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Manuscript/rsdft-cipsi-qmc.tex

@@ -157,7 +157,7 @@ fixed-node approximation is much more difficult to control than the
 finite-basis approximation, especially to compute energy differences.
 The conventional approach consists in multiplying the determinantal part of the trial wave
 function by a positive function, the Jastrow factor, which main assignment is to take into
-account the bulk of the dynamical electron correlation and reduce the statistical fluctuation without altering the location of the nodes.
+account the bulk of the dynamical electron correlation and reduce the statistical fluctuations without altering the location of the nodes.
 %electron-electron cusp and the short-range correlation effects. 
 The determinantal part of the trial wave function is then stochastically re-optimized within variational
 Monte Carlo (VMC) in the presence of the Jastrow factor (which can also be simultaneously optimized) and the nodal
@@ -248,7 +248,7 @@ applicable to large systems with a multi-configurational character is
 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.
+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} --- a scheme that we label RS-DFT-CIPSI in the following --- to obtain accurate FN-DMC energies with compact multi-determinant trial wave functions.
 An important take-home message from the present study is that the RS-DFT scheme essentially plays the role of a simple Jastrow factor by mimicking short-range correlation effects.
 Thanks to this, RS-DFT-CIPSI multi-determinant trial wave functions yield lower fixed-node energies with more compact multi-determinant expansion than CIPSI, especially for small basis sets, and can be produced in a completely deterministic and systematic way, without the burden of the stochastic optimization.
@@ -275,7 +275,7 @@ multi-determinant wave function one can wish for --- in a given basis set --- is
 FCI is the ultimate goal of post-HF methods, and there exists several systematic
 improvements on the path from HF to FCI:
 i) increasing the maximum degree of excitation of CI methods (CISD, CISDT,
-CISDTQ, \ldots), or ii) expanding the size of a complete active space
+CISDTQ,~\ldots), or ii) expanding the size of a complete active space
 (CAS) wave function until all the orbitals are in the active space.
 SCI methods take a shortcut between the HF
 determinant and the FCI wave function by increasing iteratively the
@@ -409,11 +409,11 @@ 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 $10^{-2} \times \tau_1$.
 The convergence of the algorithm was further improved
 by introducing a direct inversion in the iterative subspace (DIIS)
 step to extrapolate the one-electron density both in the outer and inner loops. \cite{Pulay_1980,Pulay_1982}
-Note that any range-separated post-HF method can be
+We emphasize that any range-separated post-HF method can be
 implemented using this scheme by just replacing the CIPSI step by the
 post-HF method of interest.
 Note that, thanks to the self-consistent nature of the algorithm,
@@ -426,13 +426,15 @@ the final trial wave function $\Psi^{\mu}$ is independent of the starting wave f
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
 All reference data (geometries, atomization
-energies, zero-point energy corrections, etc) were taken from the NIST
+energies, zero-point energy, etc) were taken from the NIST
 computational chemistry comparison and benchmark database
 (CCCBDB).\cite{nist}
+In the reference atomization energies, the zero-point vibrational
+energy was removed from the experimental atomization energies.
 
 All calculations have been performed using Burkatzki-Filippi-Dolg (BFD)
 pseudopotentials \cite{Burkatzki_2007,Burkatzki_2008} with the associated double-,
-triple-, and quadruple-$\zeta$ basis sets (VXZ-BFD).
+triple-, and quadruple-$\zeta$ basis sets (V$X$Z-BFD).
 The small-core BFD pseudopotentials include scalar relativistic effects.
 Coupled cluster with singles, doubles, and perturbative triples [CCSD(T)] \cite{Scuseria_1988,Scuseria_1989} and KS-DFT energies have been computed with
 \emph{Gaussian09},\cite{g16} using the unrestricted formalism for open-shell systems.
@@ -459,9 +461,10 @@ calculations. Simple Jastrow factors were used to reduce the
 fluctuations of the local energy (see Sec.~\ref{sec:rsdft-j} for their explicit expression).
 The FN-DMC simulations are performed with all-electron moves using the
 stochastic reconfiguration algorithm developed by Assaraf \textit{et al.},
-\cite{Assaraf_2000} with a time step of $5 \times 10^{-4}$ a.u.
+\cite{Assaraf_2000} with a time step of $5 \times 10^{-4}$ a.u. and a
+projection time of $1$ a.u.
 
-All the data related to the present study (geometries, basis sets, total energies, etc) can be found in the {\SI}.
+All the data related to the present study (geometries, basis sets, total energies, \textit{etc}) can be found in the {\SI}.
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \section{Influence of the range-separation parameter on the fixed-node error}