216
14 ADC and ISR Approaches to the Polarization Propagator
˜
D 11 = ˜
D
(0)
11 + ˜
D
(1)
11 + ˜
D
(2)
11
˜
D 12 = ˜
D
(0)
12 + ˜
D
(1)
12
˜
D 22 = ˜
D
(0)
22
(14.47)
As in the ADC(2) and ADC(3) secular equations, the configuration space comprises
the 1 p-1h and 2 p-2h excitations (classes 1 and 2), and the PT expansions in the
various sub-blocks essentially match those of the ADC(2) secular matrix.
The ISR of an operator ˆ
D can be used to compute the associated property of an
excited state | m as the expectation value
D m = = m | ˆ
D| m = X
†
m
˜
DX m
(14.48)
Here, X m is the ADC eigenvector (see Eq. 14.5) pertaining to | m . In a similar way,
transition moments involving two states are obtained according to
D mn = = m | ˆ
D| n = X
†
m
˜
DX n
(14.49)
In such applications, usually one will ensure that the approximation schemes for
the two ingredients are consistent, e.g., in combining the ISR(2) operator matrix
with ADC(2) eigenvectors. Of course, one may as well combine the more accurate
ADC(3) eigenvectors with the second-order ISR operator expressions..
Another benefit afforded by the ISR of general operators, already addressed in
Sect. 11.3, is the option to augment the original Hamiltonian ˆ
H with an additional
operator ˆ
U , for example, associated with an external field:
ˆ
H → ˆ
H
x
= ˆ
H + ˆ
U
(14.50)
In the N -electron case considered here, the intermediate states can couple to the
ground state via ˆ
U . Accordingly, the IS configurations must be enlarged by | 0 ,
and the representation of ˆ
H
x
− E 0 has an additional row (and column), the elements
being
M
x
00 = U 0 , M
x
I 0 = F I (U ), I = 0
(14.51)
Here, U 0 = = 0 | ˆ
U | 0 is the ground-state expectation value of ˆ
U , and F I (U ) denote
the IS transition moments for the operator ˆ
U ,
F I (U ) = = ˜
I | ˆ
U | 0 =
f I,rs u rs
(14.52)
supposing here a one-particle operator of the form ˆ
U =
u rs c
†
r c s . The matrix elements in the main block are given by
M
x
I J = M I J + ˜
U I J , I, J = 0
(14.53)
where ˜
U I J are the ISR matrix elements of ˆ
U .
14 ADC and ISR Approaches to the Polarization Propagator
˜
D 11 = ˜
D
(0)
11 + ˜
D
(1)
11 + ˜
D
(2)
11
˜
D 12 = ˜
D
(0)
12 + ˜
D
(1)
12
˜
D 22 = ˜
D
(0)
22
(14.47)
As in the ADC(2) and ADC(3) secular equations, the configuration space comprises
the 1 p-1h and 2 p-2h excitations (classes 1 and 2), and the PT expansions in the
various sub-blocks essentially match those of the ADC(2) secular matrix.
The ISR of an operator ˆ
D can be used to compute the associated property of an
excited state | m as the expectation value
D m = = m | ˆ
D| m = X
†
m
˜
DX m
(14.48)
Here, X m is the ADC eigenvector (see Eq. 14.5) pertaining to | m . In a similar way,
transition moments involving two states are obtained according to
D mn = = m | ˆ
D| n = X
†
m
˜
DX n
(14.49)
In such applications, usually one will ensure that the approximation schemes for
the two ingredients are consistent, e.g., in combining the ISR(2) operator matrix
with ADC(2) eigenvectors. Of course, one may as well combine the more accurate
ADC(3) eigenvectors with the second-order ISR operator expressions..
Another benefit afforded by the ISR of general operators, already addressed in
Sect. 11.3, is the option to augment the original Hamiltonian ˆ
H with an additional
operator ˆ
U , for example, associated with an external field:
ˆ
H → ˆ
H
x
= ˆ
H + ˆ
U
(14.50)
In the N -electron case considered here, the intermediate states can couple to the
ground state via ˆ
U . Accordingly, the IS configurations must be enlarged by | 0 ,
and the representation of ˆ
H
x
− E 0 has an additional row (and column), the elements
being
M
x
00 = U 0 , M
x
I 0 = F I (U ), I = 0
(14.51)
Here, U 0 = = 0 | ˆ
U | 0 is the ground-state expectation value of ˆ
U , and F I (U ) denote
the IS transition moments for the operator ˆ
U ,
F I (U ) = = ˜
I | ˆ
U | 0 =
f I,rs u rs
(14.52)
supposing here a one-particle operator of the form ˆ
U =
u rs c
†
r c s . The matrix elements in the main block are given by
M
x
I J = M I J + ˜
U I J , I, J = 0
(14.53)
where ˜
U I J are the ISR matrix elements of ˆ
U .