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%% This BibTeX bibliography file was created using BibDesk.
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%% http://bibdesk.sourceforge.net/
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%% Created for Pierre-Francois Loos at 2019-04-03 22:18:33 +0200
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%% Created for Pierre-Francois Loos at 2019-04-07 14:04:11 +0200
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%% Saved with string encoding Unicode (UTF-8)
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@misc{g09,
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Author = {M. J. Frisch and G. W. Trucks and H. B. Schlegel and G. E. Scuseria and M. A. Robb and J. R. Cheeseman and G. Scalmani and V. Barone and B. Mennucci and G. A. Petersson and H. Nakatsuji and M. Caricato and X. Li and H. P. Hratchian and A. F. Izmaylov and J. Bloino and G. Zheng and J. L. Sonnenberg and M. Hada and M. Ehara and K. Toyota and R. Fukuda and J. Hasegawa and M. Ishida and T. Nakajima and Y. Honda and O. Kitao and H. Nakai and T. Vreven and Montgomery, {Jr.}, J. A. and J. E. Peralta and F. Ogliaro and M. Bearpark and J. J. Heyd and E. Brothers and K. N. Kudin and V. N. Staroverov and R. Kobayashi and J. Normand and K. Raghavachari and A. Rendell and J. C. Burant and S. S. Iyengar and J. Tomasi and M. Cossi and N. Rega and J. M. Millam and M. Klene and J. E. Knox and J. B. Cross and V. Bakken and C. Adamo and J. Jaramillo and R. Gomperts and R. E. Stratmann and O. Yazyev and A. J. Austin and R. Cammi and C. Pomelli and J. W. Ochterski and R. L. Martin and K. Morokuma and V. G. Zakrzewski and G. A. Voth and P. Salvador and J. J. Dannenberg and S. Dapprich and A. D. Daniels and O. Farkas and J. B. Foresman and J. V. Ortiz and J. Cioslowski and D. J. Fox},
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Date-Added = {2019-04-07 14:01:06 +0200},
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Date-Modified = {2019-04-07 14:01:12 +0200},
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Note = {\uppercase{G}aussian Inc. Wallingford CT},
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Title = {Gaussian~09 \uppercase{R}evision {D}.01},
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Year = 2009}
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@article{HauJanScu-JCP-09,
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Author = {R. Haunschild and B. G. Janesko and G. E. Scuseria},
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Date-Added = {2019-04-07 13:56:11 +0200},
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Date-Modified = {2019-04-07 13:57:34 +0200},
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Doi = {10.1063/1.3247288},
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Journal = {J. Chem. Phys.},
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Pages = {154112},
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Title = {Local hybrids as a perturbation to global hybrid functionals},
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Volume = {131},
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Year = {2009}}
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@article{SceGarCafLoo-JCTC-18,
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Author = {A. Scemama and Y. Garniron and M. Caffarel and P. F. Loos},
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Date-Added = {2019-04-07 13:55:03 +0200},
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Date-Modified = {2019-04-07 13:55:23 +0200},
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Doi = {10.1021/acs.jctc.7b01250},
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Journal = {J. Chem. Theory Comput.},
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Pages = {1395},
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Title = {Deterministic construction of nodal surfaces within quantum Monte Carlo: the case of FeS},
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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.7b01250}}
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@article{QP2,
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Author = {Y. Garniron and K. Gasperich and T. Applencourt and A. Benali and A. Fert{\'e} and J. Paquier and B. Pradines and R. Assaraf and P. Reinhardt and J. Toulouse and P. Barbaresco and N. Renon and G. David and J. P. Malrieu and M. V{\'e}ril and M. Caffarel and P. F. Loos and E. Giner and A. Scemama},
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Date-Added = {2019-04-07 13:54:16 +0200},
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Date-Modified = {2019-04-07 13:54:21 +0200},
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Journal = {J. Chem. Theory Comput.},
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Title = {Quantum Package 2.0: a open-source determinant-driven suite of programs},
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Volume = {submitted},
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Year = {2019}}
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@article{PerRuzTaoStaScuCso-JCP-05,
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Author = {J. P. Perdew and A. Ruzsinszky and J. Tao and V. N. Staroverov and G. E. Scuseria and G. I. Csonka},
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Date-Added = {2019-04-03 22:17:53 +0200},
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@ -6886,8 +6926,9 @@
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@article{LooGil-WIRES-16,
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Author = {Pierre-Francois Loos and Peter M. W. Gill},
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Date-Modified = {2019-04-07 14:04:07 +0200},
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Doi = {doi: 10.1002/wcms.1257},
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Journal = {WIREs Comput. Mol. Sci.},
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Note = {doi: 10.1002/wcms.1257},
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Pages = {410},
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Title = {The uniform electron gas},
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Volume = {6},
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@ -11882,13 +11923,12 @@
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@article{FerGinTou-JCP-18,
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Author = {Fert{\'e},Anthony and Giner,Emmanuel and Toulouse,Julien},
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Date-Modified = {2019-04-07 14:03:44 +0200},
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Doi = {10.1063/1.5082638},
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Eprint = {https://doi.org/10.1063/1.5082638},
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Journal = {The Journal of Chemical Physics},
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Journal = {J. Chem. Phys.},
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Number = {8},
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Pages = {084103},
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Title = {Range-separated multideterminant density-functional theory with a short-range correlation functional of the on-top pair density},
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Url = {https://doi.org/10.1063/1.5082638},
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Volume = {150},
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Year = {2019},
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Bdsk-Url-1 = {https://doi.org/10.1063/1.5082638}}
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@ -11941,27 +11981,24 @@
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@article{LooBogSceCafJAc-JCTC-19,
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Author = {Loos, Pierre-Fran{\c c}ois and Boggio-Pasqua, Martial and Scemama, Anthony and Caffarel, Michel and Jacquemin, Denis},
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Date-Modified = {2019-04-07 14:02:34 +0200},
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Doi = {10.1021/acs.jctc.8b01205},
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Eprint = {https://doi.org/10.1021/acs.jctc.8b01205},
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Journal = {Journal of Chemical Theory and Computation},
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Journal = {J. Chem. Theory Comput.},
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Number = {3},
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Pages = {1939-1956},
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Title = {Reference Energies for Double Excitations},
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Url = {https://doi.org/10.1021/acs.jctc.8b01205},
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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.8b01205}}
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@article{LooSceBloGarCafJac-JCTC-18,
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Author = {Loos, Pierre-Fran{\c c}ois and Scemama, Anthony and Blondel, Aymeric and Garniron, Yann and Caffarel, Michel and Jacquemin, Denis},
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Date-Modified = {2019-04-07 14:03:07 +0200},
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Doi = {10.1021/acs.jctc.8b00406},
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Eprint = {https://doi.org/10.1021/acs.jctc.8b00406},
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Journal = {Journal of Chemical Theory and Computation},
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Note = {PMID: 29966098},
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Journal = {J. Chem. Theory Comput.},
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Number = {8},
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Pages = {4360-4379},
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Title = {A Mountaineering Strategy to Excited States: Highly Accurate Reference Energies and Benchmarks},
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Url = {https://doi.org/10.1021/acs.jctc.8b00406},
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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.8b00406}}
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@ -11992,30 +12029,32 @@
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Bdsk-Url-1 = {https://doi.org/10.1063/1.4992127}}
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@article{GorSav-PRA-06,
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title = {Properties of short-range and long-range correlation energy density functionals from electron-electron coalescence},
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author = {Gori-Giorgi, Paola and Savin, Andreas},
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journal = {Phys. Rev. A},
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volume = {73},
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issue = {3},
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pages = {032506},
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numpages = {9},
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year = {2006},
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month = {Mar},
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publisher = {American Physical Society},
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doi = {10.1103/PhysRevA.73.032506},
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url = {https://link.aps.org/doi/10.1103/PhysRevA.73.032506}
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}
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Author = {Gori-Giorgi, Paola and Savin, Andreas},
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Doi = {10.1103/PhysRevA.73.032506},
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Issue = {3},
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Journal = {Phys. Rev. A},
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Month = {Mar},
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Numpages = {9},
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Pages = {032506},
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Publisher = {American Physical Society},
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Title = {Properties of short-range and long-range correlation energy density functionals from electron-electron coalescence},
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Url = {https://link.aps.org/doi/10.1103/PhysRevA.73.032506},
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Volume = {73},
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Year = {2006},
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Bdsk-Url-1 = {https://link.aps.org/doi/10.1103/PhysRevA.73.032506},
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Bdsk-Url-2 = {https://doi.org/10.1103/PhysRevA.73.032506}}
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@article{HalHelJorKloKocOls-CPL-98,
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title = "Basis-set convergence in correlated calculations on Ne, N2, and H2O",
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journal = "Chemical Physics Letters",
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volume = "286",
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number = "3",
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pages = "243 - 252",
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year = "1998",
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issn = "0009-2614",
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doi = "https://doi.org/10.1016/S0009-2614(98)00111-0",
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url = "http://www.sciencedirect.com/science/article/pii/S0009261498001110",
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author = "Asger Halkier and Trygve Helgaker and Poul Jørgensen and Wim Klopper and Henrik Koch and Jeppe Olsen and Angela K. Wilson",
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abstract = "Valence and all-electron correlation energies of Ne, N2, and H2O at fixed experimental geometries are computed at the levels of second-order perturbation theory (MP2) and coupled cluster theory with singles and doubles excitations (CCSD), and singles and doubles excitations with a perturbative triples correction (CCSD(T)). Correlation-consistent polarized valence and core-valence basis sets up to sextuple zeta quality are employed. Guided by basis-set limits established by rij-dependent methods, a number of extrapolation schemes for use with the correlation-consistent basis sets are investigated. Among the schemes considered here, a linear least-squares procedure applied to the quintuple and sextuple zeta results yields the most accurate extrapolations."
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}
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Abstract = {Valence and all-electron correlation energies of Ne, N2, and H2O at fixed experimental geometries are computed at the levels of second-order perturbation theory (MP2) and coupled cluster theory with singles and doubles excitations (CCSD), and singles and doubles excitations with a perturbative triples correction (CCSD(T)). Correlation-consistent polarized valence and core-valence basis sets up to sextuple zeta quality are employed. Guided by basis-set limits established by rij-dependent methods, a number of extrapolation schemes for use with the correlation-consistent basis sets are investigated. Among the schemes considered here, a linear least-squares procedure applied to the quintuple and sextuple zeta results yields the most accurate extrapolations.},
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Author = {Asger Halkier and Trygve Helgaker and Poul J{\o}rgensen and Wim Klopper and Henrik Koch and Jeppe Olsen and Angela K. Wilson},
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Doi = {https://doi.org/10.1016/S0009-2614(98)00111-0},
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Issn = {0009-2614},
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Journal = {Chemical Physics Letters},
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Number = {3},
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Pages = {243 - 252},
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Title = {Basis-set convergence in correlated calculations on Ne, N2, and H2O},
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Url = {http://www.sciencedirect.com/science/article/pii/S0009261498001110},
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Volume = {286},
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Year = {1998},
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Bdsk-Url-1 = {http://www.sciencedirect.com/science/article/pii/S0009261498001110},
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Bdsk-Url-2 = {https://doi.org/10.1016/S0009-2614(98)00111-0}}
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@ -60,7 +60,9 @@
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\newcommand{\lr}{\text{lr}}
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\newcommand{\sr}{\text{sr}}
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\newcommand{\Nel}{N}
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\newcommand{\Ne}{N}
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\newcommand{\Nb}{N_{\Bas}}
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\newcommand{\Ng}{N_\text{grid}}
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\newcommand{\n}[2]{n_{#1}^{#2}}
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\newcommand{\Ec}{E_\text{c}}
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@ -170,7 +172,7 @@ We refer the interested reader to Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18} fo
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%=================================================================
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%\subsection{Correcting the basis set error of a general WFT model}
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%=================================================================
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Let us assume we have both the energy $\E{\modX}{\Bas}$ and density $\n{\modY}{\Bas}$ of a $\Nel$-electron system described by two methods $\modX$ and $\modY$ (potentially identical) in an incomplete basis set $\Bas$.
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Let us assume we have both the energy $\E{\modX}{\Bas}$ and density $\n{\modY}{\Bas}$ of a $\Ne$-electron system described by two methods $\modX$ and $\modY$ (potentially identical) in an incomplete basis set $\Bas$.
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According to Eq.~(15) of Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, assuming that $\E{\modX}{\Bas}$ and $\n{\modY}{\Bas}$ are reasonable approximations of the \titou{FCI} energy and density within $\Bas$, the exact ground state energy $\E{}{}$ may be written as
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\begin{equation}
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\label{eq:e0basis}
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@ -186,7 +188,7 @@ where
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- \min_{\wf{}{\Bas} \to \n{}{}} \mel*{\wf{}{\Bas}}{\hT + \hWee{}}{\wf{}{\Bas}}
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\end{equation}
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is the basis-dependent complementary density functional, $\hT$ is the kinetic operator and $\hWee{} = \sum_{i<j} r_{ij}^{-1}$ is the interelectronic repulsion operator.
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In Eq.~\eqref{eq:E_funcbasis}, $\wf{}{\Bas}$ and $\wf{}{}$ are two general $\Nel$-electron wave functions belonging to the Hilbert space spanned by $\Bas$ and the complete basis, respectively.
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In Eq.~\eqref{eq:E_funcbasis}, $\wf{}{\Bas}$ and $\wf{}{}$ are two general $\Ne$-electron wave functions belonging to the Hilbert space spanned by $\Bas$ and the complete basis, respectively.
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Both wave functions yield the same target density $\n{}{}$.
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%\alert{Provided that the functional $\bE{}{\Bas}[\n{}{}]$ is known exactly, the only approximation performed in \eqref{eq:e0basis} is that the FCI density $\n{\FCI}{\Bas}$ coincides with the exact ground state density, which in general is a reasonable approximation as the density converges rapidly with the basis set.}
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@ -486,6 +488,7 @@ In order to correct such a defect, we propose here a new ECMD functional inspire
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\end{gather}
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\end{subequations}
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The difference between the ECMD PBE functional defined in Ref.~\onlinecite{FerGinTou-JCP-18} and the present expression \eqref{eq:epsilon_cmdpbe} is that we approximate here the \textit{exact} ground-state on-top pair density of the system $\n{}{(2)}$ by its UEG version, i.e.~$\n{}{(2)} \approx \n{\UEG}{(2)} = \n{}{2} g_0(\n{}{})$, where $g_0(\n{}{})$ is the UEG correlation factor whose parametrization can be found in Eq.~(46) of Ref.~\onlinecite{GorSav-PRA-06}.
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This represents a major computational saving without loss of performance as we eschew the computation of $\n{}{(2)}$.
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Therefore, the PBE complementary functional reads
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\begin{equation}
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\label{eq:def_lda_tot}
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@ -506,6 +509,8 @@ Depending on the functional choice, the complementary functional $\bE{}{\Bas}[\n
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% \lim_{\Bas \rightarrow \infty} \ecmuapproxmurmodel = 0 \quad ,
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%\end{equation}
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%for whatever choice of density $\denmodel$, wave function $\wf{}{\Bas}$ used to define the interaction, and ECMD functional used to approximate the exact ECMD.
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The computational bottleneck of the present approach is the computation of $\f{\wf{}{\Bas}}{}(\br{})$ [See Eqs.~\eqref{eq:fbasis} and \eqref{eq:mu_of_r}] at each quadrature grid point.
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By performing successive matrix multiplications, this step can be formally performed at $\order{\Ng \Nb^4}$ computational cost and $\order{\Ng \Nb^2}$ storage (where $\Ng$ is the number of points of the quadrature grid and $\Nb$ is the number of basis functions in $\Bas$), although sparsity and/or atomic-orbital-based algorithms could further reduce this cost.
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%=================================================================
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@ -584,8 +589,8 @@ Defining $\n{\wf{}{\Bas}}{\Val}$ as the valence one-electron density, the valenc
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\begin{table*}
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\caption{
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\label{tab:diatomics}
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Dissociation energy ($\De$) in {\kcal} of the \ce{C2}, \ce{O2}, \ce{N2} and \ce{F2} molecules computed with various methods and basis sets.
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The deviation with respect to the CBS values are reported in parenthesis.
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Dissociation energy ($\De$) in {\kcal} of \ce{C2}, \ce{O2}, \ce{N2} and \ce{F2} computed with various methods and basis sets.
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The deviations with respect to the corresponding CBS values are reported in parenthesis.
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}
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\begin{ruledtabular}
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\begin{tabular}{llddddd}
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@ -672,15 +677,15 @@ We begin our investigation of the performance of the basis set correction by com
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In the case of \ce{C2} and \ce{N2}, we also perform calculations with the cc-pCVXZ family.
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\ce{N2}, \ce{O2} and \ce{F2} are weakly correlated systems and belong to the G2 test set, whereas \ce{C2} already contains a non-negligible amount of strong correlation.
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In a second time, we compute the entire atomization energies of the G2 test sets composed by 55 molecules.
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The reference values for the atomization energies are extracted from Ref.~\onlinecite{HauKlo-JCP-12} and corresponds to frozen-core non-relativistic atomization energies obtained at the CCSD(T)(F12)/cc-pVQZ-F12 level of theory corrected for higher-excitation contributions ($E_\text{CCSDT(Q)/cc-pV(D+d)Z} - E_\text{CCSD(T)/cc-pV(D+d)Z})$.
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%The reference values for the atomization energies are extracted from Ref.~\onlinecite{HauKlo-JCP-12} and corresponds to frozen-core non-relativistic atomization energies obtained at the CCSD(T)(F12)/cc-pVQZ-F12 level of theory corrected for higher-excitation contributions ($E_\text{CCSDT(Q)/cc-pV(D+d)Z} - E_\text{CCSD(T)/cc-pV(D+d)Z})$.
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As a method $\modX$ we employ either $\CCSDT$ or $\exFCI$.
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Here, exFCI stands for extrapolated FCI energies computed with the CIPSI algorithm.
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We refer the interested reader to Refs.~\onlinecite{LooSceBloGarCafJac-JCTC-18, SceBenJacCafLoo-JCP-18, LooBogSceCafJAc-JCTC-19} for more details.
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We refer the interested reader to Refs.~\onlinecite{SceGarCafLoo-JCTC-18, LooSceBloGarCafJac-JCTC-18, SceBenJacCafLoo-JCP-18, LooBogSceCafJAc-JCTC-19} for more details.
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Throughout this study, we have $\modY = \HF$ as we use the Hartree-Fock (HF) one-electron density to compute the complementary energy.
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The CCSD(T) calculations have been performed with Gaussian09 with standard threshold values. \cite{g09}
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RS-DFT and exFCI calculations are performed with {\QP}. \cite{QP2}
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\titou{For the quadrature grid, we employ ... radial and angular points.}
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Except for the carbon dimer where we have taken the experimental equilibrium bond length (\InAA{1.2425}), all geometries have been extracted from Ref.~\onlinecite{HauJanScu-JCP-2009} and have been performed at the B3LYP/6-31G(2df,p) level of theory.
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Except for the carbon dimer where we have taken the experimental equilibrium bond length (\InAA{1.2425}), all geometries have been extracted from Ref.~\onlinecite{HauJanScu-JCP-09} and have been performed at the B3LYP/6-31G(2df,p) level of theory.
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Frozen core calculations are defined as such: an \ce{He} core is frozen from \ce{B} to \ce{Mg}, while a \ce{Ne} core is frozen from \ce{Al} to \ce{Xe}.
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In the context of the basis set correction, the set of valence spinorbitals $\Val$ involved in the definition of the valence part of the effective interaction [see Eq.~\eqref{eq:Wval}] refers to the non-frozen spinorbitals.
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The ``valence'' correction was used consistently when the FC approximation was applied.
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@ -694,7 +699,7 @@ The same behaviours hold for the CCSD(T) model, and one can notice that the atom
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%\subsection{The effect of the basis set correction within the LDA and PBE approximation}
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Regarding the effect of the basis set correction, both for the CIPSI and CCSD(T) models, several observations can be done.
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First, in a given basis set, the addition of the basis set correction, both at the LDA and PBE level, improves the result even if it can overestimates the estimated CBS atomization energies by a few tens of kcal/mol (the largest deviation being 0.6 kcal/mol for N$_2$ at the (FC)CCSD(T)+PBE-val level in the cc-pv5z basis). Nevertheless, the deviations observed in the largest basis sets are typically in the range of the accuracy of the atomization energies computed with the CBS extrapolation technique.
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First, in a given basis set, the addition of the basis set correction, both at the LDA and PBE level, improves the result even if it can overestimates the estimated CBS atomization energies by a few tens of kcal/mol (the largest deviation being 0.6 kcal/mol for N$_2$ at the (FC)CCSD(T)+PBE-val level in the cc-pV5Z basis). Nevertheless, the deviations observed in the largest basis sets are typically in the range of the accuracy of the atomization energies computed with the CBS extrapolation technique.
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Also, the values obtained with the largest basis sets tends to converge toward a value close to the estimated CBS values. Also, one can observe that the sensitivity to the functional is quite large for the double- and triple-zeta basis sets, where clearly the PBE functional performs better. Nevertheless, from the quadruple-zeta basis set, the LDA and PBE functional agrees within a few tens of kcal/mol.
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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