done with Sec IV

Manuscript/rsdft-cipsi-qmc.tex

@@ -556,7 +556,7 @@ This is a key result of the present study.
 \label{sec:rsdft-j}
 %======================================================
 The data presented in Sec.~\ref{sec:fndmc_mu} evidence that, in a finite basis, RS-DFT can provide 
-trial wave functions with better nodes than FCI wave function.
+trial wave functions with better nodes than FCI wave functions.
 As mentioned in Sec.~\ref{sec:SD}, such behavior can be directly compared to the common practice of
 re-optimizing the multi-determinant part of a trial wave function $\Psi$ (the so-called Slater part) in the presence of the exponentiated Jastrow factor $e^J$. \cite{Umrigar_2005,Scemama_2006,Umrigar_2007,Toulouse_2007,Toulouse_2008}
 Hence, in the present paragraph, we would like to elaborate further on the link between RS-DFT
@@ -564,7 +564,7 @@ and wave function optimization in the presence of a Jastrow factor.
 For the sake of simplicity, the molecular orbitals and the Jastrow
 factor are kept fixed; only the CI coefficients are varied.
 
-Let us assume a fixed Jastrow factor $J(\br_1, \ldots , \br_\Nelec)$ (where $\br_i$ is the position of the $i$th electron and $\Nelec$ the total number of electrons),
+Let us then assume a fixed Jastrow factor $J(\br_1, \ldots , \br_\Nelec)$ (where $\br_i$ is the position of the $i$th electron and $\Nelec$ the total number of electrons),
 and a corresponding Slater-Jastrow wave function $\Phi = e^J \Psi$,
 where 
 \begin{equation}
@@ -597,8 +597,8 @@ To do so, we have made the following numerical experiment.
 First, we extract the 200 determinants with the largest weights in the FCI wave
 function out of a large CIPSI calculation obtained with the VDZ-BFD basis.  Within this set of determinants,
 we solve the self-consistent equations of RS-DFT [see Eq.~\eqref{rs-dft-eigen-equation}]
-for different values of $\mu$ using the srPBE functional. This gives the CI expansions $\Psi^\mu$.
-Then, within the same set of determinants we optimize the CI coefficients $c_I$ [see Eq.~\eqref{eq:Slater}] in the presence of
+for different values of $\mu$ using the srPBE functional. This gives the CI expansions of $\Psi^\mu$.
+Then, within the same set of determinants we optimize the CI coefficients in the presence of
 a simple one- and two-body Jastrow factor $e^J$ with $J = J_\text{eN} + J_\text{ee}$ and
 \begin{subequations}
 \begin{gather}
@@ -617,7 +617,7 @@ The parameters $a=1/2$
 and $b=0.89$ were fixed, and the parameters $\gamma_{\text{O}}=1.15$ and $\gamma_{\text{H}}=0.35$
 were obtained by energy minimization of a single determinant.
 The optimal CI expansion $\Psi^J$ is obtained by sampling the matrix elements
-of the Hamiltonian ($\mathbf{H}$) and the overlap ($\mathbf{S}$), in the
+of the Hamiltonian ($\mathbf{H}$) and overlap ($\mathbf{S}$) matrices in the
 basis of Jastrow-correlated determinants $e^J D_i$:
 \begin{subequations}
 \begin{gather}
@@ -678,17 +678,16 @@ report several quantities related to the one- and two-body densities of
 $\Psi^J$ and $\Psi^\mu$ with different values of $\mu$.  First, we
 report in the legend of the right panel of Fig~\ref{fig:densities} the integrated on-top pair density
 \begin{equation}
- \expval{ P } = \int d\br \,\,n_2(\br,\br),
+ \expval{ P } = \int d\br \,n_2(\br,\br),
 \end{equation}
-where $n_2(\br_1,\br_2)$ is the two-body density [normalized to $\Nelec(\Nelec-1)$]
-obtained for both $\Psi^\mu$ and $\Psi^J$.
-Then, in order to have a pictorial representation of both the on-top
-pair density and the density, we report in Fig.~\ref{fig:densities}
-the plots of the total density $n(\br)$ and on-top pair density
-$n_2(\br,\br)$ along one of the \ce{O-H} axis of the water molecule.
+obtained for both $\Psi^\mu$ and $\Psi^J$,
+where $n_2(\br_1,\br_2)$ is the two-body density [normalized to $\Nelec(\Nelec-1)$].
+Then, in order to have a pictorial representation of both the one-body density $n(\br)$ and the on-top
+pair density $n_2(\br,\br)$, we report in Fig.~\ref{fig:densities}
+the plots of $n(\br)$ and $n_2(\br,\br)$ along one of the \ce{O-H} axis of the water molecule.
 
 From these data, one can clearly notice several trends.
-First, the overall on-top pair density decreases when $\mu$ increases,
+First, the integrated on-top pair density $\expval{ P }$ decreases when $\mu$ increases,
 which is expected as the two-electron interaction increases in
 $H^\mu[n]$.
 Second, Fig.~\ref{fig:densities} shows that the relative variations of the on-top pair density with respect to $\mu$
@@ -709,10 +708,10 @@ and therefore that the operators that produced these wave functions (\ie, $H^\mu
 Considering the form of $\hat{H}^\mu[n]$ [see Eq.~\eqref{H_mu}],
 one can notice that the differences with respect to the usual bare Hamiltonian come
 from the non-divergent two-body interaction $\hat{W}_{\text{ee}}^{\text{lr},\mu}$
-and the effective one-body potential $\hat{\bar{V}}_{\text{Hxc}}^{\text{sr},\mu}[n]$ which is the functional derivative of the Hartree-exchange-correlation functional.
+and the effective one-body potential $\hat{\bar{V}}_{\text{Hxc}}^{\text{sr},\mu}[n]$ which is the functional derivative of the Hxc functional.
 The roles of these two terms are therefore very different: with respect
 to the exact ground-state wave function $\Psi$, the non-divergent two-body interaction
-increases the probability to find electrons at short distances in $\Psi^\mu$,
+increases the probability of finding electrons at short distances in $\Psi^\mu$,
 while the effective one-body potential $\hat{\bar{V}}_{\text{Hxc}}^{\text{sr},\mu}[n_{\Psi^{\mu}}]$,
 providing that it is exact, maintains the exact one-body density.
 This is clearly what has been observed in
@@ -723,7 +722,7 @@ can be non-divergent when a proper two-body Jastrow factor $J_\text{ee}$ is chos
 There is therefore a clear parallel between $\hat{W}_{\text{ee}}^{\text{lr},\mu}$ in RS-DFT and $J_\text{ee}$ in FN-DMC.
 Moreover, the one-body Jastrow term $J_\text{eN}$ ensures that the one-body density remain unchanged when the CI coefficients are re-optmized in the presence of $J_\text{ee}$.
 There is then a second clear parallel between $\hat{\bar{V}}_{\text{Hxc}}^{\text{sr},\mu}[n]$ in RS-DFT and $J_\text{eN}$ in FN-DMC.
-Thus, one can understand the similarity between the eigenfunctions of $H^\mu$ and the Slater-Jastrow optimization:
+Thus, one can understand the similarity between the eigenfunctions of $H^\mu$ and the optimization of the Slater-Jastrow wave function:
 they both deal with an effective non-divergent interaction but still
 produce a reasonable one-body density.