Molecular Modelling and Synthesis of Nanomaterials: Applications in Carbon- and Boron-based Nanotechnology [1st ed.] 9783030327255, 9783030327262

Table of contents :
Front Matter ....Pages i-xiii
Front Matter ....Pages 1-1
Molecular Modelling (Ihsan Boustani)....Pages 3-48
Front Matter ....Pages 49-49
Magic Numbers (Ihsan Boustani)....Pages 51-68
Alkali Metal Clusters (Ihsan Boustani)....Pages 69-111
Boron Clusters (Ihsan Boustani)....Pages 113-254
Carbon and Inorganic Binary Clusters (Ihsan Boustani)....Pages 255-281
Front Matter ....Pages 283-283
Two-Dimensional Sheets (Ihsan Boustani)....Pages 285-362
One-Dimensional Nanotubes (Ihsan Boustani)....Pages 363-413
Three-Dimensional Polyhedra (Ihsan Boustani)....Pages 415-501
Front Matter ....Pages 503-503
Nanocoating and Nanobattery (Ihsan Boustani)....Pages 505-553
Nanosensors and Fullerens (Ihsan Boustani)....Pages 555-589
Back Matter ....Pages 591-594

Citation preview

Springer Series in Materials Science 290

Ihsan Boustani

Molecular Modelling and Synthesis of Nanomaterials Applications in Carbon- and Boron-based Nanotechnology

Springer Series in Materials Science Volume 290

Series Editors Robert Hull, Center for Materials, Devices, and Integrated Systems, Rensselaer Polytechnic Institute, Troy, NY, USA Chennupati Jagadish, Research School of Physical, Australian National University, Canberra, ACT, Australia Yoshiyuki Kawazoe, Center for Computational Materials, Tohoku University, Sendai, Japan Jamie Kruzic, School of Mechanical & Manufacturing Engineering, UNSW Sydney, Sydney, NSW, Australia Richard M. Osgood, Department of Electrical Engineering, Columbia University, New York, USA Jürgen Parisi, Universität Oldenburg, Oldenburg, Germany Udo W. Pohl, Institute of Solid State Physics, Technical University of Berlin, Berlin, Germany Tae-Yeon Seong, Department of Materials Science & Engineering, Korea University, Seoul, Korea (Republic of) Shin-ichi Uchida, Electronics and Manufacturing, National Institute of Advanced Industrial Science and Technology, Tsukuba, Ibaraki, Japan Zhiming M. Wang, Institute of Fundamental and Frontier Sciences - Electronic, University of Electronic Science and Technology of China, Chengdu, China

The Springer Series in Materials Science covers the complete spectrum of materials research and technology, including fundamental principles, physical properties, materials theory and design. Recognizing the increasing importance of materials science in future device technologies, the book titles in this series reflect the state-of-the-art in understanding and controlling the structure and properties of all important classes of materials.

More information about this series at http://www.springer.com/series/856

Ihsan Boustani

Molecular Modelling and Synthesis of Nanomaterials Applications in Carbon- and Boron-based Nanotechnology

123

Ihsan Boustani Faculty of Mathematics and Natural Science University of Wuppertal Wuppertal, Nordrhein-Westfalen, Germany

ISSN 0933-033X ISSN 2196-2812 (electronic) Springer Series in Materials Science ISBN 978-3-030-32725-5 ISBN 978-3-030-32726-2 (eBook) https://doi.org/10.1007/978-3-030-32726-2 © Springer Nature Switzerland AG 2020 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface

This book presents carbon- and boron-based nanomaterials predicted by molecular modelling and numerical simulation tools, then confirmed by modern experimental methods. It begins by summarizing basic theoretical methods, then showing the influence of the magic numbers of the atomic alkali metal clusters on the development of the clusters into nanostructures, theoretically and experimentally. It continues into a discussion of atomic clusters and nanostructures, focusing primarily on boron and carbon. It thoroughly explores the one-, two-, and three-dimensional structures of boron and carbon, closing with their potential applications in nanotechnology. These include myriad possibilities, from nano-coating and nano-sensing to nano-battery with high borophene capacity. Materials scientists will find the broad discussion of computational modelling as well as its specific application to boron and carbon an essential reference. The evolution of small to nanosized clusters (nanosize
the width of ten small abreast atoms) in form of nanotubes, nanowires, nano-sheets, nanocages induced the epitome nanomaterials and led consequently to nanostructured Materials-by-Design. That enables us to construct nanostructures with desired properties, like definite electron conductivity or the range of band gaps for applications in semiconductors and nanoelectronics, or neutron protection, absorption, and radiation shielding for nuclear reactors and aero-space vehicles and astronauts, or coating, hardening, resistance, anti-abrasion and anti-corrosion for surfaces and interfaces, or drug delivery and cancer therapy in medicine, or energetic materials which have a high amount of chemical energy that can be stored and released, or can be used as propulsion for carrier rockets, or finally molecular sensor or wheels to attract dirty molecules. Conventionally and traditionally, the experiment observed, measured and announced always novel materials and related properties as Quasicrystals and Buckminster-Fullerenes discovered in 1984 and 1985 by Shechtman and Kroto, respectively. Furthermore, the Femtochemistry was developed by Zeweil in 1999, and the superconductivity of MgB2 was observed in 2001 by Nagamatsu. However, the theory was always lagging behind to explain and reproduce the experimental data. However, the recent development of the theory and the rapid evolution of the v

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computational tools reverse the research process and enabled the molecular modelling to establish new research fields to predict and simulate novel structures and properties, so far unknown in nature. Nowadays, there are a number of new materials, which are predicted and discovered by theory, more precisely by molecular modelling, and are already confirmed by experiments. The success of the molecular modelling can be documented with many examples. For instance, Boustani, the author of this Book, predicted in the 80s and 90s the structures of lithium and sodium clusters, and the existence of planar boron clusters and planar sheets, known as borophene, as well as the existence of boron nanotubes. The predicted nanostructures were unknown in nature at that time. Around ten years later, these species were detected and confirmed experimentally. New crystalline form of boron was proposed and predicted by Oganov via theory, named c-boron. This new boron solid was later experimentally observed. All these novel and potential applications disembogue into the term nanotechnology. The interplay between the chemical modelling and the experimental verification was in the recent years successfully performed. It also demonstrates the near-term capability and viability of controlled synthesis for specific engineering of definite material properties. Wuppertal, Germany

Ihsan Boustani

Contents

Part I 1

Molecular Modelling

Molecular Modelling . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Concepts of Molecular Modelling . . . . . . . . . . . 1.1.1 Areas of Molecular Modelling . . . . . . . . 1.1.2 The Basics of Molecular Modelling . . . . 1.2 Molecular Mechanics . . . . . . . . . . . . . . . . . . . . 1.2.1 Force Field Equation . . . . . . . . . . . . . . 1.2.2 Minimization Procedure . . . . . . . . . . . . 1.2.3 Optimization Methods . . . . . . . . . . . . . 1.2.4 Gradient Methods . . . . . . . . . . . . . . . . . 1.2.5 General Newton–Raphson Method . . . . 1.3 Molecular Dynamics . . . . . . . . . . . . . . . . . . . . . 1.3.1 Trajectory and Equation of Motion . . . . 1.3.2 Initialization of Positions and Velocities 1.3.3 Verlet Algorithm . . . . . . . . . . . . . . . . . 1.3.4 Leap-Frog Algorithm . . . . . . . . . . . . . . 1.4 Ab Initio Quantum Chemical Methods . . . . . . . . 1.4.1 Some Basics of Quantum Mechanics . . . 1.4.2 The Hamiltonian . . . . . . . . . . . . . . . . . 1.4.3 Time-Dependent Schrödinger Equation . 1.4.4 Hartree–Fock Theory . . . . . . . . . . . . . . 1.4.5 Post Hartree–Fock . . . . . . . . . . . . . . . . 1.5 Density Functional Theory . . . . . . . . . . . . . . . . 1.5.1 Kohn–Sham Equations . . . . . . . . . . . . . 1.5.2 Local (Spin) Density Approximation . . . 1.5.3 Non-local Density-Gradient Corrections . 1.5.4 Hybrid Functionals . . . . . . . . . . . . . . . .

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1.6

Semi-empirical Methods . . . . . . . . . . . . . . . . . 1.6.1 Interatomic Potentials . . . . . . . . . . . . . 1.6.2 Two-Body Potentials . . . . . . . . . . . . . 1.6.3 Three-Body Potentials . . . . . . . . . . . . 1.7 Concepts of Semi-empirical Methods . . . . . . . . 1.8 Basic Models . . . . . . . . . . . . . . . . . . . . . . . . . 1.8.1 Semiempirical Model NDDO . . . . . . . 1.8.2 Semi-empirical Model Tight-Binding . . 1.8.3 Semi-empirical Model MNDO . . . . . . 1.8.4 Semi-empirical Models AM1 and PM3 1.9 Chemistry Software . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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2

Magic Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Atomic Clusters and Magic Numbers . . . . . . . . . . . 2.2 Shell Structures in Atoms and Clusters . . . . . . . . . . 2.2.1 The Shell Structure of Electrons in Atoms . 2.2.2 Shell Structure of Nucleons . . . . . . . . . . . . 2.2.3 Shell Structure of Condensed Clusters . . . . 2.2.4 Shell Structure of Hollow Clusters . . . . . . 2.3 Magic Numbers in Nature . . . . . . . . . . . . . . . . . . . 2.3.1 Magic Numbers of Electron Shells . . . . . . 2.3.2 Magic Numbers of Nuclear Shells . . . . . . . 2.3.3 Magic Electronic Shells of Clusters . . . . . . 2.3.4 Magic Filled Shells of Clusters . . . . . . . . . 2.3.5 Magic Hollow Shells of Clusters . . . . . . . . 2.3.6 Magnetic Shells of Electrons . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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3

Alkali Metal Clusters . . . . . . . . . . . . . . . . 3.1 Alkali Metals . . . . . . . . . . . . . . . . . . 3.2 Lithium Metal Clusters . . . . . . . . . . . 3.2.1 Production of Metal Clusters . 3.2.2 Lithium Atom . . . . . . . . . . . 3.2.3 Lithium Clusters . . . . . . . . . . 3.2.4 Experimental Evidence . . . . . 3.2.5 Lin Clusters for n  6 . . . . . 3.2.6 Lin Clusters for n  14 . . . . 3.2.7 Lin Clusters for n  20 . . . . 3.2.8 Lin Clusters for n  26 . . . .

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Part II

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Magic Numbers and Clusters

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3.3

Sodium Metal Clusters . . . . . . . . . . . . . . . . . . . . 3.3.1 Sodium Atom . . . . . . . . . . . . . . . . . . . . 3.3.2 Sodium Clusters . . . . . . . . . . . . . . . . . . . 3.3.3 First Study on Small Na Clusters . . . . . . 3.3.4 Electronic Shells of Na Clusters . . . . . . . 3.3.5 Jellium Model for Na Clusters . . . . . . . . 3.3.6 Mass Spectra of Na Clusters . . . . . . . . . . 3.4 Geometrical Structures of Nan Clusters . . . . . . . . 3.4.1 Nan Clusters for n  7 . . . . . . . . . . . . . 3.4.2 The Structure of the Magic Cluster Na8 . . 3.4.3 Raman Spectra of the Magic Cluster Na8 . 3.4.4 Basics of Photoabsorption . . . . . . . . . . . . 3.4.5 Nan Clusters for 3  n  8 . . . . . . . . . 3.4.6 Nan Clusters for 9  n  14 . . . . . . . . 3.4.7 Nan Clusters for 13  n  380 . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Boron Clusters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 The Element Boron . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 History of Boron . . . . . . . . . . . . . . . . . . . . 4.1.2 Energetic Boron . . . . . . . . . . . . . . . . . . . . . 4.1.3 Boron Materials . . . . . . . . . . . . . . . . . . . . . 4.1.4 Chemistry of Boron . . . . . . . . . . . . . . . . . . 4.2 Icosahedral-Based Crystalline Boron . . . . . . . . . . . . 4.2.1 First Crystalline . . . . . . . . . . . . . . . . . . . . . 4.2.2 Rhombohedral Boron . . . . . . . . . . . . . . . . . 4.2.3 Orthorhombic and Tetragonal Boron . . . . . . 4.3 Non-icosahedral Boron Clusters Bn (2  n  14) . 4.3.1 Small Boron Clusters . . . . . . . . . . . . . . . . . 4.3.2 Prediction of Planarity and Aufbau Principle 4.3.3 Theoretical Confirmations . . . . . . . . . . . . . . 4.3.4 Experimental Confirmations . . . . . . . . . . . . 4.3.5 Hydrocarbon and Boron Clusters . . . . . . . . . 4.3.6 Coulomb Explosion . . . . . . . . . . . . . . . . . . 4.4 Neutral and Charged Boron Clusters Bn (n  40) . . 4.4.1 Boron Clusters Bn (15  n  25) . . . . . . . 4.4.2 Boron Clusters Bn (26  n  36) . . . . . . . 4.4.3 Boron Clusters Bn (37  n  40) . . . . . . . 4.5 Largest Predicted Quasi-planar Cluster B84 . . . . . . . . 4.5.1 Frequencies, IR and Raman Spectra of B84 . 4.5.2 Clusters Stability in a-Sheet . . . . . . . . . . . . 4.5.3 Electron Distribution in B84 . . . . . . . . . . . . 4.6 Boron Nitride and Carbide Clusters . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Contents

Carbon and Inorganic Binary Clusters . . . . . . . . . . . . . 5.1 History of Carbon . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.1 Chemistry of Carbon . . . . . . . . . . . . . . . . 5.1.2 Carbon Allotrope . . . . . . . . . . . . . . . . . . . 5.2 Carbon Clusters . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 Mass Spectra of Carbon Clusters . . . . . . . . 5.2.2 Small Carbon Clusters Cn (2  n  10) . 5.2.3 Small Carbon Clusters Cn (12  n  32) 5.3 Inorganic CdSe-Like Binary Clusters . . . . . . . . . . . 5.3.1 Clusters-Structures Versus Clusters-Sizes . . 5.3.2 Small Inorganic Binary Clusters . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Part III 6

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Modelling of Nanostructures

Two-Dimensional Sheets . . . . . . . . . . . . . . . . . . . . . . . 6.1 Boron-Based Nanosheets (BNSs) . . . . . . . . . . . . . 6.1.1 Nano (Concept)? . . . . . . . . . . . . . . . . . . 6.1.2 Nanostructure (Concept)? . . . . . . . . . . . . 6.1.3 The “Aufbau Principle” and Nanosheets . 6.1.4 The First Flackes of Boron Sheets . . . . . . 6.1.5 Prediction of BNSs via Theory . . . . . . . . 6.1.6 BNSs on Substrates via Theory . . . . . . . . 6.2 Confirmation of BNSs via Experiment . . . . . . . . . 6.2.1 Synthesis of Amorphous BNSs . . . . . . . . 6.2.2 Synthesis of Buckled BNSs . . . . . . . . . . 6.2.3 Synthesis of c-B28 -Sheets . . . . . . . . . . . . 6.2.4 Synthesis of b12 -Sheets (c-Sheets) . . . . . . 6.2.5 Synthesis of Honeycomb Borophene . . . . 6.3 Carbon-Based Nanosheets (CNSs) . . . . . . . . . . . . 6.3.1 Graphene: The Atom-Thick Sheet . . . . . . 6.3.2 Graphene Derivatives . . . . . . . . . . . . . . . 6.3.3 Penta-Graphene . . . . . . . . . . . . . . . . . . . 6.3.4 Penta-Hexa (Ph)-Graphene . . . . . . . . . . . 6.3.5 Penta-Hexa-Hepta (Pha)-Graphene . . . . . . 6.3.6 Synthesis of Graphene . . . . . . . . . . . . . . 6.3.7 Quantum Hall-Effect of Graphene . . . . . . 6.3.8 Graphene Nanoribbons . . . . . . . . . . . . . . 6.3.9 Half-Metals and Electric Field . . . . . . . . . 6.3.10 Halfmetallicity by Chemical Modification 6.3.11 Potential Applications of Graphene . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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One-Dimensional Nanotubes . . . . . . . . . . . . . . . . . . . . 7.1 Boron-Based Nanotubes (BNTs) . . . . . . . . . . . . . 7.1.1 Prediction of BNTs via Theory . . . . . . . . 7.1.2 Armchair and Zigzag Boron Nanotubes . . 7.1.3 Electronic and Elastic Properties of BNTs 7.2 Confirmation of BNTs via Experiment . . . . . . . . . 7.2.1 Synthesis of SWBNTs . . . . . . . . . . . . . . 7.2.2 Synthesis of BNTs via CVD . . . . . . . . . . 7.2.3 Synthesis of MWBNTs via ThEM . . . . . . 7.3 Boron-Nitride Nanotubes (BNNTs) . . . . . . . . . . . 7.3.1 History of BNNTs . . . . . . . . . . . . . . . . . 7.3.2 Properties of BNNTs . . . . . . . . . . . . . . . 7.3.3 Synthsis of BNNTs . . . . . . . . . . . . . . . . 7.3.4 Models for Double-Walled BNNTs . . . . . 7.4 Carbon-Based Nanotubes (CNTs) . . . . . . . . . . . . 7.4.1 Synthesis of CNTs . . . . . . . . . . . . . . . . . 7.4.2 CNTs Versus BNTs . . . . . . . . . . . . . . . . 7.4.3 Carbon Nanocones . . . . . . . . . . . . . . . . . 7.4.4 Growth of MWCNTs . . . . . . . . . . . . . . . 7.4.5 Cholestrol@CNTs . . . . . . . . . . . . . . . . . 7.4.6 Mechanical Properties of CNTs . . . . . . . . 7.4.7 Young’s Modulus of SWCNTs . . . . . . . . 7.4.8 Wall Defects in CNTs . . . . . . . . . . . . . . 7.4.9 X- & Y-Junctions in CNTs . . . . . . . . . . . 7.4.10 Buckling in CNTs . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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8

Three-Dimensional Polyhedra . . . . . . . . . . . . . . . . . . . . . . 8.1 Boron-Based Fullerenes . . . . . . . . . . . . . . . . . . . . . . . 8.1.1 Boron-Hydride Fullerenes . . . . . . . . . . . . . . . 8.1.2 Bare Boron Fullerenes . . . . . . . . . . . . . . . . . 8.1.3 Unusually Highly Stable B100 Fullerenes . . . . 8.1.4 The (B32 þ 8k ) and (B80 þ 8k ) Families of Boron Fullerenes . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.5 The 60n2 Family of B60 Fullerenes . . . . . . . . 8.1.6 The 80n2 Family of B80 Fullerenes . . . . . . . . 8.1.7 Condensed Boron Fullerenes . . . . . . . . . . . . . 8.1.8 The Electron Counting Rules of Fullerenes . . 8.1.9 Synthesis Smallest Boron Fullerene B40 . . . . . 8.2 Carbon-Based Fullerenes . . . . . . . . . . . . . . . . . . . . . . 8.2.1 Short History of Fullerenes . . . . . . . . . . . . . . 8.2.2 Synthesis of C60 Fullerene . . . . . . . . . . . . . . 8.2.3 Fullerene Cages . . . . . . . . . . . . . . . . . . . . . .

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415 415 417 418 424

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428 433 438 442 451 462 466 466 468 470

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Contents

8.2.4 Goldberg’s Series of Polyhedra . . . . . . . . . 8.2.5 Solid Forms of C60 Fullerene . . . . . . . . . . 8.2.6 Deposition of C60 Fullerenes on Graphene . 8.2.7 Cluster Forms of ðC60 Þn Fullerene . . . . . . . 8.3 Metal Organic Frameworks MOFs . . . . . . . . . . . . . 8.3.1 What Are MOFs? . . . . . . . . . . . . . . . . . . . 8.3.2 Chemistry of MOFs . . . . . . . . . . . . . . . . . 8.3.3 Applications of MOFs . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Part IV 9

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475 481 484 490 493 493 495 497 498

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505 505 507 508 509 510 510 511 513 514 515 521 523 524 524 531 534 536 536 541 545 549 550 551 552 552

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555 556 561 561 563

Potential Application in Nanotechnology

Nanocoating and Nanobattery . . . . . . . . . . . . . . . . . . . . 9.1 Nanocoating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1.1 Coating with Boron Clusters . . . . . . . . . . . 9.1.2 Hardness Similar to Diamond . . . . . . . . . . 9.1.3 Potential Applications in Industry . . . . . . . 9.2 Conductive Nanostructures . . . . . . . . . . . . . . . . . . 9.2.1 Conductive Boron Fullerenes . . . . . . . . . . 9.2.2 Conductive Carbon Nanotubes . . . . . . . . . 9.3 Nanobattery . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.1 A Battery (Concept) . . . . . . . . . . . . . . . . . 9.3.2 Basics of a Battery . . . . . . . . . . . . . . . . . . 9.3.3 Lithium Ion Battery . . . . . . . . . . . . . . . . . 9.3.4 Non-conventional Lithium-Ion Battery . . . . 9.4 Graphene-Based Nanobattery . . . . . . . . . . . . . . . . . 9.4.1 Lithium Atoms on Graphene . . . . . . . . . . . 9.4.2 Lithium Clusters on Graphene . . . . . . . . . . 9.4.3 Dendritic Lithium and Battery Fires . . . . . 9.5 Borophene-Based Nanobattery . . . . . . . . . . . . . . . . 9.5.1 Lithium Adsorption on Borophene . . . . . . 9.5.2 Borophene-Based Metal-Ion Battery . . . . . 9.5.3 Borophene Anode for Sodium Battery . . . . 9.6 Storage Capacity of Borophene . . . . . . . . . . . . . . . 9.6.1 Borophene Capacity Fourfold of Graphene 9.6.2 High-Capacitor b12 Borophene . . . . . . . . . 9.6.3 Super-Capacitor Honeycomb Borophene . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10 Nanosensors and Fullerens . . . . . . . . . . 10.1 Nano@Sensors . . . . . . . . . . . . . . . 10.2 Carbon-Based Nanosensors . . . . . . 10.2.1 CNT Ethanol Nanosensors 10.2.2 CNT Oxygen Nanosensors

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Contents

10.2.3 CNT Mechanical Nanosensors . . . . . . . . 10.2.4 CNT Nanomechanical Mass-Sensors . . . . 10.2.5 CNT and Graphene NH3 Nanosensors . . . 10.2.6 Graphynes Based Chemical Nanosensors . 10.3 Boron-Based Nanosensors . . . . . . . . . . . . . . . . . . 10.3.1 NH3 @B36 Ammonia@Cluster . . . . . . . . . 10.3.2 NH3 @B40 Ammonia@Fullerene . . . . . . . 10.3.3 O3 @B80 Ozone@Fullenere . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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563 565 568 570 574 574 579 584 589

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 591

Part I

Molecular Modelling

Chapter 1

Molecular Modelling

Abstract The basics of molecular modelling in molecular and quantum mechanics are presented. Starting from the classical molecular mechanics and molecular dynamics and related force field equations and algorithms, over the minimization and optimization procedures as well as the gradient and Newton–Raphson Methods, the relationship between the trajectory and the equation of motion are explained. The quantum mechanics is also the fundament of the modern quantum chemistry. The quantum chemistry methods, like the Hartree–Fock and post Hartree–Fock theory, the density functional theory and related most popular hybrid functionals, are introduced.

1.1 Concepts of Molecular Modelling Molecular Modelling (Concept): Molecular modelling is simply a general term for computer simulation of atomic, molecular, cluster or solid systems to reproduce their geometrical and electronic structures on the one side, and chemical, physical, as well as mechanical properties, on the other side. Molecular modelling can depict one-, two-, or three-dimensional geometrical structures of in nature existing molecules or solids, or can predict and simulate novel structures. The procedure proceeds through five steps: (i) the identification, purpose and definition of the problem, (ii) constructing and building the model, (iii) carrying out the calculations, (iv) analyzing the evaluating the results, and finally, (v) significance or validity of results in comparison with theoretical or experimental data.

1.1.1 Areas of Molecular Modelling Understanding and solving chemical problems, reactions, and geometrical structures of molecular systems is based on three areas. The first one is the determination of the molecular objects, which will be investigated, like molecules, clusters, DNA. The second area includes the mathematical algorithm and numerical methods to solve © Springer Nature Switzerland AG 2020 I. Boustani, Molecular Modelling and Synthesis of Nanomaterials, Springer Series in Materials Science 290, https://doi.org/10.1007/978-3-030-32726-2_1

3

4

1 Molecular Modelling

the physical laws like atomic and molecular interactions. The third area consists of the goals that to be achieved like chemical and physical properties, geometrical and electronic structures, ground and excited states and corresponding spectra, thermodynamical and activation energies and finally comparison between theoretical models and experimental evidence of real systems. Molecular Modelling and Computational Chemistry: Molecular models can help to understand their geometric and electronic structures, their behavior and shape and to visualize their electron distribution. With the help of molecular methods one can also manipulate and simulate the structures and properties of molecules and can give insights into their dynamics and reactions. Nowadays, the concept of Molecular Modelling is purely tightly bonded to “Computational Chemistry”, which uses mathematical approximations and computer software to solve chemical problems and simulate chemical reactions [1].

1.1.2 The Basics of Molecular Modelling The Principle of Molecular Modelling: Calculating the total energy of the molecular systems, is the benchmark for any investigation. The total energy is the sum of the kinetic and potential energies. In order to find the ground state energy of certain geometrical configuration of a system there are of course different procedures, like Newton–Raphson or conjugate gradient methods, which carry out the energy minimization. This procedure occurs through energy lowering via displacements of the atomic positions as long as the energy minimum is achieved. Molecular Mechanics and Quantum Mechanics: There are two well-known mathematical models in computer simulation: the classical or molecular mechanics and the quantum mechanics. The molecular mechanics is a numerical method for determining the geometrical structure of molecules or atomic clusters, while the quantum mechanics is a numerical method for calculating the motion in molecules or clusters. The inter-atomic interaction in both simulations can be calculated in two ways: (i) using classical potential function (this can be pairwise or multi center interaction) (ii) using quantum mechanical methods (this can be ab initio Hartree–Fock, density functional theory, semi-empirical or tight-binding methods, or any other quantum mechanical methods).

1.2 Molecular Mechanics The term molecular mechanics refers to the use of Newtonian mechanics to model molecular systems. The potential energy of molecules in molecular mechanics, which can be called as positional energy, is calculated by using force field concept.

1.2 Molecular Mechanics

5

1.2.1 Force Field Equation A force field is a set of functions and constants used to describe the potential energy of the molecule. This potential is composed of bonded contributions arising from bond vibrations, bond angle bending and dihedral angle or torsion around bonds, and of non-bonding contributions arising from van der Waals and Coulomb electrostatic interactions [2]:

E pot = E bond + E angle + E dihedral + E vdW + E el.static

(1.1)

This is the force field equation describing the components of the potential energy. The function of the potential energy is usually called the potential energy surface. The potential energy terms of the force field are respectively corresponding to: E bond =



K b (r − r0 )2 ..... describes bond str etching energy

(1.2)

r

E angle =



K θ (θ − θ0 )2

..........

bond angle de f or mation

(1.3)

θ

E dihedral =



  K φ 1 + cos(nφ − γ ) ... di hedral de f or mation

(1.4)

φ

E vdW =





ij

E electrostatic =

Ai j Bi j − 6 12 ri j ri j



  qi q j  ij

εri j

........

........

van der W aals interaction

Coulomb interaction

(1.5)

(1.6)

The stretching, bond angle, and torsion energies are those which deviate from the ideal bond distance, bond angle and dihedral angle of molecule. The non-bonding interaction energy contains dipole-dipole and van-der-Waals interaction energies. The force field molecular mechanics is described by a set of equations of motion, which are derived from the classical mechanics. The force field gives how the constituents of molecule relate to each other and how the forces determine the structure of the molecule. The potential energy is a function of the nuclear positions of the conformation atoms: Epot = f(r). The nuclear positions can be expressed either in Cartesian representations, as independent variables, or however, in terms of internal coordinates expressed as a Z-Matrix containing bond stretching or lengths, bond angles and torsion or dihedral angles. The function Epot of the potential energy does not include contributions of kinetic energy. The latter is made by the motion of the involved atoms

6

1 Molecular Modelling

and can be calculated using molecular dynamics based on the second Newton’s law of motion F = m x(t), ¨ which will be discussed in Sect. 1.3. The main objective or the prototypical application of molecular mechanics is to find the lowest energy conformation of a molecule.

1.2.2 Minimization Procedure The search for the lowest energy is termed as energy minimization. The minimization of energy can be obtained through the determination the forces acting on the atoms. This can be achieved with the gradient of the potentials: F = −∇ E pot = −grad E pot

(1.7)

This equation gives the link between the force and mutual potential energy and presents the force as the first derivative of the energy. The energy minimization procedure begins with the first derivative, ∂E/∂r, for all coordinates (r) of the initial structure of the system determining the energy and the slope, direction, and magnitude of the function at a definite point, telling us the steepness of the slope. If the slope is positive, it is an indication that the coordinate is too large. And if the slope is negative, then the coordinate is too small. By a positive slope, the numerical minimization technique then adjusts or rather reduces the coordinate. The calculation of the energy and the slope will be repeated until the slope is near zero. Then the minimum has converged and the derivatives are close to zero and the forces are disappeared. The search for the energy minimum on the potential energy surface uses an iterative formula to work in a step-wise fashion. This formula is: rnew = rold + corr ection

(1.8)

The second derivative of the system ∂ 2 E/∂ 2 r, known as the Hessian matrix, describes the curvature of the potential energy surface at the coordinate r. Thus, the goal of the geometry optimization algorithm is to find the coordinates which correspond to the minimum value of energy as soon as the values of both derivatives are performed: ∀ i , grad F =

∂2 E ∂E = 0 , and 2 > 0 ∂ri ∂ ri

(1.9)

That means, the force or the first derivative of energy is zero and the value of the second derivative of the energy is positive.

1.2 Molecular Mechanics

7

1.2.3 Optimization Methods There are a few numbers of optimization methods, which are broadly categorized as local and global optimization methods using diverse optimization algorithms. The local optimization methods use the information from the neighborhood of the current approximation and always converge to the nearest local extremum close to the starting point. Thus, the local optimization methods are further categorized in gradientindependent algorithms, such as Downhill simplex methods, and in gradient-based algorithms such as least-squares fitting, steepest descent and Newton–Raphson methods. The global optimization methods can be classified as deterministic, stochastic and heuristic. Stochastic optimization algorithms incorporate some probabilistic elements in the objective function or the algorithm implementation. Some of popular stochastic optimization algorithms are simulated annealing, stochastic tunnelling and parallel tempering. The most popular global optimization algorithms are heuristic and meta-heuristic algorithms, which include evolutionary algorithms (genetic algorithms and evolutionary strategies), swarm-based optimization algorithms (particle swarm optimization and ant colony optimization) and Tabu1 search [3]. Local Minimization Methods: As mentioned above, there are two local optimization categories to perform energy minimization or geometry optimization. The first category contains non-gradient methods, like Downhill and Powell simplex methods. The Downhill simplex method starts from an initial simplex. Each step of the method consists in an update of the current simplex. These updates are carried out using four operations: reflection, expansion, contraction, and multiple contraction. The second category contains the gradient-based minimization methods, which are related to the first-order and the second-order in the Taylor expansion of the potential energy surface corresponding to the first and second derivative of the energy [4].

1.2.4 Gradient Methods The well-known gradient methods are the (i) steepest descent, (ii) conjugate gradient, and (iii) Newton–Raphson methods. The steepest descent method is a first-order minimization method which uses the information of the first derivative, the gradient gk , to find local minima on the potential energy surface. This method moves along the negative gradient − gk downhill the energy landscape. The conjugate gradient requires also the first derivative and makes use of the gradient history to decide a better direction for the next optimization step. It does not exhibit a zigzag behavior during convergence, but still tends to be quite slow in very non-linear cases. The Newton–Raphson method is a second-order minimization procedure which needs the first and the second derivatives to find the minima [5]. 1 The

word Tabu comes from the Tongan word to indicate things that cannot be touched because they are sacred.

8

1 Molecular Modelling

Newton–Raphson for Dimer: The Newton–Raphson method is the most effective optimization technique. In order to take an impression of this method and to understand the tasks of the first and second derivatives, which are indicated respectively as gradient and Hessian matrix, one can introduce a simple system of a diatomic molecule of a bond length r, which is the only coordinate to be minimized [6]. In this case it is only to consider the potential energy Ebond , which can be expressed in a harmonic function term of bond stretching: E bond =

1 K b (r − r0 )2 2

(1.10)

where Kb is the force constant for the bond, and r0 is the equilibrium bond length. By calculating the first derivative of Ebond one obtains: dE bond = K b (r − r0 ) dr

(1.11)

and the second derivative of Ebond : d2 E bond = Kb dr 2

(1.12)

Solving the equation of the first derivative (1.11) for the change in bond length after (r0 − r ) one gets: 1 dE bond (1.13) (r0 − r ) = − K b dr After substitution Kb of (1.12) in the latter (1.13), a new coordinate for the harmonic potential is obtained: dE bond 1 (1.14) (r0 − r ) = − 2 d E bond dr dr 2

1.2.5 General Newton–Raphson Method Since the most force field terms are non-harmonic potentials, the derivatives of the potentials are not linear. However, many optimization steps are necessary to find minimum for the potential energy function Epot of (1.1). Therefore, in order to generalize the minimization of this potential Epot , one can consider instead an energy function f(x) and carry out the Taylor expansion around xn , where (x = xn + x):

1.2 Molecular Mechanics

9

f Taylor (xn + x) = f (xn ) + f  (xn )x +

1  f (xn )2 x + · · · 2

(1.15)

The Taylor expansion can be approximated to the second order polynomial around x. It attains its extremum when its derivative with respect to x is equal zero:

f (xn + x) − f (xn ) /x = f  (xn ) + f  (xn )x = 0

(1.16)

solving the right-hand side of the this equation for x: x = − f  (xn )/ f  (xn )

(1.17)

Since x = x − xn , then the final formula for new coordinates xn+1 is now: xn+1 = xn − f  (xn )/ f  (xn ), n = 0, 1, . . .

(1.18)

After comparison (1.8) and (1.18) and replacing r by x, we find the term corr ection = − f  (xn )/ f  (xn ). Since the first derivative f  (xn ) is the gradient ∇ f (xn ), and the second derivative f  (xn ) is the Hessian matrix H f (xn ), we introduce the inverse of the Hessian matrix as [H f (xn )]−1 and rewrite the last equation as: xn+1 = xn − [H f (xn )]−1 ∇ f (xn ), n ≥ 0.

(1.19)

The use of second derivatives in Newton–Raphson minimization is responsible for the excellent convergence properties. However, the inversion of the Hessian is time consuming. An approach, named truncated Newton–Raphson, has been developed that uses conjugate gradients to determine the directions for the minimization and then the Hessian to determine the minimum in that direction. The “direction” of the minimization determines the particular bond lengths and angles that will be changed. The minimum in that “direction” determines how much to change those bond lengths and angles. Truncated Newton–Raphson method has similar and often better convergence characteristics to the non-truncated Newton–Raphson one without a significant difference in time [6, 7].

1.3 Molecular Dynamics Equation of Motion for Dimer: The classical molecular dynamics is mainly based on the second Newton’s law of motion. This kinetic energy is called also the motional energy. We have already seen that the forces are derived from the gradients of the interaction potentials, as given in (1.7). The Newton equation has now the following form: m.a = F = −∇ E(r ) (1.20)

10

1 Molecular Modelling

For instance, a simple system for motion can be taken after considering E(r ) = k/2(r − r0 )2 , which is the equation of motion of a dimer, as given in (1.10): m.

dE(r ) dr 2 =F =− = −kr 2 dt dr

(1.21)

Thus, from the first and the fourth terms of (1.21) arises the differential equation: dr 2 = −k.r dt 2 √ for which the solution after setting w = k/m, is known: m.

r (t) = A sin(wt)

(1.22)

(1.23)

1.3.1 Trajectory and Equation of Motion In general, from the knowledge of the atomic forces and masses, it is possible to determine the acceleration of each atom in the system. Integration the equations of motion yields a solution that describes the positions, velocities and accelerations of the atoms along a series of extremely small time steps (on the order of femtoseconds = 10−15 s). The resulting series of snapshots of structural changes over time is called a trajectory. From this trajectory, the average values of properties can be determined. The method is deterministic; once the positions and velocities of each atom are known, the state of the system can be predicted at any time in the future or the past. Thus, by the molecular dynamics simulation all atoms are described by the Newton’s equation of motion:

F = m.a = m.

d2r dt 2

(1.24)

where the acceleration a is the derivatives of the position r or of the velocity v:

a=

dv d2r = 2 dt dt

(1.25)

By a constant acceleration a and after integration we obtain the following expression for the velocity: (1.26) v = a.t + v0 Since the velocity v is the first derivative of the position v = dr/dt, then by a constant velocity the position r can be obtained by an integration obtaining the following expression for the position:

1.3 Molecular Dynamics

11

r = v.t + r0

(1.27)

After insertion (1.26) in (1.27) we receive: r = a.t 2 + v0 t + r0

(1.28)

where v0 and r0 are the initial velocity, and position, respectively. The acceleration is given as the derivative of the potential energy with respect to the position r, as expressed in (1.21):

a=−

1 dE m dr

(1.29)

1.3.2 Initialization of Positions and Velocities In order to calculate a trajectory one only needs the initial positions of the atoms, an initial distribution of velocities and the acceleration, which is determined by the gradient of the potential energy function. The equations of motion are deterministic, e.g., the positions and the velocities at the time zero (t0 ) determine the positions and velocities at all other times (t). The initial positions simply correspond to the Cartesian coordinates of the initial geometry of the system, which can be obtained from experimental structures, such as the x-ray crystal structure [8]. The initial velocities, however, are generally obtained by selecting speeds from the Maxwell–Boltzmann distribution for a specified temperature, like 300 K or 1000 K, and then selecting random directions for the velocities. Thus, the Maxwell–Boltzmann distribution for the velocities, F(v), is given as:  F(v) = 4π v

2

m 2π k B T

3/2



−mv2 ex p 2k B T

 (1.30)

The maximum or the most probable velocity vmp corresponds to the maximum of the distribution, which can be determined by the derivative of F(v) after (v) has the form:   2k B T 1/2 vmp = (1.31) m However, since the distribution of the velocities is not symmetric, the average velocity vave = v of the distribution is in general relative larger than the most probable value vmp :   8 k B T 1/2 2 v = = √ vmp (1.32) π m π

12

1 Molecular Modelling

Temperature and Kinetic Energy: In the classical statistical mechanics, the law of energy equipartition remarks that the average kinetic energy of N particle is equal to the average translational kinetic energy (3/2)N k B T . Thus, the kinetic energy of a system at equilibrium at a given temperature T has the form: E kin =

1 3 m v2 = N k B T 2 2

(1.33)

The temperature can then be determined from the total kinetic energy of the system using the following formula: TK =

1  m i vi2 3N k B i

(1.34)

Note that as the kinetic energy fluctuates in time, the previous relation defines a so-called instantaneous temperature, which is also a time fluctuating quantity [9]. Taylor Expansions and Time Evolution The time evolution of a system takes place along discrete time steps t. The trajectory of a particle is not the real one but in the single time step. After considering the derivatives for velocity and acceleration as: dr = v, dt   d dr = dt dt   d dv = dt dt

(1.35) d2 r dv = a, = dt 2 dt d2 v da =b = dt 2 dt

(1.36) (1.37)

Thus, the Taylor expansion of coordinates, velocities and accelerations can be written as follows:       1 d2 r 1 d3 r dr 2 t t 3 + O(t 4 ) t + + r (t + t) = r (t) + dt 2 dt 2 3! dt 3 (1.38)  2    1 d v dv t 2 + O(t 3 ) t + v(t + t) = v(t) + (1.39) dt 2 dt 2   da t + O(t 2 ) (1.40) a(t + t) = a(t) + dt The truncation of Taylor expansions of (1.38) and (1.39) can be done at different levels. Let us consider the derivatives dr/dt = v and dr 2 /dt 2 = a for velocity and acceleration, and approximate the expansions to the first order. Then we receive the Euler’s algorithm, which is the simplest method for solving the equations of motion:

1.3 Molecular Dynamics

13

r (t + t) = r (t) + v(t) t

(1.41a)

v(t + t) = v(t) + a(t) t

(1.41b)

The Euler algorithm can determine the new atomic positions r (t1 ) starting from given initial coordinates r (t0 ) and velocities v(t0 ) as well as accelerations a(t0 ), which can be calculated from the force field potential of equation (1.29) and then it can be replaced in the second (1.41b): v(t + t) = v(t) −

1 m



dE dr

 t

(1.42)

1.3.3 Verlet Algorithm The next level of approximation is to truncate the Taylor expansion in (1.39) to the second order. Then we write down the approximate expansion of r(t) for a step forward (+t) and for a step backward (−t): r (t + t) = r (t) + v(t)t +

1 a(t)t 2 2

(1.43)

r (t − t) = r (t) − v(t)t +

1 a(t)t 2 2

(1.44)

These equations are the basics of Verlet algorithm, which is the most used class of algorithms in molecular dynamics. By adding and subtracting both equations we receive respectively: r (t + t) = 2r (t) − r (t − t) + a(t)t 2

(1.45)

v(t) = [r (t + t) − r (t − t)] /2t

(1.46)

Verlet algorithm uses the positions and accelerations at time t and the positions from the time t − t to calculate the new positions at the time t + t. These new positions of equation (1.45) are of course calculated up to the second order in t. The velocity is not explicitly calculated, but it can be derived from the knowledge of the trajectories r (t + t) and r (t − t) at every time step t, as given in (1.46). However, alternatively the velocity can be adopted from (1.40) after considering the Taylor expansion to the first order. Resolving (1.40) after da/dt and replacing it by d2 v/dt 2 in (1.39), we receive: v(t + t) = v(t) +

1 [a(t) + a(t + t)] t 2

(1.47)

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1 Molecular Modelling

1.3.4 Leap-Frog Algorithm Leapfrog integration is a first order Taylor expansion and similar to Euler integration but more accurate. The Leapfrog integration is equivalent to updating positions r (t) and velocities v(t) at interleaved time points, staggered in such a way that they ‘leapfrog’ over each other so that the position is updated at integer time steps and the velocity is updated at integer-plus-a-half time steps. However, we take the Euler equation (1.41a) and update the velocity at 21 t as follows:   1 r (t + t) = r (t) + v t + t t 2

(1.48)

The velocity v(t + t) of (1.41b) can be determined by the Taylor  considering expansion equation of equation (1.39) at the first order for t + 21 t and t − 21 t : 1 v(t + t) = v(t) + 2 1 v(t − t) = v(t) − 2

 

dv dt dv dt

 

t 2

(1.49)

t 2

(1.50)

Subtracting and rearranging both (1.49) and (1.50) then we receive: 1 1 v(t + t) = v(t − t) + a(t) t 2 2

(1.51)

The velocity of equation (1.51) can be calculated at t + 21 t by means of two quan tities, the mid-step velocity v t − 21 t and the acceleration a(t). These calculated velocities together with the stored current positions r (t) are used to determine the new positions r (t + t) of (1.48). In this way, the velocities leap over the positions, then the positions leap over the velocities. Since the velocities are updated at half time steps and leap ahead the positions, the current velocities can be obtained by adding (1.49) and (1.50) to get: v(t) =

    1 1 1 v t + t + v t − t 2 2 2

(1.52)

1.4 Ab Initio Quantum Chemical Methods Ab Initio (Latin: from the beginning) methods also called first principle methods are rigorous calculations without using any empirical parameters. The Ab Initio methods distinguish itself from other molecular mechanical methods that they are solely

1.4 Ab Initio Quantum Chemical Methods

15

based on the established laws of quantum mechanics. These methods can be applied for small systems (e.g. tens of atoms) or for periodic systems. Because of its high accuracy, the computations are expensive and extensive. Semi-empirical methods are similar to the latter one, but less rigorous, using mostly experimental parameters and more extensive approximations. They are computationally less demanding than the Ab Initio methods and are able calculate systems of some hundreds of atoms. Over the last two decades powerful algorithmic tools and program codes have been developed which are capable to predict structures, energies, reactivities and many properties of high accuracy [10].

1.4.1 Some Basics of Quantum Mechanics In classical mechanics, the goal was always to find the trajectory of a particle x(t) in relationship with a force acting on this particle and provided by the Newton’s second law F = m x(t). ¨ In quantum mechanics, the goal is to find the wavefunction (x, t) or precisely its probability density for a particular system. Thus, using the classical description of the total energy as the sum of the kinetic energy and the potential energy, the analogue of Newton’s second law is the Schrödinger’s equation for a quantum system: Hˆ = E (1.53) where Hˆ is the Hamiltonian, is the wavefunction, and E is the total energy of the system. This equation was proposed in 1926 by Erwin Schrödinger, which is the basic non-relativistic wave equation governing the motion of nuclei and electrons in molecules. It is nothing else as a differential (eigenvalue) equation for the energy E and the wavefunction of a particular state. The kinetic and the potential energies are expressed by the Hamiltonian or the Hamilton operator Hˆ (see next paragraph), while the wavefunction depends on the Cartesian and spin coordinates of the component particles. Thus, this equation can be solved using fundamental physical constants and geometry of the molecule as well. In order to determine the energy of heavy elements which incorporate a large number of core electrons, it is necessary to include the relativistic effects and spin-orbit coupling. The relativistic generalization of this equation was proposed in 1928 by Dirac. The standard Ab Initio shorthand notation outlines the theory used Hartree–Fock (HF), Configuration Interaction (CI), and the atomic orbitals (Basis Sets). Kinetic Energy in Quantum Mechanics: In classical mechanics, the kinetic energy and momentum pˆ = m.v are given as: Tcm =

1 pˆ 2 m · v2 = 2 2m

(1.54)

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1 Molecular Modelling

Thus, the correspondence of momentum between the classical and quantum mechanics is given through the operator pˆ = i ∂∂x . Thus, the kinetic energy operator in quantum mechanics has the following form:

 ∂ 2 1 2 ∂ 2 i Tˆqm = =− 2m ∂x 2m ∂ x 2

(1.55)

For a three-dimensional system becomes the kinetic energy operator the following expression: 

2 ∂ 2 ∂2 ∂2 ˆ + 2+ 2 Tqm = − (1.56) 2m ∂ x 2 ∂y ∂z

1.4.2 The Hamiltonian The Hamiltonian is composed of the kinetic and potential energies of the system. Since the molecular system contains electrons (e) and nuclei (n), which are interacting together, we introduce the terms of the Hamiltonian as the sum of the kinetic energies of nuclei and electrons, the nuclear-nuclear and electron-electron repulsion energies, and the nuclear-electron attraction energies, in a different form of energy operators:

Hˆ = Tˆn + Tˆe + Vˆnn + Vˆne + Vˆee

(1.57)

This is the energy equation describing the components of the kinetic and potential energy. The kinetic and the potential operators are respectively corresponding to: 1  2 ∇ ......... kinetic energy operator o f nuclei Tˆn = − 2M A A A 1 Tˆe = − 2 Vˆnn =

∇i2

......... kinetic energy operator o f electr ons

(1.59)

......... nuclear − nuclear r epulsion energy

(1.60)

 ZA .......... nuclear − electr on attraction energy Ri A iA

(1.61)

i

 ZAZB R AB A