170
11 Intermediate-State Representation (ISR)
can be evaluated. The result confirms the identity (11.27) of the ADC and ISR
transition amplitudes.
Of course, the ECO-IS basis can be used to represent ˆ
H (instead of ˆ
H − E 0 ),
˜
H I J = = ˜
I | ˆ
H | ˜
J = M I J + δ I J E 0
(11.50)
The second-order ISR(2) version of ˆ
H is obtained from M by adding ground-state
energy expansions to the diagonal elements in a consistent way, that is, E
(0)
0 + E
(1)
0 +
E
(2)
0 in the M 11 block, and just E
(0)
0 in the M 22 block.
In view of the ISR(2) derivation just presented, it may be expected that matters
will become rather tedious at the next higher, i.e., third-order, level. So while the
ECO-ISR is the simpler concept, the ADC approach, based on the diagrammatic
PT expansion of the electron propagator, is clearly preferable when it comes to the
practical implementation.
11.3 Intermediate-State Representation of General
Operators
Obviously, the intermediate-state representation introduced in the preceding two
sections can be applied to operators other than the hamiltonian. This equips the ISRADC approach with the full flexibility of wave-function-based methods, allowing
for applications not possible within the original propagator concept.
As an example for the additional opportunities, let us consider the treatment of
ionic-state properties. Let ˆ
D be a hermitian operator associated with the physical
property of interest, e.g., the dipole moment along a particular axis. The ISR of ˆ
D is
given by the matrix elements of ˆ
D with respect to the intermediate states | ˜
I :
˜
D I J = = ˜
I | ˆ
D| ˜
J
(11.51)
The corresponding matrix will be denoted by ˜
D. For a particular ionic energy eigenstate |
N−1
n , the desired property is obtained as the expectation value
D n = =
N−1
n | ˆ
D|
N−1
n = X
†
n
˜
DX n
where X n denotes the eigenvector associated with the ISR expansion
|
N−1
n =
I
X I n | ˜
I
(11.52)
of |
N−1
n . In s similar way, transition moments involving two ionic states can be
derived according to
11 Intermediate-State Representation (ISR)
can be evaluated. The result confirms the identity (11.27) of the ADC and ISR
transition amplitudes.
Of course, the ECO-IS basis can be used to represent ˆ
H (instead of ˆ
H − E 0 ),
˜
H I J = = ˜
I | ˆ
H | ˜
J = M I J + δ I J E 0
(11.50)
The second-order ISR(2) version of ˆ
H is obtained from M by adding ground-state
energy expansions to the diagonal elements in a consistent way, that is, E
(0)
0 + E
(1)
0 +
E
(2)
0 in the M 11 block, and just E
(0)
0 in the M 22 block.
In view of the ISR(2) derivation just presented, it may be expected that matters
will become rather tedious at the next higher, i.e., third-order, level. So while the
ECO-ISR is the simpler concept, the ADC approach, based on the diagrammatic
PT expansion of the electron propagator, is clearly preferable when it comes to the
practical implementation.
11.3 Intermediate-State Representation of General
Operators
Obviously, the intermediate-state representation introduced in the preceding two
sections can be applied to operators other than the hamiltonian. This equips the ISRADC approach with the full flexibility of wave-function-based methods, allowing
for applications not possible within the original propagator concept.
As an example for the additional opportunities, let us consider the treatment of
ionic-state properties. Let ˆ
D be a hermitian operator associated with the physical
property of interest, e.g., the dipole moment along a particular axis. The ISR of ˆ
D is
given by the matrix elements of ˆ
D with respect to the intermediate states | ˜
I :
˜
D I J = = ˜
I | ˆ
D| ˜
J
(11.51)
The corresponding matrix will be denoted by ˜
D. For a particular ionic energy eigenstate |
N−1
n , the desired property is obtained as the expectation value
D n = =
N−1
n | ˆ
D|
N−1
n = X
†
n
˜
DX n
where X n denotes the eigenvector associated with the ISR expansion
|
N−1
n =
I
X I n | ˜
I
(11.52)
of |
N−1
n . In s similar way, transition moments involving two ionic states can be
derived according to