w el ðr 1 ; . . .r i ; . . .; r j ; . . .; r N Þ ¼
1
ffiffiffiffiffi
N!
p
/ 1 ðr 1 Þ / 1 ðr 2 Þ Á Á Á / 1 ðr N Þ
/ 2 ðr 1 Þ / 2 ðr 2 Þ Á Á Á / 2 ðr N Þ
. .
.
. .
.
. .
.
/ N ðr 1 Þ / N ðr 2 Þ Á Á Á / N ðr N Þ
ð1:19Þ
In the general case, the wave function for the N-electron system can be written in
the form of a linear combination of Slater determinants (including in the determinants also partially occupied spinorbitals for excited states, empty in the ground
state):
w el ¼
X
i
d i U i ¼ d 0 U 0 þ d 1 U 1 þ d 2 U 2 þ Á Á Á
ð1:20Þ
A more detailed analysis of the electron energy operator ^
H el of the form (1.14)
shows that it consists of two types of contributions:
• One-electron contribution (depending on the coordinates of only one electron):
P n
i¼1
^ h i where ^ h i ¼ ^
T e þ ^
V Ne ¼ À
h
2
2m e
r
2
i À
P m
a¼1
Z a e
2
r ia
ð1:21Þ
• Two-electron contribution (depending on the coordinates of two electrons):
^
V ee ¼
X
j
X
i [ j
e
2
r ij
ð1:22Þ
The former can be calculated accurately, whereas the two-electron contributions
can only be approximated, due to the complexity of interelectronic interactions. The
potential energy associated with two-electron interactions is equal to:
V ee ¼ J ee À K ee ¼
X n
i
X n
j [ i
ðJ ij À K ij Þ
ð 1:23Þ
where J ij is a Coulomb integral defined as:
J ee ¼ À
ZZ
/
2
i
ð1Þ
1
r 12
/
2
j
ð2Þdr 1 dr 2
ð1:24Þ
describing Coulomb repulsion between electron 1 on orbital i and electron 2 on
orbital j, while K ij is the exchange integral (nonzero only for like-spin electrons),
1 Computational Methods in Spectroscopy
11