Molecular QUantum mechanics 4th atkins an freindman

Molecular QUantum mechanics 4th atkins an freindman Molecular QUantum mechanics 4th atkins an freindman Molecular QUantum mechanics 4th atkins an freindman Molecular QUantum mechanics 4th atkins an freindman Molecular QUantum mechanics 4th atkins an freindman

MOLECULAR QUANTUM MECHANICS, FOURTH EDITION Peter Atkins Ronald Friedman OXFORD UNIVERSITY PRESS MOLECULAR QUANTUM MECHANICS This page intentionally left blank MOLECULAR QUANTUM MECHANICS FOURTH EDITION Peter Atkins University of Oxford Ronald Friedman Indiana Purdue Fort Wayne AC 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 Sa˜o 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 19 927498 10 Typeset by Newgen Imaging Systems (P) Ltd., Chennai, India Printed in Great Britain on acid-free paper by Ashford Colour Press Table of contents Preface Introduction and orientation The foundations of quantum mechanics Linear motion and the harmonic oscillator Rotational motion and the hydrogen atom xiii 43 71 Angular momentum Group theory Techniques of approximation Atomic spectra and atomic structure An introduction to molecular structure 10 11 12 The calculation of electronic structure Molecular rotations and vibrations Molecular electronic transitions The electric properties of molecules 287 13 The magnetic properties of molecules 14 Scattering theory Further information Further reading 436 Appendix Character tables and direct products Appendix Vector coupling coefficients Answers to selected problems Index 557 98 122 168 207 249 342 382 407 473 513 553 562 563 565 This page intentionally left blank Detailed Contents Introduction and orientation The plausibility of the Schroădinger equation 36 1.22 The propagation of light 36 0.1 Black-body radiation 1.23 The propagation of particles 38 0.2 Heat capacities 1.24 The transition to quantum mechanics 39 0.3 The photoelectric and Compton effects 0.4 Atomic spectra 0.5 The duality of matter PROBLEMS 40 PROBLEMS Linear motion and the harmonic oscillator 43 The foundations of quantum mechanics The characteristics of acceptable wavefunctions 43 Some general remarks on the Schroădinger equation 44 Operators in quantum mechanics 2.1 The curvature of the wavefunction 45 1.1 Linear operators 10 2.2 Qualitative solutions 45 1.2 Eigenfunctions and eigenvalues 10 2.3 The emergence of quantization 46 1.3 Representations 12 2.4 Penetration into non-classical regions 46 1.4 Commutation and non-commutation 13 1.5 The construction of operators 14 1.6 Integrals over operators 15 1.7 Dirac bracket notation 16 1.8 Hermitian operators 17 The postulates of quantum mechanics 1.9 States and wavefunctions 19 19 1.10 The fundamental prescription 20 1.11 The outcome of measurements 20 1.12 The interpretation of the wavefunction 22 1.13 The equation for the wavefunction 23 1.14 The separation of the Schroădinger equation 23 The specification and evolution of states 25 Translational motion 47 2.5 Energy and momentum 48 2.6 The significance of the coefficients 48 2.7 The flux density 49 2.8 Wavepackets 50 Penetration into and through barriers 2.9 An infinitely thick potential wall 51 51 2.10 A barrier of finite width 52 2.11 The Eckart potential barrier 54 Particle in a box 55 2.12 The solutions 56 2.13 Features of the solutions 57 2.14 The two-dimensional square well 58 2.15 Degeneracy 59 1.15 Simultaneous observables 25 1.16 The uncertainty principle 27 1.17 Consequences of the uncertainty principle 29 1.18 The uncertainty in energy and time 30 2.16 The solutions 61 1.19 Time-evolution and conservation laws 30 2.17 Properties of the solutions 63 2.18 The classical limit 65 Matrices in quantum mechanics 32 The harmonic oscillator 60 1.20 Matrix elements 32 Translation revisited: The scattering matrix 66 1.21 The diagonalization of the hamiltonian 34 PROBLEMS 68 viii j CONTENTS Rotational motion and the hydrogen atom 71 The angular momenta of composite systems 4.9 The specification of coupled states Particle on a ring 71 3.1 The hamiltonian and the Schroădinger equation 71 3.2 The angular momentum 73 3.3 The shapes of the wavefunctions 74 3.4 The classical limit Particle on a sphere 76 76 3.5 The Schroădinger equation and its solution 76 3.6 The angular momentum of the particle 79 3.7 Properties of the solutions 81 3.8 The rigid rotor 82 Motion in a Coulombic field 3.9 The Schroădinger equation for hydrogenic atoms 112 112 4.10 The permitted values of the total angular momentum 113 4.11 The vector model of coupled angular momenta 115 4.12 The relation between schemes 117 4.13 The coupling of several angular momenta 119 PROBLEMS 120 Group theory 122 The symmetries of objects 122 5.1 Symmetry operations and elements 123 5.2 The classification of molecules 124 84 The calculus of symmetry 129 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 Schroădinger equation 85 5.5 Matrix representations 131 5.6 The properties of matrix representations 135 3.12 Probabilities and the radial distribution function 90 5.7 The characters of representations 137 3.13 Atomic orbitals 91 5.8 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 Reduced representations Angular momentum 98 The angular momentum operators 98 4.1 The operators and their commutation relations 99 142 145 5.11 The reduction of representations 146 5.12 Symmetry-adapted bases 147 The symmetry properties of functions 151 5.13 The transformation of p-orbitals 151 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 5.16 Vanishing integrals 157 5.17 Symmetry and degeneracy 159 The definition of the states 102 4.4 The effect of the shift operators 102 4.5 The eigenvalues of the angular momentum 104 4.6 The matrix elements of the angular momentum 106 4.7 The angular momentum eigenfunctions 108 4.8 Spin 110 The full rotation group 161 5.18 The generators of rotations 161 5.19 The representation of the full rotation group 162 5.20 Coupled angular momenta 164 Applications PROBLEMS 165 166 CONTENTS Techniques of approximation 168 j ix 7.10 The spectrum of helium 224 7.11 The Pauli principle 225 Time-independent perturbation theory 168 6.1 Perturbation of a two-level system 169 7.12 Penetration and shielding 229 6.2 Many-level systems 171 7.13 Periodicity 231 6.3 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 6.6 Comments on the perturbation expressions 176 7.16 Term symbols and transitions of 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 6.9 The Rayleigh ratio 6.10 The Rayleigh–Ritz method 183 7.19 The normal Zeeman effect 242 7.20 The anomalous Zeeman effect 243 7.21 The Stark effect 245 Time-dependent perturbation theory 189 6.11 The time-dependent behaviour of a two-level system 189 6.12 The Rabi formula 192 6.13 Many-level systems: the variation of constants 193 6.14 The effect of a slowly switched constant perturbation 195 6.15 The effect of an oscillating perturbation 197 6.16 Transition rates to continuum states 199 6.17 The Einstein transition probabilities 200 6.18 Lifetime and energy uncertainty 203 The spectrum of atomic hydrogen 242 185 187 Atomic spectra and atomic structure Atoms in external fields 229 183 The Hellmann–Feynman theorem PROBLEMS Many-electron atoms 204 207 207 PROBLEMS 246 An introduction to molecular structure 249 The Born–Oppenheimer approximation 249 8.1 The formulation of the approximation 250 8.2 An application: the hydrogen molecule–ion 251 Molecular orbital theory 253 8.3 Linear combinations of atomic orbitals 253 8.4 The hydrogen molecule 258 8.5 Configuration interaction 259 8.6 Diatomic molecules 261 8.7 Heteronuclear diatomic molecules 265 Molecular orbital theory of polyatomic molecules 266 7.1 The energies of the transitions 208 7.2 Selection rules 209 8.8 Symmetry-adapted linear combinations 266 7.3 Orbital and spin magnetic moments 212 8.9 Conjugated p-systems 269 7.4 Spin–orbit coupling 214 8.10 Ligand field theory 7.5 The fine-structure of spectra 216 8.11 Further aspects of ligand field theory 7.6 Term symbols and spectral details 217 7.7 The detailed spectrum of hydrogen 218 The band theory of solids 274 276 278 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 8.14 Brillouin zones 284 7.9 Excited states of helium 222 PROBLEMS 285 APPENDIX D4, 422 E C2 2C4 2C02 2C002 A1 1 1 A2 1 À1 À1 z B1 1 À1 1 x2 y2 z2, x2 ỵ y2 Rz 1 À1 À1 xy E À2 0 (x, y), (xz, yz) (Rx, Ry) D3h, 6¯2m E 'h 2C3 2S3 3C02 3'v A01 1 1 1 A02 A001 A002 1 1 À1 À1 À1 À1 À1 À1 À1 À1 z 2 À1 À1 0 (x, y), (xy, x2 À y2) 00 À2 À1 0 (xz, yz) E i h ¼ 12 z2, x2 ỵ y2 Rz 1C02 D1h E 2C0 A1g Sỵ gị 1 1 1 A1u Sỵ uị A2g Sg ị A2u S uị 1 À1 À1 À1 1sv 559 h¼8 B2 E j 2iC0 1 À1 1 À1 1 1 1 (Rx, Ry) hẳ1 z2, x2 ỵ y2 z Rz E1g ðPg Þ 2 cos 0 À2 cos 0 (xz, yz) E1u ðPu Þ 2 cos 0 À2 cos 0 (x, y) E2g ðDg Þ 2 cos 20 2 cos 20 (xy, x2 À y2) E2u ðDu Þ 2 cos 20 À2 À2 cos 20 Td, 4¯3m E 8C3 3C2 6'd 6S4 A1 1 1 A2 1 À1 À1 E À1 0 T1 À1 À1 T2 À1 1 (Rx, Ry) h ẳ 24 x2 ỵ y2 ỵ z2 (3z2 À r2, x2 À y2) (Rx, Ry, Rz) (x, y, z), (xy, xz, yz) 560 j APPENDIX O, 432 E 8C3 3C2 6C02 6C4 A1 1 1 A2 1 À1 À1 E À1 0 (x2 À y2, 3z2 À r2) T1 À1 À1 (x, y, z) T2 À1 À1 h ¼ 24 x2 þ y2 þ z2 (Rx, Ry, Rz) (xy, yz, zx) Oh, m3m E 8C3 6C2 6C4 3C2 i 6S4 8S6 3'h 6'd h ¼ 48 A1g 1 x2 þ y2 þ z2 À1 À1 1 1 1 1 À1 1 À1 2 À1 A2g Eg À1 T1g À1 T2g À1 À1 0 À1 A1u 1 A2u 1 À1 À1 Eu À1 T1u À1 T2u 0 À1 (3z2 À r2, x2 À y2) À1 À1 À1 (Rx, Ry, Rz) (xy, yz, zx) À1 À1 À1 À1 À1 À1 À1 À1 À2 À2 0 1 À1 À1 À3 À1 À1 À1 À3 (x, y, z) Direct products In general g  g ¼ g, g  u ¼ u, u  u ¼ g; G  G ¼ G , G  G00 ¼ G00 , G00  G00 ¼ G For C2, C2v, C2h; C3, C3v, C3h; D3, D3h, D3d; C6, C6v, C6h, D6, S6 A1 A2 B1 B2 E1 E2 A1 A2 B1 B2 E1 E2 A1 A2 B1 B2 E1 E2 A1 B2 B1 E1 E2 A1 A2 E2 E1 A1 E2 E1 A1 ỵ [A2] ỵ E2 B1 ỵ B2 ỵ E1 A1 ỵ [A2] ỵ E2 APPENDIX j For T, Th, Td; O, Oh: A1 A1 A2 E T1 T2 A1 A2 E T1 T2 A1 E T2 T1 A2 E A1 ỵ [A2] ỵ E T1 T1 ỵ T2 T1 ỵ T2 A1 ỵ E ỵ [T1] ỵ T2 A2 ỵ E ỵ T1 ỵ T2 T2 A1 þ E þ [T1] þ T2 For C1v, D1h: Sþ S P D ặỵ ặ ặỵ ặ ặỵ ỵẩ ặỵ ỵ [ặ] ỵ ỵ ặ þ [Ỉ ] þ Á 561 Appendix Vector coupling coefficients j1 ¼ j2 ¼ 12 jjmji mj1 mj2 j1,1i 2 À 12 À 12 À 12 À 12 j1,0i j0,0i p1 p1 p21 p2 À 12 2 j1 ¼ 1, j2 ¼ 12 jjmji mj1 mj2 j 32 ; 32i 1 À 12 À 12 À 12 1 0 À1 À1 j 32 ; 12i j 12 ; 12i q mj2 1 0 1 À1 0 À1 À1 À1 À1 À1 j2; 2i j 32 ; À 12i j 12 ; À 12i q q j 32 ; À 32i q q3 q3 À 13 q3 q3 À 23 j1 ¼ 1, j2 ¼ mj1 j1,À1i jjmji j2; 1i j1; 1i p1 p1 p21 p2 À 12 j2; 0i j1; 0i p1 p 62 p 31 p1 j0; 0i j2; À1i j1; À1i j2; À2i p1 p3 À 13 p1 p1 À p1 p1 p21 p 2 Answers to selected problems (a) 6.626  10À19 J, (b) 6.626  10À20 J, (c) 6.626  10À34 J 0.4 6000 K 0.6 (a) 3R(yE/T)2eÀyE/T, (b) 3R 0.8 2.94R, 0.23R 0.9 3.144R 0.10 2.97  1020 0.11 (a) 8.0  105 m sÀ1, (b) no electrons emitted 0.15 RH ¼ 1.097  105 cmÀ1, I ¼ 13.6 eV ¼ hcRH 0.16 (a) 6.6  10À29 m, (b) 7.3  10À40 m, (c) 0.145 nm, (d) (i) 1.23 nm, (ii) 12.3 pm 0.1 1.3 (a) Eigenfunction is (i); (b) eigenfunctions are (i), (iii), (v), (vi) 1.4 (a) À(" h2/2m)(d2/dx2) in one dimension, À("h2/2m)r2 in three dimenP sions, (b) multiplication by (1/x), (c) multiplication by ieiri, (d) ("h/ i){x(q/qy) À y(q/qx)}, (e) multiplication by x À hxi2 , À"h2 (q2 / qx2) À hpi2 1.6 " h2c2(q2C/qx2) ỵ m2c4C ẳ "h2(q2C/qt2), probability is not conserved 1.10 No 1.11 (a) 0, (b) 0, (c) i"h, (d) 2i"hx, (e) ni" hx n À1 ( a ) "h/ ( i x ) , ( b ) ( "h/ x )  (" h À ixp x ), (c) i" h(zp x À xp z ), (d) 2x (q2 /qxqy) À 2xy(q2 /qy ) 1.14 " h2 l z 1.17 (a) i" h(qV/qx), (b) (" h/im)p x ; For (i) (a) 0, (b) ("h/ im)px; For (ii) (a) i" hkx, (b) (" h/im)px; For (iii) (a) À(i"he2/4pe0)(x/r3), (b) (" h/im)px 1.21 (d/dt)hxi ¼ (1/m)hpxi, (d/dt)hpxi¼ Àkhxi 1.23 Eigenvalues (v ỵ 12)" ho 1.24 (" h/2)2(2v2 ỵ 2v ỵ 3) 1.27 N ẳ (b3/p)1/2 p 1.28 N ¼ (1/G p)1/2, 0.8427 1.29 ỈG 1.30 (a) 2.1  10À6 pmÀ3, (b) 2.9  10À7 pmÀ3; 2.1  10À6, 2.9  10À7 1.31 (a) 0.323, (b) 141 pm 2.1 (a) (i) A exp{5.123i(x/nm)}, (ii) A exp{512.3i(x/nm)}, (b) A exp{9.48  1031i (x/m)} 2.2 A2 ¼ 1/L; L ! 2.8 4g2/ {4g2 ỵ (1 g2)2sin2k L} where g ¼ k/k with k2 ¼ 2mE/"h2 and k ¼ 2m(E À V)/" h2 2.9 (a) 1, (b) {(E À V)1/2 À E1/2)}/{(E V)1/2ỵ 1/2 E )} 2.10 (a) for all n, (b) (14){1 À (2/pn) sin(np/2)}; 0.09085 for n ¼ 1, (c) (2/L){dx À (À1) n (L/2pn) sin(2npdx/L)}; (2/L){dx ỵ p L=2pị sin2pdx=Lị for n ẳp1: 2.11 lC/ 2.12 (a) n2h2/4mL3, 2 1/2 (b) p 0.49 pm 2.14 Dx ¼ (L/2 3){1 À (6/n p ) ; as n ! 1, Dx ! L/ 2.15 hpi ¼ 0, php2i ¼ n2h2/4L2, Dp ¼ nh/2L For general n, we have DxDp ¼ (np/ 3){1 À (6/n2 p2)} 1/2("h/2); For n ¼ 1, we get DxDp ¼ 1.1357(" h/2) 2.19 En ¼ n2h2/8meL2, l/nm ¼ 3.297  10À3 (R CC /pm) (N À 1) /(N ỵ 1) 2.20 (b) Cẳ2=Lị3=2 sinnx px=Lị sinny py=Lịsinnz pz=Lị, Eẳn2x ỵn2y ỵn2z ịh2 =8mL2 ị; (d) 2.27 4.57  10À20 J, 4.35  10À6 m 2.28 (a) 0.171, (b) 0.617 2.29 (a) 0, (b) (12)" ho/k, (c) 0, (d) (12)" hk/o, DxDp ¼ "h/2 E ¼ ð1:30  1022 Jịm2l ; 1:53mm: 3.5 E ẳ (2.2 1065 J) m2l , À1.5  1033 3.7 (a) N ¼ 1/(2pI0(2) )1/2 ¼ 0.2642, (b) 0, 0, 0.698"h 3:8 N ¼ 1=ð2pI0 ð2aÞÞ1=2 , hlz i ¼ a"hfI1 ð2aÞ=I0 ð2aÞg 3.9 Y00 sin yỵ Y sin y cos y À {m2l À (2IE/" h2)sin2y}Y ¼ 3.14 l ¼ 0, E ¼ 0, nondegenerate; l ¼ 1, E ¼ 2.60  10À22 J, triply degenerate; l ¼ 2, E ¼ 7.80  10À22 J, five-fold degenerate; 0.764 nm 3.15 arccos [ml/ {l(l ỵ 1)}1/2]; With angles in degrees: For l ¼ 1, 45, 90, 135; For l ¼ 2, 35.3, 65.9, 90, 114.1, 144.7; For l ¼ 3, 30, 54.7, 73.2, 90, 106.8, 125.3, 150 3.17 hTi ¼ ÀE1s, hVi ¼ 2E1s, hTi ¼ ( À 12)hVi 3.19 (a) 2a, p (b) (3Ỉ 3)(3a/2) 3.20 For 1s, (a) 3a/2Z, (b) 3(a/Z)2, (c) a/Z; For 2s, (a) 6a/Z, (b) 42(a/Z)2, (c) 5.24a/Z; For 3s, (a) 27a/2Z, (b) 2.07(a/ Z)2, (c) 13.07a/Z 3.24 For 1s, (1/p)(Z/a)3; For 2s, (1/8p)(Z/a)3; For 3s, (1/27p)(Z/a)3 3.25 (1/24)(Z/a)3 3.26 À0.357 kJ molÀ1 3.1 i" hlz 4.2 (a) i" hlz ly ỵ ly lz ị, (b) i" hlx lz ly ỵ lx ly lz ỵ lz ly lx ỵ ly lz lx ị, (c) " h2 ly : 4.4 Upon expansion of the determinant, l l ẳ i(lylz lzly) ỵ j(lxlz lzlx) þ k(lxly À lylx), which is then identified (term by term) with i" hl 4.6 (a) ½sx , sy Š ¼ i" hsz , (b) eigenvalues of s2 ¼ s2x ỵ s2y ỵ s2z are 34"h2 ẳ ss ỵ 1ị" h2 : 4.7 (a) i" hsz/2, (b) (" h/2)4sx, (c) (" h/2)6 4.9 lỵ would be a lowering operator and lÀ a raising p p h2, (e) 6" operator 4.10 (a) 0, (b) " h 6, (c) 2" h2 6, (d) 6" h2, (f) 48" h5 4.12 (a) Ài" h, (b) 0, (c) Ài" h, (d) i" h, (e) 4.19 (a) 7, 6, , 1; (b)(i) 2,1,0, (b)(ii) 4,3,2,1,0, (b)(iii) 3,2,1, (c) 2,1,1,1,0,0 4.25 hG,MLjl1zjG,MLi ¼ ML " h/2 4.1 (a) C2v, (b) D1h, (c) D2h, (d) C2v, (e) C2h, (f) D6h, (g) D2h, (h) C1, (i) C3h 5.2 (a), (d), and (h) 5.5 3A1 ỵ B1 ỵ 2B2; For A1: 1 2(H1sA ỵ H1sB), O2s, O2pz; For B1: O2px; For B2: 2(H1sB À H1sA), O2py 5.8 A1 ỵ T2; For A1: H1sA ỵ H1sB þ H1sC þ H1sD; For T2: H 1s A À H 1s B H1s C ỵ H1s D , H1s A ỵ H1s B H1s C H1s D , H1sA H1sB ỵ H1sC H1sD 5.9 (a) A1, (b) E, (c) E2, (d) A1 ỵ A2 þ E2, 2, (e) A1 þ A2 þ 2E þ 2T1 ỵ 2T2 5.11 A1 ỵ A2 ỵ B1 ỵ B2 5.13 3A1 ỵ 2A2 ỵ 2B ỵ 3B2 5.14 A ỵ B ỵ E ỵ E 2; In D 6h, it is A2u ỵ B2g þ E1g þ E2u 5.16 (a) 1A2, 3A2; (b)(i) 1E, 3E, (ii) 1A1, 3A2, 1E; (c)(i) 1E, 3E, (ii) 1T1, 3T1, 1T2, 3T2, (iii) 1A2, 3A2, 1E, 3E, 1T1, 3T1, 1T2, T2, (iv) 1A1, 1E, 1T2, 3T1, (v) 1A1, 1E, 1T2, 3T1; (d)(i) 1A1, 3A2, 1E, (ii) T1, 3T1, 1T2, 3T2, (iii) 1A1, 1E, 3T1, 1T2 5.20 (can be increased by accidental degeneracies) 5.22 A1 ỵ E 5.1 (a) 25 739.45 cmÀ1 (99.998% C1), 50 267.29 cmÀ1 (99.998% C2 ); (b) 25 699.16 cm À1 (99.835% C1 ), 50 307.58 cm À1 (99.835% C2); (c) 25 759.74 cmÀ1 (96.300% C1), 51 246.99 cmÀ1 (96.300% C2) 6.2 (a) À74.8 eV, (b) 20.4 eV 6.4 E(1) ¼ mgL/2, E(1)/ L ¼ 4.47  10À30 J mÀ1 6.5 With a ¼ mgL/(h2/8mL2), we have E ( ) ¼ À 0.010 83amgL, C( ) ẳ a{0.0600C2 ỵ 0.00096C4 ỵ 0.000 13C6 ỵ } 6.8 (a) dxz, (b) dz2, (c) fxyz 6.9 (a) B1, (b) B2 6.11 E(2) ¼ À 0.029 49e2/DE; DE ¼ 8.15(h2/8mL2) 6.13 (a) k ¼ p/L, E ¼ h2/8mL2, (b) kL ¼ 1.1331, E ¼ h2/7.9997mL2, (c) trial function inadmissible as first derivative is discontinuous p p p 6.14 12(s A À 2s B þ s C ), E ¼ a À b 2; 1/ 2(s A À s C ), E ¼ a; p p ( s ỵ s ỵ s ) , E ẳ a ỵ b at all times B C A P ( t ) % s i n ( mB bt / 0 "h) , n s V / o2 22 A ẳ 29 =37 ịpa5 c=lC ịZ4 , rB ẳ ð29 =37 Þðpa5 c=lC ÞZ4  expðÀ3hcRZ2 =4kTÞ: 6.23 A / 1/L4, B / L2 6.1 (4.3889  10 cm À1 )(1/4 À 1/n ) 7.3 (1.092  10 cm À1 ) ð1=n21 À 1=n22 Þ 7.4 (b), (c), and (e) 7.5 B4ỵ, 3.283 104 kJ molÀ1 7.9 Eso(j) À Eso(j À 1) ¼ jhcznl 7.10 z3d,mean ¼ 95.6 cmÀ1 7.12 Li 2S1/ B e S ; B E ( P / )