diff --git a/Manuscript/G2-srDFT.tex b/Manuscript/G2-srDFT.tex index 57c8786..8458ca5 100644 --- a/Manuscript/G2-srDFT.tex +++ b/Manuscript/G2-srDFT.tex @@ -161,6 +161,8 @@ % methods \newcommand{\FCI}{\text{FCI}} \newcommand{\CCSDT}{\text{CCSD(T)}} +\newcommand{\lr}{\text{lr}} +\newcommand{\sr}{\text{sr}} \newcommand{\Nel}{N} @@ -169,6 +171,9 @@ \newcommand{\bE}[2]{\Bar{E}_{#1}^{#2}} \newcommand{\wf}[2]{\Psi_{#1}^{#2}} \newcommand{\W}[2]{W_{#1}^{#2}} +\newcommand{\w}[2]{w_{#1}^{#2}} +\newcommand{\hn}[2]{\Hat{n}_{#1}^{#2}} +\newcommand{\rsmu}[2]{\mu_{#1}^{#2}} \newcommand{\SO}[2]{\phi_{#1}(\bx{#2})} \newcommand{\modX}{\text{X}} @@ -389,8 +394,8 @@ Provided that the functional $\bE{}{\Bas}[\n{}{}]$ is known exactly, the only so %================================================================= However, in addition of being unknown, the functional $\bE{}{\Bas}[\n{}{}]$ is obviously \textit{not} universal as it depends on $\Bas$. 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}]. -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}). -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}) . +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 \textit{function} $\mu(\br{})$ defined in real space. %(see Sec.~\ref{sec:weff}). +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 $\n{\modX}{\Bas}$ alongside $\mu(\br{})$.% (see Sec.~\ref{sec:ecmd}) . %================================================================= @@ -402,9 +407,9 @@ As the e-e cusp originates from the divergence of the Coulomb operator at $r_{12 Therefore, the impact of the incompleteness of $\Bas$ can be viewed as a removal of the divergence of the Coulomb interaction at $r_{12} = 0$. The present paragraph briefly describes how to obtain an effective operator $\W{\wf{}{\Bas}}{}(\bx{1},\bx{2})$ 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. -%---------------------------------------------------------------- +%================================================================= \subsection{Effective Coulomb operator} -%---------------------------------------------------------------- +%================================================================= In order to compute the effective operator $\W{\wf{}{\Bas}}{}(\bx{1},\bx{2})$ defined such that \begin{equation} \label{eq:int_eq_wee} @@ -446,42 +451,11 @@ and $\Gam{mn}{pq}[\wf{}{\Bas}] = \mel*{\wf{}{\Bas}}{ \aic{p}\aic{q}\ai{n}\ai{m} As already discussed in Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, $\W{\wf{}{\Bas}}{}(\bx{1},\bx{2})$ is symmetric, \textit{a priori} non translational nor rotational invariant if $\Bas$ does not have such symmetries and is necessarily \textit{finite} at $r_{12} = 0$ for an incomplete basis set $\Bas$. Also, as demonstrated in Appendix B of Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, $\lim_{\Bas \to \infty}\W{\wf{}{\Bas}}{}(\bx{1},\bx{2}) = r_{12}^{-1}$. -%---------------------------------------------------------------- -\subsubsection{Definition of a valence effective interaction} -%---------------------------------------------------------------- -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 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 $\f{\wf{}{\Bas}}{\Val}(\bx{1},\bx{2})$ satisfying -\begin{equation} - \label{eq:expectweebval} - \mel*{\wf{}{\Bas}}{\hWee{\Val}}{\wf{}{\Bas}} = \frac{1}{2} \iint \f{\wf{}{\Bas}}{\Val}(\bx{1},\bx{2}) \dbx{1} \dbx{2}, -\end{equation} -where $\hWee{\Val}$, the valence part of the Coulomb operator, has a similar expression as $\hWee{\Bas}$ in Eq.~\eqref{eq:WeeB}. -%\begin{equation} -% \hWee{\Val} = \frac{1}{2} \sum_{ijkl \in \Val} \vijkl \aic{k}\aic{l}\ai{j}\ai{i}, -%\end{equation} -Following the spirit of Eq.~\eqref{eq:fbasis}, the function $\f{\wf{}{\Bas}}{\Val}(\bx{1},\bx{2})$ can be defined as - \begin{multline} - \label{eq:fbasisval} - \f{\wf{}{\Bas}}{\Val}(\bx{1},\bx{2}) - \\ - = \sum_{ij \in \Bas} \sum_{klmn \in \Val} \SO{i}{1} \SO{j}{2} \vijkl \gammaklmn{\wf{}{\Bas}} \SO{n}{2} \SO{m}{1}. - \end{multline} -and, the valence part of the effective interaction is - \begin{equation} - \label{eq:def_weebasis} - \W{\wf{}{\Bas}}{\Val}(\bx{1},\bx{2}) = \frac{\f{\wf{}{\Bas}}{\Val}(\bx{1},\bx{2}) }{\n{\wf{}{\Bas},\Val}{(2)}(\bx{1},\bx{2})}, - \end{equation} -where $\n{\wf{}{\Bas},\Val}{(2)}(\bx{1},\bx{2})$ is the two body density associated to the valence electrons. -%\begin{equation} -% \twodmrdiagpsival = \sum_{klmn \in \Val} \SO{m}{1} \SO{n}{2} \gammamnkl[\wf{}{\Bas}] \SO{k}{1} \SO{l}{2} . -%\end{equation} -%It is worth noting that, in Eq.~\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 $\Basval$. - - -\subsubsection{Definition of a range-separation parameter varying in space} -To be able to approximate the complementary functional $\efuncbasis$ thanks to functionals developed in the field of RSDFT, we fit the effective interaction with a long-range interaction having a range-separation parameter \textit{varying in space}. +%================================================================= +\subsubsection{Range-separation function} +%================================================================= +To be able to approximate the complementary functional $\bE{}{\Bas}[\n{}{}]$ thanks to functionals developed in the field of RS-DFT, we fit the effective interaction with a long-range interaction having a range-separation parameter \textit{varying in space}. More precisely, if we define the value of the interaction at coalescence as \begin{equation} \label{eq:def_wcoal} @@ -490,71 +464,101 @@ More precisely, if we define the value of the interaction at coalescence as 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: \begin{equation} - \wbasiscoal{} = w^{\text{lr},\murpsi}(\bfrb{},\bfrb{}) + \wbasiscoal{} = \w{}{\lr,\rsmu{\wf{}{\Bas}}{}}(\bfrb{},\bfrb{}) \end{equation} -where the long-range-like interaction is defined as: +where the long-range-like interaction is defined as \begin{equation} - w^{\text{lr},\mur}(\bfrb{1},\bfrb{2}) = \frac{ 1 }{2} \bigg( \frac{\text{erf}\big( \murr{1} \, r_{12}\big)}{r_{12}} + \frac{\text{erf}\big( \murr{2} \, r_{12}\big)}{ r_{12}}\bigg). + \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}} }. \end{equation} -The equation \eqref{eq:def_wcoal} is equivalent to the following condition for $\murpsi$: +Equation \eqref{eq:def_wcoal} is equivalent to the following condition \begin{equation} \label{eq:mu_of_r} - \murpsi = \frac{\sqrt{\pi}}{2} \, \wbasiscoal{} \, . -\end{equation} -As we defined an effective interaction for the valence electrons, we also introduce a valence range-separation parameter as -\begin{equation} - \label{eq:mu_of_r_val} - \murpsival = \frac{\sqrt{\pi}}{2} \, \wbasiscoalval{} \, . + \rsmu{\wf{}{\Bas}}{}(\br{}) = \frac{\sqrt{\pi}}{2} \W{\wf{}{\Bas}}{}(\br{}) \end{equation} +%As we defined an effective interaction for the valence electrons, we also introduce a valence range-separation parameter as +%\begin{equation} +% \label{eq:mu_of_r_val} +% \murpsival = \frac{\sqrt{\pi}}{2} \, \wbasiscoalval{} \, . +%\end{equation} An important point to notice is that, in the limit of a complete basis set $\Bas$, as \begin{equation} - \begin{aligned} - &\lim_{\Bas \rightarrow \infty}\wbasis = 1/r_{12} \,\,\,\,\forall \,\, (\bfr{1},\bfr{2})\\ - &\lim_{\Bas \rightarrow \infty}\wbasisval = 1/r_{12} \,\,\,\,\forall \,\, (\bfr{1},\bfr{2})\,\, , - \end{aligned} -\end{equation} -one has -\begin{equation} - \begin{aligned} - &\lim_{\Bas \rightarrow \infty} \wbasiscoal{} = +\infty\,\,, \\ - &\lim_{\Bas \rightarrow \infty} \wbasiscoalval{} = +\infty\,\,, - \end{aligned} +\label{eq:lim_W} + \lim_{\Bas \rightarrow \infty}\wbasis = r_{12}^{-1} \quad \forall (\br{1},\br{2}) +% &\lim_{\Bas \rightarrow \infty}\wbasisval = 1/r_{12} \,\,\,\,\forall \,\, (\bfr{1},\bfr{2})\,\, , \end{equation} +one has $\lim_{\Bas \rightarrow \infty} \wbasiscoal{} = \infty$ +% &\lim_{\Bas \rightarrow \infty} \wbasiscoalval{} = +\infty\,\,, and therefore \begin{equation} - \label{eq:lim_mur} - \begin{aligned} - &\lim_{\Bas \rightarrow \infty} \murpsi = +\infty \,\, \\ - &\lim_{\Bas \rightarrow \infty} \murpsival = +\infty \,\, . - \end{aligned} +\label{eq:lim_mur} + \lim_{\Bas \rightarrow \infty} \rsmu{\wf{}{\Bas}}{}(\br{}) = \infty +%\lim_{\Bas \rightarrow \infty} \murpsival = +\infty \,\, . \end{equation} -\subsection{Approximations for the complementary functional $\ecompmodel$} -\subsubsection{General scheme} -\label{sec:ecmd} -In \onlinecite{GinPraFerAssSavTou-JCP-18} the authors have proposed to approximate the complementary functional $\efuncbasis$ by using a specific class of SRDFT energy functionals, namely the ECMD whose general definition is\cite{TouGorSav-TCA-05}: +%================================================================= +\subsection{Valence effective interaction} +%================================================================= +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 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 $\f{\wf{}{\Bas}}{\Val}(\bx{1},\bx{2})$ satisfying \begin{equation} - \begin{aligned} - \label{eq:ec_md_mu} - \ecmubis = & \min_{\Psi \rightarrow \denr}\elemm{\Psi}{\kinop +\weeop}{\Psi}\\-\;&\elemm{\psimu[\denr]}{\kinop+\weeop}{\psimu[\denr]}, - \end{aligned} + \label{eq:expectweebval} + \mel*{\wf{}{\Bas}}{\hWee{\Val}}{\wf{}{\Bas}} = \frac{1}{2} \iint \f{\wf{}{\Bas}}{\Val}(\bx{1},\bx{2}) \dbx{1} \dbx{2}, \end{equation} -where the wave function $\psimu[\denr]$ is defined by the constrained minimization +where $\hWee{\Val}$, the valence part of the Coulomb operator, has a similar expression as $\hWee{\Bas}$ in Eq.~\eqref{eq:WeeB}. +%\begin{equation} +% \hWee{\Val} = \frac{1}{2} \sum_{ijkl \in \Val} \vijkl \aic{k}\aic{l}\ai{j}\ai{i}, +%\end{equation} +Following the spirit of Eq.~\eqref{eq:fbasis}, the function $\f{\wf{}{\Bas}}{\Val}(\bx{1},\bx{2})$ can be defined as +\begin{multline} + \label{eq:fbasisval} + \f{\wf{}{\Bas}}{\Val}(\bx{1},\bx{2}) + \\ + = \sum_{ij \in \Bas} \sum_{klmn \in \Val} \SO{i}{1} \SO{j}{2} \vijkl \gammaklmn{\wf{}{\Bas}} \SO{n}{2} \SO{m}{1}. +\end{multline} +and, the valence part of the effective interaction is +\begin{equation} + \label{eq:def_weebasis} + \W{\wf{}{\Bas}}{\Val}(\bx{1},\bx{2}) = \frac{\f{\wf{}{\Bas}}{\Val}(\bx{1},\bx{2}) }{\n{\wf{}{\Bas},\Val}{(2)}(\bx{1},\bx{2})}, +\end{equation} +where $\n{\wf{}{\Bas},\Val}{(2)}(\bx{1},\bx{2})$ is the two body density associated to the valence electrons. +%\begin{equation} +% \twodmrdiagpsival = \sum_{klmn \in \Val} \SO{m}{1} \SO{n}{2} \gammamnkl[\wf{}{\Bas}] \SO{k}{1} \SO{l}{2} . +%\end{equation} +%It is worth noting that, in Eq.~\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 $\Basval$. +It is worth noting that, within the present definition, $\W{\wf{}{\Bas}}{\Val}(\bx{1},\bx{2})$ and $\murpsival$ fulfils Eqs.~\eqref{eq:lim_W} and \eqref{eq:lim_mur}. + + + +%================================================================= +\subsection{Complementary functional} +%================================================================= +\label{sec:ecmd} + +In Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18} the authors have proposed to approximate the complementary functional $\bE{}{\Bas}[\n{}{}]$ using a specific class of SR-DFT energy functionals, namely the ECMD whose general definition is \cite{TouGorSav-TCA-05} +\begin{multline} + \label{eq:ec_md_mu} + \ecmubis = \min_{\wf{}{} \to \n{}{}(\br{})} \mel*{\Psi}{\hT + \hWee{}}{\wf{}{}} + \\ + - \mel*{\wf{}{\rsmu{}{}}[\n{}{}(\br{})]}{\hT + \hWee{}}{\wf{}{\rsmu{}{}}[\n{}{}(\br{})]}, +\end{multline} +where the wave function $\wf{}{\rsmu{}{}}[\n{}{}(\br{})]$ is defined by the constrained minimization \begin{equation} \label{eq:argmin} -\psimu[\denr] = \arg \min_{\Psi \rightarrow \denr} \elemm{\Psi}{\kinop + \weeopmu}{\Psi}, + \wf{}{\rsmu{}{}}[\n{}{}(\br{})] = \arg \min_{\wf{}{} \to \n{}{}(\br{})} \mel*{\wf{}{}}{\hT + \hWee{\lr,\rsmu{}{}}}{\wf{}{}}, \end{equation} -where $\weeopmu$ is the long-range electron-electron interaction operator +where \begin{equation} - \label{eq:weemu} - \weeopmu = \frac{1}{2} \iint \text{d}{\bf r}_1 \text{d}{\bf r}_2 \; w^{\text{lr},\mu}(|{\bf r}_1 - {\bf r}_2|) \hat{n}^{(2)}({\bf r}_1,{\bf r}_2), +\label{eq:weemu} + \hWee{\lr,\rsmu{}{}} = \frac{1}{2} \iint \w{}{\lr,\rsmu{}{}}(r_{12}) \hn{}{(2)}(\br{1},\br{2}) \dbr{1} \dbr{2}, \end{equation} -with +is the long-range electron-electron interaction operator with \begin{equation} \label{eq:erf} -w^{\text{lr},\mu}(r_{12}) = \frac{\text{erf}(\mu r_{12})}{r_{12}}, + \w{}{\lr,\rsmu{}{}}(r_{12}) = \frac{\erf(\rsmu{}{} r_{12})}{r_{12}}, \end{equation} -and the pair-density operator $\hat{n}^{(2)}({\bf r}_1,{\bf r}_2) =\hat{n}({\bf r}_1) \hat{n}({\bf r}_2) - \delta ({\bf r}_1-{\bf r}_2) \hat{n}({\bf r}_1)$. +and the pair-density operator $\hn{}{(2)}(\br{1},\br{2}) =\hn{}{}(\br{1}) \hn{}{}(\br{2}) - \delta (\br{1}-\br{2}) \hn{}{}(\br{1})$. The ECMD functionals admit two limits as function of $\mu$ \begin{equation} \label{eq:large_mu_ecmd} @@ -581,7 +585,9 @@ It is important to notice that in the limit of a complete basis set, according t \end{equation} 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} +%-------------------------------------------- +\subsubsection{Local density approximation} +%-------------------------------------------- As done in Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, one can define an LDA-like approximation for $\ecompmodel$ as \begin{equation} \label{eq:def_lda_tot} @@ -589,7 +595,9 @@ As done in Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, one can define an LDA-li \end{equation} where $\bar{\varepsilon}^{\text{sr},\text{unif}}_{\text{c,md}}(n,\mu)$ is the multi-determinant short-range correlation energy per particle of the uniform electron gas for which a parametrization can be found in Ref.~\onlinecite{PazMorGorBac-PRB-06}. In practice, for open-shell systems, we use the spin-polarized version of this functional (i.e., depending on the spin densities) but for simplicity we will continue to use only the notation of the spin-unpolarized case. -\subsubsection{New PBE interpolated ECMD functional} +%-------------------------------------------- +\subsubsection{New PBE functional} +%-------------------------------------------- The LDA-like functional defined in \eqref{eq:def_lda_tot} relies only on the transferability of the physics of UEG which is certainly valid for large values of $\mu$ but which is known to over correlate for small values of $\mu$. In order to correct such a defect, we propose here a new ECMD functional inspired by the recently proposed functional of some of the present authors\cite{FerGinTou-JCP-18} which interpolates between the usual PBE correlation functional when $\mu \rightarrow 0$ and the exact behaviour which is known when $\mu \rightarrow \infty$. @@ -633,7 +641,9 @@ Therefore, we propose this approximation for the complementary functional $\ecom \ecompmodelpbe = \int \, \text{d}{\bf r} \,\, \bar{e}_{\text{c,md}}^\text{PBE}(n({\bf r}),\nabla n({\bf r});\,\mur) \end{equation} +%-------------------------------------------- \subsection{Valence-only approximation for the complementary functional} +%-------------------------------------------- We now introduce a valence-only approximation for the complementary functional which is needed to correct for frozen core WFT models. Defining the valence one-body spin density matrix as \begin{equation}