[Paper Review] Using tensor hypercontraction density fitting to achieve an O(L^4) CISD algorithm

Neil Shenvi, Helen van Aggelen+1 more|arXiv (Cornell University)|Sep 13, 2012

Tensor decomposition and applications3 citations

TL;DR

This paper presents an O(L⁴) configuration interaction with single and double excitations (CISD) algorithm using tensor hypercontraction density fitting (THC-DF), reducing the computational scaling from O(L⁶) in standard CISD. By decomposing electron repulsion integrals via low-rank tensor factorizations with O(L) auxiliary functions, the method achieves near-CISD accuracy at dramatically reduced cost while preserving the formal approach to the exact CISD limit.

ABSTRACT

Recently, Hohenstein et al[1] introduced tensor hypercontraction density fitting to decompose the rank-4 electron repulsion integral tensor as the product of five rank-2 tensors. In this paper, we use this methodology to construct an algorithm which calculates the approximate ground state energy in O(L^4) operations. We test our method using several small molecules and show that we quickly approach the CISD limit with a small number of auxiliary functions.

Motivation & Objective

• To overcome the O(L⁶) scaling of traditional CISD methods, which limits their application to small molecules.
• To develop a wavefunction-based method with formal O(L⁴) scaling that maintains high accuracy by leveraging tensor hypercontraction density fitting (THC-DF).
• To demonstrate that the method converges to the exact CISD energy as the number of auxiliary functions increases, preserving the physical behavior of CISD despite its known limitations.
• To enable efficient, accurate electronic structure calculations for systems where traditional correlated methods are computationally prohibitive.

Proposed method

• The method employs tensor hypercontraction density fitting (THC-DF) to decompose the rank-4 electron repulsion integral (ERI) tensor into a product of five rank-2 tensors using O(L) auxiliary functions.
• The ERI tensor is factorized as ε^{ik}_{jl} = Σ_{a,b} x_{ia} x_{ka} Z_{ab} x_{jb} x_{lb}, enabling efficient computation and transformation between bases.
• The CISD energy is computed using a parametrization of the wavefunction in terms of excitation amplitudes, with the ERI tensor expressed via THC-DF to reduce computational cost.
• The energy gradient with respect to the THC tensor Z_{ab} is derived analytically, enabling iterative optimization of the auxiliary basis parameters.
• The method assumes the ERI is already transformed into an orthonormal molecular orbital basis, enabling O(L³) transformation of the ERI tensor via THC-DF.
• Repeated tensor contractions are cached to improve performance, trading memory for computational speed.

Experimental results

Research questions

• RQ1 Can tensor hypercontraction density fitting be used to reduce the scaling of CISD from O(L⁶) to O(L⁴) while preserving accuracy?
• RQ2 How quickly does the THC-DF-based CISD method converge to the exact CISD energy as the number of auxiliary functions increases?
• RQ3 What is the computational cost and accuracy trade-off of this O(L⁴) CISD method compared to standard CISD and other correlated methods?
• RQ4 Can the THC-DF decomposition of the ERI tensor be applied effectively to configuration interaction methods without introducing significant error?

Key findings

• The proposed algorithm achieves O(L⁴) scaling for CISD energy evaluation, a dramatic reduction from the standard O(L⁶) scaling.
• The method converges to the exact CISD energy as the number of auxiliary functions increases, confirming its ability to approach the full CISD limit.
• With only a small number of auxiliary functions (O(L)), the method achieves near-CISD accuracy, demonstrating rapid convergence.
• The use of exact THC-DF decomposition (via diagonalization of the ERI tensor) ensures no approximation error from the ERI factorization during accuracy testing.
• The method enables efficient computation of CISD energies for systems previously inaccessible due to O(L⁶) scaling, particularly for larger molecules.