Exploring Hilbert space on a budget: Novel benchmark set and performance metric for testing electronic structure methods in the regime of strong correlation

Abstract

This work explores the ability of classical electronic structure methods to efficiently represent (compress) the information content of full configuration interaction (FCI) wave functions. We introduce a benchmark set of four hydrogen model systems of different dimensionality and distinctive electronic structures: a 1D chain, a 1D ring, a 2D triangular lattice, and a 3D close-packed pyramid. To assess the ability of a computational method to produce accurate and compact wave functions, we introduce the accuracy volume, a metric that measures the number of variational parameters necessary to achieve a target energy error. Using this metric and the hydrogen models, we examine the performance of three classical deterministic methods: i) selected configuration interaction (sCI) realized both via an a posteriori and variational selection of the most important determinants, ii) rank-reduced FCI, obtained by an a posteriori singular value decomposition of the FCI tensor (SVD-FCI), and iii) the matrix product state representation obtained via the density matrix renormalization group (DMRG). We find that DMRG generally gives the most efficient wave function representation for all systems, particularly in the 1D chain with a localized basis. For the 2D and 3D systems, all methods perform best with a delocalized basis, and the efficiency of sCI is closer to that of DMRG, with the former having and accuracy volume approximately twice as large in the strong correlation regime. Compared to sCI, the SVD-FCI scheme is generally found to require a slightly larger number of parameters to achieve the same energy accuracy.