10.1 ADC Representation of G
− (ω)
149
regard to the HF ground state. In the following, the designation N −1 in the notation
of the intermediate states will be skipped, as the affiliation with (N −1)-electron
states can be inferred from the configuration indices I, J, . . . . In Chap. 11, a direct
construction of the intermediate states will be presented. For the present purpose,
though, it suffices to suppose just the existence of such states, dispensing with the
need for further specification.
Analogously to the spectral representation, the ADC form (10.8) can be obtained
by using in Eq. (10.1) the resolution of the identity
ˆ
1 =
| ˜
Ψ I ˜
Ψ I |
(10.10)
in terms of the intermediate states. This allows us to define the ADC secular matrix
K + C and the matrix f of “effective” transition amplitudes as representations with
respect to the (so far hypothetical) intermediate states:
(K + C) I J = −− ˜
Ψ I ˆ
H − E 0 ˜
Ψ J
f I q = = ˜
Ψ I |c q |Ψ 0
(10.11)
In the ensuing Sect. 10.2, we discuss how the ADC matrices K + C and f can
successively be derived from the diagrammatic perturbation expansion for the
electron propagator part G
−
(ω).
Let us assume that the ADC procedure has provided approximate (or exact) expressions for K + C and f . Then, the adc secular equations,
(K + C)X = X,
X
† X = 1
(10.12)
allow one to derive the physical information of interest. Here X denotes the matrix
of eigenvectors and is the diagonal matrix of eigenvalues ω n , to be identified
with the negative ionization energies, ω n = −I n . The spectroscopic factors, x
(n)
p , are
obtained from the scalar product of the nth eigenvector and the respective columns
of the matrix f :
x
(n)
p =
J
X
∗
J n f J p
(10.13)
The eigenvector components can be viewed as the expansion coefficients
X J n = = ˜
Ψ J |Ψ
N −1
n
(10.14)
of the exact (or approximate) energy eigenstates written as linear combinations of
the hypothetical intermediate states.
− (ω)
149
regard to the HF ground state. In the following, the designation N −1 in the notation
of the intermediate states will be skipped, as the affiliation with (N −1)-electron
states can be inferred from the configuration indices I, J, . . . . In Chap. 11, a direct
construction of the intermediate states will be presented. For the present purpose,
though, it suffices to suppose just the existence of such states, dispensing with the
need for further specification.
Analogously to the spectral representation, the ADC form (10.8) can be obtained
by using in Eq. (10.1) the resolution of the identity
ˆ
1 =
| ˜
Ψ I ˜
Ψ I |
(10.10)
in terms of the intermediate states. This allows us to define the ADC secular matrix
K + C and the matrix f of “effective” transition amplitudes as representations with
respect to the (so far hypothetical) intermediate states:
(K + C) I J = −− ˜
Ψ I ˆ
H − E 0 ˜
Ψ J
f I q = = ˜
Ψ I |c q |Ψ 0
(10.11)
In the ensuing Sect. 10.2, we discuss how the ADC matrices K + C and f can
successively be derived from the diagrammatic perturbation expansion for the
electron propagator part G
−
(ω).
Let us assume that the ADC procedure has provided approximate (or exact) expressions for K + C and f . Then, the adc secular equations,
(K + C)X = X,
X
† X = 1
(10.12)
allow one to derive the physical information of interest. Here X denotes the matrix
of eigenvectors and is the diagonal matrix of eigenvalues ω n , to be identified
with the negative ionization energies, ω n = −I n . The spectroscopic factors, x
(n)
p , are
obtained from the scalar product of the nth eigenvector and the respective columns
of the matrix f :
x
(n)
p =
J
X
∗
J n f J p
(10.13)
The eigenvector components can be viewed as the expansion coefficients
X J n = = ˜
Ψ J |Ψ
N −1
n
(10.14)
of the exact (or approximate) energy eigenstates written as linear combinations of
the hypothetical intermediate states.