is the number of isolated internal coordinates, n = 3N − 6 is for nonlinear molecules, and n = 3N − 5 is for linear molecules. The abovementioned equations can be written in the matrix form 2T = Ṙ T G−1 Ṙ

(18.22)

2V = RT FR

(18.23)

|F − G−1 𝚲| = 0

(18.24)

where R and Ṙ are internal coordinate and its derivative, respectively. Ṙ T is a transposed conjugate matrix of R, and G−1 is an inverse matrix of G. Here F is the force constant matrix in the internal coordinate. The normal mode Q has a linear relation to the internal coordinate R: (18.25)

R = LQ

Thus the kinetic and potential energies in the internal coordinate can be written as 2T = Q̇ T LT G−1 LQ̇ = Q̇ T EQ̇

(18.26)

2V = QT LT FLQ = QT 𝚲Q

(18.27)

where 𝚲 is the diagonal matrix, the matrix element of which has a relation to the vibrational frequency of 𝜆k = 4π2 v2k . Here E denotes an identity matrix. Accordingly, we have LT FL = 𝚲

LT G−1 L = E, T

(18.28)

−1

By using L = L G, there is L−1 GFL = 𝚲

(18.29)

This indicates that the L matrix can be used for the diagonalization of the GF matrix. We have |GF − E𝚲| = 0

(18.30)

This is the so-called Wilson GF method to solve the Newton equation of nucleic vibrations. In summary, when we obtain the G and F matrixes in the internal coordinate, respectively, then we can do the diagonalization of the GF product matrix, and we can get the eigenvalue 𝜆k and the corresponding eigenvector L for the normal mode of studied molecules. From the vibrational analysis, the potential energy distribution can be estimated to assign the vibrational mode to certain internal coordinates. The potential energy distribution is defined according to the potential energy of the internal coordinates [50]: (PED)ij =

(Ls )2ij (Fs )ij 𝜆j

× 100

(18.31)

It is proper to do some further analysis in the view of the bonding interaction and vibrational frequency. One can relate the vibrational mode to some specific

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18 Density Functional Theoretical Study on Surface-Enhanced Raman Spectroscopy

bond stretching, angle bending, and torsion motions. Thus we can have a clear physical picture to understand the information of vibrational fingerprint at the molecular level. This is very important to build the relationship of adsorption configuration, binding interaction, and vibrational spectra of molecules in complex environments of interfaces.

18.4 Density Functional Theoretical Calculations To mimic the surface adsorption interaction of the p–π conjugated molecules containing the wagging vibration, we adopted a molecule–metallic cluster model, as shown in Figure 18.2. While for terminal olefins, there are two possible adsorption models, the weakly π-bonded configuration and the di-σ-bonded configuration [44, 51–54]. The former was considered in this chapter because the π-bonded adsorption on silver clusters is more stable thermodynamically than the latter one [47, 51, 54]. The selected metallic clusters are not only to reflect the electronic structures of surface active sites but also to follow the general binding propensity, such as the energy close, the symmetry matching, and the maximum overlap. Recently, Metiu and coworkers reported that the desorption energy of propylene from neutral Ag4 cluster was larger than that from Agn (n = 1, 2, 3, 5) [55]. Our previous studies also showed a strong binding interaction between Ag4 and pyridine [26, 56]. Accordingly, we mainly chose the Ag4 cluster to describe the surface active site. For aromatic amines, the model focuses on the interaction of the NH2 group binding to silver clusters. Figure 18.2 presents the modeling structures of terminal olefins, p-substituted aniline derivatives, and benzyl radicals interacting with Ag4 . Other silver clusters with the sizes at n = 2, 6, and 13 were also considered here to study the influence of different adsorption sites on the vibrational spectra. Density functional calculations were carried out with hybrid exchangecorrelation functional B3LYP [57, 58]. The basis sets for C, N, O, S, and H atoms of studied molecules were 6-311+G(d, p), which includes a set of polarization functions to all five kinds of atoms and a diffuse function to C, N, O, and S atoms [59, 60]. For silver atom, electrons in the valence and inner shells were described by the basis set, LANL2DZ, and the corresponding relativistic effective core potentials, respectively [61, 62]. Full geometric optimizations and analytic frequency calculations were carried out by using Gaussian 09 package [63]. To obtain the results of vibrational analysis, we first calculate the Hessian matrix (F ij ) of the optimization structures in the Cartesian coordinate. Then we transform the Cartesian force constant matrix to that in the internal coordinate according to the SQMFF method [8, 25, 48]. For comparison of theoretical and experimental vibrational frequencies, we also chose a scaling factor of 0.981 for the calculated frequencies of the vibrational modes mentioned in this paper. This is equal to scale the force constant in the form √ Fijscaled = Ci Cj Fij (18.32) where Fijscaled is the scaled force constant and C i is the scaling factor of the ith internal coordinate. This can correct the error from the defect of the theoretical

18.4 Density Functional Theoretical Calculations

PMOA

PMA

AN

PCA

PBA

PABM

PABA

PABN

PNA

(a)

(c)

(b)

Figure 18.2 Optimized structures of molecules interacting with Ag4 cluster. (a) Ethylene, propylene, 1,3-butadiene, acrolein, and vinyl amine interact with metal via double bond. (b) p-Substituted aniline derivatives interact with metal along the amino group approaching the surfaces. (c) Propylene free radical, propylene radical anion, benzyl radical, and benzyl radical anion interact with metal via methylene. From Ref. [70].

approach and the incomplete property of basis sets as well as the neglecting of anharmonic terms. In general, the SQMFF method can modify the predicted vibrational frequencies close to the observed data well. Here we used an SQMFF program scale 2.0 [64] to carry out the vibrational analysis [8, 24] for the vibrational assignment. The scaled factors are 0.981 for vibrational frequency less than 2000 cm−1 and 0.967 for vibrational frequencies higher than 2000 cm−1 . Natural bond orbital (NBO) analysis [65, 66] has been used to investigate orbital bonding interactions when the wagging mode of adsorbed molecules moves along the normal coordinate. The molecule–Ag4 model is proper not only to describe molecular adsorption at active sites on rough silver surfaces but also to well predict vibrational spectra [24]. The theoretical methods were used well to calculate the binding interaction, vibrational frequency, and Raman intensity in our previous work [25, 26, 56]. Taking the solvent effect into account, we used the solute model of density (SMD) approach [67], which considered the non-electrostatic terms and was recommended to well predict solvation Gibbs free energies (ΔG) of ions and molecules. We chose water with dielectric constant (𝜀 = 78.3) as the solvent. The

547

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18 Density Functional Theoretical Study on Surface-Enhanced Raman Spectroscopy

SMD model was also used to calculate vibrational spectra of molecule–Agn + complexes. Since the wagging vibration is sensitive to surroundings, the hydrogen bonding interaction between the molecules containing NH2 has great effect on the ωNH2 vibration. Binding energies (BE) are estimated by subtracting the energies of molecule and silver clusters from the total energy BE = −(EMOL-Mn − EMOL − EMn )

(18.33)

where n denotes the number of metal atoms in the silver cluster and EMOL-Mn , EAN , and EMOL denote the energies of the complex, free molecule, and metallic cluster, respectively. For making a correction from the full basis set associated with the entire system, the counterpoise correction of the basis set superposition error (BSSE) was considered in the binding energy in the present calculation [68, 69]. The correction decreases the overestimation of binding energies.

18.5 Raman Intensity Absolute Raman intensities are measured on top of the differential Raman scattering cross section (DRSC) from the Raman scattering factor, as published in our previous work [8]. On top of the optimized geometries, Raman intensities in on- and off-resonance Raman scattering processes were estimated in terms of the derivative of the polarizability with respect to a given normal coordinate. We employed the general formula in the harmonic approximation, where the DRSC was given by ) ( (̃ 𝜈0 − 𝜈̃i )4 (2π)4 • h • dσ S = (18.34) dΩ i 45 8π2 c̃ 𝜈i 1 − exp(−hc̃ 𝜈i ∕kB T) i where h, c, k B , and T are Planck constant, light speed, Boltzmann constant, and vi are the frequencies (in cm−1 ) of inciKelvin temperature, respectively. ̃ v0 and ̃ dent light and the ith vibrational mode. Si is the Raman scattering factor (in Å4 /amu) that can be directly calculated using the Gaussian program. In this case, the Raman scattering factor (in Å4 /amu) was calculated as ( ( )2 )2 d𝛾 d𝛼 +7 (18.35) Si = 45 dQi dQi where ′

𝛼 = 2

𝛾′ =

1 ′ ′ ′ + 𝛼ZZ ) (𝛼 + 𝛼YY 3 XX

(18.36)

} 1{ ′ ′ 2 ′ ′ 2 ′ ′ 2 ′ 2 ′ 2 ′ 2 ) + (𝛼yy − 𝛼zz ) + (𝛼xx − 𝛼zz ) + 6[(𝛼xz ) + (𝛼yz ) + (𝛼xy )] (𝛼xx − 𝛼yy 2 (18.37)

The polarizability derivative 𝛼 represents (𝜕𝛼/𝜕Q)0 in Gaussian output just to provide the values for molecule at the equilibrium position in Cartesian coordinates. For investigating the relationship of the polarizability derivatives and the ′

18.6 Modeling Molecules

geometry, we calculated the variation of the six polarizability tensor components along with the normal coordinate of the wagging vibration in the molecule and molecule–metal complexes. Each value of the normal coordinate corresponds to a molecular structure in Cartesian coordinates. Thus we can understand easily the change of different polarizability tensor elements with the variation of geometric structures, electronic property, and molecular orbital interaction in the process of the molecular vibration along the wagging mode. Then the derivatives (𝜕𝛼/𝜕Q)0 are calculated from the slope of each polarizability tensor components along with the wagging coordinate at the equilibrium position. The normal coordinate was defined according to the mass weight coordinates [70] √ √ 3N √∑ √ Qi = √ ( mi Δxi )2 (18.38) i

where Δxi and mi represent the vibrational displacement vectors in Cartesian coordinates and the atomic mass of the ith atom, respectively. To make direct comparison with the SERS experiments, the simulated Raman spectra were presented in terms of the Lorentzian expansion with a line width of 10 cm−1 , and an excitation wavelength of 514.5 nm was used.

18.6 Modeling Molecules 18.6.1

Aniline

The previous study has investigated the effect of different adsorbate–substrate interaction on the SERS through typical double-functional aromatic amines [24]. It is found that the amino group approaching a metal surface will exhibit unique Raman signals though the interaction between the para-substituted functional groups and metal clusters displays a strong binding interaction. Thus we pay our attention to the configuration of the molecules interacting with silver clusters via only amino group here. Table 18.1 presents binding energies, structural parameters, scaled frequencies, and Raman activities of aniline (AN) and its para-substituted derivatives through the NH2 group interacting with Ag4 , including p-methoxyaniline (PMOA), p-methylaniline (PMA), p-chloroaniline (PCA), p-bromoaniline (PBA), p-aminobenzoic acid methyl ester (PABM), p-aminobenzoic acid (PABA), p-aminobenzonitrile (PABN), and p-nitroaniline (PNA). The binding energies are closely associated with the property of para-substituted functional groups. The electron-withdrawing (EW) groups decrease the charge distribution at the lone pair orbital of the NH2 group and weaken the binding interaction. In contrast, the electron-donating (ED) groups increase the charge distribution at the LP orbital of the NH2 group and strengthen the binding interaction. Figure 18.3 presents that the levels of the highest occupied molecular orbital (HOMO) become lower with the stronger ability of the attracting electron substituent. So the energy gaps between the electron donor (HOMO of aromatic compounds) and the electron acceptor (LUMO

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18 Density Functional Theoretical Study on Surface-Enhanced Raman Spectroscopy

Table 18.1 Inversion angle (𝛼, ∘ ), C—NH2 bond length (Å), N—Ag bond length (Å), vibrational frequency (ω, cm−1 ), Raman activity (SR , Å4 /amu), and bonding energies (BE) of molecule–metal (kcal/mol) calculated at the B3LYP/6-311+G** level. The differential Raman scattering cross section (10−30 cm2 /mol sr) is calculated from the Raman scattering factors. Species

𝜶

C6 H4 NH2 OCH3

42.53

1.406

C6 H4 NH2 OCH3 —Ag4

51.60

1.438

C6 H4 NH2 CH3

40.52

1.401

C6 H4 NH2 CH3 —Ag4

50.95

1.435

C6 H5 NH2

39.15

1.398

C6 H5 NH2 —Ag4

50.55

1.433

C6 H4 NH2 Cl

38.41

1.396

C6 H4 NH2 Cl—Ag4

50.14

1.431

C6 H4 NH2 Br

38.04

1.395

C6 H4 NH2 Br—Ag4

50.06

1.430

C6 H4 NH2 CO2 CH3

33.11

1.386

C6 H4 NH2 CO2 CH3 —Ag4

48.99

1.425

C—N

C6 H4 NH2 CO2 H

32.25

1.384

C6 H4 NH2 CO2 H—Ag4

48.63

1.424

C6 H4 NH2 CN

31.87

1.383

C6 H4 NH2 CN—Ag4

48.18

1.422

C6 H4 NH2 NO2

28.11

1.378

C6 H4 NH2 NO2 —Ag4

47.40

1.419

N—Ag

𝝎

611

4.40

0.56

2.375

951

130.29

9.43

581

6.31

0.85

2.380 2.385 2.393 2.394

SR

(d𝛔/d𝛀)i

944

93.81

6.86

564

6.88

0.97

934

101.04

7.49

548

3.40

0.50

917

72.15

5.47

541

3.33

0.49

915

78.59

5.97

448

17.32

3.31

2.402

907

170.95

13.14

441

15.30

2.99

2.407

902

152.37

11.80

435

11.39

2.27

894

105.72

8.28

385

25.13

5.95

885

145.78

11.55

2.414 2.423

BEa)

BEb)

13.93

12.33

13.38

11.79

12.84

11.28

12.10

10.51

12.01

10.44

11.17

9.58

10.77

9.18

10.19

8.61

9.45

7.86

a) Bonding energy in gas phase. b) Considering BSSE correction.

0

–2

LUMO of Ag4

–4 Energy (ev)

550

–6

*

–8

–10

–OCH3 –CH3

–H

–Cl

–Br –CO2CH3 –CO2H –CN

–NO2

Figure 18.3 Energy levels (ranging from −11.63 to 0 in eV unit) of molecular orbitals for aniline and p-substituted derivatives of aniline. HOMOs are indicated with asterisks. The LUMO of Ag4 is marked with red dotted line. From Ref. [24].

18.6 Modeling Molecules

of Ag4 ) are larger for the EW group-substituted compounds than the ED-substituted compounds. This accounts for the binding energy increasing with the donor ability and decreasing with the withdrawing ability of para-substituted functional groups. The BE values vary in a descending order, PMOA > PMA > AN > PCA > PBA > PABM > PABA > PABN > PNA. Figure 18.4 presents dependence of amino wagging frequencies on binding energy, N—Ag force constant, inversion angle of the NH2 out-of-plane bending, C—N bond length (black), and N—Ag bond length (red) in molecule–Ag4 complexes. The dotted line connecting two points presents the same molecule. As shown in Figure 18.4a, the ωNH2 vibrational frequency strongly depends on the binding interaction between the amino group and metals. The wagging frequency exhibits a near linear relationship on the binding energies. In the EW-substituted compounds, the wagging frequencies are lower than that of aniline. On the contrary, the frequency increases in the ED-substituted compounds. This is because the EW group enhances the p–π conjugation effect between the amino group and the benzene ring, while the ED group weakens it. The p–π conjugation interaction would further change the hybridization of the amino group. In the EW(ED)-substituted compounds, the amino group is inclined to sp2 (sp3 ) hybridization. Simultaneously, the inversion angle listed in Table 18.1 can reflect the extent of the sp3 hybridization. Figure 18.4b shows a near linear relationship of the wagging frequency on the inversion angle in free molecules and their corresponding silver complexes. This clearly shows that the wagging frequency increases as the extent of sp3 hybridization becomes large. Figure 18.4c,d shows variations of the wagging frequencies with the force constants of the N—Ag and C—N bonds in different complexes. The substituent groups affect the C—N bond lengths in free molecules increasing as the enhancement of the ED ability of the substituent. When the N—Ag bond becomes stronger, in agreement with increasing the N—Ag force constant, the wagging frequency increases, and the C—N bond length becomes longer. Figure 18.5 presents simulated Raman spectra of free aniline and aniline interacting with neutral copper, silver, and gold clusters. It is noted that the Raman intensity of the wagging vibration at 564 cm−1 is relatively weak in free aniline. This agrees with our predicted RSF value of about 6.76 Å4 /amu, corresponding to the DRSC value about 0.95 × 10−30 cm2 /(sr mole) at the excitation wavelength 632.8 nm. The NH2 twisting vibration at 283 cm−1 and the rocking one at 1043 cm−1 have even smaller RSFs about 0.26 and 0.04 Å4 /amu, corresponding to the DRSC values about 0.98 × 10−31 and 0.26 × 10−32 cm2 /(sr mole), respectively. Their Raman signals can hardly be detected in the normal Raman spectrum. The adsorption interaction results in a significant change in Raman bands of the NH2 group. After adsorption, the most distinct change is that a very strong band appears at about 934 cm−1 on silver and 1029 cm−1 on gold in our simulated Raman spectra, as shown in Figure 18.5. On the basis of our normal mode analysis, this band can be attributed to the ωNH2 mode from the PED value. Surprisingly, the band remarkably changes in its vibrational frequency, which blueshifts from 564 cm-1 to 900–1000 cm−1 , simultaneously accompanying with a significant increase in the DRSC values from 0.40 × 10−30 to ∼6.36 × 10−30 cm2 /(sr mole) (RSFs change from 6.76 to ∼233.40 Å4 /amu) for

551

960 Wagging frequencies (cm–1)

Wagging frequencies (cm–1)

960 940 920 900 880 8 (a)

9

10

11

12

950 940 930 920 910 900 890 880

13

Bonding energy (kcal /mol–1)

0.44 0.46 0.48 0.50 0.52 0.54 0.56 0.58 Force constant fN-M (105 dyn cm–1)

(c)

N–Ag bond length (Å) 47

48

49

50

51

52

960

Molecule-cluster 920 880 600

Free molecule

500 400 26

(b)

Wagging frequencies (cm–1)

Wagging frequencies (cm–1)

960

28

30

32 34 36 38 Inversion angle (°)

40

42

44

2.37

2.39

2.40

2.41

2.42

2.43

PMOA PMA

940 AN PCA PBA PABM PNBA

920 900

PNBN PNA

880 1.415

(d)

2.38

1.420

1.425 1.430 1.435 C–N bond length (Å)

1.440

Figure 18.4 Dependence of amino wagging frequencies on bonding energy (a), N—Ag force constant (b), inversion angle (c), C—N bond length (black), and N—Ag bond length (red) (d) in molecule–Ag4 complexes. The dotted line connecting two points presents the same molecule. From Ref. [70].

18.6 Modeling Molecules

ν1 0.08

AN

νCN

0.04 Raman intensity (10−30 cm2/sr)

ν12

0.00 0.10 0.05

νCN

AN-Cu4

νCN

AN-Ag4

νCN

AN-Au4

0.00 0.2 0.1 0.0 0.4 0.2 0.0 400

600

800 1000 1200 Wavenumber (cm–1)

1400

1600

Figure 18.5 Calculated Raman spectra of aniline interacting with coinage metal clusters. An incident wavelength of 632.8 nm and a line width of 10 cm−1 were used in the simulated Raman spectra. Peaks labeled with star belong to NH2 wagging mode. From Ref. [24].

free aniline and AN-M4 , respectively. As seen in our previous study, the NH2 twisting mode blueshifts to 543, 494, and 600 cm−1 by binding to Cu, Ag, and Au clusters. However, the Raman intensities of the twisting mode in both free aniline and its adsorption states are quite weak. Although the vibrational frequency of the NH2 rocking vibration blueshifts to higher wavenumber by about 90 cm−1 at adsorption states, its Raman intensity is still weak. Furthermore, it is very interesting that a set of double peaks (v8a and NH2 scissoring) around 1600 cm−1 in normal Raman spectrum of free aniline change to a single band in AN-M4 . Our theoretical calculation shows that the v8a mode increases in the Raman activity, while the NH2 scissoring decreases obviously. The frequency shift of the ωNH2 mode strongly depends on the binding interaction between the amino group and metals. Here the binding interaction is stronger for gold and copper than silver. As seen in Table 18.1, the ωNH2 frequency in AN-M4 declines as the following trend: Au > Cu > Ag. As shown in Figure 18.4c, the fundamental frequency of the amino wagging mode has a nearly linear relationship on the changes of the force constants of the N—M bond. Accordingly, the frequency shift of the ωNH2 vibration can be used to correlate directly to the strength of the interaction of aniline with metal surfaces. Experimentally, the abnormal enhancement of the ωNH2 mode in SERS of aniline was not realized. For aniline adsorbed on silver [17] and gold [22], the strongest SERS band was observed at ∼1000 cm−1 [21]. Although some preternatural bumps appeared at around ∼950 and 1080 cm−1 in the SERS of aniline on [17] and gold [22], respectively, the abnormally enhanced SERS bands were overlooked. Now we assign it to the wagging vibration of the NH2 group on the basis of our present theoretical calculations [24, 70].

553

18 Density Functional Theoretical Study on Surface-Enhanced Raman Spectroscopy

0.6

AN-Ag+4

0.3 0.0 Raman intensity

554

0.6

AN-Ag+7

0.3 0.0 0.8

+

AN-Ag 13

0.4 0.0 400

600

800

1000 1200 Raman shift

1400

1600

Figure 18.6 Simulated Raman spectra of aniline adsorbed on silver electrode surfaces above the potential of zero charge (PZC) with models of aniline interacted with cationic Ag clusters (Ag4 + , Ag7 + , and Ag13 + ). An incident wavelength of 632.8 nm and a line width of 10 cm−1 were used in the simulated Raman spectra. From Ref. [24].

Figure 18.6 presents simulated Raman spectra of aniline-Agn + (n = 3, 7, and 13) complexes to describe aniline adsorbed on silver electrode [21]. Compared with aniline interacting with neutral silver clusters, the ωNH2 vibrational frequency further blueshifted to about 1000 cm−1 in aniline-Agn + . The band is responsible for a wideband near 993 cm−1 on a silver electrode [21]. It is noted that as the size of the silver clusters increases, the simulated Raman spectra are close to the SERS spectra. This indicates that the positively charged silver clusters with a large size are more appropriate to describe the Ag electrode surface. Evidence is given that SERS activity at Ag electrodes is associated with Ag+ ; most of the SERS-generating adsorbates happen to be those capable of forming Ag+ complexes in water [71]. After Roy et al. reported that the “active sites” are actually small clusters that on Ag electrodes have been identified as Ag4 + [72], a recent study suggested that the active site should be Ag3 2+ on the basis of analysis of the SERS of pyridine in colloidal silver solution [73]. Our calculated results indicated that some small positively charged silver clusters play important role on the SERS band of the wagging mode in aniline derivatives on silver surfaces. 18.6.2

Para-substituted Anilines

In this subsection, the effect of different adsorbate–substrate interactions on the SERS is studied through three typical double-functional aromatic amines, PABA, PABN, and PATP (p-aminothiophenol). It is well known that carboxyl [22], nitrile [23], and thiol [8] groups can form strong chemical interactions on metal surfaces. Recent studies especially on metal–molecule–metal junction systems [11–13] suggested that amino group also forms chemisorption interaction with gold surfaces. Thus, the identification and characterization of the amine–metal

18.6 Modeling Molecules

interaction and adsorption structures appear to be of great importance in molecular electronics. Park and coworkers studied experimentally the SERS of PABA [22] and PABN [23] on silver surfaces. It was found that these two molecules were adsorbed on the silver surfaces via both the carboxyl (nitrile) group and the amino group. Generally speaking, the former ones (—COO− and —CN groups) can form strong adsorption interaction with silver surfaces. But the latter one can be evidenced directly by the broadbands observed at 980 and 920 cm−1 , which were assigned to the amino rocking vibration in previous papers [22]. We assign the bands to the amino wagging vibration based on the PED. Recently, we also calculated the Raman spectrum of PATP adsorbed on silver surface with both single-end and double-end configurations by DFT [8]. It was found that a new peak that appeared at about 900 cm−1 should be assigned to the amino wagging vibration when amino end and thio end co-adsorbed on silver cluster [8]. However, this abnormal Raman feature did not attract great attention in other theoretical studies on PATP in metal–molecule–metal junction structures [68, 69]. Figure 18.7 presents the simulated surface Raman spectra of PABA, PABN, and PATP on silver and gold surfaces with both single-end and double-end cases. It can be seen that the Raman features of three para-substituted anilines in the single-end adsorption are different from their double-end cases. Note that the single-end adsorption takes place mainly through —COO− , —CN, or —SH. By inspecting the simulated Raman spectra of the two adsorption configurations, we can find that the major changes are from the vibrational bands related to the amino group. These variations can be summarized as the following three points. First, the amino group approaching to a metal surface will exhibit unique Raman signals. The Raman signal can be used to ascertain the adsorption configuration and the strength of the binding interaction. For example, the new Raman peaks appearing at around 900 and 1000 cm−1 for PABA, PABN, and PATP may be considered as an indicator for the formation of a double-end configuration on silver and gold surfaces, respectively. These strong Raman bands can be attributed to the ωNH2 vibration on metal surfaces. The assignment is different from the previous suggestion from Park et al., who assigned the broad and strong SERS band to the NH2 rocking vibration [22, 23]. Our calculated result shows that the NH2 rocking vibration possesses extremely low Raman signals. It was noted that such a new peak has not been observed for the SERS of PATP [8, 24], indicating that PATP may adsorb on noble metal surface mainly via a single-end adsorption. Zhou et al. studied SERS of PATP trapped in a gap site of a gold substrate and silver nanoparticles [74]. They observed three strong SERS bands appeared at 1142, 1391, and 1436 cm−1 , but no band is associated with the amino group. Our recent studies suggested that the SERS bands arise from the azobenzene-like surface species due to a surface catalytic coupling reaction of PATP molecules adsorbed on silver and gold nanoscale structural surfaces [9, 61]. In the earlier study, these new bands were assigned to the non-totally symmetric b2 vibrational modes of PATP [75]. The so-called b2 modes should be actually a1g modes of 4,4′ -dimercaptoazobenzene, an azobenzene-like due to surface catalytic coupling reaction of PATP adsorbed on nanostructured surfaces of silver and gold [9].

555

18 Density Functional Theoretical Study on Surface-Enhanced Raman Spectroscopy 0.4

Ag5-PABA

0.2 0.0 Raman intensity

1.0

Ag5-PABA-Ag4

0.5 0.0 0.4

Au5-PABA

0.2 0.0 1.5 1.0 0.5 0.0

Ag5-PABA-Au4

400

600

800

(a)

1000

1200

1400

1600

1200

1400

1600

Raman shift 0.8

Ag4-PABN

0.4 0.0 Raman intensity

1.0

Ag4-PABN-Ag4

0.5 0.0 0.8

Au4-PABN

0.4 0.0 2

Au4-PABN-Au4

1 0 400

600

800

(b)

1.5 1.0 0.5 0.0 1.5 1.0 0.5 0.0 1.5 1.0 0.5 0.0

Ag5-PATP

Ag5-PATP-Ag4

Au5-PATP

Au5-PATP-Au4 400

(c)

1000

Raman shift 1.5 1.0 0.5 0.0

Raman intensity

556

600

800

1000 1200 1400 1600

Raman shift

Figure 18.7 Simulated surface Raman spectra of PABA (a), PABN (b), and PATP (c) on silver and gold surfaces with both single end and double end at B3LYP/6-311+G(d,p)/LAN2DZ level. The excitation wavelength 632.8 nm and the line width of 10 cm−1 were used here. The peaks marked with blue and red cycles represent the amino wagging mode. From Ref. [24].

18.6 Modeling Molecules

Second, the frequencies of the C—NH2 stretching vibration of PABA, PABN, and PATP at the single-end adsorption on Ag (Au) are predicted at 1280 (1289), 1310 (1314), and 1272 (1281) cm−1 . These frequencies are very close to 1273, 1297, and 1279 cm−1 in their free molecules. However, the frequencies significantly redshift to 1229 (1218), 1252 (1234), and 1228 (1218) cm−1 when the amino group binds to silver (gold). Experimentally, the peak at 1279 cm−1 in the ordinary Raman spectrum of PABA redshifts to 1253 cm−1 in the SERS spectrum on silver [22]. Such a redshift is invoked to conclude that the amino group of PABA also interacted with silver surface. Nevertheless, the Raman intensities of the C—NH2 stretching vibration are quite weak in general; the frequency shift of this mode is hardly to be identified in SERS measurements of other aromatic amines such as PABN and PATP. Thus, the C—NH2 stretching vibration does not seem to be a good probe to study the amine–metal interaction. Third, from our simulated spectra, we can find that at single-end adsorption state, a set of double peaks appear at around 1600 cm−1 corresponding to the v8a and the NH2 scissoring vibrations. But for the double-end adsorption configuration, only one band could be observed. This phenomenon is similar to that of aniline. The above spectral feature can be explained as the amine–metal interaction decreases its relative Raman intensity of the scissoring mode. The frequency shift of the ωNH2 mode depends on the electronic properties of para-substituted functional groups. In EW-substituted aniline, the frequency of the vibration is lower than that of aniline. On the contrary, the frequency increases in ED-substituted aniline. This is because the EW group enhances the p–π conjugation effect between amino group and benzene ring, while the ED group weakens it. The p–π conjugation interaction would further change the hybridization of the amino groups. In the EW(ED)-substituted aniline, the amino group is inclined to sp2 (sp3 ). The change of hybridization property will lead to the shape change of potential energy surface along this vibration and finally results in the frequency shift of this mode. The ωNH2 frequencies of the investigated three molecules in their double-end configuration are PATP > PABA > PABN. This trend confirms with SERS experimental result: 980 and 920 cm−1 for PABA and PABN, respectively [22, 23]. 18.6.3

Benzyl Radicals and Its Anion

Table 18.2 presents calculated results of propylene free radical, propylene radical anion, benzyl radical, and benzyl radical anion interacting with silver clusters. For free propylene free radical and its anion, two C—C bonds are equivalent in chemical property. With regard to propylene free radical adsorbed on silver clusters, the C—C bond close to the silver cluster stretches. This causes a weakening C—C bond and a redshift in the wagging frequency. In the case, the CH2 inversion angle is predicted to be 23.3∘ , indicating that its hybridization changes from sp2 to sp3 for the CH2 group binding to the silver cluster. The calculated results show that the decrease in the CH2 wagging frequency is due to the C—C stretching force constant rather than the change of the hybridization. When propylene free radical anion binds to silver clusters, the C—C bond close to the silver cluster stretches significantly, and the CH2 inversion angle increases to 44.7∘ . Similar to the propylene free radical, the CH2 group with the sp3 hybridization has a large wagging frequency. Another C—C bond strengthens its double bond feature.

557

558

18 Density Functional Theoretical Study on Surface-Enhanced Raman Spectroscopy

Table 18.2 Inversion angle (𝛼, ∘ ), C—C bond length (Å), C—Ag bond length (Å), calculated vibrational frequency (𝜔, cm−1 ), Raman activity (SR , Å4 /amu), scattering cross section (10−30 cm2 /mol sr), and bonding energies (BE) of molecule–metal (kcal/mol). 𝜶

Species

C3 H5

C—C

C—Ag

1.384

C3 H5 —Ag4

−

C3 H5 —Ag4

C6 H5 CH2

C6 H5 CH2 − —Ag4

0.13

0.05

8.51

3.34 27.20

1.420

768

91.15

36.29

1.394

412

9.06

8.36

896

3354.86

1094.10

772

10.03

3.97

23.3 44.7

1.457

2.0

1.352

36.6

1.454

46.4

1.465

−

803

72.18

2.321

2.239

1.405

C6 H5 CH2 —Ag4

(d𝛔/d𝛀)i

778 1.377

C6 H5 CH2

SR

801

0

C3 H5 −

𝝎

698

6.84

0.73

2.255

831

3582.68

307.47

416

153.18

32.56

2.239

899

3879.01

301.66

1.391

BEa)

BEb)

13.24

11.88

57.91

56.51

14.64

13.03

52.78

51.27

a) Bonding energy in gas phase. b) Considering BSSE correction.

When benzyl radical and its anion interact with silver clusters via the CH2 group, the inversion angles are 36.6 and 46.4∘ , respectively. At the same time, the wagging frequencies increase dramatically. The wagging frequency is closely associated with the hybridization property. In the complexes of terminal olefins with silver clusters, the inversion angle of CH2 group has a slight change so that the hybridization of the CH2 group almost keeps at the sp2 state. In the case, the wagging frequency changes with the strength of the C=C bond. However, for benzyl radicals binding to silver clusters, the frequency shift is mainly determined by the hybridization property of the CH2 group, similar to the case of aniline derivatives. Figure 18.8 presents the simulated Raman spectra of benzyl anion-Agn with the two solvation models. The most striking difference comes from the CH2 wagging vibration in the region of 800 cm−1 wavenumber. It was noted that both of them have a redshift with respect to that in the gas phase. The relative intensity of the two intense Raman peaks of CH2 wagging vibration slightly depends on the solvation model. Take example for benzyl anion-Ag4 the Raman bands distributed at about 857 and 888 cm−1 in the PCM model, and the latter is a stronger band. While the SMD model is considered, this vibration shifts to 844 and 879 cm−1 , and the former becomes a stronger band. For benzyl anion-Ag19 , the peaks are distributed at 789 and 804 cm−1 with the PCM model, compared with 792 and 807 cm−1 by using the SMD model.

18.6.4

Terminal Olefins

The simplest terminal olefin ethylene has been considered as a prototype for the interaction of olefins with metal surfaces. The π-bonded property is preferred to

18.6 Modeling Molecules

888

857 10

359 531

795

1211

1587

Intensity (10–30 cm2/sr)

(c)

879 359 530 794 733

1209

1586

200 400 600 800 1000 1200 1400 1600 1800 Raman shift (cm–1)

(b) 40

40

825

20

334 530 579

793 868 726

1216

1584

Intensity (10–30 cm2/sr)

Intensity (10–30 cm2/sr)

200 400 600 800 1000 1200 1400 1600 1800 Raman shift (cm–1)

0

12

0

0

(a)

844

24

843

20 876

0

200 400 600 800 1000 1200 1400 1600 1800 Raman shift (cm–1)

(d)

333 530 597

731 795

Intensity (10–30 cm2/sr)

20

1017 1214

1583

200 400 600 800 1000 1200 1400 1600 1800 Raman shift (cm–1)

Intensity (10–30 cm2/sr)

Intensity (10–30 cm2/sr)

300 789 804

150 319

529 727 864

1219

807 792 120 316 529 731 868 504

1582

0

(e)

240

200 400 600 800 1000 1200 1400 1600 1800 Raman shift (cm–1)

1218

1582

0

(f)

200 400 600 800 1000 1200 1400 1600 1800 Raman shift (cm–1)

Figure 18.8 Simulated Raman spectra of benzyl anion adsorbed on silver clusters (Ag4 , Ag6 , and Ag19 ) calculated at B3LYP/6-311+G(d, p)/LAN2DZ level in aqueous solution with the PCM model (blue) and SMD model (green). The inserts are optimized structures of benzyl anion interacting with silver clusters along the methylene group approaching the surfaces. The excitation wavelength of 632.8 nm was used here with a Lorentzian line width of 10 cm−1 .

the molecule interacting with silver clusters. Table 18.3 presents the binding energies of ethylene interacting with different silver clusters. Obviously, the binding energy of ethylene-Ag4 is the largest among different complexes. This indicates that the tetramer silver Ag4 can describe well the surface active site on silver surfaces. When ethylene interacts with Ag4 through the π-bonded form, the main orbital interaction is between the HOMO orbital of ethylene and the LUMO orbital of Ag4 . The HOMO π orbital matches well with the LUMO of Ag4 itself in symmetry and energy, which belongs to s-type orbital with the electron cloud mainly populated at the obtuse angle side of the rhombus. The NBO analysis showed that the energy of the main orbital interaction is 9.08 kcal/mol. In previous studies, ethylene was proposed to chemically adsorb on metal surfaces in two

559

560

18 Density Functional Theoretical Study on Surface-Enhanced Raman Spectroscopy

Table 18.3 Selected parameters of optimized geometries, binding energy (BE), and wagging frequency for ethylene–Agn complexes. C—Ag (Å)

C—C (Å)

BE (kcal/mol)

BEa) (kcal/mol)

Frequency (cm−1 )

C2 H4 —Ag2

2.696

1.340

4.91

4.07

960/978

C2 H4 —Ag4

2.474

1.356

11.52

10.49

933/944

C2 H4 —Ag6

2.833

1.337

3.10

2.36

966/978

C2 H4 —Ag13

2.605

1.348

5.38

4.46

940/955

a) Considering BSSE correction.

different states, π- or di-σ-bonded states [44, 51–54]. The π-bonded configuration is usually proposed, the molecule adsorbed at a top site, while the di-σ bonding usually occurs at a short bridge site [47, 76, 77]. The π-bonded configuration is preferentially considered as the stable one for the terminal olefins on metal surfaces, while for the di-σ-bonded complexes, significant deformation of molecular geometries occurs [47, 54, 77]. Therefore, we only consider the π-bonded configuration in the previous paper [70]. Table 18.3 lists selected geometric parameters, binding energies, and wagging frequencies of the four structures C2 H4 -Agn + (n = 1, 3, 7, and 13) with the SMD model. The calculated results showed that the solvent effect significantly decreases the binding energies that are 11.60, 6.53, 8.46, and 3.95 kcal/mol for the four complexes, respectively. This is mainly due to the reduction of the electrostatic interaction from the solvent screening effect. After considering the solvent effect, there is a small difference in the BE values between C2 H4 -Ag4 and C2 H4 -Ag4 + , suggesting that the two surface species may coexist on Ag surfaces. This is why different wagging peaks can be observed on silver films. Especially for C2 H4 -Ag13 + , the BE is only 3.95 kcal/mol. The corresponding wagging frequency lowers to 937 cm−1 . This is analogous to ethylene adsorbed on neutral silver surfaces, indicating that the solvent effect significantly influences the binding interaction between C2 H4 and Ag13 + . The blueshift frequency for ethylene adsorbed on small cationic silver clusters can be interpreted due to the small back donation from the occupied orbital of the silver cluster to the antibonding orbital of the C=C double bond. The NBO analysis indicates that the population in the antibonding orbital for C2 H4 -Ag4 + is ∼0.04e, which is obviously smaller than ∼0.12e in neutral complexes. Table 18.4 lists the binding energies of different terminal olefins interacting with silver clusters. Our calculated results showed that the binding energies are less than 12 kcal/mol, in a fairly good agreement with the experimental values about 9–12 kcal/mol reported in previous papers [14, 17, 78–81]. This indicates that the adsorption interaction is weakly π-bonded chemisorption for ethylene derivatives on silver. Among these complexes, one can find that the vinylamine-Ag4 owns the largest binding energy. This can be interpreted due to the p–π conjugation effect from the lone pair orbital in the amino group strongly conjugating with the π orbital of the C=C double bond. The charge transfer occurs from the lone pair orbital to the π orbital. The increase in the density of

18.6 Modeling Molecules

Table 18.4 Inversion angle (𝛼, ∘ ), C=C bond length (Å), calculated vibrational frequency (𝜔, cm−1 ), Raman activity (SR , Å4 /amu), and bonding energies (BE) of molecule–Ag4 (kcal/mol). Species

𝜶

C2 H4 C2 H4 —Ag4

C(N)—Ag 𝝎

1.329 5.9

C3 H6 C3 H6 —Ag4

C=C

6.8

1.356 2.473

SR

958

2.58

956

0

944

66.09

933

24.21

1.331

925

2.88

1.355 2.473

923

74.56

1.338

921

0

920

14.13

932

22.28

BEa)

BEb)

BE (expt)

11.52 10.49 9.56 [34], 8.76 [78]

11.07 9.78

8.9 –10.1 [17] 10.8 –12.6 [14]

C4 H6 C4 H6 —Ag4

0

1.338 2.456 1.363

901

439.20

C3 H4 Ot

8.4 1.335

979

2.02

C3 H4 Ot—Ag4 9.1

1.370 2.417

915

141.71

C2 H3 NH2

1.334

0

10.23 8.85

9.6 –10.8 [79]

9.77

≤8.37 [80, 81]

8.26

813(CH2 ) 8.39

37.1

544(NH2 ) 3.86

C2 H3 NH2 — Ag4 (C—Ag)

16.1 1.364 2.424

808(CH2 ) 64.87

14.07 12.87

C2 H3 NH2 — Ag4 (N—Ag)

48.5 1.326 2.394

901(NH2 ) 58.86

12.48 11.23

0

874(CH2 ) 2.12

a) Bonding energy in gas phase. b) Considering BSSE correction.

the π electron cloud results in the strengthening of the interaction between the π orbital of olefin and the metallic orbital. On the contrary, for acrolein-Ag4 , the aldehyde group is an EW group that decreases the electron density of the π orbital on the C=C bond. Thus the interaction is the weakest in acrolein-Ag4 . The inversion angles of the CH2 group are also calculated, and the out-of-plane bending angles less than 10∘ demonstrate that these adsorbed molecules almost remain the sp2 hybridization. The phenomenon that involves charge transfer from a π orbital of the adsorbate to unoccupied metal levels leads to a decrease of the bond order and an increase of the C=C bond lengths. The CH2 wagging frequency of adsorbed olefins displays a redshift compared to that of the free molecules. Our calculation predicted that the Raman frequency of the ωCH2 mode has a redshift about 20 cm−1 for ethylene adsorbed at the neutral silver clusters. The large frequency shift is in accord with the observation in experimental Raman spectra [10, 12]. For example, for ethylene adsorbed on Ag films, a band at 917 cm−1 was observed in the SERS with a redshift about 32 cm−1 compared to that in free molecule [12]. Under very low concentration of silver, a band appeared at 920 cm−1 [10]. For propylene adsorbed on Ag surfaces, the

561

18 Density Functional Theoretical Study on Surface-Enhanced Raman Spectroscopy

Raman intensity (10–31 cm2/sr)

562

2 1 0

C2H4-Ag+

2 1 0

C2H4-Ag3+

4 2 0

C2H4-Ag7+

1314 1576 1016 1330

1600

1322

1590

1318

1586

996

990

16

C2H4-Ag13+

8

974

0 400

600

800 1000 1200 1400 Wavenumber (cm–1)

1600

1800

Figure 18.9 Simulated Raman spectra of ethylene interacted with cationic Ag clusters (Ag+ , Ag3 + , Ag7 + , and Ag13 + ). An incident wavelength of 514.5 nm and a line width of 10 cm−1 were used in the simulated Raman spectra. From Ref. [24].

ωCH2 band was observed at 909 cm−1 in the IR spectra, slightly lower than the frequency of 912 cm−1 in free molecule [14, 17]. This is in agreement with our calculated value at 910 cm−1 for propylene interacting with silver clusters. Similar phenomena were observed for 1-butene and styrene adsorbed on Ag surfaces [82, 83]. For 1,3-butadiene that is the simplest conjugated alkene containing two wagging modes, our calculated results showed that the wagging frequency of the methylene close to the silver cluster decreases as the C=C bond length increases. At the same time, the wagging frequency of the other methylene increases as the C=C bond length shortens. Accordingly, we inferred that the wagging frequencies depend on the variation of the C=C bond distances in olefin-neutral silver cluster complexes. Figure 18.9 presents simulated Raman spectra of ethylene binding to cationic Agn + clusters (n = 1, 3, 7, and 13), in analogy to the case that ethylene adsorbs on positively charged active sites of silver. The ωCH2 vibration has a blueshift to around 1000 cm−1 . When the smaller clusters are used, one can predict the larger vibrational frequency of the wagging mode. The calculated binding energies for the four complexes decrease in an order as 32.76, 18.25, 16.95, and 9.08 kcal/mol, respectively. For the smaller silver clusters, the binding energies are significantly larger than that of the corresponding neutral complexes. This is in accord with the experimental result that ethylene adsorbed on a pre-covered atomic oxygen silver surface [27] and Raman spectra of 1,3-butadiene with silver ion [30, 32].

18.7 Chemical Enhancement Effect Comparing the calculated Raman intensity for free molecules and their silver cluster complexes (see Tables 18.1 and 18.2), it is obvious that the intensity of the

18.7 Chemical Enhancement Effect

450

αxx αyx αyy αzx αzy αzz

400 Polarizability (Bohr3)

350 300 250 200 150 100 50 0 –1.0

–0.5

0.0 0.5 Normal coordinate

1.0

Figure 18.10 The polarizability components (unit in Bohr3 ) along the amino wagging vibration in aniline-Ag4 . From Ref. [70].

wagging mode has a great enhancement after the molecules adsorbed on silver surfaces. Especially for aromatic anilines and benzyl radicals, the Raman intensity is significantly enhanced by about ∼500-fold [70]. Here our purpose is to make clear the cause of the Raman enhancement of the wagging mode. Figure 18.9 presents the variation of six polarizability tensor components along the wagging mode in aniline-Ag4 complexes. The six derivative values can be directly taken from the curves in Figure 18.10. The Raman scattering factor calculated from Gaussian program can be reproduced completely by estimating these data from Eq. (18.35). Table 18.5 summarizes the related data of polarizability derivatives and Raman intensities of aniline, benzyl radical, benzyl radical anion, and their silver cluster ′ value is quite larger than complexes in the same way. It can be found that the 𝛼XX other five derivatives in aniline and the aniline–silver complex [70]. So the sig′ . nificant enhancement of Raman intensity mainly comes from the derivative 𝛼XX ′ ′ ′ For benzyl and its anion, the values of 𝛼XX , 𝛼YY , and 𝛼ZZ are zero because of ′ ′ ′ ′ ′ ′ Table 18.5 The six derivatives of polarizability components 𝛼xx , 𝛼yx , 𝛼yy , 𝛼zx , 𝛼zy , 𝛼zz , 2 1/2 isotropous polarizability (Å /amu ), anisotropic polarizability, and Raman scattering factor (SR , Å4 /amu) of wagging mode for aniline, benzyl, and their metallic complexes.

′ 𝜶zx

′ 𝜶zy

′ 𝜶zz

′

′

Species

′ 𝜶xx

′ 𝜶yx

′ 𝜶yy

C6 NH2

0.75

0.00

−0.13

−0.10

0.00

0.04

0.22

0.68

6.88

C6 NH2 —Ag4

2.83

0.00

−0.94

0.02

0.00

0.18

0.69

11.26

100.39

C6 CH2

0.00

0.00

0.00

0.57

0.00

0.00

0.00

0.97

6.76

C6 CH2 —Ag4

16.84

1.25

2.46

0.00

0.00

0.77

6.69

238.72

3686.34

C6 CH2 −

0.00

0.00

0.00

−0.03

2.73

0.00

0.00

22.29

156.01

C6 CH2 − —Ag4

−17.66

1.06

0.60

−0.35

0.15

−1.00

−6.02

310.71

3806.80

𝜶

𝜸

2

SR

563

18 Density Functional Theoretical Study on Surface-Enhanced Raman Spectroscopy

molecular planar configurations. In contrast, when they bind to silver clusters, ′ the 𝛼XX values increase significantly. So we can assume that the change of polarizability component 𝛼 xx is the decisive factor for the chemical enhancement in Raman intensity of the wagging mode from the binding interaction. Figure 18.11 presents the polarizability tensor components 𝛼 xx , 𝛼 yy , and 𝛼 zz varying along with the wagging vibration in these molecule-silver cluster complexes. Here we mainly discuss the 𝛼 xx relevant to the variation of hybridization property. From Figure 18.11 one can see that in the three complexes the 𝛼 xx is larger in the sp2 -hybridized state than that in the sp3 one. It just corresponds to the direction of the wagging vibration in standard orientation coordinate. The previous study showed that most of the molecular polarizability was contributed z

x

z

y

x

x 120

C6H5NH2

y 140

sp2

C6H5CH2

sp2

180

C6H5CH2n 2 sp

120 100

αxx αyy αzz

150 100

80

120

80 Polarizability (Bohr3)

564

60

40

40 –1.0 –0.5 0.0 0.5 1.0

400

90

60

–1.0 –0.5 0.0 0.5 1.0 600

C6H5NH2-Ag4

sp2

C6H5CH2-Ag4

60

–1.0 –0.5 0.0 0.5 1.0 C6H5CH2n-Ag4 sp2

sp2

500 500

350

400

400

300 250

300

200

200

300

200 –1.0 –0.5

0.0

0.5

1.0

–1.0 –0.5 0.0 0.5 1.0 Normal coordinate

–1.0 –0.5 0.0 0.5 1.0

Figure 18.11 The molecule-silver cluster complexes in their rectangular coordinates and dependence of the polarizability tensor components 𝛼 xx , 𝛼 yy , and 𝛼 zz (unit in Bohr3 ) on normal coordinate of wagging modes in aniline, benzyl radical, benzyl radical anion, and their silver complexes. The perpendicular red dotted lines denote plane structures of molecule parts. From Ref. [70].

18.8 The Reason of Broadbands

from the electrons in the outer valence orbitals and the higher energy orbitals are highly polarizable [84]. Therefore, the polarizability significantly varies at the frontier molecular orbitals in the silver cluster complexes upon the wagging mode vibrating periodically. In order to understand the change of the polarizability tensor, we compared the energy levels of frontier occupied orbitals in different complexes. Figure 18.12 presents the variation of the MO energies ranging from LUMO+1 to some high occupied orbitals of the silver cluster complexes along the wagging coordinate. Here, we represented the second and the third HOMOs as the HOMO-1 and HOMO-2, respectively. The bonding orbitals were HOMO-1 and HOMO-2 in AN-Ag4 , and the HOMO-1 in benzyl anion-Ag4 and benzyl-Ag4 . It is obvious that for AN-Ag4 the HOMO-1 and HOMO-2 energy levels gradually raised as the hybridization changes from sp3 to sp2 . For benzyl anion-Ag4 , the energy level of the bonding orbital changes in order, raised from HOMO-1 to HOMO as the hybridization changes from sp3 to sp2 . For benzyl-Ag4 , the HOMO-1 energy level also gradually raised as the hybridization changes from sp3 to sp2 . The lone pair electron delocalization effect and the increase of the orbital energy cause the enlargement of the orbital polarizabilities. The conjugation effect is strong in the sp2 configuration but becomes weak in the sp3 configuration. The strong adsorption interaction results in the decrease of the lone pair orbital energy and the orbital polarizability. As seen in Figure 18.12a, the energies of HOMO-1 and HOMO-2 orbitals for AN-Ag4 gradually raised when the hybridization of aniline changes from sp3 to sp2 . From Figure 18.12b there is a cross point of an orbital energies, indicating the orbital interaction existing. In [benzyl-Ag4 ]− , when the CH2 group changes the hybridization from sp3 to sp2 with the wagging vibration, the HOMO-1 from the lone pair orbital interacting with the Ag4 drastically increases in the orbital energy. In the case, the order of the frontier orbital levels changes, especially the electron configuration of Ag4 varying in a large extent. As shown in Figure 18.12c, the orbital interaction works in a similar way contributing to the increase in the 𝛼 xx polarizability in benzyl-Ag4 . It was found that the polarizability of the silver cluster plays a vital role in contributing to the total polarizability of the complexes. So the change of the electron configuration is the primary cause in increasing polarizability when the Ag4 interacts with the lone pair orbital of CH2 and NH2 groups.

18.8 The Reason of Broadbands It is an interesting problem why the vibrational signal of the wagging mode is very broad in observed SERS spectra. This is generally interpreted before due to the interaction of adsorbed molecules with metal surfaces. On the basis of our calculated results, we can summarize three factors. The first factor is the different adsorption states that contribute to the origin of the broadband. Surface-enhanced infrared spectroscopy in the ωCH2 vibration region was measured for ethylene, propylene, and acrylonitrile adsorbed on oxygen pre-covered Ag, which exhibits broad features from at least four adsorbed

565

18 Density Functional Theoretical Study on Surface-Enhanced Raman Spectroscopy

–0.05

sp2

–0.10

LOMO

E (a.u.)

–0.15 HOMO

–0.20 –0.25 –0.30 –1.0

(a)

–0.5 0.0 0.5 Normal coordinate

1.0

sp2

0.05

LUMO

E (a.u.)

0.00 –0.05

HOMO

–0.10 –0.15 –0.20 –1.5

–1.0

(b) –0.05

–0.5 0.0 0.5 Normal coordinate

1.0

1.5

sp2

–0.10

LUMO

–0.15 E (a.u.)

566

HOMO

–0.20 –0.25 –0.30 –0.35 –1.5

(c)

–1.0

–0.5 0.0 0.5 Normal coordinate

1.0

1.5

Figure 18.12 The variation of molecular orbital energies (in atomic units) for aniline-Ag4 (a), benzyl radical anion-Ag4 (b), and benzyl radical-Ag4 (c) along the normal coordinates. The lowest unoccupied molecular orbitals (LUMO) and the highest occupied molecular orbitals (HOMO) are indicated with red marks. The plots of HOMO-1 or HOMO-2 orbitals are shown in the curves. Zero value of normal coordinate corresponds to the equilibrium configurations. From Ref. [70].

18.9 Conclusions

states on surfaces [15, 27, 28, 33]. In our previous studies, we have simulated the Raman spectra of molecules adsorbed on silver clusters with different sizes and sites. As mentioned above, the peak position of the wagging mode changes strongly depending on the adsorption sites when benzyl, ethylene, or aniline adsorbed on metal surfaces. So we conclude that the band positions of both ωCH2 and ωNH2 modes are sensitive to the electronic structure of adsorption sites. This leads to the observed broadband from a superposition of the SERS wagging peaks of different adsorption states. The factor should interpret the broadband observed in olefins, aniline, and benzyl adsorbed on metal surfaces. The second factor arises from the coupling of intramolecular vibrational modes of adsorbed molecules. The reason is mainly aimed at aromatic anilines. Based on normal mode analysis of the optimized structures and the force constants calculated at the different theoretical approaches, the PED values presented for vibrational fundamentals are related to the wagging coordinates [70]. The results show that the ωNH2 vibration strongly coupled with the out-of-plane C—H bending calculated by using the larger basis sets. These vibrations also display large Raman intensities due to the mixture of the wagging coordinate. For the modes with higher frequencies, the PED values of the amino wagging coordinate are dominant, indicating that this mode mainly comes from the amino wagging vibration. The modes of the lower frequency calculated by large basis sets can be attributed to a mixed vibration of the amino wagging coordinate and the out-of-plane bending of the C—H bonds. This is another reason why the NH2 wagging band is very broadly observed in SERS experiments. The third factor comes from congeneric species adsorbed on different surface sites. A superposition of the spectra of the benzyl moieties, including benzyl radical and benzyl anion, shows that the overlap of the bands affords a broad peak between 850 and 900 cm−1 [13]. Our calculated spectra showed that there are intense Raman bands of the ωCH2 mode at 831 and 899 cm−1 for adsorbed free benzyl radical and benzyl anion, respectively [13, 70, 85]. This further showed the broadband from surface congeneric species as observed in the SERS experiment [13]. So we can make a conclusion that the broadband may come from the contribution of the three different factors.

18.9 Conclusions We have investigated the SERS of the wagging mode for terminal olefins, p-substituted anilines, and benzyl radicals adsorbed on silver surfaces. Our theoretical analysis provides further interpretation for three aspects of the SERS feature of the wagging mode by using DFT calculations. Firstly, we focus on the frequency shift. For the terminal olefins interacting with surfaces through the π-bonded form, we inferred that the redshift peaks of the wagging mode come from the molecules adsorbed on neutral silver atoms of the surfaces, while the blueshift peaks from the positively charged silver sites. For aromatic amines adsorbed on metal surfaces with the NH2 group, the frequency shifts of the wagging mode strongly depend on the hybridized property of the NH2 group.

567

568

18 Density Functional Theoretical Study on Surface-Enhanced Raman Spectroscopy

The results show that the amino wagging frequency increases with the inversion angle increasing. Secondly, we explained the reason why the Raman intensity enhancement of the wagging mode is very significant. It is concluded that the significant change of the polarizability component 𝛼 xx is a decisive factor contributing to the chemical enhancement after adsorption. The polarizability component is obviously larger than the other five ones. In molecule-silver complexes the 𝛼 xx is larger when the molecular moiety has an sp2 hybridization, and the value sharply decreases as the molecular moiety changes to an sp3 hybridization. This change is mainly affected by the variation of the electronic structure for the silver cluster moiety. Finally, we accounted for the cause of the broadband in observed SERS spectra. Our calculated results show that the wagging vibration is very sensitive to the environment factors. The broadband may come from the contribution of molecules adsorbed on several different surface adsorption sites, the coupling of vibrational modes, and the co-adsorbed congeneric species. Furthermore, similar to aniline and benzyl radical in free states, the wagging vibration displays very significant anharmonicity on silver surfaces as mentioned before. It is possible to observe the overtone and combination transitions contribution to the broadband from the wagging vibration and the related vibrational coupling. In summary, the spectral characteristic of the wagging vibration in different surface species is an interesting phenomenon in SERS. Quantum chemical calculations have provided new view for interpretation and understanding those observations of interest in the field of surface vibrational spectroscopy.

Acknowledgments We are grateful for the financial support of this work by the NSF of China (Nos. 21533006, 21373172, and 21621091) and National Basic Research Program (No. 2015CB932303). DYW is grateful for the support from Xiamen University and Fujian Science and Technology Office.

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of a weakly π-bonded C2 H4 on oxygen-covered Ag (100). J. Phys. Chem. B 110 (1): 367–376. Lyalin, A. and Taketsugu, T. (2010). Adsorption of ethylene on neutral, anionic, and cationic gold clusters. J. Phys. Chem. C 114 (6): 2484–2493. Chrétien, S., Gordon, M.S., and Metiu, H. (2004). Density functional study of the adsorption of propene on silver clusters, Ag (m= 1–5; q= 0,+ 1). J. Chem. Phys. 121: 9925. Wu, D.Y., Hayashi, M., Chang, C. et al. (2003). Bonding interaction, low-lying states and excited charge-transfer states of pyridine–metal clusters: Pyridine–M (M= Cu, Ag, Au; n= 2–4). J. Chem. Phys. 118: 4073. Lee, C., Yang, W., and Parr, R.G. (1988). Development of the Colle-Salvetti correlation-energy formula into a functional of the electron density. Phys. Rev. B 37 (2): 785. Becke, A.D. (1993). Density-functional thermochemistry. III. The role of exact exchange. J. Chem. Phys. 98: 5648. Krishnan, R., Binkley, J.S., Seeger, R., and Pople, J.A. (1980). Self-consistent molecular orbital methods. XX. A basis set for correlated wave functions. J. Chem. Phys. 72: 650. McLean, A. and Chandler, G. (1980). Contracted Gaussian basis sets for molecular calculations. I. Second row atoms, Z= 11–18. J. Chem. Phys. 72 (10): 5639–5648. Hay, P.J. and Wadt, W.R. (1985). Ab initio effective core potentials for molecular calculations. Potentials for K to Au including the outermost core orbitals. J. Chem. Phys. 82: 299. Wadt, W.R. and Hay, P.J. (1985). Ab initio effective core potentials for molecular calculations. Potentials for main group elements Na to Bi. J. Chem. Phys. 82: 284. Frisch, M.J., Trucks, G.W., Schlegel, H.B. et al. (2009). Gaussian 09, Revision A.1. Wallingford CT: Gaussian, Inc. Jarzecki, A.A. (1990). Scale 2.0. Fayetteville, AR: University of Arkansas. Carpenter, J. and Weinhold, F. (1988). Analysis of the geometry of the hydroxymethyl radical by the “different hybrids for different spins” natural bond orbital procedure. J. Mol. Struct. THEOCHEM 169: 41–62. Reed, A.E., Curtiss, L.A., and Weinhold, F. (1988). Intermolecular interactions from a natural bond orbital, donor-acceptor viewpoint. Chem. Rev. 88 (6): 899–926. Marenich, A.V., Cramer, C.J., and Truhlar, D.G. (2009). Universal solvation model based on solute electron density and on a continuum model of the solvent defined by the bulk dielectric constant and atomic surface tensions. J. Phys. Chem. B 113 (18): 6378–6396. Boys, S. and Bernardi, F. (1970). The calculation of small molecular interactions by the differences of separate total energies. Some procedures with reduced errors. Mol. Phys. 19 (4): 553–566. Schultz, G., Portalone, G., Ramondo, F. et al. (1996). Molecular structure of aniline in the gaseous phase: a concerted study by electron diffraction and ab initio molecular orbital calculations. Struct. Chem. 7 (1): 59–71.

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of rho-pi conjugated molecules containing CH2 /NH2 groups adsorbed on silver surfaces: a DFT study of wagging modes. J. Phys. Chem. C 117 (37): 18891–18903. Watanabe, T., Kawanami, O., Honda, K., and Pettinger, B. (1983). Evidence for surface Ag+ complexes as the SERS-active sites on Ag electrodes. Chem. Phys. Lett. 102 (6): 565–570. Roy, D. and Furtak, T.E. (1986). Evidence for Ag cluster vibrations in enhanced Raman-scattering from the Ag/electrolyte interface. Chem. Phys. Lett. 124 (4): 299–303. Muniz-Miranda, M., Cardini, G., Pagliai, M., and Schettino, V. (2007). DFT investigation on the SERS band at 1025cm−1 of pyridine adsorbed on silver. Chem. Phys. Lett. 436 (1): 179–183. Zhou, Q., Li, X., Fan, Q. et al. (2006). Charge transfer between metal nanoparticles interconnected with a functionalized molecule probed by surface-enhanced Raman spectroscopy. Angew. Chem. Int. Ed. 45: 3970–3973. Osawa, M., Matsuda, N., Yoshii, K., and Uchida, I. (1994). Charge transfer resonance Raman process in surface-enhanced Raman scattering from p-aminothiophenol adsorbed on silver: Herzberg-Teller contribution. J. Phys. Chem. 98: 12702–12707. Bocquet, M.L. and Sautet, P. (1998). Molecular transparence and contrast with the STM: a theoretical comparison of carbon monoxide and ethylene. Surf. Sci. 415 (1): 148–155. Itoh, K., Kiyohara, T., Shinohara, H. et al. (2002). DFT calculation analysis of the infrared spectra of ethylene adsorbed on Cu (110), Pd (110), and Ag (110). J. Phys. Chem. B 106 (41): 10714–10721. White, J. (1994). Preparation and kinetic characterization of hydrocarbon fragments on transition metals. Langmuir 10 (11): 3946–3954. Hrbek, J., Chang, Z.P., and Hoffmann, F.M. (2007). The adsorption of 1,3-butadiene on Ag(111): a TPD/IRAS study and importance of lateral interactions. Surf. Sci. 601 (5): 1409–1418. Lim, K.H., Chen, Z.X., Neyman, K.M., and Rosch, N. (2006). Adsorption of acrolein on single-crystal surfaces of silver: density functional studies. Chem. Phys. Lett. 420 (1–3): 60–64. Ferullo, R., Branda, M.M., and Illas, F. (2010). Coverage dependence of the structure of acrolein adsorbed on Ag(111). J. Phys. Chem. Lett. 1 (17): 2546–2549. Wei, W., Huang, W., and White, J. (2004). Adsorption of styrene on Ag (111). Surf. Sci. 572 (2): 401–408. Pawelacrew, J., Madix, R.J., and Vasquez, N. (1995). A Fourier-transform infrared study of the repulsive interactions and orientation of 1-butene and isobutylene on Ag(110). Surf. Sci. 340 (1–2): 119–133. Bounds, D.G. and Hinchliffe, A. (1980). The shapes of pair polarizability curves. J. Chem. Phys. 72: 298. Chen, Y.L., Panneerselvam, R., Wu, D.Y., and Tian, Z.Q. (2017). Theoretical study of normal Raman spectra and SERS of benzyl chloride and benzyl radical on silver electrodes. J. Raman Spectrosc. 48: 53–63.

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19 Modeling Plasmonic Optical Properties Using Semiempirical Electronic Structure Calculations Chelsea M. Mueller, Rebecca L.M. Gieseking, and George C. Schatz Northwestern University, Department of Chemistry, 2145 Sheridan Road, Evanston, IL 60208, USA

19.1 Introduction Noble metal nanoparticles have garnered increasing research interest for their plasmonic optical properties [1–3]. A plasmon is a collective oscillation of the conduction band electrons of these nanoparticles and is characterized by intense absorption and local electric field enhancements of several orders of magnitude. Both the frequency of the absorption and the intensity are strongly dependent on the nanoparticle size, shape, material, and dielectric environment [1]. Local field enhancements from these particles have found applications in both sensing and spectroscopy [3–6]. Surface-enhanced Raman spectroscopy (SERS) uses localized surface plasmon excitations to enhance the Raman cross section of a molecule that is adsorbed on a nanostructure. Although SERS is normally performed using nanoparticles that are 20 nm or larger in size, adsorbed molecules on smaller silver clusters show comparable enhancement to those on much larger particles, due to approximate invariance of the near-field enhancement with nanoparticle size in the quasistatic (particles much smaller than the wavelength of incident light) approximation. This allows the small clusters to serve as quality model systems for exploring the physical phenomena involved in these enhancements, theoretically using electronic structure theory and experimentally [7, 8]. The processes involved in the SERS enhancements are not thoroughly understood. It is believed that electromagnetic near-field effects dominate the interaction, with contributions from chemical interactions also playing a role. If the chemical interactions can be neglected, then a SERS spectrum can be modeled using time-dependent density functional theory (TD-DFT) or configuration interaction (CI) singles to determine excited states of the molecule and classical electrodynamics for the nanoparticle. From this, the frequency-dependent polarizability derivative and the electromagnetic near field of the nanoparticle are combined to determine the Raman spectrum. As an alternative, classical electrodynamics for the nanoparticle captures the SERS enhancement without explicitly treating each atom in the plasmonic particle quantum mechanically [9]. This approach is useful as a first approximation, as it incorporates the effects Molecular Spectroscopy: A Quantum Chemistry Approach, First Edition. Edited by Yukihiro Ozaki, Marek Janusz Wójcik, and Jürgen Popp. © 2019 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2019 by Wiley-VCH Verlag GmbH & Co. KGaA.

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of molecular orientation on SERS intensity, as well as the frequency dependence of the enhancement. The neglect of chemical interactions between the cluster and the molecule in this model is a serious consideration, so there is incentive for treating the molecule–cluster system with a quantum mechanical method such as TD-DFT, as in this case the near-field enhancement arises implicitly together with the chemical enhancement mechanism. Density functional theory (DFT) methods, however, can be prohibitively expensive for systems larger than a few hundred metal atoms, which makes them unattractive for application to more complex systems, such as biomacromolecules or tip-enhanced Raman scattering (TERS). TD-DFT results are also subject to self-interaction errors that are especially detrimental to modeling the charge-transfer states that play an important role in the chemical mechanism (CM) of enhancement [10]. By studying small metal clusters and their interactions with molecules using a semiempirical electronic structure method, we mitigate both the computational cost of modeling these systems and the loss of chemical interaction information as we explicitly treat all the valence electrons in the cluster and adsorbed molecule quantum mechanically but with an approximate (but self-interaction-free) Hamiltonian that avoids most of the computation of two-electron integrals and assumes a minimal electronic basis set. The reduction in computational cost also allows exploration of typically neglected contributions to optical processes, such as excited states with quadrupolar or doubly excited character, and their effects on the observed spectra. Here we will focus on the use of the semiempirical intermediate neglect of differential overlap Hamiltonian together with a configuration interaction (INDO/CI) [11] treatment of optical excited states to study the optical properties of bare silver nanoclusters and silver nanoclusters with adsorbed pyridine molecules. INDO/CI has recently gained a lot of attention in studies of plasmonic properties [12–15], and this chapter will highlight that work and describe the new physical insights that have been gained through the use of this method. The INDO method is one of a family of semiempirical methods in which only the valence electrons are explicitly treated, and some or all of the multicenter one- and two-electron integrals are neglected. For INDO, only the one-electron one- and two-center integrals, the two-electron one-center integrals, and the two-electron two-center Coulomb integrals are considered [16]. All three-center two-electron and four-center one- and two-electron integrals are neglected by the zero differential overlap approximation. To account for the neglected integrals, the remaining terms are parameterized to match experimental data or higher-level computations. First, we will establish a baseline comparison between INDO/CI and TD-DFT by studying small silver clusters such as tetrahedral Ag20 . Then, we discuss contributions from quadrupolar states and doubly excited electron configurations for these clusters, as well as charge-transfer processes between a nanocluster and an adsorbed molecule. Finally, we combine these to examine the contributing mechanisms to SERS for pyridine interacting with Ag20 . For each calculation unless otherwise noted, geometric structures were optimized using DFT, with the BP86 [17] functional and either a double-zeta (DZ) or a triple-zeta with polarization (TZP) Slater type basis set using the

19.2 INDO/CI vs. TD-DFT: Absorption Spectra of Ag Nanoclusters

zeroth-order regular approximation (ZORA) [18] to account for relativistic effects. Two methods for calculating excited states are considered: TD-DFT and INDO/SCI. The TD-DFT approach uses the SAOP functional [19] and the DZ basis set as implemented in the 2014 Amsterdam Density Functional (ADF) program [20–22]. For the INDO/SCI calculations, the one-electron two-center resonance integrals 𝛽 sp = − 3 eV (for s and p orbitals) and 𝛽 d = − 40 eV (for d orbitals) were used for Ag atoms unless otherwise noted. The CI matrix was constructed by considering all possible single excitations and, in the single and double excitation configuration interaction (SDCI) calculations, all possible double excitations within the 25 lowest unoccupied and 25 highest occupied orbitals. The 2000–7000 lowest energy excitations were included in the CI matrix, which was then diagonalized to obtain the 500–2000 lowest excited states. These calculations were performed using a locally modified version of Mopac 7.1 [23] and the INDO/CI code written by Jeffrey Reimers [16]. The modifications expand the active space used to generate single excitations as well as enlarging the CI matrix and improving the ground state SCF convergence for larger clusters.

19.2 INDO/CI vs. TD-DFT: Absorption Spectra of Ag Nanoclusters The INDO Hamiltonian has rarely been used for metals, so it is important to first ensure that the parameter set yields reasonable absorption spectra for Ag clusters [12]. Many of these parameters, developed by Anderson et al. [11], are considered fixed, but the one-electron two-center integrals can be tuned to reproduce the experiments of interest. To tune these integrals for a field-free nanocluster, we use single excitation configuration interaction (SCI) to calculate the absorption spectrum of the prototypical Ag20 cluster and compare against the SAOP/DZ results. We will also show how other INDO parameters, corresponding to the s, p, and d orbital energies, can be adjusted to simulate the effects of an applied static potential as well. Figure 19.1 shows a comparison of the TD-DFT and INDO/SCI results for the tetrahedral Ag20 cluster. This shows that the plasmon-like excited state involving many sp → sp transitions is highly sensitive to changing the 𝛽 sp in INDO/SCI. For a fixed 𝛽 d = − 40 eV, as |𝛽 sp | increases, the plasmon-like absorption peak shifts to higher energy. The small shoulder to the left of this peak remains, as seen in Figure 19.1a, but the higher energy features change substantially, with a second large absorption peak also shifting to higher energies with increasing |𝛽 sp | and smaller shoulders disappearing. Similar treatment with |𝛽 sp | = − 3 eV and 𝛽 d varied from −10 to −50 eV in Figure 19.1b, shows less dramatic changes in the absorption spectrum. An increase in |𝛽 d | also shifts the primary absorption peak to higher energies, but the shape of this feature is largely unchanged. Although the combination of |𝛽 sp | = − 3 eV and |𝛽 d | = − 30 eV provides the best energetic agreement with the SAOP/DZ results, the plasmon-like absorption peak is somewhat broadened compared to TD-DFT. Increasing 𝛽 d not only provides a narrower absorption but also increases the relative intensity of secondary

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19 Modeling Plasmonic Optical Properties

Absorption (a.u.)

25

βsp = –5, βd = –40 βsp = –4, βd = –40 βsp = –3, βd = –40 βsp = –2, βd = –40 SAOP/DZ

20 15 10 5 0 0

1

2

(a) 25

Absorption (a.u.)

578

3 4 Energy (eV)

5

6

βsp = –3, βd = –10 βsp = –3, βd = –20 βsp = –3, βd = –30 βsp = –3, βd = –40 βsp = –3, βd = –50 SAOP/DZ

20 15 10 5 0 0

(b)

1

2

3 Energy (eV)

4

5

6

Figure 19.1 INDO/SCI absorption spectra of the tetrahedral Ag20 cluster as a function of (a) 𝛽 sp and (b) 𝛽 d . Source: Gieseking et al. 2016 [12]. Reproduced with permission of American Chemical Society.

absorption peaks around 5 eV. Selecting 𝛽 d = − 40 eV balances these two effects and provides the best overall agreement with the TD-DFT results. In this way we have determined that the parameters 𝛽 sp = − 3 eV and 𝛽 d = − 40 eV are best at reproducing the results from an analogous calculation done at the SAOP/DZ level. Comparison of these results with other Ag nanocluster geometries shows that this parameter set works well for a variety of shapes and sizes, not just small tetrahedra. Although plasmon-like excited states have previously been identified based solely on the contributions of many single excitations to one excited state, the INDO/SCI results reveal that the plasmon-like character is also associated with collectivity among these excitations in their contributions to the transition dipole moment. An additive linear combination of these single excitations leads to a large transition dipole moment and greater absorption. A non-plasmonic state could have an equally large number of contributing single excitations but with opposite signs of their contributions to the transition dipole moment, and cancelation among these leads to a small transition dipole moment and weak absorption. To better define this additive nature, we can construct a maximum transition dipole by adding the vector magnitude of each single excitation contribution. For a plasmon-like excited state, the magnitude of the transition dipole will be near

19.3 Higher-Order Excitations: The Role of Double Excitations in Absorption

to this theoretical maximum, while for a non-plasmonic excited state, this value will be much smaller. With properly tuned parameters, INDO/SCI reproduces the major features of absorption spectra calculated with DFT. The primary absorption peaks are within a few tenths of an electron volt of their DFT counterparts, though some geometries have small shoulders not present in the DFT results. These results suggest that although INDO/CI captures the main features, the disagreement in the exact absorption energies between the methods is comparable with the disagreement of DFT spectra for different density functional choices.

19.3 Higher-Order Excitations: The Role of Double Excitations in Absorption Application of the INDO Hamiltonian to Ag nanoclusters allows us to gain a better understanding of the chemical processes involved in the plasmon-like excitation [12]. While these phenomena have largely been considered solely with single excitation calculations [24], the plasmon-like collective oscillations can also have contributions from double and higher-order excitations. Because only configuration pairs differing by one orbital contribute to the transition dipole moment, the single excitations have been assumed to dominate the absorption spectrum. Calculations including the doubly excited configurations are needed to validate this assumption, and the low cost of INDO/CI makes it favorable for extension to include higher-order excitations. To explore the effects of the inclusion of doubles, we focus on small Ag2+ 10 and Ag20 tetrahedral clusters and cuboctahedral and icosahedral Ag13 clusters with several choices of charge states. Determination of which configurations are included in the CI matrix is based only on the excitation energies. Doubly excited configuration participation in the ground state depends strongly on cluster size, shape, and charge, with double participation increasing with increasing negative charge. Comparison of equivalent icosahedral and cuboctahedral Ag5+ 13 structures shows similar double contributions, while for the same clusters with a −5 charge, the cuboctahedral cluster has a much greater contribution of double excitations to the ground state. Mixing with double excitations also stabilizes the ground state, with stabilization from 0.3 eV for the most positively charged clusters to 1.5 eV for the most negatively charged. Inclusion of double excitations has only small effects on the absorption spectra for positively charged and neutral clusters (Figure 19.2). Stabilization of the ground state shifts the absorption peaks to slightly higher energies due to ground state stabilization and reduces the oscillator strength, improving the agreement with the SAOP/DZ calculations. For the negatively charged clusters, however, the contribution of the double excitations to the ground state is greater, and so their effects on the absorption spectra are large. The ground state stabilization leads to a shift of the main absorption peak to higher energy, significantly improving the agreement with the SAOP/DZ results. Even in the negatively charged clusters, the main absorbing state is composed predominantly of single excitations.

579

18 16 14 12 10 8 6 4 2 0

Ag102+

Absorption (a.u.)

Absorption (a.u.)

19 Modeling Plasmonic Optical Properties

Ag20

0

1

2

3

4

5

16 14 12 10 8 6 4 2 0

Icosahedral

0

6

12

Ag135+

Cuboctahedral

1

2

3

4

5

6

Energy (eV)

Energy (eV) 25

Ag13–

Cuboctahedral

10

Absorption (a.u.)

Absorption (a.u.)

580

8 6 4 2

Ag135–

Cuboctahedral

20

Icosahedral

15 10 5 0

0 0

1

2

3 Energy (eV)

4

5

6

0

1

2

3

4

5

6

Energy (eV)

Figure 19.2 Absorption spectra of several Ag clusters. Solid lines indicate INDO/SDCI, dashed lines indicate INDO/SCI, and dotted lines indicate SAOP. Source: Gieseking et al. 2016 [12]. Reproduced with permission of American Chemical Society.

These results indicate that the exclusion of doubly excited configurations is generally appropriate for modeling the optical properties of plasmonic nanoclusters, as the contribution of the higher-order states primarily stabilizes the ground state. For negatively charged clusters, the inclusion of higher-order states is more important in INDO/CI calculations, as it appears to help capture some ground state correlation that is implicit in common DFT functionals. For these clusters, inclusion of the double excitations significantly improves agreement between INDO/CI and TD-DFT results.

19.4 Identification of Quadrupolar Plasmonic Excited States While the majority of research interest in plasmons and plasmon-like excitations focuses on the interaction of the dipolar plasmon resonances responding in phase with the electric field of the incident light [13], higher-order plasmon resonances such as quadrupoles are also known to occur [25]. These quadrupolar resonances are typically optically dark but can be excited by near-field radiation in small clusters [25–29] and by far-field radiation in large clusters where the retardation of light is significant [1, 30–32]. There are applications of these quadrupolar resonances in spectroscopy as they can provide greater enhancement of chemical processes, such as Raman scattering, because the excitations are both higher in energy and narrower than dipolar resonances [5, 33, 34]. In the case of small nanorods, excitations along both the short (transverse) and long (longitudinal) axis can be observed. The absorption spectra of these Ag nanorods are strongly dependent on the length and width of the nanorod. For

19.4 Identification of Quadrupolar Plasmonic Excited States

this analysis, we focus on cationic pentagonal Ag nanorods, as the absorption for these clusters is well defined and narrow, making identification of the plasmonic states easier. We will also focus on longitudinal quadrupolar modes, as these have been observed experimentally in larger Ag nanorods [35]. The longitudinal transition moment Qzz is calculated in terms of the atomic transition densities qke for each atom k in each excited state e. Values for qke are computed as ∑∑∑√ qke = 2cjb,e (aij aib ) j

b

i

where cj is the CI coefficient for an excitation from an occupied orbital i to an unoccupied orbital b in excited state e, aij and aib are the coefficients of √ atomic orbital i on atom k in molecular orbitals (MOs) j and b, and 2 is a quantum mechanical prefactor originating from wavefunction asymmetry. The transition quadrupole moment is then calculated using the general formula for quadrupoles: ∑ −r 2 ‖𝛿 ) q (3r r − ‖→ Q = ij

k,e

ik jk

k

ij

k

where i and j are the Cartesian coordinates and the position of atom k in Carte− sian coordinates is → rk = [rxk , ryk , rzk ]. The quadrupole moment is dependent on the choice of origin for systems with a nonzero dipole moment, so only excited states with negligibly small transition dipole moments are considered. Analysis of the transition quadrupole moments reveals that the Ag nanorods have few states with a large Qzz as shown in Figure 19.3. The transition densities for these states show the expected quadrupolar nature, with density of the same sign at both ends and density of the opposite sign near the center. The number of quadrupolar plasmon-like excited states decreases with increasing nanorod length, with several similar energy states in the shortest nanorods, while the larger are dominated by a single quadrupolar state; the maximum Qzz increases from 60 D Å to more than 450 D Å. The energy of the main quadrupolar state also decreases with increasing nanorod length and is consistently higher in energy than the longitudinal dipolar plasmon-like excited state. Unlike the dipolar states, the quadrupolar states only show collective character for longer nanorods. In clusters of 31 atoms or fewer, a single excitation can contribute up to three-quarters of the transition quadrupole moment, while for the corresponding transition dipole moment, no excitation contributed more than 30%. This suggests that the quadrupolar states are primarily single excitations and not plasmonic for these clusters. For the longer clusters, however, greater collectivity is observed. In Ag+37 , for example, the two states with the greatest Qzz have maximum contributions from any single excitation of only 33% (3.45 eV) and 42% (3.70 eV). For even longer nanorods, no excitation contributes more than 25% to the main quadrupolar state. In these clusters, we can conclude that the quadrupolar states are sufficiently collective to be considered plasmon-like excited states and that a larger cluster is needed to support a plasmon-like quadrupolar excited state excitation than is needed for plasmon-like dipolar excited states. The earlier justification for including only single excitations for dipolar plasmon-like excited states holds for calculation of higher-order multipoles,

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19 Modeling Plasmonic Optical Properties

Ag19+

70 50

Qzz (D Å)

Qzz (D Å)

60 40 30 20 10 0 0

1

2

3 4 Energy (eV)

5

0

2

3 4 Energy (eV)

5

6

Ag37+

250 Qzz (D Å)

Qzz (D Å)

1

300

200 150 100 50 0

Ag25+

160 140 120 100 80 60 40 20 0

6

Ag31+

250

200 150 100 50 0

0

1

2

4 3 Energy (eV)

5

0

6

Ag43+

350 350 250

Qzz (D Å)

Qzz (D Å)

200 150 100 50 0 0

1

2

3 4 Energy (eV)

Qzz (D Å)

582

500 450 400 350 300 250 200 150 100 50 0

5

6

1

2

3 4 Energy (eV)

5

6

Ag49+

400 350 300 250 200 150 100 50 0 0

1

2

3 4 Energy (eV)

5

6

Ag55+

0

1

2

3 4 Energy (eV)

5

6

Figure 19.3 Transition quadrupole moments Qzz (magnitudes) along the longitudinal axis (z) and transition densities of the states with the largest Qzz for pentagonal Ag nanorods. Source: Gieseking et al. 2016 [13]. Reproduced with permission of American Chemical Society.

including quadrupoles. Like the dipolar states, the large transition density is a key feature of the quadrupole state, and the single excitations should likewise be able to capture the dominant contributions to the transition density. We can, therefore, safely model quadrupolar plasmons using only single excitations.

19.5 Electrochemical Charge Transfer

19.5 Electrochemical Charge Transfer The electrochemical applications of plasmons in chemistry [14] are of increasing interest experimentally, and good quantum mechanical models are needed to understand electrochemical effects on the interactions between plasmons [36–43]. One of the underlying processes of these interactions is charge transfer. INDO has been shown to more accurately calculate the energy of charge-transfer states than typical DFT functionals, which tend to underestimate the charge-transfer energies due to self-interaction error [44, 45]. The ability to tune the values of the orbital energy parameters allows us to model the effect of potential on adsorption properties through the orbital energy shift approximation (OESA) [36, 46], wherein the applied potential is approximated as a fixed shift to the metal atomic orbital energies. For an applied potential of −1 V, there is a 1 eV energetic shift in the INDO orbital energy parameters. To capture the effects of a plasmonic material on an adsorbed molecule, we will examine first the cases of a tetrahedral Ag20 cluster with a pyridine molecule adsorbed on either a vertex or a face of the tetrahedron in the absence of an applied potential. We refer to these as the Vertex and Surface geometries. In these complexes, at the INDO/SCI level, the ground state MOs are primarily localized on either the Ag20 or the pyridine moiety; however, there is some wavefunction overlap between the two, and therefore some mixing occurs. The frontier MOs are all localized on the Ag20 moiety. The silver MOs are consistent with previously calculated orbitals for the bare Ag20 cluster, and many excited states involve local excitations on the Ag20 moiety. The inclusion of pyridine introduces new excited states with charge-transfer character, as shown in Figure 19.4. The first states with significant charge-transfer character occur at 3.60 and 3.71 eV for the Surface and Vertex complexes, respectively, which is 0.2–0.3 eV greater than the 10

6

> 80% Ag20 → Pyr CT

5

> 80% Pyr → Ag20 CT

20–80% Ag20 → Pyr CT

20–80% pyridine

50

< 20% pyridine

0 LUMO

–5 HOMO

–10

Excited state energy (eV)

Molecular orbital energy (eV)

> 80% pyridine

< 20% CT

4 3 2 1

–15

0

–20

(a)

20–80% Pyr → Ag20 CT

Surface

Vertex

(b)

Surface

Vertex

Figure 19.4 (a) Molecular orbital energies and (b) excited state energies for optimized geometries of the Ag20 –pyridine complex at the INDO/SCI level. Source: Gieseking et al. 2017 [14]. Reproduced with permission of Royal Society of Chemistry.

583

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19 Modeling Plasmonic Optical Properties

plasmon-like absorption at 3.4 eV. TD-DFT calculations at the BP86/TZP level predict charge-transfer states at 2.16 and 1.44 eV, highlighting the importance of self-interaction error and consequent underestimation of charge-transfer energies in the DFT results. When considering the effects of an applied potential, we are particularly interested in the range of potentials where charge-localized [Ag20 − pyridine]and [Ag+20 − pyridine− ] charge-transfer states are close in energy. Within this range of potentials, the most stable charge-localized state is the first singlet state, and the first triplet state is the most stable charge-transfer state. The formal potential, E∘ ′ , is here defined as the shift in the Ag orbital energies required to make the first charge-localized singlet state and the first charge-transfer triplet states isoenergetic. To examine the evolution of the charge-transfer and charge-localized states, we first scan the potential within the range of ±0.5 V relative to the formal potential, E∘ ′ , for the Surface complex. Across this range, the singlet–triplet energy gap has a nearly linear variation with potential. The transition from the first singlet state to the first triplet state involves nearly a full electron being transferred from the sp band of Ag20 to the first π* orbital on pyridine. As we have not referenced these potentials to a standard electrode, changes in E∘ ′ are more significant than the exact values. For our model, E∘ ′ occurs at −4.57 V in the Surface geometry and −5.11 V in the Vertex geometry. The difference between these formal potentials, 0.44 V, is more than four times the energetic difference of the first charge-transfer excited states in the absence of an applied potential. Since pyridine is 3.2 Å farther from the geometric center of the Ag20 cluster in the Vertex geometry than in the Surface geometry, a larger energy is required to overcome the electrostatic attraction and induce charge transfer in the vertex geometry. At zero applied potential, the substantial mixing between charge-transfer and local excitations dilutes this effect. The applied potential significantly changes the electronic structure of the singlet and triplet states, which is in contrast with what has been assumed in previous computational studies [47] that suggested minimal changes in adsorption properties with changes in potential.

19.6 Voltage Effects and the Chemical Mechanism of Surface-Enhanced Raman Scattering Up to this point we have discussed the application of the semiempirical INDO/CI methods to the investigation of individual phenomena that result from the plasmon-like behavior of small nanoclusters [15]. These nanoclusters, however, also serve as effective models for the behavior of their larger nanoparticle counterpart systems. In this context, it is the phenomena resulting from the interaction of plasmons and molecules that are of the most experimental interest. One such interaction effect is SERS, wherein a local surface plasmon interacts with an adsorbed molecule to enhance the observed Raman scattering intensities by as much as 1010 , which is sufficient for the study of single molecules

19.6 Voltage Effects and the Chemical Mechanism of Surface-Enhanced Raman Scattering

[48]. SERS and its counterpart, TERS, are desirable experimental methods for obtaining chemical information and, in the case of TERS, spatially resolved chemical information, about single molecules adsorbed onto a substrate. The techniques have applications to a wide array of sciences, from sensing [49–51] to electrochemistry [52, 53]. The precise mechanism of the plasmonic enhancement is not completely understood, but the primary mechanism is believed to be an electromagnetic mechanism (EM) related to the enhancement of the local electric field of light by the plasmon [3, 48, 54, 55]. Most models use either some arbitrary applied potential or electrodynamics to capture this effect. Chemical interactions between the molecule and the metal are also believed to play a part in the enhancement through two distinct but coupled mechanisms as part of the CM: changes to the ground state molecular structure and the introduction of charge-transfer excited states between the metal and molecule. Resonance of molecular excitations with the incident light also leads to strongly enhanced Raman scattering, but this is considered a separate enhancement, called surface-enhanced resonance Raman scattering or SERRS [48]. The magnitude and mechanism of the chemical enhancements are still controversial [56], with enhancements of 10–102 typically proposed and up to 107 proposed in cases where electromagnetic enhancement is insufficient to explain the observed enhancement [57–59]. SERS is known to be strongly dependent on the applied voltage [58, 59], and while this may be in part due to changes in molecular adsorption geometries and packing density in the electrochemical environment, these changes are too large to wholly attribute to these effects and changes in the electromagnetic enhancement mechanism must also be involved. As previously discussed, the energy of the charge-transfer states is sensitive to applied voltage, and so, in principle, an applied voltage can perturb the charge-transfer mechanism of chemical enhancement. In order to separate the effects of the CM and EM, the excited states are also computed with a modified INDO parameter set, where the overlap between Ag atomic orbitals and all other non-silver atomic orbitals is neglected. This modified parameter set, called INDO-EM, only provides Raman enhancements through the EM. The effects of applied potential were treated as in the charge-transfer calculations, using the OESA method previously described. As with the charge-transfer calculations, we use the model tetrahedral Ag20 cluster with pyridine adsorbed in either the Surface or Vertex position. The INDO parameters are chosen to reproduce spectroscopic properties and perform poorly for geometries and vibrational modes, so the Raman and SERS intensities at the INDO/SCI level were computed using the BP86/TZP geometries, normal coordinates, and vibrational frequencies. Additionally, the values for 𝛽 sp and 𝛽 d are retuned to reflect interactions between silver and other atoms by minimizing the error in the excited state energies for model clusters of a few Ag atoms with simple ligands using EOM-CCSD/def2-TZVP states as references. For these calculations 𝛽 sp = − 2.5 eV and 𝛽 d = − 30 eV provide the best results and are used for the following Raman calculations. The frequency-dependent polarizabilities 𝛼 ij (𝜔) were computed using a sum-over-states (SOS) approach using the INDO/SCI excited

585

586

19 Modeling Plasmonic Optical Properties

states as aij (𝜔) =

∑ ⟨g|𝜇i |e⟩⟨e|𝜇j |g⟩ c

Ege − ℏ𝜔 − iΓge

+

⟨g|𝜇i |e⟩⟨e|𝜇j |g⟩ Ege + ℏ𝜔 + iΓge

where g is the electronic ground state and e is an electronic excited state, 𝜇i represents the dipole moment operator for Cartesian axis i, Ege is the excitation energy from state g to state e, 𝜔 is the frequency of the incident light, ℏ is the reduced Planck’s constant, and Γge is the lifetime broadening of excited state e, set to 0.004 a.u. for consistency with the response function results. The differential polarizabilities were computed by numerical differentiation with respect to displacement of the system by ±0.01 Å along each normal coordinate corresponding a pyridine vibration. Using these polarizabilities, the scattering factor S was computed as ′2

2

S = 45𝛼 p + 7𝛾 ′ p where the prime indicates a derivative with respect to normal coordinate p. The ′ isotropic derivative 𝛼 p is defined by 𝛼

′2 p

=

∑ |1 |2 | (𝛼 ′ )p | | 3 ii | | i |

and the anisotropic derivative 𝛾p′ is given by 1 ′ ′ 2 ′ ′ 2 ′ ′ 2 ′ 2 ′ 2 ′ 2 | + |𝛼yy − 𝛼zz | + |𝛼zz − 𝛼xx | ) + 3(|𝛼xy | + |𝛼yz | + |𝛼zx |) (|𝛼 − 𝛼yy 2 xx The differential cross section for Stokes scattering with a 90∘ scattering angle and perpendicular plane-polarized light is [60] ( ) d𝜎 π2 1 h S = 2 (𝜔 − 𝜔p )4 2 dΩ 𝜀0 8π c𝜔p 45 1 − exp(hc𝜔p ∕kB T) 2

𝛾 ′p =

where 𝜔p is the vibrational frequency of mode p, h is Planck’s constant, k B is Boltzmann’s constant, T is the temperature, c is the speed of light in vacuum, and 𝜀0 is the permittivity of free space. To compare the overall enhancement factors between systems, the integrated Raman enhancement factors EFint were computed as [61] ∑ k IAg pyr tot IAg 20 k 20 pyr EFint = tot = ∑ j Ipyr Ipyr j k where IAg

20 pyr

is the differential Raman cross section for the kth vibrational mode j

of the Ag20 –pyridine complex and Ipyr is the differential Raman cross section for the jth vibrational mode of the isolated pyridine molecule. The summations over j and k include all modes with vibrational frequencies between 500 and 2000 cm−1 . This ensures only modes corresponding to pyridine motions are included for the [Ag20 − pyridine] complex and also neglects the high-frequency modes that are not usually measured experimentally.

19.6 Voltage Effects and the Chemical Mechanism of Surface-Enhanced Raman Scattering

Table 19.1 Enhancement factors and frequency shifts of selected Raman modes for the Ag20 –pyridine complex at photon energies of 2.00 eV (NRS) and on resonance with the plasmon (SERS). EF(NRS) −1

Mode (cm )

−1

Shift (cm )

BP86/TZP

EF(SERS)

INDO/SCI

BP86/TZP

INDO/SCI

Surface 595.3

14.6

7.9

12.8 ± 2.5

42 274.0

5298 ± 55

649.9

−0.6

0.6

1.4 ± 0.1

57.6

987.7 ± 1.5

976.8

13.1

8.8

3.6 ± 0.6

6 159.2

978.6 ± 3.8

1023.4

0.5

2.8

1.6 ± 0.5

3 904.5

448.9 ± 2.6

1205.0

−1.0

3.5

8.4 ± 2.8

3 461.0

1955 ± 22

1570.9

11.3

11.2

7.6 ± 2.9

2 509.1

2288 ± 21

EFint

—

8.0

3.9 ± 0.3

6 528.4

932.0 ± 2.5

Vertex 595.3

21.5

24.9

2.2 ± 0.7

11 887.0

1162 ± 18

649.9

−0.9

0.5

1.6 ± 0.1

327.1

24.1 ± 0.2

976.8

18.2

5.9

3.4 ± 0.6

3 002.0

62.9 ± 1.1

1023.4

3.7

22.1

1.9 ± 0.5

1 629.3

39.0 ± 1.0

1205.0

0.4

14.3

14.0 ± 3.5

12 094.2

1594 ± 27

1570.9

17.0

66.2

4.3 ± 1.8

3 309.5

106.2 ± 1.9

EFint

—

18.3

2.7 ± 0.2

3 843.2

124.3 ± 0.9

Source: Gieseking et al. 2017 [15]. Reproduced with permission of Royal Society of Chemistry.

We focus first on a comparison of off-resonance normal Raman scattering (NRS), calculated at 2.00 eV, and resonance Raman scattering, which is analogous to SERS, calculated at 3.30 eV for the Surface-adsorbed geometry and 3.44 eV for the Vertex geometry to be on resonance with the plasmon-like excited states. The results of these analyses for both the INDO/CI and BP86/TZP calculations are collected in Table 19.1. The INDO/SCI spectra show weaker enhancement than the analogous BP86/TZP calculation, which can be attributed to the higher energy charge-transfer states in the INDO/SCI calculation reducing the contributions from the charge-transfer mechanism to the overall enhancement. While these states are near resonance with the incident light at the BP86/TZP level, they are more than 1 eV higher in energy in the INDO/SCI calculation. The strongest Raman active modes in the isolated pyridine are generally the most strongly enhanced modes. The SERS enhancements are, as expected, much larger than the NRS enhancements. Enhancements for the Surface complex are also greater than those of the Vertex complex, highlighting the sensitivity of the enhancement to geometric displacements. The mode at 1205 cm−1 is particularly sensitive at both levels of theory and is more intense for the Vertex complex than the Surface in both cases. This is in good agreement with previous experiments showing this mode to be sensitive to changes in the chemical environment [62].

587

19 Modeling Plasmonic Optical Properties

Ag20–pyridine (Surface)

0.4 0.3 0.2 0.1 0.0

600

(a)

800 1000 1200 1400 1600 (b) Wavenumber (cm–1)

2000

3.30 eV

1500 1000 500 0

600

(c)

Ag20–pyridine (Vertex)

0.5 dσ/dΩ (10–30 cm2 /sr)

2.00 eV

2.00 eV

0.4 INDO/SCI INDO–EM/SCI

0.3 0.2 0.1 0.0

600

800 1000 1200 1400 1600 Wavenumber (cm–1)

500 dσ/dΩ (10–30 cm2 /sr)

dσ/dΩ (10–30 cm2 /sr)

0.5

dσ/dΩ (10–30 cm2 /sr)

588

3.44 eV

400 300 200 100

800 1000 1200 1400 1600 (d) Wavenumber (cm–1)

0

600

800 1000 1200 1400 1600 Wavenumber (cm–1)

Figure 19.5 (a, b) Normal and (c, d) resonance Raman spectra for the (a, c) Surface and (b, d) Vertex geometries of the Ag20 –pyridine complex at the INDO/SCI level. Solid lines indicate spectra computed using the standard INDO parameters, and dashed lines indicate spectra computed using the INDO-EM parameters, which neglect all overlap between orbitals localized on the Ag20 and pyridine moieties. Source: Gieseking et al. 2017 [15]. Reproduced with permission of Royal Society of Chemistry.

Implementing the INDO-EM parameter set allows us to decompose the enhanced Raman spectra into the intensity due to the electromagnetic enhancement mechanism and the portion attributable to the chemical enhancement mechanism. Spectra comparing the INDO/SCI spectra with INDO-EM results are shown in Figure 19.5 for both the NRS and SERS cases. For both the Surface and Vertex complexes, the electromagnetic enhancement is larger than the chemical enhancement at both NRS and SERS energies, and both contributions to the enhancement are larger at the SERS energy. This result is contrary to previous assumptions that the CM is the only significant contributor to Raman enhancement far from resonance with the plasmon. Table 19.2 details the decomposition of the integrated enhancement factors for both the resonance and off-resonance cases of each complex. Table 19.2 Decomposition of the integrated enhancement factors for the Ag20 –pyridine complex. EFNRS

EFSERS

Complex

EM

CM

Total

EM

CM

Total

Surface

2.2 ± 0.3

1.7 ± 0.4

3.9 ± 0.3

208.9 ± 2.1

4.5 ± 0.1

932.0 ± 2.5

Vertex

3.1 ± 0.4

0.9 ± 0.2

2.7 ± 0.2

24.7 ± 0.5

5.0 ± 0.1

124.3 ± 0.9

Source: Gieseking et al. 2017 [15]. Reproduced with permission of Royal Society of Chemistry.

19.6 Voltage Effects and the Chemical Mechanism of Surface-Enhanced Raman Scattering

Although there is a similar degree of ground state charge transfer between the two complexes, there are significant differences in the chemical enhancement. In the Surface complex, the chemical enhancement is relatively uniform across all the modes; in the Vertex complex, only a few modes, including the one at 1205 cm−1 , are enhanced, and several are actually damped by the CM. Though the enhancement from the CM is small, its contribution to the total enhancement is sufficient for the total enhancement in the Surface complex to be greater than that observed for the Vertex complex. The decomposition also unexpectedly shows that the electromagnetic enhancement is the dominant mechanism for both the resonance and far from resonance cases. In the resonance cases, the EM dominates the total enhancement, and its enhancements are much larger than in the non-resonance case, by a factor of nearly 102 in the Surface complex and 8 in the Vertex complex. The adsorbed pyridine at the vertex splits the plasmon resonance, resulting in a broader distribution of states contributing to the enhancement, and only a portion of these are contributing to the enhancement. The SOS approach makes it easy to consider a range of incident photon energies beyond just the resonance and off-resonance cases. Here we also consider the change in the total enhancement factor as well as the electromagnetic and chemical components as a function of incident photon energies, from 1 to 4 eV (Figure 19.6). For both the Surface and Vertex geometries, the electromagnetic enhancement is greatest near the absorption maxima. As observed above, the electromagnetic enhancement is greater for the Surface geometry than for the Vertex near the plasmon resonance. The chemical enhancements are also larger on average in the Surface complex than the Vertex complex because of the smaller effective distance for charge separation and the greater spatial overlap between the pyridine and Ag moieties, which lead to lowered charge-transfer state energies and more mixing of these states with lower-lying excited states. The difference in the energy dependence in both mechanisms demonstrates the need to study both mechanisms to gain a better understanding of the total enhancement of Raman intensities. Ag20–pyridine (Surface)

EM

CM

1000

Ag20–pyridine (Vertex)

10000

EM Enhancement factor

Enhancement factor

10000

Total 100 10 1 0.1

1000

CM Total

100 10 1 0.1

1

1.5

2

2.5 Energy (eV)

3

3.5

4

1

1.5

2

2.5

3

3.5

4

Energy (eV)

Figure 19.6 Integrated enhancement factors as a function of photon energy. Solid lines correspond to the total enhancement factors, dotted lines to the electromagnetic mechanism (EM) component, and dashed lines to the chemical mechanism (CM) component. Source: Gieseking et al. 2017 [15]. Reproduced with permission of Royal Society of Chemistry.

589

590

19 Modeling Plasmonic Optical Properties

We can also examine the effects of an applied potential on the SERS spectra by utilizing the same OESA that we used to study the charge-transfer states and the geometry dependence of the formal potential. As before, the potentials are referenced to the standard INDO parameters as no benchmark to standard electrode potentials has been completed. We neglect the effects of potential on the vibrational modes and geometries as they are computed at the BP86/TZP level. As demonstrated in the electrochemical studies, a negative applied potential lowers the energy of charge-transfer states. The range of potentials considered lowers the charge-transfer states from several tenths of an electron volt above the plasmon resonance to near the selected off-resonance energy at 2.00 eV. For both the off-resonance and resonance cases, the enhancement is strongly dependent on the applied potential. For the off-resonance case in the Surface complex, the chemical enhancement is less than 10 for 0.0 V applied potential, but at −2.00 V the applied potential is greater than 100. The comparatively higher energy charge-transfer states in the Vertex complex are not lowered sufficiently to see this effect in the range of applied potentials considered. For the resonance cases, the integrated enhancement factors are also potential dependent, with the greatest enhancements at potentials where charge-transfer states have been lowered nearly to resonance with the plasmon-like excited states or are significantly mixed with them. For the Vertex complex, this occurs around −1.0 V, and at −0.6 and −1.6 V for the Surface complex. Although the overall enhancement factor dependence on applied potential can be understood by the shifting of charge-transfer state energies, the potential dependence also varies between vibrational modes as shown in Figure 19.7. In both the resonance and off-resonance cases for the Surface geometry and in the resonance case for the Vertex geometry, the modes at 1205 and 1585 cm−1 are particularly strongly enhanced by near resonance of light with the charge-transfer states. This highlights the importance of considering the chemical enhancement effects, as the resultant enhancements are not uniform across all vibrational modes.

19.7 Conclusions We have described the application of the semiempirical INDO/CI method to the study of plasmon-like excited states in small metal nanoclusters. This model is comparable with TD-DFT methods in capturing the main features of absorption in these structures but allows a much larger number of excited states to be included at greatly reduced computational cost. The effects of double excitations and the quadrupolar plasmon-like excited states are now readily examined, and the limiting cases where excluding these phenomena is no longer advisable are discussed. This method has also been shown to more accurately predict charge-transfer energies than TD-DFT, which has allowed us to further investigate the enhancement mechanism for surface-enhanced Raman scattering. By removing the overlap between Ag atomic orbitals and the atomic orbitals of every other element, we

References

Ag20–pyridine (Surface) dσ/dΩ (10–30 cm2 /sr)

dσ/dΩ (10–30 cm2 /sr)

80

2.00 eV

60 40 20

1.0

0.0 V –0.6 V

2.00 eV

–1.0 V 0.8

–1.6 V –2.0 V

0.6 0.4 0.2 0.0

0 600

(a)

1200 1400 800 1000 Wavenmuber (cm–1)

1600

600

(b)

80 000

1200 800 1000 Wavenmuber (cm–1)

1400

1600

1400

1600

25000 dσ/dΩ (10–30 cm2 /sr)

dσ/dΩ (10–30 cm2 /sr)

Ag20–pyridine (Vertex)

1.2

100

3.30 eV 60 000 40 000 20 000

20 000 3.44 eV 15 000 10 000

0 600

(c)

800

1000 1200 Wavenmuber (cm–1)

1400

5000 0

1600

(d)

600

800 1000 1200 Wavenmuber (cm–1)

Figure 19.7 (a, b) Normal and (c, d) resonance Raman spectra for the (a, c) Surface and (b, d) Vertex geometries of the Ag20 –pyridine complex at the INDO/SCI level, showing the effects of applied potentials from 0.0 to −2.0 V within the OESA model. Source: Gieseking et al. 2017 [15]. Reproduced with permission of Royal Society of Chemistry.

separate the electromagnetic enhancements from chemical enhancements and show that the strong dependence of SERS on applied voltage is at least partly a result of enhancement from the charge-transfer mechanism. We have also shown that the EM dominates the enhancement across a range of incident photon energies, not only near the plasmon resonance. Application of INDO/CI to current research topics has given rise to new insights into the effects of higher-order processes previously too computationally expensive to examine and improved the understanding of the underlying mechanisms of Raman enhancement.

Acknowledgment We acknowledge support from the Air Force Office of Scientific Research MURI (FA9550-14-1-0003).

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Index a ab initio DFT simulations 462 ab initio methods 282, 330, 418, 464 ab initio molecular dynamic (AIMD) approach 173, 191 ab initio simulations 471, 487 absolute configuration (AC) 171–173, 186, 254 absorption spectrum 33, 87, 91, 92, 95, 97, 99, 411, 497, 509–511, 518, 521, 577–579, 580 acetic acid 199, 211, 212, 280, 364, 365, 367 acetylacetone 219, 220, 222, 229–232, 289 enol 231, 232 acoustic phonons 427 acrylic acid 365, 366 adenine crystal 475, 476, 479 adenosine 478, 485, 486 ad hoc broadening factor 93 adiabatic approximation 49, 62, 69, 207 adiabatic Hessian (AH) model 21, 517 adiabatic model 10 adiabatic shift (AS) 10, 517 adsorbate-substrate interaction 541, 549 adsorption interaction 538, 540, 546, 551, 555, 560, 565 adsorption models 546 Ag4 cluster 546, 547 Ag20 cluster 578, 583, 585 Ag clusters 554, 562, 577, 580 Ag Nanoclusters 577–579 Ag nanorods 580–582

Ag20 -pyridine 583, 588 algebraic diagrammatic construction (ADC) 81 α-glycine and l-alanine 476–478 amides dipole–dipole interactions and coupling 132 electron-donating effect 135 Gaussian convolution 130 hydrogen bonding interaction effects 135 Kramers–Kronig transformation 130 LR-PCM-TD-DFT calculations 131 nonequilibrium effect 134 π-π* transition 130 polarization 132 transition energies 135 amine-metal interaction 557 amino-hydroxy (AH) 290 amino-metal interaction 555 amino-oxo (AO) tautomer 290 amino wagging frequencies 551, 552 amino wagging vibration 555, 563, 567 ammonia 460 Amsterdam Density Functional (ADF) 182, 577 angle-resolved photoemission spectroscopy (ARPES) 443 angle-resolved ultraviolet photoelectron spectroscopy (ARUPS) 440 anharmonic couplings 17, 201, 203, 211, 480, 487 anharmonic effects 173, 186, 187, 228, 231, 232, 353, 355, 516

Molecular Spectroscopy: A Quantum Chemistry Approach, First Edition. Edited by Yukihiro Ozaki, Marek Janusz Wójcik, and Jürgen Popp. © 2019 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2019 by Wiley-VCH Verlag GmbH & Co. KGaA.

598

Index

anharmonic frequencies 17, 186, 337 anharmonic IR spectrum 428 anharmonicity 2, 6–8, 14, 16, 17, 147, 154, 160, 163, 182, 186–187, 191, 203–212, 234, 253, 257, 353–355, 377, 378, 380, 381, 390, 399, 404, 414, 415, 435, 524, 568 anharmonic vibrational analysis 365, 375, 378 malic acid 369 aniline (AN) 549 molecular orbitals 550 aniline-Ag4 563, 566 anisotropic contribution 11 anisotropic hyperfine coupling tensor 11 annealed scPLA 485 anthracene crystal 474, 479 anti-Stokes scattering 498, 502 aqueous malic acid, FT-NIR spectrum 370 aromatic amines 539, 540, 546, 549, 554, 557, 567 aspirin crystals 338–342 asymmetric carbon 254 atomic axial tensor (AAT) 173, 174 atomic polar tensor (APT) 174 autocorrelation function (ACF) 94, 202, 206, 232, 327, 329, 330, 336, 337, 428, 440

b basis set superposition error (BSSE) 180, 548 Beer–Lambert’s law 173 bending frequencies 147 benzyl moieties 567 benzyl radical 537, 538, 563 anion 559 anion-Ag4 566 Berry phase approach 428 β-hydroxy-α,β-unsaturated carbonyl compounds 221 β-hydroxyketone enols 231 binding energy (BE) 548, 551, 560 binding interaction 538, 540, 546, 547, 549, 551, 553, 555, 560, 564 bio-macromolecules 576 biospectroscopy 367

1,8-bis(dimethylamino)-4,5-dihydroxynaphthalene 226 bis(trifluoromethanesulfonyl)imide (H-TFSI) 282 Bloch theorem 465 B3LYP-D* functional basis set 481 Bohr frequency 499, 502, 506, 507, 509–511 Bohr magneton 11 Boltzmann coefficients 360 Boltzmann density operator 204, 206 Boltzmann factor 180, 274, 275 Boltzmann population 9, 19 Boltzmann-weighted Raman spectrum 267 bond polarity 310–313, 315 Born–Oppenheimer (BO) approximation 174, 466, 500 Born–Oppenheimer molecular dynamics (BOMD) 329, 330 broad bands 120, 126, 337, 567 butyl alcohols, NIRS 361 butyric acid 365, 366

c C-Ag bond length 558 Cambridge Structure Database (CSD) 464 carboxylic acids 207, 209, 212, 364, 369 Car–Parrinello dynamic simulations 184 Car–Parrinello formulation 225 Car–Parrinello molecular dynamics (CPMD) 173, 226, 329, 331 calculations 334 simulations 232 Cartesian coordinate system 236, 418, 460, 467–469 C–C bond length 558 C60 crystal 463, 472, 473, 479 center of mass (COM) 86, 460 channelrhodopsin 100 charge mobility 425, 436, 438, 439, 442, 443, 449, 450 charge transfer (CT) transitions 83 charge transport properties, OSCs 425, 449 chemical enhancement effect 562–565

Index

chemical interactions 554, 575, 576, 585 chemical mechanism (CM) 576, 585, 589 chemisorption interaction 554 chemometric regression vectors 370 chemometrics 369 chiral recognition, by molecular spectroscopy anharmonic approaches 186 ECD 171 ECD and CPL spectra 176–177 electronic structure methods 179–182 environment-induced CD and CPL activity 184–185 matrix-isolation vibrational circular dichroism spectra (MI-VCD) 183 molecule’s electron density 173 quantum mechanical methods 172 VCD spectra 173–175 chlorobenzaldehyde 285–286 CH3 OH 261, 374, 375 CH-stretching oscillators 404, 405, 414, 415 CH2 wagging (ωCH2 ) 538 circular dichroism (CD) 3, 171, 172, 176, 177, 183, 269–275 circularly polarized luminescence (CPL) 3, 171, 177 circularly polarized phosphorescence 177–179 clamped nucleus (CN) approximation 508 classical electrodynamics 575 classical harmonic oscillator 439 cluster-in-a-liquid model 183, 184 complete active space perturbation theory 517 complete active space SCF (CASSCF) 84, 517 computational procedure 153 condensed-phase optimized molecular potentials for atomistic simulation studies (COMPASS) 428 configuration interaction (CI) matrix 577

conformational analysis chlorobenzaldehyde 285–286 1,2-dichloroethane 283–285 vanillin 286–289 conformational diversity electronic circular dichroism 272–275 vibrational spectrum, solvation in 265–269 conventional approach 107, 389 conventional nonpolarizable QM/MM approaches 98 conventional vibrational calculations 147 core-valence (CV) correlation effects 12 Coulomb force 460 Coulombic dipole interaction 86 Coulomb integrals 576 coupled cluster singles and doubles (CCSD) method 82 coupled cluster theory like CCSD(T) 282 crotonic acid 365, 366 crystalline inorganic semiconductors 425 crystalline polymers 459 crystalline TCNQ 430, 432 crystalline tropolone 343–344 crystal packing 426, 429–431 cyclohexanol 360, 363, 377, 378

d Darling–Dennison resonances (DDRs) 8 Davydov coupling 207, 208, 211, 212 denoted degeneracy-corrected PT2 (DCPT2) 8 density functional perturbation theory (DFPT) 428 density functional theory (DFT) 48, 82, 179, 329, 358, 459, 508, 516, 537, 538, 576 calculation 253 procedures 227 density matrix renormalization group (DMRG) method 517 deperturbed VPT2 (DVPT2) 8, 354, 358

599

600

Index

DFT-D* approach 464, 481 1,8-diaminonaphthalene 291–293 di-atomic anharmonicity 354 dibenzoylmethane enol 229, 230, 232 dicarboxylic acid 369 dicyanobenzenes 432, 433 differential Raman scattering cross section (DRSC) 548, 550 1,2-dihydrobenz[cdi]indazole (DBI) 292 1,8-dihydro-1,8-naphthalenediimine (3 DND) 292 diluted methanol band assignments 357 NIRS 356 dimethyloxirane 21, 22 dioleoylphosphatidylcholine (DOPC) 158 dipalmitoylphosphatidylcholine (DPPC) 158 dipolar plasmon resonances 580 dipole–dipole coupling terms 11 dipole moment function 411 displaced harmonic oscillator model 498 diverse nonlinear spectroscopic techniques 102 donor highest occupied molecular orbital (dHOMO) 86 double harmonic approximation 29, 275, 276, 428, 501–503, 508, 516 double-hybrid functionals 13, 358 Duschinsky effects 517–519, 521 Duschinsky matrix 10, 513, 515 Duschinsky rotation effects 505, 511 Dushinsky transformation 9 dynamic disorder 93, 94, 105, 436, 437

e Eckart conditions 411, 465 Eckart transformation 542 effective unpaired electron (EUPE) 56 electric-dipole transition moment (EDTM) 174 electrochemical charge transfer 583–584 electromagnetic mechanism (EM) 585, 589 electron donating (ED) groups 549

substituted aniline 557 electron dynamics, in molecular aggregate system electron density matrix 48–49 electron dynamics calculation 45–46 energy transfer phenomena 44 Fock matrix 45 group diabatic Fock scheme Fock matrix 49–50 local excitation and electron filling 51–52 observable, Fock and density matrices 50 time propagation of density matrix, in GD representation 51 hole–electron pair 44 inter/intra molecular charge migration 44 light–electron couplings 44 20-mer ethylene system 52, 56 multi configurational time-dependent Hartree–Fock (MCTDHF) schemes 46 multi configuration time dependent Hartree (MCTDH) method 47 NPTL-TCNE dimer system 53 solar-light energy conversion 44 time-dependent configuration interaction theory (TDCI) 46 time-dependent interaction dynamics 45 electronic circular dichroism (ECD) 3, 171, 272–275, 515 electronic Hamiltonian 80, 434 electronic quadrupole moment 177 electronic structure methods 61, 71, 179–182 electronic theory 360, 365, 380 electron paramagnetic resonance (EPR) spectroscopy 10 electron-phonon coupling 48, 426, 434–444, 449 on charge tranport 437 local and non-local 434–436 molecular size 441 in organic semiconductors 439–440 in various organic semiconductors 440–443

Index

electron-phonon interaction 425, 426, 434, 436, 438, 440, 444, 446, 448, 449 electron withdrawing (EW) groups 549 substituted aniline 557 environmental coupling 94 equation-of-motion coupled cluster (EOM-CC) approach 81 equation of motion coupled cluster method restricted to single and double excitations (EOM-CCSD) 517 ethanol 275, 306, 355, 357, 358, 370 ethylene-Ag4 559 ethyl tert-butyl ether 370 exchange-correlation (XC) functional 54, 83, 179, 464, 518 excited state energies 62, 80, 81, 86, 94, 98, 103–104, 517, 583, 585 excited state intramolecular proton transfer (ESIPT) 224 explicit solvent cavity model 369

f far-infrared spectroscopy 460 far-ultraviolet (FUV) spectroscopy amides 130–136 ATR-FUV spectroscopy 119, 123–124 DFT/TD-DFT calculations 122 electronic transition 119, 120, 124–141 hydrogen bonding 137–142, 355 hydrogen bonding and coupling 122 linear response scheme (LR-PCM) 122 Monkhorst–Pack mesh 123 n-alkanes in liquid and solid phases 124–129 nylons 120, 136–141 π-π* transition 120 polarizable continuum model (PCM) 121 quantum chemical calculations 119 Rydberg transitions 120, 121, 125, 126, 139 SAC-CI 120–123, 125, 126, 133, 134

π–π* transition 138–141 Vienna abinitio simulation package (VASP) 123 Fermi-resonance coupling constant 396 Fermi resonances (FRs) 7, 186, 201, 203, 211, 396, 401 field-effect transistors 425 first order 5, 9, 204, 394, 396, 397, 401, 428, 499 Fn -TCNQ crystals 426, 427, 443 charge transport 444 low-frequency vibrations 443–449 F2n -TCNQ crystals, Raman spectroscopy 430 Fock matrix 45, 49, 50, 87 force constant 13, 17, 19, 31, 226, 227, 256, 354, 391, 515, 542, 544–546, 551, 553, 567 formic acid 211–212, 295, 364 Fourier transform 173, 206, 332, 337 Franck–Condon (FC) approximation 9 contributions 503 overlap integrals 497, 515 principle 8, 435 structure 96 vibronic couplings 497, 518–521 Franck–Condon and Herzberg–Teller (FCHT) 21, 497 Frenkel exciton Hamiltonian parameters 104 F2 -TCNQ crystals 429, 430, 444 full-width at half maximum (FWHM) 474, 475, 477, 478, 509

g gas-phase 2-(N-ethyl-α-iminoethyl)-4chloro-5-methylphenol 234 gauge invariant atomic orbital (GIAO) method 177, 234 Gaussian 229 Gaussian distributions 105 Gaussian’s 6-311G(d,p) basis set 481 generalized (GVPT2) 354, 358 GENeralized-Ensemble SImulation System (GENESIS) 148 generalized internal coordinates (GICs) 10

601

602

Index

Gibbs free energies 547 glycine crystal 477, 479 Gram–Schmidt process 81 ground state electronic-adiabatic polarizability 81, 501–503, 508 ground state electronic polarizability 502, 503, 508 ground state equilibrium geometry 502, 504, 517 G16 software 23

h half-width at half-maximum (HWHM) 483 harmonically coupled anharmonic oscillator (HCAO) model 394 harmonic approximation 17, 29, 163, 174, 176, 226–228, 505, 541, 543, 548 harmonic force field (HFF) 12, 21, 30, 174 harmonic oscillator approximation (HOA) 8, 253, 281, 393, 395 harmonic oscillators 208, 389, 502, 504, 512 harmonic potential energy function 543 harmonic vibrational frequencies 354, 516, 537 Hartree–Fock (HF) analytical approach 464 approach 180 calculations 227 exchange 83 methods 179, 358 theory 80, 254 Hartree-Fock/Kohn-Sham method 428 Hermite polynomials 512 Herzberg–Teller (HT) contributions 9, 21, 503 Herzberg–Teller vibronic couplings 497, 523 Hessian matrix 314, 429, 466, 467, 515, 517 hexanoic acid 367, 368 higher order excitations 579–580 higher-order plasmon resonances 580

higher overtone 378, 379, 411, 412 highest occupied molecular orbital (HOMO) 52, 86, 88, 343, 344, 549, 566 high-mobility OSCs 426, 443, 449 high-temperature (HT) 442, 443 4H-imidazole ligand 521, 523 Holstein–Peierls Hamiltonian 435 Holstein-type electron-phonon coupling 435 hybridization effect 537, 540 hybrid molecular dynamics (QM/MM) 329, 331 hydration 160–163 hydrogen-bonded crystals applications ab initio molecular dynamics simulations 332 aspirin crystals 338–342 Car–Parrinello molecular dynamics simulation 332 in crystalline tropolone 343–344 crystalline vitamin C 336–338 2-hydroxy-5-nitrobenzamide 332 infrared spectroscopy 333 oxalic acid dihydrate crystal, isotopic substitution effects 333–336 refinement and enhancement 332 directional intermolecular interactions 328 historical and theoretical background 329–332 quantitative reconstruction and interpretation 328 quantum effects 328 hydrogen bonded system anharmonic coupling 201 acetic acid 211 formic acid 211–212 limit situations 210–211 line shape 208–209 spectral density 203–206 infra-red transitions 200 linear response theory electric dipole interaction 201 quantitative theory 202–203

Index

nuclear dynamics theories 200 X-H stretching mode 200 hydrogen bond strength 218–220, 235, 236, 238, 241, 328, 334, 341, 406 hydroquinone 294–296 hyperpolarizability 308, 309, 311, 313–318, 320

i IMDHO model 510, 513, 517–521, 523 implicit solvation model 358 independent mode displaced harmonic oscillator (IMDHO) model 505, 506 INDO/SCI absorption spectra 578 inelastic neutron scattering (INS) 429, 440, 460, 473 inelastic scattering 497, 500 infrared (IR) 353 absorption 426 inorganic semiconductors 425 Integral Equation Formalism (IEF-PCM) method 184, 257 inter-and intra molecular vibrations, in charge transport electron-phonon coupling local and non-local 434–436 in organic semiconductors 439–440 in various organic semiconductors 440–443 electron-phonon interaction 436–439 Intermediate Neglect of Differential Overlap Hamiltonian 576 intermodal anharmonicities 369 inter-modal anharmonicity 354 intermolecular and intramolecular vibrations 459, 461, 465–467, 471, 474, 476, 477, 479–481, 485, 487 intermolecular charge delocalization 425, 436, 437, 447 intermolecular electronic coupling 425, 446, 448 intermolecular interactions 126, 131, 183, 261–265, 328, 333,

336, 337, 355, 363–368, 371, 373, 427, 443, 459, 460, 473, 475, 476, 484 intermolecular translations 459, 460, 465, 467, 468, 472–475, 478–479, 485, 486 intermolecular vibrational dynamics 460 intra-and inter molecular vibrations 430, 432, 433 intramolecular hydrogen bonds calculation of energies 220–223 Car–Parrinello molecular dynamics simulations 232–234 definition 216 donor and acceptor 215 double bond linking 216 hydrogen bond strength 218–220 IR spectra, calculation of harmonic approximation 226–228 malonaldehyde and acetylacetone 229–232 static procedure 228–229 NMR Gauge Invariant Atomic Orbital (GIAO) method 234 OH chemical shifts 235–236 resonance assisted hydrogen bonds (RAHB) 215 structural parameters, calculations of 217 tautomerism 2D approaches 223 potential energy and free energy surfaces 223–226 intramolecular interaction 261–265 intramolecular vibrations 429, 477 dynamics 460 frequency sequences 478–479 isotope shifts (IS) 474 isotopomers 373, 374 isotropic hyperfine coupling constant 11 ith parent mode 174

j julolidinemalononitrile (JM) chromophore 518, 519

603

604

Index

k Kleinpeter–Koch approach 220 Kohn–Sham (KS) density functional theory 180 determinant 64 equation 465 Fock operator 48 orbitals 463 Kramers–Kronig relations 510 Kramers–Kronig transformation 124, 130

l l-alanine crystal 462, 476–478 large amplitude motions (LAMs) 10 l-ascorbic acid 337 lattice distortion energy 436, 441, 442 lattice dynamics 459 light–electron couplings 44, 50, 51, 57 linear-response (LR) TDDFT 62 linear response theory 177, 199–212 Liouville–von Neumann equation 49, 51 lipid bilayer 157–160 local electron-phonon coupling 434–436, 438–444, 448 local (Holstein-type) electron-phonon coupling 435 local mode perturbation theory (LMPT) mode 406 local modes 150–151, 389–417 local origin gauge (LORG) method 234–235 local oscillator (LO) 310 long chain FAs (LCFAs) 364 Lorentzian expansion 549 Low barrier hydrogen bonds 216 Löwdin orthogonalizaion 45 low-energy intramolecular vibrations 487 lowest unoccupied molecular orbital (LUMO) 52, 86, 89, 343, 434, 445, 566 low-frequency Raman spectra 431 low frequency vibrations anharmonic oscillators (AO) 391 Birge–Sponer plot 393 Birge–Sponer type fit 392 CH-stretching frequencies 397–400 CH-stretching transitions 411

CH-stretching vibration 405 dimethylamine 412–413 Fermi-resonance coupling 396 ground state energy 392 harmonic oscillator approximation 393 HOH-bending mode 401 intensities 408–417 intermolecular modes 406–408 intermolecular modes in bimolecular complexes 416–417 internal energy 390 methyl torsion 414–415 Morse wavefunctions 410 NM approximation 390 off-diagonal contributions 390 OH-stretching oscillators 393 OH-stretching vibrations 394 symmetric and asymmetric fundamental transitions 395 symmetric and asymmetric transitions 412 2D LM Hamiltonian 394 water dimer 413–414 zeroth-order Hamiltonian 400 zeroth-order LM Hamiltonian 390 low-temperature (LT) 128, 183, 263, 285, 287, 294, 296, 442, 443, 471

m magnetic-dipole transition moment (MDTM) 174 magnetic spectroscopy 2, 10–12, 22–25 malic acid 369, 370 malonaldehydes 220, 229 enol 231 m-aminobenzonitrile 540 Mannich bases 224, 234, 239 mass-weighted Cartesian coordinates 389, 467, 513, 515, 544 matrix-isolation (MI) 183, 187, 255, 269, 280–287, 290, 291, 293 matrix-isolation spectroscopy acetic acid 280 adoption of theory and basis set 282–283 conformational analysis chlorobenzaldehyde 285–286 1,2-dichloroethane 283–285

Index

excitation light 289–290 vanillin 286–289 identification of chemical species cytosine, rare tautomer of 290–291 molecular complex/cluster 293–294 reversible isomerization, from 1,8-diaminonaphthalene 291–293 photoinduced transient species hydroquinone 294–296 lowest electronic excited triplet state 296–297 photolysis 282 wavenumber shifts of absorption bands 281 matrix-isolation vibrational circular dichroism spectra (MI-VCD) 183 medium chain FAs (MCFAs) 364 band assignments 368 NIR spectra 366 20-mer ethylene system 54 meta-dicyanobenzenes 431 methanol 37, 261–265, 279, 315, 316, 355, 357, 358 methyl hexanoate 319 mode decomposition method 465–468 mode-mode couplings 353, 354 molecular crystals 427–430, 440, 459–462, 464, 465, 480 molecular dynamics (MD) 35, 59–70, 94, 173, 184, 185, 217, 328, 330, 331, 345, 427 molecular hyperpolarizability 308, 313–318, 320 molecular mechanical calculation (MMC) 273 molecular orbital (MOs) 550, 583 energies 566 interaction 537 molecular phonon modes 470 adenine 475–476 anthracene 473–474 C60 471–473 α-glycine and l-alanine 476–478 rigid-body approximation 471 molecular structure refinement (MSR) 12, 16

molecular tailoring approach (MTA) 221, 222 molecular weight (MW) 479 molecule-Ag4 model 547 molecule-cluster system 576 molecule-metallic cluster model 537, 546 molecule-silver clusters complexes 564 molecule weights, comparison 479 Møller–Plesset MP2 method 369 perturbation approach 81 second order perturbation (MP2) 358 Morse wavefunctions 405, 410 Mulliken population analysis 54, 56, 57 multichromophoric system 102 multi-configurational methods 62, 517 multi-configurational SCF (MCSCF) 84 multi-configuration self-consistent field (MC SCF) model 179, 180 multi-dimensional harmonic PESs 513 multiple charge transport models 436 multi-reference approaches 83 multivariate analytical (MVA) 369

n natural bond orbital (NBO) analysis 547 2-(N-diethylamino-N-oxymethyl)-4,6dichlorophenol 224 near-infrared (NIR) spectroscopy 353, 355 band assignments 357 basic molecules 355–363 binary and ternary combination bands 375 butyl alcohols 361 combination modes 353–383 conformational isomerism 355, 357 cyclohexanol 363 diluted methanol 356 fatty acids 364–368 fundamentals 378–380 intermolecular interactions and biomolecules 363–368 miscellaneous applications 373–375 overtone 353–383, 412–415 phenol 363

605

606

Index

near-infrared (NIR) spectroscopy (contd.) rosmarinic acid 371 single mode anharmonicity by solving 1D Schrödinger equation 375–383 spectra-structure correlations 357 theoretical and analytical 368–373 vibrational self-consistent filed (VSCF) scheme 354 2-(N-ethyl-α-iminoethyl)-4-chloro-5methylphenol 232, 233 Newton equations of motion 354 Newton’s law of motion 541 NH2 group 539, 546, 549, 551, 553, 565, 567 NH stretching modes 155, 380–382 NH2 wagging mode (ωNH2 ) 538 NIRS 353 N-isopropylacetamide 263 nitroxides 35 N-methylacetamide (NMA) 122, 130, 136, 263 N-methyl-2-hydroxybenzylidene amine (HBZA) 234 2-(N-methyliminomethyl)-4,6dichlorophenol 232, 233 N,N-dimethylacetoamide (NdMAm) 130, 255–261 Noble metal nanoparticles 575 nonadiabatic coupling (NAC) 50, 61, 64–65 nonadiabatic molecular dynamics simulation 61 non-annealed scPLA 485 non-covalent interactions 100, 428, 460, 462–465, 470–481 nonlocal correlation density functionals 463 non-local (Peierls-type) coupling 435 non-local electron-phonon coupling 434–436, 438–444, 448 non-local electron-phonon interaction 444, 446 non-resonant Raman scattering 498, 501 non-vibrationally resonant anti-Stokes transition 315

normal mode (NM) 9, 10, 17, 32, 105, 151, 200, 226, 253, 256, 316, 354, 389, 428, 439, 461, 465–467, 478, 480 normal mode decomposition (NMD) 439 NTChem computational methods 43 K computer’s processing power 43 nuclear velocity perturbation theory (NVPT) 173, 191 nylon6, hydration of 160 nylons 120 ATR-FUV spectra 137 Gaussian functions 139 hydrogen bonding 139 intermolecular hydrogen bonding 141 N-methylacetamide (NMA) 136 π-π*bands 140 π-π* transition 140, 141 𝜎-Rydberg transitions 140 trintensity 137

o o-aminobenzonitrile 540 o-dimethylaminomethylphenol (DMAP) 234 off-resonance energy 590 O-H stretching modes 480 o-hydroxy aromatic aldehydes 222, 235, 241 o-hydroxyarylaldehydes 220 o-hydroxyaryl Schiff bases 223, 224, 232 o-hydroxy Schiff base 234 olefins 537, 538, 540, 546, 558–562, 567 oligoacenes 440–442 oligothiophenes 429, 440–443 optical phonon modes 427, 461, 462, 470–481 orbital Zeeman (OZ) 11 organic crystals, LF vibrations 426 organic electronic devices 425 organic light-emitting diodes (OLEDs) 425 organic semiconductor crystals (OSCs) 425, 449

Index

charge transport properties 425, 426 organic semiconductors 425, 436–443 organic semiconductorsis 425 organic solar cells 425 ortho-dicyanobenzenes 431 oxalic acid dihydrate crystal 329, 333–336

p p-aminobenzoic acid (PABA) 538–540, 549 p-aminobenzoic acid methyl ester (PABM) 549 p-aminobenzonitrile (PABN) 538–540, 549 p-aminothiophenol 540, 554 para-dicyanobenzenes 431 para-substituted anilines 556 path integral molecular dynamics (PIMD) 329 p-bromoaniline (PBA) 549 p-chloroaniline (PCA) 549 Peierls-type coupling 435 pentacene crystal 442 pentapeptide 153–157 perturbation-corrected VSCF (PT2-VSCF) 354 perturbation theory 5, 62, 148, 152, 173, 186, 229, 354, 390, 516 (–)-1-phenylethanol 270 phonon-assisted charge transport 437, 438 phonon modes 461, 487 atomic vibrations 459 center of mass 460 phonons, types 427 photoisomerization 95, 96 photolysis 282, 289, 293, 294, 411 picolinic acid N-oxide (PANO) 217, 234 π-bonded adsorption 546 π-bonded configuration 546, 560 Placzek approximation 501 Placzek polarizability theory 508 plasmon 575, 583 plasmon-like dipolar excited states 581 p-methoxyaniline (PMOA) 549 p-methylaniline (PMA) 549 p-nitroaniline (PNA) 549

polarity 311–313 polarizable continuum models (PCMs) 35, 121, 183, 184, 279, 518 polaron binding energy 435, 436, 442 poly(d-lactic acid) (PDLA) 481, 484 poly(l-lactic acid) (PLLA) 481, 484 polymer materials 160–163, 481–484 poly(methyl methacrylate) 318 poly(N,N-dimethylacrylamide) (PNdMAm) 261 poly(N-isopropylacrylamide) (PNiPAm) 262 polycrystalline TCNQ 431 poly(lactic acid) stereocomplex (scPLA) 481–483 polypeptides 153–167 portable batch system (PBS) 25 post-DFT approaches 463 post-Hartree–Fock approaches 21 post-molecular dynamics analysis 331 potential energy distribution (PED) 466, 545 potential energy surface (PES) 5, 59, 94, 147, 185, 217, 327, 330, 345, 428, 461, 502 potential of zero charge (PZC) 554 powerful graphical user interface (GUI) 3 p-π conjugated molecules 538, 540, 546 p-π conjugation effect 537, 560 P(Harm) procedure 229, 230 projector augmented wave (PAW) method 123 1-propanol 261, 357, 358 propionic acid 365, 366 propylene free radicals 547, 557 propylene radical anion 557 prototypical intermolecular interactions 459 p-substituted aniline derivatives 537, 538 Pulay’s definition of internal coordinates 541 pyrol-ethylene systems 382 pyrrole-acetylene systems 382 pyrrole–pyridine complex 381, 382 Python program VASP_RAMAN.PY 428

607

608

Index

q QM/MM exciton model Hamiltonian 105 quadrupolar plasmonic excited states 582, 590 quantum chemical approaches 439 quantum-chemical methods 12–14, 329 quantum chemistry (QC) electronic spectroscopy for absorption predictions 97–99 algebraic diagrammatic construction (ADC) 81 bacteriochlorophylls, in LH2 102–107 charge transfer (CT) transitions 83 chemical calculations 88–93 complete active space SCF (CASSCF) 84 configuration interaction singles (CIS) 80 Coulombic and exchange interactions 87 Coulombic dipole interaction 86 coupled cluster singles and doubles (CCSD) method 82 coupling, excitation energies 104 density functional theory (DFT) 82 donor highest occupied molecular orbital (dHOMO) 86 dynamics simulations 105–107 equation-of-motion coupled cluster (EOM-CC) approach 81 excited state energies 103 Hartree–Fock (HF) theory 80 isolated retinal 96–97 MD simulations, with QM/MM 99–100 Møller–Plesset (MP2) perturbation approach 81 multichromophoric systems 85 multiconfigurational SCF (MCSCF) approaches 84 multiexcitonic behaviors 88 retinal in rhodopsin 95–102 rhodopsin variants 100 Slater determinant 87

spectral lineshape 93–95 spin-flip (SF) approach 83 state-averaged CASSCF (SA-CASSCF) approach 84 symmetry-adapted-cluster configuration interaction (SAC-CI) method 82 tetracene and pentacene dimers 88 methods 497, 499 resonance raman spectra 517 quantum effects 225, 226, 328, 330, 343 quantum mechanical (QM) simulations 353–383 quantum mechanical theory 176, 499 Quantum Monte Carlo techniques 223 quartic centrifugal-distorsion Hamiltonian 5 quartic force field approximation 369 quasistatic approximation 575

r radiation absorption 353 Raman activity 19, 160, 503, 553 Raman effect 497 Raman frequency 561 Raman intensities 181, 497, 499, 537, 547, 548, 553, 568 C-NH2 stretching vibration 557 Raman intensity 172, 181, 431, 537, 548–549 Raman polarizability tensor 500–502, 508, 521 Raman scattering 305, 306, 316, 497–517, 548 (normal) Raman scattering (NRS) 587 Raman spectra 555, 556 aniline 551, 553, 554 ethylene 562 PABA 557 PATP 555 Raman spectroscopy 353, 440, 460, 568 F2n -TCNQ crystals 430 by solid-state DFT 429 techniques 95 Raman transition 521 R3c space-group symmetry 481

Index

replica-exchange molecular dynamics (REMD) method 154 resonance assisted hydrogen bonds (RAHB) 215 resonance polarizability derivatives 508–509 resonance Raman (RR) intensities 506 time-dependent formulation 508 resonance Raman (RR) scattering 498, 519, 522 resonance Raman spectra 588, 591 resonance ROA (RROA) 181 restricted active space perturbation theory 517 reverse osmosis (RO) 160 Rhodamine 6G (R6G) 518, 520 rhodopsin 79 variants 100 rigid-rotor harmonic-oscillator (RRHO) approximation 5 model 19 root-mean-square (RMS) deviation 15 root-mean-square mass-weighted atomic displacement (RMSMWAD) 470 rosmarinic acid (RA) 370, 373 band assignments 372 NIRS 371 rotational spectroscopy 4–5, 14–16 RR excitation profile (RREP) 521

s salicylaldehyde 218, 220, 221, 237 saturated hexanoic sorbic acid 367 Savin formula 506, 507 scaled quantum mechanicals force field (SQMFF)method 546 scPLA, effect of crystallinity 485 second-order Møller-Plesset (MP2) level 516 second-order perturbation corrected scheme (PT2-VSCF) 369 self-consistent field (SCF) HF solution 80 self-consistent reaction field (SCRF) 257, 358 self-localization 437, 438 semiempirical electronic structure method 576

Ser-Ile-Val-Ser-Phe (SIVSF). SIVSF 153 SFG 304–307 SFG phase measurement 310–311 short chain FAs (SCFAs) 364 molecular structures 365 short-time approximation (STA) 497, 507, 518 silver clusters 539, 540, 546, 548, 554, 555, 558, 559, 562, 564, 576 silver nanoclusters 576 simplified approximation 511 simulations of surface-enhanced ROA (SERROA) 181 single and double excitation configuration interaction (SDCI) 577 single excitations (SCI) 80, 83, 86, 577–579 site-directed labelling techniques 10 Slater determinants 80, 83, 84, 89, 90 small-molecular crystals 464 small-shift approximation 506, 511 snap shot 149 solid-state DFT (DFT) 427, 434, 459, 461, 481, 486 solute model of density (SMD) 547 solvation 265–269 sorbic acid 367, 368 spectral density 202–206 spectral lineshape 93 spectral simulations, flexible molecules Ar matrix 256 conformational diversity electronic circular dichroism 272–275 vibrational circular dichroism (VCD) spectroscopy 269–272 vibrational spectrum, solvation in 265–269 electronic circular dichroic (ECD) spectra 254 intra-and inter-molecular interaction 254 intramolecular and intermolecular interactions 261–265 scaling factor 257 in solution phase 257 ternary amide compound 254

609

610

Index

spectral simulations, flexible molecules (contd.) vibrational circular dichroism (VCD) 254 vibrational spectra 254 wavenumber linear scaling method 257 spectra-structure correlations 360 sphingomyelin (SM) bilayers 149, 157–160 spin-flip (SF) approach 70, 83 spin-forbidden circular dichroism 177–179 spin-forbidden processes 70, 71 spin orbit coupling (SOC) operators 11, 70 S-propylene oxide (S-PO) 189 state-averaged CASSCF (SA-CASSCF) 62 Stokes Raman scattering 305, 498 Stokes scattering 498, 502, 586 sum-over-states (SOS) 499, 501, 585 supermolecular approach 183 super-sonic jet technique 279 surface adsorption interaction 546 surface enhanced infrared spectroscopy 539, 565 surface enhanced Raman scattering (SERS) 537, 575 density functional theoretical calculations 546 modeling molecules aniline 549–554 benzyl radicals anion 557–558 broad bands 565–567 chemical enhancement effect 562–565 para-substituted anilines 554–557 terminal olefin 558–562 normal mode analysis 541 p-π conjugated molecules 538 Raman intensity 548–549 voltage effects and chemical mechanism 591 wagging vibrational Raman spectra 538–540 surface spectroscopy label-free method 303

nonlinear optical techniques 304 surface sum-frequency generation 537 surface vibrational optical spectroscopic techniques 537 surface vibrational spectroscopy 568 symmetry-adapted-cluster configuration interaction (SAC-CI) method 82 symmetry-adapted perturbation theory (SAPT) 462 symmetry-conservation law 481

t Tamm–Dancoff approximation (TDA) 83 tautomerism potential energy and free energy surfaces 223–226 2D approaches 223 TCNQ, charge transfer directions 445 template molecule (TM) approach 31 terahertz (THz) spectroscopy 459 terminal olefin 537–539, 546, 558–562 tetrahedral Ag20 cluster 578 tetramethylsilane (TMS) 235 thermally populated low-frequency (LF) vibrations 426 thin-film electronics 425 THz spectroscopy 459–487 time correlation function (TCF) 173, 176, 191 time-dependent complete active space self-consistent field (TDCASSCF) theory 48 time-dependent density functional theory (TDDFT) 179, 508, 516, 517, 575 time dependent Hartree–Fock (TDHF) 46, 48 time-dependent perturbation theory 497, 499 time-dependent potentials 226 time independent (TI) sum-over-states approach 10 time-resolved fluorescence 95 tip enhanced Raman scattering (TERS) 576, 585 torsion, of methyl group 414–415 total density of states (TDOS) 128

Index

trajectory surface hopping molecular dynamics simulation computational details 66 conical intersection lifetime and transition rate 61 path and role of 60–61 effects of environment 69 electronic structure methods 62–63 NAMD simulation 61 photodynamics of coumarin 65–66 results and discussion 66–69 spin-forbidden processes 70 S0 /S1 crossing 69–70 theoretical method LR-TDDFT method 63–64 nonadiabatic coupling 64–65 TSH approach 63 transfer (hopping) rate 438 transform theory 509–511 transition intensity 354 transition quadrupole moments 581, 582 transmission electron microscopy (TEM) 440 trans-porphycene 523, 524 1,3,5-triacetyl-2,4,6-trihydroxybenzene 220, 222, 223 Tully’s fewest switch surface hopping approach conical intersection, crossing point 61 electronic structure methods 63 tunnelling effect 203 two-dimensional correlation analysis (2D-COS) 358 2D electronic spectroscopy 100 two-electron one-center integrals 576

u urea, molecular structures

461

v vanillin 286–289 vertical gradient (VG) model 10, 517 vertical Hessian (VH) model 10, 517 vibrational absorption spectra (VA) 173 vibrational analysis 122, 281, 282, 284, 297, 330, 370

vibrational circular dichroism (VCD) spectroscopy 269–272 vibrational configuration interaction (VCI) method 355 vibrational coupled-cluster (VCC) method 355 vibrational energy 6, 69, 153, 470, 480 vibrational frequency 547 in crystals 433 vibrational Hamiltonian 389, 391, 418, 507, 515 vibrational optical activity 180–181 vibrational perturbation theory (VPT) 147, 186, 229, 354, 358 vibrational quasi-degenerate perturbation theory 151–152 vibrational resonance Raman spectra of molecules quantum chemistry methods 515–517 applications 518 double harmonic approximation 501–503 electronic excited states in resonance 521–523 Franck–Condon overlap integrals 511–515 Franck–Condon vibronic couplings 518–521 Herzberg–Teller vibronic couplings 523 intensities 503–506 resonance polarizability derivatives 508–509 short-time approximation 518 sum-over-state formulation 499–501 time-dependent formulation 506–508 transform theory and simplified approximation 509–511 time-dependent perturbation theory 497 vibrational second-order perturbation theory (VPT2) 229 vibrational self-consistent field (VSCF) 148, 228, 354, 369 vibrational spectral analysis 279, 540

611

612

Index

vibrational spectroscopy 16–19, 254, 357, 373, 427, 428 techniques 538 vibrational sum-frequency generation spectroscopy experimental studies 304–307 SFG phase measurement 310–311 vibronic Bohr frequencies 503 vibronic spectroscopy 8–10, 19–22 vibronic theory 510, 511 Vienna ab initio simulation package (VASP) 122–123 vinylacetic acid 365–367, 376 virtual multifrequency spectrometer (VMS) computational and experimental spectroscopic techniques 3 glycine case study 26–28 graphical user interface (GUI) 3 physical-chemical features 3 rotational spectroscopy 14–16 theoretical background magnetic spectroscopy 10–12, 22–25 quantum-chemical methods 12–14 rotational spectroscopy 4–5 vibrational spectra 5–8 vibrational spectroscopy 16–19 vibronic spectroscopy 8–10 vibronic spectroscopy, methyloxirane case study 28–30 in vitro and in silico experiments 3

w Wannier functions 335, 440 Wannier localization function 332 water dimer 413–414

331,

wavenumber linear scaling method 257 weight averaged anharmonic vibrational calculations cluster and local modes 150–151 computational procedure 153 conventional vibrational calculations 147 GENESIS 148 IR spectroscopy 147 MD simulations 148 nylon6, hydration of 160–163 pentapeptide, SIVSF 153–157 second-order vibrational perturbation theory 148 sphingomyelin (SM) bilayers 157–160 vibrational quasi-degenerate perturbation theory 151–152 vibrational self-consistent field (VSCF) 148 weight average method 149–150

x X-ray diffraction 171, 481 X-ray photoelectron spectroscopy (XPS) 443

z zero-differential overlap approximation 576 zero-point-energy-correction (ZPE) 180 zeroth-order expression 394 zeroth-order Hamiltonian 84, 397, 400, 401, 410 zeroth-order regular approximation (ZORA) 577 zwitterionic 1,8-bis(dimethylamino)4,5-dihydroxynaphthalene 227