modifications in first part of results

Manuscript/rsdft-cipsi-qmc.tex

@@ -251,7 +251,7 @@ estimate of the FCI energy, using a fixed value of the PT2 correction
 as a stopping criterion enforces a constant distance of all the
 calculations to the FCI energy. In this work, we target the chemical
 accuracy so all the CIPSI selections were made such that $\abs{\EPT} <
-1$ millihartree.
+1$ m\hartree{}.
 
 
 
@@ -279,7 +279,7 @@ 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.
 The parameter $\mu$ controls the range of the separation, and allows
 to go continuously from the KS Hamiltonian ($\mu=0$) to
-the FCI Hamiltonian ($\mu \to \infty$).
+the FCI Hamiltonian ($\mu = \infty$).
 
 To rigorously connect WFT and DFT, the universal
 Levy-Lieb density functional \cite{Lev-PNAS-79,Lie-IJQC-83} is
@@ -331,7 +331,7 @@ energy is obtained as
 Note that, for $\mu=0$, \titou{the long-range interaction vanishes}, \ie,
 $w_{\text{ee}}^{\text{lr},\mu=0}(r) = 0$, and thus
 RS-DFT reduces to standard KS-DFT and $\Psi^\mu$
-is the KS determinant. For $\mu\to\infty$, the long-range
+is the KS determinant. For $\mu = \infty$, the long-range
 interaction becomes the standard Coulomb interaction, \ie,
 $w_{\text{ee}}^{\text{lr},\mu\to\infty}(r) = r^{-1}$, and
 thus RS-DFT reduces to standard WFT and $\Psi^\mu$ is
@@ -369,12 +369,12 @@ In the outer (macro-iteration) loop (red), at the $k$th iteration, a CIPSI selec
 to obtain $\Psi^{\mu\,(k)}$ with the RS-DFT 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 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 $10^{-3}$ hartree in the present study which is consistent with the CIPSI threshold (see Sec.~\ref{sec:comp-details}).
+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 $10^{-3}$ \hartree{} in the present study which is consistent with the CIPSI threshold (see Sec.~\ref{sec:comp-details}).
 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 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 $10^{-2} \times \tau_1$ hartree.
+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 $10^{-2} \times \tau_1$ \hartree{}.
 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}
@@ -407,7 +407,7 @@ and correlation functionals defined in
 Ref.~\onlinecite{GolWerStoLeiGorSav-CP-06} (see also
 Refs.~\onlinecite{TouColSav-JCP-05,GolWerSto-PCCP-05}).
 The convergence criterion for stopping the CIPSI calculations
-has been set to $\EPT < 10^{-3}$ hartree or $ \Ndet > 10^7$.
+has been set to $\EPT < 10^{-3}$ \hartree{} or $ \Ndet > 10^7$.
 All the wave functions are eigenfunctions of the $\Hat{S}^2$ spin operator, as
 described in Ref.~\onlinecite{Applencourt_2018}.
 
@@ -431,7 +431,8 @@ with a time step of $5 \times 10^{-4}$ a.u.
 
 %%% TABLE I %%%
 \begin{table}
-  \caption{Fixed-node energies $\EDMC$ (in hartree) and number of determinants $\Ndet$ in \ce{H2O} with various trial wave functions $\Psi^{\mu}$.}
+  \caption{Fixed-node energy $\EDMC$ (in \hartree{}) and number of determinants $\Ndet$ in \ce{H2O} for various trial wave functions $\Psi^{\mu}$. 
+  \titou{srPBE?}.}
   \label{tab:h2o-dmc}
   \centering
   \begin{ruledtabular}
@@ -461,9 +462,9 @@ with a time step of $5 \times 10^{-4}$ a.u.
 \begin{figure}
   \centering
   \includegraphics[width=\columnwidth]{h2o-dmc.pdf}
-  \caption{Fixed-node energies of \ce{H2O} for different
-    values of $\mu$, using the srLDA or srPBE density
-    functionals to build the trial wave function.}
+  \caption{Fixed-node energy of \ce{H2O} as a function 
+    of $\mu$ for various levels of theory to generate
+    the trial wave function.}
   \label{fig:h2o-dmc}
 \end{figure}
 %%% %%% %%% %%%
@@ -481,31 +482,34 @@ 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},
+From Table~\ref{tab:h2o-dmc} and Fig.~\ref{fig:h2o-dmc} where we report the fixed-node energy of \ce{H2O} as a function of $\mu$ for various short-range density functionals and basis sets,
 one can clearly observe that relying on FCI trial
-wave functions ($\mu \to \infty$) give FN-DMC energies lower
+wave functions ($\mu = \infty$) 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.
+a lowering 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
 intermediate values of $\mu$, Fig.~\ref{fig:h2o-dmc} shows that
 a smooth behaviour is obtained:
-starting from $\mu=0$ (\textit{i.e.} the KS determinant),
-the FN-DMC error is reduced continuously until it reaches a minimum for an optimal value of $\mu$,
-and then the FN-DMC error raises until it reaches the $\mu=\infty$ limit (\textit{i.e.} the FCI wave function).
-For instance, with respect to the FN-DMC energy of the FCI trial wave function in the double zeta basis set,
-with the optimal value of $\mu$, one can obtain a lowering of the
-FN-DMC energy of $2.6 \pm 0.7$~m\hartree{}, with an optimal value at
-$\mu=1.75$~bohr$^{-1}$.
+starting from $\mu=0$ (\ie, the KS determinant),
+the FN-DMC error is reduced continuously until it reaches a minimum 
+for an optimal value of $\mu$ (which is obviously basis set and functional dependent),
+and then the FN-DMC error raises until it reaches the $\mu=\infty$ limit (\ie, the FCI wave function).
+For instance, with respect to the FN-DMC/VDZ-BFD energy at $\mu=\infty$, 
+one can obtain a lowering of the FN-DMC energy of $2.6 \pm 0.7$~m\hartree{}
+with an optimal value of $\mu=1.75$~bohr$^{-1}$.
 When the basis set is increased, the gain in FN-DMC energy with
 respect to the FCI trial wave function is reduced, and the optimal
 value of $\mu$ is slightly shifted towards large $\mu$.  Eventually, the nodes
-of the wave functions $\Psi^\mu$ obtained with the short-range
-LDA exchange-correlation functional give very similar FN-DMC energies with respect
-to those obtained with the short-range PBE functional, even if the
+of the wave functions $\Psi^\mu$ obtained with the srLDA 
+exchange-correlation functional give very similar FN-DMC energies with respect
+to those obtained with the srPBE functional, even if the
 RS-DFT energies obtained with these two functionals differ by several
 tens of m\hartree{}.
 
+\titou{The key fact here is that, at $\mu=1.75$~bohr$^{-1}$, $\Psi^{\mu}$ has \textit{only} $54\,540$ determinants at the srPBE/VDZ-BFD level, while the FCI wave function contains $200\,521$ determinants (see Table \ref{tab:h2o-dmc}). Even at the srPBE/VTZ-BFD level, we observe a reduction by a factor two in the number of determinants between the optimal $\mu$ value and $\mu = \infty$.
+The take-home message here is that RS-DFT trial wave functions yield a lower fixed-node energy with more compact multideterminant expansion as compared to FCI.}
+
 %======================================================
 \subsection{Link between RS-DFT and Jastrow factors }
 \label{sec:rsdft-j}
@@ -548,9 +552,9 @@ Then, within the same set of determinants we optimize the CI coefficients in the
 a simple one- and two-body Jastrow factor. This gives the CI expansion $\Psi^J$.
 Then, we can easily compare $\Psi^\mu$ and  $\Psi^J$ as they are developed
 on the same Slater determinant basis.
-In figure~\ref{fig:overlap}, we plot the overlaps
+In Fig.~\ref{fig:overlap}, we plot the overlaps
 $\braket{\Psi^J}{\Psi^\mu}$ obtained for water and the fluorine dimer,
-and in figure~\ref{dmc_small} the FN-DMC energy of the wave functions
+and in Fig.~\ref{dmc_small} the FN-DMC energy of the wave functions
 $\Psi^\mu$ together with that of $\Psi^J$.
 
 %%% FIG 3 %%%
@@ -663,7 +667,7 @@ $\mu=0.5$ and $\mu=1$~bohr$^{-1}$, constently with the FN-DMC energies
 and the overlap curve.
 
 These data suggest that the wave functions $\Psi^{0.5 \le \mu \le 1}$ and $\Psi^J$ are close,
-and therefore that the operators that produced these wave functions (\textit{i.e.} $H^\mu[n]$ and $e^{-J}He^J$) contain similar physics.
+and therefore that the operators that produced these wave functions (\ie, $H^\mu[n]$ and $e^{-J}He^J$) contain similar physics.
 Considering the form of $\hat{H}^\mu[n]$ (see Eq.~\eqref{H_mu}),
 one can notice that the differences with respect to the usual Hamiltonian come
 from the non-divergent two-body interaction $\hat{W}_{\text{ee}}^{\text{lr},\mu}$
@@ -694,7 +698,7 @@ As a conclusion of the first part of this study, we can notice that:
 \item numerical experiments (overlap $\braket{\Psi^\mu}{\Psi^J}$, 
   one-body density, on-top pair density, and FN-DMC energy) indicate
   that the RS-DFT scheme essentially plays the role of a simple Jastrow factor,
-  \textit{i.e.} mimicking short-range correlation effects.  The latter
+  \ie, mimicking short-range correlation effects.  The latter
   statement can be qualitatively understood by noticing that both RS-DFT
   and transcorrelated approaches deal with an effective non-divergent
   electron-electron interaction, while keeping the density constant.
@@ -764,7 +768,7 @@ one-electron, two-electron and one-nucleus-two-electron terms.
 The problematic part is the two-electron term, whose simplest form can
 be expressed as
 \begin{equation}
-  J_\text{ee} = \sum_i \sum_{j<i} \frac{a\, r_{ij}}{1 + b\, r_{ij}}.
+  J_\text{ee} = \sum_{i<j} \frac{a\, r_{ij}}{1 + b\, r_{ij}}.
 \end{equation}
 The parameter
 $a$ is determined by cusp conditions, and $b$ is obtained by energy
@@ -997,7 +1001,7 @@ when the trial wave function is expanded on more than a few million
 determinants.
 At the RS-DFT-CIPSI level, we can remark that with the triple-zeta
 basis set the MAE are larger for $\mu=1$~bohr$^{-1}$ than for the
-FCI. For the largest systems, as shown in figure~\ref{fig:g2-ndet}
+FCI. For the largest systems, as shown in Fig.~\ref{fig:g2-ndet}
 there are many systems which did not reach the threshold
 $\EPT<1$~m\hartree{}, and the number of determinants exceeded
 10~million so the calculation stopped. In this regime, there is a
@@ -1024,7 +1028,7 @@ $\mu=0.5$~bohr$^{-1}$ with the quadruple-zeta basis set.
 Searching for the optimal value of $\mu$ may be too costly, so we have
 computed the MAD, MSE and RMSD for fixed values of $\mu$. The results
 are illustrated in figure~\ref{fig:g2-dmc}. As seen on the figure and
-in table~\ref{tab:mad}, the best choice for a fixed value of $\mu$ is
+in Table~\ref{tab:mad}, the best choice for a fixed value of $\mu$ is
 0.5~bohr$^{-1}$ for all three basis sets. It is the value for which
 the MAE (3.74(35), 2.46(18) and 2.06(35) kcal/mol) and RMSD (4.03(23),
 3.02(06) and 2.74(13)~kcal/mol) are minimal. Note that these values