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arXiv:cond-mat/0611324v1 [cond-mat.mtrl-sci] 13 Nov 2006

Kohn-Sham calculations combined with an average pair-density

functional theory

Paola Gori-Giorgi∗and Andreas Savin

Laboratoire de Chimie Th´ eorique, CNRS UMR7616,

Universit´ e Pierre et Marie Curie, 4 Place Jussieu, F-75252 Paris, France

∗E-mail: paola.gori-giorgi@lct.jussieu.fr

www.lct.jussieu.fr/pagesequipe/DFT/gori

Abstract

A recently developed formalism in which Kohn-Sham calculations are combined with an “average

pair density functional theory” is reviewed, and some new properties of the effective electron-

electron interaction entering in this formalism are derived. A preliminary construction of a fully

self-consitent scheme is also presented in this framework.

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I.INTRODUCTION

Kohn-Sham (KS) Density Functional Theory1,2,3(DFT) is nowadays one of the most pop-

ular methods for electronic structure calculations both in chemistry and solid-state physics,

thanks to its combination of low computational cost and reasonable performances. The ac-

curacy of a KS-DFT result is limited by the approximate nature of the exchange-correlation

energy density functional Exc[n]. Typical cases in which present-day DFT fails are strongly

correlated systems, the description of van der Waals forces, the handling of near degeneracy.

Much effort is put nowadays in trying to improve the DFT performances via the construc-

tion of better approximations for the KS Exc[n] (for recent reviews, see, e.g., Refs. 2,3,4),

or via alternative routes, like, e.g., the use of non-KS options.5A popular trend in the de-

velopment of new KS Exc[n] is the use of the exact exchange functional Ex[n] (in terms of

the KS orbitals), and thus the search for an approximate, compatible, correlation functional

Ec[n].

In this work we review the basis of a theoretical framework6,7in which KS-DFT is com-

bined with an “average pair density functional theory” (APDFT) that provides an explicit

construction for Ec[n], transfering the work of finding an approximate functional to the

search of an effective particle-particle interaction. A self-consitent scheme for this approach

is presented, and some new properties of the effective interaction that enters in this combined

formalism are derived. Very preliminary applications are discussed.

II.DEFINITIONS

Our target problem is finding the ground-state energy of the standard N-electron hamilto-

nian in the Born-Oppenheimer approximation (in Hartree atomic units, ? = m = a0= e = 1,

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used throughout),

H = T + Vee+ Vne,(1)

T = −1

2

N

?

i=1

N

?

i?=j

∇2

i,(2)

Vee =1

2

1

|ri− rj|,(3)

Vne =

N

?

i=1

vne(ri),(4)

where vneis the external potential due to nuclei. Given Ψ, the exact ground-state wave-

function of H, we consider two reduced quantities that fully determine, respectively, the

expectation values ?Ψ|Vne|Ψ? and ?Ψ|Vee|Ψ?, i.e., the electronic density,

?

n(r) = N

?

σ1...σN

|Ψ(rσ1,r2σ2,...,rNσN)|2dr2...drN,(5)

and the spherically and system-averaged pair density f(r12) (APD), which is obtained by

first considering the pair density P2(r1,r2),

P2(r1,r2) = N(N − 1)

?

σ1...σN

?

|Ψ(r1σ1,r2σ2,r3σ3,...,rNσN)|2dr3...drN,(6)

and then by integrating it over all variables except r12= |r1− r2|,

f(r12) =1

2

?

P2(r1,r2)dΩr12

4π

dR,(7)

where R =

1

2(r1+ r2), r12= r2− r1. The function f(r12) is also known in chemistry as

intracule density8,9,10,11,12,13,14, and, in the special case of a an electron liquid of uniform

density n, is related to the radial pair-distribution function g(r12) by g(r) = 2f(r)/(nN).

We thus have

?Ψ|Vne|Ψ? =

?

n(r)vne(r)dr

(8)

?Ψ|Vee|Ψ? =

?

f(r12)

r12

dr12=

?∞

0

f(r12)

r12

4πr2

12dr12. (9)

In the following text we will also deal with modified systems in which the external po-

tential and/or the electron-electron interaction is changed. Thus, the notation Vneand Vee

will be used to characterize the physical system, while the modified systems will be defined

by V =?N

depends only on |ri− rj|.

i=1v(ri) and W =

1

2

?N

i?=jw(|ri− rj|), where the pairwise interaction w always

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III.THE EXCHANGE-CORRELATION FUNCTIONAL OF KS-DFT

In standard DFT one defines a universal functional of the one-electron density n(r) as

resulting from a constrained search over all the antisymmetric wavefunctions Ψ that yield

n15

˜F[n;Vee] = min

Ψ→n?Ψ|T + Vee|Ψ?,(10)

or, more completely, as a Legendre transform16

F[n;Vee] = sup

v

?

min

Ψ?Ψ|T + Vee+ V |Ψ? −

?

n(r)v(r)dr

?

.(11)

In both Eqs. (10) and (11), the dependence on the electron-electron interaction has been

explictly shown in the functional. The universality of the functional F stems exactly from

the fact that the e-e interaction is always 1/r12. The ground-state energy E0of the system

can then be obtained by minimizing the energy with respect to n,

E0= min

n

?

F[n;Vee] +

?

n(r)vne(r)dr

?

.(12)

A possible way to derive the Kohn-Sham equations in DFT is to define a set of hamilto-

nians depending on a real parameter λ17,18,19,

Hλ= T + Wλ+ Vλ, (13)

having all the same one-electron density, equal to the one of the physical system

nλ(r) = n(r)

∀λ.(14)

If Wλphys= Veeand Wλ=0= 0 (e.g., Wλ= λVee), one can slowly switch off the electron-

electron interaction, while keeping the density fixed via a suitable external potential Vλ.

Obviously, the APD f(r12) changes with λ. By the Hellmann-Feynmann theorem,

∂Eλ

∂λ

0

= ?Ψλ|∂Wλ

∂λ

+∂Vλ

∂λ|Ψλ? =

?

fλ(r12)∂wλ(r12)

∂λ

dr12+

?

n(r)∂vλ(r)

∂λ

dr,(15)

so that by directly integrating Eq. (15), and by combining it with Eq. (12), one obtains

F[n;Vee] = Ts[n] +

?λphys

0

dλ

?

dr12fλ(r12)∂wλ(r12)

∂λ

,(16)

where Ts[n] = F[n;0] is the kinetic energy of a noninteracting system of N spin-1

2fermions

with density n(r). The adiabatic connection in DFT thus naturally defines the Kohn-Sham

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non-interacting kinetic energy functional Ts[n]. The second term in the right-hand-side

of Eq. (16) is an exact expression, in terms of the APD fλ(r12), for the Hartree and the

exchange-correlation functional, EH[n] + Exc[n]. The one-body potential at λ = 0 is the

familiar Kohn-Sham potential, vλ=0(r) = vKS(r).

The traditional approach of DFT to construct approximations for Exc[n] is based on the

idea of universality. For example, the familiar local-density approximation (LDA) consists in

transfering, in each point of space, the pair density from the uniform electron gas to obtain

an approximation for fλ(r12) in Eq. (16). Our aim is to develop an alternative strategy

in which realistic APD fλ(r12) along the DFT adiabatic connection are constructed via a

formally exact theory that must be combined with the KS equations in a self-consitent way.

The formal justification for this “average pair-density functional theory” (APDFT) is the

object of the next Sec. IV.

IV.AVERAGE PAIR DENSITY FUNCTIONAL THEORY

As shown by Eqs. (8) and (9), the APD f(r12) couples to the operator Veein the same

way as the electronic density n(r) couples to Vne. In order to derive an “average pair density

functional theory” (APDFT) we thus simply repeat the steps of the previous Sec. III by

switching the roles of f and n, and of Veeand Vne.7

We thus define a system-dependent functional (i.e., a functional depending on the external

potential Vne, and thus on the specific system) of the APD f(r12) as

˜G[f;Vne] = min

Ψ→f?Ψ|T + Vne|Ψ?,(17)

where, again the minimum is over all antisymmetric wavefunction Ψ that yield a given

f(r12). We can also define the system-dependent functional G as

G[f;Vne] = sup

w

?

min

Ψ?Ψ|T + W + Vne|Ψ? −

?

f(r12)w(r12)dr12

?

. (18)

The ground-state energy could then be obtained as

E0= min

f∈Nf

?

G[f;Vne] +

?

f(r12)

r12

dr12

?

,(19)

where Nf is the space of all N-representable APD (i.e., coming from the contraction of

an N-particle antisymmetric wavefunction). The definition of the space Nf is evidently

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related to the N-representability conditions for the pair density, for which recent interesting

progresses have been made20. In our case, however, we combine APDFT with DFT so that

the minimization of Eq. (19) is never directly carried on.

In order to find the analog of the KS system for APDFT, we define an adiabatic connection

similar to the one of Eq. (13) in which, this time, we switch off the external potential. We

thus introduce a set of hamiltonians depending on a real parameter ξ,

Hξ= T + Wξ+ Vξ,(20)

in which the function f(r12) is kept fixed, equal to the one of the physical system,

fξ(r12) = f(r12)

∀ξ.(21)

If Vξphys= Vneand Vξ=0= 0 (e.g., Vξ= ξVne), we are switching continuously from the phys-

ical system, to a system of N free electrons interacting with a modified potential wξ=0(r12).

That is, f(r12) is kept fixed as ξ varies by means of a suitable electron-electron interaction

Wξwhile the one-electron density n(r) changes with ξ. Again, by the Hellmann-Feynmann

theorem, we find

∂Eξ

∂ξ

0

= ?Ψξ|∂Wξ

∂ξ

+∂Vξ

∂ξ|Ψξ? =

?

f(r12)∂wξ(r12)

∂ξ

dr12+

?

nξ(r)∂vξ(r)

∂ξ

dr,(22)

so that

G[f;Vne] = Tf[f] +

?ξphys

0

dξ

?

drnξ(r)∂vξ(r)

∂ξ

, (23)

where Tf[f] is the kinetic energy of a system of N free (zero external potential) interacting

spin-1

2fermions having the same f(r12) of the physical system. In the case of a confined

system (atoms, molecules) the effective interaction wξ=0(r12) must have an attractive tail:

the hamiltonian corresponding to ξ = 0 in Eq. (20) describes a cluster of fermions whose

center of mass is translationally invariant. The functional Tf[f] is the internal kinetic energy

of this cluster.

To fix the ideas, consider the simple case of two electrons, e.g. the He atom. When

ξ = 0, we have two fermions in a relative bound state (similar to the case of positronium,

but with a different interaction).This relative bound state is such that the square of

the wavefunction for the relative coordinate r12is equal to f(r12) of the starting physical

system. The corresponding effective interaction wξ=0(r12), obtained7by inversion from a

very accurate wavefunction,21is shown in Fig. 1, for the case of the He atom.

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-6

-4

-2

0

2

4

6

0 1 2 3 4 5

wξ(r12)

r12

1/r12 (ξ = ξphys)

ξ = 0

He

FIG. 1: The electron-electron interaction at the two ends of the APDFT adiabatic connection of

Eqs. (20)-(21) in the case of the He atom.

For more than two electrons, Tf[f] is still a complicated many-body object. Moreover,

since the corresponding wξ=0(r12) can have an attractive tail (as in the case N = 2), we may

have an “exotic” true ground-state for the cluster, i.e., the cluster state with the same f(r12)

of the physical system can be an excited state. However, we have to keep in mind that our

aim is not to solve the many-electron problem by means of APDFT alone: we want to use

APDFT to produce realistic f(r12) along the DFT adiabatic connection of Eqs. (13)-(14).

To this end, we proposed6,7an approximation for the functional Tf[f] based on a geminal

decomposition,

Tg[f] = min

{ψi}→f

?

i

ϑi?ψi| − ∇2

r12|ψi?,(24)

where ψi(r12) are some effective geminals (orbitals for two electrons, but only depending on

the relative distance r12), and ϑisome occupancy numbers to be chosen. For example, one

can always make a “bosonic” choice, by occupying only one geminal,22equal to

?f(r12).

The geminals ψi(r12) then satisfy the equations

[−∇2

?

r12+ weff(r12)]ψi(r12) = ǫiψi(r12)

iϑi|ψi(r12)|2= f(r12),

(25)

with

weff(r12) =

1

r12

+

δG[f]

δf(r12)−δTg[f]

δf(r12).(26)

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The approximation of Eq. (24) is mainly motivated by the need of having simple equations

for f(r12) (the one-dimensional character of Eqs. (25) is of course very appealing). Notice

that only in the case N = 2 we have Tg[f] = Tf[f], and thus the effective interaction weff(r12)

of Eqs. (25) becomes equal to wξ=0(r12).

V.COMBINING DFT AND APDFT IN A SELF-CONSITENT WAY

As explained in the previous sections, to compute the expectation value of the physical

hamiltonian of Eq. (1) we only need f(r12) and n(r) for Veeand Vne, and the non-interacting

KS kinetic energy Ts[n] plus the APD fλ(r12) along the DFT adiabatic connection (13) for

the expectation value of T; schematically:

?Ψ|H|Ψ? = ?Ψ|T|Ψ?

? ?? ?

Ts[n]+fλ(r12)

+?Ψ|Vee|Ψ?

?

???

f(r12)

+?Ψ|Vne|Ψ?

?

???

n(r)

. (27)

Our aim is to obtain n(r) and Ts[n] via the KS equations, and fλ(r12) via Eqs. (25), which can

be generalized to any hamiltonian along the DFT adiabatic connection, by simply replacing

the physical hamiltonian H with Hλof Eqs. (13)-(14) in the steps of Sec. IV.

A self-consitent scheme for this construction reads

(T + VKS)ΦKS= EKSΦKS

E0= min

vKS

⇒

n(r), Ts[n](28)

[−∇2

?

r12+ wλ

eff(r12;[n])]ψλ

i(r12) = ǫλ

iψλ

i(r12)

iϑi|ψλ

i(r12)|2= fλ(r12),

?

⇒

fλ(r12)(29)

Ts[n] +

?λphys

0

dλ

?

dr12fλ(r12)∂wλ(r12)

∂λ

+

?

drn(r)vne(r)

?

.(30)

The computation starts with a trial vKS(r) in the KS equations, schematically represented by

Eq. (28), where ΦKSis the Slater determinant of KS orbitals. From the KS equations we thus

get a first approximation for the density n(r) and the non-interacting kinetic energy Ts[n].

Provided that we have a prescription to build an approximate wλ

efffor a given density n(r)

(see next Sec. VI), we can obtain fλ(r12) along the DFT adiabatic connection from Eqs. (29).

In general, this step is not expensive: Eqs. (29) are unidimensional, and if the dependence

of wλ(r12) on λ is smooth, few λ values (5-20) are enough to provide a good estimate of

the coupling-constant average. The physical ground-state energy E0is then evaluated via

Eq. (30). The procedure should then be repeated by optimizing the KS potential vKS, so that

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E0is minimum. The N-representability problem of the KS exchange-correlation functional

is clearly shifted to the N-representability problem for fλ(r12). In view of the new conditions

derived for the pair density,20this seems to leave space for improvements.

VI.PROPERTIES OF THE EFFECTIVE ELECTRON-ELECTRON INTERAC-

TION

So far, we have only replaced the problem of finding an approximation for Exc[n] with

the problem of constructing wλ

eff(r12;[n]). In order to proceed further, we study here the

properties of wλ

eff(r12;[n]).

If we want our Eqs. (28)-(30) to be fully self-consitent, we should impose that for λ = 0

Eqs. (29) yield fλ=0(r12) = fKS(r12), i.e., the same APD we would obtain by inserting the

KS Slater determinant of Eq. (28) in Eqs. (6)-(7). This corresponds, in the usual DFT

language, to treat exchange exactly. The first property we should thus impose to wλ

eff(r12) is

wλ=0

eff(r12) = wKS

eff(r12).(31)

If we use only one geminal to define Tg[f] in Eq. (24), the property (31) corresponds to

eff(r12) = ∇2?fKS(r12)/?fKS(r12). For more than one geminal we need more sophisti-

cated constructions, mathematically equivalent to those used to construct the KS potential

wλ=0

vKS(r) for a given spherical density n(r).23Equation (31) also provides a very good starting

point to build wλ

eff(r12): the KS system already takes into account the fermionic structure

and part of the effect of the external potential in the physical problem. What is then left,

that needs to be approximated, is the effect of turning on the electron-electron interaction

without changing the one-electron density n(r), and the difference between Tf[f] and Tg[f].

For confined systems (atoms, molecules) another property to be imposed on wλ

eff(r12)

concerns the eigenvalue ǫλ

maxcorresponding to the highest occupied geminal in Eqs. (29).

In fact, the asymptotic behavior of the pair density P2(r1,r2) of Eq. (6) for |r1| → ∞ (or

|r2| → ∞) is, in this case,24

lim

|r1|→∞P2(r1,r2) = n(r1)nN−1(r2){ˆ r1},(32)

where nN−1(r) is one of the degenerate ground-state densities of the (N −1)-electron system

(in the same external potential Vne), with the choice depending parametrically upon the

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direction ˆ r1= r1/|r1|. A similar expansion holds for the KS pair density, obtained from the

KS Slater determinant of Eq. (28),

lim

|r1|→∞PKS

2(r1,r2) = n(r1)nKS

N−1(r2){ˆ r1},(33)

where we have used the fact that, by construction, the N-electron density is the same

for the physical system and for the KS one (while, of course, the corresponding (N − 1)-

electron densities are in general different). For a given attractive (atomic, molecular) external

potential vanishing at large distances, the N-electron density is in general more diffuse

(decaying slower at large distances) than the (N−1)-electron density, so that the asymptotic

behavior of the APD f(r12) is, for large r12, dominated by the N-electron density decay at

large distances. We thus see, from Eqs. (32) and (33), that the corresponding APD’s, f(r12)

and fKS(r12), will have the same large-r12decay, ∝ e

√−2ǫmaxr12, with a different prefactor

(which can also include a polynomial function of r12), depending on the difference between

nN−1(r) and nKS

adiabatic connection,

N−1(r). Since the same expansion holds for any Pλ

2(r1,r2) along the DFT

lim

|r1|→∞Pλ

maxin Eqs. (29) must be independent of λ and equal to the one for

2(r1,r2) = n(r1)nλ

N−1(r2){ˆ r1}, (34)

the highest eigenvalue ǫλ

the KS APD,

ǫλ

max= ǫλ=0

max= ǫKS

max.(35)

In particular, if we choose only one geminal22for the definition of Tg[f], there is only one

eigenvalue, which must be the same for every λ.

For an extended system we have scattering states in Eqs. (29). For the special case of the

uniform electron gas, Eqs. (29) become equivalent to an approach that was first proposed

by Overhauser,25and further developed by other authors in the past five years.26,27,28In

this approach, the geminal occupancy numbers ϑi are the same as the ones for a Slater

determinant: occupancy 1 for singlet states (even relative angular momentum ℓ), and occu-

pancy 3 for triplet states (odd relative angular momentum ℓ), up to N(N − 1)/2 geminals.

Rather simple approximations for the effective potential wλ

eff(r12) gave good results26,27,28

for the radial distribution function g(r12), when compared with quantum Monte Carlo data.

The long-range asymptotic behavior, in this case, corresponds to a phase-shift sum rule for

the interaction wλ

eff(r12).29The choice of one geminal for the uniform electron gas has been

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explored, with remarkable success, in Ref. 30. In this case, the formal similarity with the

Fermi-hypernetted-chain approach31(FHCN) was exploited to build a good approximation

for wλ

eff(r12) (which was split into ↑↑ and ↑↓ contributions).

Finally, the small-r12 behavior of the effective potential wλ

eff(r12) is determined by the

choice of the adiabatic connection path. For instance, if we choose wλ(r12) = λ/r12 in

Eqs. (13)-(16), then the APD fλ(r12) displays the electron-electron cusp fλ(r12 → 0) =

fλ(0)(1 + λr12+ ...), which implies, in turn, that also wλ

effmust behave, for small r12, as

wλ

eff(r12→ 0) = λ/r12+ .... In particular, for λ = λphys, we always have wλphys

eff

(r12→ 0) =

1/r12+..., as shown, for the case of the He atom in Fig. 1. If we choose a cuspless nonlinear

path, like wλ(r12) = erf(λr12)/r12, then the small r12behavior of wλ

eff(r12) is known only

when we are approaching the physical interaction.32

VII.PRELIMINARY APPLICATIONS

The construction of an approximate wλ

eff(r12;[n]) can thus start with the decomposition

wλ

eff(r12;[n]) = wKS

eff(r12) + wλ(r12) + ∆wλ

eff(r12;[n]),(36)

where the term ∆wλ

eff(r12;[n]) should take care of the fact that, when the electron-electron

interaction is turned on, the one-electron density n(r) and (for confined systems) the highest

eigenvalue ǫλ

maxdo not change.

As a starting point, we applied the method of Eqs. (28)-(30) to the He isoelectronic

series. In this simple (yet not trivial) 2-electron case, we have the advantage that we can

treat Tf[f] exactly. We developed an approximation6for ∆wλ

eff(r12;[n]) based on the one used

for the uniform electron gas.26This approximation is designed to mimic the conservation

of n(r) along the DFT adiabatic connection, but does not take into account the eigenvalue

conservation. It works remarkably well when combined with a nonlinear adiabatic connection

path wλ(r12) = erf(λr12)/r12that separates short- and long-range correlation effects, and

is reported in the Appendix of Ref. 6. Preliminary implementations of the self-consitent

procedure of Eqs. (28)-(30) yield ground-state energies within 1 mH with respect to full

configuration interaction (CI) calculations in the same basis set. However, the way we carried

out these first tests was simply based on a direct minimization of few variables parametrizing

vKS(r). This rather inefficient way to implement Eqs. (28)-(30) needs further improvment.

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Besides, the Kohn-Sham potentials we obtain in this way are very unstable (like the ones of

Ref. 33), although the corresponding total energies and electronic densities are stable, and

do not display the variational collapse of perturbation-theory-based approximate Ec[n].34

VIII. PERSPECTIVES

The generalization to many-electron systems of nonuniform density of the approximation

built in Ref. 6 for wλ

eff(r12) is not straightforward. If we simply apply it to the Be atom

case (by using only one geminal), we obtain energy errors of 300 mH. Adding the eigenvalue

conservation can improve the results, but, of course, there is not a unique way to impose it.

So far, we found that the final outcome strongly depends on how we impose the eigenvalue

conservation (i.e., if we use a functional form with one parameter adjusted to keep the

eigenvalue independent of λ, the results drastically depend on the chosen functional form). It

seems thus necessary to switch to more than one geminal, and/or to find better constructions

for wλ

eff(r12).

In particular, it may be promising to explore the possibility to construct approxima-

tions inspired to the FHNC,30,31,35and to try to include some of the new results on N-

representability conditions for the pair density.20Different approximations with respect to

the one of Eq. (24) for the functional Tf[f], and the use of Eq. (23) also deserve further

investigation.

Acknowledgments

We thank E.K.U. Gross for useful discussions and suggestions.One of the authors

(P.G.G.) gratefully aknowledges the 30th International Worksohop on Condensed Matter

Theory organizers for supporting her participiation to the meeting.

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3C. Fiolhais, F. Nogueira, and M. Marques (eds.), A Primer in Density Functional Theory

(Springer-Verlag, Berlin, 2003).

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