Highly accurate treatment of dynamical electron correlation through

197 Pages·2014·1.36 MB·English

Highly accurate treatment of
dynamical electron correlation through
R12 methods and extrapolation techniques
Hoogst nauwkeurige behandeling van dynamische
elektronencorrelatie met R12 methoden en
extrapolatietechnieken
(meteensamenvattinginhetNederlands)
Proefschrift
Ter verkrijging van de graad van doctor aan de Universiteit Utrecht, op gezag van de Rector Ma-
gni(cid:2)cus, Prof. Dr. W. H. Gispen, ingevolge het besluit van het College voor Promoties in het
openbaar teverdedigenopdinsdag12oktober2004desochtendste10:30uur
door
Claire Catherine Miche(cid:30)le Samson
geborenop9juli1976teAiresurAdour,Frankrijk.
Eerstepromotor:
Prof. Dr. F.B.vanDuijneveldt
verbondenaandeFaculteitScheikundevandeUniversiteitUtrecht.
Tweede promotor:
Prof. Dr. W.M.Klopper
verbondenaandeFaculteitScheikundevandeUniversiteitKarlsruhe,Duitsland.
The work described in this thesis was (cid:2)nancially supported jointly by the Universiteit Utrecht
and by the Deutsche Forschungsgemeinschaft through the Center for Functional Nanostructures,
Universita¤tKarlsruhe(TH).ItwasalsopartiallysponsoredbyCostChemistryActionD9.
2
Ameschers parents,
a(cid:30) JonasetSandra.
Beoordelingscommissie:
Prof. Dr. J.P. J.M.vanderEerden(UniversiteitUtrecht)
Prof. Dr. H.Rudolph(UniversiteitUtrecht)
Prof. Dr. B.M.Weckhuysen(UniversiteitUtrecht)
Paranimfen: Dr. F. Barrere
Dr. R. W.A.Havenith
Highlyaccurate treatmentofdynamicalelectroncorrelation
throughR12methodsandextrapolationtechniques
Claire CatherineMiche(cid:30)leSamson
Ph.D.thesis,UniversityofUtrecht,TheNetherlands
MeteensamenvattinginhetNederlands
PrintedbyPrintPartnersIPSKAMP
ISBN 90-393-3759-4
4
Contents
1 Ashortintroduction toQuantum Chemistry 9
1.1 Essenceofelectroncorrelation 10
1.1.1 Time-independentnon-relativisticelectronicSchro¤dingerequation 10
1.1.2 N-electronbasisfunctions 11
1.1.3 One-electronbasisfunctions 11
1.1.4 Molecular-orbital-basedstandardmodels 12
1.1.5 Considerationsoncomputationalcostversusaccuracy 16
1.2 Basis-setinvestigation 18
1.2.1 CoulombholeandCoulombcusp 18
1.2.2 Optimizationandconvergence 19
1.2.3 Extrapolationtothebasis-setlimit 20
1.2.4 Basis-setsuperpositionerror 21
1.3 Explicitlycorrelatedwavefunctions 22
1.3.1 Basicconceptandorigins 22
1.3.2 Linearcorrelationfactor 23
1.3.3 Exponentialcorrelationfactor 24
1.3.4 Advancesandchallenges 26
1.4 Compositionofthisthesis 27
2 Overviewofthe R12theory 33
2.1 R12wavefunction 34
2.1.1 Notationsandde(cid:2)nitions 34
2.1.2 Hylleraaswavefunction 36
2.1.3 Kutzelniggwavefunction 36
2.2 MP2-R12theory 38
2.2.1 ConventionalMP2scheme 38
2.2.2 DecoupledR12correction 39
2.3 CC-R12 theory 40
CONTENTS
2.3.1 GeneralCCSD(T)-R12 theory 40
2.3.2 LinktoMP2-R12workingequations 41
2.4 Computerimplementation 42
2.4.1 Standardapproximation 42
2.4.2 Integralevaluation 44
2.4.3 Numericalinstabilities 45
3 Equilibrium inversion barrier of NH3 from extrapolated coupled-cluster pair ener-
gies 49
3.1 Introduction 50
3.2 ComputationalMethods 50
3.2.1 ExtrapolationofCCSD pairenergies 50
3.2.2 Geometries,basissets,andprograms 52
3.3 ResultsandDiscussion 53
3.3.1 Hartree(cid:150)Fockresults 53
3.3.2 Valence-onlycorrelation 55
3.3.3 Core-valencecorrection 59
3.3.4 Zero-pointvibrationalenergies 59
3.3.5 Relativisticcorrections 59
3.4 Summary 61
4 Abinitiocalculationofprotonbarrier andbindingenergyofthe(H2O)OH(cid:0) complex 67
4.1 Introduction 68
4.2 Computationaldetails 69
4.3 Results 70
4.3.1 Electricmolecularproperties 70
4.3.2 Optimizedgeometries 70
4.3.3 Electronicbindingenergy 73
4.3.4 Barrier toprotonexchange 74
4.3.5 MP2-limitcorrection 75
4.3.6 Potentialenergycurve 77
4.4 Conclusion 79
5 Explicitlycorrelated second-order Młller-Plessetmethods withauxiliarybasissets 81
5.1 Introduction 82
5.2 Methodology 83
5.2.1 TheR12Ansatz 83
6
CONTENTS
5.2.2 Second-orderpairenergies 84
5.2.3 Matrixelements 85
5.2.4 WorkingequationsforAnsatz1 86
5.2.5 WorkingequationsforAnsatz2 88
5.3 Computerimplementation 90
5.3.1 ImplementationofAnsatz1 90
5.3.2 ImplementationofAnsatz2 92
5.4 OverviewofMP2-R12approaches 94
5.5 Numericalresults: TheNeatom 97
5.6 Numericalresults: Molecules 99
5.6.1 Geometriesandbasissets 99
5.6.2 Results 100
5.7 Conclusion 105
6 Computationoftwo-electronGaussianintegralsforwavefunctionsincludingthecor-
relationfactorr12exp((cid:0)(cid:13)r212 ) 113
6.1 Introduction 114
6.2 Integralevaluation 114
6.2.1 StructureoftheCartesiantwo-electrondamped-R12integrals 115
6.2.2 ExpansionofCartesianoverlapdistributionsinHermitefunctions 117
6.2.3 ExpansionofCartesianintegralsinHermiteintegrals 118
6.2.4 EvaluationofthesphericalHermiteintegrals 120
6.2.5 EvaluationofnonsphericalHermiteintegrals 122
6.3 Conclusion 125
7 Similarity-transformed Hamiltonians by means of Gaussian-damped interelectronic
distances 127
7.1 Introduction 128
7.2 Theory 129
7.2.1 Similarity-transformedHamiltonians 129
7.2.2 Two-electronsystems 131
7.2.3 Many-electronsystems 132
7.3 Computationaldetails 132
7.4 Results 135
7.5 Conclusions 137
7
CONTENTS
8 Benchmarking ethylene and ethane: Second-order Młller(cid:150)Plesset pair energies for
localizedmolecularorbitals 141
8.1 Introduction 142
8.2 ComputationalDetails 144
8.2.1 MP2-R12/AandMP2-R12/Bmethods 145
8.2.2 Geometries 147
8.2.3 Basissets 148
8.2.4 Extrapolationtechniques 148
8.2.5 Programsandprocedures 149
8.3 Resultsanddiscussion 149
8.3.1 Rawdata 149
8.3.2 Bestestimates 150
8.3.3 Extrapolateddata 154
8.4 Conclusions 158
9 Outlookonlocalized,Gaussian-damped R12wavefunctions 163
9.1 LocalcorrelationschemeforMP2-R12method 164
9.2 Integralstudyovers-typeGaussianbasisfunctions 166
9.3 Domainselectioninthealkaneseries 170
9.4 Concludingremarks 171
Summary 173
Samenvatting 177
Zusammenfassung 181
Re·sume· 187
Listofpublications 193
Acknowledgments 195
Curriculum Vitae 197
8
1
A short introduction to Quantum
Chemistry
Chapter1
1.1 Essence of electron correlation
1.1.1 Time-independent non-relativistic electronic Schro¤dinger equation
The forces that keep together the atoms in a molecule cannot be described correctly by classical
mechanics. The equation introduced by the Austrian physicist Erwin Schro¤dinger in 1926 [1]
dictates the quantum mechanical behavior of molecules at the atomic scale and allows to predict
theoreticallytheirphysicalpropertieswithrigorousaccuracy. Thisthesisaimsatthedevelopment
of improvedmethods for the approximate solutionof the non-relativisticSchro¤dinger equation in
itstime-independentform:
H^(cid:9) = E(cid:9): (1.1)
(cid:9)isthetotalwavefunctionoftheconsideredmolecularsystem. TheHamiltonianH^ istheoperator
correspondingtothetotalenergy E. Itisformulatedas:
H^ = T^ +T^ +V^ +V^ +V^ ; (1.2)
n e ee nn ne
where T^ and T^ stand for the kinetic energies of nuclei n and electrons e, and V^ , V^ and V^
n e ee nn ne
symbolizetheirpotentialenergies. TheBorn-Oppenheimerapproximation[2]is acentralconcept
for the non-linear solution of this fundamental equation. Based on the fact that nuclei are at least
1800 times heavier than electrons, it consists in uncoupling their motions, treating consequently
nuclei as stationary particles for the electronic motion. For the majority of systems, the Born-
Oppenheimer approximation introduces very small errors. We can therefore solve the electronic
Schro¤dingerequationparametericallyonthenuclearcoordinates.
H^ (cid:9) = E (cid:9) and H^ = T^ +V^ +V^ : (1.3)
e e e e e e ee ne
The electronic wave function(cid:9) describes a stationary-stateofa molecularelectronic systemand
e
dependsconcomitantlyonthecoordinatesofeachelectroninthesystem. Ifsolutionsaregenerated
without any empirical (cid:2)tting to the experimental data, the corresponding methods are baptized
abinitio, which originates from the latin expression (cid:148)from the beginning(cid:148). Energies calculated
from the electronic Schro¤dinger equation depend on the chosen nuclear coordinates and can be
used as the potential for solving the rotational-vibrational problem of a molecule, or to model
chemicalreactions.
In this context, abinitio quantum chemistry serves as a powerful tool to investigate computa-
tionally many physical properties of materials such as: geometry of molecules, electron distribu-
tions,ionizationpotentials,electronaf(cid:2)nities,multipolemomentsandvibrationalfrequencies.
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