Stochastic evaluation of fourth-order many-body perturbation energies

Doran, Alexander E.; Hirata, So

doi:10.1063/5.0047798

Title: Stochastic evaluation of fourth-order many-body perturbation energies

Abstract

A scalable, stochastic algorithm evaluating the fourth-order many-body perturbation (MP4) correction to energy is proposed. Three hundred Goldstone diagrams representing the MP4 correction are computer generated and then converted into algebraic formulas expressed in terms of Green’s functions in real space and imaginary time. They are evaluated by the direct (i.e., non-Markov, non-Metropolis) Monte Carlo (MC) integration accelerated by the redundant-walker and control-variate algorithms. The resulting MC-MP4 method is efficiently parallelized and is shown to display O(n^5.3) size-dependence of cost, which is nearly two ranks lower than the O(n⁷) dependence of the deterministic MP4 algorithm. Furthermore, it evaluates the MP4/aug-cc-pVDZ energy for benzene, naphthalene, phenanthrene, and corannulene with the statistical uncertainty of 10 mE_h (1.1% of the total basis-set correlation energy), 38 mE_h (2.6%), 110 mE_h (5.5%), and 280 mE_h (9.0%), respectively, after about 10⁹ MC steps.

Authors:

^[1];

^[1]

Univ. of Illinois at Urbana-Champaign, IL (United States)

Publication Date:: Wed Apr 07 00:00:00 EDT 2021

Research Org.:: Univ. of Illinois at Urbana-Champaign, IL (United States)

Sponsoring Org.:: USDOE Office of Science (SC), Basic Energy Sciences (BES). Chemical Sciences, Geosciences & Biosciences Division; USDOE

OSTI Identifier:: 1774417

Alternate Identifier(s):: OSTI ID: 1774704

Grant/Contract Number:: SC0006028; SPEC

Resource Type:: Accepted Manuscript

Journal Name:: Journal of Chemical Physics

Additional Journal Information:: Journal Volume: 154; Journal Issue: 13; Journal ID: ISSN 0021-9606

Publisher:: American Institute of Physics (AIP)

Country of Publication:: United States

Language:: English

Subject:: 74 ATOMIC AND MOLECULAR PHYSICS; Correlation energy; Perturbation theory; Monte Carlo methods; Many body perturbation calculations; Stochastic processes

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                    Doran, Alexander E., and Hirata, So. Stochastic evaluation of fourth-order many-body perturbation energies.  United States: N. p., 2021. 
Web.  doi:10.1063/5.0047798.

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                    Doran, Alexander E., & Hirata, So. Stochastic evaluation of fourth-order many-body perturbation energies.  United States.  https://doi.org/10.1063/5.0047798

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                    Doran, Alexander E., and Hirata, So. Wed .  
"Stochastic evaluation of fourth-order many-body perturbation energies".  United States.  https://doi.org/10.1063/5.0047798.  https://www.osti.gov/servlets/purl/1774417.

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@article{osti_1774417,

  title        = {Stochastic evaluation of fourth-order many-body perturbation energies},

  author       = {Doran, Alexander E. and Hirata, So},

  abstractNote = {A scalable, stochastic algorithm evaluating the fourth-order many-body perturbation (MP4) correction to energy is proposed. Three hundred Goldstone diagrams representing the MP4 correction are computer generated and then converted into algebraic formulas expressed in terms of Green’s functions in real space and imaginary time. They are evaluated by the direct (i.e., non-Markov, non-Metropolis) Monte Carlo (MC) integration accelerated by the redundant-walker and control-variate algorithms. The resulting MC-MP4 method is efficiently parallelized and is shown to display O(n5.3) size-dependence of cost, which is nearly two ranks lower than the O(n7) dependence of the deterministic MP4 algorithm. Furthermore, it evaluates the MP4/aug-cc-pVDZ energy for benzene, naphthalene, phenanthrene, and corannulene with the statistical uncertainty of 10 mEh (1.1% of the total basis-set correlation energy), 38 mEh (2.6%), 110 mEh (5.5%), and 280 mEh (9.0%), respectively, after about 109 MC steps.},

  doi          = {10.1063/5.0047798},

  journal      = {Journal of Chemical Physics},

  number       = 13,

  volume       = 154,

  place        = {United States},

  year         = {Wed Apr 07 00:00:00 EDT 2021},

  month        = {Wed Apr 07 00:00:00 EDT 2021}

}

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Journal Article Willow, Soohaeng Yoo ; Center for Superfunctional Materials, Department of Chemistry, Pohang University of Science and Technology, San 31, Hyojadong, Namgu, Pohang 790-784 ; Hirata, So ; ... - Journal of Chemical Physics

A new, alternative set of interpretation rules of Feynman–Goldstone diagrams for many-body perturbation theory is proposed, which translates diagrams into algebraic expressions suitable for direct Monte Carlo integrations. A vertex of a diagram is associated with a Coulomb interaction (rather than a two-electron integral) and an edge with the trace of a Green's function in real space and imaginary time. With these, 12 diagrams of third-order many-body perturbation (MP3) theory are converted into 20-dimensional integrals, which are then evaluated by a Monte Carlo method. It uses redundant walkers for convergence acceleration and a weight function for importance sampling in conjunctionmore »« less
https://doi.org/10.1063/1.4861561
Stochastic evaluation of four-component relativistic second-order many-body perturbation energies: A potentially quadratic-scaling correlation method

Journal Article Cruz, J. César ; Garza, Jorge ; Yanai, Takeshi ; ... - Journal of Chemical Physics

A second-order many-body perturbation correction to the relativistic Dirac–Hartree–Fock energy is evaluated stochastically by integrating 13-dimensional products of four-component spinors and Coulomb potentials. The integration in the real space of electron coordinates is carried out by the Monte Carlo (MC) method with the Metropolis sampling, whereas the MC integration in the imaginary-time domain is performed by the inverse-cumulative distribution function method. The computational cost to reach a given relative statistical error for spatially compact but heavy molecules is observed to be no worse than cubic and possibly quadratic with the number of electrons or basis functions. This is a vastmore »« less
https://doi.org/10.1063/5.0091973

Full Text Available
{ital Ab} {ital initio} investigation of the N{sub 2}{endash}HF complex: Accurate structure and energetics

Journal Article Woon, D E ; Dunning, Jr, T H ; Peterson, K A - Journal of Chemical Physics

Augmented correlation consistent basis sets of double (aug-cc-pVDZ), triple (aug-cc-pVTZ), and modified quadruple zeta (aug-cc-pVQZ{prime}) quality have been employed to describe the N{sub 2}{endash}HF potential energy surface at the Hartree{endash}Fock level and with single reference correlated wave functions including Mo/ller{endash}Plesset perturbation theory (MP2, MP3, MP4) and coupled cluster methods [CCSD, CCSD(T)]. The most accurate computed equilibrium binding energies {ital D}{sub {ital e}} are (with counterpoise correction) 810 cm{sup {minus}1} (MP4/aug-cc-pVQZ{prime}) and 788 cm{sup {minus}1} [CCSD(T)/aug-cc-pVQZ{prime}]. Estimated complete basis set limits of 814 cm{sup {minus}1} (MP4) and 793 cm{sup {minus}1} [CCSD(T)] indicate that the large basis set results are essentially converged.more »« less
https://doi.org/10.1063/1.471320

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Title: Stochastic evaluation of fourth-order many-body perturbation energies

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Citation Formats

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

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

Expeditious Stochastic Calculation of Random-Phase Approximation Energies for Thousands of Electrons in Three Dimensions journal, March 2013

Exploring the Accuracy Limits of Local Pair Natural Orbital Coupled-Cluster Theory journal, March 2015

Projector Monte Carlo method based on Slater determinants: a new sampling method for singlet state calculations journal, September 2011

Information guided noise reduction for Monte Carlo integration of oscillatory functions journal, December 2004

Many – Body Methods in Chemistry and Physics book, January 2009

Elimination of energy denominators in Møller—Plesset perturbation theory by a Laplace transform approach journal, June 1991

GPU-Based Implementations of the Noniterative Regularized-CCSD(T) Corrections: Applications to Strongly Correlated Systems journal, April 2011

Error estimates on averages of correlated data journal, July 1989

High accuracy benchmark calculations on the benzene dimer potential energy surface journal, October 2007

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

Gaussian basis sets for use in correlated molecular calculations. I. The atoms boron through neon and hydrogen journal, January 1989

Grid-based diffusion Monte Carlo for fermions without the fixed-node approximation journal, January 2020

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

Explicitly Correlated Electrons in Molecules journal, December 2011

A Road Map for the Calculation of Molecular Binding Energies journal, October 2000

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

Fifth‐order many‐body perturbation theory for molecular correlation energies journal, June 1989

Comparison of MBPT and coupled cluster methods with full CI. II. Polarized basis sets journal, January 1987

Tensor Contraction Engine: Abstraction and Automated Parallel Implementation of Configuration-Interaction, Coupled-Cluster, and Many-Body Perturbation Theories journal, November 2003

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

A Guided Stochastic Energy-Domain Formulation of the Second Order Møller–Plesset Perturbation Theory journal, December 2013

Hybrid stochastic-deterministic calculation of the second-order perturbative contribution of multireference perturbation theory journal, July 2017

Propagators in Quantum Chemistry book, March 2004

The calculation of higher-order energies in the many-body perturbation theory series journal, January 1985

Perspective: Explicitly correlated electronic structure theory for complex systems journal, February 2017

Is fifth-order MBPT enough? journal, January 1985

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

Multireference Stochastic Coupled Cluster journal, October 2019

Large scale parallelization in stochastic coupled cluster journal, November 2018

General recurrence formulas for molecular integrals over Cartesian Gaussian functions journal, August 1988

Sixth-order many-body perturbation theory. I. Basic theory and derivation of the energy formula journal, January 1996

Sixth-order many-body perturbation theory. III. Correlation energies of size-extensive MP6 methods journal, January 1996

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

Stochastic evaluation of second-order many-body perturbation energies journal, November 2012

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

Convergence acceleration of Monte Carlo many-body perturbation methods by direct sampling journal, September 2020

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

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

Convergence acceleration of Monte Carlo many-body perturbation methods by using many control variates journal, September 2020

Extensive regularization of the coupled cluster methods based on the generating functional formalism: Application to gas-phase benchmarks and to the SN2 reaction of CHCl3 and OH− in water journal, December 2009

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

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

Evaluating two-electron-repulsion integrals over arbitrary orbitals using zero variance Monte Carlo: Application to full configuration interaction calculations with Slater-type orbitals journal, August 2019

Parallel Calculation of Coupled Cluster Singles and Doubles Wave Functions Using Array Files journal, May 2007

Nodal hypersurfaces and Anderson’s random‐walk simulation of the Schrödinger equation journal, June 1976

Many-Body Perturbation Theory and Coupled Cluster Theory for Electron Correlation in Molecules journal, October 1981

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

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

Monte Carlo Second- and Third-Order Many-Body Green’s Function Methods with Frequency-Dependent, Nondiagonal Self-Energy journal, October 2019

One-particle many-body Green’s function theory: Algebraic recursive definitions, linked-diagram theorem, irreducible-diagram theorem, and general-order algorithms journal, July 2017

Coupled cluster algorithms for networks of shared memory parallel processors journal, May 2007

Stochastic determination of effective Hamiltonian for the full configuration interaction solution of quasi-degenerate electronic states journal, April 2013

A random‐walk simulation of the Schrödinger equation: H + 3 journal, August 1975

Quantum Monte Carlo for Ab Initio calculations of energy-relevant materials journal, August 2013

Introduction to the Variational and Diffusion Monte Carlo Methods book, January 2016

Evaluation of the Heats of Formation of Corannulene and C 60 by Means of High-Level Theoretical Procedures journal, January 2013

Comparison of MBPT and coupled-cluster methods with full CI. Importance of triplet excitation and infinite summations journal, June 1983

Metropolis Evaluation of the Hartree–Fock Exchange Energy journal, September 2014

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

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

Sixth-order many-body perturbation theory for molecular calculations journal, May 1995

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

Stochastic Self-Consistent Second-Order Green’s Function Method for Correlation Energies of Large Electronic Systems journal, October 2017

Sixth-order many-body perturbation theory. IV. Improvement of the M�ller-Plesset correlation energy series by using Pad�, Feenberg, and other approximations up to sixth order journal, January 1996

Monte Carlo explicitly correlated second-order many-body perturbation theory journal, October 2016

A fifth-order perturbation comparison of electron correlation theories journal, May 1989

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

Utilizing high performance computing for chemistry: parallel computational chemistry journal, January 2010

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

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

Stochastic multi-reference perturbation theory with application to the linearized coupled cluster method journal, January 2017

Multi-state effective Hamiltonian and size-consistency corrections in stochastic configuration interactions journal, December 2017

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

Stochastic many-body perturbation theory for electron correlation energies journal, December 2019

Finite-Size Error in Many-Body Simulations with Long-Range Interactions journal, August 2006

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

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

Expeditious Stochastic Calculation of Random-Phase Approximation Energies for Thousands of Electrons in Three Dimensions
journal, March 2013

Exploring the Accuracy Limits of Local Pair Natural Orbital Coupled-Cluster Theory
journal, March 2015

Projector Monte Carlo method based on Slater determinants: a new sampling method for singlet state calculations
journal, September 2011

Information guided noise reduction for Monte Carlo integration of oscillatory functions
journal, December 2004

Many – Body Methods in Chemistry and Physics
book, January 2009

Elimination of energy denominators in Møller—Plesset perturbation theory by a Laplace transform approach
journal, June 1991

GPU-Based Implementations of the Noniterative Regularized-CCSD(T) Corrections: Applications to Strongly Correlated Systems
journal, April 2011

Error estimates on averages of correlated data
journal, July 1989

High accuracy benchmark calculations on the benzene dimer potential energy surface
journal, October 2007

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

Gaussian basis sets for use in correlated molecular calculations. I. The atoms boron through neon and hydrogen
journal, January 1989

Grid-based diffusion Monte Carlo for fermions without the fixed-node approximation
journal, January 2020

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

Explicitly Correlated Electrons in Molecules
journal, December 2011

A Road Map for the Calculation of Molecular Binding Energies
journal, October 2000

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

Fifth‐order many‐body perturbation theory for molecular correlation energies
journal, June 1989

Comparison of MBPT and coupled cluster methods with full CI. II. Polarized basis sets
journal, January 1987

Tensor Contraction Engine: Abstraction and Automated Parallel Implementation of Configuration-Interaction, Coupled-Cluster, and Many-Body Perturbation Theories
journal, November 2003

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

A Guided Stochastic Energy-Domain Formulation of the Second Order Møller–Plesset Perturbation Theory
journal, December 2013

Hybrid stochastic-deterministic calculation of the second-order perturbative contribution of multireference perturbation theory
journal, July 2017

Propagators in Quantum Chemistry
book, March 2004

The calculation of higher-order energies in the many-body perturbation theory series
journal, January 1985

Perspective: Explicitly correlated electronic structure theory for complex systems
journal, February 2017

Is fifth-order MBPT enough?
journal, January 1985

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

Multireference Stochastic Coupled Cluster
journal, October 2019

Large scale parallelization in stochastic coupled cluster
journal, November 2018

General recurrence formulas for molecular integrals over Cartesian Gaussian functions
journal, August 1988

Sixth-order many-body perturbation theory. I. Basic theory and derivation of the energy formula
journal, January 1996

Sixth-order many-body perturbation theory. III. Correlation energies of size-extensive MP6 methods
journal, January 1996

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

Stochastic evaluation of second-order many-body perturbation energies
journal, November 2012

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

Convergence acceleration of Monte Carlo many-body perturbation methods by direct sampling
journal, September 2020

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

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

Convergence acceleration of Monte Carlo many-body perturbation methods by using many control variates
journal, September 2020

Extensive regularization of the coupled cluster methods based on the generating functional formalism: Application to gas-phase benchmarks and to the SN2 reaction of CHCl3 and OH− in water
journal, December 2009

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

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

Evaluating two-electron-repulsion integrals over arbitrary orbitals using zero variance Monte Carlo: Application to full configuration interaction calculations with Slater-type orbitals
journal, August 2019

Parallel Calculation of Coupled Cluster Singles and Doubles Wave Functions Using Array Files
journal, May 2007

Nodal hypersurfaces and Anderson’s random‐walk simulation of the Schrödinger equation
journal, June 1976

Many-Body Perturbation Theory and Coupled Cluster Theory for Electron Correlation in Molecules
journal, October 1981

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

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

Monte Carlo Second- and Third-Order Many-Body Green’s Function Methods with Frequency-Dependent, Nondiagonal Self-Energy
journal, October 2019

One-particle many-body Green’s function theory: Algebraic recursive definitions, linked-diagram theorem, irreducible-diagram theorem, and general-order algorithms
journal, July 2017

Coupled cluster algorithms for networks of shared memory parallel processors
journal, May 2007

Stochastic determination of effective Hamiltonian for the full configuration interaction solution of quasi-degenerate electronic states
journal, April 2013

A random‐walk simulation of the Schrödinger equation: H ⁺ ₃
journal, August 1975

Quantum Monte Carlo for Ab Initio calculations of energy-relevant materials
journal, August 2013

Introduction to the Variational and Diffusion Monte Carlo Methods
book, January 2016

Evaluation of the Heats of Formation of Corannulene and C ₆₀ by Means of High-Level Theoretical Procedures
journal, January 2013

Comparison of MBPT and coupled-cluster methods with full CI. Importance of triplet excitation and infinite summations
journal, June 1983

Metropolis Evaluation of the Hartree–Fock Exchange Energy
journal, September 2014

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

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

Sixth-order many-body perturbation theory for molecular calculations
journal, May 1995

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

Stochastic Self-Consistent Second-Order Green’s Function Method for Correlation Energies of Large Electronic Systems
journal, October 2017

Sixth-order many-body perturbation theory. IV. Improvement of the M�ller-Plesset correlation energy series by using Pad�, Feenberg, and other approximations up to sixth order
journal, January 1996

Monte Carlo explicitly correlated second-order many-body perturbation theory
journal, October 2016

A fifth-order perturbation comparison of electron correlation theories
journal, May 1989

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

Utilizing high performance computing for chemistry: parallel computational chemistry
journal, January 2010

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

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

Stochastic multi-reference perturbation theory with application to the linearized coupled cluster method
journal, January 2017

Multi-state effective Hamiltonian and size-consistency corrections in stochastic configuration interactions
journal, December 2017

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

Stochastic many-body perturbation theory for electron correlation energies
journal, December 2019

Finite-Size Error in Many-Body Simulations with Long-Range Interactions
journal, August 2006

Quantum Monte Carlo and Related Approaches
journal, December 2011

Stochastic Perturbation Theory: A Low-Scaling Approach to Correlated Electronic Energies
journal, October 2007

Coupled‐cluster methods with noniterative triple excitations for restricted open‐shell Hartree–Fock and other general single determinant reference functions. Energies and analytical gradients
journal, June 1993