Chemistry Reference
In-Depth Information
For the two-electron repulsion, we have
1
r
12
¼½ð
x
1
x
2
Þ
2
2
2
1
=
2
þð
y
2
y
2
Þ
þð
z
1
z
2
R
Þ
2z
1
R
þ
2z
2
R
þ
r
1
þ
r
2
2x
1
x
2
2y
1
y
2
2z
1
z
2
Þ
1
=
2
¼ð
R
2
"
#
1
=
2
r
1
r
2
1
R
1
2
z
1
R
þ
2
z
2
2
R
2
ð
x
1
x
2
þ
y
1
y
2
þ
z
1
z
2
Þ
¼
R
þ
R
2
þ
R
2
ð
11
:
20
Þ
so that, expanding according to Taylor:
"
r
1
r
2
1
r
12
1
R
z
1
R
z
2
R
1
R
2
ð
x
1
x
2
þ
y
1
y
2
þ
z
1
z
2
Þ
1
þ
2R
2
2R
2
þ
2
2
!
þ
#
3
8
2
z
1
3
8
2
z
2
R
3
8
8
z
1
z
2
þ
þ
þ
R
R
2
3z
1
r
1
2R
3
3z
2
r
2
2R
3
1
R
þ
z
1
z
2
R
2
x
1
x
2
þ
y
1
y
2
2z
1
z
2
R
3
þ
O
ð
R
4
¼
þ
þ
þ
Þ
ð
11
:
21
Þ
Adding all terms altogether with the appropriate signs, many terms do
cancel, finally giving:
1
R
3
ð
x
1
x
2
þ
y
1
y
2
2z
1
z
2
Þþ
O
ð
R
4
V
Þ
ð
11
:
22
Þ
which is the leading term, the dipole-dipole interaction, of the expanded
form of the interatomic potential V for neutral H atoms. It corresponds to
the classical electrostatic interaction of two point-like dipoles
1
located at
the two nuclei of A and B (Coulson, 1958). Expansion (11.22) is the first
term of what is known as the multipole expansion of the interatomic
potential in long range.
1
In atomic units.
