164
11 Intermediate-State Representation (ISR)
deriving completely from the PT expansion of the N -electron ground state | 0 .
Obviously, in zeroth order the intermediate states coincide with the HF states:
| ˜
(0)
I = | I
(11.15)
In this sense, the intermediate states “mediate” between the HF states and the exact
(N −1)-electron energy eigenstates.
Rather than expanding the intermediate states, it is more advantageous to deal
directly with the PT expansions of the matrix elements of the intermediate-state
representations (ISR). The ECO intermediate states | ˜
I , forming a complete basis
of the (N −1)-particle states, establish the ISR secular matrix M,
M I J = = ˜
I | ˆ
H − E 0 | ˜
J
(11.16)
representing the (shifted) hamiltonian ˆ
H − E 0 in terms of the intermediate states.
The exact eigenstates |
N−1
n can be expanded according to
|
N−1
n =
I
X I n | ˜
I
(11.17)
where X I n are the components of the nth eigenvector of M. In matrix notation, the
ISR secular equations take on the form
M X = X, X
† X = 1
(11.18)
where denotes the diagonal matrix of eigenvalues ω n , and X is the matrix of eigenvectors. According to Eq. (11.16), the eigenvalues can be identified as the ionization
energies,
ω n = E
N−1
n
− E 0
(11.19)
Moreover, introducing the matrix ˜ f of ISR transition amplitudes,
˜
f I p = = ˜
I |c p | 0
(11.20)
the spectroscopic factors (10.5) can be obtained according to
x
(n)
p =
I
X
∗
I n
˜
f I p
(11.21)
from the respective eigenvectors.
The PT expansions (11.14) of the intermediate states, based on the PT expansion
of the N -electron ground state | 0 , translate into PT expansions of the ISR secular
matrix elements,
M I J = M
(0)
I J + M
(1)
I J + M
(2)
I J + . . .
(11.22)