Theoretical Mass Spectrometry: Tracing Ions with Classical Trajectories 9783110434897, 9783110442007

Table of contents :
Acknowledgments
Contents
List of abbreviations
Physical constants
1. Introduction
2. Principles of Experimental Mass Spectrometry
3. Classical Trajectory Methods
4. Interaction Energy
5 .Principles of Chemical Kinetics
6. How to Simulate Real Systems
7.Application to Organic Molecules
8 .Application to Biological Molecules
9. Summary and Future Directions
A Mathematical Compendium
B Principles of Classical Mechanics
List of Figures
List of Tables
Bibliography
Index

Citation preview

Kihyung Song, Riccardo Spezia Theoretical Mass Spectrometry

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Kihyung Song, Riccardo Spezia

Theoretical Mass Spectrometry | Tracing Ions with Classical Trajectories

Authors Dr. Riccardo Spezia Sorbonne Université, CNRS Laboratoire de Chimie Théorique, LCT UMR 7616 CNRS 4, place Jussieu 75252 Paris Cedex 05 France

Prof. Kihyung Song Korea National University of Education Department of Chemistry 250 Taeseongtabyeon-ro Gangnae-myeon, Heungdeok-gu Cheongju, Chungbuk Korea 28173 [email protected]

Université d’Evry Val-d’Essonne Lab. Analyse et Modélisation pour la Biologie et l’Environment UMR 8587 CNRS Boulevard Francois Mitterand 91025 Évry Cedex France [email protected]

ISBN 978-3-11-044200-7 e-ISBN (PDF) 978-3-11-043489-7 e-ISBN (EPUB) 978-3-11-043356-2 Library of Congress Control Number: 2018934244 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 http://dnb.dnb.de. © 2018 Walter de Gruyter GmbH, Berlin/Boston Cover image: Pobytov/DigitalVision Vectors/Getty Images Typesetting: le-tex publishing services GmbH, Leipzig Printing and binding: CPI books GmbH, Leck www.degruyter.com

Acknowledgments This book is the fruit of many years of research done by us and, mainly, by other research groups in the world. A first particular thanks should go to Prof. William L. Hase (Texas Tech University, USA) who should be considered the pioneer of using molecular simulations to understand and thus predict reactions related to mass spectrometry. He was also very important for us, hosting us in his research group several times and giving us the opportunity to collaborate with him on a number of subjects. He is also the developer of the VENUS code, which is probably the most adapted dynamics code to be used in modeling gas phase ions reactivity and thus tandem mass spectrometry. This book would not have been possible without collaborations with different experimental groups, and in particular Dr. Jean-Yves Salpin and Prof. Jeanine Tortajada (Université d’Evry-Val-d’Essonne, France), Prof. Jos Oomens (University of Nijmegen, The Netherlands) and Dr. Debora Scuderi (Université Paris Sud, France). Collaborations and exchanges with other theoretical chemists were also very fruitful in these years for better developing concepts and methods for modeling mass spectrometry, and in particular Prof. Manuel Yanez (Universidad Autonoma de Madrid, Spain), Dr. Daniel Borgis (Ecole Normale Supérieure, Paris, France), Dr. Yannick Jeanvoine (Université d’Evry-Val-d’Essonne, France), Prof. Upakarasamy Lourderaj (NISER, India). Prof. Emilio Martinez-Nunez and Prof. Saulo Vazquez (Universidad de Santiago de Compostela, Spain). Several students and post-docs were involved in this field, and we want to thank them all: Dr. Ana Martin-Somer, Dr. Estafania Rossich-Molina, Dr. Daniel Ortiz, Dr. Ane Eizaguirre, Ms. Veronica Macaluso, Dr. Kyoyeon Park, Ms. Jae Hee Park, Ms. Ara Cho, Ms. Soo Bok Lee, Ms. Jin Joo Han, Ms. Giae Lee, Ms. Eunkyung Park, Ms. Heesun Chung, Ms. Wooyoung Shim, Ms. Jihye Jeon, and Mr. Junghyun Yoon. We would like to thank Dr. Debora Scuderi also for careful reading of some parts of the manuscript. Finally, some research grants should be also acknowledged since they were indispensable to better developing the subject: ANR DYNBIOREACT, Labex Charmmmat, Invited professorships from Université d’Evry and Korea National University of Education for us to visit our respective groups in France and Korea. In a broader sense, using classical trajectories to model and understand tandem mass spectrometry belongs to the field of molecular simulations and theoretical chemistry and we would like to thank those, who even indirectly, have stimulated us into using concepts of this field to develop the theoretical modeling of mass spectrometry: Prof. James T. Hynes and Prof. Rodolphe Vuilleumier (École Normale Supérieure, Paris, France), Dr. Sara Bonella (CECAM, Lausanne, Switzerland), Prof. Alfredo Di Nola and Prof. Giovanni Ciccotti (Università di Roma, La Sapienza, Italy), Prof. Massimiliano Aschi (Università de l’Aquila, Italy).

https://doi.org/10.1515/9783110434897-201

Contents Acknowledgments | V List of abbreviations | XI Physical constants | XII 1

Introduction | 1

2 2.1 2.2 2.2.1 2.2.2 2.3 2.3.1 2.3.2 2.3.3 2.3.4 2.4 2.4.1 2.4.2 2.4.3 2.4.4 2.4.5 2.4.6 2.5 2.5.1 2.5.2 2.5.3 2.5.4 2.5.5 2.6 2.6.1 2.6.2 2.6.3 2.6.4 2.7

Principles of Experimental Mass Spectrometry | 5 Introduction | 5 Ionization | 6 Electrospray Ionization | 6 Other Ionization Methods | 8 Activation Methods | 11 Collision Induced Dissociation | 11 Surface Induced Dissociation | 15 Action Spectroscopy | 16 Dissociation Induced by Electrons | 18 Principles of Ions Analysis | 19 Magnetic Sector | 19 Time-of-Flight | 20 Quadrupole | 21 Ion trap filter | 22 Cyclotron Resonance | 23 Detector | 23 Structural Determination | 24 Breakdown Curves | 24 Cross Section | 25 Isotopic Labeling | 26 IRMPD | 27 Theoretical Methods | 27 Examples of Experimental Apparatus for CID | 30 Triple Quadrupole and QQ-TOF Instruments | 30 Ion Trap Instrument | 31 FT-ICR Instrument | 31 MS/MS/MS | 31 Conclusions | 32

3 3.1 3.1.1

Classical Trajectory Methods | 33 Equations of Motion | 33 Basic Principles | 33

VIII | Contents

3.1.2 3.1.3 3.2 3.2.1 3.2.2 3.3 3.3.1 3.3.2 3.3.3 3.4 3.5 3.5.1 3.5.2 3.5.3 3.5.4 3.5.5 3.6

Born–Oppenheimer Approximation | 34 Newton Dynamics | 37 Numerical Integration Methods | 38 Runge–Kutta and Adams–Moulton Methods | 38 Verlet Integrators | 39 Initial Conditions Sampling for Gas Phase Systems | 40 Microcanonical Normal Mode Sampling | 41 Initial Orientation | 43 Impact Parameter | 44 Internal Activation | 45 Quantum Nuclear Effects | 47 Normal Modes | 47 Zero Point Energy | 50 Tunneling | 52 Path Integral Based Methods | 52 Quantum Thermal Bath | 53 Conclusions | 54

4 4.1 4.2 4.3 4.3.1 4.3.2 4.3.3 4.4 4.4.1 4.4.2 4.5

Interaction Energy | 55 Classical Intramolecular Potential | 55 Classical Intermolecular Potential | 58 Ab Initio Methods | 60 Wave Function Methods | 62 Density Functional Theory | 67 Semiempirical Hamiltonians | 69 Energy Transfer | 73 Impulsive Model | 74 Ergodic Collision Model | 75 Conclusions | 77

5 5.1 5.2 5.3 5.3.1 5.3.2 5.3.3 5.3.4 5.3.5 5.4 5.5 5.6

Principles of Chemical Kinetics | 78 Introduction | 78 Basic Assumptions in Unimolecular Dissociation | 78 RRKM Theory | 79 Rotational Effects | 82 Classical vs Quantum RRKM Theory | 83 Tunneling Effects in RRKM Theory | 85 Anharmonic Effects | 87 Loose Transition States | 88 Transition State Theory | 89 Phase Space Theory | 91 Quantum Rate Constant | 92

Contents | IX

5.7 5.7.1 5.7.2 5.7.3 5.7.4 5.8 5.8.1 5.8.2 5.9 5.10

Correlation Function Formalism | 93 Generalities | 93 Classical Rate Constant | 95 Quantum Mechanics Formulation | 98 The Stable State Picture | 100 Statistical and Nonstatistical Behavior in Chemical Reactions | 102 Modeling Statistical Reactivity | 103 Modeling Nonstatistical Reactivity | 106 Vibrational Energy Relaxation | 106 Conclusions | 108

6 6.1 6.2 6.2.1 6.2.2 6.3 6.4 6.4.1 6.4.2 6.5 6.6

How to Simulate Real Systems | 109 Introduction | 109 Explicit Collisions | 109 Collision Induced Dissociation | 109 Surface Induced Dissociation | 113 Thermal Activation | 115 Other Dissociation Simulations | 117 Light Induced Activation | 117 Electron Ionization | 118 Peak Intensity | 120 Conclusions | 121

7 7.1 7.2 7.3 7.4 7.4.1 7.4.2 7.4.3 7.5 7.6

Application to Organic Molecules | 122 Introduction | 122 First Pioneering Studies | 122 CID of Protonated Urea | 123 CID of Doubly Charged Ions of Metal-Neutral Complexes | 128 Nonstatistical Dissociations | 130 Statistical Dissociations | 131 Post-TS Dynamics | 136 CID of Uracil | 137 Conclusions | 143

8 8.1 8.2 8.2.1 8.2.2 8.3 8.3.1 8.3.2

Application to Biological Molecules | 144 Introduction | 144 CID of Steroids | 144 Testosterone | 145 Boldenone | 150 CID of Carbohydrates | 153 Galactose-6-Sulfate | 155 Protonated Disaccharides | 158

X | Contents

8.3.3 8.4 8.4.1 8.4.2 8.4.3 8.4.4 8.5

Perspectives | 161 CID of Peptides | 163 Singly Charged Glycines and Alanines | 166 Multiply Charged Peptides | 169 Negative Peptides | 171 Perspectives | 171 Conclusions | 173

9

Summary and Future Directions | 174

A A.1 A.1.1 A.1.2 A.2 A.2.1 A.2.2 A.2.3 A.2.4 A.3 A.3.1 A.3.2 A.3.3 A.3.4 A.4 A.4.1 A.4.2 A.4.3 A.4.4

Mathematical Compendium | 176 Linear Algebra | 176 Vector Properties | 176 Matrices | 177 Vector Analysis | 180 Differentiation | 181 Integration | 182 Theorems | 183 Changing Variables | 183 Differential Equations | 185 Direct Integration | 185 Linear First Order Differential Equations (L1 ) | 185 Exact Differential Equations | 186 Second Order Differential Equations (L2 ) | 186 Integral Transforms | 187 Fourier Transform | 188 Laplace Transform | 189 Laplace–Carson Transform | 190 Chemical Kinetics | 191

B B.1 B.2 B.3 B.4 B.5

Principles of Classical Mechanics | 193 Mechanics of a Particle | 193 Mechanics of a System of Particles | 194 Constraints and Lagrangian | 196 Hamilton Equations of Motion | 198 Energy Conservation in Lagrangian and Hamiltonian | 200

List of Figures | 203 List of Tables | 208 Bibliography | 209 Index | 229

List of abbreviations AM1 CI CID CMD CNDO CRM DFT ECD EI ESI ETD FAB FT-ICR GC-MS HF IEM INDO IRC IR(M)PD IVR KMC LDA LSC MALDI MC MM MNDO MP2 MS/MS PES PIMD PM3 PST QCISD(T) QCT QET QM QQ-TOF ReaxFF RRKM SCF SID TOF TS TSSCDS TST

Austin Model 1 Chemical Ionization Collision Induced Dissociation Centroid Molecular Dynamics Complete Neglect of Differential Overlap Charge Residue Model Density Functional Theory Electron Collision Dissociation Electron Ionization Electro Spray Ionization Electron Transfert Dissociation Fast Atom Bombardment Fourier Transform Ion Cyclotron Resonance Gas-Chromatography Mass Spectrometry Hartree–Fock Ion Evaporation Model Intermediate Neglect of Differential Overlap Intrinsic Reaction Coordinate Infra Red (Multi) Photon Dissociation Internal Vibrational Redistribution Kinetic Monte Carlo Local Density Approximation Linear Single Collision Matrix Assisted Laser Desorption Ionization Multiple Collisions Molecular Mechanics (force field) Modified Neglect of Differential Overlap Second Order Møller–Plesset perturbation theory Tandem Mass Spectrometry Potential Energy Surface Path Integral Molecular Dynamics Parametrized Model number 3 Phase Space Theory Quadratic Configuration Interaction with Single, Double and Triple excitations (implicitly) Quasi-Classical Trajectories Quasi-Equilibrium Theory Quantum Mechanics Quadrupole quadrupole Time of Flight Reactive Force Field Rice Ramsperger Kassel Marcus Self Consistent Field Surface Induced Dissociation Time-of-Flight Transition State Transition State Search using Chemical Dynamics Simulations Transition State Theory

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Physical constants kB = 1.38064852(79) ⋅ 10−23 J ⋅ K−1 R = 8.3144598(48) J ⋅ K−1 ⋅ mol−1 h = 6.626070040(81) ⋅ 10−34 J ⋅ s ℏ = (h/2π) = 1.054571800(13) ⋅ 10−34 J ⋅ s N A = 6.022140857(74) ⋅ 1023 mol−1 e = 1.6021766208(98) ⋅ 10−19 C m e = 9.10938356(11) ⋅ 10−31 kg m u = 1.660539040(20) ⋅ 10−27 kg c = 299,792,458 m ⋅ s−1 ϵ 0 = 8.854187817 ⋅ 10−12 F ⋅ m−1 μ 0 = 1.256637061 ⋅ 10−6 N ⋅ A−2 μ B = 9.274009994(57) ⋅ 10−24 J ⋅ T−1 a0 = 5.2917721067(12) ⋅ 10−11 m E h = 4.359744650(54) ⋅ 10−18 J R ∞ = 10,973,731.568508(65) m−1 F = 96,485.33289(59) C ⋅ mol−1

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(Boltzmann constant) (gas constant) (Planck constant) (reduced Planck constant) (Avogadro constant) (elementary electric charge) (electron rest mass) (atomic mass constant) (speed of light) (electric vacuum permittivity) (magnetic vacuum permittivity) (Bohr magneton) (Bohr radius) (Hartree energy) (Rydberg constant) (Faraday constant)

1 Introduction Theoretical chemistry has made enormous progress in recent years mainly thanks to advances in two areas: (i) methods and algorithms and (ii) computational power. Nowadays, many properties of molecules can be predicted by simple quantum chemistry calculations, which consist in giving initial positions to the nuclei composing a given molecule via an adapted software and obtaining requested properties as output. This apparently simple and somehow black-box procedure is composed by simple elementary steps that can be summarized as follows: 1. Given a guess of atomic arrangement, which reflects some “chemical knowledge” or “intuition,” the electronic energy is calculated following a chosen quantum chemistry method, providing the wave function and energy; 2. From the quantum mechanical energy, the gradient is also calculated and a “geometry optimization” procedure starts, leading to a topological point, which is (generally) a minimum on the potential energy surface, characterized by zero first derivatives and positive second derivatives. Algorithms were optimized in order to find the correct minimum and likely the absolute minimum. 3. Once a correct (and desired) geometry is located, it is possible to get the desired information from the wave function. This conceptually simple procedure, which hides many complex aspects as we will describe in detail, particularly in Chapter 4, is able to provide a microscopic description of a given molecule in terms of atomic distances, angles, etc . . . and more in general on its three-dimensional arrangement. Other important information obtained is the electronic energy, which corresponds to the energy to put together the N nuclei and M electrons composing the molecular system under study. Knowledge of the wave function, the geometry and the energy of the molecule, allows theoretical calculations of several spectroscopic properties, such that many experimental spectra can be obtained from standard quantum chemistry calculations. In particular, almost standard procedures are present in the literature and integrated in most common quantum chemistry software, like for example: – Infrared (IR) and Raman vibrational signatures are obtained in the harmonic approximation from the diagonalization of the Hessian matrix (which is the matrix of second derivatives, see Section 3.5.1) to locate normal modes and the evaluation of the oscillator strengths from dipole or polarizability that are accessible from the electronic density (which results from the electronic wave function) [1, 2]. Quantum chemistry represents a very powerful tool since it is possible, by reproducing the experimental vibrational spectra, to clearly characterize the geometry of a molecule. While at the beginning these calculations were limited to (small) molecules in the gas phase, nowadays it is possible to study larger molecules in solution and also to include anharmonic effects [3]. Finally, molecular dynamics https://doi.org/10.1515/9783110434897-001

2 | 1 Introduction

–

–

is emerging as an alternative tool [4] which, even if computationally expensive, is useful in case of large anharmonic effects or when the molecule under investigation is immersed in a complex environment. UV-visible spectra can be obtained from calculations of excited states and relative transition intensities from the transition dipole moments. Given a ground state geometry it is possible to calculate the excited states via wave function approaches and/or via time-dependent density functional theory. Without going into the details of these theoretical methods, the idea is that the energy between electronic states can be obtained and thus compared to experiments. As for vibrational spectroscopy, it is thus possible to make a correspondence between the molecular structure and the spectroscopic information. Calculations are performed in gas phase but also in solution and in complex environments. In this last case, molecular dynamics simulations are often performed to sample the relevant conformations (of the molecule and for molecule/environment interaction). Also nuclear magnetic resonance (NMR) chemical shifts can be evaluated from quantum chemistry, and this requires (similarly to experiments) the use of a reference molecule. The shielding tensor can be obtained formally from the second derivative of the quantum-mechanical energy with respect to the external magnetic field and the magnetic moment of the nucleus. The spin-spin coupling constants can be calculated as well as the complete anisotropic shift tensors. [5, 6] Calculations and experiments can thus be directly compared.

The list of other spectroscopies that can be modeled using quantum chemistry calculations independently from any experimental information is nowadays very long; we can add electron paramagnetic resonance (EPR) spectroscopy, Resonance Raman, and microwave spectroscopy, just to give a few further examples. The powerful use of quantum chemistry resides in its capability of reproducing experimental observables in a way that is totally independent from experiments, such that they can be (i) predictive and (ii) used to decipher many molecular properties (the first being the structure) from theory-experiment convergence. Mass spectrometry is a longstanding physical chemistry experimental approach aimed at studying molecules in the gas phase, which is very powerful in many aspects, from fundamental science to application in analytical sciences. However, a well-established theoretical approach independent from experiments aimed to provide a theoretical mass spectra was missing. The reason is that from one side, a mass spectrum is basic and simple. It gives the mass-over-charge (m/z) ratio of the species present in the mass spectrometer, so the calculations one aims to perform seem very simple (even too simple): to identify the molecule from its brute formula. The geometrical information is not given by the experiments, so one can argue that the geometry is the minimum energy structure. On the other hand simple m/z recording of molecular ions formed in the gas phase is only the first step: often these ions are activated such that they fragment, giving rise to the so called tandem mass spectrum. The result of an acti-

1 Introduction

| 3

vation process is an ensemble of chemical reactions belonging (in general) to the class of unimolecular dissociation. Sometimes, when the ion is brought to interact with a reactive species, bimolecular reactions are also performed. This means that now we have a complex information, somehow too complex for using a theoretical approach analogue to what used to model the spectroscopic signals mentioned previously. Theoretical calculations should then deal with the full set of reaction products, given the activation mode employed. In general, quantum chemistry is used to identify the reaction pathways once the products are known. This can be satisfactory for (very) small molecules in which the products as well as the possible mechanisms are few. For larger systems, the identification of the reaction pathway(s) is done but is often questionable if there are no other pathways and, from a more fundamental point of view, if the system takes such a simple minimum energy pathway (MEP) on a complex potential energy surface. Chemical dynamics was thus used in pioneering studies of Hase and coworkers [7–10] in particular to model collision induced dissociation (CID). Since then, progress in modeling CID has occurred and nowadays it is not impossible to obtain a mass spectrum from mere theoretical calculations, i.e., without information on products from experiments. This book is devoted to explaining the different aspects that must be considered when dealing with a theoretical mass spectrum based on chemical dynamics via an ensemble of trajectories. As we will see through the different chapters of this book, one needs to carefully consider the different aspects involved in the formation of a tandem mass spectrum. A first important point is to clearly decipher the underlying physics of the experiments and, in particular, the way the molecular ion is activated. This is why we detail some aspects of experimental tandem mass spectrometry in Chapter 2. Then we describe the basics of the chemical dynamics approach used to obtain a theoretical mass spectrum: trajectory methods in Chapter 3 and the way the interaction potential can be obtained in Chapter 4. As we will see, chemical dynamics simulations provide the dynamical aspects of the chemical reactivity. This can be compared and complemented by a kinetics study. The calculations of rate constants for simple chemical events (such as uni- and bimolecular reactions) were the subject of much progress in recent decades and the key points are summarized in Chapter 5. It is important to have these aspects in mind for several reasons. One important one is to understand if the dynamical description obtained in simulations follows statistical (kinetics) behavior and thus whether the time scale simulated is representative of full reactivity. It is also important to complete the reactivity obtained in the short times of explicit dynamics with long time scale reactivity, which can be obtained by a statistical (analytical) approach. Before showing some applications of theoretical mass spectrometry, we have summarized in Chapter 6 all the different aspects used (and needed) to perform such kinds of dynamical simulations. We then show in Chapters 7 and 8 recent applications to the study of organic and biological molecules, respectively.

4 | 1 Introduction

Finally, in Chapter 9 we provide some conclusions and, more importantly, some suggestions for future developments. Actually, the theoretical modeling of mass spectra is much more a reality than ten years ago, but improvements are still needed. We will review some of them and we hope that this can induce future studies in this relatively new field of theoretical chemistry. Before moving to the core of the present work, we should conclude with a nominalistic detail. Here and hereafter, we refer to “chemical dynamics” when dealing with simulations relative to theoretical mass spectrometry and not to “molecular dynamics.” There is a subtle difference between the two simulations. Molecular dynamics, which is a huge field of theoretical methods with a huge variety of applications, is intended to give equilibrium properties following the ergodic hypothesis, such that an observable is obtained as the average on a long enough “equilibrated” trajectory. Chemical dynamics, on the other hand, is aimed in studying the dynamics of a reaction that can be out-of-equilibrium, as often occurs in gas phase. In gas phase reactivity, in fact, once the products are obtained it is very unlikely that they are converted back to the reactants. This means that the statistical average should be done on the number of events and not on a single trajectory. However, apart from this kind of nominalistic difference, they have many aspects in common. A reader can enjoy finding common features and differences in the present work.

2 Principles of Experimental Mass Spectrometry 2.1 Introduction In this chapter, we will review the basic characteristics and principles of experimental mass spectrometry, with a particular aim in describing the tandem mass spectrometry (MS/MS) that is generally coupled to electrospray ionization (ESI), in particular for studying organic and biological molecules leading to the so called ESI-MS/MS. Readers interested in more experimental and application details can refer to specialized books on the field (see for example [11] and [12]). Mass spectrometry covers an ensemble of physical and analytical chemistry techniques that are based on the detection of the mass over charge ratio (m/z) of ions that are produced in the gas phase (and that, as we will see, will be subject to different kinds of activations). Besides fundamental studies on the nature and the strength of chemical or intermolecular bonds, or on the characteristics of chemical reactivity, mass spectrometry (and in particular tandem mass spectrometry) is used as an analytical tool in a broad ensemble of fields: from biological applications, like proteomics or metallomics, to doping or explosive detection. This technique, even if it is based on apparently simple events, produces phenomena that are still not completely clear and predictable. This is why theoretical and computational chemistry efforts devoted to understanding chemical reactivity in mass spectrometry are still actual. In this book we will show and discuss the theoretical methods based on chemical dynamics, and in this chapter we will provide some main ideas of the experiments that these calculations try to explain and reproduce. One important step is to introduce or produce the ions in the gas phase. To study organic or biological molecules (and more generally intact molecules) it is important that the ionization technique does not disrupt the molecule: electrospray ionization (ESI) [13, 14] is one of the most used methods, and it will be discussed in Section 2.2.1. Other methods, briefly summarized in Section 2.2.2, can be used to ionize molecular species in the gas phase, some of them preserving the structure, some not. Other than ESI, another very popular method to gently ionize molecules is matrix assisted laser desorption ionization (MALDI) [15, 16]. Once the ions are in the gas phase, they can be simply analyzed by obtaining the mass over charge ratio (m/z) or by allowing them to react. In general, the reactions are ion activation processes leading to fragmentations. Of course, it is possible to use a mass spectrometer to let the ions formed react with neutral species to synthesize new molecules. This use of mass spectrometer as a “chemical reactor” will not be considered here. The usual way to characterize ions formed is to select a given ion (the precursor ion) from its m/z and then activate it in some way in order to induce its fragmentation. The fragments are then analyzed and from this analysis, different information on https://doi.org/10.1515/9783110434897-002

6 | 2 Principles of Experimental Mass Spectrometry

the nature of the precursor ion can be obtained. Different activation methods can be used; the most common is to let the ion collide with an inert gas. This is the collision induced dissociation (CID) that will be discussed in Section 2.3.1. Making collisions with an inert gas is probably the most commonly used way of performing tandem mass spectrometry and it is also the main topic of the theoretical modeling developed by means of chemical dynamics. Other common methods are also used and here briefly discussed. Of course then the species must be detected and this is done by their m/z, as shown in Section 2.4. This experimentally limits the direct detection to ionic species. Finally we should remark that this chapter is aimed at giving an overview of mass spectrometry techniques to a theoretical chemistry readership not familiar with them and which may be interested in studying reactions occurring in mass spectrometry, thus needing the basic physical principles behind the different stages of such analytical techniques. More details are to be found in the specific literature, which is huge and growing daily.

2.2 Ionization Ion chemistry and spectroscopy is based on ions and thus an important stage is the production of such ions in the gas phase. Samples are in general in condensed phase and often neutral (a different story occurs if detecting directly ions from gas phase as for example in space) and thus they must be put in gas phase as charged species. This process is called ionization and here we will briefly recall the principal methods and basic phenomena related to that.

2.2.1 Electrospray Ionization Electrospray ionization (ESI) is a soft technique for producing ions in the gas phase without modifying their composition. The aim is, starting from a solution where molecules of one (or more) species are present at a given concentration, to ionize them and put them in the gas phase. The ESI technique, based on the works of Fenn and coworkers [13, 14, 17–19] is based on the following steps. 1. The solution is put in a capillary tube and a strong electric field is applied under atmospheric pressure. At this end a potential difference of 3–6 kV is applied between the capillary and the electrode. This produces an electric field on the order of 106 Vm−1 . 2. Charges are thus accumulated at one side of the capillary. They will then leave the capillary forming highly charged droplets after forming the so-called “Taylor cone” [20, 21]. To avoid (or anyhow limit) the dispersion in the space of these droplets, a gas is generally injected coaxially at a low flow rate. Note that the voltage from which the spray starts depends on the solvent surface tension. For example, for water, the onset voltage is 4 kV, while for methanol (which has a lower

2.2 Ionization

Hot N2

|

7

Nozzle

RF Quadrupole

Sheath Electrode Liquid

Skimmer Electrospray

Fused Silica CZE Capillary Stainless Steel Capillary

Buffer/Sheath Cone (Flow Mixing Region)

1·10 Torr

Lens

10·6 Torr

Pump Hot N2

Fig. 2.1: Details of ESI coupled to a quadrupole mass spectrometry. Reprinted from [18], with permission from Elsevier.

surface tension) it is 2.2 kV. These highly charged droplets can bear a maximum number of charges up to the so-called Rayleigh limit QR = π√8ϵ0 γD3

3. 4.

(2.1)

where QR is the total charge of the droplet, ϵ0 the vacuum permittivity, γ the surface tension and D the diameter of the supposed spherical droplets. Once this charge limit is passed, the droplet is mechanically unstable and should break. The droplets formed pass through a curtain gas (heated N2 generally) to remove the last solvent molecules and to protect the vacuum system of the apparatus. Finally, the ions are focused toward the vacuum apparatus with the help of a skimmer.

A schematic view of the ESI coupled to a quadrupole mass spectrometry is reported in Figure 2.1. The strength of this approach is the capability of linking it with a mass spectrometer. One of the most beautiful successes of ESI is the ability to produce multiply charged ions of proteins, paving the way of applying gas phase techniques to biological molecules. This is one of the main reasons why John B. Fenn received the Nobel Prize in Chemistry in 2002; the rationale reads: for their development of soft desorption ionization methods for mass spectrometric analyses of biological macromolecules.¹

1 “John B. Fenn – Facts.” Nobelprize.org. Nobel Media AB 2014. Web. 26 Dec 2015. http://www. nobelprize.org/nobel_prizes/chemistry/laureates/2002/fenn-facts.html.

8 | 2 Principles of Experimental Mass Spectrometry

More generally, in vacuo techniques can be now applied to native molecules (i.e., molecules that keep the original atomic sequence they had before ionized). Of course an always open question is whether these molecules in the gas phase have the same three-dimensional structure that they have in the original state. Thanks to these pioneering studies, fields like proteomics [22] were born and many structures from large proteins [23] to noncovalent protein complexes [24] up to whole viruses [25, 26] were ionized and studied through mass spectrometry. Beside the practical interest of ESI, understanding the ionization mechanism is important and interesting per se. The basic mechanism involved in ESI is the transition between charged droplets and bare ions. Highly charged droplets lose solvent molecules following three mechanisms: 1. The ion evaporation model (IEM) [27], that generally holds for ions with low molecular weight. The IEM is based on the fact that the electric field emanating from a charged nanodroplet is sufficiently high to cause the ejection of small solvated ions from the droplet surface. Transition state theory (which will be discussed in Section 5.4) can be applied to evaluate the ejection rate constant: k=

2.

3.

‡ k B T −∆G e kB T h

(2.2)

where ∆G‡ is the activation free energy of leaving the droplet surface. The charge residue model (CRM) [28, 29] that is evoked for large globular species. In the CRM, the highly charged nano droplets containing a single analyte lose solvent molecules one by one up to the last evaporating molecule. The chain ejection model (CEM) recently proposed by Konermann and coworkers for disordered polymers [30].

These three mechanisms are schematized in Figure 2.2. Over the years the basic technique was improved, in particular aiming at increasing the sensitivity at low flow rate. These techniques are referred as microspray [31] and nanospray [32, 33], which is what is generally employed nowadays. From a conceptual point of view of the basic physical chemistry mechanism, they are equivalent to the original ESI, such that in the following we will not make any distinction. Note that even if from a theoretical perspective they are equivalent, in experimental MS/MS the details of ionization source are very important, and the use of different techniques can deal with large differences in the ionization efficiency.

2.2.2 Other Ionization Methods Other ionization methods are used in mass spectrometry, the choice depending on the molecule under analysis. One of the most used techniques with ESI is the aforementioned MALDI (Matrix-Assisted Laser Desorption Ionization) technique [15, 16].

2.2 Ionization

| 9

Fig. 2.2: Schematic picture of the different ESI mechanisms: ion evaporation model (IEM), charge residue model (CRM) and charge ejection model (CEM). Reprinted with permission from [30]. Copyright 2013 American Chemical Society.

In this technique, the molecule is first dissolved in a solvent containing, in the MALDI source, small organic molecules that must be able to strongly absorb the laser pulse at the given wavelength. Then the mixture is dried in order to remove the liquid, resulting in a solid solution of the analyte in the matrix (which is composed by those organic molecules). Then this solid solution is put in the vacuum system and subject to laser ablation. The mechanisms are not understood in full detail [34, 35]. A huge number of macromolecules were ionized properly and studied thanks to MALDI and it is nowadays used for many applications [36–38]. Another technique, which is somehow similar to MALDI, is fast atom bombardment (FAB) [39, 40]. In this technique a current beam of neutral atoms is focused through the sample, which is dissolved in a nonvolatile liquid, generally glycerol. The energetic particles hit the surface of the solution and they induce a shock wave such that ions are ejected from the solution. This technique is thought to generate in the gas phase ions that were already present in solution. If, instead of neutral particles, ions are used in bombardment the technique is called liquid secondary ion mass spectrometry (LSIMS) [41]. FAB was used also to analyze peptides and proteins [42– 45], but nowadays it is generally replaced by ESI and MALDI which are more efficient. A schematic picture valid for both MALDI and FAB methods is reported in Figure 2.3. Beside soft ionization methods which are primarily aimed to ionize macromolecules (biological or not), electron and chemical ionizations (EI and CI, respectively [47–49]) are very common methods, especially in organic chemistry. In EI the

10 | 2 Principles of Experimental Mass Spectrometry

FAB (fast atoms Ar, Xe) MALDI (laser beam UV, IR)

Sample plate Analyte/matrix spot

++-

+ -+ + + + + + + +

To mass spectrometer

Matrix and analyte ions Matrix Analyte

Extraction grid Focussing lens

Fig. 2.3: Schematic picture of MALDI and FAB ionization techniques. Reproduced from [46] with permission from The Royal Society of Chemistry.

sample just vaporizes in the source (high vapor pressure samples are needed) where it encounters electrons with a kinetic energy normally in the 20–70 eV range. If one of the frequencies has an energy corresponding to an electronic transition of the molecule there can be energy transfer. If then there is enough energy, an electron can be expelled, forming an ion. This method is generally used for organic molecules but it can produce unstable ions. Note that (under normal conditions) only positive ions are formed. CI produce ions by ion-molecule collision, in which the molecule under analysis is put as neutral (through simple evaporation) in the source where it encounters an ion. Collisions with the ion can ionize the molecule thanks to different mechanisms, like proton transfer, electron transfer or formation of a charged adduct. CI is a complementary technique of EI and it is also generally used in organic chemistry. The ion is generally CH+5 , which is produced by EI from gaseous methane. The series of reactions involved are generally: − CH4 + e− → CH∙+ 4 + 2e

CH∙+ 4 M

+ CH+5

(2.3)

CH∙3

(2.4)

→ MH + CH4

(2.5)

+ CH4 →

CH+5 + +

2.3 Activation Methods |

11

where M is the molecule of interest. Other reactions can occur forming CH+3 , C2 H+5 or C3 H+5 , which can then react with the molecule forming ions or adducts. A variant of CI is atmospheric pressure chemical ionization (APCI) [50], which is generally used in gas-chromatography mass spectrometry (GC-MS) and able to study molecules with relatively high molecular weight (about up to 1500 Da).

2.3 Activation Methods Once the ion is formed in the gas phase the analysis is often followed by an activation that modifies the m/z of the precursor ion (mpn+ ), forming new ions, mfm+ , which are the result of the fragmentation of the precursor ion. In the case of a singly charged precursor ion, the reaction can be simply considered as m+p → m+f + mn where mn is a neutral molecule that cannot be detected (as we will see in Section 2.4, only charged species can be detected). In general if the precursor ion is multiply charged the products can be either two (or more) ions or an ion and one neutral (or of course several ions and/or several neutral molecules). An example of interesting reactivity of a doubly charged ion will be discussed in Section 7.4. Methods using activation of the precursor ion are referred to as Tandem Mass Spectrometry, MS/MS or MS2 . Here we will see in the details how it is possible to activate ions through collisions (Section 2.3.1), this method(s) taking the name of Collision Induced Dissociation (CID) or Collisional Activated Dissociation (CAD). Other activation methods will be also described, in particular Surface Induced Dissociation (SID) in Section 2.3.2, action spectroscopy (with both IR and UV-Vis lasers) in Section 2.3.3 and dissociation induced by electrons in Section 2.3.4.

2.3.1 Collision Induced Dissociation Collision induced dissociation, CID, is based on a simple physical phenomenon: an ion in vacuum can make collisions with neutral inert species and thus it increases its internal energy in terms of rotation and/or vibration. Neutral species are generally rare gas (Ar, Ne, Xe, . . . ) or molecular N2 . Their role is purely to increase the rovibrational energy of the ion, which can then dissociate. In what follows, we will consider the socalled low-energy limit in which the ion keeps its electronic ground state. Note that while often the activation is considered merely in terms of vibrational activation, the ions are activated also rotationally. Often this is called external rotation to distinguish a rotation of an internal group, which is related to a normal mode and thus considered a “vibration” from the rotation around the three principal axes of inertia.

12 | 2 Principles of Experimental Mass Spectrometry

Thus, given an ion in gas phase encountering a neutral gas, it will do a certain number of collisions that will depend on the thermodynamic conditions. In fact, an estimation of the number of collisions per time unit is given by the mean free path (L), which can be obtained from the kinetic theory of gases L=

kB T √2pσ

(2.6)

where k B is the Boltzmann constant, T the temperature, p the pressure and σ the collision cross section. This is equal to πd2 where d is the sum of the radii of the stationary and colliding molecules. In Section 2.5.2 we will discuss in more detail the evaluation of cross sections, in particular when dealing with reactive trajectories. Ions produced and able to reach the detector will form what is called a mass spectrum. Generally speaking, the CID process can be schematized as (for the simple example of a singly charged ion): M + + N → M +∗ + N M

+∗

+

→A +B

(2.7) (2.8)

It corresponds to dividing the process in two (independent) steps: a first collisional step, where the selected ion, M + , is internally activated by interaction with a neutral inert species (N) and a second fragmentation step in which products are thus formed from fragmentation of the activated molecular ion, M +∗ . This separation often implicitly assumes that the second step is independent from the first. This holds if the excess energy of the activated ion is randomly distributed through its modes before fragmentation. We now distinguish two classes of collisional activation, this distinction being important, as we will see, for the modeling of CID: 1. When the ion is accelerated linearly (for example via a quadrupole, see Section 2.6.1) in an ultravacuum and then brought to a collision room with some gas with which it makes collisions. In this case (and in the ideal limit of a single collision) it is possible to precisely set the collision energy between the ion and the neutral. We will refer in this case to “linear single collision” (LSC). 2. When the ion is trapped with some gas (at low pressures) and it is activated by means of many collisions and it is not possible to exactly know the amount of energy the ion gets due to collisions. Examples are ion traps (like Paul’s trap) or FT-ICR (see Sections 2.6.2 and 2.6.3). We will call this activation “multiple collisions” (MC). An example of MS/MS spectra for the same molecule (protonated N-acetyl O-methoxy proline) but produced through two different activation modes is shown in Figure 2.4. The two activation methods can deal with different mechanisms for the fragmentation step, particularly in the LSC, since the energy is given with a single shock and it

2.3 Activation Methods | 13

(a)

(b)

Fig. 2.4: MS/MS spectrum of protonated N-acetyl O-methoxy proline as obtained on an ion trap (a) and on a triple quadrupole (b) apparatus. Reprinted from [18] [51], with permission from Elsevier.

+ +

(a)

+

+

(b) Fig. 2.5: Schematic picture of CID activation mechanism and resulting fragmentation: (a) single collision limit (LSC in the text), (b) multiple collisions (MC in the text).

can result in local activation and bond breaking before internal energy redistribution. A schematic picture differentiating the two activation modes is reported in Figure 2.5. In fact, in LSC methods the ions are accelerated such that experimentally what an instrument reports is the collision energy in the laboratory framework, ELAB . This value can be simply set and CID done as a function of this energy. As we will see, a collisional system is defined in terms of the reference framework of the center-of-mass of the system composed by the ion and the neutral. We have in this case a collision energy relative to the center-of-mass, ECOM .

14 | 2 Principles of Experimental Mass Spectrometry

Let’s now consider an ion of mass mion hitting a static target (the neutral) of mass mgas and assume they are dimensionless particles. If Rion and R gas are the position vectors in the laboratory framework, the corresponding velocities are: dR ion dt dR gas =0 = dt

vion = vgas

(2.9) (2.10)

Note that in the laboratory framework we consider the target gas as fixed. We set the center of mass of the system composed by the ion and the gas as: RCOM = (mion Rion + mgas Rgas )/(mion + mgas ) Now, the positions of the two species in the COM framework (which is not static, as it moves with respect to the laboratory origin as a function of time) are: r ion = mgas (Rion − R gas )/(mion + mgas )

(2.11)

r gas = −mion (R ion − Rgas )/(mion + mgas )

(2.12)

with the corresponding velocities dr ion = u ion = mgas y/M dt dr gas = u gas = −mion y/M dt

(2.13) (2.14)

where y = v ion − vgas and M = mion + mgas . The kinetic energy in the COM framework can be thus written as K = (mion u 2ion + mgas u 2gas ) /2 = μy2 /2

(2.15)

where μ = mion mgas /(mion + mgas ) is the reduced mass. Since vgas = 0 one can rewrite the kinetic energy as mion mgas v2ion K= (2.16) 2M We now recognize the kinetic energy of the ion in the laboratory framework, Ek = mion v2ion /2, and thus one can obtain the expression normally used to make the energy conversion from the laboratory framework, ELAB = Ek = mion v2ion /2, which is experimentally set in triple-quadrupole mass spectrometer, for example, into the energy in the center-of-mass framework, ECOM : ECOM =

mgas ELAB mion + mgas

(2.17)

The ECOM is the upper limit of the energy that the ion and the neutral can exchange. Normally, only a fraction of them is transferred to the ion. In LSC activation method a limit model is the single-collision in which the ion gets all the energy in one shock and this amount of energy is then used to (eventually) dissociate. As we will see this

2.3 Activation Methods | 15

will not necessary imply that the ions keep a statistical distribution of internal rovibrational states during the dissociation process. In MC methods the ion is confined in a so-called ion trap, which can have different geometries; there are circular, linear, etc, . . . traps. Without detailing the different traps and more in general about the different MC technical methods, what is common is that the ion is activated with several low energy collisions and is thus “heated up” slowly such that it keeps a microcanonical distribution of its rovibrational states. In other terms fragmentation occurs via a statistical process in which internal vibrational redistribution (IVR) occurs between the different collisions. The difference between LSC and MC activation is crucial in order to model them. We will discuss the statistical activation mode in the context of statistical theory of unimolecular dissociation (see Chapter 5). One important point is that the theoretical model should be different if one is concerned with an LSC or a MC process. Finally, we should note that in both methods the energies are on the order of some eV, the low-energy regime being defined as collisions in 1–100 eV range. Note that a high-energy regime corresponds to thousands of eV and is generally attained in magnetic instruments. Here we will not discuss them; their modeling will be completely different, since the physics of ion activation will be different (first of all the Born–Oppenheimer approximation will not be valid anymore).

2.3.2 Surface Induced Dissociation An activation method similar, for some aspects, to CID is Surface Induced Dissociation (SID) [52–54]. In this technique, the ion is accelerated toward a solid surface, which is generally inert. Typical surfaces used are contaminated metals, graphite, diamond, hydrocarbon or fluorocarbon [55]. Thanks to the shock the ion gets internal energy and then it dissociates. The collision between the ion and the surface occurs at a given energy (which can be controlled and is generally in the 1–100 eV range) and impact angle (often an angle of 45° is used). Notably, the evolution of product ions (the socalled breakdown curves, see Section 2.5.1) can be obtained [56]. The ions obtained as fragmentation products are then sent to a quadrupole that is adjusted according to the impact angle (this is why 45° is the optimal value) and analyzed. One advantage of SID with respect to CID is that the fragmentation yield is much higher. Peptides, which often have a low fragmentation yield in CID, were particularly the subject of successful studies via SID [57–60], which recently were applied also to noncovalent protein complexes [61]. On the other hand, SID is still at a research stage, probably since the standard MS apparatus must be substantially modified. From a mechanistic point of view, in SID there is only one collision, such that the collision energy is given in one shot. As lately reported by simulations (see for example the review given in [62, 63]), an important amount of fragments are obtained before internal energy redistribution. Thus, the nature of the surface has an important effect on the products, since it tunes the transferred energy.

16 | 2 Principles of Experimental Mass Spectrometry

2.3.3 Action Spectroscopy Up to now, we have discussed activation methods (CID, SID) based on collisions. Another possibility of activating an ion is to let it interact with light. At this end, two light sources are mainly used: infrared (IR) and UV-visible. These techniques are also called “action” spectroscopy, since the light produces fragments that are then detected. As we will see, for example, the IR spectrum would not be an absorption spectrum, but a function of the amount of product ions at different wavelengths. In the case of IR there are two possible ways of dissociating an ion: via a single photon absorption (Infra Red Photon Dissociation, IRPD) and via multiphoton absorption (Infra Red Multiple Photon Dissociation, IRMPD). IRPD is normally used in cold ion spectroscopy to remove target rare gas atoms to an ion and it will not be discussed here. An interested reader can find explanations and examples in the specific literature [64, 65]. On the other hand, IRMPD is now largely used to characterize relatively large molecules [66–68]. We will discuss only this second method. In both cases, the photon (one or many) excites vibrationally a specific normal mode of the target ion as in IR absorption spectroscopy (see Section 3.5.1 for definition of normal modes and spectroscopy books for general details on spectroscopy [69]). In IRMPD many photons are absorbed and then the ion fragments, and thus the fragmentation yield is not supposed to be linearly connected with either photon absorption cross section nor laser power. The suggested IRMPD mechanism is as follows: a photon is absorbed at a resonant frequency, then the energy is distributed to other modes and the resonant mode relaxes to the ground state via intramolecular vibrational energy redistribution [70]. It can thus reabsorb another photon at the same wavelength and the process goes on, heating the ion up to a given energy threshold leading to dissociation. Thus, the fragmentation happens through a noncoherent process [71]. A schematic picture of the fragmentation mechanism is reported in Figure 2.6. Note that the photon absorption basically heats the ion, such that the fragmentation mechanism is normally considered a statistical dissociation and the role of the light is mainly to give internal energy to the ion without any state-specific effect. Upon irradiation, the intensity of parent Iparent and fragment Ifragment ions are then measured and the final IRMPD yield for a given value of the laser wavelength, η(ω) is given by ∑ Ifragment (ω) η(ω) = (2.18) Iparent (ω) + ∑ Ifragment (ω) In this way the typical experimental fluctuations in the intensity of the parent ion are removed. This is however doable when the intensity of the fragment ions is relevant. In some cases, it is possible that the fragment ions have a low intensity and then the IRMPD spectrum can be measured from the difference of parent ion intensity with and without the laser. This corresponds to the so-called “depletion mode.”

2.3 Activation Methods | 17

Fig. 2.6: Schematic picture of IRMPD activation mechanism considering ν 0 → ν 1 successive vibrational excitations. Reproduced from [67] with permission from The Royal Society of Chemistry.

Different regions of the IR spectrum can be investigated, by means of different instruments. For high frequency regions, 2500–4000 cm−1 , the OPO/OPA laser is used, a relatively simple table laser which can be coupled to many mass spectrometry apparatus. For low frequencies, free electron lasers are used, as in the CLIO (600–2500 cm−1 ) and FELIX (500–2000 cm−1 ) facilities, which are in Orsay (France) and Nijmegen (The Netherlands), respectively. Recently, the FELICE facility (also in Nijmegen) extended the laser range to far IR (50–500 cm−1 ) allowing the study of large amplitude modes. Furthermore, it is possible also to use a simple CO2 laser, which irradiates the ions almost indiscriminately. This is often coupled to other lasers in order to increase the fragmentation yield. By tuning the laser in the allowed wavelength region, it is thus possible to collect the IRMPD yield at different wavelength values and record a spectrum. An example is reported in Figure 2.7. The final IRMPD spectrum looks like a standard IR spectrum but one should always keep in mind that the intensity is not an absorption quantity, i.e., it is not obtained by measuring the light as resulting after being absorbed by the sample but the produced (photo-fragment) ions. Then, quantum chemistry is often used to identify the molecular structure from the spectrum. In most cases simple static calculations are enough but sometimes (especially when anharmonicities are relevant or when the ion experiences a conformational dynamics) molecular dynamics simulations are necessary [4, 73]. While IRMPD is mainly used to investigate the structure of the irradiated ion from the “pseudo” IR spectrum, its primary product is a fragmentation pattern and it is sometimes complementary to CID fragmentation. One important aspect of the IRMPD spectroscopy is that the ions must be confined in a region to be sufficiently irradiate by the laser. For this reason often ion traps and FT-ICR machines are coupled to a laser source. However, it is in principle possible to

18 | 2 Principles of Experimental Mass Spectrometry

appearance spectrum (b4)

depletion spectrum

0.0 kJ mol–1

0.4 kJ mol–1

1000

1200

1400

1600

1800

Wavenumber/cm–1 Fig. 2.7: Example of IRMPD spectrum: protonated Leuenkephalin represented as (A) appearance spectrum and (B) depletion spectrum. Comparison with theoretical spectra of two minima (C and D) is also shown within the corresponding structures. Reprinted with permission from [72]. Copyright 2007 American Chemical Society.

couple it with a linear apparatus, but in this case the incoming ion current must be trapped for irradiation (see Section 2.6.1 for more details). Ions can be activated also via UV-Vis lasers. In this case, while one still deals with action spectroscopy, the underlying mechanism is far different. In fact, when an ion absorbs an UV-Vis pulse, it is promoted to its electronically excited state. Then, excited state dynamics takes place, which can eventually lead to dissociation [74].

2.3.4 Dissociation Induced by Electrons Ions can also be activated by adding an electron, forming a metastable species that can further dissociate. Since a negative charge is added, the precursor ion should be multiple-charged (at least doubly charged). This makes the method particularly useful for studying proteins (or extended polypeptides) which, after ESI, are often present as multiple-charged ions. Two main (related) techniques are used at this aim: electron capture dissociation (ECD) [75–77] and electron transfer dissociation (ETD) [78]. In ECD, the ions are bombarded by an electron beam. These electrons have low energy (< 0.2 eV) such that the capture probability (i.e., the capture cross section) is relatively high and they are produced by a cathode. Since the capture cross section increases as the charge increases, this technique is specially suited for ions with high charge values, as notably proteins. The activation process is very fast and the fragmentation is likely to occur before the excess energy (which is normally in the order

2.4 Principles of Ions Analysis

| 19

of some eV) is randomized between the modes [75]. The process is thus nonstatistical and this is why often products are different from CID or IRMPD. In ETD, multiply-charged ions formed in the gas phase take one electron from an electron donor group, thus forming a radical cation that is generally unstable and therefore fragments. The difference with ECD is thus in the way the ion gets the extra electron. For this reason, to perform ETD, an electron donor species is present in the ion trap analyzer and thus the ion absorbs one electron, forming a radical cation species that is unstable and fragments [79]. The use of ECD and ETD is nowadays growing, in particular in the field of proteomics [80–82]. In this context, two main mechanisms are proposed for the dissociation of peptides: in one, the Cornell model [75, 77, 83, 84], the electron is captured by an X–H group (where X is normally C, O, N or S), the H is transferred to the closer backbone C=O and then the N–C α bond brakes; in the second model, the Utah–Washington model [84–87], the excess electron is captured by the backbone and then it can be followed by a proton transfer and then a N–Cα bond cleavage, or, vice versa, first by the N–Cα cleavage followed by a proton transfer step.

2.4 Principles of Ions Analysis Once the ions are produced, in MS or MS/MS, they have to be analyzed. Different methods, based on different physical properties of the ions, can be used. The choice depends on the machine architecture, on the mass range of the ion on interest and on the resolution required. In all cases, the m/z ratio is detected. Here we review some that are either most “instructive” or most used and notable: magnetic sector, time-offlight, quadrupole, ion trap and cyclotron resonance. Recently, the orbitrap has become very popular, in particular for very large systems. Readers interested in this last case can refer to the specific literature [88]. In what follows we will provide the basic principles that are related to the different ways of analyzing ions. All the mass analyzer are characterized by a different resolution, which can be defined as the ratio between the mass of the peak of interest divided by the width of the peak at a specific fraction of the peak height. For technical details one can refer to the literature devoted to mass spectrometry apparatus.

2.4.1 Magnetic Sector The magnetic sector analyzer is based on the basic properties of ions, which are deflected by a magnetic field. In particular, if the velocity of the ion is perpendicular to the magnetic induction, B, then the force acting on an ion with mass m, charge z and velocity v is: F = zv × B (2.19)

20 | 2 Principles of Experimental Mass Spectrometry

The ion feels the action of both magnetic and centrifugal forces. Thus, for a curvilinear (stable) trajectory with a given radius r we have m|v|2 = z|v||B| r m r|B| = z |v|

(2.20) (2.21)

On the other hand, an ion gets the kinetic energy for a potential difference V zV =

1 2 mv 2

(2.22)

2zV m

(2.23)

v=√

Now, combining equations (2.21) and (2.23), we get the following expression for m/z: m B 2 r2 = z 2V

(2.24)

This equation has two important consequences: (i) since r and V are fixed, one can modify the intensity of the magnetic field, B, and thus select the ion that one will monitor; (ii) not the mass of the ions but the mass-over-charge ratio, m/z, is measured. This is common for all analysis methods in mass spectrometry.

2.4.2 Time-of-Flight The principle of the time-of-flight (TOF) analyzer is based on the separation of ions by their velocity. The ions arrive with a given kinetic energy (see equation (2.22)) and thus the velocity when they come into the analyzer will depend on the mass, the charge and the acceleration potential, V: v=√

2zV m

(2.25)

Since the analyzer is an ultravacuum tube of a given length L, the ions, once accelerated, travel with a constant straight velocity and the time, t, to cover the tube distance is simply L L√m/z (2.26) t= = v √2V Note that the resolution of the instrument is proportional to its length. One can be tempted, thus, to increase the length of the tube to gain resolution. Unfortunately, increasing the length has also a drawback: it will increase the loss of ions due to scattering with some gas molecules (the vacuum is not perfect in real instruments) or due to angular dispersion (the ions are not injected in a perfect collinear way).

2.4 Principles of Ions Analysis

| 21

What we have schematically described is a simple linear TOF analyzer. Nowadays, the ions are deviated in different ways in order to increase the resolution and more generally the performances of TOF analysis. A combination of an orthogonal acceleration [89] and a reflector is what is probably most used nowadays. It is in particular the best choice for coupling with a continuous ionization source. The axial pulse, in fact, transforms a continuous source in a pulse process, thus making it possible to analyze the ions via the time they need to travel from the beginning to the detector.

2.4.3 Quadrupole Another possibility is to use a quadrupole filter, which is composed by four electrodes with hyperbolic or cylindric section. The electrodes have the same electric potential and they are separated by a distance equal to 2r0 . Generally the applied potential has the form Φ0 = U + V cos ωt (2.27) where U is the amplitude of a continuous tension and V the amplitude of a high frequency sinusoidal tension. Inside the quadrupole, the potential is (in Cartesian coordinates) (x2 − y2 ) Φ = Φ0 (2.28) r20 where z is the axis of the quadrupole. The motion of an ion is described by the equations x =0 r20 y m ÿ + 2z(U + V0 cos ωt) 2 = 0 r0

(2.29)

m z̈ = 0

(2.30)

m ẍ + 2z(U + V0 cos ωt)

and making the following substitutions 8zU mω2 r20 4zV q= mω2 r20 ωt τ= 2

a=

we obtain the Mathieu equations of motion: ẍ + (a + 2q cos 2τ)x = 0 ÿ − (a + 2q cos 2τ)y = 0

(2.31)

22 | 2 Principles of Experimental Mass Spectrometry

B m m+1 m+2 0.25 0.20 a = 4eU mr2ω2

E

0.15 0.10

S AS

N CA

LIN

S

M

0.05 0.00 0.0

0.2

0.4

0.6

0.8

q = 2eV mr2w2

Fig. 2.8: a − q stability diagram of a quadrupole filter. The shade area represents the x − y stability region. In the inset the corresponding mass allowed, showing three consecutive m/z values separated by one unit mass. Reprinted with permission from [90]. Copyright 1986 American Chemical Society.

These differential equations have two classes of solutions: 1. Stable trajectories: the ion moves in the z-axis direction, oscillating around this axis with an amplitude that is less than r0 2. Unstable trajectories: the amplitude increases exponentially The stability conditions in x and y have a common region in (a, q) space – this last defines the so-called stability diagram (see Figure 2.8). Since the ratio a/q depends on instrumental parameters (V, U, ω and r0 ) but not on m/z, the intersection between the a/q line and the stability region defines two values of masses that correspond to the range transmitted by the quadrupole filter. The resolution thus depends on the a/q ratio: the bigger this ratio, the closer the line to the vertex of the stability region and consequently the resolution increases (the mass range becoming smaller).

2.4.4 Ion trap filter On the same principles of the quadrupole filter, Paul and Steinwedel built a system composed by a circular electrode, with two ellipsoid caps on the top and the bottom [91]. A resonant frequency is applied on the z axis, thus expelling the ion with a given m/z. Ions are dynamically stored in a three-dimensional quadrupole ion storage device. The radio frequency (RF) and the direct current potentials can be scanned

2.4 Principles of Ions Analysis

|

23

to eject successive m/z from the trap into the detector (which is called mass-selective ejection). Ions are formed within the ion trap or injected into an ion trap from an external source. The ions are dynamically trapped by the applied RF potentials. The trapped ions can be manipulated by RF events to perform ion ejection, ion excitation and massselective ejection. This provides MS/MS and MS/MS/MS experiments. Finally, the ions contained within the ion trap can react with any neutral species present. At odds with a quadrupole filter, in the ion trap all the ions are present at the same time and the selected ones are expelled. This means that ions repel each other, which can result in an expansion of the trajectories in time. At this end a helium gas is injected into the instrument in order to remove the ions’ excess energy by collisions. The Mathieu equations of motion are modified with respect to the ones of the quadrupole filter, but the principle of selecting the masses are the same. More details can be found in specific literature [12, 91].

2.4.5 Cyclotron Resonance The analyzer can be based on the combined effect of an electric field with a radio frequency, ν, and of a magnetic uniform field, H, perpendicular to the electric field. The resulting trajectory is a cycloid and, given the radio frequency, only one ionic species can reach the detector, following the equation ν=

zB 2πm

(2.32)

Varying ν is thus possible to detect different ions still differentiating them by the m/z ratio. This principle is at the basis of the Fourier Transform Ion Cyclotron Resonance (FT-ICR) instrument, which will be briefly discussed in Section 2.6.3.

2.4.6 Detector In all the previous sections, we have discussed the principles for selecting and analyzing the ions obtained. At the end of all the mass spectrometers, the ions are counted in a detector, which is based on the impact of a charged particle on the surface of a metal or a semiconductor, producing the emission of an electron. The emitted electrons generate a current, which is further amplified and converted to a voltage signal, which is translated to an intensity via an analog-to-digital converter. Different technologies can be used, with different sizes and sensitivity, depending on the purposes. These details go beyond the topics of the present chapter, which is devoted to providing only the basic physical-chemical principles of the ions’ behavior in a mass spectrometer.

24 | 2 Principles of Experimental Mass Spectrometry

2.5 Structural Determination Mass spectrometry detects ions that are formed in the apparatus. This can be obtained directly after ionization or as a result of a reaction occurring, for example, upon activation in the gas phase. In all cases, what is measured is the presence or the formation of a molecular species with a given m/z ratio. Normally, from the knowledge of the initial system, it is possible to know the charge and guess the mass of the species. This allows one to precisely know the molecular weight and thus the brute formula of a molecular species. However, nothing is directly related to its structure. To have information on the structure, which is what often asked in physical-chemistry studies, a straight MS or even MS/MS is not enough but they can be used in some specific way in order to go back to the molecular structure. Here we review the most common ways of using MS to infer the structure of the molecule. Often, only one approach is not enough and they have to be coupled to provide a full picture of the molecular structure of interest.

2.5.1 Breakdown Curves In MS/MS the ions are activated with a given energy, which can be clearly known in linear instruments, like by using a quadrupole filter to select the ion we want to fragment (see Section 2.6.1 for technical details). It is thus possible to monitor the abundance of the fragmentation species as a function of the collision energy. The evolution of the ion abundance as a function of the energy gives rise to the so-called breakdown curve. An example is reported in Figure 2.9. Note that it is also possible to do an analogous curve when doing multiple-collisions by varying the potential of the trap, for example. In this case it is not possible to directly know the collision energy, but information on the appearance/disappearance of the different species as a function of energy can be also instructive. However, the structure of the parent ion is not directly accessible either via a breakdown curve, but it is an important piece of information to have more knowledge on the system under study. In fact, from the appearance of the products as a function of the energy and knowing their masses (and that of the parent ion, of course), one can start building a map of the reactivity in terms of weak/strong bonds and ion masses. Breakdown curves serve also to compare the energy needed to open each different reaction pathway and, if the ionization energy is subtracted, it is possible to correlate the ion internal energy distribution (which can be obtained by photoelectron spectroscopy, for example) with mass spectroscopy.

|

25

Relative abundance

2.5 Structural Determination

ECM (eV) Fig. 2.9: Example of a breakdown curve as obtained by CID of L-cysteine anion (m/z 200) at different collision energies. E COM are reported. We thank D. Scuderi and M.E. Crestoni for the raw data.

2.5.2 Cross Section Often in experiments, the so-called cross section is reported as a function of internal energy, in particular in Threshold-CID [92]. In the field of scattering, the cross section has a central role, which now we briefly review. In general, a collision cross section, σ, is defined as σ=

1 λnB

(2.33)

where λ is the mean free path between two collisions and nB = PB /k B T, with PB the pressure of the target gas. The collision rate constant is thus: k(v) = vσ, where v is the speed of the molecule. It is possible to get a temperature rate constant, k(T), by averaging on temperature, and, assuming that the cross section is independent on temperature, one has k(T) = ⟨k(v)⟩ = ⟨vσ⟩ ≈ ⟨v⟩σ (2.34) where the ⟨. . . ⟩ is for the temperature average. This result shows why the cross section is so important: given a thermal velocity distribution of the molecule, it reflects the rate constant.

26 | 2 Principles of Experimental Mass Spectrometry

In the case of reactive collisions, we can analogously define the rate constant from a cross section of the reaction, σ R : k(v) = vσ R

(2.35)

As before, the thermal rate constant can be evaluated from a temperature average. This is done by averaging over a Maxwell–Boltzmann velocity distribution, f(v), such that: k(T) = ⟨vσ R (v)⟩

(2.36)

= ∫ vσ R f(v)dv

(2.37)

=(

3/2 μv2 μ ) ∫ vσ R exp (− ) 4πv2 dv 2πk B T 2k B T

=(

ET ET 8k B T 1/2 ET σ R exp (− ) ∫ )d( ) πμ kB T kB T kB T

(2.38)

where μ is the reduced mass of the system composed by the molecule (the ion), m1 and the target gas molecule, m2 , μ = m1 m2 /(m1 + m2 ), and E T = μv2 /2. The reaction cross section is related to reactive events. In particular, giving collisions with an impact parameter in the range b ∈ [b, b + db] (see Section 3.3.3 for microscopic definition of the impact parameter), the differential cross section is defined in terms of the opacity function, P(b), which is the fraction of collisions leading to the given reaction: dσ R = 2πbP(b)db (2.39) where 2πbdb is the area presented to the colliding reactants when b ∈ [b, b + db]. By integrating, one gets the total reaction cross section as ∞

σ R = 2π ∫ bP(b)db

(2.40)

0

This quantity, which is related to the rate constant via equation (2.38), can be obtained both experimentally and from simulations. Note that if we set P(b) = 1, which could be the probability of making a collision which is by definition one if the integral is limited to d, the molecular size, then we have d

σ = 2π ∫ bdb = πd2

(2.41)

0

which is the result of the collision cross section we have encountered in Section 2.3.1.

2.5.3 Isotopic Labeling Mass spectroscopy directly measures the mass of a molecule, which is the sum of the mass of the nuclei composing the molecule. Nuclei are present naturally within a given isotopic distribution, such that it is possible to detect more than one peak correspond-

2.5 Structural Determination |

27

ing to the same brute formula, in particular when the abundance of two (or more) isotopes is not very different. In any case, natural isotopic abundance is known, and thus one can easily recognize when two (or more) peaks correspond to the same chemical species but from different isotopes. On the other hand, it is possible to increase the abundance of one isotope above the natural level of one (or more) atom in one (or more) position(s) of the molecule. This is the so-called isotope labeling. By comparing the MS and MS/MS of the same molecule with and without the controlled isotope labeling it is possible to know which part of the molecule reacts and which not. This, coupled for example with breakdown curve information, is a very useful tool to guess a fragmentation mechanism.

2.5.4 IRMPD Vibrational spectroscopy is, generally speaking, one of the most used techniques in chemistry to obtain information on the structure. In the gas phase, and in particular in mass spectrometers, however, the density of the molecules of interest is often too low to use absorption spectroscopy. Instead, action spectroscopy is used, which is generally IRMPD in the infrared portion of the spectrum. In Section 2.3.3 we have given some details on the principles of IRMPD (and more generally, action spectroscopy) to activate ions. IRMPD is thus one method of choice to obtain information on the structure of an ion. Since the ions have to be confined in a given portion of the instrument to let the laser efficiently act, systems trapping the ions are used. In practice, generally Paul’s ion trap or FT-ICR are used, while it is possible to use also linear traps or any other device that is able to keep the ions for a given time in a confined region. IRMPD can be done on the parent ion and also on the fragment ions, if the product abundance is sufficiently high. This makes this technique very powerful, not only to access to the structure of the parent molecule, but also on the products, thus providing a lot of information for writing down fragmentation pathways with some direct structural information. For example, IRMPD was used in peptide ion chemistry to decipher when the typical b+n products have an oxazolone either a diketopiperazine structure [93–97].

2.5.5 Theoretical Methods We have already mentioned that quantum chemistry can be used to decipher the structure of the molecule corresponding to a given spectrum, which can be obtained from vibrational or electronic excitation. In quantum chemistry, once the molecular timeindependent Schrödinger equation is solved (see Section 4.3 for details on techniques used to solve this equation for molecular systems) one can get information on the energy (which is a direct result) and on the wave function from which all the molecular properties can be obtained. In particular, quantum chemistry is used to obtain the

28 | 2 Principles of Experimental Mass Spectrometry

minimum energy structure and from it the vibrational frequencies, which is typically done from a normal mode analysis. From normal modes and by calculating the IR absorption factors, one can obtain a theoretical IR spectrum (which corresponds to an absorption IR), which can be compared to an IRMPD spectrum. Note that the intensities do not have the same meaning, since in calculations they are related to the dipole moment of the vibrational transition (thus reflecting the optical spectroscopy transition rules and intensity factors) while in IRMPD they are related to the photo-reactivity associated to the absorption from the molecule of an IR light. However, even if the two intensities are not the same quantity, they are somehow related: a vibration which has high IR intensity generally has also high photo-fragmentation probability and thus IRMPD intensity. One important condition to obtain a vibrational spectrum is to have a geometry that is a local minimum. This means that the derivatives of the energy with respect to the geometry change is zero and that the second derivative is positive. Otherwise, if the second derivative is negative the structure corresponds to a saddle point, which corresponds to a transition state connecting two minima. Note that a molecule can have more than one local minimum, the global minimum being the geometry with the smallest internal energy. In quantum chemistry literature thus a number of algorithms were developed to find local and global minima [98–101]. From each minimum it is possible to obtain a vibrational spectrum and compare it with experiments. Quantum chemistry can also be used to obtain information on the fragmentation pathway. Normally, after the fragmentation pattern for a given parent ion is obtained, the plausible products are calculated with the underlying assumption that the fragments observed are the least energy isomers. This can be a good assumption for statistical (or ergodic) reaction pathways, but it is not always the case for fast reactivity. Then the reactant and the products are connected together via intermediates and transition states (TS’s) connecting the different (local) minima. This is normally done “by hand,” and only recently some automatic algorithms were developed [102–105]. The outcome of this sometime very painful activity is a reaction pathway showing the energies of minima and TS’s connecting the reactant with the different products, as in the example given in Figure 2.10. In these plots, often the internal electronic energy is corrected by the zero point energy (ZPE), which is obtained by calculating the vibrational frequencies as well. Its theoretical origin (a nuclear quantum effect) will be given in Section 3.5.2. Simply speaking, it corresponds to adding to the electronic energy, E, which results from the solution of the time-independent Schrödinger equation, the sum of the energy of each harmonic oscillator corresponding, ω i , to the molecular normal modes at its vibrational ground state: 1 E+ZPE = E + ∑ ℏω i (2.42) 2 i

2.5 Structural Determination

| 29

(a)

(b) Fig. 2.10: Example of a fragmentation PES for (Sr-Formamide)2+ complex. Figure adapted from [106]. Reproduced by permission of the PCCP Owner Societies.

As already mentioned, this so-called “static” approach is normally done after the experiments and, more importantly, based on experimental information. Furthermore, since minima and TS’s are obtained, one supposes that the minimum energy pathways are relevant in fragmentation. Approaches based on molecular dynamics can circumvent those problems. This is the main topic of the book: details on the theoretical basis and methods and some recent applications will be found in all the next chapters.

30 | 2 Principles of Experimental Mass Spectrometry

2.6 Examples of Experimental Apparatus for CID In experiments, ionization, selection, activation and detection parts are put together in almost standard apparatus. Here we review some of them, in particular the ones corresponding more to the two limit ways (LSC and MC) of providing internal energy to the ion. Normally, the companies produce such instruments as a unique piece, such that they can be used without any manipulation from the user. However, when interested in the physical chemistry of ions, most of these instruments are modified, like, for example, to introduce a reactive neutral gas or to send a laser pulse. The ionization part is also something that is now relatively independent, in particular if ESI (with its variants, like nanospray) or MALDI are used. We thus now describe the mass spectrometers after this stage, which is somehow preliminary.

2.6.1 Triple Quadrupole and QQ-TOF Instruments Three quadrupoles can be combined together, forming the so-called triple-quadrupole (or T-quad) mass spectrometer. The three quadrupoles have different uses. The first, Q1, is a mass spectrometer that selects the ions with a given m/z ratio and sent them to the second one (Q2) with a given voltage. In this second quadrupole, where only the radio frequency is switched on, there is an inert gas that makes the CID thanks to the acceleration they got from the previous one. The ion can dissociate here. Finally, the third quadrupole, Q3, analyzes the fragments obtained by CID. As shown in Figure 2.11, they are arranged one after the other in a typical linear geometry. If the pressure of the gas is low, one will have the “single-collision limit” and the collision energy can be set in the laboratory framework by voltage modulation. This energy can be converted to the center-of-mass energy via equation (2.17). The QQ-TOF instrument is based on the same principle, but instead of the third quadrupole, the fragments are analyzed via a TOF. Q1

q2

Q3

selection

CID

analyzer

detector

ions

Fig. 2.11: Schematic picture of a triple quadrupole mass spectrometer.

2.6 Examples of Experimental Apparatus for CID | 31

2.6.2 Ion Trap Instrument An ion trap, whose working principles were given in previous Section 2.4.4, can also be used to do CID. In particular, the ions can be first selected playing with the resonant energy, such that all the ions but the selected m/z are expelled from the trap. Then, the energy is given to the selected ion by collisions with the He gas that is present in the trap. It is possible to play with the voltage in order to increase the internal energy of the fragmenting ion. Note that here the ion makes by definition several collisions with the He gas, such that in principle statistical dissociation conditions are preserved.

2.6.3 FT-ICR Instrument We have described in Section 2.4.5 the principles of the ICR analyzer. Nowadays, ICR is used in the so-called Fourier Transform mass spectrometer, leading to the FT-ICR machine. First, we should note that in the cyclotron, the magnetic field traps the ions only in two dimensions, such that they can escape on the third one, say the z-axis. A trapping potential, Vtrap is thus added, making the ICR a three-dimensional ion trap. This introduces a perturbation in the ideal cyclotron motion of the ions, which now consists of three applied trapping potentials: (i) the axial motion parallel to the magnetic field due to the trapping potential, (ii) the cyclotron motion and (iii) the magnetron motion, which results from the coupling between the magnetic and the electric field. In FT-ICR the detection is based on the measure of the currents in the detector in a well separated time sequence from the excitation. Each ion package induces at the detector an image with its individual cyclotron frequency. By accumulating over passages, a transient current (the free induction decay, FID) is recorded. This time signal is transformed into the frequency domain via a Fourier transformation, thus making the deconvolution of the FID signal composed by the different cyclotron frequencies. Equation (2.32) is then used to convert frequencies into masses.

2.6.4 MS/MS/MS Here we have discussed the CID of the precursor ion that generates fragment ions. It is also possible to make collisions to one selected m/z of these product ions. This corresponds to the so-called MS/MS/MS (or MS3 ). Different approaches can be used for this purpose.

32 | 2 Principles of Experimental Mass Spectrometry

In principle, in a T-Quad scheme one can use the Q3 for further selection, adding another quadrupole for collision just after and then analyzing the products. This is rarely done since it needs a mechanical modification of a mass spectrometer. It is possible to use a T-Quad (or a QQ-TOF) for MS/MS/MS in another way without any modification of the machine. The way of doing it is by performing the so-called “in source” fragmentation. Once one has recorded an MS/MS spectrum and has decided to do MS/MS/MS on a given product ion, it is possible to act on the first MS and try to obtain one of these products as a primary product. This is done by increasing the energetic at the ionization stage, generally increasing the curtain gas pressure. In an ion trap it is possible to do a real MS/MS/MS by selecting in the same trap one of the products, such that in principle n MS can be done (the MS n ). Of course, consecutive fragmentations will decrease the intensity of the ions and this will limit the number of consecutive CID doable. MS n can also be done in a FT-ICR mass spectrometer.

2.7 Conclusions In the present chapter, we have described the basic principles that are behind mass spectrometry of ions, and in particular, tandem mass spectrometry. This is intended for nonexperts in the field who would like to have an overview of the physical basis of the method and from which different theoretical models can be built. Somehow, it is an introduction to the heart of the matter: how it is possible to model tandem mass spectrometry, particularly thanks to chemical dynamics. Here, we have focused our attention on low-energy regime MS/MS where one ion is selected and then activated by one or more collisions with an inert gas, these collisions having the role of transforming translational energy into internal rovibrational energy of the ion. Many different experimental techniques are related to ion mass spectrometry; here we limited ourselves to the basic ones needed to understand the unimolecular fragmentation induced by collisions, which will be the center of our next theoretical development. Readers interested in more details and/or in more sophisticated techniques can (and are encouraged to) consult more specific textbooks and scientific literature.

3 Classical Trajectory Methods 3.1 Equations of Motion 3.1.1 Basic Principles A chemical reaction is, from a microscopic point of view, the evolution in time of a molecular system that follows the laws of physics. Hence nuclear motions in the process of chemical reactions can be treated either by quantum [107] or by classical mechanics [108]. In quantum mechanics, the time-dependent Schrödinger equation governing the motion of nuclei and electrons (here represented by the general variable x holding for both) is ∂Ψ(x, t) HΨ(x, t) = iℏ (3.1) ∂t where H is the quantum mechanical Hamiltonian operator, Ψ(x, t) the time-dependent total wave function and ℏ = h/2π where h is Planck’s constant. The Hamiltonian operator is the sum of a kinetic and a potential energy operator, H = K + V. Generally, none of those operators are explicitly time dependent, thus H = H(x). If we factorize the total wave function as Ψ(x, t) = ψ(x)A(t)

(3.2)

so then equation (3.1) becomes H[ψ(x)A(t)] = iℏ

∂[ψ(x)A(t)] ∂t

(3.3)

∂A(t) A(t)Hψ(x) = iℏψ(x) ∂t and dividing by A(t)ψ(x) we obtain ∂A(t) Hψ(x) iℏ ∂t = ψ(x) A(t)

(3.4)

where the left side is not time-dependent and the right side is not position-dependent. Thus, left and right sides are equal for each (x, t), i.e., they are equal to a constant value, E. From the left side we obtain the well known time-independent Schrödinger equation Hψ(x) = Eψ(x) (3.5) Solving this equation is the field of quantum chemistry and this is discussed in Section 3.1.2, and practical ways of solving the electronic part of the equation (i.e., for https://doi.org/10.1515/9783110434897-003

34 | 3 Classical Trajectory Methods

fixed nuclei) are reported in Section 4.3. The right side, on the other hand, becomes iℏ

∂A(t) = EA(t) ∂t

(3.6)

that is a simple differential equation from which A(t) is readily obtained: A(t) = e −iEt/ℏ

(3.7)

The total wave function now becomes Ψ(x, t) = ψ(x)e−iEt/ℏ

(3.8)

By changing the energy back to the Hamiltonian operator, this equation can be transformed into Ψ(x, t) = e−iHt/ℏ Ψ(x, 0) (3.9) where Ψ(x, 0) is the initial wave packet at t = 0 and e−iHt/ℏ is the so-called time-propagation operator [109]. To manipulate equation (3.9), where the operator is in the exponent, polynomial expansions are generally used thus making the solution complicate and computationally heavy for most molecular systems. Different approaches are proposed in the literature [110, 111] including semiclassical methods [112].

3.1.2 Born–Oppenheimer Approximation For molecular systems, the typical approach is to divide the motion of nuclei from the motion of electrons. This separation is justified by the difference in mass between nuclei and electrons. The total molecular Hamiltonian is, in atomic units: H(r, R) = Tel + Tn + Ven + Vee + Vnn N,M 1 N 1 M 1 2 Zk = − ∑ ∇2i (r) − ∑ ∇k (R) + ∑ − 2 i 2 k Mk |r i − Rk | i,k N

+∑ ij [ r ij ] where K r is the bond stretching constant of each bond, req the corresponding equilibrium bond distance, K θ the angle bending constant and θeq the equilibrium angle, V n , n and γ the dihedral parameters, the A ij and B ij are the Lennard–Jones parameters to take into account the interaction between atoms not directly bound and q i and q j the partial charges on each atom to consider Coulomb interaction between unconnected atoms (with ϵ the dielectric constant). It should be remarked that the last two terms (Lennard–Jones and Coulomb) are nonbonded potentials, but they should be considered also in intramolecular potential for large molecules in which, for example, two parts not directly connected by chemical bonds should interact. Generally, they are excluded for linked atoms and for neighbors up to four atoms. https://doi.org/10.1515/9783110434897-004

56 | 4 Interaction Energy

Force Fields φ

Θ

Estretching = Σ Kij(rij – rijeq )2 i,j

eq )2 Ebending = Σ Kijk(Θijk – Θrijk i,j,k

Etorsion = Σ Kijkl[1 – cos(nφ – δ)] i,j,k,l

E = Estretch + Ebend + Etorsion + Eimpropers + ECoulomb + EvanderWaals

Eimpropers = Σ Kφ(φ – φ eq)2 a,b,c,X

qmqn ECoulomb = Σ 4πε r 0 mn m,n qn = +0.51

c

qk = –0.66

X a

C12 C6 EvanderWaals = Σ r 12 – r 6 mn m,n mn

b

qm = –0.51

Fig. 4.1: Summary of typical interactions used in classical force fields.

A schematic representation of a molecular system with the different bonding, angular, dihedral, Coulomb and Lennard–Jones components is presented in Figure 4.1 When employing such classical force fields, the chemical bonds are defined at the beginning of the simulation and they cannot be broken after collisions. In the context of mass spectrometry this approach was mainly used to understand the energy transfer after collision. The use of classical potentials allows long simulation times, studying extended systems and performing a huge number of trajectories. The amount of energy transferred in ion-projectile collision is a quantity that depends on the geometry and the shape of the ion and on the nature of the projectile, and it is not known a priori. When a collision energy is set, it is assumed that most of it is given to the ion in some cases, but simulations have shown that the fraction is much less. As we will see when describing the theory of unimolecular dissociation, the knowledge of the quantity of energy given to the ion after collison(s) is a key quantity. Furthermore, the ion can be activated by collisions in two ways: vibrationally and rotationally. The partition between these two was shown to depend on the shape of the molecule: globular molecules have a small fraction of rotational activation, while in the case of systems with planar shapes, rotational activation can be important [7, 164]. In the last section of this chapter we will give some theoretical elements of energy transfer due to ionmolecule collision relevant for CID.

4.1 Classical Intramolecular Potential

| 57

When studying large and complex molecules the use of a classical potential avoids the possibility of breaking bonds, thus restricting the application of direct dynamics to the study of energy transfer. However, it is possible to use simple analytical forms in the CID of clusters. In this case, a Morse potential of the form is generally used: Vmorse = ∑ D {1 − exp[−β(r − re )]}

2

(4.3)

couples

allowing the two particles connected by this potential to be brought apart after collision. In this way, the Cr(CO)+6 complex CID was studied [166]. In this case, since the cluster is subject to successive dissociations of CO molecules, the dissociation energy, D, is a function of the extent of dissociation, Ω, Ω=∑ i

1 ri

(4.4)

and an extra term is added, ∑couples ∆D(Ω), where D(Ω) = (a1 + tanh (a2 Ω − a3 )) 2

× (a4 − a5 exp (− (

[(Ω − a6 )3 − a7 ] ) )) a8

(4.5)

The intramolecular potential was initially developed for collisions with surfaces and the different parameters obtained from DFT calculations [9] and, for the dissociation parameters, to reproduce the sequence of Cr–CO bond energies [165], as shown in Figure 4.2.

35

D(Ω)/kcal mol–1

30

Cr(CO)6+

25

Cr(CO)2+ Cr(CO)+

20 15

Cr(CO)5+ Cr(CO)3+ Cr(CO) + 4

10 0

1

2

3

4

5

Ω/Å–1 Fig. 4.2: Dissociation energy for the different Cr(CO)+6 species, written as a function of the extent of dissociation, Ω. Ω and D(Ω) are defined in equations (4.4) and (4.5), respectively. Reproduced from [165] with permission from the PCCP Owner Societies.

58 | 4 Interaction Energy

Some cluster-specific analytical functions can be used. For example, in a study of Aln clusters [7] the Al particles interact with a simple two-body Lennard–Jones V ij =

A B + 6 r12 r ij ij

(4.6)

and a three-body Axilrod–Teller function [167] V ijk = C

1 + 3 cos α 1 cos α 2 cos α 3 (r ij r jk r ki )3

(4.7)

In some special cases, analytical potentials can be used for reactivity when one (or in principle more) bond breaking is known. Toward this aim, Duchovic, Hase and Schlegel developed a CH4 potential (often called DHS potential) in which the C–H bond can be broken without explicitly considering electrons in the description [168] (i.e., without employing ab initio potentials as described in Section 4.3). To this end, Morse potentials of the form shown in equation (4.3) are used, plus some correction on other terms (in this case the pyramidal angle). Recently, the DHS potential was improved by Marques et al. [169] by taking into account H–H repulsion interaction, with an additional two-body term VH–H (R) = A exp(−bR)

(4.8)

multiplied by a switching function f i (RH–H ) =

1 2

{1 − tanh[γ(RH–H − Rcut )]}

(4.9)

in order to study the Ar + CH4 → Ar + H + CH3 reaction. This last potential can be considered a particular case of reactive force fields [170] (ReaxFF) that were developed for different applications, but never used in the context of CID (with the exception of DHS potential). While a ReaxFF approach seems intriguing, it generally needs a specific parametrization for each studied case, thus making its use less appealing.

4.2 Classical Intermolecular Potential The third term of equation (4.1) (Vion-proj ) represents the intermolecular interaction between the ion and the projectile, i.e., between the molecule that is selected for CID and the gas present in the collision room. Experimentally an inert monoatomic (He, Ar, Ne, Xe) or diatomic molecule (N2 ) gas is used. Its role is to convert the translational energy of the ion, accelerated in the collision room, into internal rovibrational energy. This can lead to fragmentation. The ion-projectile interaction is generally treated by an analytical two-body potential, where each atom of the ion interacts with the atom(s)

4.2 Classical Intermolecular Potential |

59

of the projectile. To this end, different functional forms are employed. In particular the general analytical expression V = A exp(−Br) +

C rn

(4.10)

is used, where A, B and C are generally obtained by fitting interaction energies calculated from ab initio calculations and n can be fixed or fitted. When studying relatively large molecules like peptides, a building block approach is used [10]: the interaction of the projectile with CH4 , NH3 and other elements is obtained from high level calculations (e.g., QCISD(T)). Generally, the interaction is set for each hybridization or environment: for C there are values for sp3 and sp2 , for H it depends to which atom it is attached, etc . . . Using this approach, and fixing n = 9, Meroueh and Hase have developed and reported the interaction potential between Ar and building blocks of polyglycines [10] (see Figure 4.3). Using the same approach, recently, parameters to consider aromatic rings, some metals (Li+ , Ca2+ , Sr2+ ) and sulfur atoms were developed, thus allowing the study of a large variety of biomolecules and some cationized species [171–174]. The same functional form was used in the case of N2 interacting with the building block of polyglycines and to study the CID of protonated urea [175]. Values for A, B and C parameters (fixing n = 9) for Ar interacting with different atom types are reported in Table 4.1. At the Tab. 4.1: Parameters used intermolecular potential (equation (4.10) with n = 9) between Ar and different atom types. −1

9

atom

A (kcal/mol)

B (Å )

C (kcal Å /mol)

Reference

C (CH4 ) H (CH4 ) N (NH3 ) H (NH3 ) N (NH+4 ) H (NH+4 ) O (OH; HCO2 H) H (OH; HCO2 H) C (CO; HCO2 H) O (CO; HCO2 H) O (ROR󸀠 ) O (SO4 ) C (COO− ) O (COO− ) S (SO4 ) Ca2+ Sr2+ C (C=C) H (C=C)

11,202.65 8,668.195 8,186.600 4,220.855 13,609.85 10,803.06 15,387.06 8,696.623 8,471.329 12,914.72 21,672.61 15,473.036 10,403.577 13,659.01 49,444.7528 168,360.00 168,349.00 9,714.05 5,678.231

2.399515 3.801426 2.328971 2.982401 2.433643 4.406716 2.698321 4.196012 4.648228 2.681826 3.201106 2.876424 3.063062 2.701491 3.355363 3.99773 3.78322 2.622328 3.033125

152.7291 1.727232 218.8906 3.719138 101.5290 2.066664 90.09528 5.277458 304.6066 99.56698 541.8338 105.7033 159.8867 105.09595 90.95155 1.91387 9.59811 142.4527 1.547723

[10] [10] [10] [10] [10] [10] [10] [10] [10] [10] [171] [171] [130] [130] [171] [172] [173] [174] [174]

60 | 4 Interaction Energy present moment the functional form A exp(−Br) + C/r9 is the most employed one and it can become the standard for building a library useful in simulating every system. A possible tempting option would be to use Lennard–Jones parameters with Lorenz–Berthelot combination rules as generally employed in molecular dynamics based on classical potentials: A ij B ij V ij = − (4.11) 12 (r ij ) (r ij )6 A ij = √ϵ i ϵ j (R ij )12

(4.12)

B ij = 2√ϵ i ϵ j (R ij )

(4.13)

6

where ϵ are the Lennard–Jone’s well depths and R ij the radii. This was tested by Meroueh and Hase [10], using experimental Ar-Ar parameters and Amber force fields for other atoms. Unfortunately, the energy transfer was found to be significantly different with respect to what was obtained by accurate parametrization, and this route was abandoned. It would be, in any case, useful to have a simple rule to obtain interaction parameters without doing ab initio parametrization for each new system and/or projectile not yet parametrized. In the literature, other expressions different from equation (4.10) are present. For example, for the interaction of Ar with Al clusters, the form a b + + c exp(−dr) (4.14) r12 r6 was used by de Sainte Claire et al. [7] and its generalization to study Ar colliding with [Li(uracil)]+ [176]: C E V = A exp(−Br) + D + F (4.15) r r While using parameters from standard intermolecular potential plus Lorenz–Berthelot combination rules seemed to fail, the use of (slightly) different analytical forms, with parameters obtained by high level ab initio calculations, did not show particular differences, as reported by Knyazev and Stein who investigated the reactivity of n-Butylbenzene in CID by employing the expression of equation (4.10) with n = 9 and n = 6 [177]. V=

4.3 Ab Initio Methods Up to now, we have considered classical potential expressions used for inter- and intramolecular interactions. To explicitly treat bond breaking, the straightforward approach is to use quantum chemistry, allowing molecular ions to dissociate after collisions (we have discussed the particular case of specific reactive potentials in Section 4.1). Quantum chemistry is based on the approximate solution of the timeindependent Schrödinger equation HΨ = EΨ

(4.16)

4.3 Ab Initio Methods |

61

Potential Energy (kcal/mol)

2500 H

2000 Ar

C

H

H

1500

H

1000

500

0 0.5

1

1.5

2

2.5

3

rAr–C (angstroms) Fig. 4.3: Example of ab initio fit of equation (4.10) fixing n = 9, as reported by Meroueh and Hase. Reprinted with permission from [10]. Copyright 1999 American Chemical Society.

where H is the molecular Hamiltonian, Ψ the wave function and E the electronic energy. Then forces are needed to integrate equations of motion. The exact force is the first derivative of the energy, which, in quantum mechanics is the expectation value of the Hamiltonian operator: ∇⟨Ψ | H | Ψ⟩ = ⟨Ψ | ∇H | Ψ⟩ + 2⟨∇Ψ | H | Ψ⟩

(4.17)

where the first term, ⟨Ψ|∇H|Ψ⟩, is known as the Hellmann–Feynman force and the second term, ⟨∇Ψ|H|Ψ⟩, is the wave function force or Pulay force.¹ The Hellmann– Feynman theorem [178, 179] states that the Hellmann–Feynman force equals the exact force for a stationary wave function that satisfies the Schrödinger equation. The Hellmann–Feynman force is expressed in an extremely convenient form since ∇H is a one-electron operator and the force is thus readily calculated. Unfortunately, the Hellmann–Feynman theorem is limited by the fact that it is applied only to true stationary state wave functions, like, for example, in the Hartree–Fock limit. In practice, quantum chemistry methods do not provide the true solution of the Schrödinger equation and the correction (the Pulay term) should be calculated. To avoid calculations of such corrections, one possibility is to have an exact eigenstate of the Hamiltonian

1 ∇ is the Laplacian operator, defined as ∇=(

∂ ∂ ,..., ) ∂x 1 ∂x n

(4.18)

The bra, ⟨|, and ket, |⟩, Dirac notation is here used for simplicity. See quantum mechanics textbooks for details.

62 | 4 Interaction Energy

operator or to use basis functions on which to expand Ψ that do not depend explicitly on the nuclear coordinates (like plane waves or one-center orbitals). In fact the Hellmann–Feynman theorem was shown [180] to be satisfied by a variationally determined wave function only if the parameter with respect to which the energy is differentiated is itself a valid variational parameter of the wave function. Another option to avoid the calculation of the Pulay correction term is to obtain forces numerically. In the molecular Hamiltonian, H, used in equation (4.16), to be solved by some quantum mechanics (QM) approximate method, often only intramolecular interactions are considered, while explicit potentials are used for the ion-molecule interaction, as described in Section 4.2, thus leading to a QM+MM approach. It is possible to use, of course, a full QM approach, treating the whole system composed by the ion and the projectile at the same QM level. Anyway, this has some disadvantages: (i) calculations are slower; (ii) convergence can be problematic and (iii) the accuracy is not necessarily better, since MM parameters are obtained by calculations done at a higher level of what can be used at the present time in the QM part of the on-the-fly dynamics. This is particularly true for noble gas atoms, whose interactions with molecules are poorly treated at Hartree–Fock, DFT or MP2 levels of theory, but which, as we will see, are generally used in CID dynamics. In the case of N2 projectiles a full semiempirical approach can be used. Three classes of methods are used for the QM part: (a) wave function based; (b) density functional theory (DFT); (c) semiempirical Hamiltonians. We now review the basis of such methods, with particular emphasis on the advantages and disadvantages and on their use in chemical dynamics for CID. More details can be found in several good textbooks of quantum chemistry [181–183].

4.3.1 Wave Function Methods The simpler (and in quantum chemistry, historically earliest) method to provide approximate solutions of the Schrödinger equation for a molecular system is the Hartree–Fock (HF) method. The solution corresponds to finding a reliable wave function, Ψ, that fulfills equation (4.16), from which it is possible to get energies and forces. The molecular Hamiltonian for a many-electron system (disregarding spin orbit coupling and relativistic effects) can be written in terms of zero-, one- and two-electron terms as follows: H = H0 + ∑ Hi + ∑ Gij (4.19) i

where H0 = ∑ μa

occ. occ. vir. vir.

E(2) = ∑ ∑ ∑ ∑ i

j>i a

2

should be calculated², thus making the MP2 calculations relatively fast. ∗ 2 Where (ij|ab) = ∫ ϕ ∗i (1)ϕ j (1)r −1 12 ϕ a (2)ϕ b (2)dr 1 dr 2 and ∗ (ia|jb) = ∫ ϕ ∗i (1)ϕ a (1)r −1 12 ϕ j (2)ϕ b (2)dr 1 dr 2 .

(4.42)

4.3 Ab Initio Methods | 67

Other more sophisticated methods are based also considering excited states; the most known and used are the Configuration Interaction (CI) and the Coupled Cluster (CC) methods [187]. In CID chemical dynamics, and more in general in direct dynamics, Hartree–Fock and MP2 methods are used to calculate on the fly energies and forces and integrate the equations of motion, as shown in Table 4.2. The more sophisticated methods are computationally too heavy to be applied in dynamics nowadays, and they are generally used to perform parametrizations of semiempirical parameters.

4.3.2 Density Functional Theory Hartre–Fock theory does not take into account the electronic correlation: two electrons with opposite spins are treated as noninteracting particles. We have discussed in the previous section the wave function way of circumventing this problem (Møller– Plesset perturbation theory (MP-n) or couple cluster (CC) methods). In direct dynamics, only MP2 can be sometimes used, while other methods, like MP4 or CCSD(T) are far too expensive for any practical use. An alternative way, very popular in recent decades, and that allows for treating relatively large systems with fair reliable results, is to use density functional theory, DFT [183]. In this theory, the complicated N-electron wave function, Ψ(x 1 , x 2 , . . . , x N ), is replaced by the much simpler electron density, ρ(r). In this way the electron correlation is taken into account (as we will see, the theory is exact in principle, but needs assumptions on the functionals in the practical use). In the original idea of Thomas [188] and Fermi [189] the theory is derived from the following assumptions: (1) electrons are distributed uniformly in the six-dimensional phase space for the motion of an electron at the rate of two for each h3 volume and (2) there is an effective potential field that is itself determined by the nuclear charge and this distribution of electrons. The practical use of DFT was made possible by the two theorems of Hohenberg and Kohn [190] whose statements we report in the following. Theorem 4.3.2 (First Hoheneberg–Kohn theorem). The electron density ρ(r) determines the external potential. It allows us to represent the energy as a function of the density: E[ρ] = Vne [ρ] + T[ρ] + Vee [ρ] = ∫ ρ(r)v(r) + T[ρ] + Vee [ρ]

(4.43)

where v(r) is the external potential, T[ρ] the kinetic energy and Vee [ρ] the electronelectron interaction energy that contains the Coulomb interaction, J[ρ].

68 | 4 Interaction Energy Theorem 4.3.3 (Second Hoeneberg–Kohn theorem). For a trial density ρ̃ (r) ≥ 0 and ∫ ρ̃ (r)dr = N, where N is the number of electrons, it holds that E0 ≤ E[ρ̃ ]

(4.44)

where E[ρ̃ ] is the energy functional. It introduces the variational principle for the ground state, thus restricting DFT theory to the ground state. The practical importance of this theorem is similar to the variational principle of general quantum mechanics: if we assume to have a good functional representation, then we can minimize it (with respect to its parameters) to get the best electronic structure. The use of DFT in quantum chemistry is due to the idea of Kohn and Sham [191] in considering the determinant wave function of N noninteracting electrons in N orbitals, ϕ i . From that, they derived the so-called Kohn–Sham equations for the Kohn– Sham orbitals, ϕ i , used to solve the DFT problem for molecular systems. From these equations, one obtains the exact density once the exact exchange-correlation functional has been determined. Moreover, these equations look like the SCF equations, such that the orbitals can be expanded in terms of atomic basis sets of “standard” quantum chemistry. This is why formally DFT and wave function methods look similar to each other and why they are both present in common quantum chemistry packages. Kohn–Sham equations The Kohn–Sham (KS) orbital equations in their canonical form read [− 12 ∇2 + veff ] ψ i = ϵ i ψ i

(4.45)

veff (r) = v(r) + ∫

ρ(r󸀠 ) |r − r󸀠 |

dr󸀠 + vxc (r)

(4.46)

N

ρ(r) = ∑ ∑ |ψ i (r, s)| i

(4.47)

s

where veff is the effective potential, ρ(r) the electron density, v(r) the external potential and vxc (r) the exchange-correlation potential that depends on the exchange correlation energy, Exc , and density, ρ: vxc (r) =

δExc [ρ] δρ(r)

(4.48)

The procedure to solve the problem is the following: first one solves the Euler equation δT s [ρ] μ = veff (r) + (4.49) δρ(r) where μ are the Lagrange multipliers with the constraint ∫ ρ(r)dr = N. This is what one obtains when applying it to a system of noninteracting electrons moving in

4.3 Ab Initio Methods | 69

the external potential veff (r). Then, one obtains the density and can solve equation (4.45) with the constraint of equation (4.47). Now, veff (r) is a function of the density so equations (4.45), (4.46) and (4.47) are solved via a self-consistent procedure, similarly to what is done in the Hartree–Fock method. Here the starting point will be a guess density from which the effective potential is built, thus obtaining a new density and so on. The energy can be thus expressed as N

E = ∑ ϵi − i

1 ρ(r)ρ(r󸀠 ) ∫ drdr󸀠 + Exc [ρ] − ∫ vxc (r)ρ(r)dr 2 |r − r󸀠 |

(4.50)

Then, the iterations go on until a requested convergence is reached. The way of describing the exchange-correlation functional is a key point, and different ways are present in the literature, like the local density approximation, LDA [183, 192], the Becke exchange correction to the LDA [193] or the Lee–Yang–Parr, LYP [194], correlation potential. In general, we can distinguish different classes of functionals, based on the “correction” to the LDA: the GGA functionals and the metafunctional. Furthermore, it is possible to use the HF exact exchange within the DFT formalism, with a tunable percentage. Typical examples of functionals are the popular B3LYP [193–195] or the new M06 and M06-2X [196] (in this last, which seems to provide very good results in a number of examples, the HF exchange percentage is the double of the original M06, for example). In any case, DFT is in some sense a semiempirical theory, probably the best semiempirical theory. which performs much better than other semiempirical methods, because the parameters, which define the exchange-correlation functional, are not molecule/system specific. This is why DFT was applied to a huge number of different chemical problems, obtaining very good results. In the context of direct dynamics, and more specifically of unimolecular fragmentation induced by collision like in CID, it was applied successfully to relatively small molecules (from three to about ten atoms). In Table 4.2 we report some of the studies performed by using some DFT functionals in the context of theoretical CID.

4.3.3 Semiempirical Hamiltonians Hartree–Fock, post-Hartree Fock and DFT methods, while now applicable to a relatively large set of systems for both static and dynamical calculations, are still computationally too heavy for large systems, particularly when performing chemical dynamics. Regarding the specific problem of addressing the chemical dynamics of CID (as detailed in previous Chapter 3) one has to perform a statistically representative ensemble of trajectories, each one long enough to allow the system to react. Thus, there are two factors that are at the basis of increasing computational time. First, the num-

70 | 4 Interaction Energy

Tab. 4.2: Unimolecular dissociation induced by collisions with inert gases studied by wave function and DFT methods. System

Method

Basis Set

Reference

C2 H5 → C2 H4 + H (Ca-Urea)2+ + Ar Urea-H+ + Ar Urea-H+ + N2 NO+2 + X → NO + O + X (X = Ne, Ar, Kr, Xe) H2 CO+ + Ne N-formylalanylamide + Ar (Ca-formamide)2+ + Ar (Ca-formamide)2+ + Ar (Sr-formamide)2+ + Ar (Sr-formamide)2+ + Ar Uracil-H+ + Ar

UHF MP2 MP2 MP2 B3LYP

4-31G 6-31G* 6-31G* 6-31G* LanLDZ

[197] [172] [198] [175] [199]

B3LYP B3LYP BLYP G96LYP BLYP G96LYP B3LYP and BLYP

6-31G* 6-31G* 6-31G* 6-31G* 6-31+G** 6-31+G** 6-31G and 6-31G*

[200] [201] [173] [173] [173] [173] [174]

ber of trajectories grows with the size of the molecule, since collisions have to be done on the entire surface of the molecule: the bigger the molecule, the larger the number of trajectories. This is reflected, in the methodology described in Chapter 3 by the increasing of the maximum value of the impact parameter sampled. The second point is the simulation time. Even if trajectories are limited in time to some picoseconds, and the full internal vibrational relaxation (IVR) is not attained in simulations, the fragmentation processes are, on average, slower for larger molecules. Full IVR is slower for larger molecules and then also to obtain fragmentations where energy flows (even not within full IVR), longer simulation times are necessary. Thus, when dealing with larger molecules, more and longer simulations are needed. For that reason, HF, postHF and DFT methods are actually computationally too heavy to obtain results that are statistically reliable. An option for relatively large systems is to use the methods based on semiempirical Hamiltonians: these methods are faster and they allow for bond breaking and making (even if, of course, the quality is lower than the previously described methods). They were developed in the “early days” of quantum chemistry when solving Hartree–Fock equations even for organic molecules was not possible. Now for static calculations (i.e., localization of minima and transition states – stationary points) of small-to-medium size molecules they are no longer used, since more reliable calculations are allowed (in particular DFT was applied to systems up to small peptides). But in the context of direct dynamics they were used for a relatively important set of systems (see Table 4.3 for an overview). Furthermore, due to the new parametrizations their accuracy is increasing and thus they are very promising to be used in applications of CID dynamics for larger molecules.

4.3 Ab Initio Methods | 71

Tab. 4.3: Systems studied by semiempirical methods in chemical dynamics of CID. System

Method

Reference

N-protonated glycine + Ar (Li-Uracil)+ + Ar PHB− + N2 (PHB = poly[(R, S)-3-hydroxybutanoic acid]) Uracil-H+ + Ar Galactose-6-sulfate + Ar N-formylalanylamide + Ar C-terminal amidated protonated diglycine + Ar C-terminal amidated protonated triglycine + Ar Uracil-H+ + Ar H+ -Gly5 -R + Ar (R = OH, NH2 ) H+ -Gly8 -R + Ar (R = OH, NH2 ) Pro−2 + Ar TIK(H+ )2 + N2

AM1 AM1 AM1 AM1 PM3 PM3 PM3 PM3 PM3 PM3 PM3 PM3 RM1

[202] [176] [203] [174] [171] [201] [204] [204] [174] [205] [205] [130] [131]

Here we review quickly the basis of semiempirical methods. For a deeper description and discussion of the methods the reader is referred to textbooks [181] and specific articles of each different semiempirical method and parametrization. The basic idea of semiempirical methods is to provide a recipe to quickly obtain the terms of equation (4.41). In particular, one has to obtain expressions for the overlap matrix, S μν and elements of the Fock matrix, Fμν , which is composed by one-electron and two-electron terms. The starting method was proposed by Pople and coworkers in 1965 [206, 207] and it is called complete neglect of differential overlap, CNDO. It is based on: 1. The basis set is composed by one Slater type orbital (STO) per valence orbital; 2. The overlap matrix of equation (4.41) is simply: S μν = δ μν ; 3. The two-electron integrals are parametrized by defining³: (μν|λσ) = δ μν δ λσ (μμ|λλ)

(4.51)

and the surviving integrals are: (μμ|λλ) = γ AB . μ functions are located on atom A and λ functions on atom B. The γ AB values can be explicitly calculated or obtained from some parametrization. One typical parametrization is the Pariser–Parr approximation [208, 209]: γ AA = IP A − EA A , where IP and EA are the ionization potential and electron affinity, respectively. For the two-center term, one popular expression is [210]: γ AA + γ BB γ AB = (4.52) 2 + r AB (γ AA + γ BB )

3 Here we use the compact notation: (μν|λσ) = ∫ ϕ μ (1)ϕ ν (1)r −1 12 ϕ λ (2)ϕ σ (2)dr(1)dr(2).

72 | 4 Interaction Energy

4. For the one-electron integrals, the diagonal elements are simply: 󵄨󵄨 1 Z k 󵄨󵄨󵄨󵄨 󵄨 ⟨μ 󵄨󵄨󵄨 − ∇2 − ∑ 󵄨 μ⟩ = −IP μ − ∑(Z k − δ Z A Z k )γ Ak 󵄨󵄨 2 r k 󵄨󵄨󵄨 k k

(4.53)

and the off-diagonal terms 󵄨󵄨 1 (β A + β B )S μν Z k 󵄨󵄨󵄨󵄨 󵄨 ⟨μ 󵄨󵄨󵄨 − ∇2 − ∑ 󵄨󵄨 ν⟩ = 󵄨󵄨 2 r k 󵄨󵄨 2 k

(4.54)

where β are the semiempirical parameters that are adjusted to reproduce some experimental quantities. The CNDO method was quickly improved by Pople and coworkers with the intermediate neglect of differential overlap (INDO) [211] method and later by Dewar and coworkers who developed the modified neglect of differential overlap (MNDO) [212] method. In MNDO the elements of the Fock matrix are: – For diagonal terms Fμμ = U μ − ∑ Z AB (μμ|s B s B ) + ∑ P νν [(μμ|νν) − 12 (μν|μν)] B=A ̸

ν∈A

+ ∑ ∑ ∑ P λσ (μμ|λσ)

(4.55)

B λ∈B σ∈B

–

where μ is on A, U μ is the atomic orbital ionization potential, s B is the s-type orbital on atom B and P λσ is the density matrix. For off-diagonal terms in the case of two basis functions on the same atom A, then Fμν = − ∑ Z B (μν|s B s B ) + P μν [ 32 (μν|μν) − 12 (μμ|νν)] B=A ̸

+ ∑ ∑ ∑ P λσ (μν|λσ)

(4.56)

B λ∈B σ∈B

–

For off-diagonal terms in the case of two basis functions on different atoms, and notably μ on A and ν on B Fμν =

1 2

(β μ β ν ) S μν −

1 2

∑ ∑ P λσ (μλ|νσ)

(4.57)

λ∈A σ∈B

The crucial point is how to calculate the two-electron integrals. In the MNDO method the continuous charge is replaced with a classical multiple, such that ss product becomes a point charge, sp a dipole (expressed as two point charges) and pp a quadrupole. The energy is completed by the nuclear repulsion energy, that has the form nuclei

V MNDO = ∑ Z k Z l (s k s k |s l s l ) (1 + τe−α Zk r kl + e−α Zl r kl )

(4.58)

k Er ) the reactivity.

5.3.2 Classical vs Quantum RRKM Theory The density and sum of states are key quantities in applying RRKM theory. Let us focus here on density of states, ρ(E), and consider for simplicity just the vibrational part. It is defined as the inverse Laplace transform¹ of the partition function, Q(β), where β = 1/k B T. The important point is thus how we represent the partition function. In the case of a set of s classical harmonic oscillators, the vibrational partition function is s

s

i

i

qv (β) = ∏[βhν i ]−1 = β −s ∏(hν i )−1

(5.20)

from which it can be derived the classical expression for vibrational (harmonic) density of states s E s−1 (5.21) ρ(E) = ∏(hν i )−1 (s − 1)! i Before considering the quantum case, we can now obtain the sum of states, N(E), for the classical case. By definition it is the integration of the density of states over energy,

1 The Laplace transform, L of a function f(t) is a particular integral transform defined as: ∞

L[f(t)] = ∫ f(t)e −st dt = F(P) 0

See more details in Appendix A.

84 | 5 Principles of Chemical Kinetics and from the integration theorem, it is the inverse Laplace transform, L, of Q(β)/β. Thus in the case of s classical harmonic oscillators, we can obtain an analytical expression: s Es N(E) = ∏(hν i )−1 L−1 [(β)−(s+1) ] = ∏(hν i )−1 (5.22) s! i We should note that when applying classical RRKM theory we have to use the classical energy differences, i.e., in the energy of reactants and transition states we do not have to include the zero point vibrational energy (ZPE). On the other hand, if we consider s quantum harmonic oscillators, the partition function is s

−1

qv (β) = ∏ [1 − e−hν i β ]

(5.23)

i=1

and the density of states can be formally written as its inverse Laplace transform integral c+∞

ρ(E) =

1 ∫ qv (β)e βE dβ 2πi

(5.24)

c−i∞

This integral can be solved numerically, for example, by using a steepest descent approximation. There are problems arising when calculating this integral, for example since, in this numerical method, we consider that ρ(E) is a smooth function, while it consists of a series of δ functions. This is why one of the most used ways of calculating the quantum vibrational density of states is by means of the so-called direct count algorithm. In fact, it calculates ρ(E) by dividing the energy scale into a series of cells and counting how many vibrational bands are in each cell. A simplified description of the algorithm as developed by Beyer and Swinehart [247] is:

Algorithm 2: Beyer–Swinehart algorithm to count the density of vibrational states. s is the number of vibrational modes and M sets the maximum energy range. 1: ρ(I) = [1, 0, 0, . . . ] (initialize the ρ vector) 2: for J = 1 to s do

for I = ω(J) to M do ρ(I) = ρ(I) + ρ(I − ω(J)) 5: end for 6: end for 3:

4:

The density of states obtained by this algorithm approximates very well the exact quantum vibrational density of states. The sum of states is calculated in a similar fashion:

5.3 RRKM Theory | 85

Algorithm 3: Beyer–Swinehart algorithm to count the sum of vibrational states. As before, s is the number of vibrational modes and M sets the maximum energy range. 1: N(I) = [1, 1, 1, . . . ] (initialize the N vector) 2: for J = 1 to s do

for I = ω(J) to M do 4: N(I) = N(I) + N(I − ω(J)) 5: end for 6: end for 3:

One drawback of direct counting is that as the system grows, the number of vibrations also grows and the algorithm can become very hard computationally. This was particularly critical in the past, such that the semiclassical state counting of Whitten and Rabinovitch [248] was used: ρ(E) =

(E + aEZPE )s−1 dω(ϵ) ] [1 − β (s − 1)! ∏ hν i dϵ

(5.25)

N(E) =

(E + aEZPE )s s! ∏ hν i

(5.26)

where ϵ = E/EZPE , ω(ϵ) is parametrized, a = 1 − βω(ϵ) and β = ((s − 1)/s)⟨ν2 ⟩/⟨ν⟩2 with ν the frequency of the transition state curvature. It is important to note that with actual computer power, it is possible to use the direct count method for relatively large molecules. When increasing the size of the molecule the critical aspect will not be the calculation of density and sum of states but the localization of the saddle points relevant to the molecular reactivity. If we are interested in simple and well-defined isomerization, like proton transfer between two adjacent sites, the location of the transition state will be also well-defined and, once it it is located, the rate constant can be obtained. As the system grows, a computational problem can arise in using the direct count method. On the other hand, if we are interested in more complex reactions, as the system grows, what will be more challenging is the location (and definition) of a series of minima and transition states determining the pathway connecting the reactants with the desired products univocally. This is where chemical dynamics will play an important role. More details on the complexity of reactivity (and in particular on unimolecular decomposition relevant to mass spectrometry) will be given in Chapters 7 and 8 when discussing the fragmentation of organic molecules and peptides.

5.3.3 Tunneling Effects in RRKM Theory In quantum RRKM theory, as exposed so far, the quantum dynamics effects are considered only in terms of ZPE and partition function (and as a consequence density

86 | 5 Principles of Chemical Kinetics

and sum of states). Another quantum effect can have an impact in calculating rate constant, the well-known tunneling. Tunneling is a quantum mechanical property because there is a finite probability of passing a potential energy barrier even if the energy is less than the barrier itself. In chemistry, tunneling is generally relevant in the case of proton transfers, since, as we will see, the tunneling probability depends on the curvature of the barrier. In the context of RRKM theory it is possible to formally derive the rate constant as a function of the tunneling probability, κ(ϵt ), which depends on the translation energy, ϵt . The rate constant now reads [245]: E−E 0

k(E) = ∫ κ(ϵt )k(E, ϵt )dϵt −E 0 E−E 0

1 = ∫ κ(ϵt )ρ ‡ (E − E0 − ϵt )dϵt hρ(E)

(5.27)

−E 0

For chemical reactions the barrier considered is generally modeled as an Eckart barrier. In this case the transmission coefficient, κ(ϵt ), has an analytical form. Here enters the dependence on the curvature at the barrier. An example of the effect of including tunneling for an Eckart barrier model is reported in Figure 5.1, showing how, as expected, the importance of tunneling is for energies close to or just lower than the energy barrier.

k (s–1)

1e+11

with tunneling without tunneling

1e+10

1e+09

32,000

34,000

36,000 38,000 Energy (cm–1)

40,000

42,000

Fig. 5.1: Rate constant for a model Eckart barrier in which the barrier energy is 32,737 cm−1 and the curvature at the barrier is 2288 cm−1 . In black are rate constants obtained taking into account the tunneling via equation (5.27), while in red we report results without tunneling. Frequencies for minima and TS are the ones for proton transfer in formaldehyde as reported by Miller [245].

5.3 RRKM Theory |

87

5.3.4 Anharmonic Effects In all the derivation exposed so far we assumed that vibrational modes are harmonic. This approximation, while largely employed and working relatively well in many cases, is surely very crude: anharmonicity is important not only for floppy molecules and/or systems at high temperatures, but they are present by definition during dissociation processes (and they are another way of describing the coupling that is necessary in order for IVR to take place; IVR that is a basic assumption of statistical theory). Different approaches are used to consider them. Here we will present some: 1. In principle, one easy way is to do the anharmonic state counting using the modified version of the direct counting method of Bayer and Swinehart reported by Stein and Rabinovitch [249]. For this purpose, it is necessary to define which are the anharmonic modes. While obtaining harmonic modes from quantum chemistry is easily doable via the diagonalization of the Hessian matrix, calculating anharmonic modes is more complex and computationally heavy. Algorithms were developed toward this end in recent years, like the second-order perturbation theory of Bloino and Barone [250], thus paving the way for a broader use of direct calculation of anharmonic density of states. 2. Another way of considering anharmonicity is to obtain the density of states from the partition function via the inverse Laplace transform. Then the point is to obtain the partition function considering the anharmonicities too. This can be done in principle via a Monte Carlo algorithm (or even molecular dynamics), which samples the phase space via an anharmonic potential. Building an analytical anharmonic potential is doable only for simple systems, i.e., systems for which quantum chemistry calculations of anharmonic vibrations are doable. 3. A more straightforward way is to use a global anharmonicity correction factor. This value is difficult to estimate, and it depends on the energy, i.e., fanh = fanh (E). Once estimated it is possible to obtain anharmonic rate constant from: k anh (E) = fanh (E)k har (E)

4.

(5.28)

Some values of fanh are reported in the literature in the 0.2–0.5 range depending on the system and on the energy [140, 197, 251–254]. While this approach is of a simple application, it is clear that it has a high degree of arbitrariness due to the choice of the anharmonicity factor. Another possibility of taking into account anharmonicities in a simple way is to use in density and sum of states calculations, scaled frequencies that take into account the anharmonicity in some way. This is not a full account of anharmonicity effects, but it should be preferable to the use of harmonic frequencies from purely harmonic calculations.

In principle, dynamics can be used to evaluate the anharmonicity and more generally anharmonic rate constants. This was reported in a recent work of Paul and Hase [140].

88 | 5 Principles of Chemical Kinetics

The basic idea is to obtain unimolecular rate constants from chemical dynamics simulations: they will be anharmonic rate constants directly. Unfortunately if the dynamic is Newtonian (as is often in the case of molecular systems) then the resulting rate constant will be a classical rate constant. To obtain the quantum rate constant then two possibilities are available: (i) to perform quantum simulations or in any case to consider the ZPE at the transition state, or (ii) to obtain the anharmonicity factor by comparing classical anharmonic (issued from dynamics) and harmonic (from analytical theory) rate constants. Unfortunately it is not clear how to obtain fanh (E) without performing simulations at the given internal energy (see [140] for a deeper discussion on this point).

5.3.5 Loose Transition States So far we have considered reactivity through a tight TS. In this case the TS is located geometrically, analogous to the minimum location (with the difference that the topology has now one imaginary frequency). As described previously, it is possible that the TS is a loose TS (this occurs in particular in evaporation mechanisms), and thus its location cannot be known as easily as in the case of a tight TS. Formally a loose TS is defined as: (i) a TS for which ∆S‡ < 0 and (ii) a reaction with no reverse activation energy. A more general definition of a loose TS location is strictly related to the variational-TST theory (VTST) and will be practical since it gives a direct procedure to locate it. In fact a TS (loose or tight) corresponds to the minimum in the sum of states along the reaction coordinate. This means that the TS can be obtained by locating the minimum flux by setting the derivative of the sum of states to zero dN ‡ (E, R) =0 dR

(5.29)

The distance, R, which fulfills the minimum condition of equation (5.29), will provide the position of the loose TS, R‡ . We should note that in the case of loose TS the geometry will depend on the internal energy. In practice, to locate the loose TS, more laborious and often delicate calculations are necessary. In particular, one should first know the reaction coordinate (RC) and then obtain structures that are a minimum in all the other coordinates but the RC. Then frequencies are calculated and projected on the RC, to be used to calculate the sum of states along the RC and locating the minimum. This last step needs to be repeated at the energies of interest and a set of transition states will be located in this way. They can be used to calculate the density of states and finally to get the corresponding rate constant. Figure 5.2 shows the way used to obtain loose TS and rate constants graphically. We should note that the more delicate part is locating the structures along the RC correctly. Often, in fact, they diverge from the desired RC and a careful inspection of the initial geometries that are supposed to be close to the particular stationary

5.4 Transition State Theory

| 89

3.17

3.37

Sum of States (N)

3.57

1e+14

250 ×

200

×

×

3.77

×

×

150

×

×

×

1e+12 1e+10 1e+08

3.97

k (s–1)

300 ×

1e+06 ×

100

Reaction Coordinate

10000 0

100 200 E (kcal/mol)

300

Fig. 5.2: Example of loose TS location and rate constant calculation for the reaction: (Ca-formamide)2+ → Ca2+ + formamide. [255] Reproduced by permission of the PCCP Owner Societies.

points along the RC is necessary. This procedure is somehow an “art” and often fails (for example when other reaction channels are close, like proton transfer). In general it is useful for evaporation as well, in a broad sense, for the detachment of an atom (or more likely a group of atoms), like recently reported in the case of Ca2+ -formamide complex. [255]

5.4 Transition State Theory Up to now, we have formulated the rate constant in the microcanonical ensemble (constant energy). In chemical kinetics the way used more often is to formulate it in the canonical ensemble (constant temperature). While the microcanonical formulation seems more straightforward for gas phase reactivity, the canonical ensemble one is also useful. The most common formulation is the transition state theory (TST), known also as the Eyring theory [256, 257]. We have already discussed the statistical assumptions in RRKM theory, which are the same as in TST. Now, let us consider a quasiequilibrium between reactants (A) and the transition state (A‡ ), such that we can formally write an equilibrium constant: [A]‡ K‡ = (5.30) [A] As an assumption of TST, once the system has reached the TS, the velocity of product (P) formation will depend on the amount of species at TS, [A]‡ , and a constant, k ‡ , which is proportional to the frequency corresponding to the TS coordinate, ν, such

90 | 5 Principles of Chemical Kinetics that k ‡ = κν and

d[P] = k ‡ [A]‡ = k ‡ K ‡ [A] = k[A] dt

(5.31)

where k is the TST rate constant we want to evaluate, which results to be k = k ‡ K ‡ . K ‡ is formally an equilibrium constant between A and A‡ , which differ, at a first approximation, only for the frequency at the TS, ν, such that we can assume the expression K‡ =

k B T −∆G‡ /RT e hν

(5.32)

and thus the rate constant will have the simple expression k B T −∆G‡/RT e h k B T ∆S‡ /R −∆H ‡ /RT =κ e e h

k(T) = κ

(5.33) (5.34)

where the equation (5.34) is the so-called Eyring formula. In Section 5.3 we have discussed the microcanonical rate constant, k(E). The canonical rate constant, k(T), can be derived from it as follows. Given that the reactants are in thermal equilibrium, by integrating k(E) over the density of states at given temperature, β = 1/k B T, we have: ∞ −1

k(T) = Q(T)

∫ k(E)ρ(E)e−βE dE

(5.35)

0

where Q(T) is the partition function, which can also be expressed as an average with Boltzmann weight of the density of states ∞

Q(T) = ∫ ρ(E)e−βE dE

(5.36)

0

if now we use the equation (5.11) to express k(E) we have ∞ −1

k(T) = Q(T)

∫

N ‡ (E − E0 ) ρ(E)e−βE dE hρ(E)

0 ∞

=

=

1 ∫ N ‡ (E − E0 )e−βE dE hQ(T) e−βE0 hQ(T)

E0 ∞

∫ N ‡ (E)e−βE dE

(5.37)

0

Note that the last passages were possible since N ‡ assumes nonzero values only for E > E0 by definition, since it is the sum of states at the TS. We can then rewrite the last

5.5 Phase Space Theory

| 91

integral as a Laplace transform, such that ∞

∫ N ‡ (E)e−βE dE = L[N ‡ (E)] 0 E

= L [∫ ρ ‡ (E)dE] = [0 ] = Q‡ (T)k B T

L[ρ ‡ (E)] β (5.38)

The thermal rate constant, k(T), can be rewritten as k(T) =

k B TQ‡ (T) −E0 /kB T e hQ(T)

(5.39)

which is the general formulation of transition state theory.

5.5 Phase Space Theory Reactions with no potential energy barrier on the minimum-energy path (MEP) curve have been one of the most active subjects in TST studies [258–260]. Conventional TST, which assumes the transition state is located on the top of the barrier, cannot be applied to this kind of system. An early approach to this problem is the orbiting transition state (OTS), which assumes the transition state to be on top of the effective potential, which is the sum of the potential energy and centrifugal term [261]. This will be briefly discussed below. Phase space theory (PST) was originally developed by Pechukas and Light [262– 265] for atom-diatom reactions. This was further generalized by Klots [266–268] for general systems without linear fragments. This is based on a “vertical” approximation also known as the Langevin model for the total cross section [266]. As discussed by Chesnavich and coworkers [269, 270], the method developed by Klots can be used for systems with linear fragments in the K → 0 limit, which is valid for rotationally cold systems, such as in the unimolecular reaction. The generalization of PST theory was made by Chesnavich and coworkers [269, 270] by introducing an approximation that can be used for systems with linear fragments for a wide range of total angular momentum. The basic assumption of PST is that the transition state occurs at a large distance between fragments. Due to this large distance, only the long-range potential term can be considered, which is a “physical” force, e.g., an ion-induced dipole interaction term. Also, the transition state degrees of freedom, and hence the sum of states, are usually well-defined, since they are the degrees of freedom of the fragments themselves. The transition state of PST is called the orbiting transition state (OTS), which assumes that the fragments rotate freely and that two fragments make an orbiting motion with respect to each other. The location of the OTS is determined by the maximum

92 | 5 Principles of Chemical Kinetics

of the effective potential [261]. The expression for the OTS sum of states as a function of total energy E and angular momentum L is N(E, K) = ∫ ρ J (E∗r , E j )ρ v (Ev )dLdE J dEv

(5.40)

where K = L + J, L is the orbiting angular moment and J is the angular momentum of the molecule, ∗ (L) E∗r = E − Ev − Veff

EJ = ρ J (E∗r , E J )dE J

=

(5.41)

Br J r2

(5.42)

Nr (E∗r , J r )dJ r

(5.43)

Nr (E∗r , J r ) being the sum of rotational states, such that ρ J (E∗r , E J ) is the density of such states, and ρ v (Ev ) is the vibrational density of states. The reduced rotational constant Br is defined as the rotational constant of the molecule itself for an atom-molecule −1 −1 pair, or, for a molecule-molecule pair, it is defined as B−1 r = B 1 + B 2 , where B 1 and B2 are the rotational constants of the molecule. The expressions for the rotational sum of states, Nr (E∗r , J r ), for every combination of fragments are given by Chesnavich and Bowers [269, 271, 272]. This theory was successfully applied to (H2 O)OH− + CO2 → HCO−3 + H2 O reaction, for example [270]. Recently Calvo and coworkers [273, 274] have extensively applied PST to cluster evaporation and they included quantum effects too.

5.6 Quantum Rate Constant In Section 5.4 we have provided the TST rate constant in classical mechanics formulation. We now give some basic idea of how it should be translated in quantum mechanics pictures (see [275] for more details and derivations). Given that in quantum mechanics the partition function of a species A has the expression Z A = Tr [e−βH0 ]

(5.44)

if we consider a A → B transformation, the associated rate constant has the expression under the TST approximation: k QM,TST =

Tr [e−βH δ(s) mp ss h(p s )] ZA

(5.45)

where s is the vector perpendicular to the dividing surface, with p s and m s being the associated momentum and mass, H the Hamiltonian operator and δ the Dirac delta function. h(p s ) is a step function that reflects the TST approximation: all the trajectories moving from the dividing surface will lead to products (in mathematical terms this corresponds to positive momenta). A more extensive use of the step (or Heaviside)

5.7 Correlation Function Formalism

|

93

function will be described in Section 5.7 when the rate constant will be expressed in terms of correlation function. Note that since the h(p s ) function is an approximation of the exact projection operator, it does not commute with the total Hamiltonian, making the choice of the ordering in operators not evident. If one considers two coordinates, s, which is the previously defined motion perpendicular to the dividing surface and u as a motion perpendicular to s, for which the one-dimensional Hamiltonians are h s and h u , respectively, then the rate constant can be written as 1 ∂ ℏ 󵄨 󵄨 k QM,TST = ∫ (− ) ⟨su 󵄨󵄨󵄨󵄨 e−βH 󵄨󵄨󵄨󵄨 − s, u⟩ dsdu Z A 4πm s s ∂s =Γ

kB T Z u h ZA

(5.46)

where 1 d ℏ2 β 󵄨 󵄨 ∫ (− ) ⟨s 󵄨󵄨󵄨󵄨 e−βh s 󵄨󵄨󵄨󵄨 − s⟩ ds 2m s s ds 󵄨 󵄨 Z u = ∫ ⟨u 󵄨󵄨󵄨󵄨 e−βh u 󵄨󵄨󵄨󵄨 u⟩ du Γ=

(5.47) (5.48)

The expression in equation (5.46) can be considered analogous to the TST rate constant of equation (5.39) in which Γ is a tunneling correction for a motion along s and the partition functions are quantum.

5.7 Correlation Function Formalism 5.7.1 Generalities A powerful expression of rate constant uses the time-correlation function approach. This is based on the Onsager regression theory and the fluctuation-dissipation theorem. Briefly, if we have a quantity A, its instantaneous deviation from the equilibrium condition, δA(t), can be expressed as: δA(t) = A(t) − ⟨A⟩

(5.49)

where ⟨A⟩ is the equilibrium (time-independent) average. The equilibrium of such a quantity is zero if A is not a constant of motion, while more interesting is the correlation with a fluctuation at t = 0, i.e., C(t) = ⟨δA(0)δA(t)⟩ = ⟨A(0)A(t)⟩ − ⟨A⟩2

(5.50)

where the average is done on the initial conditions. C(t) is the time correlation function, which for a classical system, is C(t) = ∫ Φ(r, p)δA(r, p; t = 0)δA(r, p; t)drdp where Φ(r, p) is the classical equilibrium distribution in the phase space.

(5.51)

94 | 5 Principles of Chemical Kinetics

In classical mechanics, the following properties hold for the correlation function: C(t) = ⟨δA(0)δA(t)⟩ = ⟨δA(−t)δA(0)⟩

(5.52)

C(t) = ⟨δA(0)δA(−t)⟩ = C(−t)

(5.53)

2

(5.54)

C(0) = ⟨δA(0)δA(0)⟩ = ⟨(δA) ⟩ lim C(t) = ⟨δA(0)⟩⟨δA(t)⟩

t→∞

=0

(5.55)

Note that the equation (5.53) implies that A(0) and A(−t) commutes, which is not necessary in quantum mechanics, while the last property (equation (5.55)) is nothing more than the Onsager regression of spontaneous fluctuations. The Fluctuation-Dissipation Theorem The fluctuation-dissipation theorem is a crucial theorem in statistical mechanics that can be formulated in various forms. One possible expression is [276]: Theorem 5.7.1 (The fluctuation-dissipation theorem). The linear response of a given system to an external perturbation is expressed in terms of fluctuation properties of the system in thermal equilibrium. Here we will briefly provide two possible manifestations in statistical mechanics. 1. One is related to the Einstein relation [277] between the diffusion constant, D, of a Brownian particle and the friction, γ: kB T D= (5.56) mγ In the presence of a potential, V(x), the particle flows with a velocity (the drift velocity) 1 dV(x) v=− (5.57) mγ dx such that being t

x(t) − x(0) = ∫ v(τ)dτ

(5.58)

0

the diffusion coefficient can be expressed as 1 ⟨[x(t) − x(0)]2 ⟩ D = lim t→∞ 2t

(5.59)

∞

= ∫ ⟨v(t0 )v(t0 + t)⟩dt

(5.60)

0

and thus the Einstein relation (equation (5.56)) becomes ∞

1 D 1 ∫ ⟨v(t0 )v(t0 + t)⟩dt = = mγ k B T k B T

(5.61)

0

which says that the diffusion is related to the fluctuation of the velocity of the Brownian particle.

5.7 Correlation Function Formalism

2.

|

95

A second and powerful manifestation comes into the classical Langevin equation of motion. Given a Brownian particle, its motion in the presence of an external field, V(x), can be expressed as ̇ = mγv − m v(t)

∂V + R(t) ∂x

(5.62)

where γ is the friction as in the Einstein relation (see equation (5.56)) and R(t) is the random force due to random collisions of the surrounding molecules. The random force is assumed to be in classical statistical mechanics a Gaussian process with an infinitely short correlation time ⟨R(t1 )R(t2 )⟩ = 2πG R δ(t1 − t2 )

(5.63)

where G R is a constant. Combining these assumptions on the random force with the Fokker–Planck for the transition probability from v0 to v t we obtain some important results [276]. First, the diffusion coefficient in velocity space, Dv , can be determined by the random force, ∞

m2 Dv 1 mγ = ∫ ⟨R(t0 )R(t0 + t)dt = kB T kB T

(5.64)

0

Second, that the mk B T γ (5.65) π and if we define R(ω) as the power spectrum of the random force (i.e., its Fourier transform), we obtain GR =

⟨R(ω)R(ω󸀠 )⟩ =

k B Tmγ δ(ω + ω󸀠 ) π

(5.66)

In other words, the friction is connected with how the external work is dissipated into microscopic thermal energy and the reverse process is the generation of random force, which is a consequence of thermal fluctuation.

5.7.2 Classical Rate Constant In the field of chemical kinetics, a typical way of expressing the rate constant via the time-correlation function approach can be elucidated with the simple case of isomerization. Given two species A and B, whose time dependent concentrations are c A (t) and c B (t), respectively, for which the rate constants for forward and backward transformation are k BA and k AB , respectively, then the kinetic equations can be written as dc A { { dt = −k BA c A (t) + k AB c B (t) { { dc B { dt = k BA c A (t) − k AB c B (t)

(5.67)

96 | 5 Principles of Chemical Kinetics

The following conditions must be preserved: c A (t) + c B (t) = const. −k BA ⟨c A ⟩ + k AB ⟨c B ⟩ = 0 Keq =

⟨c B ⟩ k BA = ⟨c A ⟩ k AB

(5.68) (5.69) (5.70)

in which ⟨c A ⟩ and ⟨c B ⟩ are the equilibrium concentrations and Keq is the thermodynamic equilibrium constant. These equations can be solved giving ∆c A (t) = c A (t) − ⟨c1 ⟩ = ∆c A (0)e−t/τ

(5.71)

where τ−1 = k BA + k AB . If we can express the time evolution of the concentrations, c J (where J = A, B) via a dynamical variable, n J (t), then from the fluctuation-dissipation theorem, we have ∆c A (t) ⟨δn A (0)δn A (t)⟩ = (5.72) ∆c A (0) ⟨(δn A )2 ⟩ ⟨δn A (0)δn A (t)⟩ e−t/τ = (5.73) ⟨(δn A )2 ⟩ Equation (5.73) gives a direct connection between the rate constants and the microscopic dynamics. In the TST theory, the dividing surface can be used to define this quantity. In particular, if we consider a reaction coordinate, Q, and the position at the transition state as Q‡ , for which A corresponds to Q < Q‡ , then we can define the population (Heaviside) function, H A (t), as { H A (Q) = 1 , { =0, {

Q < Q‡ Q > Q‡

(5.74)

and trivially for B H B (Q) = 1 − H A (Q)

(5.75)

Using this definition, the average fraction of A and B, x A and x B , fulfills the following relationships: x A = ⟨H A ⟩ =

⟨c A ⟩ ⟨c A ⟩ + ⟨c A ⟩

(5.76)

xB = 1 − xA

(5.77)

2

(5.78)

⟨(δH A ) ⟩ = x A (1 − x A ) = x A x B

From the fluctuation dissipation theorem we can thus express the rate constant in terms of the H A function as e−t/τ = (x A x B )−1 [⟨H A (0)H A (t)⟩ − x2A ]

(5.79)

This expression can be further developed if we consider the time derivatives² d ̇ Ḣ A [Q] = Q̇ − Q‡ ) H A [Q] = −Qδ(Q dQ ̇ 2 Note that ⟨ Q(0)δ[Q(0) − Q ‡ ]⟩ = 0.

(5.80)

5.7 Correlation Function Formalism

| 97

̇ − Q‡ ]H A [Q(t)]⟩ − ⟨H A (0)Ḣ A (t)⟩ = − ⟨Q(0)δ[Q(0)

(5.81)

̇ = − ⟨Q(0)δ[Q(0) − Q‡ ]H B [Q(t)]⟩

(5.82)

Now if we consider the time derivative of equation (5.79), τ −1 e−t/τ = −(x A x B ) − ⟨H A (0)Ḣ A (t)⟩

(5.83)

τe−t/τ = (x A x B )−1 ⟨v(0)δ[Q(0) − Q‡ ]H B [Q(t)]⟩

(5.84)

it can be expressed as

̇ where v(0) = Q(0). The right hand side of equation (5.84) is the average flux crossing the surface identified by Q = Q‡ . At short times the behavior should not correspond to the macroscopic exponential decay, since the phenomenological rate constants are correct only on longer time scales. If we notice as τ short the time scale corresponding to the short relaxation times, for time windows, ∆t, for which ∆t ≫ τshort

(5.85)

∆t ≪ τ

(5.86)

then the exponential of equation (5.84) can be approximated as e−∆t/τ ≈ 1. Thus for time scales that are longer than the short relaxation time but shorter than the macroscopic times, we have τ−1 = (x A x B )−1 ⟨v(0)δ[Q(0) − Q‡ ]H B [Q(∆t)]⟩ k BA =

x−A 1 ⟨v(0)δ[Q(0)

‡

− Q ]H B [Q(∆t)]⟩

(5.87) (5.88)

This is the well-known result developed by Chandler [278, 279] expressing the rate constant in terms of correlation function of a Heaviside function. While it is not used in theoretical mass spectrometry (up to now), it is at the basis of the so-called transition path sampling method that is used to study rare events [280]. Generally k BA has the behavior shown in Figure 5.3. For t < τshort the reaction decay deviates from the phenomenological rate constant, which is reached as a plateau. If a system does not show any plateau, it means that the typical kinetic description, for which the rate constant is time independent, does not hold. In TST all the trajectories that are directed from Q‡ toward B region will end up in B and vice versa. The Heaviside function will be {H TST B [Q(t)] = 1 , { =0, {

v(0) > 0 v(0) < 0

(5.89)

and thus the rate constant becomes −1 ‡ TST k TST BA = (x A ) ⟨v(0)δ[Q(0) − Q ]H B [Q(t)]⟩

(5.90)

98 | 5 Principles of Chemical Kinetics

Fig. 5.3: Schematic picture of time evolution of rate constant as from equation (5.88). The short time scale, τ short , and the long phenomenological rate time, τ, are also sketched. Note that for t = 0 the rate constant coincides with the TST one (see equation (5.90)).

which coincides with the rate constant of equation (5.88) at t = 0, i.e., k BA (0) =

1 ⟨|v|⟩ ⟨δ(Q − Q‡ )⟩ 2x A

= k TST BA

(5.91)

We should remark here a result that we already knew about TST approximation: it includes no information about the reaction dynamics and it is entirely dependent on the equilibrium conditions of the system.

5.7.3 Quantum Mechanics Formulation In quantum mechanics the rate constant can be developed analogously (see for example [156]). If we now note Z the quantum mechanics partition function (defined as in equation (5.44)), every equilibrium macroscopic observable is ⟨̂ A⟩QM =

Tr [̂ Ae−βH ] Z

(5.92)

where ̂ A is the quantum mechanical operator associated to the observable A and Tr is the trace operation. In general, if we have two quantities A and B for which the classical time correlation function is the average on the classical ensemble τ

1 C(t) = ⟨A(t0 )B(t + t0 )⟩ = lim ∫ dtA(t0 )B(t0 + t) τ→∞ τ 0

1 = ∫ e−βH(r,p) A[r(t), p(t)]B[r(t), p(t)]drdp Z

(5.93)

5.7 Correlation Function Formalism

| 99

which is the same as equation (5.51) where now we have expressed the equilibrium distribution in standard classical mechanics, Φ = e−βH(r,p) /Z, and then the quantum mechanical expression is C(t) =

̂ Tr [e−βH ̂ A B(t)] Z

(5.94)

̂ = e−iHt/ℏ Be ̂ in the Heisenberg representation. ̂ −iHt/ℏ is the operator B where B(t) If we now use the quantum mechanical operators for the Heaviside function used to identify when the system is in state A or B, we end up with: ̂ A e−iHt/ℏ ̂ A (t) = e iHt/ℏ h h and the equilibrium fraction xA =

̂ A e−βH ] Tr [h

Z As previously, for a time scale ∆t for which τshort ≪ ∆t ≪ τ we have [281]

(5.95)

(5.96)

βℏ

k BA

1 =− ∫ ⟨h A (−iϵ)ḣ A (∆t)⟩ dϵ βℏx1

(5.97)

0

Since the stationary condition implies ⟨h A (−iϵ)ḣ A (∆t)⟩ = − ⟨ḣ A (−iϵ)h A (∆t)⟩

(5.98)

the integration in ϵ will provide the following expression for the rate constant [156] k BA =

1 h A (0) − h A (−iβℏ) ⟨[ ] h A (∆t)⟩ xA iβℏ

(5.99)

where the term in [. . . ] is the equivalent of a flux operator. By expanding in Taylor series βℏ and expressing the rate constant in terms of matrix elements, we can write k BA =

1 h A (Q) − h A (Q󸀠 ) 󵄨 󵄨 󵄨 󵄨 ∫[ ] ⟨Q󸀠 󵄨󵄨󵄨󵄨 e−βH 󵄨󵄨󵄨󵄨 Q⟩ ⟨Q 󵄨󵄨󵄨󵄨 h A (∆t) 󵄨󵄨󵄨󵄨 Q󸀠 ⟩ dQdQ󸀠 ZA iβℏ

(5.100)

󵄨 󵄨 Z A = Tr [h A e−βH ] = ∫ h A (Q) ⟨Q 󵄨󵄨󵄨󵄨 e−βH 󵄨󵄨󵄨󵄨 Q⟩ dQ

(5.101)

where

Since the term ⟨Q|e−βH |Q󸀠 ⟩ = ρ(Q, Q󸀠 ; β) is the thermal density, which is real and even on interchanging Q with Q󸀠 and since the term ⟨Q|h A (∆t)|Q󸀠 ⟩ is Hermitian, then the rate constant can be expressed as [156] k B T 4π ∫ h A (Q)h B (Q󸀠 )ρ(Q󸀠 , Q; β)z(Q, Q󸀠 ; ∆t)dQdQ󸀠 ) h ZA

(5.102)

󵄨 󵄨 󵄨 󵄨 z(Q, Q󸀠 ; ∆t) = ℑ ⟨Q 󵄨󵄨󵄨󵄨 h A (∆t) 󵄨󵄨󵄨󵄨 Q󸀠 ⟩ = −ℑ ⟨Q 󵄨󵄨󵄨󵄨 h B (∆t) 󵄨󵄨󵄨󵄨 Q󸀠 ⟩

(5.103)

k BA = ( where

100 | 5 Principles of Chemical Kinetics

Note that the z term is purely dynamical and does not depend on temperature, which is contained in the density ρ. Alternative expressions can be worked out. We can mention one that comes from another factorization: βℏ

k BA

1 = ∫ ⟨ḣ B h B (∆t + iϵ)⟩ dϵ βℏx A

(5.104)

0

where h B (∆t + iϵ) = e−ϵH/ℏ h B (∆t)e ϵH/ℏ = h B (∆t) + . . .

(5.105)

The omitted terms are negligible if h B is almost conserved, which corresponds to the fact that transitions are rare events. Thanks to the separation of time scales, which makes the rate constant independent on ϵ, we can express it as [275, 282] k BA =

1 ̂ 1 ̂ B (∆t)] ⟨Fh B (∆t)⟩ = Tr [e−βH Fh xA ZA

(5.106)

̂ = (i/ℏ)[H, h B ], which can be written as where F̂ is the flux operator, F ̂ + Pδ(Q ̂ ̂ = 1 [δ(Q − Q‡ )P − Q‡ )] F 2M

(5.107)

̂ are the mass and momentum operator associated to Q. Expressing the where M and P matrix elements as 1 󵄨 ̂ 󵄨󵄨 󸀠 󵄨 ̂ 󵄨󵄨 󸀠 󵄨󵄨 Q ⟩ = 󵄨󵄨 Q ⟩ [δ(Q − Q‡ ) + δ(Q󸀠 − Q‡ )] ⟨Q 󵄨󵄨󵄨󵄨 F ⟨Q 󵄨󵄨󵄨󵄨 P 󵄨 󵄨 2M

(5.108)

after some algebra, the rate constant can be written as k BA =

ℏ ∂ρ(Q, Q‡ ; β) ∫[ ] z(Q‡ , Q; ∆t)dQ MQ A ∂Q‡

(5.109)

=

∂z(Q‡ , Q; ∆t) ℏ ∫ ρ(Q, Q‡ ; β) [ ] dQ MQ A ∂Q‡

(5.110)

5.7.4 The Stable State Picture Another formulation of the rate constant in terms of correlation functions was given by Hynes and coworkers [283, 284]. The starting point is the Stable State Picture (SSP) of a chemical reaction. It states that a reaction is a net transition between stable reactant and product states [283]. It is based only on regions in configurational or phase spaces that are far from the high energy zone, which makes a large difference with transition state theory and all formulation focusing on the TS. Three regions are identified: the

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reactant (R), the product (P) and the intermediate (I), this last not being (by definition) at the TS surface. Assuming that the time for the internal equilibration of each stable region is much faster than the reaction time, the rate constant can be expressed as a flux from a stable state (being in internal equilibrium) and an irreversible flux into the opposite state. Note that in this way the events occurring at the saddle point do not intervene explicitly. The rate constant for the A → B process thus assumes the expression ∞

k BA = ∫ ⟨J i (S R , t = 0) J o∗ (S P , t)⟩ R dt

(5.111)

0

in which J i (S R ) is the incoming flux across the surface, S R , dividing the R and I states at t = 0, J 0 (S P , t) is the outgoing flux from the surface, S P , dividing I and P, while ∗ means that the dynamics must be calculated with absorption upon crossing S R or S P and the ⟨. . . ⟩ is the average on equilibrium distribution normalized by reactant partition function. In this way, the recrossing trajectories are accounted for nonreactive events. This expression can be rewritten as ∞

k BA = ⟨J i (S R )⟩R + ∫ ⟨J i (S R , t = 0) J o∗ (S R , t)⟩ R dt

(5.112)

0

where now J o∗ (S R , t) is the negative flux outgoing from I to R across S R . This expression is composed by a first term, determined by the initial flux moving from R to I. In the second integral the nonreactive trajectories assume that the negative value of ∞ ∫0 ⟨J i (S R , t = 0)J o∗ (S R , t)⟩R dt will cancel the positive value of the first term. Note that now reactive trajectories crossing S P do not appear explicitly. The definition of R, P and I is thus crucial and somehow problematic. In the case of unimolecular decomposition the definition of the three regions is shown in Figure 5.4. In this description, the I state is a region of the vibrational level of the decomposing molecules with energy higher than the threshold energy, E0 .

Fig. 5.4: Schematic view of the SSP picture for unimolecular decomposition. Only a few states are shown for simplicity.

102 | 5 Principles of Chemical Kinetics

In the low density limit, the rate constant is determined only by the rate of activating R to I: all molecules in I will decompose before a deactivation collision. The second term in equation (5.112) vanishes and the rate constant depends only by the initial flux k uni ≈ ⟨J i (S R )⟩R

(5.113)

This holds in the strong coupling limit in which the internal equilibrium is maintained for R below E0 . The SSP model can also be formulated for weak collisions, corresponding to heavy molecules in a gas of small and light molecules [285], gas phase ion recombination [286] or molecules on surfaces [287]. Historically, it was the starting point for the development of the Grote–Hynes theory [284] for barrier crossing reactions in solution, which couples the SSP picture with a generalized (non-Markovian) Langevin description of the dynamics. This description provided an important improvement from the Kramers theory, which considered only constant friction [288]. Recently, for example, the SSP model was employed to describe the self-exchange reaction of water molecules around halide ions [289].

5.8 Statistical and Nonstatistical Behavior in Chemical Reactions RRKM theory is a powerful tool for obtaining rate constants in a simple way, but it rests on the assumption that the dissociation is statistical (there are other hypothesis as described previously, but the statistical one is probably the most relevant). Thus, it is important to be able to identify if a system follows a statistical dissociation or not. In order for a system to be statistical, it should have the following properties [235]. a. The phase space needs to be fully accessible. b. The system needs to have enough time to explore the accessible phase space region before dissociation occurs. c. The molecular motion needs to be chaotic, not periodic. There are systems that do not satisfy these conditions. Here we list three possibilities. 1. In some cases a subset of the vibrational modes does not couple with others such that the phase space region occupied by those uncoupled modes is not accessible. In this case, the number of active vibrational modes can be less than 3N − 6 (3N −5 for linear molecules). Note that the coupling with rotational motion should also be evaluated. For example, the z-axis component of rotational quantum number, K, could be treated as active (freely exchangeable rotational energy) or adiabatic (conserved) [290]. 2. If the system cannot spend enough time before dissociation, it cannot explore the phase space uniformly. One example would be reactions that do not follow the minimum energy path [291]. Dissociation occurs before the system reaches minimum energy path.

5.8 Statistical and Nonstatistical Behavior in Chemical Reactions

3.

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103

If dissociation occurs through the so-called shattering mechanism. This corresponds to a dissociation occurring just after the collision so that the system has only a few vibrational periods and does not have time for intramolecular vibrational relaxation to occur. In shattering mechanism, bond cleavage almost always occurs on or near the impact point. This can occur in either surface-induced dissociation (SID) [292] or collision-induced dissociation (CID). [198] Using explicit collision trajectories is the best approach to have evidence of this phenomenon, which is accompanied by fragmentation products that cannot be explained via statistical theories.

In general, trajectory simulations are an excellent tool to study the nonstatistical behavior of chemical reactions. Hence, we show how classical trajectories can be used to simulate ESI-MS/MS of protonated molecules such as organic and biological ones.

5.8.1 Modeling Statistical Reactivity As we have described in detail up to now, a direct way to model statistical reactivity is by means of RRKM theory. This theory can provide all the rate constants (with advantages and limitations resulting from what exposed in the present chapter) for all the elementary steps. If a simple direct fragmentation reaction is involved, then the rate of product formation is straightforward. More in general, a unimolecular dissociation proceeds through multiple minima and transition states connecting the reactant structure to the product(s). In this case, kinetic networks must be solved. This can be done in different ways. 1. A first possibility is to solve the resulting system of kinetic differential equations. This is generally done for small kinetic networks, and it is more appropriate when the question is to identify different exit channels “close” to the reactants. In fact this assumes that once the intermediate is obtained then it randomizes instantaneously before proceeding to the next step. This was applied to systems like fragmentation of [Ca-urea]2+ [172] or kinetic stability of water clusters. [293] 2. Solving a kinetic Monte Carlo (KMC) scheme [294]. This is particularly suited when dealing with complex reaction networks and gives the appearance of the product at the thermodynamic limit, i.e., for t → ∞. This was applied, for example, to studying the statistical fragmentation of protonated uracil [232] or s-transpropenal [104]. 3. Using a master equation approach. In this way the thermalization effects and the IVR time of the intermediates can be taken into account. This approach is often used for bimolecular reactivity [295], but also for unimolecular fragmentation [296] and isomer equilibrium [297]. A detailed description of the method can be found in the specific literature [298, 299].

104 | 5 Principles of Chemical Kinetics

In all the approaches described so far, a crucial point is the determination of the stationary points connecting reactants with products. For small systems this can be done manually, i.e., by inspecting the molecular shape and, knowing the structure(s) of the product(s), guessing the molecular deformation needed to connect the reactant with the products. Minima and TS are then located using the standard quantum chemistry algorithms for searching stationary points [98–101, 300]. When the size of the system grows, it becomes extremely complex to find the stationary points in this way. Some authors have thus proposed algorithms to automatically locate minima and TS on the potential energy surface (PES) [102, 103, 105, 301]. In particular, the method proposed by Martinez-Nunez [104], which makes use of molecular dynamics to sample the PES, is suited to studying unimolecular dissociations and it was successfully applied to the aforementioned statistical fragmentation of protonated uracil. [232] Also these methods, anyhow, fail when dealing with very large systems. As we will detail in the following, molecular dynamics can also be used to study the statistical reactivity. In particular, from classical vibrational partition function, it is possible to make a connection between the internal energy and the number of vibrational modes, s: E = sk B T

(5.114)

where k B is the Boltzmann constant and T the target temperature. For a sufficiently large molecule, assuming that s ≈ (s − 1) and the unimolecular dissociation energy E0 is much less than the reactants energy, E, that is E0 /E ≪ 1, the classical microcanonical RRKM unimolecular rate constant k(E) at E becomes identical to the classical canonical transition state theory (TST) rate constant k(T) at T [302, 303]. This allows for using simulations at constant energy to obtain rate constants as a function of temperature. In this way it is possible to study the population decay of the initial system as a function of time: if the system has an RRKM behavior it will show an exponential decay from which it is possible to extract the rate constant. By repeating for different temperatures (and verifying the RRKM behavior each time) one can extract the rate constants as a function of temperature and, if they show an Arrhenius behavior, extracting activation energies and pre-exponential factors, such that it is possible to obtain temperature (and energy) dependence of the rate constant: k(T) = Ae−Ea /kB T

(5.115)

This approach was initially applied to organic systems [304, 305] and more recently to peptide fragmentation. [130, 131] In Figure 5.5 we report an example of decay of initial population at different temperatures and the following Arrhenius plot from which rate constants, barrier energies and pre-exponential factors can be obtained is shown in Figure 5.6.

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| 105

Fig. 5.5: Initial population decay for internal activation of Pro−2 dipeptide as from chemical dynamics simulations at different energies (here expressed as temperature thanks to relation in equation (5.114). Data are shown as black dots, while the results of the exponential fitting are shown as red dashed curves. [130]. Reproduced by permission of The Royal Society of Chemistry.

Fig. 5.6: Arrhenius plot as from rate constants obtained in Figure 5.5 for unimolecular fragmentation of Pro−2 . [130]. Reproduced by permission of The Royal Society of Chemistry.

106 | 5 Principles of Chemical Kinetics

5.8.2 Modeling Nonstatistical Reactivity We can distinguish two general classes of nonstatistical behavior in unimolecular decompositions [306]: 1. Even though the energy is randomized initially, systems do not dissociate while populating the phase space microcanonically. These systems are intrinsically non-RRKM. This occurs because the phase space is partitioned. 2. Systems that are activated by a localized excess energy and the subsequent reaction is faster than IVR thus resulting in a nonexponential behavior. This last behavior occurs often in surface-induced dissociation (SID) and collision induced dissociation (CID) in which each single collision is relatively high in energy (some eV). Physically what happens is that the energy is provided to a specific part of the molecule (for example a given bond) and the dissipation of that energy from this particular bond to the other modes (i.e., the microcanonical randomization) is not fast enough with respect to the dissociation time scale. To model this particular fragmentation one has to model first the way the energy is induced to the molecule and then the following dynamics. For CID the procedure is what is described in Chapter 6, while for SID it is slightly different. Instead of a projectile-ion collision, the ion is sent (with a given energy and impact angle) to a model surface. More details of SID simulations are reported in a recent review paper by Hase and coworkers [62]. A limitation of the direct approach is mainly in the computing time, which does not allow for having access to processes occurring on long time-scales. We will provide applications of this method in Chapters 7 and 8. This is, to date, the way used most often to take into account the nonstatistical effects in CID.

5.9 Vibrational Energy Relaxation A key phenomenon in statistical dissociation is the vibrational relaxation: the energy initially localized on a given molecular mode is distributed along the different molecular modes after a certain time. Here we will give a short account on the molecular basis of such phenomenon. If we consider a molecule as a set of harmonic oscillators, ω i , as is generally the case when calculating harmonic frequencies, then they are independent and once one is excited, the energy is kept in the given harmonic oscillator. To account for the energy flow from one given excited mode to the other molecular modes the simpler way of expressing the overall Hamiltonian is H = HS + HB + Hcoupling

(5.116)

5.9 Vibrational Energy Relaxation

|

107

where HS is the harmonic Hamiltonian of this given mode, ̂2 HS = 12 ω2 q̂ 2 + 12 p

(5.117)

the Hamiltonian of the other modes³, N−1

HB = VB (Q) + ∑

1 ̂2 2 Pi

(5.118)

i=1

and Hcoupling is a coupling term. Following a quantum mechanical treatment, it results that the allowed transitions relative to vibrational relaxation of the system are n + 1 → n, and the respective rate constants are k n+1→n =

(n + 1) ̃ CFF (ω) 2ℏω

(5.119)

̃ FF (ω) is the Fourier transform of the force-force correlation function where C CFF (t) = Tr [ρ B Fe iHt/ℏ Fe−iHt/ℏ ] eq

̃ FF (ω) = ∫ CFF (t)e iωt dt C

(5.120) (5.121)

where F is the force exerted by the surrounding system on the relaxing mode when the mode is at its equilibrium bond length. As we have previously shown, given the connection between classical and quantum time correlation functions, it is thus possible to consider the classical limit, such that the classical force-force correlation function can be expressed as CCl FF (t) =

1 ∫ e−βHB (Q,P) F(t)F(0)dQ dP ZB

(5.122)

where Z B and HB are the (classical) partition function and total Hamiltonian of the surrounding system. More details on the several theoretical aspects related to vibrational energy relaxations can be found elsewhere [226]. Here we can conclude by noticing that theoretical description of this phenomenon are generally done by assuming a model Hamiltonian of the system and the surrounding molecule, which can be a bath or the other part of the molecule. This was done, for example, to study vibrational relaxation in solution [307] or proteins [308]. Often these studies are directly connected with vibrational time-resolved experiments, but the vibrational energy relaxation phenomenon is also crucial (even if not quantitatively measured) in mass spectrometry. Even if its direct calculation is not necessary to model a mass spectrum theoretically, the theoretical basis of it are important for interpreting simulations and correctly connecting their results with experiments.

3 Here we refer to the other modes as B in analogy with the typical treatment that considers the vibrational energy relaxation from a system and a bath. The bath is often associated with an external environment, but it can be, as in the present case, the surrounding part of a molecule.

108 | 5 Principles of Chemical Kinetics

5.10 Conclusions In this chapter we have recalled the basic principles of chemical kinetics, with a particular aim in describing unimolecular dissociation, which is the physical phenomenon occurring to an ion in mass spectrometry experiments. In particular, when the dissociation is statistically controlled, kinetic theories like RRKM can be applied. We have given some basic ingredients here to apply this theory with its important assumptions. One interesting point is that in some cases the collisional process induces a fragmentation process that is too fast with respect to the equilibration of the other modes, thus resulting in a fast fragmentation that does not correspond to a statistical process. In this case, which occurs in general when the shock provides to the molecule a relatively large amount of energy, the statistical theories fail and only an explicit chemical dynamics approach can address the reactivity correctly. Finally, some description of a general formulation of rate constants (both classical and quantum) as can be obtained by correlation functions is shown. Correlation functions can be obtained from dynamics and the resulting expressions have both equilibrium and dynamical terms. Even if they have not yet been used in theoretical mass spectrometry of complex molecules, they should be investigated for future and more developed treatments.

6 How to Simulate Real Systems 6.1 Introduction In the previous chapters, we have described in detail different aspects related to the theoretical modeling of mass spectrometry using a dynamical approach aimed at reproducing physical conditions, leading to what is observed in experiments. Of course, there are many different experimental apparatus, which differ in ionization, activation, detection, etc. . . . . The aim of a theoretical mass spectrometry approach is not to model a specific instrument in detail, but to provide a framework that is able to reproduce what is observed in classes of experiments at best. Here we first specify how we can classify the different experiments, based on how the energy is given to the ion, and then we summarize the way a model of such experiments can be built. When performing simulations, in fact, it is always important to clearly identify the limit conditions behind the model employed (explicit collision, thermal activation, etc. . . . ). Here we will summarize some possible approaches to building different models that will be related to experimental conditions. Of course, in a real experiment, many factors come into play; then a simulation is aimed (as always) in providing a framework to understand and rationalize what is observed, providing some microscopic details that are not directly accessible in experiments.

6.2 Explicit Collisions As we have detailed in Chapter 3, molecular dynamics can be set such that bimolecular collision can be performed. In a similar fashion, collision between a molecule and a surface can be performed. Here we review the way these two (similar) simulations can be set up. They will correspond to Collision Induced Dissociation (CID) and Surface Induced Dissociation (SID), respectively.

6.2.1 Collision Induced Dissociation As described in Chapter 2, electrospray ionization (ESI) produces almost intact ions in the gas phase. These ions can then be analyzed and selected by the mass spectrometer. This analysis gives the exact mass-over-charge ratio (m/z), but any information is provided on the structure. The ion then collides with an inert gas. Here we first describe, by explicit collision between the ion and an inert gas (Ar, Ne, N2 , Xe, etc. . . . ), the limit case in which one single energetic collision takes place, this collision being (in principle) sufficient (of course if enough energy is given) to make the precursor ion

https://doi.org/10.1515/9783110434897-006

110 | 6 How to Simulate Real Systems

dissociating and providing one or more products. Here we describe the steps necessary to simulate such a single collision in a proper way. The first step, thus, is to determine the structure of the ion. In some cases, experiments can help, in particular if some ion-spectroscopies (like IRMPD, see Section 2.3.3) are performed. Theoretical chemistry can be used to build the initial structure: this can be done “by hand” (in particular in the case of small systems) or by using conformational searching tools, like dynamics at high internal energy or temperature. In some cases, more developed conformational searching tools, like for example the basin hopping method [309], can be employed. One important aspect to be determined is the charge localization: protonation or deprotonation site in case of positive or negative ions, respectively. In many cases, a given ion can have several possible protonation (or deprotonation) sites, as shown for the simple case of testosterone in Figure 6.1. The energetic stability (proton affinity) can be obtained through quantum chemistry. While in principle the lowest energy tautomer should be considered, it is recommended to consider different tautomers (more details will be given in the examples of uracil or diglycine fragmentations reported in Sections 7.5 and 8.4.1, respectively). There are several aspects that contribute to the importance of using different protonation sites for a given molecule. One is connected with the ESI process, in which the charged ion is produced from the liquid phase. It is not uncommon that a tautomer, which is the most stable in the gas phase, is not in the liquid phase and vice versa. The ESI is a complex process that can produce a tautomer, which is not the most stable one due to a kinetic trapping (see for example the case of protonated uracil [310, 311]).

Fig. 6.1: Protonated testosterone with its two possible tautomers. In red we underline the charged portion of the molecule.

Once the initial structure is set, one has to determine the interaction with the projectile. Ar and N2 are the most studied, for which are already present MM interaction parameters for many atom types (see Table 4.1 in Section 4.2). In case one needs to study other rare gases (Ne or Xe for example, for which only a few parametrizations are reported) or a system for which Ar and N2 parameters are not present, then they must be calculated. This is generally done by fitting high level quantum chemistry calculations with an explicit interaction potential. Note that for N2 a full semiempir-

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ical Hamiltonian description seems to be reasonable [131], while this is not possible for rare gas atoms. Full DFT/MP2 descriptions will be too computationally expensive (and DFT cannot be safely used for rare gas atoms). The next step is to set the collision energy. It can be given by experiments in the case of linear collisions (triple-quadrupole experiments, for example), and it is important to check that the collision energy is in the center-of-mass framework. In case it is given in the laboratory framework it is necessary to convert it in the center-of-mass one for the simulations. Equation (2.17) must be employed at this aim. Typically such energies are in the 1–10 eV range. In simulations it is somehow useful to increase the collision energy in order to increase the fragmentation probability. Before running the simulations one has to set the impact parameter, b. To reproduce CID, the best option is to sample it between 0 and bmax , where this last value is obtained by preliminary simulations. These preliminary simulations are done at fixed b values (generally with a spacing of 0.5 Å between two values) and the % of energy transfer after the collision is reported. In this case a relatively small number of trajectories are enough, since we are not interested in the fragmentation pathways, but only in the average energy transferred upon collision. Of course the number of trajectories should be enough to correctly sample all possible collisional orientations: the larger the molecule, the larger the number of trajectories needed. The b max value is generally set to the value for which the % of energy transfer is less then 10% of the collision energy [130, 131, 198, 312]. An example is given in Figure 6.2. One should remark that increasing the b max value will increase the number of trajectories needed 70

% Energy transfer

60 50 40 30 20 10 0 0

0.5

1

1.5

2

2.5

3 3.5 b (Å)

4

4.5

5

5.5

6

6.5

Fig. 6.2: Percentage of energy transfer as a function of impact parameter, b, as obtained from the β-aminoethylcellobiose + Ar collisional system at E coll = 19.5 eV. (Data elaborated from results shown in [312]).

112 | 6 How to Simulate Real Systems

to correctly sample the collisional process. On the other hand, in general the higher the b value, the lower is the reaction probability. It should be thus tempting to drastically reduce the b max : this can be very dangerous since, in particular for shattering fragmentations, it can be very important to consider the collisions with the external part of the molecules. We should also note that in general the b max value determined from the energy transfer is similar to what could be determined by simply looking for the molecular size. Once the initial structure is determined, along with the neutral partner and its interaction with the fragmenting ion, the collision energy and the impact parameter, it is possible to run trajectories to study the collision and the subsequent fragmentation. The number of trajectories is determined by the computational power available but it should be good to have a number of trajectories such that the reactive ones are at least 100. In fact, when collecting the ratios of fragmentation products, it is important to have in mind that the variance associated with each pathway probability is given by: σi = √

P i (1 − P i ) N

(6.1)

where P i is the probability of a given event obtained from N realizations. If N are the reactive trajectories and P i the probability of the fragmentation (or pathway) i, it is important to have large N in order to provide statistically converged values of the reactive pathways. In Table 6.1 we report an example of the behavior of variance for the case of a fragmentation giving four reaction products. Note that, even if, as is well known, the maximum standard deviation is for P = 50%, the uncertainties for small P i are similar to the P i , such that it is important to increase the statistics if one needs converged values for less occurrent processes. Generally, having 500– 1000 reactive trajectories ensures statistical convergence of the results also for less probable pathways, and 100 trajectories can be considered a best compromise if one needs a relatively low variance for less probable pathways. Unfortunately, the CID fragmentation probability is often very small, and, in particular when doing MP2 or DFT trajectories it is difficult to have hundreds of reactive trajectories. Finally, the time Tab. 6.1: Standard deviations associated with different reaction probabilities as a function of the number of reactive trajectories (N).

N N N N N N

= 20 = 50 = 100 = 500 = 1000 = 10,000

P1 = 50%

P2 = 30%

P3 = 15%

P4 = 5%

11.2 7.1 5.0 2.2 1.6 0.5

10.2 6.5 4.6 2.0 1.4 0.5

8.0 5.0 3.6 1.6 1.1 0.4

4.9 3.1 2.2 1.0 0.7 0.2

6.2 Explicit Collisions

| 113

length is set as the function of the system and trajectories are generally propagated for few ps and/or up to fragmentation. The sampling of the ion-neutral relative orientations is generally done automatically by randomly rotating the molecule around its Euler angles (see Section 3.3.2 for details). Once all the trajectories are obtained, it is first necessary to distinguish between reactive and nonreactive ones and then to look more into detail at the products and the mechanisms corresponding to the different reactions. Recently, automatic tools based on graph theory have been developed [130], in particular to distinguish between reactive and nonreactive and to automatically determine the reactive products. The mechanistic details, on the contrary, are generally obtained by direct observation. A more extended use of graph theory, as recently developed by Martinez-Nunez and coworkers for example [104, 105], which could be coupled to the set of obtained trajectories in order to minimize the visual observations, is surely needed. However, while watching many trajectories can be sometimes very tedious, it is an optimum way to deeply understand the chemical processes obtained in simulations. Finally, by counting the number of products obtained as the final result of the ensemble of trajectories for each given m/z value, it is possible to obtain a (timedependent) theoretical mass spectrum. A flowchart summarizing how to model CID in the single collision limit is given in Figure 6.3.

6.2.2 Surface Induced Dissociation Another method analogous to collisional activation is surface induced dissociation (SID). To model such a process, the approach is almost the same as for collisional simulations, with the difference that instead of the impact parameter, what governs the way the impact between the ion and the surface is done is the impact angle. Another important aspect is the interaction potential between the ion and the surface. Up to now, few surfaces have been studied over the possible variety of surfaces that can be used, in particular, diamond [313], self-assembled monolayers (SAM) [314], and organic surfaces [62, 63]. In Figure 6.4, a snapshot of surface-induced dissociation of octaglycine-H+ with diamond surface is shown. Note that in some cases (in particular SAM and organic surfaces) the surface can react with the ion thus making the computational approach more complex. A recent overview of the computational progresses and future perspectives is reported by Hase and coworkers [62, 63]. Here, from a computational point of view, we should only remark the few differences in setup with respect to ion-molecule collision, while applications and practical issues can be very different. However, experimentally, SID is not largely used, and much progress is needed also from the experimental point of view to make it a valid alternative to CID.

114 | 6 How to Simulate Real Systems

Structure of the ion

Protonation state Equilibrium geometry Manual or automatic searching of relevant structures

Definition of the collisional partner (inert gas)

Present in the MM interaction database Determine MM parameters For N2 is possible to use full QM with semi-empirical Hamiltonians

Setting Collision Energy

Determining bmax

Running trajectories at fixed b Plotting ET ransf vs b Setting bmax for ET ransf ≤ 10% E Coll

Running trajectories

Number of trajectories depend on the system and on the reaction probabilities

Analyzing results

Theoretical Mass Spectrum can be obtained

Fig. 6.3: Schematic flowchart for the collisional simulations aimed to mimic CID in the single collision limit.

Fig. 6.4: A snapshot of dissociation of octaglycine-H+ by collision with diamond surface.

6.3 Thermal Activation

|

115

6.3 Thermal Activation Experimentally, collisional activation is often done in ion traps, where the ion is subject to many (low) energy collisions up to its dissociation. In this case it is assumed that after each collision the energy flows through the internal modes and the next one occurs after IVR. These conditions are considered to correspond to statistical unimolecular dissociation, for which theories like RRKM hold. In principle, thus, from the knowledge of the transition states it would be possible to reconstruct a mass spectrum at a given temperature. However, this can be difficult in particular for large and flexible molecules. Chemical dynamics can be useful and accomplished as follows. As previously, one has to first determine the equilibrium structure. The ion is thus activated by microcanonical sampling. When dealing with large enough molecules, where s is the number of normal modes, if s ≈ (s − 1) and the dissociation energy is much less than the reactant’s energy, then the microcanonical rate constant k(E) at energy E becomes identical to the classical canonical transition state theory rate constant k(T) with E = skB T. We thus have a relation between E and T, and trajectories can be set at a given internal energy. We have thus to set the simulation time length, which should be long enough to ensure the appearance of the products. The number of trajectories can be reduced with respect to collisional simulations since one does not have to sample either orientation collisions nor impact parameter. We need, however, to have enough trajectories to have statistically converging results. Once the trajectories are determined, it is possible to construct a theoretical mass spectrum by counting the occurrence of each ion as in collisional simulations. Moreover, by plotting the fraction of initial ions as a function of time it is possible to obtain the rate constant if an exponential behavior is obtained. From the total rate constant and the occurrence of each path, it is possible to estimate (the quality of the results will largely depend on the uncertainty on the product probability) the rate constant of each pathway, simply from k i (T) = p i k(T), where k i (T) is the rate constant of product i to which we have associated an occurrence probability p i . Finally, by running simulations at different temperatures (i.e., internal energies) it is possible to obtain the rate constant as a function of temperature and, if an Arrhenius behavior is verified, to obtain the activation energy by simply fitting the Arrhenius law: k(T) = Ae−E a /kB T (6.2) where E a is the activation energy and A is the pre-exponential factor which, according to transition state theory, reads A = (k B T/h)e(1+∆S

‡

/R)

(6.3)

116 | 6 How to Simulate Real Systems

Fig. 6.5: Population decay and Arrhenius plot for thermal fragmentation of doubly protonated TIK peptide. Reprinted with permission from [131]. Copyright 2016 American Chemical Society.

in which ∆S‡ is the entropy of activation. This approach was successfully applied to fragmentation of diproline anion [130] and positively doubly charged TIK peptide [131]. An important point is verifying that: (i) the decay of initial population is exponential (such that the rate constant can be obtained); (ii) the Arrhenius behavior is obtained. Examples are reported in Figure 6.5. A flowchart for thermal activation simulations is reported in Figure 6.6.

6.4 Other Dissociation Simulations

Structure of the ion

Protonation state Equilibrium geometry Manual or automatic searching of relevant structures

Setting internal energy and temperature

If the system is large enough: E = skB T s is the number of internal modes

Running trajectories at different temperatures

Number of trajectories depend on the system and on the reaction probabilities

Analyzing results

Theoretical Mass Spectrum can be obtained

Determining the population decay

Rate constant can be obtained by fitting: N (t) = N (0)e−kt

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From Arrhenius plot it is possible to obtain activation energy and pre-exponential factor

Fig. 6.6: Schematic flowchart for the thermal activation simulations.

6.4 Other Dissociation Simulations As we have reported previously, other activation methods are possible. While the modeling of such other processes goes beyond the aim of the present work, we will give here some main aspects that must be considered if one is interested in simulating them.

6.4.1 Light Induced Activation One possibility is to irradiate with a laser an ion, which can then undergo dissociation. Generally, infrared (IR) and UV-VIS lasers are used for this aim. The UV-VIS light promotes the system in electronically excited state(s) and thus the subsequent dynamics should be modeled using the advanced techniques of excited state dynamics. Note that in this case the pure Born–Oppenheimer description (see Section 3.1.2) does not hold, and nonadiabatic couplings should be considered. In the field of classical trajectories, one of the most used methods is Tully’s fewest switching surface

118 | 6 How to Simulate Real Systems

hopping, in which a classical wave packet is propagated following Newton’s equation of motion [141]. Each trajectory has a finite probability of changing electronic state, depending on the energy gap and the velocity vector. These simulations are computationally heavy since excited states energies must be calculated on-the-fly and to have statistically converged results, one has to run thousands of trajectories (the delicate point is in the hopping probability). However, recently it was applied in the TD-DFT framework, producing qualitative results in terms of photoproducts [315–317]. The other possibility is IR activation, as in IRPD and IRMPD experiments (see Section 2.3.3). In this case a vibrational excitation is produced. These spectroscopies are generally aimed at studying (and authors only report mainly) vibrational modes, which can be modeled either in the harmonic approximation via diagonalization of Hessian matrix or by dynamics simulations, when anharmonic properties are relevant [4]. Studying the photoproduct via chemical dynamics is problematic due to the zero-point energy leakage [135, 318, 319] and, more in general, due to the fact that vibrational modes are not quantized in Newtonian dynamics. Surely more developments will be presented in the next years thanks to advances in considering quantum nuclear effects on classical trajectories. Often, these methods are tailored to avoid the leak of zero-point energy by referring to normal modes [134, 136, 137, 139]: since during a dissociation process the normal modes are largely modified this is an obstacle in applying them to IR induced fragmentation. Recently, Paul and Hase have proposed a method that assures that the products in unimolecular dissociation have the correct ZPE [140]: while this method seems simple and general, it requires a detailed knowledge of the transition states and generally does not assure that the intramolecular vibrational energy flow is correct. More recent methods, like for example those based on path integral theory [148, 149, 152], quantum Langevin [157] or Bohmian [320] dynamics, can be used in Cartesian coordinates and lead to modification of reference normal modes during a trajectory. However, they are defined on the basis of equilibrium correlation functions and they are computationally expensive, such that they are scarcely applied to irreversible chemical reactions. They should be considered and eventually adapted to study IR induced dissociation in the future.

6.4.2 Electron Ionization The other possible activation way is by using electron ionization. This produces metastable species that can dissociate. Recently, Grimme and coworkers have developed an approach based on Born–Oppenheimer dynamics (called quantum chemistry electron ionization mass spectra, QCEIMS) to study electron impact mass spectra of molecules [321–323]. This approach is also based on running an ensemble of trajectories and finally accumulating fragmentation products. One important difference with respect to the methods described for CID is in the internal energy that is given to the

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119

initial structure. In fact, while in collisional or thermal activation the excess energy is a quantity that can be obtained either by mimicking explicit single collision either by considering statistical collisions, in electron ionization it is not obvious how to determine the ion excess energy. In the QCEIMS method the excess energy is assigned randomly to each initial geometry from a Poisson distribution P(E) =

exp[cE(1 + ln(b/cE)) − b] √aE + 1

(6.4)

where P(E) is the probability of having a given excess energy E, b and c are two parameters which were optimized to be 1 and 1/(aNel ), respectively, in which a ≈ 2 eV and Nel is the number of valence electrons. This excess energy is assigned to the species prior to ionization, and then one electron is removed forming the unstable ion, thus assuming that the ionization is instantaneous. Velocities are scaled to conserve the excess energy assigned in the previous step. Trajectories are then run generally employing DFT, tight binding DFT or semiempirical MNDO methods. This method was successfully applied to organic molecules [323–327]. An example is shown in Figure 6.7, where calculated and experimental spectra of four organic molecules are reported.

Fig. 6.7: Calculated and experimental electronic ionization mass spectra using the QCEIMS method. Reprinted with permission from [322]. Copyright 2016 American Chemical Society.

120 | 6 How to Simulate Real Systems

As shown from QCEIMS simulations (and as we will see more extensively in subsequent sections), the simulations are generally able to provide the peaks obtained experimentally, while the intensities are not often quite well reproduced. We will discuss in general the subtle point of peak intensity in the next section.

6.5 Peak Intensity In experimental mass spectra two pieces of information are provided: the ions formed, in terms of the m/z ratio, and their intensity. This last value is generally obtained from an ion counting and what is finally reported is some normalized intensity, where, for example, the most intense peak is set to one and the others scaled accordingly. The intensity thus depends on many experimental parameters and not only on the way the activation is done, which is what we have largely discussed up to now. For example, if the ions are lost from the fragmentation sector and the detector, they will not be counted. This ion loss is not uniform in all the m/z ranges, such that the resulting relative intensities do not necessarily reflect the product ratio. Other more subtle experimental details can influence the relative intensity, like, for example, the detection range. This is connected with the electronics of each machine. Without going into these technical and experimental details, from the theoretical point of view it is impossible (and even not desirable) to take into account all these details. In fact, what simulations are aimed at providing is a general and an as universal as possible (i.e., independent on the specificity of the apparatus) picture of the fragmentation. This could be a source of difficulties, since a quantitative comparison in terms of peaks and their intensity seems to be almost impossible. However, much information can be obtained, and evaluation of the quality of the calculations is possible. The most important aspect, in fact, is the presence (or absence) of the observed peaks. Then their relative intensity behavior is often a source of discussion. When available, having experiments done on different experimental apparatus is certainly very useful. Another aspect to take into account is the intrinsic limitation in simulations’ time scales. In fact, while simulations are generally run on picosecond time scales, the fragmentation in the experiments is a process that can last for micro- to milliseconds. Therefore, the results obtained from collisional, thermal or QCEIMS simulations should be considered as time-dependent results. The fast processes are necessarily overestimated. As we will show, it is possible to complete this dynamical picture with kinetics analysis, which will provide information on fragmentations at longer time scales. This is particularly important if, for example, an experimentally very intense peak is not observed (or better observed only with low probability) in simulations. Unfortunately, to be able to provide kinetics at longer time scales, information on important minima and saddle points along the fragmentation pathway are necessary. In some cases they can be evaluated by analyzing simulations, in particular in the case of extended systems, while for small ones standard potential energy surface ex-

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121

plorations can be performed. Of course, if a product is totally absent in the simulation results and any other information on the characteristic transition states are absent it will be almost impossible to predict its appearance. In this last case, one first aspect to take into account is a failure in the electronic structure description employed, and to check if it can be obtained by using another method. A second option (if the first fails) is to increase the energetics, thus speeding up the process, such that, possibly, the desired pathway would be obtained and some information (like energy barriers and/or transition state structure) would become available.

6.6 Conclusions In this chapter we have summarized some practical recipes for using molecular dynamics to obtain mass spectrometry. In particular we have described in details the way simulations should be set up to model CID in two limiting case: (i) single collision with given collision energy (this last being a parameter which can be get from experiments); (ii) internal energy activation mimicking the multiple-collision activation. Furthermore, we have provided basic information to simulate SID and electron ionization, still based on molecular dynamics simulations. How to simulate dissociation induced by interaction with a laser is also discussed. These methods should go beyond the Born–Oppenheimer approximation, which is employed in other approaches and generally they should more severely take into account nuclear quantum effects.

7 Application to Organic Molecules 7.1 Introduction In the present chapter we will show some applications of chemical dynamics to the study of collision induced dissociation (CID) of relatively small organic molecules. For these molecules the ion can be treated at the MP2 or DFT level of theory, since its size is relatively small and consequently the number of trajectories needed for a correct collisional sampling is small too. Normally, hundreds of trajectories are enough to provide reasonable results. We will first provide an overview of early studies in which a direct comparison with experiments was not done. Also, because the fragmentation channels are few, they were important for establishing a first basis for further developments. Then, we will summarize some of the approaches used to study urea, doubly-charged complexes and uracil. The aim is not only to summarize the results, but also to give to the reader a road map on how the different approaches can be combined and how they should be compared with experiments. In this way, we will suggest possible approaches to study other systems in the future.

7.2 First Pioneering Studies One of the first chemical dynamics studies focusing on ion fragmentation products and mechanisms was performed by the Hase group on N-protonated glycine [202]. Both CID and SID simulations were performed and results compared. Argon was used for CID with a relative energy of 300 kcal/mol. In SID simulations, glycine-H+ was bombarded with diamond {111} at a 70° angle with a collision energy of 70 eV. For intermolecular potential between peptide-Ar and peptide-diamond, purely repulsive two body interactions of equation (4.10) was used. The AM1 semiempirical method was used for intramolecular potential for the ion. For both CID and SID simulations, 100 trajectories were run and frequencies of product channel occurrences were determined. About 55% of the 42 fragmentations of SID occurred by the shattering mechanism, where the dissociation occurred at impact point or with strong interactions with the surface. The SID of Cr(CO)+6 with diamond {111} and n-hexyl thiolate SAM surfaces [328] was studied using analytical potential functions with different dissociation energies of CO depending on the number of COs remaining. The effect of collision energies on the energy transfer to the surfaces and lifetimes of Cr(CO)+6 ion was investigated. The CID of H2 CO by the Ne atom [200] was also studied using the Hessian update trajectory integration method incorporated in Gaussian at B3LYP/6-31G** level of theory. Effects of collision energy, vibrational excitation of reactant ion, impact pahttps://doi.org/10.1515/9783110434897-007

7.3 CID of Protonated Urea

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rameter, and orientation on energy transfer and branching ratio were investigated. The results of the direct dynamics simulation were in good agreement with previous experimental ones.

7.3 CID of Protonated Urea The fragmentation of protonated urea was studied by coupling collisional simulations and experiments in a triple quadrupole apparatus recently [175, 198]. This is, probably, the first CID study in which both experiments and chemical dynamics simulations are discussed together. This relatively small system is one of the most instructive examples for showing some particular features of CID. Experimentally, two main fragmentation pathways are observed: UreaH+ → NH3 + CONH+2 +

UreaH →

NH+4

+ CONH

(7.1) (7.2)

Both ions, NH+4 (m/z 18) and CONH+2 (m/z 44), were observed as a collision product, even if products of reaction (7.2) are about 35 kcal/mol more stable than those of reaction (7.1). Protonated urea is present in two tautomeric forms, protonated on the oxygen and the nitrogen, respectively (see Figure 7.1), where the former is about 10 kcal/mol more stable than the latter.

Fig. 7.1: Two protonated forms of urea considered. Relative energies as obtained from MP2/aug-cc-pVTZ and MP2/6-31G* (in parenthesis) calculations are also shown.

Simulations were done using a MP2/6-31G* representation of the ion and an explicit interaction potential between the urea and Ar atom, for three collision energies: 101.5, 130.5 and 145.5 kcal/mol, which fall into the experimental range. While collisions with the most stable protonated form did not lead to any product in the simulation time length (some ps), the collisions with the nitrogen protonated tautomer lead to both products observed experimentally. Product formation as a function of collision energy was reported showing a behavior that qualitatively agrees with experiments (see Figure 7.2).

124 | 7 Application to Organic Molecules 100 90

Reactants + NH3 + CONH2

80

+

NH4 + CONH

70

%

60 50 40 30 20 10 0 80

90

100

110 120 130 140 Collision Energy (kcal/mol)

150

160

170

Fig. 7.2: Products as a function of collision energy for UreaH+ + Ar collisions (nitrogen-protonated tautomer of urea results are reported). Data are taken from [198].

8 +

NH3 + CONH2

7 shattering / non-shattering

+

NH4 + CONH

6 5 4 3 2 1 0 80

90

100

110 120 130 140 Collision Energy (kcal/mol)

150

160

170

Fig. 7.3: Shattering/nonshattering ratio obtained in UreaH+ + Ar collisions (nitrogen-protonated tautomer of urea results are reported). Data are taken from [198].

Furthermore, by inspecting the reactive trajectories, it was possible to determine which products are formed via a shattering mechanism. Results are collected as a function of collision energy showing that: (i) the high energy path (reaction (7.1)) is mainly formed through a shattering mechanism; and (ii) increasing the collision energy, the shattering probability for both reactions increases. These data are reported in Figure 7.3.

7.3 CID of Protonated Urea

| 125

1 Reactants + NH3 + CONH2

Probability

0.8

+

NH4 + CONH

0.6

0.4

0.2

0 0

1000 time (fs)

2000

Fig. 7.4: Probabilities of forming reaction (7.1) and reaction (7.2) products and reactants (urea protonated on nitrogen) as a function of time. Results are reported for E coll = 145.5 kcal/mol. Data are taken from [198].

Other than watching the trajectories, the different time scales in forming products were noticed by plotting as a function of time the fraction of trajectories leading to reactions (7.1) and (7.2) (and also those that did not react). In Figure 7.4 we show an example at high collision energy (145.5 kcal/mol), from which we can notice that the highenergy reaction (7.1) pathway occurs faster than the lower energy one (reaction (7.2)) because while the former occurs basically only via shattering mechanism, in the second there is a copresence of shattering and nonshattering (which does not proceed through a full IVR necessarily) mechanisms. Simulations can give, as we have shown, direct information on fragmentation products and mechanisms. However, most collisions do not lead to any product, because they are limited in time scale and because the transferred energy is sometimes not enough to overcome the reaction barriers. In the study of protonated urea, one striking point was that the most stable tautomer, the one protonated on oxygen, did not react. Nonreactive simulations were then used to calculate the transferred energy, total, vibrational and rotational. They were used in different ways: 1. In the case of nitrogen protonated nonreactive trajectories, we have obtained the fraction of trajectories, which have enough energy to pass the barriers leading to the two different products. We can consider that in the long time scale they will react and thus we report the probability of passing such barriers (see Figure 7.5). The trajectories with enough energy to form products of reaction (7.2) are not negligible, thus compensating the relatively low fraction of such products obtained from direct collisions. This will increase the fraction of reaction (7.2) products providing better agreement with experiments.

126 | 7 Application to Organic Molecules 30 +

% of trajectories

25

NH4 + CONH Isomerization

20 15 10 5 0 80

100

120 140 Collision Energy (kcal/mol)

160

Fig. 7.5: Fraction of trajectories with enough internal energy to pass the barrier to: form NH+4 + CONH products (from trajectories using as initial structure the nitrogen protonated tautomer) and isomerize the most stable oxygen-protonated tautomer to the nitrogen protonated one (from trajectories using as initial structure the oxygen protonated tautomer). Data are taken from [198].

2.

The same was done for the oxygen-protonated trajectories, for which the isomerization barrier leading to nitrogen-protonated species was considered. Results are reported in the same Figure 7.5. 3. For both activated ions, RRKM rate constants have been calculated. Based on the transferred energy calculations, the characteristic time scales for the reactivity have been considered, resulting in the 10–100 ns time range for fragmentation of nitrogen protonated urea and in ns time scale for isomerization of oxygen-protonated ion to the nitrogen protonated one (which will subsequently dissociate). 4. Information on the rotational energy activation was used at first glance to qualitatively address the rotational activation effect on the reactivity of nitrogen-protonated urea (in the case of the other tautomer the reactant and transition state structures are very similar such that the RRKM rate constant does not depend on rotational energy). In this case, moving to high rotational energies leads to modifications in the RRKM rate constant of more than one order of magnitude (see Figure 7.6). Later, the study was done also using N2 as colliding neutral species, leading to very similar results [175]. One interesting difference was that when using N2 as a projectile at the same collision energy, the transferred energy and thus reactivity decrease with respect to simulations done with Ar at the same collision energy values. In Table 7.1 we report a comparison between N2 and Ar collisions as obtained at Ecoll = 145.1 kcal/mol. We should notice that rotational activation also decreases when moving from Ar to N2 as well as the fraction of shattering trajectories.

7.3 CID of Protonated Urea

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1e+16 1e+15

–1

k (sec )

1e+14 Erot = 10 kcal/mol Erot = 50 kcal/mol

1e+13 1e+12 1e+11 1e+10 1e+09 0

20

40

60

80 100 120 140 160 180 200 220 240 E (kcal/mol)

Fig. 7.6: RRKM rate constant as a function of energy for the reactivity of nitrogen protonated urea towards formation of NH+4 + CONH products at two different rotational energies. Tab. 7.1: Comparison between UreaH+ + N2 and UreaH+ + Ar collisional systems results. Data are collected from [198] and [175].

Reactivity (%) NH3 + CONH+2 (%) NH+4 + CONH (%) % of shattering reactivity % of average transferred energy % of average transferred energy to rotation

N2

Ar

31 12 19 14 19 13

56 19 37 41 35 21

More interestingly, the study was extended to the reactivity of activated oxygen-protonated urea as follows. The transition state corresponding to the isomerization reaction (a proton transfer) between the two tautomers have been considered and simulations have been run using the transition state geometry as the initial structure and tuning vibrational and rotational energies in values obtained from inelastic scattering trajectories. In particular three vibrational and two rotational initial internal energy values were considered. Results show an interesting dependence on rotational activation; in particular, increasing the rotational energy has a clear enhancing in the formation of the high energy pathways for low vibrational energies, while when increasing the vibrational energy, both pathways assume a similar probability. Results are summarized in Figure 7.7. Dynamics starting from the transition state (called post-TS dynamics) will be very useful for investigating fragmentation dynamics of a slightly more complex system as we will report in the next section. These simulations are a variant of microcanonical internal energy activation, in which the energy is given in all the modes but the one

128 | 7 Application to Organic Molecules

+

Erot = 20 kcal/mol, NH3 + CONH2

100

+

Erot = 20 kcal/mol, NH4 + CONH +

Erot = 50 kcal/mol, NH3 + CONH2 Erot = 50 kcal/mol,

%

80

+ NH4

+ CONH

60

40

20

0 50

55

60 65 70 Vibrational Energy (kcal/mol)

75

80

Fig. 7.7: Ratio (in relative %) between NH3 + CONH+2 and NH+4 + CONH products as obtained from post-TS dynamics (the initial TS is the one connecting the two tautomeric for of protonated urea, Figure 7.1) as a function of internal vibrational energy at a different total rotational energy. Data are taken from [175].

corresponding to the reaction coordinate (i.e., the one corresponding to the imaginary frequency). It is thus possible to set the vibrational and rotational energy of choice in the other modes and study the subsequent dynamics and product formation as a function of the given energy. More details can be found in [329]. Its use in theoretical mass spectrometry can be a source of very interesting findings, but it requires the knowledge of the saddle points on the potential energy surface. It is thus limited to systems for which these points are known. At the present time it has been used, in the field of theoretical mass spectrometry, for fragmentation of protonated urea [175] and [Ca-formamide]2+ complex [330]. A limitation in its use for large biomolecules resides not only in the difficulty of finding the transition states but, even if they are known, in choosing the important ones: such systems can have a huge number of saddle points and the characterization of all can be a very long (and difficult) task.

7.4 CID of Doubly Charged Ions of Metal-Neutral Complexes Doubly charged ions being a complex between a cation (generally a metal, M) and a neutral molecule (acting as base, B), present a particular interest in the mass spectrometry (and more generally in gas phase reaction) community. First, it is difficult to stabilize them, since in many cases they are less stable than some of their products. This occurs because most of these M-B complexes are either thermochemically or kinetically unstable [331]. This is particularly the case when M is a doubly-charged transition metal, like Cu2+ , Pb2+ or Co2+ . While some species have been detected in

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129

Fig. 7.8: Schematic pictures showing the two general mechanisms of doubly charged ions fragmentation: (a) neutral loss and (b) Coulomb explosion.

the gas phase [332, 333], generally, the system spontaneously forms the monocation (loosing H+ ), as observed in different experiments [334–336]. It has been proposed that the reason resides in the high recombination energy of such dicationic transition metals [337]. Alkaline-earth dications, like Ca2+ , have a lower recombination energy and this should be at the basis of the stabilization of the [CaB]2+ complex, which can be detected in the gas phase. For example, recombination energy of Cu2+ and Ni2+ are 20.3 and 18.2 eV, respectively, while the one of Ca2+ is 12 eV [338]. Secondly, they can dissociate leading to two general pathways: (i) neutral loss and (ii) Coulomb explosion (see Figure 7.8 for a schematic picture). If a doubly charged ion has the general formula [M − B]2+ , where M is the metal and B is the neutral (organic) molecule, then the neutral loss is often the simple dissociation forming back M2+ and neutral B. More in general (as we will see in next examples) the metal can fragment B, forming a doubly charged ion [M − (B − X)]2+ and the neutral molecule X. The Coulomb explosion implies a reorganization of the charge distribution inside the complex up to a situation in which there are two positive charges inside the same molecule. This will make the system very unstable and due to simple Coulomb repulsion the system will fragment into two positively charged species. Recently, the fragmentation dynamics of three systems, [Ca-Urea]2+ , [Ca-Formamide]2+ and [Sr-Formamide]2+ , was studied by chemical dynamics simulations [172, 173, 255, 330, 339]. For these three systems, we will discuss different aspects of the fragmentation, which were understood by employing methods based on chemical dynamics. In particular, explicit collision dynamics was used to understand the formation of some high energy products in terms of nonstatistical dissociation, while coupling these results with the information on the potential energy surfaces (minima and TS’s along the reaction pathways), it was possible to explain the reaction mechanisms occurring statistically [172, 173, 255]. In this way it was also possible to understand the differences between Ca2+ and Sr2+ formamide complexes. Finally, by a systematic study of the dynamics from the different TS’s connecting the [Ca-Formamide]2+ complex with the different reaction pathways, it was possible to show that fragmentation dynamics cannot be described only in terms of the minimum energy path [330]. We will now discuss each of these points separately.

130 | 7 Application to Organic Molecules

7.4.1 Nonstatistical Dissociations The first approach was to study the collisions between the complexes and an inert Ar atom explicitly. Previous studies have reported experimental CID data and the potential energy surfaces of [Ca-Urea]2+ , [Ca-Formamide]2+ and [Sr-Formamide]2+ in terms of minima and TS’s leading to the observed fragments [106, 340, 341]: the minimum energy structures of each system were thus used as initial structures for the dynamics. In the case of [Ca-Urea]2+ the MP2/6-31G* method was used, being the same used for the fragmentation of protonated urea [175, 198] and effectively reproducing the main features of the potential energy surface. For [Ca-Formamide]2+ and [Sr-Formamide]2+ DFT was employed, and the functional was chosen based on its performances with respect to the energetics and kinetics of the system [255]. G96LYP/ 6-31G(d) and BLYP/6-31G(d) were used for [Ca-Formamide]2+ while G96LYP with two basis sets, 6-31G(d) and 6-31+G(d,p) for [Sr-Formamide]2+ . Since the Hamiltonian is calculated from MP2 or DFT, about hundreds of trajectories were run for each “point” (i.e., each system, method and value of collision energy). For initial conditions the minimum energy structure was considered and positions and momenta were sampled according to normal mode sampling, in which the vibrational quantum numbers were obtained from Boltzmann probability at room temperature (see details in Section 3.3). A summary of the fragments obtained in the simulations compared with the experimentally observed ions is reported in Table 7.2. The most intense fragmentation observed in all the simulations is the M2+ + B one (where M is Ca or Sr and B is the neutral molecule, urea or formamide). This reaction is very fast and corresponds to the direct collision of Ar with the M atom (or to the M–O bond; M is linked to the neutral molecule by interacting with an oxygen atom of the C=O group). Once the collision happens with enough energy, the metal can leave the molecule as M2+ : this is a typical example of the shattering mechanism. Interestingly, while this is the highest exit channel in energy, it is observed as dominant in simulations and in experiments. From simulations, the fast channels are generally overestimated: in this case the observation of these fragmentations explained why in experiments these are very intense, while other pathways have lower activation energies. In Figure 7.9 we show an example of such shattering trajectories. Other pathways are obtained with much less probability. By comparing the reaction times obtained from simulations with RRKM time scales¹ it was possible to quantitatively distinguish between statistical and nonstatistical dynamics (see Figure 7.10). As shown in Table 7.2 most of the trajectories did not react. However many of them had enough energy to pass the barriers and react. In the next section we report how 1 From the rate constant, k, expressed in [time]−1 units, the half-life time is t1/2 = ln 2/k. This assumes a single exponential decay.

7.4 CID of Doubly Charged Ions of Metal-Neutral Complexes |

131

Tab. 7.2: Summary of collisional products obtained from chemical dynamics simulations of three doubly charged complexes with Ar, as from [172] and [173]: U is for Urea and F for Formamide. For [Ca-Urea]2+ we show MP2/6-31G* results, while for [Ca-Formamide]2+ and [Sr-Formamide]2+ we show G96LYP/6-31G* ones. Collision energies (CE) are given in kcal/mol. Experimental data are taken from Refs. [340, 341], and [106] for [Ca-Urea]2+ , [Ca-Formamide]2+ and [Sr-Formamide]2+ , respectively. System

Products

Simulations

[Ca − U]2+ nonreactive Ca2+ + U [Ca(NH2 )]+ + H2 NCO+ [Ca(NH3 )]2+ + HNCO (CaNHCO)2+ + NH3 [Ca − F]2+

Experiments

CE = 210

CE = 300

90% 7% 3% – –

65% 29% 4.5% 1% 0.5%

Intense Low Medium Medium Intense

CE = 180

CE = 230

CE = 280

nonreactive Ca2+ + F [Ca(NH2 )]+ + [HCO]+ [Ca(NH3 )]2+ + CO [Ca(H2 O)]2+ + HCN Ca(OH)+ + HNCH+

76.0% 24.0% – – – –

62.7% 35.8% 1.3% 0.3% – –

59.7% 39.3% 1.0% – – –

Intense Intense Low Low Low Medium

nonreactive Sr2+ + F [Sr(NH2 )]+ + [HCO]+ [Sr(NH3 )]2+ + CO [Sr(H2 O)]2+ + HCN Sr(OH)+ + HNCH+

80.0% 20.0% – – – –

72.9% 27.1% – – – –

68.1% 31.9% – – – –

Intense Intense – – Medium Low

[Sr − F]2+

the direct dynamics data can be completed thanks to information on the internal activation of the ion in order to improve the understanding of unimolecular dissociation and finally of the whole CID experiment.

7.4.2 Statistical Dissociations For these doubly charged systems composed by a cation and a relatively small neutral molecule, it is possible to reconstruct all the series of minima and TS’s connecting the reactant with the products. From the energies and vibrational frequencies of reactant and TS’s it is possible to obtain the RRKM rate constants as a function of the internal energy. While in general it is not known the quantity of energy that the ions get in collisions, this is exactly the kind of information that simulations can provide. Rate

132 | 7 Application to Organic Molecules

Fig. 7.9: Example of neutral loss from [Ca(Urea)]2+ + Ar collisions. Ca–O and Ca–Ar distances as a function of time are shown with corresponding structures (snapshots from one trajectory). Ar atom is in green, Ca in cyan. Data are taken from simulations of [172]. 2000

2+

2+

[Ca(formamide)] --> Ca

2000

+

1500

time (fs)

time (fs)

+

[Ca(formamide)] --> [Ca(NH2)] + HCO

1500

1000

500

0

2+

+ formamide

1000

500

0

50

100 200 150 Energy (kcal/mol)

250

300

0

0

50

100 200 150 Energy (kcal/mol)

250

300

Fig. 7.10: Characteristic times as a function of ion energy as obtained from collisional simulations and RRKM analysis for the [Ca(formamide)]2+ system. Reprinted with permission from [173]. Copyright 2014 American Chemical Society.

constants and transferred energies can then be combined together. We have shown previously (Section 7.3) how this is possible in the simple case of protonated urea. Now we show how it is possible in the case of more complex systems. RRKM assumptions state that the energy is randomized in the reactants and that, while reaching the TS, the microcanonical ensemble is kept (see Section 5.3 for more details). We can thus consider that trajectories that did not react in the fast time scales of simulations (some ps) but have enough energy to reach the TS’s can react statisti-

7.4 CID of Doubly Charged Ions of Metal-Neutral Complexes |

133

cally. In particular, they can do the first step statistically. After reaching the TS the pathways to the products can involve passing through other minima: in this case it is not evident that these intermediates are randomized in position and momenta, in particular in the triple-quadrupole experiments. The first statistical analysis thus concerns the study of the kinetics connecting the reactant with the different fragmentation pathways. We show now two different ways of doing the analysis, as reported for [Ca(Urea)]2+ , [Ca(Formamide)]2+ and [Sr(Formamide)]2+ systems. 1. One first possibility is to solve the kinetic problem associated and obtain the evolution of the population as a function of time. A kinetic system is in general not easy to solve analytically: the Laplace transform method (see Appendix A) or numerical integration using Runge–Kutta can be employed [119]. In the case of simple systems, analytical solutions or steady state approximation solutions are also available. In the case of [Ca(Urea)]2+ first a time-dependent solution was obtained by restricting the system to the reactants, one intermediate and two products. Other pathways were excluded by considering the ratio between the different rate constants. In this case it is possible to consider the probability of each internal energy of the ion, build weighted k(E) sets and then use them in the solution of the kinetic problem. An example of time evolution is shown in Figure 7.11, from which one can see that in about 15 ps these products will also be formed in the statistical limit. It is also important to note that considering the full distribution of transferred energy at a given collision energy instead of a fixed value of energy has a huge impact on results. The full solution of the kinetic system is rather complicated and assumes a randomization of each intermediate. We will show it in the case of uracil fragmentation (see Section 7.5), but we should note that if there is not any external bath which rerandomizes the system, trajectories reaching an intermediate will easily react much before IVR, as remarked recently from an initially thermalized simulations (an example can be found in [130]). 2. In the case of [Ca(Formamide)]2+ and [Sr(Formamide)]2+ the RRKM rate constants were analyzed in terms of the energy distribution and results for the two systems were compared. In this way it was shown, for example, why [CaNH2 ]+ + OCH+ channel is present experimentally while the analogous [SrNH2 ]+ + OCH+ one is not (see Figure 7.12). To understand why in [Ca(Formamide)]2+ the [Ca(OH)]+ + [HCNH]+ products are more abundant than the Sr analogous ones, while the opposite holds for [Ca(H2 O)]2+ + HCN, statistical analysis can be employed, assuming that the intermediate that is in common between the two pathways is equilibrated and reactivity is governed by RRKM kinetics. At the values corresponding to average energy transfer the process leading to [Ca(OH)]+ + [HCNH]+ is much faster than the one leading to [Ca(H2 O)]2+ + HCN. Note that for this last, the kinetics connecting the intermediate to another intermediate, which is unique to this reaction pathway, was analyzed: in this way we assume that randomization holds only for one intermediate and that once the barrier connecting to the second in-

134 | 7 Application to Organic Molecules 1

1

0.8

+

0.8

+

CaNH2 + H2NCO

0.6

probability

probability

Reactants

0.4 +

+

0.6

0.4

CaNH2 + H2NCO

Reactants

0.2

0.2

2+

Ca

Int

Int 0

2+

Ca 0

1

2

3

4

+ H2NCONH2

5

6

7 8 9 time (ps)

+ H2NCONH2

0 10 11 12 13 14 15

0

1

2

3

4

5

6

7 8 9 time (ps)

10 11 12 13 14 15

Fig. 7.11: Evolution in time of the fraction of reactant, products and intermediate (which lies between reactant and CaNH+2 + H2 NCO+ product) obtained from RRKM analysis of [Ca(Urea)]2+ fragmentation. Left panel: results considering the energy transfer distribution (at E coll = 300 kcal/mol); right panel: results at a fixed value of internal energy (200 kcal/mol), corresponding to the upper limit of the transferred energy.

0.8

G [NH2-Ca]+ m/z 55.98 + [OCH]+ m/z 29.01

m/z 19.98

1 0.9

10

[Ca(Formamide)]2+

Ca2+ formamide

12

0.7 9

0.5 0.4

int5

0.3

k1G

0 0 1

100

50

10

6

k15

10

int10

k101 k21

int2

3

G 10

[Sr(Formamide)]2+

k110

min1

k51 k12

k01 k 12 k1102+ [Sr(formamide)]

0.1

0.8

k15

k1G

0.2

0.9

k1

-1

10

k (s )

P(E)

0.6

[NH2-Sr]+ + [OCH]+

m/z 43,95 Sr2+

12

formamide

m/z 103,92 m/z 29.01

0.7

0.5 0.4 0.3

k01

k1G

0.2 0.1 0 0

k1

9

-1

10

k (s )

P(E)

0.6

k12

k15

int5

k1G

k15 k51

10

k110

min1

int10 k101

6

k12

k21

k110

50

100

10

3

int2

E (kcal/mol)

Fig. 7.12: RRKM rate constants and internal energy distributions for [Ca(Formamide)]2+ and [Sr(Formamide)]2+ (left panels). A diagrammatic sketch of the reaction pathways accessible and not accessible is on the right. Data were obtained from RRKM data and simulations of [173] and [255].

7.4 CID of Doubly Charged Ions of Metal-Neutral Complexes |

[Ca(Formamide)]2+ 10

×

k23

12

×

×

×

×

×

×

A k2A

-1

k (s )

k2A ×

10

9

k21

min1

k12

int2

k21

10

135

[CaOH]+ + [HCNH]+

m/z 56.97 m/z 28.03 D

k23 k23

int2

m/z 28.99

[Ca(H2O)]2+

6

k32

k12

10

3

[Sr(Formamide)]2+

-1

k (s )

10

10

10

k21

k23

12

×

9

×

k2A

×

×

×

6

×

× ×

A k2A

min1

k12 k21

int2

[SrOH] + + [HCNH]+

m/z 28.03 D

k23 k23

m/z 104.90

int3

m/z 52.96

[Sr(H2O)]2+

k32

k12

10

3

0

50 100 Internal vibrational energy (kcal/mol)

Fig. 7.13: RRKM rate constants and internal energy distributions for [Ca(Formamide)]2+ and [Sr(Formamide)]2+ (left panels) for the reactivity leading to [M(OH)]+ + HCNH+ vs [M(H2 O]2+ + HCN (M = Ca, Sr). A diagrammatic sketch of the reaction pathways is shown on the right. Data were obtained from RRKM data and simulations of [173] and [255].

termediate passed the products are obtained. This last assumption seems a strong approximation, but considering further thermalization of the second intermediate is probably a stronger approximation, in particular when modeling triplequadrupole experiments. For [Sr(Formamide)]2+ , the opposite behavior was obtained for the energy transfer values got from collisional simulations: the kinetics leading to [Ca(H2 O)]2+ + HCN (or better, to the intermediate, which will lately evolve into these products) is faster than the one leading to [Sr(OH)]+ + [HCNH]+ . Results are graphically shown in Figure 7.13. Finally, the statistical behavior is determined by the amount of transferred energy, which largely explains differences between two systems that are very similar (not only from qualitative considerations on Ca2+ vs Sr2+ , but also from the quantitative analysis of minima and TS’s). By combining collisional simulations and kinetic analysis (in which the rate constants were calculated via RRKM theory), it was possible to explain the experimental CID spectra. More details on some subtle reactivity behavior is shown in the next section, in which the post-TS dynamics for the [Ca(Formamide)]2+ are reported.

136 | 7 Application to Organic Molecules

7.4.3 Post-TS Dynamics RRKM theory (following transition state theory) assumes that once the TS is reached then products are formed. One well-known underlying approximation is that there is no “recrossing:” when the TS is passed the trajectory does not come back into the reactant region. Many refinements are present in the literature to improve this aspect, by taking into account the recrossing, which can be due to frictional effects. For example, the Grote–Hynes theory ends up in modifying the transmission coefficient due to timedependent friction [284]. Another phenomenon, which is not considered in TST and RRKM theory, is that the system can diverge from the minimum energy path connecting TS with products such that it can take a pathway leading to other products. One typical example is given by bifurcations [342–344]. While they were found to be important in many organic reactions [345–349], only recently it was shown that they can be involved in CID fragmentation [330]. Post-TS dynamics are generally employed to understand and point out this phenomenon. These particular dynamics were applied to the aforementioned case of [Ca(Formamide)]2+ fragmentation. We now summarize how these dynamics can be employed in order to better understand the formation of different CID products: 1. Identify all the pertinent transition states. If it is not possible to identify all the TS along the complex pathways leading to the products, at least the ones directly related to the global minimum (the initial structure of collisional dynamics) must be identified. 2. From collisional simulations, extract the pertinent ranges of transferred energy in terms of vibrational and rotational energies of the ion. 3. Select a reasonable number of rotational and vibrational activation values and run simulations with different combinations from the different TS’s. Generally, post-TS trajectories are not long, since the system has an important amount of initial acceleration and it quickly goes to products. Note that, as from TST theory, since we start from the TS, about 50% of the trajectories are expected to form back reactants. 4. Analyze the occurrence of different products as a function of the way the TS is activated. If the minimum energy path connecting the TS to the products (or to an intermediate) is known, look in particular if some of the TS’s lead to other products and how this observation eventually evolves as a function of the activation energy. 5. If one TS provides other products except those resulting from preliminary intrinsic reaction coordinate (IRC, see [350]) calculations, then one should suspect that the system has one (or more) bifurcation(s). Note that from a strictly topological point of view a bifurcation is characterized by a ridge inflection point: in this case the multiple-product behavior should also persist if little activation energy is released (for example no rotational energy and vibrational energy less then zeropoint energy).

7.5 CID of Uracil

| 137

The application of this approach to the [Ca(Formamide)]2+ system, for which the minima and TS’s are known as well as the final internal energy of the ions after collisions in terms of internal vibration and rotation, provided the following main interesting points: 1. Non-IRC dynamics were observed when starting from the TS, which connects the minimum to the [Ca(NH3 )]2+ + CO product. In fact also the non-IRC product CaNH+2 + HCO+ was found. When the TS is excited, the path leading to other products is opened. However, the final product distribution does not depend on how the energy is injected: same results are obtained independently if the energy is given to rotational or vibrational activation. 2. A bifurcation was observed when starting from the TS connecting the reactant to the [Ca(NH2 )]+ + HCO+ product. In fact, also in this case the product [Ca(NH3 )]2+ + CO was found, but with two additional features: (1) trajectories go to both pathways also when the initial vibrational energy is less than ZPE; (2) giving the same amount of energy in vibrational or rotational excitation has an important impact on the products. 3. Another non-IRC dynamics is observed when starting from the TS connecting the minimum to the [Ca(H2 O)]2+ + HNC product, since also the [CaOH]+ + [HCNH]+ product is observed. The ratio between the two pathways strongly depends on the activation energy. As discussed previously, fewer trajectories are needed in post-TS dynamics. For the present case, which was treated at the DFT level of theory, about 50 trajectories per “point” (i.e., TS and set of vibrational and rotational energy) each being no more than 2 ps long, were run. In Figure 7.14 we show an example of bifurcation obtained from [Ca(Formamide)]2+ post-TS dynamics. In this last, two relevant coordinates of the system are reported and the potential energy is calculated on-the-fly on the structures sampled by the dynamics.

7.5 CID of Uracil Uracil was studied in collisional simulations in its protonated (UracilH+ ) [174, 232] and cationized (with Li+ , [Li(Uracil)]+ ) [176] forms. The size of the molecule is already relatively too big to be studied with a sufficient statistical accuracy with DFT (and even less affordable is the use of MP2), such that most of the conclusions were obtained by employing semiempirical Hamiltonians, like PM3 and AM1. Note that in the case of [Li(Uracil)]+ the AM1 Hamiltonian was reparametrized to correctly describe the interaction potential. However, while a full statistically converged study is not doable at the DFT level, some trajectories can be run with DFT and the main issues compared with results from AM1 and PM3. This was done in particular to verify if some features observed using semiempirical Hamiltonians is a result of the approximation used in

138 | 7 Application to Organic Molecules

Fig. 7.14: Evolution of post TS dynamics as from TS3 (corresponding to the TS connecting [Ca(Formamide)]2+ with [Ca[NH2 ]]+ + HCO+ (marked as G in the figure). B is the other product obtained as from the dynamics: [Ca(NH3 )]2+ + CO. Other minima (min1 corresponds to the reactants, int10 and int11 to intermediate) and TS are also shown. Reprinted (adapted) with permission from [330]. Copyright 2016 American Chemical Society.

these methods or something that is also obtained using more precise approaches. This was done in the case of protonated uracil. As for protonated urea, protonated uracil has different tautomers – and much more than urea. Furthermore each tautomer can have different conformations. In the study of uracil different tautomers were considered (and for each tautomer the most stable conformer) and trajectories run at both AM1 and PM3 level of theory. For each structure about 10,000 trajectories were run, using Ar as a colliding projectile and employing the quasiclassical Boltzmann normal mode sampling. The six tautomers considered are shown in Figure 7.15 and results summarized in Table 7.3 where also experimental results are reported. Note that the simulations were done at collision energies higher than in experiments. This is a common computational trick to enhance the reactivity, which can be used if one does not do a study of the fragmentation as a function of collision energy. Results show that the two semiempirical methods globally show the same results and are in relatively good agreement with DFT results. We should note that in the case of DFT only 50 trajectories per system were run, such that products which were found as 100% in DFT were also obtained as dominant in AM1 or PM3. Surely, semiempirical Hamiltonian results should be considered as qualitative information; while results are robust concerning the peak distribution obtained, their abundance can also vary considerably. Considering collisional dynamics results, two main aspects are relevant: (i) the appearance of the products is due to the initial tautomer and (ii) trajectories can be analyzed to determine the underlying mechanisms.

7.5 CID of Uracil

| 139

Fig. 7.15: Tautomers of uracil considered in collisional simulations (see [174] for details).

Tab. 7.3: Fragmentation products as obtained from UracilH+ CID simulations with a collision energy of 13 eV (49.7 eV in the laboratory framework). T1–T6 are the six tautomers considered and shown in Figure 7.15. DFT is at B3LYP/6-31G level except for T6 for which BLYP/6-31G results are reported. Experiments reported correspond to experimental (normalized) abundances as obtained for a collision energy of 25 eV (in the laboratory framework). Data are extracted from [174]. Ions (m/z)

96

Experiments T1-PM3 T1-AM1 T1-DFT T2-PM3 T2-AM1 T2-DFT T3-PM3 T3-AM1 T3-DFT T4-PM3 T4-AM1 T4-DFT T5-PM3 T5-AM1 T5-DFT T6-PM3 T6-AM1 T6-DFT

38.6 – – – – – – – – – 3 2.4 – – – – 5 2 –

95

85

70

60.0 – – – 3 24 100 1 1 – – – – – – – 1 1.5

– – – – 77 5 – 13 2 – – 1 – 1 1 – 88 78 100

50.2 100 – 100 8 25 – 9 3 40 6 3.6 15 97 98 100 2 1.5 –

68 2.1 – – – – – – – – – – – – – – – 0, ∀α. Here n = 2 such that ∂T ∑ q̇ j = 2T (B.78) ∂ q̇ j j And finally we obtain the well-known expression of the Hamiltonian as H = 2T − (T − V) = T + V

(B.79)

The Hamiltonian is crucial in classical mechanics and its quantum operator analogue is the well-known key operator in both the time-independent and time-dependent Schrödinger equation. Finally, we should notice that the Lagrangian (equation (B.49)) and Hamiltonian (equations (B.63) and (B.64)) are coordinate invariant (the form is equal if we use polar coordinates). The (q i , p i ) pair, which is connected by Hamiltonian equations of motion, is called a canonical conjugate. In quantum mechanics, the uncertainty principle holds for this pair.

List of Figures Fig. 2.1 Fig. 2.2

Fig. 2.3 Fig. 2.4

Fig. 2.5

Fig. 2.6

Fig. 2.7

Fig. 2.8

Fig. 2.9

Fig. 2.10 Fig. 2.11

Details of ESI coupled to a quadrupole mass spectrometry. Reprinted from [18], with permission from Elsevier. | 7 Schematic picture of the different ESI mechanisms: ion evaporation model (IEM), charge residue model (CRM) and charge ejection model (CEM). Reprinted with permission from [30]. Copyright 2013 American Chemical Society. | 9 Schematic picture of MALDI and FAB ionization techniques. Reproduced from [46] with permission from The Royal Society of Chemistry. | 10 MS/MS spectrum of protonated N-acetyl O-methoxy proline as obtained on an ion trap (a) and on a triple quadrupole (b) apparatus. Reprinted from [18] [51], with permission from Elsevier. | 13 Schematic picture of CID activation mechanism and resulting fragmentation: (a) single collision limit (LSC in the text), (b) multiple collisions (MC in the text). | 13 Schematic picture of IRMPD activation mechanism considering ν 0 → ν 1 successive vibrational excitations. Reproduced from [67] with permission from The Royal Society of Chemistry. | 17 Example of IRMPD spectrum: protonated Leuenkephalin represented as (A) appearance spectrum and (B) depletion spectrum. Comparison with theoretical spectra of two minima (C and D) is also shown within the corresponding structures. Reprinted with permission from [72]. Copyright 2007 American Chemical Society. | 18 a − q stability diagram of a quadrupole filter. The shade area represents the x − y stability region. In the inset the corresponding mass allowed, showing three consecutive m/z values separated by one unit mass. Reprinted with permission from [90]. Copyright 1986 American Chemical Society. | 22 Example of a breakdown curve as obtained by CID of L-cysteine anion (m/z 200) at different collision energies. E COM are reported. We thank D. Scuderi and M.E. Crestoni for the raw data. | 25 Example of a fragmentation PES for (Sr-Formamide)2+ complex. Figure adapted from [106]. Reproduced by permission of the PCCP Owner Societies. | 29 Schematic picture of a triple quadrupole mass spectrometer. | 30

Fig. 3.1 Fig. 3.2

Example of rotation around molecular Euler angles. | 44 Definition of the impact parameter, b, for two spherical particles colliding (the horizontal parallel lines represent the trajectory of the particles) at a distance d. | 45

Fig. 4.1 Fig. 4.2

Summary of typical interactions used in classical force fields. | 56 Dissociation energy for the different Cr(CO)+6 species, written as a function of the extent of dissociation, Ω. Ω and D(Ω) are defined in equations (4.4) and (4.5), respectively. Reproduced from [165] with permission from the PCCP Owner Societies. | 57 Example of ab initio fit of equation (4.10) fixing n = 9, as reported by Meroueh and Hase. Reprinted with permission from [10]. Copyright 1999 American Chemical Society. | 61

Fig. 4.3

https://doi.org/10.1515/9783110434897-012

204 | List of Figures

Fig. 5.1

Fig. 5.2

Fig. 5.3

Fig. 5.4 Fig. 5.5

Fig. 5.6

Fig. 6.1 Fig. 6.2

Fig. 6.3 Fig. 6.4 Fig. 6.5

Fig. 6.6 Fig. 6.7

Fig. 7.1

Fig. 7.2 Fig. 7.3

Rate constant for a model Eckart barrier in which the barrier energy is 32,737 cm−1 and the curvature at the barrier is 2288 cm−1 . In black are rate constants obtained taking into account the tunneling via equation (5.27), while in red we report results without tunneling. Frequencies for minima and TS are the ones for proton transfer in formaldehyde as reported by Miller [245]. | 86 Example of loose TS location and rate constant calculation for the reaction: (Ca-formamide)2+ → Ca2+ + formamide. [255] Reproduced by permission of the PCCP Owner Societies. | 89 Schematic picture of time evolution of rate constant as from equation (5.88). The short time scale, τ short , and the long phenomenological rate time, τ, are also sketched. Note that for t = 0 the rate constant coincides with the TST one (see equation (5.90)). | 98 Schematic view of the SSP picture for unimolecular decomposition. Only a few states are shown for simplicity. | 101 Initial population decay for internal activation of Pro−2 dipeptide as from chemical dynamics simulations at different energies (here expressed as temperature thanks to relation in equation (5.114). Data are shown as black dots, while the results of the exponential fitting are shown as red dashed curves. [130]. Reproduced by permission of The Royal Society of Chemistry. | 105 Arrhenius plot as from rate constants obtained in Figure 5.5 for unimolecular fragmentation of Pro−2 . [130]. Reproduced by permission of The Royal Society of Chemistry. | 105 Protonated testosterone with its two possible tautomers. In red we underline the charged portion of the molecule. | 110 Percentage of energy transfer as a function of impact parameter, b, as obtained from the β-aminoethylcellobiose + Ar collisional system at E coll = 19.5 eV. (Data elaborated from results shown in [312]). | 111 Schematic flowchart for the collisional simulations aimed to mimic CID in the single collision limit. | 114 A snapshot of dissociation of octaglycine-H+ by collision with diamond surface. | 114 Population decay and Arrhenius plot for thermal fragmentation of doubly protonated TIK peptide. Reprinted with permission from [131]. Copyright 2016 American Chemical Society. | 116 Schematic flowchart for the thermal activation simulations. | 117 Calculated and experimental electronic ionization mass spectra using the QCEIMS method. Reprinted with permission from [322]. Copyright 2016 American Chemical Society. | 119 Two protonated forms of urea considered. Relative energies as obtained from MP2/aug-cc-pVTZ and MP2/6-31G* (in parenthesis) calculations are also shown. | 123 Products as a function of collision energy for UreaH+ + Ar collisions (nitrogenprotonated tautomer of urea results are reported). Data are taken from [198]. | 124 Shattering/nonshattering ratio obtained in UreaH+ + Ar collisions (nitrogenprotonated tautomer of urea results are reported). Data are taken from [198]. | 124

List of Figures

Fig. 7.4

Fig. 7.5

Fig. 7.6

Fig. 7.7

Fig. 7.8 Fig. 7.9

Fig. 7.10

Fig. 7.11

Fig. 7.12

Fig. 7.13

Fig. 7.14

Fig. 7.15 Fig. 7.16

| 205

Probabilities of forming reaction (7.1) and reaction (7.2) products and reactants (urea protonated on nitrogen) as a function of time. Results are reported for E coll = 145.5 kcal/mol. Data are taken from [198]. | 125 Fraction of trajectories with enough internal energy to pass the barrier to: form NH+4 + CONH products (from trajectories using as initial structure the nitrogen protonated tautomer) and isomerize the most stable oxygen-protonated tautomer to the nitrogen protonated one (from trajectories using as initial structure the oxygen protonated tautomer). Data are taken from [198]. | 126 RRKM rate constant as a function of energy for the reactivity of nitrogen protonated urea towards formation of NH+4 + CONH products at two different rotational energies. | 127 Ratio (in relative %) between NH3 + CONH+2 and NH+4 + CONH products as obtained from post-TS dynamics (the initial TS is the one connecting the two tautomeric for of protonated urea, Figure 7.1) as a function of internal vibrational energy at a different total rotational energy. Data are taken from [175]. | 128 Schematic pictures showing the two general mechanisms of doubly charged ions fragmentation: (a) neutral loss and (b) Coulomb explosion. | 129 Example of neutral loss from [Ca(Urea)]2+ + Ar collisions. Ca–O and Ca–Ar distances as a function of time are shown with corresponding structures (snapshots from one trajectory). Ar atom is in green, Ca in cyan. Data are taken from simulations of [172]. | 132 Characteristic times as a function of ion energy as obtained from collisional simulations and RRKM analysis for the [Ca(formamide)]2+ system. Reprinted with permission from [173]. Copyright 2014 American Chemical Society. | 132 Evolution in time of the fraction of reactant, products and intermediate (which lies between reactant and CaNH+2 + H2 NCO+ product) obtained from RRKM analysis of [Ca(Urea)]2+ fragmentation. Left panel: results considering the energy transfer distribution (at E coll = 300 kcal/mol); right panel: results at a fixed value of internal energy (200 kcal/mol), corresponding to the upper limit of the transferred energy. | 134 RRKM rate constants and internal energy distributions for [Ca(Formamide)]2+ and [Sr(Formamide)]2+ (left panels). A diagrammatic sketch of the reaction pathways accessible and not accessible is on the right. Data were obtained from RRKM data and simulations of [173] and [255]. | 134 RRKM rate constants and internal energy distributions for [Ca(Formamide)]2+ and [Sr(Formamide)]2+ (left panels) for the reactivity leading to [M(OH)]+ + HCNH+ vs [M(H2 O]2+ + HCN (M = Ca, Sr). A diagrammatic sketch of the reaction pathways is shown on the right. Data were obtained from RRKM data and simulations of [173] and [255]. | 135 Evolution of post TS dynamics as from TS3 (corresponding to the TS connecting [Ca(Formamide)]2+ with [Ca[NH2 ]]+ + HCO+ (marked as G in the figure). B is the other product obtained as from the dynamics: [Ca(NH3 )]2+ + CO. Other minima (min1 corresponds to the reactants, int10 and int11 to intermediate) and TS are also shown. Reprinted (adapted) with permission from [330]. Copyright 2016 American Chemical Society. | 138 Tautomers of uracil considered in collisional simulations (see [174] for details). | 139 Fragmentation mechanisms corresponding to the formation of m/z 70 and 96 as obtained from collisional simulations (see [174] for details). | 141

206 | List of Figures

Fig. 7.17

Fig. 7.18

Fig. 8.1 Fig. 8.2

Fig. 8.3 Fig. 8.4

Fig. 8.5

Fig. 8.6

Fig. 8.7

Fig. 8.8 Fig. 8.9 Fig. 8.10 Fig. 8.11 Fig. 8.12 Fig. 8.13 Fig. 8.14 Fig. 8.15

Fig. 8.16

Average energy transfer as obtained from collisional simulations (dots) of UracilH+ with Ar as a function of the collision energy (in the center-of-mass framework) and from energy transfer fitting (see equation (7.3)). Int: total; Vib: vibrational; Rot: rotational. [232]. Reproduced by permission of the PCCP Owner Societies. | 142 Fragmentation mechanisms corresponding to the formation of m/z 95 as obtained from collisional (schemes (a) and (b)) and KMC simulations (scheme (a)). This suggests that mechanism (b) occurs via a nonstatistical process (see [174] and [232] for details). | 143 Structure of testosterone with the associated nomenclature for cycles and atom numbers. | 145 Theoretical mass spectrum of protonated testosterone as obtained by MSINDO collisional simulations from the most stable tautomer. The structures corresponding to the most relevant products are also shown. Reprinted from [366], with permission from Elsevier. | 146 Structure and formation mechanisms of m/z 97 ion as determined by direct dynamics simulations. Reprinted from [366], with permission from Elsevier. | 148 Time evolution of some relevant atomic distances (C labels are those of Figure 8.1) as obtained by trajectories resulting in product m/z 97 through mechanism b. Data are those of simulations reported in [366]. | 149 Final structures (a) and mechanisms corresponding to formation of structure I (the most abundant pathway) for the formation of ion m/z 123 as revealed by collisional simulations of protonated testosterone. Data are extracted from [366]. | 149 Final structures for ion m/z 109 as from CID of protonated testosterone: I is the structure proposed by Williams et al. [367]; II is the structure as obtained from chemical dynamics simulations (see [366]). In both cases different resonance structures are shown. | 150 Structure and major formation mechanisms of m/z 109 ion obtained as a result of collisional dynamics of protonated testosterone. Reprinted from [366], with permission from Elsevier. | 150 The structure of boldenone, androst-1,4-diene-17β-ol-3-one. | 151 Theoretical mass spectrum of boldenone as obtained from PM3 collisional simulations. | 151 Most relevant mechanisms responsible for the formation of the m/z 121 ion as result of collisional dynamics of protonated boldenone. | 152 Most important fragmentation pathway responsible for the formation of ion m/z 135 as result of collisional dynamics of protonated boldenone. | 153 Generic structure of a six-member based polysaccharide. | 154 Domon and Costello nomenclature for carbohydrate fragmentation [371]. | 154 Structure of the Galactose-6-Sulfate anion used as initial condition for collisional simulations (see [171]). | 155 Mechanisms responsible for the formation of m/z 139 (a) and 97 (b) from CID of Galactose-6-sulfate anion as obtained by coupling collisional dynamics simulations with experiments (see [171]). | 157 Mechanism suggested for the formation of ion m/z 119 from CID of Galactose-6-sulfate anion as from collisional dynamics of the precursor ion and one of its primary fragmentation products (ion m/z 199) (see [171]). | 158

List of Figures

Fig. 8.17 Fig. 8.18

Fig. 8.19 Fig. 8.20 Fig. 8.21 Fig. 8.22 Fig. 8.23 Fig. 8.24 Fig. 8.25

| 207

β-amino-ethyl derivatives of D-cellobiose (a), D-maltose (b) and D-gentobiose (c) studied by collisional simulations (see [312]). | 159 Theoretical mass spectra of β-amino-ethyl-cellobiose (upper panel) and β-amino-ethyl-gentobiose (lower panel). For cellobiose in red are shown the fragments (except the parent ion m/z 386 which are in common with experiments with the associated nomenclature. Data are extracted from [312]. | 160 Fragmentation sites corresponding to the formation of Y1 and Y0 ions (respectively m/z 224 and 62) for β-amino-ethyl cellobiose and β-amino-ethyl-gentobiose. | 161 Peptide fragmentation with the nomenclature of Roepstorff and Fohlman [390] for an example of generic singly protonated tetrapeptide. | 164 Schematic picture of suggested mechanisms as in [391] involved in peptide sequence loss (scrambling). | 164 Fragmentation mechanisms involved in the formation of diketopiperazine or oxazolone b ion. | 165 Structure of N-formylalanylamide with the four different protonation state and the pathways showing the observed product ions. Results extracted from [201]. | 167 Mechanism leading to the formation of ion b1 as from chemical dynamics simulations of [201]. | 168 Theoretical mass spectra of protonated penta- and octaglycine (panels a and b, respectively). Results are extracted from [205]. | 169

List of Tables Tab. 4.1 Tab. 4.2 Tab. 4.3

Parameters used intermolecular potential (equation (4.10) with n = 9) between Ar and different atom types. | 59 Unimolecular dissociation induced by collisions with inert gases studied by wave function and DFT methods. | 70 Systems studied by semiempirical methods in chemical dynamics of CID. | 71

Tab. 6.1

Standard deviations associated with different reaction probabilities as a function of the number of reactive trajectories (N). | 112

Tab. 7.1

Comparison between UreaH+ + N2 and UreaH+ + Ar collisional systems results. Data are collected from [198] and [175]. | 127 Summary of collisional products obtained from chemical dynamics simulations of three doubly charged complexes with Ar, as from [172] and [173]: U is for Urea and F for Formamide. For [Ca-Urea]2+ we show MP2/6-31G* results, while for [Ca-Formamide]2+ and [Sr-Formamide]2+ we show G96LYP/6-31G* ones. Collision energies (CE) are given in kcal/mol. Experimental data are taken from Refs. [340, 341], and [106] for [Ca-Urea]2+ , [Ca-Formamide]2+ and [Sr-Formamide]2+ , respectively. | 131 Fragmentation products as obtained from UracilH+ CID simulations with a collision energy of 13 eV (49.7 eV in the laboratory framework). T1–T6 are the six tautomers considered and shown in Figure 7.15. DFT is at B3LYP/6-31G level except for T6 for which BLYP/6-31G results are reported. Experiments reported correspond to experimental (normalized) abundances as obtained for a collision energy of 25 eV (in the laboratory framework). Data are extracted from [174]. | 139

Tab. 7.2

Tab. 7.3

Tab. 8.1 Tab. 8.2

Tab. 8.3

Tab. A.1 Tab. A.2 Tab. A.3 Tab. A.4

Peaks observed in the mass spectrometry of boldenone as obtained from experiments ([370]) and collisional simulations. | 152 Fragments obtained in theoretical and experimental MS/MS and MS/MS/MS of protonated β-amino-ethyl-cellobiose. For MS/MS we report in “( )” results as from m/z 368 ion while in “[ ]” as from m/z 224. In all cases “X” means that the fragment is observed, while “–” that it is not. Data are extracted from [312]. | 162 Activation energies (in kJ/mol) corresponding to fragmentation pathways of TIK doubly protonated peptide (data are extracted from [131]). Values obtained from Arrhenius plot and PES analysis are shown. | 170 Some important and useful Fourier transforms using definition (A.96). | 189 Some important and useful Laplace transforms. | 190 Some important and useful Laplace–Carson transforms. | 190 Table of some Laplace–Carson transforms and corresponding original functions pertinent to chemical kinetics. | 192

https://doi.org/10.1515/9783110434897-013

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Index B Born–Oppenheimer 36, 37 C Collision Induced Dissociation 11, 12, 14, 103, 106, 109 – Chemical dynamics – Application 130, 138, 146, 151, 156, 160, 167, 170 – Methods 43–45, 67, 69, 70, 73 – Energy transfer 74–76 – Modeling flowchart 114 Cross section 91 – Collision 12, 25, 73 – Reaction 26, 45 E ECD 18 – Cornell model 19 – Utah–Washington model 19 Electrospray Ionization 6 – CEM 8 – CRM 8 – IEM 8 ETD 19 F FAB 9 Fluctuation dissipation theorem 94 Force Field 56 FT-ICR 23, 31 H Hamiltonian – Classical 38, 53 – Electrostatic 34 – Harmonic oscillators 42, 107 – Molecular 34, 61 – Operator 33, 92 Hartree–Fock 62 – Canonical equations 65 – Fock operator 65 – General equation 65

https://doi.org/10.1515/9783110434897-015

– LCAO 66 – Roothan–Hall equations 66 Hellmann Feynman theorem 61 I IRMPD 16 Isotopic labeling 27, 140, 147, 168 L Linear single collision 12 M MALDI 8 Morse function 57 P Pauli principle 63 Pulay force 61 R Rate constant – Arrhenius 104 – Grote Hynes 101 RRKM theory 79, 80, 115 – Anharmonicity 87 – Application 126, 130, 131, 133 – Rotational effect 126 – Derivation 80 – Formula 81 – Rotational effects 82 – Tunneling 86 – Loose transition state 88 S Slater determinant 64 Surface Induced Dissociation 15, 113 T Tandem mass spectrometry 11 Transition state theory – Eyring formula 90 V Variational theorem 64