166
11 Intermediate-State Representation (ISR)
It should be noted though that equivalence not necessarily means identity here. In
fact, we have seen in Chap. 10 that the ADC procedure does not completely determine
the secular matrix, but leaves some room for algebraic transformations. By contrast,
for a given ground state and one-particle basis, the ECO-ISR procedure as defined
by Eqs. (11.11)–(11.13) results in a unique IS representation. Here, however, the
second step, that is, the symmetric orthonormalization of the respective precursor
states, is a mere convention, not necessary for constituting the two basic features. In
fact, any orthonormalization scheme could be used here. So there is some flexibility
in the ECO-ISR concept, which may be seen as the counterpart to the residual nonuniqueness in the ADC expressions. Recognizing the general equivalence, we will
use the terms ECO-ISR and ISR-ADC synonymously in the following.
11.2 Explicit ISR Procedure Through Second Order
In the following, we shall derive the explicit PT expressions of the ISR secular matrix
M as needed for a consistent treatment of the 1h (main) ionic states through second
order, to be referred to as ISR(2) scheme.
The explicit configuration space required at the ISR(2) level is spanned by the
1h and 2h-1 p intermediate states. The coupling of 1h and 3h-2 p states is already of
second order (see Sect. 12.1). The PT expansions extend through second order in the
1h diagonal block M 11 ,
M kl = − k δ kl + M
(1)
kl + M
(2)
kl
(11.28)
through first-order in the M 12 block,
M k,a k l = M
(1)
k,a k l
(11.29)
while only the zeroth-order contributions
M akl,a k l = ( a − k − l )δ aa δ kk δ ll
(11.30)
are needed in the M 22 block.
At the ISR(2) level, the PT expansion of the N -electron ground state underlying
the ISR construction is needed through second order:
| 0 = | 0 + |
(1)
0 + |
(2)
0 + O(3)
(11.31)
The first-order term,
|
(1)
0 =
a
(11.32)
11 Intermediate-State Representation (ISR)
It should be noted though that equivalence not necessarily means identity here. In
fact, we have seen in Chap. 10 that the ADC procedure does not completely determine
the secular matrix, but leaves some room for algebraic transformations. By contrast,
for a given ground state and one-particle basis, the ECO-ISR procedure as defined
by Eqs. (11.11)–(11.13) results in a unique IS representation. Here, however, the
second step, that is, the symmetric orthonormalization of the respective precursor
states, is a mere convention, not necessary for constituting the two basic features. In
fact, any orthonormalization scheme could be used here. So there is some flexibility
in the ECO-ISR concept, which may be seen as the counterpart to the residual nonuniqueness in the ADC expressions. Recognizing the general equivalence, we will
use the terms ECO-ISR and ISR-ADC synonymously in the following.
11.2 Explicit ISR Procedure Through Second Order
In the following, we shall derive the explicit PT expressions of the ISR secular matrix
M as needed for a consistent treatment of the 1h (main) ionic states through second
order, to be referred to as ISR(2) scheme.
The explicit configuration space required at the ISR(2) level is spanned by the
1h and 2h-1 p intermediate states. The coupling of 1h and 3h-2 p states is already of
second order (see Sect. 12.1). The PT expansions extend through second order in the
1h diagonal block M 11 ,
M kl = − k δ kl + M
(1)
kl + M
(2)
kl
(11.28)
through first-order in the M 12 block,
M k,a k l = M
(1)
k,a k l
(11.29)
while only the zeroth-order contributions
M akl,a k l = ( a − k − l )δ aa δ kk δ ll
(11.30)
are needed in the M 22 block.
At the ISR(2) level, the PT expansion of the N -electron ground state underlying
the ISR construction is needed through second order:
| 0 = | 0 + |
(1)
0 + |
(2)
0 + O(3)
(11.31)
The first-order term,
|
(1)
0 =
a
(11.32)