Chemistry Reference
In-Depth Information
In a short notation, properties (7.4)-(7.6) can be symbolically written
as
2
0
tr
r ¼
N
;
r
¼ r;
r
¼ r
ð
7
:
10
Þ
It is often said that Equation 7.3 provides a spin-orbital representation
of the projector
r
.
7.1.2 Electronic Energy Expression
The general expression for the electronic energy of the many-electron
system was given in Chapter 6 as
ð
dx
1
h
1
ð
dx
1
dx
2
1
2
1
r
12
r
2
ð
x
1
x
2
x
0
1
Þj
x
0
1
¼
x
1
þ
E
e
¼
r
1
ð
x
1
;
;
x
1
x
2
Þ
7
:
11
Þ
But, Lennard-Jones (1931) showed that in HF theory
x
0
Þ¼rð
x
x
0
Þ
r
1
ð
x
;
;
ð
7
:
12
Þ
x
0
1
Þ
x
0
2
Þ
x
0
1
x
0
2
Þ¼
rð
x
1
;
rð
x
1
;
r
2
ð
x
1
x
2
;
x
0
1
Þ
x
0
2
Þ
rð
x
2
;
rð
x
2
;
ð
7
:
13
Þ
x
0
1
Þrð
x
2
x
0
2
Þrð
x
1
x
0
2
Þrð
x
2
x
0
1
Þ
¼ rð
x
1
;
;
;
;
Hence, in HF theory, the electronic energy assumes the characteristic
one-electron form (no electron correlation
)
IPM):
ð
dx h
rð
x
ð
dx
½
J
ð
x
Þ
K
ð
x
Þ rð
x
1
2
x
0
Þj
x
0
¼
x
þ
x
0
Þj
x
0
¼
x
E
e
¼
;
;
ð
7
:
14
Þ
where both one-electron and two-electron components of E
e
are ex-
pressed in terms of the fundamental invariant
r
and of Coulomb and
exchange one-electron potentials:
ð
dx
2
rð
x
2
x
2
Þ
r
12
;
J
ð
x
1
Þ¼
ð
7
:
15
Þ
