Chemistry Reference
In-Depth Information
We usually say that the first of Equations (1.22) expresses the diago-
nalization of the symmetric matrix A through a transformation with the
complete matrix of its eigenvectors, while the second equations express
the normalization of the coefficients (i.e., the resulting vectors are chosen
to have modulus 1).
2
Equations (18-20) simplify noticeably in the caseA
22
¼
A
11
¼ a
. Then,
putting A
12
¼
A
21
¼ b
, we obtain:
8
<
l
¼ aþb;
l
¼ ab
1
2
!
;
!
p
p
2
1
=
1
=
ð
1
:
23
Þ
:
c
1
¼
c
2
¼
p
p
2
1
=
1
=
Occasionally, we shall need to solve the so called pseudosecular
equation for the symmetric matrix A arising from the pseudoeigenvalue
equation:
¼
A
11
l
A
12
l
S
Ac
¼
l
Sc
Y
j
A
l
S
j¼
0
ð
1
:
24
Þ
A
21
l
SA
22
l
where S is the overlap matrix:
!
!
S
11
S
12
1 S
S 1
S
¼
¼
ð
1
:
25
Þ
S
21
S
22
Solution of Equation (1.24) then gives:
<
A
11
þ
A
22
2A
12
S
D
l
¼
1
2
ð
1
S
2
Þ
2
ð
1
S
2
Þ
ð
1
:
26
Þ
:
A
11
þ
A
22
2A
12
S
D
l
2
¼
þ
2
ð
1
S
2
Þ
2
ð
1
S
2
Þ
h
i
1
=
2
2
D¼ð
A
22
A
11
Þ
þ
4
ð
A
12
A
11
S
Þð
A
12
A
22
S
Þ
>
0
ð
1
:
27
Þ
The eigenvectors corresponding to the roots (Equations 1.26) are rather
complicated (Magnasco, 2007), so we shall content ourselves here by
giving only the results for A
22
¼
A
11
¼ a
and A
21
¼
A
12
¼ b:
2
The length of the vectors. A matrix satisfying the second of Equations (1.22) is said to be an
orthogonal matrix.
