Table Of ContentMolecular Modelling for Beginners
Molecular Modelling
for Beginners
Second Edition
ALAN HINCHLIFFE
The University of Manchester
A John Wiley and Sons, Ltd, Publication
Thiseditionfirstpublished2008
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LibraryofCongressCataloginginPublicationData
Hinchliffe,Alan.
Molecularmodellingforbeginners/AlanHinchliffe.—2nded.
p. cm.
Includesbibliographicalreferencesandindex.
ISBN978-0-470-51313-2(cloth)—ISBN978-0-470-51314-9(pbk.:alk.paper)
1.Molecules—Mathematicalmodels. 2.Molecules—Computersimulation.
I.Title.
QD480.H582008
541(cid:2).22015118—dc22
2008024099
BritishLibraryCataloguinginPublicationData
AcataloguerecordforthisbookisavailablefromtheBritishLibrary
CLOTH:978-0-470-51313-2
PAPER:978-0-470-51314-9
Setin10/12ptTimesbyIntegraSoftwareServicesPvt.Ltd,Pondicherry,India
PrintedandboundinGreatBritainbyAntonyRoweLtd,Chippenham,Wiltshire
Contents
PrefacetotheSecondEdition xv
PrefacetotheFirstEdition xvii
1 ElectricChargesandtheirProperties 1
1.1 PointCharges 1
1.2 Coulomb’sLaw 3
1.3 PairWiseAdditivity 4
1.4 ElectricField 5
1.5 Work 6
1.6 ChargeDistributions 8
1.7 MutualPotentialEnergy,U 8
1.8 RelationshipbetweenForceandMutualPotentialEnergy 9
1.9 ElectricMultipoles 10
1.9.1 ContinuousChargeDistributions 12
1.9.2 ElectricSecondMoment 13
1.9.3 HigherElectricMoments 15
1.10 ElectrostaticPotential 15
1.11 PolarizationandPolarizability 16
1.12 DipolePolarizability 17
1.12.1 PropertiesofPolarizabilities 18
1.13 Many-BodyForces 19
1.14 ProblemSet 20
2 TheForcesbetweenMolecules 21
2.1 PairPotential 21
2.2 MultipoleExpansion 23
2.3 Charge–DipoleInteraction 23
2.4 Dipole–DipoleInteraction 25
2.5 TakingAccountoftheTemperature 26
2.6 InductionEnergy 27
2.7 DispersionEnergy 28
2.8 RepulsiveContributions 29
2.9 CombinationRules 31
2.10 ComparisonwithExperiment 32
2.10.1 GasImperfections 32
2.10.2 MolecularBeams 32
vi Contents
2.11 ImprovedPairPotentials 33
2.12 ANumericalPotential 33
2.13 Site–SitePotentials 34
2.14 ProblemSet 35
References 35
3 BallsonSprings 37
3.1 VibrationalMotion 38
3.2 TheForceLaw 41
3.3 ASimpleDiatomic 41
3.4 ThreeProblems 43
3.5 TheMorsePotential 45
3.6 MoreAdvancedPotentials 46
References 47
4 MolecularMechanics 49
4.1 MoreAboutBallsonSprings 49
4.2 LargerSystemsofBallsonSprings 51
4.3 ForceFields 53
4.4 MolecularMechanics(MM) 53
4.4.1 BondStretching 54
4.4.2 BondBending 54
4.4.3 DihedralMotions 55
4.4.4 Out-of-PlaneAnglePotential(Inversion) 56
4.4.5 NonbondedInteractions 56
4.4.6 CoulombInteractions 57
4.5 ModellingtheSolvent 57
4.6 Time-andMoney-SavingTricks 57
4.6.1 UnitedAtoms 57
4.6.2 Cut-Offs 58
4.7 ModernForceFields 58
4.7.1 VariationsonaTheme 59
4.8 SomeCommercialForceFields 60
4.8.1 Dreiding 60
4.8.2 MM1 60
4.8.3 MM2(ImprovedHydrocarbonForceField) 61
4.8.4 AMBER 61
4.8.5 OPLS(OptimizedPotentialsforLiquidSimulations) 62
4.8.6 Johnson 62
4.8.7 UnifiedForceField(UFF) 63
References 63
5 TheMolecularPotentialEnergySurface 65
5.1 MultipleMinima 65
5.2 SaddlePoints 66
5.3 Characterization 68
Contents vii
5.4 FindingMinima 68
5.5 MultivariateGridSearch 69
5.5.1 UnivariateSearch 69
5.6 DerivativeMethods 70
5.7 First-OrderMethods 70
5.7.1 SteepestDescent 70
5.7.2 ConjugateGradients 71
5.8 Second-OrderMethods 72
5.8.1 Newton–Raphson 72
5.8.2 BlockDiagonalNewton–Raphson 75
5.8.3 Quasi-Newton–Raphson 75
5.8.4 Fletcher–PowellAlgorithm 75
5.9 ChoiceofMethod 76
5.10 TheZ-Matrix 77
5.11 TheEndoftheZ-Matrix 78
5.12 RedundantInternalCoordinates 80
References 80
6 MolecularMechanicsExamples 81
6.1 GeometryOptimization 81
6.2 ConformationSearches 82
6.3 AminoAcids 84
6.3.1 MolecularStructureDatabases 85
6.3.2 The.pdbFileFormat 85
6.4 QSAR 86
6.4.1 AtomicPartialCharges 86
6.4.2 Polarizabilities 87
6.4.3 MolecularVolumeandSurfaceArea 90
6.4.4 Log(P) 92
6.5 ProblemSet 93
References 93
7 SharingOuttheEnergy 95
7.1 GamesofChance 97
7.1.1 TwoDice 99
7.2 Enumeration 99
7.3 BoltzmannProbability 100
7.4 SafetyinNumbers 104
7.5 PartitionFunction 106
7.6 Two-LevelQuantumSystem 107
7.7 Lindemann’sTheoryofMelting 109
7.8 ProblemSet 111
8 IntroductiontoStatisticalThermodynamics 113
8.1 TheEnsemble 114
8.2 InternalEnergy,U 116
th
viii Contents
8.3 HelmholtzEnergy,A 116
8.4 EntropyS 117
8.5 EquationofStateandPressure 117
8.6 PhaseSpace 117
8.7 ConfigurationalIntegral 119
8.8 VirialofClausius 120
9 MonteCarloSimulations 123
9.1 AnEarlyPaper 125
9.2 TheFirst‘Chemical’MonteCarloSimulation 127
9.3 ImportanceSampling 128
9.4 PeriodicBox 129
9.5 Cut-Offs 131
9.6 MCSimulationofRigidMolecules 132
9.7 FlexibleMolecules 133
References 133
10 MolecularDynamics 135
10.1 RadialDistributionFunction 136
10.2 PairCorrelationFunctions 139
10.3 MolecularDynamicsMethodology 140
10.3.1 HardSpherePotential 140
10.3.2 FiniteSquareWell 140
10.3.3 Lennardjonesium 142
10.4 AlgorithmsforTimeDependence 142
10.4.1 LeapfrogAlgorithm 142
10.4.2 VerletAlgorithm 143
10.5 MoltenSalts 144
10.6 LiquidWater 144
10.6.1 OtherWaterPotentials 147
10.7 DifferentTypesofMolecularDynamics 147
10.8 UsesinConformationalStudies 148
References 150
11 IntroductiontoQuantumModelling 151
11.1 TheSchrödingerEquation 151
11.2 TheTime-IndependentSchrödingerEquation 153
11.3 ParticlesinPotentialWells 154
11.3.1 One-DimensionalInfiniteWell 154
11.4 CorrespondencePrinciple 156
11.5 Two-DimensionalInfiniteWell 157
11.6 Three-DimensionalInfiniteWell 159
11.7 TwoNoninteractingParticles 161
11.8 FiniteWell 162
11.9 UnboundStates 164
Contents ix
11.10 FreeParticles 164
11.11 VibrationalMotion 165
12 QuantumGases 169
12.1 SharingOuttheEnergy 170
12.2 RayleighCounting 172
12.3 Maxwell–BoltzmannDistributionofAtomicKineticEnergies 173
12.4 BlackBodyRadiation 175
12.5 ModellingMetals 177
12.5.1 TheDrudeModel 178
12.5.2 ThePauliTreatment 180
12.6 Indistinguishability 181
12.7 Spin 185
12.8 FermionsandBosons 187
12.9 PauliExclusionPrinciple 187
12.10 Boltzmann’sCountingRule 188
References 189
13 One-ElectronAtoms 191
13.1 AtomicSpectra 192
13.1.1 Bohr’sTheory 192
13.2 CorrespondencePrinciple 194
13.3 InfiniteNucleusApproximation 194
13.4 Hartree’sAtomicUnits 195
13.5 SchrödingerTreatmentoftheHydrogenAtom 195
13.6 RadialSolutions 197
13.7 AtomicOrbitals 199
13.7.1 l=0(s-orbitals) 200
13.7.2 Thep-orbitals 202
13.7.3 Thed-orbitals 205
13.8 TheStern–GerlachExperiment 205
13.9 ElectronSpin 208
13.10 TotalAngularMomentum 209
13.11 DiracTheoryoftheElectron 210
13.12 MeasurementintheQuantumWorld 212
References 213
14 TheOrbitalModel 215
14.1 One-andTwo-ElectronOperators 216
14.2 Many-BodyProblem 216
14.3 OrbitalModel 217
14.4 PerturbationTheory 219
14.5 VariationMethod 221
14.6 LinearVariationMethod 223
14.7 SlaterDeterminants 226
14.8 Slater–Condon–ShortleyRules 228
