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

International Symposium on Molecular Spectroscopy (ISMS)

DIFFUSION MONTE CARLO USING MACHINE LEARNING POTENTIAL ENERGY SURFACES

Ryan J. DiRisio, Anne B. McCoy, Fenris Lu, Jacob M. Finney, Mark Boyer

Published Jun 23, 2021• 0 citations
Publisher

Abstract

Diffusion Monte Carlo (DMC) is a technique for obtaining the ground-state solution to the vibrational time-independent Schrödinger equation based on a stochastic sampling of an electronic potential energy surface (PES). Ideally, the electronic energies used in DMC are calculated at the CCSD(T) level of theory. Recently, Tom Miller’s group at Caltech has developed a technique for calculating accurate CCSD(T) corrections to the electronic energy at the cost of a standard Hartree-Fock calculation using Gaussian process regression, a machine learning technique\footnote{Welborn, M. , Cheng, L., Miller, T. M. JCTC. 2018 14 (9), 4772-4779}\footnote{Cheng, L., Kovachki, N. B., Welborn, M., Miller, T. M. JCTC. 2019 15 (12), 6668-6677}. DMC is a well-suited technique to validate these PESs, referred to as MOB-ML surfaces, since DMC samples the entire region of the PES in which the vibrational ground state wave function has amplitude. Despite the speed of these MOB-ML PESs relative to an \emph{ab initio} calculation, calculating the Hartree-Fock energy for every one of thousands of DMC configurations over tens of thousands of time-steps makes the use of these surfaces computationally challenging. To this end, we combined two techniques to bring these calculations into computational feasibility. First, we implemented a DMC algorithm to take advantage of massively parallel high performance computing environments. Second, we use this DMC code to collect training data for a neural network (NN). We use a generic algorithm to train the NN to learn the MOB-ML surface based on the points sampled by the DMC. Using TensorFlow, the evaluation of the NN can then easily be done using graphics processing units (GPUs). We have performed small-scale DMC calculations on \chem{H2O} and \chem{CH5+} using these MOB-ML surfaces, and have also run large-scale NN+MOB-ML DMC simulations to obtain a high resolution ground state vibrational wave function and measure of the zero-point energy. Due to the generality of both the MOB-ML surface generation and the NN fitting workflow, we can easily extend this work to larger systems, such as large ion-water clusters or a Criegee intermediate, where running the DMC with traditional electronic structure would be intractable.

DOI
10.15278/isms.2021.wk02

Related papers

UTILIZATION OF NEURAL NETWORKS FOR GPU-ACCELERATED DIFFUSION MONTE CARLO FOR VIBRATIONAL PROBLEMS

Fenris Lu, Anne B. McCoy, Ryan J. DiRisio

In recent years, there has been increasing interest in combining machine learning with computational chemistry. Many machine learning applications in theoretical chemistry have focused on the development of potential energy surfaces (PES) based on \emph{ab initio} data points\footnote{Manzhos, S.; Carrington, T. Chem. Rev. Article ASAP. https://doi.org/10.1021/acs.chemrev.0c00665}. Another computational resource that has gained popularity in recent years is the use of Graphics Processing Units (GPUs) to accelerate calculations such as molecular dynamics simulations \footnote{Abraham, M. J.; Murtola, T.; Schulz, R.; Páll, S.; Smith, J. C.; Hess, B.; Lindahl, E. SoftwareX 2015, 1–2, 19–25}. In this work, we take advantage of the knowledge gained from both of these areas, and apply these approaches to the evaluations of the potential energy for Diffusion Monte Carlo (DMC) calculations. Diffusion Monte Carlo is a stochastic simulation approach through which an ensemble of localized functions (walkers) converges to the ground state vibrational wave function at long propagation times. This is accomplished through the ensemble randomly sampling a potential energy surface. These calculations require on the order of 10$^8$ to 10$^10$ potential energy evaluations, making the call to a potential energy surface the most expensive part of these calculations. To this end, we have developed a general technique for “learning” already existing potential energy surfaces using neural networks, which can then easily be evaluated on GPUs using TensorFlow. The technique entails running a small-scale DMC simulation to collect training data, coordinates and energies, and using this data to fit a subspace of the potential energy surface that is relevant for the DMC simulations. We find that using the neural network potential energy surface maintains the fidelity of the wave function while also enabling the use of ensemble sizes orders of magnitude larger than previously used for the same amount of wall time. We test this method on small systems such as \chem{CH5+}, \chem{H2O}, and water dimer.

International Symposium on Molecular Spectroscopy (ISMS)•Jun 22, 2021•0 citations

Beyond Single-Reference Fixed-Node Approximation in Ab Initio Diffusion Monte Carlo Using Antisymmetrized Geminal Power Applied to Systems with Hundreds of Electrons

Kousuke Nakano, Sandro Sorella, Dario Alfè +1 more

High Resolution Image Download MS PowerPoint Slide Diffusion Monte Carlo (DMC) is an exact technique to project out the ground state (GS) of a Hamiltonian. Since the GS is always bosonic, in Fermionic systems, the projection needs to be carried out while imposing antisymmetric constraints, which is a nondeterministic polynomial hard problem. In practice, therefore, the application of DMC on electronic structure problems is made by employing the fixed-node (FN) approximation, consisting of performing DMC with the constraint of having a fixed, predefined nodal surface. How do we get the nodal surface? The typical approach, applied in systems having up to hundreds or even thousands of electrons, is to obtain the nodal surface from a preliminary mean-field approach (typically, a density functional theory calculation) used to obtain a single Slater determinant. This is known as single reference. In this paper, we propose a new approach, applicable to systems as large as the C 60 fullerene, which improves the nodes by going beyond the single reference. In practice, we employ an implicitly multireference ansatz (antisymmetrized geminal power wave function constraint with molecular orbitals), initialized on the preliminary mean-field approach, which is relaxed by optimizing a few parameters of the wave function determining the nodal surface by minimizing the FN-DMC energy. We highlight the improvements of the proposed approach over the standard single-reference method on several examples and, where feasible, the computational gain over the standard multireference ansatz, which makes the methods applicable to large systems. We also show that physical properties relying on relative energies, such as binding energies, are affordable and reliable within the proposed scheme.

arXiv:2402.01458•Journal of Chemical Theory and Computation•May 24, 2024•6 citations

A brief introduction to the diffusion Monte Carlo method and the fixed-node approximation

Alfonso Annarelli, Dario Alfè, Andrea Zen

Quantum Monte Carlo (QMC) methods represent a powerful family of computational techniques for tackling complex quantum many-body problems and performing calculations of stationary state properties. QMC is among the most accurate and powerful approaches to the study of electronic structure, but its application is often hindered by a steep learning curve; hence it is rarely addressed in undergraduate and postgraduate classes. This tutorial is a step toward filling this gap. We offer an introduction to the diffusion Monte Carlo (DMC) method, which aims to solve the imaginary time Schrödinger equation through stochastic sampling of the configuration space. Starting from the theoretical foundations, the discussion leads naturally to the formulation of a step-by-step algorithm. To illustrate how the method works in simplified scenarios, examples such as the harmonic oscillator and the hydrogen atom are provided. The discussion extends to the fixed-node approximation, a crucial approach for addressing the fermionic sign problem in multi-electron systems. In particular, we examine the influence of trial wave function nodal surfaces on the accuracy of DMC energy by evaluating results from a non-interacting two-fermion system. Extending the method to excited states is feasible in principle, but some additional considerations are needed, supported by practical insights. By addressing the fundamental concepts from a hands-on perspective, we hope this tutorial will serve as a valuable guide for researchers and students approaching DMC for the first time.

arXiv:2412.06006•The Journal of Chemical Physics•Dec 28, 2024•9 citations

QUANTUM MONTE CARLO SIMULATIONS OF FERMIONS: A MATHEMATICAL ANALYSIS OF THE FIXED-NODE APPROXIMATION

Eric Cancès B, Benjamin Jourdain A, Tony Lelièvre A

The Diffusion Monte Carlo (DMC) method is a powerful strategy to estimate the ground state energy E0 of an N-body Schrödinger Hamiltonian H = -½Δ + V with high accuracy. It consists of writing E0 as the long-time limit of an expectation value of a drift-diffusion process with a source term, and numerically simulating this process by means of a collection of random walkers. As for a number of stochastic methods, a DMC calculation makes use of an importance sampling function ψI which hopefully approximates some ground state ψ0 of H. In the fermionic case, it has been observed that the DMC method is biased, except in the special case when the nodal surfaces of ψI coincide with those of a ground state of H. The approximation due to the fact that, in practice, the nodal surfaces of ψI differ from those of the ground states of H, is referred to as the Fixed Node Approximation (FNA). Our purpose in this paper is to provide a mathematical analysis of the FNA. We prove that, under convenient hypotheses, a DMC calculation performed with the importance sampling function ψI, provides an estimation of the infimum of the energy 〈ψ, Hψ〉 on the set of the fermionic test functions ψ that exactly vanish on the nodal surfaces of ψI.

Mathematical Models and Methods in Applied Sciences•Aug 23, 2006•27 citations

Anwendung und Weiterentwicklung der Diffusions-Quanten-Monte-Carlo-Methode

Raphaël Berner, Arne Lüchow

The diffusion quantum Monte Carlo method (DMC) provides a numeric solution of the time dependent Schrödinger equation by means of a random walk process. To fulfil the pauli principle the nodes of a so called trial wavefunction, obtained with standard ab initio quantum chemistry methods like HF, DFT, or CASSCF, are considered as additional boundary conditions. In case of such nodes coincide with those of the exact wavefunction DMC yields the exact solution of the Schrödinger equation. Generally, this is not true and one ends up with an systematic bias, referred to as the node location error. Theoretical chemistry aims the prediction of reaction, excitation, or activation energies, i.e. energy differences. Therefore, the node location error has to be seen under the light of efficient error cancellation. One can distinguish dynamic and non-dynamic electron correlation. The first one accounts for the instantaneous, short distance interaction among the electrons. The non-dynamic correlation addresses the interaction between different states of the same symmetry. Both kinds of correlation have a significant impact on the nodal structure and therefore govern the node location error. In this thesis the isomerization of bicylo[1.1.0]butane has been studied by exploiting the DMC method. The underlying reaction mechanism poses a great challenge to quantum chemistry because it comprises species with different amounts of non-dynamical correlation. Indeed, very sophisticated methods like for example CCSD(T) fail in giving a proper description. For the DMC method, the construction of the trial function is of crucial importance. Hence, the latter is generated by means of CASSCF calculations. With regard to efficient error cancellation it is most reasonable to take only those CSFs into account which contribute significantly to the proper description of the non-dynamic correlation. Since the DMC method is variational for the fermionic ground state, the incorporation of such a CSFs leads to a decrease of the energy compared with the output of a computation just based on the reference determinant. In the present case, only up to three CSFs very necessary. As a consequence, the direct optimization of the DMC energy with respect to the CASSCF coefficients is possible. By means of this construction scheme, dynamic and non-dynamic electron correlation effects are accounted for in equal measure for every molecule comprising the reaction path. Hence, the error cancellation is excellent. The obtained reaction, and activation energies are in very good agreement with experimental data and completely renormalized coupled cluster calculations. The presented CSF selection scheme yields very short linear combinations of CSFs and therefore should be applicable even for larger organic molecules. As it has already pointed out, quantum chemistry aims the prediction of energy differences rather than absolute values. A very efficient approach to calculate energy differences within a Monte Carlo framework is correlated sampling. The main idea behind this procedure is to describe the two different systems whose energy difference is to be calculated by means of the same random walk process. The resulting statistical correlation gives rise to a reduction of the statistical uncertainty of the energy difference. As far as we know, until now no implementation of the correlated sampling formalism into the DMC method has been reported in the literature. To implement correlated sampling in the DMC method a mathematical formalism based on the theory of stochastic differential equations has been devised. Based on this formalism an appropriate algorithm has been implemented for test systems like the particle in a one dimensional box, and the harmonic oszillator. Correlated sampling is very well suited for the computation of small energy differences with considerable small statistical uncertainties. A possible application would be the calculation of the weak interaction between DNA base pairs. For this purpose the corresponding formalism has to be implemented in our quantum Monte Carlo program Amolqc. In doing so, a major task is the implementation of routines which account for the propagation with respect to a positive guide function what has already been finished in the course of this work.

•Jan 1, 2010•0 citations

Towards the ground state of molecules via diffusion Monte Carlo on neural networks

Weiluo Ren, Weizhong Fu, Xiaojie Wu +1 more

Diffusion Monte Carlo (DMC) based on fixed-node approximation has enjoyed significant developments in the past decades and become one of the go-to methods when accurate ground state energy of molecules and materials is needed. However, the inaccurate nodal structure hinders the application of DMC for more challenging electronic correlation problems. In this work, we apply the neural-network based trial wavefunction in fixed-node DMC, which allows accurate calculations of a broad range of atomic and molecular systems of different electronic characteristics. Our method is superior in both accuracy and efficiency compared to state-of-the-art neural network methods using variational Monte Carlo (VMC). We also introduce an extrapolation scheme based on the empirical linearity between VMC and DMC energies, and significantly improve our binding energy calculation. Overall, this computational framework provides a benchmark for accurate solutions of correlated electronic wavefunction and also sheds light on the chemical understanding of molecules.

arXiv:2204.13903•Nature Communications•Apr 3, 2023•60 citations
Diffusion Monte Carlo (DMC) is a technique for obtaining the ground-state solution to the vibrational time-independent Schrödinger equation based on a stochastic sampling of an electronic potential energy surface (PES). Ideally, the electronic energies used in DMC are calculated at the CCSD(T) level of theory. Recently, Tom Miller’s group at Caltech has developed a technique for calculating accurate CCSD(T) corrections to the electronic energy at the cost of a standard Hartree-Fock calculation using Gaussian process regression, a machine learning technique\footnote{Welborn, M. , Cheng, L., Miller, T. M. JCTC. 2018 14 (9), 4772-4779}\footnote{Cheng, L., Kovachki, N. B., Welborn, M., Miller, T. M. JCTC. 2019 15 (12), 6668-6677}. DMC is a well-suited technique to validate these PESs, referred to as MOB-ML surfaces, since DMC samples the entire region of the PES in which the vibrational ground state wave function has amplitude. Despite the speed of these MOB-ML PESs relative to an \emph{ab initio} calculation, calculating the Hartree-Fock energy for every one of thousands of DMC configurations over tens of thousands of time-steps makes the use of these surfaces computationally challenging. To this end, we combined two techniques to bring these calculations into computational feasibility. First, we implemented a DMC algorithm to take advantage of massively parallel high performance computing environments. Second, we use this DMC code to collect training data for a neural network (NN). We use a generic algorithm to train the NN to learn the MOB-ML surface based on the points sampled by the DMC. Using TensorFlow, the evaluation of the NN can then easily be done using graphics processing units (GPUs). We have performed small-scale DMC calculations on \chem{H2O} and \chem{CH5+} using these MOB-ML surfaces, and have also run large-scale NN+MOB-ML DMC simulations to obtain a high resolution ground state vibrational wave function and measure of the zero-point energy. Due to the generality of both the MOB-ML surface generation and the NN fitting workflow, we can easily extend this work to larger systems, such as large ion-water clusters or a Criegee intermediate, where running the DMC with traditional electronic structure would be intractable.
In recent years, there has been increasing interest in combining machine learning with computational chemistry. Many machine learning applications in theoretical chemistry have focused on the development of potential energy surfaces (PES) based on \emph{ab initio} data points\footnote{Manzhos, S.; Carrington, T. Chem. Rev. Article ASAP. https://doi.org/10.1021/acs.chemrev.0c00665}. Another computational resource that has gained popularity in recent years is the use of Graphics Processing Units (GPUs) to accelerate calculations such as molecular dynamics simulations \footnote{Abraham, M. J.; Murtola, T.; Schulz, R.; Páll, S.; Smith, J. C.; Hess, B.; Lindahl, E. SoftwareX 2015, 1–2, 19–25}. In this work, we take advantage of the knowledge gained from both of these areas, and apply these approaches to the evaluations of the potential energy for Diffusion Monte Carlo (DMC) calculations. Diffusion Monte Carlo is a stochastic simulation approach through which an ensemble of localized functions (walkers) converges to the ground state vibrational wave function at long propagation times. This is accomplished through the ensemble randomly sampling a potential energy surface. These calculations require on the order of 108^88 to 1010^1010 potential energy evaluations, making the call to a potential energy surface the most expensive part of these calculations. To this end, we have developed a general technique for “learning” already existing potential energy surfaces using neural networks, which can then easily be evaluated on GPUs using TensorFlow. The technique entails running a small-scale DMC simulation to collect training data, coordinates and energies, and using this data to fit a subspace of the potential energy surface that is relevant for the DMC simulations. We find that using the neural network potential energy surface maintains the fidelity of the wave function while also enabling the use of ensemble sizes orders of magnitude larger than previously used for the same amount of wall time. We test this method on small systems such as \chem{CH5+}, \chem{H2O}, and water dimer.
High Resolution Image Download MS PowerPoint Slide Diffusion Monte Carlo (DMC) is an exact technique to project out the ground state (GS) of a Hamiltonian. Since the GS is always bosonic, in Fermionic systems, the projection needs to be carried out while imposing antisymmetric constraints, which is a nondeterministic polynomial hard problem. In practice, therefore, the application of DMC on electronic structure problems is made by employing the fixed-node (FN) approximation, consisting of performing DMC with the constraint of having a fixed, predefined nodal surface. How do we get the nodal surface? The typical approach, applied in systems having up to hundreds or even thousands of electrons, is to obtain the nodal surface from a preliminary mean-field approach (typically, a density functional theory calculation) used to obtain a single Slater determinant. This is known as single reference. In this paper, we propose a new approach, applicable to systems as large as the C 60 fullerene, which improves the nodes by going beyond the single reference. In practice, we employ an implicitly multireference ansatz (antisymmetrized geminal power wave function constraint with molecular orbitals), initialized on the preliminary mean-field approach, which is relaxed by optimizing a few parameters of the wave function determining the nodal surface by minimizing the FN-DMC energy. We highlight the improvements of the proposed approach over the standard single-reference method on several examples and, where feasible, the computational gain over the standard multireference ansatz, which makes the methods applicable to large systems. We also show that physical properties relying on relative energies, such as binding energies, are affordable and reliable within the proposed scheme.
Quantum Monte Carlo (QMC) methods represent a powerful family of computational techniques for tackling complex quantum many-body problems and performing calculations of stationary state properties. QMC is among the most accurate and powerful approaches to the study of electronic structure, but its application is often hindered by a steep learning curve; hence it is rarely addressed in undergraduate and postgraduate classes. This tutorial is a step toward filling this gap. We offer an introduction to the diffusion Monte Carlo (DMC) method, which aims to solve the imaginary time Schrödinger equation through stochastic sampling of the configuration space. Starting from the theoretical foundations, the discussion leads naturally to the formulation of a step-by-step algorithm. To illustrate how the method works in simplified scenarios, examples such as the harmonic oscillator and the hydrogen atom are provided. The discussion extends to the fixed-node approximation, a crucial approach for addressing the fermionic sign problem in multi-electron systems. In particular, we examine the influence of trial wave function nodal surfaces on the accuracy of DMC energy by evaluating results from a non-interacting two-fermion system. Extending the method to excited states is feasible in principle, but some additional considerations are needed, supported by practical insights. By addressing the fundamental concepts from a hands-on perspective, we hope this tutorial will serve as a valuable guide for researchers and students approaching DMC for the first time.
The Diffusion Monte Carlo (DMC) method is a powerful strategy to estimate the ground state energy E0 of an N-body Schrödinger Hamiltonian H = -½Δ + V with high accuracy. It consists of writing E0 as the long-time limit of an expectation value of a drift-diffusion process with a source term, and numerically simulating this process by means of a collection of random walkers. As for a number of stochastic methods, a DMC calculation makes use of an importance sampling function ψI which hopefully approximates some ground state ψ0 of H. In the fermionic case, it has been observed that the DMC method is biased, except in the special case when the nodal surfaces of ψI coincide with those of a ground state of H. The approximation due to the fact that, in practice, the nodal surfaces of ψI differ from those of the ground states of H, is referred to as the Fixed Node Approximation (FNA). Our purpose in this paper is to provide a mathematical analysis of the FNA. We prove that, under convenient hypotheses, a DMC calculation performed with the importance sampling function ψI, provides an estimation of the infimum of the energy 〈ψ, Hψ〉 on the set of the fermionic test functions ψ that exactly vanish on the nodal surfaces of ψI.
The diffusion quantum Monte Carlo method (DMC) provides a numeric solution of the time dependent Schrödinger equation by means of a random walk process. To fulfil the pauli principle the nodes of a so called trial wavefunction, obtained with standard ab initio quantum chemistry methods like HF, DFT, or CASSCF, are considered as additional boundary conditions. In case of such nodes coincide with those of the exact wavefunction DMC yields the exact solution of the Schrödinger equation. Generally, this is not true and one ends up with an systematic bias, referred to as the node location error. Theoretical chemistry aims the prediction of reaction, excitation, or activation energies, i.e. energy differences. Therefore, the node location error has to be seen under the light of efficient error cancellation. One can distinguish dynamic and non-dynamic electron correlation. The first one accounts for the instantaneous, short distance interaction among the electrons. The non-dynamic correlation addresses the interaction between different states of the same symmetry. Both kinds of correlation have a significant impact on the nodal structure and therefore govern the node location error. In this thesis the isomerization of bicylo[1.1.0]butane has been studied by exploiting the DMC method. The underlying reaction mechanism poses a great challenge to quantum chemistry because it comprises species with different amounts of non-dynamical correlation. Indeed, very sophisticated methods like for example CCSD(T) fail in giving a proper description. For the DMC method, the construction of the trial function is of crucial importance. Hence, the latter is generated by means of CASSCF calculations. With regard to efficient error cancellation it is most reasonable to take only those CSFs into account which contribute significantly to the proper description of the non-dynamic correlation. Since the DMC method is variational for the fermionic ground state, the incorporation of such a CSFs leads to a decrease of the energy compared with the output of a computation just based on the reference determinant. In the present case, only up to three CSFs very necessary. As a consequence, the direct optimization of the DMC energy with respect to the CASSCF coefficients is possible. By means of this construction scheme, dynamic and non-dynamic electron correlation effects are accounted for in equal measure for every molecule comprising the reaction path. Hence, the error cancellation is excellent. The obtained reaction, and activation energies are in very good agreement with experimental data and completely renormalized coupled cluster calculations. The presented CSF selection scheme yields very short linear combinations of CSFs and therefore should be applicable even for larger organic molecules. As it has already pointed out, quantum chemistry aims the prediction of energy differences rather than absolute values. A very efficient approach to calculate energy differences within a Monte Carlo framework is correlated sampling. The main idea behind this procedure is to describe the two different systems whose energy difference is to be calculated by means of the same random walk process. The resulting statistical correlation gives rise to a reduction of the statistical uncertainty of the energy difference. As far as we know, until now no implementation of the correlated sampling formalism into the DMC method has been reported in the literature. To implement correlated sampling in the DMC method a mathematical formalism based on the theory of stochastic differential equations has been devised. Based on this formalism an appropriate algorithm has been implemented for test systems like the particle in a one dimensional box, and the harmonic oszillator. Correlated sampling is very well suited for the computation of small energy differences with considerable small statistical uncertainties. A possible application would be the calculation of the weak interaction between DNA base pairs. For this purpose the corresponding formalism has to be implemented in our quantum Monte Carlo program Amolqc. In doing so, a major task is the implementation of routines which account for the propagation with respect to a positive guide function what has already been finished in the course of this work.
Diffusion Monte Carlo (DMC) based on fixed-node approximation has enjoyed significant developments in the past decades and become one of the go-to methods when accurate ground state energy of molecules and materials is needed. However, the inaccurate nodal structure hinders the application of DMC for more challenging electronic correlation problems. In this work, we apply the neural-network based trial wavefunction in fixed-node DMC, which allows accurate calculations of a broad range of atomic and molecular systems of different electronic characteristics. Our method is superior in both accuracy and efficiency compared to state-of-the-art neural network methods using variational Monte Carlo (VMC). We also introduce an extrapolation scheme based on the empirical linearity between VMC and DMC energies, and significantly improve our binding energy calculation. Overall, this computational framework provides a benchmark for accurate solutions of correlated electronic wavefunction and also sheds light on the chemical understanding of molecules.