Reference and error bars in FCI
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commit
a3298e82ec
@ -935,3 +935,19 @@
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Url = {https://arxiv.org/abs/1812.06902v1},
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Year = {2018},
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Bdsk-Url-1 = {https://arxiv.org/abs/1812.06902v1}}
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@software{qp2_2020,
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author = {Anthony Scemama and
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Emmanuel Giner and
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Anouar Benali and
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Thomas Applencourt and
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Kevin Gasperich},
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title = {QuantumPackage/qp2: Version 2.1.2},
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month = feb,
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year = 2020,
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publisher = {Zenodo},
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version = {2.1.2},
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doi = {10.5281/zenodo.3677565},
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url = {https://doi.org/10.5281/zenodo.3677565}
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}
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@ -13,10 +13,19 @@
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]{hyperref}
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\urlstyle{same}
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\newcommand{\ie}{\textit{i.e.}}
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\newcommand{\eg}{\textit{e.g.}}
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\newcommand{\alert}[1]{\textcolor{red}{#1}}
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\newcommand{\eg}[1]{\textcolor{blue}{#1}}
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\definecolor{darkgreen}{HTML}{009900}
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\usepackage[normalem]{ulem}
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\newcommand{\titou}[1]{\textcolor{red}{#1}}
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\newcommand{\trashPFL}[1]{\textcolor{red}{\sout{#1}}}
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\newcommand{\PFL}[1]{\titou{(\underline{\bf PFL}: #1)}}
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\newcommand{\mc}{\multicolumn}
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\newcommand{\fnm}{\footnotemark}
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\newcommand{\fnt}{\footnotetext}
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\newcommand{\tabc}[1]{\multicolumn{1}{c}{#1}}
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\newcommand{\EPT}{E_{\text{PT2}}}
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\newcommand{\EDMC}{E_{\text{FN-DMC}}}
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@ -93,20 +102,20 @@ sets because the constraints imposed by the FN approximation
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are less severe than the constraints imposed by the finite-basis
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approximation.
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\alert{However, it is usually harder to control the FN error in DMC, and this might affect energy differences such as atomization energies.
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\titou{However, it is usually harder to control the FN error in DMC, and this might affect energy differences such as atomization energies.
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Moreover, improving systematically the nodal surface of the trial wave function can be a tricky job as there is no variational principle for the nodes.}
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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. 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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density-functional theory (DFT) and coupled cluster theory.
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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, its very low memory requirements and
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its adequacy with massively parallel architectures make it a
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serious alternative for high-accuracy simulations on large systems.
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serious alternative for high-accuracy simulations of large systems.
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As it is not possible to minimize directly the FN-DMC energy with respect
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to the variational parameters of the trial wave function, the
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@ -115,7 +124,7 @@ finite-basis approximation.
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The conventional approach consists in multiplying the trial wave
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function by a positive function, the \emph{Jastrow factor}, taking
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account of the electron-electron cusp and the short-range correlation
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effects. The wave function is then re-optimized within Variational
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effects. The wave function is then re-optimized within variational
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Monte Carlo (VMC) in the presence of the Jastrow factor and the nodal
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surface is expected to be improved. Using this technique, it has been
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shown that the chemical accuracy could be reached within
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@ -125,13 +134,13 @@ Another approach consists in considering the FN-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 (sCI)
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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 gets closer
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\titou{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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improved.\cite{Caffarel_2016} WRONG}
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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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But this technique cannot be applied to large systems because of the
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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 FN-DMC energies
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obtained with FCI wave functions,\cite{Scemama_2018} and other authors
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@ -339,9 +348,9 @@ post-HF method of interest.
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All the calculations were made using BFD
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pseudopotentials\cite{Burkatzki_2008} with the associated double,
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triple and quadruple zeta basis sets (BFD-V$n$Z).
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CCSD(T) and DFT calculations were made with
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pseudopotentials\cite{Burkatzki_2008} with the associated double-,
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triple-, and quadruple-$\zeta$ basis sets (BFD-VXZ).
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CCSD(T) and KS-DFT calculations were made with
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\emph{Gaussian09},\cite{g16} using an unrestricted Hartree-Fock
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determinant as a reference for open-shell systems.
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@ -361,7 +370,7 @@ in the determinant localization approximation (DLA),\cite{Zen_2019}
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where only the determinantal component of the trial wave
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function is present in the expression of the wave function on which
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the pseudopotential is localized. Hence, in the DLA the fixed-node
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energy is independent of the Jatrow factor, as in all-electron
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energy is independent of the Jastrow factor, as in all-electron
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calculations. Simple Jastrow factors were used to reduce the
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fluctuations of the local energy.
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@ -372,36 +381,37 @@ fluctuations of the local energy.
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error}
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\label{sec:mu-dmc}
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\begin{table}
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\caption{Fixed-node energies and number of determinants in the water
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molecule and the fluorine dimer with different trial wave functions.}
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\caption{Fixed-node energies (in hartree) and number of determinants in \ce{H2O} and \ce{F2} with various trial wave functions.}
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\label{tab:h2o-dmc}
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\centering
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\begin{ruledtabular}
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\begin{tabular}{crlrl}
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& \multicolumn{2}{c}{BFD-VDZ} & \multicolumn{2}{c}{BFD-VTZ} \\
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$\mu$ & $\Ndet$ & $\EDMC$ & $\Ndet$ & $\EDMC$ \\
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\begin{tabular}{ccrlrl}
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& & \multicolumn{2}{c}{BFD-VDZ} & \multicolumn{2}{c}{BFD-VTZ} \\
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\cline{3-4} \cline{5-6}
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System & $\mu$ & $\Ndet$ & $\EDMC$ & $\Ndet$ & $\EDMC$ \\
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\hline
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& \multicolumn{4}{c}{H$_2$O} \\
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$0.00$ & $11$ & $-17.253\,59(6)$ & $23$ & $-17.256\,74(7)$ \\
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$0.20$ & $23$ & $-17.253\,73(7)$ & $23$ & $-17.256\,73(8)$ \\
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$0.30$ & $53$ & $-17.253\,4(2)$ & $219$ & $-17.253\,7(5)$ \\
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$0.50$ & $1\,442$ & $-17.253\,9(2)$ & $16\,99$ & $-17.257\,7(2)$ \\
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$0.75$ & $3\,213$ & $-17.255\,1(2)$ & $13\,362$ & $-17.258\,4(3)$ \\
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$1.00$ & $6\,743$ & $-17.256\,6(2)$ & $256\,73$ & $-17.261\,0(2)$ \\
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$1.75$ & $54\,540$ & $-17.259\,5(3)$ & $207\,475$ & $-17.263\,5(2)$ \\
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$2.50$ & $51\,691$ & $-17.259\,4(3)$ & $858\,123$ & $-17.264\,3(3)$ \\
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$3.80$ & $103\,059$ & $-17.258\,7(3)$ & $1\,621\,513$ & $-17.263\,7(3)$ \\
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$5.70$ & $102\,599$ & $-17.257\,7(3)$ & $1\,629\,655$ & $-17.263\,2(3)$ \\
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$8.50$ & $101\,803$ & $-17.257\,3(3)$ & $1\,643\,301$ & $-17.263\,3(4)$ \\
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$\infty$ & $200\,521$ & $-17.256\,8(6)$ & $1\,631\,982$ & $-17.263\,9(3)$ \\
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& \multicolumn{3}{c}{F$_2$} \\
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$0.00$ & $23$ & $-48.419\,5(4)$ \\
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$0.25$ & $8$ & $-48.421\,9(4)$ \\
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$0.50$ & $1743$ & $-48.424\,8(8)$ \\
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$1.00$ & $11952$ & $-48.432\,4(3)$ \\
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$2.00$ & $829438$ & $-48.441\,0(7)$ \\
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$5.00$ & $5326459$ & $-48.445(2)$ \\
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$\infty$ & $8302442$ & $-48.437(3)$ \\
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\ce{H2O}
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& $0.00$ & $11$ & $-17.253\,59(6)$ & $23$ & $-17.256\,74(7)$ \\
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& $0.20$ & $23$ & $-17.253\,73(7)$ & $23$ & $-17.256\,73(8)$ \\
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& $0.30$ & $53$ & $-17.253\,4(2)$ & $219$ & $-17.253\,7(5)$ \\
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& $0.50$ & $1\,442$ & $-17.253\,9(2)$ & $16\,99$ & $-17.257\,7(2)$ \\
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& $0.75$ & $3\,213$ & $-17.255\,1(2)$ & $13\,362$ & $-17.258\,4(3)$ \\
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& $1.00$ & $6\,743$ & $-17.256\,6(2)$ & $256\,73$ & $-17.261\,0(2)$ \\
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& $1.75$ & $54\,540$ & $-17.259\,5(3)$ & $207\,475$ & $-17.263\,5(2)$ \\
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& $2.50$ & $51\,691$ & $-17.259\,4(3)$ & $858\,123$ & $-17.264\,3(3)$ \\
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& $3.80$ & $103\,059$ & $-17.258\,7(3)$ & $1\,621\,513$ & $-17.263\,7(3)$ \\
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& $5.70$ & $102\,599$ & $-17.257\,7(3)$ & $1\,629\,655$ & $-17.263\,2(3)$ \\
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& $8.50$ & $101\,803$ & $-17.257\,3(3)$ & $1\,643\,301$ & $-17.263\,3(4)$ \\
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& $\infty$ & $200\,521$ & $-17.256\,8(6)$ & $1\,631\,982$ & $-17.263\,9(3)$ \\
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\\
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\ce{F2}
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& $0.00$ & $23$ & $-48.419\,5(4)$ \\
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& $0.25$ & $8$ & $-48.421\,9(4)$ \\
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& $0.50$ & $1743$ & $-48.424\,8(8)$ \\
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& $1.00$ & $11952$ & $-48.432\,4(3)$ \\
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& $2.00$ & $829438$ & $-48.441\,0(7)$ \\
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& $5.00$ & $5326459$ & $-48.445(2)$ \\
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& $\infty$ & $8302442$ & $-48.437(3)$ \\
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\end{tabular}
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\end{ruledtabular}
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\end{table}
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@ -515,7 +525,7 @@ the density and of electron correlation, but also reduces the
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imbalance in the quality of the description of the atoms and the
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molecule, leading to more accurate atomization energies.
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\subsection{Size-consistency}
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\subsection{Size consistency}
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An extremely important feature required to get accurate
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atomization energies is size-consistency (or strict separability),
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@ -589,12 +599,12 @@ Ref.~\onlinecite{Scemama_2015}).
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%\begin{squeezetable}
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\begin{table}
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\caption{FN-DMC Energies of the fluorine atom and the dissociated fluorine
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dimer, and size-consistency error.}
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\caption{FN-DMC energies (in hartree) of the fluorine atom and the dissociated fluorine
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dimer, and size-consistency error. \titou{BASIS?}}
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\label{tab:size-cons}
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\begin{ruledtabular}
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\begin{tabular}{cccc}
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$\mu$ & F & Dissociated F$_2$ & Size-consistency error \\
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$\mu$ & \ce{F} & Dissociated \ce{F2} & Size-consistency error \\
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\hline
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0.00 & $-24.188\,7(3)$ & $-48.377\,7(3)$ & $-0.000\,3(4)$ \\
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0.25 & $-24.188\,7(3)$ & $-48.377\,2(4)$ & $+0.000\,2(5)$ \\
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@ -619,7 +629,7 @@ size-consistent FN-DMC energies for all values of $\mu$ (within
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$2\times$ statistical error bars).
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\subsection{Spin-invariance}
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\subsection{Spin invariance}
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Closed-shell molecules often dissociate into open-shell
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fragments. To get reliable atomization energies, it is important to
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@ -644,7 +654,7 @@ So in the context of RS-DFT, the determinantal expansions will be
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impacted by this spurious effect, as opposed to FCI.
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\begin{table}
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\caption{FN-DMC Energies of the triplet carbon atom (BFD-VDZ) with
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\caption{FN-DMC energies (in hartree) of the triplet carbon atom (BFD-VDZ) with
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different values of $m_s$.}
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\label{tab:spin}
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\begin{ruledtabular}
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@ -687,38 +697,39 @@ noticeable with $\mu=5$~bohr$^{-1}$.
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\begin{squeezetable}
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\begin{table*}
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\caption{Mean absolute error (MAE), mean signed errors (MSE) and
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standard deviations (RMSD) obtained with the different methods and
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\caption{Mean absolute errors (MAE), mean signed errors (MSE) and
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standard deviations (RMSD) obtained with various methods and
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basis sets.}
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\label{tab:mad}
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\begin{ruledtabular}
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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{} & VQZ-BFD & \phantom{} \\
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\phantom{} & \phantom{} & MAE & MSE & RMSD & MAE & MSE & RMSD & MAE & MSE & RMSD \\
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\begin{tabular}{ll ddd ddd ddd}
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& & \mc{3}{c}{VDZ-BFD} & \mc{3}{c}{VTZ-BFD} & \mc{3}{c}{VQZ-BFD} \\
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\cline{3-5} \cline{6-8} \cline{9-11}
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Method & $\mu$ & \tabc{MAE} & \tabc{MSE} & \tabc{RMSD} & \tabc{MAE} & \tabc{MSE} & \tabc{RMSD} & \tabc{MAE} & \tabc{MSE} & \tabc{RMSD} \\
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\hline
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PBE & 0 & 5.02 & -3.70 & 6.04 & 4.57 & 1.00 & 5.32 & 5.31 & 0.79 & 6.27 \\
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BLYP & 0 & 9.53 & -9.21 & 7.91 & 5.58 & -4.44 & 5.80 & 5.86 & -4.47 & 6.43 \\
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PBE0 & 0 & 11.20 & -10.98 & 8.68 & 6.40 & -5.78 & 5.49 & 6.28 & -5.65 & 5.08 \\
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B3LYP & 0 & 11.27 & -10.98 & 9.59 & 7.27 & -5.77 & 6.63 & 6.75 & -5.53 & 6.09 \\
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\hline
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\\
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CCSD(T) & \(\infty\) & 24.10 & -23.96 & 13.03 & 9.11 & -9.10 & 5.55 & 4.52 & -4.38 & 3.60 \\
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\hline
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\\
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RS-DFT-CIPSI & 0 & 4.53 & -1.66 & 5.91 & 6.31 & 0.91 & 7.93 & 6.35 & 3.88 & 7.20 \\
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\phantom{} & 1/4 & 5.55 & -4.66 & 5.52 & 4.58 & 1.06 & 5.72 & 5.48 & 1.52 & 6.93 \\
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\phantom{} & 1/2 & 13.42 & -13.27 & 7.36 & 6.77 & -6.71 & 4.56 & 6.35 & -5.89 & 5.18 \\
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\phantom{} & 1 & 17.07 & -16.92 & 9.83 & 9.06 & -9.06 & 5.88 & --- & --- & --- \\
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\phantom{} & 2 & 19.20 & -19.05 & 10.91 & --- & --- & --- & --- & --- & --- \\
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\phantom{} & 5 & 22.93 & -22.79 & 13.24 & --- & --- & --- & --- & --- & --- \\
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\phantom{} & \(\infty\) & 23.63(4) & -23.49(4) & 12.81(4) & 8.43(39) & -8.43(39) & 4.87(7) & 4.51(78) & -4.18(78) & 4.19(20) \\
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\hline
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DMC@RS-DFT-CIPSI & 0 & 4.61(\phantom{0.}34) & -3.62(\phantom{0.}34) & 5.30(\phantom{0.}09) & 3.52(19) & -1.03(19) & 4.39(04) & 3.16(26) & -0.12(26) & 4.12(03) \\
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\phantom{} & 1/4 & 4.04(\phantom{0.}37) & -3.13(\phantom{0.}37) & 4.88(\phantom{0.}10) & 3.39(77) & -0.59(77) & 4.44(34) & 2.90(25) & 0.25(25) & 3.745(5) \\
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\phantom{} & 1/2 & 3.74(\phantom{0.}35) & -3.53(\phantom{0.}35) & 4.03(\phantom{0.}23) & 2.46(18) & -1.72(18) & 3.02(06) & 2.06(35) & -0.44(35) & 2.74(13) \\
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\phantom{} & 1 & 5.42(\phantom{0.}29) & -5.14(\phantom{0.}29) & 4.55(\phantom{0.}03) & 4.38(94) & -4.24(94) & 5.11(31) & --- & --- & --- \\
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\phantom{} & 2 & 5.98(\phantom{0.}83) & -5.91(\phantom{0.}83) & 4.79(\phantom{0.}71) & --- & --- & --- & --- & --- & --- \\
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\phantom{} & 5 & 6.18(\phantom{0.}84) & -6.13(\phantom{0.}84) & 4.87(\phantom{0.}55) & --- & --- & --- & --- & --- & --- \\
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\phantom{} & \(\infty\) & 7.38(1.08) & -7.38(1.08) & 5.67(\phantom{0.}68) & --- & --- & --- & --- & --- & --- \\
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\phantom{} & Opt. & 5.85(1.75) & -5.63(1.75) & 4.79(1.11) & --- & --- & --- & --- & --- & --- \\
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& 1/4 & 5.55 & -4.66 & 5.52 & 4.58 & 1.06 & 5.72 & 5.48 & 1.52 & 6.93 \\
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& 1/2 & 13.42 & -13.27 & 7.36 & 6.77 & -6.71 & 4.56 & 6.35 & -5.89 & 5.18 \\
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& 1 & 17.07 & -16.92 & 9.83 & 9.06 & -9.06 & 5.88 & & & \\
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& 2 & 19.20 & -19.05 & 10.91 & & & & & & \\
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& 5 & 22.93 & -22.79 & 13.24 & & & & & & \\
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& \(\infty\) & 23.63(4) & -23.49(4) & 12.81(4) & 8.43(39) & -8.43(39) & 4.87(7) & 4.51(78) & -4.18(78) & 4.19(20) \\
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\\
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DMC@ & 0 & 4.61(34) & -3.62(34) & 5.30(09) & 3.52(19) & -1.03(19) & 4.39(04) & 3.16(26) & -0.12(26) & 4.12(03) \\
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RS-DFT-CIPSI & 1/4 & 4.04(37) & -3.13(37) & 4.88(10) & 3.39(77) & -0.59(77) & 4.44(34) & 2.90(25) & 0.25(25) & 3.745(5) \\
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& 1/2 & 3.74(35) & -3.53(35) & 4.03(23) & 2.46(18) & -1.72(18) & 3.02(06) & 2.06(35) & -0.44(35) & 2.74(13) \\
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& 1 & 5.42(29) & -5.14(29) & 4.55(03) & 4.38(94) & -4.24(94) & 5.11(31) & & & \\
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& 2 & 5.98(83) & -5.91(83) & 4.79(71) & & & & & & \\
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& 5 & 6.18(84) & -6.13(84) & 4.87(55) & & & & & & \\
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& \(\infty\) & 7.38(1.08) & -7.38(1.08) & 5.67(68) & & & & & & \\
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& Opt. & 5.85(1.75) & -5.63(1.75) & 4.79(1.11) & & & & & & \\
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\end{tabular}
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\end{ruledtabular}
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\end{table*}
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@ -874,7 +885,7 @@ large systems.
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%%---------------------------------------
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%%
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\begin{acknowledgments}
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This work was performed using HPC resources from GENCI-TGCC (Grand
|
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Challenge 2019-gch0418) and from CALMIP (Toulouse) under allocation
|
||||
|
||||
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