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

Yueyang Zhang, Xuewei Xiong, Wei Wu, Peifeng Su

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.

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

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 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.

Paper Structure

This paper contains 11 sections, 23 equations, 8 figures, 1 table.

Table of Contents

  1. Introduction
  2. Methodology
  3. Introduction of THC-RI
  4. Block tensor decomposition
  5. Grids generation scheme based on Hilbert curve and pivoted Cholesky decomposition
  6. Implementation of exchange and SOS-MP2
  7. Results and discussions
  8. BTD kernel generation
  9. Accuracy and efficiency for HF exchange
  10. Accuracy and efficiency for SOS-MP2
  11. Conclusion

Figures (8)

  • Figure 1: Conceptual diagram of BTD's approximation procedures.
  • Figure 2: 2D Hilbert curve in order from 1 to 4.
  • Figure 3: The algorithm of pruning redundant grids by diagonal-blocked pivoted Cholesky decomposition in an iterative way.
  • Figure 4: The algorithm of calculation for THC kernel. The time scale of each step is shown on the right hand and the asymptotic time scale is shown by right arrow.
  • Figure 5: Training process with 20 samples one cycle. Calculation begins with random guesses and runs 45 steps.
  • ...and 3 more figures