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arXiv:2005.12338 · The Journal of Chemical Physics

Range-separated stochastic resolution of identity: Formulation and application to second-order Green’s function theory

Wenjie Dou, Ming Chen, Tyler Y. Takeshita, Roi Baer, Daniel Neuhauser, Eran Rabani

Published Aug 21, 2020• 14 citations
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Abstract

We develop a range-separated stochastic resolution of identity (RS-SRI) approach for the four-index electron repulsion integrals, where the larger terms (above a predefined threshold) are treated using a deterministic RI and the remaining terms are treated using a SRI. The approach is implemented within a second-order Green’s function formalism with an improved O(N3) scaling with the size of the basis set, N. Moreover, the RS approach greatly reduces the statistical error compared to the full stochastic version [T. Y. Takeshita et al., J. Chem. Phys. 151, 044114 (2019)], resulting in computational speedups of ground and excited state energies of nearly two orders of magnitude, as demonstrated for hydrogen dimer chains and water clusters.

DOI
10.1063/5.0015177

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We develop a range-separated stochastic resolution of identity (RS-SRI) approach for the four-index electron repulsion integrals, where the larger terms (above a predefined threshold) are treated using a deterministic RI and the remaining terms are treated using a SRI. The approach is implemented within a second-order Green’s function formalism with an improved O(N3) scaling with the size of the basis set, N. Moreover, the RS approach greatly reduces the statistical error compared to the full stochastic version [T. Y. Takeshita et al., J. Chem. Phys. 151, 044114 (2019)], resulting in computational speedups of ground and excited state energies of nearly two orders of magnitude, as demonstrated for hydrogen dimer chains and water clusters.
We develop a stochastic resolution of identity approach to the real-time second-order Green’s function (real-time sRI-GF2) theory, extending our recent work for imaginary-time Matsubara Green’s function [Takeshita et al. J. Chem. Phys. 2019, 151, 044114 ]. The approach provides a framework to obtain the quasi-particle spectra across a wide range of frequencies and predicts ionization potentials and electron affinities. To assess the accuracy of the real-time sRI-GF2, we study a series of molecules and compare our results to experiments as well as to a many-body perturbation approach based on the GW approximation, where we find that the real-time sRI-GF2 is as accurate as self-consistent GW. The stochastic formulation reduces the formal computatinal scaling from O ( N e 5 ) down to O ( N e 3 ) where N e is the number of electrons. This is illustrated for a chain of hydrogen dimers, where we observe a slightly lower than cubic scaling for systems containing up to N e ≈ 1000 electrons.
We develop a stochastic resolution of identity representation to the second-order Matsubara Green’s function (sRI-GF2) theory. Using a stochastic resolution of the Coulomb integrals, the second order Born self-energy in GF2 is decoupled and reduced to matrix products/contractions, which reduces the computational cost from O(N5) to O(N3) (with N being the number of atomic orbitals). The current approach can be viewed as an extension to our previous work on stochastic resolution of identity second order Møller-Plesset perturbation theory [T. Y. Takeshita et al., J. Chem. Theory Comput. 13, 4605 (2017)] and offers an alternative to previous stochastic GF2 formulations [D. Neuhauser et al., J. Chem. Theory Comput. 13, 5396 (2017)]. We show that sRI-GF2 recovers the deterministic GF2 results for small systems, is computationally faster than deterministic GF2 for N > 80, and is a practical approach to describe weak correlations in systems with 103 electrons and more.
A stochastic orbital approach to the resolution of identity (RI) approximation for 4-index 2-electron electron repulsion integrals (ERIs) is presented. The stochastic RI-ERIs are then applied to Møller-Plesset perturbation theory (MP2) utilizing a \textit{multiple stochastic orbital approach}. The introduction of multiple stochastic orbitals results in an N3N^3N3 scaling for both the stochastic RI-ERIs and stochastic RI-MP2. We demonstrate that this method exhibits a small prefactor and an observed scaling of N2.4N^{2.4}N2.4 for a range of water clusters, already outperforming MP2 for clusters with as few as 21 water molecules.
We present a real-time second-order Green’s function (GF) method for computing excited states in molecules and nanostructures, with a computational scaling of O ( N e 3 ), where N e is the number of electrons. The cubic scaling is achieved by adopting the stochastic resolution of the identity to decouple the 4-index electron repulsion integrals. To improve the time propagation and the spectral resolution, we adopt the dynamic mode decomposition technique and assess the accuracy and efficiency of the combined approach for a chain of hydrogen dimer molecules of different lengths. We find that the stochastic implementation accurately reproduces the deterministic results for the electronic dynamics and excitation energies. Furthermore, we provide a detailed analysis of the statistical errors, bias, and long-time extrapolation. Overall, the approach offers an efficient route to investigate excited states in extended systems with open or closed boundary conditions.
Stochastic orbital techniques offer reduced computational scaling and memory requirements to describe ground and excited states at the cost of introducing controlled statistical errors. Such techniques often rely on two basic operations, stochastic trace estimation and stochastic resolution of identity, both of which lead to statistical errors that scale with the number of stochastic realizations ( N ξ ) as N ξ − 1 . Reducing the statistical errors without significantly increasing N ξ has been challenging and is central to the development of efficient and accurate stochastic algorithms. In this work, we build upon recent progress made to improve stochastic trace estimation based on the ubiquitous Hutchinson’s algorithm and propose a two-step approach for the stochastic resolution of identity, in the spirit of the Hutch++ method. Our approach is based on employing a randomized low-rank approximation followed by a residual calculation, resulting in statistical errors that scale much better than N ξ − 1 . We implement the approach within the second-order Born approximation for the self-energy in the computation of neutral excitations and discuss three different low-rank approximations for the two-body Coulomb integrals. Tests on a series of hydrogen dimer chains with varying lengths demonstrate that the Hutch++-like approximations are computationally more efficient than both deterministic and purely stochastic (Hutchinson) approaches for low error thresholds and intermediate system sizes. Notably, for arbitrarily large systems, the Hutchinson-like approximation outperforms both deterministic and Hutch++-like methods.
We explore the use of the stochastic resolution-of-the-identity (sRI) with the phaseless auxiliary-field quantum Monte Carlo (ph-AFQMC) method. sRI is combined with four existing local energy evaluation strategies in ph-AFQMC, namely, (1) the half-rotated electron repulsion integral tensor (HR), (2) Cholesky decomposition (CD), (3) tensor hypercontraction (THC), or (4) low-rank factorization (LR). We demonstrate that HR–sRI achieves no scaling reduction, CD–sRI scales as O(N3), and THC–sRI and LR–sRI scale as O(N2), albeit with a potentially large prefactor. Furthermore, the walker-specific extra memory requirement in CD is reduced from O(N3) to O(N2) with sRI, while sRI-based THC and LR algorithms lead to a reduction from O(N2) extra memory to O(N). Based on numerical results for one-dimensional hydrogen chains and water clusters, we demonstrated that, along with the use of a variance reduction technique, CD–sRI achieves cubic-scaling without overhead. In particular, we find that for the systems studied, the observed scaling of standard CD is O(N3–4), while for CD–sRI, it is reduced to O(N2–3). Once a memory bottleneck is reached, we expect THC–sRI and LR–sRI to be preferred methods due to their quadratic-scaling memory requirements and their quadratic-scaling of the local energy evaluation (with a potentially large prefactor). The theoretical framework developed here should facilitate large-scale ph-AFQMC applications that were previously difficult or impossible to carry out with standard computational resources.