saving work

Manuscript/rsdft-cipsi-qmc.tex

@@ -249,7 +249,7 @@ still an active field of research. The present paper falls
 within this context.
 
 The central idea of the present work, and the launch-pad for the remainder of this study, is that one can combine the various strengths of WFT, DFT, and QMC in order to create a new hybrid method with more attractive features and higher accuracy.
-In particular, we show here that one can combine CIPSI and KS-DFT via the range separation (RS) of the interelectronic Coulomb operator \cite{Sav-INC-96a,Toulouse_2004} to obtain accurate FN-DMC energies with compact multi-determinant trial wave functions.
+In particular, we show here that one can combine CIPSI and KS-DFT via the range separation (RS) of the interelectronic Coulomb operator \cite{Sav-INC-96a,Toulouse_2004} --- a scheme that we label RS-DFT-CIPSI in the following --- to obtain accurate FN-DMC energies with compact multi-determinant trial wave functions.
 
 
 The present manuscript is organized as follows.
@@ -1029,21 +1029,22 @@ going to the quadruple-$\zeta$ basis.
 This large imbalance at the VDZ-BFD level affects the nodal
 surfaces, because although the FN-DMC energies obtained with near-FCI
 trial wave functions are much lower than the single-determinant FN-DMC
-energies, the MAE obtained with FCI (7.38~$\pm$ 1.08~kcal/mol) is
-larger than the single-determinant MAE (4.61~$\pm$ 0.34 kcal/mol).
+energies, the MAE obtained with FCI ($7.38\pm1.08$ kcal/mol) is
+larger than the single-determinant MAE ($4.61\pm 0.34$ kcal/mol).
 Using the FCI trial wave function the MSE is equal to the
 negative MAE which confirms that all the atomization energies are
 underestimated. This confirms that some of the basis-set
 incompleteness error is transferred in the fixed-node error.
 
-Within the double-$\zeta$ basis set, the calculations could be done for the
+Within the double-$\zeta$ basis set, the calculations could be performed for the
 whole range of values of $\mu$, and the optimal value of $\mu$ for the
 trial wave function was estimated for each system by searching for the
 minimum of the spline interpolation curve of the FN-DMC energy as a
 function of $\mu$.
-This corresponds the the line of the table labelled by the \emph{Opt}
-value of $\mu$. Using the optimal value of $\mu$ clearly improves the
-MAE, the MSE an the RMSD compared the the FCI wave function. This
+This corresponds the line of Table~\ref{tab:mad} labelled as ``Opt.''
+\titou{The optimal $\mu$ value for each system is reported in the \SI.}
+Using the optimal value of $\mu$ clearly improves the
+MAE, the MSE an the RMSD compared to the FCI wave function. This
 result is in line with the common knowledge that re-optimizing
 the determinantal component of the trial wave function in the presence
 of electron correlation reduces the errors due to the basis set incompleteness.
@@ -1051,8 +1052,8 @@ These calculations were done only for the smallest basis set
 because of the expensive computational cost of the QMC calculations
 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
+At the RS-DFT-CIPSI level, one can see that with the VTZ-BFD
+basis the MAEs are larger for $\mu=1$~bohr$^{-1}$ than for the
 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
@@ -1077,13 +1078,13 @@ $\mu=0.5$~bohr$^{-1}$ with the quadruple-$\zeta$ basis set.
 \end{figure*}
 %%% %%% %%% %%%
 
-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 Fig.~\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
+Searching for the optimal value of $\mu$ may be too costly and time consuming, so we have
+computed the MAD, MSE and RMSD for fixed values of $\mu$. 
+As illustrated in Fig.~\ref{fig:g2-dmc} and 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
+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
 are even lower than those obtained with the optimal value of
 $\mu$. Although the FN-DMC energies are higher, the numbers show that
 they are more consistent from one system to another, giving improved
@@ -1106,28 +1107,28 @@ The number of determinants in the trial wave functions are shown in
 Fig.~\ref{fig:g2-ndet}. As expected, the number of determinants
 is smaller when $\mu$ is small and larger when $\mu$ is large.
 It is important to remark that the median of the number of
-determinants when $\mu=0.5$~bohr$^{-1}$ is below 100~000 determinants
-with the quadruple-$\zeta$ basis set, making these calculations feasilble
+determinants when $\mu=0.5$~bohr$^{-1}$ is below $100\,000$ determinants
+with the VQZ-BFD basis, making these calculations feasible
 with such a large basis set. At the double-$\zeta$ level, compared to the
 FCI trial wave functions the median of the number of determinants is
 reduced by more than two orders of magnitude.
 Moreover, going to $\mu=0.25$~bohr$^{-1}$ gives a median close to 100
-determinants at the double-$\zeta$ level, and close to 1~000 determinants
+determinants at the VDZ-BFD level, and close to $1\,000$ determinants
 at the quadruple-$\zeta$ level for only a slight increase of the
 MAE. Hence, RS-DFT-CIPSI trial wave functions with small values of
 $\mu$ could be very useful for large systems to go beyond the
 single-determinant approximation at a very low computational cost
-while keeping the size-consistency.
+while ensuring size-consistency.
 
 Note that when $\mu=0$ the number of determinants is not equal to one because
-we have used the natural orbitals of a first CIPSI calculation, and
+we have used the natural orbitals of a preliminary CIPSI calculation, and
 not the srPBE orbitals.
 So the Kohn-Sham determinant is expressed as a linear combination of
-determinants built with natural orbitals. It is possible to add
-an extra step to the algorithm to compute the natural orbitals from the
-RS-DFT/CIPSI wave function, and re-do the RS-DFT/CIPSI calculation with
+determinants built with NOs. It is possible to add
+an extra step to the algorithm to compute the NOs from the
+RS-DFT-CIPSI wave function, and re-do the RS-DFT-CIPSI calculation with
 these orbitals to get an even more compact expansion. In that case, we would
-have converged to the Kohn-Sham orbitals with $\mu=0$, and the
+have converged to the KS orbitals with $\mu=0$, and the
 solution would have been the PBE single determinant.
 
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