Appendix
309
By construction, the intermediate state | ˜
I is orthogonal to the CE states |
0
K
of lower classes, [K ] < [I ]. Therefore, the summation over K can be restricted to
[K ] ≥ [I ]. An opposite restriction, namely [L] ≤ [J ], applies to the summation over
L. To see this, consider the matrix element
L | ˜
J = = 0 | ˆ
C
†
L e
− ˆ
T
| ˜
J
(A.6.37)
and recall that by construction | ˜
J is of the form of
| ˜
J =
[L ]≤[J ]
z L ˆ
C L | 0
(A.6.38)
where the excitation classes of the operators ˆ
C L are restricted to [L
] ≤ [J ]. As a
consequence,
L | ˜
J =
[L ]≤[J ]
z L 0 | ˆ
C
†
L
ˆ
C L | 0 = 0, for [L] > [J ]
(A.6.39)
Here, e
− ˆ
T has been commuted to the right, yielding e
− ˆ
T
| 0 = | 0 . Altogether,
the summations in Eq. (A.6.36) are restricted according to [K ] ≥ [I ] > [J ] ≥ [L].
Since K | ˆ
H |
0
L ∼ O([K ] − [L]) according to the BCC order relations (A.6.16),
we may conclude that every summation term and thus the entire sum (A.6.36) is at
least of the order [I ] − [J ]. The minimum value [I ] − [J ] is associated with the case
[K ] = [I ] and [L] = [J ], as here the overlap integrals ˜
I |
0
I and J | ˜
J are of
zeroth order (in fact, of the form 1 + O(2)).
ISR of Operators and Transition Moments
The foregoing proof can easily be transferred to the ISR of an arbitrary one-particle
operator, ˆ
D,
˜
D I J = = ˜
I | ˆ
D| ˜
J
(A.6.40)
as considered in Sects. 11.3 and 12.1 for (N − 1)-electron excitations (Eq. 11.51)
and Sect. 14.3 for N -electron excitations (Eq. 14.46). As above, one may insert the
biorthogonal RI before and after the operator ˆ
D and use the order structure (A.6.30)
in the related BCC matrix ˆ
D
cc . The order relations thereby established read (see
Eq. 12.17 and Fig. 12.2)
˜
D I J ∼ O(|[I ] − [J ]| − 1), [I ] = [J ]
(A.6.41)
The transition moments
F I (D) = = ˜
I | ˆ
D| 0 ∼ O([I ] − 1)
(A.6.42)
309
By construction, the intermediate state | ˜
I is orthogonal to the CE states |
0
K
of lower classes, [K ] < [I ]. Therefore, the summation over K can be restricted to
[K ] ≥ [I ]. An opposite restriction, namely [L] ≤ [J ], applies to the summation over
L. To see this, consider the matrix element
L | ˜
J = = 0 | ˆ
C
†
L e
− ˆ
T
| ˜
J
(A.6.37)
and recall that by construction | ˜
J is of the form of
| ˜
J =
[L ]≤[J ]
z L ˆ
C L | 0
(A.6.38)
where the excitation classes of the operators ˆ
C L are restricted to [L
] ≤ [J ]. As a
consequence,
L | ˜
J =
[L ]≤[J ]
z L 0 | ˆ
C
†
L
ˆ
C L | 0 = 0, for [L] > [J ]
(A.6.39)
Here, e
− ˆ
T has been commuted to the right, yielding e
− ˆ
T
| 0 = | 0 . Altogether,
the summations in Eq. (A.6.36) are restricted according to [K ] ≥ [I ] > [J ] ≥ [L].
Since K | ˆ
H |
0
L ∼ O([K ] − [L]) according to the BCC order relations (A.6.16),
we may conclude that every summation term and thus the entire sum (A.6.36) is at
least of the order [I ] − [J ]. The minimum value [I ] − [J ] is associated with the case
[K ] = [I ] and [L] = [J ], as here the overlap integrals ˜
I |
0
I and J | ˜
J are of
zeroth order (in fact, of the form 1 + O(2)).
ISR of Operators and Transition Moments
The foregoing proof can easily be transferred to the ISR of an arbitrary one-particle
operator, ˆ
D,
˜
D I J = = ˜
I | ˆ
D| ˜
J
(A.6.40)
as considered in Sects. 11.3 and 12.1 for (N − 1)-electron excitations (Eq. 11.51)
and Sect. 14.3 for N -electron excitations (Eq. 14.46). As above, one may insert the
biorthogonal RI before and after the operator ˆ
D and use the order structure (A.6.30)
in the related BCC matrix ˆ
D
cc . The order relations thereby established read (see
Eq. 12.17 and Fig. 12.2)
˜
D I J ∼ O(|[I ] − [J ]| − 1), [I ] = [J ]
(A.6.41)
The transition moments
F I (D) = = ˜
I | ˆ
D| 0 ∼ O([I ] − 1)
(A.6.42)