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QMCPACK: Advances in the development, efficiency, and application of auxiliary field and real-space variational and diffusion Quantum Monte Carlo

P. R. C. Kent, Abdulgani Annaberdiyev, Anouar Benali, M. Chandler Bennett, Edgar Josue Landinez Borda, Peter Doak, Kenneth D. Jordan, Jaron T. Krogel, Ilkka Kylanpaa, Joonho Lee, Ye Luo, Fionn D. Malone, Cody A. Melton, Lubos Mitas, Miguel A. Morales, Eric Neuscamman, Fernando A. Reboredo, Brenda Rubenstein, Kayahan Saritas, Shiv Upadhyay, Hongxia Hao, Guangming Wang, Shuai Zhang, Luning Zhao

arXiv:2003.01831v2physics.comp-ph

TL;DR

The paper addresses how to extend the efficiency, accuracy, reproducibility, and scope of ab initio QMC calculations. It reviews QMCPACK and Nexus advances spanning AFQMC, real-space methods, correlation-consistent effective core potentials, and workflow integration. These developments expand accessible systems and properties, including more accurate band gaps and systematically improvable nodal surfaces.

  • Problem

    Ab initio QMC calculations require improved efficiency, accuracy, reproducibility, and methods for treating systems, properties, and approximations across molecules and materials.

  • Method

    The paper reviews QMCPACK and Nexus developments, including AFQMC enhancements, real-space QMC techniques, new correlation-consistent effective core potentials, and integrated workflows.

  • Results

    The advances expand QMC applicability and improve access to accurate band gaps, systematically improved nodal surfaces, and broader many-body calculations.

  • Takeaways & Limitations

    QMC is becoming easier to apply while its higher accuracy and many-body character support studies of more systems and phenomena.

Abstract

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We review recent advances in the capabilities of the open source ab initio Quantum Monte Carlo (QMC) package QMCPACK and the workflow tool Nexus used for greater efficiency and reproducibility. The auxiliary field QMC (AFQMC) implementation has been greatly expanded to include k-point symmetries, tensor-hypercontraction, and accelerated graphical processing unit (GPU) support. These scaling and memory reductions greatly increase the number of orbitals that can practically be included in AFQMC calculations, increasing accuracy. Advances in real space methods include techniques for accurate computation of band gaps and for systematically improving the nodal surface of ground state wavefunctions. Results of these calculations can be used to validate application of more approximate electronic structure methods including GW and density functional based techniques. To provide an improved foundation for these calculations we utilize a new set of correlation-consistent effective core potentials (pseudopotentials) that are more accurate than previous sets; these can also be applied in quantum-chemical and other many-body applications, not only QMC. These advances increase the efficiency, accuracy, and range of properties that can be studied in both molecules and materials with QMC and QMCPACK.

I. INTRODUCTION

QMC methods offer scalable, parallel approaches to ab initio electronic-structure problems, while QMCPACK advances their efficiency, applicability, accuracy, and reproducibility. The updates span AFQMC, real-space QMC, effective core potentials, workflow tools, and applications across molecules and materials.

  • I. INTRODUCTION: QMC methods combine statistical solution of the Schrödinger equation, systematically controllable approximations, low power scaling, and strong parallel suitability.Their computational cost prefactor remains large despite demonstrated scaling to millions of compute cores.
  • I. INTRODUCTION: QMCPACK updates expand the systems, properties, and accuracies accessible to QMC calculations.The package and its ecosystem include workflow tools, wavefunction converters, AFQMC enhancements, real-space methods, and new effective core potentials.
  • I. INTRODUCTION: AFQMC now supports GPU acceleration and k-point symmetries, extending its practical range beyond real-space VMC and DMC methods.The AFQMC solver is orbitally based and complementary to real-space methods.
  • I. INTRODUCTION: Nexus simplifies QMC applications and supports reproducibility across complex, multi-step research workflows.The workflow package integrates electronic-structure workflows rather than relying on manual invocation of individual calculations.
  • I. INTRODUCTION: New correlation-consistent effective core potentials provide an improved foundation for QMC and other ab initio many-body calculations.Their use is motivated by the infeasibility of all-electron QMC calculations for general systems.
  • I. INTRODUCTION: Real-space QMC advances improve ground-state energies and solid-state band-gap calculations while enabling tests of long-used nodal surfaces.Multiple-determinant trial wavefunctions and new algorithms support these developments.

II. OPEN DEVELOPMENT AND TESTING

QMCPACK combines open development practices with expanded testing, documentation, and coding standards to improve software quality and contributor accessibility. Deterministic tests make broad feature coverage fast enough for continuous integration, while longer statistical tests provide follow-up validation.

  • Open development: The development process routinely catches untested cases, reduces bugs and developer effort, and saves reviewer time through early review and continuous integration.Examples include complex-valued builds and accelerated GPU support.
  • Testing: QMCPACK expanded testing from limited unit, integration, and performance categories to approximately 25 machine and build combinations running around 1000 labeled tests each.Most tests cover multiple features, and combinations are run automatically on a nightly basis.
  • Testing: Longer statistical integration tests remain necessary because QMC stochasticity makes reliable pass/fail judgments impractical for short runs and many build combinations.The shortest integration test set takes around one hour on a 16-core machine and can still experience occasional statistical failures.
  • Testing: Deterministic integration tests use short QMC runs, few walkers, and fixed random seeds to provide reproducible coverage of major QMCPACK features.Unit and deterministic integration tests take approximately one minute, enabling iterative development and continuous integration.
  • Documentation and readability: Coding standards, automatic formatting, expanded documentation, and clearer source code help new contributors connect implementation with textbook equations more easily.The changes are described as providing long-term benefits for QMCPACK and potentially other scientific applications.

III. IMPROVING QMC WORKFLOWS WITH NEXUS

Nexus addresses the complexity of multi-stage QMC workflows by unifying electronic-structure codes, reducing user inputs, and automating execution and monitoring. Its use across systems from laptops to leadership computing facilities supports broader practical use of QMCPACK.

  • Workflow motivation: A realistic fixed-node DMC calculation requires many convergence, optimization, and extrapolation steps across SCF, NSCF, VMC, optimization, and DMC calculations.The example spans orbital cutoffs, B-spline meshes, twist grids, Jastrow forms, timesteps, and supercell sizes.
  • Workflow motivation: Nexus reduces QMC workflow complexity by combining required electronic-structure codes, simplifying inputs, and managing execution and monitoring under one framework.The workflow system is designed to transfer responsibility for correct execution from users to usable computational infrastructure.
  • Adoption: Nexus has been used successfully on workstations, laptops, institutional clusters, university high-performance computing centers, and Department of Energy Leadership Computing Facilities.The passage also reports high uptake of QMCPACK among new users.
  • Nexus implementation: Nexus represents workflows as sequences of simplified function-call blocks, generates expanded inputs and job-submission files, and monitors execution across target architectures.It also automates transfer of optimized wavefunctions to downstream calculations such as diffusion Monte Carlo.
  • Future development: Future Nexus improvements could encapsulate recurring convergence studies and dynamically create and monitor workflows for requested tolerances.These capabilities are presented as possible additional productivity gains rather than current functionality.

IV. EFFECTIVE CORE POTENTIALS

The paper introduces correlation-consistent effective core potentials to address the mixed fidelity of existing core approximations in high-accuracy QMC. Atomic and molecular benchmarks show ccECPs achieve smaller or comparable errors across tested spectra, geometries, and transition-metal molecules.

  • Construction and motivation: ccECPs are valence-only Hamiltonians constructed using many-body electron correlations to improve upon existing ECPs with mixed fidelity to all-electron calculations.The construction emphasizes accurate many-body valence spectra alongside norm conservation and shape consistency.
  • Validation scope: The ccECP reference and benchmark program covers atomic spectra, molecular bonds, total and kinetic energies, and fixed-node DMC energies for elements H–Kr.The tests were designed to probe accuracy and transferability, with plans to extend coverage across the periodic table.
  • Validation scope: ccECPs maintain close agreement with all-electron results over large valence excitation and ionization energy windows and across molecular properties such as binding energies and vibrational frequencies.The development also prioritizes compressed bond lengths and provides smaller cores in cases where larger errors were observed.
  • Benchmark results: ccECPs achieve smaller or comparable average errors than other ECPs in atomic and molecular tests, with improvements consistent across elements and varying geometries.The authors characterize ccECP as the best accuracy compromise for atomic spectral and molecular properties.
  • Molecular properties: In FeH, FeO, VH, and VO binding curves, ccECPs are the only valence Hamiltonians consistently within chemical accuracy relative to scalar-relativistic all-electron CCSD(T).The comparison includes a broad range of geometries, including short bond lengths relevant to high pressures.

C. ccECP Database and Website

The ccECP effort provides improved effective core potentials, associated basis sets, and multiple code formats for correlated quantum-chemical and solid-state calculations. QMCPACK also adds factorization and symmetry techniques that reduce AFQMC storage or scaling and improve GPU suitability.

  • C. ccECP Database and Website: Basis sets from DZ to 6Z, augmented variants, and code-specific formats support correlated calculations across quantum-chemistry and solid-state software.XML is directly usable in QMCPACK, while transformed Kleinman–Bylander potentials support plane-wave codes such as Quantum Espresso.
  • C. ccECP Database and Website: ccECPs target improved treatment of systematic errors and are supplied for broad correlated-calculation use.The paper describes the potentials as effective valence Hamiltonians and plans continued adaptation for plane-wave, 4d, and 5d applications.
  • V. AUXILIARY FIELD QUANTUM MONTE CARLO: AFQMC offers three factorization approaches, including modified-Cholesky, tensor-hypercontraction, and explicitly k-point-dependent forms for different system sizes and symmetries.The approaches are presented as alternatives whose suitability depends on system size, periodicity, and available computational resources.
  • V. AUXILIARY FIELD QUANTUM MONTE CARLO: Explicit k-point symmetry reduces many-operation scaling and storage by 1/N_k while enabling batched dense linear algebra on GPUs.The THC and k-point-symmetric factorizations can be combined, although that combination had not yet been used at the time described.

VI. TOWARDS SYSTEMATIC CONVERGENCE OF REAL-SPACE QMC CALCULATIONS

Systematic convergence of real-space QMC trial wavefunctions and their nodal surfaces remains difficult for general many-electron systems. More flexible multi-determinant and related wavefunctions are needed, but their construction and evaluation are computationally demanding.

  • VI. TOWARDS SYSTEMATIC CONVERGENCE OF REAL-SPACE QMC CALCULATIONS: Systematically converging trial wavefunctions and nodal surfaces remains a central challenge for real-space QMC in general many-electron systems.Improved convergence is needed both for higher property accuracy and to reduce starting-point dependence.
  • VI. TOWARDS SYSTEMATIC CONVERGENCE OF REAL-SPACE QMC CALCULATIONS: Optimizing VMC energy or variance does not guarantee minimization of the fixed-node energy used in DMC.This makes indirect nodal optimization insufficient as a general route to systematically improved DMC results.
  • VI. TOWARDS SYSTEMATIC CONVERGENCE OF REAL-SPACE QMC CALCULATIONS: Single-determinant orbital optimization cannot provide exact nodes for general systems because its nodal surface has limited (3N − 1)-dimensional flexibility.It can nevertheless serve as a simple, independent starting point for trial-wavefunction improvement.
  • VI. TOWARDS SYSTEMATIC CONVERGENCE OF REAL-SPACE QMC CALCULATIONS: Backflow, iterative backflow, and antisymmetrized geminal-product wavefunctions provide alternative ways to increase trial-wavefunction flexibility.The paper frames these approaches as retaining simplicity while improving the nodal surface.
  • VI. TOWARDS SYSTEMATIC CONVERGENCE OF REAL-SPACE QMC CALCULATIONS: Multi-Slater or configuration-interaction expansions offer a systematically improvable route toward exact wavefunctions, but direct configuration interaction is prohibitively costly beyond the smallest systems.Efficient determinant-selection and wavefunction-evaluation algorithms are therefore required.

A. Ground state calculations

Selected configuration-interaction methods, especially CIPSI, provide practical trial wavefunctions for QMC in molecules and solids. Their increasing determinant counts improve convergence but impose large computational costs.

  • A. Ground state calculations: CIPSI supports high-accuracy QMC trial wavefunctions for both molecular and solid systems, while effective core potentials reduce variance in DMC calculations.The paper describes the method as practical across these two application classes.
  • A. Ground state calculations: sCI methods avoid expert active-space selection by constructing trial wavefunctions with a single threshold parameter and interfacing with QMCPACK and Nexus.CIPSI is implemented in Quantum Package 2.0 and connected to the QMC workflow.
  • A. Ground state calculations: CIPSI trial wavefunctions can be used directly, reoptimized with a Jastrow factor, or projected through DMC, and the procedure applies to molecules and solids.The solid-state application requires fully treating k-points and their symmetries.
  • A. Ground state calculations: The largest reported CIPSI calculations reached convergence with fewer than 5M determinants, while DMC coefficients were not reoptimized with a Jastrow factor.The reported DMC cost scales with determinant number and the squared variance ratio.

B. Molecular Lithium Fluoride

LiF calculations compare single- and multi-determinant trial wavefunctions across basis sets and at the complete-basis-set limit. CIPSI-based DMC recovers additional correlation energy and yields basis-set-stable ionization potentials that agree with CIPSI and CCSD(T) at the CBS limit.

  • B. Molecular Lithium Fluoride: Experimental comparisons remain limited because temperature, zero-point motion, and other environmental effects are omitted from the calculations.These omitted effects may explain remaining discrepancies with the experimental ionization energy.
  • B. Molecular Lithium Fluoride: CIPSI recovers approximately 0.24 eV for the ground state and 0.13 eV for the cation relative to the reference comparison described.Its CBS total energy converges to the DMC(CIPSI) energy, whereas CCSD(T) converges to the single-determinant DMC energy.
  • B. Molecular Lithium Fluoride: Less than approximately 10 meV is the worst-case DMC basis-set dependence, and all tested single-determinant LiF nodal surfaces are essentially equivalent within error bars.This weak dependence holds across the tested single-determinant starting methods and basis sets.
  • B. Molecular Lithium Fluoride: DMC(CIPSI), CIPSI, and CCSD(T) agree at the CBS limit for LiF’s vertical ionization potential.The CIPSI-based DMC ionization potential shows almost no dependence on basis-set size.

C. Solid-state Lithium Fluoride

The LiF example examines basis-set effects and systematic DMC-energy convergence with selected determinants. Larger bases require more determinants, while cc-pVTZ and cc-pVQZ converge to agreeing energies.

  • The study uses a four-atom (LiF)2 cell at the Gamma point with ccECPs and associated cc-pVDZ, cc-pVTZ, and cc-pVQZ bases.
  • About 700K, 6M, and 9M determinants are needed for approximate convergence with cc-pVDZ, cc-pVTZ, and cc-pVQZ, respectively.The determinant requirement increases substantially with basis-set size.
  • The converged cc-pVTZ and cc-pVQZ energies agree, indicating that the basis set is sufficiently converged.
  • DMC energy converges faster for cc-pVTZ, requiring about 700K rather than 6M determinants.The slower cc-pVQZ convergence suggests that important static-correlation determinants enter late in selection.

E. Summary

The paper combines selected-configuration-interaction wavefunctions with VMC and DMC to improve challenging molecular and solid-state calculations. Applications include non-valence anions, localized excitations, and systematically improved nodal surfaces.

  • DMC used after sCI can improve molecular and solid-state calculations beyond single-determinant wavefunctions and converge faster than sCI alone.
  • Nexus fully interfaces PySCF, Quantum Package, and QMCPACK, automating multistep and finite-size-scaling workflows.
  • The non-valence-anion result demonstrates that DMC can characterize systems whose HF trial orbital collapses onto a discretized continuum.

B. Excitation energies of localized defects

DMC is applied to localized Mn4+ excitations in insulating hosts, where standard DFT approaches have difficulty reproducing emission energies. The calculations reproduce experimental emission energies and reveal strongly localized excited-electron density.

  • The approach is applicable to similar systems when the excitation is sufficiently localized.
  • The emission energy is defined as Eem = E(2Eg) − E(4A2g) for the localized Mn4+ excitation.
  • DMC reproduces experimental emission energies for Mn4+-doped insulating host compounds, whereas DFT+U and hybrid DFT substantially underestimate them.
  • The excited electron density is strongly localized on the Mn impurity, with n(r) approaching zero as radius increases.
  • DMC and LDA+U produce nearly identical radial densities despite substantially different emission energies.The comparison underscores difficulties in obtaining accurate emission energies with DFT functionals.

C. Calculation of the many-body properties: the momentum distribution

QMCPACK computes momentum distributions and related Compton profiles from many-body wavefunctions, providing information beyond mean-field observables. Algorithmic optimizations reduce estimator cost and enable analysis of VO2 across its metal-insulator transition.

  • The momentum distribution is obtained from the Fourier transform of the one-body density matrix and can reveal Fermi-surface and correlation properties.
  • The estimator is computationally expensive because it requires many wavefunction evaluations and can double QMC cost in a naive implementation.
  • Optimizing vectorized momentum-distribution evaluation reduces the overhead for a 48-atom VO2 cell from 150% to a 50% cost increase over an estimator-free DMC run.
  • Compton profiles and dynamical structure factors are related to projections of the momentum distribution under the impulse approximation.
  • For VO2, momentum-distribution analysis identifies non-Fermi-liquid signatures in the metallic phase and back-scattering characteristics linked to anomalously low electronic thermal conductivity.

VIII. SUMMARY

QMCPACK has expanded its QMC capabilities alongside improvements to Nexus, open effective core potentials, and GPU-oriented development. These advances broaden QMC’s applicability while making calculations easier to perform and supporting higher-accuracy studies.

  • Capabilities and workflow: QMCPACK now combines enhanced real-space and auxiliary-field QMC methods with ecosystem improvements for broader and more efficient studies.The surrounding workflow and software infrastructure were improved alongside both QMC approaches.
  • Capabilities and workflow: Nexus reduces the complexity of research studies involving tens to hundreds of individual calculations.
  • Effective core potentials: A new open database of effective core potentials provides increased accuracy for QMC, quantum-chemical, and other many-body calculations.The reported accuracy improvements include stretched bonds and excited states.
  • Future development: Ongoing work targets GPU-accelerated machines and portable performance from a single code base.The new design is intended to be described after validation on diverse GPUs.
  • Impact: QMC’s applicability continues to expand as higher accuracy and its many-body character become warranted and usable for more systems and phenomena.
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