Chapter 12
Order Relations and Separability
The equivalence of the direct ADC approach and the ECO-ISR formulation rests on
two common key features. The first is the so-called canonical PT order structure of the
secular matrix [1], establishing the compactness of the nth-order ISR-ADC approximation schemes. The other is the separability [2] of the secular matrix with respect to
two (or more) non-interacting sub-systems, which warrants size-consistency. While
in the ADC context these features could have been substantiated using diagrammatic
arguments, the ECO-ISR concept established in the preceding chapter allows for a
stringent formulation and rigorous proofs. This is the topic of the present chapter.
Section 12.1 (together with Appendix A.6) discusses the order structure of the secular matrix and the characteristic truncation errors thus entailed. The separability
property will be treated in Sect. 12.2. Finally, in Sect. 12.3, we take a comparative
look at the standard CI method, where the secular structure is simple but neither
canonical nor separable.
12.1 Canonical Order Relations
In the ISR-ADC approach, the secular matrix elements are given in the form
of PT expansions (see Eqs. 10.15, 11.22). Here, the remarkable finding is that
these expansions do not necessarily begin at zeroth order. Rather, the lowest nonvanishing order in a matrix element M I J depends on the “distance” [I ] − [J ] of the
excitation classes, [I ], [J ], to which the configurations I and J belong. The rules
fulfilled by the ISR-ADC secular matrix elements are as follows:
M I J ∼ O(|[I ] − [J ]|)
(12.1)
© 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_12
177
Order Relations and Separability
The equivalence of the direct ADC approach and the ECO-ISR formulation rests on
two common key features. The first is the so-called canonical PT order structure of the
secular matrix [1], establishing the compactness of the nth-order ISR-ADC approximation schemes. The other is the separability [2] of the secular matrix with respect to
two (or more) non-interacting sub-systems, which warrants size-consistency. While
in the ADC context these features could have been substantiated using diagrammatic
arguments, the ECO-ISR concept established in the preceding chapter allows for a
stringent formulation and rigorous proofs. This is the topic of the present chapter.
Section 12.1 (together with Appendix A.6) discusses the order structure of the secular matrix and the characteristic truncation errors thus entailed. The separability
property will be treated in Sect. 12.2. Finally, in Sect. 12.3, we take a comparative
look at the standard CI method, where the secular structure is simple but neither
canonical nor separable.
12.1 Canonical Order Relations
In the ISR-ADC approach, the secular matrix elements are given in the form
of PT expansions (see Eqs. 10.15, 11.22). Here, the remarkable finding is that
these expansions do not necessarily begin at zeroth order. Rather, the lowest nonvanishing order in a matrix element M I J depends on the “distance” [I ] − [J ] of the
excitation classes, [I ], [J ], to which the configurations I and J belong. The rules
fulfilled by the ISR-ADC secular matrix elements are as follows:
M I J ∼ O(|[I ] − [J ]|)
(12.1)
© 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_12
177