300
Appendix
A.6 Proofs of Order Relations
A.6.1 Diagrammatic Perturbation Theory for Ground-State
CI and CC Amplitudes
To explain the order relations (17.8),
t I ∼ O([I ] − 1), [I ] > 1
(A.6.1)
for the CC amplitudes we have to take a closer look at the diagrammatic PT for
the ground state. In Sect. 4.4, we have considered the Gell-Mann and Low expression (4.42) for the ground state,
| 0 = lim
→0
ˆ
U (0, −∞)| 0
0 | ˆ
U (0, −∞)| 0
(A.6.2)
and discussed how the time integrations in the time-evolution operator ,
ˆ
U (0, −∞) =
∞
n=0
(−i)
n
n!
0
−∞
dt 1 e
−|t 1 |
. . .
0
−∞
dt n e
−|t n | ˆ
T T T
ˆ
H I (t 1 ) . . . ˆ
H I (t n )
(A.6.3)
could explicitly be performed resulting in the closed-form expression (4.70). Then,
the connection to the familiar RSPT series (see Sect. A.1) could be established by
inserting these results both in the numerator and the denominator in Eq. (A.6.2)
and expanding their ratio. Alternatively, one could have proceeded along the lines
of Chaps. 5–7 to devise a diagrammatic PT formulation for the ground-state or the
ground-state energy. Let us consider in particular the diagrammatic PT expansions
for individual ground-state amplitudes (CI coefficients),
x I = = 0 | ˆ
C
†
I | 0 = lim
→0
0 | ˆ
C
†
I
ˆ
U (0, −∞)| 0
0 | ˆ
U (0, −∞)| 0
(A.6.4)
in the ground-state CI expansion (17.10),
| 0 = | 0 +
I
x I ˆ
C I | 0
(A.6.5)
The respective excitation operator, ˆ
C
†
I , I ≡ ab . . . kl, can be supplied with the time
argument t = 0, that is, ˆ
C
†
I (0) = c
†
l (0)c
†
k (0) . . . c b (0)c a (0) (see Eq. 3.50). Now ˆ
C
†
I (0)
can be incorporated in the time-ordered products of the time-evolution operator (4.37)
according to