In physics and quantum chemistry, specifically density functional theory, the Kohn–Sham equation is the Schrödinger equation of a fictitious system (the "Kohn–Sham system") of noninteracting particles (typically electrons) that generate the same density as any given system of interacting particles. The Kohn–Sham equation is defined by a local effective (fictitious) external potential in which the noninteracting particles move, typically denoted as v_{s}(r) or v_{eff}(r), called the Kohn–Sham potential. As the particles in the Kohn–Sham system are noninteracting fermions, the Kohn–Sham wavefunction is a single Slater determinant constructed from a set of orbitals that are the lowest energy solutions to
(
−
ℏ
2
2
m
∇
2
+
v
e
f
f
(
r
)
)
ϕ
i
(
r
)
=
ε
i
ϕ
i
(
r
)
This eigenvalue equation is the typical representation of the Kohn–Sham equations. Here, ε_{i} is the orbital energy of the corresponding Kohn–Sham orbital, φ_{i}, and the density for an Nparticle system is
ρ
(
r
)
=
∑
i
N

ϕ
i
(
r
)

2
.
The Kohn–Sham equations are named after Walter Kohn and Lu Jeu Sham (沈呂九), who introduced the concept at the University of California, San Diego in 1965.
In KohnSham density functional theory, the total energy of a system is expressed as a functional of the charge density as
E
[
ρ
]
=
T
s
[
ρ
]
+
∫
d
r
v
e
x
t
(
r
)
ρ
(
r
)
+
E
H
[
ρ
]
+
E
x
c
[
ρ
]
where T_{s} is the Kohn–Sham kinetic energy which is expressed in terms of the Kohn–Sham orbitals as
T
s
[
ρ
]
=
∑
i
=
1
N
∫
d
r
ϕ
i
∗
(
r
)
(
−
ℏ
2
2
m
∇
2
)
ϕ
i
(
r
)
,
v_{ext} is the external potential acting on the interacting system (at minimum, for a molecular system, the electronnuclei interaction), E_{H} is the Hartree (or Coulomb) energy,
E
H
=
e
2
2
∫
d
r
∫
d
r
′
ρ
(
r
)
ρ
(
r
′
)

r
−
r
′

.
and E_{xc} is the exchangecorrelation energy. The Kohn–Sham equations are found by varying the total energy expression with respect to a set of orbitals, subject to constraints on those orbitals, to yield the Kohn–Sham potential as
v
e
f
f
(
r
)
=
v
e
x
t
(
r
)
+
e
2
∫
ρ
(
r
′
)

r
−
r
′

d
r
′
+
δ
E
x
c
[
ρ
]
δ
ρ
(
r
)
.
where the last term
v
x
c
(
r
)
≡
δ
E
x
c
[
ρ
]
δ
ρ
(
r
)
is the exchangecorrelation potential. This term, and the corresponding energy expression, are the only unknowns in the Kohn–Sham approach to density functional theory. An approximation that does not vary the orbitals is Harris functional theory.
The Kohn–Sham orbital energies ε_{i}, in general, have little physical meaning (see Koopmans' theorem). The sum of the orbital energies is related to the total energy as
E
=
∑
i
N
ε
i
−
E
H
[
ρ
]
+
E
x
c
[
ρ
]
−
∫
δ
E
x
c
[
ρ
]
δ
ρ
(
r
)
ρ
(
r
)
d
r
Because the orbital energies are nonunique in the more general restricted openshell case, this equation only holds true for specific choices of orbital energies (see Koopmans' theorem).