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arXiv:1909.00307 · The Journal of Chemical Physics

Stochastic many-body perturbation theory for electron correlation energies

Zhendong Li

Published Dec 27, 2019• 12 citations
arXivHTMLPDF

Abstract

Treating electron correlation more accurately and efficiently is at the heart of the development of electronic structure methods. In the present work, we explore the use of stochastic approaches to evaluate high-order electron correlation energies, whose conventional computational scaling is unpleasantly steep, being O(Nn+3) with respect to the system size N and the perturbation order n for the Møller-Plesset (MP) series. To this end, starting from Goldstone’s time-dependent formulation of ab initio many-body perturbation theory (MBPT), we present a reformulation of MBPT, which naturally leads to a Monte Carlo scheme with O(nN2 + n2N + f(n)) scaling at each step, where f(n) is a function of n depending on the specific numerical scheme. Proof-of-concept calculations demonstrate that the proposed quantum Monte Carlo algorithm successfully extends the previous Monte Carlo approaches for MP2 and MP3 to higher orders by overcoming the factorial scaling problem. For the first time, Goldstone’s time-dependent formulation is made useful numerically for electron correlation energies, not only being purely as a theoretical tool.

DOI
10.1063/1.5128719

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The Journal of Chemical Physics•Apr 7, 2021•8 citations
Treating electron correlation more accurately and efficiently is at the heart of the development of electronic structure methods. In the present work, we explore the use of stochastic approaches to evaluate high-order electron correlation energies, whose conventional computational scaling is unpleasantly steep, being O(Nn+3) with respect to the system size N and the perturbation order n for the Møller-Plesset (MP) series. To this end, starting from Goldstone’s time-dependent formulation of ab initio many-body perturbation theory (MBPT), we present a reformulation of MBPT, which naturally leads to a Monte Carlo scheme with O(nN2 + n2N + f(n)) scaling at each step, where f(n) is a function of n depending on the specific numerical scheme. Proof-of-concept calculations demonstrate that the proposed quantum Monte Carlo algorithm successfully extends the previous Monte Carlo approaches for MP2 and MP3 to higher orders by overcoming the factorial scaling problem. For the first time, Goldstone’s time-dependent formulation is made useful numerically for electron correlation energies, not only being purely as a theoretical tool.
A new, alternative set of interpretation rules of Feynman-Goldstone diagrams for many-body perturbation theory is proposed, which translates diagrams into algebraic expressions suitable for direct Monte Carlo integrations. A vertex of a diagram is associated with a Coulomb interaction (rather than a two-electron integral) and an edge with the trace of a Green's function in real space and imaginary time. With these, 12 diagrams of third-order many-body perturbation (MP3) theory are converted into 20-dimensional integrals, which are then evaluated by a Monte Carlo method. It uses redundant walkers for convergence acceleration and a weight function for importance sampling in conjunction with the Metropolis algorithm. The resulting Monte Carlo MP3 method has low-rank polynomial size dependence of the operation cost, a negligible memory cost, and a naturally parallel computational kernel, while reproducing the correct correlation energies of small molecules within a few mEh after 10(6) Monte Carlo steps.
We present a low-scaling diagrammatic Monte Carlo approach to molecular correlation energies. Using combinatorial graph theory to encode many-body Hugenholtz diagrams, we sample the M\o{}ller-Plesset (MPn\mathrm{MP}nMPn) perturbation series, obtaining accurate correlation energies up to n=5n=5n=5, with quadratic scaling in the number of basis functions. Our technique reduces the computational complexity of the molecular many-fermion correlation problem, opening up the possibility of low-scaling, accurate stochastic computations for a wide class of many-body systems described by Hugenholtz diagrams.
High-order perturbative ab initio\textit{ab initio}ab initio calculations are challenging due to the rapidly growing configuration space and the difficulty of assessing convergence. In this letter, we introduce perturbation theory quantum Monte Carlo (PTQMC), a stochastic approach designed to compute high-order many-body perturbative corrections. By representing the perturbative wave function with random walkers in configuration space, PTQMC avoids the exponential scaling inherent to conventional constructions of high-rank excitation operators. Benchmark calculations for the Richardson pairing model demonstrate that PTQMC accurately reproduces exact many-body perturbation theory (MBPT) coefficients up to 16th order, even in strongly divergent regimes. We further show that combining PTQMC with series resummation techniques yields stable and precise energy estimates in cases where the straightforward perturbative series fails. Finally, we propose the effective number of configurations, eSe^{S}eS, as a global measure of perturbative wave-function complexity that can be directly extracted within PTQMC. We demonstrate that the saturation behavior of eSe^{S}eS provides a more reliable indicator of the validity of perturbative expansions than energy convergence alone.
This thesis is devoted to the development of numerical methods for correlated electrons in realistic systems based on perturbation theories using Feynman diagrams. Electron correlations in realistic systems give rise to a wide range of interesting physical properties. Diagrammatic expansions and its low-order approximations are key building blocks of theoretical and numerical methods for studying correlated electrons. Recent developments of the diagrammatic quantum Monte Carlo (DiagMC) methods enable efficient stochastic evaluations of diagrams and does not suffer from numerical sign problem which worsens with increasing system size. It is therefore a promising direction to apply DiagMC techniques to realistic electron systems in the pursuit of ab initio calculations of correlated materials. This thesis starts by reviewing basic concepts and definitions of realistic electron systems, the perturbation theory based on Feynman diagrams, Monte Carlo evaluations of the series, as well as self-consistent diagrammatic methods. Low-order diagrammatic methods are applied to realistic calculations and benchmarked against large collections of state-of-the-art many-body methods. The thesis then proposes a diagrammatic Monte Carlo solver for realistic molecules and impurities based on a recursive evaluation of diagrams, which is tested on realistic molecules and impurities with varying temperature, system size, and interaction strength. We observe that this method is ideal for problems with many orbitals with moderate correlations, but may suffer from series divergence at strong correlations. Next, we propose the inchworm Monte Carlo algorithm for interaction expansion, which overcomes limitations of divergent diagrammatic series by replacing the standard perturbation series with a sequence of incremental series expansions. A prototype implementation is presented for system with on-site Hubbard interactions. Finally, we discuss technical aspects of numerical representations of the Green's function, which has significant impact on the efficiency of diagrammatic methods. We review several common numerical representations, and introduce our development of the sparse sampling method, which provides compact representations of the Green's function in both imaginary time and frequency as well as fast transformation formula between the two.
Post-Hartree-Fock methods are wavefunction-based quantum chemistry approaches that explicitly account for the electron correlation beyond the mean-field approximation, allowing accurate determination of electron orbitals and energies in a given system. However, their computational cost scales as the fourth power or higher with respect to the number of atoms, severely limiting their practical applications. This paper proposes an efficient GPU implementation of the Møller-Plesset method, a post-Hartree-Fock approach based on perturbation theory. We focus on the computation of second- and thirdorder perturbation energies (MP2 and MP3) for closed-shell systems, which incorporate electron correlation on top of the Hartree-Fock energy. To accelerate the extensive contraction operations arising in the Møller-Plesset method, our implementation employs a formulation that maximizes the use of highly optimized matrix multiplication routines. Furthermore, we effectively utilize computational capabilities of GPU architecture through a hierarchical reduction using warp-level primitives and shared memory, as well as contiguous coalesced access to matrix elements. Performance evaluation on an NVIDIA A100 GPU shows up to 255.5 and 71.1 times speedups for the MP2 correlation energy computation, compared with the widely used quantum chemistry software package PySCF and its GPU plugin GPU4PySCF, respectively. In the MP3 calculations, our DGEMM-based implementation achieves a maximum speedup of 96.2 times over PySCF.
A scalable, stochastic algorithm evaluating the fourth-order many-body perturbation (MP4) correction to energy is proposed. Three hundred Goldstone diagrams representing the MP4 correction are computer generated and then converted into algebraic formulas expressed in terms of Green’s functions in real space and imaginary time. They are evaluated by the direct (i.e., non-Markov, non-Metropolis) Monte Carlo (MC) integration accelerated by the redundant-walker and control-variate algorithms. The resulting MC-MP4 method is efficiently parallelized and is shown to display O(n5.3) size-dependence of cost, which is nearly two ranks lower than the O(n7) dependence of the deterministic MP4 algorithm. It evaluates the MP4/aug-cc-pVDZ energy for benzene, naphthalene, phenanthrene, and corannulene with the statistical uncertainty of 10 mEh (1.1% of the total basis-set correlation energy), 38 mEh (2.6%), 110 mEh (5.5%), and 280 mEh (9.0%), respectively, after about 109 MC steps.