162
11 Intermediate-State Representation (ISR)
Here, ˆ
C J denote physical excitation (ionization) operators of the manifold
{ ˆ
C J } =
c k ; c
†
a c k c l , k < l; c
†
a c
†
b c j c k c l , a < b, j < k < l; . . .
(11.2)
where the index notations conform to those used in Eq. (10.9). As in Sect. 10.2, the
successive 1h, 2h-1 p, 3h-2 p, . . . excitation classes are numbered μ = 1, 2, 3, . . . .
To specify the class of a given configuration, J , we will use the symbol [J ], that is,
[J ] = μ if J designates a μh-(μ − 1) p configuration. The operators ˆ
C J are called
physical, since, when acting on the HF ground state, they create the HF configurations
| J = ˆ
C J | 0
(11.3)
forming the CI expansion manifold for the (N −1)-electron system.
In contrast to the HF configurations, the CE states are not orthonormal:
S I J = =
0
I |
0
J = = 0 | ˆ
C
†
I
ˆ
C J | 0 = δ I,J
(11.4)
Here, S I J denote the matrix elements of the CE-state overlap matrix S, which also
can be seen as a generalized density matrix. The CE states form a complete set
of (N −1)-electron states [3, 4]. This suggests to generate proper basis states by
applying a suitable orthonormalization procedure to the CE states. However, the
most obvious choice, namely symmetric orthonormalization, according to
| J =
I
|
0
I (S
−1/2
) I J
(11.5)
has to be discarded because the resulting | J states lead to a secular problem
that is neither compact nor size-consistent, as explained in Sects. 12.1 and 12.2. By
contrast, the ADC features are recovered by Gram–Schmidt orthogonalization with
respect to successively higher CE-state excitation classes μ, augmented by symmetric
orthonormalization within each class.
For illustration, we shall construct the intermediate states in the two lowest excitation classes, μ = 1 and 2. In case of the 1h states, being the lowest class, only
symmetric orthonormalization is needed, and the intermediate states are obtained
from the CE states, |
0
k = c k | 0 , by symmetric orthonormalization,
| ˜
k =
i
c i | 0 (S
−1/2
1
) ik
(11.6)
where
(S 1 ) i j = = 0 |c
†
i c j | 0
(11.7)
defines the overlap matrix S 1 of the CE states of class 1. Note that S 1 is the transpose
of the h-h block of the one-particle density matrix as defined in Eq. (3.26).
11 Intermediate-State Representation (ISR)
Here, ˆ
C J denote physical excitation (ionization) operators of the manifold
{ ˆ
C J } =
c k ; c
†
a c k c l , k < l; c
†
a c
†
b c j c k c l , a < b, j < k < l; . . .
(11.2)
where the index notations conform to those used in Eq. (10.9). As in Sect. 10.2, the
successive 1h, 2h-1 p, 3h-2 p, . . . excitation classes are numbered μ = 1, 2, 3, . . . .
To specify the class of a given configuration, J , we will use the symbol [J ], that is,
[J ] = μ if J designates a μh-(μ − 1) p configuration. The operators ˆ
C J are called
physical, since, when acting on the HF ground state, they create the HF configurations
| J = ˆ
C J | 0
(11.3)
forming the CI expansion manifold for the (N −1)-electron system.
In contrast to the HF configurations, the CE states are not orthonormal:
S I J = =
0
I |
0
J = = 0 | ˆ
C
†
I
ˆ
C J | 0 = δ I,J
(11.4)
Here, S I J denote the matrix elements of the CE-state overlap matrix S, which also
can be seen as a generalized density matrix. The CE states form a complete set
of (N −1)-electron states [3, 4]. This suggests to generate proper basis states by
applying a suitable orthonormalization procedure to the CE states. However, the
most obvious choice, namely symmetric orthonormalization, according to
| J =
I
|
0
I (S
−1/2
) I J
(11.5)
has to be discarded because the resulting | J states lead to a secular problem
that is neither compact nor size-consistent, as explained in Sects. 12.1 and 12.2. By
contrast, the ADC features are recovered by Gram–Schmidt orthogonalization with
respect to successively higher CE-state excitation classes μ, augmented by symmetric
orthonormalization within each class.
For illustration, we shall construct the intermediate states in the two lowest excitation classes, μ = 1 and 2. In case of the 1h states, being the lowest class, only
symmetric orthonormalization is needed, and the intermediate states are obtained
from the CE states, |
0
k = c k | 0 , by symmetric orthonormalization,
| ˜
k =
i
c i | 0 (S
−1/2
1
) ik
(11.6)
where
(S 1 ) i j = = 0 |c
†
i c j | 0
(11.7)
defines the overlap matrix S 1 of the CE states of class 1. Note that S 1 is the transpose
of the h-h block of the one-particle density matrix as defined in Eq. (3.26).