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

Yueyang Zhang, Wei Wu, Peifeng Su

TL;DR

This work tackles the high computational cost of GW calculations for large molecules by marrying a block tensor decomposition (BTD) with an imaginary-time, Laplace-transform framework and real-space screening. The resulting BTD-GW algorithm achieves formal $O(N^3)$ scaling, approaching $O(N^2)$ in practice thanks to sparsity, while a BTD-RPA path demonstrates near-quadratic efficiency. Accuracy benchmarks (GW100, S66x8) show reasonable quasiparticle energies and reliable noncovalent energies, though starting-point dependence remains for ev$GW$ in some cases. The approach enables ev$GW$ for systems with over 3000 basis functions and points toward scalable BSE@$GW$ workflows, with implications for large-scale excited-state calculations in chemistry.

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.

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

TL;DR

This work tackles the high computational cost of GW calculations for large molecules by marrying a block tensor decomposition (BTD) with an imaginary-time, Laplace-transform framework and real-space screening. The resulting BTD-GW algorithm achieves formal scaling, approaching in practice thanks to sparsity, while a BTD-RPA path demonstrates near-quadratic efficiency. Accuracy benchmarks (GW100, S66x8) show reasonable quasiparticle energies and reliable noncovalent energies, though starting-point dependence remains for ev in some cases. The approach enables ev for systems with over 3000 basis functions and points toward scalable BSE@ workflows, with implications for large-scale excited-state calculations in chemistry.

Abstract

Within the framework of many-body perturbation theory based on Green's functions, the 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 -scaling algorithm. By integrating BTD with an imaginary-time formalism and introducing a real space screening strategy for the polarizability, we achieve an observed scaling of approximately 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 scaling, and eigenvalue-self-consistent calculations become feasible for systems with over 3000 basis functions. This work establishes BTD as an efficient and scalable approach for large-scale calculations in molecular systems.
Paper Structure (9 sections, 19 equations, 6 figures, 1 table)

This paper contains 9 sections, 19 equations, 6 figures, 1 table.

Table of Contents

  1. Introduction
  2. Methodology
  3. Imaginary-Time $GW$ Formalism with Laplace Transformation
  4. Block tensor decomposition
  5. Formal $O(N^3)$ scaling BTD-$GW$ Algorithm
  6. Results and discussions
  7. Accuracy Assessment: $GW$100 and S66 Benchmarks
  8. Performance Analysis: Scaling and Efficiency
  9. Conclusion

Figures (6)

  • Figure 1: Workflow of the BTD-ev$GW$ algorithm
  • Figure 2: The procedure of BTD-RPA and BTD-$GW$
  • Figure 3: Deviations of $G_0W_0$ and ev$GW$ HOMO energies from experimental ionization potentials for the $GW$100 set
  • Figure 4: (a) Training process on S66 test, (b) Error distribution of BTD-RPA compared to RI-RPA on S66$\times$8 with the BLYP reference
  • Figure 5: (a,b) The time costs of glycine chains and water clusters by RI-RPA and BTD-RPA (time of each step is shown); (c,d) the error per atom of BTD compared to RI-RPA.
  • ...and 1 more figures