diff --git a/Manuscript/G2-srDFT.tex b/Manuscript/G2-srDFT.tex index 194da8c..ac8612a 100644 --- a/Manuscript/G2-srDFT.tex +++ b/Manuscript/G2-srDFT.tex @@ -54,107 +54,15 @@ % -% energies -\newcommand{\Ec}{E_\text{c}} -\newcommand{\EexFCI}{E_\text{exFCI}} -\newcommand{\EexFCIbasis}{E_\text{exFCI}^{\Bas}} -\newcommand{\EexFCIinfty}{E_\text{exFCI}^{\infty}} -\newcommand{\Ead}{\Delta E_\text{ad}} -\newcommand{\efci}[0]{E_{\text{FCI}}^{\Bas}} -\newcommand{\emodel}[0]{E_{\model}^{\Bas}} -\newcommand{\emodelcomplete}[0]{E_{\model}^{\infty}} -\newcommand{\efcicomplete}[0]{E_{\text{FCI}}^{\infty}} -\newcommand{\ecccomplete}[0]{E_{\text{CCSD(T)}}^{\infty}} -\newcommand{\ecc}[0]{E_{\text{CCSD(T)}}^{\Bas}} -\newcommand{\efuncbasisfci}[0]{\bar{E}^\Bas[\denfci]} -\newcommand{\efuncbasis}[0]{\bar{E}^\Bas[\den]} -\newcommand{\efuncden}[1]{\bar{E}^\Bas[#1]} -\newcommand{\ecompmodel}[0]{\bar{E}^\Bas[\denmodel]} -\newcommand{\ecmubis}[0]{\bar{E}_{\text{c,md}}^{\text{sr}}[\denr;\,\mu]} -\newcommand{\ecmubisldapbe}[0]{\bar{E}_{\text{c,md}\,\text{PBE}}^{\text{sr}}[\denr;\,\mu]} -\newcommand{\ecmuapprox}[0]{\bar{E}_{\text{c,md-}\mathcal{X}}^{\text{sr}}[\den;\,\mu]} -\newcommand{\ecmuapproxmur}[0]{\bar{E}_{\text{c,md-}\mathcal{X}}^{\text{sr}}[\den;\,\mur]} -\newcommand{\ecmuapproxmurfci}[0]{\bar{E}_{\text{c,md-}\mathcal{X}}^{\text{sr}}[\denfci;\,\mur]} -\newcommand{\ecmuapproxmurmodel}[0]{\bar{E}_{\text{c,md-}\mathcal{X}}^{\text{sr}}[\denmodel;\,\mur]} -\newcommand{\ecompmodellda}[0]{\bar{E}_{\text{LDA}}^{\Bas,\wf{}{\Bas}}[\denmodel]} -\newcommand{\ecompmodelldaval}[0]{\bar{E}_{\text{LDA, val}}^{\Bas,\wf{}{\Bas}}[\den]} -\newcommand{\ecompmodelpbe}[0]{\bar{E}_{\text{PBE}}^{\Bas,\wf{}{\Bas}}[\den]} -\newcommand{\ecompmodelpbeval}[0]{\bar{E}_{\text{PBE, val}}^{\Bas,\wf{}{\Bas}}[\den]} -\newcommand{\emulda}[0]{\bar{\varepsilon}^{\text{sr},\text{unif}}_{\text{c,md}}\left(\denr;\mu({\bf r};\wf{}{\Bas})\right)} -\newcommand{\emuldamodel}[0]{\bar{\varepsilon}^{\text{sr},\text{unif}}_{\text{c,md}}\left(\denmodelr;\mu({\bf r};\wf{}{\Bas})\right)} -\newcommand{\emuldaval}[0]{\bar{\varepsilon}^{\text{sr},\text{unif}}_{\text{c,md}}\left(\denval ({\bf r});\murval;\wf{}{\Bas})\right)} - % numbers -%\newcommand{\rnum}[0]{{\rm I\!R}} \newcommand{\bfr}[1]{{\bf x}_{#1}} -\newcommand{\bfrb}[1]{{\bf r}_{#1}} -\newcommand{\dr}[1]{\text{d}\bfr{#1}} -\newcommand{\rr}[2]{\bfr{#1}, \bfr{#2}} -\newcommand{\rrrr}[4]{\bfr{#1}, \bfr{#2},\bfr{#3},\bfr{#4} } - - - -% effective interaction -\newcommand{\twodm}[4]{\elemm{\Psi}{\psixc{#4}\psixc{#3} \psix{#2}\psix{#1}}{\Psi}} -\newcommand{\murpsi}[0]{\mu({\bf r};\wf{}{\Bas})} -\newcommand{\mur}[0]{\mu({\bf r})} -\newcommand{\murr}[1]{\mu({\bf r}_{#1})} -\newcommand{\murval}[0]{\mu_{\text{val}}({\bf r})} -\newcommand{\murpsival}[0]{\mu_{\text{val}}({\bf r};\wf{}{\Bas})} -\newcommand{\murrval}[1]{\mu_{\text{val}}({\bf r}_{#1})} -\newcommand{\weeopmu}[0]{\hat{W}_{\text{ee}}^{\text{lr},\mu}} - - -\newcommand{\wbasis}[0]{W_{\wf{}{\Bas}}(\bfr{1},\bfr{2})} \newcommand{\wbasisval}[0]{W_{\wf{}{\Bas}}^{\text{val}}(\bfr{1},\bfr{2})} -\newcommand{\fbasis}[0]{f_{\wf{}{\Bas}}(\bfr{1},\bfr{2})} -\newcommand{\fbasisval}[0]{f_{\wf{}{\Bas}}^{\text{val}}(\bfr{1},\bfr{2})} - \newcommand{\ontop}[2]{ n^{(2)}_{#1}({\bf #2}_1)} - \newcommand{\twodmrpsi}[0]{ n^{(2)}_{\wf{}{\Bas}}(\rrrr{1}{2}{2}{1})} - \newcommand{\twodmrdiagpsi}[0]{ n^{(2)}_{\wf{}{\Bas}}(\rr{1}{2})} - \newcommand{\twodmrdiagpsival}[0]{ n^{(2)}_{\wf{}{\Bas},\,\text{val}}(\rr{1}{2})} - \newcommand{\gammamnpq}[1]{\Gamma_{mn}^{pq}[#1]} - \newcommand{\gammamnkl}[0]{\Gamma_{mn}^{kl}} - \newcommand{\gammaklmn}[1]{\Gamma_{kl}^{mn}[#1]} -\newcommand{\wbasiscoal}[1]{W_{\wf{}{\Bas}}({\bf r}_{#1})} -\newcommand{\ontoppsi}[1]{ n^{(2)}_{\wf{}{\Bas}}(\bfr{#1},\barr{#1},\barr{#1},\bfr{#1})} -\newcommand{\wbasiscoalval}[1]{W_{\wf{}{\Bas}}^{\text{val}}({\bf r}_{#1})} -\newcommand{\ontoppsival}[1]{ n^{(2)}_{\wf{}{\Bas}}^{\text{val}}(\bfr{#1},\barr{#1},\barr{#1},\bfr{#1})} - -\newcommand{\ex}[4]{$^{#1}#2_{#3}^{#4}$} -\newcommand{\ra}{\rightarrow} -\newcommand{\De}{D_\text{e}} - -% MODEL -\newcommand{\model}[0]{\mathcal{Y}} - % densities -\newcommand{\denmodel}[0]{\den_{\model}^\Bas} -\newcommand{\denmodelr}[0]{\den_{\model}^\Bas ({\bf r})} -\newcommand{\denfci}[0]{\den_{\psifci}} -\newcommand{\denhf}[0]{\den_{\text{HF}}^\Bas} -\newcommand{\denrfci}[0]{\denr_{\psifci}} -\newcommand{\dencipsir}[0]{{n}_{\text{CIPSI}}^\Bas({\bf r})} -\newcommand{\dencipsi}[0]{{n}_{\text{CIPSI}}^\Bas} -\newcommand{\den}[0]{{n}} -\newcommand{\denval}[0]{{n}^{\text{val}}} -\newcommand{\denr}[0]{{n}({\bf r})} \newcommand{\onedmval}[0]{\rho_{ij,\sigma}^{\text{val}}} -% wave functions -\newcommand{\psifci}[0]{\Psi^{\Bas}_{\text{FCI}}} -\newcommand{\psimu}[0]{\Psi^{\mu}} -% operators -\newcommand{\weeopbasis}[0]{\hat{W}_{\text{ee}}^\Bas} -\newcommand{\kinop}[0]{\hat{T}} - -\newcommand{\weeopbasisval}[0]{\hat{W}_{\text{ee}}^{\Basval}} -\newcommand{\weeop}[0]{\hat{W}_{\text{ee}}} - - % units \newcommand{\IneV}[1]{#1 eV} \newcommand{\InAU}[1]{#1 a.u.} @@ -177,7 +85,9 @@ \newcommand{\Nel}{N} + \newcommand{\n}[2]{n_{#1}^{#2}} +\newcommand{\Ec}{E_\text{c}} \newcommand{\E}[2]{E_{#1}^{#2}} \newcommand{\bE}[2]{\Bar{E}_{#1}^{#2}} \newcommand{\bEc}[1]{\Bar{E}_\text{c}^{#1}} @@ -212,6 +122,9 @@ \newcommand{\dbr}[1]{d\br{#1}} \newcommand{\dbx}[1]{d\bx{#1}} +\newcommand{\ra}{\rightarrow} +\newcommand{\De}{D_\text{e}} + \newcommand{\LCPQ}{Laboratoire de Chimie et Physique Quantiques (UMR 5626), Universit\'e de Toulouse, CNRS, UPS, France} \newcommand{\LCT}{Laboratoire de Chimie Th\'eorique, Universit\'e Pierre et Marie Curie, Sorbonne Universit\'e, CNRS, Paris, France} @@ -483,7 +396,7 @@ Also, as demonstrated in Appendix B of Ref.~\onlinecite{GinPraFerAssSavTou-JCP-1 To be able to approximate the complementary functional $\bE{}{\Bas}[\n{}{}]$ thanks to functionals developed in the field of RS-DFT, we associate the effective interaction to a long-range interaction characterized by a range-separation function $\rsmu{}{}(\br{})$. Although this choice is not unique, the long-range interaction we have chosen is \begin{equation} - \w{}{\lr,\rsmu{}{}}(\br{1},\br{2}) = \frac{1}{2} \qty{ \frac{\erf[ \murr{1} \, r_{12}]}{r_{12}} + \frac{\erf[ \murr{2} r_{12}]}{ r_{12}} }. + \w{}{\lr,\rsmu{}{}}(\br{1},\br{2}) = \frac{1}{2} \qty{ \frac{\erf[ \rsmu{}{}(\br{1}) r_{12}]}{r_{12}} + \frac{\erf[ \rsmu{}{}(\br{2}) r_{12}]}{ r_{12}} }. \end{equation} Ensuring that $\w{}{\lr,\rsmu{}{}}(\br{1},\br{2})$ and $\W{\wf{}{\Bas}}{}(\bx{1},\bx{2})$ have the same value at coalescence of opposite-spin electron pairs yields \begin{equation}