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%% Created for Pierre-Francois Loos at 2020-08-09 15:41:06 +0200
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%% Created for Pierre-Francois Loos at 2020-08-16 14:01:03 +0200
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%% Saved with string encoding Unicode (UTF-8)
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@article{Becke_2014,
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Author = {A. D. Becke},
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@book{ParrBook,
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Address = {New York},
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|
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Title = {Density-Functional Theory of Atoms and Molecules},
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@article{Hohenberg_1964,
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Author = {P. Hohenberg and W. Kohn},
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@article{Reynolds_1982,
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Author = {Reynolds, P. J. and Ceperley, D. M. and Alder, B. J. and Lester, W. A.},
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Doi = {10.1063/1.443766},
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Month = {Jun},
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Title = {Fixed‐node quantum Monte Carlo for molecules},
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Volume = {77},
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Bdsk-Url-1 = {https://doi.org/10.1063/1.443766}}
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@article{Kent_2020,
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Author = {P. R. C. Kent and Abdulgani Annaberdiyev and Anouar Benali and M. Chandler Bennett and Edgar Josu{\'{e}} Landinez Borda and Peter Doak and Hongxia Hao and Kenneth D. Jordan and Jaron T. Krogel and Ilkka Kyl{\"{a}}np{\"{a}}{\"{a}} and Joonho Lee and Ye Luo and Fionn D. Malone and Cody A. Melton and Lubos Mitas and Miguel A. Morales and Eric Neuscamman and Fernando A. Reboredo and Brenda Rubenstein and Kayahan Saritas and Shiv Upadhyay and Guangming Wang and Shuai Zhang and Luning Zhao},
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Journal = {J. Chem. Phys.},
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Title = {{QMCPACK}: Advances in the development, efficiency, and application of auxiliary field and real-space variational and diffusion quantum Monte Carlo},
|
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Volume = {152},
|
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Year = {2020},
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Bdsk-Url-1 = {https://doi.org/10.1063/5.0004860}}
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@article{Kim_2018,
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||||
Annote = {QMCPACK is an open source quantum Monte Carlo package for ab initio electronic structure calculations. It supports calculations of metallic and insulating solids, molecules, atoms, and some model Hamiltonians. Implemented real space quantum Monte Carlo algorithms include variational, diffusion, and reptation Monte Carlo. QMCPACK uses Slater--Jastrow type trial wavefunctions in conjunction with a sophisticated optimizer capable of optimizing tens of thousands of parameters. The orbital space auxiliary-field quantum Monte Carlo method is also implemented, enabling cross validation between different highly accurate methods. The code is specifically optimized for calculations with large numbers of electrons on the latest high performance computing architectures, including multicore central processing unit and graphical processing unit systems. We detail the program's capabilities, outline its structure, and give examples of its use in current research calculations. The package is available at http://qmcpack.org.},
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Author = {Jeongnim Kim and Andrew D Baczewski and Todd D Beaudet and Anouar Benali and M Chandler Bennett and Mark A Berrill and Nick S Blunt and Edgar Josu{\'{e}} Landinez Borda and Michele Casula and David M Ceperley and Simone Chiesa and Bryan K Clark and Raymond C Clay and Kris T Delaney and Mark Dewing and Kenneth P Esler and Hongxia Hao and Olle Heinonen and Paul R C Kent and Jaron T Krogel and Ilkka Kyl{\"a}np{\"a}{\"a} and Ying Wai Li and M Graham Lopez and Ye Luo and Fionn D Malone and Richard M Martin and Amrita Mathuriya and Jeremy McMinis and Cody A Melton and Lubos Mitas and Miguel A Morales and Eric Neuscamman and William D Parker and Sergio D Pineda Flores and Nichols A Romero and Brenda M Rubenstein and Jacqueline A R Shea and Hyeondeok Shin and Luke Shulenburger and Andreas F Tillack and Joshua P Townsend and Norm M Tubman and Brett Van Der Goetz and Jordan E Vincent and D ChangMo Yang and Yubo Yang and Shuai Zhang and Luning Zhao},
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Date-Added = {2020-08-16 13:48:38 +0200},
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Doi = {10.1088/1361-648x/aab9c3},
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Journal = {J. Phys.: Condens. Matter},
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Number = {19},
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Pages = {195901},
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Title = {{QMCPACK}: an open source ab initio quantum Monte Carlo package for the electronic structure of atoms, molecules and solids},
|
||||
Volume = {30},
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Year = 2018,
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Bdsk-Url-1 = {https://doi.org/10.1088%2F1361-648x%2Faab9c3},
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Bdsk-Url-2 = {https://doi.org/10.1088/1361-648x/aab9c3}}
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@article{Needs_2020,
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Author = {R. J. Needs and M. D. Towler and N. D. Drummond and P. L{\'{o}}pez R{\'{\i}}os and J. R. Trail},
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Doi = {10.1063/1.5144288},
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Journal = {The Journal of Chemical Physics},
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Month = {apr},
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Number = {15},
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Pages = {154106},
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Publisher = {{AIP} Publishing},
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Title = {Variational and diffusion quantum Monte Carlo calculations with the {CASINO} code},
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Volume = {152},
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Year = {2020},
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Bdsk-Url-1 = {https://doi.org/10.1063/1.5144288}}
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@article{Nakano_2020,
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Author = {Nakano, Kousuke and Attaccalite, Claudio and Barborini, Matteo and Capriotti, Luca and Casula, Michele and Coccia, Emanuele and Dagrada, Mario and Genovese, Claudio and Luo, Ye and Mazzola, Guglielmo and Zen, Andrea and Sorella, Sandro},
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Date-Added = {2020-08-16 13:47:39 +0200},
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Eprint = {2002.07401},
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Journal = {arXiv},
|
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Month = {Feb},
|
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Title = {{TurboRVB: a many-body toolkit for {$\lbrace$}{\ifmmode\backslash\else\textbackslash\fi}it ab initio{$\rbrace$} electronic simulations by quantum Monte Carlo}},
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Url = {https://arxiv.org/abs/2002.07401},
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Year = {2020},
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Bdsk-Url-1 = {https://arxiv.org/abs/2002.07401}}
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@article{Austin_2012,
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Author = {Austin, Brian M. and Zubarev, Dmitry Yu. and Lester, William A.},
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Doi = {10.1021/cr2001564},
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Journal = {Chem. Rev.},
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Number = {1},
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Pages = {263--288},
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Title = {Quantum Monte Carlo and Related Approaches},
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Volume = {112},
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Year = {2012},
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Bdsk-Url-1 = {http://dx.doi.org/10.1021/cr2001564}}
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@article{Foulkes_2001,
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Author = {Foulkes, W. M. C. and Mitas, L. and Needs, R. J. and Rajagopal, G.},
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Journal = {Rev. Mod. Phys.},
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Pages = {33--83},
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Title = {Quantum Monte Carlo simulations of solids},
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Volume = {73},
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Year = {2001},
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Bdsk-Url-1 = {https://link.aps.org/doi/10.1103/RevModPhys.73.33},
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Bdsk-Url-2 = {https://doi.org/10.1103/RevModPhys.73.33}}
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@article{Xu_2018,
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Author = {Xu, E. and Uejima, M. and Ten-no, S. L.},
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Date-Added = {2020-08-16 13:38:49 +0200},
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Doi = {10.1103/PhysRevLett.121.113001},
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Journal = {Phys. Rev. Lett.},
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Pages = {113001},
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Title = {Full Coupled-Cluster Reduction for Accurate De- scription of Strong Electron Correlation},
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Volume = {121},
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Year = {2018},
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Bdsk-Url-1 = {https://doi.org/10.1103/PhysRevLett.121.113001}}
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@article{Eriksen_2019,
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Author = {J. J. Eriksen and J. Gauss},
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Date-Added = {2020-08-16 13:35:02 +0200},
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Doi = {10.1021/acs.jctc.9b00456},
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Journal = {J. Chem. Theory Comput.},
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Pages = {4873},
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Title = {Many-Body Expanded Full Configuration Interaction. II. Strongly Correlated Regime},
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Volume = {15},
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Year = {2019},
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Bdsk-Url-1 = {https://doi.org/10.1021/acs.jctc.9b00456}}
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@article{Eriksen_2018,
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Author = {J. J. Eriksen and J. Gauss},
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Date-Added = {2020-08-16 13:34:04 +0200},
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Date-Modified = {2020-08-16 13:36:21 +0200},
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Doi = {10.1021/acs.jctc.8b00680},
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Journal = {J. Chem. Theory Comput.},
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Pages = {5180},
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Title = {Many-Body Expanded Full Configuration Interaction. I. Weakly Correlated Regime},
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Volume = {14},
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Year = {2018},
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Bdsk-Url-1 = {https://doi.org/10.1021/acs.jctc.8b00680}}
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@misc{Eriksen_2020,
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Archiveprefix = {arXiv},
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Author = {Janus J. Eriksen and Tyler A. Anderson and J. Emiliano Deustua and Khaldoon Ghanem and Diptarka Hait and Mark R. Hoffmann and Seunghoon Lee and Daniel S. Levine and Ilias Magoulas and Jun Shen and Norman M. Tubman and K. Birgitta Whaley and Enhua Xu and Yuan Yao and Ning Zhang and Ali Alavi and Garnet Kin-Lic Chan and Martin Head-Gordon and Wenjian Liu and Piotr Piecuch and Sandeep Sharma and Seiichiro L. Ten-no and C. J. Umrigar and J{\"u}rgen Gauss},
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Eprint = {2008.02678},
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Primaryclass = {physics.chem-ph},
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Title = {The Ground State Electronic Energy of Benzene},
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Year = {2020}}
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@article{Williams_2020,
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Author = {Williams, Kiel T. and Yao, Yuan and Li, Jia and Chen, Li and Shi, Hao and Motta, Mario and Niu, Chunyao and Ray, Ushnish and Guo, Sheng and Anderson, Robert J. and Li, Junhao and Tran, Lan Nguyen and Yeh, Chia-Nan and Mussard, Bastien and Sharma, Sandeep and Bruneval, Fabien and van Schilfgaarde, Mark and Booth, George H. and Chan, Garnet Kin-Lic and Zhang, Shiwei and Gull, Emanuel and Zgid, Dominika and Millis, Andrew and Umrigar, Cyrus J. and Wagner, Lucas K.},
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Collaboration = {Simons Collaboration on the Many-Electron Problem},
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Date-Added = {2020-08-16 13:29:13 +0200},
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Date-Modified = {2020-08-16 13:29:33 +0200},
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Doi = {10.1103/PhysRevX.10.011041},
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Journal = {Phys. Rev. X},
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Pages = {011041},
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Title = {Direct Comparison of Many-Body Methods for Realistic Electronic Hamiltonians},
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Volume = {10},
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Year = {2020},
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Bdsk-Url-1 = {https://link.aps.org/doi/10.1103/PhysRevX.10.011041},
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Bdsk-Url-2 = {https://doi.org/10.1103/PhysRevX.10.011041}}
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@article{Motta_2017,
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Author = {Motta, Mario and Ceperley, David M. and Chan, Garnet Kin-Lic and Gomez, John A. and Gull, Emanuel and Guo, Sheng and Jim\'enez-Hoyos, Carlos A. and Lan, Tran Nguyen and Li, Jia and Ma, Fengjie and Millis, Andrew J. and Prokof'ev, Nikolay V. and Ray, Ushnish and Scuseria, Gustavo E. and Sorella, Sandro and Stoudenmire, Edwin M. and Sun, Qiming and Tupitsyn, Igor S. and White, Steven R. and Zgid, Dominika and Zhang, Shiwei},
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Collaboration = {Simons Collaboration on the Many-Electron Problem},
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Date-Added = {2020-08-16 13:27:37 +0200},
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Doi = {10.1103/PhysRevX.7.031059},
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Journal = {Phys. Rev. X},
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Numpages = {28},
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Pages = {031059},
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Publisher = {American Physical Society},
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Title = {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},
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Url = {https://link.aps.org/doi/10.1103/PhysRevX.7.031059},
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Volume = {7},
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Year = {2017},
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Bdsk-Url-1 = {https://link.aps.org/doi/10.1103/PhysRevX.7.031059},
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Bdsk-Url-2 = {https://doi.org/10.1103/PhysRevX.7.031059}}
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@article{Scemama_2006c,
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Author = {Scemama, Anthony and Filippi, Claudia},
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@ -569,13 +793,10 @@
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@article{Scemama_2013,
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Author = {Scemama, Anthony and Caffarel, Michel and Oseret, Emmanuel and Jalby, William},
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Date-Modified = {2020-08-16 13:53:13 +0200},
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Number = {11},
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Pages = {938--951},
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Publisher = {John Wiley {\&} Sons, Ltd},
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Title = {{Quantum Monte Carlo for large chemical systems: Implementing efficient strategies for petascale platforms and beyond}},
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Volume = {34},
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Year = {2013},
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@ -1058,16 +1279,13 @@
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Bdsk-Url-1 = {https://doi.org/10.1016/S1380-7323(96)80091-4}}
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@incollection{Sav-INC-96a,
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author = {A. Savin},
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title = {Beyond the Kohn-Sham Determinant},
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booktitle = {Recent Advances in Density Functional Theory},
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||||
publisher = {World Scientific},
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address = {},
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editor = {D. P. Chong},
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pages = {129-148},
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year = {1996}
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}
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Author = {A. Savin},
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Booktitle = {Recent Advances in Density Functional Theory},
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Editor = {D. P. Chong},
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Pages = {129-148},
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Title = {Beyond the Kohn-Sham Determinant},
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@article{Toulouse_2004,
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Author = {Toulouse, Julien and Colonna, Fran{\c c}ois and Savin, Andreas},
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@ -1142,12 +1360,9 @@
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Bdsk-Url-1 = {https://doi.org/10.5281/zenodo.3677565}}
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@article{TouSavFla-IJQC-04,
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author = {J. Toulouse and A. Savin and H.-J. Flad},
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title = {Short-range exchange-correlation energy of a uniform electron gas with modified electron-electron interaction},
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journal = {Int. J. Quantum Chem.},
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volume = {100},
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pages = {1047},
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year = {2004},
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note = {}
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}
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Author = {J. Toulouse and A. Savin and H.-J. Flad},
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Journal = {Int. J. Quantum Chem.},
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Pages = {1047},
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Title = {Short-range exchange-correlation energy of a uniform electron gas with modified electron-electron interaction},
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Volume = {100},
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Year = {2004}}
|
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@ -83,36 +83,43 @@ As the WFT method is relieved from describing the short-range part of the correl
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Solving the Schr\"odinger equation for atoms and molecules is a complex task that has kept theoretical and computational chemists busy for almost hundred years now. \cite{Schrodinger_1926}
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In order to achieve this formidable endeavour, various strategies have been carefully designed and implemented in quantum chemistry software packages.
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One of this strategies consists in relying on wave function theory and, in particular, on the full configuration interaction (FCI) method.
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However, FCI delivers only an approximate solution of the Schr\"odinger equation within a finite one-electron basis.
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This solution is the eigenpair of an approximate Hamiltonian defined as
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One of this strategies consists in relying on wave function theory (WFT) and, in particular, on the full configuration interaction (FCI) method.
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However, FCI delivers only an approximate solution of the Schr\"odinger equation within a finite basis (FB) of one-electron basis functions.
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The FB-FCI wave function and its corresponding energy form the eigenpair of an approximate Hamiltonian defined as
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the projection of the exact Hamiltonian onto the finite many-electron basis of
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all possible Slater determinants generated within this finite one-electron basis.
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The FCI wave function can be interpreted as a constrained solution of the
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true Hamiltonian forced to span the restricted space provided by the one-electron basis.
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In the complete basis set (CBS) limit, the constraint is lifted and the
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exact solution is recovered.
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Hence, the accuracy of a FCI calculation can be systematically improved by increasing the size of the one-electron basis set.
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Nevertheless, its exponential scaling with the number of electrons and with the size of the basis is prohibitive for most chemical systems.
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In recent years, the introduction of new algorithms \cite{Booth_2009} and the
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The FB-FCI wave function can then be interpreted as a constrained solution of the
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true Hamiltonian forced to span the restricted space provided by the finite one-electron basis.
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In the complete basis set (CBS) limit, the constraint is lifted and the exact solution is recovered.
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Hence, the accuracy of a FB-FCI calculation can be systematically improved by increasing the size of the one-electron basis set.
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Nevertheless, the exponential growth of its computational scaling with the number of electrons and with the basis set size is prohibitive for most chemical systems.
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In recent years, the introduction of new algorithms \cite{Booth_2009,Xu_2018,Eriksen_2018,Eriksen_2019} and the
|
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revival \cite{Abrams_2005,Bytautas_2009,Roth_2009,Giner_2013,Knowles_2015,Holmes_2016,Liu_2016,Garniron_2018}
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of selected configuration interaction (SCI)
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methods \cite{Bender_1969,Huron_1973,Buenker_1974} pushed the limits of
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the sizes of the systems that could be computed at the FCI level. \cite{Booth_2010,Cleland_2010,Daday_2012,Chien_2018,Loos_2018a,Loos_2019,Loos_2020b,Loos_2020c}
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However, the scaling remains exponential unless some bias is introduced leading
|
||||
methods \cite{Bender_1969,Huron_1973,Buenker_1974} significantly expanded the range of applicability of this family of methods.
|
||||
Importantly, one can now routinely compute the ground- and excited-state energies of small- and medium-sized molecular systems with near-FCI accuracy. \cite{Booth_2010,Cleland_2010,Daday_2012,Motta_2017,Chien_2018,Loos_2018a,Loos_2019,Loos_2020b,Loos_2020c,Williams_2020,Eriksen_2020}
|
||||
However, although the prefactor is reduced, the overall computational scaling remains exponential unless some bias is introduced leading
|
||||
to a loss of size consistency. \cite{Evangelisti_1983,Cleland_2010,Tenno_2017}
|
||||
|
||||
Diffusion Monte Carlo (DMC) is another numerical scheme to obtain
|
||||
Another route to solve the Schr\"odinger equation is density-functional theory (DFT). \cite{Hohenberg_1964}
|
||||
Present-day DFT calculations are almost exclusively done within the so-called Kohn-Sham (KS) formalism, \cite{Kohn_1965} which
|
||||
transfers the complexity of the many-body problem to the exchange-correlation (xc) functional thanks to a judicious mapping between a non-interacting reference system and its interacting analog which both have exactly the same one-electron density.
|
||||
KS-DFT \cite{Hohenberg_1964,Kohn_1965} is now the workhorse of electronic structure calculations for atoms, molecules and solids thanks to its very favorable accuracy/cost ratio. \cite{ParrBook}
|
||||
As compared to WFT, DFT has the indisputable advantage of converging much faster with respect to the size of the basis set. \cite{FraMusLupTou-JCP-15}
|
||||
However, one faces the unsettling choice of the \emph{approximate} xc functional which makes inexorably KS-DFT hard to systematically improve. \cite{Becke_2014}
|
||||
|
||||
Diffusion Monte Carlo (DMC) is yet another numerical scheme to obtain
|
||||
the exact solution of the Schr\"odinger equation with a different
|
||||
constraint. In DMC, the solution is imposed to have the same nodes (or zeroes)
|
||||
as a given trial (approximate) wave function.
|
||||
Within this so-called \emph{fixed-node} (FN) approximation,
|
||||
constraint. \cite{Foulkes_2001,Austin_2012,Needs_2020}
|
||||
In DMC, the solution is imposed to have the same nodes (or zeroes)
|
||||
as a given trial (approximate) wave function. \cite{Reynolds_1982,Ceperley_1991}
|
||||
Within this so-called fixed-node (FN) approximation,
|
||||
the FN-DMC energy associated with a given trial wave function is an upper
|
||||
bound to the exact energy, and the latter is recovered only when the
|
||||
nodes of the trial wave function coincide with the nodes of the exact
|
||||
wave function.
|
||||
The polynomial scaling with the number of electrons and with the size
|
||||
of {\manu{in what sense is it polynomial?}the trial wave function makes the FN-DMC method particularly attractive.
|
||||
of {\manu{in what sense is it polynomial?}the trial wave function makes the FN-DMC method particularly attractive.\cite{Nakano_2020,Scemama_2013,Needs_2020,Kim_2018,Kent_2020}
|
||||
In addition, the total energies obtained are usually far below
|
||||
those obtained with the FCI method in computationally tractable basis
|
||||
sets because the constraints imposed by the FN approximation
|
||||
@ -159,10 +166,10 @@ method such as CIPSI for instance.\cite{Giner_2013,Giner_2015,Caffarel_2016_2}
|
||||
\titou{When the basis set is increased, the trial wave function gets closer
|
||||
to the exact wave function, so the nodal surface can be systematically
|
||||
improved.\cite{Caffarel_2016} WRONG}
|
||||
This technique has the advantage \manu{of using the} FCI nodes in a given basis
|
||||
set \manu{, which is perfectly well defined and therefore makes the calculations} reproducible in a
|
||||
This technique has the advantage of using the FCI nodes in a given basis
|
||||
set, which is perfectly well defined and therefore makes the calculations reproducible in a
|
||||
black-box way without needing any expertise in QMC.
|
||||
\manu{Nevertheless,} this technique cannot be applied to large systems because of the
|
||||
Nevertheless, this technique cannot be applied to large systems because of the
|
||||
exponential scaling of the size of the trial wave function.
|
||||
Extrapolation techniques have been used to estimate the FN-DMC energies
|
||||
obtained with FCI wave functions,\cite{Scemama_2018} and other authors
|
||||
@ -190,7 +197,7 @@ In single-determinant DMC calculations, the degrees of freedom used to
|
||||
reduce the fixed-node error are the molecular orbitals on which the
|
||||
Slater determinant is built.
|
||||
Different molecular orbitals can be chosen:
|
||||
Hartree-Fock (HF), Kohn-Sham (KS), natural (NO) orbitals of a
|
||||
Hartree-Fock (HF), Kohn-Sham (KS), natural orbitals (NOs) of a
|
||||
correlated wave function, or orbitals optimized under the
|
||||
presence of a Jastrow factor.
|
||||
The nodal surfaces obtained with the KS determinant are in general
|
||||
@ -225,8 +232,8 @@ of the wave functions is required.
|
||||
\subsection{CIPSI}
|
||||
%====================
|
||||
Beyond the single-determinant representation, the best
|
||||
multi-determinant wave function one can obtain \manu{in a given basis set} is the FCI.
|
||||
FCI is \manu{the ultimate goal of} \emph{post-Hartree-Fock} methods, and there exists several systematic
|
||||
multi-determinant wave function one can obtain in a given basis set is the FCI.
|
||||
FCI is the ultimate goal of \emph{post-Hartree-Fock} methods, and there exists several systematic
|
||||
improvements between the Hartree-Fock and FCI wave functions:
|
||||
increasing the maximum degree of excitation of CI methods (CISD, CISDT,
|
||||
CISDTQ, \emph{etc}), or increasing the complete active space
|
||||
@ -263,7 +270,7 @@ accuracy so all the CIPSI selections were made such that $\abs{\EPT} <
|
||||
\label{sec:rsdft}
|
||||
%=================================
|
||||
|
||||
\manu{The range-separated DFT (RS-DFT)} was introduced in the seminal work of Savin,\cite{Sav-INC-96a,Toulouse_2004}
|
||||
Range-separated DFT (RS-DFT) was introduced in the seminal work of Savin,\cite{Sav-INC-96a,Toulouse_2004}
|
||||
where the Coulomb operator entering the electron-electron repulsion is split into two pieces:
|
||||
\begin{equation}
|
||||
\frac{1}{r}
|
||||
@ -279,7 +286,7 @@ where
|
||||
are the singular short-range (sr) part and the non-singular long-range (lr) part, respectively, $\mu$ is the range-separation parameter which controls how rapidly the short-range part decays, $\erf(x)$ is the error function, and $\erfc(x) = 1 - \erf(x)$ is its complementary version.
|
||||
|
||||
The main idea behind RS-DFT is to treat the short-range part of the
|
||||
interaction \manu{using a density functional}, and the long-range part within a WFT method like FCI in the present case.
|
||||
interaction using a density functional, and the long-range part within a WFT method like FCI in the present case.
|
||||
The parameter $\mu$ controls the range of the separation, and allows
|
||||
to go continuously from the KS Hamiltonian ($\mu=0$) to
|
||||
the FCI Hamiltonian ($\mu = \infty$).
|
||||
@ -296,8 +303,8 @@ $\mathcal{F}^{\text{lr},\mu}$ is a long-range universal density
|
||||
functional and $\bar{E}_{\text{Hxc}}^{\text{sr,}\mu}$ is the
|
||||
complementary short-range Hartree-exchange-correlation (Hxc) density
|
||||
functional. \cite{Savin_1996,Toulouse_2004}
|
||||
\manu{The exact ground state energy can be therefore obtained as a minimization
|
||||
over a multi-determinant wave function as follows}:
|
||||
The exact ground state energy can be therefore obtained as a minimization
|
||||
over a multi-determinant wave function as follows:
|
||||
\begin{equation}
|
||||
\label{min_rsdft} E_0= \min_{\Psi} \qty{
|
||||
\mel{\Psi}{\hat{T}+\hat{W}_\text{{ee}}^{\text{lr},\mu}+\hat{V}_{\text{ne}}}{\Psi}
|
||||
@ -414,7 +421,7 @@ has been set to $\EPT < 10^{-3}$ \hartree{} or $ \Ndet > 10^7$.
|
||||
All the wave functions are eigenfunctions of the $\Hat{S}^2$ spin operator, as
|
||||
described in Ref.~\onlinecite{Applencourt_2018}.
|
||||
|
||||
Quantum Monte Carlo calculations were made with QMC=Chem,\cite{scemama_2013}
|
||||
Quantum Monte Carlo calculations were made with QMC=Chem,\cite{Scemama_2013}
|
||||
in the determinant localization approximation (DLA),\cite{Zen_2019}
|
||||
where only the determinantal component of the trial wave
|
||||
function is present in the expression of the wave function on which
|
||||
@ -479,7 +486,7 @@ For this purpose, we consider a weakly correlated molecular system, namely the w
|
||||
molecule \titou{near its equilibrium geometry.} \cite{Caffarel_2016}
|
||||
We then generate trial wave functions $\Psi^\mu$ for multiple values of
|
||||
$\mu$, and compute the associated fixed-node energy keeping all the
|
||||
parameters having an impact on the nodal surface fixed \manu{such as CI coefficients and molecular orbitals}.
|
||||
parameters having an impact on the nodal surface fixed such as CI coefficients and molecular orbitals.
|
||||
|
||||
%======================================================
|
||||
\subsection{Fixed-node energy of $\Psi^\mu$}
|
||||
@ -501,7 +508,7 @@ and then the FN-DMC error raises until it reaches the $\mu=\infty$ limit (\ie, t
|
||||
For instance, with respect to the FN-DMC/VDZ-BFD energy at $\mu=\infty$,
|
||||
one can obtain a lowering of the FN-DMC energy of $2.6 \pm 0.7$~m\hartree{}
|
||||
with an optimal value of $\mu=1.75$~bohr$^{-1}$.
|
||||
\manu{This lowering in FN-DMC energy is to be compared with the $3.2 \pm 0.7$~m\hartree{} of gain in FN-DMC energy between the KS wave function ($\mu=0$) and the FCI wave function ($\mu=\infty$)}.
|
||||
This lowering in FN-DMC energy is to be compared with the $3.2 \pm 0.7$~m\hartree{} of gain in FN-DMC energy between the KS wave function ($\mu=0$) and the FCI wave function ($\mu=\infty$).
|
||||
When the basis set is increased, the gain in FN-DMC energy with
|
||||
respect to the FCI trial wave function is reduced, and the optimal
|
||||
value of $\mu$ is slightly shifted towards large $\mu$.
|
||||
@ -511,9 +518,9 @@ to those obtained with the srPBE functional, even if the
|
||||
RS-DFT energies obtained with these two functionals differ by several
|
||||
tens of m\hartree{}.
|
||||
|
||||
\manu{An other important aspect here regards the compactness of the trial wave functions $\Psi^\mu$:}
|
||||
\titou{at $\mu=1.75$~bohr$^{-1}$, $\Psi^{\mu}$ has \textit{only} $54\,540$ determinants at the srPBE/VDZ-BFD level, while the FCI wave function contains $200\,521$ determinants (see Table \ref{tab:h2o-dmc}). Even at the srPBE/VTZ-BFD level, we observe a reduction by a factor two in the number of determinants between the optimal $\mu$ value and $\mu = \infty$.
|
||||
The take-home message of this numerical study is that RS-DFT trial wave functions can yield a lower fixed-node energy with more compact multideterminant expansion as compared to FCI.}
|
||||
An other important aspect here regards the compactness of the trial wave functions $\Psi^\mu$:
|
||||
at $\mu=1.75$~bohr$^{-1}$, $\Psi^{\mu}$ has \textit{only} $54\,540$ determinants at the srPBE/VDZ-BFD level, while the FCI wave function contains $200\,521$ determinants (see Table \ref{tab:h2o-dmc}). Even at the srPBE/VTZ-BFD level, we observe a reduction by a factor two in the number of determinants between the optimal $\mu$ value and $\mu = \infty$.
|
||||
The take-home message of this numerical study is that RS-DFT trial wave functions can yield a lower fixed-node energy with more compact multideterminant expansion as compared to FCI.
|
||||
|
||||
%======================================================
|
||||
\subsection{Link between RS-DFT and Jastrow factors }
|
||||
@ -828,7 +835,7 @@ Ref.~\onlinecite{Scemama_2015}).
|
||||
In this section, we make a numerical verification that the produced
|
||||
wave functions are size-consistent for a given range-separation
|
||||
parameter.
|
||||
We have computed the \manu{FN-DMC} energy of the dissociated fluorine dimer, where
|
||||
We have computed the FN-DMC energy of the dissociated fluorine dimer, where
|
||||
the two atoms are at a distance of 50~\AA. We expect that the energy
|
||||
of this system is equal to twice the energy of the fluorine atom.
|
||||
The data in table~\ref{tab:size-cons} shows that it is indeed the
|
||||
@ -845,7 +852,7 @@ Closed-shell molecules often dissociate into open-shell
|
||||
fragments. To get reliable atomization energies, it is important to
|
||||
have a theory which is of comparable quality for open-shell and
|
||||
closed-shell systems. A good test is to check that all the components
|
||||
of a spin multiplet are degenerate\manu{, as expected from exact solutions}.
|
||||
of a spin multiplet are degenerate, as expected from exact solutions.
|
||||
FCI wave functions have this property and give degenerate energies with
|
||||
respect to the spin quantum number $m_s$, but the multiplication by a
|
||||
Jastrow factor introduces spin contamination if the parameters
|
||||
@ -1085,7 +1092,7 @@ solution would have been the PBE single determinant.
|
||||
\section{Conclusion}
|
||||
%%%%%%%%%%%%%%%%%%%%
|
||||
|
||||
\manu{In the present work} we have shown that introducing short-range correation via
|
||||
In the present work, we have shown that introducing short-range correlation via
|
||||
a range-separated Hamiltonian in a full CI expansion yields improved
|
||||
nodal surfaces, especially with small basis sets. The effect of sr-DFT
|
||||
on the determinant expansion is similar to the effect of re-optimizing
|
||||
|
||||
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Reference in New Issue
Block a user