In density functional theory (DFT), the Harris energy functional is a non-self-consistent approximation to the Kohn-Sham density functional theory. It gives the energy of a combined system as a function of the electronic densities of the isolated parts. The energy of the Harris functional varies much less than the energy of the Kohn-Sham functional as the density moves away from the converged density.
Assuming that we have an approximate electron density
ρ
(
r
→
)
, which is different from the exact electron density
ρ
0
(
r
→
)
. We construct exchange-correlation potential
v
x
c
(
r
→
)
and the Hartree potential
v
H
(
r
→
)
based on the approximate electron density
ρ
(
r
→
)
. Kohn-Sham equations are then solved with the XC and Hartree potentials and eigenvalues are then obtained. The sum of eigenvalues is often called the band energy:
E
b
a
n
d
=
∑
i
ϵ
i
,
where
i
loops over all occupied Kohn-Sham orbitals. Harris energy functional is defined as
E
H
a
r
r
i
s
=
∑
i
ϵ
i
−
∫
d
r
3
v
x
c
(
r
→
)
ρ
(
r
→
)
−
1
2
∫
d
r
3
v
H
(
r
→
)
ρ
(
r
→
)
+
E
x
c
[
ρ
]
It was discovered by Harris that the difference between the Harris energy
E
H
a
r
r
i
s
and the exact total energy is to the second order of the error of the approximate electron density, i.e.,
O
(
(
ρ
−
ρ
0
)
2
)
. Therefore, for many systems the accuracy of Harris energy functional may be sufficient. The Harris functional was originally developed for such calculations rather than self-consistent convergence, although it can be applied in a self-consistent manner in which the density is changed. Many density-functional tight-binding methods, such as DFTB+, Fireball, and Hotbit, are built based on the Harris energy functional. In these methods, one often does not perform self-consistent Kohn-Sham DFT calculations and the total energy is estimated using the Harris energy functional. These codes are often much faster than conventional Kohn-Sham DFT codes that solve Kohn-Sham DFT in a self-consistent manner.
While the Kohn-Sham DFT energy is Variational method (never lower than the ground state energy), the Harris DFT energy was originally believed to be anti-variational (never higher than the ground state energy). This was however conclusively demonstrated to be incorrect.
