Block Tensor Decomposition: A dual grid scheme with formal O(N3) for THC decomposition of molecular systems

TL;DR

This paper introduces a block tensor decomposition algorithm with a dual grid scheme that achieves strict O(N^3) scaling for tensor hyper-contraction (THC) kernel construction, significantly improving efficiency in electronic structure calculations.

Contribution

The authors develop a novel BTD algorithm combining Hilbert sort and Cholesky decomposition to generate compact grids, enabling efficient THC/ISDF kernel construction with optimized parameters.

Findings

Achieves strict O(N^3) scaling for THC kernel construction.

Enables quadratic scaling in MP2 electron correlation calculations.

Provides a robust framework for efficient electronic structure computations.

Abstract

Accurate and fast treatment of electron-electron interactions remains a central challenge in electronic structure theory because post-Hartree-Fock methods often suffered from the computational cost for 4-index electron repulsion integrals (ERIs). Low-rank approaches such as tensor hyper-contraction (THC) and interpolative separable density fitting (ISDF) have been proposed for Hartree-Fock exchange and correlation's calculations. Their application to molecular systems remains inefficient due to the construction of THC kernel whose time scale increases as quartic with the number of basis functions. In this work, we present an algorithm named block tensor decomposition (BTD) based on a dual grid scheme that combines Hilbert sort and pivoted Cholesky decomposition to generate compact interpolative grids, allowing strict $O(N^3)$ scaling for THC/ISDF kernel construction. The key parameters in BTD are optimized via differential evolution, balancing efficiency and accuracy. Furthermore, we apply BTD in scaled opposite-spin MP2 (SOS-MP2), leveraging sparse mapping in real space to achieve quadratic scaling for electron correlation calculation and linear scaling for exchange calculation. This work advances low-scaling THC/ISDF methodologies for molecular systems, offering a robust framework for efficient and accurate electronic structure computations.