Sparsity of the Electron Repulsion Integral Tensor Using Different Localized Virtual Orbital Representations in Local Second Order Møller-Plesset Theory

Utilizing localized orbitals, local correlation theory can reduce the unphysically high system-size scaling of post-Hartree-Fock (post-HF) methods all the way to linear scaling in insulating molecules. Revealing the numerical sparsity of the 4-index electron repulsion (ERI) tensor is central to achieving linear scaling in local correlation methods. For second order Møller-Plesset theory (MP2), one of the simplest post-HF methods, only the $(ia|jb)$ ERIs are needed, coupling occupied orbitals $i,j$, and virtuals $a,b$. In this paper, we compare the numerical sparsity (called the "ragged list representation") and the low-rank sparsity of the $(ia|jb)$ ERI tensor. The ragged list requires only one set of (localized) virtual orbitals, and we find that the orthogonal valence virtual-hard virtual (VV-HV) set of virtuals originally proposed by Subotnik et al. gives the sparsest ERI tensor relative to 4 other good choices: projected atomic orbitals (PAOs), Boys localized virtuals, Pipek-Mezey localized virtuals, and Cholesky localized virtuals. To further compress the $(ia|jb)$ ERI tensor, the low-rank (pair-natural orbital) representation uses different sets of virtual orbitals for different occupied orbital pairs. Our results indicate that while the low-rank representation achieves significant rank reduction, it also requires more memory than the ragged list representation. An approximation (called the "fixed sparsity pattern") for solving the local MP2 equations using the numerically sparse ERI tensor is proposed and tested to be sufficiently accurate and to have highly controllable error. A low-scaling local MP2 algorithm based on the ragged list and the fixed sparsity pattern using this representation is therefore promising.