2nd screening of Sec IV done
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@article{Pack_1966,


Author = {{R. T. Pack and W. ByersBrown}},


DateAdded = {20200818 09:51:18 +0200},


DateModified = {20200818 09:51:34 +0200},


Journal = {J. Chem. Phys.},


Pages = {556},


Title = {{Cusp conditions for molecular wavefunctions}},


Volume = {45},


Year = {1966}}




@article{Kato_1957,


Author = {T. Kato},


DateAdded = {20200818 09:50:39 +0200},


DateModified = {20200818 09:50:46 +0200},


Doi = {10.1002/cpa.3160100201},


Journal = {Comm. Pure Appl. Math.},


Pages = {151},


Title = {{On the eigenfunctions of manyparticle systems in quantum mechanics}},


Volume = {10},


Year = {1957},


BdskUrl1 = {https://doi.org/10.1002/cpa.3160100201}}




@article{Loos_2015b,


Author = {Loos, PierreFran{\c c}ois and Bressanini, Dario},


DateAdded = {20200817 12:54:30 +0200},


@ 1487,8 +1509,9 @@


Year = {2004},


BdskUrl1 = {https://doi.org/10.1063/1.1757439}}




@article{Tenno2000Nov,


@article{Tenno_2000,


Author = {Tenno, Seiichiro},


DateModified = {20200818 09:50:06 +0200},


Doi = {10.1016/S00092614(00)010666},


Issn = {00092614},


Journal = {Chem. Phys. Lett.},



@ 568,7 +568,7 @@ Such a wave function satisfies the generalized Hermitian eigenvalue equation


e^{J} \hat{H} \qty( e^{J} \Psi^J ) = E \, e^{2J} \Psi^J,


\label{eq:cij}


\end{equation}


but also the nonHermitian transcorrelated eigenvalue problem\cite{BoyHanPRSLA69,BoyHanLin1PRSLA69,BoyHanLin2PRSLA69,Tenno2000Nov,LuoJCP10,YanShiJCP12,CohLuoGutDowTewAlaJCP19}


but also the nonHermitian transcorrelated eigenvalue problem\cite{BoyHanPRSLA69,BoyHanLin1PRSLA69,BoyHanLin2PRSLA69,Tenno_2000,LuoJCP10,YanShiJCP12,CohLuoGutDowTewAlaJCP19}


\begin{equation}


\label{eq:transcor}


e^{J} \hat{H} \qty( e^{J} \Psi^J) = E \, \Psi^J,


@ 647,7 +647,7 @@ This is yet another key result of the present study.


\begin{table}


\caption{\ce{H2O}, doublezeta basis set. Integrated ontop pair density $\expval{ P }$


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


\titou{Please remove table and merge data in Fig. 5.}}


\titou{Please remove table and merge data in Fig. 4.}}


\label{tab:table_on_top}


\begin{ruledtabular}


\begin{tabular}{cc}


@ 697,52 +697,56 @@ From these data, one can clearly notice several trends.


First, from Table~\ref{tab:table_on_top}, we can observe that the overall


ontop pair density decreases when $\mu$ increases, which is expected


as the twoelectron interaction increases in $H^\mu[n]$.


Second, the relative variations of the ontop pair density with $\mu$


Second, Fig.~\ref{fig:densities} shows that the relative variations of the ontop pair density with respect to $\mu$


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


In the highdensity region of the \ce{OH} bond, the value of the ontop


pair density obtained from $\Psi^J$ is superimposed with


$\Psi^{\mu=0.5}$, and at a large distance the ontop pair density is


$\Psi^{\mu=0.5}$, and at a large distance the ontop pair density of $\Psi^J$ is


the closest to $\mu=\infty$. The integrated ontop pair density


obtained with $\Psi^J$ lies between the values obtained with


$\mu=0.5$ and $\mu=1$~bohr$^{1}$, consistently with the FNDMC energies


$\mu=0.5$ and $\mu=1$~bohr$^{1}$ (see Table~\ref{tab:table_on_top}), consistently with the FNDMC energies


and the overlap curve depicted in Fig.~\ref{fig:overlap}.




These data suggest that the wave functions $\Psi^{0.5 \le \mu \le 1}$ and $\Psi^J$ are close,


and therefore that the operators that produced these wave functions (\ie, $H^\mu[n]$ 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


one can notice that the differences with respect to the usual bare 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 roles of these two terms are therefore very different: with respect


to the exact groundstate wave function $\Psi$, the nondivergent twobody 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


providing that it is exact, maintains the exact onebody density.


This is clearly what has been observed in


Fig.~\ref{fig:densities}.


Regarding now the transcorrelated Hamiltonian $e^{J}He^J$, as pointed out by Tenno,\cite{Tenno2000Nov}


Regarding now the transcorrelated Hamiltonian $e^{J}He^J$, as pointed out by Tenno,\cite{Tenno_2000}


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


can be nondivergent when a proper Jastrow factor is chosen.


can be nondivergent when a proper Jastrow factor is chosen, \ie, the Jastrow factor must fulfill the socalled electronelectron cusp conditions. \cite{Kato_1957,Pack_1966}


\titou{T2: I think we are missing the point here that the onebody Jastrow mimics the effective onebody potential which makes the onebody density fixed.


The twobody Jastrow makes the interaction nondivergent like the nondivergent twobody interaction in RSDFT.


Therefore, the onebody terms take care of the onebody properties and the twobody terms take care of the twobody properties. QED.}


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


they both deal with an effective nondivergent interaction but still


produce a reasonable onebody density.




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


%============================


\subsection{Intermediate conclusion}


%============================




As a conclusion of the first part of this study, we can highlight the following observations:


\begin{itemize}


\item with respect to the nodes of a KS determinant or a FCI wave function,


\item With respect to the nodes of a KS determinant or a FCI wave function,


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


fixednode error by properly choosing an optimal value of $\mu$


in RSDFT calculations,


\item the optimal value of $\mu$ depends on the system and the


basis set, and the larger the basis set, the larger the optimal value


of $\mu$,


\item numerical experiments (overlap $\braket*{\Psi^\mu}{\Psi^J}$,


fixednode error by properly choosing an optimal value of $\mu$.


\item The optimal $\mu$ value is system and basissetdependent, and it grows with basis set size.


\item Numerical experiments (overlap $\braket*{\Psi^\mu}{\Psi^J}$,


onebody density, ontop pair density, and FNDMC energy) indicate


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


\ie, mimicking shortrange correlation effects. The latter


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


by mimicking shortrange correlation effects. This latter


statement can be qualitatively understood by noticing that both RSDFT


and the transcorrelated approach deal with an effective nondivergent


electronelectron interaction, while keeping the density constant.



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