A simple and operational test for external connectivity of tensors in many-body methods

Abstract

Size extensivity of the correlation energy is a crucial property of many-body methods (e.g., coupled cluster, Rayleigh–Schrödinger perturbation theory). It is related to the connectivity of the energy expression which itself relies on the connectivity of the amplitude tensors. Before the introduction of the concept of “generalized extensivity” by Nooijen et al. (Mol Phys 103:2277, 2005) there was no operational test to check for extensivity. The latter probes for connectivity of the energy by testing for separability using an artificial manipulation of the two-electron integrals according to varying partitionings of the orbitals. The present work tests for external connectivity of the wave operator and does not require specially crafted two-electron integrals but access to the values of the individual amplitudes and Coulomb integrals. External connectivity refers to the R-scaling of tensor entries with respect to the distance between the centers of the localized orbital indices. In contrast to “generalized extensivity” the present test does not detect disconnected closed diagrams within a tensor. Since standard coupled cluster tensors are always connected a diagrammatic analysis focuses on tensors related to various configuration interaction methods. In this respect the present work is complementary to that of Lyakh and Bartlett (Mol Phys 112:213, 2014) using algebraic techniques. The theoretical results are confirmed by the proposed operational external connectivity check. The proposed ansatz gives some insight into the property of connectivity especially the meaning of cluster lines and may be easily applied to other, particularly multi-reference coupled cluster, methods to check numerically for proper external connectivity of the amplitudes.