Grid-based diffusion Monte Carlo for fermions without the fixed-node approximation
Abstract
A diffusion Monte Carlo algorithm is introduced that can determine the correct nodal structure of the wave function of a few-fermion system and its ground-state energy without an uncontrolled bias. This is achieved by confining signed random walkers to the points of a uniform infinite spatial grid, allowing them to meet and annihilate one another to establish the nodal structure without the fixed-node approximation. An imaginary-time propagator is derived rigorously from a discretized Hamiltonian, governing a non-Gaussian, sign-flipping, branching, and mutually annihilating random walk of particles. The accuracy of the resulting stochastic representations of a fermion wave function is limited only by the grid and imaginary-time resolutions and can be improved in a controlled manner. Here, the method is tested for a series of model problems including fermions in a harmonic trap as well as the He atom in its singlet or triplet ground state. For the latter case, the energies approach from above with increasing grid resolution and converge within 0.015 Eh of the exact basis-set-limit value for the grid spacing of 0.08 a.u. with a statistical uncertainty of 10–5 Eh without an importance sampling or Jastrow factor.
- Authors:
-
- Univ. of Illinois at Urbana-Champaign, IL (United States)
- Publication Date:
- Research Org.:
- Univ. of Illinois at Urbana-Champaign, IL (United States)
- Sponsoring Org.:
- USDOE Office of Science (SC), Basic Energy Sciences (BES)
- OSTI Identifier:
- 1594980
- Grant/Contract Number:
- SC0006028
- Resource Type:
- Accepted Manuscript
- Journal Name:
- Physical Review E
- Additional Journal Information:
- Journal Volume: 101; Journal Issue: 1; Journal ID: ISSN 2470-0045
- Publisher:
- American Physical Society (APS)
- Country of Publication:
- United States
- Language:
- English
- Subject:
- 74 ATOMIC AND MOLECULAR PHYSICS
Citation Formats
Kunitsa, Alexander A., and Hirata, So. Grid-based diffusion Monte Carlo for fermions without the fixed-node approximation. United States: N. p., 2020.
Web. doi:10.1103/PhysRevE.101.013311.
Kunitsa, Alexander A., & Hirata, So. Grid-based diffusion Monte Carlo for fermions without the fixed-node approximation. United States. https://doi.org/10.1103/PhysRevE.101.013311
Kunitsa, Alexander A., and Hirata, So. Mon .
"Grid-based diffusion Monte Carlo for fermions without the fixed-node approximation". United States. https://doi.org/10.1103/PhysRevE.101.013311. https://www.osti.gov/servlets/purl/1594980.
@article{osti_1594980,
title = {Grid-based diffusion Monte Carlo for fermions without the fixed-node approximation},
author = {Kunitsa, Alexander A. and Hirata, So},
abstractNote = {A diffusion Monte Carlo algorithm is introduced that can determine the correct nodal structure of the wave function of a few-fermion system and its ground-state energy without an uncontrolled bias. This is achieved by confining signed random walkers to the points of a uniform infinite spatial grid, allowing them to meet and annihilate one another to establish the nodal structure without the fixed-node approximation. An imaginary-time propagator is derived rigorously from a discretized Hamiltonian, governing a non-Gaussian, sign-flipping, branching, and mutually annihilating random walk of particles. The accuracy of the resulting stochastic representations of a fermion wave function is limited only by the grid and imaginary-time resolutions and can be improved in a controlled manner. Here, the method is tested for a series of model problems including fermions in a harmonic trap as well as the He atom in its singlet or triplet ground state. For the latter case, the energies approach from above with increasing grid resolution and converge within 0.015 Eh of the exact basis-set-limit value for the grid spacing of 0.08 a.u. with a statistical uncertainty of 10–5 Eh without an importance sampling or Jastrow factor.},
doi = {10.1103/PhysRevE.101.013311},
journal = {Physical Review E},
number = 1,
volume = 101,
place = {United States},
year = {Mon Jan 27 00:00:00 EST 2020},
month = {Mon Jan 27 00:00:00 EST 2020}
}
Web of Science
Works referenced in this record:
Size-consistent variational approaches to nonlocal pseudopotentials: Standard and lattice regularized diffusion Monte Carlo methods revisited
journal, April 2010
- Casula, Michele; Moroni, Saverio; Sorella, Sandro
- The Journal of Chemical Physics, Vol. 132, Issue 15
On the product of semi-groups of operators
journal, April 1959
- Trotter, H. F.
- Proceedings of the American Mathematical Society, Vol. 10, Issue 4
Nodal hypersurfaces and Anderson’s random‐walk simulation of the Schrödinger equation
journal, June 1976
- Klein, D. J.; Pickett, H. M.
- The Journal of Chemical Physics, Vol. 64, Issue 11
Quantum chemistry by random walk: Exact treatment of many‐electron systems
journal, November 1991
- Anderson, James B.; Traynor, Carol A.; Boghosian, Bruce M.
- The Journal of Chemical Physics, Vol. 95, Issue 10
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
Quantum Monte Carlo methods: Quantum Monte Carlo methods
journal, April 2011
- Lüchow, Arne
- Wiley Interdisciplinary Reviews: Computational Molecular Science, Vol. 1, Issue 3
Numerical Solution of Quantum‐Mechanical Pair Equations
journal, June 1968
- McKoy, Vincent; Winter, N. W.
- The Journal of Chemical Physics, Vol. 48, Issue 12
Exact Monte Carlo Method for Continuum Fermion Systems
journal, October 2000
- Kalos, M. H.; Pederiva, Francesco
- Physical Review Letters, Vol. 85, Issue 17
A random‐walk simulation of the Schrödinger equation: H + 3
journal, August 1975
- Anderson, James B.
- The Journal of Chemical Physics, Vol. 63, Issue 4
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
Mirror potentials and the fermion problem
journal, November 1985
- Carlson, J.; Kalos, M. H.
- Physical Review C, Vol. 32, Issue 5
Correlated pairs in model fermion Monte Carlo calculations
journal, May 1996
- Kalos, M. H.
- Physical Review E, Vol. 53, Issue 5
Diffusion Monte Carlo Method with Lattice Regularization
journal, September 2005
- Casula, Michele; Filippi, Claudia; Sorella, Sandro
- Physical Review Letters, Vol. 95, Issue 10
Error estimates on averages of correlated data
journal, July 1989
- Flyvbjerg, H.; Petersen, H. G.
- The Journal of Chemical Physics, Vol. 91, Issue 1
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
Coupled-Cluster in Real Space. 1. CC2 Ground State Energies Using Multiresolution Analysis
journal, November 2017
- Kottmann, Jakob S.; Bischoff, Florian A.
- Journal of Chemical Theory and Computation, Vol. 13, Issue 12
Monte Carlo explicitly correlated second-order many-body perturbation theory
journal, October 2016
- Johnson, Cole M.; Doran, Alexander E.; Zhang, Jinmei
- The Journal of Chemical Physics, Vol. 145, Issue 15
Computing molecular correlation energies with guaranteed precision
journal, September 2013
- Bischoff, Florian A.; Valeev, Edward F.
- The Journal of Chemical Physics, Vol. 139, Issue 11
Monte Carlo Calculations of the Ground State of Three- and Four-Body Nuclei
journal, November 1962
- Kalos, M. H.
- Physical Review, Vol. 128, Issue 4
Model fermion Monte Carlo with correlated pairs. II
journal, October 1997
- Kalos, M. H.; Schmidt, K. E.
- Journal of Statistical Physics, Vol. 89, Issue 1-2
Numerical solution of the Sinanoǧlu equation using a multicentre radial-angular grid
journal, June 2016
- Hirata, So; Shiozaki, Toru; Johnson, Cole M.
- Molecular Physics, Vol. 115, Issue 5
Fixed-node Monte Carlo study of the two-dimensional Hubbard model
journal, November 1991
- An, Guozhong; van Leeuwen, J. M. J.
- Physical Review B, Vol. 44, Issue 17
Remedy for the fermion sign problem in the diffusion Monte Carlo method for few fermions with antisymmetric diffusion process
journal, February 2006
- Mishchenko, Yuriy
- Physical Review E, Vol. 73, Issue 2
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
Unexpected Symmetry in the Nodal Structure of the He Atom
journal, September 2005
- Bressanini, Dario; Reynolds, Peter J.
- Physical Review Letters, Vol. 95, Issue 11
Computing many-body wave functions with guaranteed precision: The first-order Møller-Plesset wave function for the ground state of helium atom
journal, September 2012
- Bischoff, Florian A.; Harrison, Robert J.; Valeev, Edward F.
- The Journal of Chemical Physics, Vol. 137, Issue 10
Stochastic evaluation of second-order many-body perturbation energies
journal, November 2012
- Willow, Soohaeng Yoo; Kim, Kwang S.; Hirata, So
- The Journal of Chemical Physics, Vol. 137, Issue 20
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
Many‐Electron Theory of Atoms and Molecules. II
journal, June 1962
- Sinanoğlu, Oktay
- The Journal of Chemical Physics, Vol. 36, Issue 12
Quantum simulation of systems with nodal surfaces
journal, August 1986
- Coker, D. F.; Watts, R. O.
- Molecular Physics, Vol. 58, Issue 6
Quantum Monte Carlo Methods
journal, January 2002
- Kawashima, Naoki
- Progress of Theoretical Physics Supplement, Vol. 145
Size-consistent variational approaches to non-local pseudopotentials: standard and lattice regularized diffusion Monte Carlo methods revisited
text, January 2010
- Casula, Michele; Moroni, Saverio; Sorella, Sandro
- arXiv
An Exact Monte Carlo Method for Continuum Fermion Systems
text, January 2000
- Kalos, M. H.; Pederiva, Francesco
- arXiv