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arXiv:2605.28750

Formal O(N3)-Scaling Second-Order Perturbation Theory by Block Tensor Decomposition: Implementation on MP2 and rPT2

Yueyang Zhang, Wei Wu, Peifeng Su

Published May 27, 2026• 0 citations
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Abstract

Block tensor decomposition (BTD) and canonical polyadic decomposition (CPD) are combined into a unified $O(N^3)$-scaling framework for second-order perturbation theory (PT2), demonstrated on MP2 and renormalized PT2 (rPT2). BTD constructs the tensor hyper-contraction kernel at $O(N^3)$ via a dual-grid scheme; CPD factorizes the exchange channel through a block-based two-stage ALS. An asymmetric half-kernel design applies bare Coulomb to one vertex and coupling-constant-averaged screening to the other, capturing the SOSEX component of rPT2 without a frequency-dependent CPD. For MP2, BTD-CPD reproduces canonical RI-MP2 to 0.058~kcal/mol per heavy atom. For rPT2@PBE0 on the S66x8 benchmark, the mean absolute error is 0.36~kcal/mol (ME $-$0.19, RMSE 0.46) over 528 data points. The CPD-compressed intermediates yield $O(N^2)$ storage alongside $O(N^3)$ scaling.

DOI
10.48550/arxiv.2605.28750

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We investigate the efficient combination of the canonical polyadic decomposition (CPD) and tensor hyper-contraction (THC) approaches. We first present a novel low-cost CPD solver that leverages a precomputed THC factorization of an order-4 tensor to efficiently optimize the order-4 CPD with O ( N R 2 ) scaling. With the matrix-free THC-based optimization strategy in hand, we can efficiently generate CPD factorizations of the order-4 two-electron integral tensors and develop novel electronic structure methods that take advantage of both the THC and CPD approximations. Next, we investigate the application of a combined CPD and THC approximation of the Laplace transform (LT) second-order Mo̷ller-Plesset (MP2) method. We exploit the ability to switch efficiently between the THC and CPD factorizations of the two-electron integrals to reduce the computational complexity of the LT MP2 method while preserving the accuracy of the approach. Furthermore, we take advantage of the robust fitting approximation to eliminate the leading-order error in the CPD approximated tensor networks. Finally, we show that modest values of THC and CPD rank preserve the accuracy of the LT MP2 method and that this CPD + THC LT MP2 strategy realizes a performance advantage over canonical LT MP2 in both computational wall-times and memory resource requirements.
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We investigate the efficient combination of the canonical polyadic decomposition (CPD) and tensor hyper-contraction (THC) approaches. We first present a novel low-cost CPD solver which leverages a precomputed THC factorization of an order-444 tensor to efficiently optimize the order-444 CPD with O(NR2)\mathcal{O}(NR^2)O(NR2) scaling. With the matrix-free THC-based optimization strategy in hand we can: efficiently generate CPD factorizations of the order-4 two-electron integral tensors; and develop novel electronic structure methods which take advantage of both the THC and CPD approximations. Next, we investigate the application of a combined CPD and THC approximation of the Laplace transform (LT) second-order Møller-Plesset (MP2) method. We exploit the ability to switch efficiently between the THC and CPD factorizations of the two electron integrals to reduce the computational complexity of the LT MP2 method while preserving the accuracy of the approach. Furthermore we take advantage of the robust fitting approximation to eliminate leading order error in the CPD approximated tensor networks. Finally, we show that modest values of THC and CPD rank preserve the accuracy of the LT MP2 method and that this CPD+THC LT MP2 strategy realizes a performance advantage over canonical LT MP2 in both computational wall-times and memory resource requirements.
In a previous publication, we have discussed the usage of tensor decomposition in the canonical polyadic (CP) tensor format for electronic structure methods. There, we focused on two-electron integrals and second order Møller-Plesset perturbation theory (MP2). In this work, we discuss the CP format for Coupled Cluster (CC) theory and present a pilot implementation for the Coupled Cluster Doubles method. We discuss the iterative solution of the CC amplitude equations using tensors in CP representation and present a tensor contraction scheme that minimizes the effort necessary for the rank reductions during the iterations. Furthermore, several details concerning the reduction of complexity of the algorithm, convergence of the CC iterations, truncation errors, and the choice of threshold for chemical accuracy are discussed.
We derive and implement a new way of solving coupled cluster equations with lower computational scaling. Our method is based on the decomposition of both amplitudes and two electron integrals, using a combination of tensor hypercontraction and canonical polyadic decomposition. While the original theory scales as O(N6) with respect to the number of basis functions, we demonstrate numerically that we achieve sub-millihartree difference from the original theory with O(N4) scaling. This is accomplished by solving directly for the factors that decompose the cluster operator. The proposed scheme is quite general and can be easily extended to other many-body methods.