Simple model for the spherically- and system-averaged pair density: Results for two-electron atoms

Abstract

As shown by Overhauser and others, accurate pair densities for the uniform electron gas may be found by solving a two-electron scattering problem with an effective screened electron-electron repulsion. In this work we explore the extension of this approach to nonuniform systems, and we discuss its potential for density functional theory. For the spherically- and system-averaged pair density of two-electron atoms we obtain very accurate short-range properties, including, for nuclear charge $Z\ge 2$, ``on-top'' values (zero electron-electron distance) essentially indistinguishable from those coming from precise variational wavefunctions. By means of a nonlinear adiabatic connection that separates long- and short-range effects, we also obtain Kohn-Sham correlation energies whose error is less than 4 mHartree, again for $Z\ge 2$, and short-range-only correlation energies whose accuracy is one order of magnitude better. Comment: 9 pages, 6 figures (14 .eps files); revised version, to appear in Phys. Rev. A

Full text

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arXiv:cond-mat/0411179v2 [cond-mat.mtrl-sci] 8 Feb 2005
Simple model for the spherically- and system-averaged pair density:
Results for two-electron atoms
Paola Gori-Giorgi and Andreas Savin
Laboratoire de Chimie Th´ eorique, CNRS, Universit´ e Pierre et Marie Curie, 4 Place Jussieu, F-75252 Paris, France
(Dated: February 2, 2008)
As shown by Overhauser and others, accurate pair densities for the uniform electron gas may
be found by solving a two-electron scattering problem with an effective screened electron-electron
repulsion. In this work we explore the extension of this approach to nonuniform systems, and we
discuss its potential for density functional theory. For the spherically- and system-averaged pair
density of two-electron atoms we obtain very accurate short-range properties, including, for nuclear
charge Z ≥ 2, “on-top” values (zero electron-electron distance) essentially indistinguishable from
those coming from precise variational wavefunctions. By means of a nonlinear adiabatic connection
that separates long- and short-range effects, we also obtain Kohn-Sham correlation energies whose
error is less than 4 mHartree, again for Z ≥ 2, and short-range-only correlation energies whose
accuracy is one order of magnitude better.
I. INTRODUCTION AND SUMMARY OF
RESULTS
Density Functional Theory (DFT) [1, 2, 3] is nowa-
days the most widely used method for electronic struc-
ture calculations, in both condensed matter physics and
quantum chemistry, thanks to a combination of low com-
putational cost and reasonable accuracy.
In the application of this theory within the Kohn-Sham
(KS) formalism [4], one deals with a model system (the
KS system) of N noninteracting electrons in a local po-
tential vKS(r) that forces them to yield the same density
n(r) of the physical system. The energy of the physi-
cal system is then obtained from that of the KS system
via a functional of the density, whose only term not ex-
plicitly known is the exchange-correlation energy Exc[n].
Correspondingly, in the local potential vKS(r) there is an
unknown term, vxc(r) = δExc[n]/δn(r).
The success of KS DFT is mostly due to the fact that
even simple physical approximations of Exc[n], like the
local density approximation (LDA) [4], already give ac-
ceptable results for many purposes. This spurred fun-
damental research in the field, and led to a wealth of
more and more sophisticated exchange-correlation func-
tionals [2, 3, 5], and to the development of different ap-
proaches to DFT [6, 7].
Recently, in the search for accurate Exc[n], the fo-
cus of a large part of the DFT community has shifted
from seeking explicit functionals of the density like the
generalized gradient approximations (GGA) [8], to im-
plicit functionals, tipically using the Kohn-Sham orbital
kinetic energy density [9] or the Kohn-Sham orbitals
(see, e.g., [3, 10, 11]). The so-called “third generation”
of exchange-correlation functionals is based on the ex-
act exchange of the noninteracting (KS) system, sim-
ply obtained by putting in the formal expression for the
Hartee-Fock exchange the Kohn-Sham orbitals ϕiσ(r).
Such expression corresponds to an implicit functional
of the density, Ex[n] = Ex[{ϕiσ[n]}]. The local poten-
tial vx(r) = δEx[n]/δn(r) that generates the orbitals
ϕiσ(r)[n] can be obtained via the optimized effective po-
tential method (OEP) [12].
In this broad context, sketchily summarized here, we
propose a simplified method to build the “bridge” be-
tween the physical and the KS system, or, more gener-
ally, with a reference model system of partially interact-
ing electrons. We focus on a quantity which is known to
play a crucial role in DFT and has an intuitive physical
meaning, the spherically and system-averaged electronic
pair density f(r12) (also known in chemistry as spherical
average of the intracule density, see e.g. [13, 14, 15, 16],
and especially [17, 18]). Given the spin-resolved diagonal
of the two-body reduced density matrix,
γ(2)
σ1σ2(r1,r2) =
?
σ3...σN
?
|Ψ(r1σ1,...,rNσN)|2dr3...drN,
(1)
we define the spin-summed pair density n2(r1,r2),
n2(r1,r2) =N(N − 1)
2
?
σ1σ2
γ(2)
σ1σ2(r1,r2), (2)
and we integrate it over all variables but r12= |r2− r1|
by switching, e.g., to center-of-mass coordinates, R =
1
2(r1+ r2), r12= r2− r1,
?
4π
f(r12) =dRdΩr12
n2
?
R −r12
2,R +r12
2
?
. (3)
The function f(r12) times the volume element 4πr2
is proportional to the probability density for the particle-
particle distance in a system of N electrons in the state
Ψ, and is normalized to the number of electron pairs,
N(N − 1)/2 .
pectation value of the electronic Coulomb repulsion (in
Hartree atomic units used throughout),
12dr12
This quantity fully determines the ex-
?Vee? ≡ ?Ψ|Vee|Ψ? =
?∞
0
4πr2
12
f(r12)
r12
dr12, (4)
and is a measurable quantity, being essentially the
Fourier transform of the electronic static structure fac-
tor [19]. By construction, the one-electron density n(r) is

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2
0
0.002
0.004
0.006
0.008
0.01
0.012
0.014
0 1 2 3
r12
4 5 6
f(r12)
’exact’
this work
Kohn-Sham
H-
0.04
0.08
0.12
0.16
0 0.5 1 1.5 2 2.5
f(r12)
r12
’exact’
this work
Kohn-Sham
He
0
0.2
0.4
0.6
0.8
0 0.2 0.4 0.6 0.8 1 1.2
f(r12)
r12
’exact’
this work
Kohn-Sham
Li+
0
0.5
1
1.5
2
0 0.2 0.4 0.6 0.8 1
f(r12)
r12
’exact’
this work
Kohn-Sham
Be2+
FIG. 1: Spherically and system-averaged pair densities for two-electron atoms: ’exact’ results [40] are compared with the values
obtained for the Kohn-Sham system and with the present approach, which is designed to get realistic f(r12) starting from the
Kohn-Sham ones.
the same in the KS and in the physical system, whereas
f(r12) will be different in the two cases, as shown, e.g., in
Fig. 1 for some two-electron atoms. In the physical sys-
tem f(r12) has a much lower “on-top” value f(r12= 0)
than in the KS system, and it has a cusp [20], as expected
from the fact that the electrons repel each other via the
Coulomb interaction. Roughly speaking, in the classic
DFT approach to correlation, the difference in energy
arising when we evaluate the r.h.s. of Eq. (4) with the
two f(r12), the physical and the KS, is what one tries to
describe with a universal functional of the density [21].
Here we follow a different approach: we try to build re-
alistic f(r12) from a set of simple radial equations, to be
solved for each system, and eventually coupled to a DFT
calculation.
Our approach is inspired by the seminal work of Over-
hauser [22] and its subsequent extension [23], in which the
function f(r12) for the uniform electron gas is obtained
from a set of geminals, solutions of a radial Schr¨ odinger
equation with an effective electron-electron (e-e) poten-
tial. Simple approximations for such effective e-e po-
tential give indeed accurate results at all relevant densi-
ties [23, 24, 25]. Here we try to generalize this approach
to systems of nonuniform density to get accurate f(r12).
The main goal of the present work is understanding
whether the method is promising, and whether it is worth
developing and refining it. To this purpose, we define the
formalism (Sec. II), we give a physically-motivated pre-
scription for the effective e-e potential (Sec. III), and we
test it on the simple but not trivial case of two-electron
atoms (Sec. IV). The prescription for the effective e-e
potential used here is not very sophisticated. Improve-
ments along the lines of what has been done for the uni-
form electron gas [23, 24, 25] will be the subject of future
work. Yet, even at this simple first stage of the theory
we already obtain rather accurate results, especially for
the short-range part of f(r12) (see Fig. 1 and Table I).
In Sec. V we show that with the present approach we can
also recover the difference in kinetic energy between the
physical and the KS system. Finally, Sec. VI is devoted
to conclusions, perspectives and open questions.
II.FORMALISM
In addition to the work on the “Overhausermodel” [22,
23, 24], the approach described here takes advantage of
inspiring papers on the possibility of constructing a pair-
density functional theory [26, 27, 28, 29], and a local-
density-of-states functional theory [30].
Our starting point is a constrained search over1
1) “effective” orthonormal geminals ψi(r12) that mini-
mize the electron-electron relative kinetic energy T12=
−∇2
2N(N−
r12(the reduced mass for the relative motion is 1/2)

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3
and yield the exact f,?
i|ψi(r12)|2= f(r12),
?
i
min
{ψi}→f
?ψi| − ∇2
r12|ψi?, (5)
thus leading to a set of radial equations formally similar
to the KS ones,
[−∇2
N(N−1)/2
?
i=1
r12+ veff(r12)]ψi(r12) = ǫiψi(r12) (6)
|ψi(r12)|2= f(r12). (7)
These equations imply that an expansion in spherical
harmonics of f(r12) has been done, so that the kinetic
energy operator also contains the usual ℓ(ℓ+1)/r2term.
To fully define these equations we need a rule for the oc-
cupancy of the effective geminals. In analogy with what
has been done for the uniform electron gas [23, 24], we
can assign spin degeneracy 1 to even-angular-momentum
states (singlet) and spin degeneracy 3 to odd-angular-
momentum states (triplet), up to N(N − 1)/2 occupied
states. More generally, for open shell systems it could
be better to develop the formalism for the spin-resolved
quantities, starting from Eq. (1). This will be investi-
gated in future work.
The effective electron-electron potential veff(r12) of
Eq. (6) is the Lagrange parameter for f(r12), and is a
functional of f itself and of the electron-nucleus external
potential Vne (or, equivalently, of the density n(r)). To
see this, we can rewrite our Eqs. (6)-(7) in terms of a
minimization of the total energy in two steps, using the
constrained search formalism [31, 32] for the ground state
energy E = minΨ?Ψ|T + Vee+ Vne|Ψ?,
E = min
f
min
Ψ→f
?
min
{ψi}→f
?
i
?ψi| − ∇2
r12|ψi? +
?
f
r12dr12
+?Ψ|T + Vne|Ψ? − min
{ψi}→f
?
i
?ψi| − ∇2
r12|ψi?
?
.(8)
Defining the kinetic and external-potential functional as
FKE[f;Vne] =
?ψi| − ∇2
min
Ψ→f?Ψ|T + Vne|Ψ? − min
{ψi}→f
?
i
r12|ψi?, (9)
we can rewrite
E = min
f
?
min
{ψi}→f
?
i
?
?ψi| − ∇2
r12|ψi? +
?
f
r12dr12
+FKE[f;Vne].(10)
Searching this minimum by directly varying the ψi(with
given, fixed, Vne) leads to Eqs. (6)-(7) with the identifi-
cation
veff(r12) =
1
r12
+δFKE[f;Vne]
δf(r12)
.(11)
Thus, in principle we could recover the whole ground-
state energy via the (unknown) system-dependent func-
tional FKE[f;Vne]. In practice, it seems much more fea-
sible to combine Eqs. (6)-(7) with a DFT calculation,
that yields the complementary information (the density,
and thus ?Ψ|Vne|Ψ?). The steps of Eqs. (8)-(11) can be
repeated for arbitrary electron-electron interaction and
external one-body potential. In particular, we can set
Vλ
tential that keeps the density equal to the one of the
physical system. One could thus obtain fλat each cou-
pling strength λ between 0 and 1 from Eqs. (6)-(7) with
a suitable vλ
eff. The correlation energy of KS theory is
then simply given by [6, 33, 34]
ee= λVeeand Vne= Vλ, where Vλis an external po-
Ec[n] =
?1
0
dλ
?
dr12fλ(r12) − fλ=0(r12)
r12
. (12)
Alternatively, this procedure (usually called adiabatic
connection [34]) can be performed along a nonlinear path,
e.g., by setting [6, 35, 36, 37] vλ
erf(x) is the error function (see Sec. V). Eventually, the
two sets of equations, KS and (6)-(7) plus (12), could
be solved together self-consistently. This last issue is dis-
cussed in Sec. VI. Notice that if we combine Eqs. (6), (7)
and (12) with a DFT calculation, we only need to approx-
imate the potential vλ
eff(r12) and not the whole functional
FKEsince the remaining information is provided by DFT.
It is also worth to stress at this point that there is no
wavefunction behind our Eqs. (6)-(7): the effective gem-
inals ψiare defined via Eq. (5), and by specifying their
occupancy (e.g., triplet and singlet). A bosonic version
of the theory, in which only one geminal (proportional
to
?f(r12)) is occupied can also be considered [29, 38].
In this work we only focus on two-electron systems for
which the two choices are equivalent. A careful com-
parison of performances of the “fermion-like” and of the
“boson-like” occupancy in the uniform electron gas is the
subject of current investigations [39].
As for KS DFT, the formalism just described can be
useful only if simple approximations for veff(r12) yield ac-
curate results. This is what we start to check in the rest
of this paper. First, we construct a physically-motivated
veff for two-electron atoms for the fully-interacting sys-
tem, and we compare our results with “exact” ones [40].
Then, we generalize our construction to build veff along
the adiabatic connection, and we calculate the KS corre-
lation energy.
ee= erf(λr)/r, where
III.
POTENTIAL: THE OVERHAUSER MODEL
EFFECTIVE ELECTRON-ELECTRON
For the interacting electron gas of uniform density n,
Overhauser [22] proposed a simple and reasonable effec-
tive potential veff(r12): he took the sphere of volume n−1
around a given electron as the boundary within which
the other electrons are excluded, due to exchange and

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4
correlation effects. In the standard uniform-electron-gas
model, a rigid positively-charged background maintains
the electrical neutrality. Thus the exclusion region (or
“hole”) around a given electron, modeled with a sphere
of radius rs= (4πn/3)−1/3, uncovers the background of
positive charge, leading to an effective screened Coulomb
potential with screening length rs,
vOv
eff(r12;rs) =
1
r12
−
?
|r|≤rs
n
|r − r12|dr, (13)
equal to
vOv
eff(r12;rs) =
vOv
eff(r12;rs) =
1
r12+
r2
2r3
12
s−
0
3
2rs
r12≤ rs
r12> rs. (14)
Equations (6)-(7), combined with the Overhauser effec-
tive potential of Eq. (14) gave extremely accurate results
for the short-range part (r12≤ rs) of the function f(r12)
in the uniform electron gas at all relevant densities [23].
A more sophisticated effective potential, based on a self-
consistent Hartree approximation, extended such accu-
racy to the long-range part of f(r12) at metallic den-
sities [24]. Other approximate veff(r12) for the uniform
electron gas have also been proposed [25], and exact prop-
erties have been derived [41].
To produce realistic f(r12) for nonuniform systems
from Eqs. (6)-(7), here we generalize the original idea
of Overhauser [22, 23] to two-electron atoms, and show
that it gives rather accurate results, especially for the
short-rangepart of f(r12). We start from the effective po-
tential v(0)
eff(r12) that generates fKS(r12), the spherically-
and system averaged pair density of the Kohn-Sham sys-
tem. In the special case of a spin-compensated two-
electron system, the KS wavefunction is simply equal to
1
2
?n(r1)?n(r2). Because, at this first stage, we are in-
terested in testing our method as a “bridge” between
the KS and the real system, here we use the “exact”
Kohn-Sham system. We thus take accurate one-electron
densities [40], and construct fKS(r12),
fKS(r12) =1
4
?
n
?
R −r12
2
?
n
?
R +r12
2
?
dRdΩr12
4π
,
(15)
and the corresponding “exact” potential v(0)
can be calculated by inverting Eqs. (6)-(7),
eff(r12), that
v(0)
eff=∇2√fKS
√fKS
+ const.(16)
For systems with more than two electrons, the poten-
tial v(0)
effcould be calculated, e.g., with the methods of
Refs. [42, 43]. In practice, it would be much more efficient
to build approximations also for v(0)
amples of functions fKSfor nuclear charges Z = 1,2,3,4
are given in Fig. 1: they have a maximum at r12 = 0,
as expected in a system of two non-interacting electrons
with antiparallel spins in a confining one-body external
eff(see Sec. VI). Ex-
H−
2.1
He
0.86
Li+
0.54
Be2+
0.39
Ne8+
0.15rs
f(0)
“exact”
LDA
0.0021 0.104 0.528 1.526
0.0027 0.106 0.534 1.523
0.0047 0.119 0.563 1.587
32.6
32.7
33.0
rmax
12
“exact”
0.835
0.927
0.193 0.083 0.0465 0.0074
0.194 0.083 0.0465 0.0074
f(rmax
“exact”
12 )0.0031 0.114 0.55
0.0040 0.117 0.56
1.56
1.56
32.74
32.74
?Vee? − ?Vee?KS -0.12
“exact”
-0.097 -0.10
-0.078 -0.082 -0.089 -0.09
-0.10-0.10
-0.07
TABLE I: Our results for the function f(r12) for two-electron
atoms (first line for each property) compared with the corre-
sponding “exact” quantities [40]. In the first line of the table
we report the average rsas defined by Eqs. (17) and (19). For
the “on-top” value f(0) we also show the LDA result (with
f(0) for the uniform electron gas from Ref. [23]). All values
are in Hartree atomic units.
potential. When the interaction is turned on, the aver-
age distance between the two electrons increases, with
the constraint that n(r) is kept fixed. We can thus imag-
ine that, with respect to the Kohn-Sham system, in the
physical system the Coulomb repulsion between the elec-
trons creates, on average, a screening “hole” around the
reference electron of volume (n)−1, where n is an average
density (i.e., n(r) integrated over the wavefunction),
n =
1
N
?
drn(r)2. (17)
An approximate veff(r12) could thus be simply con-
structed as
veff(r12) ≈ v(0)
eff(r12) + vOv
eff(r12;rs)(18)
with an average rsin vOv
effof Eq. (14),
rs=?4π
3n?−1/3.(19)
The Overhauser-like potential vOv
relation potential to be added to the one that generates
fKS. It describes the correlation between pairs of elec-
trons due to Coulomb interaction, and keeps the informa-
tion on the one-electron density in an approximate way,
via the average n of Eq. (17). Of course, for more com-
plicated systems we expect to need a more sophisticated
construction for rs.
eff(r12;rs) is thus a cor-
IV. RESULTS
We have inserted the potential of Eq. (18) into Eqs. (6)-
(7), and solved them for several two-electron atoms. Our
results are shown in Fig. 1 and summarized in Table I.

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5
We see that the simple effective potential of Eq. (18) gives
already reasonable results for Z = 1 and 2, and that the
accuracy of the results increases with Z (as the system
becomes less and less correlated). The “on-top” value
f(0) is essentially exact for Z ≥ 2, and is much better
than the LDA estimate (normally regarded as accurate)
for all Z. This feature is appealing, since the on-top value
plays an important role in DFT [44], and accurate f(0)
are not easy to obtain from ab initio methods (see, e.g.,
Ref. [45] and references therein). The term 1/r12in the
effective potential ensures that the calculated f(r12) sat-
isfies the exact cusp condition f′(0) = f(0). Table I also
shows that the position rmax
12
the maximum of f is very well predicted by the present
approach. The presence of this maximum is essentially
due to the combined effect of the Coulomb repulsion be-
tween the electrons and the confining external potential.
In Fig. 2 we consider He and Ne8+, and we compare
the correlated part of our f, fc = f − fKS, with the
“exact” result [40] and with the corresponding quantity
calculated within LDA, i.e.,
and the height f(rmax
12 ) of
fLDA
c
(r12) =1
2
?
n(r)2gc(r12;n(r))dr, (20)
where gc is the pair-correlation function of the uni-
form electron gas at full coupling strength, taken from
Ref. [46]. (For an extended system of uniform density n,
we have gc= 2fc/nN.) Figure 3 shows the same quanti-
ties multiplied by 4πr12, i.e. the integrand of Eq. (4) for
the correlation part of ?Vee?: the area under each curve
gives ?Vee?−?Vee?KS. In the last line of Table I we report
quantitative results for ?Vee?−?Vee?KS. This quantity is
less accurate than the short-range properties, but it is
still encouraging. Moreover, it saturates for large Z as in
the exact case.
V. ADIABATIC CONNECTION AND
CORRELATION ENERGY
For the calculation of the energy of the physical sys-
tem, in addition to Vc[n] = ?Vee?−?Vee?KS, one needs to
know the kinetic-energy difference, Tc[n] = ?T?−?T?KS,
that can be obtained via the adiabatic connection for-
malism [6, 33, 34]. By varying a parameter λ, the in-
teraction vλ
ee(r12) between the electrons is switched on
continuously from zero to 1/r12, while the density is
kept fixed by an external one-body potential Vλ.
vλ=0
ee
= 0 and vλ=a
ee
= 1/r12, the KS correlation energy
Ec[n] = Tc[n] + Vc[n] is given by [6, 34]
If
Ec[n] =
?a
0
dλ
?∞
0
dr124π r2
12fλ
c(r12)∂vλ
ee(r12)
∂λ
, (21)
where fλ
Usually, the adiabatic connection is performed along a
linear “path” [11, 33], by setting vλ
to Eq. (12). If one is able to compute the exact fλ
c= fλ− fKS.
ee= λ/r12, which leads
c, the
-0.08
-0.06
-0.04
-0.02
0
0 0.5 1 1.5 2 2.5
fc(r12)
r12
He
’exact’
this work
LDA
-4
-3.5
-3
-2.5
-2
-1.5
-1
-0.5
0
0.5
0 0.1 0.2 0.3 0.4 0.5
fc(r12)
r12
Ne8+
’exact’
this work
LDA
FIG. 2: The correlated part of the spherically- and system-
averaged pair density, fc(r12) = f(r12)−fKS(r12). Our results
for He and Ne8+are compared with the “exact” ones and with
the LDA result (the hole for the uniform electron gas is taken
from Ref. [46]).
resulting Ecfrom Eq. (21) is independent of the choice
of vλ
ee. However, when approximations are made some
“paths” can give much better results than others [6]. As
we shall see, this is the case with the present approach.
We build an Overhauser-like potential for interaction
vλ
eff) as
ee(to be added to v(0)
vOv, λ
eff
(r12;rs) = vλ
ee(r12)−
?
|r|≤rs
nvλ
ee(|r−r12|)dr. (22)
That is, the average density n of Eq. (17) (and thus
the average rs) is kept fixed to mimic the fact that the
one-electron density does not change along the adiabatic
connection. The modified interaction vλ
a sphere of radius rs and of positive uniform charge of
density n that attracts the electrons with the same mod-
ified interaction.This attractive background approxi-
mates the effect of the external potential Vλon f.
eeis screened by
A. Linear adiabatic connection
If we choose vλ
vOv, λ
eff
(r12;rs) = λvOv
given by Eq. (14).
The results for ?Vλ
shown in Fig. 4, and are compared with the “exact”
ones of Ref. [42].The correlation energy Ec can be
ee
= λ/r12
we simply obtain
eff(r12;rs) is
eff(r12;rs), where vOv
ee? − ?Vee?KS for He and Ne8+are