new pbe functional

Manuscript/G2-srDFT.tex

@@ -58,6 +58,7 @@
 \newcommand{\efuncbasisfci}[0]{\bar{E}^\basis[\denfci]}
 \newcommand{\efuncbasis}[0]{\bar{E}^\basis[\den]}
 \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]}
@@ -376,7 +377,44 @@ Therefore, one can define an LDA-like functional for $\efuncbasis$ as
 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}
-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 bias, we propose here a new ECMD functional inspired by the recently proposed functional of some of the present authors\cite{pbeontop} which interpolates between the usual PBE correlation functional when $\mu \rightarrow 0$ and the exact behaviour which is known when $\mu \rightarrow \infty$. 
+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{pbeontop} which interpolates between the usual PBE correlation functional when $\mu \rightarrow 0$ and the exact behaviour which is known when $\mu \rightarrow \infty$. 
+
+The exact behaviour of the $\ecmubis$ is known in the large $\mu$ limit\cite{pbeontop}:
+\begin{equation}
+ \label{eq:ecmd_large_mu}
+  \ecmubis = \frac{2\sqrt{\pi}\left(1 - \sqrt{2}\right)}{3\,\mu^3} + \int \text{d}{\bf r} \,\, n^{(2)}({\bf} r)
+\end{equation}
+where $ n^{(2)}({\bf r}) $ is the \textit{exact} on-top pair density for the ground state of the system. 
+As the exact ground state on-top pair density $n^{(2)}({\bf} r)$ is not known, we propose here to approximate it by that of the UEG at the density of the system:
+\begin{equation}
+ \label{eq:ueg_ontop}
+ n^{(2)}({\bf} r) \approx n^{(2)}_{\text{UEG}}(n_{\uparrow}({\bf} r) , \, n_{\downarrow}({\bf} r))
+\end{equation}
+where $n_{\uparrow}({\bf} r)$ and $ n_{\downarrow}({\bf} r)$ are, respectively, the up and down spin densities of the physical system at ${\bf} r$, $n^{(2)}_{\text{UEG}}(n_{\uparrow} \, n_{\downarrow})$ is the UEG on-top pair density
+\begin{equation}
+ \label{eq:ueg_ontop}
+ n^{(2)}_{\text{UEG}}(n_{\uparrow} \, n_{\downarrow}) = 4\, n_{\uparrow} \, n_{\downarrow} \, g_0(n_{\uparrow}(,\, n_{\downarrow})
+\end{equation}
+and $g_0(n_{\uparrow} ,\, n_{\downarrow})$ is the correlation factor of the UEG whose parametrization can be found in \cite{ueg_ontop}. 
+
+As such a form diverges for small values of $\mu$ as $1/\mu^3$, we follow the work proposed in \cite{pbeontop} and interpolate with the Kohn-Sham correlation functional at $\mu = 0$. 
+More precisely, we propose the following expression for the 
+\begin{equation}
+ \label{eq:ecmd_large_mu}
+  \ecmubis = \int \text{d}{\bf r} \,\, \bar{e}_{\text{c,md}}^\text{PBE}(n({\bf} r),\nabla n({\bf} r);\,\mu)
+\end{equation}
+with 
+\begin{equation}
+ \label{eq:epsilon_cmdpbe}
+ \bar{e}_{\text{c,md}}^\text{PBE}(n,\nabla n;\,\mu) = \frac{e_c^{PBE}(n,\nabla n)}{1 + \beta_{\text{c,md}\,\text{PBE}}(n,\nabla n;\,\mu)\mu^3 }
+and 
+\end{equation}
+\begin{equation}
+ \label{eq:epsilon_cmdpbe}
+ \beta(n,\nabla n;\,\mu) = \frac{3 e_c^{PBE}(n,\nabla n)}{2\sqrt{\pi}\left(1 - \sqrt{2}\right)n^{(2)}_{\text{UEG}}(n_{\uparrow} \, n_{\downarrow})}.
+\end{equation}
+
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 \section{Results}
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