15.2 ADC Formulation of the RPA
235
For given approximations of M
R P A and f
R P A , the spectral information is obtained
by solving the hermitian eigenvalue problem (ADC secular equation)
M
R P A Y = Y , Y
† Y = 1
(15.57)
where Y and denote the eigenvector matrix and the diagonal matrix of eigenvalues,
respectively.
The RPA spectroscopic amplitudes are obtained according to
(X
+
)
†
= Y
† f
R P A
(15.58)
so that the RPA-ADC transition moments are given by
T
R P A
m
= Y
†
m f
R P A d
(15.59)
Here, Y m denotes the mth eigenvector of M
R P A .
At the RPA-ADC(n) level, the results of Eqs. (15.57) and (15.58) provide approximations to the full RPA energies and transition moments, being consistent through
nth order and expected to converge to the RPA results in the limit n → ∞.
Finally, we take a look at the explicit RPA-ADC expressions in the first- and
second-order schemes. Through first order, the RPA diagrams are complete so that
the derivation of the first-order RPA-ADC scheme does not differ from that for the
polarization propagator in Sect. 14.2. According to Eqs. (14.25, 14.26), the ADC(1)
expressions read
M
R P A
ak,a k = ( a − k )δ aa δ kk − V ak [a k]
f
R P A
ak,bl = δ ab δ kl
f
R P A
ak,lb =
V ab[lk]
a + b − k − l
(15.60)
In second order, the RPA diagram (2C) in Fig. 13.3 is one of altogether five
Feynman diagrams for the full propagator. There are 24 Goldstone diagrams deriving
from (C), of which the 12 diagrams contributing to
+
(ω) are shown in Fig. 14.1.
In Sect. 14.2, we have addressed the ADC(2) contributions deriving from these 12
diagrams, and the findings here can directly be transferred to the case of the secondorder RPA-ADC scheme. Accordingly, the second-order contribution to M
R P A is
given by the expression C
(C) of Eq. (14.39):
M
R P A (2)
ak,a k = C
(C)
ak,a k
(15.61)
The second-order part of f
R P A consists of three distinct contributions
f
R P A
= f
(C)
1 + f
(2,5)
1
+ f
(2,6)
1
(15.62)
235
For given approximations of M
R P A and f
R P A , the spectral information is obtained
by solving the hermitian eigenvalue problem (ADC secular equation)
M
R P A Y = Y , Y
† Y = 1
(15.57)
where Y and denote the eigenvector matrix and the diagonal matrix of eigenvalues,
respectively.
The RPA spectroscopic amplitudes are obtained according to
(X
+
)
†
= Y
† f
R P A
(15.58)
so that the RPA-ADC transition moments are given by
T
R P A
m
= Y
†
m f
R P A d
(15.59)
Here, Y m denotes the mth eigenvector of M
R P A .
At the RPA-ADC(n) level, the results of Eqs. (15.57) and (15.58) provide approximations to the full RPA energies and transition moments, being consistent through
nth order and expected to converge to the RPA results in the limit n → ∞.
Finally, we take a look at the explicit RPA-ADC expressions in the first- and
second-order schemes. Through first order, the RPA diagrams are complete so that
the derivation of the first-order RPA-ADC scheme does not differ from that for the
polarization propagator in Sect. 14.2. According to Eqs. (14.25, 14.26), the ADC(1)
expressions read
M
R P A
ak,a k = ( a − k )δ aa δ kk − V ak [a k]
f
R P A
ak,bl = δ ab δ kl
f
R P A
ak,lb =
V ab[lk]
a + b − k − l
(15.60)
In second order, the RPA diagram (2C) in Fig. 13.3 is one of altogether five
Feynman diagrams for the full propagator. There are 24 Goldstone diagrams deriving
from (C), of which the 12 diagrams contributing to
+
(ω) are shown in Fig. 14.1.
In Sect. 14.2, we have addressed the ADC(2) contributions deriving from these 12
diagrams, and the findings here can directly be transferred to the case of the secondorder RPA-ADC scheme. Accordingly, the second-order contribution to M
R P A is
given by the expression C
(C) of Eq. (14.39):
M
R P A (2)
ak,a k = C
(C)
ak,a k
(15.61)
The second-order part of f
R P A consists of three distinct contributions
f
R P A
= f
(C)
1 + f
(2,5)
1
+ f
(2,6)
1
(15.62)