Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2012.12228 · The Journal of Chemical Physics

A phaseless auxiliary-field quantum Monte Carlo perspective on the uniform electron gas at finite temperatures: Issues, observations, and benchmark study

Joonho Lee, Miguel A. Morales, Fionn D. Malone

Published Feb 12, 2021• 56 citations
arXivHTMLPDF

Abstract

We investigate the viability of the phaseless finite-temperature auxiliary-field quantum Monte Carlo (ph-FT-AFQMC) method for ab initio systems using the uniform electron gas as a model. Through comparisons with exact results and FT coupled cluster theory, we find that ph-FT-AFQMC is sufficiently accurate at high to intermediate electronic densities. We show, both analytically and numerically, that the phaseless constraint at FT is fundamentally different from its zero-temperature counterpart (i.e., ph-ZT-AFQMC), and generally, one should not expect ph-FT-AFQMC to agree with ph-ZT-AFQMC in the low-temperature limit. With an efficient implementation, we are able to compare exchange-correlation energies to the existing results in the thermodynamic limit and find that the existing parameterizations are highly accurate. In particular, we found that ph-FT-AFQMC exchange-correlation energies are in better agreement with a known parameterization than is restricted path-integral MC in the regime of Θ ≤ 0.5 and rs ≤ 2, which highlights the strength of ph-FT-AFQMC.

DOI
10.1063/5.0041378

Related papers

An auxiliary-Field quantum Monte Carlo perspective on the ground state of the dense uniform electron gas: An investigation with Hartree-Fock trial wavefunctions

Joonho Lee, Fionn D. Malone, Miguel A. Morales

We assess the utility of Hartree-Fock (HF) trial wavefunctions in performing phaseless auxiliary-field quantum Monte Carlo (ph-AFQMC) on the uniform electron gas (UEG) model. The combination of ph-AFQMC with spin-restricted HF (RHF+ph-AFQMC) was found to be highly accurate and efficient for systems containing up to 114 electrons in 2109 orbitals, particularly for rs ≤ 2.0. Compared to spin-restricted coupled-cluster (RCC) methods, we found that RHF+ph-AFQMC performs better than CC with singles, doubles, and triples (RCCSDT) and similarly to or slightly worse than CC with singles, doubles, triples, and quadruples (RCCSDTQ) for rs ≤ 3.0 in the 14-electron UEG model. With the 54-electron, we found RHF+ph-AFQMC to be nearly exact for rs ≤ 2.0 and pointed out potential biases in existing benchmarks. Encouraged by these, we performed RHF+ph-AFQMC on the 114-electron UEG model for rs ≤ 2.0 and provided new benchmark data for future method development. We found that the UEG models with rs = 5.0 remain to be challenging for RHF+ph-AFQMC. Employing nonorthogonal configuration expansions or unrestricted HF states as trial wavefunctions was also found to be ineffective in the case of the 14-electron UEG model with rs = 5.0. We emphasize the need for a better trial wavefunction for ph-AFQMC in simulating strongly correlated systems. With the 54-electron and 114-electron UEG models, we stress the potential utility of RHF+ph-AFQMC for simulating dense solids.

arXiv:1905.04361•The Journal of Chemical Physics•Aug 14, 2019•24 citations

An Auxiliary-Field Quantum Monte Carlo Perspective on the Ground State of the Uniform Electron Gas: An Investigation with Spin-Restricted Hartree-Fock Trial Wavefunctions

Joonho Lee, Fionn D. Malone, Miguel A. Morales

We assess the utility of spin-restricted Hartree-Fock (RHF) trial wavefunctions in performing phaseless auxiliary-field quantum Monte Carlo (ph-AFQMC) on the uniform electron gas (UEG) model. This particular combination, RHF+ph-AFQMC, was found to be highly accurate and efficient for systems containing up to 114 electrons in 2109 orbitals, particularly for rs $\le$ 2.0. Compared to spin-restricted coupled-cluster (RCC) methods, we found that RHF+ph-AFQMC performs better than CC with singles, doubles, and triples (RCCSDT) and similarly to or slightly worse than CC with singles, doubles, triples, and quadruples (RCCSDTQ) for rs $\le$ 3.0 in the 14-electron UEG model. With the 54-electron, we found RHF+ph-AFQMC to be nearly exact for rs $\le$ 2.0 and pointed out potential biases in existing benchmarks. Encouraged by these, we performed RHF+ph-AFQMC on the 114-electron UEG model for rs $\le$ 2.0 and provided new benchmark data for future method development. As the UEG models with rs = 5.0 remain to be challenging for the RHF+ph-AFQMC, we emphasize the need for a better trial wavefunction for ph-AFQMC in simulating strongly correlated systems. With the 54-electron and 114-electron UEG models, we stress the potential utility of RHF+ph-AFQMC for simulating dense solids.

•May 10, 2019•1 citation

Twenty Years of Auxiliary-Field Quantum Monte Carlo in Quantum Chemistry: An Overview and Assessment on Main Group Chemistry and Bond-Breaking

Joonho Lee, Hung Q. Pham, David R. Reichman

In this work, we present an overview of the phaseless auxiliary-field quantum Monte Carlo (ph-AFQMC) approach from a computational quantum chemistry perspective and present a numerical assessment of its performance on main group chemistry and bond-breaking problems with a total of 1004 relative energies. While our benchmark study is somewhat limited, we make recommendations for the use of ph-AFQMC for general main-group chemistry applications. For systems where single determinant wave functions are qualitatively accurate, we expect the accuracy of ph-AFQMC in conjunction with a single-determinant trial wave function to be between that of coupled-cluster with singles and doubles (CCSD) and CCSD with perturbative triples (CCSD(T)). For these applications, ph-AFQMC should be a method of choice when canonical CCSD(T) is too expensive to run. For systems where multireference (MR) wave functions are needed for qualitative accuracy, ph-AFQMC is far more accurate than MR perturbation theory methods and competitive with MR configuration interaction (MRCI) methods. Due to the computational efficiency of ph-AFQMC compared to MRCI, we recommended ph-AFQMC as a method of choice for handling dynamic correlation in MR problems. We conclude with a discussion of important directions for future development of the ph-AFQMC approach.

arXiv:2208.01280•Journal of Chemical Theory and Computation•Oct 18, 2022•76 citations

Benchmark Phaseless Auxiliary-Field Quantum Monte Carlo Method for Small Molecules

Zoran Sukurma, Martin Schlipf, Moritz Humer +2 more

High Resolution Image Download MS PowerPoint Slide We report a scalable Fortran implementation of the phaseless auxiliary-field quantum Monte Carlo (ph-AFQMC) and demonstrate its excellent performance and beneficial scaling with respect to system size. Furthermore, we investigate modifications of the phaseless approximation that can help to reduce the overcorrelation problems common to the ph-AFQMC. We apply the method to the 26 molecules in the HEAT set, the benzene molecule, and water clusters. We observe a mean absolute deviation of the total energy of 1.15 kcal/mol for the molecules in the HEAT set, close to chemical accuracy. For the benzene molecule, the modified algorithm despite using a single-Slater-determinant trial wavefunction yields the same accuracy as the original phaseless scheme with 400 Slater determinants. Despite these improvements, we find systematic errors for the CN, CO 2, and O 2 molecules that need to be addressed with more accurate trial wavefunctions. For water clusters, we find that the ph-AFQMC yields excellent binding energies that differ from CCSD(T) by typically less than 0.5 kcal/mol.

arXiv:2303.04256•Journal of Chemical Theory and Computation•Jul 20, 2023•17 citations

Selecting optimal unrestricted Hartree-Fock trial wavefunctions for phaseless auxiliary-field quantum Monte Carlo: Accuracy and limitations in modeling three iron-sulfur clusters

Don Danilov, Brad Ganoe, Leon Otis +3 more

Phaseless auxiliary-field quantum Monte Carlo (ph-AFQMC) has emerged as a promising electronic structure method for correlated electronic systems. However, the quality of its predictions depends critically on the choice of trial wavefunction, and it is not obvious how to make an optimal choice especially for strongly correlated states of large systems. Mean-field wavefunctions are compelling trial wavefunction candidates as they map directly to chemical concepts and can be obtained with $O(N^4)$ cost. Yet in the strongly correlated regime one faces a symmetry dilemma and the existence of multiple nearly-degenerate solutions. In this work we investigate active space models of [2Fe-2S]$^{2+}$, mixed-valent [4Fe-4S]$^{2+}$, and [4Fe-4S]$^{4+}$ and explore the sensitivity of ph-AFQMC to the choice of unrestricted Hartree-Fock trial wavefunction. We find that chemical properties and physical symmetries, rather than the variational energy, ought to guide the choice of mean-field trial for ph-AFQMC (or reference state for coupled cluster models), and show that surprisingly accurate ground-state energies for these systems can be obtained. However, in all cases we find a rapidly vanishing overlap between the stochastic wavefunction and the UHF trial, indicating that the trials are suboptimal importance functions. By analogy to a similar situation in the stretched helium dimer cation, we show how this sampling bias pushes ph-AFQMC towards artificially negative energies, which evidently can be compensated for by the phaseless bias in certain cases.

physics.chem-phCross-listedarXiv:2605.03981•May 6, 2026•0 citations

Ab Initio Finite Temperature Auxiliary Field Quantum Monte Carlo

Yuan Liu, Minsik Cho, Brenda M. Rubenstein

We present an ab initio auxiliary field quantum Monte Carlo method for studying the electronic structure of molecules, solids, and model Hamiltonians at finite temperature. The algorithm marries the ab initio phaseless auxiliary field quantum Monte Carlo algorithm known to produce high accuracy ground state energies of molecules and solids with its finite temperature variant, long used by condensed matter physicists for studying model Hamiltonian phase diagrams, to yield a phaseless, ab initio finite temperature method. We demonstrate that the method produces internal energies within chemical accuracy of exact diagonalization results across a wide range of temperatures for H 2 O (STO-3G), C 2 (STO-6G), the one-dimensional hydrogen chain (STO-6G), and the multiorbital Hubbard model. Our method effectively controls the phase problem through importance sampling, often even without invoking the phaseless approximation, down to temperatures at which the systems studied approach their ground states and may therefore be viewed as exact over wide temperature ranges. This technique embodies a versatile tool for studying the finite temperature phase diagrams of a plethora of systems whose properties cannot be captured by a Hubbard U term alone. Our results moreover illustrate that the severity of the phase problem for model Hamiltonians far exceeds that for many molecules at all of the temperatures studied.

arXiv:1806.02848•Journal of Chemical Theory and Computation•Aug 13, 2018•51 citations
We investigate the viability of the phaseless finite-temperature auxiliary-field quantum Monte Carlo (ph-FT-AFQMC) method for ab initio systems using the uniform electron gas as a model. Through comparisons with exact results and FT coupled cluster theory, we find that ph-FT-AFQMC is sufficiently accurate at high to intermediate electronic densities. We show, both analytically and numerically, that the phaseless constraint at FT is fundamentally different from its zero-temperature counterpart (i.e., ph-ZT-AFQMC), and generally, one should not expect ph-FT-AFQMC to agree with ph-ZT-AFQMC in the low-temperature limit. With an efficient implementation, we are able to compare exchange-correlation energies to the existing results in the thermodynamic limit and find that the existing parameterizations are highly accurate. In particular, we found that ph-FT-AFQMC exchange-correlation energies are in better agreement with a known parameterization than is restricted path-integral MC in the regime of Θ ≤ 0.5 and rs ≤ 2, which highlights the strength of ph-FT-AFQMC.
We assess the utility of Hartree-Fock (HF) trial wavefunctions in performing phaseless auxiliary-field quantum Monte Carlo (ph-AFQMC) on the uniform electron gas (UEG) model. The combination of ph-AFQMC with spin-restricted HF (RHF+ph-AFQMC) was found to be highly accurate and efficient for systems containing up to 114 electrons in 2109 orbitals, particularly for rs ≤ 2.0. Compared to spin-restricted coupled-cluster (RCC) methods, we found that RHF+ph-AFQMC performs better than CC with singles, doubles, and triples (RCCSDT) and similarly to or slightly worse than CC with singles, doubles, triples, and quadruples (RCCSDTQ) for rs ≤ 3.0 in the 14-electron UEG model. With the 54-electron, we found RHF+ph-AFQMC to be nearly exact for rs ≤ 2.0 and pointed out potential biases in existing benchmarks. Encouraged by these, we performed RHF+ph-AFQMC on the 114-electron UEG model for rs ≤ 2.0 and provided new benchmark data for future method development. We found that the UEG models with rs = 5.0 remain to be challenging for RHF+ph-AFQMC. Employing nonorthogonal configuration expansions or unrestricted HF states as trial wavefunctions was also found to be ineffective in the case of the 14-electron UEG model with rs = 5.0. We emphasize the need for a better trial wavefunction for ph-AFQMC in simulating strongly correlated systems. With the 54-electron and 114-electron UEG models, we stress the potential utility of RHF+ph-AFQMC for simulating dense solids.
We assess the utility of spin-restricted Hartree-Fock (RHF) trial wavefunctions in performing phaseless auxiliary-field quantum Monte Carlo (ph-AFQMC) on the uniform electron gas (UEG) model. This particular combination, RHF+ph-AFQMC, was found to be highly accurate and efficient for systems containing up to 114 electrons in 2109 orbitals, particularly for rs ≤\le≤ 2.0. Compared to spin-restricted coupled-cluster (RCC) methods, we found that RHF+ph-AFQMC performs better than CC with singles, doubles, and triples (RCCSDT) and similarly to or slightly worse than CC with singles, doubles, triples, and quadruples (RCCSDTQ) for rs ≤\le≤ 3.0 in the 14-electron UEG model. With the 54-electron, we found RHF+ph-AFQMC to be nearly exact for rs ≤\le≤ 2.0 and pointed out potential biases in existing benchmarks. Encouraged by these, we performed RHF+ph-AFQMC on the 114-electron UEG model for rs ≤\le≤ 2.0 and provided new benchmark data for future method development. As the UEG models with rs = 5.0 remain to be challenging for the RHF+ph-AFQMC, we emphasize the need for a better trial wavefunction for ph-AFQMC in simulating strongly correlated systems. With the 54-electron and 114-electron UEG models, we stress the potential utility of RHF+ph-AFQMC for simulating dense solids.
In this work, we present an overview of the phaseless auxiliary-field quantum Monte Carlo (ph-AFQMC) approach from a computational quantum chemistry perspective and present a numerical assessment of its performance on main group chemistry and bond-breaking problems with a total of 1004 relative energies. While our benchmark study is somewhat limited, we make recommendations for the use of ph-AFQMC for general main-group chemistry applications. For systems where single determinant wave functions are qualitatively accurate, we expect the accuracy of ph-AFQMC in conjunction with a single-determinant trial wave function to be between that of coupled-cluster with singles and doubles (CCSD) and CCSD with perturbative triples (CCSD(T)). For these applications, ph-AFQMC should be a method of choice when canonical CCSD(T) is too expensive to run. For systems where multireference (MR) wave functions are needed for qualitative accuracy, ph-AFQMC is far more accurate than MR perturbation theory methods and competitive with MR configuration interaction (MRCI) methods. Due to the computational efficiency of ph-AFQMC compared to MRCI, we recommended ph-AFQMC as a method of choice for handling dynamic correlation in MR problems. We conclude with a discussion of important directions for future development of the ph-AFQMC approach.
High Resolution Image Download MS PowerPoint Slide We report a scalable Fortran implementation of the phaseless auxiliary-field quantum Monte Carlo (ph-AFQMC) and demonstrate its excellent performance and beneficial scaling with respect to system size. Furthermore, we investigate modifications of the phaseless approximation that can help to reduce the overcorrelation problems common to the ph-AFQMC. We apply the method to the 26 molecules in the HEAT set, the benzene molecule, and water clusters. We observe a mean absolute deviation of the total energy of 1.15 kcal/mol for the molecules in the HEAT set, close to chemical accuracy. For the benzene molecule, the modified algorithm despite using a single-Slater-determinant trial wavefunction yields the same accuracy as the original phaseless scheme with 400 Slater determinants. Despite these improvements, we find systematic errors for the CN, CO 2, and O 2 molecules that need to be addressed with more accurate trial wavefunctions. For water clusters, we find that the ph-AFQMC yields excellent binding energies that differ from CCSD(T) by typically less than 0.5 kcal/mol.
Phaseless auxiliary-field quantum Monte Carlo (ph-AFQMC) has emerged as a promising electronic structure method for correlated electronic systems. However, the quality of its predictions depends critically on the choice of trial wavefunction, and it is not obvious how to make an optimal choice especially for strongly correlated states of large systems. Mean-field wavefunctions are compelling trial wavefunction candidates as they map directly to chemical concepts and can be obtained with O(N4)O(N^4)O(N4) cost. Yet in the strongly correlated regime one faces a symmetry dilemma and the existence of multiple nearly-degenerate solutions. In this work we investigate active space models of [2Fe-2S]2+^{2+}2+, mixed-valent [4Fe-4S]2+^{2+}2+, and [4Fe-4S]4+^{4+}4+ and explore the sensitivity of ph-AFQMC to the choice of unrestricted Hartree-Fock trial wavefunction. We find that chemical properties and physical symmetries, rather than the variational energy, ought to guide the choice of mean-field trial for ph-AFQMC (or reference state for coupled cluster models), and show that surprisingly accurate ground-state energies for these systems can be obtained. However, in all cases we find a rapidly vanishing overlap between the stochastic wavefunction and the UHF trial, indicating that the trials are suboptimal importance functions. By analogy to a similar situation in the stretched helium dimer cation, we show how this sampling bias pushes ph-AFQMC towards artificially negative energies, which evidently can be compensated for by the phaseless bias in certain cases.
We present an ab initio auxiliary field quantum Monte Carlo method for studying the electronic structure of molecules, solids, and model Hamiltonians at finite temperature. The algorithm marries the ab initio phaseless auxiliary field quantum Monte Carlo algorithm known to produce high accuracy ground state energies of molecules and solids with its finite temperature variant, long used by condensed matter physicists for studying model Hamiltonian phase diagrams, to yield a phaseless, ab initio finite temperature method. We demonstrate that the method produces internal energies within chemical accuracy of exact diagonalization results across a wide range of temperatures for H 2 O (STO-3G), C 2 (STO-6G), the one-dimensional hydrogen chain (STO-6G), and the multiorbital Hubbard model. Our method effectively controls the phase problem through importance sampling, often even without invoking the phaseless approximation, down to temperatures at which the systems studied approach their ground states and may therefore be viewed as exact over wide temperature ranges. This technique embodies a versatile tool for studying the finite temperature phase diagrams of a plethora of systems whose properties cannot be captured by a Hubbard U term alone. Our results moreover illustrate that the severity of the phase problem for model Hamiltonians far exceeds that for many molecules at all of the temperatures studied.