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@ 131,7 +131,7 @@ $$


set xrange [0.0:6]


set xtics 0.5


set xlabel "1/$\mu$"


set ylabel "E(srPBE)E(FCI)"


set ylabel "E(srPBE)E(FCI)"


plot data u (1./$1):($3+17.164669) w p notitle , data u (1./$1):($3+17.164669) sm cs title "H_{2}O", \


data u (1./$1):($7+2.*$5+15.83002472+2.*0.49904523) w p notitle , data u (1./$1):($7+2.*$5+15.83002472+2.*0.49904523) sm cs title "O"


#+END_SRC


@ 308,7 +308,7 @@ $$


set xrange [0.0:6]


set xtics 0.5


set xlabel "1/$\mu$"


set ylabel "E(srPBE)E(FCI)"


set ylabel "E(srPBE)E(FCI)"


plot data u (1./$1):($3+17.23217115) w p notitle , data u (1./$1):($3+17.23217115) sm cs title "H_{2}O", \


data u (1./$1):($7+2.*$5+15.88257866+2.*0.499042917477163) w p notitle , data u (1./$1):($7+2.*$5+15.88257866+2.*0.499042917477163) sm cs title "O"


#+END_SRC


@ 523,7 +523,7 @@ $$


set xrange [0.0:6]


set xtics 0.5


set xlabel "1/$\mu$"


set ylabel "E(srPBE)E(FCI)"


set ylabel "E(srPBE)E(FCI)"




plot data u (1./$1):($3+17.25606450) w p notitle , data u (1./$1):($3+17.25606450) sm cs title "H_{2}O", \


data u (1./$1):($7+2.*$5+2.*0.499915948575730+15.89763971) w p notitle , data u (1./$1):($7+2.*$5+2.*0.499915948575730+15.89763971) sm cs title "O"


@ 5065,7 +5065,7 @@ D 1


set xrange [0.0:6]


set xtics 0.5


set xlabel "1/$\mu$"


set ylabel "E(srPBE)E(FCI)"


set ylabel "E(srPBE)E(FCI)"


plot data u (1./$1):($3+11.02347855) w p notitle , data u (1./$1):($3+11.02347855) sm cs title "C_{2}", \


data u (1./$1):(2.*$5+2.*5.40960365) w p notitle , data u (1./$1):(2.*$5+2.*5.40960365) sm cs title "C"


#+END_SRC


@ 5199,7 +5199,7 @@ F 1


set xrange [0.0:6]


set xtics 0.5


set xlabel "1/$\mu$"


set ylabel "E(srPBE)E(FCI)"


set ylabel "E(srPBE)E(FCI)"


plot data u (1./$1):($3+11.02347855) w p notitle , data u (1./$1):($3+11.02347855) sm cs title "C_{2}", \


data u (1./$1):(2.*$5+2.*5.42800915) w p notitle , data u (1./$1):(2.*$5+2.*5.42800915) sm cs title "C"


#+END_SRC


@ 5630,21 +5630,21 @@ vdz = data["X"]


vdz$"E" = data$"E.DMC."


vdz$"E.lo" = data$"E.DMC."  data$"Error"


vdz$"E.hi" = data$"E.DMC." + data$"Error"


vdz$Basis = "DZ, srPBE"


vdz$Basis = "VDZBFD, srPBE"


vdz.spline_int < as.data.frame(spline(vdz$X, vdz$E))




vtz = data["X"]


vtz$"E" = data$"E.DMC..1"


vtz$"E.lo" = data$"E.DMC..1"  data$"Error.1"


vtz$"E.hi" = data$"E.DMC..1" + data$"Error.1"


vtz$Basis = "TZ, srPBE"


vtz$Basis = "VTZBFD, srPBE"


vtz.spline_int < as.data.frame(spline(vtz$X, vtz$E))




lda = data["X"]


lda$"E" = data$"E.DMC..2"


lda$"E.lo" = data$"E.DMC..2"  data$"Error.2"


lda$"E.hi" = data$"E.DMC..2" + data$"Error.2"


lda$Basis = "DZ, srLDA"


lda$Basis = "VDZBFD, srLDA"


lda.spline_int < as.data.frame(spline(lda$X, lda$E))




d = rbind(lda,vdz,vtz)


@ 5662,13 +5662,13 @@ p < p + geom_point()


#p < p + geom_line(data = vdz.spline_int, aes(x = x, y = y))


p < p + scale_x_continuous(name=TeX("$\\mu$ (bohr$^{1}$)"), breaks=breaks, labels=labels)


p < p + scale_y_continuous(name = TeX("Energy (a.u.)"), breaks=seq(17.266,17.252,0.002))


p < p + theme(text = element_text(size = 20, family="Times"), legend.position = c(.15, .15))


p < p + theme(text = element_text(size = 20, family="Times"), legend.position = c(.20, .20), legend.title = element_blank())


p




#+end_src




#+RESULTS:


[[file:/tmp/babeleZHQur/figureN5OKd9.png]]


[[file:/tmp/babeleZHQur/figureu3DiHX.png]]




#+begin_src R :results output :session *R* :exports both


pdf("../Manuscript/h2odmc.pdf", family="Times", width=8, height=5)


@ 5818,6 +5818,59 @@ H2O


1.0000000


#+end_example




**** DMC energies




#+NAME:tabh2o200


 Mu  E  Error 


++


 0.00  17.2537075442  0.0003272719 


 0.25  17.2540586404  0.0002976313 


 0.50  17.2550687726  0.0002976167 


 1.00  17.2552742178  0.0002854467 


 2.00  17.2529369035  0.0002926223 


 5.00  17.2501268640  0.0003236742 


 Inf  17.2492453661  0.0003067415 


   


17.2550782462 +/ 0.0002578873


#+begin_src R :results output :session *R* :exports both :var data=tabh2o200


data


#+end_src




#+RESULTS:


:


: Mu E Error


: 1 0.00 17.25371 0.0003272719


: 2 0.25 17.25406 0.0002976313


: 3 0.50 17.25507 0.0002976167


: 4 1.00 17.25527 0.0002854467


: 5 2.00 17.25294 0.0002926223


: 6 5.00 17.25013 0.0003236742


: 7 Inf 17.24925 0.0003067415




#+begin_src R :results output graphics :file (orgbabeltempfile "figure" ".png") :exports both :width 600 :height 400 :session *R* :var data=tabh2o200


Mu < c(0., 0.25, 0.5, 1., 2., 5., Inf)


data$X < Mu/(Mu+1.)


data$X[7] < 1.


Mu < as.character(Mu)


Mu[7] < TeX("$\\infty$")




H2O.spline_int < as.data.frame(spline(data$X, data$E, n=101))




p < ggplot()


p < p + geom_point(aes(x=X,y=data$E), color='blue')


p < p + geom_line(data = H2O.spline_int, aes(x=x, y=y))


p < p + geom_errorbar(aes(x=X, y=data$E, ymin=data$E  data$Error, ymax=data$E + data$Error), width=.2


, position=position_dodge(0.05))


p < p + scale_x_continuous(name=TeX("$\\mu$ (bohr$^{1}$)"), breaks=X, labels=Mu)


p < p + scale_y_continuous(name = TeX("E_{DMC}"))


p < p + theme(text = element_text(size = 20, family="Times") )


p




#+end_src




#+RESULTS:


[[file:/tmp/babeleZHQur/figureDoO3ml.png]]




*** F2


Parameters of the Jastrow:


a r_{12} / (1 + b r_{12})


@ 6065,6 +6118,41 @@ dev.off()


: png


: 2




** Ontop pair density


#+begin_src gnuplot :file output.png


#set terminal pdf


#set output "ontopmu.pdf"


set xlabel "distance from O"


set ylabel "ontop pair density"


plot "H2O_1.e6.density" w l title "n2(r,r), {/Symbol m}=0", \


"H2O_0.25.density" w l title "n2(r,r), {/Symbol m}=0.25", \


"H2O_0.5.density" w l title "n2(r,r), {/Symbol m}=0.5", \


"H2O_1.0.density" w l title "n2(r,r), {/Symbol m}=1.0", \


"H2O_1e6.density" w l title "n2(r,r), {/Symbol m}=10^6", \


"H2O.density" w l title "n2(r,r), Jastrow"


#+end_src




#+RESULTS:


[[file:output.png]]




** Onebody density




#+begin_src gnuplot :file output.png


#set terminal pdf


#set output "density.pdf"


set xlabel "distance from O"


set ylabel "Density"


plot "H2O_1.e6.density" u 1:3 w l title "n2(r,r), {/Symbol m}=0", \


"H2O_0.25.density" u 1:3 w l title "n2(r,r), {/Symbol m}=0.25", \


"H2O_0.5.density" u 1:3 w l title "n2(r,r), {/Symbol m}=0.5", \


"H2O_1.0.density" u 1:3 w l title "n2(r,r), {/Symbol m}=1.0", \


"H2O_1e6.density" u 1:3 w l title "n2(r,r), {/Symbol m}=10^6", \


"H2O.density" u 1:3 w l title "n2(r,r), Jastrow"


#+end_src




#+RESULTS:


[[file:output.png]]




** Optimal mu


*** Table


Find the optimal $\mu$ in VDZBFD for each molecule and atom, making


@ 6582,3 +6670,5 @@ dev.off()


: null device


: 1









Binary file not shown.
@ 99,7 +99,7 @@ to a loss of size consistency. \cite{Evangelisti_1983,Cleland_2010,Tenno_2017}




Diffusion Monte Carlo (DMC) is another numerical scheme to obtain


the exact solution of the Schr\"odinger equation with a different


constraint. In DMC, the solution is imposed to have the same nodes (or zeroes)


constraint. In DMC, the solution is imposed to have the same nodes (or zeroes)


as a given trial (approximate) wave function.


Within this socalled \emph{fixednode} (FN) approximation,


the FNDMC energy associated with a given trial wave function is an upper


@ 205,7 +205,7 @@ Jastrow factor can be interpreted as a selfconsistent field procedure


with an effective Hamiltonian,\cite{Filippi_2000} similarly to DFT.


So KSDFT can be viewed as a very cheap way of introducing the effect of


correlation in the orbital parameters determining the nodal surface


of a single Slater determinant.


of a single Slater determinant.


Nevertheless, even when using the exact exchange correlation potential at the


CBS limit, a fixednode error necessarily remains because the


singledeterminant ansätz does not have enough flexibility to describe the


@ 256,9 +256,9 @@ accuracy so all the CIPSI selections were made such that $\EPT <


Following the seminal work of Savin,\cite{Savin_1996,Toulouse_2004}


the Coulomb operator entering the interelectronic repulsion is split into two pieces:


\begin{equation}


\frac{1}{r}


= w_{\text{ee}}^{\text{sr}, \mu}(r)


+ w_{\text{ee}}^{\text{lr}, \mu}(r)


\frac{1}{r}


= w_{\text{ee}}^{\text{sr}, \mu}(r)


+ w_{\text{ee}}^{\text{lr}, \mu}(r)


\end{equation}


where


\begin{align}


@ 266,7 +266,7 @@ where


&


w_{\text{ee}}^{\text{lr},\mu}(r) & = \frac{\erf \qty( \mu\, r)}{r}


\end{align}


are the singular shortrange (sr) part and the nonsingular longrange (lr) part, respectively, $\mu$ is the rangeseparation parameter which controls how rapidly the shortrange part decays, $\erf(x)$ is the error function, and $\erfc(x) = 1  \erf(x)$ is its complementary version.


are the singular shortrange (sr) part and the nonsingular longrange (lr) part, respectively, $\mu$ is the rangeseparation parameter which controls how rapidly the shortrange part decays, $\erf(x)$ is the error function, and $\erfc(x) = 1  \erf(x)$ is its complementary version.




The main idea behind RSDFT is to treat the shortrange part of the


interaction within KSDFT, and the long range part within a WFT method like FCI in the present case.


@ 291,15 +291,15 @@ electronic energy


\begin{equation}


\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]


+ \bar{E}^{\text{sr},\mu}_{\text{Hxc}}[n_\Psi]


},


\end{equation}


with $\hat{T}$ the kinetic energy operator,


with $\hat{T}$ the kinetic energy operator,


$\hat{W}_\text{ee}^{\text{lr}}$ the longrange


electronelectron interaction,


$n_\Psi$ the oneelectron density associated with $\Psi$,


and $\hat{V}_{\text{ne}}$ the electronnucleus potential.


The minimizing multideterminant wave function $\Psi^\mu$


The minimizing multideterminant wave function $\Psi^\mu$


can be determined by the selfconsistent eigenvalue equation


\begin{equation}


\label{rsdfteigenequation}


@ 345,12 +345,12 @@ will always provide reduced fixednode errors compared to the path


connecting HF to FCI which consists in increasing the number of determinants.




We follow the KStoFCI path by performing FCI calculations using the


RSDFT Hamiltonian with different values of $\mu$.


Our algorithm, depicted in Fig.~\ref{fig:algo}, starts with a


RSDFT Hamiltonian with different values of $\mu$.


Our algorithm, depicted in Fig.~\ref{fig:algo}, starts with a


single and multideterminant 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 RSDFT Hamiltonians $\hat{H}^\mu$. \cite{GinPraFerAssSavTouJCP18}


with the RSDFT Hamiltonians $\hat{H}^\mu$. \cite{GinPraFerAssSavTouJCP18}


In the outer loop (red), a CIPSI selection is performed with a RS Hamiltonian


parameterized using the current oneelectron density.


An inner loop (blue) is introduced to accelerate the


@ 364,7 +364,7 @@ Note that any rangeseparated postHF method can be


implemented using this scheme by just replacing the CIPSI step by the


postHF method of interest.


\titou{T2: introduce $\tau_1$ and $\tau_2$. More description of the algorithm needed.}


Note that, thanks to the selfconsistent nature of the algorithm,


Note that, thanks to the selfconsistent nature of the algorithm,


the final trial wave function $\Psi^{\mu}$ is independent of the starting wave function $\Psi^{(0)}$.






@ 420,7 +420,7 @@ Allelectron move DMC.}


\cline{34} \cline{56}


System & $\mu$ & $\Ndet$ & $\EDMC$ & $\Ndet$ & $\EDMC$ \\


\hline


\ce{H2O}


\ce{H2O}


& $0.00$ & $11$ & $17.253\,59(6)$ & $23$ & $17.256\,74(7)$ \\


& $0.20$ & $23$ & $17.253\,73(7)$ & $23$ & $17.256\,73(8)$ \\


& $0.30$ & $53$ & $17.253\,4(2)$ & $219$ & $17.253\,7(5)$ \\


@ 434,10 +434,10 @@ System & $\mu$ & $\Ndet$ & $\EDMC$ & $\Ndet$ & $\ED


& $8.50$ & $101\,803$ & $17.257\,3(3)$ & $1\,643\,301$ & $17.263\,3(4)$ \\


& $\infty$ & $200\,521$ & $17.256\,8(6)$ & $1\,631\,982$ & $17.263\,9(3)$ \\


\\


\ce{F2}


\ce{F2}


& $0.00$ & $23$ & $48.419\,5(4)$ \\


& $0.25$ & $8$ & $48.421\,9(4)$ \\


& $0.50$ & $1743$ & $48.424\,8(8)$ \\


& $0.50$ & $1743$ & $48.424\,8(8)$ \\


& $1.00$ & $11952$ & $48.432\,4(3)$ \\


& $2.00$ & $829438$ & $48.441\,0(7)$ \\


& $5.00$ & $5326459$ & $48.445(2)$ \\


@ 452,8 +452,7 @@ System & $\mu$ & $\Ndet$ & $\EDMC$ & $\Ndet$ & $\ED


\includegraphics[width=\columnwidth]{h2odmc.pdf}


\caption{Fixednode 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 srLDA and srPBE.}}


functionals to build the trial wave function.}


\label{fig:h2odmc}


\end{figure}




@ 482,35 +481,35 @@ wave functions ($\mu = \infty$) gives FNDMC energies which are lower


than the energies obtained with a single KohnSham determinant ($\mu=0$):


a gain of $3.2 \pm 0.6$~m\hartree{} at the doublezeta level and $7.2 \pm


0.3$~m\hartree{} at the triplezeta level are obtained for water, and


a gain of $18 \pm 3$~m\hartree{} for F$_2$.


Coming now to the nodes of the trial wave functions $\Psi^{\mu}$ with intermediate values of $\mu$,


the figures~\ref{fig:h2odmc} show that a smooth behaviour is obtained:


starting from $\mu=0$ (\textit{i.e.} the KS determinant),


the FNDMC error is reduced continuously until it reaches a minimum for an optimal value of $\mu$,


and then the FNDMC error raises until it reaches the $\mu=\infty$ limit (\textit{i.e.} the FCI wave function).


For instance, with respect to the FNDMC energy of the FCI trial wave function in the double zeta basis set,


with the optimal value of $\mu$, one can obtain a lowering of the FNDMC energy of $2.6 \pm 0.7$~m\hartree{}


and $8 \pm 3$~m\hartree{} for the water and difluorine molecules, respectively.


The optimal value of $\mu$ is $\mu=1.75$~bohr$^{1}$ and $\mu=5$~bohr$^{1}$ for the water and fluorine dimer, respectively.


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 shifted towards large $\mu$.


Eventually, the nodes of the wave functions $\Psi^\mu$ obtained with shortrange


LDA exchangecorrelation functional give very similar FNDMC energy with respect


to those obtained with the shortrange PBE functional, even if the RSDFT energies obtained


with these two functionals differ by several tens of m\hartree{}.


a gain of $18 \pm 3$~m\hartree{} for F$_2$.


Coming now to the nodes of the trial wave functions $\Psi^{\mu}$ with intermediate values of $\mu$,


the figures~\ref{fig:h2odmc} show that a smooth behaviour is obtained:


starting from $\mu=0$ (\textit{i.e.} the KS determinant),


the FNDMC error is reduced continuously until it reaches a minimum for an optimal value of $\mu$,


and then the FNDMC error raises until it reaches the $\mu=\infty$ limit (\textit{i.e.} the FCI wave function).


For instance, with respect to the FNDMC energy of the FCI trial wave function in the double zeta basis set,


with the optimal value of $\mu$, one can obtain a lowering of the FNDMC energy of $2.6 \pm 0.7$~m\hartree{}


and $8 \pm 3$~m\hartree{} for the water and difluorine molecules, respectively.


The optimal value of $\mu$ is $\mu=1.75$~bohr$^{1}$ and $\mu=5$~bohr$^{1}$ for the water and fluorine dimer, respectively.


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 shifted towards large $\mu$.


Eventually, the nodes of the wave functions $\Psi^\mu$ obtained with shortrange


LDA exchangecorrelation functional give very similar FNDMC energy with respect


to those obtained with the shortrange PBE functional, even if the RSDFT energies obtained


with these two functionals differ by several tens of m\hartree{}.




\subsection{Link between RSDFT and Jastrow factors }


\label{sec:rsdftj}


The data obtained in \ref{sec:fndmc_mu} show that RSDFT can provide CI coefficients


The data obtained in \ref{sec:fndmc_mu} show that RSDFT can provide CI coefficients


giving trial wave functions with better nodes than FCI wave functions.


Such behaviour can be compared to the common practice of


reoptimizing the Slater part of a trial wave function in the presence of the Jastrow factor.


In the present paragraph, we would like therefore to elaborate some more on the link between RSDFT


and wave function optimization within the presence of a jastrow factor.


Such behaviour can be compared to the common practice of


reoptimizing the Slater part of a trial wave function in the presence of the Jastrow factor.


In the present paragraph, we would like therefore to elaborate some more on the link between RSDFT


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




Let us assume a fixed jastrow factor $J({\bf{r}_1}, \hdots , {\bf{r}_N})$,


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


where $\Psi = \sum_I c_I D_I$ is a general linear combination of Slater determinants $D_I$.


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


Let us assume a fixed Jastrow factor $J({\bf{r}_1}, \hdots , {\bf{r}_N})$,


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


where $\Psi = \sum_I c_I D_I$ is a general linear combination of Slater determinants $D_I$.


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{ \langle \Psi  e^{J} H e^{J} \Psi \rangle}{\langle \Psi  e^{2J} \Psi \rangle}.


@ 519,29 +518,29 @@ Such a wave function $\Psi^J$ satisfies the generalized hermitian eigenvalue equ


\begin{equation}


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


\end{equation}


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


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


\begin{equation}


\label{eq:transcor}


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


\end{equation}


which is much easier to handle despite its nonhermicity.


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


which is much easier to handle despite its nonhermicity.


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




To do so, we have made the following numerical experiment.


First, we extract the 200 determinants with the largest weights in the FCI wave


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 $\Psi^\mu$.


To do so, we have made the following numerical experiment.


First, we extract the 200 determinants with the largest weights in the FCI wave


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


on the same Slater determinant basis.


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


on the same Slater determinant basis.


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


$\braket{\Psi^J}{\Psi^\mu}$ obtained for water and the fluorine dimer,


and in figure~\ref{dmc_small} the FNDMC energy of the wave functions


$\Psi^\mu$ together with that of $\Psi^J$.


and in figure~\ref{dmc_small} the FNDMC energy of the wave functions


$\Psi^\mu$ together with that of $\Psi^J$.




\begin{figure}


\centering


@ 561,54 +560,57 @@ $\Psi^\mu$ together with that of $\Psi^J$.






In the case of H$_2$O, there is a clear maximum of overlap at


$\mu=1$~bohr$^{1}$, which coincide with the minimum of the FNDMC energy of $\Psi^\mu$.


???? hypothetic: Also, it is interesting to notice that the FNDMC energy of $\Psi^J$ is very close to that of $\Psi^\mu$ with $\mu=1$~bohr$^{1}$.


This confirms that introducing shortrange correlation with DFT has


an impact on the CI coefficients similar to the Jastrow factor.


$\mu=1$~bohr$^{1}$, which coincide with the minimum of the FNDMC energy of $\Psi^\mu$.


Also, it is interesting to notice that the FNDMC energy of $\Psi^J$ is very


close to that of $\Psi^\mu$ with $0.5 < \mu < 1$~bohr$^{1}$. This confirms that


introducing shortrange correlation with DFT has


an impact on the CI coefficients similar to the Jastrow factor.


In the case of F$_2$, the Jastrow factor has


very little effect on the CI coefficients, as the overlap


$\braket{\Psi^J}{\Psi^{\mu=\infty}}$ is very close to


$1$.


$1$.


Nevertheless, a slight maximum is obtained for


$\mu=5$~bohr$^{1}$.


In order to refine the comparison between $\Psi^\mu$ and $\Psi^J$,


we report, in the case of the water molecule in the doublezeta basis set,


several quantities related to the one and twobody density of $\Psi^J$ and $\Psi^\mu$ with different values of $\mu$.


First, we report in table~\ref{table_on_top} the integrated ontop pair density $\langle n_2({\bf r},{\bf r}) \rangle$


$\mu=5$~bohr$^{1}$.


In order to refine the comparison between $\Psi^\mu$ and $\Psi^J$,


we report, in the case of the water molecule in the doublezeta basis set,


several quantities related to the one and twobody density of $\Psi^J$ and $\Psi^\mu$ with different values of $\mu$.


First, we report in table~\ref{table_on_top} the integrated ontop pair density $\langle n_2({\bf r},{\bf r}) \rangle$


\begin{equation}


\langle n_2({\bf r},{\bf r}) \rangle = \int \text{d}{\bf r} \,\,n_2({\bf r},{\bf r})


\end{equation}


where $n_2({\bf r}_1,{\bf r}_2)$ is the twobody density (normalized to $N(N1)$ where $N$ is the number of electrons)


obtained for both $\Psi^\mu$ and $\Psi^J$.


Then, in order to have a pictorial representation of both the ontop pair density and the density, we report in figures~\ref{fig:n1} and ~\ref{fig:n2}


the plots of the total density $n({\bf r})$ and ontop pair density $n_2({\bf r},{\bf r})$ along the OH axis of the water molecule.


From these data, one can clearly observe several trends.


First, from table~\ref{table_on_top}, we can observe that the overall ontop pair density decreases


when one increases $\mu$, which is expected as the twoelectron interaction increases in $H^\mu[n]$.


Second, the relative variation of the ontop pair density with $\mu$ are much more important than that of the onebody density, the latter being essentially unchanged between $\mu=0$ and $\mu=\infty$ while the former can vary by about 10$\%$ in some regions.


??? Hypothetic: the value of the ontop pair density in $\Psi^\mu$ are closer for certain values of $\mu$ to that of $\Psi^J$ than the


FCI wave function.


where $n_2({\bf r}_1,{\bf r}_2)$ is the twobody density (normalized to $N(N1)$ where $N$ is the number of electrons)


obtained for both $\Psi^\mu$ and $\Psi^J$.


Then, in order to have a pictorial representation of both the ontop pair density and the density, we report in figures~\ref{fig:n1} and ~\ref{fig:n2}


the plots of the total density $n({\bf r})$ and ontop pair density $n_2({\bf r},{\bf r})$ along the OH axis of the water molecule.


From these data, one can clearly observe several trends.


First, from table~\ref{table_on_top}, we can observe that the overall ontop pair density decreases


when one increases $\mu$, which is expected as the twoelectron interaction increases in $H^\mu[n]$.


Second, the relative variation of the ontop pair density with $\mu$ are much more important than that of the onebody density, the latter being essentially unchanged between $\mu=0$ and $\mu=\infty$ while the former can vary by about 10$\%$ in some regions.


% TODO TOTO


The value of the ontop pair density in $\Psi^\mu$ are closer for


certain values of $\mu$ to that of $\Psi^J$ than the FCI wave


function.




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 nondivergent twobody interaction $\hat{W}_{\text{ee}}^{\text{lr},\mu}$


and the effective onebody potential $\hat{\bar{V}}_{\text{Hxc}}^{\text{sr},\mu}[n]$ which is the functional derivative of the Hartreeexchangecorrelation 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 onebody potential $\hat{\bar{V}}_{\text{Hxc}}^{\text{sr},\mu}[n_{\Psi^{\mu}}]$,


provided that it is exact, maintains the exact onebody 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 TenNo\cite{Tenno2000Nov},


the effective twobody interaction induced by the presence of a jastrow factor


can be nondivergent when a proper Jastrow factor is chosen.


Therefore, one can understand the similarity between the eigenfunctions of $H^\mu$ and the jastrowSlater optimization:


they both deal with an effective nondivergent interaction but still produce reasonable onebody density.


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 nondivergent twobody interaction $\hat{W}_{\text{ee}}^{\text{lr},\mu}$


and the effective onebody potential $\hat{\bar{V}}_{\text{Hxc}}^{\text{sr},\mu}[n]$ which is the functional derivative of the Hartreeexchangecorrelation 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 onebody potential $\hat{\bar{V}}_{\text{Hxc}}^{\text{sr},\mu}[n_{\Psi^{\mu}}]$,


provided that it is exact, maintains the exact onebody 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 TenNo\cite{Tenno2000Nov},


the effective twobody interaction induced by the presence of a jastrow factor


can be nondivergent when a proper Jastrow factor is chosen.


Therefore, one can understand the similarity between the eigenfunctions of $H^\mu$ and the JastrowSlater optimization:


they both deal with an effective nondivergent interaction but still produce reasonable onebody density.




\begin{table}


\caption{H$_2$O, doublezeta basis set. Integrated ontop pair density $\langle n_2({\bf r},{\bf r}) \rangle$


\caption{H$_2$O, doublezeta basis set. Integrated ontop pair density $\langle n_2({\bf r},{\bf r}) \rangle$


for $\Psi^J$ and $\Psi^\mu$ with different values of $\mu$. }


\label{table_on_top}


\begin{ruledtabular}


@ 622,7 +624,7 @@ they both deal with an effective nondivergent interaction but still produce rea


2.00 & 1.325 \\


5.00 & 1.288 \\


$\infty$ & 1.277 \\


\hline


\hline


$\Psi^J$ & \\


\end{tabular}


\end{ruledtabular}


@ 642,17 +644,17 @@ they both deal with an effective nondivergent interaction but still produce rea


\end{figure}






As a conclusion of the first part of this study, we can notice that:


i) with respect to the nodes of a KS determinant or a FCI wave function,


one can obtain a multi determinant trial wave function $\Psi^\mu$ with a smaller


fixed node error by properly choosing an optimal value of $\mu$


in RSDFT calculations, ii) the value of the optimal $\mu$ depends


on the system and the basis set, and the larger the basis set, the larger the optimal value of $\mu$,


iii) numerical experiments (such as computation of overlap, FNDMC energies) indicates


that the RSDFT scheme essentially plays the role of a simple Jastrow factor,


\textit{i.e.} mimicking shortrange correlation effects.


The latter statement can be qualitatively understood by noticing that both RSDFT


and transcorrelated approach deal with an effective nondivergent electronelectron interaction, while keeping the density constant.


As a conclusion of the first part of this study, we can notice that:


i) with respect to the nodes of a KS determinant or a FCI wave function,


one can obtain a multi determinant trial wave function $\Psi^\mu$ with a smaller


fixed node error by properly choosing an optimal value of $\mu$


in RSDFT calculations, ii) the value of the optimal $\mu$ depends


on the system and the basis set, and the larger the basis set, the larger the optimal value of $\mu$,


iii) numerical experiments (such as computation of overlap, FNDMC energies) indicates


that the RSDFT scheme essentially plays the role of a simple Jastrow factor,


\textit{i.e.} mimicking shortrange correlation effects.


The latter statement can be qualitatively understood by noticing that both RSDFT


and transcorrelated approach deal with an effective nondivergent electronelectron interaction, while keeping the density constant.








@ 695,7 +697,7 @@ functions. But ultimately the exact energy will be reached.


In the context of selected CI calculations, when the variational energy is


extrapolated to the FCI energy\cite{Holmes_2017} there is no


sizeconsistency error. But when the truncated SCI wave function is used


as a reference for postHartreeFock methods such as SCI+PT2


as a reference for postHartreeFock methods such as SCI+PT2


or for QMC calculations, there is a residual sizeconsistency error


originating from the truncation of the wave function.




@ 721,7 +723,7 @@ The parameter


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


or variance minimization.\cite{Coldwell_1977,Umrigar_2005}


One can easily see that this parameterization of the twobody


interation is not sizeconsistent: the dissociation of a


interaction is not sizeconsistent: the dissociation of a


diatomic molecule $AB$ with a parameter $b_{AB}$


will lead to two different twobody Jastrow factors, each


with its own optimal value $b_A$ and $b_B$. To remove the


@ 788,7 +790,7 @@ closedshell systems. A good test is to check that all the components


of a spin multiplet are degenerate.


FCI wave functions have this property and give degenrate energies with


respect to the spin quantum number $m_s$, but the multiplication by a


Jastrow factor introduces spin contamination if the parameters


Jastrow factor introduces spin contamination if the parameters


for the samespin electron pairs are different from those


for the oppositespin pairs.\cite{Tenno_2004}


Again, when pseudopotentials are used this tiny error is transferred


@ 909,7 +911,7 @@ accuracy. Thanks to the singlereference character of these systems,


the CCSD(T) energy is an excellent estimate of the FCI energy, as


shown by the very good agreement of the MAE, MSE and RMSD of CCSDT(T)


and FCI energies.


The imbalance of the quality of description of molecules compared


The imbalance of the quality of description of molecules compared


to atoms is exhibited by a very negative value of the MSE for


CCSD(T) and FCI/VDZBFD, which is reduced by a factor of two


when going to the triplezeta basis, and again by a factor of two when


@ 945,7 +947,7 @@ basis set the MAE are larger for $\mu=1$~bohr$^{1}$ than for the


FCI. For the largest systems, as shown in figure~\ref{fig:g2ndet}


there are many systems which did not reach the threshold


$\EPT<1$~m\hartree{}, and the number of determinants exceeded


10~million so the calculation stopped. In this regime, there is a


10~million so the calculation stopped. In this regime, there is a


small sizeconsistency error originating from the imbalanced


truncation of the wave functions, which is not present in the


extrapolated FCI energies. The same comment applies to


@ 1037,10 +1039,10 @@ energy via the variation of the parameter $\mu$.


The second method is for the computation of energy differences, where


the target is not the lowest possible FNDMC energies but the best


possible cancellation of errors. Using a fixed value of $\mu$


increases the consistency of the trial wave functions, and we have found


increases the consistency of the trial wave functions, and we have found


that $\mu=0.5$~bohr$^{1}$ is the value where the cancellation of


errors is the most effective.


Moreover, such a small value of $\mu$ gives extermely


Moreover, such a small value of $\mu$ gives extremely


compact wave functions, making this recipe a good candidate for


the accurate description of the whole potential energy surfaces of


large systems. If the number of determinants is still too large, the


@ 1064,70 +1066,6 @@ The data that support the findings of this study are available within the articl




\bibliography{rsdftcipsiqmc}




\begin{enumerate}


\item Total energies and nodal quality:


\begin{itemize}


% \item Facts: KS occupied orbitals closer to NOs than HF


% \item Even if exact functional, complete basis set, still approximated nodes for KS


\item KS > exponentially fast convergence (as HF) with basis because of non divergence of effective KS potential (citer le papier de Gill)


% \item With correlation consistent basis set, FCI nodes (which include correlation) are better than KS


% \item With FCI, good limit at CBS ==> exact energy


% \item But slow convergence with basis set because of divergence of the ee interaction not well represented in atom centered basis set


% \item Exponential increase of number of Slater determinants


\item Cite papiers RSDFT: there exists an hybrid scheme combining fast convergence wr to basis set (non divergent basis set) and short expansion in SCI (cite papier Ferté)


\item Question: does such a scheme provide better nodal quality ?


%<<<<<<< HEAD


\item In RSDFT we cannot optimize energy with $\mu$ , but in FNDMC


\item Factual stuffs: with optimal $\mu$, lower FNDMC energy than HF/KS/FCI


\begin{itemize}


\item less determinants $\Rightarrow$ large systems


\item only one parameter to optimize $\Rightarrow$ deterministic


\item $\Rightarrow$ reproducible


\end{itemize}


\item with the optimal $\mu$:


\begin{itemize}


\item Direct optimization of FNDMC with one parameter


\item Do we improve energy differences ?


\item system dependent


\item basis set dependent: $\mu \rightarrow \infty$ when $\mathcal{B}\rightarrow \text{CBS}$


\item large wave functions


\end{itemize}


% \item Invariance with m_s


%=======


\item In RSDFT we cannot optimize energy with $\mu$ , but in FNDMC


% \item Factual stuffs: with optimal $\mu$, lower FNDMC energy than HF/KS/FCI


% \begin{itemize}


% \item less determinants $\Rightarrow$ large systems


% \item only one parameter to optimize $\Rightarrow$ deterministic


% \item $\Rightarrow$ reproducible


% \end{itemize}


% \item with the optimal $\mu$:


% \begin{itemize}


% \item Direct optimization of FNDMC with one parameter


% \item Do we improve energy differences ?


% \item system dependent


% \item basis set dependent: $\mu \rightarrow \infty$ when $\mathcal{B}\rightarrow \text{CBS}$


% \item large wave functions


% \end{itemize}


%>>>>>>> 44470b89936d1727b638aefd982ce83be9075cc8




\end{itemize}


\end{enumerate}








\end{document}






% * Recouvrement avec Be : Optimization tous electrons


% impossible. Abandon. On va prendre H2O.


% * Manu doit faire des programmes pour des plots de ensite a 1 et 2


% corps le long des axes de liaison, et l'integrale de la densite a


% 2 corps a coalescence.


% 5 Toto optimise les coefs en presence e jastrow


% 6 Toto renvoie a manu psicoef


% 7 Manu fait tourner ses petits programmes avec psi_J


% 8 Toto calcule les energies FNDMC de


% +) toutes les fonctions d'ondes psi_mu


% +) la fonction d'onde psi_J





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