A density functional theory study on structure and mechanism of some isomerization and cycloaddition reactions

Pople and Walter Kohn for their respective work in developing computational chemistry methods Pople and density functional theory Kohn.4 In recent years, involvement in computational che

Vrije Universiteit Brussel Faculteit Wetenschaid Alge

Onderzoeksgroep Algemene Chemie

A Density Functional Theory

Study on Structure and Mechanism of some Isomerization

and Cycloaddition Reactions

Loc Thanh Nguyen

Promotors:

Prof Dr Paul Geerlings,

Vrije Universiteit Brussel

Prof Dr Minh Tho Nguyen,

Katholieke Universiteit Leuven

October 2002

Proefschrift voorgelegd tot het behalen van de wettelijke graad van Doctor in de Wetenschappen

Acknowledgements

The story of this thesis started in September 1997 with the “Interuniversity Program for Education in Computational Chemistry in Vietnam” supported by the Flemish Government (project VIET/97-4) Professor Minh Tho Nguyen at the Katholieke Universiteit Leuven (KULeuven), the main promoter of this project, together with professor Paul Geerlings at the Vrije Universiteit Brussel (VUB) and professor Kris Van Alsenoy of the Universiteit van Antwerpen (UIA), has opened a door for me to enter the fascinating field of computational chemistry when providing me an opportunity to join two of the research groups involved, first the KULeuven group, then the VUB one

At the beginning, every quantum theory was new for me But, from that time, both professors had tried hard not only to give me a sound background but also to look for financial resources Finally they used their own research funds, namely the Geconcerteerde Onderzoeksacties (GOA) and the DFT Research community of the Fund for Scientific Research (FWO-Vlaanderen), along with the kind support of their colleagues (professor Luc Vanquickenborne, professor Arnout Ceulemans, professor Kristine Pierloot, professor Marc Hendrickx (KULeuven) and professor Frank De Proft (VUB)), to give me a unique chance to perform research at the doctoral level in the department of chemistry, VUB, but on a joint research project between both VUB and KULeuven quantum chemistry groups The last doctoral year fellowship was granted by the Vrije Universiteit Brussel Therefore, I would first like to express my sincere thanks

to both universities, the research funds, professors Nguyen and Geerlings as well as their colleagues for their kind support I would like to express my deep gratitude to professor Paul Geerlings and professor Minh Tho Nguyen for their constant encouragement, scientific guidance and patient supervision of my research work

This work would not have been achievable without the friendly support and efficient help from many other people In particular, I wish to acknowledge professor Frank De Proft (VUB), Dr Wilfried Langenaeker (VUB; now at Janssen Pharmaceutica) and Dr Asit Kumar Chandra (KULeuven, now in India) for patiently answering my technical problems and valuable help I would also like to thank professor Kalidas Sen (India) for uncomplainingly explaining my theoretical questions during his short visiting day at VUB A special thank is also sent to professor Kris Van Alsenoy for helping me with the Hirshfeld charge calculations I am particularly grateful to Dr Hans Vansweevelt (KULeuven) for computational help and to the VUB Computer Center for support Mrs Rita Jungbluth (KULeuven) and Mrs Martine De Valck (VUB) are also acknowledged for administrative help

The days would have passed far more slowly without the support of my friends, both at the KULeuven and VUB, providing me such a rich source of conversation, education and entertainment My warmest gratitude goes to my friends Trung Ngoc Le, Hung Thanh Le, Hue Minh Thi Nguyen, Thanh Lam Nguyen, Nam Cam Pham, Nguyen

Nguyen Pham-Tran, David Delaere, Dr Annelies Delabie, Dr Steven Creve, Dr Raman Sumathy (KULeuven), Ricardo Vivas-Reyes, Bennasser Safi, Dr Gregory Van Lier, Pierre Mignon, Dr Frederik J C Tielens, Dr Robert Balawender, Dr Stefan Loverix, Greet Boon, Goedele Roos, Jan Baert, Montserrat Cases Amat (VUB) … and many more I could not have asked for a better working environment and friendship Furthermore I also owe a debt of thanks to Mr Diet Van Tran and his family, Mrs Mai Phuong Le and her two nice daughters, my friends (Nho Hao Dinh, Chau Ngan Nguyen-Vo, Ngoc Lien Truong, Phuong Khuong Ong, Minh Tri Nhan, Chi Thanh Truong, Thu Phong Phan-Vo, Lam Thanh Nguyen, Thai An Mai, Thi Xuan Tran … and many more), who have no direct relation with my research, but they have given me much concern and useful help during my stays in Belgium

My thanks also extend to my home university in Vietnam, the Faculty of Chemical Engineering, HoChiMinh City University of Technology (HUT), for administrative support I would like to acknowledge professor Van Luong Dao for scientific guidance and valuable advice of my research work in Vietnam I am indebted to professor Van Bon Pham, professor Huu Nieu Nguyen, professor Van Lua Nguyen, professor Van Hang Tong, professor Huu Khiem Mai, professor Thuong Truong Le, professor Khac Chuong Tran, professor Viet Hoa Thi Tran, professor Minh Tan Phan, Msc Dinh Pho Nguyen, Mrs Thi Dung Huynh, Msc Ba Minh Vu, Mr Hung Dung Tran, Msc Minh Nam Hoang, Msc Thanh Son Thanh Do, Mr Van Co Ngo, Msc Thanh Trung Duong, Msc Huu Thao Vo, Dr Dac Thanh Nguyen, Dr Van Phuoc Nguyen, Dr Ngoc Hanh Nguyen, Mrs Thi Thu Nguyen, Mrs Kim Anh Thi Lam, Mrs Ngoc Phu Thi Nguyen (HUT) and Mr Cat Si Thanh Le (HoChiMinh City) for their continuous support and valuable help Many thanks also go to my colleagues and my friends in HUT for their cooperation, friendship and encouragement

Especially, I would express heartfelt thanks to my parents, my parents-in-law, my brothers, my sisters and their families, for their love, invaluable help and support throughout my life

Finally, I would like to give my special thanks to my wife, Dieu Chan Thi Truong, and our two children, Huong Lan Ngoc Nguyen and Thanh Triet Nguyen, for their love, patience and encouragement that enabled me to complete this work

Table of Contents

Acknowledgments i

Table of Contents iii

Summary vii

Samenvatting ix

Chapter 1 Introduction 1

1.1 Computational Chemistry 1

1.1.1 Current Situation 1

1.1.2 Methods 2

1.2 Structures and reaction mechanism in organic chemistry 4

1.3 Scope of the Thesis 10

1.4 References 11

Chapter 2 Theoretical Background 13

2.1 Wave function Ab Initio methods 13

2.1.1 Schrödinger equation 13

2.1.2 The Hartree-Fock theory 14

2.1.2.1 Restricted closed-shell Hartree-Fock: The Roothaan-Hall equations 16

2.1.2.2 Unrestricted open-shell Hartree-Fock: The Pople-Nesbet equations 17

2.1.3 Post Hartree-Fock methods 18

2.1.3.1 The Configuration Interaction method 19

2.1.3.2 The Coupled Cluster method 20

2.1.3.3 The Møller-Plesset Perturbation method 20

2.1.4 Basis sets 22

2.1.4.1 Minimal basis sets 22

2.1.4.2 Scaling the orbital by splitting the minimal basis set 23

2.1.4.2.1 Split valence basis sets 23

2.1.4.2.2 Double zeta basis sets 23

2.1.4.3 Extended basis sets 24

2.1.4.3.1 Polarization basis functions 24

2.1.4.3.2 Diffuse basis functions 24

2.1.4.4 Dunning's correlation consistent basis sets 24

2.1.5 Molecular quantities 25

2.1.5.1 The electron density function 25

2.1.5.2 Atomic charges 25

2.1.5.2.1 The Mulliken population analysis method 25

2.1.5.2.2 The natural population analysis 26

2.1.5.2.3 The electrostatic potential derived charges 27

2.2 Density Functional Theory 27

2.2.1 The Thomas-Fermi-Dirac theory 28

2.2.2 The Kohn-Sham method 28

2.2.3 The exchange-correlation energy functional 30

2.2.3.1 Local Density methods 30

2.2.3.2 Gradient Corrected methods 30

2.2.3.3 Hybrid methods 30

2.2.4 DFT-based chemical concepts 31

2.2.4.1 The chemical potential 31

2.2.4.2 Hardness and Softness 32

2.2.4.3 The Fukui function and local softness 33

2.2.4.4 The Local Hard and Soft Acids and Bases principle 34

2.3 Solvent effect 34

2.3.1 Introduction 34

2.3.2 Solvation models 35

2.3.2.1 Explicit solvation models 35

2.3.2.2 Implicit solvation models 36

2.3.3 The PCM model 37

2.3.3.1 Introduction 37

2.3.3.2 Model Implementation 37

2.4 References 39

Chapter 3 Computational Details 41

3.1 Software and Hardware 41

3.2 References 43

Chapter 4 Application of Density Functional Theory (DFT) in constructing the Potential Energy Surface for Simple Isomerization and Fragmentation Reactions 45

4.1 Introduction 45

4.2 Theoretical study of the CH 3 + NS and related reactions: mechanism of HCN formation 48

4.2.1 Introduction 48

4.2.2 Methods of Calculations 48

4.2.3 Results and Discussion 48

4.2.4 Conclusions 55

4.3 Theoretical Study of the Potential Energy Surface Related to NH 2 + NS Reaction: N 2 versus H 2 Elimination 56

4.3.1 Introduction 56

4.3.2 Methods of Calculations 56

4.3.3 Results and Discussion 57

4.3.4 Conclusions 65

4.4 General Conclusion 66

4.5 References 67

Chapter 5 Application of Density Functional Theory (DFT) in studying Cycloaddition Reactions 71

5.1 Introduction 71

Table of Contents v

5.2 Mechanism of [2+1] Cycloadditions of Hydrogen Isocyanide to Acetylenes 76

5.2.1 Introduction 76

5.2.2 Methods of Calculation 77

5.2.3 Results and Discussion 77

5.2.3.1 Preliminary analysis of frontier orbital interactions 77

5.2.3.2 Addition of the unsubstituted system HN≡C + HC≡CH (Reaction H) 78

5.2.3.3 Addition of HN≡C to HC≡C-CH3 (Reaction M) 82

5.2.3.4 Addition of HN≡C to HC≡C-NH2 (Reaction A) 84

5.2.3.5 Addition of HN≡C to HC≡C-F (Reaction F) 87

5.2.3.6 Asynchronism in Addition 90

5.2.4 Conclusions 98

5.3 [2+1] Cycloadditions of CO and CS to Acetylenes 99

5.3.1 Cyclopropenones and cyclopropenethiones: decomposition via intermediates 99

5.3.1.1 Introduction 99

5.3.1.2 Methods of Calculation 100

5.3.1.3 Results and Discussion 100

5.3.1.3.1 Analysis of the nature of the reaction partners 100

5.3.1.3.2 Reaction of H-C≡C-H with C=X (X = O, S) 101

5.3.1.3.2.1 Potential energy surfaces 101

5.3.1.3.2.2 Solvent effect 107

5.3.1.3.2.3 Estimation of the vertical first excitation energies 107

5.3.1.3.3 Reaction of H-C≡C-F with C=X (X = O, S) 108

5.3.1.3.4 Reaction of F-C≡C-F with C=X (X = O, S) 111

5.3.1.3.5 Profiles of hardness, polarizability and activation energy along an IRC path 113

5.3.1.4 Conclusions 115

5.3.2 [2 + 1] Cycloaddition of CO and CS to Acetylenes forming Cyclopropenones and Cyclopropenethiones 116

5.3.2.1 Introduction 116

5.3.2.2 Methods of Calculation 117

5.3.2.3 Results and Discussion 118

5.3.2.3.1 Classification of the reactants as nucleophile or electrophile 118

5.3.2.3.2 Potential Energy Surfaces 119

5.3.2.3.2.1 Reaction of H-C≡C-CH3 with CX (X = O, S) 119

5.3.2.3.2.2 Reaction of H-C≡C-OH 123

5.3.2.3.2.3 Reaction of H-C≡C-NH2 124

5.3.2.3.2.4 Reaction of H-C≡C-C6H5 125

5.3.2.3.2.5 Reaction of HO-C≡C-CH3 127

5.3.2.3.2.6 Reaction of HO-C≡C-C6H5 128

5.3.2.3.3 Effects of substituents on the aromaticity of cyclo-propenones and cyclopropenethiones 132

5.3.2.3.4 Site selectivity in the initial attack of the addition 133

5.3.2.4 Conclusions 136

5.4 1,3-Dipolar cycloadditions of thionitroso compounds (R–N=S) 138

5.4.1 Introduction 138

5.4.2 Details of calculations 138

5.4.3 Results and discussion 139

5.4.3.1 Structure and energetics 139

5.4.3.1.1 The HC≡N+ –O- + HN=S addition (A) 139

5.4.3.1.2 The HC≡N+ –O- + H2N–N=S addition (B) 140

5.4.3.1.3 The HN=N+=N- + HN=S addition (C) 141

5.4.3.1.4 Additions of substituted systems 143

5.4.3.2 Regiochemistry of the addition: testing the local HSAB principle 144

5.4.3.3 Testing the maximum hardness principle 147

5.4.4 Conclusions 149

5.5 Nitrous Oxide as a 1,3-Dipole: A Study of Its Cycloaddition Mechanism 150

5.5.1 Introduction 150

5.5.2 Details of Calculation 152

5.5.3 Results and Discussion 152

5.5.3.1 Frontier Molecular Orbital Analysis 152

5.5.3.2 The 1,3-DC of N2O to acetylene 154

5.5.3.3 The 1,3-DC of N2O to substituted alkynes 157

5.5.3.3.1 Geometries 157

5.5.3.3.2 Energy barriers and solvent effect 159

5.5.3.3.3 Regioselectivity 161

5.5.4 Conclusions 166

5.6 1,3-Dipolar cycloadditions of diazoalkanes, hydrazoic acid and nitrous oxide to acetylenes, phosphaalkynes and cyanides: a regioselectivity study 168

5.6.1 Introduction 168

5.6.2 Details of Calculation 171

5.6.3 Results and Discussion 171

5.6.3.1 The 1,3-DC of Diazoalkanes 171

5.6.3.2 The 1,3-DC of Hydrazoic acid and Nitrous Oxide 178

5.6.4 Conclusions 182

5.7 General Conclusion 184

5.8 References 187

Chapter 6 General Conclusions 197

Appendices 199

A1 List of Symbols and Abbreviations 199

A2 List of supplementary Tables and Figures in §5.3.2 and §5.6 201

A3 List of Publications 212

Summary

In this thesis we apply the Density Functional Theory (DFT) in its Kohn Sham formulation using the B3LYP functional, for constructing of the potential energy surface (PES) for some isomerization and fragmentation reactions and studying a number of [2+1] and 1,3-dipolar cycloadditions

The PES constructions for the isomerization and fragmentation reactions involving two

NS moieties, [CH3NS] and [NH2NS] show that, with respect to the CCSD(T) values, the B3LYP method tends to overestimate the energy gaps between equilibrium structures relative to the starting structures (CH3NS or NH2NS) The energy ordering however remains almost unchanged Moreover, the most significant chemical results of the theoretical studies are a prediction on the preferential formation of HCN in the CH3 +

NS reaction and the fact that both radicals NH2 and NS can go through an initial free nitrogen-nitrogen association giving NH2NS, which in turn tends to follow a low-energy two-step path leading to the stable products, N2 and H2S A one-step elimination

barrier-of H2 seems to be a more energy-demanding process

The theoretical studies of the [2+1] cycloaddition of hydrogen isocyanide (HN≡C), CX (X = O, S) to acetylenes demonstrates that these reactions proceed in two steps: addition

of the carbon atom in HN≡C or CX to a carbon atom of the acetylenes giving rise to an intermediate, followed by a ring closure step of the latter to form at last the cycloadducts The intermediate has the properties of a semi-carbene, semi-zwitterion and its structure is best described as a resonance hybrid between a carbene and a zwitterion In all cases acetylenes behave as nucleophiles The investigation of the hardness and polarizability profiles along the IRC reaction paths shows that there is a maximum in the polarizability profile besides an inverse relationship between hardness and polarizability

In the cycloadditions of CX to acetylenes, it is also shown that the promotion of an electron from the ground state to an excited state for any reaction partner requires a large amount of energy As such, all investigated reactions are expected to take place in the ground state rather than in an excited state We also show that the solvent effect is small on the reactions, and tends to stabilize all the isomers

Different reactivity criteria such as Frontier Molecular Orbital (FMO) coefficients, local softness, hardness, polarizability and nucleus-independent chemical shifts (NICS) are used to predict the site selectivity in all studied cases, and the NICS, FMO coefficients, local softness seem to yield the best results among them

The 1,3-dipolar cycloadditions (1,3-DC) of fulminic acid (HCNO) and the simple azides (XNNN, X=H, CH3, NH2) to thionitroso compounds (R-N=S, R = H, NH2) are generally characterized by their rather low energy barriers In the cases of azides, the reaction is not stereospecific In all cases, they show a certain regioselectivity favoring the formation of a cycloadduct

The 1,3-DC reactions of diazoalkanes, hydrazoic acid and nitrous oxide to the polar

dipolarophiles considered are concerted but asynchronous processes When approaching

a polar dipolarophile partner either the C-end of a diazo derivative, or the N(R) of an azide or the O-atom of nitrous oxide, consistently acts as a new bond donor and the other molecular terminus being the new bond acceptor As a consequence, the lone pair

of the central nitrogen, formed upon bending of the dipole, originates from the triple N≡N bond Those cycloaddition reactions are essentially orbital-controlled, which is supported by the successful prediction of the regioselectivity based on reactivity criteria that are basically generalized forms of FMO theory The local softness differences and FMO coefficient products remain the criteria of choice in predicting the regioselectivity

of cycloaddition reactions Among available population analysis methods to define the atomic charges, the Natural Population Analysis (NPA) seems to give the best support

to the local Hard and Soft Acids and Bases (HSAB) principle

In the cycloadditions of nitrous oxide to acetylenes, in general, the shape of the potential energy surface appears not to be affected by the polarity of the solvent Although all Ts’s are aromatic, their aromaticity does not influence the regioselectivity

of the reactions In this study the less aromatic, more polar and more asynchronous Ts is the Ts-normal

Samenvatting

In deze thesis wordt Density Functional Theory (DFT) toegepast in de Kohn Sham formulering, met gebruik van de B3LYP functionaal, om het Potentiële Energie Oppervlak (PES) van een aantal isomerisatie en fragmentatiereacties te bestuderen, alsook een aantal [2+1] en 1,3-dipolaire cycloaddities

De PES constructie voor de isomerisatie en fragmentatiereacties voor twee NS entiteiten bevattende species, [CH3NS] en [NH2NS], toont aan dat, in vergelijking met CCSD(T), de B3LYP methode de energieverschillen overschat tussen evenwichtsstructuren, transitietoestanden en de startstructuren (CH3NS of NH2NS) De ordening van de energieën daarentegen is bijna steeds onveranderd De meest significante chemische resultaten van de theoretische studie zijn enerzijds een voorspelling van de voorkeur van vorming van HCN in de CH3 + NS reactie, anderzijds het feit dat beide radicalen, NH2 en NS, een initiële barrièrevrije stikstof-stikstof associatie vertonen, aanleiding gevend tot NH2NS Op zijn beurt volgt NH2NS een laag-energetisch tweestapsmechanisme leidend tot de stabiele eindproducten N2 en H2S Een eenstapseliminatie van H2 blijkt een energetisch minder gunstig proces te zijn

De theoretische studie van de [2+1] cycloadditie van waterstofisocyanide (HN≡C) en

CX (X = O, S) aan acetylenen toont aan dat deze reacties in twee stappen verlopen : additie van het koolstofatoom in HN≡C of CX aan een koolstofatoom van de acetylenen wat aanleiding geeft tot een intermediair, gevolgd door een ringsluiting met finaal vorming van het cycloadduct Het intermediair blijkt de eigenschappen van een semi-carbeen, semi-zwitterion te hebben Zijn structuur wordt het best beschreven als een resonantiehybride tussen een carbeen en een zwitterion In alle gevallen blijken acetylenen zich als nucleofiel te gedragen Het onderzoek van de hardheid en de polariseerbaarheidsprofielen langsheen het IRC reactiepad toont aan dat er een maximum in het polariseerbaarheidsprofiel is Een inverse relatie tussen hardheid en polarisabiliteit wordt vastgesteld

Bij de cycloaddities van CX op acetylenen wordt ook aangetoond dat de overgang van

de electronische grondtoestand naar een aangeslagen toestand voor elk van de reactiepartners veel energie vergt Men kan daarom verwachten dat alle reacties plaatsgrijpen vanuit de grondtoestand en niet vanuit een aangeslagen toestand Ook wordt aangetoond dat het solvent effect op deze reacties klein is en dat het alle isomeren stabiliseert

Verschillende reactiviteitscriteria zoals de "Frontier Molecular Orbital" (FMO) coëfficiënten en de lokale zachtheid, hardheid, polariseerbaarheid en de "nucleus-independent chemical shifts" (NICS) worden gebruikt om de "site selectivity" te voorspellen in alle beschouwde gevallen De NICS, FMO coëfficiënten en de lokale zachtheid blijken hierbij de beste resultaten op te leveren

De 1,3-dipolaire cycloaddities (1,3-DC) van HCNO en eenvoudige azides (XNNN, X =

H, CH3, NH2) op thionitroso verbindingen (R-N=S, R = H, NH2) worden over het

algemeen gekarakteriseerd door vrij lage energiebarrières In het geval van azides is de reactie niet stereospecifiek In alle gevallen vertonen deze reacties een zekere regioselectiviteit in de vorming van de cycloadducten

De 1,3-DC reacties van diazoalkanen, waterstofazide en distikstofmonoxide aan de beschouwde polaire dipolarofielen zijn geconcerteerde maar asynchrone processen Bij nadering van een polair dipolarofiel zal ofwel het terminale C-atoom van een diazoderivaat, het N(R) atoom van een azide of het O-atoom van distikstofmonoxide, onveranderlijk dienst doen als donorcentrum voor een nieuwe binding terwijl het andere uiteinde van het molecule als bindingsacceptor zal fungeren Bijgevolg vindt het Niet Gebonden Elektronenpaar van het centrale stikstofatoom, dat ontstaat bij "bending" van het dipool, zijn oorsprong in de drievoudige N≡N binding Deze cycloadditiereacties zijn essentieel orbitaal-gecontroleerd wat bevestigd wordt door de succesvolle voorspelling van regioselectiviteit gebaseerd op reactiviteitscriteria die als veralgemening van de FMO theorie kunnen beschouwd worden Verschillen in lokale zachtheid en producten van FMO coëfficiënten blijken de beste criteria te zijn om de regioselectiviteit van cycloadditiereacties te voorspellen Binnen de voorhanden zijnde populatie analysemethodes die toelaten atomaire ladingen te genereren blijkt de

"Natural Population Analysis" (NPA) het sterkst het lokale "Hard and Soft Acids and Bases" (HSAB) principe te ondersteunen

In de cycloaddities van de distikstofmonoxide aan acetylenen blijkt over het algemeen dat het potentiële energieoppervlak niet vervormd wordt door de polariteit van het solvent Hoewel alle transitietoestanden aromatisch zijn, blijkt aromaticiteit de regioselectiviteit van de reacties niet te bẹnvloeden De meest voordelige ("normale") transitietoestand (Ts-normal) blijkt in deze studie steeds de meest aromatische, meest polaire en meest asynchrone transitietoestand te zijn

To meet industry goals for the 21st century, the R&D should be conducted in a number

of areas,1 consisting of new chemical science and engineering technology; supply chain management; information systems; and manufacturing and operations New chemical science and engineering technology, that will promote more cost-efficient and higher performance products and processes, comprises chemical science and enabling technology The latter identified as essential to the industry’s future includes process science and engineering (e.g., engineering scale-up and design, thermodynamics and kinetics, reaction engineering); chemical measurement; and computational technologies (e.g., computational chemistry, simulation of processes and operations, smart systems, computational fluid dynamics).1,2

Computational chemistry is usually referred to as a series of mathematical methods, well enough developed so that they can be automatically implemented on a computer3

to solve chemical problems mostly at the molecular level It initially began in chemistry and physics with the development of quantum mechanics in the 1920s and considerable efforts have been done in the development of methods and codes Nobel Prize in chemistry has been awarded to Linus Carl Pauling in 1954 for his research into the nature of the chemical bond and its application to the elucidation of the structure of complex substances Robert Sanderson Mulliken received the Nobel Prize in 1966 for his fundamental work concerning chemical bonds and the electronic structure of molecules by the molecular orbital method William N Lipscomb won the prize in 1976 for his studies on the structure of boranes illuminating problems of chemical bonding

In 1981 the prize was given jointly to Kenichi Fukui and Roald Hoffmann for the

Frontier orbital theory of chemical reactivity, developed independently, concerning the course of chemical reactions The 1998 Nobel Prize in chemistry went to John A Pople and Walter Kohn for their respective work in developing computational chemistry methods (Pople) and density functional theory (Kohn).4 In recent years, involvement in computational chemistry activities by organizations over the world has also risen dramatically.2

A variety of chemical systems with a wide range of complexity can be described by computational chemistry By predicting the characteristics and behavior of a system, computational chemistry can powerfully be used to improve the efficiency of existing operating systems or the design of new systems It is being used to complete, guide and sometimes replace experimental methods, reducing the amount of time and money spent

on research to bring ideas from the lab to practical application.2

In chemistry, computational chemistry can play an important role in the design of new chemical products, materials, and catalysts For example, by calculating the energy associated with a chemical reaction, it is possible to study the reaction pathways to determine whether a reaction is thermodynamically allowed Computational chemistry can also be used to reliably predict a wide range of spectroscopic properties from Ultraviolet and Visible Spectroscopy (UV), Infrared and Fourier Transform Infrared Spectroscopy (IR), Nuclear Magnetic Resonance Spectroscopy (NMR)…) to help in the identification of chemical species such as reaction intermediates Electronic structure calculations are also able to provide useful understanding of bonding, orbital energies and shapes, which can be used to design new molecules with selective reactivity Computational tools have also been applied with varying degrees of success in adhesives, coatings, polymers, and surfactants and in the prediction of the toxicity of chemicals.2

1.1.2 Methods

The goal of computational chemistry is to solve complex equations such as the Schrödinger equation HΨ = EΨ (H is the Hamilton operator or Hamiltonian for a system of electrons and nuclei, Ψ is the wave function and E is the energy) for electronic and nuclear motion, which accurately describe phenomena at the atomic or molecular level

Computational chemistry includes calculations at the quantum, atomistic or molecular, mesoscales, as well as methods that form bridges between scales At the quantum scale, computations try to solve the Schrödinger equationand obtain the ground state (or the excited state) energies and other properties (such as molecular geometry, vibrational and NMR spectroscopic data, multipolar moments…) of chemical species The

atomistic or molecular scale involves a wide variety of computations, usually done by

molecular dynamics or Monte Carlo methods using classical force fields At this scale, thermodynamic properties (critical points, pressures), transport properties (mass and

heat transfer) and phase equilibria can be expressed The aims of mesoscale calculations

are to identify the qualitative trends in a system given specific chemical structures, compositions, and process conditions; to quantitatively predict the continuum properties

of the system on scales as large as 10 microns with accuracies similar to atomistic level calculations; and to accurately model larger systems on the physical timescale much greater than 100 nanoseconds Finally, bridging techniques provide continuity and

interface between the various scales; so the results of calculations at one scale can be

Chapter 1 3

used as input parameters to calculations at another scale.2 In this thesis, we only focus

on the quantum scale computations

A variety of methods, each having their own specific approximations and accuracies,

have been invented, developed and widely used at the quantum scale Roughly, these

methods can be divided into two main categories: the semi-empirical methods, which

use simplified Hamiltonian together with sets of parameters directly taken from

experimental data and the ab initio methods which, in contrast, use the correct

molecular Hamiltonian and no experimental data, except for the values of the

fundamental physical constants

In the semi-empirical calculations, certain pieces of information such as two electron

integrals are approximated or completely omitted To compensate the errors introduced

by omitting part of the calculation and to give the best possible agreement with

experimental data, the method is parameterized by curve fitting in a few parameters or

numbers A number of semi-empirical methods are available today such as Complete

Neglect of Differential Overlap (CNDO), Intermediate Neglect of Differential Overlap

(INDO), Neglect of Diatomic Differential Overlap (NDDO), Modified INDO

(MINDO/3), (Including parameters for transition metals (ZINDO)), Modified NDO

(MNDO), Austin Model 1 (AM1), Parametric Model 3 (PM3)… Semi-empirical

calculations have been very successful in the description of organic chemistry, where

there are only a few elements used extensively and the molecules are of moderate size

However, semi-empirical methods have been devised specifically for the description of

inorganic chemistry as well.3

The good side of semi-empirical calculations is that they are much faster than the ab

initio approaches The bad side of semi-empirical calculations is that the results can be

erratic If the molecule being computed is similar to molecules in the database used to

parameterize the method, then the results may be very good If the molecule being

computed is considerably different from anything in the parameterization set, the

answers may be very poor.3,5,7

The ab initio methods again can be divided into two separate categories: the

wave-functional ab initio methods,5-8 where the wave function Ψ is used as the basic source of

information for an atomic or molecular system and Density Functional Theory

(DFT),9,10 where the electron density is used for that purpose

The most common wave-functional ab initio technique is the Hartree-Fock (HF) level

based on the use of one-electron functions (orbitals) to construct the many-electron

wave function, obeying the Pauli principle A single determinantal wave function is

used for this purpose The coulombic electron-electron repulsion is only accounted for

in an average fashion; therefore, the HF method is also referred to as mean field

approximation The molecular orbitals are formed from linear combinations of atomic

orbitals or usually from linear combinations of basis functions Due to these

approximations, HF calculations give a computed energy greater than the Hartree-Fock

limit (the best single determinantal wave function that can be obtained, which is

however not the exact solution to the Schrödinger equation due to the incomplete

treatment of electron correlation).3

Several types of correlated calculations beginning with a HF calculation then correcting

for the explicit electron-electron repulsion are available today Some of these methods

are Møller-Plesset perturbation theory (MPn, where n is the order of correction),

Multi-Configuration Self Consistent Field (MCSCF), Multi-Configuration Interaction (CI) and

Coupled Cluster theory (CC)

The good side of ab initio methods is that they eventually converge to the exact solution

if all of the approximations are made sufficiently small in magnitude The bad side of ab initio methods is that they are expensive as considering enormous amounts of computer CPU time, memory and disk space The HF method originally scaled as about N4, where N is the number of basis functions, so doubling the basis set will take a calculation 16 times as long to complete This situation is much worse in the correlated calculation scales In practice, extremely accurate solutions can only be obtained when the molecule contains less than or equal to half a dozen electrons.3 In recent years, the factor “4” has been dramatically reduced thanks to the progress in computer power and algorithms

An alternative ab initio method is the Density Functional Theory (DFT) This method is based on the electron density, which for an N-electron system, only depends on three coordinates, independently of the number of electrons as compared to 4N coordinates (including spin coordinate) of the wave function in the wave-functional ab initio approaches The complexity of a wave function increases with the number of electrons, whereas the electron density has the same number of variables, independently of the system size; therefore, the significance of the DFT method is the reduction of calculation cost Moreover, for about the same cost of doing a HF calculation, DFT includes a significant part of the electron correlation.10 The disadvantage of DFT is that the explicit form of the Hamiltonian written in terms of the electron density is not known

At the present moment, there are three main lines of research in DFT:

1) Fundamental DFT: extends the objective of DFT to excited states, external fields, time-dependent process… or finds new physical knowledge about atomic or molecular systems

2) Conceptual DFT: concentrates on the applications of chemical concepts derived from DFT, particularly in explaining the reactivity of reactants Those concepts include electronegativity, global hardness, global and local softness, Fukui functions, …

3) Computational DFT: develops new generations of functionals to be able to compute faster and more precisely various atomic, molecular or solid state properties

In this thesis we concentrate on the Density Functional Theory method and, particularly,

on the application of DFT in studying product structures and mechanisms of some organic chemical reactions Its objectives belong to the Conceptual DFT field The Computations themselves are also done with DFT methods, the techniques not being the focus of our research, standard as they are

1.2 Structures and reaction mechanism in organic chemistry

The reaction mechanism is a microscopic description of the course of a reaction, showing the transformation of starting material into products as a series of discreet steps, each of which may produce a distinct intermediate This description makes it possible to understand why a reaction takes place, thus providing a procedure to predict the influences of changing reaction conditions and enables us to estimate the results of

Chapter 1 5

related reactions The insights into how and why a given reaction occurs often reveal

close relationships between reactions that originally might be thought to be unrelated

Moreover, the study of reaction mechanisms can be used as a basis to develop new

transformations and improve existing procedures However, explaining the ways by

which the reagents in a reaction mixture are converted to the observed products requires

careful interpretation of painstaking experiments.11

In traditional organic chemistry, the experimental types providing data and the methods

used to extract information about reaction mechanisms from the data can be

summarized as follows.11

1 Identification of starting material, intermediates and products

Starting materials are tested for purity, whereas reaction products are separated by

distillation, crystallization or chromatography, and then identified by using chemical

tests, infrared, mass (MS), and NMR spectroscopy Additional starting materials are

often designed and synthesized to test various aspects of the mechanism The proposed

mechanism should be able to explain all of the products, the dependence of the reaction

products on starting material structure (substrate, nucleophile or electrophile), and any

observed regioselectivity Normally, little information is gained about how the reaction

occurred by looking at the products, so additional experiments are necessary

On the other hand, in a multistep reaction, the identification of the intermediates is also

a main objective of studies of reaction mechanism The intermediate may be isolated by

interrupting the reaction (lowering the temperature rapidly or adding a reagent that stops

the reaction) or trapped by adding a compound that is expected to react specifically with

the intermediate Because of its low concentration, the intermediate is normally studied

by spectroscopic methods such as ultraviolet-visible (UV-VIS), infrared (IR), nuclear

magnetic resonance (NMR), and electron paramagnetic resonance (EPR) spectroscopy

2 Thermodynamic data

Any reaction is always accompanied by a change in enthalpy (ΔH), entropy (ΔS), and

free energy (ΔG) The equilibrium constant K relates these changes by the fundamental

equation ΔG0 = - RTlnK, with ΔG = ΔH – TΔS and the superscript 0

referring to standard state

These quantities are characteristics of the reactants and products, but are independent of

the reaction path; hence they cannot provide insight into mechanisms However,

information about ΔG, ΔH, and ΔS may indicate the feasibility of any specific reaction

The enthalpies and free energies of formation for many compounds can be obtained

from tabulated thermodynamic data

3 Kinetic data

Kinetic data can provide much detailed insight into reaction mechanisms The rate of a

given reaction is determined by measuring the concentration of products or reactants as

a function of time (about 20 measurements) for 10-20 concentrations of each reagent

The presence or absence of equilibria between reactants and products is also tested by

addition of products or product analogues Generally, any method (such as

spectroscopic techniques, continuous pH measurement, acid-base titration, conductance

measurement, polarimetry…) based on the properties relating to the concentration of

reactants or products can be used to determine the reaction rate

The purpose of a kinetic investigation is to set up quantitative relationships between the

concentration of reactants, catalysts and the rate of the reaction, which are summarized

in the rate law The relationship between a kinetic expression and a reaction mechanism can be evaluated by considering the rates for the successive steps in a multistep reaction The overall rate of a reaction will depend on the rate of the step, which is slow relative to other steps, and this step is called the rate-determining step Normally, kinetic data provide information only about the rate-determining step and steps preceding it The steps following the rate-determining step are bypassed since their rates

do not affect the overall rate

A kinetic study normally starts from postulating possible mechanisms, then comparing the observed rate law with the proposed mechanisms, and finally eliminating those mechanisms that are incompatible with the observed kinetics However, sometimes, several mechanisms give rise to identical predicted rate expressions In this case, the mechanisms are called kinetically equivalent, and it is not possible to choose between them on the basis of kinetic data

4 Substituent effects and linear free-energy relationship

Between substituent groups and chemical properties, there are a number of important relationships, which can be quantitatively expressed in some cases The most widely applied of these relationships is the Hammett equation, which relates rates and equilibria for many reactions of compounds containing substituted phenyl groups

It was noted in the 1930s that there is a linear correlation between the ratio of the rate constant for hydrolysis of ethyl benzoate (k1) to the rate constant for the substituted esters (k2) and the ratio of the corresponding acid dissociation constants (K1 and K2).12Similar relationships are also observed for many other reactions of aromatic compounds Furthermore, from this linear correlation it can be shown that the change in the free energy of activation for hydrolysis of substituted benzoates is directly proportional to the change in the free energy of ionization caused by the same substituents on benzoic acid The correlations due to the directly proportional changes

in free energies are called linear free-energy relationships

The Hammett free-energy relationship is expressed in the following equations:

log(K2/K1) = log(k2/k1) = σρ

The values of σ and ρ are empirically defined by the selection of the reference reaction,

in this case, the ionization of benzoic acids The reaction constant ρ is arbitrarily assigned the value 1 and the substituent constant σ is determined for a series of substituent groups by measuring the corresponding acid dissociation constants The σ values are then used in the correlation of other reactions, and the ρ values of the reactions are thus determined While the value of ρ reflects the sensitivity of the particular reaction to substituent effects, the value of σ indicates the effect of the substituent group on the free-energy ionization of the substituted benzoic acid

Beside the resonance and field (including inductive) effects, which are common in reactions of aromatic compounds, electronegativity and polarizability are also included

in the substituent effect The general form of the Hammett free-energy relationship can

be written as:

log(K2/K1) = log(k2/k1) = σFρF + σRρR + σχρχ+ σαρα

where σF, ρF are the field; σR, ρR the resonance; σχ, ρχ the electronegativity; and σα, ρα

the polarizability substituent constants and reaction constants, respectively

The linear free-energy relationships can provide insight into reaction mechanisms and

Chapter 1 7

enable us to predict reaction rates and equilibria When the ionization of benzoic acid is

chosen as a reference reaction for the Hammett equation, it leads to σ > 0 for

electron-withdrawing groups and σ < 0 for electron-donating groups, since the former groups

favor the ionization of the acid and the latter groups have the opposite effect Moreover,

further consideration of the Hammett equation shows that ρ will be positive for all

reactions favored by electron-withdrawing groups and negative for all reactions favored

by electron-donating groups If the reaction rates for a series of substituents show a

suitable correlation, both the sign and the magnitude of ρ will give information (such as

the distribution of charge) about the transition state for the reaction

It should be noted that not all reactions could be fitted by the Hammett equation or its

modified forms, which is commonly due to the change in mechanism as substituents

vary For example, in a multistep reaction one step may be rate-determining in the

region of electron-withdrawing substituents, but a different step may become

rate-limiting as the substituents become electron-donating

5 Isotope effects

The replacement of an atom by one of its isotopes is a useful tool in the study of

reaction mechanisms Isotopic substitution often involves replacing protium by

deuterium (or tritium) but the principle is applicable to nuclei other than hydrogen,

however, the quantitative differences are largest for hydrogen Isotopic substitution does

not qualitatively affect the course of the reaction, but it has a measurable effect on the

reaction rates If the bond to the isotopically substituted atom is broken in the

rate-determining step, the rate will be affected by isotopic substitution, which is called the

primary kinetic isotope effect In this case, due to different masses, the contributions to

the zero-point energy of the vibrations associated with the bond are not the same

leading to different activation energies and reaction rates Isotope effects may also be

observed even when the substituent hydrogen atom is not directly involved in the

reaction Such effects are called secondary kinetic isotope effects, which result from a

tightening or loosening of the bond at the transition state On the other hand, isotopes

are used as tracers to determine the route that a particular atom takes during the

reaction Determination of the location of an isotope is usually done by NMR or MS,

and does not require techniques based on radioactivity The proposed mechanism will

explain both the location and the effect of isotopes on the reaction rate

6 Catalysis

Catalysts do not affect the reaction equilibrium but they increase the rate of one or more

steps in a reaction mechanism by lowering the corresponding activation energies

Reaction rates and rate laws are determined to verify if the suspect catalyst affects the

rate Moreover, the products are examined to ensure that catalysts do not incorporate

into the products The proposed mechanism should include the role of the catalyst in a

chemically reasonable manner

7 Stereochemistry

Stereochemistry is the study of the three dimensional arrangement in space of the atoms

in molecules and the way it changes upon reaction Different compounds that have the

same molecular formula are called isomers, which can be classified as constitutional

isomers and stereoisomers Constitutional isomers will have the same number and types

of atoms, but they are connected in a different order In stereoisomers, the atoms are

connected sequentially in the same way, but the isomers differ in the way the atoms are

arranged in space There are two major sub-classes of stereoisomers; conformational isomers, which interconvert through rotations around single bonds, and configurational isomers, which differ in the arrangement of their atoms in space and therefore cannot interconvert Configurational isomers are divided into enantiomers and diastereomers Enantiomers are comprised of a chiral compound, which cannot superimpose on its mirror image, and its mirror image Stereoisomers, which are not enantiomers, are called diastereomers A process wherein enantiomers are separated is called a resolution A collection containing equal amounts of two enantiomers is called a racemic mixture or racemate A reaction that forms a racemate is called a racemization The study of the stereo-chemical course of organic reactions, which can be determined

by using instrumental techniques such as IR and NMR spectroscopy, optical rotatory dispersion, and circular dichroism, often leads to detailed insight into reaction mechanism Normally, mechanistic postulates are made to predict the stereochemical outcome of the reaction and then compared with the observed products

8 Solvent effect

Solvents can affect the identity of the products, the course and the rate of reactions Solvents can be classified as protic solvents, which contain relatively mobile protons such as those bonded to oxygen, nitrogen, or sulphur; and aprotic solvents, in which all hydrogen is bonded to carbon They are also classified as polar solvents, which have high dielectric constants and do have effects on reaction rates, and non-polar solvents Furthermore, it is important to distinguish between the macroscopic effects related to the properties of the bulk solvent and the effects based on the details of structure For example, the dielectric constant is a measure of the ability of the bulk material to increase the capacity of a condenser In terms of structure, the dielectric constant is proportional to the dipole moment and the polarizability of the molecule Polarizability,

in turn, refers to the ease of electron density distortion of the molecule One important property of solvent molecules is the response of a solvent to changes in charge distribution as the reaction occurs The dielectric constant indicates the ability of the solvent to accommodate the separation of charge However, being a macroscopic property, it conveys little information about the ability of the solvent molecules to interact with the solute molecules at close range The direct solute – solvent interactions will depend on the specific structures of the molecules The mechanism must explain the effect of different solvents on the reaction rate and any incorporation of solvent into the reaction products

9 Other reaction characteristics

Occasionally a reaction rate or outcome depends on the size or material of the container,

as it often does for free radical chain reactions In this case the mechanism must take into account the effect of hidden reagents or catalysts like water and oxygen, which divert reactions, especially those involving organometallic compounds

In summary, experimental methods give data, which allow only indirect conclusions to the overall reaction pathway because they all are based on the studies of only the initial and the final state of every elementary step of the reaction

Chapter 1 9

Computational chemistry at quantum scale opens up new possibilities of studying

chemical reactions and enables the researchers to calculate all critical parameters of the

mechanism of a reaction

In order to generally describe the structural changes in a reacting system, it is necessary

to solve the time-dependent Schrödinger equation However, even approximate

solutions to this equation for a system only containing several atoms are extraordinarily

complicated On the other hand, most chemical reactions do not significantly exhibit

quantum effects at room temperature Hence, to describe the dynamics of a chemical

reaction, another approach is employed, namely, the calculation of the potential energy

surface (PES).13

The PES of a system is a geometric surface describing the variation of its potential

energy (the sum of electronic energy and nuclear repulsion energy) as a function of the

coordinates of all nuclei in the system In case the system contains N nuclei, there are

3N coordinates defining the geometry Of these coordinates, three describe the overall

translation of the molecule, and three describe the overall rotation of the molecule with

respect to the three principal axes of inertia For a linear molecule, only two coordinates

are necessary for describing the rotation Therefore, the number of the independent

coordinates (degrees of freedom) that fully determine the PES is 3N – 6 (or 3N – 5 in

the case of a linear molecule).13

The energetically easiest passage from reactant to products on the potential energy

contour map defines the potential energy profile on which the potential energy is plotted

as a function of one geometric coordinate For an elementary reaction such as A-B + C

→ A-C + B, that geometric coordinate is the reaction coordinate, whereas for a stepwise

reaction it is the succession of reaction coordinates for the successive individual

reaction steps The reaction coordinate is defined as the geometric parameter (bond

length, bond angle…) that changes smoothly from the configuration of the reactants

through that of the transition state to the configuration of the products Typically, the

reaction coordinate is chosen to follow the path along the gradient (path of shallowest

ascent or deepest descent) of potential energy from reactants to products.13

In practice, to have enough information on the mechanism and the kinetics of a

chemical reaction, it is not necessary to know the full function but only some portions

of the PES, mainly those corresponding to the minima (reactants, intermediates,

products) and to the saddle points (transition structure).13

Studying the PES of a system, one can obtain various important characteristics of the

reaction: relative energies of the reactants and the products (energy of reaction); relative

energies of the reactants and the transition state (activation energy); the curvature of the

PES in the minima zone or the saddle point region, which can be used to determine the

vibration spectrum, the entropy and the kinetic isotopic effects (ratio between the

reaction rate constant of the compound with the light isotope and that of the compound

containing the heavy isotope); geometrical characteristics of the reactants, the products

and the transition state Moreover, based on the activation energies from the PES

containing more than one reaction pathway, one can determine and then explain which

path is energetically favored In mass spectrometry studies, using the calculated

energies of the isomers and the energy barriers between them, one can predict and

explain which isomer is more stable than the others Furthermore, by verifying the

existence of the intermediates in the reaction pathway, one can determine whether the

reaction is concerted or stepwise

Besides the information from the PES, one tries to use the information on the starting structures to explain and predict the first stages of the reaction, which can be done by using a variety of reactivity indices In the wave function ab intio approaches, the Frontier Molecular Orbital (FMO) theory14 is widely applied The coefficients and the shapes5,15 of the Highest Occupied Molecular Orbital (HOMO) and of the Lowest Unoccupied Molecular Orbital (LUMO) are used to explain why a reaction is favored over another In the DFT framework, the global and local softness in conjunction with the hard and soft acids and bases (HSAB) principle16 become the useful tools10,17-21 to predict the favored product on the basis of the electronic properties of the isolated reactants The idea of the aromaticity of the transition state22-23 is also applied for this purpose

1.3 Scope of the Thesis

“Structure and Mechanism in Organic Chemistry” has been the title of a very influential book in physical organic chemistry, written by Ingold in the early fifties.24 It has been used by several generations of organic chemists as a guide in the sometimes bewildering forest of organic reactions

Since then the field has known impressive developments and in recent years quantum chemical/computational methods turned out to be an important tool to elucidate reaction mechanisms as discussed in §1.2 It is in this direction that in our thesis we concentrate

on the “Structure and Mechanism” of some isomerization, [2+1] and 1,3-dipolar cycloaddition reactions, using Density Functional Theory methods In each kind of reaction, the structures and relative energies of reactants, transition structures, intermediates and final products will be determined to construct the potential energy surface Besides, the DFT-based reactivity descriptors such as hardness, global and local softness, Fukui functions and indices of aromaticity (if possible) are also calculated From those parameters, we will analyze the reaction steps, the favored site in the initial attack, the stability of intermediates and final products, the effect of substituents and solvents on the reacting system

Starting with a general overview of the current situation and the methods used in computational chemistry, the first chapter discusses the ways to determine “Structure and Mechanism” in traditional organic chemistry and in computational chemistry Chapter 2 presents a general introduction to the most currently used methods in computational/theoretical chemistry and provides DFT-based reactivity criteria together with others as tools for studying “Structure and Mechanism” of chemical reactions The computational details including software and hardware used in this thesis are discussed

in chapter 3 Chapter 4 uses Density Functional Theory methods to construct the potential energy surface for simple isomerization and fragmentations reactions involving two NS moieties, [CH3NS] and [NH2NS] The [2+1] cycloaddition reactions

of hydrogen isocyanide (HN≡C), CX (X = O, S) to acetylenes are reported in chapter 5 Besides, the 1,3-dipolar cycloaddition (1,3-DC) of fulminic acid (HCNO) and the simple azides (XNNN, X=H, CH3, NH2) to thionitroso compounds (R-N=S, R = H,

NH2); the 1,3-DC of diazoalkanes, hydrazoic acid and nitrous oxide to polar dipolarophiles are also included Finally, chapter 6 gives the general conclusion of this work and further development of Density Functional Theory methods

Chapter 1 11

1.4 References

1 Technology Vision 2020: The U.S Chemical Industry,

The American Chemical Society, American Institute of Chemical Engineers,

The Chemical Manufacturers Association, The Council for Chemical

Research, and The Synthetic Organic Chemical Manufacturers Association,

December, 1996

http://www.chemicalvision2020.org/pdfs/chem_vision.pdf

2 Technology Roadmap for Computational Chemistry

Dixon, D A et al., The Council for Chemical Research, 1999

http://www.chemicalvision2020.org/pdfs/compchem.pdf

3 Young, D Computational Chemistry: A Practical Guide for Applying

Techniques to Real World Problems; John Wiley & Son: Chichester, 2001

4 Nobel Prize in Chemistry Winners 2001-1901

7 Levine, I N Quantum Chemistry (Fourth Edition); Prentice Hall, Englewood

Cliffs: New Jersey, 1991

8 Hehre, W J.; Radom, L.; Schleyer, P v R.; Pople, J A Ab Initio Molecular

Orbital Theory; Wiley: New York, 1986

9 Hohenberg, P.; Kohn, W Phys Rev B 1964, 136, 864

10 Parr, R G.; Yang, W Density Functional Theory of Atoms and Molecules;

Oxford University Press: New York, 1989

11 Carey, F A.; Sundberg, R J Advanced Organic Chemistry, Part A:

Structure and Mechanisms; Plenum Press: New York, 1990

12 Hammett, L P J Am Chem Soc 1937, 59, 96

13 Minkin, V I.; Simkin, B Ya.; Minyaev, R M Quantum Chemistry of

Organic Compound, Mechanisms of Reaction; Springer-Verlag: Berlin, 1990

14 Woodward, R B.; Hoffmann, R The Conservation of Orbital Symmetry;

Verlag Chemie: Weinheim, 1970

15 Fleming, I Frontier Orbitals and Organic Chemical Reactions; Wiley:

Chichester, 1978

16 Pearson, R G J Am Chem Soc 1963, 85, 3533

17 Geerlings, P.; De Proft, F.; Langenaeker, W Adv Q Chem 1999, 33, 303

18 Chattaraj, P K.; Lee, H.; Parr, R G J Am Chem Soc 1991, 113, 1855

19 Gázquez, J L.; Méndez, F J Phys Chem 1994, 98, 4591

20 Chandra, A K.; Geerlings, P.; Nguyen, M T J Org Chem 1997, 62, 6417

21 Damoun, S.; Van de Woude, G.; Méndez, F.; Geerlings, P J Phys Chem

1997, 101, 886

22 De Proft, F.; Geerlings, P Chem Rev 2001, 101, 1451

23 Cossío, F P.; Morao, I.; Jiao, H.; Schleyer, P v R J Am Chem Soc 1999,

121, 6737

24 Ingold, C K Structure and Mechanism in Organic Chemistry; 2d ed.;

Cornell University Press: Ithaca, New York, 1969

2 Theoretical Background

2.1 Wave function Ab Initio methods

2.1.1 Schrödinger equation

Electrons are very light particles and display both particle and wave characteristics;

therefore, they can be described in terms of a wave function Ψ The wave function

concept and the equation describing its change with time were put forward in 1926 by

Erwin Schrödinger This equation, known as the non-relativistic time-dependent

Schrödinger equation, can be written as follows1,2

where H is the Hamilton operator (Hamiltonian), and η is the Planck constant divided

by 2π

If the Hamiltonian does not contain the time variable explicitly, the time dependence of

the wave function can be separated out as a simple phase factor Denoting r as the

position vector, in the one particle case, one obtains

η / iEt

e)()t,

r =Ψ −

Consequently, the energies and wave functions of stationary states of the system are

given by the solution of the time-independent Schrödinger equation

HΨ(r) = EΨ(r) (2.1.3) For a general N-particle system, the Hamilton operator contains kinetic (T) and

potential (V) energy operators for all particles (e.g electrons and nuclei).1-4

∂

∂+

2 2 i

2 N

1

2 N

1 i

2 i i

2 N

1 i i

zyxm2m

2T

i ij

j i N

1 i N

i ij

r

qqV

with mi the mass, qi the charge of particle i, and rij the distance between particles i and j

Nuclei are much heavier than electrons, and thus move much slower Hence, the

electrons will adjust rapidly to any change in nuclear positions Consequently, the

Schrödinger equation can be approximately separated into one part describing the

electronic wave function for a fixed nuclear geometry, and another part expressing the

nuclear wave function, in which the energy from the electronic wave function plays the

role of the potential energy (Born-Oppenheimer approximation) Accordingly, the

electronic wave function depends only on the position of the nuclei, not their momenta

Denoting nuclear coordinates with R and subscript n, and electron coordinates with r

and e, the Schrödinger equation can be written in the following way

HtotΨtot(R,r) = EtotΨtot(R,r) (2.1.7)

Htot = He + Tn

He = Te + Vne + Vee + Vnn (2.1.8)

Ψtot(R,r) = Ψn(R)Ψe(r;R)

Note that the role of R in Ψe(r;R) is that of a parameter

The electronic Schrödinger equation becomes

HeΨe(r;R)= Ee(R)Ψe(r;R) (2.1.9)

Finally, the nuclear Schrödinger equation has the form

{Tn + Ee(R)}Ψn(R) = EtotΨn(R) (2.1.10)

In this thesis we only concern with the electronic Schrödinger equation (2.1.9)

2.1.2 The Hartree-Fock theory

The goal of the wave function ab initio methods is to find the wave function Ψ, which

satisfies the equation (2.1.9) and thus determines the electronic energy of the molecule

One approach is the Molecular Orbital (MO) theory, which uses one-electron functions

or orbitals to approximate the full wave function

The spatial function termed molecular orbital, ψ(x, y, z), is a function of the cartesian

coordinates x, y, z of a single electron Its square, ψ2 (or ⏐ψ⏐2 if ψ is complex) is

interpreted as the probability distribution of the electron in space To describe the spin

of an electron, it is necessary to specify a complete set of two orthonormal spin

functions α(ξ) for spin up and β(ξ) for spin down The full wave function for a single

electron is the product of a molecular orbital and a spin function, ψ(x, y, z)α(ξ) or ψ(x,

y, z)β(ξ) It is termed a spin orbital, χ(x, y, z, ξ) One important property of the wave

function is that it must satisfy the anti-symmetry principle (the Pauli exclusion

principle), which states that a wave function must change sign when the spatial and spin

components of any two electrons are exchanged

To account for this problem, in the Hartree-Fock theory, the spin orbitals are arranged

in a determinantal wave function, called a Slater determinant

Ψ(χ1,χ2,…, χN) =

)N(

)N( )N(

)2(

)2( )2(

)1(

)1( )1(

!N1

N 2

1

N 2

1

N 2

1

χχ

χ

χχ

χ

χχ

χ

ΜΜ

The (N!)-1/2 factor is a normalization constant For convenience, a shorthand notation is

often used for (2.1.11) This notation uses the anti-symmetry operator A to represent the

determinant and explicitly normalizes the wave function. 1,3

Ψ = A[χ1(1)χ2(2)…χN(N)] = AΠ (2.1.12)

A = ∑− − = −∑ +∑ −

ijk ij

ij

1 N

0 p

p

]

PP

1[

!N

1P)1(

!N1

The 1 operator is the identity, whereas Pij generates all possible permutations of two

Chapter 2 15

electron coordinates, Pijk all possible permutations of three electron coordinates etc

On the other hand, as can be seen in the series of equations from (2.1.3) to (2.1.9), the

electronic Hamiltonian contains two terms depending only on one electron coordinate

(the kinetic energy Te and the nuclear-electron attraction Vne), a term depending on two

electron coordinates (the electron-electron repulsion Vee), and the nuclear-nuclear

repulsion Vnn The latter does not depend on electron coordinates and is constant for a

given nuclear geometry Therefore, with the use of atomic units, the Hamiltonian may

be collected according to the number of electron indices.1

Hi = − ∇ −∑ −

a 2

i

rR

Z2

i ij N

1 i

1 i N

1 j

ij ij N

1 i

2

1h

HE

+

−+

=

〉ΨΨ

The two-electron integrals, Coulomb integral Jij and the Exchange integral Kij, are

defined as:

)j()i(G)j()i(K

)j()i(G)j()i(J

i j ij j i ij

j i ij j i ij

χχχ

χ

=

χχχ

χ

=

(2.1.16)

Applying the variational principle, which states that the best wave function of the form

(2.1.11) is the one giving the lowest possible energy, and introducing Lagrange

multipliers in the constraints that the spin orbitals remain orthogonal and normalized,

gives the final set of canonical Hartree-Fock equations:

' i i ' i i

where the Fock operator Fi is defined as:

( ) ( ) ( ) ( ) ( )2 ( )1 G ( ) ( )1 2K

21G12

J

)KJ(HF

j i 12 j i

j

i j 12 j i

j

N

j

j j i

i

χχχ

=χ

χχχ

=χ

−+

(2.1.18)

The corresponding spin orbitals of the canonical Hartree-Fock equations are the

canonical Hartree-Fock spin orbitals, and the eigenvalues εi are referred to as spin

orbital energies, which can be written as (dropping the prime notation and letting χ be

the canonical orbitals):

+

=χχ

=

j

ij ij i

i i i

The total energy can then be written either as (2.1.14) or in terms of spin orbital

energies:

nn N

ij

ij ij N

There are two types of spin orbitals: restricted and unrestricted spin orbitals.3 The

restricted spin orbitals are constrained to have the same spatial function for α (spin up)

and β (spin down) spin functions A set of K orthonormal spatial orbitals can form a set

of 2K spin orbitals by multiplying each spatial orbital by either the α or β spin function

)()z,y,x(),z,y,x(

)()z,y,x(),z,y,x(

i i

2

i 1

i

2

ξβψ

=ξχ

ξαψ

=ξ

χ −

i=1,2, ,K (2.1.21) The unrestricted spin orbitals; in contrast, have different spatial functions for different

spins A set of K orthonormal spatial orbitals {ψi α

} and a different set of K orthonormal spatial orbitals {ψi β

}, such that the two sets are not orthogonal, can form an orthonormal set of 2K unrestricted spin orbitals as:

)()z,y,x(),z,y,x(

)()z,y,x(),z,y,x(

i i

2

i 1

i

2

ξβψ

=ξχ

ξαψ

=ξχ

β

α

−

i=1,2, ,K (2.1.22)

Here we only consider the restricted formalism for closed-shell ground states in which

each spatial orbital is doubly occupied, and the unrestricted formalism for open-shell

(unpaired) ground states and also for open-shell excited states

2.1.2.1 Restricted closed-shell Hartree-Fock: The

Roothaan-Hall equations

For most computational tasks, the spin orbital formulations must be converted to the

ones involving only the spatial functions and spatial integrals by integrating out the spin

functions α and β Therefore the calculation of spin orbitals turns out to be equivalent to

the problem of solving the spatial integro-differential equation3

i i i i

Here the closed-shell Fock operator fi is expressed as:

( ) ( ) ( ) ( ) ( )2 ( )1 G ( ) ( )1 2K

21G12

J

rR

Z2

1H

)KJ2(Hf

j i 12 j i

j

i j 12 j i

j

a 2

i i

2 / N

j

j j i

i

ψψψ

=ψ

ψψψ

=ψ

=

∑

∑

(2.1.24)

These equations are quite analogous to those for spin orbitals, except for the factor of 2

occurring with the coulomb operator and the sum is over N/2 occupied orbitals

Chapter 2 17

By introducing a set of K known basis functions and expanding the unknown spatial

(molecular) orbitals in the linear expansion

i C (2.1.25)

the problem of calculating the Hartree-Fock molecular orbitals reduces to the problem

of calculating the set of expansion coefficients Cμi This finally leads to the

FC = SCε (2.1.27) The overlap matrix S has elements

ν μ

The Fock matrix F contains a one-electron part Hcore and a two-electron part G, which

depends on the density matrix P and a set of two-electron integrals

( )

CC2P

|

HH

2

1P

HGHF

2 / N

i

i i

i core

core core

∑

∑

∗ σ λ λσ

σ λ ν μ

ν μ μν

λσ λσμν

μν μν μν

=

φφφφ

=λσμν

φφ

=+

=

(2.1.29)

Finally, the total energy E is obtained via the formula:

nn K

1 K

1

H(P2

In this case, the α and β electrons are assigned to different molecular orbitals, which are

expanded in the same set of basis functions:

∑

=

α μ

ψ K

1 i

ψ K

1 i

The coefficients C and μiα Cβμi are then varied separately, leading to the Pople-Nesbet

equations

β β μν

=

α μν α

=

α α μν

K

1

vi

CSC

F

CSC

μν

α

μν

σνμλ

−σλμν+

=

σνμλ

−σλμν+

=

)]

(P)(P[H

F

)]

(P)(P[H

F

T core

T core

(2.1.34)

with the expressions for the density matricesPλσα Pλσβ and the total density matrix PλσT

β λσ

α λσ λσ

=

∗ β σ

β λ

β

λσ

=

∗ α σ

α λ

PPP

)C(CP

)C(CP

T

N

1 i

i i

N

1 i

i i

(2.1.35)

Finally, the total energy becomes:

nn K

1 K

1

core

P[2

1

μν

β νμ

α μν

α νμ

=

The UHF wave function is not an eigenfunction of the spin operator S2, so it can be

contaminated by components of higher multiplicity Although its expectation value is

always too high, due to larger values of S of the contaminants, it is almost exclusively

used as a first approximation to doublet and triplet states

2.1.3 Post Hartree-Fock methods

The primary shortcoming of Hartree-Fock (HF) theory is the inadequate treatment of

the correlation between motions of electrons The HF wave functions partially take

account of the correlation of electrons with the same spin by virtue of their single

determinant form, whereas they neglect the correlation between electrons with opposite

spin.4

Electron correlation is defined as the adjustment of electron motion to the instantaneous

positions of all the electrons in a molecular entity According to the Coulomb’s law, in

reality, electrons repel each other, and the instantaneous position of each electron forms

the center of a region in space, which other electrons will avoid When one electron

changes its position, the Coulomb hole for other electrons will move with it, and their

motions are correlated In HF method, the instantaneous electron-electron repulsion is

replaced by the averaged intra-electron repulsion; therefore, each electron does not

know the instantaneous position of others, only its average value, and thus motions are

uncorrelated The electrons in reality are thus further apart than estimated by the HF

method This limitation leads to calculated HF energies being above the exact values.1

The correlation energy Ecor is currently defined as the difference between the exact,

non-relativistic ground state energy E of the system within the Born-Oppenheimer

approximation and the Hartree-Fock energy E in a complete basis set:1-4

Chapter 2 19

Another shortcoming of HF theory is that it is often difficult to converge excited states

Unless the excited state has a different overall symmetry than the ground state, the HF

calculations generally collapse to the ground state upon orbital optimization This

prevents HF theory from providing chemically useful information about excitation

energies and charge densities of excited states Moreover, the restricted HF cannot

describe the dissociation of molecules into open-shell fragments (e.g., H2 → 2H),

whereas the unrestricted version gives a qualitatively correct prediction of such

dissociations but the resulting potential energy surfaces are not accurate.1

Despite its deficiencies, the HF method still provides the best one-determinant trial

wave function and its solution usually gives ~99% of the exact energy Therefore, the

starting point for improvements must be a many-determinantal trial wave function

starting from the HF wave function as a reference This also means that the mental

picture of electrons residing in orbitals has to be abandoned Several available

approaches to calculate the correlation energy after Hartree-Fock calculations are

briefly discussed in the following sections.1-5

2.1.3.1 The Configuration Interaction method

With the use of second quantization techniques and indices i, j, … for occupied HF spin

orbitals and a, b, … for unoccupied spin orbitals, an configuration such as the doubly

excited determinant can be written as:

HF i a j b ab

(2.1.38) where Xi, Xj are the annihilation operators which remove one electron from the spin

orbitals χi, χj and X , +a X are the creation operators which create one electron in the +b

spin orbitals χa, χb

The configuration interaction (CI) wave function can be generated from the HF wave

function through the action of the operator (1 + C):

ΨCI = (1 + C)ΨHF

C = C1 + C2 + C3 + … (2.1.39)

XXXXcC

XXcC

b

a i j

i a j b ab ij 2

a i

i a a i

> >

+ +

The operator C1 generates the singly excited determinants, C2 doubly excited

determinants, and so on Therefore, the operator C generates all possible excited

determinants, which can be used as a basis to expand the real wave function The

coefficients c , ai cabij , … are computed by means of the variational principle A full CI

calculation, in which all configurations are taken into consideration, with a complete set

of basis functions leads in principle to the exact solution of the many-electron problem

However, the full CI is a computationally impractical procedure because even for

relative small systems and minimal basis sets, the number of determinants included in

the full CI expansion becomes extremely large In practice, a common way to truncate

the CI expansion is to consider only singly and doubly excited configurations, which

yields the CISD method with ΨCISD = (1 + C1 + C2)ΨHF

In general, the CI method is not practical for the calculation of the correlation energy

because full CI is not possible, the convergence of the CI expansion is slow, and the

integral transformation is time-consuming Furthermore, truncated CI is not

size-consistent, which means that the calculation of a system containing several molecules at

finite separation does not give the same energy as the sum of the calculations on

individual molecules This is because a different selection of excited configurations is

made in the two kinds of calculations An advantage of the CI method is that it is

variational, so the calculated energy is always greater than the exact energy Although

CI is not recommendable as a method for ground states, CI-singles (CIS) has been

advocated as an approach to computation of excited state potential energy surfaces

Besides, by adding terms to the CISD wave function to restore size consistency,6 the

quadratic configuration interaction method (QCISD), which can be seen as an

approximation to the CCSD method, is also applied

2.1.3.2 The Coupled Cluster method

The coupled cluster (CC) methods use an exponential approach instead of the linear

expression for CI:

)T24

1TT2

1T2

1TTT(

)T6

1TTT()T2

1T(T1)Texp(

)Texp(

4 1 2

1 2 2 2 1 3 4

3 1 2 1 3 2 1 2 1

HF CC

++

+++

++

+++

++

=

Ψ

=Ψ

(2.1.40)

If all levels of excitation (T1 up to TN) are included, the CC wave function is equivalent

to full CI, which is impossible for all but the smallest system; therefore the cluster

operator must be truncated at some excitation level Among the possible truncated

forms, the Coupled Cluster Single and Double excitations (CCSD), in which T = T1 +

T2, is the only generally applicable CC method In this case, the form (2.1.40) becomes:

)T24

1TT2

1T2

1(

)T6

1TT()T2

1T(T1)Texp(

4 1 2

1 2 2 2

3 1 2 1 2 1 2 1

++

+

++

++

++

a i

i a a i

=

j

i k l a b c d

k c l d i a j b cd kl ab ij 2

T

Alternatively, the triples contribution may be evaluated by perturbation theory and

added to the CCSD results, as done in the CCSD(T) method

2.1.3.3 The Møller-Plesset Perturbation method

The idea in perturbation methods is that a given system only differs slightly from a

system, which has already been solved (exactly or approximately) Hence, the solution

to the given system, in some sense, should be close to that of the already known system

Mathematically, that idea can be described by defining a Hamilton operator which

involves a reference H0 and a perturbation H’ The premise of perturbation methods is

that the H’ operator in some sense is small compared to H0

Chapter 2 21

where λ is a parameter determining the strength of the perturbation

H0Φ0 = E0Φ0 with Φ0 = A[χ1(1)χ2(2)…χN(N)]

(see (2.1.12) for the notation)

The perturbed Schrödinger equation is

HΨ = WΨ (2.1.43)

If λ = 0, then H = H0, Ψ = Φ0 and W = E0

Since the perturbation is increased from zero to a finite value, the new energy and wave

function must also change continuously and they can be expressed as:

For λ = 0, then W0 = E0, Ψ0 = Φ0 These are called the unperturbed or zero-order energy

and wave function The W1, W2 … and Ψ1, Ψ2 … are the first-, second-, etc order

corrections

Substituting (2.1.42), (2.1.44) into (2.1.43) and collecting terms with the same power of

λ gives the zero-, first-, second-, nth-order perturbation equations

i n i

W

In order to apply perturbation theory to calculate the correlation energy, the unperturbed

wave function is taken as the Hartree-Fock function and the unperturbed Hamiltonian as

a sum over Fock operators, leading to the Møller-Plesset (MP) perturbation theory

The perturbation H’ is the difference between the true molecular electronic Hamiltonian

H and H0, where H is defined as in eq (2.1.13) and H0 as

nn N

1 i

N

1 j

ij ij i

N

1 i i

1 j

ij ij N

1 i N

i

Hence, the perturbation H’ is the difference between the true inter-electronic repulsions

and the HF inter-electronic potential, which is an average potential Using the notation

E(MPn) to indicate the correction at order n, and MPn the total energy up to order n

The zero-order wave function is the ground state HF wave function Φ0, which is the

Slater determinant of spin orbitals, and the zero-order energy is the sum of spin orbital

energies

MP1 = MP0 + E(MP1) = E(HF) (2.1.49)

The first-order energy is exactly the HF energy Thus, the electron correlation energy

starts at order 2 The second-order energy correction has the form

E(MP2) = ∑∑

< < ε +ε −ε −ε

χχχχ

−χχχχ

occ

j i vir

b

2 a b j i b a j i

(2.1.50)

MP2 = E(MP0) + E(MP1) + E(MP2) = E(HF) + E(MP2)

The formulas for higher-order corrections become more and more complex The MP2

typically accounts for ~80-90% of the correlation energy, and it can be seen as the most

economical method for including electron correlation.1

MP calculations truncated at any order can be shown to be size consistent and are much

faster than CI calculations However, since the MP method is non-variational, it can

produce energies below the true energy For open shell systems, MP calculations are, in

most cases, based on the unrestricted HF wave function, giving calculations which are

denoted as UMP2, UMP3 UMP4, However, the presence of spin contamination in

the unrestricted Hartree-Fock reference function can produce serious errors in UMP

calculated properties, which can be solved by projecting the spin contaminants out of

the contaminated wave function

2.1.4 Basis sets

As can be seen in eq (2.1.25), the molecular orbitals are expanded as a linear

combination of atomic orbitals or basis functions The basis functions are collected to

form a basis set There are two guidelines for choosing the basis functions One is that

they should agree with the physics of the problem, which means that the expansion

(2.1.25) will require the fewest possible terms for an accurate representation of the

molecular orbitals Second, the chosen functions should make the calculations of the

necessary integrals easier and faster Slater functions (Slater type orbitals, STO,

Aexp(-ξr)) are best under the first criterion but are very difficult to handle computationally

Gaussian functions (Gaussian type orbitals, GTO, Nxmynzpexp(-αr2

)) are much easier to evaluate but need more to accurately represent the true wave function Therefore, in

practice, a number of GTO (called primitive functions) is contracted in a linear

combination to form a basis function A basis function is defined as

n

m

iNx y z exp( br )

c , where the values of c and b are fixed and not varied in the

variational calculation The number of GTO can be possibly expanded, but usually less

than six are used On the other hand, it is common to add some basis functions, which

are single GTO The most common basis sets devised by John Pople and his group are

outlined as follows.1-4

2.1.4.1 Minimal basis sets

The essential idea of the minimal basis set is selecting one basis function for every

atomic orbital including all sub shells Thus for hydrogen, the minimum basis set is just

one 1s orbital For carbon, the minimum basis set consists of a 1s orbital, a 2s orbital

and a set of three 2p orbitals The minimum basis set for the methane molecule consists

of 4 1s orbitals, one per hydrogen atom, and the set of 1s, 2s and 2p described above for

carbon, in total, this set comprises 9 basis functions

Among the most common minimum basis sets of STO-nG types, the STO-3G, where

three GTOs are combined to fit to a STO orbital, is widely applied The STO-3G basis

Chapter 2 23

set for methane thus consists of a total of 9 contracted functions built from 27 primitive

functions

Another feature of the STO-nG basis set, as indeed of most of the basis sets devised by

the Pople group, is that the exponents b (the constant weight of r2 inside the exponential

part of the function) are kept the same for both the 2s and the 2p orbitals, whereas the

coefficients c are different The STO-nG basis sets are available for almost all elements

in the periodic table

2.1.4.2 Scaling the orbital by splitting the minimal basis set

In order to contract the orbitals differently in different molecular environments by a

flexible way, each minimal basis set orbital is replaced by two orbitals, one large (small

exponent) and one small (large exponent) In each molecular orbital, both orbitals of the

set appear and they will mix in the ratio that gives the lowest energy The combination

of a large orbital and a small orbital is equivalent to an orbital of intermediate size,

which best fits the molecular environment since it is obtained from minimizing the

energy Scaling only the valence orbitals of the minimal basis set in this manner, gives

rise to the split valence basis set, whereas scaling all the orbitals of the minimal basis

set leads to double-zeta basis sets

2.1.4.2.1 Split valence basis sets

The most commonly used split valence basis sets are the 3-21G and 6-31G In these

basis sets, the inner shell orbitals (1s for the first row atoms and 1s, 2s and 2p for the

second row atoms) are represented by a combination of 3 and 6 GTOs respectively The

valence orbitals are represented by two basis functions; one is a linear combination of 2

and 3 primitive GTOs respectively and the other consists of a single GTO The 3-21G

basis set is available for all atoms up to Xe, while the 6-31G basis set is only available

for atoms up to Cl Results obtained with split valence basis sets are a significant

improvement on those obtained with a minimum basis set

For hydrogen these basis sets consist of two 1s basis functions

The 3-21G basis sets for carbon comprise a single 1s basis function, two 2s functions

and 6 2p functions (two 2px, two 2py and two 2pz), giving 9 basis functions in all Thus

the total 3-21G basis set for CH4 consists of 17 basis functions

2.1.4.2.2 Double zeta basis sets

In the double zeta basis set, every member of a minimum basis set is replaced by two

functions, so that both core and valence orbitals are scaled in size For some heavier

atoms, the number of basis functions in the double zeta basis sets may be slightly less

than double the number of minimum basis set orbitals For example, some double zeta

basis sets for the atoms Ga - Br have 7 (rather than 8) s basis functions and 5 (instead of

6) p basis functions

Normally the symbol DZ is represented for a double zeta basis set, but the D95 basis

built into Gaussian program is also a double zeta basis set When using the abbreviation

DZ, it is necessary to be clear which author constructed the basis set

It is also quite common to use split valence basis sets where the valence orbitals are

split into three functions An example is the 6-311G, in which the core consists of 6

GTOs and the valence orbitals are described by three basis functions; one expanded in 3

GTOs and the others two, each in one GTO Basis sets where all orbitals are split into three basis functions are called triple zeta functions and referred to as TZ, TZP, TZ2P,

etc

2.1.4.3 Extended basis sets

The most important additions to basis sets are polarization functions and diffuse basis functions

2.1.4.3.1 Polarization basis functions

It is clear that the influence of the other nucleus will distort or polarize the electron density near them; therefore, the basis functions must have more flexible shapes in a molecule than the s, p, d, etc types in the free atoms This can be done by mixing the spherical 1s orbital on hydrogen with an orbital having p symmetry, and thus the positive lobe at one side increases the value of the orbital whereas the negative one at the other side decreases the orbital In this way the 1s orbital has been polarized Similarly the p orbitals are polarized by mixing with an orbital of d symmetry These additional basis functions, normally single GTOs, are called polarization functions For instance, the polarization functions to the 6-31G basis set are as follows:

6-31G* or 6-31G(d) - adds a set of d orbitals to the atoms in the first and second rows (Li - Cl)

6-31G** or 6-31G(d,p) - adds a set of d orbitals to the atoms Li- Cl and a set of p functions to hydrogen

6-31G(3df,pd) adds 3 d-type GTOs and 1 f-type GTO to atoms Li – Cl; and one p-type and 1 d-type function to H

2.1.4.3.2 Diffuse basis functions

In the case of excited states and anions where the electronic density is spread out more over the molecule, the normal basis functions are not adequate This can be corrected by using some basis functions, GTOs with small exponents, which themselves are more spread out These additional basis functions, normally single GTOs, are called diffuse functions

For example, the diffuse functions to the 6-31G basis set are as follows:

6-31+G - adds a set of diffuse s and p orbitals to the atoms in the first and second rows 6-31++G - adds a set of diffuse s and p orbitals to the atoms in the first and second rows and a set of diffuse s functions to hydrogen

Diffuse functions can also be added along with polarization functions, leading, for example, to the 6-31+G*, 6-31++G*, 6-31+G** and 6-31++G** basis sets

2.1.4.4 Dunning's correlation consistent basis sets

These basis sets are specifically designed for high quality calculations using correlation methods They comprise of 4 basis sets such as the correlation consistent valence double-zeta (cc-pVDZ), valence triple-zeta (cc-pVTZ), valence quadruple-zeta (cc-pVQZ), and the valence quintuple zeta (cc-pV5Z) Each of these can be increased by a single diffuse function of each type s, p, f, g, h, etc, for example, the AUG- cc-pVDZ

Chapter 2 25

2.1.5 Molecular quantities

Some chemically interesting molecular quantities, based mainly on the results of the

RHF approximation, are briefly discussed in this section

2.1.5.1 The electron density function

The electron density function ρ(r) is the probability for finding an electron in the

neighborhood of point (x, y, z) with no regard for spin ξ

ρ(r) = ρ(x, y, z) = ∑∫ ∫

ξ

ξξΨ

all

N 2 2 N 1 N

2, ,z , , ) dx dz x

z,y,(x,

=

∗ μ ν

∗ μ

φφ

=

φφ

=ψ

=ρ

K

1 K

1

K

1 K

1 i i 2

i i i

)()(P

)()(CC)

(n)(

(2.1.52)

where ni is the number of electrons (occupation number) in the spatial orbitals ψi and

Pμν is the element of the density matrix P

The total number of electrons within a molecule is given by

1 K

1

SPr

d)(

where Sμν are the elements of the overlap matrix S

2.1.5.2 Atomic charges

2.1.5.2.1 The Mulliken population analysis method

In the Mulliken population analysis method,7-10 the total number of electrons in the

molecule is divided into components that can be assigned to individual atoms

∑

=

= Natom

1 Q , P PQ

This equation contains both one- and two-center contributions, but in this method, those

contributions are equally partitioned between the two atoms Thus the electron density

can be assigned to pairs of atoms ( = bonds) as well as to the atoms themselves

The quantity qPQ for P ≠ Q is called the overlap population between atoms P and Q

The net charge of an atom P is defined as

PP P net

where ZP is the nuclear charge of the atom

The gross atomic charge can be written as

gross

The sum over all gross atomic charges is equal to the total net charge of the molecule,

so it is obvious that the Mulliken population analysis, in fact, provides a breakdown of

the total charge into atomic components However, it has been shown11 that this method

can give unphysical negative values, extremely depends on the basis set, and seems to

produce an unreliable physical picture of the charge distribution in compounds having

significant ionic character Therefore, the Mulliken population analysis should be

applied with necessary cautions

2.1.5.2.2 The natural population analysis

The first order reduced density matrix r,r')

1 1

1

1,r) N r r , ,r ) r r , ,r )dr dr

It should be noted that the coordinates for Ψ and Ψ*

are different, and integrating (2.1.58) over coordinate “1” yields the number of electrons N This matrix may be

diagonalized, and its eigenvectors and eigenvalues are called natural orbitals (NO) and

occupation numbers For a single determinant RHF wave function, the first order

density matrix is identical to the density matrix in eq (2.1.29), and the occupation

numbers of NOs have the values of either 0 or 2 exactly For a multi-determinant wave

function of post HF methods or UHF wave functions (when different from RHF), the

occupation numbers may have fractional values between 0 and 2 The concept of NOs

can then be applied for distributing electrons into atomic and molecular orbitals, and

consequently for deriving atomic charges

The idea in the natural population analysis method1,11,12 is using the one-electron

density matrix to construct a set of natural atomic orbitals (NAO) in the molecular

environment Assuming that the basis functions have been arranged so that all orbitals

located on atom A are before those on atom B, which are before those on atom C etc

,

,,, ,,

,, ,,, A2 3A Bk Bk 1 Bk 2 Cn Cn 1 Cn 2

P P P

P P P

P P P

P

CC BC AC

BC BB AB

AC AB AA

(2.1.60)

The NAOs for atom A in the molecule are defined as those diagonalizing the PAA block,

NAOs for atom B as those diagonalizing the PBB block etc Generally, those NAOs will

not be orthogonal, so the sum of orbital occupation numbers will not give the total

number of electrons Those NAOs are then treated by an orthogonalization

procedure11,12 to give a set of orthogonal orbitals, and the diagonal elements of the

density matrix in this basis are the orbital populations The sum of all contributions

from orbitals belonging to a specific atom produces the atomic charge Although this

Chapter 2 27

scheme is computationally more demanding, the additional cost as compared to the

Mulliken method is not so large, making this method widely applicable

2.1.5.2.3 The electrostatic potential derived charges

The electrostatic interactions among fragments having asymmetrical electron

distribution play a significant role in the non-bonded interactions of polar molecules.1

The main interaction is between the electrostatic potential generated by one molecule

(or fraction of) and the charged particles of another

The partial atomic charges are derived by choosing a set of parameters, which generate

the best fit to the actual electrostatic potential as calculated from an electronic wave

function, in a least squares sense.13 A suitable grid of points, in a form of a regular

rectangular mesh, is placed around each nucleus with distances from just outside to

about twice the Van der Waals radius The atomic charges are obtained as those

parameters reproducing the electrostatic potential as closely as possible at these points,

subject to the constraint that the sum is equal to the total molecular charge

The electrostatic potential VESP(r) at position r is defined as

A

rr

)rR

r

Z)

(

where ρ(ri) is the electronic charge density at point ri; ZA, RA is the nuclear charge and

position of atom A, respectively The sum runs over all atoms, and the integral runs

over all space

The potential from the point charges is written in the form

∑ −

=

A q

Rr

q)

(

where qA is the partial charge assigned to atom A

The partial charges are determined by minimizing the least-squares difference between

the electrostatic potential of (2.1.61) and the potential from the point charges of

(2.1.62), with the condition that the partial charges sum up to the total charge of the

molecule Q Thus minimizing the expression

∑

A A i

2 i q i

V

with respect to each qA , over a set of grid points will determine the atomic charges As

there is no an analytic representation for VESP, it is necessary to determine the difference

VESP – Vq by numerical integration

The various schemes for deriving atomic charges differ in the number and location of

points used in the fitting, and in the additional subjected constraints beyond

preservation of charges The electrostatic potential derived charges used in this work are

obtained by using the MK (Merz-Kollman)14-15 option in the Gaussian program

2.2 Density Functional Theory

The basis idea of Density Functional Theory (DFT), based on the Hohenberg-Kohn

theorems,16 is that the ground-state electronic energy can be completely determined by

the electron density ρ, defined in (2.1.51) Each different density will yield a different

ground state energy, and thus the goal of DFT methods is to design functionals relating

the electron density to the energy.1,5,17,18

The energy functional E[ρ] can be written as:

E[ρ] = T[ρ] + Ene[ρ] + Eee[ρ] (2.2.1)

where T[ρ] is the kinetic energy, Ene[ρ] the attraction between the nuclei and electrons,

and Eee[ρ] the electron-electron repulsion, which can be divided into a Coulomb (J[ρ])

and an Exchange (K[ρ]) part The nuclear-nuclear repulsion being a constant in the

Born-Oppenheimer approximation is omitted

2.2.1 The Thomas-Fermi-Dirac theory

The Thomas-Fermi-Dirac (TFD)19 theory is the first approximation to the energy

functional by considering a non-interacting uniform electron gas

−

=ρ

−

=ρ

−

ρρ

=ρ

−

ρ

=ρ

ρπ

=ρ

=ρ

rd)()

3(4

3rd)(C

][K

'rdrd'rr

)'r)(2

1][

J

rdrR

)(Z]

[E

rd)()

3(10

3rd)(C

][T

3 / 4 3 / 1 3

/ 4 x D

a ne

3 / 5 3 / 2 2 3

/ 5 F TF

(2.2.2)

The energy functional of the Thomas-Fermi (TF) theory is ETF[ρ] = TTF[ρ] + Ene[ρ] +

J[ρ], and when including the exchange part KD[ρ], it belongs to the

Thomas-Fermi-Dirac model In general, it has been shown that those models and their improved forms

cannot give results comparable to those obtained by wave function methods

2.2.2 The Kohn-Sham method

The main idea in the Kohn-Sham (KS)20 method is mapping the many electron problem

onto a system of non-interacting electrons with the same ground state density as the

original many electron system The kinetic energy functional is thus divided into two

parts, one can be calculated exactly and the other, a small correction term

Considering the Slater determinantal wave function for N non-interacting electrons in N

orbitals χi, the kinetic energy and the electron density are exactly given by

χ

∇

−χ

=ρ

N

1 i

2 i

N

1 i

i 2 i s

)()

(

2

1]

)]

rv2

1

where v(r) is the external potential, usually just the potential due to the nuclei

Z)

=

ρ] T[ ] v ( ) ( )dr

[

In the real (interacting electrons) system, the difference between the exact kinetic

energy and that calculated by assuming non-interacting orbitals is absorbed into an

exchange-correlation term, and an exact DFT energy expression can be written as

)()()]

r[)(

][E)(

xc xc

δερ+ρε

=δρ

ρδ

with εxc the exchange-correlation energy per particle (energy density)

Therefore, the problem is recast into one involving non-interacting electrons in N

orbitals, which obey the Kohn-Sham equations

i i i eff

= dr' v ( )

r'r

)'r)

(v)(

Using a procedure similar to that in HF method, a set of canonical Kohn-Sham orbitals

can be determined by numerical methods or expanded in a set of basis functions

Accordingly, the Kohn-Sham equations are nonlinear and need to be solved iteratively

Computationally, solving the Kohn-Sham equations is not much more demanding than

solving the HF equations, but if the exact Exc[ρ] is known, DFT will provide the exact

total energy, including electron correlation Unfortunately, this functional has remained

intact in the Kohn-Sham equations: an explicit form for this functional is necessary to

solve the equations The difference between DFT methods is the choice of the

functional form of the exchange-correlation energy

In practical calculations, it is common to separate Exc into a pure exchange Ex and a

correlation part Ec Each of these energies can be written in terms of the energy density,

εx and εc

],[E][E][E][E

][E][E][E

rd)]

r[)(rd)]

r[)(

][E][E][E

c c

c c

x x

x

c x

c x

xc

β α

αβ β

ββ α αα

β

β α α

ρρ+ρ+ρ

=ρ

ρ+ρ

=ρ

ρερ+ρ

ερ

=

ρ+ρ

=ρ

∫

The total density is the sum of the α and β contributions, ρ = ρα + ρβ, and for a