15.1 Derivation and Properties of the RPA Equations
229
The transition moments derive from the RPA eigenvector components in an analogous manner to Eq. (13.6),
T
R P A
m
=
r,s
X
∗
rs,m d rs ( ¯
n r n s + n r ¯
n s ), m ∈ {+}
(15.30)
where d rs are the one-particle matrix elements of a transition operator ˆ
D. This expression may be written more compactly as
T
R P A
m
= X
†
m d
(15.31)
where
d =
d ph
d hp
=
d ph
d
∗
ph
(15.32)
denotes the (column) vector of p-h and h- p matrix elements d rs .
Perturbation Theoretical Analysis of the RPA
As already mentioned, the RPA energies and transition moments are consistent with
those obtained at the TDA (or CIS and ADC(1)) level, since the 1 p-1h sub-block
A of the RPA secular matrix is identical with the TDA secular matrix. However,
the RPA treatment goes beyond the TDA, the differences beginning at second order,
as a result of the off-diagonal blocks B and B
∗ , which couple the (physical) 1 p-1h
excitations and the (unphysical) 1h-1 p de-excitations. How can this curious coupling
be understood and what is the quality of the results thereby obtained? As a way to
better understand the situation, we shall analyze, like in Sect. 8.3, the RPA results
using perturbation theory, which, moreover, will allow us to discuss the relevant
physics of the excitation process.
Let us consider a single excitation a ← k from the occupied orbital k to the
virtual (unoccupied) orbital a, represented by the zeroth-order (HF) state | ak =
c
†
a c k | 0 . The formal PT expansion of the excitation energy, E ak = E ak − E 0 ,
through second order can be written as
E ak = a − k − V ak[ak]
+ U
(2)
ak (1 p-1h) + U
(2)
ak (2 p-2h) + U
(2)
ak (3 p-3h) − E
(2)
0 + O(3)
(15.33)
Here, the three second-order terms, U
(2)
ak (ν p-νh), ν = 1, 2, 3, are due to the (firstorder) coupling of | ak with 1 p-1h, 2p-2h, and 3 p-3h excitations, respectively,
and E
(2)
0 is the second-order ground-state energy, as in Eq. (A.1.18).
In zeroth order, the excitation energy is just the difference of the HF orbital
energies, a − k . The HF orbitals, though, reflect the N -electron charge distribution
in the HF ground state. This means, the orbital energy a does not account for the
instance that there is an electron vacancy in orbital k; nor does k account for the
229
The transition moments derive from the RPA eigenvector components in an analogous manner to Eq. (13.6),
T
R P A
m
=
r,s
X
∗
rs,m d rs ( ¯
n r n s + n r ¯
n s ), m ∈ {+}
(15.30)
where d rs are the one-particle matrix elements of a transition operator ˆ
D. This expression may be written more compactly as
T
R P A
m
= X
†
m d
(15.31)
where
d =
d ph
d hp
=
d ph
d
∗
ph
(15.32)
denotes the (column) vector of p-h and h- p matrix elements d rs .
Perturbation Theoretical Analysis of the RPA
As already mentioned, the RPA energies and transition moments are consistent with
those obtained at the TDA (or CIS and ADC(1)) level, since the 1 p-1h sub-block
A of the RPA secular matrix is identical with the TDA secular matrix. However,
the RPA treatment goes beyond the TDA, the differences beginning at second order,
as a result of the off-diagonal blocks B and B
∗ , which couple the (physical) 1 p-1h
excitations and the (unphysical) 1h-1 p de-excitations. How can this curious coupling
be understood and what is the quality of the results thereby obtained? As a way to
better understand the situation, we shall analyze, like in Sect. 8.3, the RPA results
using perturbation theory, which, moreover, will allow us to discuss the relevant
physics of the excitation process.
Let us consider a single excitation a ← k from the occupied orbital k to the
virtual (unoccupied) orbital a, represented by the zeroth-order (HF) state | ak =
c
†
a c k | 0 . The formal PT expansion of the excitation energy, E ak = E ak − E 0 ,
through second order can be written as
E ak = a − k − V ak[ak]
+ U
(2)
ak (1 p-1h) + U
(2)
ak (2 p-2h) + U
(2)
ak (3 p-3h) − E
(2)
0 + O(3)
(15.33)
Here, the three second-order terms, U
(2)
ak (ν p-νh), ν = 1, 2, 3, are due to the (firstorder) coupling of | ak with 1 p-1h, 2p-2h, and 3 p-3h excitations, respectively,
and E
(2)
0 is the second-order ground-state energy, as in Eq. (A.1.18).
In zeroth order, the excitation energy is just the difference of the HF orbital
energies, a − k . The HF orbitals, though, reflect the N -electron charge distribution
in the HF ground state. This means, the orbital energy a does not account for the
instance that there is an electron vacancy in orbital k; nor does k account for the