saving work in Jastrow part
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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 2020-08-08 08:15:47 +0200
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%% Created for Pierre-Francois Loos at 2020-08-09 15:41:06 +0200
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%% Saved with string encoding Unicode (UTF-8)
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@article{Scemama_2006c,
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Author = {Scemama, Anthony and Filippi, Claudia},
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Date-Added = {2020-08-09 15:41:04 +0200},
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Date-Modified = {2020-08-09 15:41:04 +0200},
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Doi = {10.1103/physrevb.73.241101},
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Issn = {1550-235X},
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Journal = {Phys. Rev. B},
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Month = {Jun},
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Number = {24},
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Publisher = {American Physical Society (APS)},
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Title = {Simple and efficient approach to the optimization of correlated wave functions},
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Url = {http://dx.doi.org/10.1103/PhysRevB.73.241101},
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Volume = {73},
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Year = {2006},
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Bdsk-Url-1 = {http://dx.doi.org/10.1103/PhysRevB.73.241101},
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Bdsk-Url-2 = {http://dx.doi.org/10.1103/physrevb.73.241101}}
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@article{Umrigar_2005,
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Author = {Umrigar, C. J. and Filippi, Claudia},
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Date-Added = {2020-08-09 15:37:17 +0200},
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Date-Modified = {2020-08-09 15:37:17 +0200},
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Doi = {10.1103/physrevlett.94.150201},
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Issn = {1079-7114},
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Journal = {Phys. Rev. Lett.},
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Month = {Apr},
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Number = {15},
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Publisher = {American Physical Society (APS)},
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Title = {Energy and Variance Optimization of Many-Body Wave Functions},
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Url = {http://dx.doi.org/10.1103/PhysRevLett.94.150201},
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Volume = {94},
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Year = {2005},
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Bdsk-Url-1 = {http://dx.doi.org/10.1103/PhysRevLett.94.150201},
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Bdsk-Url-2 = {http://dx.doi.org/10.1103/physrevlett.94.150201}}
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@article{Toulouse_2007,
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Author = {Toulouse, Julien and Umrigar, C. J.},
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Date-Added = {2020-08-09 15:36:54 +0200},
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Date-Modified = {2020-08-09 15:36:54 +0200},
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Doi = {10.1063/1.2437215},
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Issn = {1089-7690},
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Journal = {J. Chem. Phys.},
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Month = {Feb},
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Number = {8},
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Pages = {084102},
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Publisher = {AIP Publishing},
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Title = {Optimization of quantum Monte Carlo wave functions by energy minimization},
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Url = {http://dx.doi.org/10.1063/1.2437215},
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Volume = {126},
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Year = {2007},
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Bdsk-Url-1 = {http://dx.doi.org/10.1063/1.2437215}}
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@article{Toulouse_2008,
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Author = {Toulouse, Julien and Umrigar, C. J.},
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Date-Added = {2020-08-09 15:36:54 +0200},
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Date-Modified = {2020-08-09 15:36:54 +0200},
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Doi = {10.1063/1.2908237},
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Issn = {1089-7690},
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Journal = {J. Chem. Phys.},
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Month = {May},
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Number = {17},
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Pages = {174101},
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Publisher = {AIP Publishing},
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Title = {Full optimization of Jastrow--Slater wave functions with application to the first-row atoms and homonuclear diatomic molecules},
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Url = {http://dx.doi.org/10.1063/1.2908237},
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Volume = {128},
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Year = {2008},
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Bdsk-Url-1 = {http://dx.doi.org/10.1063/1.2908237}}
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@article{Umrigar_2007,
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Author = {Umrigar, C. J. and Toulouse, Julien and Filippi, Claudia and Sorella, S. and Hennig, R. G.},
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Date-Added = {2020-08-09 15:36:54 +0200},
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Date-Modified = {2020-08-09 15:36:54 +0200},
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Doi = {10.1103/physrevlett.98.110201},
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Issn = {1079-7114},
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Journal = {Phys. Rev. Lett.},
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Month = {Mar},
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Number = {11},
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Publisher = {American Physical Society (APS)},
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Title = {Alleviation of the Fermion-Sign Problem by Optimization of Many-Body Wave Functions},
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Url = {http://dx.doi.org/10.1103/PhysRevLett.98.110201},
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Volume = {98},
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Year = {2007},
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Bdsk-Url-1 = {http://dx.doi.org/10.1103/PhysRevLett.98.110201},
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Bdsk-Url-2 = {http://dx.doi.org/10.1103/physrevlett.98.110201}}
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@article{Pulay_1980,
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Author = {Pulay, P{\'e}ter},
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Date-Added = {2020-08-08 08:15:42 +0200},
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@ -622,20 +707,6 @@
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Year = {1977},
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Bdsk-Url-1 = {https://doi.org/10.1002/qua.560120826}}
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@article{Umrigar_2005,
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Author = {Umrigar, C. J. and Filippi, Claudia},
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Doi = {10.1103/PhysRevLett.94.150201},
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Issn = {1079-7114},
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Journal = {Phys. Rev. Lett.},
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Month = {Apr},
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Number = {15},
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Pages = {150201},
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Publisher = {American Physical Society},
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Title = {{Energy and Variance Optimization of Many-Body Wave Functions}},
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Volume = {94},
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Year = {2005},
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Bdsk-Url-1 = {https://doi.org/10.1103/PhysRevLett.94.150201}}
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@article{Huron_1973,
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Author = {Huron, B. and Malrieu, J. P. and Rancurel, P.},
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Doi = {10.1063/1.1679199},
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@ -30,6 +30,8 @@
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\newcommand{\fnt}{\footnotetext}
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\newcommand{\tabc}[1]{\multicolumn{1}{c}{#1}}
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\newcommand{\br}{\mathbf{r}}
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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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@ -514,14 +516,15 @@ The take-home message here is that RS-DFT trial wave functions yield a lower fix
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\subsection{Link between RS-DFT and Jastrow factors }
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\label{sec:rsdft-j}
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%======================================================
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The data obtained in Sec.~\ref{sec:fndmc_mu} show that RS-DFT can provide CI coefficients
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giving trial wave functions with better nodes than FCI wave functions.
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Such behaviour can be compared to the common practice of
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re-optimizing the Slater part of a trial wave function in the presence of the Jastrow factor.
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In the present paragraph, we would like therefore to elaborate some more on the link between RS-DFT
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and wave function optimization with the presence of a Jastrow factor.
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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 directty compared to the common practice of
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re-optimizing the multideterminant 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_2006c,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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\titou{T2: maybe we should mention that we only reoptimize the CI coefficients as it is of common practice to re-optimize more than this.}
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Let us assume a fixed Jastrow factor $J({\bf{r}_1}, \hdots , {\bf{r}_N})$,
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Let us assume a fixed Jastrow factor $J(\br_1, \ldots , \br_N)$,
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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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The only remaining variational parameters in $\Phi$ are therefore the Slater part $\Psi$.
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@ -533,7 +536,7 @@ Such a wave function $\Psi^J$ satisfies the generalized hermitian eigenvalue equ
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\begin{equation}
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e^{J} H e^{J} \Psi^J = E e^{2J} \Psi^J,
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\end{equation}
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but also the non-hermitian transcorrelated eigenvalue problem\cite{many_things}
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but also the non-hermitian transcorrelated eigenvalue problem \cite{many_things}
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\begin{equation}
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\label{eq:transcor}
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e^{-J} H e^{J} \Psi^J = E \Psi^J,
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@ -591,12 +594,12 @@ an impact on the CI coefficients similar to the Jastrow factor.
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%%% TABLE II %%%
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\begin{table}
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\caption{\ce{H2O}, double-zeta basis set. Integrated on-top pair density $\expval{ n_2({\bf r},{\bf r}) }$
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\caption{\ce{H2O}, double-zeta basis set. Integrated on-top pair density $\expval{ n_2(\br,\br) }$
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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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$\mu$ & $\expval{ n_2({\bf r},{\bf r}) }$ \\
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$\mu$ & $\expval{ n_2(\br,\br) }$ \\
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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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@ -617,7 +620,7 @@ an impact on the CI coefficients similar to the Jastrow factor.
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\centering
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\includegraphics[width=\columnwidth]{on-top-mu.pdf}
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\caption{\ce{H2O}, double-zeta basis set. On-top pair
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density $n_2({\bf r},{\bf r})$ along the \ce{O-H} axis,
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density $n_2(\br,\br)$ along the \ce{O-H} axis,
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for $\Psi^J$ (dashed curve) 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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@ -627,7 +630,7 @@ an impact on the CI coefficients similar to the Jastrow factor.
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\begin{figure}
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\centering
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\includegraphics[width=\columnwidth]{density-mu.pdf}
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\caption{\ce{H2O}, double-zeta basis set. Density $n({\bf r})$ along
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\caption{\ce{H2O}, double-zeta basis set. Density $n(\br)$ along
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the \ce{O-H} axis, for $\Psi^J$ (dashed curve) and $\Psi^\mu$ with different values of $\mu$. }
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\label{fig:n1}
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\end{figure}
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@ -639,14 +642,14 @@ report several quantities related to the one- and two-body density of
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$\Psi^J$ and $\Psi^\mu$ with different values of $\mu$. First, we
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report in Table~\ref{table_on_top} the integrated on-top pair density
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\begin{equation}
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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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\expval{ n_2(\br,\br) } = \int d\br \,\,n_2(\br,\br)
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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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where $n_2(\br_1,\br_2$ 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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Then, in order to have a pictorial representation of both the on-top
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pair density and the density, we report in Fig.~\ref{fig:n1} and Fig.~\ref{fig:n2}
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the plots of the total density $n({\bf r})$ and on-top pair density
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$n_2({\bf r},{\bf r})$ along one \ce{O-H} axis of the water molecule.
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the plots of the total density $n(\br)$ and on-top pair density
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$n_2(\br,\br)$ along one \ce{O-H} axis of the water molecule.
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From these data, one can clearly notice several trends.
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First, from Table~\ref{table_on_top}, we can observe that the overall
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@ -694,7 +697,7 @@ As a conclusion of the first part of this study, we can notice that:
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\item the optimal value of $\mu$ depends on the system and the
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basis set, and the larger the basis set, the larger the optimal value
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of $\mu$,
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\item numerical experiments (overlap $\braket{\Psi^\mu}{\Psi^J}$,
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\item numerical experiments (overlap $\braket*{\Psi^\mu}{\Psi^J}$,
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one-body density, on-top pair density, and FN-DMC energy) indicate
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that the RS-DFT scheme essentially plays the role of a simple Jastrow factor,
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\ie, mimicking short-range correlation effects. The latter
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@ -793,15 +796,15 @@ only depends on the parameters of the determinantal component. Using a
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non-size-consistent Jastrow factor, or a non-optimal Jastrow factor will
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not introduce an additional error in FN-DMC calculations, although it
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will reduce the statistical errors by reducing the variance of the
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local energy. Moreover, the integrals involved in the pseudo-potential
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local energy. Moreover, the integrals involved in the pseudopotential
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are computed analytically and the computational cost of the
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pseudo-potential is dramatically reduced (for more detail, see
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pseudopotential is dramatically reduced (for more detail, see
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Ref.~\onlinecite{Scemama_2015}).
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%%% TABLE III %%%
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\begin{table}
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\caption{FN-DMC energies (in hartree) using the VDZ-BFD basis set
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and pseudo-potential of the fluorine atom and the dissociated fluorine
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and pseudopotential of the fluorine atom and the dissociated fluorine
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dimer, and size-consistency error. }
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\label{tab:size-cons}
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\begin{ruledtabular}
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@ -846,7 +849,7 @@ respect to the spin quantum number $m_s$, but the multiplication by a
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Jastrow factor introduces spin contamination if the parameters
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for the same-spin electron pairs are different from those
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for the opposite-spin pairs.\cite{Tenno_2004}
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Again, when pseudo-potentials are used this tiny error is transferred
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Again, when pseudopotentials are used this tiny error is transferred
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in the FN-DMC energy unless the determinant localization approximation
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is used.
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@ -882,7 +885,7 @@ impacted by this spurious effect, as opposed to FCI.
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In this section, we investigate the impact of the spin contamination
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due to the short-range density functional on the FN-DMC energy. We have
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computed the energies of the carbon atom in its triplet state
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with BFD pseudo-potentials and the corresponding double-zeta basis
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with BFD pseudopotentials and the corresponding double-zeta basis
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set. The calculation was done with $m_s=1$ (3 $\uparrow$ electrons
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and 1 $\downarrow$ electrons) and with $m_s=0$ (2 $\uparrow$ and 2
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$\downarrow$ electrons).
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