On the cluster structure of linear-chain fermionic wave functions

Using the model of cyclic polyenes \(\hbox {C}_N\hbox {H}_N\) with a nondegenerate ground state, \(N = 4 \nu + 2 \; (\nu = 1, 2, \ldots )\), as a prototype of extended linear metallic-like systems we explore the cluster structure of the relevant wave functions. Based on the existing configuration interaction and coupled cluster (CC) results, as obtained with the Hubbard and Pariser–Parr–Pople Hamiltonians in the entire range of the coupling constant extending from the uncorrelated Huckel limit to the fully correlated limit, we recall the breakdown of the CCD or CCSD methods as the size of the system increases and the strongly correlated regime is approached. We introduce the concept of the indecomposable quadruply-excited clusters which arise for \(\nu > 1\) and represent those connected quadruples that do not possess any corresponding disconnected cluster component. It is shown via explicit enumeration that the ratio of the number of these indecomposables relative to that of the decomposables depends linearly on the size of the polyene \(N\), so that the limit of the ratio of the number of indecomposables relative to the total number of quadruples approaches unity as \(N \rightarrow \infty \). We then briefly outline the implications of these results for the applicability of CC approaches to extended systems and provide a qualitative argument for an even more extreme behavior of hexa-excited, octa-excited, etc., clusters as \(N \rightarrow \infty \).