262
17 Coupled-Cluster Methods for Generalized Excitations
F
(r )
I = = 0 | ˆ
D|
0
I
(17.43)
and write the right transition moments as
T
(r )
n = x n 0 | ˆ
D|
cc
0 + F
(r )† X n
(17.44)
Here, the first term derives from a possible ground-state admixture in |
(x)
n according to Eq. (17.35). We note that the biorthonormality relations 0 |
cc
0 = 1 and
(l)
n |
(x)
n = 1 ensure the proper normalization of the product (17.38).
17.3 Order Relations and Separability Properties
The different quality of the two expansion manifolds becomes apparent in the order
structure of the BCC secular matrix M
cc shown in Fig. 17.1. As explained in Chap. 12,
the entries in the M
cc
μν sub-blocks associated with the excitation-class partitioning
denote the lowest (non-vanishing) PT orders of the matrix elements in these blocks.
The upper right (UR) triangular part of M
cc shows the characteristic CI order structure
as in Fig. 12.5 (there for the case of (N −1)-electron excitations). By contrast, the
lower left (LL) triangular part (μ > ν) displays the canonical order relations (see
Eqs. (12.1), (14.44))
M
cc
μν ∼ O(μ − ν)
(17.45)
The CI structure of the UR part can be inferred by using the expansions (17.17),
(17.18) in the BCC matrix elements M
cc
I J , [I ] < [J ]:
I | ˆ
H − E 0 |
0
J = = I | ˆ
H | J +
[K ]<[I ]
[L]>[J ]
z
(I )
K z
(J )
L K | ˆ
H | L (17.46)
Obviously, there are no contributions from the double sum, because the excitation
classes of the states K and L differ at least by a triple excitation, that is, [L] − [K ] ≥
3, so that all matrix elements K | ˆ
H | L vanish. In fact, this shows that the BCC
and CI secular matrix elements in the UR blocks are identical:
M
cc
I J = H I J , for [I ] < [J ]
(17.47)
For matrix elements M
cc
I J of a diagonal block, [I ] = [J ], we obtain
M
cc
I J = = I | ˆ
H − E 0 | J +
[K ]=[I ]−1
[L]=[I ]+1
z
(I )
K z
(J )
L K | ˆ
H | L
(17.48)
Obviously, the BCC and CI expressions differ beyond first order, which of course
does not affect the trivial order relation, M
cc
μμ ∼ O(0), in the diagonal blocks.
17 Coupled-Cluster Methods for Generalized Excitations
F
(r )
I = = 0 | ˆ
D|
0
I
(17.43)
and write the right transition moments as
T
(r )
n = x n 0 | ˆ
D|
cc
0 + F
(r )† X n
(17.44)
Here, the first term derives from a possible ground-state admixture in |
(x)
n according to Eq. (17.35). We note that the biorthonormality relations 0 |
cc
0 = 1 and
(l)
n |
(x)
n = 1 ensure the proper normalization of the product (17.38).
17.3 Order Relations and Separability Properties
The different quality of the two expansion manifolds becomes apparent in the order
structure of the BCC secular matrix M
cc shown in Fig. 17.1. As explained in Chap. 12,
the entries in the M
cc
μν sub-blocks associated with the excitation-class partitioning
denote the lowest (non-vanishing) PT orders of the matrix elements in these blocks.
The upper right (UR) triangular part of M
cc shows the characteristic CI order structure
as in Fig. 12.5 (there for the case of (N −1)-electron excitations). By contrast, the
lower left (LL) triangular part (μ > ν) displays the canonical order relations (see
Eqs. (12.1), (14.44))
M
cc
μν ∼ O(μ − ν)
(17.45)
The CI structure of the UR part can be inferred by using the expansions (17.17),
(17.18) in the BCC matrix elements M
cc
I J , [I ] < [J ]:
I | ˆ
H − E 0 |
0
J = = I | ˆ
H | J +
[K ]<[I ]
[L]>[J ]
z
(I )
K z
(J )
L K | ˆ
H | L (17.46)
Obviously, there are no contributions from the double sum, because the excitation
classes of the states K and L differ at least by a triple excitation, that is, [L] − [K ] ≥
3, so that all matrix elements K | ˆ
H | L vanish. In fact, this shows that the BCC
and CI secular matrix elements in the UR blocks are identical:
M
cc
I J = H I J , for [I ] < [J ]
(17.47)
For matrix elements M
cc
I J of a diagonal block, [I ] = [J ], we obtain
M
cc
I J = = I | ˆ
H − E 0 | J +
[K ]=[I ]−1
[L]=[I ]+1
z
(I )
K z
(J )
L K | ˆ
H | L
(17.48)
Obviously, the BCC and CI expressions differ beyond first order, which of course
does not affect the trivial order relation, M
cc
μμ ∼ O(0), in the diagonal blocks.