diff --git a/Manuscript/rsdft-cipsi-qmc.bib b/Manuscript/rsdft-cipsi-qmc.bib index 4b5bc95..cbbe2a3 100644 --- a/Manuscript/rsdft-cipsi-qmc.bib +++ b/Manuscript/rsdft-cipsi-qmc.bib @@ -1,13 +1,47 @@ %% This BibTeX bibliography file was created using BibDesk. %% http://bibdesk.sourceforge.net/ -%% Created for Pierre-Francois Loos at 2020-08-03 21:30:38 +0200 +%% Created for Pierre-Francois Loos at 2020-08-07 13:35:24 +0200 %% Saved with string encoding Unicode (UTF-8) +@article{Burkatzki_2007, + Author = {Burkatzki, M. and Filippi, C. and Dolg, M.}, + Date-Added = {2020-08-07 13:35:15 +0200}, + Date-Modified = {2020-08-07 13:35:15 +0200}, + Doi = {10.1063/1.2741534}, + Issn = {1089-7690}, + Journal = {J. Chem. Phys.}, + Month = {Jun}, + Number = {23}, + Pages = {234105}, + Publisher = {AIP Publishing}, + Title = {Energy-consistent pseudopotentials for quantum Monte Carlo calculations}, + Url = {http://dx.doi.org/10.1063/1.2741534}, + Volume = {126}, + Year = {2007}, + Bdsk-Url-1 = {http://dx.doi.org/10.1063/1.2741534}} + +@article{Burkatzki_2008, + Author = {Burkatzki, M. and Filippi, Claudia and Dolg, M.}, + Date-Added = {2020-08-07 13:35:15 +0200}, + Date-Modified = {2020-08-07 13:35:15 +0200}, + Doi = {10.1063/1.2987872}, + Issn = {1089-7690}, + Journal = {J. Chem. Phys.}, + Month = {Oct}, + Number = {16}, + Pages = {164115}, + Publisher = {AIP Publishing}, + Title = {Energy-consistent small-core pseudopotentials for 3d-transition metals adapted to quantum Monte Carlo calculations}, + Url = {http://dx.doi.org/10.1063/1.2987872}, + Volume = {129}, + Year = {2008}, + Bdsk-Url-1 = {http://dx.doi.org/10.1063/1.2987872}} + @article{Schrodinger_1926, Author = {Schr\"odinger, Erwin}, Date-Added = {2020-08-03 21:29:29 +0200}, @@ -17,7 +51,8 @@ Pages = {1049--1070}, Title = {An Undulatory Theory of the Mechanics of Atoms and Molecules}, Volume = {28}, - Year = {1926}} + Year = {1926}, + Bdsk-Url-1 = {https://doi.org/10.1103/PhysRev.28.1049}} @article{Evangelista_2014, Author = {Evangelista, Francesco A.}, @@ -367,21 +402,6 @@ Title = {Gaussian˜16 {R}evision {C}.01}, Year = {2016}} -@article{Burkatzki_2008, - Author = {Burkatzki, M. and Filippi, Claudia and Dolg, M.}, - Doi = {10.1063/1.2987872}, - Issn = {1089-7690}, - Journal = {The Journal of Chemical Physics}, - Month = {Oct}, - Number = {16}, - Pages = {164115}, - Publisher = {AIP Publishing}, - Title = {Energy-consistent small-core pseudopotentials for 3d-transition metals adapted to quantum Monte Carlo calculations}, - Url = {http://dx.doi.org/10.1063/1.2987872}, - Volume = {129}, - Year = {2008}, - Bdsk-Url-1 = {http://dx.doi.org/10.1063/1.2987872}} - @article{Garniron_2019, Author = {Garniron, Yann and Applencourt, Thomas and Gasperich, Kevin and Benali, Anouar and Fert{\'e}, Anthony and Paquier, Julien and Pradines, Barth{\'e}l{\'e}my and Assaraf, Roland and Reinhardt, Peter and Toulouse, Julien and Barbaresco, Pierrette and Renon, Nicolas and David, Gr{\'e}goire and Malrieu, Jean-Paul and V{\'e}ril, Micka{\"e}l and Caffarel, Michel and Loos, Pierre-Fran{\c c}ois and Giner, Emmanuel and Scemama, Anthony}, Doi = {10.1021/acs.jctc.9b00176}, @@ -924,45 +944,42 @@ Bdsk-Url-1 = {https://doi.org/10.13140/RG.2.1.3187.9766}} @article{Tenno_2004, - author = {Ten-no, Seiichiro}, - title = {{Explicitly correlated second order perturbation theory: Introduction of a - rational generator and numerical quadratures}}, - journal = {J. Chem. Phys.}, - volume = {121}, - number = {1}, - pages = {117--129}, - year = {2004}, - month = {Jul}, - issn = {0021-9606}, - publisher = {American Institute of Physics}, - doi = {10.1063/1.1757439} -} + Author = {Ten-no, Seiichiro}, + Doi = {10.1063/1.1757439}, + Issn = {0021-9606}, + Journal = {J. Chem. Phys.}, + Month = {Jul}, + Number = {1}, + Pages = {117--129}, + Publisher = {American Institute of Physics}, + Title = {{Explicitly correlated second order perturbation theory: Introduction of a rational generator and numerical quadratures}}, + Volume = {121}, + Year = {2004}, + Bdsk-Url-1 = {https://doi.org/10.1063/1.1757439}} @article{Ten-no2000Nov, - author = {Ten-no, Seiichiro}, - title = {{A feasible transcorrelated method for treating electronic cusps using a frozen - Gaussian geminal}}, - journal = {Chem. Phys. Lett.}, - volume = {330}, - number = {1}, - pages = {169--174}, - year = {2000}, - month = {Nov}, - issn = {0009-2614}, - publisher = {North-Holland}, - doi = {10.1016/S0009-2614(00)01066-6} -} + Author = {Ten-no, Seiichiro}, + Doi = {10.1016/S0009-2614(00)01066-6}, + Issn = {0009-2614}, + Journal = {Chem. Phys. Lett.}, + Month = {Nov}, + Number = {1}, + Pages = {169--174}, + Publisher = {North-Holland}, + Title = {{A feasible transcorrelated method for treating electronic cusps using a frozen Gaussian geminal}}, + Volume = {330}, + Year = {2000}, + Bdsk-Url-1 = {https://doi.org/10.1016/S0009-2614(00)01066-6}} @article{Applencourt_2018, - author = {Applencourt, Thomas and Gasperich, Kevin and - Scemama, Anthony}, - title = {{Spin adaptation with determinant-based selected configuration interaction}}, - journal = {arXiv}, - year = {2018}, - month = {Dec}, - eprint = {1812.06902}, - url = {https://arxiv.org/abs/1812.06902v1} -} + Author = {Applencourt, Thomas and Gasperich, Kevin and Scemama, Anthony}, + Eprint = {1812.06902}, + Journal = {arXiv}, + Month = {Dec}, + Title = {{Spin adaptation with determinant-based selected configuration interaction}}, + Url = {https://arxiv.org/abs/1812.06902v1}, + Year = {2018}, + Bdsk-Url-1 = {https://arxiv.org/abs/1812.06902v1}} @software{qp2_2020, Author = {Anthony Scemama and Emmanuel Giner and Anouar Benali and Thomas Applencourt and Kevin Gasperich}, diff --git a/Manuscript/rsdft-cipsi-qmc.tex b/Manuscript/rsdft-cipsi-qmc.tex index d5a8546..b3dd70a 100644 --- a/Manuscript/rsdft-cipsi-qmc.tex +++ b/Manuscript/rsdft-cipsi-qmc.tex @@ -38,6 +38,7 @@ \newcommand{\ANL}{Argonne Leadership Computing Facility, Argonne National Laboratory, Argonne, IL 60439, United States} \newcommand{\LCPQ}{Laboratoire de Chimie et Physique Quantiques (UMR 5626), Universit\'e de Toulouse, CNRS, UPS, France} +\DeclareMathOperator{\erfc}{erfc} \begin{document} @@ -247,49 +248,52 @@ accuracy so all the CIPSI selections were made such that $|\EPT| < \label{sec:rsdft} Following the seminal work of Savin,\cite{Savin_1996,Toulouse_2004} -the Coulomb electron-electron interaction is split into a short-range -(sr) and a long range (lr) interaction as +the Coulomb operator entering the interelectronic repulsion is split into two pieces: \begin{equation} - \frac{1}{r_{ij}} = w_{\text{ee}}^{\text{lr}, \mu}(r_{ij}) + \qty( - \frac{1}{r_{ij}} - w_{\text{ee}}^{\text{lr}, \mu}(r_{ij}) ) + \frac{1}{r} + = w_{\text{ee}}^{\text{sr}, \mu}(r) + + w_{\text{ee}}^{\text{lr}, \mu}(r) \end{equation} where -\begin{equation} - w_{\text{ee}}^{\text{lr},\mu}(r_{ij}) = \frac{\erf \qty( \mu\, r_{ij})}{r_{ij}} -\end{equation} -The main idea is to treat the short-range electron-electron -interaction with DFT, and the long range with wave function theory. -The parameter $\mu$ controls the range of the separation, and allows -to go continuously from the Kohn-Sham Hamiltonian ($\mu=0$) to -the FCI Hamiltinoan ($\mu = \infty$). +\begin{align} + w_{\text{ee}}^{\text{sr},\mu}(r) & = \frac{\erfc \qty( \mu\, r)}{r}, + & + w_{\text{ee}}^{\text{lr},\mu}(r) & = \frac{\erf \qty( \mu\, r)}{r} +\end{align} +are the singular short-range (sr) part and the non-singular long-range (lr) part, respectively, $\mu$ is the range-separation parameter which controls how rapidly the short-range part decays, $\erf(x)$ is the error function, and $\erfc(x) = 1 - \erf(x)$ is its complementary version. -To rigorously connect wave function theory and DFT, the universal -Levy-Lieb density functional\cite{Lev-PNAS-79,Lie-IJQC-83} is +The main idea behind RS-DFT is to treat the short-range part of the +interaction within KS-DFT, and the long range part within a WFT method like FCI in the present case. +The parameter $\mu$ controls the range of the separation, and allows +to go continuously from the KS Hamiltonian ($\mu=0$) to +the FCI Hamiltonian ($\mu = \infty$). + +To rigorously connect WFT and DFT, the universal +Levy-Lieb density functional \cite{Lev-PNAS-79,Lie-IJQC-83} is decomposed as \begin{equation} - \mathcal{F}[n] = \mathcal{F}^{\mathrm{lr},\mu}[n] + \bar{E}_{\mathrm{Hxc}}^{\mathrm{sr,}\mu}[n], + \mathcal{F}[n] = \mathcal{F}^{\text{lr},\mu}[n] + \bar{E}_{\text{Hxc}}^{\text{sr,}\mu}[n], \label{Fdecomp} \end{equation} -where $n$ is a one-particle density, -$\mathcal{F}^{\mathrm{lr},\mu}$ is a long-range universal density -functional and $\bar{E}_{\mathrm{Hxc}}^{\mathrm{sr,}\mu}$ is the +where $n$ is a one-electron density, +$\mathcal{F}^{\text{lr},\mu}$ is a long-range universal density +functional and $\bar{E}_{\text{Hxc}}^{\text{sr,}\mu}$ is the complementary short-range Hartree-exchange-correlation (Hxc) density -functional\cite{Savin_1996,Toulouse_2004}. +functional. \cite{Savin_1996,Toulouse_2004} One obtains the following expression for the ground-state electronic energy \begin{equation} - \label{min_rsdft} E_0= \min_{\Psi} \left\{ -\left - \langle\Psi|\hat{T}+\hat{W}_\mathrm{{ee}}^{\mathrm{lr},\mu}+\hat{V}_{\mathrm{ne}}|\Psi\right -\rangle -+ \bar{E}^{\mathrm{sr},\mu}_{\mathrm{Hxc}}[n_\Psi]\right\} + \label{min_rsdft} E_0= \min_{\Psi} \qty{ + \mel{\Psi}{\hat{T}+\hat{W}_\text{{ee}}^{\text{lr},\mu}+\hat{V}_{\text{ne}}}{\Psi} + + \bar{E}^{\text{sr},\mu}_{\text{Hxc}}[n_\Psi] + }, \end{equation} with $\hat{T}$ the kinetic energy operator, -$\hat{W}_\mathrm{ee}^{\mathrm{lr}}$ the long-range +$\hat{W}_\text{ee}^{\text{lr}}$ the long-range electron-electron interaction, -$n_\Psi$ the one-particle density associated with $\Psi$, -and $\hat{V}_{\mathrm{ne}}$ the electron-nucleus potential. -The minimizing multi-determinant wave function $\Psi^\mu$ +$n_\Psi$ the one-electron density associated with $\Psi$, +and $\hat{V}_{\text{ne}}$ the electron-nucleus potential. +The minimizing multideterminant wave function $\Psi^\mu$ can be determined by the self-consistent eigenvalue equation \begin{equation} \label{rs-dft-eigen-equation} @@ -298,50 +302,51 @@ can be determined by the self-consistent eigenvalue equation with the long-range interacting Hamiltonian \begin{equation} \label{H_mu} - \hat{H}^\mu[n_{\Psi^{\mu}}] = \hat{T}+\hat{W}_{\mathrm{ee}}^{\mathrm{lr},\mu}+\hat{V}_{\mathrm{ne}}+ \hat{\bar{V}}_{\mathrm{Hxc}}^{\mathrm{sr},\mu}[n_{\Psi^{\mu}}], + \hat{H}^\mu[n_{\Psi^{\mu}}] = \hat{T}+\hat{W}_{\text{ee}}^{\text{lr},\mu}+\hat{V}_{\text{ne}}+ \hat{\bar{V}}_{\text{Hxc}}^{\text{sr},\mu}[n_{\Psi^{\mu}}], \end{equation} where -$\hat{\bar{V}}_{\mathrm{Hxc}}^{\mathrm{sr},\mu}$ +$\hat{\bar{V}}_{\text{Hxc}}^{\text{sr},\mu}$ is the complementary short-range Hartree-exchange-correlation potential operator. Once $\Psi^{\mu}$ has been calculated, the electronic ground-state -energy is obtained by +energy is obtained as \begin{equation} \label{E-rsdft} - E_0= \mel{\Psi^{\mu}}{\hat{T}+\hat{W}_\mathrm{{ee}}^{\mathrm{lr},\mu}+\hat{V}_{\mathrm{ne}}}{\Psi^{\mu}}+\bar{E}^{\mathrm{sr},\mu}_{\mathrm{Hxc}}[n_{\Psi^\mu}]. + 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 $\mu=0$, the long-range interaction vanishes, -$w_{\mathrm{ee}}^{\mathrm{lr},\mu=0}(r_{12}) = 0$, and thus -range-separated DFT (RS-DFT) reduces to standard KS-DFT and $\Psi^\mu$ +Note that, for $\mu=0$, \titou{the long-range interaction vanishes}, \ie, +$w_{\text{ee}}^{\text{lr},\mu=0}(r) = 0$, and thus +RS-DFT reduces to standard KS-DFT and $\Psi^\mu$ is the KS determinant. For $\mu\to\infty$, the long-range -interaction becomes the standard Coulomb interaction, -$w_{\mathrm{ee}}^{\mathrm{lr},\mu\to\infty}(r_{12}) = 1/r_{12}$, and -thus RS-DFT reduces to standard wave-function theory and $\Psi^\mu$ is +interaction becomes the standard Coulomb interaction, \ie, +$w_{\text{ee}}^{\text{lr},\mu\to\infty}(r) = r^{-1}$, and +thus RS-DFT reduces to standard WFT and $\Psi^\mu$ is the FCI wave function. \begin{figure*} \centering \includegraphics[width=0.7\linewidth]{algorithm.pdf} \caption{Algorithm showing the generation of the RS-DFT wave - function.} + function $\Psi^{\mu}$ starting from the .} \label{fig:algo} \end{figure*} -Hence we have a continuous path connecting the KS determinant to the -FCI wave function, and as the KS nodes are of higher quality than the +Hence, range separation creates a continuous path connecting smoothly the KS determinant to the +FCI wave function. Because the KS nodes are of higher quality than the HF nodes, we expect that using wave functions built along this path will always provide reduced fixed-node errors compared to the path -connecting HF to FCI using an increasing number of selected -determinants. +connecting HF to FCI which consists in increasing the number of determinants. -We can follow this path by performing FCI calculations using the -RS-DFT Hamiltonian with different values of $\mu$. In this work, we +We follow the KS-to-FCI path by performing FCI calculations using the +RS-DFT Hamiltonian with different values of $\mu$. +Our algorithm, depicted in Fig.~\ref{fig:algo}, starts with a +single- and multi-determinant wave function $\Psi^{(0)}$. +One of the particularity of the present work is that we have used the CIPSI algorithm to perform approximate FCI calculations -with the RS-DFT Hamiltonians,\cite{GinPraFerAssSavTou-JCP-18} -$\hat{H}^\mu$ as shown in figure~\ref{fig:algo}. In the outer loop -(red), a CIPSI selection is performed with a RS-Hamiltonian -parameterized using the current density. +with the RS-DFT Hamiltonians $\hat{H}^\mu$. \cite{GinPraFerAssSavTou-JCP-18} +In the outer loop (red), a CIPSI selection is performed with a RS Hamiltonian +parameterized using the current one-electron density. An inner loop (blue) is introduced to accelerate the convergence of the self-consistent calculation, in which the set of determinants is kept fixed, and only the diagonalization of the @@ -349,31 +354,37 @@ RS-Hamiltonian is performed iteratively with the updated density. The convergence of the algorithm was further improved by introducing a direct inversion in the iterative subspace (DIIS) step to extrapolate the density both in the outer and inner loops. -Note that any range-separated post-Hartree-Fock method can be +Note that any range-separated post-HF method can be implemented using this scheme by just replacing the CIPSI step by the post-HF method of interest. +\titou{T2: introduce $\tau_1$ and $\tau_2$. More description of the algorithm needed.} +Note that, thanks to the self-consistent nature of the algorithm, +the final trial wave function $\Psi^{\mu}$ is independent of the starting wave function $\Psi^{(0)}$. + \section{Computational details} \label{sec:comp-details} +\titou{The geometries for the G2 data set.} -All the calculations were made using BFD -pseudopotentials\cite{Burkatzki_2008} with the associated double-, + +All the calculations have been performed using Burkatzki-Filippi-Dolg (BFD) +pseudopotentials \cite{Burkatzki_2007,Burkatzki_2008} with the associated double-, triple-, and quadruple-$\zeta$ basis sets (BFD-VXZ). -CCSD(T) and KS-DFT calculations were made with -\emph{Gaussian09},\cite{g16} using an unrestricted Hartree-Fock -determinant as a reference for open-shell systems. +The small-core BFD pseudopotentials include scalar relativistic effects. +CCSD(T) and KS-DFT energies have been computed with +\emph{Gaussian09},\cite{g16} using the unrestricted formalism for open-shell systems. -All the CIPSI calculations were made with \emph{Quantum +All the CIPSI calculations have been performed with \emph{Quantum Package}.\cite{Garniron_2019,qp2_2020} We used the short-range version -of the Perdew-Burke-Ernzerhof (PBE)~\cite{PerBurErn-PRL-96} exchange -and correlation functionals of +of the Perdew-Burke-Ernzerhof (PBE) \cite{PerBurErn-PRL-96} exchange +and correlation functionals defined in Ref.~\onlinecite{GolWerStoLeiGorSav-CP-06} (see also Refs.~\onlinecite{TouColSav-JCP-05,GolWerSto-PCCP-05}). The convergence criterion for stopping the CIPSI calculations -was $\EPT < 1$~m\hartree{} $\vee \Ndet > 10^7$. -All the wave functions are eigenfunctions of the $S^2$ operator, as -described in ref~\onlinecite{Applencourt_2018}. +has been set to $\EPT < 1$~m\hartree{} or $ \Ndet > 10^7$. +All the wave functions are eigenfunctions of the $\Hat{S}^2$ spin operator, as +described in Ref.~\onlinecite{Applencourt_2018}. Quantum Monte Carlo calculations were made with QMC=Chem,\cite{scemama_2013} in the determinant localization approximation (DLA),\cite{Zen_2019} @@ -383,15 +394,18 @@ the pseudopotential is localized. Hence, in the DLA the fixed-node energy is independent of the Jastrow factor, as in all-electron calculations. Simple Jastrow factors were used to reduce the fluctuations of the local energy. - +\titou{time step $5 \times 10^{-4}$. +All-electron move DMC.} \section{Influence of the range-separation parameter on the fixed-node error} \label{sec:mu-dmc} + +%%% TABLE I %%% \begin{table} - \caption{Fixed-node energies (in hartree) and number of determinants in \ce{H2O} and \ce{F2} with various trial wave functions.} + \caption{Fixed-node energies $\EDMC$ (in hartree) and number of determinants $\Ndet$ in \ce{H2O} and \ce{F2} with various trial wave functions.} \label{tab:h2o-dmc} \centering \begin{ruledtabular} @@ -425,21 +439,23 @@ System & $\mu$ & $\Ndet$ & $\EDMC$ & $\Ndet$ & $\ED \end{tabular} \end{ruledtabular} \end{table} +%%% %%% %%% %%% \begin{figure} \centering \includegraphics[width=\columnwidth]{h2o-dmc.pdf} - \caption{Fixed-node energies of the water molecule for different - values of $\mu$, using the sr-LDA or sr-PBE short-range density - functionals to build the trial wave function.} + \caption{Fixed-node energies of \ce{H2O} for different + values of $\mu$, using the srLDA or srPBE density + functionals to build the trial wave function. + \titou{Toto: please remove hyphens in sr-LDA and sr-PBE.}} \label{fig:h2o-dmc} \end{figure} \begin{figure} \centering \includegraphics[width=\columnwidth]{f2-dmc.pdf} - \caption{Fixed-node energies of difluorine for different - values of $\mu$.} + \caption{Fixed-node energies of \ce{F2} for different + values of $\mu$ using the srPBE density functional to build the trial wave function.} \label{fig:f2-dmc} \end{figure} The first question we would like to address is the quality of the @@ -449,12 +465,12 @@ 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. We considered two weakly correlated molecular systems: the water -molecule and the fluorine dimer, near their equilibrium -geometry\cite{Caffarel_2016}. +molecule and the fluorine dimer, near their equilibrium. +geometry\cite{Caffarel_2016} -\subsection{Fixed node energy of $\Psi^\mu$} +\subsection{Fixed-node energy of $\Psi^\mu$} \label{sec:fndmc_mu} -From table~\ref{tab:h2o-dmc} and figures~\ref{fig:h2o-dmc} +From Table~\ref{tab:h2o-dmc} and Figs.~\ref{fig:h2o-dmc} and~\ref{fig:f2-dmc}, one can clearly observe that using FCI trial wave functions ($\mu = \infty$) gives FN-DMC energies which are lower than the energies obtained with a single Kohn-Sham determinant ($\mu=0$): @@ -476,7 +492,7 @@ LDA exchange-correlation functional give very similar FN-DMC energy with respect to those obtained with the short-range PBE functional, even if the RS-DFT energies obtained with these two functionals differ by several tens of m\hartree{}. -\subsection{Link between RS-DFT and jastrow factors } +\subsection{Link between RS-DFT and Jastrow factors } \label{sec:rsdft-j} The data obtained in \ref{sec:fndmc_mu} show that RS-DFT can provide CI coefficients giving trial wave functions with better nodes than FCI wave functions. @@ -571,12 +587,12 @@ These data suggest that the wave functions $\Psi^\mu$ and $\Psi^J$ are similar, and therefore that the operators that produced these wave functions (\textit{i.e.} $H^\mu[n]$ and $e^{-J}He^J$) contain similar physics. Considering the form of $\hat{H}^\mu[n]$ (see Eq.~\eqref{H_mu}), one can notice that the differences with respect to the usual Hamiltonian come -from the non-divergent two-body interaction $\hat{W}_{\mathrm{ee}}^{\mathrm{lr},\mu}$ -and the effective one-body potential $\hat{\bar{V}}_{\mathrm{Hxc}}^{\mathrm{sr},\mu}[n]$ which is the functional derivative of the Hartree-exchange-correlation functional. +from the non-divergent two-body interaction $\hat{W}_{\text{ee}}^{\text{lr},\mu}$ +and the effective one-body potential $\hat{\bar{V}}_{\text{Hxc}}^{\text{sr},\mu}[n]$ which is the functional derivative of the Hartree-exchange-correlation functional. 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 one-body potential $\hat{\bar{V}}_{\mathrm{Hxc}}^{\mathrm{sr},\mu}[n_{\Psi^{\mu}}]$, +while the effective one-body potential $\hat{\bar{V}}_{\text{Hxc}}^{\text{sr},\mu}[n_{\Psi^{\mu}}]$, provided that it is exact, maintains the exact one-body density. This is clearly what has been observed from the plots in figures ~\ref{fig:n1} and~\ref{fig:n2} in the case of the water molecule. Regarding now the transcorrelated Hamiltonian $e^{-J}He^J$, as pointed out by Ten-No\cite{Ten-no2000Nov},