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Manuscript/G2-srDFT.bib
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%% This BibTeX bibliography file was created using BibDesk. %% This BibTeX bibliography file was created using BibDesk.
%% http://bibdesk.sourceforge.net/ %% http://bibdesk.sourceforge.net/
%% Created for Pierre-Francois Loos at 2019-04-07 21:16:12 +0200 %% Created for Pierre-Francois Loos at 2019-04-11 14:20:29 +0200
%% Saved with string encoding Unicode (UTF-8) %% Saved with string encoding Unicode (UTF-8)
@article{BarLoo-JCP-17,
Author = {Barca, Giuseppe MJ and Loos, Pierre-Fran{\c c}ois},
Date-Added = {2019-04-11 14:20:15 +0200},
Date-Modified = {2019-04-11 14:20:29 +0200},
File = {/Users/loos/Zotero/storage/DCFUMHWZ/56.pdf},
Journal = {J. Chem. Phys.},
Number = {2},
Pages = {024103},
Shorttitle = {Three-and Four-Electron Integrals Involving {{Gaussian}} Geminals},
Title = {Three-and Four-Electron Integrals Involving {{Gaussian}} Geminals: {{Fundamental}} Integrals, Upper Bounds, and Recurrence Relations},
Volume = {147},
Year = {2017}}
@article{FelPet-JCP-09, @article{FelPet-JCP-09,
Author = {D. Feller and K. A. Peterson}, Author = {D. Feller and K. A. Peterson},
Date-Added = {2019-04-07 20:41:03 +0200}, Date-Added = {2019-04-07 20:41:03 +0200},
@ -3824,20 +3837,20 @@
Year = {1977}} Year = {1977}}
@article{FroCimJen-PRA-10, @article{FroCimJen-PRA-10,
title = {Merging multireference perturbation and density-functional theories by means of range separation: Potential curves for ${\mathrm{Be}}_{2}$, ${\mathrm{Mg}}_{2}$, and ${\mathrm{Ca}}_{2}$}, Author = {Fromager, Emmanuel and Cimiraglia, Renzo and Jensen, Hans J\o{}rgen Aa.},
author = {Fromager, Emmanuel and Cimiraglia, Renzo and Jensen, Hans J\o{}rgen Aa.}, Doi = {10.1103/PhysRevA.81.024502},
journal = {Phys. Rev. A}, Issue = {2},
volume = {81}, Journal = {Phys. Rev. A},
issue = {2}, Month = {Feb},
pages = {024502}, Numpages = {4},
numpages = {4}, Pages = {024502},
year = {2010}, Publisher = {American Physical Society},
month = {Feb}, Title = {Merging multireference perturbation and density-functional theories by means of range separation: Potential curves for ${\mathrm{Be}}_{2}$, ${\mathrm{Mg}}_{2}$, and ${\mathrm{Ca}}_{2}$},
publisher = {American Physical Society}, Url = {https://link.aps.org/doi/10.1103/PhysRevA.81.024502},
doi = {10.1103/PhysRevA.81.024502}, Volume = {81},
url = {https://link.aps.org/doi/10.1103/PhysRevA.81.024502} Year = {2010},
} Bdsk-Url-1 = {https://link.aps.org/doi/10.1103/PhysRevA.81.024502},
Bdsk-Url-2 = {https://doi.org/10.1103/PhysRevA.81.024502}}
@article{Fro-JCP-11, @article{Fro-JCP-11,
Author = {E. Fromager}, Author = {E. Fromager},
@ -5198,9 +5211,9 @@
@article{HedKneKieJenRei-JCP-15, @article{HedKneKieJenRei-JCP-15,
Author = {E. D. Hedeg{\aa}rd and S. Knecht and J. S. Kielberg and H. J. Aa. Jensen and M. Reiher}, Author = {E. D. Hedeg{\aa}rd and S. Knecht and J. S. Kielberg and H. J. Aa. Jensen and M. Reiher},
title = {Density matrix renormalization group with efficient dynamical electron correlation through range separation},
Journal = {J. Chem. Phys.}, Journal = {J. Chem. Phys.},
Pages = {224108}, Pages = {224108},
Title = {Density matrix renormalization group with efficient dynamical electron correlation through range separation},
Volume = {142}, Volume = {142},
Year = {2015}} Year = {2015}}
@ -12241,29 +12254,29 @@
Bdsk-Url-2 = {https://doi.org/10.1016/S0009-2614(98)00111-0}} Bdsk-Url-2 = {https://doi.org/10.1016/S0009-2614(98)00111-0}}
@article{DasHer-JCC-17, @article{DasHer-JCC-17,
author = {Dasgupta, Saswata and Herbert, John M.}, Author = {Dasgupta, Saswata and Herbert, John M.},
title = {Standard grids for high-precision integration of modern density functionals: SG-2 and SG-3}, Doi = {10.1002/jcc.24761},
journal = {Journal of Computational Chemistry}, Eprint = {https://onlinelibrary.wiley.com/doi/pdf/10.1002/jcc.24761},
volume = {38}, Journal = {Journal of Computational Chemistry},
number = {12}, Number = {12},
pages = {869-882}, Pages = {869-882},
doi = {10.1002/jcc.24761}, Title = {Standard grids for high-precision integration of modern density functionals: SG-2 and SG-3},
eprint = {https://onlinelibrary.wiley.com/doi/pdf/10.1002/jcc.24761}, Volume = {38},
year = {2017} Year = {2017},
} Bdsk-Url-1 = {https://doi.org/10.1002/jcc.24761}}
@article{rs_dft_toul_colo_savin, @article{rs_dft_toul_colo_savin,
title = {Long-range--short-range separation of the electron-electron interaction in density-functional theory}, Author = {J. Toulouse and F. Colonna and A. Savin},
author = {J. Toulouse and F. Colonna and A. Savin}, Doi = {10.1103/PhysRevA.70.062505},
journal = {Phys. Rev. A}, Issue = {6},
volume = {70}, Journal = {Phys. Rev. A},
issue = {6}, Month = {Dec},
pages = {062505}, Numpages = {16},
numpages = {16}, Pages = {062505},
year = {2004}, Publisher = {American Physical Society},
month = {Dec}, Title = {Long-range--short-range separation of the electron-electron interaction in density-functional theory},
publisher = {American Physical Society}, Url = {https://link.aps.org/doi/10.1103/PhysRevA.70.062505},
doi = {10.1103/PhysRevA.70.062505}, Volume = {70},
url = {https://link.aps.org/doi/10.1103/PhysRevA.70.062505} Year = {2004},
} Bdsk-Url-1 = {https://link.aps.org/doi/10.1103/PhysRevA.70.062505},
Bdsk-Url-2 = {https://doi.org/10.1103/PhysRevA.70.062505}}

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Manuscript/G2-srDFT.tex
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@ -145,34 +145,38 @@ These values drop below 1 {\kcal} with the cc-pVQZ basis set.
\section{Introduction} \section{Introduction}
%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%
Contemporary quantum chemistry has developed in two directions --- wave function theory (WFT) \cite{Pop-RMP-99} and density-functional theory (DFT). \cite{Koh-RMP-99} Contemporary quantum chemistry has developed in two directions --- wave function theory (WFT) \cite{Pop-RMP-99} and density-functional theory (DFT). \cite{Koh-RMP-99}
Although both spring from the same Schr\"odinger equation, each of these philosophies has its own advantages and shortcomings. Although both spring from the same Schr\"odinger equation, each of these philosophies has its own \textit{pros} and \textit{cons}.
WFT is attractive as it exists a well-defined path for systematic improvement and powerful tools, such as perturbation theory, to guide the development of new attractive WFT models. WFT is attractive as it exists a well-defined path for systematic improvement as well as powerful tools, such as perturbation theory, to guide the development of new attractive WFT \textit{ans\"atze}.
The coupled-cluster (CC) family of methods are a typical example of the WFT philosophy for the description of weakly correlated systems and is well regarded as the gold standard of quantum chemistry. The coupled cluster (CC) family of methods is a typical example of the WFT philosophy and is well regarded as the gold standard of quantum chemistry for weakly correlated systems.
By increasing the excitation degree of the CC expansion, one can systematically converge, for a given basis set, to the exact, full configuration-interaction (FCI) limit, although the computational cost associated with such improvement is usually pricey. By increasing the excitation degree of the CC expansion, one can systematically converge, for a given basis set, to the exact, full configuration interaction (FCI) limit, although the computational cost associated with such improvement is usually pricey.
One of the most fundamental drawback of conventional WFT methods is the slow convergence of energies and properties with respect to the size of the one-electron basis set. One of the most fundamental drawback of conventional WFT methods is the slow convergence of energies and properties with respect to the size of the one-electron basis set.
This undesirable feature was put into light by Kutzelnigg more than thirty years ago. \cite{Kut-TCA-85} This undesirable feature was put into light by Kutzelnigg more than thirty years ago. \cite{Kut-TCA-85}
To palliate this, in Hylleraas' footsteps, \cite{Hyl-ZP-29} Kutzelnigg proposed to introduce explicitly the interelectronic distance $r_{12} = \abs{\br{1} - \br{2}}$ as a basis function. \cite{Kut-TCA-85, KutKlo-JCP-91, NogKut-JCP-94} To palliate this, following Hylleraas' footsteps, \cite{Hyl-ZP-29} Kutzelnigg proposed to introduce explicitly the interelectronic distance $r_{12} = \abs{\br{1} - \br{2}}$ to properly describe the electronic wave function around the coalescence of two electrons. \cite{Kut-TCA-85, KutKlo-JCP-91, NogKut-JCP-94}
The resulting F12 methods yields a prominent improvement of the energy convergence, and achieve chemical accuracy for small organic molecules with relatively small Gaussian basis sets. \cite{Ten-TCA-12, TenNog-WIREs-12, HatKloKohTew-CR-12, KonBisVal-CR-12} The resulting F12 methods yields a prominent improvement of the energy convergence, and achieve chemical accuracy for small organic molecules with relatively small Gaussian basis sets. \cite{Ten-TCA-12, TenNog-WIREs-12, HatKloKohTew-CR-12, KonBisVal-CR-12}
For example, at the CCSD(T) level, it is advertised that one can obtain quintuple-zeta quality correlation energies with a triple-zeta basis, \cite{TewKloNeiHat-PCCP-07} although computational overheads are introduced by the large auxiliary basis used to resolve three- and four-electron integrals. Except for these computational considerations, a possible drawback of F12 theory is its quite complicated formulation which requires a deep knowledge in this field in order to adapt F12 theory to a new WFT model. For example, at the CCSD(T) level, it is advertised that one can obtain quintuple-$\zeta$ quality correlation energies with a triple-$\zeta$ basis, \cite{TewKloNeiHat-PCCP-07} although computational overheads are introduced by the large auxiliary basis used to resolve three- and four-electron integrals. \cite{BarLoo-JCP-17}
Approximated schemes\cite{TorVal-JCP-09, KonVal-JCP-10, KonVal-JCP-11, BooCleAlaTew-JCP-2012, IrmHumGru-arXiv-2019, IrmGru-arXiv-2019} have emerged in order to reduce the computational cost and simplify the transferability of F12 theory. %\trashPFL{Except for these computational considerations, a possible drawback of F12 theory is its quite complicated formulation which requires a deep knowledge in this field in order to adapt F12 theory to a new WFT model.}
To reduce further the computational cost and/or ease the transferability of the F12 correction, approximated and/or universal schemes have recently emerged. \cite{TorVal-JCP-09, KonVal-JCP-10, KonVal-JCP-11, BooCleAlaTew-JCP-2012, IrmHumGru-arXiv-2019, IrmGru-arXiv-2019}
Regarding present-day DFT calculations, these are almost exclusively done within the so-called Kohn-Sham (KS) formalism, which corresponds to an exact dressed one-electron theory. \cite{KohSha-PR-65} Present-day DFT calculations are almost exclusively done within the so-called Kohn-Sham (KS) formalism, which corresponds to an exact dressed one-electron theory. \cite{KohSha-PR-65}
DFT's attractivity originates from its very favorable cost/efficient ratio as it can provide accurate energies and properties at a relatively low computational cost. DFT's attractiveness originates from its very favorable cost/efficiency ratio as it can provide accurate energies and properties at a relatively low computational cost.
Thanks to this, KS-DFT \cite{HohKoh-PR-64, KohSha-PR-65} has become the workhorse of electronic structure calculations for atoms, molecules and solids. \cite{ParYan-BOOK-89} Thanks to this, KS-DFT \cite{HohKoh-PR-64, KohSha-PR-65} has become the workhorse of electronic structure calculations for atoms, molecules and solids. \cite{ParYan-BOOK-89}
To obtain accurate results within DFT, one only requires an exchange and correlation functionals, which can be classified in various families depending on their physical input quantities. \cite{Bec-JCP-14} In the present context, one of the interesting feature of density-based methods is their much faster convergence with respect to the size of the basis set. \cite{FraMusLupTou-JCP-15}
Although there is no clear way on how to systematically improve density-functional approximations (DFAs), climbing the Jacob's ladder of DFT is potentially the most satisfactory way forward (or upward in that case). \cite{PerSch-AIPCP-01, PerRuzTaoStaScuCso-JCP-05} %especially in the range-separated (RS) context where the WFT method is relieved from describing the short-range part of the correlation hole. \cite{TouColSav-PRA-04, FraMusLupTou-JCP-15}
%To obtain accurate results within DFT, one must develop the art of selecting the adequate exchange-correlation functional, which can be classified in various families depending on their physical input quantities. \cite{Bec-JCP-14}
Although there is no clear way on how to systematically improve density-functional approximations, \cite{Bec-JCP-14} climbing the Jacob's ladder of DFT is potentially the most satisfactory way forward. \cite{PerSch-AIPCP-01, PerRuzTaoStaScuCso-JCP-05}
%The local-density approximation (LDA) sits on the first rung of the Jacob's ladder and only uses as input the electron density $n$. \cite{Dir-PCPRS-30, VosWilNus-CJP-80} %The local-density approximation (LDA) sits on the first rung of the Jacob's ladder and only uses as input the electron density $n$. \cite{Dir-PCPRS-30, VosWilNus-CJP-80}
%The generalized-gradient approximation (GGA) corresponds to the second rung and adds the gradient of the electron density $\nabla n$ as an extra ingredient.\cite{Bec-PRA-88, PerWan-PRA-91, PerBurErn-PRL-96} %The generalized-gradient approximation (GGA) corresponds to the second rung and adds the gradient of the electron density $\nabla n$ as an extra ingredient.\cite{Bec-PRA-88, PerWan-PRA-91, PerBurErn-PRL-96}
In the present context, one of the interesting feature of density-based methods is their much faster convergence with respect to the size of the basis set. \cite{FraMusLupTou-JCP-15}
Progress toward unifying these two approaches are on-going thanks to a more general formulation of DFT, the so-called range-separated DFT (RS-DFT) (see Ref.~\onlinecite{TouColSav-PRA-04} and references therein) which rigorously combines WFT and DFT. Progress toward unifying WFT and DFT are on-going.
In such a formalism the electron-electron interaction is split into a non divergent long-range part which is treated using WFT and a complementary short-range part treated with DFT. As the wave function part only deals with a non-diverging electron-electron interaction, it is free from the problematic electron cusp condition and the convergence with respect to the one-particle basis set is greatly improved\cite{FraMusLupTou-JCP-15}. Therefore, a number of approximate RS-DFT schemes have been developed using either single-reference WFT approaches (such as M{\o}ller-Plesset perturbation theory\cite{AngGerSavTou-PRA-05}, coupled cluster\cite{GolWerSto-PCCP-05}, random-phase approximations\cite{TouGerJanSavAng-PRL-09,JanHenScu-JCP-09}) or multi-reference WFT approaches (such as multi-reference CI\cite{LeiStoWerSav-CPL-97}, multiconfiguration self-consistent field\cite{FroTouJen-JCP-07}, multi-reference perturbation theory\cite{FroCimJen-PRA-10}, density-matrix renormalization group\cite{HedKneKieJenRei-JCP-15}, selected CI\cite{FerGinTou-JCP-18}). In particular, range-separated DFT (RS-DFT) (see Ref.~\onlinecite{TouColSav-PRA-04} and references therein) rigorously combines these two approaches via a decomposition of the electron-electron (e-e) interaction into a smooth long-range part and a (complementary) short-range part treated with WFT and DFT, respectively.
As the WFT method is relieved from describing the short-range part of the correlation hole around the e-e coalescence points, the convergence with respect to the one-electron basis set is greatly improved. \cite{FraMusLupTou-JCP-15}
Therefore, a number of approximate RS-DFT schemes have been developed using either single-reference \cite{AngGerSavTou-PRA-05, GolWerSto-PCCP-05, TouGerJanSavAng-PRL-09,JanHenScu-JCP-09} or multi-reference \cite{LeiStoWerSav-CPL-97, FroTouJen-JCP-07, FroCimJen-PRA-10, HedKneKieJenRei-JCP-15, FerGinTou-JCP-18} WFT approaches.
The present work proposes the extension of a recently proposed basis set correction scheme based on RS-DFT\cite{GinPraFerAssSavTou-JCP-18} together with the first numerical tests on molecular systems. %Therefore, a number of approximate RS-DFT schemes have been developed using either single-reference WFT approaches (such as M{\o}ller-Plesset perturbation theory\cite{AngGerSavTou-PRA-05}, coupled cluster\cite{GolWerSto-PCCP-05}, random-phase approximations\cite{TouGerJanSavAng-PRL-09,JanHenScu-JCP-09}) or multi-reference WFT approaches (such as multi-reference CI\cite{LeiStoWerSav-CPL-97}, multiconfiguration self-consistent field\cite{FroTouJen-JCP-07}, multi-reference perturbation theory\cite{FroCimJen-PRA-10}, density-matrix renormalization group\cite{HedKneKieJenRei-JCP-15}, selected CI\cite{FerGinTou-JCP-18}).
%The present manuscript is organized as follows. The present work proposes an extension of a recently proposed basis set correction scheme based on RS-DFT \cite{GinPraFerAssSavTou-JCP-18} together with the first numerical tests on molecular systems.
Unless otherwise stated, atomic used are used. Unless otherwise stated, atomic units are used.
%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%
\section{Theory} \section{Theory}