Development and Implementation of Ab Initio Quantum Chemistry

130 Pages·1998·0.56 MB·English

Ab Initio
Development and Implementation of
Quantum Chemistry Techniques for Application
to Large Molecules
Thesis by
Richard P. Muller
in partial ful(cid:12)llment of the requirements
of the Degree of Doctor of Philosophy,
California Institute of Technology,
Pasadena, California.
Submitted May 4, 1994.
(cid:13)c 1994
Richard P. Muller
Second Revision
(cid:13)c 1997 All Rights Reserved.
April 23, 1997
Contents
1 Introduction 5
2 Overview of Ab Initio Electronic Structure Theory 10
2.1 Introduction : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 10
2.2 The Electronic Hamiltonianand Hartree-Fock Wave Functions : 11
2.3 Open Shell Hartree-Fock Wave Functions : : : : : : : : : : : : : 18
2.4 Interpretation of Hartree-Fock Theory Results : : : : : : : : : : : 21
2.5 Shortcomings of Hartree-Fock Theory : : : : : : : : : : : : : : : 21
2.6 Generalized Valence Bond Theory : : : : : : : : : : : : : : : : : 23
2.7 Summaryof Equations for General HF/GVB WaveFunctions : : 25
2.8 Con(cid:12)guration Interaction : : : : : : : : : : : : : : : : : : : : : : 28
2.9 Multi-Con(cid:12)gurationalSelf-Consistent Field Wave Functions : : : 29
2.10 Conclusion : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 30
3 Pseudospectral- GeneralizedValenceBond(PS-GVB)Theory 31
3.1 Introduction : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 31
3.2 The Scaling of Hartree-Fock and GVB Calculations: : : : : : : : 32
3.3 The Pseudospectral Method : : : : : : : : : : : : : : : : : : : : : 33
3.4 Naive Grid-based Calculation of Coulomband Exchange Matrices 33
3.5 Pseudospectral Coulomband Exchange Operators : : : : : : : : 36
3.6 Grids, Dealiasing Functions, and AtomicCorrections : : : : : : : 38
3.7 The Structure of the PS-GVB Program : : : : : : : : : : : : : : 39
3.8 PS-GVB Applications : : : : : : : : : : : : : : : : : : : : : : : : 41
3.8.1 Methylene : : : : : : : : : : : : : : : : : : : : : : : : : : : 42
3.8.2 Silylene : : : : : : : : : : : : : : : : : : : : : : : : : : : : 44
3.8.3 Ethylene : : : : : : : : : : : : : : : : : : : : : : : : : : : : 45
3.9 Conclusion : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 46
4 PseudospectralParametersforCalculationsonNickelClusters 49
4.1 Introduction : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 49
4.2 Nickel Cluster and Metal Bonding : : : : : : : : : : : : : : : : : 50
4.3 Grid Parameters : : : : : : : : : : : : : : : : : : : : : : : : : : : 52
4.4 DealiasingFunction Parameters : : : : : : : : : : : : : : : : : : : 53
4.5 OptimizationProcedure : : : : : : : : : : : : : : : : : : : : : : : 57
2
4.6 Application to Ni4 Clusters : : : : : : : : : : : : : : : : : : : : : 59
4.7 Conclusions : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 60
5 Direct Inversion in the Iterative Subspace Convergence for
GeneralizedValence Bond Wave Functions 63
5.1 Introduction : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 63
5.2 Closed-Shell HF-DIIS : : : : : : : : : : : : : : : : : : : : : : : : 65
5.3 General Fock Operators : : : : : : : : : : : : : : : : : : : : : : : 67
5.3.1 Simplest Approach: CombineSeparate Fock Operators : : 68
5.3.2 The Pulay Multi-shell Fock Operator : : : : : : : : : : : : 69
5.3.3 The Page and McIver Fock Operator : : : : : : : : : : : : 69
5.3.4 The GVB-DIIS Multi-shell Fock Operator : : : : : : : : : 69
5.4 Choice of the Error Vector : : : : : : : : : : : : : : : : : : : : : : 70
5.5 Related Convergence Issues : : : : : : : : : : : : : : : : : : : : : 71
5.6 Results: : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 73
5.6.1 One pair GVB wave functions : : : : : : : : : : : : : : : : 73
5.6.2 Multiple GVB pair wave functions : : : : : : : : : : : : : 76
5.7 Conclusions : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 82
6 Ab Initio Calculation of PorphyrinExcited States 83
6.1 Introduction : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 83
6.2 The Four-Orbital Model and Four-Orbital Excited States : : : : 85
6.3 The Multi-Con(cid:12)gurationalNature of FOES : : : : : : : : : : : : 87
6.4 The Frozen Core Four Orbital Excited State Method : : : : : : : 89
6.5 Results of FC-FOES Porphine Calculations : : : : : : : : : : : : 90
6.6 Self-Consistent Orbital Optimizationin FOES : : : : : : : : : : : 92
6.6.1 Core-Active Mixing : : : : : : : : : : : : : : : : : : : : : 94
6.6.2 Active-Active Mixing : : : : : : : : : : : : : : : : : : : : 95
6.6.3 Occupied-Virtual Mixing : : : : : : : : : : : : : : : : : : 95
6.7 PS-GVB CalculationofMulti-Con(cid:12)gurationalOperators for SC-
FOES : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 96
6.8 Convergence Acceleration in SC-FOES : : : : : : : : : : : : : : : 97
6.9 SC-FOES Results for Porphine : : : : : : : : : : : : : : : : : : : 97
6.9.1 Orbital Energies : : : : : : : : : : : : : : : : : : : : : : : 98
6.9.2 SC-FOES Porphine Calculations : : : : : : : : : : : : : : 99
6.10 Frozen Core Corrections to SC-FOES: SC-FOES-CI : : : : : : : 100
6.11 Reduced Porphyrins : : : : : : : : : : : : : : : : : : : : : : : : : 101
6.12 Conclusion : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 105
A A Technique for Evaluating HamiltonianMatrix Elements 106
A.1 Introduction : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 106
A.2 The Method : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 106
A.2.1 Break Into Individual Determinants : : : : : : : : : : : : 106
A.2.2 Transformto Spin Orbitals : : : : : : : : : : : : : : : : : 107
A.2.3 Order and Classify Determinants : : : : : : : : : : : : : : 107
A.2.4 Evaluate Matrix Elements between Operators : : : : : : : 108
3
A.2.5 TransformBack to Spatial Orbitals: : : : : : : : : : : : : 108
A.2.6 Example: One Pair GVB Energy : : : : : : : : : : : : : : 109
A.3 Conclusion : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 109
B The Self-Consistent Optimization of Restricted Con(cid:12)guration
Interaction Wave Functions: Two-Pair, Closed-Shell Special
Case 111
B.1 Introduction : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 111
B.2 The GVB-RCI WaveFunction: : : : : : : : : : : : : : : : : : : : 112
B.3 RCI Spin Coupling : : : : : : : : : : : : : : : : : : : : : : : : : : 113
B.4 Determinants in the GVB-RCI Wave Function : : : : : : : : : : 114
B.5 Matrix Elements Between Determinants in the GVB-RCI Wave
Function : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 115
B.6 The GVB-RCI Energy Expression : : : : : : : : : : : : : : : : : 117
B.7 Energy Coe(cid:14)cients and the Ya(cid:11)e-Goddard Form : : : : : : : : : 119
B.8 GVB-RCI Orbital Optimization: : : : : : : : : : : : : : : : : : : 121
B.8.1 Active-Core Mixing : : : : : : : : : : : : : : : : : : : : : 122
B.8.2 Active-Active Mixing : : : : : : : : : : : : : : : : : : : : 123
B.8.3 Occupied-Virtual Mixing : : : : : : : : : : : : : : : : : : 123
B.9 Operators Required For GVB-RCI : : : : : : : : : : : : : : : : : 124
B.10Conclusions : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 124
Bibliography 125
4
Chapter 1
Introduction
This thesis describes new methods for extending ab initio electronic structure
theory calculations to larger molecules. Ab initio methods are important for
investigation of the chemicalproperties of large moleculesbecause they remain
the most accurate method of computing the electronic structure of a molecule.
Forthepurposesofthisthesis,thetermlargemoleculesreferstothosemolecules
that require more than 200 basis functions (vide infra). Molecules of this size
are di(cid:14)cult to describe with standard methods in ab initio electronic structure
theory because they require large amounts of CPU time, disk storage space,
and physical memory. Subsequent chapters will describe ab initio calculations
in greater detail, but to summarize brie(cid:13)y, the principal expense of ab initio
methods arises from the basis function techniques used to evaluate the two-
electron integrals. Basis function methods, also known as spectral methods,
expand the one- and two-electron integrals as a sum of integrals that can be
evaluatedanalytically. Abinitiocalculationsusebasisfunctionmethodsbecause
these methods provideahighdegree ofaccuracy, especially forthe one-electron
terms where the fact that basis functions are analytically di(cid:11)erentiable yields
highlyaccurate integration. These methodsbecomeexpensive asthe size ofthe
molecule increases. In particular, evaluation of the two-electron integrals using
4
Nbf basis functions scales as (Nbf), whichbecomes prohibitivelyexpensive as
O
Nbf increases. Evaluationoftheone-electronintegralsusingNbf basisfunctions
2
is much less expensive, scaling only as (Nbf).
4 O
The notation (Nbf) denotes that when Nbf is doubled, the computation
4 O
requires 2 = 16 times as much CPU time to complete; similarly, the nota-
2
tion (Nbf) meansthat when Nbf is doubled the computationrequires 4times
O
as much CPU time to complete. When Nbf is large|the assumption made
throughout the rest of this thesis for scaling comparisons|the higher-order
scalings dominate the smaller order ones, so that a calculation that depends
4 2 4
upondi(cid:11)erent elementsscalingas (Nbf) and (Nbf)scales overallas (Nbf).
O O O
Clearly, one of the requirements in making ab initio electronic structure cal-
culations possible on larger molecules is to determine methods that scale with
4
molecular size better than (Nbf).
O
5
Numerical methods provide a less expensive alternative to integral evalu-
ations. Numerical techniques evaluate one- and two-electron integrals over a
grid of points in three-dimensional space. Numerical methods scale roughly as
3
(N ), where N is related to the number of grid points (vide infra). Tradi-
O
tionally,numerical methods have not been used for electronic structure theory
because the accuracy required by electronic structure theory can be achieved
usingnumericalmethodsonlywithanimpracticallylargenumberofgridpoints.
The one-electron integrals are particularlydi(cid:14)culttointegrate usingnumerical
methods.
For electronic structure theory calculations there has traditionally been a
tradeo(cid:11) between the accuracy of basis function methods and the speed of nu-
merical methods. Because ab initio calculations on smallmolecules are not as
computationally intensive as those on larger molecules, basis function meth-
ods have predominated. As the size of the molecule|andthus Nbf|increases,
4
the (Nbf) scaling of basis function methods makesthese methods impossible.
O
Thus, tosolve the ab initio electronic structure oflargemoleculesnew methods
must be developed.
Thesolutiontothe problemofapplyingab initio electronic structure theory
tolargemoleculesistouse ahybridapproach, thepseudospectral method. The
pseudospectralmethodutilizesthespeedofnumericaltechniquestoevaluatethe
two-electron integrals; and it utilizes the accuracy of basis function techniques
to evaluate the one-electron integrals. Evaluation of the two electron integrals
usingthepseudospectral methodwithNgrid gridpointsandNbf basisfunctions
2 3
scales as (NgridNbf)|approximately (Nbf). At the sametimethe accuracy
O O
achieved by basis function techniques for one electron integrals is preserved
4
with the pseudospectral method. The improvementin scaling from (Nbf) to
3 O
(Nbf)reduces theexpense ofcalculationsonlargermolecules,andpromisesto
O
make ab initio electronic theory calculations possible on larger molecules that
are impossiblewith pure basis function techniques.
Theworkpresented inthisthesisdevelopsprogramsthatuse pseudospectral
operator construction to calculate the electronic structure of real chemical sys-
tems. Chapter 2 presents an overview of ab initio electronic structure theory,
describing not only the manner in which basis function techniques are used to
evaluatetherequisiteone-andtwo-electronoperatorsinelectronicstructurethe-
ory, but also how these operators are used to obtain converged wave functions.
Chapter 3 describes the formationof ab initio one- and two-electron operators
using the pseudospectral method; and it outlines the procedure for using these
operators to calculate the electronic structure of molecules with a wide variety
of compositions, geometries, orbital occupancies, and wave functions. The or-
bitaloptimizationtechniques used instandard basis function methodsare used
withthe pseudospectral method;thiscombinationresults inaprogramthathas
the (cid:13)exibilityofthebasisfunctionmethodsandthe speed ofthe pseudospectral
operator construction. The accuracy of the pseudospectral method is demon-
stratedusingexamplesofcalculationsonmethylene,silylene,andethylene,with
avarietyofgeometries,spinstates,andwavefunctions. Chapter3demonstrates
6
thatthespeedandaccuracy pseudospectral methodsdisplayedwithclosed-shell
Hartree-Fockwavefunctionscanbeextendedto: (i)generalwavefunctionswith
arbitrary numbers of doubly-occupied core orbitals, singly-occupied open-shell
orbitals, and variably occupied GVB natural orbitals; (ii) the calculation of
physical properties of moleculesrather than total electronic energies; (iii)a va-
riety of quality basis sets, including ones with di(cid:11)use functions; (iv) elements
notinthe(cid:12)rstrowoftheperiodictable,inparticularSilicon;(v)moleculesthat
use e(cid:11)ective core potentials. This work demonstrates that the pseudospectral
method can be applied toany ofthe moleculescontainingmaingroup elements
that the standard basis function methods can describe.
Chapter 4 extends the methods of Chapter 3 to metallic elements, which
present particular problems for the pseudospectral method. The nature of the
chemical bonding is qualitatively di(cid:11)erent between metallic elements than be-
tween non-metallic maingroup elements. Metallic systems have a wider range
ofbondangles,bondlengthsandcoordinationsthandonon-metallicmolecules.
Moreover, the electrons in metallic systems localize between atoms in metallic
systems, whereas the electrons in molecules composed of main-group elements
localizeontheatomsthemselves. Chapter 4demonstratesthatthepseudospec-
tral method can be used to form the ab initio one- and two-electron operators
forNickelclusters withoutlossofaccuracy. Chapter4alsodescribes the adjust-
ments to the parameters used by the pseudospectral method that are needed
to describe these systems. These parameters are optimized using Ni3 clusters
with several di(cid:11)erent geometries. The generality of the optimized parameters
is demonstrated by using them to determine a chemicalproperty of another Ni
system. Chapter 4 demonstrates that on the scale of chemical properties the
error thatthepseudospectral methodmakesrelativetothe standardbasisfunc-
tion method is negligible. This work demonstrates that with the appropriate
modi(cid:12)cations to the pseudospectral parameters to re(cid:13)ect the di(cid:11)erent nature
of the metallic bonding, the pseudospectral method may be used to describe
metallicsystems with virtuallyno loss of accuracy.
Chapters3and4present methodstospeedtheconstructionoftheoperators
required in ab initio electronic structure theory. Operator construction is the
most computationallyintensive part of an ab initio calculation, and fast oper-
ator construction is essential for performing this level of calculation on large
molecules. Even with fast operator construction ab initio calculations can be
di(cid:14)cultonlargemoleculesbecauseofslowconvergenceofthewavefunctionthat
describes the electronic structure. The electronic wave function is obtained in
an iterative fashion. A large number of iterations does not pose a serious dif-
(cid:12)culty for small molecules because little work is required for each iteration; as
the molecular size increases the work required for each iteration becomes sig-
ni(cid:12)cant, and an e(cid:14)cient method for converging the electronic wave function is
essential. Chapter 5 introduces the Direct Inversion in the Iterative Subspace
method, developed by Pulay for converging Hartree-Fock wave functions, and
extends ittogeneralwavefunctions witharbitrarynumbersofdoubly-occupied
core orbitals, singly-occupied open-shell orbitals, and variably occupied GVB
natural orbitals. These general wave functions are necessary to describe physi-
7
cal properties of real chemical systems. The convergence method described in
Chapter5rapidlyconverges these wavefunctions,ofteninafractionofthetime
required by standard convergence methods.
The combination of the fast operator construction described in Chapters 3
and 4 and the rapid convergence algorithm described in Chapter 5 provides
a method of accurately computing the electronic structure of large molecules.
Chapter 6 describes an application of these methods to porphyrin molecules.
Porphyrin moleculesare tetrapyrrole rings thatappear inavarietyofbiological
applications including the photosynthetic reaction center and the heme group,
as well as manyapplications in chemical catalysis. Because much of the chem-
istry of porphyrin rings proceeds through the excited states, and because por-
phyrins are primarily characterized through their optical absorption spectra,
the porphyrinexcited states havelongbeenanareaofintense experimentaland
theoreticalinvestigations. Approximatemethodsofelectronicstructure calcula-
tions have suggested that the porphyrin excited states are composedofcoupled
single excitations from the ground state. The combination of the large size of
the porphyrin rings and the multi-con(cid:12)gurationalnature of the excited states
have prevented ab initio calculations on the porphyrin excited states. Chapter
6 presents two di(cid:11)erent approaches to solve this di(cid:14)culty, both of which use
the pseudospectral method of operator construction. The (cid:12)rst method takes
advantage of the planar geometry of many porphyrin rings to separate the s
and p orbitals of the molecules. A potential to replace the s electrons is cal-
culated using the pseudospectral method. This potential is incorporated into
standard basis function methods for calculating the multi-con(cid:12)gurational ex-
cited states in the p space only. The second method calculates explicitly the
multi-con(cid:12)gurational excited state energies and optimum orbitals. All of the
operators required forthe energy expression and orbitaloptimizationequations
are computed using the pseudospectral method. Both methods yield excellent
agreement with experimental results.
This thesis also includes two appendices. The (cid:12)rst appendix describes an
e(cid:11)ective method to evaluate Hamiltonian matrix elements between electronic
wave functions. Such evaluations are necessary in deriving energy expressions
for Hartree-Fock, Generalized Valence Bond, and other types of wave func-
tions. Theyalsocanbeusefulincomputingcouplingtermsbetween groundand
excited determinants in Con(cid:12)guration Interaction and properties calculations.
The second appendix describes the energy expression and orbital optimization
equations for the Restricted Con(cid:12)guration Interaction wave function, a more
accurate way to include electron correlation in a wave function that does not
require afulltransformationofthetwo-electronintegrals. These appendices are
included because they maybe of help to other researchers in this (cid:12)eld.
The work presented in this thesis develops new methods forcalculating and
converging electronic wave functions. These methods allow ab initio electronic
structure theory calculations on molecules much larger than those that can be
addressed by standard basis function methods. The use of larger molecules,
in turn, allows theoretical chemists to use more accurate models to investigate
chemicalinteractions. This thesis alsopresents anapplicationofthese methods
8
to the study of porphyrin excited states. The calculations on the porphyrins
are signi(cid:12)cant not only because they demonstrate the types of molecules that
can be studied withthe pseudospectral method,but alsobecause these calcula-
tions show good agreement with experimental results for an importantclass of
molecules.
9
Chapter 2
Ab Initio
Overview of
Electronic Structure Theory
2.1 Introduction
Chapter 2 presents an overview of ab initio electronic structure theory and it
reviews the standard methods that relate to topics later in this thesis. Section
2.2 describes the nature of electronic wave functions in general, and the closed-
shellHartree-Fock(HF)wavefunctionsinparticular. Closed-shellHartree-Fock
wavefunctionsareimportantbecause theyarethesimplestwavefunctionsused
inelectronicstructure theory. Muchoftheworkdescribedinthisthesisinvolves
extending methods developed for closed-shell Hartree-Fock wave functions to
the moregeneral wave functions described in Section 2.7. Section 2.3describes
open-shell Hartree-Fock wave functions, an extension of closed-shell Hartree-
Fock theory to wave functions where some orbitals are singly-occupied. These
wave functions are signi(cid:12)cant because they are the simplest wave functions
where orbital optimizationbetween occupied orbitals is necessary.
Hartree-Fockcalculationsgenerateagreatdealofinformationabouttheelec-
tronic energy and density of the molecules they describe; Section 2.4 describes
how this informationis interpreted to yield chemically important insights. Al-
thoughHartree-Fockcalculationscanprovideagreatdealofimportantchemical
information,there aremanytypes ofsystemsforwhichthesewavefunctionsare
impractical or inappropriate, for example distorted or dissociated bonds or ex-
cited states. Section 2.5describes these shortcomings,andSection2.6describes
a convenient solution to many of them, namely the generalized valence bond
wave (GVB) function. Generalized valence bond wave functions introduce de-
grees offunctionalfreedomtotheelectronicwavefunctionandallowittoadjust
more accurately to its particular environment. In addition, the equations for
optimizingthis type of wave function are presented in this section.
Section 2.7 summarizes Sections 2.2{2.6 by presenting the form of the gen-
eralwavefunctionhavinganarbitrarynumberofdoubly-occupiedcore orbitals,
10