Origin of the field-counteracting term of the Kohn-Sham exchange-correlation potential
of molecular chains in an electric field
O. V. Gritsenko, S. J. A. van Gisbergen, P. R. T. Schipper, and E. J. Baerends
Section Theoretical Chemistry, Vrije Universiteit, De Boelelaan 1083, 1081 HV Amsterdam, The Netherlands
Received 29 December 1999; published 14 June 2000
The origin and the mechanisms responsible for an ultranonlocal field-counteracting term in the Kohn-Sham
exchange-correlation potential
xc
of molecular chains in an external electric field are established for prototype
systems. Using various analysis tools—conditional probability amplitude analysis, construction of the nearly
exact
xc
, one-electron perturbation theory, and Krieger-Li-Iafrate KLI calculations—it is shown that a
field-counteracting term emerges in the ‘‘response’’ part
resp
of
xc
. For systems A
2
of two open-shell units
A, with H
2
as a prototype, the left-right electron Coulomb correlation generates a step in
resp
, which effec-
tively compensates for an electric field in the limit of large interatomic distances. For systems A
n
of closed-
shell units A, with He
2
and H
2
+H
2
as prototypes, a field-counteracting term is generated in
resp
by the Pauli
repulsion of electrons in the occupied Kohn-Sham orbitals. An analytical estimate of the field-counteracting
step is obtained for He
2
with the exchange-only KLI model of
resp
, and the existence of an ultranonlocal
linear term in
resp
is established with KLI calculations on the prototype molecular chain H
18
.
PACS numbers: 31.15.Ew, 31.25.-v
I. INTRODUCTION
The theoretical originality and practical efficiency of the
Kohn-Sham KS approach of density-functional theory
DFT derive from the fact that in this theory all the effects
of electron correlation are embodied in the single local KS
exchange-correlation xc potential
xc
( r) 1. Together with
the external potential
ext
and the Hartree potential of the
electrostatic electron repulsion
H
,
xc
determines the KS
orbitals
i
,
-
1
2
2
+
ext
r +
H
r +
xc
r
i
r =
i
r
i
r ,
1.1
and the electron density ( r) of a many-electron system:
r =
i =1
N
|
i
r |
2
. 1.2
In the absence of additional external fields,
ext
is the poten-
tial
N
of the pointlike nuclei. The features of
xc
which
distinguish nuclear positions are the relatively deep wells
localized around each nucleus 2–6, which arise mostly
from the self-interaction correction term in the exchange po-
tential.
However,
xc
develops a different feature in response to a
static electric extra field
E
( r),
ext
( r) =
N
( r) +
E
( r),
which is of a crucial importance for accurate evaluation of
the electronic polarization of molecular chains, semiconduc-
tors and insulating crystals within DFT 7–16. As was first
established for model spontaneously polarized insulator sys-
tems 8,
xc
builds up an ultranonlocal linear term, which
counteracts the long-wavelength potential of the internal
electric field in the polarized system. Such a linear term can-
not be taken into account in standard band-structure codes.
The possible existence of such a linear term implies that
xc
does not depend solely on the periodic bulk density the only
density available in a band-structure code with periodic
boundary conditions. In addition, it has an ‘‘ultranonlocal’’
density dependence on the surface density. For polarized sys-
tems this means a dependence of
xc
on the macroscopic
polarization of a system, in addition to the periodic density.
This extension of DFT for polarized periodic systems has
been called ‘‘density-polarization functional theory.’’ This
topic, which touches upon fundamentals of DFT, has at-
tracted the attention of many authors, which has led to an
ongoing period of intense discussion on the origin and
mechanism behind the linear term, its importance in practical
applications, and the possibility of finding improved xc func-
tionals 7–13.
An analogous field-counteracting term in
xc
was also an-
ticipated for finite molecular clusters in an external electric
field 11, and a linear term in
xc
was indeed established in
Ref. 15 for polarized prototype molecular chains. This lin-
ear term spans the entire molecular system, and counteracts
the external electric field. It was recognized that the absence
or presence of a field-counteracting term in
xc
is a matter of
practical importance for DFT calculations. In particular, the
local-density approximation LDA and standard generalized
gradient approximations GGA’s, which all lack this term,
gave quite poor results for the linear and nonlinear polariz-
abilities of finite polyacetylene PA chains of varying length
and also of the hydrogen chains H
n
14,15. The correspond-
ing LDA/GGA errors are much larger by orders of magni-
tude than similar calculations on ‘‘standard molecules,’’ and
also much larger than estimates in Ref. 17 for dielectric
constants and nonlinear susceptibilities. Except for the size
of the errors, further discomforting outcomes of Refs. 14,
15 were the increase of the LDA/GGA errors for larger
chain lengths and the incorrect behavior of the calculated
hyperpolarizability as a function of the difference between
the single and double C-C bonds in PA the bond-length
alternation. On the other hand, the orbital-dependent
Krieger-Li-Iafrate KLI approximation 18, which exhibits
a field-counteracting term in
xc
, improves considerably
upon the LDA 15.
PHYSICAL REVIEW A, VOLUME 62, 012507
1050-2947/2000/621/01250710/$15.00 ©2000 The American Physical Society 62 012507-1