MOLECULAR QUANTUM MECHANICS, FOURTH EDITION Peter Atkins Ronald Friedman OXFORD UNIVERSITY PRESS www.com MOLECULAR QUANTUM MECHANICS www.com This page intentionally left blank www.com MOLECULAR QUANTUM MECHANICS FOURTH EDITION Peter Atkins University of Oxford Ronald Friedman Indiana Purdue Fort Wayne AC www.com AC 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 Bangkok Buenos Aires Cape Town Chennai Dar es Salaam Delhi Hong Kong Istanbul Karachi Kolkata Kuala Lumpur Madrid Melbourne Mexico City Mumbai Nairobi São Paulo Shanghai Taipei Tokyo Toronto 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 2005 The moral rights of the authors have been asserted. Database right Oxford University Press (maker) First published 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 this same condition on any acquirer British Library Cataloguing in Publication Data Data available Library of Congress Cataloging in Publication Data Data available ISBN 0--19--927498--3 10 9 8 7 6 5 4 3 2 1 Typeset by Newgen Imaging Systems (P) Ltd., Chennai, India Printed in Great Britain on acid-free paper by Ashford Colour Press www.com Table of contents Preface xiii Introduction and orientation 1 1 The foundations of quantum mechanics 9 2 Linear motion and the harmonic oscillator 43 3 Rotational motion and the hydrogen atom 71 4 Angular momentum 98 5 Group theory 122 6 Techniques of approximation 168 7 Atomic spectra and atomic structure 207 8 An introduction to molecular structure 249 9 The calculation of electronic structure 287 10 Molecular rotations and vibrations 342 11 Molecular electronic transitions 382 12 The electric properties of molecules 407 13 The magnetic properties of molecules 436 14 Scattering theory 473 Further information 513 Further reading 553 Appendix 1 Character tables and direct products 557 Appendix 2 Vector coupling coefficients 562 Answers to selected problems 563 Index 565 www.com This page intentionally left blank www.com Detailed Contents Introduction and orientation 1 The plausibility of the Schrödinger equation 36 1.22 The propagation of light 36 0.1 Black-body radiation 1 1.23 The propagation of particles 38 0.24 The transition to quantum mechanics 39 0.3 The photoelectric and Compton effects 4 PROBLEMS 40 0.5 The duality of matter 6 2 Linear motion and the harmonic PROBLEMS 8 oscillator 43 1 The foundations of quantum mechanics 9 The characteristics of acceptable wavefunctions 43 Some general remarks on the Schrödinger equation 44 Operators in quantum mechanics 9 2.1 The curvature of the wavefunction 45 1.2 Eigenfunctions and eigenvalues 10 2.3 The emergence of quantization 46 1.4 Penetration into non-classical regions 46 1.4 Commutation and non-commutation 13 Translational motion 47 1.5 The construction of operators 14 2.5 Energy and momentum 48 1.6 Integrals over operators 15 2.6 The significance of the coefficients 48 1.7 Dirac bracket notation 16 2.7 The flux density 49 1.8 Wavepackets 50 The postulates of quantum mechanics 19 Penetration into and through barriers 51 1.9 States and wavefunctions 19 2.9 An infinitely thick potential wall 51 1.10 The fundamental prescription 20 2.10 A barrier of finite width 52 1.11 The outcome of measurements 20 2.11 The Eckart potential barrier 54 1.12 The interpretation of the wavefunction 22 1.13 The equation for the wavefunction 23 Particle in a box 55 1.14 The separation of the Schrödinger equation 23 2.13 Features of the solutions 57 The specification and evolution of states 25 2.14 The two-dimensional square well 58 1.16 The uncertainty principle 27 1.17 Consequences of the uncertainty principle 29 The harmonic oscillator 60 1.18 The uncertainty in energy and time 30 2.19 Time-evolution and conservation laws 30 2.17 Properties of the solutions 63 2.18 The classical limit 65 Matrices in quantum mechanics 32 1.20 Matrix elements 32 Translation revisited: The scattering matrix 66 1.21 The diagonalization of the hamiltonian 34 PROBLEMS 68 www.com viii j CONTENTS 3 Rotational motion and the hydrogen atom 71 The angular momenta of composite systems 112 4.9 The specification of coupled states 112 Particle on a ring 71 4.10 The permitted values of the total angular momentum 113 3.1 The hamiltonian and the Schrödinger equation 71 4.11 The vector model of coupled angular momenta 115 3.2 The angular momentum 73 4.12 The relation between schemes 117 3.3 The shapes of the wavefunctions 74 4.13 The coupling of several angular momenta 119 3.4 The classical limit 76 PROBLEMS 120 Particle on a sphere 76 3.5 The Schrödinger equation and 5 Group theory 122 its solution 76 3.6 The angular momentum of the particle 79 The symmetries of objects 122 3.7 Properties of the solutions 81 5.1 Symmetry operations and elements 123 3.8 The rigid rotor 82 5.2 The classification of molecules 124 Motion in a Coulombic field 84 The calculus of symmetry 129 3.9 The Schrödinger equation for hydrogenic atoms 84 5.3 The definition of a group 129 3.10 The separation of the relative coordinates 85 5.4 Group multiplication tables 130 3.11 The radial Schrödinger equation 85 5.12 Probabilities and the radial 5.6 The properties of matrix representations 135 distribution function 90 5.7 The characters of representations 137 3.8 Characters and classes 138 3.14 The degeneracy of hydrogenic atoms 94 5.9 Irreducible representations 139 PROBLEMS 96 5.10 The great and little orthogonality theorems 142 Reduced representations 145 5.11 The reduction of representations 146 4 Angular momentum 98 5.12 Symmetry-adapted bases 147 The angular momentum operators 98 The symmetry properties of functions 151 4.1 The operators and their commutation 5.13 The transformation of p-orbitals 151 relations 99 5.14 The decomposition of direct-product bases 152 4.2 Angular momentum observables 101 5.15 Direct-product groups 155 4.3 The shift operators 101 5.17 Symmetry and degeneracy 159 The definition of the states 102 4.4 The effect of the shift operators 102 The full rotation group 161 4.5 The eigenvalues of the angular momentum 104 5.18 The generators of rotations 161 4.6 The matrix elements of the angular 5.19 The representation of the full rotation group 162 momentum 106 5.20 Coupled angular momenta 164 4.7 The angular momentum eigenfunctions 108 Applications 165 4.8 Spin 110 PROBLEMS 166 www.com CONTENTS j ix 6 Techniques of approximation 168 7.10 The spectrum of helium 224 7.11 The Pauli principle 225 Time-independent perturbation theory 168 Many-electron atoms 229 6.1 Perturbation of a two-level system 169 7.12 Penetration and shielding 229 6.2 Many-level systems 171 7.3 The first-order correction to the energy 172 7.14 Slater atomic orbitals 233 6.4 The first-order correction to the wavefunction 174 7.15 Self-consistent fields 234 6.5 The second-order correction to the energy 175 7.16 Term symbols and transitions of 6.6 Comments on the perturbation expressions 176 many-electron atoms 236 6.7 The closure approximation 178 7.17 Hund’s rules and the relative energies of terms 239 6.8 Perturbation theory for degenerate states 180 7.18 Alternative coupling schemes 240 Variation theory 183 Atoms in external fields 242 6.9 The Rayleigh ratio 183 7.19 The normal Zeeman effect 242 6.10 The Rayleigh–Ritz method 185 7.20 The anomalous Zeeman effect 243 7.21 The Stark effect 245 The Hellmann–Feynman theorem 187 Time-dependent perturbation theory 189 PROBLEMS 246 6.11 The time-dependent behaviour of a two-level system 189 6.12 The Rabi formula 192 8 An introduction to molecular structure 249 6.13 Many-level systems: the variation of constants 193 6.14 The effect of a slowly switched constant The Born–Oppenheimer approximation 249 perturbation 195 8.1 The formulation of the approximation 250 6.15 The effect of an oscillating perturbation 197 8.2 An application: the hydrogen molecule–ion 251 6.16 Transition rates to continuum states 199 6.17 The Einstein transition probabilities 200 Molecular orbital theory 253 6.18 Lifetime and energy uncertainty 203 8.3 Linear combinations of atomic orbitals 253 8.4 The hydrogen molecule 258 PROBLEMS 204 8.5 Configuration interaction 259 7 Atomic spectra and atomic structure 207 8.7 Heteronuclear diatomic molecules 265 The spectrum of atomic hydrogen 207 Molecular orbital theory of polyatomic 7.1 The energies of the transitions 208 molecules 266 7.8 Symmetry-adapted linear combinations 266 7.3 Orbital and spin magnetic moments 212 8.4 Spin–orbit coupling 214 8.10 Ligand field theory 274 7.5 The fine-structure of spectra 216 8.11 Further aspects of ligand field theory 276 7.6 Term symbols and spectral details 217 The band theory of solids 278 7.7 The detailed spectrum of hydrogen 218 8.12 The tight-binding approximation 279 The structure of helium 219 8.13 The Kronig–Penney model 281 7.8 The helium atom 219 8.9 Excited states of helium 222 PROBLEMS 285 www.com x j CONTENTS 9 The calculation of electronic structure 287 10.3 Rotational energy levels 345 10.4 Centrifugal distortion 349 The Hartree–Fock self-consistent field method 288 10.5 Pure rotational selection rules 349 9.1 The formulation of the approach 288 10.6 Rotational Raman selection rules 351 9.2 The Hartree–Fock approach 289 10.3 Restricted and unrestricted Hartree–Fock The vibrations of diatomic molecules 357 calculations 291 10.8 The vibrational energy levels of diatomic 9.4 The Roothaan equations 293 molecules 357 9.5 The selection of basis sets 296 10.6 Calculational accuracy and the basis set 301 10.10 Vibrational selection rules 360 Electron correlation 302 10.11 Vibration–rotation spectra of diatomic molecules 362 9.7 Configuration state functions 303 10.12 Vibrational Raman transitions of diatomic molecules 364 9.9 CI calculations 305 The vibrations of polyatomic molecules 365 9.10 Multiconfiguration and multireference methods 308 10.11 Møller–Plesset many-body perturbation theory 310 10.14 Vibrational selection rules for polyatomic 9.12 The coupled-cluster method 313 molecules 368 10.15 Group theory and molecular vibrations 369 Density functional theory 316 10.16 The effects of anharmonicity 373 9.13 Kohn–Sham orbitals and equations 317 10.14 Exchange–correlation functionals 319 10.18 Inversion doubling 377 Gradient methods and molecular properties 321 Appendix 10.15 Energy derivatives and the Hessian matrix 321 PROBLEMS 380 9.16 Analytical derivatives and the coupled perturbed equations 322 11 Molecular electronic transitions 382 Semiempirical methods 325 9.17 Conjugated p-electron systems 326 The states of diatomic molecules 382 9.18 Neglect of differential overlap 329 11.1 The Hund coupling cases 382 Molecular mechanics 332 11.2 Decoupling and L-doubling 384 9.20 Quantum mechanics–molecular mechanics 334 Vibronic transitions 386 Software packages for 11.4 The Franck–Condon principle 386 electronic structure calculations 336 11.5 The rotational structure of vibronic transitions 389 PROBLEMS 339 The electronic spectra of polyatomic molecules 390 10 Molecular rotations and vibrations 342 11.7 Chromophores 391 Spectroscopic transitions 342 11.8 Vibronically allowed transitions 393 10.1 Absorption and emission 342 11.9 Singlet–triplet transitions 395 10.2 Raman processes 344 The fate of excited species 396 Molecular rotation 344 11.10 Non-radiative decay 396 www.com CONTENTS j xi 11.11 Radiative decay 397 Magnetic resonance parameters 452 11.12 The conservation of orbital symmetry 399 13.12 The diamagnetic contribution to shielding 456 11.13 The paramagnetic contribution to shielding 458 11.15 Photochemically induced electrocyclic reactions 403 13.16 Photochemically induced cycloaddition reactions 404 13.15 Spin–spin coupling 462 PROBLEMS 406 13.17 Nuclear spin–spin coupling 467 12 The electric properties of molecules 407 PROBLEMS 471 The response to electric fields 407 14 Scattering theory 473 12.1 Molecular response parameters 407 12.2 The static electric polarizability 409 The formulation of scattering events 473 12.3 Polarizability and molecular properties 411 14.1 The scattering cross-section 473 12.4 Polarizabilities and molecular spectroscopy 413 14.2 Stationary scattering states 475 12.5 Polarizabilities and dispersion forces 414 12.6 Retardation effects 418 Partial-wave stationary scattering states 479 14.3 Partial waves 479 Bulk electrical properties 418 14.4 The partial-wave equation 480 12.7 The relative permittivity and the electric 14.