1
|
Abstract
Diffusion Monte Carlo (DMC) is one of the most accurate techniques available for calculating the electronic properties of molecules and materials, yet it often remains a challenge to economically compute forces using this technique. As a result, ab initio molecular dynamics simulations and geometry optimizations that employ Diffusion Monte Carlo forces are often out of reach. One potential approach for accelerating the computation of "DMC forces" is to machine learn these forces from DMC energy calculations. In this work, we employ Behler-Parrinello Neural Networks to learn DMC forces from DMC energy calculations for geometry optimization and molecular dynamics simulations of small molecules. We illustrate the unique challenges that stem from learning forces without explicit force data and from noisy energy data by making rigorous comparisons of potential energy surface, dynamics, and optimization predictions among ab initio density functional theory (DFT) simulations and machine-learning models trained on DFT energies with forces, DFT energies without forces, and DMC energies without forces. We show for three small molecules─C2, H2O, and CH3Cl─that machine-learned DMC dynamics can reproduce average bond lengths and angles within a few percent of known experimental results at one hundredth of the typical cost. Our work describes a much-needed means of performing dynamics simulations on high-accuracy, DMC PESs and for generating DMC-quality molecular geometries given current algorithmic constraints.
Collapse
Affiliation(s)
- Cancan Huang
- Department of Chemistry, Brown University, Providence, Rhode Island02912, United States
| | - Brenda M Rubenstein
- Department of Chemistry, Brown University, Providence, Rhode Island02912, United States
| |
Collapse
|
2
|
Hanindriyo AT, Yadav AKS, Ichibha T, Maezono R, Nakano K, Hongo K. Diffusion Monte Carlo evaluation of disiloxane linearisation barrier. Phys Chem Chem Phys 2022; 24:3761-3769. [PMID: 35080527 DOI: 10.1039/d1cp01471d] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/21/2022]
Abstract
The disiloxane molecule is a prime example of silicate compounds containing the Si-O-Si bridge. The molecule is of significant interest within the field of quantum chemistry, owing to the difficulty in theoretically predicting its properties. Herein, the linearisation barrier of disiloxane is investigated using a fixed-node diffusion Monte Carlo (FNDMC) approach, which is one of the most reliable ab initio methods in accounting for the electronic correlation. Calculations utilizing the density functional theory (DFT) and the coupled cluster method with single and double substitutions, including noniterative triples (CCSD(T)) are carried out alongside FNDMC for comparison. It is concluded that FNDMC successfully predicts the disiloxane linearisation barrier and does not depend on the completeness of the basis-set as much as DFT or CCSD(T), thus establishing its suitability.
Collapse
Affiliation(s)
- Adie Tri Hanindriyo
- School of Materials Science, JAIST, Asahidai 1-1, Nomi, Ishikawa, 923-1292, Japan.
| | - Amit Kumar Singh Yadav
- Department of Electrical Engineering, Indian Institute of Technology Gandhinagar, Palaj 382355, Gujarat, India
| | - Tom Ichibha
- Materials Science and Technology Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA
| | - Ryo Maezono
- School of Information Science, JAIST, Asahidai 1-1, Nomi, Ishikawa, 923-1292, Japan
| | - Kousuke Nakano
- School of Information Science, JAIST, Asahidai 1-1, Nomi, Ishikawa, 923-1292, Japan.,Scuola Internazionale Superiore di Studi Avanzati (SISSA), via Bonomea, 265-34136 Trieste, Italy
| | - Kenta Hongo
- Research Center for Advanced Computing Infrastructure, JAIST, Asahidai 1-1, Nomi, Ishikawa 923-1292, Japan.
| |
Collapse
|
3
|
Tiihonen J, Clay RC, Krogel JT. Toward quantum Monte Carlo forces on heavier ions: Scaling properties. J Chem Phys 2021; 154:204111. [PMID: 34241166 DOI: 10.1063/5.0052266] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
Quantum Monte Carlo (QMC) forces have been studied extensively in recent decades because of their importance with spectroscopic observables and geometry optimization. Here, we benchmark the accuracy and computational cost of QMC forces. The zero-variance zero-bias (ZVZB) force estimator is used in standard variational and diffusion Monte Carlo simulations with mean-field based trial wavefunctions and atomic pseudopotentials. Statistical force uncertainties are obtained with a recently developed regression technique for heavy tailed QMC data [P. Lopez Rios and G. J. Conduit, Phys. Rev. E 99, 063312 (2019)]. By considering selected atoms and dimers with elements ranging from H to Zn (1 ≤ Zeff ≤ 20), we assess the accuracy and the computational cost of ZVZB forces as the effective pseudopotential valence charge, Zeff, increases. We find that the costs of QMC energies and forces approximately follow simple power laws in Zeff. The force uncertainty grows more rapidly, leading to a best case cost scaling relationship of approximately Zeff 6.5(3) for diffusion Monte Carlo. We find that the accessible system size at fixed computational cost scales as Zeff -2, insensitive to model assumptions or the use of the "space warp" variance-reduction technique. Our results predict the practical cost of obtaining forces for a range of materials, such as transition metal oxides where QMC forces have yet to be applied, and underscore the importance of further developing force variance-reduction techniques, particularly for atoms with high Zeff.
Collapse
Affiliation(s)
- Juha Tiihonen
- Materials Science and Technology Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA
| | - Raymond C Clay
- Sandia National Laboratories, Albuquerque, New Mexico 87185, USA
| | - Jaron T Krogel
- Materials Science and Technology Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA
| |
Collapse
|
4
|
Needs RJ, Towler MD, Drummond ND, López Ríos P, Trail JR. Variational and diffusion quantum Monte Carlo calculations with the CASINO code. J Chem Phys 2020; 152:154106. [DOI: 10.1063/1.5144288] [Citation(s) in RCA: 35] [Impact Index Per Article: 8.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Affiliation(s)
- R. J. Needs
- TCM Group, Cavendish Laboratory, University of Cambridge, 19 J. J. Thomson Avenue, Cambridge CB3 0HE, United Kingdom
| | - M. D. Towler
- University College London, London WC1E 6BT, United Kingdom
| | - N. D. Drummond
- Department of Physics, Lancaster University, Lancaster LA1 4YB, United Kingdom
| | - P. López Ríos
- Max Planck Institute for Solid State Research, Heisenbergstraße 1, 70569 Stuttgart, Germany
| | - J. R. Trail
- TCM Group, Cavendish Laboratory, University of Cambridge, 19 J. J. Thomson Avenue, Cambridge CB3 0HE, United Kingdom
| |
Collapse
|
5
|
Ríos PL, Conduit GJ. Tail-regression estimator for heavy-tailed distributions of known tail indices and its application to continuum quantum Monte Carlo data. Phys Rev E 2019; 99:063312. [PMID: 31330629 DOI: 10.1103/physreve.99.063312] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/18/2019] [Indexed: 11/07/2022]
Abstract
Standard statistical analysis is unable to provide reliable confidence intervals on expectation values of probability distributions that do not satisfy the conditions of the central limit theorem. We present a regression-based estimator of an arbitrary moment of a probability distribution with power-law heavy tails that exploits knowledge of the exponents of its asymptotic decay to bypass this issue entirely. Our method is applied to synthetic data and to energy and atomic force data from variational and diffusion quantum Monte Carlo calculations, whose distributions have known asymptotic forms [J. R. Trail, Phys. Rev. E 77, 016703 (2008)PLEEE81539-375510.1103/PhysRevE.77.016703; A. Badinski et al., J. Phys.: Condens. Matter 22, 074202 (2010)JCOMEL0953-898410.1088/0953-8984/22/7/074202]. We obtain convergent, accurate confidence intervals on the variance of the local energy of an electron gas and on the Hellmann-Feynman force on an atom in the all-electron carbon dimer. In each of these cases the uncertainty on our estimator is 45% and 60 times smaller, respectively, than the nominal (ill-defined) standard error.
Collapse
Affiliation(s)
- Pablo López Ríos
- Max-Planck Institute for Solid State Research, Heisenbergstraße 1, 70569 Stuttgart, Germany.,Theory of Condensed Matter Group, Cavendish Laboratory, 19 J. J. Thomson Avenue, Cambridge CB3 0HE, United Kingdom
| | - Gareth J Conduit
- Theory of Condensed Matter Group, Cavendish Laboratory, 19 J. J. Thomson Avenue, Cambridge CB3 0HE, United Kingdom
| |
Collapse
|
6
|
Filippi C, Assaraf R, Moroni S. Simple formalism for efficient derivatives and multi-determinant expansions in quantum Monte Carlo. J Chem Phys 2016; 144:194105. [DOI: 10.1063/1.4948778] [Citation(s) in RCA: 41] [Impact Index Per Article: 5.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/15/2022] Open
Affiliation(s)
- Claudia Filippi
- MESA+ Institute for Nanotechnology, University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands
| | - Roland Assaraf
- Sorbonne Universités, UPMC Univ Paris 06, CNRS, Laboratoire de Chimie Théorique CC 137-4, place Jussieu F-75252 Paris Cedex 05, France
| | - Saverio Moroni
- CNR-IOM DEMOCRITOS, Istituto Officina dei Materiali, and SISSA Scuola Internazionale Superiore di Studi Avanzati, Via Bonomea 265, I-34136 Trieste, Italy
| |
Collapse
|
7
|
Affiliation(s)
- Brian M. Austin
- Kenneth S. Pitzer Center for Theoretical Chemistry, Department of Chemistry, University of California at Berkeley, Berkeley, California 94720, United States
| | - Dmitry Yu. Zubarev
- Kenneth S. Pitzer Center for Theoretical Chemistry, Department of Chemistry, University of California at Berkeley, Berkeley, California 94720, United States
| | - William A. Lester
- Kenneth S. Pitzer Center for Theoretical Chemistry, Department of Chemistry, University of California at Berkeley, Berkeley, California 94720, United States
- Chemical Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, United States
| |
Collapse
|
8
|
Lüchow A. Quantum Monte Carlo methods. WILEY INTERDISCIPLINARY REVIEWS-COMPUTATIONAL MOLECULAR SCIENCE 2011. [DOI: 10.1002/wcms.40] [Citation(s) in RCA: 43] [Impact Index Per Article: 3.3] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 12/27/2022]
|
9
|
Sorella S, Capriotti L. Algorithmic differentiation and the calculation of forces by quantum Monte Carlo. J Chem Phys 2010; 133:234111. [DOI: 10.1063/1.3516208] [Citation(s) in RCA: 72] [Impact Index Per Article: 5.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
|
10
|
Wagner LK, Grossman JC. Quantum Monte Carlo calculations for minimum energy structures. PHYSICAL REVIEW LETTERS 2010; 104:210201. [PMID: 20867077 DOI: 10.1103/physrevlett.104.210201] [Citation(s) in RCA: 15] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/08/2009] [Revised: 03/24/2010] [Indexed: 05/29/2023]
Abstract
We present an efficient method to find minimum energy structures using energy estimates from accurate quantum Monte Carlo calculations. This method involves a stochastic process formed from the stochastic energy estimates from Monte Carlo calculations that can be averaged to find precise structural minima while using inexpensive calculations with moderate statistical uncertainty. We demonstrate the applicability of the algorithm by minimizing the energy of the H2O-OH- complex and showing that the structural minima from quantum Monte Carlo calculations affect the qualitative behavior of the potential energy surface substantially.
Collapse
Affiliation(s)
- Lucas K Wagner
- Department of Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA.
| | | |
Collapse
|
11
|
Badinski A, Haynes PD, Trail JR, Needs RJ. Methods for calculating forces within quantum Monte Carlo simulations. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2010; 22:074202. [PMID: 21386380 DOI: 10.1088/0953-8984/22/7/074202] [Citation(s) in RCA: 12] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/30/2023]
Abstract
Atomic force calculations within the variational and diffusion quantum Monte Carlo methods are described. The advantages of calculating diffusion quantum Monte Carlo forces with the 'pure' rather than the 'mixed' probability distribution are discussed. An accurate and practical method for calculating forces using the pure distribution is presented and tested for the SiH molecule. The statistics of force estimators are explored and violations of the central limit theorem are found in some cases.
Collapse
Affiliation(s)
- A Badinski
- Theory of Condensed Matter Group, Cavendish Laboratory, Cambridge CB3 0HE, UK
| | | | | | | |
Collapse
|
12
|
Needs RJ, Towler MD, Drummond ND, López Ríos P. Continuum variational and diffusion quantum Monte Carlo calculations. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2010; 22:023201. [PMID: 21386247 DOI: 10.1088/0953-8984/22/2/023201] [Citation(s) in RCA: 163] [Impact Index Per Article: 11.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/30/2023]
Abstract
This topical review describes the methodology of continuum variational and diffusion quantum Monte Carlo calculations. These stochastic methods are based on many-body wavefunctions and are capable of achieving very high accuracy. The algorithms are intrinsically parallel and well suited to implementation on petascale computers, and the computational cost scales as a polynomial in the number of particles. A guide to the systems and topics which have been investigated using these methods is given. The bulk of the article is devoted to an overview of the basic quantum Monte Carlo methods, the forms and optimization of wavefunctions, performing calculations under periodic boundary conditions, using pseudopotentials, excited-state calculations, sources of calculational inaccuracy, and calculating energy differences and forces.
Collapse
Affiliation(s)
- R J Needs
- Theory of Condensed Matter Group, Cavendish Laboratory, Cambridge CB3 0HE, UK
| | | | | | | |
Collapse
|
13
|
Badinski A, Trail JR, Needs RJ. Energy derivatives in quantum Monte Carlo involving the zero-variance property. J Chem Phys 2008; 129:224101. [DOI: 10.1063/1.3013817] [Citation(s) in RCA: 14] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
|
14
|
Trail JR. Heavy-tailed random error in quantum Monte Carlo. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2008; 77:016703. [PMID: 18351956 DOI: 10.1103/physreve.77.016703] [Citation(s) in RCA: 20] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/11/2007] [Indexed: 05/26/2023]
Abstract
The combination of continuum many-body quantum physics and Monte Carlo methods provide a powerful and well established approach to first principles calculations for large systems. Replacing the exact solution of the problem with a statistical estimate requires a measure of the random error in the estimate for it to be useful. Such a measure of confidence is usually provided by assuming the central limit theorem to hold true. In what follows it is demonstrated that, for the most popular implementation of the variational Monte Carlo method, the central limit theorem has limited validity, or is invalid and must be replaced by a generalized central limit theorem. Estimates of the total energy and the variance of the local energy are examined in detail, and shown to exhibit uncontrolled statistical errors through an explicit derivation of the distribution of the random error. Several examples are given of estimated quantities for which the central limit theorem is not valid. The approach used is generally applicable to characterizing the random error of estimates, and to quantum Monte Carlo methods beyond variational Monte Carlo.
Collapse
Affiliation(s)
- J R Trail
- University of Cambridge, Cambridge, CB3 0HE, United Kingdom.
| |
Collapse
|
15
|
Trail JR. Alternative sampling for variational quantum Monte Carlo. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2008; 77:016704. [PMID: 18351957 DOI: 10.1103/physreve.77.016704] [Citation(s) in RCA: 12] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/11/2007] [Indexed: 05/26/2023]
Abstract
Expectation values of physical quantities may accurately be obtained by the evaluation of integrals within many-body quantum mechanics, and these multidimensional integrals may be estimated using Monte Carlo methods. In a previous publication it has been shown that for the simplest, most commonly applied strategy in continuum quantum Monte Carlo, the random error in the resulting estimates is not well controlled. At best the central limit theorem is valid in its weakest form, and at worst it is invalid and replaced by an alternative generalized central limit theorem and non-normal random error. In both cases the random error is not controlled. Here we consider a new "residual sampling strategy" that reintroduces the central limit theorem in its strongest form, and provides full control of the random error in estimates. Estimates of the total energy and the variance of the local energy within variational Monte Carlo are considered in detail, and the approach presented may be generalized to expectation values of other operators, and to other variants of the quantum Monte Carlo method.
Collapse
Affiliation(s)
- J R Trail
- University of Cambridge, JJ Thomson Avenue, Cambridge, CB3 0HE, United Kingdom.
| |
Collapse
|