11.1 Correlated Excited States and Excitation Class Orthogonalization
163
Now, we may turn to the 2h-1 p excitations. As the first step, the CE states
c
†
a c k c l | 0 have to be orthogonalized to the intermediate states | ˜
i of class 1. Applying the Gram–Schmidt procedure,
|
#
akl = c
†
a c k c l | 0 −
i
| ˜
i ˜
i |c
†
a c k c l | 0
(11.8)
we obtain “precursor” states being orthogonal with respect to the intermediate states
of class 1 but not yet orthonormal. Thus, in a second step, we apply symmetric
orthonormalization to the precursor states. To that end, we introduce the overlap
matrix
S 2 ≡
#
akl |
#
a k l
(11.9)
of the precursor states of class 2, and the final 2h-1 p intermediate states are obtained
according to
| ˜
akl =
a ,k ,l
|
#
a k l (S
−1/2
2
) a k l ,akl
(11.10)
The extension to the higher classes, μ = 3, 4, . . . , is obvious, and we may formulate the general excitation class orthogonalization (ECO) procedure for intermediate states as follows:
(1) Assume that the intermediate states | ˜
K of the classes 1, . . . , ν − 1 have been
constructed. Then, orthogonalize the CE states |
0
J of class ν with respect to
the intermediate states of class 1, . . . , ν − 1 according to
|
#
J = |
0
J −
[K ]<ν
| ˜
K ˜
K |
0
J , [J ] = ν
(11.11)
(2) The “precursor states” |
#
J of class ν may then be orthonormalized symmetrically, yielding
| ˜
J =
[I ]=ν
|
#
I (S
−1/2
ν
) I J
(11.12)
where S ν is the overlap matrix of the precursor states of class ν,
(S ν ) I J = =
#
I |
#
J , [I ] = [J ] = ν
(11.13)
It should be noted that the N -electron ground state | 0 underlying the ECO-IS
construction needs not be normalized to unity. A convenient choice is intermediate normalization, 0 | 0 = 1 (see Appendix A.1), which will be supposed in the
following.
The ECO construction allows one to establish a PT expansion of the intermediate
states,
| ˜
I = | ˜
(0)
I + | ˜
(1)
I + | ˜
(2)
I + . . .
(11.14)