IDEAS OF QUANTUM CHEMISTRY pdf

IDEAS OF
QUANTUM CHEMISTRY
IDEAS OF
QUANTUM CHEMISTRY
by
LUCJAN PIELA
Department of Chemistry University of Warsaw
Warsaw, Poland
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ELSEVIER
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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
X
Contents
6.8.3 Approximation:decouplingofrotationandvibrations 244
6.8.4 The kinetic energy operators of translation, rotation and vibrations . 245
6.8.5 Separationoftranslational,rotationalandvibrationalmotions 246
6.9 Non-boundstates 247
6.10Adiabatic,diabaticandnon-adiabaticapproaches 252
6.11Crossingofpotentialenergycurvesfordiatomics 255
6.11.1Thenon-crossingrule 255
6.11.2 Simulating the harpooning effect in the NaCl molecule 257
6.12Polyatomicmoleculesandconicalintersection 260
6.12.1Conicalintersection 262
6.12.2Berryphase 264
6.13Beyondtheadiabaticapproximation 268
6.13.1Muoncatalyzednuclearfusion 268
6.13.2“Russiandolls”–oramoleculewithinmolecule 270
7. MotionofNuclei 275

7.1 Rovibrational spectra – an example of accurate calculations: atom – di-
atomicmolecule 278
7.1.1 CoordinatesystemandHamiltonian 279
7.1.2 Anisotropy of the potential V 280
7.1.3 Addingtheangularmomentainquantummechanics 281
7.1.4 ApplicationoftheRitzmethod 282
7.1.5 Calculationofrovibrationalspectra 283
7.2 Forcefields(FF) 284
7.3 LocalMolecularMechanics(MM) 290
7.3.1 Bondsthatcannotbreak 290
7.3.2 Bondsthatcanbreak 291
7.4 Globalmolecularmechanics 292
7.4.1 Multipleminimacatastrophe 292
7.4.2 Isittheglobalminimumwhichcounts? 293
7.5 Smallamplitudeharmonicmotion–normalmodes 294
7.5.1 Theoryofnormalmodes 295
7.5.2 Zero-vibrationenergy 303
7.6 MolecularDynamics(MD) 304
7.6.1 TheMDidea 304
7.6.2 WhatdoesMDofferus? 306
7.6.3 Whattoworryabout? 307
7.6.4 MD of non-equilibrium processes 308
7.6.5 Quantum-classicalMD 308
7.7 Simulatedannealing 309
7.8 LangevinDynamics 310
7.9 MonteCarloDynamics 311
7.10Car–Parrinellodynamics 314
7.11Cellularautomata 317
8. ElectronicMotionintheMeanField:AtomsandMolecules 324
8.1 Hartree–Fock method – a bird’s eye view 329

8.1.1 Spinorbitals 329
Contents
XI
8.1.2 Variables 330
8.1.3 Slaterdeterminants 332
8.1.4 What is the Hartree–Fock method all about? 333
8.2 TheFockequationforoptimalspinorbitals 334
8.2.1 DiracandCoulombnotations 334
8.2.2 Energyfunctional 334
8.2.3 Thesearchfortheconditionalextremum 335
8.2.4 ASlaterdeterminantandaunitarytransformation 338
8.2.5 Invariance of the
ˆ
J and
ˆ
K operators 339
8.2.6 DiagonalizationoftheLagrangemultipliersmatrix 340
8.2.7 The Fock equation for optimal spinorbitals (General Hartree–Fock
method–GHF) 341
8.2.8 The closed-shell systems and the Restricted Hartree–Fock (RHF)
method 342
8.2.9 Iterative procedure for computing molecular orbitals: the Self-
ConsistentFieldmethod 350
8.3 Total energy in the Hartree–Fock method 351
8.4 Computational technique: atomic orbitals as building blocks of the molecu-
larwavefunction 354
8.4.1 Centringoftheatomicorbital 354
8.4.2 Slater-typeorbitals(STO) 355
8.4.3 Gaussian-typeorbitals(GTO) 357
8.4.4 LinearCombinationofAtomicOrbitals(LCAO)Method 360

8.4.5 BasissetsofAtomicOrbitals 363
8.4.6 The Hartree–Fock–Roothaan method (SCF LCAO MO) 364
8.4.7 PracticalproblemsintheSCFLCAOMOmethod 366
RESULTS OF THE HARTREE–FOCK METHOD 369
8.5 Backtofoundations 369
8.5.1 WhendoestheRHFmethodfail? 369
8.5.2 Fukutomeclasses 372
8.6 MendeleevPeriodicTableofChemicalElements 379
8.6.1 Similartothehydrogenatom–theorbitalmodelofatom 379
8.6.2 Yettherearedifferences 380
8.7 Thenatureofthechemicalbond 383
8.7.1 H
+
2
intheMOpicture 384
8.7.2 Canweseeachemicalbond? 388
8.8 Excitation energy, ionization potential, and electron affinity (RHF approach) 389
8.8.1 Approximateenergiesofelectronicstates 389
8.8.2 Singletortripletexcitation? 391
8.8.3 Hund’srule 392
8.8.4 Ionization potential and electron affinity (Koopmans rule) 393
8.9 LocalizationofmolecularorbitalswithintheRHFmethod 396
8.9.1 Theexternallocalizationmethods 397
8.9.2 Theinternallocalizationmethods 398
8.9.3 Examplesoflocalization 400
8.9.4 Computationaltechnique 401
8.9.5 The σ, π, δ bonds 403
8.9.6 Electronpairdimensionsandthefoundationsofchemistry 404
8.9.7 Hybridization 407
XII

Contents
8.10Aminimalmodelofamolecule 417
8.10.1ValenceShellElectronPairRepulsion(VSEPR) 419
9. ElectronicMotionintheMeanField:PeriodicSystems 428
9.1 Primitivelattice 431
9.2 Wavevector 433
9.3 Inverselattice 436
9.4 FirstBrillouinZone(FBZ) 438
9.5 PropertiesoftheFBZ 438
9.6 AfewwordsonBlochfunctions 439
9.6.1 Wavesin1D 439
9.6.2 Wavesin2D 442
9.7 Theinfinitecrystalasalimitofacyclicsystem 445
9.8 Atripleroleofthewavevector 448
9.9 Bandstructure 449
9.9.1 Born–vonKármánboundaryconditionin3D 449
9.9.2 CrystalorbitalsfromBlochfunctions(LCAOCOmethod) 450
9.9.3 SCFLCAOCOequations 452
9.9.4 Bandstructureandbandwidth 453
9.9.5 Fermi level and energy gap: insulators, semiconductors and metals . 454
9.10Solidstatequantumchemistry 460
9.10.1Whydosomebandsgoup? 460
9.10.2Whydosomebandsgodown? 462
9.10.3Whydosomebandsstayconstant? 462
9.10.4Howcanmorecomplexbehaviourbeexplained? 462
9.11 The Hartree–Fock method for crystals 468
9.11.1Secularequation 468
9.11.2IntegrationintheFBZ 471
9.11.3Fockmatrixelements 472
9.11.4Iterativeprocedure 474

9.11.5Totalenergy 474
9.12Long-rangeinteractionproblem 475
9.12.1Fockmatrixcorrections 476
9.12.2Totalenergycorrections 477
9.12.3MultipoleexpansionappliedtotheFockmatrix 479
9.12.4Multipoleexpansionappliedtothetotalenergy 483
9.13Backtotheexchangeterm 485
9.14Choiceofunitcell 488
9.14.1Fieldcompensationmethod 490
9.14.2Thesymmetryofsubsystemchoice 492
10.CorrelationoftheElectronicMotions 498
VARIATIONAL METHODS USING EXPLICITLY CORRELATED WAVE FUNC-
TION 502
10.1 Correlationcuspcondition 503
10.2 TheHylleraasfunction 506
10.3 TheHylleraasCImethod 506
10.4 Theharmonicheliumatom 507
Contents
XIII
10.5 James–CoolidgeandKołos–Wolniewiczfunctions 508
10.5.1 Neutrinomass 511
10.6 MethodofexponentiallycorrelatedGaussianfunctions 513
10.7 Coulombhole(“correlationhole”) 513
10.8 Exchangehole(“Fermihole”) 516
VARIATIONAL METHODS WITH SLATER DETERMINANTS 520
10.9 Valencebond(VB)method 520
10.9.1 Resonancetheory–hydrogenmolecule 520
10.9.2 Resonancetheory–polyatomiccase 523
10.10Configurationinteraction(CI)method 525
10.10.1Brillouintheorem 527

10.10.2ConvergenceoftheCIexpansion 527
10.10.3 Example of H
2
O 528
10.10.4Whichexcitationsaremostimportant? 529
10.10.5Naturalorbitals(NO) 531
10.10.6Sizeconsistency 532
10.11DirectCImethod 533
10.12MultireferenceCImethod 533
10.13MulticonfigurationalSelf-ConsistentFieldmethod(MCSCF) 535
10.13.1ClassicalMCSCFapproach 535
10.13.2UnitaryMCSCFmethod 536
10.13.3Completeactivespacemethod(CASSCF) 538
NON-VARIATIONAL METHODS WITH SLATER DETERMINANTS 539
10.14 Coupled cluster (CC) method 539
10.14.1Waveandclusteroperators 540
10.14.2RelationshipbetweenCIandCCmethods 542
10.14.3SolutionoftheCCequations 543
10.14.4 Example: CC with double excitations 545
10.14.5SizeconsistencyoftheCCmethod 547
10.15Equation-of-motionmethod(EOM-CC) 548
10.15.1Similaritytransformation 548
10.15.2DerivationoftheEOM-CCequations 549
10.16Manybodyperturbationtheory(MBPT) 551
10.16.1UnperturbedHamiltonian 551
10.16.2Perturbationtheory–slightlydifferentapproach 552
10.16.3 Reduced resolvent or the “almost” inverse of (E
(0)
0
−

ˆ
H
(0)
) 553
10.16.4MBPTmachinery 555
10.16.5Brillouin–Wignerperturbationtheory 556
10.16.6Rayleigh–Schrödingerperturbationtheory 557
10.17Møller–PlessetversionofRayleigh–Schrödingerperturbationtheory 558
10.17.1ExpressionforMP2energy 558
10.17.2ConvergenceoftheMøller–Plessetperturbationseries 559
10.17.3 Special status of double excitations 560
11.ElectronicMotion:DensityFunctionalTheory(DFT) 567
11.1 Electronicdensity–thesuperstar 569
11.2 Baderanalysis 571
11.2.1 Overall shape of ρ 571
XIV
Contents
11.2.2 Criticalpoints 571
11.2.3 Laplacianoftheelectronicdensityasa“magnifyingglass” 575
11.3 TwoimportantHohenberg–Kohntheorems 579
11.3.1 Equivalence of the electronic wave function and electron density . 579
11.3.2 Existence of an energy functional minimized by ρ
0
580
11.4 TheKohn–Shamequations 584
11.4.1 TheKohn–Shamsystemofnon-interactingelectrons 584
11.4.2 Totalenergyexpression 585
11.4.3 DerivationoftheKohn–Shamequations 586
11.5 What to take as the DFT exchange–correlation energy E
xc

? 590
11.5.1 Localdensityapproximation(LDA) 590
11.5.2 Non-localapproximations(NLDA) 591
11.5.3 The approximate character of the DFT vs apparent rigour of ab
initio computations 592
11.6 Onthephysicaljustificationfortheexchangecorrelationenergy 592
11.6.1 Theelectronpairdistributionfunction 592
11.6.2 Thequasi-staticconnectionoftwoimportantsystems 594
11.6.3 Exchange–correlation energy vs 
aver
596
11.6.4 Electronholes 597
11.6.5 Physicalboundaryconditionsforholes 598
11.6.6 Exchangeandcorrelationholes 599
11.6.7 PhysicalgroundsfortheDFTapproximations 601
11.7 ReflectionsontheDFTsuccess 602
12.TheMoleculeinanElectricorMagneticField 615
12.1 Hellmann–Feynmantheorem 618
ELECTRIC PHENOMENA 620
12.2 The molecule immobilized in an electric field 620
12.2.1 Theelectricfieldasaperturbation 621
12.2.2 Thehomogeneouselectricfield 627
12.2.3 The inhomogeneous electric field: multipole polarizabilities and
hyperpolarizabilities . 632
12.3 Howtocalculatethedipolemoment 633
12.3.1 Hartree–Fock approximation 633
12.3.2 Atomicandbonddipoles 634
12.3.3 WithintheZDOapproximation 635
12.4 How to calculate the dipole polarizability 635
12.4.1 SumOverStatesMethod 635

12.4.2 Finitefieldmethod 639
12.4.3 Whatisgoingonathigherelectricfields 644
12.5 A molecule in an oscillating electric field 645
MAGNETIC PHENOMENA 647
12.6 Magneticdipolemomentsofelementaryparticles 648
12.6.1 Electron 648
12.6.2 Nucleus 649
12.6.3 Dipolemomentinthefield 650
12.7 Transitions between the nuclear spin quantum states – NMR technique . . 652
12.8 Hamiltonianofthesystemintheelectromagneticfield 653
Contents
XV
12.8.1 Choiceofthevectorandscalarpotentials 654
12.8.2 RefinementoftheHamiltonian 654
12.9 EffectiveNMRHamiltonian 658
12.9.1 Signalaveraging 658
12.9.2 EmpiricalHamiltonian 659
12.9.3 Nuclearspinenergylevels 664
12.10TheRamseytheoryoftheNMRchemicalshift 666
12.10.1Shieldingconstants 667
12.10.2Diamagneticandparamagneticcontributions 668
12.11TheRamseytheoryofNMRspin–spincouplingconstants 668
12.11.1Diamagneticcontributions 669
12.11.2Paramagneticcontributions 670
12.11.3Couplingconstants 671
12.11.4TheFermicontactcouplingmechanism 672
12.12Gaugeinvariantatomicorbitals(GIAO) 673
12.12.1Londonorbitals 673
12.12.2Integralsareinvariant 674
13.IntermolecularInteractions 681

THEORY OF INTERMOLECULAR INTERACTIONS 684
13.1 Interactionenergyconcept 684
13.1.1 Naturaldivisionanditsgradation 684
13.1.2 Whatismostnatural? 685
13.2 Bindingenergy 687
13.3 Dissociationenergy 687
13.4 Dissociationbarrier 687
13.5 Supermolecular approach 689
13.5.1 Accuracyshouldbethesame 689
13.5.2 Basissetsuperpositionerror(BSSE) 690
13.5.3 Good and bad news about the supermolecular method 691
13.6 Perturbationalapproach 692
13.6.1 Intermoleculardistance–whatdoesitmean? 692
13.6.2 Polarizationapproximation(twomolecules) 692
13.6.3 Intermolecularinteractions:physicalinterpretation 696
13.6.4 Electrostatic energy in the multipole representation and the pene-
trationenergy 700
13.6.5 Inductionenergyinthemultipolerepresentation 703
13.6.6 Dispersionenergyinthemultipolerepresentation 704
13.7 Symmetryadaptedperturbationtheories(SAPT) 710
13.7.1 Polarization approximation is illegal 710
13.7.2 Constructingasymmetryadaptedfunction 711
13.7.3 The perturbation is always large in polarization approximation . . 712
13.7.4 Iterative scheme of the symmetry adapted perturbation theory . . 713
13.7.5 Symmetryforcing 716
13.7.6 A link to the variational method – the Heitler–London interaction
energy 720
13.7.7 When we do not have at our disposal the ideal ψ
A0
and ψ

B0
. . 720
13.8 Convergenceproblems 721
XVI
Contents
13.9 Non-additivityofintermolecularinteractions 726
13.9.1 Many-bodyexpansionofinteractionenergy 727
13.9.2 Additivityoftheelectrostaticinteraction 730
13.9.3 Exchangenon-additivity 731
13.9.4 Inductionenergynon-additivity 735
13.9.5 Additivityofthesecond-orderdispersionenergy 740
13.9.6 Non-additivityofthethird-orderdispersioninteraction 741
ENGINEERING OF INTERMOLECULAR INTERACTIONS 741
13.10Noblegasinteraction 741
13.11VanderWaalssurfaceandradii 742
13.11.1PaulihardnessofthevanderWaalssurface 743
13.11.2Quantumchemistryofconfinedspace–thenanovessels 743
13.12Synthonsandsupramolecularchemistry 744
13.12.1Boundornotbound 745
13.12.2 Distinguished role of the electrostatic interaction and the valence
repulsion 746
13.12.3Hydrogenbond 746
13.12.4Coordinationinteraction 747
13.12.5Hydrophobiceffect 748
13.12.6Molecularrecognition–synthons 750
13.12.7“Key-lock”,templateand“hand-glove”synthoninteractions 751
14.IntermolecularMotionofElectronsandNuclei:ChemicalReactions 762
14.1 Hypersurfaceofthepotentialenergyfornuclearmotion 766
14.1.1 Potentialenergyminimaandsaddlepoints 767
14.1.2 Distinguishedreactioncoordinate(DRC) 768

14.1.3 Steepestdescentpath(SDP) 769
14.1.4 Ourgoal 769
14.1.5 Chemicalreactiondynamics(apioneers’approach) 770
14.2 Accuratesolutionsforthereactionhypersurface(threeatoms) 775
14.2.1 CoordinatesystemandHamiltonian 775
14.2.2 SolutiontotheSchrödingerequation 778
14.2.3 Berryphase 780
14.3 Intrinsicreactioncoordinate(IRC)orstatics 781
14.4 ReactionpathHamiltonianmethod 783
14.4.1 EnergyclosetoIRC 783
14.4.2 Vibrationallyadiabaticapproximation 785
14.4.3 Vibrationallynon-adiabaticmodel 790
14.4.4 Application of the reaction path Hamiltonian method to the reac-
tion H
2
+OH →H
2
O +H 792
14.5 Acceptor–donor(AD)theoryofchemicalreactions 798
14.5.1 Mapsofthemolecularelectrostaticpotential 798
14.5.2 Wheredoesthebarriercomefrom? 803
14.5.3 MO,ADandVBformalisms 803
14.5.4 Reactionstages 806
14.5.5 Contributionsofthestructuresasreactionproceeds 811
14.5.6 Nucleophilic attack H
−
+ ETHYLENE → ETHYLENE + H
−
. . 816
14.5.7 Electrophilic attack H

+
+ H
2
→ H
2
+ H
+
818
Contents
XVII
14.5.8 Nucleophilic attack on the polarized chemical bond in the VB pic-
ture 818
14.5.9 Whatisgoingoninthechemist’sflask? 821
14.5.10Roleofsymmetry 822
14.5.11Barriermeansacostofopeningtheclosed-shells 826
14.6 Barrierfortheelectron-transferreaction 828
14.6.1 Diabaticandadiabaticpotential 828
14.6.2 Marcustheory 830
15.InformationProcessing–theMissionofChemistry 848
15.1 Complexsystems 852
15.2 Self-organizingcomplexsystems 853
15.3 Cooperative interactions 854
15.4 Sensitivityanalysis 855
15.5 Combinatorialchemistry–molecularlibraries 855
15.6 Non-linearity 857
15.7 Attractors 858
15.8 Limitcycles 859
15.9 Bifurcationsandchaos 860
15.10Catastrophes 862
15.11Collectivephenomena 863

15.11.1Scalesymmetry(renormalization) 863
15.11.2Fractals 865
15.12Chemicalfeedback–non-linearchemicaldynamics 866
15.12.1Brusselator–dissipativestructures 868
15.12.2Hypercycles 873
15.13Functionsandtheirspace-timeorganization 875
15.14Themeasureofinformation 875
15.15Themissionofchemistry 877
15.16Molecularcomputersbasedonsynthoninteractions 878
APPENDICES 887
A. AREMAINDER:MATRICESANDDETERMINANTS 889
1.Matrices 889
2.Determinants 892
B. AFEWWORDSONSPACES,VECTORSANDFUNCTIONS 895
1.Vectorspace 895
2.Euclideanspace 896
3.Unitaryspace 897
4.Hilbertspace 898
5.Eigenvalueequation 900
C. GROUPTHEORYINSPECTROSCOPY 903
1.Group 903
2.Representations 913
XVIII
Contents
3.Grouptheoryandquantummechanics 924
4.Integralsimportantinspectroscopy 929
D. ATWO-STATEMODEL 948
E. DIRACDELTAFUNCTION 951
1. Approximations to δ(x) 951
2. Properties of δ(x) 953

3.AnapplicationoftheDiracdeltafunction 953
F. TRANSLATION vs MOMENTUM and ROTATION vs ANGULAR MOMENTUM 955
1. The form of the
ˆ
U operator 955
2.TheHamiltoniancommuteswiththetotalmomentumoperator 957
3. The Hamiltonian,
ˆ
J
2
and
ˆ
J
z
docommute 958
4.Rotationandtranslationoperatorsdonotcommute 960
5.Conclusion 960
G. VECTORANDSCALARPOTENTIALS 962
H. OPTIMALWAVEFUNCTIONFORAHYDROGEN-LIKEATOM 969
I. SPACE-ANDBODY-FIXEDCOORDINATESYSTEMS 971
J. ORTHOGONALIZATION 977
1.Schmidtorthogonalization 977
2.Löwdinsymmetricorthogonalization 978
K. DIAGONALIZATIONOFAMATRIX 982
L. SECULAR EQUATION (H −εS)c =0 984
M.SLATER–CONDONRULES 986
N. LAGRANGEMULTIPLIERSMETHOD 997
O. PENALTYFUNCTIONMETHOD 1001
P. MOLECULARINTEGRALSWITHGAUSSIANTYPEORBITALS1s 1004
Q. SINGLETANDTRIPLETSTATESFORTWOELECTRONS 1006

Contents
XIX
R. THE HYDROGEN MOLECULAR ION IN THE SIMPLEST ATOMIC BASIS
SET 1009
S. POPULATIONANALYSIS 1015
T. THEDIPOLEMOMENTOFALONEELECTRONPAIR 1020
U. SECONDQUANTIZATION 1023
V. THE HYDROGEN ATOM IN THE ELECTRIC FIELD – VARIATIONAL AP-
PROACH 1029
W.NMRSHIELDINGANDCOUPLINGCONSTANTS–DERIVATION 1032
1.Shieldingconstants 1032
2.Couplingconstants 1035
X. MULTIPOLEEXPANSION 1038
Y. PAULIDEFORMATION 1050
Z. ACCEPTOR–DONOR STRUCTURE CONTRIBUTIONS IN THE MO CON-
FIGURATION 1058
NameIndex 1065
SubjectIndex 1077
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“toseeaworldinagrainofsand
and Heaven in a wildflower
hold infinity in the palm of your hand
andeternityinanhour ”
William Blake “Auguries of Innocence”
I
NTRODUCTION
Our wonderful world
Colours! The most beautiful of buds – an apple bud in my garden changes colour
from red to rosy after a few days. Why? It then explodes into a beautiful pale rosy
flower. After a few months what was once a flower looks completely different: it

has become a big, round and red apple. Look at the apple skin. It is pale green,
but moving along its surface the colour changes quite abruptly to an extraordinary
vibrant red. The apple looks quite different when lit by full sunlight, or when placed
in the shade.
Touch the apple, you will feel it smooth as silk.
How it smells! An exotic mixture of subtle scents.
What a taste: a fantastic juicy pulp!
Sounds theamazingmelodyofafinchisrepeatedwithremarkableregularity.
My friend Jean-Marie André says it is the same here as it is in Belgium. The same?
Is there any program that forces finches to make the same sound in Belgium as
in Poland? A woodpecker hits a tree with the regularity of a machine gun, my
Kampinos forest echoes that sound. Has the woodpecker also been programmed?
What kind of program is used by a blackbird couple that forces it to prepare, with
enormous effort and ingenuity, a nest necessary for future events?
What we do know
Our senses connect us to what we call the Universe. Using them we feel its pres-
ence, while at the same time we are a part of it. Sensory operations are the direct
result of interactions, both between molecules and between light and matter. All
of these phenomena deal with chemistry, physics, biology and even psychology.
In these complex events it is impossible to discern precisely where the disciplines
of chemistry, physics, biology, and psychology begin and end. Any separation of
these domains is artificial. The only reason for making such separations is to focus
XXI
XXII
Introduction
our attention on some aspects of one indivisible phenomenon. Touch, taste, smell,
sight, hearing, are these our only links and information channels to the Universe?
How little we know about it! To feel that, just look up at the sky. A myriad of stars
around us points to new worlds, which will remain unknown forever. On the other
hand, imagine how incredibly complicated the chemistry of friendship is.

We try to understand what is around us by constructing in our minds pictures
representing a “reality”, which we call models. Any model relies on our perception
of reality (on the appropriate scale of masses and time) emanating from our expe-
rience, and on the other hand, on our ability to abstract by creating ideal beings.
Many such models will be described in this book.
It is fascinating that man is able to magnify the realm of his senses by using so-
phisticated tools, e.g., to see quarks sitting in a proton,
1
to discover an amazingly
simple equation of motion
2
that describes both cosmic catastrophes, with an inten-
sity beyond our imagination, as well as the flight of a butterfly. A water molecule
has exactly the same properties in the Pacific as on Mars, or in another galaxy. The
conditions over there may sometimes be quite different from those we have here
in our laboratory, but we assume that if these conditions could be imposed on the
lab, the molecule would behave in exactly the same way. We hold out hope that a
set of universal physical laws applies to the entire Universe.
The set of these basic laws is not yet complete or unified. Given the progress and
important generalizations of physics in the twentieth century, much is currently un-
derstood. For example, forces with seemingly disparate sources have been reduced
to only three kinds:
• those attributed to strong interactions (acting in nuclear matter),
• those attributed to electroweak interactions (the domain of chemistry, biology, as
well as β-decay),
• those attributed to gravitational interaction (showing up mainly in astrophysics).
Many scientists believe other reductions are possible, perhaps up to a single
fundamental interaction, one that explains Everything (quoting Feynman: the frogs
as well as the composers). This assertion is based on the conviction, supported by
developments in modern physics, that the laws of nature are not only universal, but

simple.
Which of the three basic interactions is the most important? This is an ill con-
ceived question. The answer depends on the external conditions imposed (pres-
sure, temperature) and the magnitude of the energy exchanged amongst the in-
teracting objects. A measure of the energy exchanged
3
may be taken to be the
percentage of the accompanying mass deficiency according to Einstein’s relation
E =mc
2
. At a given magnitude of exchanged energies some particles are stable.
1
Aprotonis10
15
times smaller than a human being.
2
Acceleration is directly proportional to force. Higher derivatives of the trajectory with respect to time
do not enter this equation, neither does the nature or cause of the force. The equation is also invariant
with respect to any possible starting point (position, velocity, and mass). What remarkable simplicity
and generality (within limits, see Chapter 3)!
3
This is also related to the areas of operation of particular branches of science.
Introduction
XXIII
Strong interactions produce the huge pressures that accompany the gravitational
collapse of a star and lead to the formation of neutron stars, where the mass de-
ficiency approaches 40%. At smaller pressures, where individual nuclei may exist
and undergo nuclear reactions (strong interactions
4
), the mass deficiency is of the

order of 1%. At much smaller pressures the electroweak forces dominate, nuclei
are stable, atomic and molecular structures emerge. Life (as we know it) becomes
possible. The energies exchanged are much smaller and correspond to a mass de-
ficiency of the order of only about 10
−7
%. The weakest of the basic forces is gravi-
tation. Paradoxically, this force is the most important on the macro scale (galaxies,
stars, planets, etc.). There are two reasons for this. Gravitational interactions share
with electric interactions the longest range known (both decay as 1/r). However,
unlike electric interactions
5
those due to gravitation are not shielded. For this rea-
son the Earth and Moon attract each other by a huge gravitational force
6
while
their electric interaction is negligible. This is how David conquers Goliath, since at
any distance electrons and protons attract each other by electrostatic forces, about
40 orders of magnitude stronger than their gravitational attraction.
Gravitation does not have any measurable influence on the collisions of mole-
cules leading to chemical reactions, since reactions are due to much stronger elec-
tric interactions.
7
A narrow margin
Due to strong interactions, protons overcome mutual electrostatic repulsion and
form (together with neutrons) stable nuclei leading to the variety of chemical ele-
ments. Therefore, strong interactions are the prerequisite of any chemistry (except
hydrogen chemistry). However, chemists deal with already prepared stable nuclei
8
and these strong interactions have a very small range (of about 10
−13

cm) as com-
pared to interatomic distances (of the order of 10
−8
cm). This is why a chemist
may treat nuclei as stable point charges that create an electrostatic field. Test tube
conditions allow for the presence of electrons and photons, thus completing the
set of particles that one might expect to see (some exceptions are covered in this
book). This has to do with the order of magnitude of energies exchanged (under
the conditions where we carry out chemical reactions, the energies exchanged ex-
clude practically all nuclear reactions).
4
With a corresponding large energy output; the energy coming from the fusion D +D→He taking
place on the Sun makes our existence possible.
5
In electrostatic interactions charges of opposite sign attract each other while charges of the same
sign repel each other (Coulomb’s law). This results in the fact that large bodies (built of a huge num-
ber of charged particles) are nearly electrically neutral and interact electrically only very weakly. This
dramatically reduces the range of their electrical interactions.
6
Huge tides and deformations of the whole Earth are witness to that.
7
It does not mean that gravitation has no influence on reagent concentration. Gravitation controls the
convection flow in liquids and gases (and even solids) and therefore a chemical reaction or even crystal-
lization may proceed in a different manner on the Earth’s surface, in the stratosphere, in a centrifuge
or in space.
8
At least in the time scale of a chemical experiment. Instability of some nuclei is used in nuclear
chemistry and radiation chemistry.
XXIV
Introduction

On the vast scale of attainable temperatures
9
chemical structures may exist in
the narrow temperature range of 0 K to thousands of K. Above this range one
has plasma, which represents a soup made of electrons and nuclei. Nature, in its
vibrant living form, requires a temperature range of about 200–320 K, a margin
of only 120 K. One does not require a chemist for chemical structures to exist.
However, to develop a chemical science one has to have a chemist. This chemist
can survive a temperature range of 273 K ±50 K, i.e. a range of only 100 K. The
reader has to admit that a chemist may think of the job only in the narrow range
10
of 290–300 K, only 10 K.
A fascinating mission
Suppose our dream comes true and the grand unification of the three remaining
basic forces is accomplished one day. We would then know the first principles of
constructing everything. One of the consequences of such a feat would be a cat-
alogue of all the elementary particles. Maybe the catalogue would be finite, per-
haps it would be simple.
11
We might have a catalogue of the conserved symme-
tries (which seem to be more elementary than the particles). Of course, knowing
such first principles would have an enormous impact on all the physical sciences.
It could, however, create the impression that everything is clear and that physics is
complete. Even though structures and processes are governed by first principles,
it would still be very difficult to predict their existence by such principles alone.
The resulting structures would depend not only on the principles, but also on the
initial conditions, complexity, self-organization, etc.
12
Therefore, if it does happen,
the Grand Unification will not change the goals of chemistry.

Chemistry currently faces the enormous challenge of information processing,
quite different to this posed by our computers. This question is discussed in the
last chapter of this book.
BOOK GUIDELINES
TREE
Any book has a linear appearance, i.e. the text goes from page to page and the page
numbers remind us of that. However, the logic of virtually any book is non-linear,
and in many cases can be visualized by a diagram connecting the chapters that
9
Millions of degrees.
10
The chemist may enlarge this range by isolation from the specimen.
11
None of this is certain. Much of elementary particle research relies on large particle accelerators.
This process resembles discerning the components of a car by dropping it from increasing heights from
a large building. Dropping it from the first floor yields five tires and a jack. Dropping from the second
floor reveals an engine and 11 screws of similar appearance. Eventually a problem emerges: after land-
ing from a very high floor new components appear (having nothing to do with the car) and reveal that
some of the collision energy has been converted to the new particles!
12
The fact that Uncle John likes to drink coffee with cream at 5 p.m. possibly follows from first princi-
ples, but it would be very difficult to trace that dependence.