Solving The Schrödinger Equation: Has Everything Been Tried?

375 Pages·2011·1.988 MB·English

by  Paul Popelier (ed.)

Solving the
Schrödinger
Equation
Has Everything Been Tried?
P780.9781848167247-tp.indd 1 7/25/11 11:49 AM
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Solving the
Schrödinger
Equation
Has Everything Been Tried?
Editor
Paul Popelier
Imperial College Press
ICP
P780.9781848167247-tp.indd 2 7/25/11 11:49 AM
Published by
Imperial College Press
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Covent Garden
London WC2H 9HE
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British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library.
SOLVING THE SCHRÖDINGER EQUATION
Has Everything Been Tried?
Copyright © 2011 by Imperial College Press
All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means,
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ISBN-13 978-1-84816-724-7
ISBN-10 1-84816-724-5
Typeset by Stallion Press
Email: [email protected]
Printed in Singapore.
Catherine - Solving the Schrodinger Eqn.pmd 1 9/7/2011, 5:06 PM
July20,2011 9:6 9inx6in b1189-fm SolvingtheSchro¨dingerEquation
ToD.P.B.
v
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‘Therichestinteractionsoccurbetweentwoalmostidenticalbut
opposingconstituents.’
vi
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Contents
Preface xv
1. IntraculeFunctionalTheory 1
DeborahL.CrittendenandPeterM.W.Gill
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Intracules . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 ElectronCorrelationModels . . . . . . . . . . . . . . 13
1.4 DynamicandStaticCorrelation . . . . . . . . . . . . . 16
1.5 DispersionEnergies . . . . . . . . . . . . . . . . . . . 18
1.6 FutureProspects . . . . . . . . . . . . . . . . . . . . . 21
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2. ExplicitlyCorrelatedElectronicStructureTheory 25
FrederickR.Manby
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 25
2.1.1 Basis-setexpansions . . . . . . . . . . . . . . 25
2.2 F12Theory . . . . . . . . . . . . . . . . . . . . . . . 28
2.2.1 MP2-F12 . . . . . . . . . . . . . . . . . . . . 29
2.2.2 Explicitlycorrelatedcoupled-cluster
theory . . . . . . . . . . . . . . . . . . . . . 30
2.3 FiveThoughtsforF12Theory . . . . . . . . . . . . . . 31
2.3.1 Thought1:Doweneed(productsof)
virtuals? . . . . . . . . . . . . . . . . . . . . 31
2.3.2 Thought2:Aretherebettertwo-electron
basissets? . . . . . . . . . . . . . . . . . . . 34
vii
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viii Contents
2.3.3 Thought3:Doweneedtheresolution
oftheidentity? . . . . . . . . . . . . . . . . . 35
2.3.4 Thought4:Couldwehaveexplicitcorrelation
forhigherexcitations? . . . . . . . . . . . . . 38
2.3.5 Thought5:Canweavoidthree-electronerrors
intwo-electronsystems? . . . . . . . . . . . . 39
2.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . 40
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3. SolvingProblemswithStrongCorrelationUsing
theDensityMatrixRenormalizationGroup(DMRG) 43
GarnetKin-LicChanandSandeepSharma
3.1 TheProblemofStrongCorrelation . . . . . . . . . . . 43
3.2 TheDensityMatrixRenormalizationGroup
Wavefunction . . . . . . . . . . . . . . . . . . . . . . 46
3.3 LocalityandEntanglementintheDMRG . . . . . . . . 47
3.4 OtherPropertiesoftheDMRG . . . . . . . . . . . . . 50
3.5 RelationtotheRenormalizationGroup . . . . . . . . . 51
3.6 DynamicCorrelation—theRoleofCanonical
Transformations . . . . . . . . . . . . . . . . . . . . . 53
3.7 WhatCantheDMRGDo?ABriefHistory . . . . . . . 54
3.8 TheFuture:HigherDimensionalAnalogues . . . . . . 57
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4. Reduced-Density-MatrixTheoryforMany-electronCorrelation 61
DavidA.Mazziotti
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 62
4.2 Variational2-RDMMethod . . . . . . . . . . . . . . . 63
4.2.1 Energyasa2-RDMfunctional . . . . . . . . 63
4.2.2 Positivityconditions . . . . . . . . . . . . . . 64
4.2.3 Semidefiniteprogramming . . . . . . . . . . . 67
4.2.4 Applications . . . . . . . . . . . . . . . . . . 69
4.3 ContractedSchro¨dingerTheory . . . . . . . . . . . . . 73
4.3.1 ACSEandcumulantreconstruction . . . . . . 74
4.3.2 SolvingtheACSEforground
andexcitedstates. . . . . . . . . . . . . . . . 75
4.3.3 Applications . . . . . . . . . . . . . . . . . . 77
4.4 Parametric2-RDMMethod . . . . . . . . . . . . . . . 80
4.4.1 Parametrizationofthe2-RDM . . . . . . . . . 81
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Contents ix
4.4.2 Applications . . . . . . . . . . . . . . . . . . 83
4.5 LookingAhead . . . . . . . . . . . . . . . . . . . . . 85
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . 87
5. FiniteSizeScalingforCriticalityoftheSchro¨dingerEquation 91
SabreKais
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 92
5.2 CriticalityforLarge-dimensionalModels . . . . . . . . 93
5.3 FiniteSizeScaling:ABriefHistory . . . . . . . . . . . 95
5.4 FiniteSizeScalingfortheSchro¨dingerEquation . . . . 97
5.5 TheHulthenPotential . . . . . . . . . . . . . . . . . . 100
5.5.1 Analyticalsolution . . . . . . . . . . . . . . . 100
5.5.2 Basissetexpansion . . . . . . . . . . . . . . 101
5.5.3 Finiteelementmethod . . . . . . . . . . . . . 101
5.5.4 Finitesizescalingresults . . . . . . . . . . . 102
5.6 FiniteSizeScalingandCriticality
ofM-electronAtoms. . . . . . . . . . . . . . . . . . . 105
5.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . 107
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . 108
6. TheGeneralizedSturmianMethod 111
JamesAveryandJohnAvery
6.1 DescriptionoftheMethod . . . . . . . . . . . . . . . . 111
6.1.1 TheintroductionofSturmians
intoquantumtheory . . . . . . . . . . . . . . 111
6.1.2 GeneralizedSturmians . . . . . . . . . . . . . 114
6.1.3 ThegeneralizedSturmianmethodapplied
toatoms . . . . . . . . . . . . . . . . . . . . 117
6.1.4 Goscinskianconfigurations . . . . . . . . . . 118
6.1.5 Goscinskiansecularequationsforatoms
andatomicions . . . . . . . . . . . . . . . . 120
6.2 Advantages:SomeIllustrativeExamples . . . . . . . . 120
6.2.1 Thelarge-Zapproximation:restriction
ofthebasissettoanR-block . . . . . . . . . 121
6.2.2 Validityofthelarge-Zapproximation . . . . . 126
6.2.3 Coreionizationenergies . . . . . . . . . . . . 129
6.3 LimitationsoftheMethod;ProspectsfortheFuture . . 130
6.3.1 CanthegeneralizedSturmianmethod
beappliedtoN-electronmolecules?. . . . . . 133