intro
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%% This BibTeX bibliography file was created using BibDesk.
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%% Created for Pierre-Francois Loos at 2019-04-07 21:16:12 +0200
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%% Created for Pierre-Francois Loos at 2019-04-11 14:20:29 +0200
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@article{BarLoo-JCP-17,
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Author = {Barca, Giuseppe MJ and Loos, Pierre-Fran{\c c}ois},
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Date-Added = {2019-04-11 14:20:15 +0200},
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Date-Modified = {2019-04-11 14:20:29 +0200},
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File = {/Users/loos/Zotero/storage/DCFUMHWZ/56.pdf},
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Journal = {J. Chem. Phys.},
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Number = {2},
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Pages = {024103},
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Shorttitle = {Three-and Four-Electron Integrals Involving {{Gaussian}} Geminals},
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Title = {Three-and Four-Electron Integrals Involving {{Gaussian}} Geminals: {{Fundamental}} Integrals, Upper Bounds, and Recurrence Relations},
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Volume = {147},
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Year = {2017}}
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@article{FelPet-JCP-09,
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@article{FelPet-JCP-09,
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Author = {D. Feller and K. A. Peterson},
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Author = {D. Feller and K. A. Peterson},
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Date-Added = {2019-04-07 20:41:03 +0200},
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Date-Added = {2019-04-07 20:41:03 +0200},
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@ -3824,20 +3837,20 @@
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Year = {1977}}
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Year = {1977}}
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@article{FroCimJen-PRA-10,
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@article{FroCimJen-PRA-10,
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title = {Merging multireference perturbation and density-functional theories by means of range separation: Potential curves for ${\mathrm{Be}}_{2}$, ${\mathrm{Mg}}_{2}$, and ${\mathrm{Ca}}_{2}$},
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Author = {Fromager, Emmanuel and Cimiraglia, Renzo and Jensen, Hans J\o{}rgen Aa.},
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author = {Fromager, Emmanuel and Cimiraglia, Renzo and Jensen, Hans J\o{}rgen Aa.},
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Doi = {10.1103/PhysRevA.81.024502},
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journal = {Phys. Rev. A},
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Issue = {2},
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volume = {81},
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Journal = {Phys. Rev. A},
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issue = {2},
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Month = {Feb},
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pages = {024502},
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Numpages = {4},
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numpages = {4},
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Pages = {024502},
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year = {2010},
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Publisher = {American Physical Society},
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month = {Feb},
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Title = {Merging multireference perturbation and density-functional theories by means of range separation: Potential curves for ${\mathrm{Be}}_{2}$, ${\mathrm{Mg}}_{2}$, and ${\mathrm{Ca}}_{2}$},
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publisher = {American Physical Society},
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Url = {https://link.aps.org/doi/10.1103/PhysRevA.81.024502},
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doi = {10.1103/PhysRevA.81.024502},
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Volume = {81},
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url = {https://link.aps.org/doi/10.1103/PhysRevA.81.024502}
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Year = {2010},
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}
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Bdsk-Url-1 = {https://link.aps.org/doi/10.1103/PhysRevA.81.024502},
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Bdsk-Url-2 = {https://doi.org/10.1103/PhysRevA.81.024502}}
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@article{Fro-JCP-11,
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@article{Fro-JCP-11,
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Author = {E. Fromager},
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Author = {E. Fromager},
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@ -5198,9 +5211,9 @@
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@article{HedKneKieJenRei-JCP-15,
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@article{HedKneKieJenRei-JCP-15,
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Author = {E. D. Hedeg{\aa}rd and S. Knecht and J. S. Kielberg and H. J. Aa. Jensen and M. Reiher},
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Author = {E. D. Hedeg{\aa}rd and S. Knecht and J. S. Kielberg and H. J. Aa. Jensen and M. Reiher},
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title = {Density matrix renormalization group with efficient dynamical electron correlation through range separation},
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Journal = {J. Chem. Phys.},
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Journal = {J. Chem. Phys.},
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Pages = {224108},
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Pages = {224108},
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Title = {Density matrix renormalization group with efficient dynamical electron correlation through range separation},
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Volume = {142},
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Volume = {142},
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Year = {2015}}
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Year = {2015}}
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@ -12241,29 +12254,29 @@
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Bdsk-Url-2 = {https://doi.org/10.1016/S0009-2614(98)00111-0}}
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Bdsk-Url-2 = {https://doi.org/10.1016/S0009-2614(98)00111-0}}
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@article{DasHer-JCC-17,
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@article{DasHer-JCC-17,
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author = {Dasgupta, Saswata and Herbert, John M.},
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Author = {Dasgupta, Saswata and Herbert, John M.},
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title = {Standard grids for high-precision integration of modern density functionals: SG-2 and SG-3},
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Doi = {10.1002/jcc.24761},
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journal = {Journal of Computational Chemistry},
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Eprint = {https://onlinelibrary.wiley.com/doi/pdf/10.1002/jcc.24761},
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volume = {38},
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Journal = {Journal of Computational Chemistry},
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number = {12},
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Number = {12},
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pages = {869-882},
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Pages = {869-882},
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doi = {10.1002/jcc.24761},
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Title = {Standard grids for high-precision integration of modern density functionals: SG-2 and SG-3},
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eprint = {https://onlinelibrary.wiley.com/doi/pdf/10.1002/jcc.24761},
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Volume = {38},
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year = {2017}
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Year = {2017},
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}
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Bdsk-Url-1 = {https://doi.org/10.1002/jcc.24761}}
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@article{rs_dft_toul_colo_savin,
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@article{rs_dft_toul_colo_savin,
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title = {Long-range--short-range separation of the electron-electron interaction in density-functional theory},
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Author = {J. Toulouse and F. Colonna and A. Savin},
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author = {J. Toulouse and F. Colonna and A. Savin},
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Doi = {10.1103/PhysRevA.70.062505},
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journal = {Phys. Rev. A},
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Issue = {6},
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volume = {70},
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Journal = {Phys. Rev. A},
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issue = {6},
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Month = {Dec},
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pages = {062505},
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Numpages = {16},
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numpages = {16},
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Pages = {062505},
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year = {2004},
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Publisher = {American Physical Society},
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month = {Dec},
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Title = {Long-range--short-range separation of the electron-electron interaction in density-functional theory},
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publisher = {American Physical Society},
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Url = {https://link.aps.org/doi/10.1103/PhysRevA.70.062505},
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doi = {10.1103/PhysRevA.70.062505},
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Volume = {70},
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url = {https://link.aps.org/doi/10.1103/PhysRevA.70.062505}
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Year = {2004},
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}
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Bdsk-Url-1 = {https://link.aps.org/doi/10.1103/PhysRevA.70.062505},
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Bdsk-Url-2 = {https://doi.org/10.1103/PhysRevA.70.062505}}
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@ -145,34 +145,38 @@ These values drop below 1 {\kcal} with the cc-pVQZ basis set.
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\section{Introduction}
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\section{Introduction}
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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}
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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}
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Although both spring from the same Schr\"odinger equation, each of these philosophies has its own advantages and shortcomings.
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Although both spring from the same Schr\"odinger equation, each of these philosophies has its own \textit{pros} and \textit{cons}.
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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.
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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}.
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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.
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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.
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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.
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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.
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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.
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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.
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This undesirable feature was put into light by Kutzelnigg more than thirty years ago. \cite{Kut-TCA-85}
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This undesirable feature was put into light by Kutzelnigg more than thirty years ago. \cite{Kut-TCA-85}
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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}
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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}
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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}
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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}
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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.
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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}
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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.
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%\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.}
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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}
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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}
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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}
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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.
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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.
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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}
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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}
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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}
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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}
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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}
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%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}
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%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}
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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}
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%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}
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%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}
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%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}
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%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}
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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}
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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.
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Progress toward unifying WFT and DFT are on-going.
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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}).
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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.
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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}
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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.
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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.
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%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}).
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%The present manuscript is organized as follows.
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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.
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Unless otherwise stated, atomic used are used.
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Unless otherwise stated, atomic units are used.
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%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Theory}
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\section{Theory}
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