Clustered Low-Rank Tensor Format: Introduction and Application to Fast Construction of Hartree-Fock Exchange

TL;DR

This paper introduces the Clustered Low-Rank (CLR) tensor format, which efficiently reduces storage and computation in Hartree-Fock exchange calculations by controlling precision with two parameters, outperforming traditional methods.

Contribution

The paper presents a novel CLR tensor framework that eliminates ad-hoc heuristics, enabling efficient and accurate tensor computations in quantum chemistry.

Findings

CLR reduces storage and computational complexity below $\\mathcal{O}(N^3)$ and $\\mathcal{O}(N^4)$.

CLR-based density fitting HF is more efficient than standard DF and non-DF HF for small systems.

Negligible impact on molecular energies and properties.

Abstract

Clustered Low Rank (CLR) framework for block-sparse and block-low-rank tensor representation and computation is described. The CLR framework depends on 2 parameters that control precision: one controlling the CLR block rank truncation and another that controls screening of small contributions in arithmetic operations on CLR tensors. As these parameters approach zero CLR representation and arithmetic become exact. There are no other ad-hoc heuristics, such as domains. Use of the CLR format for the order-2 and order-3 tensors that appear in the context of density fitting (DF) evaluation of the Hartree-Fock (exact) exchange significantly reduced the storage and computational complexities below their standard $\mathcal{O}(N^3)$ and $\mathcal{O}(N^4)$ figures. Even for relatively small systems and realistic basis sets CLR-based DF HF becomes more efficient than the standard DF approach, and significantly more efficient than the conventional non-DF HF, while negligibly affecting molecular energies and properties.