VOLUME 76, NUMBER 11 PHYSICAL REVIEW LETTERS 11 MARCH 1996
Constraint Satisfaction in Local and Gradient Susceptibility Approximations:
Application to a van der Waals Density Functional
John F. Dobson and Bradley P. Dinte
Faculty of Science and Technology, Griffith University, Nathan, Queensland 4111, Australia
(Received 31 October 1995)
We show how charge conservation and reciprocity can be built into local density or gradient
approximations for density-density response functions (susceptibilities). We apply these ideas to derive
from first principles a variant of the Rapcewicz-Ashcroft formula for the van der Waals interaction. We
also discuss how improved formulas may be obtained.
PACS numbers: 31.15.Ew, 34.20.–b, 71.45.Gm
Overall, electron density functional theory (DFT) has
been gratifyingly successful in approximating the energies
of inhomogeneous interacting electronic systems [1,2].
Part of this success is due to the satisfaction of the
exchange-correlation (xc) hole normalization condition,
often achieved with considerable effort [3]. The usual lo-
cal density approximation (LDA) [4] and its various gra-
dient extensions [2] do not, however, give an adequate
description of dispersion or van der Waals (vdW) forces
[5]. The approach to be introduced here both simplifies
the problem of achieving hole normalization, and facili-
tates the derivation of van der Waals functionals.
The difficulty of describing vdW forces in the LDA or
gradient approaches is not surprising since these forces
depend on correlations between distant density fluctua-
tions, which may be different from those in the uniform or
near-uniform electron gas upon which the above approxi-
mations are based. Rather general methods have been
proposed [6,7] for treating these long-ranged correlations
by explicit solution of nonlocal screening equations, while
still making the local density approximation for a suit-
able intermediate quantity. These methods are expected
to work well for a wide variety of situations including
both overlapping and nonoverlapping electron distribu-
tions. Such approaches will usually require substantial
computation, however.
We therefore first consider a less ambitious problem,
the efficient calculation of the vdW interaction between
a pair of nonoverlapping neutral systems, using as input
only the ground-state electron densities n
1
r and n
2
r
of the two systems. Rapcewicz and Ashcroft [8] have al-
ready given an expression for this limit of the vdW inter-
action, using arguments based on Feynman diagrams and
three-point functions, plus a conjecture about an appropri-
ate average density computed between two distant points.
We will here derive a similar but not identical expres-
sion from a straightforward local density approximation
for a suitable quantity. The result will depend crucially
on satisfaction of some constraints, and the methods used
to achieve this should be useful elsewhere.
Consider a pair of nonoverlapping many-electron
systems so that electrons in system 1 can be considered
distinguishable from those in system 2. Under these
circumstances it is well known that second-order per-
turbation theory in the Coulomb interaction between the
two systems yields a good approximation to the van der
Waals or dispersion interaction. Zaremba and Kohn [9]
reexpressed this second-order energy, without further
approximation, in the form
E
2
2
¯ h
2p
Z
dr
1
dr
2
dr
0
1
dr
0
2
e
2
jr
1
2 r
2
j
e
2
jr
0
1
2 r
0
2
j
3
Z `
0
x
1
r
1
, r
0
1
, iux
2
r
2
, r
0
2
, iu du . (1)
Here x
1
r , r
0
, v and x
2
r , r
0
, v are the exact density-
density response functions (in the Kubo sense) of each
separate system in the absence of the other. x
1
is
defined by the linear density response dn
1
r exput of
the electrons in system 1 to an externally applied electron
potential energy perturbation dV
ext
r exput ,
dn
1
r
Z
x
1
r , r
0
, iudV
ext
r
0
dr
0
, (2)
and similarly for x
2
. It is important to note that x
1
includes the electron-electron interaction amongst the
electrons of system 1 to all orders, and similarly for x
2
.
[Note also that, unlike Ref. [9], we have referred the space
arguments of x
1
and x
2
in (1) to a common origin.]
To simplify (1), we will approximate x
1
and x
2
via
a form of local density approximation. That is, we will
appeal to a simple model of the density-density response
of a homogeneous electron gas, and modify it suitably to
approximate the response of the inhomogeneous gas in
each system. The simplest such homogeneous response
model is a nondispersive form of hydrodynamics in which
we eliminate the fluid velocity v between the linearized
continuity equation, Newton’s second law (with no pres-
sure term), and Poisson’s equation. In Fourier repre-
sentation, this gives a density perturbation dnq, v
x q, vdV
ext
q, v where dV
ext
is the bare external po-
tential energy and
x
hom
q ! 0, v
q
2
n
0
mv
2
2v
2
P
n
0
, (3)
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