Chemistry Reference
In-Depth Information
10.5 LINEAR PSEUDOSTATES AND MOLECULAR
PROPERTIES
A convenient way to proceed is to apply the Ritz method to E
2
. We start
from a convenient set of basis functions
x
written as the (1
N) row
vector:
x ¼ðx
1
x
2
...x
N
Þ
ð
10
:
36
Þ
possibly orthonormal in themselves but necessarily orthogonal to
c
0
.We
shall assume that
x
x ¼
1
; x
c
0
¼
0
ð
10
:
37
Þ
s are not orthogonal then they must be preliminarily orthogo-
nalized by the Schmidt method. Then, we construct the matrices
If the
x
M
¼ x
ð
H
0
E
0
Þx
ð
10
:
38
Þ
the (N
N) Hermitian matrix of the excitation energies, and
m ¼ x
ð
H
1
c
0
Þ
ð
10
:
39
Þ
the (N
1) column vector of the transition moments.
By expanding
c
1
in the finite set of the
x
s, we can write
C
¼
X
N
k¼
1
x
k
C
k
c
1
¼ x
ð
10
:
40
Þ
E
2
¼
C
MC
þ
C
mþm
C
ð
10
:
41
Þ
which is minimum for
d
E
2
d
C
¼
MC
þm ¼
0
)
C
ð
best
Þ¼
M
1
m
ð
10
:
42
Þ
giving as best variational approximation to E
2
E
2
ð
best
Þ¼m
M
1
m
ð
10
:
43
Þ
