Merge branch 'master' of git.irsamc.ups-tlse.fr:scemama/RSDFT-CIPSI-QMC
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%% This BibTeX bibliography file was created using BibDesk.
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%% http://bibdesk.sourceforge.net/
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%% Created for Pierre-Francois Loos at 2020-08-07 13:35:24 +0200
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%% Created for Pierre-Francois Loos at 2020-08-07 20:17:07 +0200
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%% Saved with string encoding Unicode (UTF-8)
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@article{Assaraf_2000,
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Author = {Assaraf, Roland and Caffarel, Michel and Khelif, Anatole},
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Date-Added = {2020-08-07 20:12:45 +0200},
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Date-Modified = {2020-08-07 20:12:45 +0200},
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Doi = {10.1103/physreve.61.4566},
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Issn = {1095-3787},
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Journal = {Phys. Rev. E},
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Month = {Apr},
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Number = {4},
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Pages = {4566--4575},
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Publisher = {American Physical Society (APS)},
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Title = {Diffusion Monte Carlo methods with a fixed number of walkers},
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Url = {http://dx.doi.org/10.1103/PhysRevE.61.4566},
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Volume = {61},
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Year = {2000},
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Bdsk-Url-1 = {http://dx.doi.org/10.1103/PhysRevE.61.4566},
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Bdsk-Url-2 = {http://dx.doi.org/10.1103/physreve.61.4566}}
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@article{Assaraf_2007,
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Author = {Assaraf, Roland and Caffarel, Michel and Scemama, Anthony},
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Date-Added = {2020-08-07 20:12:45 +0200},
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Date-Modified = {2020-08-07 20:12:45 +0200},
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Doi = {10.1103/physreve.75.035701},
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Issn = {1550-2376},
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Journal = {Phys. Rev. E},
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Month = {Mar},
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Number = {3},
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Publisher = {American Physical Society (APS)},
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Title = {Improved Monte Carlo estimators for the one-body density},
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Url = {http://dx.doi.org/10.1103/PhysRevE.75.035701},
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Volume = {75},
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Year = {2007},
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Bdsk-Url-1 = {http://dx.doi.org/10.1103/PhysRevE.75.035701},
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Bdsk-Url-2 = {http://dx.doi.org/10.1103/physreve.75.035701}}
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@article{Burkatzki_2007,
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Author = {Burkatzki, M. and Filippi, C. and Dolg, M.},
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Date-Added = {2020-08-07 13:35:15 +0200},
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@ -65,7 +65,7 @@
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By combining density-functional theory (DFT) and wave function theory (WFT) via the range separation (RS) of the interelectronic Coulomb operator, we obtain accurate fixed-node diffusion Monte Carlo (FN-DMC) energies with compact multideterminant trial wave functions.
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These compact trial wave functions are generated via the diagonalization of the RS-DFT Hamiltonian.
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In particular, we combine here short-range correlation functionals with selected configuration interaction (SCI).
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As the WFT method is relieved from describing the short-range part of the correlation hole around the electron?electron coalescence points, the number of determinants in the trial wave function required to reach a given accuracy is significantly reduced as compared to a conventional SCI calculation.
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As the WFT method is relieved from describing the short-range part of the correlation hole around the electron-electron coalescence points, the number of determinants in the trial wave function required to reach a given accuracy is significantly reduced as compared to a conventional SCI calculation.
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\titou{T2: work in progress.}
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\end{abstract}
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@ -400,9 +400,10 @@ the pseudopotential is localized. Hence, in the DLA the fixed-node
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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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\titou{time step $5 \times 10^{-4}$.
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All-electron move DMC.}
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The FN-DMC simulations are performed with the stochastic reconfiguration
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algorithm developed by Assaraf \textit{et al.}, \cite{Assaraf_2000}
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with a time step of $5 \times 10^{-4}$ a.u.
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\titou{All-electron move DMC?}
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\section{Influence of the range-separation parameter on the fixed-node
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@ -497,7 +498,7 @@ where $\Psi = \sum_I c_I D_I$ is a general linear combination of Slater determin
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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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\begin{equation}
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\Psi^J = \text{argmin}_{\Psi}\frac{ \langle \Psi | e^{J} H e^{J} |\Psi \rangle}{\langle \Psi | e^{2J} |\Psi \rangle}.
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\Psi^J = \text{argmin}_{\Psi}\frac{ \mel{ \Psi }{ e^{J} 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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\begin{equation}
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@ -558,16 +559,16 @@ introducing short-range correlation with DFT has
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an impact on the CI coefficients similar to the Jastrow factor.
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In the case of F$_2$, the Jastrow factor has
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very little effect on the CI coefficients, as the overlap
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$\braket{\Psi^J}{\Psi^{\mu=\infty}}$ is very close to
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$\braket*{\Psi^J}{\Psi^{\mu=\infty}}$ is very close to
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$1$.
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Nevertheless, a slight maximum is obtained for
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$\mu=5$~bohr$^{-1}$.
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In order to refine the comparison between $\Psi^\mu$ and $\Psi^J$,
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we report, in the case of the water molecule in the double-zeta basis set,
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several quantities related to the one- and two-body density of $\Psi^J$ and $\Psi^\mu$ with different values of $\mu$.
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First, we report in table~\ref{table_on_top} the integrated on-top pair density $\langle n_2({\bf r},{\bf r}) \rangle$
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First, we report in table~\ref{table_on_top} the integrated on-top pair density $\expval{ n_2({\bf r},{\bf r}) }$
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\begin{equation}
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\langle n_2({\bf r},{\bf r}) \rangle = \int \text{d}{\bf r} \,\,n_2({\bf r},{\bf r})
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\expval{ n_2({\bf r},{\bf r}) } = \int \text{d}{\bf r} \,\,n_2({\bf r},{\bf r})
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\end{equation}
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where $n_2({\bf r}_1,{\bf r}_2)$ is the two-body density (normalized to $N(N-1)$ where $N$ is the number of electrons)
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obtained for both $\Psi^\mu$ and $\Psi^J$.
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@ -601,12 +602,12 @@ Therefore, one can understand the similarity between the eigenfunctions of $H^\m
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they both deal with an effective non-divergent interaction but still produce reasonable one-body density.
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\begin{table}
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\caption{H$_2$O, double-zeta basis set. Integrated on-top pair density $\langle n_2({\bf r},{\bf r}) \rangle$
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\caption{H$_2$O, double-zeta basis set. Integrated on-top pair density $\expval{ n_2({\bf r},{\bf r}) }$
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for $\Psi^J$ and $\Psi^\mu$ with different values of $\mu$. }
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\label{table_on_top}
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\begin{ruledtabular}
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\begin{tabular}{cc}
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$\mu$ & $\langle n_2({\bf r},{\bf r}) \rangle$ \\
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$\mu$ & $\expval{ n_2({\bf r},{\bf r}) }$ \\
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\hline
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0.00 & 1.443 \\
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0.25 & 1.438 \\
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@ -624,7 +625,7 @@ they both deal with an effective non-divergent interaction but still produce rea
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\begin{figure}
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\centering
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\includegraphics[width=\columnwidth]{on-top-mu.pdf}
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\caption{H$_2$O, double-zeta basis set. Integrated on-top pair density $\langle n_2({\bf r},{\bf r}) \rangle$ along the OH axis, for $\Psi^J$ and $\Psi^\mu$ with different values of $\mu$. }
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\caption{H$_2$O, double-zeta basis set. Integrated on-top pair density $\expval{ n_2({\bf r},{\bf r}) }$ along the OH axis, for $\Psi^J$ and $\Psi^\mu$ with different values of $\mu$. }
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\label{fig:n2}
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\end{figure}
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\begin{figure}
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