Comment on “Functional derivative of the universal density functional in Fock space”
Espen Sagvolden,
1,2,
*
John P. Perdew,
2
and Mel Levy
3,4
1
Institut für Physikalische Chemie, Universität Karlsruhe, Kaiserstraße 12, D-76128 Karlsruhe, Germany
2
Department of Physics and Quantum Theory Group, Tulane University, New Orleans, Louisiana 70118, USA
3
Department of Chemistry and Quantum Theory Group, Tulane University, New Orleans, Louisiana 70118, USA
4
Department of Chemistry, Duke University, Durham, North Carolina 27708, USA
Received 7 July 2008; published 2 February 2009
Zahariev and Wang Phys. Rev. A, 70, 042503 2004 discuss the density functional theory of noninteger
average particle numbers. Among their many results is one we dispute: that the exact exchange-correlation
potential more precisely, the exact functional derivative of the exchange-correlation energy with respect to the
density cannot have a discontinuity as the particle number crosses an integer, in contradiction to works by two
of us J.P.P. and M.L. in collaboration with co-workers Phys. Rev. Lett. 49, 1691 1982; 51, 1884 1983
and by Sham and Schlüter Phys. Rev. Lett. 51, 1888 1983. We point to a counterexample to Zahariev and
Wang’s claim, which two of us E.S. and J.P.P. have presented in a separate paper: A rigorous proof that, in the
absence of external magnetic fields, the exchange-correlation potential jumps by the difference between the
ionization potential I and electron affinity A when the particle number crosses 1, given that I A. We point
out that Zahariev and Wang’s derivation neglects an order-of-limits problem. We also prove that I A for any
one-electron system in the absence of magnetic fields.
DOI: 10.1103/PhysRevA.79.026501 PACS numbers: 31.15.E-, 02.30.Sa, 02.30.Xx
It was discovered in the early 1980s 1–3 that the func-
tional derivative
E
xc
nr
n of the exact exchange-correlation
energy of a system free to exchange energy and electrons
with its surroundings may undergo a discontinuous jump as
the mean particle number of the system crosses an integer.
This discovery explained a troubling discrepancy in the
density-functional theory DFT of the time: Based on Jan-
ak’s theorem 4 it was believed that the fundamental band
gap at integer particle number J should equal the Kohn-Sham
band gap,
J+1
-
J
, where
J+1
and
J
are the exact Jth and
J +1th Kohn-Sham eigenvalues evaluated at the same par-
ticle number—e.g., J. It was, however, known that the Kohn-
Sham band gap found from density-functional calculations
using the local density approximation underestimated the
fundamental band gap by as much as 50%. However, if
E
xc
nr
n jumps by some spatially constant number CJ when
the particle number expectation value of the system crosses
the integer J from below, then the fundamental band gap at
the integer J is
J+1
-
J
+ CJ. Hence, the discovery of the
discontinuous jump implied that the Kohn-Sham band gap of
even the unknown exact exchange-correlation functional
should underestimate the fundamental band gap by an
amount CJ. The discovery of the derivative discontinuity is
also important because it explains why local or semilocal
functionals often produce a qualitatively incorrect dissocia-
tion limit e.g., 5–8 and why they underestimate charge-
transfer excitation energies in time-dependent DFT 9.
In their article “Functional derivative of the universal den-
sity functional in Fock space” 10, Zahariev and Wang dis-
cuss the DFT of noninteger average particle numbers. Many
of the statements made in the article are correct, and some of
these statements have been cited by us 11,12. However, we
strongly disagree with their Eq. 112, which asserts that in
all systems CJ = 0 at all integer particle numbers J.
Let us begin with a few definitions. In the DFT of nonin-
teger particle numbers, the system is assumed to be able to
exchange energy and particles with an infinitely large, distant
reservoir. The state of the system is described by an en-
semble
ˆ
where the probability is p
i
that the system can be
found in the pure quantum mechanical state
i
:
ˆ
=
i
|
i
p
i
i
| . 1
The expectation value of any observable O
ˆ
is given by
O
ˆ
= Tr
ˆ
O
ˆ
=
i
p
i
i
|O
ˆ
|
i
. 2
We define the ensemble constrained-search functional of a
density nr
of any real particle number as
Gn = min
ˆ
→nr
Tr
ˆ
T
ˆ
+ V
ˆ
ee
, 3
where one searches over all
ˆ
that have nr
as their density
expectation value. T
ˆ
and V
ˆ
ee
are the kinetic energy and
electron-electron Coulomb repulsion energy operators, re-
spectively. This definition differs from the conventional defi-
nition of the pure-state constrained-search functional Fn
13 in the DFT of integer particle numbers in that the search
in Eq. 3 goes over all ensembles yielding nr
, not just all
wave functions yielding nr
.
By the same token, the noninteracting kinetic energy is
defined
*
Present address: University of California, Irvine, Department of
Chemistry, 1102 Natural Sciences II, Irvine, CA 92697-2025, USA.
esagvold@uci.edu
PHYSICAL REVIEW A 79, 026501 2009
1050-2947/2009/792/0265014 ©2009 The American Physical Society 026501-1