minor corrections

Manuscript/rsdft-cipsi-qmc.tex

@@ -23,7 +23,7 @@
 \newcommand{\titou}[1]{\textcolor{red}{#1}}
 \newcommand{\trashPFL}[1]{\textcolor{red}{\sout{#1}}}
 \newcommand{\PFL}[1]{\titou{(\underline{\bf PFL}: #1)}}
-\newcommand{\SI}{supplementary material}
+\newcommand{\SI}{\textcolor{blue}{supplementary material}}
 
 \newcommand{\mc}{\multicolumn}
 \newcommand{\fnm}{\footnotemark}
@@ -269,10 +269,10 @@ where
 are the singular short-range (sr) part and the non-singular long-range (lr) part, respectively, $\mu$ is the range-separation parameter which controls how rapidly the short-range part decays, $\erf(x)$ is the error function, and $\erfc(x) = 1 - \erf(x)$ is its complementary version.
 
 The main idea behind RS-DFT is to treat the short-range part of the
-interaction within KS-DFT, and the long range part within a WFT method like FCI in the present case.
+interaction within KS-DFT, and the long-range part within a WFT method like FCI in the present case.
 The parameter $\mu$ controls the range of the separation, and allows
 to go continuously from the KS Hamiltonian ($\mu=0$) to
-the FCI Hamiltonian ($\mu = \infty$).
+the FCI Hamiltonian ($\mu \to \infty$).
 
 To rigorously connect WFT and DFT, the universal
 Levy-Lieb density functional \cite{Lev-PNAS-79,Lie-IJQC-83} is
@@ -295,7 +295,7 @@ electronic energy
   },
 \end{equation}
 with $\hat{T}$ the kinetic energy operator,
-$\hat{W}_\text{ee}^{\text{lr}}$ the long-range
+$\hat{W}_\text{ee}^{\text{lr},\mu}$ the long-range
 electron-electron interaction,
 $n_\Psi$ the one-electron density associated with $\Psi$,
 and $\hat{V}_{\text{ne}}$ the electron-nucleus potential.
@@ -360,12 +360,12 @@ In the outer (macro-iteration) loop (red), at the $k$th iteration, a CIPSI selec
 to obtain $\Psi^{\mu\,(k)}$ with the RS Hamiltonian $\hat{H}^{\mu\,(k)}$
 parameterized using the current one-electron density $n^{(k)}$.
 At each iteration, the number of determinants in $\Psi^{\mu\,(k)}$ increases.
-One exits the macro-iteration loop when the absolute energy difference between two successive iterations $\Delta E^{(k)}$ is below a threshold $\tau_1$ that has been set to \titou{???} in the present study.
+One exits the outer loop when the absolute energy difference between two successive macro-iterations $\Delta E^{(k)}$ is below a threshold $\tau_1$ that has been set to \titou{???} in the present study.
 An inner (micro-iteration) loop (blue) is introduced to accelerate the
 convergence of the self-consistent calculation, in which the set of
 determinants in $\Psi^{\mu\,(k,l)}$ is kept fixed, and only the diagonalization of
 $\hat{H}^{\mu\,(k,l)}$ is performed iteratively with the updated density $n^{(k,l)}$.
-The micro-iteration loop is exited when the absolute energy difference between two successive iterations $\Delta E^{(k,l)}$ is below a threshold $\tau_2$ that has been set to \titou{???} in the present study.
+The inner loop is exited when the absolute energy difference between two successive micro-iterations $\Delta E^{(k,l)}$ is below a threshold $\tau_2$ that has been here set to \titou{???}.
 The convergence of the algorithm was further improved
 by introducing a direct inversion in the iterative subspace (DIIS)
 step to extrapolate the density both in the outer and inner loops. \cite{Pulay_1980,Pulay_1982}
@@ -379,8 +379,8 @@ the final trial wave function $\Psi^{\mu}$ is independent of the starting wave f
 \section{Computational details}
 \label{sec:comp-details}
 
-\titou{The geometries for the G2 data set are provided as {\SI}.}
-
+For all the systems considered here, experimental geometries have been considered and they have been extracted from the NIST website.
+Geometries for each system are reported in the {\SI}.
 
 All the calculations have been performed using Burkatzki-Filippi-Dolg (BFD)
 pseudopotentials \cite{Burkatzki_2007,Burkatzki_2008} with the associated double-,
@@ -413,6 +413,7 @@ algorithm developed by Assaraf \textit{et al.}, \cite{Assaraf_2000}
 with a time step of $5 \times 10^{-4}$ a.u.
 \titou{All-electron move DMC?}
 
+\titou{Missing details and references about srLDA and srPBE functionals.}
 
 \section{Influence of the range-separation parameter on the fixed-node
   error}
@@ -457,19 +458,19 @@ with a time step of $5 \times 10^{-4}$ a.u.
 
 The first question we would like to address is the quality of the
 nodes of the wave functions $\Psi^{\mu}$ obtained with an intermediate
-range separation parameter (\textit{i.e.} $0 < \mu  < +\infty$).
-We generated trial wave functions $\Psi^\mu$ with multiple values of
-$\mu$, and computed the associated fixed node energy keeping all the
-parameters having an impact on the nodal surface fixed.
-We considered a weakly correlated molecular systems: the water
+range separation parameter (\ie, $0 < \mu  < +\infty$).
+For this purpose, we consider a weakly correlated molecular system: the water
 molecule near its equilibrium geometry.\cite{Caffarel_2016}
+We then generate trial wave functions $\Psi^\mu$ for multiple values of
+$\mu$, and compute the associated fixed-node energy keeping all the
+parameters having an impact on the nodal surface fixed (\titou{such as ??}).
 
 \subsection{Fixed-node energy of $\Psi^\mu$}
 \label{sec:fndmc_mu}
 From Table~\ref{tab:h2o-dmc} and Fig.~\ref{fig:h2o-dmc},
-one can clearly observe that using a FCI trial
-wave functions ($\mu = \infty$) give an FN-DMC energies lower
-than the energies obtained with a single Kohn-Sham determinant ($\mu=0$):
+one can clearly observe that using FCI trial
+wave functions ($\mu \to \infty$) give FN-DMC energies lower
+than the energies obtained with a single KS determinant ($\mu=0$):
 a gain of $3.2 \pm 0.6$~m\hartree{} at the double-zeta level and $7.2 \pm
 0.3$~m\hartree{} at the triple-zeta level are obtained.
 Coming now to the nodes of the trial wave functions $\Psi^{\mu}$ with