12.2 Separability of the ISR-ADC Secular Matrix
185
M I A ,J B = = ˜
I A | ˆ
H | ˜
J B = = ˜
A
I A
||
B
0 |( ˆ
H A + ˆ
H B )| ˜
B
J B
|
A
0
= 0
(12.35)
as, for example, the states of fragment A, | ˜
I A
and |
A
0
, are orthogonal (even relate
to different electron numbers). The case of the coupling block M A,AB is less trivial:
M I A ,J AB == ˜
I A | ˆ
H | ˜
J AB
== ˜
A
I A
||
B
0 |( ˆ
H A + ˆ
H B )| ˜
A
J A
| ˜
B
J B
== ˜
A
I A
| ˆ
H A | ˜
A
J A
B
0 | ˜
B
J B
+ + ˜
A
I A
| ˜
A
J A
B
0 | ˆ
H B | ˜
B
J B
= 0
(12.36)
In the last line, the first term vanishes because, by construction, | ˜
B
J B
is orthogonal
to |
B
0
, that is,
B
0 | ˜
B
J B
= 0. The second term vanishes according to
B
0 | ˆ
H B | ˜
B
J B
= E
B
0
B
0 | ˜
B
J B
= 0
(12.37)
as |
B
0
is the ground state of ˆ
H B , and, again,
B
0 | ˜
B
J B
= 0.
It is important to note that Eq. (12.36) holds through all orders of PT. As a consequence, the separability property not only applies to the exact ISR but also to the
systematic ISR(n) approximation schemes.
It remains to show that the diagonal block M A A of M is identical with the secular
matrix M
A of fragment A. This is easily seen as follows:
M I A ,J A == ˜
I A | ˆ
H A + ˆ
H B | ˜
J A − δ I A ,J A E 0
== ˜
A
I A
| ˆ
H A | ˜
A
J A
+ δ I A ,J A E
B
0 − δ I A ,J A E 0
== ˜
A
I A
| ˆ
H A − E
A
0 | ˜
A
J A
= M
A
I A ,J A
(12.38)
The full separability structure of the ECO-ISR secular matrix guarantees sizeconsistent results: For local excitations (ionizations) on one of the fragments, say
A, the eigenvalues (ionization energies) of the secular matrix M of the composite
system are identical with those of the fragment secular matrix M
A .
Of course, this finding applies also to the spectroscopic factors. According to the
separability structure of M, the eigenvector of a local excitation, say on A, has the
form
X n =
⎛
⎝
X
A
n
0
0
⎞
⎠
(12.39)
where X
A
n denotes the corresponding eigenvector of M
A . For an orbital p of fragment
A, the vector f p of transition amplitudes f I p takes on the same form,
f p =
⎛
⎝
f
A
p
0
0
⎞
⎠
(12.40)
185
M I A ,J B = = ˜
I A | ˆ
H | ˜
J B = = ˜
A
I A
||
B
0 |( ˆ
H A + ˆ
H B )| ˜
B
J B
|
A
0
= 0
(12.35)
as, for example, the states of fragment A, | ˜
I A
and |
A
0
, are orthogonal (even relate
to different electron numbers). The case of the coupling block M A,AB is less trivial:
M I A ,J AB == ˜
I A | ˆ
H | ˜
J AB
== ˜
A
I A
||
B
0 |( ˆ
H A + ˆ
H B )| ˜
A
J A
| ˜
B
J B
== ˜
A
I A
| ˆ
H A | ˜
A
J A
B
0 | ˜
B
J B
+ + ˜
A
I A
| ˜
A
J A
B
0 | ˆ
H B | ˜
B
J B
= 0
(12.36)
In the last line, the first term vanishes because, by construction, | ˜
B
J B
is orthogonal
to |
B
0
, that is,
B
0 | ˜
B
J B
= 0. The second term vanishes according to
B
0 | ˆ
H B | ˜
B
J B
= E
B
0
B
0 | ˜
B
J B
= 0
(12.37)
as |
B
0
is the ground state of ˆ
H B , and, again,
B
0 | ˜
B
J B
= 0.
It is important to note that Eq. (12.36) holds through all orders of PT. As a consequence, the separability property not only applies to the exact ISR but also to the
systematic ISR(n) approximation schemes.
It remains to show that the diagonal block M A A of M is identical with the secular
matrix M
A of fragment A. This is easily seen as follows:
M I A ,J A == ˜
I A | ˆ
H A + ˆ
H B | ˜
J A − δ I A ,J A E 0
== ˜
A
I A
| ˆ
H A | ˜
A
J A
+ δ I A ,J A E
B
0 − δ I A ,J A E 0
== ˜
A
I A
| ˆ
H A − E
A
0 | ˜
A
J A
= M
A
I A ,J A
(12.38)
The full separability structure of the ECO-ISR secular matrix guarantees sizeconsistent results: For local excitations (ionizations) on one of the fragments, say
A, the eigenvalues (ionization energies) of the secular matrix M of the composite
system are identical with those of the fragment secular matrix M
A .
Of course, this finding applies also to the spectroscopic factors. According to the
separability structure of M, the eigenvector of a local excitation, say on A, has the
form
X n =
⎛
⎝
X
A
n
0
0
⎞
⎠
(12.39)
where X
A
n denotes the corresponding eigenvector of M
A . For an orbital p of fragment
A, the vector f p of transition amplitudes f I p takes on the same form,
f p =
⎛
⎝
f
A
p
0
0
⎞
⎠
(12.40)