Analytic form of the correlation energy of the uniform electron gas
Emil Proynov
*
and Jing Kong
Q-Chem, Inc., 5001 Baum Boulevard, Suite 690, Pittsburgh, Pennsylvania 15213, USA
Received 1 October 2008; published 26 January 2009
An expression for the correlation energy density of the uniform electron gas is derived based on the adiabatic
connection method. It covers with a single form the transition between high-density and low-density regions in
the range 0.1 r
s
30, parsing the entire spin-polarization range 0 1. The pair-correlation function used
to generate the result has been used previously to describe mainly finite systems. We argue that the universality
implied by the short-wavelength hypothesis goes both ways, and a model that works well for finite systems
may be adapted to describe the uniform electron gas as well.
DOI: 10.1103/PhysRevA.79.014103 PACS numbers: 03.65.Db, 03.65.Fd, 31.15.eg
The uniform electron gas UEG is an important subject
in the quest for accurate exchange and correlation functionals
in density functional theory. Local local spin density1,2
and semilocal generalized gradient approximation GGA
and meta GGA3–6 approximations were created from a
detailed knowledge of the UEG properties. The success of
these approximations for systems much different from the
UEG can in part be explained by the short-wavelength hy-
pothesis 7. It suggests that some degree of universality may
exist in the way electrons correlate in different systems, es-
pecially at short interelectronic distances. Alternatively, effi-
cient functionals for atoms and molecules have been devel-
oped without referring to properties of the UEG 8–12. In
this Brief Report, we argue that the universality implied by
the short-wavelength hypothesis goes both ways, and a
model that works well for finite systems can be used to de-
rive approximations for the UEG correlation energy. Some
evidence along this line has been given previously 13.
Exact or nearly exact first-principles results are still in
demand for the UEG correlation energy. Interpolations based
on quantum Monte Carlo QMC data 14,15 combined with
known exact limits and constraints have reached a reasonable
accuracy 2,16–18, while feasible first-principles solutions
are still lacking it.
In this Brief Report we use the adiabatic connection
method 19–22, employing a spin-polarized, opposite-spin
pair-correlation function PCF of Jastrow type, which de-
pends on the coupling strength parameter :
g
↑↓
r = 1 - exp- k
↑↓
2
r
2
2-
↑↓
2+ r - exp-2k
↑↓
2
r
2
1-
↑↓
2+ r +
↑↓
2
1+ r +
2
r
2
4
, 1
where r = |r
1
- r
2
| is the interelectronic distance. This PCF
obeys automatically the known opposite-spin, -dependent
cusp condition 23 for any k
↑↓
and , and has the following
cusp value:
lim
r→0
g
↑↓
r =
↑↓
k
↑↓
2
0 for any 0 1. 2
In the noninteracting limit =0, the opposite-spin correla-
tion vanishes, which imposes the conditions
lim
=0
g
↑↓
r = 1, lim
=0
↑↓
k
↑↓
=1 for any r, k
↑↓
. 3
This kind of PCF has been used in the past to derive func-
tionals mainly for finite systems 11,12,24.
The opposite-spin correlation energy density is next deter-
mined by the adiabatic connection formula ACF20,22
applied to the considered case n = n
↑
+ n
↓
:
c
opp
=2
n
↑
n
↓
n
0
1
d
0
r drg
↑↓
r -1 . 4
To this end we have to specify the form of the cusp factor
↑↓
k
↑↓
. We deduce it from the exact PCF normalization
condition, which in the case of the UEG reads opposite
spins only
0
r
2
drg
↑↓
r -1 =0. 5
Solution of the above equation first at =1 with respect to
↑↓
gives
↑↓
=1
k =
1
2Ak
- Bk + Bk
2
-4AkC
1/2
, k k
↑↓
,
6
Ak =
2
4
+
3
2
64k
2
+
1
2k
, 7
Bk = 2-
2
2
+
3
2k
, C =
2
4
-2. 8
The form of
↑↓
k
↑↓
at arbitrary is then obtained from Eq.
6 by the substitution k
↑↓
→ k
↑↓
/ , following the coordinate-
scaling rules of Levy 22. This form of
↑↓
obeys automati-
cally the noninteracting limit, Eq. 3.
Next we solve analytically the ACF, Eq. 4, using Eq. 1
with the above form of
↑↓
k
↑↓
, Eq. 6. The resulting ex-
pression is decomposed into a sum of several different types
of integral, each handled separately. After some tedious and
lengthy algebra that we do not present here due to a lack of
space, we arrive at the following expression for the opposite-
spin correlation energy density, up to an arbitrary form of the
correlation wave vector k
↑↓
:
c
opp
=
n
↑
n
↓
n
Q
1
↑↓
k
↑↓
+ Q
2
↑↓
k
↑↓
+ Q
3
↑↓
k
↑↓
, 9
*
emil@q-chem.com
PHYSICAL REVIEW A 79, 014103 2009
1050-2947/2009/791/0141034 ©2009 The American Physical Society 014103-1