work in progress
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@ -25,9 +25,6 @@
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\newcommand{\mr}{\multirow}
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\newcommand{\SI}{\textcolor{blue}{supporting information}}
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% Titou's macros
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\newcommand{\br}{\mathbf{r}}
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% second quantized operators
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\newcommand{\psix}[1]{\hat{\Psi}\left({\bf X}_{#1}\right)}
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\newcommand{\psixc}[1]{\hat{\Psi}^{\dagger}\left({\bf X}_{#1}\right)}
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@ -74,13 +71,13 @@
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\newcommand{\ecmuapproxmur}[0]{\bar{E}_{\text{c,md-}\mathcal{X}}^{\text{sr}}[\den;\,\mur]}
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\newcommand{\ecmuapproxmurfci}[0]{\bar{E}_{\text{c,md-}\mathcal{X}}^{\text{sr}}[\denfci;\,\mur]}
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\newcommand{\ecmuapproxmurmodel}[0]{\bar{E}_{\text{c,md-}\mathcal{X}}^{\text{sr}}[\denmodel;\,\mur]}
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\newcommand{\ecompmodellda}[0]{\bar{E}_{\text{LDA}}^{\Bas,\psibasis}[\denmodel]}
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\newcommand{\ecompmodelldaval}[0]{\bar{E}_{\text{LDA, val}}^{\Bas,\psibasis}[\den]}
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\newcommand{\ecompmodelpbe}[0]{\bar{E}_{\text{PBE}}^{\Bas,\psibasis}[\den]}
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\newcommand{\ecompmodelpbeval}[0]{\bar{E}_{\text{PBE, val}}^{\Bas,\psibasis}[\den]}
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\newcommand{\emulda}[0]{\bar{\varepsilon}^{\text{sr},\text{unif}}_{\text{c,md}}\left(\denr;\mu({\bf r};\psibasis)\right)}
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\newcommand{\emuldamodel}[0]{\bar{\varepsilon}^{\text{sr},\text{unif}}_{\text{c,md}}\left(\denmodelr;\mu({\bf r};\psibasis)\right)}
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\newcommand{\emuldaval}[0]{\bar{\varepsilon}^{\text{sr},\text{unif}}_{\text{c,md}}\left(\denval ({\bf r});\murval;\psibasis)\right)}
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\newcommand{\ecompmodellda}[0]{\bar{E}_{\text{LDA}}^{\Bas,\wf{}{\Bas}}[\denmodel]}
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\newcommand{\ecompmodelldaval}[0]{\bar{E}_{\text{LDA, val}}^{\Bas,\wf{}{\Bas}}[\den]}
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\newcommand{\ecompmodelpbe}[0]{\bar{E}_{\text{PBE}}^{\Bas,\wf{}{\Bas}}[\den]}
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\newcommand{\ecompmodelpbeval}[0]{\bar{E}_{\text{PBE, val}}^{\Bas,\wf{}{\Bas}}[\den]}
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\newcommand{\emulda}[0]{\bar{\varepsilon}^{\text{sr},\text{unif}}_{\text{c,md}}\left(\denr;\mu({\bf r};\wf{}{\Bas})\right)}
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\newcommand{\emuldamodel}[0]{\bar{\varepsilon}^{\text{sr},\text{unif}}_{\text{c,md}}\left(\denmodelr;\mu({\bf r};\wf{}{\Bas})\right)}
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\newcommand{\emuldaval}[0]{\bar{\varepsilon}^{\text{sr},\text{unif}}_{\text{c,md}}\left(\denval ({\bf r});\murval;\wf{}{\Bas})\right)}
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@ -97,30 +94,30 @@
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% effective interaction
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\newcommand{\twodm}[4]{\elemm{\Psi}{\psixc{#4}\psixc{#3} \psix{#2}\psix{#1}}{\Psi}}
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\newcommand{\murpsi}[0]{\mu({\bf r};\psibasis)}
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\newcommand{\murpsi}[0]{\mu({\bf r};\wf{}{\Bas})}
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\newcommand{\mur}[0]{\mu({\bf r})}
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\newcommand{\murr}[1]{\mu({\bf r}_{#1})}
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\newcommand{\murval}[0]{\mu_{\text{val}}({\bf r})}
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\newcommand{\murpsival}[0]{\mu_{\text{val}}({\bf r};\psibasis)}
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\newcommand{\murpsival}[0]{\mu_{\text{val}}({\bf r};\wf{}{\Bas})}
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\newcommand{\murrval}[1]{\mu_{\text{val}}({\bf r}_{#1})}
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\newcommand{\weeopmu}[0]{\hat{W}_{\text{ee}}^{\text{lr},\mu}}
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\newcommand{\wbasis}[0]{W_{\psibasis}(\bfr{1},\bfr{2})}
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\newcommand{\wbasisval}[0]{W_{\psibasis}^{\text{val}}(\bfr{1},\bfr{2})}
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\newcommand{\fbasis}[0]{f_{\psibasis}(\bfr{1},\bfr{2})}
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\newcommand{\fbasisval}[0]{f_{\psibasis}^{\text{val}}(\bfr{1},\bfr{2})}
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\newcommand{\wbasis}[0]{W_{\wf{}{\Bas}}(\bfr{1},\bfr{2})}
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\newcommand{\wbasisval}[0]{W_{\wf{}{\Bas}}^{\text{val}}(\bfr{1},\bfr{2})}
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\newcommand{\fbasis}[0]{f_{\wf{}{\Bas}}(\bfr{1},\bfr{2})}
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\newcommand{\fbasisval}[0]{f_{\wf{}{\Bas}}^{\text{val}}(\bfr{1},\bfr{2})}
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\newcommand{\ontop}[2]{ n^{(2)}_{#1}({\bf #2}_1)}
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\newcommand{\twodmrpsi}[0]{ n^{(2)}_{\psibasis}(\rrrr{1}{2}{2}{1})}
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\newcommand{\twodmrdiagpsi}[0]{ n^{(2)}_{\psibasis}(\rr{1}{2})}
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\newcommand{\twodmrdiagpsival}[0]{ n^{(2)}_{\psibasis,\,\text{val}}(\rr{1}{2})}
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\newcommand{\twodmrpsi}[0]{ n^{(2)}_{\wf{}{\Bas}}(\rrrr{1}{2}{2}{1})}
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\newcommand{\twodmrdiagpsi}[0]{ n^{(2)}_{\wf{}{\Bas}}(\rr{1}{2})}
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\newcommand{\twodmrdiagpsival}[0]{ n^{(2)}_{\wf{}{\Bas},\,\text{val}}(\rr{1}{2})}
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\newcommand{\gammamnpq}[1]{\Gamma_{mn}^{pq}[#1]}
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\newcommand{\gammamnkl}[0]{\Gamma_{mn}^{kl}}
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\newcommand{\gammaklmn}[1]{\Gamma_{kl}^{mn}[#1]}
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\newcommand{\wbasiscoal}[1]{W_{\psibasis}({\bf r}_{#1})}
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\newcommand{\ontoppsi}[1]{ n^{(2)}_{\psibasis}(\bfr{#1},\barr{#1},\barr{#1},\bfr{#1})}
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\newcommand{\wbasiscoalval}[1]{W_{\psibasis}^{\text{val}}({\bf r}_{#1})}
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\newcommand{\ontoppsival}[1]{ n^{(2)}_{\psibasis}^{\text{val}}(\bfr{#1},\barr{#1},\barr{#1},\bfr{#1})}
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\newcommand{\wbasiscoal}[1]{W_{\wf{}{\Bas}}({\bf r}_{#1})}
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\newcommand{\ontoppsi}[1]{ n^{(2)}_{\wf{}{\Bas}}(\bfr{#1},\barr{#1},\barr{#1},\bfr{#1})}
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\newcommand{\wbasiscoalval}[1]{W_{\wf{}{\Bas}}^{\text{val}}({\bf r}_{#1})}
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\newcommand{\ontoppsival}[1]{ n^{(2)}_{\wf{}{\Bas}}^{\text{val}}(\bfr{#1},\barr{#1},\barr{#1},\bfr{#1})}
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@ -146,7 +143,6 @@
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% wave functions
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\newcommand{\psifci}[0]{\Psi^{\Bas}_{\text{FCI}}}
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\newcommand{\psibasis}[0]{\Psi^{\Bas}}
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\newcommand{\psimu}[0]{\Psi^{\mu}}
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% operators
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\newcommand{\weeopbasis}[0]{\hat{W}_{\text{ee}}^\Bas}
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@ -161,11 +157,40 @@
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\newcommand{\InAU}[1]{#1 a.u.}
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\newcommand{\InAA}[1]{#1 \AA}
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\newcommand{\pis}{\pi^\star}
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\newcommand{\si}{\sigma}
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\newcommand{\sis}{\sigma^\star}
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% methods
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\newcommand{\FCI}{\text{FCI}}
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\newcommand{\CCSDT}{\text{CCSD(T)}}
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\newcommand{\Nel}{N}
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\newcommand{\n}[2]{n_{#1}^{#2}}
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\newcommand{\E}[2]{E_{#1}^{#2}}
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\newcommand{\bE}[2]{\Bar{E}_{#1}^{#2}}
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\newcommand{\wf}[2]{\Psi_{#1}^{#2}}
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\newcommand{\W}[2]{W_{#1}^{#2}}
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\newcommand{\SO}[2]{\phi_{#1}(\bx{#2})}
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\newcommand{\modX}{\text{X}}
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\newcommand{\modY}{Y}
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% basis sets
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\newcommand{\Bas}{\mathcal{B}}
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\newcommand{\Basval}{\mathcal{B}_\text{val}}
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\newcommand{\Val}{\mathcal{V}}
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\newcommand{\Cor}{\mathcal{C}}
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% operators
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\newcommand{\hT}{\Hat{T}}
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\newcommand{\hWee}[1]{\Hat{W}_\text{ee}^{#1}}
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\newcommand{\f}[2]{f_{#1}^{#2}}
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\newcommand{\Gam}[2]{\Gamma_{#1}^{#2}}
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% coordinates
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\newcommand{\br}[1]{\mathbf{r}_{#1}}
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\newcommand{\bx}[1]{\mathbf{x}_{#1}}
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\newcommand{\dbr}[1]{d\br{#1}}
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\newcommand{\dbx}[1]{d\bx{#1}}
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\newcommand{\LCPQ}{Laboratoire de Chimie et Physique Quantiques (UMR 5626), Universit\'e de Toulouse, CNRS, UPS, France}
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\newcommand{\LCT}{Laboratoire de Chimie Th\'eorique, Universit\'e Pierre et Marie Curie, Sorbonne Universit\'e, CNRS, Paris, France}
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@ -203,7 +228,7 @@ For example, the coupled cluster (CC) family of methods offers a powerful WFT ap
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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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This undesirable feature was put into light by Kutzelnigg more than thirty years ago, \cite{Kut-TCA-85}
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who proposed, to palliate this, to introduce explicitly the interelectronic distance $r_{12} = \abs{\br_1 - \br_2}$ as a basis function. \cite{Kut-TCA-85, KutKlo-JCP-91, TerKloKut-JCP-91, KloKut-JCP-91, KloRohKut-CPL-91, NogKut-JCP-94}
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who proposed, to palliate this, to introduce explicitly the interelectronic distance $r_{12} = \abs{\br{1} - \br{2}}$ as a basis function. \cite{Kut-TCA-85, KutKlo-JCP-91, TerKloKut-JCP-91, KloKut-JCP-91, KloRohKut-CPL-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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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.
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@ -229,35 +254,16 @@ Unless otherwise stated, atomic used are used.
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%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Theory}
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%%%%%%%%%%%%%%%%%%%%%%%%
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The basis set correction investigated here uses the RS-DFT formalism to capture the part of the short-range correlation effects missing from the description of the WFT in a finite basis set.
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Here, we briefly explain the working equations and notations needed for this work, and the interested reader can find the detailed formal derivation of the theory in Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}.
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The basis set correction employed here relies on the RS-DFT formalism to capture the missing part of the short-range correlation effects, a consequence of the non-completeness of the one-electron basis set.
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Here, we provide the main working equations.
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We refer the interested reader to Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18} for further details about the formal derivation of the theory.
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\newcommand{\FCI}{\text{FCI}}
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\newcommand{\CCSDT}{\text{CCSD(T)}}
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\newcommand{\Nel}{N}
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\newcommand{\n}[2]{n_{#1}^{#2}}
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\newcommand{\E}[2]{E_{#1}^{#2}}
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\newcommand{\bE}[2]{\Bar{E}_{#1}^{#2}}
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\newcommand{\wf}[2]{\Psi_{#1}^{#2}}
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\newcommand{\modX}{\text{X}}
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\newcommand{\modY}{Y}
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% basis sets
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\newcommand{\Bas}{\mathcal{B}}
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\newcommand{\Basval}{\mathcal{B}_\text{val}}
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% operators
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\newcommand{\hT}{\Hat{T}}
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\newcommand{\hWee}{\Hat{W}_\text{ee}}
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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 density $\n{\modX}{\Bas}$ and energy $\E{\modX}{\Bas}$ of a $\Nel$-electron system described by a method $\modX$ in an incomplete basis set $\Bas$.
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According to Eq.~(15) of Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, assuming that $\n{\modX}{\Bas}$ is a good approximation of the \textit{exact} ground state density $\n{}{}$, one may approximate the \textit{exact} ground state energy as
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According to Eq.~(15) of Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, assuming that $\n{\modX}{\Bas}$ is a resonable approximation of the \textit{exact} ground state density $\n{}{}$, one may approximate the \textit{exact} ground state energy as
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\begin{equation}
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\label{eq:e0basis}
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\E{}{}
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@ -268,24 +274,23 @@ where
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\begin{equation}
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\label{eq:E_funcbasis}
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\bE{}{\Bas}[\n{}{}]
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= \min_{\wf{}{} \to \n{}{}} \mel*{\wf{}{}}{\hT + \hWee}{\wf{}{}}
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- \min_{\wf{}{\Bas} \to \n{}{}} \mel*{\wf{}{\Bas}}{\hT + \hWee}{\wf{}{\Bas}}
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= \min_{\wf{}{} \to \n{}{}} \mel*{\wf{}{}}{\hT + \hWee{}}{\wf{}{}}
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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 complementary density functional defined in Eq.~(8) of Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, and $\hT$ and $\hWee$ are the kinetic and interelectronic repulsion operators, respectively.
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In Eq.~\eqref{eq:E_funcbasis}, $\wf{}{\Bas}$ is the wave function obtained from the $\Nel$-electron Hilbert space spanned by $\Bas$, and $\wf{}{}$ is a general $\Nel$-electron wave function being obtained in a complete basis.
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is the complementary density functional defined in Eq.~(8) of Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, and $\hT$ and $\hWee{}$ are the kinetic and interelectronic repulsion operators, respectively.
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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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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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An important aspect of such a theory is that, in the limit of a complete basis set $\Bas$ (which we refer to as $\Bas \to \infty$), we have, for any density $\n{}{}$, $\lim_{\Bas \to \infty} \bE{}{\Bas}[\n{}{}] = 0$, which implies that
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An important aspect of such theory is that, in the limit of a complete basis set $\Bas$ (which we refer to as $\Bas \to \infty$), we have, for any density $\n{}{}$, $\lim_{\Bas \to \infty} \bE{}{\Bas}[\n{}{}] = 0$, which implies that
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\begin{equation}
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\label{eq:limitfunc}
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\lim_{\Bas \rightarrow \infty} \qty( \E{\modX}{\Bas} + \bE{}{\Bas}[\n{\wf{\modX}{\Bas}}{}] ) = \E{\modX}{\infty} \approx E,
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\lim_{\Bas \to \infty} \qty( \E{\modX}{\Bas} + \bE{}{\Bas}[\n{\wf{\modX}{\Bas}}{}] ) = \E{\modX}{} \approx E,
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\end{equation}
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where $\E{\modX}{\infty}$ is the energy associated with the method $\modX$ in complete basis set.
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In the case of $\modX = \FCI$, we $\E{\FCI}{\infty} = E$.
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Provided that the functional $\bE{}{\Bas}[\n{}{}]$ is known exactly, at this stage, the only source of error lies in the potential approximate nature of the method $\modX$.
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where $\E{\modX}{}$ is the energy associated with the method $\modX$ in the complete basis set.
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In the case of $\modX = \FCI$, we have as strict equality as $\E{\FCI}{} = E$.
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Provided that the functional $\bE{}{\Bas}[\n{}{}]$ is known exactly, the only source of error at this stage lies in the potential approximate nature of the method $\modX$.
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%Here we propose to generalize such approach to a general WFT model, referred here as $\model$, projected in a basis set $\Bas$ which must provides a density $\denmodel$ and an energy $\emodel$.
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%As any wave function model is necessary an approximation to the FCI model, one can write
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@ -384,18 +389,18 @@ Provided that the functional $\bE{}{\Bas}[\n{}{}]$ is known exactly, at this sta
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%=================================================================
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However, in addition of being unknown, the functional $\bE{}{\Bas}[\n{}{}]$ is obviously \textit{not} universal as it depends on $\Bas$.
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Following Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, we approximate $\bE{}{\Bas}[\n{}{}]$ following a two-step procedure which guarantees the correct behaviour in the limit $\Bas \to \infty$ [see Eq.~\eqref{eq:limitfunc}].
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First, we define a real-space representation of the Coulomb operator projected in $\Bas$, which is then fitted with a long-range interaction thanks to a range-separation parameter $\mu(\br)$ varying in space (see Sec.~\ref{sec:weff}).
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Then, we choose a specific class of short-range density functionals, namely the short-range correlation functionals with multi-determinantal reference (ECMD) introduced by Toulouse \textit{et al.} \cite{TouGorSav-TCA-05}, that we evaluate at $\n{\modX}{\Bas}$ (see Sec.~\ref{sec:ecmd}) with $\mu(\br)$.
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First, we define a real-space representation of the Coulomb operator projected in $\Bas$, which is then fitted with a long-range interaction thanks to a range-separation parameter $\mu(\br{})$ varying in space. %(see Sec.~\ref{sec:weff}).
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Then, we choose a specific class of short-range density functionals, namely the short-range correlation functionals with multi-determinantal reference (ECMD) introduced by Toulouse \textit{et al.} \cite{TouGorSav-TCA-05}, that we evaluate at $\n{\modX}{\Bas}$ with $\mu(\br{})$.% (see Sec.~\ref{sec:ecmd}) .
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%=================================================================
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\subsection{Definition of a real-space representation of the coulomb operator truncated in a basis-set $\Bas$}
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\label{sec:weff}
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%\subsection{Definition of a real-space representation of the coulomb operator truncated in a basis-set $\Bas$}
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%\label{sec:weff}
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%=================================================================
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One of the consequences of the incompleteness of $\Bas$ is that $\wf{}{\Bas}$ does not have a cusp (i.e. a discontinuous derivative) at the electron-electron coalescence points.
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As the electron-electron cusp originates from the divergence of the Coulomb operator at $r_{12} = 0$, a cuspless wave function could also originate from a Hamiltonian with a non-divergent electron-electron interaction.
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Therefore, the impact of the incompleteness of a finite basis set $\Bas$ can be viewed as a removal of the divergence of the coulomb interaction at $r_{12} = 0$.
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The present paragraph briefly describes how to obtain an effective interaction $\wbasis$ which i) is finite at the electron-electron coalescence points as long as an incomplete basis set is used, and ii) tends to the genuine, unbounded $r_{12}^{-1}$ Coulomb operator in the limit of a complete basis set.
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One of the consequences of the incompleteness of $\Bas$ is that $\wf{}{\Bas}$ does not have a cusp (i.e.~a discontinuous derivative) at the electron-electron (e-e) coalescence points.
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As the e-e cusp originates from the divergence of the Coulomb operator at $r_{12} = 0$, a cuspless wave function could also originate from a Hamiltonian with a non-divergent Coulomb interaction.
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Therefore, the impact of the incompleteness of $\Bas$ can be viewed as a removal of the divergence of the Coulomb interaction at $r_{12} = 0$.
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The present paragraph briefly describes how to obtain an effective interaction $\W{\wf{}{\Bas}}{}(\bx{1},\bx{2})$ which i) is finite at the e-e coalescence points as long as an incomplete basis set is used, and ii) tends to the genuine, unbounded $r_{12}^{-1}$ Coulomb operator in the limit of a complete basis set.
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%----------------------------------------------------------------
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%\subsubsection{General definition of an effective interaction for the basis set $\Bas$}
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@ -406,34 +411,34 @@ Consider the Coulomb operator projected in $\Bas$
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\weeopbasis = \frac{1}{2} \sum_{ijkl \in \Bas} \vijkl \aic{k}\aic{l}\ai{j}\ai{i},
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\end{aligned}
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\end{equation}
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where the indices run over all orthonormal spin-orbitals in $\Bas$ and $\vijkl$ are the usual coulomb two-electron integrals.
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Consider now the expectation value of $\weeopbasis$ over a general wave function $\psibasis$ belonging to the $N-$electron Hilbert space spanned by the basis set $\Bas$.
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After a few mathematical work (see appendix A of \onlinecite{GinPraFerAssSavTou-JCP-18} for a detailed derivation), such an expectation value can be rewritten as an integral over the two-electron spin and space coordinates:
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where the indices run over all orthonormal spin-orbitals in $\Bas$ and $\vijkl$ are the usual two-electron Coulomb integrals.
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Consider now the expectation value of $\weeopbasis$ over a general wave function $\wf{}{\Bas}$ belonging to the $\Nel$-electron Hilbert space spanned by $\Bas$.
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One can show (see Appendix A of Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}) that such an expectation value can be rewritten as an integral over the two-electron spin and space coordinates:
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\begin{equation}
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\label{eq:expectweeb}
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\mel*{\psibasis}{\weeopbasis}{\psibasis} = \frac{1}{2}\,\,\iint \dr{1}\,\dr{2} \,\, \fbasis\,\,,
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\mel*{\wf{}{\Bas}}{\hWee{\Bas}}{\wf{}{\Bas}} = \frac{1}{2} \iint \f{\wf{}{\Bas}}{}(\bx{1},\bx{2}) \dbx{1} \dbx{2}
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\end{equation}
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where
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\begin{equation}
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\begin{multline}
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\label{eq:fbasis}
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\begin{aligned}
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\fbasis = \sum_{ijklmn\,\,\in\,\,\Bas} & \vijkl \,\, \gammaklmn{\psibasis} \\& \phix{n}{2} \phix{m}{1} \phix{i}{1} \phix{j}{2}\,\,,
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\end{aligned}
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\end{equation}
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\f{\wf{}{\Bas}}{}(\bx{1},\bx{2})
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\\
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= \sum_{ijklmn \in \Bas} \vijkl \Gam{mn}{pq}[\wf{}{\Bas}] \SO{n}{2} \SO{m}{1} \SO{i}{1} \SO{j}{2},
|
||||
\end{multline}
|
||||
and
|
||||
\begin{equation}
|
||||
\gammamnpq{\psibasis} = \mel*{\psibasis}{ \aic{p}\aic{q}\ai{n}\ai{m} }{\psibasis}\,\,,
|
||||
\Gam{mn}{pq}[\wf{}{\Bas}] = \mel*{\wf{}{\Bas}}{ \aic{p}\aic{q}\ai{n}\ai{m} }{\wf{}{\Bas}},
|
||||
\end{equation}
|
||||
is the two-body density tensor of $\psibasis$ and $\bfr{} = \qty(\br,\sigma)$ collects the space and spin variables, $\int \, \dr{} = \sum_{\sigma}\,\int_{{\rm I\!R}^3} \, \text{d}{\bf r}$.
|
||||
Then, consider the expectation value of the exact coulomb operator over $\psibasis$
|
||||
is the two-body density tensor of $\wf{}{\Bas}$, $\bfr{} = \qty(\br,\sigma)$ collects the space and spin variables, $\int \dbx{} = \sum_{\sigma}\,\int_{\mathbb{R}^3} \dbr{}$.
|
||||
Then, consider the expectation value of the exact Coulomb operator over $\wf{}{\Bas}$
|
||||
\begin{equation}
|
||||
\label{eq:expectwee}
|
||||
\mel*{\psibasis}{\weeop}{\psibasis} = \frac{1}{2} \iint r_{12}^{-1} \twodmrdiagpsi \dr{1} \dr{2}
|
||||
\mel*{\wf{}{\Bas}}{\weeop}{\wf{}{\Bas}} = \frac{1}{2} \iint r_{12}^{-1} \twodmrdiagpsi \dr{1} \dr{2}
|
||||
\end{equation}
|
||||
where $\twodmrdiagpsi$ is the two-body density associated to $\psibasis$.
|
||||
Because $\psibasis$ belongs to $\Bas$, such an expectation value coincides with the expectation value of $\weeopbasis$
|
||||
where $\n{\wf{}{\Bas}}{(2)}$ is the two-body density associated with $\wf{}{\Bas}$.
|
||||
Because $\wf{}{\Bas}$ belongs to $\Bas$, such an expectation value coincides with the expectation value of $\weeopbasis$
|
||||
\begin{equation}
|
||||
\mel*{\psibasis}{\weeopbasis}{\psibasis} = \mel*{\psibasis}{\weeop}{\psibasis},
|
||||
\mel*{\wf{}{\Bas}}{\weeopbasis}{\wf{}{\Bas}} = \mel*{\wf{}{\Bas}}{\weeop}{\wf{}{\Bas}},
|
||||
\end{equation}
|
||||
which can be rewritten as:
|
||||
\begin{multline}
|
||||
@ -445,23 +450,24 @@ which can be rewritten as:
|
||||
where we introduced $\wbasis$
|
||||
\begin{equation}
|
||||
\label{eq:def_weebasis}
|
||||
\wbasis = \frac{\fbasis}{\twodmrdiagpsi},
|
||||
\W{\wf{}{\Bas}}{}(\bx{1},\bx{2}) = \frac{\f{\wf{}{\Bas}}{}(\bx{1},\bx{2})}{\n{\wf{}{\Bas}}{(2)}(\bx{1},\bx{2})},
|
||||
\end{equation}
|
||||
which is the effective interaction in the basis set $\Bas$.
|
||||
|
||||
As already discussed in \onlinecite{GinPraFerAssSavTou-JCP-18}, such an effective interaction is symmetric, \textit{a priori} non translational nor rotational invariant if the basis set $\Bas$ does not have such symmetries and is necessary \textit{finite} at the electron coalescence point for an incomplete basis set $\Bas$.
|
||||
Also, as demonstrated in the appendix B of \onlinecite{GinPraFerAssSavTou-JCP-18}, $\wbasis$ tends to the regular coulomb interaction $1/r_{12}$ for all points $(\bfr{1},\bfr{2})$ and any choice of $\psibasis$ in the limit of a complete basis set $\Bas$.
|
||||
Also, as demonstrated in the appendix B of \onlinecite{GinPraFerAssSavTou-JCP-18}, $\W{\wf{}{\Bas}}{}(\bx{1},\bx{2})$ tends to the regular coulomb interaction $r_{12}^{-1}$ for all points $(\bx{1},\bx{2})$ and any choice of $\wf{}{\Bas}$ in the limit of a complete basis set.
|
||||
|
||||
|
||||
%----------------------------------------------------------------
|
||||
\subsubsection{Definition of a valence effective interaction}
|
||||
%----------------------------------------------------------------
|
||||
As most of the WFT calculations are done using a frozen core approximation, it is important to define an effective interaction within a general subset of molecular orbitals that we refer as $\Basval$.
|
||||
As most WFT calculations are performed within the frozen-core (FC) approximation, it is important to define an effective interaction within a general subset of molecular orbitals.
|
||||
We then split the basis set as $\Bas = \Cor \bigcup \Val$, where $\Cor$ and $\Val$ are its core and valence parts, respectively, and $\Cor \bigcap \Val = \O$.
|
||||
|
||||
According to \eqref{eq:def_weebasis} and \eqref{eq:expectweeb}, the effective interaction is defined by the expectation value of the coulomb operator over a wave function $\psibasis$. Therefore, to define an effective interaction accounting only for the valence electrons, one needs to define a function $\fbasisval$ satisfying
|
||||
According to Eqs.~\eqref{eq:expectweeb} and \eqref{eq:def_weebasis} , the effective interaction is defined by the expectation value of the coulomb operator over a wave function $\wf{}{\Bas}$. Therefore, to define an effective interaction accounting only for the valence electrons, one needs to define a function $\fbasisval$ satisfying
|
||||
\begin{equation}
|
||||
\label{eq:expectweebval}
|
||||
\mel*{\psibasis}{\weeopbasisval}{\psibasis} = \frac{1}{2}\,\,\iint \dr{1}\,\dr{2} \,\, \fbasisval,
|
||||
\mel*{\wf{}{\Bas}}{\weeopbasisval}{\wf{}{\Bas}} = \frac{1}{2}\,\,\iint \dr{1}\,\dr{2} \,\, \fbasisval,
|
||||
\end{equation}
|
||||
where $\weeopbasisval$ is the valence coulomb operator defined as
|
||||
\begin{equation}
|
||||
@ -474,7 +480,7 @@ Following the spirit of \eqref{eq:fbasis}, the function $\fbasisval$ can be defi
|
||||
\begin{equation}
|
||||
\label{eq:fbasisval}
|
||||
\begin{aligned}
|
||||
\fbasisval = \sum_{ij\,\,\in\,\,\Bas} \,\, \sum_{klmn\,\,\in\,\,\Basval} & \vijkl \,\, \gammaklmn{\psibasis} \\& \phix{n}{2} \phix{m}{1} \phix{i}{1} \phix{j}{2}.
|
||||
\fbasisval = \sum_{ij\,\,\in\,\,\Bas} \,\, \sum_{klmn\,\,\in\,\,\Basval} & \vijkl \,\, \gammaklmn{\wf{}{\Bas}} \\& \phix{n}{2} \phix{m}{1} \phix{i}{1} \phix{j}{2}.
|
||||
\end{aligned}
|
||||
\end{equation}
|
||||
|
||||
@ -485,7 +491,7 @@ Then, the effective interaction associated to the valence $\wbasisval$ is simply
|
||||
\end{equation}
|
||||
where $\twodmrdiagpsival$ is the two body density associated to the valence electrons:
|
||||
\begin{equation}
|
||||
\twodmrdiagpsival = \sum_{klmn\,\,\in\,\,\Basval} \gammamnkl[\psibasis] \,\, \phix{m}{1} \phix{n}{2} \phix{k}{1} \phix{l}{2} .
|
||||
\twodmrdiagpsival = \sum_{klmn\,\,\in\,\,\Basval} \gammamnkl[\wf{}{\Bas}] \,\, \phix{m}{1} \phix{n}{2} \phix{k}{1} \phix{l}{2} .
|
||||
\end{equation}
|
||||
It is important to notice in \eqref{eq:fbasisval} the difference between the set of orbitals for the indices $(i,j)$, which span the full set of MOs within $\Bas$, and the $(k,l,m,n)$, which span only the valence space $\Basval$. Only with such a definition, one can show (see annex) that $\fbasisval$ fulfills \eqref{eq:expectweebval} and tends to the exact interaction $1/r_{12}$ in the limit of a complete basis set $\Bas$, whatever the choice of subset $\Basval$.
|
||||
|
||||
@ -495,7 +501,7 @@ To be able to approximate the complementary functional $\efuncbasis$ thanks to f
|
||||
More precisely, if we define the value of the interaction at coalescence as
|
||||
\begin{equation}
|
||||
\label{eq:def_wcoal}
|
||||
\wbasiscoal{} = W_{\psibasis}(\bfr{},\bar{{\bf x}}_{}).
|
||||
\wbasiscoal{} = W_{\wf{}{\Bas}}(\bfr{},\bar{{\bf x}}_{}).
|
||||
\end{equation}
|
||||
where $(\bfr{},\bar{{\bf x}}_{})$ means a couple of anti-parallel spins at the same point in $\bfrb{}$,
|
||||
we propose a fit for each point in $\rnum^3$ of $\wbasiscoal{ }$ with a long-range-like interaction:
|
||||
@ -589,7 +595,7 @@ It is important to notice that in the limit of a complete basis set, according t
|
||||
\begin{equation}
|
||||
\lim_{\Bas \rightarrow \infty} \ecmuapproxmurmodel = 0 \quad ,
|
||||
\end{equation}
|
||||
for whatever choice of density $\denmodel$, wave function $\psibasis$ used to define the interaction, and ECMD functional used to approximate the exact ECMD.
|
||||
for whatever choice of density $\denmodel$, wave function $\wf{}{\Bas}$ used to define the interaction, and ECMD functional used to approximate the exact ECMD.
|
||||
|
||||
\subsubsection{LDA approximation for the complementary functional}
|
||||
As done in Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, one can define an LDA-like approximation for $\ecompmodel$ as
|
||||
@ -648,13 +654,13 @@ We now introduce a valence-only approximation for the complementary functional w
|
||||
Defining the valence one-body spin density matrix as
|
||||
\begin{equation}
|
||||
\begin{aligned}
|
||||
\onedmval[\psibasis] & = \elemm{\psibasis}{a^{\dagger}_{i,\sigma} a_{j,\sigma}}{\psibasis} \qquad \text{if }(i,j)\in \Basval \\
|
||||
\onedmval[\wf{}{\Bas}] & = \elemm{\wf{}{\Bas}}{a^{\dagger}_{i,\sigma} a_{j,\sigma}}{\wf{}{\Bas}} \qquad \text{if }(i,j)\in \Basval \\
|
||||
& = 0 \qquad \text{in other cases}
|
||||
\end{aligned}
|
||||
\end{equation}
|
||||
then one can define the valence density as:
|
||||
\begin{equation}
|
||||
\denval_{\sigma}({\bf r}) = \sum_{i,j} \onedmval[\psibasis] \phi_i({\bf r}) \phi_j({\bf r})
|
||||
\denval_{\sigma}({\bf r}) = \sum_{i,j} \onedmval[\wf{}{\Bas}] \phi_i({\bf r}) \phi_j({\bf r})
|
||||
\end{equation}
|
||||
Therefore, we propose the following valence-only approximations for the complementary functional
|
||||
\begin{equation}
|
||||
@ -674,7 +680,7 @@ Therefore, we propose the following valence-only approximations for the compleme
|
||||
We begin the investigation of the behavior of the basis-set correction by the study of the atomization energies of the C$_2$, N$_2$, O$_2$, F$_2$ homo-nuclear diatomic molecules in the Dunning cc-pVXZ and cc-pCVXZ (X=D,T,Q,5) using both the CIPSI algorithm and the CCSD(T). All through this work, we follow the frozen core (FC) convention of Klopper \textit{et. al}\cite{HauKlo-JCP-12} which consists in all-electron calculations for Li-Be, a He core for B-Na atoms and a Ne core for the Al-Cl series. In the context of the DFT correction for the basis-set, this implies that, for a given system in a given basis set $\Bas$, the set of valence orbitals $\Basval$ involved in the definition of the valence interaction $\wbasisval$ and density $\onedmval$ refers to all MOs except the core.
|
||||
%\subsubsection{CIPSI calculations and the basis-set correction}
|
||||
%All CIPSI calculations were performed in two steps. First, a CIPSI calculation was performed until the zeroth-order wave function reaches $10^6$ Slater determinants, from which we extracted the natural orbitals. From this set of natural orbitals, we performed CIPSI calculations until the $\EexFCIbasis$ reaches about $0.1$ mH convergence for each systems. Such convergence criterion is more than sufficient for the CIPSI densities $\dencipsi$.
|
||||
%Regarding the wave function $\psibasis$ chosen to define the local range-separation parameter $\mur$, we take a single Slater determinant built with the natural orbitals of the first CIPSI calculation.
|
||||
%Regarding the wave function $\wf{}{\Bas}$ chosen to define the local range-separation parameter $\mur$, we take a single Slater determinant built with the natural orbitals of the first CIPSI calculation.
|
||||
\subsubsection{CCSD(T) calculations and the basis-set correction}
|
||||
|
||||
\begin{table*}
|
||||
|
||||
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Reference in New Issue
Block a user