Ideas of Quantum Chemistry P1 ppsx

IDEAS OF
QUANTUM CHEMISTRY
IDEAS OF
QUANTUM CHEMISTRY
by
LUCJAN PIELA
Department of Chemistry University of Warsaw
Warsaw, Poland
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First edition 2007
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C

ONTENTS
Introduction XXI
1. TheMagicofQuantumMechanics 1
1.1 Historyofarevolution 4
1.2 Postulates 15
1.3 TheHeisenberguncertaintyprinciple 34
1.4 TheCopenhageninterpretation 37
1.5 How to disprove the Heisenberg principle? The Einstein–Podolsky–Rosen
recipe 38
1.6 Istheworldreal? 40
Bilocation 40
1.7 TheBellinequalitywilldecide 43
1.8 Intriguingresultsofexperimentswithphotons 46
1.9 Teleportation 47
1.10Quantumcomputing 49
2. TheSchrödingerEquation 55
2.1 SymmetryoftheHamiltoniananditsconsequences 57
2.1.1 The non-relativistic Hamiltonian and
conservationlaws 57
2.1.2 Invariancewithrespecttotranslation 61
2.1.3 Invariancewithrespecttorotation 63
2.1.4 Invariance with respect to permutation of identical particles (fermi-
onsandbosons) 64
2.1.5 Invarianceofthetotalcharge 64
2.1.6 Fundamentalandlessfundamentalinvariances 65
2.1.7 Invariancewithrespecttoinversion–parity 65
2.1.8 Invariancewithrespecttochargeconjugation 68
2.1.9 Invariance with respect to the symmetry of the nuclear framework . . 68
2.1.10Conservationoftotalspin 69
2.1.11Indicesofspectroscopicstates 69

2.2 Schrödingerequationforstationarystates 70
2.2.1 WavefunctionsofclassQ 73
2.2.2 Boundaryconditions 73
2.2.3 Ananalogy 75
VII
VIII
Contents
2.2.4 Mathematicalandphysicalsolutions 76
2.3 Thetime-dependentSchrödingerequation 76
2.3.1 Evolutionintime 77
2.3.2 Normalizationispreserved 78
2.3.3 ThemeanvalueoftheHamiltonianispreserved 78
2.3.4 Linearity 79
2.4 Evolutionafterswitchingaperturbation 79
2.4.1 Thetwo-statemodel 81
2.4.2 First-orderperturbationtheory 82
2.4.3 Time-independentperturbationandtheFermigoldenrule 83
2.4.4 Themostimportantcase:periodicperturbation 84
3. BeyondtheSchrödingerEquation 90
3.1 Aglimpseofclassicalrelativitytheory 93
3.1.1 Thevanishingofapparentforces 93
3.1.2 The Galilean transformation 96
3.1.3 TheMichelson–Morleyexperiment 96
3.1.4 The Galilean transformation crashes 98
3.1.5 TheLorentztransformation 100
3.1.6 Newlawofaddingvelocities 102
3.1.7 TheMinkowskispace-timecontinuum 104
3.1.8 How do we get E =mc
2
? 106

3.2 Reconciling relativity and quantum mechanics 109
3.3 TheDiracequation 111
3.3.1 TheDiracelectronicsea 111
3.3.2 TheDiracequationsforelectronandpositron 115
3.3.3 Spinorsandbispinors 115
3.3.4 Whatnext? 117
3.3.5 Largeandsmallcomponentsofthebispinor 117
3.3.6 HowtoavoiddrowningintheDiracsea 118
3.3.7 From Dirac to Schrödinger – how to derive the non-relativistic
Hamiltonian? 119
3.3.8 Howdoesthespinappear? 120
3.3.9 Simplequestions 122
3.4 Thehydrogen-likeatominDiractheory 123
3.4.1 Step by step: calculation of the ground state of the hydrogen-like
atomwithinDiractheory 123
3.4.2 Relativisticcontractionoforbitals 128
3.5 Largersystems 129
3.6 Beyond the Dirac equation 130
3.6.1 TheBreitequation 130
3.6.2 Afewwordsaboutquantumelectrodynamics(QED) 132
4. ExactSolutions–OurBeacons 142
4.1 Freeparticle 144
4.2 Particleinabox 145
4.2.1 Boxwithends 145
4.2.2 Cyclicbox 149
Contents
IX
4.2.3 Comparisonoftwoboxes:hexatrieneandbenzene 152
4.3 Tunnelling effect 153
4.3.1 Asinglebarrier 153

4.3.2 Themagicoftwobarriers 158
4.4 The harmonic oscillator 164
4.5 Morse oscillator 169
4.5.1 Morsepotential 169
4.5.2 Solution 170
4.5.3 Comparison with the harmonic oscillator 172
4.5.4 Theisotopeeffect 172
4.5.5 Bondweakeningeffect 174
4.5.6 Examples 174
4.6 Rigidrotator 176
4.7 Hydrogen-likeatom 178
4.8 Harmonicheliumatom(harmonium) 185
4.9 Whatdoallthesesolutionshaveincommon? 188
4.10Beaconsandpearlsofphysics 189
5. TwoFundamentalApproximateMethods 195
5.1 Variationalmethod 196
5.1.1 Variationalprinciple 196
5.1.2 Variationalparameters 200
5.1.3 RitzMethod 202
5.2 Perturbationalmethod 203
5.2.1 Rayleigh–Schrödingerapproach 203
5.2.2 Hylleraasvariationalprinciple 208
5.2.3 Hylleraasequation 209
5.2.4 Convergenceoftheperturbationalseries 210
6. SeparationofElectronicandNuclearMotions 217
6.1 Separationofthecentre-of-massmotion 221
6.1.1 Space-fixedcoordinatesystem(SFCS) 221
6.1.2 Newcoordinates 221
6.1.3 Hamiltonianinthenewcoordinates 222
6.1.4 Afterseparationofthecentre-of-massmotion 224

6.2 Exact(non-adiabatic)theory 224
6.3 Adiabaticapproximation 227
6.4 Born–Oppenheimerapproximation 229
6.5 Oscillations of a rotating molecule 229
6.5.1 Onemoreanalogy 232
6.5.2 The fundamental character of the adiabatic approximation – PES . . 233
6.6 Basicprinciplesofelectronic,vibrationalandrotationalspectroscopy 235
6.6.1 Vibrationalstructure 235
6.6.2 Rotationalstructure 236
6.7 Approximateseparationofrotationsandvibrations 238
6.8 Polyatomicmolecule 241
6.8.1 Kineticenergyexpression 241
6.8.2 SimplifyingusingEckartconditions 243