In recent years there has been a rapid growth in the development and application of new stochastic methods in electronic structure. These methods are quite diverse, from many-body wave function techniques in real space or determinant space to being used to sum perturbative expansions. This growth has been spurred by the more favorable scaling with the number of electrons and often better parallelization over large numbers of central processing unit (CPU) cores or graphical processing units (GPUs) than for high-end non-stochastic wave function based methods. This special issue of the Journal of Chemical Physics includes 33 papers that describe recent developments and applications in this area. As seen from the articles in the issue, stochastic electronic structure methods are applicable to both molecules and solids and can accurately describe systems with strong electron correlation. This issue was motivated, in part, by the 2019 Telluride Science Research Center workshop on Stochastic Electronic Structure Methods that we organized. Below we briefly describe each of the papers in the special issue, dividing the papers into six subtopics.
I. AUXILIARY-FIELD QUANTUM MONTE CARLO
The auxiliary-field quantum Monte Carlo (AFQMC) method is an important tool in the study of correlated quantum many-body systems. While traditionally popular in the study of lattice Hamiltonians in condensed matter physics, the method is receiving increased attention in the study of atoms, molecules, and solids. This special issue provides a collection of recent developments of the method that aim to improve its efficiency and applicability, with particular emphasis on its use in the study of realistic materials.
In the paper by Lee and Reichman,1 the authors use a stochastic resolution-of-the-identity approach in phaseless AFQMC calculations and demonstrate reductions in computational scaling and memory requirements when combined with low-rank factorizations of the two-electron integral tensor. These advances will help us extend the range of applicability of AFQMC to larger systems.
Shi and Zhang provide a comprehensive description of the AFQMC method, with particular emphasis on the formulation specialized for real materials.2 The paper also describes a proposed approach to generate trial wave functions for real materials with self-consistent constraints and a detailed presentation of their efficient implementation of large multi-determinant trial wave functions. This paper provides an excellent pedagogical starting point for anyone interested in learning more about AFQMC approaches for real materials.
In the paper by Lee, Malone, and Reichman,3 the authors report phaseless AFQMC calculations on the ground state energy of benzene, the focus of a recent blind test that included several accurate quantum many-body approaches. The authors showed that phaseless AFQMC is capable of producing accurate results, comparable to some of the most accurate methods in the blind test. As argued by the authors, the favorable scaling of AFQMC, combined with its accuracy, makes it an important and complementary tool in the study of correlated many-body problems.
Morales and Malone investigate the use of basis sets optimized using second-order Møller–Plesset perturbation (MP2) theory in insulating solids and their impact on the convergence of AFQMC calculations to the complete basis set limit.4 The authors also introduce and validate MP2-based basis set correction schemes for correlated many-body methods in solids, which together with the optimized basis sets allow them to obtain well-converged results in a series of materials, with significant reductions in computational cost compared to alternative approaches.
Shen and co-workers present a novel approach to finite temperature AFQMC calculations in the canonical ensemble.5 The extension of the finite temperature method, which is traditionally formulated in the grand canonical ensemble, to the canonical ensemble introduces several advantages, which the authors describe in detail and demonstrate in applications to the Bose and Fermi Hubbard models. This collection of articles shows great promise in the use of AFQMC as a general purpose many-body approach. At the same time, it also suggests that there is still significant room for further improvements and developments in the method, further strengthening the prospects of the method.
II. REAL-SPACE METHODS
One of the reasons for the continued maturation of real-space quantum Monte Carlo (QMC) is the vibrancy of the community improving its core methodology. In this special issue, several groups present their work to improve the central approximations, both in core-valence interaction and in the ubiquitous fixed node approximation. In addition, an area that continues to receive attention is how to efficiently and effectively calculate excited state properties. The work presented here looks at both how to approach calculations of larger groups of excited states and how to approach core excitations.
Krogel and Reboredo revisit the pseudo-Hamiltonian approach to developing effective, efficient, and accurate approximations of the core-valence interaction in diffusion Monte Carlo (DMC) calculations.6 They demonstrate how this technique can be used to reduce the attendant approximations in the application of effective core-valence potentials that are necessary for efficient DMC calculations involving heavier elements.
Garner and Neuscamman7 show how to predict core excitation energies that can be utilized for comparison to x-ray absorption spectroscopy. Their approach relies on utilizing flexible optimization of descriptive many-body wave functions in such a way as to preserve cancellation of errors between the ground and excited state calculations.
Pathak and co-workers’ paper presents a new approach, borrowing from the density matrix renormalization group (DMRG) literature, to optimize wave functions to target excited states without an adjustable parameter in the method.8 Such work strengthens the ability of QMC to make predictions that can be directly compared to experimental results. Here, they show a computationally tractable approach by calculating the rich manifold of low-lying excited states of benzene.
In their paper, Scemama and co-workers9 demonstrate an advance in the use of the selected-configuration interaction (CI) approach for generating highly accurate multi-determinant trial wave functions that can be used to arbitrarily reduce the fixed node error in DMC. In this work, they show that using range-separated density functional theory (DFT) in the generation of determinants can result in a considerably smaller multi-determinant expansion for a given accuracy level and therefore a more efficient trial wave function.
Benali and co-workers show how QMC can be applied in a systematically improvable way to condensed matter in the thermodynamic limit.10 This showcases the ability of quantum Monte Carlo methods to quantify their errors in a way that is not generally possible outside of high scaling quantum chemistry approaches.
III. PERTURBATION THEORY
In recent years, there has been an explosion of Monte Carlo methods applied to the evaluation of perturbative and diagrammatic expansions. In some cases, the many terms in these expansions can be summed more efficiently using stochastic methods instead of deterministic methods. In this collection, a common thread is to use a partially deterministic evaluation of the series, in which the “long tails” of small terms are treated stochastically. This idea works because the efficiency of Monte Carlo methods is heavily dependent on variance, so removing a few large terms can result in large gains in efficiency.
Dou and co-workers11 apply an idea that has been termed “semistochastic,” in which the large terms of the series expansion of Green’s function (GF2) are treated deterministically, while the small terms are treated stochastically. They present an application of this idea to the second-order Matsubara–Green’s function (GF2) method, demonstrating calculations on a system with up to 500 electrons.
Stochastic methods often reduce the scaling of a calculation with system size and improve the ease of parallelization, but at the cost of increasing the prefactor. Hirata has pioneered the Monte Carlo evaluation of MP2 and MP3 methods using a stochastic integral in real space. Two papers by Doran and Hirata12,13 focus on using sampling techniques to reduce the prefactor of these methods. With these new techniques, the performance crossover between the deterministic and stochastic methods is achieved at smaller systems.
Blunt, Mahajan, and Sharma14 derive an algorithm to implement the perturbation theory evaluation of dynamic correlations from a complex multi-determinantal reference using only the two-body reduced density matrices. Deterministic algorithms to do this require higher body density matrices, which have a number of terms as (N2n), where n is the number of bodies in the density matrix. They also form a semistochastic method, in which some terms are evaluated deterministically and some stochastically. The key insight in this paper is that they evaluate the necessary perturbative integrals by sampling in determinant space, instead of using the high-order reduced density matrices, resulting in a much simpler algorithm that can also be applied to orbital-space variational Monte Carlo wave functions. In this case, the stochastic algorithm is used to expand the reach of the perturbative methods to more starting wave functions, rather than improving the scaling.
In their paper, Romanova and Vlček15 focus on using stochastic methods to solve problems in the GW approximation, also using a semistochastic approach. They find that treating localized states deterministically and the remaining stochastically offers a good balance between cost and variance. They find significant speedups upon using a stochastic algorithm for the GW Green’s function, for large systems. The stochastic GW approach scales well enough that they were able to treat defect states in graphene/hexagonal boron nitride bilayers.
IV. QUANTUM CHEMISTRY METHODS
The use of stochastic sampling has greatly extended the size of systems that can be treated with “quantum chemical” methods such as perturbation theory, configuration interaction, and coupled cluster theory. This has proven especially valuable in generating benchmark results for assessing the performance of more approximate electronic structure methods.
Filip and Thom describe a stochastic implementation of the unitary coupled cluster method.16 They show that their algorithm gives an energy close to that of variational unitary coupled cluster calculations but without explicit variational optimization of the cluster coefficients.
Scott et al. present details of their implementation of a novel stochastic approach to diagrammatic coupled cluster theory.17 A key aspect of their approach is the imaginary time propagation of the linked coupled cluster diagrams. The algorithm is walkerless and scales quadratically in system size for any truncation level.
Yang and co-workers introduce a new approach, based on the analogy of the dynamics to the damped harmonic oscillator problem, to improve population control in the full configuration interaction quantum Monte Carlo (FCIQMC) method.18 They also introduce a so-called growth witness that detects the annihilation threshold. The new population control algorithm has the potential to more efficiently use computational resources in FCIQMC calculations.
Ladóczki and co-workers introduce a third-order Epstein–Nesbit (EN) perturbation theory correction to the initiator correction to the FCIQMC algorithm.19 A major motivation for this extension is to make FCIQMC more size consistent. In addition, going to a higher-order (third rather than second) EN correction can reduce the size of the variational space needed to achieve a given accuracy, reducing the memory demands of such calculations.
Mihm et al. demonstrate the applicability of their connectivity twist averaging (cTA) procedure in initiator FCIQMC calculations on uniform electron gas.20 Unlike traditional twist averaging calculations, which require carrying out calculations at multiple twist angles, the cTA method requires use of a single twist angle. It is demonstrated that cTA works as well for initiator FCIQMC (iFCIQMC) as for coupled cluster doubles considered in the authors’ prior work. It is also demonstrated that one can obtain comparable results by the use of a twist angle determined from MP2 calculations.
V. APPLICATION TO FINITE SYSTEMS
Stochastic methods, both those that operate in real space and those that operate in the space of Slater determinants, are increasingly being applied to challenging chemical problems. Several such applications are described in articles in this special issue.
DMC, in part, because of its relative insensitivity to the basis set used to express the trial wave functions, has proven particularly useful in describing dispersion interactions. The paper by Benali and co-workers21 reports well-converged DMC interaction energies for the L7 benchmark set of molecular dimers. These results should prove especially valuable to researchers testing more approximate electronic structure methods.
In their paper, Upadhyay et al.22 explore the applicability of DMC and AFQMC to the non-valence correlation-bound (NVCB) anion of a model (H2O)4 cluster. Both approaches are shown to be capable of yielding accurate electron binding energies, provided that one employs trial wave functions that avoid the collapse onto the continuum problem.
Genovese and Sorella present a very interesting analysis of the nature of bonding in the C2 diatomic molecule.23 Their variational Monte Carlo calculations using trial wave functions comprised of antisymmetrized geminals and spin-dependent Jastrow factors reveal that C2 has a considerable antiferromagnetic character, with the two atoms having opposite spins.
Ito et al. present the results of DMC calculations using multiconfigurational trial wave functions on the H−e+H− system.24 It is demonstrated that a complex between H2 and Ps−, where Ps denotes positronium, is energetically more stable than the structure with the positron being localized between two H− ions.
The paper of Wang et al. presents the results of DMC, CI, and coupled cluster calculations on the ground and low-lying electronically excited singlet and triplet states of SiH2, SiH4, and Si2H6.25 The DMC results obtained using single-reference trial wave functions are in excellent agreement with the CI and coupled cluster results when the latter include extrapolation to the complete basis set limit.
Yao et al.26 apply the semistochastic heat bath CI (SHCI) method to atomization energies of the molecules in the G2 test set. The calculations are done for systematic sequences of basis sets, allowing extrapolation to the complete basis set limit, and employed the frozen core approximation. Corrections are applied for core-valence correlation, relativistic effects, and vibrational zero-point energy. Overall, excellent agreement is obtained with the experimental values of the atomization energies.
VI. APPLICATION TO INFINITE SYSTEMS
Quantum Monte Carlo has an important role in its application to condensed matter. It has long been applied to materials and has the potential to provide reference quality results to which other methods can be compared. This collection presents several works both showing this utility to problems of interest and expanding the frontiers where the method may be applied.
Militzer et al. look to understand how to calculate properties of matter in the extremely high pressure and temperature conditions common in astrophysical phenomena.27 They utilize a combination of path integral Monte Carlo and density functional theory to develop accurate equations of state of compound materials that can be compared with an ideal mixing approximation.
In their paper, Yilmaz et al. introduce two new restricted formulations of the configuration path integral Monte Carlo method, named RCPIMD and RCPIMD+, aimed at alleviating the effects of the sign problem in Fermionic simulations and extending their reach of applicability.28 The authors demonstrate the accuracy and efficacy of the new formulations in the homogeneous electron gas at intermediate temperatures and low densities.
Dubecky et al. investigate the disagreement between experimental and theoretical approaches for the fundamental gap of fluorographene.29 By examining the methodology in detail, they determine that the fixed node DMC and GW approaches agree and they arrive at a larger value for the fundamental gap than deduced from experiments. Furthermore, they find a large value for the exciton binding energy.
Similarly, Wines et al. apply diffusion Monte Carlo to two-dimensional (2D) GaSe to resolve theoretical discrepancies in the structural, energetic, and ground state charge densities in the material.30 They also calculate the excitation energy, finding that the exciton binding energy is low.
In their paper, Azadi et al. consider several different approaches to determine the equation of state of body centered cubic solid hydrogen.31 Specifically, they present a systematic investigation of the energy vs volume for the material calculated using a combination of DMC, FCIQMC, and the coupled cluster technique. By combining the approaches, they seek to show a route toward mitigating the respective weaknesses of the methods and provide a tractable route toward a generally applicable technique.
Continuing the theme of developing tractable methodologies for the many-body investigation of condensed matter, Powell et al.32 advance the state of the art in applying DMC to the adsorption of H2 on the Al (110) surface with the goal of developing a benchmark procedure upon which empirical density functional theory (DFT) functionals could be based in the absence of unambiguous experiments.
Finally, Gorelov et al. present a method to better account for the effects of the quantum motion of nuclei.33 This phenomenon is particularly important for light elements at lower temperatures. They consider how their methodological improvements can pave the way for a more controlled and thermodynamically consistent analysis of electronic structure from a many-body theory perspective.
As seen from the articles in this special issue, important advances are continuing to be made in “traditional” stochastic electronic structure methods such as diffusion Monte Carlo and auxiliary-field quantum Monte Carlo, enabling accurate calculations on increasingly complex systems. In addition, stochastic approaches are now being increasingly applied to more traditional electronic structure methods including perturbation theory, configuration interaction, coupled cluster, and Green’s function and diagrammatic methods, extending the size of systems that can be treated with those methods.
We wish to thank all the contributing authors and the journal editors and staff who made this issue possible. The work was performed, in part, under the auspices of the U.S. DOE by LLNL under Contract No. DE-AC52-07NA27344. Sandia National Laboratories is a multimission laboratory managed and operated by National Technology & Engineering Solutions of Sandia, LLC, a wholly owned subsidiary of Honeywell International Inc., for the U.S. Department of Energy’s National Nuclear Security Administration under contract DE-NA0003525. This paper describes objective technical results and analysis. Any subjective views or opinions that might be expressed in the paper do not necessarily represent the views of the U.S. Department of Energy or the United States Government.