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.
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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):224101. http://link.aip.org/link/?JCP/124/224101/1
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, http://link.aip.org/link/?JCP/126/194105/1
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)
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, http://link.aip.org/link/?JCP/109/6219/1
Bartlett RJ, MusiałM (2007) Coupled-cluster theory in quantum chemistry. Rev Mod Phys 79(1):291. https://doi.org/10.1103/RevModPhys.79.291
Blankenbecler R, Scalapino DJ, Sugar RL (1981) Monte Carlo calculations of coupled Boson-Fermion systems. I. Phys Rev D 24:2278
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, http://scitation.aip.org/content/aip/journal/jcp/131/5/10.1063/1.3193710
Car R, Parrinello M (1985) Unified approach for molecular dynamics and density functional theory. Phys Rev Lett 55:2471
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
Ceperley DM (1995) Path integrals in the theory of condensed helium. Rev Mod Phys 67:279, and references therein
Crawford TD, Schaefer HF III (2000) An introduction to coupled cluster theory for computational chemists. Rev Comput Chem 14:33–136
Diedrich DL, Anderson JB (1992) An accurate quantum monte carlo calculation of the barrier height for the reaction h + h2 → h2 + 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
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
Fahy SB, Hamann DR (1990) Positive-projection Monte Carlo simulation: a new variational approach to strongly interacting fermion systems. Phys Rev Lett 65:3437
Foulkes WMC, Mitas L, Needs RJ, Rajagopal G (2001) Quantum Monte Carlo simulations of solids. Rev Mod Phys 73:33, and references therein
Hamann DR, Fahy SB (1990) Energy measurement in auxiliary-field many-electron calculations. Phys Rev B 41(16):11352
Kalos MH, Whitlock PA (1986) Monte Carlo methods, vol I. Wiley, New York
Kalos MH, Levesque D, Verlet L (1974) Helium at zero temperature with hard-sphere and other forces. Phys Rev A 9:2178
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, http://link.aip.org/link/?JCP/118/9481/1
Kohn W (1999) Nobel lecture: electronic structure of matter – wave functions and density functionals. Rev Mod Phys 71:1253, and references therein
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
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 41:9301
Ma F, Zhang S, Krakauer H (2013) Excited state calculations in solids by auxiliary-field quantum Monte Carlo. New J Phys. arXiv:1211.4635
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
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
Martin RM (2004) Electronic structure: basic theory and practical methods. Cambridge University Press, Cambridge
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 77:349
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:29053270
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
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
Negele JW, Orland H (1998) Quantum many-particle systems. Advanced book classics. Perseus Books, Reading
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
Purwanto W, Zhang S (2004) Quantum Monte Carlo method for the ground state of many-boson systems. Phys Rev E 70:056702
Purwanto W, Zhang S (2005) Correlation effects in the ground state of trapped atomic bose gases. Phys Rev A 72(5):053610
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
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, http://link.aip.org/link/?JCP/130/094107/1
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 135:164105
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
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
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
Schmidt KE, Kalos MH (1984) Few- and many-Fermion problems. In: Binder K (ed) Applications of the Monte Carlo method in statistical physics. Springer, Heidelberg
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: 28481546
Shi H, Zhang S (2013) Symmetry in auxiliary-field quantum Monte Carlo calculations. Phys Rev B 88:125132
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
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
Sorella S, Baroni S, Car R, Parrinello M (1989) A novel technique for the simulation of interacting fermion systems. Europhys Lett 8:663
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 and Materials Physics) 75(24):245123. https://doi.org/10.1103/PhysRevB.75.245123
Sugiyama G, Koonin SE (1986) Auxiliary field Monte-Carlo for quantum many-body ground states. Ann Phys (NY) 168:1
Szabo A, Ostlund N (1989) Modern quantum chemistry. McGraw-Hill, New York
Umrigar CJ, Nightingale MP, Runge KJ (1993) A diffusion Monte Carlo algorithm with very small time-step errors. J Chem Phys 99(4):2865
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
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
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
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):506
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/9909090
Zhang S (1999b) Finite-temperature Monte Carlo calculations for systems with fermions. Phys Rev Lett 83:2777
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 York. At http://physics.wm.edu/~/shiwei
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, Jülich
Zhang S, Ceperley DM (2008) Hartree-Fock ground state of the three-dimensional electron gas. Phys Rev Lett 100:236404
Zhang S, Kalos MH (1991) Exact Monte Carlo calculations for few-electron systems. Phys Rev Lett 67:3074
Zhang S, Krakauer H (2003) Quantum Monte Carlo method using phase-free random walks with Slater determinants. Phys Rev Lett 90:136401
Zhang S, Carlson J, Gubernatis JE (1997) Constrained path Monte Carlo method for fermion ground states. Phys Rev B 55:7464
Zhang S, Krakauer H, Al-Saidi WA, Suewattana M (2005) Quantum simulations of realistic systems by auxiliary fields. Comput Phys Commun 169:394
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
Acknowledgements
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.
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Zhang, S. (2018). Ab Initio Electronic Structure Calculations by Auxiliary-Field Quantum Monte Carlo. In: Andreoni, W., Yip, S. (eds) Handbook of Materials Modeling . Springer, Cham. https://doi.org/10.1007/978-3-319-42913-7_47-1
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