introduced properly the PBEot

Manuscript/Ex-srDFT.tex

@@ -65,6 +65,7 @@
 \newcommand{\ROHF}{\text{ROHF}}
 \newcommand{\LDA}{\text{LDA}}
 \newcommand{\PBE}{\text{PBE}}
+\newcommand{\PBEot}{\text{PBEot}}
 \newcommand{\FCI}{\text{FCI}}
 \newcommand{\CBS}{\text{CBS}}
 \newcommand{\exFCI}{\text{exFCI}}
@@ -100,6 +101,9 @@
 
 \newcommand{\tX}{\text{X}}
 
+\newcommand{\pbeotint}[0]{\be{\text{c,md}}{\sr,\PBEot}(\br{})\,\n{}{}(\br{})}
+\newcommand{\pbeint}[0]{\be{\text{c,md}}{\sr,\PBE}(\br{})\,\n{}{}(\br{})}
+
 % basis sets 
 \newcommand{\Bas}{\mathcal{B}}
 \newcommand{\BasFC}{\mathcal{A}}
@@ -307,27 +311,46 @@ The local-density approximation (LDA) of the ECMD complementary functional is de
 \end{equation}
 where $\zeta = (\n{\uparrow}{} - \n{\downarrow}{})/\n{}{}$ is the spin polarization and $\be{\text{c,md}}{\sr,\LDA}(\n{}{},\zeta,\rsmu{}{})$ is the ECMD short-range correlation energy per electron of the uniform electron gas (UEG) \cite{LooGil-WIRES-16} parameterized in Ref.~\citenum{PazMorGorBac-PRB-06}.
 
-To go beyond the LDA and cure its over correlation at small $\mu$, some of the authors recently proposed a Perdew-Burke-Ernzerhof (PBE)-based ECMD functional \cite{FerGinTou-JCP-18}, 
+To go beyond the LDA and cure its over correlation at small $\mu$, some of the authors recently proposed a Perdew-Burke-Ernzerhof (PBE)-based ECMD functional\cite{LooPraSceTouGin-JPCL-19}, 
 \begin{equation}
 	\label{eq:def_pbe_tot}
 	\bE{\PBE}{\Bas}[\n{}{},\rsmu{}{\Bas}] =
 	\int \n{}{}(\br{}) \be{\text{c,md}}{\sr,\PBE}\qty(\n{}{}(\br{}),s(\br{}),\zeta(\br{}),\rsmu{}{\Bas}(\br{})) \dbr{},
 \end{equation}
 where $s = \abs{\nabla \n{}{}}/\n{}{4/3}$ is the reduced density gradient. 
-$\be{\text{c,md}}{\sr,\PBE}\qty(\n{}{},s,\zeta,\rsmu{}{})$ interpolates between the usual PBE correlation functional, \cite{PerBurErn-PRL-96} $\e{\text{c}}{\PBE}(\n{}{},s,\zeta)$, at $\rsmu{}{} = 0$ (DFT limit) and the exact large-$\rsmu{}{}$ behavior (WFT limit). \cite{TouColSav-PRA-04, GorSav-PRA-06, PazMorGorBac-PRB-06}
-It reads
+$\be{\text{c,md}}{\sr,\PBE}\qty(\n{}{},s,\zeta,\rsmu{}{})$ interpolates between the usual PBE correlation functional, \cite{PerBurErn-PRL-96} $\e{\text{c}}{\PBE}(\n{}{},s,\zeta)$, at $\rsmu{}{} = 0$ (DFT limit) and the exact large-$\rsmu{}{}$ behavior where the on-top pair density appears. \cite{TouColSav-PRA-04, GorSav-PRA-06, PazMorGorBac-PRB-06} In order to avoid the computation of the exact on-top pair density, the authors proposed to use that of the UEG which leads to
 \begin{subequations}
 \begin{gather}
 	\label{eq:epsilon_cmdpbe}
 	\be{\text{c,md}}{\sr,\PBE}(\n{}{},s,\zeta,\rsmu{}{}) = \frac{\e{\text{c}}{\PBE}(\n{}{},s,\zeta)}{1 + \beta(\n{}{},s,\zeta) \rsmu{}{3} },
 	\\
 	\label{eq:beta_cmdpbe}
-	\beta(\n{}{},s,\zeta) = \frac{3}{2\sqrt{\pi} (1 - \sqrt{2} )} \frac{\e{\text{c}}{\PBE}(\n{}{},s,\zeta)}{\n{2}{}(\br{},\br{})/\n{}{}}.
+	\beta(\n{}{},s,\zeta) = \frac{3}{2\sqrt{\pi} (1 - \sqrt{2} )} \frac{\e{\text{c}}{\PBE}(\n{}{},s,\zeta)}{\n{2}{\UEG}(\n{}{},\zeta)/\n{}{}}.
+\end{gather}
+\end{subequations}
+where $\n{2}{\UEG}(\n{}{},\zeta) \approx \n{}{2} (1-\zeta^2) g_0(n)$ with the parametrization of the UEG on-top pair-distribution function $g_0(n)$ given in Eq.~(46) of Ref.~\citenum{GorSav-PRA-06}. 
+We will refer to this functional as the PBE ECMD functional.
+
+As the UEG on-top pair density might not be suitable to treat strongly correlated systems, we propose here to use an extrapolation of the exact on-top pair density based on the on-top pair density: 
+\begin{equation}
+ \n{2}{\text{extr}}(\br{},\n{2}{\Bas},\rsmu{}{\Bas}) = \frac{\n{2}{\Bas}(\br{},\br{})}{1+ \frac{2}{\sqrt{\pi}\rsmu{}{\Bas}(\br{})}}. 
+\end{equation}
+Therefore, we introduce the ``on top'' PBE (PBEot) ECMD functional which reads: 
+\begin{equation}
+	\label{eq:def_pbe_tot}
+	\bE{\PBEot}{\Bas}[\n{}{},\rsmu{}{\Bas}] =
+	\int \n{}{}(\br{}) \be{\text{c,md}}{\sr,\PBEot}\qty(\n{}{}(\br{}),s(\br{}),\zeta(\br{}),\n{2}{\Bas}(\br{},\br{}),\rsmu{}{\Bas}(\br{})) \dbr{},
+\end{equation}
+with
+\begin{subequations}
+\begin{gather}
+	\label{eq:epsilon_cmdpbe}
+	\be{\text{c,md}}{\sr,\PBEot}(\n{}{},s,\zeta,\n{2}{\Bas},\rsmu{}{}) = \frac{\e{\text{c}}{\PBE}(\n{}{},s,\zeta)}{1 + \beta(\n{}{},s,\zeta,\n{2}{}) \rsmu{}{3} },
+	\\
+	\label{eq:beta_cmdpbe}
+	\beta(\n{}{},s,\zeta,\n{2}{}) = \frac{3}{2\sqrt{\pi} (1 - \sqrt{2} )} \frac{\e{\text{c}}{\PBE}(\n{}{},s,\zeta)}{\n{2}{\text{extr}}(\n{2}{\Bas},\rsmu{}{\Bas})/\n{}{}}.
 \end{gather}
 \end{subequations}
-We will refer to this functional as the ``on top'' PBE (PBEot) ECMD functional.
-More recently, \cite{LooPraSceTouGin-JPCL-19} we have also proposed a simplified version of the PBEot functional where we replaced the on-top pair density by its UEG version, i.e.~$\n{2}{\Bas}(\br{},\br{})  \approx \n{2}{\UEG}(\n{}{}(\br{}),\zeta(\br{}))$, where $\n{2}{\UEG}(\n{}{},\zeta) \approx \n{}{2} (1-\zeta^2) g_0(n)$ with the parametrization of the UEG on-top pair-distribution function $g_0(n)$ given in Eq.~(46) of Ref.~\citenum{GorSav-PRA-06}. 
-This computationally-lighter functional will be referred to as PBE.
 
 
 %%%%%%%%%%%%%%%%%%%%%%%%
@@ -674,28 +697,10 @@ However, these results also clearly evidence that special care has to be taken f
 %%% %%% %%%
 
 %=======================
-\subsection{Range-Separation Function in Carbon Monoxyde}
-\label{sec:CO}
+%\subsection{Carbon Monoxyde}
+%\label{sec:CO}
 %=======================
 
-It is interesting to study the behavior of $\rsmu{}{\Bas}(\br{})$ for different states as the basis set incompleteness error is obviously state specific.
-To do so, we consider the ground state (${}^{1}\Sigma^+$) of carbon monoxide as well as its lowest singlet excited state (${}^{1}\Pi$). 
-The values of the vertical excitation energies obtained for various methods and basis sets are reported in Table \ref{tab:Mol}.
-Figure \ref{fig:CO} represents $\rsmu{}{}(z)$ along the nuclear axis ($z$) for these two electronic states computed with the AVDZ, AVTZ and AVQZ basis sets. 
-\manu{These figures illustrate several important things: 
-i) the maximal values of $\rsmu{}{\Bas}(\br{})$ are systematically close to the nuclei, a signature of the atom-centered basis set, 
-ii) the overall values of $\rsmu{}{\Bas}(\br{})$ increase with the basis set, which reflects the improvement of the description of the correlation effects when enlarging the basis set, 
-iii) the value of $\rsmu{}{\Bas}(\br{})$ are slightly larger near the oxygen atom, which traduces the fact that the inter-electronic distance is higher than close to the carbon atom due to a higher nuclear charge. }
-
-%%% FIG 4 %%%
-\begin{figure}
-	\includegraphics[width=\linewidth]{CO}
-	\caption{$\rsmu{}{\Bas}(z)$ along the molecular axis ($z$) for the ground state ${}^{1}\Sigma^+$ (black curve) and first singlet excited state ${}^{1}\Pi$ (red curve) of \ce{CO} for various basis sets $\Bas$.
-	The carbon and oxygen nuclei are located at $z=-1.249$ and $z=0.893$ bohr, respectively, and are represented by the thin black lines.}
-	\label{fig:CO}
-\end{figure}
-%%% %%% %%%
-
 %=======================
 \subsection{Doubly-Excited States of the Carbon Dimer}
 \label{sec:C2}
@@ -718,6 +723,48 @@ In other words, the UEG on-top density used in the LDA and PBE functionals (see
 \end{figure}
 %%% %%% %%%
 
+It is interesting to study the behavior of \manu{the key quantities involved in the basis set correction} for different states as the basis set incompleteness error is obviously state specific. 
+%\manu{To do so, we report the value of the range separation parameter in real space $\rsmu{}{\Bas}(\br{})$, the value of the energetic correction $\be{\text{c,md}}{\sr,\PBE}\qty(\n{}{}(\br{}),s(\br{}),\zeta(\br{}),\rsmu{}{\Bas}(\br{}) $ and the on-top pair density $\n{2}{\Bas}(\br{},\br{})$ computed with different basis sets for the ground state and second excited state of the carbon dimer which are both of $\Sigma_g^+$ symmetry, in Figures . } 
+We report $\rsmu{}{\Bas}(z)$,  along the nuclear axis ($z$) for these two electronic states computed with the AVDZ, AVTZ and AVQZ basis sets. 
+\manu{These figures illustrate several important things: 
+i) the maximal values of $\rsmu{}{\Bas}(\br{})$ are systematically close to the nuclei, a signature of the atom-centered basis set, 
+ii) the overall values of $\rsmu{}{\Bas}(\br{})$ increase with the basis set, which reflects the improvement of the description of the correlation effects when enlarging the basis set, 
+iii) the value of $\rsmu{}{\Bas}(\br{})$ are slightly larger near the oxygen atom, which traduces the fact that the inter-electronic distance is higher than close to the carbon atom due to a higher nuclear charge. }
+
+%%% FIG 4 %%%
+\begin{figure}
+	\includegraphics[width=\linewidth]{C2_mu}
+	\caption{ C$_2$: $\rsmu{}{\Bas}(z)$ along the molecular axis ($z$) for the ground state (black curve) and second singlet excited state (red curve) which are both of $\Sigma_g^+$ symmetry for various basis sets $\Bas$. The two carbon nuclei are located at $z=-1.180$ and $z=1.180$ bohr, respectively, and are represented by the thin black lines.}
+	\label{fig:C2_mu}
+\end{figure}
+%%% %%% %%%
+
+%%% FIG 4 %%%
+\begin{figure}
+	\includegraphics[width=\linewidth]{C2_PBEot.pdf}
+	\caption{ C$_2$: $\pbeotint$ along the molecular axis ($z$) for the ground state (black curve) and second singlet excited state (red curve) which are both of $\Sigma_g^+$ symmetry for various basis sets $\Bas$. The two carbon nuclei are located at $z=-1.180$ and $z=1.180$ bohr, respectively, and are represented by the thin black lines.}
+	\label{fig:C2_PBEot}
+\end{figure}
+%%% %%% %%%
+
+%%% FIG 4 %%%
+\begin{figure}
+	\includegraphics[width=\linewidth]{C2_PBE.pdf}
+	\caption{ C$_2$: $\pbeint$ along the molecular axis ($z$) for the ground state (black curve) and second singlet excited state (red curve) which are both of $\Sigma_g^+$ symmetry for various basis sets $\Bas$. The two carbon nuclei are located at $z=-1.180$ and $z=1.180$ bohr, respectively, and are represented by the thin black lines.}
+	\label{fig:C2_PBE}
+\end{figure}
+%%% %%% %%%
+
+%%% FIG 4 %%%
+\begin{figure}
+	\includegraphics[width=\linewidth]{C2_n2.pdf}
+	\caption{ C$_2$: $\n{2}{\Bas}(\br{})$ along the molecular axis ($z$) for the ground state (black curve) and second singlet excited state (red curve) which are both of $\Sigma_g^+$ symmetry for various basis sets $\Bas$. The two carbon nuclei are located at $z=-1.180$ and $z=1.180$ bohr, respectively, and are represented by the thin black lines.}
+	\label{fig:C2_n2}
+\end{figure}
+%%% %%% %%%
+
+
+
 %=======================
 \subsection{Ethylene}
 \label{sec:C2H4}