NON-EMPIRICAL CALCULATIONS ON THE ELECTRONIC STRUCTURE OF OLEFINS AND AROMATICS by Robert H. Findlay, B.Sc. Thesis presented for the Degree of Doctor of philosophy University of Edinburgh December 1973 U N /),, cb CIV 3 ACKNOWLEDGEMENTS I Wish to express my gratitude to Dr. M.H. Palmer for his advice and encouragement during this period of study. I should also like to thank Professor J.I.G. Cadogan and Professor N. Campbell for the provision of facilities, and the Carnegie Institute for the Universities of Scotland for a Research Scholarship. SUMMARY Non-empirical, self-consistent field, molecular orbital calculations, with the atomic orbitals represented by linear combinations of Gaussian-type functions have been carried out on the ground state electronic structures of some nitrogen-, oxygen-, sulphur- and phosphorus-containing heterocycles. Some olefins and olefin derivatives have also been studied. Calculated values of properties have been compared with the appropriate experimental quantities, and in most cases the agreement is good, with linear relationships being established; these are found to have very small standard deviations. Extensions to molecules for which there is no experimental data have been made. In many cases it has been iôtrnd possible to relate the molecular orbitals to the simplest member of a series, or to the hydrocarbon analogue. Predictions of the preferred geometry of selected molecules have been made; these have been used to predict inversion barriers and reaction mechanisms. / / The extent of d-orbital participation in molecules containing second row atoms has been investigated and found to be of trivial importance except in molecules containing high valence states of the second row atoms. C 0 N T E N T S Page CHAPTER ONE: Theory The Necessity of Quantum Theory 1 The Schroedinger Wave Equation 2 The Hydrogen Atom 4 The Hydrogen Molecule .7 The Triplet State of the Hydrogen Molecule 12 Koopmanst Theorem 13 The Variational Method 14 The Hartree-Fock Method 15 The Hartree-Fock Procedure Applied to the LCAO Approximation 18 Matrix Diagonalisation 21 The Self-Consistent Field 23 Electron Density Distribution 24 Molecular and Atomic Properties 26 The Nature of the Atomic Orbitals 32 Minimising Ccmputer Time . 35 CHAPTER TWO: Results and Discussion Ozonolysis Introduction .45 Geometries and Calculation Data 47 The Ozone + Ethyleae System 48 Formation and Stru'ture of 1,2,4-Trioxolane 51 . 56 The Structure of C 24 02 Orbital Energies and Ionisation Potentials 60 One-Electron Properties of Ethylene Oxide - arid 1,2,4-Trioxolane . 65 The Azoles Introduction 69 Total Energies 71 Orbital Energies and Ionisation Potentials 72 Correlation of Molecular OrhitaLi of Azoles 86 One-Electron Properties of the Azoles 90 Dipole Moments and Population Analysis 94 Tautomerism in the Triazoles and Tetrazole 102 Protonation of the Triazoles and Tetrazole 106 The Azines Introduction . 116 Total Energies . 118 Correlation of the Calculated Energy Levels 124 Correlation of Theoretical and Experimental Energy Levels 139 Dipole Moments and Charge S Distributions 145, One-Electron Properties of the Azines 150 Page Norbornadi.ene and Its Cations Introduction 155 Norbornadiene 156 Scaled Norbornadiene 161 The Nature of the Norbornadiene Valency Shell Orbitals 166 Inversion Barrier of the 7--Norbornadienyl Cation 169 The 1-Norbornadienyl Cation 174 The Norbornadiene .Cations 175 Cationib Heterocycles Introductidn . 183 Calculations on the Pyryliurn Cation 184 Thiopyrylium Cation . 194 The Dithiolium Cations . 201 d-Orbitals and their Participation in Bonding . 207 Phosphorous Heterocycles. Introduction . 218 Phosphorin 219 Phosphole . 228 Phosphole:- Inversion Barrier and Diels- AlderS Reactivity 233 Orbitals and Orbital Energies 237 Thiophene and Its S-Oxides Introduction 248 Thiophene . 249 Thiophene-S-Oxide . 258 Thiophene-S-Dioxide . 265 Thiathiophthen and Its Isosteres Introduction 275. The Bonding in Thiathiophthen 279 Isosteres of Thiathiophthen 287 CHAPTER THREE: Appendices Gaussian Basis Sets 305 Tables of Gaussian Basis Sets 318 Geometric Parametexs and Symmetry Orbitals . 328 Integrals over Gaussian Type Functions 353 PUBLICATIONS CHAPTER ONE Theory I The Necessity of Quantum Theory In classical mechanics the energy E of a system of interacting particles is the sum of the kinetic energy (T 0 ) and potential energy CV) E = T0 + V0 (1) In this approach both the energy terms are infinitely variable resulting in an inability to explain the spectrum of atomic hydrogen. The classical concept would give rise to a continuous spectrum, while the allowed spectrum was found to consist of a series of lines. In 1885, J.J. Balmer' discovered a regular relationship between the frequencies of those lines found in the visible region of the spectrum; this was V = - 2) (2) where v is the frequency in wave numbers (cm), Ft is a constant called the Rydberg constant and n takes the values 3,4,5.... i.e.,.integers greater than 2. The problem of the interpretation of atomic spectra was solved by Neils Bohr1 who postulated that the angular momentum (p) of an electron was quantised, i.e., that rim the expression below could only take integral values > 1. p = n (h/air) (3) where h is Planck's constant. This integer n is called the principal quantum number. The angular momentum of an electron of mass in moving in a circular path of radius r with a velocity v has an angular momentum below:- p= mvr = n(h/2u) (4) 2 Bohr was then able to evaluate both the radius and the energy of the electron moving round a nucleus of charge Ze; these were -h2 r = . n2 (5) 4u2 e2 Z (6) In the case of the hydrogen atom Z = 1 and the smallest orbit is that with n = 1. Insertion of these values and those of the constants into Equation 5 gives a value of 0.529k; this is usually given the identifier a 0 and is called the Bohr radius. The Bohr theory was able to predict the Balmer series, these being the result of the movement of an electron in an orbit with n = 2 to orbits with n > 2. Further it predicted the existence of other series similar to the Balmer series, this being the first example found of the series. v=R(—-.---) n2 >n2. (7) n2 The _Schran er Wave Equation The Bohr Theory of atoms assumed that electrons were particles. However a beam of electrons directed at a si3fficiently narrow slit, as well as going straight through is diffracted to the sides (below) 3 This behaviour is characteristic of waves rather than particles and exhibits the wave/particle duality of electrons. Schroedinger suggested3 that the proper way to 'describe the wave character of electrons was to replace the classical kinetic and potential energy functions T0 and V0 with linear operators T and V. These were used to set up an equation of the form - (T+V)I =E.± (8) - where I is a wave function describing the movement of electrons. In the hydrogen atom the potential energy operator is identical to the classical potential energy (-e 2 /r), while the classical kinetic energy (T = p 2 /2m) is replaced by the linear differential operator 2 2 T = (0" 8u2m 17 where = + + ( io) x2 y2 3z 2 The Schroedinger equation for the hydrogen atom takes the form h2 2 1-i\ = E - Ias) - ZE There is considerable advantage if one works in atomic units, which take as fundamental quantities the mass of the electron m as the unit of mass the charge of the electron e as the unit of charge the Bohr radius a0 as the unit of length e0 as the unit of energy I 4 Equation (n) now becomes (v2_)i =E (12) The Hydrogen Atom The wave function is a mathematical function just like any other. To be physically meaningful it must be restricted in several ways:- (1) it must be quadratically integrable (i.e., it must be possible to integrate 'I. twice); (2) it must be single-valued at all points in space; (3) it must go to zero at 1nfiiiitr; (4) Ainy wave function which is a solution of equation (12) would still be a valid solution if it was multiplied by c where c is any number. To fix the value of c a normalisation condition is imposed, i.e. dT = 1 (13) 1 1 2 dT =1 (14) where the superscript asterisk refers to the complex conjugate of Jr } is real then the superscript is redundant, giving Equation (14). For the hydrogen atom equation (12) is more conveniently written in spherical polar co-ordinates. When this is done1 the solutions can be written in the form (r, o, 4) = Rnl(P) Ylm (G, 4) (15) where 1, m, n are integers, and r, Q, 0 are the spherical polar co-ordinates centred on the atom. The angular parts 0) are called spherical harmoni s and are defined as lm' c) = Aim (9) B(4) (16) 5 Alm (49) where = [( 2+m1* P (cos g) (17) Bm(G) = (27) -* cos m , m 0 - (isa) 006 m Q , M 'j 0 (lab) B-M= (if) sin m (lSc) with PT being associated Leg endre Polynomi?.is. 4 Some - examples of these are shown in Table I. TABLE Angular Parts of Hydrogen-Like-Functions t = 0(s) £ = 1(p) £ = 2(d) S - - p = ()2 cos o d3ag=(y)2(3cosQ_1) Py=(7YsinGcos(Fdxz=(7;)2 sin 9 cos Qcos 4 = ()sin 0 sin ( d = (l 5 )Usin Q cos 0 sin yz 7r 4) 1 2 - - dxy2 = sin 9 cos 2Q = ()sin 2 -9 sin 20 The radial part of the atomic functions are polynomials in £ multiplied by an exponential term exp-ar) where a is known as the orbital exponent; some of these are shown in Table 2. This exponent is 1 for the hydrogen atom is orbital and for the general case is given b? Z/n where Z is the nuclear charge and n is the principal quantum number.