Modifs in conclusion

Manuscript/rsdft-cipsi-qmc.tex

@@ -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:g2-dmc}. 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:g2-ndet}. 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 quadruple-zeta basis set, making these calculations feasilble
+with such a large basis set. At the double-zeta 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 double-zeta level, and close to 1~000 determinants
+at the quadruple-zeta level for only a slight increase of the
+MAE. Hence, RS-DFT-CIPSI trial wave functions with small values of
+$\mu$ could be very useful for large systems to go beyond the
+single-determinant approximation at a very low computational cost
+while keeping the size-consistency.
 
 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 re-optimizing
 the CI coefficients in the presence of a Jastrow factor, but without
 the burden of performing a stochastic optimization.
 
-Varying the range-separation parameter $\mu$ and aproaching the
+Varying the range-separation parameter $\mu$ and approaching the
 RS-DFT-FCI with CIPSI provides a way to adapt the number of
-determinants in the trial wave function, leading to size-consistent
-FN-DMC energies.
+determinants in the trial wave function, leading always to
+size-consistent FN-DMC energies.
 We propose two methods. The first one is for the computation of
 accurate total energies by a one-parameter optimization of the FN-DMC
 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 FN-DMC 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 2019-gch0418) 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 e-e 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 e-e interaction not well represented in atom centered basis set
+%   \item Exponential increase of number of Slater determinants
    \item Cite papiers RS-DFT: 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 RS-DFT 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 2019-gch0418) and from CALMIP (Toulouse) under allocation
 \end{document}
 
 
-% * Recouvrement avec Be
-% * Tester sr-LDA avec H2O-DZ
+% * 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 cc-pvdz tous electrons: FCI -> NOs -> FCI ->
 %   qp edit -n 200
 % 2 Manu calcule qp_cipsi_rsh avec mu = [ 1.e-6 , 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