Correlated optimized effective-potential treatment of the derivative discontinuity
and of the highest occupied Kohn-Sham eigenvalue: A Janak-type theorem
for the optimized effective-potential model
Mark E. Casida*
De ´partement de Chimie, Universite ´ de Montre ´al, Case Postale 6128, Succursale Centre-Ville, Montre ´al, Que ´bec, Canada H3C 3J7
Received 27 July 1998
A Janak theorem is derived for the correlated optimized effective-potential model of the Kohn-Sham
exchange-correlation potential v
xc
. It is used to evaluate the derivative discontinuity DD and to show that the
highest occupied Kohn-Sham eigenvalue,
H
-I , the negative of the ionization potential, when relaxation
and correlation effects are included. This reconciles an apparent inconsistency between the ensemble theory
and fractional occupation number approaches to noninteger particle number in density-functional theory. For
finite systems,
H
=-I implies that v
xc
=0 independent of particle number, and that the DD vanishes asymp-
totically as 1/r . The difference in behavior of the DD in the bulk and asymptotic regions means that the DD
affects the shape of v
xc
, even at fixed, integer particle number. S0163-18299904907-3
I. INTRODUCTION
Hohenberg-Kohn-Sham density-functional theory
1,2
DFT is an important workhorse for electronic structure cal-
culations in both physics and chemistry. Since the quality of
the results is determined by the approximation used for the
exchange-correlation functional, the development of im-
proved practical, approximate functionals is of central im-
portance in DFT. For example, the functionals widely used
for molecular calculations yield exchange-correlation poten-
tials v
xc
, whose asymptotic behavior is qualitatively incor-
rect. This is critical for the calculation of high-lying bound
excitations from time-dependent DFT Refs. 3 and 4 and is
important for other properties that are sensitive to the outer
part of the charge density, such as static and dynamic polar-
izabilities. Furthermore, the shortcomings of approximate
functionals have meant that molecular first ionization poten-
tials and band gaps in solids are, in practice, not computed
from the Kohn-Sham eigenvalues but instead from total-
energy-difference-based procedures and post-DFT Green-
function approximations, respectively. Work on improving
approximate functionals is often guided by properties dem-
onstrated to hold for the exact functional.
The concept that the exact DFT exchange-correlation
energy E
xc
has a discontinuity in its derivative at integer
particle electron number,
v
xc
+
r
E
xc
r
N+0
+
E
xc
r
N-0
+
v
xc
-
r , 1.1
was originally introduced to explain the discrepancy between
calculated and measured band gaps in solid-state physics
5,6
and the dissociation of diatomic molecules into neutral
atoms.
7–9
In contrast to the well-established place of the de-
rivative discontinuity DD in the theory for infinite systems,
the problem of the DD in finite systems has often been re-
garded as abstruse, due to questions of consistency between
different methods of introducing noninteger particle number
into DFT, and as irrelevant for practical applications. How-
ever, several recent works suggest that the DD should be
taken into account in designing functionals with the correct
asymptotic behavior.
10,11,3,12,13
Indeed, the derivatives with
respect to particle number are intimately connected to the
relation between the highest occupied Kohn-Sham eigen-
value
H
and the ionization potential I and, hence, to the
asymptotic value v
xc
lim
r →
v
xc
( r) of the exchange-
correlation potential. The proof of these latter relations has
been the subject of recent controversy.
14,9,15
This paper re-
conciles the apparent inconsistencies in the DD obtained
from different methods of handling noninteger particle num-
ber and gives an independent proof confirming that
H
=
-I . For finite systems, this means v
xc
=0 and that the DD
affects the shape of v
xc
. Thus, a proper consideration of the
DD is important for the design of improved practical func-
tionals.
There are two main approaches for introducing noninteger
particle number into DFT. In the fractional occupation num-
ber approach, the usual Kohn-Sham equation, which was de-
rived only for integer occupation number, is used, but the
orbital occupation numbers f
i
entering into the total-energy
expression
E =-
1
2
i
f
i
i
|
2
|
i
+
v r r d r
+
1
2
r
1
r
2
| r
1
-r
2
|
d r
1
d r
2
+E
xc
1.2
through the kinetic energy and the charge density ( r)
=
j
f
j
|
j
( r) |
2
are allowed to be fractional. In this formal-
ism, Janak’s theorem states that
E
f
i
v
=
i
, 1.3
wherever the derivative exists.
16
The fractional occupation number approach can be
viewed as simply providing a well-defined smooth extension
of the charge density and of the energy expression, which
PHYSICAL REVIEW B 15 FEBRUARY 1999-I VOLUME 59, NUMBER 7
PRB 59 0163-1829/99/597/46945/$15.00 4694 ©1999 The American Physical Society