Chapter 11
Intermediate-State Representation (ISR)
The direct ADC procedure for the electron propagator part G
−
(ω) considered in
the preceding chapter allows one to construct successively an in principle exact hermitian secular matrix, K + C, where (i) the eigenvalues are the negative ionization
energies −I n ; (ii) the matrix index labels are given by the HF configurations (10.9)
of N −1 electrons; (iii) the secular matrix elements can be expanded in regular PT
expansions. As already anticipated in Chap. 10, these features suggest that K + C
is essentially a representation of the hamiltonian, or, more specifically, of ˆ
H − E 0 ,
deriving from a set of (N −1)-electron “intermediate” basis states. But what actually
are these presumed intermediate states underlying the ADC secular matrix? There
is a surprisingly simple solution to this issue, establishing an alternative closedform version of the ADC secular equations, completely independent of the original
diagrammatic derivation [1, 2]. Being a wave-function approach, the new formulation overcomes certain limitations inherent to the propagator concept. Of the three
sections of this chapter, Sect. 11.1 presents the general procedure for constructing
the intermediate states and the consequent intermediate-state representation (ISR);
Sect. 11.2 demonstrates the explicit derivation of the second-order ISR equations;
and, finally, Sect. 11.3 discusses how the ISR concept can be applied to general
operators.
11.1 Correlated Excited States and Excitation Class
Orthogonalization
The starting point for the construction of the desired intermediate states are the
so-called correlated excited (CE) states (here of N −1 electrons),
|
0
J = ˆ
C J | 0
(11.1)
© Springer Nature Switzerland AG 2018
J. Schirmer, Many-Body Methods for Atoms, Molecules and Clusters, Lecture
Notes in Chemistry 94, https://doi.org/10.1007/978-3-319-93602-4_11
161
Intermediate-State Representation (ISR)
The direct ADC procedure for the electron propagator part G
−
(ω) considered in
the preceding chapter allows one to construct successively an in principle exact hermitian secular matrix, K + C, where (i) the eigenvalues are the negative ionization
energies −I n ; (ii) the matrix index labels are given by the HF configurations (10.9)
of N −1 electrons; (iii) the secular matrix elements can be expanded in regular PT
expansions. As already anticipated in Chap. 10, these features suggest that K + C
is essentially a representation of the hamiltonian, or, more specifically, of ˆ
H − E 0 ,
deriving from a set of (N −1)-electron “intermediate” basis states. But what actually
are these presumed intermediate states underlying the ADC secular matrix? There
is a surprisingly simple solution to this issue, establishing an alternative closedform version of the ADC secular equations, completely independent of the original
diagrammatic derivation [1, 2]. Being a wave-function approach, the new formulation overcomes certain limitations inherent to the propagator concept. Of the three
sections of this chapter, Sect. 11.1 presents the general procedure for constructing
the intermediate states and the consequent intermediate-state representation (ISR);
Sect. 11.2 demonstrates the explicit derivation of the second-order ISR equations;
and, finally, Sect. 11.3 discusses how the ISR concept can be applied to general
operators.
11.1 Correlated Excited States and Excitation Class
Orthogonalization
The starting point for the construction of the desired intermediate states are the
so-called correlated excited (CE) states (here of N −1 electrons),
|
0
J = ˆ
C J | 0
(11.1)
© Springer Nature Switzerland AG 2018
J. Schirmer, Many-Body Methods for Atoms, Molecules and Clusters, Lecture
Notes in Chemistry 94, https://doi.org/10.1007/978-3-319-93602-4_11
161