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@ 13,27 +13,40 @@


]{hyperref}


\urlstyle{same}




\newcommand{\ie}{\textit{i.e.}}


\newcommand{\eg}{\textit{e.g.}}


\newcommand{\alert}[1]{\textcolor{red}{#1}}


\newcommand{\eg}[1]{\textcolor{blue}{#1}}


\definecolor{darkgreen}{HTML}{009900}


\usepackage[normalem]{ulem}


\newcommand{\toto}[1]{\textcolor{blue}{#1}}


\newcommand{\trashAS}[1]{\textcolor{blue}{\sout{#1}}}


\newcommand{\titou}[1]{\textcolor{red}{#1}}


\newcommand{\trashPFL}[1]{\textcolor{red}{\sout{#1}}}


\newcommand{\PFL}[1]{\titou{(\underline{\bf PFL}: #1)}}




\newcommand{\mc}{\multicolumn}


\newcommand{\fnm}{\footnotemark}


\newcommand{\fnt}{\footnotetext}


\newcommand{\tabc}[1]{\multicolumn{1}{c}{#1}}




\newcommand{\EPT}{E_{\text{PT2}}}


\newcommand{\EDMC}{E_{\text{FNDMC}}}


\newcommand{\Ndet}{N_{\text{det}}}


\newcommand{\hartree}{$E_h$}




\newcommand{\LCT}{Laboratoire de Chimie Théorique (UMR 7616), Sorbonne Université, CNRS, Paris, France}


\newcommand{\LCT}{Laboratoire de Chimie Th\'eorique (UMR 7616), Sorbonne Universit\'e, CNRS, Paris, France}


\newcommand{\ANL}{Argonne Leadership Computing Facility, Argonne National Laboratory, Argonne, IL 60439, United States}


\newcommand{\LCPQ}{Laboratoire de Chimie et Physique Quantiques (UMR 5626), Université de Toulouse, CNRS, UPS, France}


\newcommand{\LCPQ}{Laboratoire de Chimie et Physique Quantiques (UMR 5626), Universit\'e de Toulouse, CNRS, UPS, France}






\begin{document}




\title{Enabling high accuracy diffusion Monte Carlo calculations with


rangeseparated density functional theory and selected configuration interaction}


\title{Taming the fixednode error in diffusion Monte Carlo via range separation}


%\title{Enabling high accuracy diffusion Monte Carlo calculations with


% rangeseparated density functional theory and selected configuration interaction}




\author{Anthony Scemama}


\email{scemama@irsamc.upstlse.fr}


\affiliation{\LCPQ}


\author{Emmanuel Giner}


\email{emmanuel.giner@lct.jussieu.fr}


@ 55,58 +68,64 @@


\section{Introduction}


\label{sec:intro}




The full configuration interaction (FCI) method within a finite atomic


basis set leads to an approximate solution of the Schrödinger


equation.


This solution is the eigenpair of an approximate Hamiltonian, which is


the projection of the exact Hamiltonian onto the finite basis of all


possible Slater determinants.


The FCI wave function can be interpreted as the constrained solution of the


true Hamiltonian, where the solution is forced to span the space


provided by the basis.


At the complete basis set (CBS) limit, the constraint vanishes and the


exact solution is obtained.


Hence the FCI method enables a systematic improvement of the


calculations by improving the quality of the basis set.


Nevertheless, its exponential scaling with the number of electrons and


with the size of the basis is prohibitive for large systems.


In recent years, the introduction of new algorithms\cite{Booth_2009}


and the


revival\cite{Abrams_2005,Bytautas_2009,Roth_2009,Giner_2013,Knowles_2015,Holmes_2016,Liu_2016}


of selected configuration interaction (sCI)


methods\cite{Bender_1969,Huron_1973,Buenker_1974} pushed the limits of


the sizes of the systems that could be computed at the FCI level, but


the scaling remains exponential unless some bias is introduced leading


to a loss of size consistency.


Solving the Schr\"odinger equation for atoms and molecules is a complex task that has kept theoretical and computational chemists busy for almost hundred years now. \cite{Schrodinger_1926}


In order to achieve this formidable endeavour, various strategies have been carefully designed and implemented in quantum chemistry software packages.




The Diffusion Monte Carlo (DMC) method is a numerical scheme to obtain


the exact solution of the Schrödinger equation with an additional


constraint, imposing the solution to have the same nodal hypersurface


as a given trial wave function.


One of this strategies consists in relying on wave function theory and, in particular, on the full configuration interaction (FCI) method.


However, FCI delivers only an approximate solution of the Schr\"odinger equation within a finite oneelectron basis.


This solution is the eigenpair of an approximate Hamiltonian defined as


the projection of the exact Hamiltonian onto the finite manyelectron basis of


all possible Slater determinants generated within this finite oneelectron basis.


The FCI wave function can be interpreted as a constrained solution of the


true Hamiltonian forced to span the restricted space provided by the oneelectron basis.


In the complete basis set (CBS) limit, the constraint is lifted and the


exact solution is recovered.


Hence, the accuracy of a FCI calculation can be systematically improved by increasing the size of the oneelectron basis set.


Nevertheless, its exponential scaling with the number of electrons and with the size of the basis is prohibitive for most chemical systems.


In recent years, the introduction of new algorithms \cite{Booth_2009} and the


revival \cite{Abrams_2005,Bytautas_2009,Roth_2009,Giner_2013,Knowles_2015,Holmes_2016,Liu_2016,Garniron_2018}


of selected configuration interaction (SCI)


methods \cite{Bender_1969,Huron_1973,Buenker_1974} pushed the limits of


the sizes of the systems that could be computed at the FCI level. \cite{Booth_2010,Cleland_2010,Daday_2012,Chien_2018,Loos_2018a,Loos_2019,Loos_2020b,Loos_2020c}


However, the scaling remains exponential unless some bias is introduced leading


to a loss of size consistency. \cite{Evangelisti_1983,Cleland_2010,Tenno_2017}




Diffusion Monte Carlo (DMC) is another numerical scheme to obtain


the exact solution of the Schr\"odinger equation with a different


constraint. In DMC, the solution is imposed to have the same nodes (or zeroes)


as a given trial (approximate) wave function.


Within this socalled \emph{fixednode} (FN) approximation,


the FNDMC energy associated with a given trial wave function is an upper


bound to the exact energy, and the latter is recovered only when the


nodes of the trial wave function coincide with the nodes of the exact


wave function.


The polynomial scaling with the number of electrons and with the size


of the trial wave function makes the FNDMC method attractive.


of the trial wave function makes the FNDMC method particularly attractive.


In addition, the total energies obtained are usually far below


those obtained with the FCI method in computationally tractable basis


sets because the constraints imposed by the fixednode approximation


sets because the constraints imposed by the FN approximation


are less severe than the constraints imposed by the finitebasis


approximation.




%However, it is usually harder to control the FN error in DMC, and this


%might affect energy differences such as atomization energies.


%Moreover, improving systematically the nodal surface of the trial wave


%function can be a tricky job as \trashAS{there is no variational


%principle for the nodes}\toto{the derivatives of the FNDMC energy


%with respect to the variational parameters of the wave function can't


%be computed}.




The qualitative picture of the electronic structure of weakly


correlated systems, such as organic molecules near their equilibrium


geometry, is usually well represented with a single Slater


determinant. This feature is in part responsible for the success of


density functional theory (DFT) and coupled cluster.


densityfunctional theory (DFT) and coupled cluster theory.


DMC with a singledeterminant trial wave function can be used as a


singlereference postHatreeFock method, with an accuracy comparable


to coupled cluster.\cite{Dubecky_2014,Grossman_2002}


The favorable scaling of QMC, its very low memory requirements and


its adequation with massively parallel architectures make it a


serious alternative for highaccuracy simulations on large systems.


its adequacy with massively parallel architectures make it a


serious alternative for highaccuracy simulations of large systems.




As it is not possible to minimize directly the FNDMC energy with respect


to the variational parameters of the trial wave function, the


@ 115,7 +134,7 @@ finitebasis approximation.


The conventional approach consists in multiplying the trial wave


function by a positive function, the \emph{Jastrow factor}, taking


account of the electronelectron cusp and the shortrange correlation


effects. The wave function is then reoptimized within Variational


effects. The wave function is then reoptimized within variational


Monte Carlo (VMC) in the presence of the Jastrow factor and the nodal


surface is expected to be improved. Using this technique, it has been


shown that the chemical accuracy could be reached within


@ 125,13 +144,13 @@ Another approach consists in considering the FNDMC method as a


\emph{postFCI method}. The trial wave function is obtained by


approaching the FCI with a selected configuration interaction (sCI)


method such as CIPSI for instance.\cite{Giner_2013,Caffarel_2016_2}


When the basis set is increased, the trial wave function gets closer


\titou{When the basis set is increased, the trial wave function gets closer


to the exact wave function, so the nodal surface can be systematically


improved.\cite{Caffarel_2016}


improved.\cite{Caffarel_2016} WRONG}


This technique has the advantage that using FCI nodes in a given basis


set is well defined, so the calculations are reproducible in a


blackbox way without needing any expertise in QMC.


But this technique can't be applied to large systems because of the


But this technique cannot be applied to large systems because of the


exponential scaling of the size of the trial wave function.


Extrapolation techniques have been used to estimate the FNDMC energies


obtained with FCI wave functions,\cite{Scemama_2018} and other authors


@ 318,7 +337,7 @@ determinants.




We can follow this path by performing FCI calculations using the


RSDFT Hamiltonian with different values of $\mu$. In this work, we


have used the CIPSI algorithm to peform approximate FCI calculations


have used the CIPSI algorithm to perform approximate FCI calculations


with the RSDFT Hamiltonians,\cite{GinPraFerAssSavTouJCP18}


$\hat{H}^\mu$ as shown in figure~\ref{fig:algo}. In the outer loop


(red), a CIPSI selection is performed with a RSHamiltonian


@ 339,9 +358,9 @@ postHF method of interest.






All the calculations were made using BFD


pseudopotentials\cite{Burkatzki_2008} with the associated double,


triple and quadruple zeta basis sets (BFDV$n$Z).


CCSD(T) and DFT calculations were made with


pseudopotentials\cite{Burkatzki_2008} with the associated double,


triple, and quadruple$\zeta$ basis sets (BFDVXZ).


CCSD(T) and KSDFT calculations were made with


\emph{Gaussian09},\cite{g16} using an unrestricted HartreeFock


determinant as a reference for openshell systems.




@ 361,7 +380,7 @@ in the determinant localization approximation (DLA),\cite{Zen_2019}


where only the determinantal component of the trial wave


function is present in the expression of the wave function on which


the pseudopotential is localized. Hence, in the DLA the fixednode


energy is independent of the Jatrow factor, as in allelectron


energy is independent of the Jastrow factor, as in allelectron


calculations. Simple Jastrow factors were used to reduce the


fluctuations of the local energy.




@ 372,36 +391,37 @@ fluctuations of the local energy.


error}


\label{sec:mudmc}


\begin{table}


\caption{Fixednode energies and number of determinants in the water


molecule and the fluorine dimer with different trial wave functions.}


\caption{Fixednode energies (in hartree) and number of determinants in \ce{H2O} and \ce{F2} with various trial wave functions.}


\label{tab:h2odmc}


\centering


\begin{ruledtabular}


\begin{tabular}{crlrl}


& \multicolumn{2}{c}{BFDVDZ} & \multicolumn{2}{c}{BFDVTZ} \\


$\mu$ & $\Ndet$ & $\EDMC$ & $\Ndet$ & $\EDMC$ \\


\begin{tabular}{ccrlrl}


& & \multicolumn{2}{c}{BFDVDZ} & \multicolumn{2}{c}{BFDVTZ} \\


\cline{34} \cline{56}


System & $\mu$ & $\Ndet$ & $\EDMC$ & $\Ndet$ & $\EDMC$ \\


\hline


& \multicolumn{4}{c}{H$_2$O} \\


$0.00$ & $11$ & $17.253\,59(6)$ & $23$ & $17.256\,74(7)$ \\


$0.20$ & $23$ & $17.253\,73(7)$ & $23$ & $17.256\,73(8)$ \\


$0.30$ & $53$ & $17.253\,4(2)$ & $219$ & $17.253\,7(5)$ \\


$0.50$ & $1\,442$ & $17.253\,9(2)$ & $16\,99$ & $17.257\,7(2)$ \\


$0.75$ & $3\,213$ & $17.255\,1(2)$ & $13\,362$ & $17.258\,4(3)$ \\


$1.00$ & $6\,743$ & $17.256\,6(2)$ & $256\,73$ & $17.261\,0(2)$ \\


$1.75$ & $54\,540$ & $17.259\,5(3)$ & $207\,475$ & $17.263\,5(2)$ \\


$2.50$ & $51\,691$ & $17.259\,4(3)$ & $858\,123$ & $17.264\,3(3)$ \\


$3.80$ & $103\,059$ & $17.258\,7(3)$ & $1\,621\,513$ & $17.263\,7(3)$ \\


$5.70$ & $102\,599$ & $17.257\,7(3)$ & $1\,629\,655$ & $17.263\,2(3)$ \\


$8.50$ & $101\,803$ & $17.257\,3(3)$ & $1\,643\,301$ & $17.263\,3(4)$ \\


$\infty$ & $200\,521$ & $17.256\,8(6)$ & $1\,631\,982$ & $17.263\,9(3)$ \\


& \multicolumn{3}{c}{F$_2$} \\


$0.00$ & $23$ & $48.419\,5(4)$ \\


$0.25$ & $8$ & $48.421\,9(4)$ \\


$0.50$ & $1743$ & $48.424\,8(8)$ \\


$1.00$ & $11952$ & $48.432\,4(3)$ \\


$2.00$ & $829438$ & $48.441\,0(7)$ \\


$5.00$ & $5326459$ & $48.445(2)$ \\


$\infty$ & $8302442$ & $48.437(3)$ \\


\ce{H2O}


& $0.00$ & $11$ & $17.253\,59(6)$ & $23$ & $17.256\,74(7)$ \\


& $0.20$ & $23$ & $17.253\,73(7)$ & $23$ & $17.256\,73(8)$ \\


& $0.30$ & $53$ & $17.253\,4(2)$ & $219$ & $17.253\,7(5)$ \\


& $0.50$ & $1\,442$ & $17.253\,9(2)$ & $16\,99$ & $17.257\,7(2)$ \\


& $0.75$ & $3\,213$ & $17.255\,1(2)$ & $13\,362$ & $17.258\,4(3)$ \\


& $1.00$ & $6\,743$ & $17.256\,6(2)$ & $256\,73$ & $17.261\,0(2)$ \\


& $1.75$ & $54\,540$ & $17.259\,5(3)$ & $207\,475$ & $17.263\,5(2)$ \\


& $2.50$ & $51\,691$ & $17.259\,4(3)$ & $858\,123$ & $17.264\,3(3)$ \\


& $3.80$ & $103\,059$ & $17.258\,7(3)$ & $1\,621\,513$ & $17.263\,7(3)$ \\


& $5.70$ & $102\,599$ & $17.257\,7(3)$ & $1\,629\,655$ & $17.263\,2(3)$ \\


& $8.50$ & $101\,803$ & $17.257\,3(3)$ & $1\,643\,301$ & $17.263\,3(4)$ \\


& $\infty$ & $200\,521$ & $17.256\,8(6)$ & $1\,631\,982$ & $17.263\,9(3)$ \\


\\


\ce{F2}


& $0.00$ & $23$ & $48.419\,5(4)$ \\


& $0.25$ & $8$ & $48.421\,9(4)$ \\


& $0.50$ & $1743$ & $48.424\,8(8)$ \\


& $1.00$ & $11952$ & $48.432\,4(3)$ \\


& $2.00$ & $829438$ & $48.441\,0(7)$ \\


& $5.00$ & $5326459$ & $48.445(2)$ \\


& $\infty$ & $8302442$ & $48.437(3)$ \\


\end{tabular}


\end{ruledtabular}


\end{table}


@ 410,7 +430,8 @@ $\infty$ & $8302442$ & $48.437(3)$ \\


\centering


\includegraphics[width=\columnwidth]{h2odmc.pdf}


\caption{Fixednode energies of the water molecule for different


values of $\mu$.}


values of $\mu$, using the srLDA or srPBE shortrange density


functionals to build the trial wave function.}


\label{fig:h2odmc}


\end{figure}




@ 423,7 +444,7 @@ $\infty$ & $8302442$ & $48.437(3)$ \\


\end{figure}


The first question we would like to address is the quality of the


nodes of the wave functions $\Psi^{\mu}$ obtained with an intermediate


range separation parameter $\mu$ (\textit{i.e.} $0 < \mu < +\infty$).


range separation parameter (\textit{i.e.} $0 < \mu < +\infty$).


We generated trial wave functions $\Psi^\mu$ with multiple values of


$\mu$, and computed the associated fixed node energy keeping all the


parameters having an impact on the nodal surface fixed.


@ 433,80 +454,76 @@ geometry\cite{Caffarel_2016}.




From table~\ref{tab:h2odmc} and figures~\ref{fig:h2odmc}


and~\ref{fig:f2dmc}, one can clearly observe that using FCI trial


wave functions gives FNDMC energies which are lower than the energies


obtained with a single KohnSham determinant:


wave functions ($\mu = \infty$) gives FNDMC energies which are lower


than the energies obtained with a single KohnSham determinant ($\mu=0$):


a gain of $3.2 \pm 0.6$~m\hartree{} at the doublezeta level and $7.2 \pm


0.3$~m\hartree{} at the triplezeta level are obtained for water, and


a gain of $18 \pm 3$~m\hartree{} for F$_2$. Interestingly, with the


doublezeta basis one can obtain for water a FNDMC energy $2.6 \pm


0.7$~m\hartree{} lower than the energy obtained with the FCI


trial wave function, using the RSDFTCIPSI wave function $\Psi^{\mu}$ with a rangeseparation


parameter of $\mu=1.75$~bohr$^{1}$. This can be explained by the


inability of the basis set to properly describe the shortrange


correlation effects, shifting the nodes from their optimal


position. Using DFT to take account of shortrange correlation frees


the determinant expansion from describing shortrange effects, and


enables a placement of the nodes closer to the optimum. In the case


of F$_2$, a similar behavior with a gain of $8 \pm 4$ m\hartree{} is


observed for $\mu\sim 5$~bohr$^{1}$.


The optimal value of $\mu$ is larger than in the case of water. This


is probably the signature of the fact that the average


electronelectron distance in the valence is smaller in F$_2$ than in


H$_2$O due to the larger nuclear charge shrinking the electron


density. At the triplezeta level, the shortrange correlations can be


better described by the determinant expansion, and the improvement due


to DFT is insignificant. However, it is important to note that the


same FNDMC energy can be obtained with a CI expansion which is eight


times smaller when srDFT is introduced. One can also remark that the


minimum has been slightly shifted towards the FCI, which is consistent


with the fact that in the CBS limit we expect the minimum of the


FNDMC energy to be obtained for the FCI wave function, i.e. at


$\mu=\infty$.


a gain of $18 \pm 3$~m\hartree{} for F$_2$.


Coming now to the nodes of the trial wave functions $\Psi^{\mu}$ with intermediate values of $\mu$,


the figures~\ref{fig:h2odmc} show that a smooth behaviour is obtained:


starting from $\mu=0$ (\textit{i.e.} the KS determinant),


the FNDMC error is reduced continuously until it reaches a minimum for an optimal value of $\mu$,


and then the FNDMC error raises until it reaches the $\mu=\infty$ limit (\textit{i.e.} the FCI wave function).


For instance, with respect to the FNDMC 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 FNDMC energy of $2.6 \pm 0.7$~m\hartree{}


and $8 \pm 3$~m\hartree{} for the water and difluorine molecules, respectively.


The optimal value of $\mu$ is $\mu=1.75$~bohr$^{1}$ and $\mu=5$~bohr$^{1}$ for the water and fluorine dimer, respectively.


When the basis set is increased, the gain in FNDMC energy with respect to the FCI trial wave function is reduced, and the optimal value of $\mu$ is shifted towards large $\mu$.


Eventually, the nodes of the wave functions $\Psi^\mu$ obtained with shortrange


LDA exchangecorrelation functional give very similar FNDMC energy with respect


to those obtained with the shortrange PBE functional, even if the RSDFT energies obtained


with these two functionals differ by several tens of m\hartree{}.


These observations call some important comments for the present study.




\begin{figure}


\centering


\includegraphics[width=\columnwidth]{overlap.pdf}


\caption{Overlap of the RSDFT CI expansion with the


\caption{Overlap of the RSDFT CI expansion with the


CI expansion optimized in the presence of a Jastrow factor.}


\label{fig:overlap}


\end{figure}




This data confirms that RSDFTCIPSI can give improved CI coefficients


This data confirms that RSDFTCIPSI can give improved CI coefficients


with small basis sets, similarly to the common practice of


reoptimizing the wave function in the presence of the Jastrow


factor. To confirm that the introduction of RSDFT has the same impact


that the Jastrow factor on the CI coefficients, we have made the following


numerical experiment. First, we extract the 200 determinants with the


largest weights in the FCI wave function out of a large CIPSI calculation.


Within this set of determinants, we diagonalize selfconsistently the


RSDFT Hamiltonian with different values of $\mu$. This gives the CI


expansions $\Psi^\mu$. Then, within the same set of determinants we


optimize the CI coefficients in the presence of a simple one and


twobody Jastrow factor. This gives the CI expansion $\Psi^J$.


In figure~\ref{fig:overlap}, we plot the overlaps


reoptimizing the trial wave function in the presence of the Jastrow


factor.


To confirm that the introduction of srDFT has an impact on


the CI coefficients similar to the Jastrow factor, we have made the


following numerical experiment. First, we extract the 200 determinants


with the largest weights in the FCI wave function out of a large CIPSI


calculation. Within this set of determinants, we diagonalize


selfconsistently the RSDFT Hamiltonian with different values of


$\mu$. This gives the CI expansions $\Psi^\mu$. Then, within the same


set of determinants we optimize the CI coefficients in the presence of


a simple one and twobody Jastrow factor. This gives the CI expansion


$\Psi^J$. In figure~\ref{fig:overlap}, we plot the overlaps


$\braket{\Psi^J}{\Psi^\mu}$ obtained for water and the fluorine dimer.




In the case of H$_2$O, there is a clear maximum of overlap at


$\mu=1$~bohr$^{1}$. This confirms that introducing shortrange


correlation with DFT has the same impact on the CI coefficients than


with the Jastrow factor. In the case of F$_2$, the Jastrow factor has


correlation with DFT has the an impact on the CI coefficients similar to


the Jastrow factor. In the case of F$_2$, the Jastrow factor has


very little effect on the CI coefficients, as the overlap


$\braket{\Psi^J}{\Psi^{\mu=\infty}}$ is very close to


$1$.


Nevertheless, a slight maximum is obtained for


$\mu=5$~bohr$^{1}$.


These data suggest that the RSDFT effective Hamiltonian is somehow similar


to the effective Hamiltonian obtained with a usual JastrowSlater


optimization and a one and twobody Jastrow factor.


These data suggest that the eigenfunctions of $H^\mu$ and that of the effective


Hamiltonian obtained with a simple one and twobody jastrow factor are similar, and therefore that the operators


themselves contain similar physics.


Considering the form of $\hat{H}^\mu[n_{\Psi^{\mu}}]$ (see Eq.~\eqref{H_mu}),


one can notice that the differences with respect to the usual Hamiltonian come


from the nondivergent twobody interaction $\hat{W}_{\mathrm{ee}}^{\mathrm{lr},\mu}$


and the effective onebody potential $\hat{\bar{V}}_{\mathrm{Hxc}}^{\mathrm{sr},\mu}[n_{\Psi^{\mu}}]$ which is the functional derivative of the Hartreeexchangecorrelation functional.


The role of these two terms are therefore very different: the non divergent two body interaction tend to make electrons closer at shortrange distance while the effective onebody potential, provided that it is exact, maintains the exact onebody density. Considering that


The role of these two terms are therefore very different: with respect


to the exact ground state wave function $\Psi$, the non divergent two body interaction


increases the probability to find electrons at short distances in $\Psi^\mu$,


while the effective onebody potential $\hat{\bar{V}}_{\mathrm{Hxc}}^{\mathrm{sr},\mu}[n_{\Psi^{\mu}}]$,


provided that it is exact, maintains the exact onebody density.


As pointed out by TenNo in the context of transcorrelated approaches\cite{Tenno2000Nov},


the effective twobody interaction induced by the presence of a Jastrow factor


can be nondivergent when a proper Jastrow factor is chosen.


Therefore, it is


Therefore, one can understand the similarity between the eigenfunctions of $H^\mu$ and the explicitely correlated parts.






As a conclusion of the first part of this study, we can notice that:


@ 518,6 +535,8 @@ on the system and the basis set, and the larger the basis set, the larger the op


iii) numerical experiments (such as computation of overlap) indicates


that the RSDFT scheme essentially plays the role of a simple Jastrow factor,


\textit{i.e.} mimicking shortrange correlation effects.


Nevertheless, a slight maximum is obtained for


$\mu=5$~bohr$^{1}$.






\section{Atomization energies}


@ 531,14 +550,14 @@ atoms than molecules and atomization energies usually tend to be


underestimated with variational methods.


In the context of FNDMC calculations, the nodal surface is imposed by


the trial wavefunction which is expanded on an atomcentered basis


set. So we expect the fixednode error to be also tightly related to


set, so we expect the fixednode error to be also tightly related to


the basis set incompleteness error.


Increasing the size of the basis set improves the description of


the density and of electron correlation, but also reduces the


imbalance in the quality of the description of the atoms and the


molecule, leading to more accurate atomization energies.




\subsection{Sizeconsistency}


\subsection{Size consistency}




An extremely important feature required to get accurate


atomization energies is sizeconsistency (or strict separability),


@ 579,7 +598,7 @@ oneelectron, twoelectron and onenucleustwoelectron terms.


The problematic part is the twoelectron 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 \sum_{j<i} \frac{a\, r_{ij}}{1 + b\, r_{ij}}.


\end{equation}


The parameter


$a$ is determined by cusp conditions, and $b$ is obtained by energy


@ 610,104 +629,216 @@ are computed analytically and the computational cost of the


pseudopotential is dramatically reduced (for more detail, see


Ref.~\onlinecite{Scemama_2015}).




In this section, we make a numerical verification that the produced


wave functions are sizeconsistent. We have computed the energy of the


dissocited fluorine dimer, where the two atoms are at a distance of 50~\AA.


We expect that the energy of this system is equal to twice the energy


of the fluorine atom.




%\begin{squeezetable}


\begin{table}


\caption{FNDMC Energies of the fluorine atom and the dissociated fluorine


dimer, and sizeconsistency error.}


\caption{FNDMC energies (in hartree) using the VDZBFD basis set


and pseudopotential of the fluorine atom and the dissociated fluorine


dimer, and sizeconsistency error. }


\label{tab:sizecons}


\begin{ruledtabular}


\begin{tabular}{clll}


$\mu$ & F & Dissociated F$_2$ & Sizeconsistency error \\


\begin{tabular}{cccc}


$\mu$ & \ce{F} & Dissociated \ce{F2} & Sizeconsistency error \\


\hline


0.00 & & & \\


0.25 & & & \\


0.50 & & & \\


1.00 & & & \\


2.00 & & & \\


5.00 & & & \\


$\infty$ & & & \\


0.00 & $24.188\,7(3)$ & $48.377\,7(3)$ & $0.000\,3(4)$ \\


0.25 & $24.188\,7(3)$ & $48.377\,2(4)$ & $+0.000\,2(5)$ \\


0.50 & $24.188\,8(1)$ & $48.376\,9(4)$ & $+0.000\,7(4)$ \\


1.00 & $24.189\,7(1)$ & $48.380\,2(4)$ & $0.000\,8(4)$ \\


2.00 & $24.194\,1(3)$ & $48.388\,4(4)$ & $0.000\,2(5)$ \\


5.00 & $24.194\,7(4)$ & $48.388\,5(7)$ & $+0.000\,9(8)$ \\


$\infty$ & $24.193\,5(2)$ & $48.386\,9(4)$ & $+0.000\,1(5)$ \\


\end{tabular}


\end{ruledtabular}


\end{table}




In this section, we make a numerical verification that the produced


wave functions are sizeconsistent for a given rangeseparation


parameter.


We have computed the energy of the dissociated fluorine dimer, where


the two atoms are at a distance of 50~\AA. We expect that the energy


of this system is equal to twice the energy of the fluorine atom.


The data in table~\ref{tab:sizecons} shows that it is indeed the


case, so we can conclude that the proposed scheme provides


sizeconsistent FNDMC energies for all values of $\mu$ (within


$2\times$ statistical error bars).




\subsection{Spininvariance}




Closedshell molecules usually dissociate into openshell


\subsection{Spin invariance}




Closedshell molecules often dissociate into openshell


fragments. To get reliable atomization energies, it is important to


have a theory which is of comparable quality for openshell and


closedshell systems.


FCI wave functions are invariant with respect to the spin quantum


number $m_s$, but the introduction of a


Jastrow factor breaks this spininvariance if the parameters


closedshell systems. A good test is to check that all the components


of a spin multiplet are degenerate.


FCI wave functions have this property and give degenrate energies with


respect to the spin quantum number $m_s$, but the multiplication by a


Jastrow factor introduces spin contamination if the parameters


for the samespin electron pairs are different from those


for the oppositespin pairs.\cite{Tenno_2004}


Again, using pseudopotentials this error is transferred in the DMC


calculation unless the determinant localization approximation is used.


Again, when pseudopotentials are used this tiny error is transferred


in the FNDMC energy unless the determinant localization approximation


is used.




Within DFT, the common density functionals make a difference for


samespin and oppositespin interactions. As DFT is a


singledeterminant theory, the density functionals are designed to be


used with the highest value of $m_s$, and therefore different values


of $m_s$ lead to different energies.


So in the context of RSDFT, the determinantal expansions will be


impacted by this spurious effect, as opposed to FCI.




\begin{table}


\caption{FNDMC energies (in hartree) of the triplet carbon atom (BFDVDZ) with


different values of $m_s$.}


\label{tab:spin}


\begin{ruledtabular}


\begin{tabular}{cccc}


$\mu$ & $m_s=1$ & $m_s=0$ & Spininvariance error \\


\hline


0.00 & $5.416\,8(1)$ & $5.414\,9(1)$ & $+0.001\,9(2)$ \\


0.25 & $5.417\,2(1)$ & $5.416\,5(1)$ & $+0.000\,7(1)$ \\


0.50 & $5.422\,3(1)$ & $5.421\,4(1)$ & $+0.000\,9(2)$ \\


1.00 & $5.429\,7(1)$ & $5.429\,2(1)$ & $+0.000\,5(2)$ \\


2.00 & $5.432\,1(1)$ & $5.431\,4(1)$ & $+0.000\,7(2)$ \\


5.00 & $5.431\,7(1)$ & $5.431\,4(1)$ & $+0.000\,3(2)$ \\


$\infty$ & $5.431\,6(1)$ & $5.431\,3(1)$ & $+0.000\,3(2)$ \\


\end{tabular}


\end{ruledtabular}


\end{table}




In this section, we investigate the impact of the spin contamination


due to the shortrange density functional on the FNDMC energy. We have


computed the energies of the carbon atom in its triplet state


with BFD pseudopotentials and the corresponding doublezeta basis


set. The calculation was done with $m_s=1$ (3 $\uparrow$ electrons


and 1 $\downarrow$ electrons) and with $m_s=0$ (2 $\uparrow$ and 2


$\downarrow$ electrons).




The results are presented in table~\ref{tab:spin}.


Although using $m_s=0$ the energy is higher than with $m_s=1$, the


bias is relatively small, more than one order of magnitude smaller


than the energy gained by reducing the fixednode error going from the single


determinant to the FCI trial wave function. The highest bias, close to


2~m\hartree, is obtained for $\mu=0$, but the bias decreases quickly


below 1~m\hartree{} when $\mu$ increases. As expected, with $\mu=\infty$


there is no bias (within the error bars), and the bias is not


noticeable with $\mu=5$~bohr$^{1}$.




To check that the RSDFTCIPSI are spininvariant, we compute the


FNDMC energies of the ?? dimer with different values of the spin


quantum number $m_s$.








\subsection{Benchmark}




The 55 molecules of the benchmark for the Gaussian1


theory\cite{Pople_1989,Curtiss_1990} were chosen to test the quality


of the RSDFTCIPSI trial wave functions for energy differences.








%\begin{squeezetable}


\begin{squeezetable}


\begin{table*}


\caption{Mean absolute error (MAE), mean signed errors (MSE) and


standard deviations (RMSD) obtained with the different methods and


\caption{Mean absolute errors (MAE), mean signed errors (MSE) and


standard deviations (RMSD) obtained with various methods and


basis sets.}


\label{tab:mad}


\begin{ruledtabular}


\begin{tabular}{ll rrr rrr rrr}


Method & \(\mu\) & \phantom{} & VDZBFD & \phantom{} & \phantom{} & VTZBFD & \phantom{} & \phantom{} & VQZBFD & \phantom{} \\


\phantom{} & \phantom{} & MAE & MSE & RMSD & MAE & MSE & RMSD & MAE & MSE & RMSD \\


\begin{tabular}{ll ddd ddd ddd}


& & \mc{3}{c}{VDZBFD} & \mc{3}{c}{VTZBFD} & \mc{3}{c}{VQZBFD} \\


\cline{35} \cline{68} \cline{911}


Method & $\mu$ & \tabc{MAE} & \tabc{MSE} & \tabc{RMSD} & \tabc{MAE} & \tabc{MSE} & \tabc{RMSD} & \tabc{MAE} & \tabc{MSE} & \tabc{RMSD} \\


\hline


PBE & 0 & 5.02 & 3.70 & 6.04 & 4.57 & 1.00 & 5.32 & 5.31 & 0.79 & 6.27 \\


BLYP & 0 & 9.53 & 9.21 & 7.91 & 5.58 & 4.44 & 5.80 & 5.86 & 4.47 & 6.43 \\


PBE0 & 0 & 11.20 & 10.98 & 8.68 & 6.40 & 5.78 & 5.49 & 6.28 & 5.65 & 5.08 \\


B3LYP & 0 & 11.27 & 10.98 & 9.59 & 7.27 & 5.77 & 6.63 & 6.75 & 5.53 & 6.09 \\


\hline


CCSD(T) & \(\infty\) & 24.10 & 23.96 & 13.03 & 9.11 & 9.10 & 5.55 & 4.52 & 4.38 & 3.60 \\


\hline


RSDFTCIPSI & 0 & 4.53 & 1.66 & 5.91 & 6.31 & 0.91 & 7.93 & 6.35 & 3.88 & 7.20 \\


\phantom{} & 1/4 & 5.55 & 4.66 & 5.52 & 4.58 & 1.06 & 5.72 & 5.48 & 1.52 & 6.93 \\


\phantom{} & 1/2 & 13.42 & 13.27 & 7.36 & 6.77 & 6.71 & 4.56 & 6.35 & 5.89 & 5.18 \\


\phantom{} & 1 & 17.07 & 16.92 & 9.83 & 9.06 & 9.06 & 5.88 &  &  &  \\


\phantom{} & 2 & 19.20 & 19.05 & 10.91 &  &  &  &  &  &  \\


\phantom{} & 5 & 22.93 & 22.79 & 13.24 &  &  &  &  &  &  \\


\phantom{} & \(\infty\) & 23.62 & 23.48 & 12.81 &  &  &  &  &  &  \\


\hline


DMC@RSDFTCIPSI & 0 & 4.61(34) & 3.62(\phantom{0.}34) & 5.30 & 3.52(19) & 1.03(19) & 4.39 & 3.16(26) & 0.12(26) & 4.12 \\


\phantom{} & 1/4 & 4.04(37) & 3.13(\phantom{0.}37) & 4.88 & 3.39(77) & 0.59(77) & 4.44 & 2.90(25) & 0.25(25) & 3.74 \\


\phantom{} & 1/2 & 3.74(35) & 3.53(\phantom{0.}35) & 4.03 & 2.46(18) & 1.72(18) & 3.02 & 2.06(35) & 0.44(35) & 2.74 \\


\phantom{} & 1 & 5.42(29) & 5.14(\phantom{0.}29) & 4.55 & 4.38(94) & 4.24(94) & 5.11 &  &  &  \\


\phantom{} & 2 & 5.98(83) & 5.91(\phantom{0.}83) & 4.79 &  &  &  &  &  &  \\


\phantom{} & 5 & 6.18(84) & 6.13(\phantom{0.}84) & 4.87 &  &  &  &  &  &  \\


\phantom{} & \(\infty\) & 7.38(1.08) & 7.38(1.08) & 5.67 &  &  &  &  &  &  \\


\phantom{} & Opt. & 5.85(1.75) & 5.63(1.75) & 4.79 &  &  &  &  &  &  \\


PBE & 0 & 5.02 & 3.70 & 6.04 & 4.57 & 1.00 & 5.32 & 5.31 & 0.79 & 6.27 \\


BLYP & 0 & 9.53 & 9.21 & 7.91 & 5.58 & 4.44 & 5.80 & 5.86 & 4.47 & 6.43 \\


PBE0 & 0 & 11.20 & 10.98 & 8.68 & 6.40 & 5.78 & 5.49 & 6.28 & 5.65 & 5.08 \\


B3LYP & 0 & 11.27 & 10.98 & 9.59 & 7.27 & 5.77 & 6.63 & 6.75 & 5.53 & 6.09 \\


\\


CCSD(T) & \(\infty\) & 24.10 & 23.96 & 13.03 & 9.11 & 9.10 & 5.55 & 4.52 & 4.38 & 3.60 \\


\\


RSDFTCIPSI & 0 & 4.53 & 1.66 & 5.91 & 6.31 & 0.91 & 7.93 & 6.35 & 3.88 & 7.20 \\


& 1/4 & 5.55 & 4.66 & 5.52 & 4.58 & 1.06 & 5.72 & 5.48 & 1.52 & 6.93 \\


& 1/2 & 13.42 & 13.27 & 7.36 & 6.77 & 6.71 & 4.56 & 6.35 & 5.89 & 5.18 \\


& 1 & 17.07 & 16.92 & 9.83 & 9.06 & 9.06 & 5.88 & & & \\


& 2 & 19.20 & 19.05 & 10.91 & & & & & & \\


& 5 & 22.93 & 22.79 & 13.24 & & & & & & \\


& \(\infty\) & 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) \\


\\


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) \\


RSDFTCIPSI & 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) \\


& 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) \\


& 1 & 5.42(29) & 5.14(29) & 4.55(03) & 4.38(94) & 4.24(94) & 5.11(31) & & & \\


& 2 & 5.98(83) & 5.91(83) & 4.79(71) & & & & & & \\


& 5 & 6.18(84) & 6.13(84) & 4.87(55) & & & & & & \\


& \(\infty\) & 7.38(1.08) & 7.38(1.08) & 5.67(68) & & & & & & \\


& Opt. & 5.85(1.75) & 5.63(1.75) & 4.79(1.11) & & & & & & \\


\end{tabular}


\end{ruledtabular}


\end{table*}


%\end{squeezetable}


\end{squeezetable}




The 55 molecules of the benchmark for the Gaussian1


theory\cite{Pople_1989,Curtiss_1990} were chosen to test the


performance of the RSDFTCIPSI trial wave functions in the context of


energy differences. Calculations were made in the double, triple


and quadruplezeta basis sets with different values of $\mu$, and using


natural orbitals of a preliminary CIPSI calculation.


For comparison, we have computed the energies of all the atoms and


molecules at the DFT level with different density functionals, and at


the CCSD(T) level. Table~\ref{tab:mad} gives the corresponding mean


absolute errors (MAE), mean signed errors (MSE) and standard


deviations (RMSD). For FCI (RSDFTCIPSI, $\mu=\infty$) we have


given extrapolated values at $\EPT\rightarrow 0$, and the error bars


correspond to the difference between the energies computed with a


twopoint and with a threepoint linear extrapolation.




In this benchmark, the great majority of the systems are well


described by a single determinant. Therefore, the atomization energies


calculated at the DFT level are relatively accurate, even when


the basis set is small. The introduction of exact exchange (B3LYP and


PBE0) make the results more sensitive to the basis set, and reduce the


accuracy. Thanks to the singlereference character of these systems,


the CCSD(T) energy is an excellent estimate of the FCI energy, as


shown by the very good agreement of the MAE, MSE and RMSD of CCSDT(T)


and FCI energies.


The imbalance of the quality of description of molecules compared


to atoms is exhibited by a very negative value of the MSE for


CCSD(T) and FCI/VDZBFD, which is reduced by a factor of two


when going to the triplezeta basis, and again by a factor of two when


going to the quadruplezeta basis.




This large imbalance at the doublezeta level affects the nodal


surfaces, because although the FNDMC energies obtained with nearFCI


trial wave functions are much lower than the singledeterminant FNDMC


energies, the MAE obtained with FCI (7.38~$\pm$ 1.08~kcal/mol) is


larger than the singledeterminant 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 basisset


incompleteness error is transferred in the fixednode error.




Within the doublezeta basis set, the calculations could be done 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 FNDMC 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


result is in line with the common knowledge that reoptimizing


the determinantal component of the trial wave function in the presence


of electron correlation reduces the errors due to the basis set incompleteness.


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 RSDFTCIPSI level, we can remark that with the triplezeta


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:g2ndet}


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


small sizeconsistency error originating from the imbalanced


truncation of the wave functions, which is not present in the


extrapolated FCI energies. The same comment applies to


$\mu=0.5$~bohr$^{1}$ with the quadruplezeta basis set.






\begin{figure}


\centering


\includegraphics[width=\columnwidth]{g2dmc.pdf}


\caption{Errors in the DMC atomization energies with the different


\caption{Errors in the FNDMC atomization energies with the different


trial wave functions. Each dot corresponds to an atomization


energy.


The boxes contain the data between first and third quartiles, and


@ 716,6 +847,18 @@ DMC@RSDFTCIPSI & 0 & 4.61(34) & 3.62(\phantom{0.}34) & 5.30


\label{fig:g2dmc}


\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 figure~\ref{fig:g2dmc}. As seen on the figure and


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


are even lower than those obtained with the optimal value of


$\mu$. Although the FNDMC energies are higher, the numbers show that


they are more consistent from one system to another, giving improved


cancellations of errors.




\begin{figure}


\centering


\includegraphics[width=\columnwidth]{g2ndet.pdf}


@ 727,61 +870,75 @@ DMC@RSDFTCIPSI & 0 & 4.61(34) & 3.62(\phantom{0.}34) & 5.30


\label{fig:g2ndet}


\end{figure}




The number of determinants in the wave functions are shown in


figure~\ref{fig:n2ndet}. For all the calculations, the stopping


criterion of the CIPSI algorithm was $\EPT < 1$~m\hartree{} or $\Ndet >


10^7$.


For FCI, we have given extrapolated values at $\EPT\rightarrow 0$.


At $\mu=0$ the number of determinants is not equal to one because


we have used the natural orbitals of a first CIPSI calculation.




The number of determinants in the trial wave functions are shown in


figure~\ref{fig:g2ndet}. 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 quadruplezeta basis set, making these calculations feasilble


with such a large basis set. At the doublezeta 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 doublezeta level, and close to 1~000 determinants


at the quadruplezeta level for only a slight increase of the


MAE. Hence, RSDFTCIPSI trial wave functions with small values of


$\mu$ could be very useful for large systems to go beyond the


singledeterminant approximation at a very low computational cost


while keeping the sizeconsistency.




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


not the srPBE orbitals.


So the KohnSham determinant is expressed as a linear combination of


determinants built with natural orbitals. Note that it is possible to add


determinants built with natural orbitals. It is possible to add


an extra step to the algorithm to compute the natural orbitals from the


RSDFT/CIPSI wave function, and redo the RSDFT/CIPSI calculation with


these orbitals to get an even more compact expansion. In that case, we would


have obtained the KohnSham orbitals with $\mu=0$, and the solution would have


been the KS single determinant.


have converged to the KohnSham orbitals with $\mu=0$, and the


solution would have been the PBE single determinant.




We could have obtained


singledeterminant wave functions by using the natural orbitals of a first






\section{Conclusion}




We have seen that introducing shortrange correation via


a rangeseparated Hamiltonian in a full CI expansion yields improved


nodes, especially with small basis sets. The effect is similar to the


effect of reoptimizing the CI coefficients in the presence of a


Jastrow factor, but without the burden of performing a stochastic


optimization.


The proposed procedure provides a method to optimize the


FNDMC energy via a single parameter, namely the rangeseparation


parameter $\mu$. The sizeconsistency error is controlled, as well as the


invariance with respect to the spin projection $m_s$.


Finding the optimal value of $\mu$ gives the lowest FNDMC energies


within basis set. However, if one wants to compute an energy


difference, one should not minimize the


FNDMC energies of the reactants independently. It is preferable to


choose a value of $\mu$ for which the fixednode errors are well


balanced, leading to a good cncellation of errors. We found that a


value of $\mu=0.5$~bohr${^1}$ is the value where the errors are the


smallest. Moreover, such a small value of $\mu$ gives extermely


nodal surfaces, especially with small basis sets. The effect of srDFT


on the determinant expansion is similar to the effect of reoptimizing


the CI coefficients in the presence of a Jastrow factor, but without


the burden of performing a stochastic optimization.




Varying the rangeseparation parameter $\mu$ and approaching the


RSDFTFCI with CIPSI provides a way to adapt the number of


determinants in the trial wave function, leading always to


sizeconsistent FNDMC energies.


We propose two methods. The first one is for the computation of


accurate total energies by a oneparameter optimization of the FNDMC


energy via the variation of the parameter $\mu$.


The second method is for the computation of energy differences, where


the target is not the lowest possible FNDMC energies but the best


possible cancellation of errors. Using a fixed value of $\mu$


increases the consistency of the trial wave functions, and we have found


that $\mu=0.5$~bohr$^{1}$ is the value where the cancellation of


errors is the most effective.


Moreover, such a small value of $\mu$ gives extermely


compact wave functions, making this recipe a good candidate for


accurate calcultions of large systems with a multireference character.


the accurate description of the whole potential energy surfaces of


large systems. If the number of determinants is still too large, the


value of $\mu$ can be further reduced to $0.25$~bohr$^{1}$ to get


extremely compact wave functions at the price of less efficient


cancellations of errors.








%%




%%


\begin{acknowledgments}


An award of computer time was provided by the Innovative and Novel


Computational Impact on Theory and Experiment (INCITE) program. This


research has used resources of the Argonne Leadership Computing


Facility, which is a DOE Office of Science User Facility supported


under Contract DEAC0206CH11357. AB, was supported by the


U.S. Department of Energy, Office of Science, Basic Energy Sciences,


Materials Sciences and Engineering Division, as part of the


Computational Materials Sciences Program and Center for Predictive


Simulation of Functional Materials.


This work was performed using HPC resources from GENCITGCC (Grand


Challenge 2019gch0418) and from CALMIP (Toulouse) under allocation


20190510.


\end{acknowledgments}






@ 790,15 +947,16 @@ Simulation of Functional Materials.


\begin{enumerate}


\item Total energies and nodal quality:


\begin{itemize}


\item Facts: KS occupied orbitals closer to NOs than HF


\item Even if exact functional, complete basis set, still approximated nodes for KS


% \item Facts: KS occupied orbitals closer to NOs than HF


% \item Even if exact functional, complete basis set, still approximated nodes for KS


\item KS > exponentially fast convergence (as HF) with basis because of non divergence of effective KS potential (citer le papier de Gill)


\item With correlation consistent basis set, FCI nodes (which include correlation) are better than KS


\item With FCI, good limit at CBS ==> exact energy


\item But slow convergence with basis set because of divergence of the ee interaction not well represented in atom centered basis set


\item Exponential increase of number of Slater determinants


% \item With correlation consistent basis set, FCI nodes (which include correlation) are better than KS


% \item With FCI, good limit at CBS ==> exact energy


% \item But slow convergence with basis set because of divergence of the ee interaction not well represented in atom centered basis set


% \item Exponential increase of number of Slater determinants


\item Cite papiers RSDFT: there exists an hybrid scheme combining fast convergence wr to basis set (non divergent basis set) and short expansion in SCI (cite papier Ferté)


\item Question: does such a scheme provide better nodal quality ?


%<<<<<<< HEAD


\item In RSDFT we cannot optimize energy with $\mu$ , but in FNDMC


\item Factual stuffs: with optimal $\mu$, lower FNDMC energy than HF/KS/FCI


\begin{itemize}


@ 815,6 +973,23 @@ Simulation of Functional Materials.


\item large wave functions


\end{itemize}


% \item Invariance with m_s


%=======


\item In RSDFT we cannot optimize energy with $\mu$ , but in FNDMC


% \item Factual stuffs: with optimal $\mu$, lower FNDMC energy than HF/KS/FCI


% \begin{itemize}


% \item less determinants $\Rightarrow$ large systems


% \item only one parameter to optimize $\Rightarrow$ deterministic


% \item $\Rightarrow$ reproducible


% \end{itemize}


% \item with the optimal $\mu$:


% \begin{itemize}


% \item Direct optimization of FNDMC with one parameter


% \item Do we improve energy differences ?


% \item system dependent


% \item basis set dependent: $\mu \rightarrow \infty$ when $\mathcal{B}\rightarrow \text{CBS}$


% \item large wave functions


% \end{itemize}


%>>>>>>> 44470b89936d1727b638aefd982ce83be9075cc8




\end{itemize}


\end{enumerate}


@ 822,3 +997,18 @@ Simulation of Functional Materials.






\end{document}






% * Recouvrement avec Be : Optimization tous electrons


% impossible. Abandon. On va prendre H2O.


% * Manu doit faire des programmes pour des plots de ensite a 1 et 2


% corps le long des axes de liaison, et l'integrale de la densite a


% 2 corps a coalescence.


% 1 Manu calcule Be en ccpvdz tous electrons: FCI > NOs > FCI >


% qp edit n 200


% 2 Manu calcule qp_cipsi_rsh avec mu = [ 1.e6 , 0.25, 0.5, 1.0, 2.0, 5.0, 1e6 ]


% 3 Manu fait tourner ses petits programmes


% 4 Manu envoie a toto un tar avec tous les ezfio


% 5 Toto optimise les coefs en presence e jastrow


% 6 Toto renvoie a manu psicoef


% 7 Manu fait tourner ses petits programmes avec psi_J



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