15.1 Derivation and Properties of the RPA Equations
233
with the zeroth- and first-order terms in the PT expansion (14.82) of the exact spectral
moment is valid. Recalling the result of Eq. (14.84), this proves the dipole sum rule
S
R P A
1
(Z ) =
1
2
N
2
/m e
(15.46)
for the RPA excitation spectrum.
For RPA excitations, the identity (14.64) applying to exact states would read
ω m X
†
m d = −i X
†
m p z /m e
(15.47)
where p z is the vector of the 1 p-1h and 1h-1 p matrix elements of the momentum
operator ˆ
p z , analogous to d, being the vector of matrix elements of ˆ
z. This equation,
so far only asserted, can be replaced by a more abstract version not relating to a
particular excitation. To this end, we insert the S matrix twice on the left side of
Eq. (15.47), and make use of the RPA secular equation (15.10), which yields
ω m X
†
m d = ω m SX
†
m Sd = X
†
m MSd
(15.48)
Abstracting the eigenvector X
†
m both on the left- and right-hand side of Eq. (15.47)
leads to the “global” identity
MSd = −i p z /m e
(15.49)
In fact, the RPA secular matrices M and S fulfill that identity [6, 7], thus ensuring
the equivalence of the length and velocity forms of the RPA transition moments. The
identity (15.49) can be shown in a straightforward, if somewhat tedious way (see
Exercise 15.4), using the general length–velocity relation at the level of one-particle
operators,
[ ˆ
f , ˆ
z] = [ ˆ
w, ˆ
z] − i ˆ
p z /m e
(15.50)
according to Eqs. (14.73)–(14.75).
15.2 ADC Formulation of the RPA
The RPA version of the polarization propagator is constituted by a specific class
of diagrams, namely the RPA diagrams shown in Fig. 15.1. Obviously, the ADC
procedure can be confined to that particular class of diagrams. This allows one to
convert the original RPA scheme into ADC-type secular equations [8], as will briefly
be discussed in this section.
As in the general case, we start with the spectral representation (15.27):
R P A +
(ω) = X
+
(ω1 −
+
)
−1
(X
+
)
†
233
with the zeroth- and first-order terms in the PT expansion (14.82) of the exact spectral
moment is valid. Recalling the result of Eq. (14.84), this proves the dipole sum rule
S
R P A
1
(Z ) =
1
2
N
2
/m e
(15.46)
for the RPA excitation spectrum.
For RPA excitations, the identity (14.64) applying to exact states would read
ω m X
†
m d = −i X
†
m p z /m e
(15.47)
where p z is the vector of the 1 p-1h and 1h-1 p matrix elements of the momentum
operator ˆ
p z , analogous to d, being the vector of matrix elements of ˆ
z. This equation,
so far only asserted, can be replaced by a more abstract version not relating to a
particular excitation. To this end, we insert the S matrix twice on the left side of
Eq. (15.47), and make use of the RPA secular equation (15.10), which yields
ω m X
†
m d = ω m SX
†
m Sd = X
†
m MSd
(15.48)
Abstracting the eigenvector X
†
m both on the left- and right-hand side of Eq. (15.47)
leads to the “global” identity
MSd = −i p z /m e
(15.49)
In fact, the RPA secular matrices M and S fulfill that identity [6, 7], thus ensuring
the equivalence of the length and velocity forms of the RPA transition moments. The
identity (15.49) can be shown in a straightforward, if somewhat tedious way (see
Exercise 15.4), using the general length–velocity relation at the level of one-particle
operators,
[ ˆ
f , ˆ
z] = [ ˆ
w, ˆ
z] − i ˆ
p z /m e
(15.50)
according to Eqs. (14.73)–(14.75).
15.2 ADC Formulation of the RPA
The RPA version of the polarization propagator is constituted by a specific class
of diagrams, namely the RPA diagrams shown in Fig. 15.1. Obviously, the ADC
procedure can be confined to that particular class of diagrams. This allows one to
convert the original RPA scheme into ADC-type secular equations [8], as will briefly
be discussed in this section.
As in the general case, we start with the spectral representation (15.27):
R P A +
(ω) = X
+
(ω1 −
+
)
−1
(X
+
)
†