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Title: Monte Carlo explicitly correlated second-order many-body perturbation theory

Abstract

A stochastic algorithm is proposed and implemented that computes a basis-set-incompleteness (F12) correction to an ab initio second-order many-body perturbation energy as a short sum of 6- to 15-dimensional integrals of Gaussian-type orbitals, an explicit function of the electron-electron distance (geminal), and its associated excitation amplitudes held fixed at the values suggested by Ten-no. The integrals are directly evaluated (without a resolution-of-the-identity approximation or an auxiliary basis set) by the Metropolis Monte Carlo method. Applications of this method to 17 molecular correlation energies and 12 gas-phase reaction energies reveal that both the nonvariational and variational formulas for the correction give reliable correlation energies (98% or higher) and reaction energies (within 2 kJ mol-1 with a smaller statistical uncertainty) near the complete-basis-set limits by using just the aug-cc-pVDZ basis set. The nonvariational formula is found to be 2–10 times less expensive to evaluate than the variational one, though the latter yields energies that are bounded from below and is, therefore, slightly but systematically more accurate for energy differences. Being capable of using virtually any geminal form, the method confirms the best overall performance of the Slater-type geminal among 6 forms satisfying the same cusp conditions. Lastly, not having to precompute lower-dimensionalmore » integrals analytically, to store them on disk, or to transform them in a nonscalable dense-matrix-multiplication algorithm, the method scales favorably with both system size and computer size; the cost increases only as O(n4) with the number of orbitals (n), and its parallel efficiency reaches 99.9% of the ideal case on going from 16 to 4096 computer processors.« less

Authors:
 [1];  [1];  [2]; ORCiD logo [2]; ORCiD logo [1]
  1. Univ. of Illinois, Urbana-Champaign, IL (United States). Dept. of Chemistry
  2. Virginia Polytechnic Inst. and State Univ. (Virginia Tech), Blacksburg, VA (United States). Dept. of Chemistry
Publication Date:
Research Org.:
Univ. of Illinois at Urbana-Champaign, IL (United States)
Sponsoring Org.:
USDOE
OSTI Identifier:
1473885
Alternate Identifier(s):
OSTI ID: 1329495
Grant/Contract Number:  
FG02-12ER46875; SC0008692; FG02-11ER16211; SC0006028
Resource Type:
Accepted Manuscript
Journal Name:
Journal of Chemical Physics
Additional Journal Information:
Journal Volume: 145; Journal Issue: 15; Journal ID: ISSN 0021-9606
Publisher:
American Institute of Physics (AIP)
Country of Publication:
United States
Language:
English
Subject:
97 MATHEMATICS AND COMPUTING; 37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CHEMISTRY

Citation Formats

Johnson, Cole M., Doran, Alexander E., Zhang, Jinmei, Valeev, Edward F., and Hirata, So. Monte Carlo explicitly correlated second-order many-body perturbation theory. United States: N. p., 2016. Web. doi:10.1063/1.4964854.
Johnson, Cole M., Doran, Alexander E., Zhang, Jinmei, Valeev, Edward F., & Hirata, So. Monte Carlo explicitly correlated second-order many-body perturbation theory. United States. https://doi.org/10.1063/1.4964854
Johnson, Cole M., Doran, Alexander E., Zhang, Jinmei, Valeev, Edward F., and Hirata, So. Fri . "Monte Carlo explicitly correlated second-order many-body perturbation theory". United States. https://doi.org/10.1063/1.4964854. https://www.osti.gov/servlets/purl/1473885.
@article{osti_1473885,
title = {Monte Carlo explicitly correlated second-order many-body perturbation theory},
author = {Johnson, Cole M. and Doran, Alexander E. and Zhang, Jinmei and Valeev, Edward F. and Hirata, So},
abstractNote = {A stochastic algorithm is proposed and implemented that computes a basis-set-incompleteness (F12) correction to an ab initio second-order many-body perturbation energy as a short sum of 6- to 15-dimensional integrals of Gaussian-type orbitals, an explicit function of the electron-electron distance (geminal), and its associated excitation amplitudes held fixed at the values suggested by Ten-no. The integrals are directly evaluated (without a resolution-of-the-identity approximation or an auxiliary basis set) by the Metropolis Monte Carlo method. Applications of this method to 17 molecular correlation energies and 12 gas-phase reaction energies reveal that both the nonvariational and variational formulas for the correction give reliable correlation energies (98% or higher) and reaction energies (within 2 kJ mol-1 with a smaller statistical uncertainty) near the complete-basis-set limits by using just the aug-cc-pVDZ basis set. The nonvariational formula is found to be 2–10 times less expensive to evaluate than the variational one, though the latter yields energies that are bounded from below and is, therefore, slightly but systematically more accurate for energy differences. Being capable of using virtually any geminal form, the method confirms the best overall performance of the Slater-type geminal among 6 forms satisfying the same cusp conditions. Lastly, not having to precompute lower-dimensional integrals analytically, to store them on disk, or to transform them in a nonscalable dense-matrix-multiplication algorithm, the method scales favorably with both system size and computer size; the cost increases only as O(n4) with the number of orbitals (n), and its parallel efficiency reaches 99.9% of the ideal case on going from 16 to 4096 computer processors.},
doi = {10.1063/1.4964854},
journal = {Journal of Chemical Physics},
number = 15,
volume = 145,
place = {United States},
year = {Fri Oct 21 00:00:00 EDT 2016},
month = {Fri Oct 21 00:00:00 EDT 2016}
}

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Works referenced in this record:

Cusp Conditions for Molecular Wavefunctions
journal, July 1966

  • Pack, Russell T.; Brown, W. Byers
  • The Journal of Chemical Physics, Vol. 45, Issue 2
  • DOI: 10.1063/1.1727605

Møller-plesset calculations taking care of the correlation CUSP
journal, February 1987


New correlation factors for explicitly correlated electronic wave functions
journal, August 2005

  • Tew, David P.; Klopper, Wim
  • The Journal of Chemical Physics, Vol. 123, Issue 7
  • DOI: 10.1063/1.1999632

Thermodynamic limit of the energy density in a crystal
journal, January 2012

  • Hirata, So; Ohnishi, Yu-ya
  • Physical Chemistry Chemical Physics, Vol. 14, Issue 21
  • DOI: 10.1039/c2cp23958b

Second-order Møller–Plesset theory with linear R12 terms (MP2-R12) revisited: Auxiliary basis set method and massively parallel implementation
journal, July 2004

  • Valeev, Edward F.; Janssen, Curtis L.
  • The Journal of Chemical Physics, Vol. 121, Issue 3
  • DOI: 10.1063/1.1759319

Stochastic algorithm for size-extensive vibrational self-consistent field methods on fully anharmonic potential energy surfaces
journal, December 2014

  • Hermes, Matthew R.; Hirata, So
  • The Journal of Chemical Physics, Vol. 141, Issue 24
  • DOI: 10.1063/1.4904220

General orbital invariant MP2-F12 theory
journal, April 2007

  • Werner, Hans-Joachim; Adler, Thomas B.; Manby, Frederick R.
  • The Journal of Chemical Physics, Vol. 126, Issue 16
  • DOI: 10.1063/1.2712434

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

Explicitly Correlated Electrons in Molecules
journal, December 2011

  • Hättig, Christof; Klopper, Wim; Köhn, Andreas
  • Chemical Reviews, Vol. 112, Issue 1
  • DOI: 10.1021/cr200168z

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
  • DOI: 10.1063/1.4820404

Accurate quantum-chemical calculations using Gaussian-type geminal and Gaussian-type orbital basis sets: applications to atoms and diatomics
journal, January 2007

  • Dahle, P?l; Helgaker, Trygve; Jonsson, Dan
  • Physical Chemistry Chemical Physics, Vol. 9, Issue 24
  • DOI: 10.1039/b616488a

R12 methods in explicitly correlated molecular electronic structure theory
journal, July 2006

  • Klopper, Wim; Manby, Frederick R.; Ten-No, Seiichiro
  • International Reviews in Physical Chemistry, Vol. 25, Issue 3
  • DOI: 10.1080/01442350600799921

Accuracy of atomization energies and reaction enthalpies in standard and extrapolated electronic wave function/basis set calculations
journal, June 2000

  • Bak, Keld L.; Jørgensen, Poul; Olsen, Jeppe
  • The Journal of Chemical Physics, Vol. 112, Issue 21
  • DOI: 10.1063/1.481544

A new functional for variational calculation of atomic and molecular second-order correlation energies
journal, September 1982

  • Szalewicz, Krzystof; Jeziorski, Bogumil; Monkhorst, Hendrik J.
  • Chemical Physics Letters, Vol. 91, Issue 3
  • DOI: 10.1016/0009-2614(82)83634-8

Stochastic evaluation of second-order Dyson self-energies
journal, April 2013

  • Willow, Soohaeng Yoo; Kim, Kwang S.; Hirata, So
  • The Journal of Chemical Physics, Vol. 138, Issue 16
  • DOI: 10.1063/1.4801862

Accurate quantum‐chemical calculations: The use of Gaussian‐type geminal functions in the treatment of electron correlation
journal, October 1996

  • Persson, B. Joakim; Taylor, Peter R.
  • The Journal of Chemical Physics, Vol. 105, Issue 14
  • DOI: 10.1063/1.472432

Evaluation of molecular integrals over Gaussian basis functions
journal, July 1976

  • Dupuis, Michel; Rys, John; King, Harry F.
  • The Journal of Chemical Physics, Vol. 65, Issue 1
  • DOI: 10.1063/1.432807

Investigation of the full configuration interaction quantum Monte Carlo method using homogeneous electron gas models
journal, June 2012

  • Shepherd, James J.; Booth, George H.; Alavi, Ali
  • The Journal of Chemical Physics, Vol. 136, Issue 24
  • DOI: 10.1063/1.4720076

Convergence Acceleration of Parallel Monte Carlo Second-Order Many-Body Perturbation Calculations Using Redundant Walkers
journal, September 2013

  • Willow, Soohaeng Yoo; Hermes, Matthew R.; Kim, Kwang S.
  • Journal of Chemical Theory and Computation, Vol. 9, Issue 10
  • DOI: 10.1021/ct400557z

Wave functions with terms linear in the interelectronic coordinates to take care of the correlation cusp. I. General theory
journal, February 1991

  • Kutzelnigg, Werner; Klopper, Wim
  • The Journal of Chemical Physics, Vol. 94, Issue 3
  • DOI: 10.1063/1.459921

Rates of convergence of the partial‐wave expansions of atomic correlation energies
journal, March 1992

  • Kutzelnigg, Werner; Morgan, John D.
  • The Journal of Chemical Physics, Vol. 96, Issue 6
  • DOI: 10.1063/1.462811

Impact of electron–electron cusp on configuration interaction energies
journal, July 2001

  • Prendergast, David; Nolan, M.; Filippi, Claudia
  • The Journal of Chemical Physics, Vol. 115, Issue 4
  • DOI: 10.1063/1.1383585

Improving on the resolution of the identity in linear R12 ab initio theories
journal, September 2004


Symmetry-adapted integrals over many-electron basis functions and operators
journal, May 2001

  • Dahle, P.; Taylor, P. R.
  • Theoretical Chemistry Accounts, Vol. 105, Issue 6
  • DOI: 10.1007/PL00012384

Explicitly Correlated R12/F12 Methods for Electronic Structure
journal, December 2011

  • Kong, Liguo; Bischoff, Florian A.; Valeev, Edward F.
  • Chemical Reviews, Vol. 112, Issue 1
  • DOI: 10.1021/cr200204r

On the eigenfunctions of many-particle systems in quantum mechanics
journal, January 1957


Erratum: “Stochastic many-body perturbation theory for anharmonic molecular vibrations” [J. Chem. Phys. 141, 084105 (2014)]
journal, September 2015

  • Hermes, Matthew R.; Hirata, So
  • The Journal of Chemical Physics, Vol. 143, Issue 12
  • DOI: 10.1063/1.4932101

Variational formulation of perturbative explicitly-correlated coupled-cluster methods
journal, January 2008

  • Torheyden, Martin; Valeev, Edward F.
  • Physical Chemistry Chemical Physics, Vol. 10, Issue 23
  • DOI: 10.1039/b803620a

Monte Carlo MP2 on Many Graphical Processing Units
journal, September 2016

  • Doran, Alexander E.; Hirata, So
  • Journal of Chemical Theory and Computation, Vol. 12, Issue 10
  • DOI: 10.1021/acs.jctc.6b00588

The lowest wave function of the symmetrical many particles system
journal, November 1940


Gaussian Geminals for Electron Pair Correlation
journal, December 1970

  • Pan, Kee‐Chuan; King, Harry F.
  • The Journal of Chemical Physics, Vol. 53, Issue 11
  • DOI: 10.1063/1.1673955

Revisiting the use of explicitly correlated geminals in coupled pair calculations: new perspectives and a proposal
journal, August 2005


Stochastic many-body perturbation theory for anharmonic molecular vibrations
journal, August 2014

  • Hermes, Matthew R.; Hirata, So
  • The Journal of Chemical Physics, Vol. 141, Issue 8
  • DOI: 10.1063/1.4892614

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


�ber den Grundterm der Zweielektronenprobleme von H?, He, Li+, Be++ usw.
journal, March 1930


Note on an Approximation Treatment for Many-Electron Systems
journal, October 1934


Explicitly correlated local second-order perturbation theory with a frozen geminal correlation factor
journal, March 2006

  • Manby, Frederick R.; Werner, Hans-Joachim; Adler, Thomas B.
  • The Journal of Chemical Physics, Vol. 124, Issue 9
  • DOI: 10.1063/1.2173247

Explicitly correlated plane waves: Accelerating convergence in periodic wavefunction expansions
journal, August 2013

  • Grüneis, Andreas; Shepherd, James J.; Alavi, Ali
  • The Journal of Chemical Physics, Vol. 139, Issue 8
  • DOI: 10.1063/1.4818753

Density functional theory for systems of very many atoms
journal, November 1995


Communication: Stochastic evaluation of explicitly correlated second-order many-body perturbation energy
journal, January 2014

  • Willow, Soohaeng Yoo; Zhang, Jinmei; Valeev, Edward F.
  • The Journal of Chemical Physics, Vol. 140, Issue 3
  • DOI: 10.1063/1.4862255

Stochastic, real-space, imaginary-time evaluation of third-order Feynman–Goldstone diagrams
journal, January 2014

  • Willow, Soohaeng Yoo; Hirata, So
  • The Journal of Chemical Physics, Vol. 140, Issue 2
  • DOI: 10.1063/1.4861561

Calculation of the molecular integrals with the range-separated correlation factor
journal, March 2015

  • Silkowski, Michał; Lesiuk, Michał; Moszynski, Robert
  • The Journal of Chemical Physics, Vol. 142, Issue 12
  • DOI: 10.1063/1.4915272

Quantum Monte Carlo simulations of solids
journal, January 2001


Explicitly correlated second-order Møller–Plesset methods with auxiliary basis sets
journal, April 2002

  • Klopper, Wim; Samson, Claire C. M.
  • The Journal of Chemical Physics, Vol. 116, Issue 15
  • DOI: 10.1063/1.1461814

Molecular integrals over Gaussian-type geminal basis functions
journal, October 1997

  • Persson, B. Joakim; Taylor, Peter R.
  • Theoretical Chemistry Accounts: Theory, Computation, and Modeling (Theoretica Chimica Acta), Vol. 97, Issue 1-4
  • DOI: 10.1007/s002140050258

Second-Order Many-Body Perturbation Theory: An Eternal Frontier
journal, December 2013

  • Hirata, So; He, Xiao; Hermes, Matthew R.
  • The Journal of Physical Chemistry A, Vol. 118, Issue 4
  • DOI: 10.1021/jp410587b

Communication: Excited states, dynamic correlation functions and spectral properties from full configuration interaction quantum Monte Carlo
journal, November 2012

  • Booth, George H.; Chan, Garnet Kin-Lic
  • The Journal of Chemical Physics, Vol. 137, Issue 19
  • DOI: 10.1063/1.4766327

Error estimates on averages of correlated data
journal, July 1989

  • Flyvbjerg, H.; Petersen, H. G.
  • The Journal of Chemical Physics, Vol. 91, Issue 1
  • DOI: 10.1063/1.457480

Central Fields and Rydberg Formulas in Wave Mechanics
journal, March 1928


Projector Monte Carlo method based on configuration state functions. Test applications to the H4 system and dissociation of LiH
journal, October 2008


Extensivity of Energy and Electronic and Vibrational Structure Methods for Crystals
journal, May 2012


Numerical solution of the Sinanoǧlu equation using a multicentre radial-angular grid
journal, June 2016


Expeditious Stochastic Approach for MP2 Energies in Large Electronic Systems
journal, December 2012

  • Neuhauser, Daniel; Rabani, Eran; Baer, Roi
  • Journal of Chemical Theory and Computation, Vol. 9, Issue 1
  • DOI: 10.1021/ct300946j

Initiation of explicitly correlated Slater-type geminal theory
journal, November 2004


Analysis of the errors in explicitly correlated electronic structure theory
journal, January 2005

  • May, Andrew J.; Valeev, Edward; Polly, Robert
  • Physical Chemistry Chemical Physics, Vol. 7, Issue 14
  • DOI: 10.1039/b507781h

Accuracy of electronic wave functions in quantum Monte Carlo: The effect of high-order correlations
journal, August 1997

  • Huang, Chien-Jung; Umrigar, C. J.; Nightingale, M. P.
  • The Journal of Chemical Physics, Vol. 107, Issue 8
  • DOI: 10.1063/1.474658

Combining explicitly correlated R12 and Gaussian geminal electronic structure theories
journal, December 2006

  • Valeev, Edward F.
  • The Journal of Chemical Physics, Vol. 125, Issue 24
  • DOI: 10.1063/1.2403852

NWChem: A comprehensive and scalable open-source solution for large scale molecular simulations
journal, September 2010

  • Valiev, M.; Bylaska, E. J.; Govind, N.
  • Computer Physics Communications, Vol. 181, Issue 9, p. 1477-1489
  • DOI: 10.1016/j.cpc.2010.04.018

New implementation of second-order Møller-Plesset perturbation theory with an analytic Slater-type geminal
journal, January 2007

  • Ten-no, Seiichiro
  • The Journal of Chemical Physics, Vol. 126, Issue 1
  • DOI: 10.1063/1.2403853

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
  • DOI: 10.1063/1.4768697

Efficient recursive computation of molecular integrals over Cartesian Gaussian functions
journal, April 1986

  • Obara, S.; Saika, A.
  • The Journal of Chemical Physics, Vol. 84, Issue 7
  • DOI: 10.1063/1.450106

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
  • DOI: 10.1063/1.4747538

LI. The zero-point energy of a system of particles
journal, May 1949

  • Dingle, R. B.
  • The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, Vol. 40, Issue 304
  • DOI: 10.1080/14786444908521743

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

Elimination of Coulombic infinities through transformation of the Hamiltonian
journal, November 1998

  • Nooijen, Marcel; Bartlett, Rodney J.
  • The Journal of Chemical Physics, Vol. 109, Issue 19
  • DOI: 10.1063/1.477485

One- and two-electron integrals over cartesian gaussian functions
journal, February 1978


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

Local configuration interaction: An efficient approach for larger molecules
journal, January 1985


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


Explicitly correlated second order perturbation theory: Introduction of a rational generator and numerical quadratures
journal, January 2004

  • Ten-no, Seiichiro
  • The Journal of Chemical Physics, Vol. 121, Issue 1
  • DOI: 10.1063/1.1757439

Explicitly correlated electronic structure theory from R12/F12 ansätze: Explicitly correlated electronic structure theory
journal, May 2011

  • Ten-no, Seiichiro; Noga, Jozef
  • Wiley Interdisciplinary Reviews: Computational Molecular Science, Vol. 2, Issue 1
  • DOI: 10.1002/wcms.68

Diagrammatic Monte Carlo for correlated fermions
journal, April 2010


M ONTE C ARLO M ETHODS IN E LECTRONIC S TRUCTURES FOR L ARGE S YSTEMS
journal, October 2000


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


Computation of two-electron Gaussian integrals for wave functions including the correlation factor r12exp(−γr122)
journal, November 2002


r 12-Dependent terms in the wave function as closed sums of partial wave amplitudes for large l
journal, December 1985


The Normal State of Helium
journal, September 1928


Basis-set convergence of correlated calculations on water
journal, June 1997

  • Helgaker, Trygve; Klopper, Wim; Koch, Henrik
  • The Journal of Chemical Physics, Vol. 106, Issue 23
  • DOI: 10.1063/1.473863

Prediction of Reaction Barriers and Thermochemical Properties with Explicitly Correlated Coupled-Cluster Methods: A Basis Set Assessment
journal, August 2012

  • Zhang, Jinmei; Valeev, Edward F.
  • Journal of Chemical Theory and Computation, Vol. 8, Issue 9
  • DOI: 10.1021/ct3005547

Local Treatment of Electron Correlation
journal, October 1993


Atomic and molecular correlation energies with explicitly correlated Gaussian geminals. I. Second‐order perturbation treatment for He, Be, H 2 , and LiH
journal, February 1983

  • Szalewicz, Krzysztof; Jeziorski, Bogumil; Monkhorst, Hendrik J.
  • The Journal of Chemical Physics, Vol. 78, Issue 3
  • DOI: 10.1063/1.444884

A general formulation for the efficient evaluation of n -electron integrals over products of Gaussian charge distributions with Gaussian geminal functions
journal, June 2011

  • Komornicki, Andrew; King, Harry F.
  • The Journal of Chemical Physics, Vol. 134, Issue 24
  • DOI: 10.1063/1.3600745

Second-order Møller-Plesset perturbation theory with terms linear in the interelectronic coordinates and exact evaluation of three-electron integrals
journal, March 2002

  • Wind, Peter; Klopper, Wim; Helgaker, Trygve
  • Theoretical Chemistry Accounts: Theory, Computation, and Modeling (Theoretica Chimica Acta), Vol. 107, Issue 3
  • DOI: 10.1007/s00214-001-0318-6

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journal, November 2018

  • Johnson, Cole M.; Doran, Alexander E.; Ten-no, Seiichiro L.
  • The Journal of Chemical Physics, Vol. 149, Issue 17
  • DOI: 10.1063/1.5054610