Chemistry Reference
In-Depth Information
method is intimately connected with the problem of the matrix diago-
nalization of Chapter 2.
We shall limit ourselves to consideration of a finite basis set of N
orthonormal functions
, the problem being best treated in matrix form.
The Rayleigh ratio (4.1) can be written as
x
« ¼
HM
1
ð
4
:
35
Þ
where
H
¼hwj
H
jwi;
M
¼hwjwi
ð
4
:
36
Þ
If we introduce the set of N orthonormal functions as the row
matrix
x ¼ðx
1
x
2
...x
N
Þ
ð
4
:
37
Þ
and the corresponding set of variational coefficients as the column
matrix
0
@
1
A
c
1
c
2
c
N
c ¼
ð
4
:
38
Þ
then H and M in (4.36) can be written in terms of the (N
N)Hermitian
matrices
H
and
M
:
H
¼ c
x
H
xc ¼ c
Hc;
M
¼ c
x
xc ¼ c
Mc ¼ c
1c
ð
4
:
39
Þ
where
M ¼ 1
is the metric matrix of the basis functions
x
. The matrix
elements of matrices
H
and
M
are
H
mn
¼hx
m
j
H
jx
n
i;
M
mn
¼hx
m
jx
n
i¼
1
mn
¼ d
mn
ð
4
:
40
Þ
An infinitesimal first variation in the linear coefficients will induce an
infinitesimal change in the energy functional (4.35):
d« ¼ d
H
M
1
H
M
2
d
M
¼
M
1
ðd
H
«d
M
Þ
ð
4
:
41
Þ
