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arXiv:2203.12666 · Physical review. B./Physical review. B

Diagrammatic Monte Carlo for electronic correlation in molecules: High-order many-body perturbation theory with low scaling

Giacomo Bighin, Quoc P. Ho, M. Lemeshko, Timur V. Tscherbul

Published Jul 11, 2023• 1 citation
arXivHTMLPDF

Abstract

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 ($\mathrm{MP}n$) perturbation series, obtaining accurate correlation energies up to $n=5$, 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.

DOI
10.1103/physrevb.108.045115

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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.
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.
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We show that Monte Carlo sampling of the Feynman diagrammatic series (DiagMC) can be used for tackling hard fermionic quantum many-body problems in the thermodynamic limit by presenting accurate results for the repulsive Hubbard model in the correlated Fermi liquid regime. Sampling Feynman's diagrammatic series for the single-particle self-energy, we can study moderate values of the on-site repulsion ( U / t ∼4) and temperatures down to T / t =1/40. We compare our results with high-temperature series expansions and with single-site and cluster dynamical mean-field theory.
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.
Convergent hierarchies of theories for calculating many-body vibrational ground and excited-state wave functions, such as M{o}ller-Plesset perturbation theory or coupled cluster theory, tend to rely on matrix-algebraic manipulations of large, high-dimensional arrays of anharmonic force constants, tasks which require large amounts of computer storage space and which are very difficult to implement in a parallel-scalable fashion. On the other hand, existing quantum Monte Carlo (QMC) methods for vibrational wave functions tend to lack robust techniques for obtaining excited-state energies, especially for large systems. By exploiting analytical identities for matrix elements of position operators in a harmonic oscillator basis, we have developed stochastic implementations of the size-extensive vibrational self-consistent field (MC-XVSCF) and size-extensive vibrational M{o}ller-Plesset second-order perturbation (MC-XVMP2) theories which do not require storing the potential energy surface (PES). The programmable equations of MC-XVSCF and MC-XVMP2 take the form of a small number of high-dimensional integrals evaluated using Metropolis Monte Carlo techniques. The associated integrands require independent evaluations of only the value, not the derivatives, of the PES at many points, a task which is trivial to parallelize. However, unlike existing vibrational QMC methods, MC-XVSCF and MC-XVMP2 can calculate anharmonic frequencies directly, rather than as a small difference between two noisy total energies, and do not require user-selected coordinates or nodal surfaces. MC-XVSCF and MC-XVMP2 can also directly sample the PES in a given approximation without analytical or grid-based approximations, enabling us to quantify the errors induced by such approximations.