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arXiv:2512.21022

Formal O(N3) scaling GW calculations by block tensor decomposition for large molecule systems

Yueyang Zhang, Wei Wu, Peifeng Su

Published Dec 24, 2025• 0 citations
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Abstract

Within the framework of many-body perturbation theory based on Green's functions, the $GW$ approximation has emerged as a pivotal method for computing quasiparticle energies and excitation spectra. However, its high computational cost and steep scaling present significant challenges for applications to large molecular systems. In this work, we extend the block tensor decomposition (BTD) algorithm, recently developed in our previous work [J. Chem. Phys. 163, 174109 (2025)] for low-rank tensor compression, to enable a formally $O(N^3)$-scaling $GW$ algorithm. By integrating BTD with an imaginary-time $GW$ formalism and introducing a real space screening strategy for the polarizability, we achieve an observed scaling of approximately $O(N^2)$ in test systems. Key parameters of the algorithm are optimized on the S66 dataset using the JADE algorithm, ensuring a balanced compromise between accuracy and efficiency. Our BTD-based random phase approximation also exhibits $O(N^2)$ scaling, and eigenvalue-self-consistent $GW$ calculations become feasible for systems with over 3000 basis functions. This work establishes BTD as an efficient and scalable approach for large-scale $GW$ calculations in molecular systems.

DOI
10.48550/arxiv.2512.21022

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arXiv:2301.09107•The Journal of Chemical Physics•Mar 2, 2023•28 citations
Within the framework of many-body perturbation theory based on Green's functions, the GWGWGW approximation has emerged as a pivotal method for computing quasiparticle energies and excitation spectra. However, its high computational cost and steep scaling present significant challenges for applications to large molecular systems. In this work, we extend the block tensor decomposition (BTD) algorithm, recently developed in our previous work [J. Chem. Phys. 163, 174109 (2025)] for low-rank tensor compression, to enable a formally O(N3)O(N^3)O(N3)-scaling GWGWGW algorithm. By integrating BTD with an imaginary-time GWGWGW formalism and introducing a real space screening strategy for the polarizability, we achieve an observed scaling of approximately O(N2)O(N^2)O(N2) in test systems. Key parameters of the algorithm are optimized on the S66 dataset using the JADE algorithm, ensuring a balanced compromise between accuracy and efficiency. Our BTD-based random phase approximation also exhibits O(N2)O(N^2)O(N2) scaling, and eigenvalue-self-consistent GWGWGW calculations become feasible for systems with over 3000 basis functions. This work establishes BTD as an efficient and scalable approach for large-scale GWGWGW calculations in molecular systems.
Within the framework of many-body perturbation theory based on Green's functions, the GW approximation has emerged as a pivotal method for computing quasiparticle energies and excitation spectra. However, its high computational cost and steep scaling present significant challenges for applications to large molecular systems. In this work, we extend the block tensor decomposition (BTD) algorithm, recently developed in our previous work [Zhang et al., J. Chem. Phys. 163, 174109 (2025)] for low-rank tensor compression, to enable a formally O(N3)-scaling GW algorithm. By integrating BTD with an imaginary-time GW formalism and introducing a real space screening strategy for the polarizability, we achieve an observed scaling of approximately O(N2) in test systems. Key parameters of the algorithm are optimized on the S66 dataset using the JADE algorithm, ensuring a balanced compromise between accuracy and efficiency. Our BTD-based random phase approximation also exhibits O(N2) scaling, and eigenvalue-self-consistent GW calculations become feasible for systems with over 3000 basis functions. This work establishes BTD as an efficient and scalable approach for large-scale GW calculations in molecular systems.
Block tensor decomposition (BTD) and canonical polyadic decomposition (CPD) are combined into a unified O(N3)O(N^3)O(N3)-scaling framework for second-order perturbation theory (PT2), demonstrated on MP2 and renormalized PT2 (rPT2). BTD constructs the tensor hyper-contraction kernel at O(N3)O(N^3)O(N3) 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(N2)O(N^2)O(N2) storage alongside O(N3)O(N^3)O(N3) scaling.
The GW formalism, a Green’s function many-body perturbation theory, is growing in popularity for the description of the electronic properties of condensed matter systems in solid-state physics, and more recently chemistry. Unfortunately, its application to complex systems of interest in nanosciences, chemistry, or even biology, is hampered by the large associated computing cost, in particular in the case of disordered systems, or systems immersed in an opened environment (a solvent, a molecular medium, an electrode, etc.) The goal of the present PhD thesis is to focus on the development of multiscale techniques, merging high-level many-body treatments of the subsystem of interest, with a simplified but fully ab initio description of the electrostatic and dielectric environment. Such approaches aim to go beyond classical parametrized models, mainly developed in the quantum chemistry community, which are based on a continuum (“polarizable continuum model”) or discrete (QM/MM) description of the environment.To reach such a goal, we adopt a divide-and-conquer fragmentation scheme for the environment, particularly suited to molecular systems. This leads to a block-diagonal non- interacting electron susceptibility, decreasing the algorithmic complexity from quartic to cubic. To reduce the prefactor associated with the inversion of the Dyson equation for the screened Coulomb potential W, we have further developed a compression algorithm for the susceptibility operator. The automatic computation of an extremely compact polarization basis set allows a large reduction of the size of the susceptibility blocks, associated to the fragments in the environment. Such a method enables us to compute the dielectric response of systems made of several hundred thousand atoms, with an excellent accuracy when it comes to reproduce the effect of the environment as a response to an excitation in the immersed subsystem. This approach is presented through the study of fullerene bulk, surface and subsurface crystals.While the GW formalism is dynamical, with a frequency-dependent screened Coulomb potential W, a first study is done adopting a static approximation (low-frequency limit) for the screening properties of the environment. Such an approach follows the traditional semi-empirical models of a polarizable environment. This PhD thesis assesses the validity of such an approximation, which assumes an instantaneous response (adiabatic limit) of the environment to an electronic excitation, thanks to an explicit comparison with a fully dynamical dielectric response of the environment. The study of a surface of fullerenes, as well as a water molecule inside a metallic carbon nanotube, show that a static description of the environment leads to errors on the polarization energy below 10%, provided that the “folding” of the environment is treated in a proper way.The fragment approach is also applied to covalent insulator crystals, and more particularly to hexagonal boron nitride (h-BN). We explain how to compute the energy levels of point defects in h-BN, in the true dilute limit, and we give the asymptotic scaling laws for the renormalization of these energy levels, from the monolayer to a (n)-layer system. This study highlights thus the possibility to apply the fragment approach to covalent insulator systems, a possibility hinging probably on the short range behavior of the susceptibility in these systems.All of these developments, extending ab initio many-body methods to increasingly complex systems, have been implemented in the massively parallel code beDeft, dedicated to the study of the electronic properties of large scale systems.
We present a many-body GW formalism for quantum subsystems embedded in discrete polarizable environments containing up to several hundred thousand atoms described at a fully ab initio random phase approximation level. Our approach is based on a fragment approximation in the construction of the Green's function and independent-electron susceptibilities. Further, the environing fragments susceptibility matrices are reduced to a minimal but accurate representation preserving low order polarizability tensors through a constrained minimization scheme. This approach dramatically reduces the cost associated with inverting the Dyson equation for the screened Coulomb potential W, while preserving the description of short to long-range screening effects. The efficiency and accuracy of the present scheme is exemplified in the paradigmatic cases of fullerene bulk, surface, subsurface, and slabs with varying number of layers.
Background: The computational resources needed to generate the ab initio solution of the nuclear many-body problem for increasing mass number and/or accuracy necessitates innovative developments to improve upon (i) the storage of many-body operators and (ii) the scaling of many-body methods used to evaluate nuclear observables. The storing and efficient handling of many-body operators with high particle ranks is currently one of the major bottlenecks limiting the applicability range of ab initio studies with respect to mass number and accuracy. Recently, the application of tensor decomposition techniques to many-body tensors has proven highly beneficial to reduce the computational cost of ab initio calculations in quantum chemistry and solid-state physics.Purpose: The impact of applying state-of-the-art tensor factorization techniques to modern nuclear Hamiltonians derived from chiral effective field theory is investigated. Subsequently, the error induced by the tensor decomposition of the input Hamiltonian on ground-state energies of closed-shell nuclei calculated via second-order many-body perturbation theory is benchmarked.Methods: The first proof-of-principles application of tensor-decomposition techniques to the nuclear Hamiltonian is performed. Two different tensor formats are investigated by systematically benchmarking the approximation error on matrix elements stored in various bases of interest. The analysis is achieved while including normal-ordered three-nucleon interactions that are nowadays used as input to the most advanced ab initio calculations in medium-mass nuclei. With the aid of the factorized Hamiltonian, the second-order perturbative correction to ground-state energies is decomposed and the scaling properties of the underlying tensor network are discussed.Results: The employed tensor formats are found to lead to an efficient data compression of two-body matrix elements of the nuclear Hamiltonian. In particular, the sophisticated tensor hypercontraction scheme yields low tensor ranks with respect to both harmonic-oscillator and Hartree-Fock single-particle bases. It is found that the tensor rank depends on the two-body total angular momentum JJJ for which one performs the decomposition, which is itself directly related to the sparsity of the corresponding tensor. Furthermore, including normal-ordered two-body contributions originating from three-body interactions does not compromise the efficient data compression. Ultimately, the use of factorized matrix elements authorizes controlled approximations of the exact second-order ground-state energy corrections. In particular, a small enough error is obtained from low-rank factorizations in 4He,16O^{4}\mathrm{He},^{16}\mathrm{O}4He,16O, and 40Ca^{40}\mathrm{Ca}40Ca.Conclusions: It is presently demonstrated that tensor-decomposition techniques can be efficiently applied to systematically approximate the nuclear many-body Hamiltonian in terms of lower-rank tensors. Beyond the input Hamiltonian, tensor-decomposition techniques can be envisioned to scale down the cost of state-of-the-art nonperturbative many-body methods in order to extend ab initio studies to (i) higher precisions, (ii) larger masses, and (iii) nuclei of doubly open-shell character.
We show how to construct an effective Hamiltonian whose dimension scales linearly with system size, and whose eigenvalues systematically approximate the excitation energies of GW theory. This is achieved by rigorously expanding the self-energy in order to exactly conserve a desired number of frequency-independent moments of the self-energy dynamics. Recasting GW in this way admits a low-scaling O[N4] approach to build and solve this Hamiltonian, with a proposal to reduce this further to O[N3]. This relies on exposing a novel recursive framework for the density response moments of the random phase approximation, where the efficient calculation of its starting point mirrors the low-scaling approaches to compute RPA correlation energies. The frequency integration of GW, which distinguishes so many different GW variants, can be performed without approximation directly in this moment representation. Furthermore, the solution to the Dyson equation can be performed exactly, avoiding analytic continuation, diagonal approximations, or iterative solutions to the quasiparticle equation, with the full-frequency spectrum obtained from the complete solution of this effective static Hamiltonian. We show how this approach converges rapidly with respect to the order of the conserved self-energy moments and is applied across the GW100 benchmark dataset to obtain accurate GW spectra in comparison to traditional implementations. We also show the ability to systematically converge all-electron full-frequency spectra and high-energy features beyond frontier excitations, as well as avoiding discontinuities in the spectrum, which afflict many other GW approaches.