12.3 A Look at the CI Method
187
12.3 A Look at the CI Method
The benefit of the ISR-ADC approach presented in the preceding chapters has to
be seen in comparison to conventional quantum-chemical methods, most notably
the standard configuration–interaction (CI) treatment. Therefore, the performance of
the CI method with regard to truncation errors and size-consistency shall briefly be
inspected in the following.
In the CI treatment, here of an (N −1)-electron system, the Schrödinger equation
ˆ
H |
N−1
n
= E
N−1
n |
N−1
n
(12.45)
is recast into an algebraic eigenvalue problem by expanding the ionic states
according to
|
N−1
n
=
J
X J n | J
(12.46)
in terms of the CI states
| J
= ˆ
C J | 0
(12.47)
as introduced by Eqs. (11.2), (11.3). In matrix notation, the algebraic eigenvalue
equations read
H X = X, X
† X = 1
(12.48)
Here, H denotes the CI secular matrix,
H I J = = I | ˆ
H | J
(12.49)
is the diagonal matrix of ionic-state energies E
N−1
n , and X is the matrix of (column)
eigenvectors X n . To form ionization energies, I n = E
N−1
n
− E 0 , one has to obtain E 0
from a separate CI computation for the N -electron system.
Structure of the CI Matrix
Approximate CI treatments are obtained by limited CI expansions as opposed to full
(FCI) expansions. In the following, we will be concerned with systematic truncations
of the CI expansions, that is, expansions being complete through a given excitation
class μ. These systematic truncation schemes can be examined with respect to the
PT order of the induced error in the CI results. For this purpose, one has to inspect
the structure of the CI secular matrix H.
Being a fully variational method, the CI secular matrix H has a rather basic “order
structure,” as shown in Fig. 12.5. Owing to the ˆ
H 0 part of the hamiltonian, the diagonal
matrix elements are of zeroth order, as indicated by the zeros in the diagonal blocks.
Each excitation class μ is coupled through terms linear in the two-electron Coulomb
integrals (first order) to the excitation classes μ ± 1 and μ ± 2, which is indicated
by the entries 1 in the respective matrix blocks. There is no coupling between states
differing by more than two excitation classes (entry “-”).
187
12.3 A Look at the CI Method
The benefit of the ISR-ADC approach presented in the preceding chapters has to
be seen in comparison to conventional quantum-chemical methods, most notably
the standard configuration–interaction (CI) treatment. Therefore, the performance of
the CI method with regard to truncation errors and size-consistency shall briefly be
inspected in the following.
In the CI treatment, here of an (N −1)-electron system, the Schrödinger equation
ˆ
H |
N−1
n
= E
N−1
n |
N−1
n
(12.45)
is recast into an algebraic eigenvalue problem by expanding the ionic states
according to
|
N−1
n
=
J
X J n | J
(12.46)
in terms of the CI states
| J
= ˆ
C J | 0
(12.47)
as introduced by Eqs. (11.2), (11.3). In matrix notation, the algebraic eigenvalue
equations read
H X = X, X
† X = 1
(12.48)
Here, H denotes the CI secular matrix,
H I J = = I | ˆ
H | J
(12.49)
is the diagonal matrix of ionic-state energies E
N−1
n , and X is the matrix of (column)
eigenvectors X n . To form ionization energies, I n = E
N−1
n
− E 0 , one has to obtain E 0
from a separate CI computation for the N -electron system.
Structure of the CI Matrix
Approximate CI treatments are obtained by limited CI expansions as opposed to full
(FCI) expansions. In the following, we will be concerned with systematic truncations
of the CI expansions, that is, expansions being complete through a given excitation
class μ. These systematic truncation schemes can be examined with respect to the
PT order of the induced error in the CI results. For this purpose, one has to inspect
the structure of the CI secular matrix H.
Being a fully variational method, the CI secular matrix H has a rather basic “order
structure,” as shown in Fig. 12.5. Owing to the ˆ
H 0 part of the hamiltonian, the diagonal
matrix elements are of zeroth order, as indicated by the zeros in the diagonal blocks.
Each excitation class μ is coupled through terms linear in the two-electron Coulomb
integrals (first order) to the excitation classes μ ± 1 and μ ± 2, which is indicated
by the entries 1 in the respective matrix blocks. There is no coupling between states
differing by more than two excitation classes (entry “-”).