14.2 Explicit ADC Schemes for the Polarization Propagator
209
where the respective 1 p-1h matrix blocks are indicated by the subscript 1. The three
terms on the right-hand side are matched by the first-order Goldstone diagrams (b),
(c), and (a) shown in Fig. 13.4, which allows one to simply read off the quantities
C
(1)
11 and f
(1)
1 from the diagrammatic expressions. In particular, the expressions for
diagram (a) and diagram (b) (see Eq. 13.22) yield
C
(1)
ak,a k = −V ak [a k]
(14.25)
and
f
(1)
ak,rs =
V as[rk] n r n s
a + s − k − r
(14.26)
Note that f
(1)
ak,rs vanishes unless the index pair rs is of h- p type.
Of course, these first-order results could have been obtained equally well using
the ISR formulation. Through first order, the 1 p-1h intermediate states simply read
| ˜
ak = | ak + c
†
a c k |
(1)
0 + O(2)
(14.27)
Consequently, the secular matrix
M ak,a k = = ˜
ak | ˆ
H − E 0 | ˜
a k = K ak,a k + + ak | ˆ
H I − E
(1)
0 | a k + O(2)
(14.28)
can easily be expanded through first order, reproducing the ADC result of Eq. (14.25).
In a similar way, the expansion of the IS transition amplitudes
f ak,rs = = ˜
ak |c
†
r c s | 0
= = ak |c
†
r c s | 0 + + ak |c
†
r c s |
(1)
0 + O(2)
(14.29)
recovers Eqs. (14.23) and (14.26).
The ADC(1) scheme, constituted by the secular matrix
M ak,a k = (( a − k )δ aa δ kk − V ak [a k]
(14.30)
and the effective transition amplitudes given by Eqs. (14.23) and (14.26), can be compared to the CIS (single excitation configuration interaction) method (see Sect. 12.3),
also referred to as Tamm–Dancoff approximation (TDA). Eq. (14.28) shows that the
CIS secular matrix is identical with the ADC(1) matrix (up to the shift by −E 0 (1)
in the diagonal). The quality of the ADC(1) or CIS computational schemes is rather
modest, as the excitation energies of the 1 p-1h states are treated consistently through
first order only. With regard to the transition moments, the CIS description based on
the expression
T m =
a k
X
∗
a k ,m d a k
(14.31)
209
where the respective 1 p-1h matrix blocks are indicated by the subscript 1. The three
terms on the right-hand side are matched by the first-order Goldstone diagrams (b),
(c), and (a) shown in Fig. 13.4, which allows one to simply read off the quantities
C
(1)
11 and f
(1)
1 from the diagrammatic expressions. In particular, the expressions for
diagram (a) and diagram (b) (see Eq. 13.22) yield
C
(1)
ak,a k = −V ak [a k]
(14.25)
and
f
(1)
ak,rs =
V as[rk] n r n s
a + s − k − r
(14.26)
Note that f
(1)
ak,rs vanishes unless the index pair rs is of h- p type.
Of course, these first-order results could have been obtained equally well using
the ISR formulation. Through first order, the 1 p-1h intermediate states simply read
| ˜
ak = | ak + c
†
a c k |
(1)
0 + O(2)
(14.27)
Consequently, the secular matrix
M ak,a k = = ˜
ak | ˆ
H − E 0 | ˜
a k = K ak,a k + + ak | ˆ
H I − E
(1)
0 | a k + O(2)
(14.28)
can easily be expanded through first order, reproducing the ADC result of Eq. (14.25).
In a similar way, the expansion of the IS transition amplitudes
f ak,rs = = ˜
ak |c
†
r c s | 0
= = ak |c
†
r c s | 0 + + ak |c
†
r c s |
(1)
0 + O(2)
(14.29)
recovers Eqs. (14.23) and (14.26).
The ADC(1) scheme, constituted by the secular matrix
M ak,a k = (( a − k )δ aa δ kk − V ak [a k]
(14.30)
and the effective transition amplitudes given by Eqs. (14.23) and (14.26), can be compared to the CIS (single excitation configuration interaction) method (see Sect. 12.3),
also referred to as Tamm–Dancoff approximation (TDA). Eq. (14.28) shows that the
CIS secular matrix is identical with the ADC(1) matrix (up to the shift by −E 0 (1)
in the diagonal). The quality of the ADC(1) or CIS computational schemes is rather
modest, as the excitation energies of the 1 p-1h states are treated consistently through
first order only. With regard to the transition moments, the CIS description based on
the expression
T m =
a k
X
∗
a k ,m d a k
(14.31)