150
10 Direct ADC Procedure for the Electron Propagator
10.2 Explicit ADC Procedure Through Second Order
So far the ADC form (10.8) is merely a formal construct yet to be related to the
diagrammatic perturbation expansion of G
−
(ω). To this end, one may expand the
ADC form in a (still formal) perturbation series supposing the same MP partitioning
of the hamiltonian as in the diagrammatic expansion of the electron propagator. Both
the ADC secular matrix and the f matrix are subject to perturbation expansions
K + C = K + C
(1)
+ C
(2)
+ C
(3)
+ . . .
(10.15)
f = f
(0)
+ f
(1)
+ f
(2)
+ . . .
(10.16)
Here K denotes the zeroth-order part of the secular matrix, that is, the diagonal
matrix of (negative) HF ionization energies,
K kk = k
K akl,akl = k + l − a
. . .
(10.17)
The partitioning of the ADC matrix into a zeroth-order part K and a remainder C
allows one to expand the resolvent matrix (ω − K − C)
−1 as a geometric series,
˜
G(ω) = f
†
(ω − K − C)
−1 f = f
†
(ω − K )
−1
∞
ν=0
C
ω − K
ν
f
(10.18)
Together with the PT expansions for C and f , this establishes the formal ADC series
for ˜
G(ω).
The ADC expansion can further be specified by considering the block structure of
the ADC matrices associated with the partitioning of the configurations J into classes
μ = 1, 2, 3, . . . of 1h, 2h-1 p, 3h-2 p, . . . excitations. The respective matrix blocks
will be denoted by K μ , C μμ , and f μ , where μ denotes the class of μh-(μ − 1) p
excitations. The f μ matrix blocks may be distinguished further according to the
second matrix index, f I q , as particle (n q = 1) or hole terms (n q = 1).
The ADC expansion (10.18) is to be compared with the diagrammatic PT expansion for G
−
(ω). According to Chap. 7, the latter is constituted directly from the class
of time-ordered (Goldstone) diagrams with t < t
. As an instructive demonstration,
let us perform the ADC procedure for G
−
(ω) through second order of PT.
Zeroth Order:
The zeroth-order ADC matrix
˜
G
(0) (ω) = f
(0)†
(ω − K )
−1 f
(0)
(10.19)
is to be compared with the explicit zeroth-order expression
10 Direct ADC Procedure for the Electron Propagator
10.2 Explicit ADC Procedure Through Second Order
So far the ADC form (10.8) is merely a formal construct yet to be related to the
diagrammatic perturbation expansion of G
−
(ω). To this end, one may expand the
ADC form in a (still formal) perturbation series supposing the same MP partitioning
of the hamiltonian as in the diagrammatic expansion of the electron propagator. Both
the ADC secular matrix and the f matrix are subject to perturbation expansions
K + C = K + C
(1)
+ C
(2)
+ C
(3)
+ . . .
(10.15)
f = f
(0)
+ f
(1)
+ f
(2)
+ . . .
(10.16)
Here K denotes the zeroth-order part of the secular matrix, that is, the diagonal
matrix of (negative) HF ionization energies,
K kk = k
K akl,akl = k + l − a
. . .
(10.17)
The partitioning of the ADC matrix into a zeroth-order part K and a remainder C
allows one to expand the resolvent matrix (ω − K − C)
−1 as a geometric series,
˜
G(ω) = f
†
(ω − K − C)
−1 f = f
†
(ω − K )
−1
∞
ν=0
C
ω − K
ν
f
(10.18)
Together with the PT expansions for C and f , this establishes the formal ADC series
for ˜
G(ω).
The ADC expansion can further be specified by considering the block structure of
the ADC matrices associated with the partitioning of the configurations J into classes
μ = 1, 2, 3, . . . of 1h, 2h-1 p, 3h-2 p, . . . excitations. The respective matrix blocks
will be denoted by K μ , C μμ , and f μ , where μ denotes the class of μh-(μ − 1) p
excitations. The f μ matrix blocks may be distinguished further according to the
second matrix index, f I q , as particle (n q = 1) or hole terms (n q = 1).
The ADC expansion (10.18) is to be compared with the diagrammatic PT expansion for G
−
(ω). According to Chap. 7, the latter is constituted directly from the class
of time-ordered (Goldstone) diagrams with t < t
. As an instructive demonstration,
let us perform the ADC procedure for G
−
(ω) through second order of PT.
Zeroth Order:
The zeroth-order ADC matrix
˜
G
(0) (ω) = f
(0)†
(ω − K )
−1 f
(0)
(10.19)
is to be compared with the explicit zeroth-order expression