Quantum Mechanical Electronic Structure Calculations with Chemical Accuracy

450 Pages·1995·35.02 MB·English

by  James B. Anderson (auth.)

QUANTUM MECHANICAL ELECTRONIC STRUCTURE CALCULATIONS
WITH CHEMICALACCURACY
Understanding Chemical Reactivity
Volume 13
SeriesEditor
PaulG. Mezey, UniversityofSaskatchewan, Saskatoon, Canada
EditorialAdvisoryBoard
R. Stephen Berry, UniversityofChicago, IL, USA
John I. Brauman, StanfordUniversity, CA, USA
A. WelfordCastleman, Jr., PennsylvaniaStateUniversity, PA, USA
EnricoClementi, IBMCorporation, Kingston, NY, USA
Stephen R. Langhoff, NASA AmesResearchCenter, MoffettField, CA, USA
K. Morokuma, Institute forMolecularScience, Okazaki, Japan
PeterJ. Rossky, UniversityofTexasatAustin, TX, USA
ZdenekSlanina, CzechAcademyofSciences, Prague, Czech Republic
DonaldG. Truhlar, UniversityofMinnesota, Minneapolis, MN, USA
IvarUgi, Technische Universitat, MOnchen, Germany
The titlespublishedin thisseriesarelistedatthe endofthis volume.
Quantum Mechanical
Electronic Structure Calculations
with Chemical Accuracy
edited by
Stephen R. Langhoff
NASA Ames Research Genter,
Moffett Fie/d, Califomia, U.S.A.
SPRINGER-SCIENCE+BUSINESS MEDIA, BV. .
Library of Congress Cataloging-in-Publication Data
Cuantum mechanlcal electronIc structure calculatlons wlth chemlcal
accuracy / edlted by Stephen R. Langhoff.
p. cm. -- (Understandlng chemlcal reactlvlty ; v. 13)
ISBN 978-94-010-4087-7 ISBN 978-94-011-0193-6 (eBook)
DOI 10.1007/978-94-011-0193-6
1. Cuantum chemlstry. r. Langhoff. Stephen R. II. Serles.
CD462.C347 1995
541.2' 8--dc20 94-39289
ISBN 978-94-010-4087-7
Printed an acid-free paper
AII Rights Reserved
© 1995 Springer Science+Business Media Dordrecht
Originally published by Kluwer Academic Publishers in 1995
Softcover reprint of the hardcover 1s t edition 1995
No part of the material protected by this copyright notice may be reproduced or
utilized in any farm ar by any means, electronic ar mechanical,
including photocopying, recording ar by any inlormation starage and
retrieval system, without written permission lrom the copyright owner.
Contents
JamesB.Anderson/ExactQuantumChemistrybyMonteCarloMethods
TimothyJ.LeeandGustavoE.Scuseria/AchievingChemicalAccuracy with
Coupled-ClusterTheory 47
DanielM.Chipman/MagneticHyperfineCouplingConstantsinFreeRadicals 109
Larry A.Curtissand KrishnanRaghavachari/CalculationofAccurateBond
Energies,ElectronAffinities,andIonizationEnergies 139
KrishnanRaghavachariandLarry A.Curtiss/ AccurateTheoreticalStudiesof
SmallElementalClusters 173
HarryPartridge,StephenR. LanghoffandCharlesW. Bauschlicher,Jr./
ElectronicSpectroscopyofDiatomicMolecules 209
M. Peric,B.Engels,andS.D.Peyerimhoff/TheoreticalSpectoscopyofSmall
Molecules: AbInitioInvestigationsofVibronicStructure,Spin-Orbit
SplittingsandMagneticHyperfineEffectsintheElectronicSpectraof
TriatomicMolecules 261
BjornO.Roos, MarkusFulscher,Per-AkeMalmqvist,ManuelaMerchan,and
LuisSerrano-Andres/TheoreticalStudiesoftheElectronicSpectraof
OrganicMolecules 357
Index 439
Exact Quantum Chemistry by Monte Carlo Methods
James B. Anderson
DepartmentofChemi6try, The Penn6ylvaniaState Univer6ity, 15£DaveyLaboratory, Univer6ity
Park, Penn6ylvania1680£
June-1994
Abstract
WithMonte Carlomethodsitis nowpossible to solvethe Schrodingerequation
for systems ofa few electrons without systematicerror. The first quantum calcu
lationtoachieveanabsoluteaccuracyof1.0microhartreefor a polyatomicsystem
Ht
was a quantum Monte Carlo calculation for the molecular ion. The first to
achieveanabsoluteaccuracy of0.01kcal/molefor the H-H-H systemwas a Monte
+ +
Carlo calculationfor the barrier for the reaction H H H H. The first to
2 -t 2
achieve an absolute accuracy of0.1 K for the stable dimer He-He was a quantum
Monte Carlo calculation. In this chapter we review the exact quantum Monte
Carlo method, its successful applications thus far to systems of a few electrons,
and its prospects for successful applications to larger systems.
S.R.Langhojf!Ed.),QuantumMechanicalElectronicStructureCalculationswithChemicalAccuracy, 1--45.
©1995KluwerAcademicPublishers.PrintedintheNetherlands.
2 J.B.ANDERSON
Contents
1 Introduction
2 Overview ofQuantumMonte Carlo
3 Node Structure
4 Solutions to the Node Problem
5 Exact Cancellation Schemes: General
6 Exact Cancellation: A Simple Example
7 Exact Cancellation: Stability
8 Exact Cancellation: ComputationalDetails
9 The Molecular Ion Hs+
10 The Molecule H2
+ +
11 Potential Energy Surface for the Reaction H H2 -+ H2 H
12 The Helium Dimer He2
13 The H-He Interaction
14Prognostication
15 Acknowledgments
EXACTQUANTUMCHEMISTRYBYMONTECARLOMETHODS 3
1 Introduction
The theoretical chemist is accustomed to judging the success ofa theoretical
prediction according to how well it agrees with an experimental measurement.
Since the object of theory is the prediction of the results of experiment, that
would appear to be an entirely satisfactory state ofaffairs. However, ifit is true
that "theunderlyingphysicallaws ... for the wholeofchemistryare ... completely
known" (1),thenitshouldbepossible,atleastinprinciple,topredicttheresultsof
experimentmoreaccurately than they canbemeasured. Ifthe theoreticalchemist
could obtainexact solutions ofthe Schrodinger equationfor many-body systems,
then the experimental chemist would soon become accustomed to judging the
success ofan experimental measurement by how well it agrees with a theoretical
prediction.
Infact, itis now possible to obtainexact solutions ofthe Schrodinger equation
Ht,
for systems ofa few electrons(2-8). These systems include the molecularion
the molecule H2, the reaction intermediate H-H-H, the unstable pair H-He, the
stable dimer He2' and the trimer He3. The quantum Monte Carlo method used
in solving the time-independent Schrodinger equation for these systems is exact
in that it requires no physical or mathematicalassumptions beyond those ofthe
Schrodinger equation. As in most Monte Carlo methods there is a statistical or
samplingerror whichis readily estimated.
We use the term 'exact' as it has been described by Zabolitzky(9) in the sense
that 211"r is the 'exact' circumference ofa circle ofradius r. The numerical value
of is not known exactly but may be computed to any precision desired and the
11"
same is true for the circumference ofa circle ofa given r. A calculation method
is 'exact'ifthereexist finite amounts ofcomputer time such that a result may be
determined with any arbitraryprecision.
The exact solution of the Schrodinger equation for many-electron systems is
one of the Grand Challenges to Computational Science (10). More specifically,
the challenge is that of overcoming 'the node problem' or 'the sign problem in
quantumMonte Carlo' (11,12). This challenge has now beenmetfor systems ofa
few electrons.
Inthis chapterwereviewtheexactquantumMonteCarlomethod,itssuccessful
applicationsthusfar tosystemsofafew electrons, andtheprospectsfor successful
applications tolargersystems. Approximatemethods applicable tolarger systems
are mentioned, but the emphasis is on 'exact' quantum chemistry.
2 Overview ofQuantum Monte Carlo
There are at least a dozen different Monte Carlo methods which might be used
for solvingthe Schrodingerequation, but onlythreeofthese are currentlypopular
in calculations of electronic structure. The three are collectively referred to as
quantumMonte Carlomethods (QMC). They are the variationalquantumMonte
4 J.B.ANDERSON
Carlo method (VQMC), the diffusion quantum Monte Carlo method (DQMC),
and the Green's function quantumMonte Carlomethod (GFQMC).
Thevariationalmethodisthesameasconventionalanalyticvariationalmethods
except that the integralsrequired areevaluatedby Monte Carloprocedures. The
first applications in quantum chemistry were those by Conroy(13) in the 1960's.
The expectation value ofthe energy is determined for a trial function WT using
Metropolis samplingbasedon w~. The expectation value <E > is given by
'flo 'flo
<E >= f f o W202-dX =N "L.'JI~_Nl 0 ' (1)
Wo dX L:i=11
where the summationsarefor samplesofequalweights selectedwithprobabilities
proportional to w~. As in analytic variational calculations the expectation value
<E > is an upper limit to the true valueofthe energy E,
<E> ~ E . (2)
The term ~ftx is a local energy E,oc• In determining < E > it is not necessary
~alytic
to carry out integrations; and, since only differentiationofthe trial wave
function is required to evaluate the localenergy, the trialwavefunction may take
anydesiredfunctionalform. Itmayevenincludeinter-electrondistancesrij explic
itly. Thus, relatively simple trial functions may incorporate electron correlation
effects rather accurately andproduce expectation values ofthe energy well below
those ofthe Hartree-Fock limit. Except in the limit ofa large number ofterms
the VQMC methodis not an exact method.
The diffusion and Green's function methods are based on the similarityofthe
Schrodinger equation and the diffusion equation and the use of a random walk
process to simulate diffusion. The relation of the Schrodinger equation to the
diffusionequationand to diffusion was notedas earlyas 1932by Wigner(14), and
arandomwalksimulationofthe SchrodingerequationwasdiscussedbyMetropolis
and Ulam(15) and by King(16) in 1949.
Hw =Ew,
The time-independent Schrodingerequation, for a singleparticleis
given by
(3)
where the potentialenergy V is a function ofposition. The equationhas as solu
tions the eigenfunctions w(X) together with theireigenvalues Ei.
Inthe solutionofpartialdifferentialequationsit isinsomecaseseasier tosolve
thecorrespondingtransientequationthantosolveasteady-stateequationdirectly.
Ifwe introduce an imaginarytimeT and define the transient function
(4)
EXACTQUANTUMCHEMISTRYBYMONTECARLOMETHODS 5
~(i,T) obeys the relations
a~(i,T) =-E~(i,T)
(5)
aT
and
a~ =~V2~_ V~
(6)
aT 2m
The wave function ~(i,T) consists of a spatial function ~(i) changing expo
nentially with time. The function ~(.i) is a solution to the time-independent
equation. Thereis aninfinitenumberofsolutionshavingenergieshigher than the
lowest-energyor ground-state solution. Ifaninitialfunction contains components
ofthe ground-state wavefunction and higher-state wavefunctions the morerapid
exponential decay ofthe higher-energy wave functions, as indicated by Eq. (4),
ensures that the solutionat large values of is the lowest-energysolution.
T
Thus, we have
~(i,o) =ECi~i(i) , (7)
(8)
~(i,T -+ 00) =co~o(i)e-EoT , (9)
where ~o(i) is the ground-state wavefunction.
The Schrodinger equation in imaginary time, Eq. (6), is identical in form to
Fick's diffusionequation to whicha first-order reactionrate termis added,
=
aC(i,t) DV2C(i,t) _ kC(i,t) . (10)
at
The concentration C is equivalent to the wave function W, the diffusion coeffi
cient D is equivalent to the group :~ , and the rate constant k is equivalent to
the potential energy V. In the language of mathematics the two equations are
'isomorphic'.
The diffusion ofmolecules occurs physically by a random walk process. The
molecules are subject to randommotion, movingin a timeintervalllt a distance
llz in one direction or the other in each ofthe three dimensions. The time and
distance steps are relatedto the diffusioncoefficient by the Einstein equation(17),
D =(~z)2 .
(11)
2~T
Usually, the diffusionequationis used tosimulatea randomwalk process (namely
diffusion)inaphysicalsystem. InquantumMonteCarlotheprocedureisreversed:
the random walk process is used to simulatethe differentialequation.