Performance of the Harris functional for extended basis sets at the Hartree–Fock and density...

Performance of the Harris Functional for Extended BasisSets at the Hartree–Fock and Density Functional Levels

JOHN CULLENDepartment of Chemistry, University of Manitoba, Winnipeg, Manitoba R3T 2N2, Canada

Received 7 July 2003; Accepted 27 October 2003

Abstract: The Harris functional is a noniterative variational procedure that uses an input charge density to producean energy that is surprisingly accurate compared to the converged Kohn–Sham self-consistent result. We adapted andgeneralized this functional for the Hartree–Fock closed- and open-shell cases as well as examined its use for hybriddensity functional methods such as B3LYP. Analysis of extended basis set calculations shows that at the B3LYP levelan input density formed from a double zeta � polarization orbital basis is accurate enough to reproduce the energy oftriple zeta � double polarization � diffuse orbital basis. For large molecules this translates into a computational speedthat can be an order of magnitude faster. In the case of Hartree–Fock calculations a “bootstrapping technique” thatemploys successive applications of the Harris functional can further reduce computational times while retainingsufficient accuracy.

© 2004 Wiley Periodicals, Inc. J Comput Chem 25: 637–648, 2004

Key words: Harris functional; Hartree–Fock; restricted open shell; B3LYP

Introduction

Most ab initio calculations are a practical compromise betweencomputational speed and accuracy. Although molecular geometrycan often be determined at lower levels of theory, obtainingsufficient accuracy in the energy usually means increasing thebasis set size by splitting atomic orbitals into finer segments oradding higher angular momentum functions or more diffuse orbit-als. This naturally increases the computational time. In iterativedirect calculations where integrals are generated “on the fly” as, forexample, Hartree–Fock (HF) and Kohn–Sham (KS) density func-tional theory (DFT) methods, this time is linearly proportional tothe number of cycles required to achieve self-consistency even forthe class of methods that scale linearly with system size.1 Oneattractive approximation that avoids iteration is the Harris func-tional. This variational procedure uses an input charge density toproduce in a “single shot” calculation an energy that is surprisinglyaccurate compared to the converged self-consistent result. Intro-duced independently by Harris2 and Foulkes3 the Harris functionalemploying a superposition of input atomic densities has beenextensively applied within the local density approximation for theinvestigation of very large systems in solid-state physics4 and alsobeen used to justify and develop semiempirical tight-binding mod-els.5 In comparison, investigations of the Harris functional appliedto higher-order gradient-corrected functionals or other self-consis-tent methods, not density functional based, have been far lessstudied. One exception has been the work of Chizmeshya and

Zaremba, who used a closed-shell HF adaptation to model therepulsive interactions of the physisorption of rare gas atoms withmetal surfaces.6

In this article we investigate the utility of the Harris functionalapproach for extended basis set calculations both at the HF andB3LYP levels.7 We begin by rederiving the theory for the closed-shell HF method and then generalize these results to the multide-terminant restricted open-shell HF (ROHF). As well, the Harrisfunctional formulation for the hybrid DFT method B3LYP is alsopresented. Finally, numerical calculations are examined to ascer-tain the accuracy and efficiency of the Harris functional approach.For completeness the unrestricted HF (UHF) version of the Harrisfunctional is also presented as an appendix.

Restricted Hartree–Fock Harris Functional

For an initial set of approximate orthonormal molecular orbitals{�i} a Fock matrix can be constructed for the closed-shell casewith matrix elements given by

��i�F��j� � ��i�h��j� � 2 �k

occ.

��i�j��k�k� � 1/2��i�k��k�j��

(1a)

© 2004 Wiley Periodicals, Inc.

where h is the usual one-electron operator containing kineticenergy and nuclear–electron attraction terms and Mulliken nota-tion is used for the two electron integrals, i.e.,

��i�j��k�l� � �� *i�r�1��j�r�1�1

r12�*k�r�2��l�r�2�dr�1dr�2

Diagonalization of the Fock matrix produces a new set of molecularorbitals {��i} with corresponding orbital energies {��i} given by

��i � ��i�h��i� � 2 �j

occ.

��i��i��j�j� � 1/2��i�j��j��i�� (1b)

The corresponding variational energy formed from the singleSlater determinant of closed-shell orbitals {��i} is

ERHF � 2 �i

occ.

��i�h��i� � �ij

occ.

�2��i��i��j��j� � ��i��j��j��i�� Enuc

(2a)

where Enuc is the classical nuclear repulsion energy. Using eq.(1b), core Hamiltonian matrix elements, ��i�h��i� can be replacedin (2a) to produce

ERHF � 2 �i

occ.

��i � 2�C � �X � Enuc (2b)

Here �C, �X are defined by

�C � �ij

occ.

��i��i��j��j� � 2��i��i��j�j�� (2c)

�X � �ij

occ.

��i��j��j��i� � 2��i�j��j��i�� (2d)

The total Fock operator is a sum of one-electron operators F(r�k)with ground-state eigensolution

�k

F�r�k��0� � E0��0� (3a)

��0� �1

�2n!��1�� 1��2�� 2 · · · ��n�� n� (3b)

E0 � 2 �i

occ.

��i (3c)

Because E0, which is also the first term found in eq. (2b), repre-sents a variational bound to eq. (3) above, it is therefore invariantto any first-order change in ��i. The resulting density matrix �(r�1,r�2) is given by

��r�1, r�2� � 2 �i

occ.

��i�r�1��i�r�2� (4a)

Similarly, the input density matrix, (r�1, r�2), can be defined by

�r�1, r�2� � 2 �i

occ.

�i�r�1��i�r�2� (4b)

with the difference (r�1, r�2) between the two density matricesgiven by

��r�1, r�2� � �r�1, r�2� � �r�1, r�2� (4c)

The Coulombic term [eq. (2c)], �C, can then be reexpressed as

�C � �ij

occ.

��i��i��j��j � 2�j�j� �1

4 �� r�1, r�1�

� �r�1, r�1��1

r12��r�2, r�2� � �r�2, r�2� � 2�r�2, r�2��dr�1dr�2

� 1

4 �� r�1, r�1�1

r12�r�2, r�2�dr�1dr�2

�1

4 �� r�1, r�1��r�2, r�2� � �r�1, r�1��r�2, r�2��1

r12dr�1dr�2

�1

4 �� r�1, r�1�1

r12�r�2, r�2�dr�1dr�2 (5)

The last term above is of order 2 while the second last termvanishes, as can be seen by a simple interchange of the dummyvariables of integration. Therefore, to first order in �C becomes

�C � �1

4 �� (r�1, r�1)1

r12(r�2, r�2)dr�1dr�2�

ij

occ.

��i�i��j�j� (6a)

In similar fashion �X can be expanded as

�X �1

4 �� r�1, r�2� � �r�1, r�2��1

r12��r�1, r�2�

� �r�1, r�2��dr�1dr�2 �1

4 �� r�1, r�2�1

r12�r�1, r�2�dr�1dr�2

� O��r�1, r�2�2� � �

ij

occ.

��i�j��i�j� (6b)

Thus, the Harris functional for RHF found in eq. (2) is invariant toany first-order changes from the initial input density and reduces to

638 Cullen • Vol. 25, No. 5 • Journal of Computational Chemistry

ERHFHarris � 2 �

i

occ.

��i � �ij

occ.

�2��i�i��j�j� � ��i�j��i�j��

� Enuc � O�2� (7)

Comparison of eq. (7) with the true variational energy of eq. (2b)shows a difference of a contribution that is a second-order changein the input density, O(2). Because O(2) can be either pos-itive or negative the Harris functional does not in general providean upper bound to the energy.

ROHF Harris Functional

In the case of open molecules one has the option of applying at the HFlevel for open-shell systems the unrestricted or restricted open-shellmethods. The single determinant UHF wave function composed ofdifferent spatial orbitals for different spins yields a Harris functionalfor the energy that is easily derived in a manner similar to the RHFcase. Details of this derivation are given in the appendix. In compar-ison the ROHF wave function, unlike UHF, is a proper spin eigen-function of S2 and SZ. It is in general constructed from a linearcombination of several Slater determinants, each containing a com-mon set of doubly occupied closed-shell molecular orbitals, as well aseach containing a subset of partially occupied open-shell molecularorbitals. The arbitrary nature of the off-diagonal Lagrange multipliersused to ensure orthonormality between orbitals of different shells hasspawned several variants of ROHF theory.8,9 Here, we use the pro-jection method of Edwards and Zerner,10 which is similar to thatdeveloped earlier by Davidson.11

We begin by grouping orbitals into shells {�; � 1, 2, . . . N}according to their common occupancy �� where 0 � �� 2. Thevariational ROHF energy can then be written in compact form as

EROHF � ��

N

�� i��

��i��h��i

�� Enuc

�1

2 ��

N

��

N

� �i��

�j�

�a� ��i��i

��j ��j

��

�1

2b� ��i

��j ��j

��i�� (8)

where a� and b� are generalizations of the coupling coefficientsoriginally introduced by Roothaan.8 Coefficient values for the mostcommon open-shell cases are tabulated in the Edwards–Zerner arti-cle10 as well as recipes for their construction. Here, we extend thisformulation choosing the first shell, � 1, to be the closed shell with�1 2 and defining a 1 a1 b 1 b1 1.

Energy minimization with respect to variations of the molecu-lar orbitals produces a set of operators {R�}, each associated witha given shell with

R� � F� �1

��

�j�

��j ��j

�F � F ��j ��j

�� (9)

Here, F� is the Fock operator of the �th shell. The remainingterm* above arises from the requirement that the off-diagonalLagrange multipliers that enter into ROHF theory must be Hermi-tian. Due to orthogonality constraints among orbitals of differentshells, simply diagonalizing each R� within the given atomicorbital basis is insufficient. In addition, a method of successiveprojection is used to impose intershell orthogonality. This proce-dure begins with a set of orthonormal input orbitals from theprevious iteration or starting guess. The closed shell is then ini-tially solved and the remaining open shells that are arranged inorder of decreasing number of electrons are sequentially solved. Ateach step occupied orbitals are projected out of the basis set andthis reduced basis is then used to solve for the R�th eigenvalueequation of the next open shell. This results in

R��i�� i

��i�� i

��F��i�� Qi

��i�� (10a)

where ��i��F��i

�� and Qi�� are defined for real orbitals by

��i��F��i

�� i��h��i

��

�j�

� �a� ��i��i

��j �j

�

�1

2b� ��i

��j ��j

�i�� (10b)

Qi��

2

��

�j�

��i��F ��j

��j ��i

�� (10c)

In general, ¥j� ��j ��j

� represents the projector, P, onto theorbital subspace of the �th shell and any unitary transformationwithin the same subspace will produce an equivalent projector.Therefore, we choose the unitary transformation that diagonalizesF . This produces

F ��j � � �i

��j � (11a)

P � �j�

��j ��j

� � �j�

��j ��j

� (11b)

Expanding �i��

��i�� i

�� i�� (12a)

and noting that orbitals {�i�} are orthogonal to those in the

subspace of the �th shell allows eq. (10c) to be rewritten as

Qi��

2

��

�j�

�i ��i

��j �2 �

2

��

�j�

�i ��i

��j �2 (12b)

Thus,

*The arbitrary but nonzero constants �� are set to one in eq. (17b) of ref.10.

Harris Functional for Extended Basis Sets 639

�i�� i

��F��i�� O��i

��2� (13)

Substitution of eq. (13) into the ROHF variational energy, eq. (8),yields

EROHF � ��

N

�� i��

�i��

1

2 ��

N

��

N

� �a� �c� �

1

2b� �X

� �

� Enuc � ��

N �i��

O��i��2� (14)

Although the above is invariant to first-order changes in the inputorbitals {�i

�}, omitting the last term, ¥�N ¥i�� O([�i

�]2),

does not still guarantee an upper bound to the energy because thesigns of the second-order changes are unknown. In fact, one mightexpect that the eigenvalues, �i

, from eq. (12b) are normallynegative and therefore ¥�

N ¥i�� O([�i�]2) is positive. The

intermediate quantities �C��, �X

�� in eq. (14) are generalizations ofeqs. (2c) and (2d) and are defined by

�C� � �

i��

�j�

��i��i

��j ��j

�� 2��i��i

��j �j

�� (15a)

�X� � �

i��

�j�

��i��j

��j ��i

�� 2��i��j

��j �i

�� (15b)

Pairing up the Coulombic terms and expanding produces

�� C� � �C

�� i��

�j�

��i��i

��j ��j

� � 2�j �j

� � ��j ��j

��i��i

�� 2�i��i

�� r�1, r�1�

� ��r�1, r�1��1

r12� �r�2, r�2� � �r�2, r�2� � 2 �r�2, r�2��dr�1dr�2 � �� r�1, r�1� � �r�1, r�1��

1

r12��r�2, r�2�

� ��r�2, r�2� � 2 �r�2, r�2��dr�1dr�2 � 2 �� r�1, r�1�1

r12 �r�2, r�2�dr�1dr�2 �� r�1, r�1�

1

r12 �r�2, r�2�dr�1dr�2�

�� r�1, r�1� �r�2, r�2� � ��r�1, r�1� �r�2, r�2��

1

r12dr�1dr�2 �� r�1, r�1�

��r�2, r�2� � �r�1, r�1��r�2, r�2��1

r12dr�1dr�2 (16)

Here, �(r�1, r�2) is the difference between the output density andinput density for the �th shell, with the input density given as

��r�1, r�2� � �� i��

�i��r�1��i

��r�2� (17)

The final two terms in eq. (16) cancel against each other, leaving

�� C� � �C

�� 2�� i��

�j�

��i��i

��j �j

� � O��, �2�

(18)

In similar fashion

�� X� � �X

�� i��

�j�

�� i��r1��i

��r2�1

r12��j

��r1��j ��r2� � 2�j

�r1��j �r2��dr1dr2

� �� i��

�j�

�� j ��r1��j

��r2�1

r12��i

��r1��i��r2� � 2�i

��r1��i��r2��dr1dr2 � �� r�1, r�2� � ��r�1, r�2��

1

r12� �r�1, r�2� � �r�1, r�2�

� 2 �r�1, r�2��dr�1dr�2 �� r�1, r�2� � �r�1, r�2��1

r12��r�1, r�2� � ��r�1, r�2� � 2��r�1, r�2��dr�1dr�2

� 2�� r�1, r�2�1

r12 �r�1, r�2�dr�1dr�2 �� r�1, r�2�

1

r12 �r�1, r�2�dr�1dr�2� �� r�1, r�1�

�r�1, r�2� � ��r�1, r�2� �r�1, r�2��1

r12dr�1dr�2

�� r�1, r�2��r�1, r�2� � �r�1, r�2��r�1, r�2��

1

r12dr�1dr�2 � 2��

i��

�j�

��i��j

��j �i

�� O��, �2� (19)


The ROHF Harris functional energy is finally obtained when theresults of eqs. (18) and (19) are substituted back into eq. (14).

EROHFHarris � �

�

N

�� m��

�m��

1

2 ��

N

��

N

� �i��

�j�

�a� ��i��i

��j �j

�

�1

2b� ��i

��j ��j

�i�� Enuc � �

�

N �

N

O��, �2� (20)

Generalized Harris Functional

As previously pointed out, the Harris functional, although varia-tional in nature, does not in general provide an upper or lowerbound to the energy. If the input density is converged to itsself-consistent limit, however, then the functional must yield anenergy identical to that found for the self-consistent method onwhich it has been modeled. This suggests a more general approachfor constructing the Harris-like functional for a given quantummechanical method is to first write down the exact energy expres-sion and then decompose it into an invariant part comprised ofoptimized orbital energies {��i} and noninvariant part formed fromCoulombic and exchange molecular integrals over the set of opti-mized orbitals {��i}. This latter part as shown in the sequence ofeqs. (2)–(6) can be made invariant by introducing integrals involv-ing combinations of the input orbitals �i with the optimizedorbitals ��i. The net result, however, is the simple replacement ofthe optimized molecular orbitals with input orbitals in the Cou-lombic and exchange molecular integrals. This suggests that ageneralization of the Harris functional is to simply expand theexact energy into an invariant one-electron component and anoninvariant remainder that becomes a constant when input orbit-als are substituted in. Ideally, one should try to minimize the roleplayed by the noninvariant component. For example, in the RHFcase we have

ERHF � �i

occ.

��i � ��i�h��i� � Enuc (21a)

One could define the approximation

E � �i

occ.

��i � ��i�h��i� � Enuc (21b)

However, a better approximation is to expand the core Hamilto-nian term above into an invariant and noninvariant part:

��i�h��i� � ��i � 2 �ij

occ. ��i��i��j�j� �1

2��i�j��i�j�� (21c)

Because in general both terms on the right side of eq. (21c) arenegative in value as is the original ��i�h��i�, the magnitude and

therefore the role played by the noninvariant part is indeed re-duced. Substitution back into eq. (21a) followed by replacement of��i orbitals in the resulting molecular two-electron integrals with�i orbitals reproduces eq. (7).

In a similar fashion an ROHF Harris-like functional can bederived. Here, as a test case, consider Roothaan’s ROHF singleFock matrix formulation.9 Unlike the Edwards–Zerner ROHFmethod given in the previous section, the Roothaan formulationcannot be converted into a Harris functional in the strict sense.

We begin by defining f, the fractional occupancy, as the num-ber of occupied open-shell orbitals divided by the number ofavailable open-shell orbitals. Indices m, n will be used to labelopen-shell orbitals while indices i, j will be reserved for closed-shell orbitals. According to Roothaan the total self-consistentenergy becomes

EROHF � �i

��i � ��i�h��i�� f �m

��m � ��m�h��m��

� f �im

occ.

�2��i��i��m��m� � ��i��m��i��m��

� f3 �nm

occ.

�2��n��n��m��m� � ��n��m��n��m�� Enuc (22)

where � and � are numerical constants that depend upon theparticular spin state being calculated. For example, in the high-spincase, i.e., all spins of open-shell orbitals are parallel, � 0, � 2, and f 1/ 2. In general, the corresponding closed- andopen-shell orbital energies ��i, ��m are given, respectively, by

��i � ��i�h��i� � 2 �j

��i��i��j��j� � 2f�1 � �� n

��i��i��n��n�

� �j

��i��j��j��i� � f�1 � �� j

��i��n��n��i� (22a)

��m � ��m�h��m� � 2 �j

��m��m��j��j� � 2f�1 � ��2f � 1��

� �n

��m��m��n��n� � �j

��m��j��j��m�

� f�1 � ��2f � 1�� j

��i��n��n��i� (22b)

Using the above equations to replace core Hamiltonian matrixelements in eq. (22) followed by substitution of input orbitals intothe resulting two-electron integrals produces a generalized Harrisfunctional for this ROHF method:

EROHF � 2 �i

��i � 2f �m

��m � 2 �ij

��i�i��j�j�

� �ij

��i�j��j�i� � 4f�1 � �� im

��i�i��m�m� � 2f �1 � ��


� �im

��i�m��m�i� � 2f2�1 � ��3f � 1�� nm

��n�n��m�m�

� f2�1 � ��3f � 1�� nm

��n�m��m�n� � Enuc (23)

B3LYP Harris Functional

In KS DFT,12 one begins with the assumption of an independentparticle system that has the same charge density as the real system.The Hamiltonian that characterizes this fictious system has theform

HKS � �k

nelec.

H�r�k� (24a)

where

H�r�k��i�r�k� � �1/2 k2 � Vnuc-el � VC � VXC��i�r�k� � ��i�r�k�

(24b)

Vnuc-el, VC, VXC represent the nuclear–electronic attraction, Cou-lombic repulsion, and exchange correlation potentials, respec-tively. These are defined in relation to the charge density �(r�1, r�2)by

Vnuc-el�r�� ¥A ZA�r�, r��

�r� � R� A�(25a)

VC�r�� r�1, r�2�

�r� � r�1�dr�1 (25b)

VXC�r�� EXC��

��(25c)

EXC[�] found in eq. (25c) is the exchange correlation energyfunctional. In the case of B3LYP7 this functional is a fittedcombination of several other functionals, namely,

EXC�� 0.2EHF�� EXSlater�� EX

B88�� ECLYP�� EC

VWN��

(26)

EHF is the HF functional while EXSlater and EX

B88 are exchangefunctionals devised, respectively, by Slater13 and Becke.14 Thecorrelation functionals EC

LYP, ECVWN were developed respectively

by Lee, Yang, and Parr (LYP)15 and Vosko, Wilk, and Nusair(VWN).16 The latter, EC

VWN, actually refers to two local densityfunctionals: one found from fitting to the random-phase approxi-mation and the other to Monte Carlo calculations of the homoge-neous electron gas. This duplicity has caused some come confu-sion in the literature.17 The total KS DFT energy for the realsystem is given by

EKS � �i

��i�h��i� � 1/2 �ij

��i��i��j��j� � EXC�� Enuc (27)

Following Foulkes and Haydock5 we expand the exchange corre-lation energy functional about the input density �(r�1, r�2)

EXC�� EXC�� EXC��

�� dr � O�2� (28a)

In addition, from the KS eq. (24) we obtain

��i�h��i� � ��i � �j

��i��i��j�j� � � �EXC��

��dr (28b)

Substitution of these results into eq. (27) yields

EKS � �i

��i � 1/2 �ij

��i��i��j��j � 2�j�j� � EXC��

�� EXC��

��dr � Enuc � O�2� (29a)

Noting that the second term above is the Coulombic term, �C, ofeq. (5), we obtain the final form of the DFT Harris functional as

EKS � �i

��i � 1/2 �ij

��i�i��j�j� � EXC��

�� EXC��

��dr � Enuc (29b)

Implementation

Harris functional routines were implemented into the GAMESS18

program. The corresponding orbital method19 was employed totransform computed lower-level basis set molecular orbital solu-tions into higher-level basis set representations that were then usedto construct input densities. For Harris functional calculationsinvolving the union of a lower-level normalized basis {��} and anauxiliary normalized set {�a} the density of the converged lower-level calculation can be used directly. This is accomplished bySchmidt orthogonalizing the auxiliary basis {�a} against {��} to

form a modified basis ��a˜ � given by

�a˜ � �a��a � �

�

S�a�� (30a)

where �a is a normalization constant and S�a� represents apseudooverlap. These factors are defined by

S�a� � ��

Sa T �1 (30b)


�a � �1 � ��

S�a�S�a�1/ 2

(30c)

Here, S is the overlap matrix for the entire basis {��, �a} while Tis the corresponding overlap matrix for the subset {��}. Becausein this case the input density has no contributions from the Schmidt

orthogonalized auxiliary basis ��a˜ � the noninvariant part of the

Harris functional can be taken directly from the lower-level cal-culation’s converged results. Matrix elements from the lower-levelcalculation can also be directly used to form part of the new Fockor KS matrix. To complete these matrices one must also include

��a˜ �F�� and ��a

˜ �F��b˜ �. These are given, respectively, by

��a˜�F�� a��a�F��

�

S�a��F�� (31a)

��a˜�F��b

˜� � �a�b��a�F��b� � ��

�S�a��F��b� � ��a�F��S�b��

� �a�b ��

S�a��F�� S�b (31b)

In addition, new overlap matrix elements ��a˜ ��b˜ � are also needed to

solve the eigenvalue equation for the orbital energies. These aregiven by

Table 1. Harris Functional Statistics.

Harris functional typea

Relative error � 105 in energy Computational speedup

Mean Maximum Minimum Mean Maximum Minimum

RHF resultsb

dz f tzp 5.48 12.63 0.04 6.34 8.00 4.50dz f tzdp� 8.67 19.73 4.19 9.74 28.75 6.00dzp f tzp 0.94 3.75 0.20 1.96 2.98 1.15dzp f tzdp� 2.10 6.45 0.85 5.81 15.50 3.62tzp f tzdp� 0.89 3.03 0.13 3.61 12.29 2.25dz f f tzp 1.03 3.87 0.12 4.63 5.92 2.50dz f f f tzdp� 0.88 3.59 0.06 7.25 22.65 3.60tzp� f tzqp�� 0.80 1.13 0.53 7.18 9.83 5.36

B3LYP resultsc

dz f dzp 3.51 10.38 0.53 1.75 3.17 0.90dz f tzp 3.66 14.06 0.02 2.64 5.60 1.55dz f tzdp� 3.13 12.86 0.12 5.19 10.28 2.20dzp f tzp 0.23 0.92 0.00 1.51 4.18 0.98dzp f tzdp� 0.52 1.46 0.00 3.37 6.70 1.79tzp f tzdp� 0.25 0.92 0.04 2.31 4.64 0.73dz f f tzp 2.84 9.01 0.36 2.31 4.88 1.40dz f f f tzdp� 5.85 37.30 0.37 4.11 8.35 1.79

ROHF resultsd

dz f dzp 5.03 7.83 0.94 5.27 6.88 3.333.86 7.03 0.58 4.71 5.87 3.00

dz f tzp 6.51 10.58 2.79 8.70 10.34 6.254.95 8.07 1.67 7.51 9.35 5.31

dz f tzdp� 0.84 1.82 0.29 2.22 2.74 1.630.61 1.00 0.16 2.17 2.52 1.71

dzp f tzp 7.29 12.08 2.93 12.83 15.49 7.794.76 8.62 1.75 12.14 16.45 7.78

dzp f tzdp� 1.60 3.44 0.45 6.13 7.73 4.960.77 1.42 0.08 6.46 7.78 4.93

tzp f tzdp� 0.45 0.86 0.10 3.15 4.03 2.500.31 0.97 0.09 3.47 4.33 2.64

dz f f tzp 0.93 2.20 0.23 6.99 8.33 5.000.46 0.90 0.11 5.92 7.56 4.25

dz f f f tzdp� 0.53 1.14 0.04 9.87 12.24 5.640.28 0.72 0.02 9.27 12.71 5.60

adz 3-21G basis, dzp 6-31G(d) basis, tzp 6-311G(d,p) basis, tzdp� 6-311�G(2d,2p), tzp� 6-311�G(d,p), tzqp�� 6-311��G(3d1f,3p1d) basis.b66-molecule sample size.c60-molecule sample size.d12-molecule sample size, with generalized Harris functional results given in second rows.


��a˜��b˜� � �a�b�Sab � �

�

Sb T �1S�a� (32)

All calculations were performed on an Athlon 1.3-GHz per-sonal computer within a Linux operating environment.

Results and Discussion

To assess the accuracy and performance of the Harris functionalRHF, ROHF, and B3LYP calculations were carried out usingPople-type basis sets for a wide variety of systems ranging fromstrained molecules such as cubane, to delocalized ones such asnaphthalene, to transition metal complexes such as hexaaquairo-n(II) ion, to polarized systems such as pentafluoropyridine. Table1 provides an overview of these results. Here, the acronyms dz,dzp, tzp, tzdp�, and tzqp�� represent the corresponding basissets 3-21G, 6-31G(d), 6-311G(d,p), 6-311G�(2d2p), and6-311��G(3d1f,3p1d), respectively. The notation X f Y de-

notes that the density found from an X basis set calculation is usedin the Harris functional to produce a Y basis set energy estimate.The RHF Harris functional tzp� f tzqp�� provides an exam-ple of the union of a lower-level basis, 6-311�G(d,p), with anauxiliary set of diffuse and polarization functions to produce ahigher-level 6-311��G(3d1f,3p1d) basis set. In general, thisapproach is less flexible but allows the use of a Schmidt orthogo-nalization procedure that pares down the number of two-electronintegrals needed overall, providing a slight gain in computationalspeed. A “bootstrapping” technique is also examined that beginswith a 3-21G self-consistent field (SCF) calculation. The corre-sponding orbital method19 is then used to convert the 3-21Gorbitals into a 6-31G(d) basis and the resulting density is then usedto generate a Harris functional solution at the 6-31G(d) level. Thesingle diagonalization step that produces the orbital eigenvalues{��i} needed in this Harris functional also produces a new im-proved set of orbitals {��i} at the 6-31G(d) level. Again, thecorresponding orbital method is employed now with the orbitals{��i} and a new density is constructed at the 6-311G(d,p) level

Table 2. ROHF Energies and Computation Times.

Molecule Spin state E[tzdp�]a T (s) �E1b T (s) �E2

c T (s) �E3d T (s) �E4

e T (s)

Ethylene Triplet 77.9609 26.5 9.42 3.4 2.68 5.1 0.67 10.5 0.89 4.730.2 3.98 3.5 0.93 5.0 0.76 10.0 0.56 4.9

Ethylene (twisted 90°) Triplet 77.9855 32.9 8.73 3.5 2.29 5.6 0.55 12.4 0.71 4.728.0 4.38 3.6 0.09 5.1 0.28 10.2 0.19 5.0

Cyclopropenyl radical Doublet 115.1341 120.8 10.28 7.8 2.2 17.0 0.5 36.4 0.75 10.396.7 2.76 8.0 0.09 13.2 0.19 26.5 0.68 10.9

Cyclopropenyl radical Quartet 115.2245 94.5 11.37 7.5 2.81 15.2 0.82 31.1 0.94 10.088.5 7.94 7.8 1.26 13.1 0.13 24.5 0.07 10.6

HC4N (linear) Triplet 206.3739 236.0 14.49 15.4 3.03 41.7 1.05 65.2 1.08 20.5174.5 9.47 15.6 1.97 35.4 0.64 56.8 0.05 20.4

Pyridine cation Doublet 246.4432 1095.4 19.06 78.1 2.68 178.2 0.98 330.3 1.07 100.7796.4 16.65 77.6 2.45 145.4 0.87 244.9 0.56 100.9

Benzyl radical Doublet 269.1795 2347.3 25.33 159.0 4.7 303.7 1.48 582.1 1.58 201.42324.8 23.21 176.3 3.69 298.8 1.16 546.9 1.18 220.3

2,5 Didehydrophenol Triplet 304.3455 1776.9 26.59 130.3 5.41 255.5 2.22 461.6 2.25 145.21648.8 22.50 117.0 4.32 225.4 1.78 380.7 1.22 148.4

Manganocene Doublet 1534.1750 13,929.9 47.87 1047.2 8.07 2408.0 1.56 4583.5 3.42 1419.417,989.6 43.75 1198.2 7.47 2789.6 1.52 5437.2 6.43 1555.1

Manganocene Sextet 1534.4133 11,495.9 45.08 922.6 7.78 1970.3 2.11 4072.2 1.87 1210.814,958.9 26.81 1036.4 4.81 2070.3 1.75 4243.4 0.54 1329.7

[Fe(H2O)6]�2 Quintet 1718.4294 3168.6 59.62 280.1 15.06 638.7 2.92 1267.0 3.62 372.33418.4 42.00 303.3 13.37 659.5 2.58 1293.6 3.28 394.9

Cobaltocene Doublet 1765.8472 14,531.4 51.78 1048.3 7.87 2376.4 1.87 4684.7 0.77 1378.519,609.5 46.68 1192.0 6.94 2602.5 1.66 4604.2 0.87 1543.2

Harris functional and generalized Harris functional results are given in the first and second rows, respectively.aTotal 6-311G�(2d,2p) energy in Hartrees.b�E1 is the energy difference in millihartrees between the exact 6-311G�(2d,2p) ROHF result and the Harrisfunctional energy using a 3-21G ROHF density.c�E2 is the energy difference in millihartrees between the exact 6-311G�(2d,2p) ROHF result and the Harrisfunctional energy using a 6-31G(d) ROHF density.d�E3 is the energy difference in millihartrees between the exact 6-311G�(2d,2p) ROHF result and the Harrisfunctional energy using a 6-311G(d,p) ROHF density.e�E4 is the energy difference in millihartrees between the exact 6-311G�(2d,2p) ROHF result and the Harrisbootstrapping functional energy using an initial 3-21G ROHF density.


that is finally used to generate a Harris functional solution at thisbasis set level. This is indicated by the notation dz f f tzp. Afurther step to the 6-311G�(2d,2p) level is indicated by dz f ff tzdp�. For both RHF and ROHF calculations this latterbootstrapping approach provides an optimal compromise betweenaccuracy and speed. For example, in comparison to the exactself-consistent RHF calculation, for every 100 a.u. of energy theerror made by the Harris functional is 0.8 millihartrees or less onaverage while the gain in computational speed ranges from 3.6 forsmall molecules to almost a factor of 23 for large ones.

In contrast to the RHF and ROHF results the bootstrappingmethod fails for B3LYP calculations. Here, the Harris functionaldzp f tzdp� appears to be optimal with mean errors of 0.52millihartrees per 100 a.u. and gains in computational speed rangingfrom 1.79–6.70. There seems to be a less dramatic increase in

speed when B3LYP Harris functional results are compared to theRHF ones. Due to convergence difficulties KS self-consistentcalculations need much better starting input densities than HFones, where semiempirical Huckel densities can be used. Wetherefore employed precalculated RHF densities as guesses andthis has reduced the number of iterations required to reach con-vergence. The time required to compute these guess densities hasnot been included.

For the ROHF calculations, high-spin cases have only beenexamined. Results both for the exact Harris functional derivedfrom the Edwards–Zerner method10 and the generalized Harrisfunctional based on Roothaan’s single matrix formulation8 aregiven in Table 1. The latter is clearly superior with relative errorsoften half those found from the exact ROHF Harris functional.This is also seen in more detail in Table 2. Computational times

Table 3. RHF and B3LYP Energies and Computational Times.

Molecule

RHF B3LYP

E[tzdp�]a T (s) �E1b T (s) �E2

c T (s) E[tzdp�]a T (s) �E1b T (s) �E2

c T (s)

Acetonitrile 131.9624 57.7 2.14 10.7 0.87 8.8 132.7194 145.3 0.61 46.7 1.39 40.1trans Butene 156.1608 339.3 3.17 59.8 0.82 55.8 157.1675 634.3 0.99 166.4 6.37 175.2Acetic acid 227.8883 173.7 4.03 33.5 1.47 26.1 229.0551 303.2 0.96 94.9 19.29 84.6Borazine 241.2310 672.8 5.15 137.1 2.00 111.7 242.6071 1135.3 1.35 300.5 11.03 235.5Cubane 307.4677 4524.7 3.37 458.7 0.87 384.7 309.3395 4179.9 1.08 810 46.31 703.2Trioxane 341.7565 791.4 4.32 151.7 2.03 119.7 343.5018 1461.4 1.18 364.5 18.31 272.7BF3NH3 379.5486 228.1 5.67 48.0 2.82 36.5 381.1409 432.5 0.49 138.1 7.39 100.9Hexafluorocyclohexane 629.4734 3883.1 9.06 700.6 4.32 576.4 632.3543 7354.3 1.40 1359.1 24.58 1043.3Pentafluoropyridine 740.7925 2059.6 10.06 434.3 6.80 283.6 743.9750 3715.9 0.87 810.4 17.69 499.6Trinitrobenzene 841.3036 10,333.6 7.82 1303.3 1.64 902.3 845.5619 12,404.5 1.60 1850.6 213.96 1521.6Dibenzo[e,1]pyrene 917.2775 67,367.9 10.88 7929.1 4.15 6988.9 922.7127 90,892.1 3.11 14,415.4 29.46 10,883.1

aTotal 6-311G�(2d,2p) energy in Hartrees.b�E1 is the energy difference in millihartrees between the exact 6-311G�(2d,2p) result and the Harris functionalenergy using a 6-31G(d) density.c�E2 is the energy difference in millihartrees between the exact 6-311G�(2d,2p) result and the Harris functional 3-21Gf f f 6-311G�(2d,2p).

Table 4. 6-311G�(2d,2p) Structural Isomer Energy Differences (kcal/mol).

Low-energy isomer High-energy isomer �ERHF �E1a �E2

b �EB3LYP �E1a

Hydrogen cyanide Hydrogen isocyanide 10.74 11.28 10.79 14.20 14.27Formaldehyde Hydroxymethylene 48.46 49.52 48.64 52.43 52.67Formamide Nitrosomethane 69.97 68.99 69.66 65.21 65.10Nitromethane Methyl nitrite 0.85 1.08 0.55 4.10 4.10Dimethyl ether Ethanol 11.33 10.62 11.20 11.22 11.091,2-Ethanediol Ethanehydroperoxide 60.79 59.49 60.34 50.97 50.74

Dimethylperoxide 69.24 67.39 68.71 59.18 58.83Vinylsilane 1-Methylsilaethylene 27.53 28.04 28.90 9.40 9.57Propyne Allene 2.06 2.13 2.19 1.71 1.62

Cyclopropene 28.69 28.49 28.63 24.80 24.84Acetone Prop-1-ene-2-ol 18.68 19.68 19.08 15.80 16.01trans Butene 1-Methylcyclopropene 9.51 9.13 9.21 8.97 8.87

a�E1 is the Harris functional 6-31G(d) f 6-311G�(2d,2p) result.b�E2 is the Harris functional 3-21G f f f 6-311G�(2d,2p) result.


between the two methods vary because the exact SCF calculationsthat are used to generate initial input guess densities or exactbenchmarks converge differently. For example, the6-311G�(2d,2p) energy of hexaaquairon(II) ion, 1718.4294Hartrees, requires 3418.4 s for Roothaan’s single matrix formula-tion and 3165.6 s for the Edwards–Zerner procedure. Comparingthe exact Harris functional results with the generalized versiongiven in parentheses one finds that Harris functional energies dzftzdp�, dzp f tzdp�, tzp f tzdp�, and dz f f f tzdp�,respectively, differ from the ROHF value by 59.6 (42), 15.1 (13.4),2.9 (2.6), and 3.6 (3.3) millihartrees for this compound. Thecorresponding computational times are 280.1 (303.3), 638.7(659.5), 1267.0 (1293.6), and 372.3 (394.9) s. Similarly, the dou-blet-sextet energy difference for manganocene is 238.3 millihar-trees at the 6-311G�(2d,2p) level. The Harris (generalized) func-tional dz f f f tzdp� predicts the difference to be 233.0(231.3) millihartrees. The error of several millihartrees is unusu-ally high but is a much better estimate than the 3-21G result of270.9 millihartrees for the starting point of this Harris functional.

In the case of RHF and B3LYP, representative results for theHarris functionals dzp f tzdp� and dz f f f tzdp� arepresented in Table 3. For HF the worst errors occur for highlyfluorinated species such as hexafluorocyclohexane and pentaflu-oropyridine. In comparison the B3LYP dzfff tzdp� resultsappear to be insensitive to polarization and delocalization effects.For ordinary B3LYP self-consistent calculations with extendedbasis sets with diffuse functions on unusual molecules as, forexample, octafluorocubane convergence problems do arise. Inthese cases the use of the Harris functional is in particular advan-tageous.

In Table 4 relative energies are examined for several structuralisomers whose geometries were first optimized at the 3-21G RHFlevel. The B3LYP dzp f tzdp� Harris functional shows excel-lent agreement with the KS B3LYP 6-311�G(2d,2p) results witherrors of 0.3 millihartrees or less. The agreement is not as good forthe dz f f f tzdp� bootstrapping Harris functional at the HFlevel, where errors in the energy can be as high as 1.4 millihartreesfor the methyl nitrite–nitromethane case. Unusual polarized sys-tems appear to be problematic. Experimentally, the nitromethaneisomer is the more stable one by 3.7 kcal/mol,20 which contradictsthe RHF values but does agree with the B3LYP 4.1-kcal/molprediction. The use of low-level 3-21G RHF-optimized geometriesprevents one from obtaining an exact comparison with experimentand may explain why the B3LYP calculations for the propyne–allene isomer pair appear to not agree with literature values.21

Finally, there is the question of whether a Harris functional hasa basis set superposition error (BSSE) similar to the extended basisit approximates or whether it is similar to the lower-level basis setused to generate its input density. To resolve this, calculationswere performed at the HF level on the “T-shaped” benzene dimerfor varying separation distances between the centers of the ben-zene rings. These results, plotted in Figure 1 show good agree-ment, to within 0.1 millihartrees, between the 6-311G�(2d,2p)energy profile and the dz f f f tzdp� Harris functionalresults. The computing times of the Harris functional, though, areon average 9.5 times faster.

Conclusions

The noniterative Harris functional method provides a computation-ally cheap approach for obtaining accurate energy estimates forself-consistent-type calculations using extended basis sets. InB3LYP calculations, for example, using a double zeta � polariza-tion density, relative energies at the triple zeta � double polariza-tion � diffuse basis level are well produced (Table 4). In the caseof RHF and ROHF calculations a bootstrapping technique thatfurther reduces computational times can be used for molecules thatare not highly polarized. Finally, BSSEs of the Harris functionalare as small as those found in the exact extended basis set calcu-lations that they mimic.

Acknowledgments

This work was initiated a few years ago during the author’ssabbatical at NRC in Ottawa. The author is grateful for the supportand stimulating conversations that took place during his stay withJohn Tse and Dennis Klug. He also gratefully acknowledges DanEdwards (University of Idaho) for helpful discussions on the finerpoints of ROHF theory. Finally, he thanks his chemistry depart-ment for providing the funding for the Athlon personal computeron which all calculations were performed.

Appendix: UHF Harris Functional

The UHF variational energy formed from the single Slater deter-minant composed of orbitals {�i

��, i 1, 2, . . . , N�; �i��, i

1, 2, . . . , N�} for N� electrons of � spin and N� electrons of �spin, respectively, is given by22

EUHF � �i

N�

��i��h��i

�� i

N�

��i��h��i

��

Figure 1. T-shaped benzene dimer.


�1

2 �i

N�

�j

N�

��i��i

��j��j

�� i��j

��j��i

��

�1

2 �i

N�

�j

N�

��i��i

��j��j

�� i��j

��j��i

��

� �i

N�

�j

N�

��i��i

��j��j

�� Enuc (A1)

The corresponding orbital energies are

�i�� i

��h��i��

j

N�

��i��i

��j��j

�� i��j

��j��i

��

� �j

N�

��i��i

��j��j

�� (A2)

�i�� i

��h��i��

j

N�

��i��i

��j��j

�� i��j

��j��i

��

� �j

N�

��i��i

��j��j

�� (A3)

One can eliminate the core Hamiltonian terms in eq. (A1) bysubstitution of the orbital energies given above. In similar fashionto the RHF Harris functional derivation we first define the inter-mediate quantities

�C��

i

N�

�j

N�

��i��i

��j��j

�� 2��i��i

��j��j

�� (A4)

�X��

i

N�

�j

N�

��i��j

��j��i

�� 2��i��j

��j��i

�� (A5)

�C��

i

N�

�j

N�

��i��i

��j��j

�� 2��i��i

��j��j

�� (A6)

By a simple interchange of � and �, terms �C��, �X

��, �C�� can also

be defined. The UHF variational energy, eq. (A1), then reduces tothe simple form

EUHF � �i

N�

�i��

i

N�

�i�� C

�� X�� C

�� X��

� �C�� C

�� Enuc (A7)

The total UHF Fock operator F is a sum of one-electron operators¥i

N�

F�(r�k) � ¥iN�

F�(r�k) of � and � spin with ground-stateeigensolution:

F��0� � E0��0� (A8)

��0� �1

��N� � N��!��1

��2�� · · · �N�

� ��1��2

�� · · · �N�� (A9)

E0 � �i

N�

�i��

i

N�

�i�� (A10)

E0 is a variational bound and therefore the sum of the first twoterms in eq. (A7) is invariant to any first-order change in theorbitals {�i

��, �i��}. The two-electron quantities �C

��, �X�� are

isomorphic to �C, �X defined in eqs. (2c) and (2d). However,instead of functionals of the total electron density matrix, �C

��, �X��

are functionals of the � spin density matrix given by

��r�1, r�2� � �i

N�

�i��r�1��i

��r�2� (A11)

Expanding back into the old input density produces similar resultsto those found in eqs. (6a) and (6b), with

�C��

i

N�

�j

N�

��i��i

��j��j

�� O��r�, r��2� (A12)

�X��

i

N�

�j

N�

��i��j

��j��i

�� O��r�1, r�2�2� (A13)

In the same manner �C��, �X

�� can also be reduced. The finalCoulombic terms, �C

�� and �C��, found in eq. (A7) sum to become

�C�� C

�� i

N�

�j

N�

��i��i

��j��j

�� 2�j��j

�� j��j

��i��i

�� 2�i��i

�� r�1, r�1� � ��r�1, r�1��1

r12��r�2, r�2�

� ��r�2, r�2� � 2��r�2, r�2��dr�1dr�2 � �� r�1, r�1� � ��r�1, r�1��1

r12��r�2, r�2� � ��r�2, r�2� � 2��r�2, r�2��dr�1dr�2

� 2 �� r�1, r�1�1

r12��r�2, r�2�dr�1dr�2 �� r�1, r�1�

1

r12��r�2, r�2�dr�1dr�2� �� r�1, r�1�

��r�2, r�2�

� ��r�1, r�1��r�2, r�2��1

r12dr�1dr�2 �� r�1, r�1�

��r�2, r�2� � ��r�1, r�1��r�2, r�2��1

r12dr�1dr�2 (A14)


�(r1, r2) and �(r1, r2) are the respective differences betweenthe output and input densities for � and � spin. The last two termsin eq. (A14) cancel against each other, leaving as a final result

�C�� C

�� 2 �i

N�

�j

N�

��i��i

��j��j

�� O��, ��2 (A15)

Combining all these results into eq. (A7) finally produces the UHFHarris functional:

EUHFHarris � �

i

N�

�i��

i

N�

�i��

1

2 �i

N�

�j

N�

��i��i

��j��j

��

� ��i��j

��j��i

�� 1

2 �i

N�

�j

N�

��i��i

��j��j

�� i��j

��j��i

��

� �i

N�

�j

N�

��i��i

��j��j

�� O��, ��2 (A16)

References

1. Goedecker, S. Rev Mod Phys 1999, 71, 1085–1123.2. Harris, J. Phys Rev B 1985, 31, 1770–1779.3. Foulkes, W. M. C. Pre-Ph.D. thesis (1985) and Ph.D. thesis; University

of Cambridge: Cambridge, UK, 1988.4. Sankey, O. T.; Demkov, A. A.; Windl, W.; Fritsch, J. H.; Lewis, J. P.;

Fuentes-Cabrera, M. Int J Quantum Chem 1998, 69, 327–349.5. Foulkes, W. M. C.; Haydock, R. Phys Rev B 1989, 39, 12520–12536.

6. Chizmeshya, A.; Zaremba, E. Surf Sci 1992, 268, 432–456;Chizmeshya, A. Ph.D. thesis; Queens University: Kingston, Ontario,Canada, 1992.

7. Stephens, P. J.; Devlin, J. F.; Chabalowski, C. F.; Frisch, M. J. J PhysChem 1994, 98, 11623.

8. Roothaan, C. C. J. Rev Modern Phys 1960, 32, 179–185.9. See, e.g., Berthier, G. In Molecular Orbitals in Chemistry, Physics and

Biology; Lowdin, P. O.; Pullman, B., Eds.; Academic Press: NewYork, 1964, pp. 57–85.

10. Edwards, W. D.; Zerner, M. C. Theor Chim Acta 1987, 72, 347–361.11. Davidson, E. R. Chem Phys Lett 1973, 21, 565–567.12. Kohn, W.; Sham, L. J Phys Rev A 1965, 140, 1133.13. Slater, J. C. Quantum Theory of Atomic Structure, vol. II; McGraw-

Hill: Toronto, 1960; Chapter 17, appendix 22.14. Becke, A. D. Phys Rev A 1988, 38, 3098.15. Lee, C.; Yang, W.; Parr, R. G. Phys Rev B 1988, 37, 785–789.16. Vosko, S. J.; Willk, L.; Nusair, M. Can J Phys 1980, 58, 1200–1211.17. Hertwig, R. H.; Koch, W. Chem Phys Lett 1997, 268, 345–351.18. Schmidt, M. W.; Baldridge, K. K.; Boatz, J. A.; Elbert, S. T.; Gordon,

M. S.; Jensen, J. H.; Koseki, S.; Matsunaga, N.; Nguyen, K. A.; Su,S. J.; Windus, T. L.; Dupuis, M.; Montgomery, J. A. J Comput Chem1993, 14, 1347–1363.

19. King, H. F.; Stanton, R. E.; Kim, H.; Wyatt, R. E.; Parr, R. G. J ChemPhys 1967, 47, 1936–1941.

20. Ray, J. D.; Geshon, A. A. J Phys Chem 1962, 66, 1750–1752; Knobel,Y. K.; Miroshnichenko, E. A.; Lebedev, Y. A. Bull Acad Sci USSRDiv Chem Sci 1971, 425–428; NIST Chemistry Web Book; http://webbook.nist.gov/chemistry.

21. Cox, J. O.; Pilcher, G. Thermochemistry of Organic and Organome-tallic Compounds; Academic Press: New York, 1970.

22. See, e.g., Szabo, A.; Ostlund, N. S. Modern Quantum Chemistry,Introduction to Advanced Electronic Structure Theory; Macmillan:New York, 1982, pp. 214–216.


Performance of the Harris functional for extended basis sets at the Hartree–Fock and density...

Documents

Transcript of Performance of the Harris functional for extended basis sets at the Hartree–Fock and density...