Ideas of Quantum Chemistry P2 ppt

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