with no classical interpretation—this integral is entirely a quantum mechanical
effect resulting from fermions’ antisymmetry.
K ee ¼ À
ZZ
/
Ã
i
ð1Þ/
Ã
j
ð2Þ
1
r 12
/ i ð2Þ/ j ð1Þdr 1 dr 2
ð1:25Þ
In order to be able to solve this Schrödinger equation, we must replace the
electron–electron operator ^
V ee with an approximate, effective, one-electron operator
^
V eff :
^
V eff i
ð Þ ¼
X n
j
2 ^ J j i
ð Þ À ^
K j i
ð Þ
À
Á
ð1:26Þ
where
^ J j 1
ð Þ ¼
Z
u
Ã
j 2
ð Þ
1
r 12
u j 2
ð Þdr 2
ð1:27Þ
^
K j 1
ð Þu i 1
ð Þ ¼
Z
u
Ã
j 2
ð Þ
1
r 12
u i 2
ð Þdr 2
!
u j 1
ð Þ
ð1:28Þ
and as a result, the so-called one-electron Fock operator is obtained:
^ f i ¼ À
h
2
2m e
r
2
i À
X m
a¼1
Z a e
2
r ia
þ ^
V eff i
ð Þ
ð1:29Þ
Using variational principle:
E trial
h
i ¼
w
Ã
trial
^
Hw trial
w
Ã
trial w trial
! E exact ; d
w
Ã
trial
^
Hw trial
w
Ã
trial w trial
¼ 0
ð1:30Þ
to the problem of energy minimization, we get a set of appropriate one-electron
Hartree–Fock equations:
^ f i /ðiÞ ¼ e i /ðiÞ;
i ¼ 1; 2; . . .; N
ð1:31Þ
One-electron Fock operator ^ f i depends, through effective potential ^
V eff i
ð Þ, on the
one-electron wave functions /ðiÞ of all other electrons, and the Hartree–Fock
equations, despite the form resembling a classical eigenvalue problem, are not
possible to be solved directly; hence, it is necessary to perform calculations iteratively (using the so-called self-consistent field—SCF method):
• A set of one-electron initial functions (orbitals) is defined and respective Fock
operator is calculated;
12
A. Koleżyński