problem faced by DFT (finding an approximate function) results directly from the
complexity of the problem of interacting electrons in many-body system.
Kohn and Sham [59] have reformulated the above problem by replacing the
system of N interacting electrons with a special reference system—the system of
N non-interacting electrons, having the same ground electron density. In the case of
such a reference system, this electron density is equal to the sum over the
one-electron w i (r) orbitals (Kohn–Sham orbitals):
nðrÞ ¼ 2
X N=2
i¼1
w i ðrÞ
j
j
2
ð1:37Þ
being a solution to the following Schrödinger equations:
À
1
2
r
2
þ V KS ðrÞ
w i ðrÞ ¼ e i w i ðrÞ
ð 1:38Þ
and satisfying the orthonormality condition:
Z
w
Ã
i ðrÞ
w j ðrÞdr ¼ d ij
ð1:39Þ
The existence of a uniquely determined V KS (r) potential for the ground state
electron density n(r) is a direct consequence of the first Hohenberg–Kohn theorem.
Therefore, we have a situation in which the problem of finding a universal functional F[n(r)] was replaced by the problem of finding an effective V KS [n(r)] functional. This problem can be solved using the variational principle. Lets write down
the energy in the form:
E ¼ T S ½nðrÞ þ E H ½nðrÞ þ E XC ½nðrÞ þ
Z
nðrÞV ext ðrÞdr
ð1:40Þ
The first part is the kinetic energy of the system of non-interacting electrons:
T S ½nðrÞ ¼ À
h
2
2m
2
X N=2
i¼1
Z
w
Ã
i ðrÞr
2 w i ðrÞdr
ð1:41Þ
The second is Hartree energy, classic electrostatic energy:
E H ½nðrÞ ¼
e
2
2
Z nðrÞnðr
0
Þ
r À r 0
j
j
drdr
0
ð1:42Þ
and the last term is the classical Coulomb energy of electron interactions with
external potential. The key element in the expression for the energy of the system is
the third term whose exact form is unknown—this is the exchange and correlation
16
A. Koleżyński
complexity of the problem of interacting electrons in many-body system.
Kohn and Sham [59] have reformulated the above problem by replacing the
system of N interacting electrons with a special reference system—the system of
N non-interacting electrons, having the same ground electron density. In the case of
such a reference system, this electron density is equal to the sum over the
one-electron w i (r) orbitals (Kohn–Sham orbitals):
nðrÞ ¼ 2
X N=2
i¼1
w i ðrÞ
j
j
2
ð1:37Þ
being a solution to the following Schrödinger equations:
À
1
2
r
2
þ V KS ðrÞ
w i ðrÞ ¼ e i w i ðrÞ
ð 1:38Þ
and satisfying the orthonormality condition:
Z
w
Ã
i ðrÞ
w j ðrÞdr ¼ d ij
ð1:39Þ
The existence of a uniquely determined V KS (r) potential for the ground state
electron density n(r) is a direct consequence of the first Hohenberg–Kohn theorem.
Therefore, we have a situation in which the problem of finding a universal functional F[n(r)] was replaced by the problem of finding an effective V KS [n(r)] functional. This problem can be solved using the variational principle. Lets write down
the energy in the form:
E ¼ T S ½nðrÞ þ E H ½nðrÞ þ E XC ½nðrÞ þ
Z
nðrÞV ext ðrÞdr
ð1:40Þ
The first part is the kinetic energy of the system of non-interacting electrons:
T S ½nðrÞ ¼ À
h
2
2m
2
X N=2
i¼1
Z
w
Ã
i ðrÞr
2 w i ðrÞdr
ð1:41Þ
The second is Hartree energy, classic electrostatic energy:
E H ½nðrÞ ¼
e
2
2
Z nðrÞnðr
0
Þ
r À r 0
j
j
drdr
0
ð1:42Þ
and the last term is the classical Coulomb energy of electron interactions with
external potential. The key element in the expression for the energy of the system is
the third term whose exact form is unknown—this is the exchange and correlation
16
A. Koleżyński