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Title: Observations on variational and projector Monte Carlo methods

Abstract

In this work, variational Monte Carlo and various projector Monte Carlo (PMC) methods are presented in a unified manner. Similarities and differences between the methods and choices made in designing the methods are discussed. Both methods where the Monte Carlo walk is performed in a discrete space and methods where it is performed in a continuous space are considered. It is pointed out that the usual prescription for importance sampling may not be advantageous depending on the particular quantum Monte Carlo method used and the observables of interest, so alternate prescriptions are presented. The nature of the sign problem is discussed for various versions of PMC methods. A prescription for an exact PMC method in real space, i.e., a method that does not make a fixed-node or similar approximation and does not have a finite basis error, is presented. This method is likely to be practical for systems with a small number of electrons. Lastly, approximate PMC methods that are applicable to larger systems and go beyond the fixed-node approximation are also discussed.

Authors:
 [1]
  1. Cornell Univ., Ithaca, NY (United States). Laboratory of Atomic and Solid State Physics
Publication Date:
Research Org.:
Cornell Univ., Ithaca, NY (United States)
Sponsoring Org.:
USDOE Office of Science (SC)
OSTI Identifier:
1469560
Alternate Identifier(s):
OSTI ID: 1224329
Grant/Contract Number:  
SC0006650
Resource Type:
Accepted Manuscript
Journal Name:
Journal of Chemical Physics
Additional Journal Information:
Journal Volume: 143; Journal Issue: 16; Journal ID: ISSN 0021-9606
Publisher:
American Institute of Physics (AIP)
Country of Publication:
United States
Language:
English
Subject:
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS

Citation Formats

Umrigar, C. J. Observations on variational and projector Monte Carlo methods. United States: N. p., 2015. Web. doi:10.1063/1.4933112.
Umrigar, C. J. Observations on variational and projector Monte Carlo methods. United States. doi:10.1063/1.4933112.
Umrigar, C. J. Mon . "Observations on variational and projector Monte Carlo methods". United States. doi:10.1063/1.4933112. https://www.osti.gov/servlets/purl/1469560.
@article{osti_1469560,
title = {Observations on variational and projector Monte Carlo methods},
author = {Umrigar, C. J.},
abstractNote = {In this work, variational Monte Carlo and various projector Monte Carlo (PMC) methods are presented in a unified manner. Similarities and differences between the methods and choices made in designing the methods are discussed. Both methods where the Monte Carlo walk is performed in a discrete space and methods where it is performed in a continuous space are considered. It is pointed out that the usual prescription for importance sampling may not be advantageous depending on the particular quantum Monte Carlo method used and the observables of interest, so alternate prescriptions are presented. The nature of the sign problem is discussed for various versions of PMC methods. A prescription for an exact PMC method in real space, i.e., a method that does not make a fixed-node or similar approximation and does not have a finite basis error, is presented. This method is likely to be practical for systems with a small number of electrons. Lastly, approximate PMC methods that are applicable to larger systems and go beyond the fixed-node approximation are also discussed.},
doi = {10.1063/1.4933112},
journal = {Journal of Chemical Physics},
number = 16,
volume = 143,
place = {United States},
year = {2015},
month = {10}
}

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Cited by: 6 works
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Works referenced in this record:

Full optimization of Jastrow–Slater wave functions with application to the first-row atoms and homonuclear diatomic molecules
journal, May 2008

  • Toulouse, Julien; Umrigar, C. J.
  • The Journal of Chemical Physics, Vol. 128, Issue 17
  • DOI: 10.1063/1.2908237

Monte-Carlo solution of Schrödinger's equation
journal, February 1971


Exact Monte Carlo Method for Continuum Fermion Systems
journal, October 2000


Stable Liquid Hydrogen at High Pressure by a Novel Ab Initio Molecular-Dynamics Calculation
journal, March 2008


A diffusion Monte Carlo algorithm with very small time‐step errors
journal, August 1993

  • Umrigar, C. J.; Nightingale, M. P.; Runge, K. J.
  • The Journal of Chemical Physics, Vol. 99, Issue 4
  • DOI: 10.1063/1.465195

The effect of quantization on the full configuration interaction quantum Monte Carlo sign problem
journal, January 2013

  • Kolodrubetz, M. H.; Spencer, J. S.; Clark, B. K.
  • The Journal of Chemical Physics, Vol. 138, Issue 2
  • DOI: 10.1063/1.4773819

Quantum chemistry by random walk. H 2 P , H + 3 D 3 h 1 A1 , H 2 3 Σ + u , H 4 1 Σ + g , Be 1 S
journal, November 1976

  • Anderson, James B.
  • The Journal of Chemical Physics, Vol. 65, Issue 10
  • DOI: 10.1063/1.432868

Quantum Monte Carlo simulations of solids
journal, January 2001


Prospects for release-node quantum Monte Carlo
journal, November 2011

  • Tubman, Norm M.; DuBois, Jonathan L.; Hood, Randolph Q.
  • The Journal of Chemical Physics, Vol. 135, Issue 18
  • DOI: 10.1063/1.3659143

Correlated pairs in model fermion Monte Carlo calculations
journal, May 1996


Model fermion Monte Carlo method with antithetical pairs
journal, October 1994


Green Function Monte Carlo with Stochastic Reconfiguration
journal, May 1998


Green’s function Monte Carlo for few fermion problems
journal, December 1982

  • Arnow, D. M.; Kalos, M. H.; Lee, Michael A.
  • The Journal of Chemical Physics, Vol. 77, Issue 11
  • DOI: 10.1063/1.443762

Fixed‐node quantum Monte Carlo for molecules a) b)
journal, December 1982

  • Reynolds, Peter J.; Ceperley, David M.; Alder, Berni J.
  • The Journal of Chemical Physics, Vol. 77, Issue 11
  • DOI: 10.1063/1.443766

Semistochastic Projector Monte Carlo Method
journal, December 2012


Fermion Monte Carlo for continuum systems
journal, May 2000


The calculation of excited state properties with quantum Monte Carlo
journal, November 1988

  • Ceperley, D. M.; Bernu, B.
  • The Journal of Chemical Physics, Vol. 89, Issue 10
  • DOI: 10.1063/1.455398

Optimization of Ground- and Excited-State Wave Functions and van der Waals Clusters
journal, July 2001


Optimum and efficient sampling for variational quantum Monte Carlo
journal, November 2010

  • Trail, J. R.; Maezono, Ryo
  • The Journal of Chemical Physics, Vol. 133, Issue 17
  • DOI: 10.1063/1.3488651

Alleviation of the Fermion-Sign Problem by Optimization of Many-Body Wave Functions
journal, March 2007


Equation of State Calculations by Fast Computing Machines
journal, June 1953

  • Metropolis, Nicholas; Rosenbluth, Arianna W.; Rosenbluth, Marshall N.
  • The Journal of Chemical Physics, Vol. 21, Issue 6
  • DOI: 10.1063/1.1699114

Weak binding between two aromatic rings: Feeling the van der Waals attraction by quantum Monte Carlo methods
journal, July 2007

  • Sorella, Sandro; Casula, Michele; Rocca, Dario
  • The Journal of Chemical Physics, Vol. 127, Issue 1
  • DOI: 10.1063/1.2746035

Monte Carlo sampling methods using Markov chains and their applications
journal, April 1970


Unbiased reduced density matrices and electronic properties from full configuration interaction quantum Monte Carlo
journal, December 2014

  • Overy, Catherine; Booth, George H.; Blunt, N. S.
  • The Journal of Chemical Physics, Vol. 141, Issue 24
  • DOI: 10.1063/1.4904313

Fermion Monte Carlo without fixed nodes: A game of life, death, and annihilation in Slater determinant space
journal, January 2009

  • Booth, George H.; Thom, Alex J. W.; Alavi, Ali
  • The Journal of Chemical Physics, Vol. 131, Issue 5
  • DOI: 10.1063/1.3193710

Exact Monte Carlo calculation for few-electron systems
journal, November 1991


Quantum Monte Carlo Method using Phase-Free Random Walks with Slater Determinants
journal, April 2003


Symmetry in auxiliary-field quantum Monte Carlo calculations
journal, September 2013


Optimization of quantum Monte Carlo wave functions by energy minimization
journal, February 2007

  • Toulouse, Julien; Umrigar, C. J.
  • The Journal of Chemical Physics, Vol. 126, Issue 8
  • DOI: 10.1063/1.2437215

An auxiliary-field quantum Monte Carlo study of the chromium dimer
journal, February 2015

  • Purwanto, Wirawan; Zhang, Shiwei; Krakauer, Henry
  • The Journal of Chemical Physics, Vol. 142, Issue 6
  • DOI: 10.1063/1.4906829

Convergence of many-body wave-function expansions using a plane-wave basis: From homogeneous electron gas to solid state systems
journal, July 2012


Communications: Survival of the fittest: Accelerating convergence in full configuration-interaction quantum Monte Carlo
journal, January 2010

  • Cleland, Deidre; Booth, George H.; Alavi, Ali
  • The Journal of Chemical Physics, Vol. 132, Issue 4
  • DOI: 10.1063/1.3302277

Applications of quantum Monte Carlo methods in condensed systems
journal, January 2011


Diffusion Monte Carlo methods with a fixed number of walkers
journal, April 2000


Proof for an upper bound in fixed-node Monte Carlo for lattice fermions
journal, May 1995

  • ten Haaf, D. F. B.; van Bemmel, H. J. M.; van Leeuwen, J. M. J.
  • Physical Review B, Vol. 51, Issue 19
  • DOI: 10.1103/PhysRevB.51.13039

Energy and Variance Optimization of Many-Body Wave Functions
journal, April 2005


A very high accuracy potential energy surface for H3
journal, January 1999

  • Mark Wu, Y. -S; Kuppermann, Aron; Anderson, James B.
  • Physical Chemistry Chemical Physics, Vol. 1, Issue 6
  • DOI: 10.1039/a808797k

Ground State of the Electron Gas by a Stochastic Method
journal, August 1980


The sign problem and population dynamics in the full configuration interaction quantum Monte Carlo method
journal, February 2012

  • Spencer, J. S.; Blunt, N. S.; Foulkes, W. M. C.
  • The Journal of Chemical Physics, Vol. 136, Issue 5
  • DOI: 10.1063/1.3681396

Path integrals in the theory of condensed helium
journal, April 1995


Linear-scaling and parallelisable algorithms for stochastic quantum chemistry
journal, January 2014


Towards an exact description of electronic wavefunctions in real solids
journal, December 2012

  • Booth, George H.; Grüneis, Andreas; Kresse, Georg
  • Nature, Vol. 493, Issue 7432
  • DOI: 10.1038/nature11770

Quantum Monte Carlo for molecules: Green’s function and nodal release
journal, December 1984

  • Ceperley, D. M.; Alder, B. J.
  • The Journal of Chemical Physics, Vol. 81, Issue 12
  • DOI: 10.1063/1.447637

Partial node configuration-interaction Monte Carlo as applied to the Fermi polaron
journal, August 2012


Effects of three-body and backflow correlations in the two-dimensional electron gas
journal, October 1993


Fermion Monte Carlo
journal, September 1999


    Works referencing / citing this record:

    Complementary first and second derivative methods for ansatz optimization in variational Monte Carlo
    journal, January 2019

    • Otis, Leon; Neuscamman, Eric
    • Physical Chemistry Chemical Physics, Vol. 21, Issue 27
    • DOI: 10.1039/c9cp02269d