Chemistry Reference
In-Depth Information
1
r
12
r
1
2
r
2
O
Figure 4.4
The atomic two-electron system
the sumof the two one-electron hydrogenicHamiltonians and the electron
repulsion term (a two-electron operator). This last term hinders separa-
tion of the two-electron Schroedinger eigenvalue equation into two
separate equations, one for each hydrogenic electron. However, approx-
imations are possible through the variation method using the simple
two-electron product variational function:
wð
1
;
2
Þ¼w
1
ð
1
Þw
2
ð
2
Þ
ð
4
:
24
Þ
where the normalized one-electron functions are given by
1
=
2
c
0
p
w
1s
¼
exp
ð
c
0
r
Þ
ð
4
:
25
Þ
Calculation then gives the variational bound
w
1
w
2
1
r
12
w
1
w
2
h
1
þ
h
2
þ
¼
2h
1s1s
þð
1s
2
j
1s
2
«
w
¼
Þ
ð
4
:
26
Þ
