Chemistry Reference
In-Depth Information
*
+
ð
dx
1
dx
2
O
12
Yj
X
N
O
ij
jY
¼
r
2
ð
x
1
x
2
;
x
1
x
2
Þ
ð
6
:
52
Þ
i
;
j
¼
1
ð
j
=
i
Þ
is the N-electron wavefunction normalized to 1 and O
i
and O
ij
are the one- and two-electron operators respectively. In (6.51) we have
used the one-electron density matrix (6.13) instead of the one-electron
distribution function, leaving the possibility that O
1
could be a differential
operator (like
r
where
Y
2
), which acts on the first set of variables in
r
1
but not on
the second. The notation in (6.51) should by now be clear: first, let the
operator
O
1
act on
r
1
, then put x
0
1
¼
x
1
in the resulting integrand and
integrate over x
1
. In (6.52) we have assumed that the operator
O
12
is a
simple multiplier (like the electron repulsion 1/r
12
).
The electronic Hamiltonian H
e
contains two such symmetrical sums,
and its average value over the N-electron wavefunction
Y
can be written
as
*
+
hYj
H
e
jYi¼ Y
X
2
X
N
i
¼
1
h
i
þ
N
1
1
r
ij
Y
ð
6
:
53
Þ
i
;
j
ð
j
=
i
Þ
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
¼
r
1
ð
x
1
;
;
x
1
x
2
Þ
where
V
1
¼
X
a
1
2
r
Z
a
r
a
1
h
1
¼
2
1
þ
V
1
;
ð
6
:
54
Þ
is the one-electron bare nuclei Hamiltonian, V
1
is the attraction of the
electron by all nuclei of charge
þ
Z
a
in themolecule, and 1/r
12
the electron
repulsion.
Hence, the electronic energy E
e
consists of the following three terms:
ð
dx
1
ð
dx
1
V
1
r
1
ð
x
1
;
x
1
Þ
1
2
r
2
1
x
0
1
Þj
x
0
1
¼
x
1
E
e
¼
r
1
ð
x
1
;
þ
ð
dx
1
dx
2
ð
6
:
55
Þ
1
2
1
r
12
r
2
ð
x
1
x
2
þ
;
x
1
x
2
Þ
which have the following simple, physically transparent, interpretation.
The first term in (6.55) is the average kinetic energy of the electron
distribution
r
1
, the second is the average potential energy of the electron
