added Ref from Julien

Manuscript/G2-srDFT.bib

@@ -11997,3 +11997,19 @@
 	Volume = {147},
 	Year = {2017},
 	Bdsk-Url-1 = {https://doi.org/10.1063/1.4992127}}
+
+@article{GorSav-PRA-06,
+  title = {Properties of short-range and long-range correlation energy density functionals from electron-electron coalescence},
+  author = {Gori-Giorgi, Paola and Savin, Andreas},
+  journal = {Phys. Rev. A},
+  volume = {73},
+  issue = {3},
+  pages = {032506},
+  numpages = {9},
+  year = {2006},
+  month = {Mar},
+  publisher = {American Physical Society},
+  doi = {10.1103/PhysRevA.73.032506},
+  url = {https://link.aps.org/doi/10.1103/PhysRevA.73.032506}
+}
+

Manuscript/G2-srDFT.tex

@@ -88,7 +88,7 @@
 
 % numbers
 \newcommand{\rnum}[0]{{\rm I\!R}}
-\newcommand{\bfr}[1]{{\bf X}_{#1}}
+\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}}
@@ -238,10 +238,10 @@ The basis-set correction investigated here proposes to use the RSDFT formalism t
 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 \onlinecite{GinPraFerAssSavTou-JCP-18}.  
 
 \subsection{Correcting the basis set error of a general WFT model}
-Consider a $N-$electron physical system described in an incomplete basis-set $\basis$ and for which we assume to have both the FCI density $\denfci$ and energy $\efci$. Assuming that $\denfci$ is a good approximation of the \textit{exact} ground state density, according to equation (15) of \onlinecite{GinPraFerAssSavTou-JCP-18}, one can approximate the exact ground state energy $E_0$ as
+Consider a $N-$electron physical system described in an incomplete basis-set $\basis$ and for which we assume to have both the FCI density $\denfci$ and energy $\efci$. Assuming that $\denfci$ is a good approximation of the \textit{exact} ground state density, according to equation (15) of \onlinecite{GinPraFerAssSavTou-JCP-18}, one can approximate the exact ground state energy $E$ as
 \begin{equation}
   \label{eq:e0basis}
-  E_0 \approx \efci + \efuncbasisfci
+  E \approx \efci + \efuncbasisfci
 \end{equation}
 where $\efuncbasis$ is the complementary density functional defined in equation (8) of \onlinecite{GinPraFerAssSavTou-JCP-18}  
 \begin{equation}
@@ -262,7 +262,7 @@ An important aspect of such a theory is that, in the limit of a complete basis s
 which implies that the exact ground state energy coincides with the FCI energy in complete basis set (which we refer as $\efcicomplete$)
 \begin{equation}
   \label{eq:limitfunc}
-        \lim_{\basis \rightarrow \infty} \efci + \efuncbasisfci = \efcicomplete\,\,.
+        \lim_{\basis \rightarrow \infty} \efci + \efuncbasisfci = E \,\,.
 \end{equation}
 
 Here we propose to generalize such approach to a general WFT model, referred here as $\model$, projected in a basis set $\basis$ which must provides a density $\denmodel$ and an energy $\emodel$. 
@@ -468,9 +468,9 @@ 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_{\psibasis}(\bfr{},\bar{{\bf x}}_{}).
 \end{equation}
-where $(\bfr{},\bar{{\bf X}}_{})$ means a couple of anti-parallel spins at the same point in $\bfrb{}$, 
+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{})
@@ -592,7 +592,7 @@ where $n_{\uparrow}({\bf} r)$ and $ n_{\downarrow}({\bf} r)$ are, respectively,
  \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}. 
+and $g_0(n_{\uparrow} ,\, n_{\downarrow})$ is the correlation factor of the UEG whose parametrization can be found in equation (46) of \onlinecite{GorSav-PRA-06}. 
 
 As the form in \eqref{eq:ecmd_large_mu} diverges for small values of $\mu$ as $1/\mu^3$, we follow the work proposed in \cite{FerGinTou-JCP-18} and interpolate between the large-$\mu$ limit and the $\mu = 0$ limit where the $\ecmubis$ reduces to the Kohn-Sham correlation functional (see equation \eqref{eq:small_mu_ecmd}), for which we take the PBE approximation as in \cite{FerGinTou-JCP-18}. 
 More precisely, we propose the following expression for the