Modifs in conclusion
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@ 826,7 +826,7 @@ Searching for the optimal value of $\mu$ may be too costly, so we have


computed the MAD, MSE and RMSD for fixed values of $\mu$. The results


are illustrated in figure~\ref{fig:g2dmc}. As seen on the figure and


in table~\ref{tab:mad}, the best choice for a fixed value of $\mu$ is


0.5~bohr$^{1}$ for all three basis sets. Is is the value for which


0.5~bohr$^{1}$ for all three basis sets. It is the value for which


the MAE (3.74(35), 2.46(18) and 2.06(35) kcal/mol) and RMSD (4.03(23),


3.02(06) and 2.74(13)~kcal/mol) are minimal. Note that these values


are even lower than those obtained with the optimal value of


@ 849,6 +849,19 @@ cancellations of errors.


The number of determinants in the trial wave functions are shown in


figure~\ref{fig:g2ndet}. As expected, the number of determinants


is smaller when $\mu$ is small and larger when $\mu$ is large.


It is important to remark that the median of the number of


determinants when $\mu=0.5$~bohr$^{1}$ is below 100~000 determinants


with the quadruplezeta basis set, making these calculations feasilble


with such a large basis set. At the doublezeta level, compared to the


FCI trial wave functions the median of the number of determinants is


reduced by more than two orders of magnitude.


Moreover, going to $\mu=0.25$~bohr$^{1}$ gives a median close to 100


determinants at the doublezeta level, and close to 1~000 determinants


at the quadruplezeta level for only a slight increase of the


MAE. Hence, RSDFTCIPSI trial wave functions with small values of


$\mu$ could be very useful for large systems to go beyond the


singledeterminant approximation at a very low computational cost


while keeping the sizeconsistency.




Note that when $\mu=0$ the number of determinants is not equal to one because


we have used the natural orbitals of a first CIPSI calculation, and


@ 872,10 +885,10 @@ on the determinant expansion is similar to the effect of reoptimizing


the CI coefficients in the presence of a Jastrow factor, but without


the burden of performing a stochastic optimization.




Varying the rangeseparation parameter $\mu$ and aproaching the


Varying the rangeseparation parameter $\mu$ and approaching the


RSDFTFCI with CIPSI provides a way to adapt the number of


determinants in the trial wave function, leading to sizeconsistent


FNDMC energies.


determinants in the trial wave function, leading always to


sizeconsistent FNDMC energies.


We propose two methods. The first one is for the computation of


accurate total energies by a oneparameter optimization of the FNDMC


energy via the variation of the parameter $\mu$.


@ 883,12 +896,15 @@ 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


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


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


compact wave functions, making this recipe a good candidate for


the accurate description of the whole potential energy surfaces of


large systems.


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


value of $\mu$ can be further reduced to $0.25$~bohr$^{1}$ to get


extremely compact wave functions at the price of less efficient


cancellations of errors.








@ 906,31 +922,30 @@ Challenge 2019gch0418) and from CALMIP (Toulouse) under allocation


\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 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 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 ?


\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 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}




\end{itemize}


\end{enumerate}


@ 940,15 +955,15 @@ Challenge 2019gch0418) and from CALMIP (Toulouse) under allocation


\end{document}






% * Recouvrement avec Be


% * Tester srLDA avec H2ODZ


% * 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.


% 1 Manu calcule Be en ccpvdz tous electrons: FCI > NOs > FCI >


% qp edit n 200


% 2 Manu calcule qp_cipsi_rsh avec mu = [ 1.e6 , 0.25, 0.5, 1.0, 2.0, 5.0, 1e6 ]


% 3 Manu fait tourner es petits programmes


% 3 Manu fait tourner ses petits programmes


% 4 Manu envoie a toto un tar avec tous les ezfio


% 5 Toto optimise les coefs en presence e jastrow


% 6 Toto renvoie a manu psicoef



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