15.1 Derivation and Properties of the RPA Equations
231
The corresponding PT expansion of the RPA excitation energy is given by
E
R P A
ak
= a − k − V ak[ak] + U
(2)
ak (1 p-1h) + U
(2)
ak (1h-1 p) + O(3)
(15.36)
As expected, the RPA result reproduces the exact excitation energy through first order
and, moreover, correctly accounts for the CI singles contribution, U
(2)
ak (1 p-1h). As
a specific RPA feature, the other second-order term, U
(2)
ak (1h-1 p), results from the
coupling of the considered excitation a ← k with de-excitation configurations of
block A
∗ via the coupling matrix elements of B. What is its physical content? The
explicit expression reads
U
(2)
ak (1h-1 p) = −
j,b
|V ab[k j] |
2
a + b − k − j
(15.37)
This result can be obtained, for example, using ordinary matrix perturbation theory
for the eigenvalue problem of M
= SM. Alternatively, one can resort to the ADC
formulation of the RPA presented in Sect. 15.2 and identify U
(2)
ak (1h-1 p) with the
diagonal element M
R P A (2)
ak,ak
of the RPA-ADC secular matrix in second order (see
Eqs. 15.61 and 14.39). The comparison with Eq. (15.35) shows that U
(2)
ak (1h-1 p) is
identical with the third term in the latter equation. This means that the RPA treatment
recovers one of the three correlation contributions to the excitation energy in second
order, however, the one lowering the excitation energy, whereas, as we just have
argued, the full second-order correlation increases the excitation energy.
Obviously, the gravest deficiency is the absence of the U
(2)
ak (2 p-2h) contribution,
indicating that in the RPA model the 2 p-2h (and higher) excitations are excluded
from the outset. Accordingly, the physically important effects of relaxation and polarization are beyond the RPA description. Therefore, the RPA cannot be seen as a
satisfactory approach to the treatment of excitation energies.
In a similar way, one may analyze the RPA transition moments, where the signature of the 1h-1 p de-excitations can already be seen at first order. The exact transition
moment for the a ← k excitation can be expanded as
T ak = = ak | ˆ
D| 0 = = ak | ˆ
D| 0 + +
(1)
ak | ˆ
D| 0 + + ak | ˆ
D|
(1)
0 + O(2)
(15.38)
There are two first-order terms, related to the first-order excited state and the firstorder ground state, respectively. They are matched by corresponding terms in the
RPA expansion, reading
T
R P A
ak
= d ak +
b, j
X
(1)
bj,ak d bj +
j,b
X
(1)
jb,ak d jb + O(2)
(15.39)
where X
(1)
rs,ak are the components of the first-order RPA eigenvector, X
(1)
ak . In the
second term, the sum runs over the 1 p-1h components of X
(1)
ak , reflecting the (firstorder) CI singles mixing in |
(1)
ak . The third term, to be identified with ak | ˆ
D|
(1)
0 ,
231
The corresponding PT expansion of the RPA excitation energy is given by
E
R P A
ak
= a − k − V ak[ak] + U
(2)
ak (1 p-1h) + U
(2)
ak (1h-1 p) + O(3)
(15.36)
As expected, the RPA result reproduces the exact excitation energy through first order
and, moreover, correctly accounts for the CI singles contribution, U
(2)
ak (1 p-1h). As
a specific RPA feature, the other second-order term, U
(2)
ak (1h-1 p), results from the
coupling of the considered excitation a ← k with de-excitation configurations of
block A
∗ via the coupling matrix elements of B. What is its physical content? The
explicit expression reads
U
(2)
ak (1h-1 p) = −
j,b
|V ab[k j] |
2
a + b − k − j
(15.37)
This result can be obtained, for example, using ordinary matrix perturbation theory
for the eigenvalue problem of M
= SM. Alternatively, one can resort to the ADC
formulation of the RPA presented in Sect. 15.2 and identify U
(2)
ak (1h-1 p) with the
diagonal element M
R P A (2)
ak,ak
of the RPA-ADC secular matrix in second order (see
Eqs. 15.61 and 14.39). The comparison with Eq. (15.35) shows that U
(2)
ak (1h-1 p) is
identical with the third term in the latter equation. This means that the RPA treatment
recovers one of the three correlation contributions to the excitation energy in second
order, however, the one lowering the excitation energy, whereas, as we just have
argued, the full second-order correlation increases the excitation energy.
Obviously, the gravest deficiency is the absence of the U
(2)
ak (2 p-2h) contribution,
indicating that in the RPA model the 2 p-2h (and higher) excitations are excluded
from the outset. Accordingly, the physically important effects of relaxation and polarization are beyond the RPA description. Therefore, the RPA cannot be seen as a
satisfactory approach to the treatment of excitation energies.
In a similar way, one may analyze the RPA transition moments, where the signature of the 1h-1 p de-excitations can already be seen at first order. The exact transition
moment for the a ← k excitation can be expanded as
T ak = = ak | ˆ
D| 0 = = ak | ˆ
D| 0 + +
(1)
ak | ˆ
D| 0 + + ak | ˆ
D|
(1)
0 + O(2)
(15.38)
There are two first-order terms, related to the first-order excited state and the firstorder ground state, respectively. They are matched by corresponding terms in the
RPA expansion, reading
T
R P A
ak
= d ak +
b, j
X
(1)
bj,ak d bj +
j,b
X
(1)
jb,ak d jb + O(2)
(15.39)
where X
(1)
rs,ak are the components of the first-order RPA eigenvector, X
(1)
ak . In the
second term, the sum runs over the 1 p-1h components of X
(1)
ak , reflecting the (firstorder) CI singles mixing in |
(1)
ak . The third term, to be identified with ak | ˆ
D|
(1)
0 ,