Single graph for MADs
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@ -78,7 +78,7 @@ The Diffusion Monte Carlo (DMC) method is a numerical scheme to obtain
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the exact solution of the Schrödinger equation with an additional
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constraint, imposing the solution to have the same nodal hypersurface
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as a given trial wave function.
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Within this so-called \emph{fixed-node} approximation,
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Within this so-called \emph{fixed-node} approximation,
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the DMC energy associated with a given trial wave function is an upper
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bound to the exact energy, and the latter is recovered only when the
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nodes of the trial wave function coincide with the nodes of the exact
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@ -94,16 +94,18 @@ approximation.
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The qualitative picture of the electronic structure of weakly
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correlated systems, such as organic molecules near their equilibrium
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geometry, is usually well represented with a single Slater
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determinant.
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determinant. This feature is in part responsible for the success of
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density functional theory (DFT) and coupled cluster.
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DMC with a single-determinant trial wave function can be used as a
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single-reference post-Hatree-Fock method, with an accuracy comparable
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to coupled cluster.\cite{Dubecky_2014,Grossman_2002}
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The favorable scaling of QMC and its adequation with massively
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parallel architectures makes it a favourable alternative for large
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systems.
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The Slater determinant is determined by the nature of the molecular
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orbitals. Three main options are commonly used: Hartree-Fock (HF),
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Kohn-Sham (KS) or natural orbitals (NO) of a correlated wave function.
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The favorable scaling of QMC, its very low memory requirements and
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its adequation with massively parallel architectures make it a
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serious alternative for high-accuracy simulations on large systems.
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Different molecular orbitals can be chosen to build the single Slater
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determinant. Three main options are commonly chosen: Hartree-Fock (HF),
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Kohn-Sham (KS) or natural (NO) orbitals of a correlated wave function.
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The nodal surfaces obtained with the KS determinant are in general
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better than those obtained with the HF determinant,\cite{Per_2012} and
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of comparable quality to those obtained with a Slater determinant
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@ -125,19 +127,30 @@ Another approach consists in considering the DMC method as a
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\emph{post-FCI method}. The trial wave function is obtained by
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approaching the FCI with a selected configuration interaction
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method such as CIPSI for instance.\cite{Giner_2013,Caffarel_2016_2}
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When the basis set is increased, the trial wave function tends to the
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exact wave function, so the nodal surface can be systematically
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When the basis set is increased, the trial wave function gets closer
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to the exact wave function, so the nodal surface can be systematically
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improved.\cite{Caffarel_2016}
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This technique has the advantage that using FCI nodes in a given basis
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set is well defined, so the calculations are reproducible in a
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black-box way without needing any expertise in QMC.
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But this technique can't be applied to large systems because of the
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exponential scaling of the size of the wave function.
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Extrapolation techniques have been used to estimate the DMC energy of
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FCI wave functions in a large basis sets,\cite{Scemama_2018} and other
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authors have used a combination of the two approaches where CIPSI
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trial wave functions are re-optimized under the presence of a Jastrow
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factor.\cite{Giner_2016,Dash_2018,Dash_2019}
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exponential scaling of the size of the trial wave function.
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Extrapolation techniques have been used to estimate the DMC energies
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obtained with FCI wave functions,\cite{Scemama_2018} and other authors
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have used a combination of the two approaches where highly truncated
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CIPSI trial wave functions are re-optimized under the presence of a
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Jastrow factor to keep the number of determinants
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small,\cite{Giner_2016} and where the consistency between the
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different wave functions is kept by imposing a constant energy
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difference between the estimated FCI energy and the variational energy
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of the CI wave function.\cite{Dash_2018,Dash_2019}
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Nevertheless, finding a robust protocol to obtain high accuracy
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calculations which can be reproduced systematically, and which is
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applicable for large systems with a multi-configurational character is
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still an active field of research. The present paper falls
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within this context.
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\begin{enumerate}
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\item Total energies and nodal quality:
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@ -401,7 +414,7 @@ implemented.\cite{GinPraFerAssSavTou-JCP-18}
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\begin{figure}
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\centering
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\includegraphics[width=\columnwidth]{h2o-dmc.pdf}
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\caption{Fixed-node energies of the water molecule for different
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\caption{Fixed-node energies of the water molecule for different
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values of $\mu$.}
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\label{fig:h2o-dmc}
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\end{figure}
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@ -409,7 +422,7 @@ implemented.\cite{GinPraFerAssSavTou-JCP-18}
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\begin{figure}
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\centering
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\includegraphics[width=\columnwidth]{f2-dmc.pdf}
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\caption{Fixed-node energies of difluorine for different
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\caption{Fixed-node energies of difluorine for different
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values of $\mu$.}
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\label{fig:f2-dmc}
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\end{figure}
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@ -567,64 +580,56 @@ theory\cite{Pople_1989,Curtiss_1990} were chosen to test the quality
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of the RSDFT-CIPSI trial wave functions for energy differences.
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\begin{squeezetable}
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\begin{table}
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%\begin{squeezetable}
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\begin{table*}
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\caption{Mean absolute deviations obtained with the different
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methods and basis sets.}
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\label{tab:mad}
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\begin{ruledtabular}
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\begin{tabular}{lrrrr}
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Method & \(\mu\) & VDZ-BFD & VTZ-BFD & VQZ-BFD\\
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\hline
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PBE & 0 & 5.02 & 4.57 & 5.31\\
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BLYP & 0 & 9.53 & 5.58 & 5.86\\
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PBE0 & 0 & 11.20 & 6.40 & 6.28\\
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B3LYP & 0 & 11.27 & 7.27 & 6.75\\
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\hline
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CCSD(T) & \(\infty\) & 24.10 & 9.11 & 4.52\\
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\hline
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RSDFT-CIPSI & 0 & 10.08 & 6.31 & 6.35\\
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& 1/4 & 5.55 & 4.58 & 5.48\\
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& 1/2 & 13.42 & 6.77 & 6.35\\
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& 1 & 17.07 & 9.06 & \\
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& 2 & 19.20 & & \\
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& 5 & 22.93 & & \\
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& \(\infty\) & 23.62 & & \\
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\hline
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DMC@RSDFT-CIPSI & 0 & 5.07 \(\pm\) 0.44 & 3.52 \(\pm\) 0.19 & 3.16 \(\pm\) 0.26\\
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& 1/4 & 4.04 \(\pm\) 0.37 & 3.39 \(\pm\) 0.77 & 2.90 \(\pm\) 0.25\\
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& 1/2 & 3.74 \(\pm\) 0.35 & 2.46 \(\pm\) 0.18 & 2.06 \(\pm\) 0.35\\
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& 1 & 5.42 \(\pm\) 0.29 & 4.38 \(\pm\) 0.94 & \\
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& 2 & 5.98 \(\pm\) 0.83 & & \\
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& 5 & 6.18 \(\pm\) 0.84 & & \\
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& \(\infty\) & 7.38 \(\pm\) 1.08 & & \\
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& Opt. & 5.84 \(\pm\) 1.75 & & \\
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\begin{tabular}{ll rrr rrr rrr}
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Method & \(\mu\) & \phantom{} & VDZ-BFD & \phantom{} & \phantom{} & VTZ-BFD & \phantom{} & \phantom{} & VTZ-BFD & \phantom{} \\
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\phantom{} & \phantom{} & MAD & MUD & SD & MAD & MUD & SD & MAD & MUD & SD \\
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\hline
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PBE & 0 & 5.02 & & & 4.57 & & & 5.31 & & \\
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BLYP & 0 & 9.53 & & & 5.58 & & & 5.86 & & \\
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PBE0 & 0 & 11.20 & & & 6.40 & & & 6.28 & & \\
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B3LYP & 0 & 11.27 & & & 7.27 & & & 6.75 & & \\
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\hline
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CCSD(T) & \(\infty\) & 24.10 & & & 9.11 & & & 4.52 & & \\
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\hline
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RSDFT-CIPSI & 0 & 10.08 & & & 6.31 & & & 6.35 & & \\
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\phantom{} & 1/4 & 5.55 & & & 4.58 & & & 5.48 & & \\
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\phantom{} & 1/2 & 13.42 & & & 6.77 & & & 6.35 & & \\
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\phantom{} & 1 & 17.07 & & & 9.06 & & & \phantom{} & & \\
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\phantom{} & 2 & 19.20 & & & \phantom{} & & & \phantom{} & & \\
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\phantom{} & 5 & 22.93 & & & \phantom{} & & & \phantom{} & & \\
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\phantom{} & \(\infty\) & 23.62 & & & \phantom{} & & & \phantom{} & & \\
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\hline
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DMC@RSDFT-CIPSI & 0 & 5.07\(\pm\)0.44 & & & 3.52\(\pm\)0.19 & & & 3.16\(\pm\)0.26 & & \\
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\phantom{} & 1/4 & 4.04\(\pm\)0.37 & & & 3.39\(\pm\)0.77 & & & 2.90\(\pm\)0.25 & & \\
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\phantom{} & 1/2 & 3.74\(\pm\)0.35 & & & 2.46\(\pm\)0.18 & & & 2.06\(\pm\)0.35 & & \\
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\phantom{} & 1 & 5.42\(\pm\)0.29 & & & 4.38\(\pm\)0.94 & & & \phantom{} & & \\
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\phantom{} & 2 & 5.98\(\pm\)0.83 & & & \phantom{} & & & \phantom{} & & \\
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\phantom{} & 5 & 6.18\(\pm\)0.84 & & & \phantom{} & & & \phantom{} & & \\
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\phantom{} & \(\infty\) & 7.38\(\pm\)1.08 & & & \phantom{} & & & \phantom{} & & \\
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\phantom{} & Opt. & 5.84\(\pm\)1.75 & & & \phantom{} & & & \phantom{} & & \\
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\end{tabular}
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\end{ruledtabular}
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\end{table}
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\end{squeezetable}
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\end{table*}
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%\end{squeezetable}
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\begin{figure}
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\centering
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\includegraphics[width=\columnwidth]{g2-dmc-dz.pdf}
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\caption{Histogram of the errors in atomization energies with the double-zeta basis set.}
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\includegraphics[width=\columnwidth]{g2-dmc.pdf}
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\caption{Errors in the DMC atomization energies with the different
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trial wave functions. Each dot corresponds to an atomization
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energy.
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The boxes contain the data between first and third quartiles, and
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the line in the box represents the median. The outliers are shown
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with a cross.}
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\label{fig:g2-dmc-dz}
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\end{figure}
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\begin{figure}
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\centering
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\includegraphics[width=\columnwidth]{g2-dmc-tz.pdf}
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\caption{Histogram of the errors in atomization energies with the triple-zeta basis set.}
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\label{fig:g2-dmc-tz}
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\end{figure}
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\begin{figure}
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\centering
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\includegraphics[width=\columnwidth]{g2-dmc-qz.pdf}
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\caption{Histogram of the errors in atomization energies with the quadruple-zeta basis set.}
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\label{fig:g2-dmc-qz}
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\end{figure}
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