172
11 Intermediate-State Representation (ISR)
and limit ourselves to some guiding remarks. For a more detailed account of the
procedure, the reader is referred to Ref. [5] where an analogous derivation for N -
electron excitations has been presented. For simplicity, we shall confine ourselves to
one-body operators, being of the form
ˆ
D =
d rs c
†
r c s
The ISR construction can be applied to two-body (and higher-rank) operators as well,
but obviously this is more cumbersome.
1. In deriving the ISR(2) expressions of a general one-body operator, the secondorder term in the ground-state PT expansion,
|
(2)
0 = |
(2)
1 p−1h + |
(2)
2 p−2h + |
(2)
3 p−3h + |
(2)
4 p−4h
(11.60)
comes into play, though only with the 1 p-1h part:
|
(2)
1 p−1h =
ak
x
(2)
ak c
†
a c k | 0 .
(11.61)
Here, x
(2)
ak denote the second-order expansion coefficients of the 1 p-1h admixtures
to the ground state,
x
(2)
ak = = 0 |c
†
k c a |
(2)
0
(11.62)
Note that the x
(2)
ak coefficients through second order can be identified with the p-h
components of the one-particle density matrix,
ρ ak = = 0 |c
†
k c a | 0 = x
(2)
ak + O(3)
(11.63)
This relation can be used to replace the 1 p-1h PT coefficient in the final ISR(2)
expressions. As should be recalled, the density matrix elements ρ ak can be derived
from the electron propagator part G
−
(ω), for example, according to Eqs. (3.34)
or (10.37).
2. The IS matrix elements turn out to be of the form
˜
D I J = D 0 δ I J + ˜
D
I J
(11.64)
where
D 0 = I
−1
0 0 | ˆ
D| 0
(11.65)
is the expectation value of ˆ
D with respect to the (normalized) N -electron ground
state, and the matrix ˜
D
is a representation of the subtracted operator ˆ
D
= ˆ
D −
D 0 . In the diagonal ISR(2) matrix elements of ˜
D 11 , the PT expansion of D 0
11 Intermediate-State Representation (ISR)
and limit ourselves to some guiding remarks. For a more detailed account of the
procedure, the reader is referred to Ref. [5] where an analogous derivation for N -
electron excitations has been presented. For simplicity, we shall confine ourselves to
one-body operators, being of the form
ˆ
D =
d rs c
†
r c s
The ISR construction can be applied to two-body (and higher-rank) operators as well,
but obviously this is more cumbersome.
1. In deriving the ISR(2) expressions of a general one-body operator, the secondorder term in the ground-state PT expansion,
|
(2)
0 = |
(2)
1 p−1h + |
(2)
2 p−2h + |
(2)
3 p−3h + |
(2)
4 p−4h
(11.60)
comes into play, though only with the 1 p-1h part:
|
(2)
1 p−1h =
ak
x
(2)
ak c
†
a c k | 0 .
(11.61)
Here, x
(2)
ak denote the second-order expansion coefficients of the 1 p-1h admixtures
to the ground state,
x
(2)
ak = = 0 |c
†
k c a |
(2)
0
(11.62)
Note that the x
(2)
ak coefficients through second order can be identified with the p-h
components of the one-particle density matrix,
ρ ak = = 0 |c
†
k c a | 0 = x
(2)
ak + O(3)
(11.63)
This relation can be used to replace the 1 p-1h PT coefficient in the final ISR(2)
expressions. As should be recalled, the density matrix elements ρ ak can be derived
from the electron propagator part G
−
(ω), for example, according to Eqs. (3.34)
or (10.37).
2. The IS matrix elements turn out to be of the form
˜
D I J = D 0 δ I J + ˜
D
I J
(11.64)
where
D 0 = I
−1
0 0 | ˆ
D| 0
(11.65)
is the expectation value of ˆ
D with respect to the (normalized) N -electron ground
state, and the matrix ˜
D
is a representation of the subtracted operator ˆ
D
= ˆ
D −
D 0 . In the diagonal ISR(2) matrix elements of ˜
D 11 , the PT expansion of D 0