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Ab Initio Electronic Structure Calculations by Auxiliary-Field Quantum Monte Carlo

  • Shiwei ZhangEmail author
Reference work entry
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Abstract

The auxiliary-field quantum Monte Carlo (AFQMC) method provides a computational framework for solving the time-independent Schrödinger equation in atoms, molecules, solids, and a variety of model systems by stochastic sampling. We introduce the theory and formalism behind this framework, briefly discuss the key technical steps that turn it into an effective and practical computational method, present several illustrative results, and conclude with comments on the prospects of ab initio computation by this framework.

Notes

Acknowledgments

I thank the many colleagues and outstanding students and postdocs whose contributions to the work discussed here are invaluable, among whom I would especially like to mention W. Al-Saidi, H. Krakauer, F. Ma, M. Motta, W. Purwanto, and H. Shi. Support from the National Science Foundation (NSF), the Simons Foundation, and the Department of Energy (DOE) is gratefully acknowledged. Computing was done via XSEDE supported by NSF, on the Oak Ridge Leadership Computing Facilities, and on the HPC facilities at William & Mary.

References

  1. Al-Saidi WA, Zhang S, Krakauer H (2006) Auxiliary-field quantum Monte Carlo calculations of molecular systems with a Gaussian basis. J Chem Phys 124(22):224101ADSCrossRefGoogle Scholar
  2. Al-Saidi WA, Krakauer H, Zhang S (2007) A study of H +  H2 and several H-bonded molecules by phaseless auxiliary-field quantum Monte Carlo with plane wave and Gaussian basis sets. J Chem Phys 126(19):194105. https://doi.org/10.1063/1.2735296 ADSCrossRefGoogle Scholar
  3. Aquilante F, De Vico L, Ferre N, Ghigo G, Malmqvist P, Neogrady P, Pedersen T, Pitonak M, Reiher M, Roos B, Serrano-Andres L, Urban M, Veryazov V, Lindh R (2010) J Comput Chem 31(1):224–247.  https://doi.org/10.1002/jcc.21318. The information about affiliations in this record was updated in December 2015. The record was previously connected to the following departments: Theoretical Chemistry (S) (011001039)
  4. Baer R, Head-Gordon M, Neuhauser D (1998) Shifted-contour auxiliary field Monte Carlo for ab initio electronic structure: straddling the sign problem. J Chem Phys 109(15):6219–6226. https://doi.org/10.1063/1.477300 ADSCrossRefGoogle Scholar
  5. Bartlett RJ, MusiałM (2007) Coupled-cluster theory in quantum chemistry. Rev Mod Phys 79(1):291.  https://doi.org/10.1103/RevModPhys.79.291
  6. Blankenbecler R, Scalapino DJ, Sugar RL (1981) Monte Carlo calculations of coupled Boson-Fermion systems. I. Phys Rev D 24:2278ADSCrossRefGoogle Scholar
  7. Booth GH, Thom AJW, Alavi A (2009) Fermion Monte Carlo without fixed nodes: a game of life, death, and annihilation in slater determinant space. J Chem Phys 131(5):054106. https://doi.org/10.1063/1.3193710 ADSCrossRefGoogle Scholar
  8. Car R, Parrinello M (1985) Unified approach for molecular dynamics and density functional theory. Phys Rev Lett 55:2471ADSCrossRefGoogle Scholar
  9. Carlson J, Gandolfi S, Schmidt KE, Zhang S (2011) Auxiliary-field quantum Monte Carlo method for strongly paired fermions. Phys Rev A 84:061602.  https://doi.org/10.1103/PhysRevA.84.061602 ADSCrossRefGoogle Scholar
  10. Ceperley DM (1995) Path integrals in the theory of condensed helium. Rev Mod Phys 67:279, and references thereinGoogle Scholar
  11. Crawford TD, Schaefer HF III (2000) An introduction to coupled cluster theory for computational chemists. Rev Comput Chem 14:33–136Google Scholar
  12. Diedrich DL, Anderson JB (1992) An accurate quantum monte carlo calculation of the barrier height for the reaction h + h 2 → h 2 + h. Science 258(5083):786–788.  https://doi.org/10.1126/science.258.5083.786, http://science.sciencemag.org/content/258/5083/786.full.pdf
  13. Esler KP, Kim J, Ceperley DM, Purwanto W, Walter EJ, Krakauer H, Zhang S, Kent PRC, Hennig RG, Umrigar C, Bajdich M, Kolorenc J, Mitas L, Srinivasan A (2008) Quantum Monte Carlo algorithms for electronic structure at the petascale; the Endstation project. J Phys Conf Ser 125:012057 (15pp). http://stacks.iop.org/1742-6596/125/012057 CrossRefGoogle Scholar
  14. Fahy SB, Hamann DR (1990) Positive-projection Monte Carlo simulation: a new variational approach to strongly interacting fermion systems. Phys Rev Lett 65: 3437ADSCrossRefGoogle Scholar
  15. Foulkes WMC, Mitas L, Needs RJ, Rajagopal G (2001) Quantum Monte Carlo simulations of solids. Rev Mod Phys 73:33, and references thereinGoogle Scholar
  16. Hamann DR, Fahy SB (1990) Energy measurement in auxiliary-field many-electron calculations. Phys Rev B 41(16):11352ADSCrossRefGoogle Scholar
  17. Kalos MH, Whitlock PA (1986) Monte Carlo methods, vol I. Wiley, New YorkzbMATHCrossRefGoogle Scholar
  18. Kalos MH, Levesque D, Verlet L (1974) Helium at zero temperature with hard-sphere and other forces. Phys Rev A 9:2178ADSCrossRefGoogle Scholar
  19. Koch H, de Merás AS, Pedersen TB (2003) Reduced scaling in electronic structure calculations using Cholesky decompositions. J Chem Phys 118(21):9481–9484. https://doi.org/10.1063/1.1578621 ADSCrossRefGoogle Scholar
  20. Kohn W (1999) Nobel lecture: Electronic structure of matter – wave functions and density functionals. Rev Mod Phys 71:1253, and references thereinGoogle Scholar
  21. LeBlanc JPF, Antipov AE, Becca F, Bulik IW, Chan GKL, Chung CM, Deng Y, Ferrero M, Henderson TM, Jiménez-Hoyos CA, Kozik E, Liu XW, Millis AJ, Prokof’ev NV, Qin M, Scuseria GE, Shi H, Svistunov BV, Tocchio LF, Tupitsyn IS, White SR, Zhang S, Zheng BX, Zhu Z, Gull E (2015) Solutions of the two-dimensional hubbard model: benchmarks and results from a wide range of numerical algorithms. Phys Rev X 5:041041.  https://doi.org/10.1103/PhysRevX.5.041041 Google Scholar
  22. Loh EY Jr, Gubernatis JE, Scalettar RT, White SR, Scalapino DJ, Sugar R (1990) Sign problem in the numerical simulation of many-electron systems. Phys Rev B 41:9301ADSCrossRefGoogle Scholar
  23. Ma F, Zhang S, Krakauer H (2013) Excited state calculations in solids by auxiliary-field quantum Monte Carlo. New J 15:093017. https://doi.org/10.1088/1367-2630/15/9/093017 Google Scholar
  24. Ma F, Purwanto W, Zhang S, Krakauer H (2015) Quantum Monte Carlo calculations in solids with downfolded hamiltonians. Phys Rev Lett 114:226401.  https://doi.org/10.1103/PhysRevLett.114.226401 ADSCrossRefGoogle Scholar
  25. Ma F, Zhang S, Krakauer H (2017) Auxiliary-field quantum Monte Carlo calculations with multiple-projector pseudopotentials. Phys Rev B 95:165103.  https://doi.org/10.1103/PhysRevB.95.165103 ADSCrossRefGoogle Scholar
  26. Martin RM (2004) Electronic structure: basic theory and practical methods. Cambridge University Press, CambridgezbMATHCrossRefGoogle Scholar
  27. Moskowitz JW, Schmidt KE, Lee MA, Kalos MH (1982) A new look at correlation energy in atomic and molecular systems. II. The application of the Green’s function Monte Carlo method to LiH. J Chem Phys 77:349Google Scholar
  28. Motta M, Zhang S (2017) Computation of ground-state properties in molecular systems: back-propagation with auxiliary-field quantum Monte Carlo. J Chem Theory Comput 13(11):5367–5378.  https://doi.org/10.1021/acs.jctc.7b00730, PMID:29053270CrossRefGoogle Scholar
  29. Motta M, Zhang S (2018, in press) Ab initio computations of molecular systems by the auxiliary-field quantum Monte Carlo method. WIREs Comput Mol Sci.  https://doi.org/10.1002/wcms.1364
  30. Motta M, Ceperley DM, Chan GKL, Gomez JA, Gull E, Guo S, Jiménez-Hoyos CA, Lan TN, Li J, Ma F, Millis AJ, Prokof’ev NV, Ray U, Scuseria GE, Sorella S, Stoudenmire EM, Sun Q, Tupitsyn IS, White SR, Zgid D, Zhang S (2017) Towards the solution of the many-electron problem in real materials: equation of state of the hydrogen chain with state-of-the-art many-body methods. Phys Rev X 7:031059.  https://doi.org/10.1103/PhysRevX.7.031059 Google Scholar
  31. Negele JW, Orland H (1998) Quantum many-particle systems. Advanced book classics. Perseus Books, ReadingGoogle Scholar
  32. Nguyen H, Shi H, Xu J, Zhang S (2014) CPMC-lab: a matlab package for constrained path Monte Carlo calculations. Comput Phys Commun 185(12):3344–3357. https://doi.org/10.1016/j.cpc.2014.08.003, http://www.sciencedirect.com/science/article/pii/S0010465514002707
  33. Purwanto W, Zhang S (2004) Quantum Monte Carlo method for the ground state of many-boson systems. Phys Rev E 70:056702ADSCrossRefGoogle Scholar
  34. Purwanto W, Zhang S (2005) Correlation effects in the ground state of trapped atomic bose gases. Phys Rev A 72(5):053610ADSCrossRefGoogle Scholar
  35. Purwanto W, Krakauer H, Zhang S (2009a) Pressure-induced diamond to β -tin transition in bulk silicon: a quantum Monte Carlo study. Phys Rev B 80(21):214116.  https://doi.org/10.1103/PhysRevB.80.214116 ADSCrossRefGoogle Scholar
  36. Purwanto W, Zhang S, Krakauer H (2009b) Excited state calculations using phaseless auxiliary-field quantum Monte Carlo: potential energy curves of low-lying C2 singlet states. J Chem Phys 130(9):094107. https://doi.org/10.1063/1.3077920 ADSCrossRefGoogle Scholar
  37. Purwanto W, Krakauer H, Virgus Y, Zhang S (2011) Assessing weak hydrogen binding on Ca+ centers: an accurate many-body study with large basis sets. J Chem Phys 135:164105ADSCrossRefGoogle Scholar
  38. Purwanto W, Zhang S, Krakauer H (2013) Frozen-orbital and downfolding calculations with auxiliary-field quantum Monte Carlo. J Chem Theory Comput. https://doi.org/10.1021/ct4006486 CrossRefGoogle Scholar
  39. Qin M, Shi H, Zhang S (2016) Coupling quantum Monte Carlo and independent-particle calculations: self-consistent constraint for the sign problem based on the density or the density matrix. Phys Rev B 94:235119.  https://doi.org/10.1103/PhysRevB.94.235119 ADSCrossRefGoogle Scholar
  40. Rosenberg P, Shi H, Zhang S (2017) Accurate computations of Rashba spin-orbit coupling in interacting systems: from the Fermi gas to real materials. J Phys Chem Solids. https://doi.org/10.1016/j.jpcs.2017.12.026, 1710.00887
  41. Schmidt KE, Kalos MH (1984) Few- and many-Fermion problems. In: Binder K (ed) Applications of the Monte Carlo method in statistical physics. Springer, HeidelbergGoogle Scholar
  42. Shee J, Zhang S, Reichman DR, Friesner RA (2017) Chemical transformations approaching chemical accuracy via correlated sampling in auxiliary-field quantum Monte Carlo. J Chem Theory Comput 13(6):2667–2680.  https://doi.org/10.1021/acs.jctc.7b00224, PMID: 28481546CrossRefGoogle Scholar
  43. Shi H, Zhang S (2013) Symmetry in auxiliary-field quantum Monte Carlo calculations. Phys Rev B 88:125132ADSCrossRefGoogle Scholar
  44. Shi H, Zhang S (2016) Infinite variance in fermion quantum Monte Carlo calculations. Phys Rev E 93:033303.  https://doi.org/10.1103/PhysRevE.93.033303 ADSMathSciNetCrossRefGoogle Scholar
  45. Shi H, Zhang S (2017) Many-body computations by stochastic sampling in Hartree-Fock-Bogoliubov space. Phys Rev B 95:045144.  https://doi.org/10.1103/PhysRevB.95.045144 ADSCrossRefGoogle Scholar
  46. Sorella S, Baroni S, Car R, Parrinello M (1989) A novel technique for the simulation of interacting fermion systems. Europhys Lett 8:663ADSCrossRefGoogle Scholar
  47. Suewattana M, Purwanto W, Zhang S, Krakauer H, Walter EJ (2007) Phaseless auxiliary-field quantum Monte Carlo calculations with plane waves and pseudopotentials: applications to atoms and molecules. Phys Rev B (Condensed Matter Mater Phys) 75(24):245123.  https://doi.org/10.1103/PhysRevB.75.245123 ADSCrossRefGoogle Scholar
  48. Sugiyama G, Koonin SE (1986) Auxiliary field Monte-Carlo for quantum many-body ground states. Ann Phys (NY) 168:1ADSCrossRefGoogle Scholar
  49. Szabo A, Ostlund N (1989) Modern quantum chemistry. McGraw-Hill, New YorkGoogle Scholar
  50. Umrigar CJ, Nightingale MP, Runge KJ (1993) A diffusion Monte Carlo algorithm with very small time-step errors. J Chem Phys 99(4):2865ADSCrossRefGoogle Scholar
  51. Virgus Y, Purwanto W, Krakauer H, Zhang S (2014) Stability, energetics, and magnetic states of cobalt adatoms on graphene. Phys Rev Lett 113:175502.  https://doi.org/10.1103/PhysRevLett.113.175502 ADSCrossRefGoogle Scholar
  52. Vitali E, Shi H, Qin M, Zhang S (2016) Computation of dynamical correlation functions for many-fermion systems with auxiliary-field quantum Monte Carlo. Phys Rev B 94:085140.  https://doi.org/10.1103/PhysRevB.94.085140 ADSCrossRefGoogle Scholar
  53. Wei ZC, Wu C, Li Y, Zhang S, Xiang T (2016) Majorana positivity and the fermion sign problem of quantum Monte Carlo simulations. Phys Rev Lett 116:250601.  https://doi.org/10.1103/PhysRevLett.116.250601 ADSCrossRefGoogle Scholar
  54. White SR, Scalapino DJ, Sugar RL, Loh EY, Gubernatis JE, Scalettar RT (1989) Numerical study of the two-dimensional Hubbard model. Phys Rev B 40(1):506ADSCrossRefGoogle Scholar
  55. Zhang S (1999a) Constrained path Monte Carlo for fermions. In: Nightingale MP, Umrigar CJ (eds) Quantum Monte Carlo methods in physics and chemistry. Kluwer Academic Publishers, Dordrech, cond-mat/9909090Google Scholar
  56. Zhang S (1999b) Finite-temperature Monte Carlo calculations for systems with fermions. Phys Rev Lett 83:2777ADSCrossRefGoogle Scholar
  57. Zhang S (2003) Quantum Monte Carlo methods for strongly correlated fermions. In: Sénéchal D, Tremblay AM, Bourbonnais C (eds) Theoretical methods for strongly correlated electrons. CRM series in mathematical physics, and references therein. Springer, New YorkGoogle Scholar
  58. Zhang S (2013) Auxiliary-Field quantum monte carlo for correlated electron systems. In: Pavarini E, Koch E, Schollwöck U (eds) Emergent phenomena in correlated matter: modeling and simulation, vol 3. Verlag des Forschungszentrum Jülich, JülichGoogle Scholar
  59. Zhang S, Ceperley DM (2008) Hartree-Fock ground state of the three-dimensional electron gas. Phys Rev Lett 100:236404ADSCrossRefGoogle Scholar
  60. Zhang S, Kalos MH (1991) Exact Monte Carlo calculations for few-electron systems. Phys Rev Lett 67:3074ADSCrossRefGoogle Scholar
  61. Zhang S, Krakauer H (2003) Quantum Monte Carlo method using phase-free random walks with Slater determinants. Phys Rev Lett 90:136401ADSCrossRefGoogle Scholar
  62. Zhang S, Carlson J, Gubernatis JE (1997) Constrained path Monte Carlo method for fermion ground states. Phys Rev B 55:7464ADSCrossRefGoogle Scholar
  63. Zhang S, Krakauer H, Al-Saidi WA, Suewattana M (2005) Quantum simulations of realistic systems by auxiliary fields. Comput Phys Commun 169:394ADSCrossRefGoogle Scholar
  64. Zheng BX, Chung CM, Corboz P, Ehlers G, Qin MP, Noack RM, Shi H, White SR, Zhang S, Chan GKL (2017) Stripe order in the underdoped region of the two-dimensional Hubbard model. Science 358(6367):1155–1160.  https://doi.org/10.1126/science.aam7127 ADSMathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.Center for Computational Quantum PhysicsFlatiron InstituteNew YorkUSA
  2. 2.Department of PhysicsCollege of William and MaryWilliamsburgUSA

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