Merge branch 'master' of https://git.irsamc.upstlse.fr/scemama/RSDFTCIPSIQMC
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071a0232e2
@ 1366,3 +1366,20 @@


Title = {Shortrange exchangecorrelation energy of a uniform electron gas with modified electronelectron interaction},


Volume = {100},


Year = {2004}}








@article{Nightingale_2001,


author = {Nightingale, M. P. and MelikAlaverdian, Vilen},


title = {{Optimization of Ground and ExcitedState Wave Functions and van der Waals


Clusters}},


journal = {Phys. Rev. Lett.},


volume = {87},


number = {4},


pages = {043401},


year = {2001},


month = {Jul},


issn = {10797114},


publisher = {American Physical Society},


doi = {10.1103/PhysRevLett.87.043401}


}



@ 13,6 +13,8 @@


]{hyperref}


\urlstyle{same}




\DeclareMathOperator*{\argmin}{arg\,min}




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


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


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


@ 36,6 +38,8 @@


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


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


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


\newcommand{\Nelec}{N_{\text{elec}}}


\newcommand{\Nat}{N_{\text{atoms}}}


\newcommand{\hartree}{$E_h$}




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


@ 164,9 +168,9 @@ 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,Giner_2015,Caffarel_2016_2}


\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} WRONG}


\toto{When the basis set is enlarged, the trial wave function gets closer to


the exact wave function, so we expect the nodal surface to be


improved.\cite{Caffarel_2016} }


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


set, which is perfectly well defined and therefore makes the calculations reproducible in a


blackbox way without needing any expertise in QMC.


@ 339,14 +343,12 @@ energy is obtained as


E_0= \mel{\Psi^{\mu}}{\hat{T}+\hat{W}_\text{{ee}}^{\text{lr},\mu}+\hat{V}_{\text{ne}}}{\Psi^{\mu}}+\bar{E}^{\text{sr},\mu}_{\text{Hxc}}[n_{\Psi^\mu}].


\end{equation}




Note that, for $\mu=0$, \titou{the longrange interaction vanishes}, \ie,


$w_{\text{ee}}^{\text{lr},\mu=0}(r) = 0$, and thus


RSDFT reduces to standard KSDFT and $\Psi^\mu$


is the KS determinant. For $\mu = \infty$, the longrange


Note that for $\mu=0$ the longrange interaction vanishes, \ie,


$w_{\text{ee}}^{\text{lr},\mu=0}(r) = 0$, and thus RSDFT reduces to standard


KSDFT and $\Psi^\mu$ is the KS determinant. For $\mu = \infty$, the longrange


interaction becomes the standard Coulomb interaction, \ie,


$w_{\text{ee}}^{\text{lr},\mu\to\infty}(r) = r^{1}$, and


thus RSDFT reduces to standard WFT and $\Psi^\mu$ is


the FCI wave function.


$w_{\text{ee}}^{\text{lr},\mu\to\infty}(r) = r^{1}$, and thus RSDFT reduces


to standard WFT and $\Psi^\mu$ is the FCI wave function.




%%% FIG 1 %%%


\begin{figure*}


@ 430,10 +432,9 @@ the pseudopotential is localized. Hence, in the DLA the fixednode


energy is independent of the Jastrow factor, as in allelectron


calculations. Simple Jastrow factors were used to reduce the


fluctuations of the local energy.


The FNDMC simulations are performed with the stochastic reconfiguration


algorithm developed by Assaraf \textit{et al.}, \cite{Assaraf_2000}


with a time step of $5 \times 10^{4}$ a.u.


\titou{Allelectron move DMC?}


The FNDMC simulations are performed with allelectron moves using the


stochastic reconfiguration algorithm developed by Assaraf \textit{et al.},


\cite{Assaraf_2000} with a time step of $5 \times 10^{4}$ a.u.




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\section{Influence of the rangeseparation parameter on the fixednode error}


@ 442,8 +443,7 @@ with a time step of $5 \times 10^{4}$ a.u.




%%% TABLE I %%%


\begin{table}


\caption{Fixednode energy $\EDMC$ (in \hartree{}) and number of determinants $\Ndet$ in \ce{H2O} for various trial wave functions $\Psi^{\mu}$.


\titou{srPBE?}.}


\caption{Fixednode energy $\EDMC$ (in \hartree{}) and number of determinants $\Ndet$ in \ce{H2O} for various trial wave functions $\Psi^{\mu}$ obtained with the srPBE density functional.}


\label{tab:h2odmc}


\centering


\begin{ruledtabular}


@ 484,10 +484,11 @@ The first question we would like to address is the quality of the


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


range separation parameter (\ie, $0 < \mu < +\infty$).


For this purpose, we consider a weakly correlated molecular system, namely the water


molecule \titou{near its equilibrium geometry.} \cite{Caffarel_2016}


molecule near its equilibrium geometry. \cite{Caffarel_2016}


We then generate trial wave functions $\Psi^\mu$ for multiple values of


$\mu$, and compute the associated fixednode energy keeping all the


parameters having an impact on the nodal surface fixed such as CI coefficients and molecular orbitals.


\toto{$\mu$, and compute the associated fixednode energy keeping fixed all the


parameters such as the CI coefficients and molecular orbitals impacting the


nodal surface.}




%======================================================


\subsection{Fixednode energy of $\Psi^\mu$}


@ 509,10 +510,11 @@ and then the FNDMC error raises until it reaches the $\mu=\infty$ limit (\ie, t


For instance, with respect to the FNDMC/VDZBFD energy at $\mu=\infty$,


one can obtain a lowering of the FNDMC energy of $2.6 \pm 0.7$~m\hartree{}


with an optimal value of $\mu=1.75$~bohr$^{1}$.


This lowering in FNDMC energy is to be compared with the $3.2 \pm 0.7$~m\hartree{} of gain in FNDMC energy between the KS wave function ($\mu=0$) and the FCI wave function ($\mu=\infty$).


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 slightly shifted towards large $\mu$.


This lowering in FNDMC energy is to be compared with the $3.2 \pm


0.7$~m\hartree{} gain in FNDMC energy between the KS wave function ($\mu=0$)


and the FCI wave function ($\mu=\infty$). 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 slightly shifted towards large $\mu$.


Last but not least, the nodes of the wave functions $\Psi^\mu$ obtained with the srLDA


exchangecorrelation functional give very similar FNDMC energies with respect


to those obtained with the srPBE functional, even if the


@ 529,11 +531,12 @@ The takehome message of this numerical study is that RSDFT trial wave function


%======================================================


The data presented in Sec.~\ref{sec:fndmc_mu} evidence that, in a finite basis, RSDFT can provide


trial wave functions with better nodes than FCI wave function.


Such behaviour can be directty compared to the common practice of


Such behaviour can be directly compared to the common practice of


reoptimizing the multideterminant part of a trial wave function $\Psi$ (the socalled Slater part) in the presence of the exponentiated Jastrow factor $e^J$. \cite{Umrigar_2005,Scemama_2006c,Umrigar_2007,Toulouse_2007,Toulouse_2008}


Hence, in the present paragraph, we would like to elaborate further on the link between RSDFT


and wave function optimization in the presence of a Jastrow factor.


\titou{T2: maybe we should mention that we only reoptimize the CI coefficients as it is of common practice to reoptimize more than this.}


\toto{For simplicity in the comparison, the molecular orbitals and the Jastrow


factor are kept fixed: only the CI coefficients are modified.}




Let us assume a fixed Jastrow factor $J(\br_1, \ldots , \br_N)$,


and a corresponding SlaterJastrow wave function $\Phi = e^J \Psi$,


@ 541,18 +544,19 @@ where $\Psi = \sum_I c_I D_I$ is a general linear combination of Slater determin


The only remaining variational parameters in $\Phi$ are therefore the Slater part $\Psi$.


Let us call $\Psi^J$ the linear combination of Slater determinant minimizing the variational energy


\begin{equation}


\Psi^J = \text{argmin}_{\Psi}\frac{ \mel{ \Psi }{ e^{J} H e^{J} }{ \Psi } }{\mel{ \Psi }{ e^{2J} }{ \Psi } }.


\Psi^J = \argmin_{\Psi}\frac{ \mel{ \Psi }{ e^{J} \hat{H} e^{J} }{ \Psi } }{\mel{ \Psi }{ e^{2J} }{ \Psi } }.


\end{equation}


Such a wave function $\Psi^J$ satisfies the generalized hermitian eigenvalue equation


Such a wave function $\Psi^J$ satisfies the generalized Hermitian eigenvalue equation


\begin{equation}


e^{J} H e^{J} \Psi^J = E e^{2J} \Psi^J,


e^{J} \hat{H} \qty( e^{J} \Psi^J ) = E e^{2J} \Psi^J,


\label{eq:cij}


\end{equation}


but also the nonhermitian transcorrelated eigenvalue problem \cite{many_things}


but also the nonHermitian \manu{transcorrelated eigenvalue problem \cite{many_things} MANU:CITATIONS}


\begin{equation}


\label{eq:transcor}


e^{J} H e^{J} \Psi^J = E \Psi^J,


e^{J} \hat{H} \qty( e^{J} \Psi^J) = E \Psi^J,


\end{equation}


which is much easier to handle despite its nonhermicity.


which is much easier to handle despite its nonHermiticity.


Of course, the FNDMC energy of $\Phi$ depends therefore only on the nodes of $\Psi^J$.


In a finite basis set and with a quite accurate Jastrow factor, it is known that the nodes


of $\Psi^J$ may be better than that of the FCI wave function, and therefore, we would like to compare $\Psi^J$ and $\Psi^\mu$.


@ 563,8 +567,27 @@ function out of a large CIPSI calculation. Within this set of determinants,


we solve the selfconsistent equations of RSDFT [see Eq.~\eqref{rsdfteigenequation}]


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 $\Psi^J$.


Then, we can easily compare $\Psi^\mu$ and $\Psi^J$ as they are developed


a simple one and twobody Jastrow factor \toto{$e^J$ of the form $\exp(J_{eN} + J_{ee})$ with


\begin{eqnarray}


J_\text{eN} & = &  \sum_{A=1}^{\Nat} \sum_{i=1}^{\Nelec} \left( \frac{\alpha_A\, r_{iA}}{1 + \alpha_A\, r_{iA}} \right)^2


\label{eq:jasteN} \\


J_\text{ee} & = & \sum_{i=1}^{\Nelec} \sum_{j=1}^{i1} \frac{a\, r_{ij}}{1 + b\, r_{ij}}. \label{eq:jastee}


\end{eqnarray}


$J_\text{eN}$ contains the electronnucleus terms with a single parameter


$\alpha_A$ per atom, and $J_\text{ee}$ contains the electronelectron terms


where the indices $i$ and $j$ loop over all electrons. The parameters $a=1/2$


and $b=0.89$ were fixed, and the parameters $\gamma_O=1.15$ and $\gamma_H=0.35$


were obtained by energy minimization with a single determinant.


The optimal CI expansion $\Psi^J$ is obtained by sampling the matrix elements


of the Hamiltonian ($\mathbf{H}$) and the overlap ($\mathbf{S}$), in the


basis of Jastrowcorrelated determinants $e^J D_i$:


\begin{eqnarray}


H_{ij} & = & \left \langle \frac{e^J D_i}{\Psi^J}\, \frac{\hat{H}\, (e^J D_j)}{\Psi^J} \right \rangle \\


S_{ij} & = & \left \langle \frac{e^J D_i}{\Psi^J}\, \frac{e^J D_j}{\Psi^J} \right \rangle


\end{eqnarray}


and solving Eq.~\eqref{eq:cij}.\cite{Nightingale_2001}}




We can easily compare $\Psi^\mu$ and $\Psi^J$ as they are developed


on the same Slater determinant basis.


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


$\braket*{\Psi^J}{\Psi^\mu}$ obtained for water,


@ 779,10 +802,7 @@ Another source of sizeconsistency error in QMC calculations originates


from the Jastrow factor. Usually, the Jastrow factor contains


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<j} \frac{a\, r_{ij}}{1 + b\, r_{ij}}.


\end{equation}


be expressed as in Eq.\eqref{eq:jastee}.


The parameter


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


or variance minimization.\cite{Coldwell_1977,Umrigar_2005}



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