Table Of Content

Molecular Quantum Mechanics This page intentionally left blank Molecular Quantum Mechanics Fifth edition Peter Atkins Ronald Friedman and University of Oxford Indiana Purdue Fort Wayne 1 3 Great Clarendon Street, Oxford ox2 6dp Oxford University Press is a department of the University of Oxford. It furthers the University’s objective of excellence in research, scholarship, and education by publishing worldwide in Oxford New York Auckland Cape Town Dar es Salaam Hong Kong Karachi Kuala Lumpur Madrid Melbourne Mexico City Nairobi New Delhi Shanghai Taipei Toronto With ofﬁces in Argentina Austria Brazil Chile Czech Republic France Greece Guatemala Hungary Italy Japan Poland Portugal Singapore South Korea Switzerland Thailand Turkey Ukraine Vietnam Oxford is a registered trade mark of Oxford University Press in the UK and in certain other countries Published in the United States by Oxford University Press Inc., New York © Peter Atkins and Ronald Friedman, 2011 The moral rights of the authors have been asserted Database right Oxford University Press (maker) Second edition 1983 Third edition 1997 Fourth edition 2005 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, without the prior permission in writing of Oxford University Press, or as expressly permitted by law, or under terms agreed with the appropriate reprographics rights organization. Enquiries concerning reproduction outside the scope of the above should be sent to the Rights Department, Oxford University Press, at the address above You must not circulate this book in any other binding or cover and you must impose the same condition on any acquirer British Library Cataloguing in Publication Data Data available Library of Congress Cataloging in Publication Data Data available Typeset by Graphicraft Limited, Hong Kong Printed in Italy on acid-free paper by L.E.G.O. S.p.A. ISBN 978–0–19–954142–3 10 9 8 7 6 5 4 3 2 1 Brief contents Introduction and orientation 1 1 The foundations of quantum mechanics 9 Mathematical background 1 Complex numbers 35 2 Linear motion and the harmonic oscillator 37 Mathematical background 2 Differential equations 66 3 Rotational motion and the hydrogen atom 69 4 Angular momentum 99 Mathematical background 3 Vectors 121 5 Group theory 125 Mathematical background 4 Matrices 166 6 Techniques of approximation 170 7 Atomic spectra and atomic structure 210 8 An introduction to molecular structure 258 9 Computational chemistry 295 10 Molecular rotations and vibrations 338 Mathematical background 5 Fourier series and Fourier transforms 379 11 Molecular electronic transitions 382 12 The electric properties of molecules 407 13 The magnetic properties of molecules 437 Mathematical background 6 Scalar and vector functions 474 14 Scattering theory 476 Resource section 513 Answers to selected exercises and problems 523 Index 529 This page intentionally left blank DDeettaaiilleedd ccoonntteennttss Introduction and orientation 1 2.2 Some general remarks on the Schrödinger equation 38 (a) The curvature of the wavefunction 38 (b) Qualitative solutions 39 0.1 Black-body radiation 1 (c) The emergence of quantization 40 0.2 Heat capacities 2 (d) Penetration into non-classical regions 40 0.3 The photoelectric and Compton effects 3 Translational motion 41 0.4 Atomic spectra 4 2.3 Energy and momentum 41 0.5 The duality of matter 5 2.4 The significance of the coefficients 42 2.5 The flux density 43 1 The foundations of quantum mechanics 9 2.6 Wavepackets 44 Penetration into and through barriers 44 Operators in quantum mechanics 9 2.7 An infinitely thick potential wall 45 1.1 Linear operators 10 2.8 A barrier of finite width 46 1.2 Eigenfunctions and eigenvalues 10 (a) The case E < V 46 1.3 Representations 12 (b) The case E > V 48 1.4 Commutation and non-commutation 13 2.9 The Eckart potential barrier 48 1.5 The construction of operators 14 Particle in a box 49 1.6 Integrals over operators 15 1.7 Dirac bracket and matrix notation 16 2.10 The solutions 50 (a) Dirac brackets 16 2.11 Features of the solutions 51 (b) Matrix notation 17 2.12 The two-dimensional square well 52 1.8 Hermitian operators 17 2.13 Degeneracy 53 (a) The definition of hermiticity 18 (b) The consequences of hermiticity 19 The harmonic oscillator 54 The postulates of quantum mechanics 20 2.14 The solutions 55 1.9 States and wavefunctions 20 2.15 Properties of the solutions 57 1.10 The fundamental prescription 21 2.16 The classical limit 58 1.11 The outcome of measurements 22 Further information 60 1.12 The interpretation of the wavefunction 24 2.1 The motion of wavepackets 60 1.13 The equation for the wavefunction 24 2.2 The harmonic oscillator: solution by factorization 61 1.14 The separation of the Schrödinger equation 25 2.3 The harmonic oscillator: the standard solution 62 The specification and evolution of states 26 2.4 The virial theorem 62 1.15 Simultaneous observables 27 Mathematical background 2 Differential equations 66 1.16 The uncertainty principle 28 MB2.1 The structure of differential equations 66 1.17 Consequences of the uncertainty principle 30 MB2.2 The solution of ordinary differential equations 66 1.18 The uncertainty in energy and time 31 MB2.3 The solution of partial differential equations 67 1.19 Time-evolution and conservation laws 31 Mathematical background 1 Complex numbers 35 3 Rotational motion and the hydrogen atom 69 MB1.1 Definitions 35 MB1.2 Polar representation 35 Particle on a ring 69 MB1.3 Operations 36 3.1 The hamiltonian and the Schrödinger equation 69 3.2 The angular momentum 70 2 Linear motion and the harmonic oscillator 37 3.3 The shapes of the wavefunctions 71 3.4 The classical limit 72 3.5 The circular square well 73 The characteristics of wavefunctions 37 (a) The separation of variables 73 2.1 Constraints on the wavefunction 37 (b) The radial solutions 73 viii | DETAILED CONTENTS Particle on a sphere 75 Mathematical background 3 Vectors 121 3.6 The Schrödinger equation and its solution 75 MB3.1 Definitions 121 (a) The wavefunctions 77 MB3.2 Operations 121 (b) The allowed energies 78 MB3.3 The graphical representation of vector operations 122 3.7 The angular momentum of the particle 78 MB3.4 Vector differentiation 123 3.8 Properties of the solutions 80 3.9 The rigid rotor 81 3.10 Particle in a spherical well 83 5 Group theory 125 Motion in a Coulombic field 84 The symmetries of objects 125 3.11 The Schrödinger equation for hydrogenic atoms 84 5.1 Symmetry operations and elements 126 3.12 The separation of the relative coordinates 85 5.2 The classification of molecules 127 3.13 The radial Schrödinger equation 86 (a) The solutions close to the nucleus for l = 0 86 The calculus of symmetry 131 (b) The solutions close to the nucleus for l ≠ 0 86 (c) The complete solutions 87 5.3 The definition of a group 131 (d) The allowed energies 89 5.4 Group multiplication tables 132 3.14 Probabilities and the radial distribution function 89 5.5 Matrix representations 133 3.15 Atomic orbitals 90 5.6 The properties of matrix representations 136 (a) s-orbitals 91 5.7 The characters of representations 138 (b) p-orbitals 91 (c) d- and f-orbitals 93 5.8 Characters and classes 139 (d) The radial extent of orbitals 93 5.9 Irreducible representations 140 3.16 The degeneracy of hydrogenic atoms 94 5.10 The great and little orthogonality theorems 142 Further information 95 Reduced representations 146 3.1 The angular wavefunctions 95 5.11 The reduction of representations 146 3.2 Reduced mass 95 5.12 Symmetry-adapted bases 147 3.3 The radial wave equation 96 (a) Projection operators 148 (b) The generation of symmetry-adapted bases 149 4 Angular momentum 99 The symmetry properties of functions 151 5.13 The transformation of p-orbitals 151 The angular momentum operators 99 5.14 The decomposition of direct-product bases 152 4.1 The operators and their commutation relations 99 5.15 Direct-product groups 154 (a) The angular momentum operators 100 5.16 Vanishing integrals 156 (b) The commutation relations 100 5.17 Symmetry and degeneracy 158 4.2 Angular momentum observables 101 4.3 The shift operators 102 The full rotation group 159 5.18 The generators of rotations 159 The definition of the states 102 5.19 The representation of the full rotation group 161 4.4 The effect of the shift operators 103 5.20 Coupled angular momenta 162 4.5 The eigenvalues of the angular momentum 104 Applications 163 4.6 The matrix elements of the angular momentum 106 4.7 The orbital angular momentum eigenfunctions 108 Mathematical background 4 Matrices 166 4.8 Spin 110 MB4.1 Definitions 166 (a) The properties of spin 110 (b) The matrix elements of spin operators 111 MB4.2 Matrix addition and multiplication 166 MB4.3 Eigenvalue equations 167 The angular momenta of composite systems 111 4.9 The specification of coupled states 111 6 Techniques of approximation 170 4.10 The permitted values of the total angular momentum 112 4.11 The vector model of coupled angular momenta 114 The semiclassical approximation 170 4.12 The relation between schemes 115 Time-independent perturbation theory 174 (a) Singlet and triplet coupled states 115 (b) The construction of coupled states 116 6.1 Perturbation of a two-level system 174 (c) States of the configuration d2 117 6.2 Many-level systems 176 4.13 The coupling of several angular momenta 118 (a) Formulation of the problem 177 DETAILED CONTENTS | ix (b) The first-order correction to the energy 177 (a) The Hartree–Fock equations 235 (c) The first-order correction to the wavefunction 178 (b) One-electron energies 237 (d) The second-order correction to the energy 180 7.17 Restricted and unrestricted Hartree–Fock calculations 238 6.3 Comments on the perturbation expressions 181 7.18 Density functional procedures 239 (a) The role of symmetry 182 (a) The Thomas–Fermi method 239 (b) The closure approximation 183 (b) The Thomas–Fermi–Dirac method 242 6.4 Perturbation theory for degenerate states 185 7.19 Term symbols and transitions of many-electron atoms 243 (a) Russell–Saunders coupling 243 Variation theory 187 (b) Excluded terms 244 6.5 The Rayleigh ratio 187 (c) Selection rules 245 6.6 The Rayleigh–Ritz method 189 7.20 Hund’s rules and Racah parameters 245 7.21 Alternative coupling schemes 247 The Hellmann–Feynman theorem 191 Time-dependent perturbation theory 192 Atoms in external fields 248 6.7 The time-dependent behaviour of a two-level system 192 7.22 The normal Zeeman effect 248 (a) The solutions 193 7.23 The anomalous Zeeman effect 249 (b) The Rabi formula 195 7.24 The Stark effect 251 6.8 Many-level systems: the variation of constants 196 (a) The general formulation 196 Further information 253 (b) The effect of a slowly switched constant perturbation 198 7.1 The Hartree–Fock equations 253 (c) The effect of an oscillating perturbation 199 7.2 Vector coupling schemes 253 6.9 Transition rates to continuum states 201 7.3 Functionals and functional derivatives 254 6.10 The Einstein transition probabilities 202 6.11 Lifetime and energy uncertainty 204 7.4 Solution of the Thomas–Fermi equation 255 Further information 206 6.1 Electric dipole transitions 206 8 An introduction to molecular structure 258 7 Atomic spectra and atomic structure 210 The Born–Oppenheimer approximation 258 8.1 The formulation of the approximation 258 The spectrum of atomic hydrogen 210 8.2 An application: the hydrogen molecule-ion 260 (a) The molecular potential energy curves 260 7.1 The energies of the transitions 210 (b) The molecular orbitals 261 7.2 Selection rules 211 (a) The Laporte selection rule 211 Molecular orbital theory 262 (b) Constraints on Dl 212 8.3 Linear combinations of atomic orbitals 262 (c) Constraints on Dm 212 l (a) The secular determinant 263 (d) Higher-order transitions 213 (b) The Coulomb integral 263 7.3 Orbital and spin magnetic moments 214 (c) The resonance integral 265 (a) The orbital magnetic moment 214 (d) The LCAO-MO energy levels for the hydrogen (b) The spin magnetic moment 215 molecule-ion 265 7.4 Spin–orbit coupling 215 (e) The LCAO-MOs for the hydrogen molecule-ion 266 7.5 The fine-structure of spectra 217 8.4 The hydrogen molecule 266 7.6 Term symbols and spectral details 218 8.5 Configuration interaction 268 7.7 The detailed spectrum of hydrogen 219 8.6 Diatomic molecules 269 (a) Criteria for atomic orbital overlap and bond formation 269 The structure of helium 221 (b) Homonuclear diatomic molecules 270 (c) Heteronuclear diatomic molecules 272 7.8 The helium atom 221 (a) Atomic units 221 Molecular orbital theory of polyatomic molecules 274 (b) The orbital approximation 222 8.7 Symmetry-adapted linear combinations 274 7.9 Excited states of helium 224 (a) The HO molecule 274 7.10 The spectrum of helium 225 (b) The N2H molecule 276 3 7.11 The Pauli principle 227 8.8 Conjugated p-systems and the Hückel approximation 276 Many-electron atoms 229 8.9 Ligand field theory 282 (a) The SALCs of the octahedral complex 282 7.12 Penetration and shielding 230 (b) The molecular orbitals of the octahedral complex 282 7.13 Periodicity 232 (c) The ground-state configuration: low- and high-spin complexes 283 7.14 Slater atomic orbitals 233 (d) Tanabe–Sugano diagrams 284 7.15 Slater determinants and the Condon–Slater rules 234 (e) Jahn–Teller distortion 284 7.16 Self-consistent fields 235 (f) Metal–ligand p bonding 285