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Hutcheon M. Stochastic nodal surfaces in quantum Monte Carlo calculations. Phys Rev E 2020; 102:042105. [PMID: 33212637 DOI: 10.1103/physreve.102.042105] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/17/2020] [Accepted: 09/16/2020] [Indexed: 11/07/2022]
Abstract
Treating the fermionic ground state-problem as a constrained stochastic optimization problem, a formalism for fermionic quantum Monte Carlo is developed that makes no reference to a trial wave function. Exchange symmetry is enforced by nonlocal terms appearing in the Green's function corresponding to an additional walker propagation channel. Complemented by a treatment of diffusion that encourages the formation of a stochastic nodal surface, we find that an approximate long-range extension of walker cancellations can be employed without introducing significant bias, reducing the number of walkers required for a stable calculation. A proof-of-concept implementation is shown to give a stable fermionic ground state for simple harmonic and atomic systems.
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Affiliation(s)
- Michael Hutcheon
- Theory of Condensed Matter Group, Cavendish Laboratory, J. J. Thomson Avenue, Cambridge CB3 0HE, United Kingdom
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Tubman NM, Freeman CD, Levine DS, Hait D, Head-Gordon M, Whaley KB. Modern Approaches to Exact Diagonalization and Selected Configuration Interaction with the Adaptive Sampling CI Method. J Chem Theory Comput 2020; 16:2139-2159. [DOI: 10.1021/acs.jctc.8b00536] [Citation(s) in RCA: 56] [Impact Index Per Article: 14.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/30/2022]
Affiliation(s)
- Norm M. Tubman
- Kenneth S. Pitzer Center for Theoretical Chemistry, Department of Chemistry, University of California, Berkeley, California 94720, United States
- Chemical Sciences Division, Lawrence Berkeley National Laboratory Berkeley, California 94720, United States
| | - C. Daniel Freeman
- Kenneth S. Pitzer Center for Theoretical Chemistry, Department of Chemistry, University of California, Berkeley, California 94720, United States
- Chemical Sciences Division, Lawrence Berkeley National Laboratory Berkeley, California 94720, United States
| | - Daniel S. Levine
- Kenneth S. Pitzer Center for Theoretical Chemistry, Department of Chemistry, University of California, Berkeley, California 94720, United States
- Chemical Sciences Division, Lawrence Berkeley National Laboratory Berkeley, California 94720, United States
| | - Diptarka Hait
- Kenneth S. Pitzer Center for Theoretical Chemistry, Department of Chemistry, University of California, Berkeley, California 94720, United States
- Chemical Sciences Division, Lawrence Berkeley National Laboratory Berkeley, California 94720, United States
| | - Martin Head-Gordon
- Kenneth S. Pitzer Center for Theoretical Chemistry, Department of Chemistry, University of California, Berkeley, California 94720, United States
- Chemical Sciences Division, Lawrence Berkeley National Laboratory Berkeley, California 94720, United States
| | - K. Birgitta Whaley
- Kenneth S. Pitzer Center for Theoretical Chemistry, Department of Chemistry, University of California, Berkeley, California 94720, United States
- Chemical Sciences Division, Lawrence Berkeley National Laboratory Berkeley, California 94720, United States
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Calcavecchia F, Holzmann M. Fermion sign problem in imaginary-time projection continuum quantum Monte Carlo with local interaction. Phys Rev E 2016; 93:043321. [PMID: 27176442 DOI: 10.1103/physreve.93.043321] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/13/2016] [Indexed: 06/05/2023]
Abstract
We use the shadow wave function formalism as a convenient model to study the fermion sign problem affecting all projector quantum Monte Carlo methods in continuum space. We demonstrate that the efficiency of imaginary-time projection algorithms decays exponentially with increasing number of particles and/or imaginary-time propagation. Moreover, we derive an analytical expression that connects the localization of the system with the magnitude of the sign problem, illustrating this behavior through numerical results. Finally, we discuss the computational complexity of the fermion sign problem and methods for alleviating its severity.
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Affiliation(s)
- Francesco Calcavecchia
- LPMMC, UMR 5493 of CNRS, Université Grenoble Alpes, 38042 Grenoble, France; Institute of Physics, Johannes Gutenberg University, Staudingerweg 7, D-55128 Mainz, Germany; and Graduate School of Excellence Materials Science in Mainz, Staudingerweg 9, D-55128 Mainz, Germany
| | - Markus Holzmann
- LPMMC, UMR 5493 of CNRS, Université Grenoble Alpes, 38042 Grenoble, France and Institut Laue Langevin, BP 156, F-38042 Grenoble Cedex 9, France
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Umrigar CJ. Observations on variational and projector Monte Carlo methods. J Chem Phys 2015; 143:164105. [DOI: 10.1063/1.4933112] [Citation(s) in RCA: 22] [Impact Index Per Article: 2.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/22/2022] Open
Affiliation(s)
- C. J. Umrigar
- Laboratory of Atomic and Solid State Physics, Cornell University, Ithaca, New York 14853, USA
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Yang Y, Kylänpää I, Tubman NM, Krogel JT, Hammes-Schiffer S, Ceperley DM. How large are nonadiabatic effects in atomic and diatomic systems? J Chem Phys 2015; 143:124308. [DOI: 10.1063/1.4931667] [Citation(s) in RCA: 13] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Affiliation(s)
- Yubo Yang
- Department of Physics, University of Illinois, Urbana, Illinois 61801, USA
| | - Ilkka Kylänpää
- Department of Physics, University of Illinois, Urbana, Illinois 61801, USA
- Department of Physics, Tampere University of Technology, P.O. Box 692, FI-33101 Tampere, Finland
| | - Norm M. Tubman
- Department of Physics, University of Illinois, Urbana, Illinois 61801, USA
| | - Jaron T. Krogel
- Oak Ridge National Laboratory, Materials Sciences & Technology Division, Oak Ridge, Tennessee 37831, USA
| | | | - David M. Ceperley
- Department of Physics, University of Illinois, Urbana, Illinois 61801, USA
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Tubman NM, Liberatore E, Pierleoni C, Holzmann M, Ceperley DM. Molecular-Atomic Transition along the Deuterium Hugoniot Curve with Coupled Electron-Ion Monte Carlo Simulations. PHYSICAL REVIEW LETTERS 2015; 115:045301. [PMID: 26252690 DOI: 10.1103/physrevlett.115.045301] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/21/2014] [Indexed: 06/04/2023]
Abstract
We have performed simulations of the principal deuterium Hugoniot curve using coupled electron-ion Monte Carlo calculations. Using highly accurate quantum Monte Carlo methods for the electrons, we study the region of maximum compression along the Hugoniot, where the system undergoes a continuous transition from a molecular fluid to a monatomic fluid. We include all relevant physical corrections so that a direct comparison to experiment can be made. Around 50 GPa we find a maximum compression of 4.85. This compression is approximately 5.5% higher than previous theoretical predictions and 15% higher than the most accurate experimental data. Thus first-principles simulations encompassing the most advanced techniques are in disagreement with the results of the best experiments.
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Affiliation(s)
- Norm M Tubman
- Department of Physics, University of Illinois, Urbana, Illinois 61801, USA
| | | | - Carlo Pierleoni
- Department of Physical and Chemical Sciences, University of L'Aquila and CNISM UdR L'Aquila, Via Vetoio 10, I-67010 L'Aquila, Italy
| | - Markus Holzmann
- LPTMC, Université Pierre et Marie Curie and CNRS, 75005 Paris, France and LPMMC, Université Grenoble I and CNRS, 38042 Grenoble, France
| | - David M Ceperley
- Department of Physics, University of Illinois, Urbana, Illinois 61801, USA
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Nakatsuji H, Nakashima H. Free-complement local-Schrödinger-equation method for solving the Schrödinger equation of atoms and molecules: Basic theories and features. J Chem Phys 2015; 142:084117. [DOI: 10.1063/1.4909520] [Citation(s) in RCA: 17] [Impact Index Per Article: 1.9] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/14/2023] Open
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Abstract
The most commonly employed diffusion Monte Carlo algorithm and some of its variants afford a way to sample configuration space from a so-called “mixed distribution”, the product of an input trial solution to the Schrödinger equation for the ground state and its unknown exact solution. This mixed distribution is sufficient to compute the ground state energy and other properties represented by operators that commute with the Hamiltonian. These energy-related properties are exact, save for a small bias introduced by the input trial function’s incorrect exchange nodes, the so-called “fixed-node error”. However, properties represented by operators that commute with the position operator are also of interest. When calculated by sampling from the mixed distribution, these properties are much more strongly biased by the input trial function. Our objective is to review methods that allow sampling from the desired “pure” distribution, one that is unbiased except for the exchange node error. Thereby, one accurately calculates physical properties such as the dipole and other electrical moments, electrical response properties of molecules, and particle distribution functions for clusters. We survey the results of calculations that employ pure-sampling methods through what has been published in year 2012. Our review also touches on truly exact sampling methods.
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Affiliation(s)
- Stuart M. Rothstein
- Departments of Chemistry and Physics, Brock University, St. Catharines, ON L2S 3A1, Canada
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