Tensor Hypercontraction Error Correction Using Regression

Ishna Satyarth; Eric C. Larson; Devin A. Matthews

physics.chem-phFeb 27, 2026

Tensor Hypercontraction Error Correction Using Regression

Authors: Ishna Satyarth, Eric C. Larson, Devin A. Matthews

Organizations: Department of Computer Science, Southern Methodist University, Dallas, TX 75275, USA · Department of Chemistry, Southern Methodist University, Dallas, TX 75275, USA

Abstract

Wavefunction-based quantum methods are some of the most accurate tools for predicting and analyzing the electronic structure of molecules, in particular for accounting for dynamical electron correlation. However, most methods of including dynamical correlation beyond the simple second-order Møller-Plesset perturbation theory (MP2) level are too computationally expensive to apply to large molecules. Approximations which reduce scaling with system size are a potential remedy, such as the tensor hyper-contraction (THC) technique of Hohenstein et al., but also result in additional sources of error. In this work, we correct errors in THC-approximated methods using machine learning. Specifically, we apply THC to third-order Møller-Plesset theory (MP3) as a simplified model for coupled cluster with single and double excitations (CCSD), and train several regression models on observed THC errors from the Main Group Chemistry Database (MGCDB84). We compare performance of multiple linear regression models and non-linear Kernel Ridge regression models. We also investigate correlation procedures using absolute and relative corrections and evaluate the corrections for both molecule and reaction energies. We discuss the potential for using regression techniques to correct THC-MP3 errors by comparing it to the "canonical" MP3 reference values and find the optimum technique based on accuracy. We find that non-linear regression models reduced root mean squared errors between THC- and canonical MP3 by a factor of 6-9 $\times$ for total molecular energies and 2-3 $\times$ for reaction energies.

Figures & tables

Figure 1: Goldstone diagrammatic depiction of the 10 components of the MP3 energy. Solid horizontal lines denote first-order amplitudes while dashed lines denote two-electron integrals. Note that

E_{9}

and

E_{10}

are each the sum of two Hermitian conjugate diagrams.

$\delta$	Model	Molecule		$\Delta$ Molecule
		%IMP	RMSE	%IMP	RMSE
1	MP3b	—	$0.1337$	—	$0.1337$
	SCS	61%	$0.0517$	61%	$0.0517$
	MLR	78%	$0.0292$	78%	$0.0292$
	KRR	85%	$0.0206$	89%	0.0153
1.25	MP3b	—	$0.0956$	—	$0.0956$

Table 1: Comparison of root mean squared error (RMSE) and improvement (%IMP) results for SCS, MLR and KRR models on the Molecule and

\Delta

Molecule data sets (

\mathrm{k}\mathrm{c}\mathrm{a}\mathrm{l}\,\mathrm{m}\mathrm{o}\mathrm{l}^{-1}\,\mathrm{e}^{-1}

). Boldface numbers indicate the lowest RMSE for each value of

\delta

Panel of Figure 2 — Figure 2: RMSE %IMP for KRR-Molecule and KRR- $\Delta$ Molecule model with $\delta=1$ as a function of hyper-parameters $\alpha$ (regularization strength) and $\gamma$ (inverse length scale).

Figure 3: Comparative analysis of absolute energy RMSE for all models. Whiskers denote one standard deviation across the ten cross-validation sets.

$\delta$	Model	%IMP	RMSE	%IMP	MAE	%IMP	MAPE
1	MP3b	—	$1.5711$	—	0.9374	—	27.51%
	R	38%	$0.9835$	29%	$0.6625$	34%	18.14%
	$\Delta$ R	64%	0.5732	59%	0.3885	70%	8.31%
1.25	MP3b	—	$0.8869$	—	0.4853	—	11.33%
	R	55%	$0.4050$	45%	$0.2675$	27%	8.31%
	$\Delta$ R	65%	0.3078	56%	0.2153	53%	5.36%

Table 2: Comparison of RMSE (

\mathrm{k}\mathrm{c}\mathrm{a}\mathrm{l}\,\mathrm{m}\mathrm{o}\mathrm{l}^{-1}

), MAE (

\mathrm{k}\mathrm{c}\mathrm{a}\mathrm{l}\,\mathrm{m}\mathrm{o}\mathrm{l}^{-1}

), MAPE (%) and their fractional improvements (%IMP) for KRR-Reaction and KRR-

\Delta

Reaction models. Here, R represents KRR-Reaction and

\Delta

R represents KRR-

\Delta

Reaction . Boldface values denote the best model for each value of

\delta

, if evident.

Computational Quantum Chemistry

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Quantum Chemistry Computational Quantum Chemistry Neural Quantum States

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Density functional theory (DFT) is the most widely used method for calculating molecular properties; however, its accuracy is often insufficient for quantitative predictions. Coupled-cluster (CC) theory is the most successful method for achieving accuracy beyond DFT and for predicting properties that closely align with experiment. It is known as the ''gold standard'' of quantum chemistry. Unfortunately, the high computational cost of CC limits its widespread applicability. In this work, we present the Molecular Orbital Learning (MōLe) architecture, an equivariant machine learning model that directly predicts CC's core mathematical objects, the excitation amplitudes, from the mean-field Hartree-Fock molecular orbitals as inputs. We test various aspects of our model and demonstrate its remarkable data efficiency and out-of-distribution generalization to larger molecules and off-equilibrium geometries, despite being trained only on small equilibrium geometries. Finally, we also examine its ability to reduce the number of cycles required to converge CC calculations. MōLe can set the foundations for high-accuracy wavefunction-based ML architectures to accelerate molecular design and complement force-field approaches.

Abstract

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