fix mix up in basis names
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@ -39,8 +39,8 @@
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\newcommand{\EPT}{E_{\text{PT2}}}
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\newcommand{\EDMC}{E_{\text{FN-DMC}}}
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\newcommand{\Ndet}{N_{\text{det}}}
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\newcommand{\Nelec}{N_{\text{elec}}}
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\newcommand{\Nat}{N_{\text{atoms}}}
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\newcommand{\Nelec}{N}
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\newcommand{\Nat}{M}
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\newcommand{\hartree}{$E_h$}
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\newcommand{\LCT}{Laboratoire de Chimie Th\'eorique (UMR 7616), Sorbonne Universit\'e, CNRS, Paris, France}
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@ -300,7 +300,7 @@ In RS-DFT, the Coulomb operator entering the electron-electron repulsion is spli
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\begin{equation}
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\frac{1}{r}
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= w_{\text{ee}}^{\text{sr}, \mu}(r)
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+ w_{\text{ee}}^{\text{lr}, \mu}(r)
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+ w_{\text{ee}}^{\text{lr}, \mu}(r),
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\end{equation}
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where
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\begin{align}
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@ -428,24 +428,24 @@ Geometries for each system are reported in the {\SI}.
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All the calculations have been performed using Burkatzki-Filippi-Dolg (BFD)
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pseudopotentials \cite{Burkatzki_2007,Burkatzki_2008} with the associated double-,
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triple-, and quadruple-$\zeta$ basis sets (BFD-VXZ).
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triple-, and quadruple-$\zeta$ basis sets (VXZ-BFD).
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The small-core BFD pseudopotentials include scalar relativistic effects.
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Coupled cluster with singles, doubles, and perturbative triples [CCSD(T)] and KS-DFT energies have been computed with
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\emph{Gaussian09},\cite{g16} using the unrestricted formalism for open-shell systems.
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All the CIPSI calculations have been performed with \emph{Quantum
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Package}.\cite{Garniron_2019,qp2_2020} We used the short-range version
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of the local-density approximation (LDA)\cite{Sav-INC-96a,TouSavFla-IJQC-04} and Perdew-Burke-Ernzerhof (PBE) \cite{PerBurErn-PRL-96} exchange
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and correlation functionals defined in
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Package}.\cite{Garniron_2019,qp2_2020} We consider the short-range version
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of the local-density approximation (LDA)\cite{Sav-INC-96a,TouSavFla-IJQC-04} and Perdew-Burke-Ernzerhof (PBE) \cite{PerBurErn-PRL-96}
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xc functionals defined in
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Ref.~\onlinecite{GolWerStoLeiGorSav-CP-06} (see also
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Refs.~\onlinecite{TouColSav-JCP-05,GolWerSto-PCCP-05}).
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Refs.~\onlinecite{TouColSav-JCP-05,GolWerSto-PCCP-05}) that we label srLDA and srPBE respectively in the following.
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In this work, we target chemical accuracy, so
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the convergence criterion for stopping the CIPSI calculations
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has been set to $\EPT < 10^{-3}$ \hartree{} or $ \Ndet > 10^7$.
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All the wave functions are eigenfunctions of the $\Hat{S}^2$ spin operator, as
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described in Ref.~\onlinecite{Applencourt_2018}.
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QMC calculations have been performed with QMC=Chem,\cite{Scemama_2013}
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QMC calculations have been performed with \textit{QMC=Chem},\cite{Scemama_2013}
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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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@ -464,12 +464,12 @@ stochastic reconfiguration algorithm developed by Assaraf \textit{et al.},
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%%% TABLE I %%%
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\begin{table}
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\caption{Fixed-node energy $\EDMC$ (in \hartree{}) and number of determinants $\Ndet$ in \ce{H2O} for various trial wave functions $\Psi^{\mu}$ obtained with the sr-PBE density functional.}
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\caption{FN-DMC 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.}
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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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& \multicolumn{2}{c}{VDZ-BFD} & \multicolumn{2}{c}{VTZ-BFD} \\
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\cline{2-3} \cline{4-5}
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$\mu$ & $\Ndet$ & $\EDMC$ & $\Ndet$ & $\EDMC$ \\
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\hline
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@ -494,7 +494,7 @@ stochastic reconfiguration algorithm developed by Assaraf \textit{et al.},
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\begin{figure}
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\centering
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\includegraphics[width=\columnwidth]{h2o-dmc.pdf}
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\caption{Fixed-node energy of \ce{H2O} as a function
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\caption{FN-DMC energy of \ce{H2O} as a function
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of $\mu$ for various levels of theory to generate
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the trial wave function.}
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\label{fig:h2o-dmc}
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@ -502,14 +502,13 @@ stochastic reconfiguration algorithm developed by Assaraf \textit{et al.},
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%%% %%% %%% %%%
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The first question we would like to address is the quality of the
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nodes of the wave function $\Psi^{\mu}$ obtained with an intermediate
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nodes of the wave function $\Psi^{\mu}$ obtained for intermediate values of the
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range separation parameter (\ie, $0 < \mu < +\infty$).
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For this purpose, we consider a weakly correlated molecular system, namely the water
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molecule near its equilibrium geometry. \cite{Caffarel_2016}
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molecule \titou{at its experimental geometry. \cite{Caffarel_2016}}
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We then generate trial wave functions $\Psi^\mu$ for multiple values of
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$\mu$, and compute the associated fixed-node energy keeping fixed all the
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parameters such as the CI coefficients and molecular orbitals impacting the
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nodal surface.
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$\mu$, and compute the associated FN-DMC energy keeping fixed all the
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parameters impacting the nodal surface, such as the CI coefficients and the molecular orbitals.
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%======================================================
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\subsection{Fixed-node energy of $\Psi^\mu$}
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@ -520,7 +519,7 @@ one can clearly observe that relying on FCI trial
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wave functions ($\mu = \infty$) give FN-DMC energies lower
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than the energies obtained with a single KS determinant ($\mu=0$):
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a lowering of $3.2 \pm 0.6$~m\hartree{} at the double-$\zeta$ level and $7.2 \pm
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0.3$~m\hartree{} at the triple-$\zeta$ level are obtained.
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0.3$~m\hartree{} at the triple-$\zeta$ level are obtained with the srPBE functional.
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Coming now to the nodes of the trial wave functions $\Psi^{\mu}$ with
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intermediate values of $\mu$, Fig.~\ref{fig:h2o-dmc} shows that
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a smooth behaviour is obtained:
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@ -528,7 +527,7 @@ starting from $\mu=0$ (\ie, the KS determinant),
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the FN-DMC error is reduced continuously until it reaches a minimum
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for an optimal value of $\mu$ (which is obviously basis set and functional dependent),
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and then the FN-DMC error raises until it reaches the $\mu=\infty$ limit (\ie, the FCI wave function).
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For instance, with respect to the FN-DMC/VDZ-BFD energy at $\mu=\infty$,
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For instance, with respect to the fixed-node energy associated with the srPBE/VDZ-BFD trial wave function at $\mu=\infty$,
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one can obtain a lowering of the FN-DMC energy of $2.6 \pm 0.7$~m\hartree{}
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with an optimal value of $\mu=1.75$~bohr$^{-1}$.
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This lowering in FN-DMC energy is to be compared with the $3.2 \pm
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@ -537,12 +536,12 @@ and the FCI wave function ($\mu=\infty$). When the basis set is increased, the
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gain in FN-DMC energy with respect to the FCI trial wave function is reduced,
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and the optimal value of $\mu$ is slightly shifted towards large $\mu$.
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Last but not least, the nodes of the wave functions $\Psi^\mu$ obtained with the srLDA
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exchange-correlation functional give very similar FN-DMC energies with respect
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to those obtained with the srPBE functional, even if the
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functional give very similar FN-DMC energies with respect
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to those obtained with srPBE, even if the
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RS-DFT energies obtained with these two functionals differ by several
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tens of m\hartree{}.
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An other important aspect here regards the compactness of the trial wave functions $\Psi^\mu$:
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Another important aspect here is the compactness of the trial wave functions $\Psi^\mu$:
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at $\mu=1.75$~bohr$^{-1}$, $\Psi^{\mu}$ has \textit{only} $54\,540$ determinants at the srPBE/VDZ-BFD level, while the FCI wave function contains $200\,521$ determinants (see Table \ref{tab:h2o-dmc}). Even at the srPBE/VTZ-BFD level, we observe a reduction by a factor two in the number of determinants between the optimal $\mu$ value and $\mu = \infty$.
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The take-home message of this numerical study is that RS-DFT trial wave functions can yield a lower fixed-node energy with more compact multi-determinant expansion as compared to FCI.
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@ -552,22 +551,27 @@ The take-home message of this numerical study is that RS-DFT trial wave function
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%======================================================
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The data presented in Sec.~\ref{sec:fndmc_mu} evidence that, in a finite basis, RS-DFT can provide
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trial wave functions with better nodes than FCI wave function.
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Such behaviour can be directly compared to the common practice of
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As mentioned in Sec.~\ref{sec:SD}, such behavior can be directly compared to the common practice of
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re-optimizing the multi-determinant part of a trial wave function $\Psi$ (the so-called Slater part) in the presence of the exponentiated Jastrow factor $e^J$. \cite{Umrigar_2005,Scemama_2006,Umrigar_2007,Toulouse_2007,Toulouse_2008}
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Hence, in the present paragraph, we would like to elaborate further on the link between RS-DFT
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and wave function optimization in the presence of a Jastrow factor.
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For simplicity in the comparison, the molecular orbitals and the Jastrow
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factor are kept fixed: only the CI coefficients are modified.
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For the sake of simplicity, the molecular orbitals and the Jastrow
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factor are kept fixed; only the CI coefficients are varied.
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Let us assume a fixed Jastrow factor $J(\br_1, \ldots , \br_N)$,
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Let us assume a fixed Jastrow factor $J(\br_1, \ldots , \br_N)$ (where $\br_i$ is the position of the $i$th electron),
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and a corresponding Slater-Jastrow wave function $\Phi = e^J \Psi$,
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where $\Psi = \sum_I c_I D_I$ is a general linear combination of Slater determinants $D_I$.
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where
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\begin{equation}
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\label{eq:Slater}
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\Psi = \sum_I c_I D_I
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\end{equation}
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is a general linear combination of Slater determinants $D_I$.
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The only remaining variational parameters in $\Phi$ are therefore the Slater part $\Psi$.
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Let us call $\Psi^J$ the linear combination of Slater determinant minimizing the variational energy
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Let us define $\Psi^J$ as the linear combination of Slater determinants minimizing the variational energy associated with $\Phi$, \ie,
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\begin{equation}
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\Psi^J = \argmin_{\Psi}\frac{ \mel{ \Psi }{ e^{J} \hat{H} e^{J} }{ \Psi } }{\mel{ \Psi }{ e^{2J} }{ \Psi } }.
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\end{equation}
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Such a wave function $\Psi^J$ satisfies the generalized Hermitian eigenvalue equation
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Such a wave function satisfies the generalized Hermitian eigenvalue equation
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\begin{equation}
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e^{J} \hat{H} \qty( e^{J} \Psi^J ) = E e^{2J} \Psi^J,
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\label{eq:ci-j}
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@ -578,27 +582,34 @@ but also the non-Hermitian transcorrelated eigenvalue problem\cite{BoyHan-PRSLA-
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e^{-J} \hat{H} \qty( e^{J} \Psi^J) = E \Psi^J,
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\end{equation}
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which is much easier to handle despite its non-Hermiticity.
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Of course, the FN-DMC energy of $\Phi$ depends therefore only on the nodes of $\Psi^J$.
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Of course, the FN-DMC energy of $\Phi$ depends only on the nodes of $\Psi^J$.
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In a finite basis set and with a quite accurate Jastrow factor, it is known that the nodes
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of $\Psi^J$ may be better than that of the FCI wave function, and therefore, we would like to compare $\Psi^J$ and $\Psi^\mu$.
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of $\Psi^J$ may be better than the nodes of the FCI wave function.
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Hence, we would like to compare $\Psi^J$ and $\Psi^\mu$.
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To do so, we have made the following numerical experiment.
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First, we extract the 200 determinants with the largest weights in the FCI wave
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function out of a large CIPSI calculation. Within this set of determinants,
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function out of a large CIPSI calculation obtained with the VDZ-BFD basis. Within this set of determinants,
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we solve the self-consistent equations of RS-DFT [see Eq.~\eqref{rs-dft-eigen-equation}]
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with different values of $\mu$. This gives the CI expansions $\Psi^\mu$.
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Then, within the same set of determinants we optimize the CI coefficients in the presence of
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a simple one- and two-body Jastrow factor $e^J$ of the form $\exp(J_{eN} + J_{ee})$ with
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\begin{eqnarray}
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J_\text{eN} & = & - \sum_{A=1}^{\Nat} \sum_{i=1}^{\Nelec} \left( \frac{\alpha_A\, r_{iA}}{1 + \alpha_A\, r_{iA}} \right)^2
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Then, within the same set of determinants we optimize the CI coefficients $c_I$ [see Eq.~\eqref{eq:Slater}] in the presence of
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a simple one- and two-body Jastrow factor $e^J$ with $J = J_\text{eN} + J_\text{ee}$ and
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\begin{subequations}
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\begin{gather}
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J_\text{eN} = - \sum_{A=1}^{\Nat} \sum_{i=1}^{\Nelec} \qty( \frac{\alpha_A\, r_{iA}}{1 + \alpha_A\, r_{iA}} )^2,
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\label{eq:jast-eN} \\
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J_\text{ee} & = & \sum_{i=1}^{\Nelec} \sum_{j=1}^{i-1} \frac{a\, r_{ij}}{1 + b\, r_{ij}}. \label{eq:jast-ee}
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\end{eqnarray}
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$J_\text{eN}$ contains the electron-nucleus terms with a single parameter
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$\alpha_A$ per atom, and $J_\text{ee}$ contains the electron-electron terms
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where the indices $i$ and $j$ loop over all electron pairs. The parameters $a=1/2$
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J_\text{ee} = \sum_{i < j}^{\Nelec} \frac{a\, r_{ij}}{1 + b\, r_{ij}}.
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\label{eq:jast-ee}
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\end{gather}
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\end{subequations}
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The one-body Jastrow factor $J_\text{eN}$ contains the electron-nucleus terms (where $\Nat$ is the number of nuclei) with a single parameter
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$\alpha_A$ per nucleus.
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The two-body Jastrow factor $J_\text{ee}$ contains the electron-electron terms
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where the sum over $i < j$ loops over all electron pairs.
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In Eqs.~\eqref{eq:jast-eN} and \eqref{eq:jast-ee}, $r_{iA}$ is the distance between the $i$th electron and the $A$th nucleus while $r_{ij}$ is the interlectronic distance between electrons $i$ and $j$.
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The parameters $a=1/2$
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and $b=0.89$ were fixed, and the parameters $\gamma_{\text{O}}=1.15$ and $\gamma_{\text{H}}=0.35$
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were obtained by energy minimization with a single determinant.
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were obtained by energy minimization with a single \titou{HF?} determinant.
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The optimal CI expansion $\Psi^J$ is obtained by sampling the matrix elements
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of the Hamiltonian ($\mathbf{H}$) and the overlap ($\mathbf{S}$), in the
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basis of Jastrow-correlated determinants $e^J D_i$:
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@ -611,7 +622,7 @@ S_{ij} = \expval{ \frac{e^J D_i}{\Psi^J}\, \frac{e^J D_j}{\Psi^J} }
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and solving Eq.~\eqref{eq:ci-j}.\cite{Nightingale_2001}
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We can easily compare $\Psi^\mu$ and $\Psi^J$ as they are developed
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on the same Slater determinant basis.
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on the same set of Slater determinants.
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In Fig.~\ref{fig:overlap}, we plot the overlaps
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$\braket*{\Psi^J}{\Psi^\mu}$ obtained for water,
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and in Fig.~\ref{dmc_small} the FN-DMC energy of the wave functions
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@ -621,10 +632,9 @@ $\Psi^\mu$ together with that of $\Psi^J$.
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\begin{figure}
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\centering
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\includegraphics[width=\columnwidth]{overlap.pdf}
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\caption{\ce{H2O}, double-zeta basis set, 200 most important
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determinants of the FCI expansion (see Sec.~\ref{sec:rsdft-j}).
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Overlap of the RS-DFT CI expansions $\Psi^\mu$ with the CI
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expansion optimized in the presence of a Jastrow factor $\Psi^J$.}
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\caption{Overlap between $\Psi^\mu$ and $\Psi^J$ as a function of $\mu$ for \ce{H2O}.
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For these two trial wave functions, the CI expansion consists of the 200 most important
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determinants of the FCI expansion obtained with the VDZ-BFD basis (see Sec.~\ref{sec:rsdft-j} for more details).}
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\label{fig:overlap}
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\end{figure}
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%%% %%% %%% %%%
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@ -916,7 +926,7 @@ impacted by this spurious effect, as opposed to FCI.
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%%% TABLE IV %%%
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\begin{table}
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\caption{FN-DMC energies (in hartree) of the triplet carbon atom (BFD-VDZ) with
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\caption{FN-DMC energies (in hartree) of the triplet carbon atom (VDZ-BFD) 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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@ -1124,7 +1134,7 @@ while keeping the size-consistency.
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Note that when $\mu=0$ the number of determinants is not equal to one because
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we have used the natural orbitals of a first CIPSI calculation, and
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not the sr-PBE orbitals.
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not the srPBE orbitals.
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So the Kohn-Sham determinant is expressed as a linear combination of
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determinants built with natural orbitals. It is possible to add
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an extra step to the algorithm to compute the natural orbitals from the
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