17.3 Order Relations and Separability Properties
265
As a consequence, both the CC states and the biorthogonal states can be written as
products of fragment states. For a non-local excitation, J ≡ J AB , the CC state takes
on the form
|
0
J AB
= |
A
J A
|
B
J B
(17.52)
where
|
A
J A
= ˆ
C J A e
ˆ
T A |
A
0
(17.53)
denotes a CC state for fragment A. The corresponding biorthogonal state reads
J AB | = =
A
J A
||
B
J B
|
(17.54)
where the fragment biorthogonal state, say for A, is given by
A
J A
| = =
A
0 | ˆ
C
†
J A
e
− ˆ
T A
(17.55)
Using the factorization of the left and right BCC basis states, the block structure of
M
cc can readily be established (Exercise 17.4). As an example, we consider a matrix
element in the M
cc
A,AB block:
M
cc
I A ,J AB
== I A | ˆ
H A + ˆ
H B |
0
J AB
==
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
While the first term in the last line vanishes due to the orthogonality of |
B
0 and
|
B
J B
, the second term may not vanish for I A = J A , since |
B
0 is not an eigenstate
of ˆ
H B . In a similar way, one may verify that
M
cc
A A = M
cc
(A)
(17.56)
where M
cc
(A) denotes the BCC secular matrix of fragment A.
What are the consequences of the semi-separable structure of the BCC secular
matrix? There are no ramifications for the excitation energies. The characteristic
polynomial for the fragment secular matrix, M
cc
A A , is a factor of the characteristic
polynomial of the full M
cc matrix, so that the eigenvalues of M
cc
A A are a subset of
all eigenvalues. This means that the energies are size-consistent: the BCC results for
local excitations do not depend on whether the method is applied to the respective
fragment or to the composite.
For the transition moments, the situation is more nuanced. The left transition
moments are size-consistent, although the left eigenvectors are non-separable: The
eigenvector Y n for a local excitation in one of the fragments has non-vanishing
non-local components, Y J AB ,n = 0. However, this does not matter because in the
vector of the left basis set transition moments, F
(l) , all non-local components vanish:
F
(l)
J AB
= 0. The right transition moments (17.40) are not size-consistent, even though
265
As a consequence, both the CC states and the biorthogonal states can be written as
products of fragment states. For a non-local excitation, J ≡ J AB , the CC state takes
on the form
|
0
J AB
= |
A
J A
|
B
J B
(17.52)
where
|
A
J A
= ˆ
C J A e
ˆ
T A |
A
0
(17.53)
denotes a CC state for fragment A. The corresponding biorthogonal state reads
J AB | = =
A
J A
||
B
J B
|
(17.54)
where the fragment biorthogonal state, say for A, is given by
A
J A
| = =
A
0 | ˆ
C
†
J A
e
− ˆ
T A
(17.55)
Using the factorization of the left and right BCC basis states, the block structure of
M
cc can readily be established (Exercise 17.4). As an example, we consider a matrix
element in the M
cc
A,AB block:
M
cc
I A ,J AB
== I A | ˆ
H A + ˆ
H B |
0
J AB
==
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
While the first term in the last line vanishes due to the orthogonality of |
B
0 and
|
B
J B
, the second term may not vanish for I A = J A , since |
B
0 is not an eigenstate
of ˆ
H B . In a similar way, one may verify that
M
cc
A A = M
cc
(A)
(17.56)
where M
cc
(A) denotes the BCC secular matrix of fragment A.
What are the consequences of the semi-separable structure of the BCC secular
matrix? There are no ramifications for the excitation energies. The characteristic
polynomial for the fragment secular matrix, M
cc
A A , is a factor of the characteristic
polynomial of the full M
cc matrix, so that the eigenvalues of M
cc
A A are a subset of
all eigenvalues. This means that the energies are size-consistent: the BCC results for
local excitations do not depend on whether the method is applied to the respective
fragment or to the composite.
For the transition moments, the situation is more nuanced. The left transition
moments are size-consistent, although the left eigenvectors are non-separable: The
eigenvector Y n for a local excitation in one of the fragments has non-vanishing
non-local components, Y J AB ,n = 0. However, this does not matter because in the
vector of the left basis set transition moments, F
(l) , all non-local components vanish:
F
(l)
J AB
= 0. The right transition moments (17.40) are not size-consistent, even though