252
16 Algebraic Propagator Methods
the ground state in a consistent way. Of special interest is the RPA scheme, which
we have extensively reviewed in Chap. 15. In fact, the RPA secular equations (15.5),
(15.6) result from Eqs. (16.46), (16.47) upon restricting the EOM operator manifold
to the 1 p-1h and 1h-1 p configurations and replacing | 0 with the HF ground state
| 0 (see Exercise 16.4). The derivation as an EOM approximation supplies the
RPA solutions with state representations according to Eqs. (16.54), (16.55); a brief
inspection of these RPA states should be of interest.
As in Sect. 15.1, we consider the RPA solution for a single excitation, a ← k,
According to Eq. (16.54), the related state representation derives from
|
R P A
ak ∼
b,l
X bl,ak c
†
b c l + X lb,ak c
†
l c b
| 0
(16.60)
where X rs,ak denote the components of the RPA eigenvector, the zeroth-order contribution being X
(0)
rs,ak = δ ra δ sk . This expression combines an approximate excitation
operator, ˆ
R P A
ak , with the exact ground state, | 0 , and one may ask whether there is
more consistent approximation for the latter. To address this question, it is useful to
inspect the KC relations (16.55) for the RPA excitation,
0 ∼
b,l
X
∗
bl,ak c
†
l c b + X
∗
lb,ak c
†
b c l
| 0
(16.61)
Using first-order PT expansions, | 0 = | 0 + |
(1)
0 + . . . , and X rs,ak = δ ra δ sk +
X
(1)
rs,ak + . . . , the KC expression can be expanded through first order,
c
†
k c a |
(1)
0 +
b,l
X
(1)∗
lb,ak c
†
b c l | 0 + O(2) = 0 + O(2)
(16.62)
In zeroth order, the KC relation is trivially fulfilled, and also the two first-order
terms cancel each other, as can be seen by comparing X
(1)
lb,ak (see Eq. 15.40) and the
2 p-2h coefficients in |
(1)
0 . Obviously, the RPA excitations satisfy the KC relations
consistently through first order, which also shows that the first-order ground state,
| 0 + |
(1)
0 , is consistent with the RPA level of theory. One may try to go beyond
the first-order ground state and devise an RPA ground state, |
R P A
0
, being consistent
at second and higher order, but since the RPA errors are of first and second order
in the states and excitation energies, respectively, the relevance of such constructs is
questionable.
For the RPA state (16.60), the PT expansion through first order reads
|
R P A
ak = c
†
a c k | 0 +
b,l
X
(1)
bl,ak c
†
b c l | 0 + c
†
a c k |
(1)
0 + O(2)
(16.63)
There are two first-order terms, of which the former simply accounts for the mixing
with other 1 p-1h excitations. The second term correctly describes the (first-order)
16 Algebraic Propagator Methods
the ground state in a consistent way. Of special interest is the RPA scheme, which
we have extensively reviewed in Chap. 15. In fact, the RPA secular equations (15.5),
(15.6) result from Eqs. (16.46), (16.47) upon restricting the EOM operator manifold
to the 1 p-1h and 1h-1 p configurations and replacing | 0 with the HF ground state
| 0 (see Exercise 16.4). The derivation as an EOM approximation supplies the
RPA solutions with state representations according to Eqs. (16.54), (16.55); a brief
inspection of these RPA states should be of interest.
As in Sect. 15.1, we consider the RPA solution for a single excitation, a ← k,
According to Eq. (16.54), the related state representation derives from
|
R P A
ak ∼
b,l
X bl,ak c
†
b c l + X lb,ak c
†
l c b
| 0
(16.60)
where X rs,ak denote the components of the RPA eigenvector, the zeroth-order contribution being X
(0)
rs,ak = δ ra δ sk . This expression combines an approximate excitation
operator, ˆ
R P A
ak , with the exact ground state, | 0 , and one may ask whether there is
more consistent approximation for the latter. To address this question, it is useful to
inspect the KC relations (16.55) for the RPA excitation,
0 ∼
b,l
X
∗
bl,ak c
†
l c b + X
∗
lb,ak c
†
b c l
| 0
(16.61)
Using first-order PT expansions, | 0 = | 0 + |
(1)
0 + . . . , and X rs,ak = δ ra δ sk +
X
(1)
rs,ak + . . . , the KC expression can be expanded through first order,
c
†
k c a |
(1)
0 +
b,l
X
(1)∗
lb,ak c
†
b c l | 0 + O(2) = 0 + O(2)
(16.62)
In zeroth order, the KC relation is trivially fulfilled, and also the two first-order
terms cancel each other, as can be seen by comparing X
(1)
lb,ak (see Eq. 15.40) and the
2 p-2h coefficients in |
(1)
0 . Obviously, the RPA excitations satisfy the KC relations
consistently through first order, which also shows that the first-order ground state,
| 0 + |
(1)
0 , is consistent with the RPA level of theory. One may try to go beyond
the first-order ground state and devise an RPA ground state, |
R P A
0
, being consistent
at second and higher order, but since the RPA errors are of first and second order
in the states and excitation energies, respectively, the relevance of such constructs is
questionable.
For the RPA state (16.60), the PT expansion through first order reads
|
R P A
ak = c
†
a c k | 0 +
b,l
X
(1)
bl,ak c
†
b c l | 0 + c
†
a c k |
(1)
0 + O(2)
(16.63)
There are two first-order terms, of which the former simply accounts for the mixing
with other 1 p-1h excitations. The second term correctly describes the (first-order)