The exchange-correlation potential in ab initio density functional theory

The exchange-correlation potential in ab initio density functional theoryRodney J. BartlettQuantum Theory Project, Departments of Chemistry and Physics, University of Florida, Gainesville,Florida 32611

Ireneusz GrabowskiInstitute of Physics, Nicolaus Copernicus University, 87-100 Torun´, Poland

So Hirata and Stanislav Ivanova)

Quantum Theory Project, Department of Chemistry, University of Florida, Gainesville, Florida 32611

~Received 8 April 2004; accepted 2 September 2004; published online 28 December 2004!

From coupled-cluster theory and many-body perturbation theory we derive the localexchange-correlation potential of density functional theory in an orbital dependent form. We showthe relationship between the coupled-cluster approach and density functional theory, andconnections and comparisons with our previous second-order correlation potential@OEP-MBPT~2!~OEP—optimized effective potential!# @I. Grabowski, S. Hirata, S. Ivanov, and R. J. Bartlett, J.Chem. Phys.116, 4415~2002!#. Starting from a general theoretical framework based on the densitycondition in Kohn–Sham theory, we define a rigorous exchange-correlation functional, potential andorbitals. Specifying initially to second-order terms, we show that ourab initio correlation potentialprovides the correct shape compared to those from reference quantum Monte Carlo calculations, andwe demonstrate the superiority of using Fock matrix elements or more general infinite-ordersemicanonical transformations. This enables us to introduce a method that is guaranteed to convergeto the right answer in the correlation and basis set limit, just as doesab initio wave function theory.We also demonstrate that the energies obtained from this generalized second-order method@OEP-MBPT~2!-f# and @OEP-MBPT~2!-sc# are often of coupled-cluster accuracy and substantiallybetter than ordinary Hartree–Fock based second-order MBPT5MP2. © 2005 American Instituteof Physics. @DOI: 10.1063/1.1809605#

I. INTRODUCTION

There are two independent exact theories of the elec-tronic structure of atoms, molecules, and solids: the wavefunction theory ~WFT! and the density functional theory~DFT!. The former provides results which are guaranteed toconverge to the solutions of the nonrelativistic Schro¨dingerequation, as electron correlation and basis set are ex-tended. This provides the now well-known paradigm ofconverging, size-extensive results in electronic structuretheory, MBPT(2),CCD,CCSD,CCSD(T),CCSDT,CCSDT(Qf),CCSDTQ,Full CI ~CI—configuration in-dicator!, where after second-order perturbation theory, CCmeans coupled-cluster theory,1,2 and S indicates single exci-tations; D, double excitations; T, triple excitations; and Q,quadruple excitations (Qf indicates the factorizedapproximation3,4!. The full configuration interaction solution,which containsn-fold excitations forn electrons, is the bestpossible solution in a given basis set.5 DFT replaces themany-electron wave function theory, which in particular isexplicitly dependent on two-particle interactions, by whatwould have to be a complex, effective, local one-particletheory, where all unknown information is contained in anexchange-correlation functionalExc(r) and its associatedpotential Vxc(x1)5dExc@r#/dr(x1). Here x1 indicates the

space-spin coordinates of one electron. Whereas WFT is atheory that provides a prescription for obtaining increasinglymore accurate solutions of Schro¨dinger equation, DFT estab-lishes the existence ofExc , but does not provide the energyfunctional, nor systematic converging series of approxima-tions to it. Hence unlike WFT, DFT methods, as used today,whether local density, gradient corrected, or hybrid,6–9 donot offer guaranteed convergence to the right answer. How-ever, their one-particle structure makes it possible to treat,computationally, much larger systems than in correlatedWFT.

Previously10 we derived rigorous exchange-only andexchange-correlation11 KS-DFT ~KS—Kohn–Sham! meth-ods in a Gaussian basis set by variationally optimizing therespective orbital-dependent energy expression with respectto electron density under the constraint that KS orbitals arederived from a single multiplicative potential. This leads tooptimized effective potential method~OEP! for theexchange-only case10 and OEP-MBPT~2!-KS11 for the low-est order exchange correlation. Similar methods can bededuced12 by applying the Sham–Schlu¨ter equation13 to dif-ferent, converging approximations to Dyson’s equation. Inthis paper, however, we will start with a general theoreticalframework, which is based on the density condition in KStheory and which defines all orders of rigorous exchange-correlation functionals, potentials, and orbitals in a transpar-ent and robust fashion.

a!Present address: Analytic Solutions, 422 Park Avenue, Suite 200, Claren-don Hills, Illinois 60514.

THE JOURNAL OF CHEMICAL PHYSICS122, 034104 ~2005!

122, 034104-10021-9606/2005/122(3)/034104/12/$22.50 © 2005 American Institute of Physics

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http://dx.doi.org/10.1063/1.1809605

http://dx.doi.org/10.1063/1.1809605

In the following, we show the interconnections betweenthe CC and MBPT approach and DFT. This enables us tointroduceab initio DFT11,14as a method that is guaranteed toconverge to the right answer in the correlation and basis setlimit, just as doesab initio WFT. Furthermore, we apply thetheory in its initial MBPT~2! form to define a multiplicativecorrelation potential and demonstrate that unlike any of thecurrently used correlation potentials, thisab initio one hasthe correct shape and behavior. As a consequence, it alsoprovides improved results for energies. Furthermore, weshow that much better results for the exchange and correla-tion potentials are obtained when diagonal Fock matrix ele-ments are used@OEP-MBPT~2!-f# and@OEP-MBPT~2!-sc# inplace of the usual Kohn–Sham eigenvalues@OEP-MBPT~2!-KS#, as has to be the case from the WFT view point. Finally,we demonstrate that OEP-MBPT~2!-f offers excellent qualityresults for the energy that exceed those based upon aHartree–Fock reference and, in some cases, are close to CCaccuracy.

II. THEORY

Consider the Hamiltonian,

H5(i

@ h~ i !1u~ i !#1(i , j

1/r i j 2(i

u~ i !, ~1!

whereh( i )5 t ( i )1 v( i ), is the usual one-particle part of theHamiltonian composed of the kinetic energy operator and the~external! potential, andu( i ) is an arbitrarily chosen one-particle potential. It can be local or nonlocal, but in our casewe will make it a local potential to relate it to theVxc ofKohn–Sham density functional theory.15 The Hamiltoniancan be written in second-quantized, normal ordered form16 as

H5(p,q

^pu f uq&$p†q%11

4 (p,q,r ,s

^pquurs&$p†q†sr%

1^0uHu0&, ~2!

where f 5h1 vHF, is the usual Fock operator,vHF

5( j*w j* (2) (12 P12)/r 12w j (2)d25 J2K. Note, thisoperator arises from normal ordering and does not implyany dependence on HF orbitals. The antisymmetrizedtwo-electron integrals are ^pquurs&5*d1*d2fp* (1)3fq* (2)r 12

21(12P12)f r(1)fs(2) where the integration isover spin and space coordinates.

Once we introduce the eigenfunctions in either canonicalor noncanonical form,

heff~ i !5h~ i !1u~ i !, ~3!

heffup&5(q

lpquq&5epup&, ~4!

we recognize that the vacuumu0& consists of the occupiedorbitals, 1,2,. . . ,n which we indicate byi , j ,k,l . The unoc-cupied orbitals area,b,c,d. When orbitals are unspecified,we usep,q,r ,s. Using the canonical eigenfunction proper-ties, we have

^pu f uq&5^puheff1 vHF2uuq&

5^puheffuq&dpq1^puvHF2uuq&

5^puheffuq&dpq2^puK1Vxcuq&, ~5!

where we used

u5 J1Vxc . ~6!

H then becomes

H5(p

^pu f up&$p†p%1 (pÞq

^puvHF2uuq&$p†q%

11

4 (p,q,r ,s

^pquurs&$p†q†sr%1^0uHu0&. ~7!

We will consider different choices for the separation ofthe Hamiltonian,

H5H01V, ~8!

to develop the tools to solve the orbital dependent OEP equa-tions for some orderExc

(n) , which will be our orbital depen-dent functional, and to define the potentialu5u(1)1u(2)

1 . . . 1u(n). Note the infinite-order solution is necessarilyindependent ofu, so we choose to use perturbation theory todefineu to some order. In the limit of MBPT(n) which givesthe energy(m51

n Exc(m) we will obtain the potential for the

basis set represented in the second-quantized Hamiltonianthroughu(n).

We would potentially have several different choices forthe separation of the Hamiltonian, but we will limit ourselvesto only the most likely choices here. In the first case,

H05(p

f pp$p†p%1(iÞ j

f i j $ i† j %1 (

aÞbf ab$a

†b%, ~9!

V5(i ,a

f ai@$a†i %1$ i †a%#1W, ~10!

whereW indicates the two-particle term. In the second case,which is customary in Kohn–Sham theoryH085(pep$p†p%is composed of the orbital energies ofheff and

V85V1(p

^puvHF2uup&$p†p%

1(iÞ j

^ i uvHF2uu j &$ i † j %

1 (aÞb

^auvHF2uub&$a†b%. ~11!

This choice would cause the diagonal contribution^puvHF2uup& to become a part of the perturbation, alongwith the off-diagonali , j anda,b mixing. This is not likely tobe the best choice, although this is the choice natural to KStheory and to Go¨rling–Levy PT.17 Unlike this choice, theunprimedH0 in Eq. ~9!, is invariant to any transformation

034104-2 Bartlett et al. J. Chem. Phys. 122, 034104 (2005)


among the occupied or excited orbital space2 which wewould argue is an essential condition for a mathematicallyconsistent approach. ThoughH0 is formally the same for anychoice of orbitals once a separation is made into an occupiedand unoccupied space, its effect on the convergence of theperturbation theory will change with the choice ofu sincethe orbitals andH0 will reflect that choice.

In practice when we do generalized MBPT~GMBPT!with arbitrary orbitals,2,18 we usually exploit the freedomto rotate the occupied and unoccupied orbitals to make asemicanonical transformation of thef matrix to eliminate theoff-diagonal terms,f i j 5 f ab50. ThenH05(pf pp$ p†p% andby virtue of usingf pp in H0 , which will appear as denomi-nators in the following, the above diagonal contributions^ puvHF2uu p& are selectively summed to all orders in evenlow orders of perturbation theory. Of course,f ia remains aspart of the perturbation because the orbitals are not HF or-bitals. In particular,

H5H01V1^0uHu0& ~12!

5(p

f pp$ p†p%1(i ,a

f ai$a† i 1 i †a%1W1^0uHu0& ~13!

5(p

^ puheff2K2Vxcu p&$ p†p% ~14!

2(i ,a

^auK1Vxcu i &$a† i 1 i †a%1W1^0uHu0&. ~15!

Here, as our orbitals are not completely arbitrary but arethe solutions to the KS equations, we can accomplish thesame thing by simply making an intelligent choice of theoff-diagonal Lagrangian multipliers,l i j andlab , which, ofcourse, are at our disposal. By choosing these so thatl i j

5^ i uheffuj& and the same forlab we achieve the semicanoni-cal form quite simply. In practice, subject to the canonicalKS solutions, we can insert a unitary transformation,U†heffUU†CU5U†eUU†CU wherel5U†eU with U chosento make the Fock matrix semicanonical. If this is imposed ateach iteration we need never consider the off-diagonalf i j

and f ab elements inH0 . If we insist upon using one set of~computational! orbitals rather than their unitary transforma-tion, we can achieve the same simplification by iterative so-lution where allf i j and f ab terms are summed to all ordersinto H0 . This is used elsewhere for ROHF-MBPT~2!hessians.19

Alternatively, we can retain these off-diagonal Fock ele-ments at the cost of their participation in the perturbation andlack of invariance inH0 . This is one alternative to the KSchoice we will make in this paper, namely,

H5H091V91^0uHu0&. ~16!

H095(p

f pp$p†p%, ~17!

V95(iÞ j

f i j $ i† j %1 (

aÞbf ab$a

†b%1(ai

f ai$a†i 1 i †a%1W.

~18!

The semicanonical results, though reported here for com-pleteness for energies, will be discussed in detail in otherpapers.20,21

III. COUPLED-CLUSTER DENSITY MATRIX

Coupled-cluster theory offers a convenient nonperturba-tive framework from which perturbation approximations areeasily obtained by iteration of the CC equations. The exactmany-body wave function can be conveniently written interms of CC theory1,2,16,22–24as

C5exp~T!u0&, ~19!

T5T11T21T31 ¯ , ~20!

T15(i ,a

t ia$a†i %, ~21!

T25 (i , j ,a,b

t i jab$a†ib† j %, ~22!

¯ . ~23!

The functional of CC theory2,24–26is

E5^0u~11L!e2THeTu0&5^0u~11L!Hu0&, ~24!

L5L11L21L3 ¯ , ~25!

L15(i ,a

l ia$ i †a%, ~26!

L25 (i , j ,a,b

l i jab$ i †a j†b%, ~27!

¯ , ~28!

whereL is a deexcitation operator. When the functional isvaried with respect toL, it yields the CC equations for thet ia ,t i j

ab , . . . amplitudes

^ i j ¯ab¯uHu0&50, ~29!

while variation with respect toT gives thel i j ¯ab¯ amplitudes,

^0u~11L!~H2E!u i j ¯ab¯&50. ~30!

The quantity$a†ib† j . . . %u0&5u i j ¯ab¯&, etc., are the various,

single, double, triple, . . . spin-orbital excitations from thereference determinant.

Writing the density operator in normal ordered form,

d~x1!5(p,q

^pud~x1!uq&$p†q%1^0ud~x1!u0&, ~31!

034104-3 Exchange-correlation potential J. Chem. Phys. 122, 034104 (2005)


where the second term isrKS, in CC theory we have thedensity matrix correction,2

Dgqp5^0u~11L!e2T$p†q%eTu0&

5^0ueT†$p†q%eTu0&/^0ueT†

eTu0&

5^0u@eT†$p†q%eT#Cu0&, ~32!

from which

DrKS~x1!5(p,q

wq~x1!Dgqpwp* ~x1!. ~33!

The different forms for the density matrix are equivalent inthe limit, but not necessarily order by order. TheL form hasthe advantage that it can be written in closed form.26 How-ever,L5T† through linear terms, and for low-order pertur-bative considerations the symmetric form has the advantagethat no solution forL is necessary and the Hermitian struc-ture of the density matrix is explicit.

The density may be immediately interpreted in terms ofour usual antisymmetrized CC diagrams,27 to give represen-tative diagrams in terms of increasing orders inV, diagrams1–8. T1 andT2 first appear in first order, whileT3 is sec-ond order,

IV. DENSITY OPTIMIZATION

The fundamental idea of Kohn–Sham DFT15 is thatthere exists an exact energy functionalExc@r# from which alocal, multiplicative potential can be defined,

Vxc@r~x1!#5dExc@r#/dr~x1!, ~34!

such that then lowest energy solutions of the KS equations,

heff5h1 J1Vxc , ~35!

heff~1!u i &5e i u i &, ~36!

compose a single determinantFKS whose density,

rKS~x1!5E FKSFKS* dx2 dx3¯5(i

n

w i~x1!w i* ~x1!,

~37!

is exact.

2u52F(i ,a

âuuu i &$a†i %1(i ,a

^ i uuua&$ i †a%

1(a,b

âuuub&$a†b%1(i , j

^ i uuu j &$ i † j %

1^F0uuuF0&G , ~38!

which is our objective. However, this form of operator doesnot distinguish between a local or nonlocal operator. For KStheory, we will be able to further simplify the equations sinceas shown elsewhere, only thea,i matrix elements or its ad-joint are required to completely specifyVxc(r ),29 fixingâuuub& and ^ i uuu j &.

We choose to expand the operatoru5 J1Vxc by ordersin the perturbationV,

u5u(1)1u(2)1 ¯5 J1Vxc(1)1Vxc

(2)1 ¯ ~39!

to give

vHF2u52K2Vxc(1)2Vxc

(2)2 ¯ . ~40!

These orders inV are independent of the particular KS or-bitals involved in their determination, which will be deter-mined self-consistently. The two particular choices forV arethe V8 in Eq. ~11!, the KS choice, andV9 in Eq. ~18!.

Expanding the density corrections from CC theory byorders in the perturbationV, we have in the first order



while in second order

Algebraically, the first- and second-order terms in thegeneral perturbationV, derived from the CCT1 amplitudesare in order~diagrams 9–15!,

wa~r 1!w i* ~r 1!@ t ia(1)1t i

a(2)1 ¯#5wa~r 1!w i* ~r 1!F âuvHF2uu i &/dia

11

2â j uucb&^cbuu i j &/di jcbdia2

1

2^k j uu ib&âbuuk j&/dk jabdia

1 (aÞb

âuvHF2uub&^buvHF2uu i &/dibdia2(iÞ j

^ j uvHF2uu i &âuvHF2uu j &/djadia

1^ j uvHF2uub&âbuu i j &/di jabdia1â j uu ib&^buvHF2uu j &/djbdia1 ¯G , ~41!

where summation over all repeated indices is assumed, except where indicated. Note restricted summations are required in thefourth and fifth terms when diagonal Fock elements are used in the denominators; but these summations are unrestricted whenusing KS orbital energies in the denominator as in Eq.~11!. All first- and second-order terms can invokeL5T† withoutrestriction, which provides the Hermitian adjoint to the above equations. Differences betweenL andT† will occur in higherorders.

Setting the density diagrams that derive fromT1 to zero and movinguia to the left side, and indicating by order inV,

u(1)5J1Vxc(1) gives in first order the well-known exchange only OEP result when KS denominators are used,10,17,30

wa~x1!w i* ~x1!uia(1)/dia5wa~x1!w i* ~x1!âuvHFu i &/dia . ~42!

In second-order perturbation theory,11,17

wa~x1!w i* ~x1!uia(2)/dia5wa~x1!w i* ~x1!F1

2â j uucb&^cbuu i j &/di jcbdia

21

2^k j uu ib&âbuuk j&/dk jabdia1 (

aÞbâuvHF2u(1)ub&^buvHF2u(1)u i &/dibdia

2(iÞ j

^ j uvHF2u(1)u i &âuvHF2u(1)u j &/djadia1^ j uvHF2u(1)ub&âbuu i j &/di jabdia

1â j uu ib&^buvHF2u(1)u j &/djbdia1 ¯G . ~43!



Notice the fact that to obtainuia(2) we only requireu(1) on

the right-hand side~RHS! to ensure a consistent order inV,and this would continue to higher orders. This is the mainreason we separateu into orders and this order is enforced inthe iterative solution of the equations and in the associatedfunctional,

E(1)51

2 (i , j

^ i j u j i &, ~44!

ES(2)5(

a,i

u f ai(1)u2

f i i 2 f aa, ~45!

ED(2)5

1

4 (i , j ,a,b

u^ i j uuab&u2

f i i 1 f j j 2 f aa2 f bb. ~46!

There are additional terms that contribute to the second-order density that do not relate as simply to the usual CCTamplitudes. Of necessity, these diagrams must have thedfunction interaction between those due toT andT†. In par-ticular, we have the second-order diagrams

Diagrams 20 and 21 are complex conjugates, and dia-gram 20 contributes to thewaw i* part just as those above,with value^buvHF2u(1)u j &^bauu j i &/di jabdjb while its conju-gate contributes toT1

† . Using the factorization theorem, wecan combine diagram 20 with diagram 14 above to removethe di jab denominator, to give ^buvHF2u(1)u j &3^bauu j i &/diadjb , as the penultimate term in the workingequations for theai part. However, it should be recognizedthat this extra term is not part of thet i

a amplitude of standardCC theory.

Diagrams 16, 17 and 18, 19 contribute to thewa* wb andthe w i* w j parts of the second-order density. Explicitly, wehave

wa* ~1!wb~1!@^ i uvHF2u(1)ua&^buvHF2u(1)u i &/dibdia

1 12 ^ i j uuca&^cbuu i j &/di jacdi jbc1 ¯#, ~47!

2w i* ~1!w j~1!@^ j uvHF2u(1)ua&âuvHF2u(1)u i &/diadja

1 12 ^k j uuab&âbuuki&/dk jabdikab1 ¯#, ~48!

respectively. Despite their ‘‘pattern’’ being different, whichwould point to different parts of the operator in Eq.~38!, thedensity condition and local operator requirement mandatesthese forms be added to RHS of Eq.~43! in the final equa-tions.

For first order we have

(a,i

wa~x1!w i* ~x1!@âuvHF2u(1)u i &/dia#50. ~49!

We show the summation explicitly for emphasis. This equa-tion defines the local, optimized exchange potentialVxc

(1)

5u(1)2J.30 The exchange only OEP equations have beenconsidered by many, but only a few10,31,32in Gaussian func-tions, essential to general molecular applications. This is apointwise identity. It is also the best fit in a weighted least-squares sense of the local exchange potentialVx5Vxc

(1) to thenonlocal HF exchange potentialK in the basis set, where theweight factor is the denominator. The necessity of a least-squares fit is due to the fact that this equation is pointwiseand linearly dependent, so the only way it can vanish issubject to the fit in the basis set,wa(x1)w i* (x1). Otherwise,we would force the condition thatâuvHF2u(1)u i &50, whichmakes the orbitals HF ones. This emphasizes the role of thebasis in the OEP procedure and why our definition ofui

a iscontingent on its use in the OEP equation, rather than as ageneral operator form.

In matrix form, we can write the OEP equations as

^0ud~x1!uFia&R0^Fi

auVxc1Ku0&50, ~50!

where the resolvent matrix,R05^FiauE02H0uFi

a&21 is writ-ten in terms of single excitations,@ uF i

a&5$a†i %u0&], andonly has diagonal elements,^F i

au(E02H0)uF ia&215dia

21 .Clearly, this expression is invariant to any transformationamong just occupied or unoccupied orbitals, though the di-agonal property of the resolvent is lost for more generalcases.

As we want to haveVxc(1) represented in terms of a basis

set for application, we projectVxc(1) into a space spanned by

um&,Vxc(1)5um&D21^muVxc

(1)& ~Ref. 33! which can be an auxil-iary basis set withD5^mum&. Writing the OEP equation interms of the response function

X~1,2!5(i ,a

w i* ~1!wa~1!wa* ~2!w i~2!/dia1c.c.

5^0ud~x1!uFia&R0^Fi

aud~x2!u0&1c.c. ~51!

whose matrix representation inum& is X, we obtain the OEPequation in matrix form,29

XV xc(1)5Y(1), ~52!

where the elements inY(1) are the weighted exchange inte-grals, (mu ia)(Kia)/dia;,a. (mu ia) means the overlap inte-gral of the expansion functionm with the orbitalsw i* andwa . The^ i uVxc

(1)ua& follows from Hermitian symmetry in thisorder. The explicit form forVxc

(1)(1) can be obtained frominverting X subject to singular value decomposition, viaVxc

(1)(r )5um&X21Y(1). This is what is done to plot thepotentials.11,29 The âuVxc

(1)u i & matrix elements, after spin-integration, completely define theVxc

(1)(r ) operator up to aconstant, as any transformation within the occupied or virtualorbital space is allowed. In particular, from the resolution ofthe identity,



05(a

ua&âuh1 J1Vxcu i &5S 12(j

u j &^ j u D heffu i &,

~53!

we regain the noncanonical equations34

heffu i &5(j

u j &^ j uheffu i & ~54!

and, similarly, we can obtain its counterpart in the virtualspace. These can be transformed, without restriction, to thecanonical form of Eq.~4!. Hence, we can define the other,off-diagonal, components ofVxc

(1)(r ), ^ i uVxc(1)u j &, and

âuVxc(1)ub& from using the potentialVxc(r )5um&X21Y and

evaluate these matrix elements as done previously.11

The remaining freedom inVxc(1) is that it is only deter-

mined up to a constant. Note that the exact conditionfor the highest occupied molecular orbitaln is

^nuK1Vxc(1)un&50,35,36 which has been used to fix the arbi-

trary constant.10 However, if the OEP exchange potential hasthe correct21/r long-range behavior, the constant should be0. By grafting a Slater potential to the basis set, the correctlong-range behavior is improved despite potential basis setdeficiencies.37,38 Other potential improvements are alsopossible.32

V. CORRELATION POTENTIAL

Our objective now is to introduce the correlation poten-tial part of the operatorVxc in a given order that will mix anyinitial ua& with ui& until we have satisfied the density conditionthrough that order, and to do so self-consistently.

Invoking the condition that the second-order density cor-rection vanishes in addition to the first order, the first term inthe correlation potentialVxc

(2) is given by

(a,i

wa~r1!w i* ~r1!âuu(2)u i &/dia5(a,i

wa~r1!w i* ~r1!F1

2â j uucb&^cbuu i j &/di jcbdia2

1

2^k j uu ib&âbuuk j&/dk jabdia

1 (bÞa

âuK1Vxub&^buK1Vxu i &/dibdia2(j Þ i

^ j uK1Vxu i &âuK1Vxu j &/djadia

2^ j uK1Vxub&âbuu i j &/djbdia2^ jauu ib&^buK1Vxu j &/djbdia1 ¯G1wa* ~1!wb~1!@^ i uvHF2u(1)ua&^buvHF2u(1)u i &/dibdia1 1

2 ^ i j uuca&

3^cbuu i j &/di jacdi jbc1 ¯#2w i* ~1!w j~1!@^ j uvHF2u(1)ua&

3âuvHF2u(1)u i &/diadja1 12 ^k j uuab&âbuuki&/dk jabdikab1 ¯#, ~55!

where the sum over the target indices,a and i , is writtenexplicitly. Note that onlyVx5Vxc

(1) appears on the right-handside of Eq.~55!. This ensures a precise definition of second-order in V. However, unlike the situation in first order, wenow have nonvanishing contributions to the second-orderdensity that arise from thei , j anda,b sectors, Eqs.~47! and~48!. In the case of a general, nonlocal, one-particle operator,we would have the flexibility to independently define theub

a(2) anduji (2) operator components, perhaps to satisfy a dif-

ferent desirable condition; but the fact that only thea,i ma-trix element definesVxc in any order, this cannot be the casefor a local potential. Instead, we have the terms in Eqs.~47!and ~48!, which simply represent a value at some r, contrib-uting as part of the equation forui

a(2) and its conjugate.Then the equations can be expressed in the matrix form,XV xc

(2)5Y(2) to solve for Vxc(2) . Transforming to the KS or-

bitals, this equation definesa,i matrix elements ofVxc(2) .

Generalizing to any order of perturbation theory,

XV xc5Y. ~56!

The above condition defines the operatoru to any order. Wecan equally well rewrite the above equation to include the

Hartree potentialJ asXu5Y8 as we have done throughoutthis paper, to avoid any possible lack of cancellation betweenthe finite-basis representation ofJ and Vxc .32

VI. ENERGY VARIATION

As it also follows that the KS determinant minimizes thenoninteracting kinetic energy at fixed density, and thatdE/dr(x1) has to be a minimum for the exact functionalE

5^0u(11L)Hu0& or a stationary point for its perturbationapproximations; equivalent equations forVxc can be derivedfrom this view point. Our prior work11 used the functionalderivative chain rule

Vxc~1!5dExc

dr~1!

5(pE E dExc

dfp~2!

dfp~2!

du~3!

du~3!

dr~1!d2 d3

5E dExc

du~3!

du~3!

dr~1!d3, ~57!



for E5E(0)1E(1)1E(2) to derive equivalent equations, buttheir form is less compact than those shown here as the cur-rent ones benefit from diagrammatic simplifications.

Furthermore, recognizing that the final~infinite-order!energy should be independent of the operatoru @dE/dr(1)50# we can insist on this conditionfor a given approxima-tion to the energy, such as that through second order.39 In thisway, we obtain a kind of ‘‘best’’ possibleu operator in termsof its second-order approximation, implying that we shouldobtain a best second-orderu(2) from those equations.~Werecognize the energy through second order is not necessarilybounded for pathological cases.! Generalized second-orderMBPT ~Ref. 2! in V gives the expressions

E(0)5(i 51

n

f ii , ~58!

E(1)5^F0uVuF0&, ~59!

E(2)5^F0uVR0VuF0&

5(i ,a

u f iau2/dia11

4 (i , j ,a,b

u^ i j uuab&u2/di jab , ~60!

VII. DISCUSSION

We want to contrast theab initio DFT approach fromthat of Zhao–Morrison–Parr~ZMP!.43 In the ZMP method, adensity as a function ofr would be taken from anab initiowave function calculationa posteriori, and a KS potential

would be created that would reproduce it. If we computed adensity from a MBPT~2! wave function, e.g.,C01C (1) @thisis not the same asr01r (1)1r (2) ~Ref. 44!#; we would ob-

tain a one-particleVxc which gave that density. However,unlike ZMP,ab initio DFT provides an energy functional and

equations that explicitly defineVxc as an integral part of thecalculation. In other words, we choose an orbital dependentexpression for the correlation energy such as in Eq.~60! andthis defines, unambiguously, theEc@w(r)#. This is a majorelement inab initio DFT as unlike all other practical DFTimplementations, there is no uncertainty, other than the orderof perturbation theory, about the correlation or exchangefunctional. This leads to a self-contained computational ap-proach that does not depend upon having already obtainedthe correct wave function density, or its fit to many mol-ecules before subsequent application.45 In this sense, wewould hope to provide a method whose error bars could becalibrated at various levels of approximation leading to theexact solution, as inab initio wave function methods. Onceorbital dependent expressions are introduced, however, theuniversality that is often said to be a cornerstone of DFT iscompromised.

Operationally, we will solve the KS equations for aninitial choice of potential, whose variational condition re-

quires that auheffui&50. This separates the space into an oc-cupied and an unoccupied part. Then in thattrial set of KSorbitals, we build the Fock matrix of Eq.~5!, where we insistthat in thef ai block we have au f u i &52^auK1Vxu i & and inthe diagonal block^pu f up&52^puK1Vx1Vcup&. Theseequations then depend upon a given order ofu, from whichwe develop an orbital dependent perturbation approximation

for the perturbed energiesExc(n) and potentialsVxc

(n) . We start

with a solution of the OEP~1! equation to defineVx5Vxc(1) .

Then using that value on the right-hand side of Eq.~55!, we

obtain the second-orderincrement to define Vxc5Vxc(1)

1Vxc(2) . Then returning to the Kohn–Sham equations, we

obtain a new set of KS orbitals, from which we obtain a new

Vx and use it to compute a newVxc(2) . We continue until these

equations are satisfied to some tolerance. A check is obtainedby ensuring thatrKS5r01r (1)1r (2) at convergence. We dothis by computingDgpq from Eq. ~32! limited to second-order terms.

As an alternative to the strict order dependent procedurejust described, we could attempt to solve thequadraticequa-tion in u. That arises from having the unknown potential onboth sides of the OEP equations as occur in diagrams 12–16,18, 20, and 21. This would define a ‘‘quadratic OEP.’’

Taking the density from a full CI wave function wouldbe equivalent to imposing the condition thatDr(x1)50 inall orders of perturbation theory, but adding theu to theHamiltonian would be meaningless. As is well known, thefull CI is completely invariant to all orbital transformations,thus there would be no freedom to rotate the virtual spaceinto the occupied space to define improved results; so forthat case our order-by-order approach has to result in a seriesof trivially satisfied ‘‘higher-order’’ equations to give thesame results as ZMP. For CCSD,46 another infinite order



method, we would also have trouble getting an improvement.However, at the MBPT~2! level, we still have substantialfreedom to rotate the virtual space into the occupied space togenerate an improved density, which suggests that such anabinitio DFT based upon even a second-order density, will givesuperior results to ordinary MBPT~2!, where that degree offreedom is not exploited. We actually see this in studies ofweak interactions20 and in treatments of potential energycurves with OEP-MBPT~2! where we have much improvedresults over ordinary MP2 that are almost of CCSD~T! orCCSDT accuracy.21 In this sense, OEP-MBPT~2! is a kind ofbest second-order approximation that is vastly less computa-tionally expensive (;n5) than CCSD~T! (;n7) or CCSDT(;n8).

VIII. NUMERICAL RESULTS

The numerical results shown below will be mostly lim-ited to using KS denominators and diagonal Fock matrixelements to connect with our previous work.11 Solving theiterative equations without the semicanonical transformation~discussed in depth elsewhere20,21! recognizes that the re-maining off-diagonal elements in that case contribute to theperturbation. To achieve self-consistency, the above equa-tions must be solved several times in a computation and therate determining step~diagram 17! is ;nocc

2 nvirt3 . In time, it

should be possible to develop anab initio DFT procedurewhose equations can be solved within an order of magnitudeor so of a normal DFT calculation, butab initio DFT has thefundamental advantage over the currently applied DFT meth-ods that it has to provide the right answer at convergence inV, like ab initio WFT.

To illustrateab initio DFT and particularly the choice ofunperturbed problem, we compare the correlation potentialsfor He and Ne obtained with the standard KS denominators,H08 using canonical KS orbitals presented previously11 andthat for Fock operatorsH09 . We also include the VWN ver-sion of local-density approximation~LDA !,47 LYP,8 and KLI~Ref. 37! with the Colle–Salvetti correlation potential48 forcomparison, along with the exact results from Umrigar andGonze ~see Fig. 1!.49 Note that the latter actually usesinfinite-order KS orbitals since they take exact densities andcreate aVxc from a ZMP type procedure. Using the orbitalsso obtained, they then solve the exchange only OEP equa-tions to defineVx . Then by subtractingVx from their Vxc

they obtain aVc . In our case, we generate our KS orbitals inthe same finite order as our potentials. For KS orbitals, thereare no single excitations in the two-electron functional,50 norin the potential. As one would expect from the theory, theform using the Fock matrix elements in the denominator con-verges better and provides closer results to quantum MonteCarlo ~QMC!. Of course, as is now well known, all the stan-dard approximations for correlation potentials even have thewrong sign~see Figs. 2 and 3!.51 Furthermore, this behavioris not ameliorated by the exchange potential. It is true thatthe effect of the exchange potential is at least an order ofmagnitude more important than the correlation potential, butit has errors in it, too,52 and the numerical consequences ofthe errors in the correlation potentials are not alleviated; and

indeed converging DFT calculations would require that bothbe correct.

Quite similar results are obtained for Ne. All the struc-tural features of the exact potentials are reproduced by theabinitio ones. But once again, we see considerable improve-ment in the Fock based results, which doubtless reflect thesuperior infinite summation of terms compared to using the

FIG. 1. Comparison of the exact correlation potential of the helium atomwith the correlation potentials obtained from the OEP-MBPT~2!-KS, OEP-MBPT~2!-f, KLICS, SVWN, and BLYP functionals. The exact correlationpotential from Ref. 49.

FIG. 2. Comparison of the exact~QMC! correlation potential of the neonatom with the correlation potentials obtained from the OEP-MBPT~2!-KS,OEP-MBPT~2!-f, SVWN, and BLYP functionals. The exact~QMC! corre-lation potential from Ref. 51.



KS denominators. As the diagonal contribution is put into theperturbation in that case, it will strongly affect the higherorders of perturbation theory, and will likely lead to a poorPT expansion.12,53However, it should be realized that even adivergent perturbation expansion can be frequently resumedvia infinite-order methods such as CC theory or rationalfunction ~Padeapproximants!, etc.52

The more significant improvement over OEP-MBPT~2!-KS results can be found on Fig. 3, where we plotcorrelation potentials for the beryllium atom. Unfortunately,because we could not even converge the OEP-MBPT~2!-KS~i.e., GLPT! calculations in this particular case, the correla-tion potential must be obtained in a one step OEPprocedure,54 where the potential is simply calculated~with-out full self-consistency! using converged exchange-only or-bitals. This one-step procedure gives in most cases a goodapproximation to the fully converged correlation potentials.54

Contrary to the GLPT calculations, the correlation potential

obtained from OEP-MBPT~2!-f is fully converged and inmuch better agreement with the exact one than the OEP-MBPT~2!-KS correlation potential.

Clearly, if the above is only a complicated way to get asecond-order perturbation theory result, there is little to gainover standard MBPT~2! with a HF reference~MP2!. How-ever, as we see from Table I this does not seem to be thecase. Our best results, OEP-MBPT~2!-f where the -f indi-cates the Fock denominators shows better agreement withCCSDT55 ~essentially the full CI results for these examples!.The average absolute error in the total energy is 0.0060,compared to MP2’s error of 0.0155. As we would expect,OEP-MBPT~2!-f total energies are significantly improvedover the OEP-MBPT~2!-KS ones. For completeness, we alsoshow the OEP-MBPT~2!-sc ~semicanonical! results, whichthough more rigorious, are not quite as good. Such resultsare considered in depth elsewhere.20,21 However, any suchresults can be criticized that the second-order result can befortuitous, until documented by higher-order results; and thisis likely to be the case for the Fock diagonal choice. How-ever, since our exchange correlation potential is tailored toparticular approximation to the correlation energy as dis-cussed above, we expect that second order will provide abetter result than usual. This is further verified in the ex-tremely hard to describe Van der Waals systems such as He2

and Ne2 where the slightest error would destroy any kind ofagreement. Our results indicate that OEP-MBPT~2!-sc is ca-pable of recovering these effects with surprising accuracy.The quality of the results is of MP2 quality for dimers com-posed of two-electron monomers, but results of CCSD orCCSD~T! accuracy are obtained for larger systems such asNe2 .20 Nonetheless, higher-order approximations will bepursued to document the range of validity of OEP-MBPT~2!-sc.

Many types of approximations can be made in theseequations. If we forego the requirement that the potential be

fully local, we can easily choose to makeVx5K and use theHF determinant and nonlocal potential in place of the exact,local OEP exchange; with onlyVc being determined from theOEP correlation equations. We can also use a fixed set oforbitals from HF or exchange only OEP, and forego orbitalself-consistency in the equations.54 Another possibility is touse a constant energy denominator approximation as is donein the KLI method,37 or its improved constant energy de-

FIG. 3. Comparison of the exact~QMC! correlation potential of the beryl-lium atom with the correlation potentials obtained from the OEP-MBPT~2!-KS ~one step procedure~Ref. 54!, OEP-MBPT~2!-f, SVWN, andBLYP functionals. The exact~QMC! correlation potential from Ref. 51.

TABLE I. Total energies~in hartrees! and average absolute errors with respect to CCSDT obtained fromHF-MBPT(2), CCSDT, OEP-MBPT~2!-KS, OEP-MBPT~2!-f and OEP-MBPT-sc calculations for a few atomsand molecules. Calculations were performed using even-tempered 20s10p2d basis set for He, uncontractedROOS-ATZP basis set for Ne, Mg, Ar, and an uncontracted cc-PVTZ basis set for the molecules.

HF1E(2) OEP-MBPT~2!-KS OEP-MBPT~2!-f OEP-MBPT~2!-sc CCSDT

He 22.8970 22.9078 22.9000 22.8970 22.9025Ne 2128.8924 2128.9886 2128.9119 2128.8941 2128.8999Mg 2199.8166 2199.8836 2199.8283 2199.8182 2199.8291Ar 2527.2059 2527.3332 2527.2249 2527.2074 2527.2307H2O 276.3720 276.5028 276.3898 276.3743 276.3870N2 2109.4562 2109.7503 2109.4879 2109.4647 2109.4760CO 2113.2375 2113.5156 2113.2669 2113.2467 2113.2610Average absolute error 0.0155 0.1280 0.0060 0.0120 0.0000



nominator approximation~CEDA! form.56,57The latter notesthat only the occupied-virtual part of the Green’s function isrequired in the response function as we originally used,10 andbase their constant energy denominator approximation onthat, rather than in the unseparated Green’s function as isdone in KLI method. We can also exploit auxiliary bases asformulated above, but not used in the numerical resultsshown here where only one Gaussian atomic orbital basiswas used to express both the potentials and the KS results.We have studied this problem elsewhere.29 We can also in-troduce selective infinite summations of terms to obtain bet-ter approximations. All of these will be considered in futurework.

IX. CONCLUSIONS

Of all the current implementations of DFT, local, gradi-ent corrected, or hybrid;ab initio DFT is arguably muchcloser to a realization of ‘‘exact’’ DFT, in that it assumes theDFT local potential form, but uses nonlocal, orbital depen-dent functionals, namely, the exact expressions known fromwave function theory. As presented elsewhere, such resultsderived fromVxc

(2) properly include dispersion effects, withOEP-MBPT~2!-sc showing excellent agreement with theCCSDT results.20 Ab initio DFT correctly describes self-interaction, has the formally correct long-range behavior forVx and Vc , is evaluated in a Gaussian basis set bypassingany need for numerical quadrature, and as will be shownelsewhere,21 provides good ionization potentials for all orbit-als and excitation energies.38

In comparison to ordinary MP2 results,ab initio DFTresults are tailored to get the best result possible at a givenorder of PT. We have correlation included in the orbitals,potentials, and orbital energies, soab initio DFT is an ex-ample of a correlated orbital~or correlated ‘‘independent’’particle! theory. This also suggests a universal DFT/TDDFTab initio way of improving ionization potentials~IPs!, exci-tation energies, polarizabilities, or essentially any propertydescribed by WFT.

Unlike most DFT calculations where the theory bypassesmost finite basis set aspects~though, of course, the actualapplications might be basis set dependent for the one-particlespace!, OEP-MBPT~2! is explicitly basis set dependent forthe one- and two-particle space. This follows from the factthat we have to evaluateE(1)1E(2) in a basis of occupiedand virtual orbitals, just as in wave function theory, and thensolve the resulting one-particle equations self-consistently.Hence, we introduce a basis set error for the two-particleterms that lead to the effective one-particle theory. One of thegreat advantages of DFT compared toab initio correlatedmethods is that we only have to span the one-particle space,while in the latter we suffer from correlation cusp~short-range correlation! problems58 because our product one-particle basis is nowhere near complete in the two-particlespace. However, as a route toward understanding the detailedrelationships between WFT and DFT, and being able tochoose the best of both worlds, it is first necessary to ensurethat a DFT method will converge to the right answerin thefinite and complete basis set limit. And ab initio DFT for-

mally achieves that, being equivalent to full CI in the finitebasis case.

As shown above, we are able to generate very accurateexchange and correlation potentials for a wide range of at-oms and molecules with less numerical cost than extremelyaccurate WFT~or QMC! calculations. Hence, our potentialscan be used as reference ones for testing or developing newexchange and correlation potentials for general atoms andmolecules.

Within a basis set, the CC equations offer a systematicroute toward obtaining the DFT solution in any finite-orderof perturbation theory, and point the way to nonperturbativesolutions. As shown, the solution forces the condition thatthe density obtained from the KS determinant be exact up tosome order of perturbation theory. Note, there is no need forthe kerneldVxc /dr(x1) in any of the foregoing.40 Also, un-like Gorling-Levy PT, which uses the KS denominators~equivalent to OEP-MBPT~2!-KS in our terminology!, weuse a differentH0 based upon Fock matrix elements or itseven more general, semicanonical forms.21 It is demonstratedthat the results obtained for potentials are superior to thoseobtained from standard approximations in KS theory, as theyshould be. Having the right potentials should make it pos-sible to do essentially everything that can be done with WFTwithin the ab initio DFT framework. Alsothe correlated abinitio DFT KS orbitals obtained from that make the densityobtained from a single determinant exact~and the kineticenergy a minimum! can take their place in wave functiontheory with HF orbitals that minimize the energy of a singledeterminant, natural orbitals that give the best possible singledeterminant approximation to the one particle densitymatrix,59 and Brueckner60 orbitals whose single determinantoffers maximum overlap with the exact wave function.34,61

Each has their assets.

ACKNOWLEDGMENTS

This work was supported by the U.S. Air Force Office ofScientific Research under Grant No. AFOSR-F49620-95-1-0130 and by the Committee for Scientific Research KBNunder Grant Nos. 4-T09A-187-25 and 7-T09A-068-21. Weappreciate many informative discussions with Dr. VictorLotrich and Mr. Igor Schweigert and their help in obtainingthe results in Table I.

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The exchange-correlation potential in ab initio density functional theory

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