work done on algorithm

Manuscript/rsdft-cipsi-qmc.bib

@@ -1,13 +1,44 @@
 %% This BibTeX bibliography file was created using BibDesk.
 %% http://bibdesk.sourceforge.net/
 
-%% Created for Pierre-Francois Loos at 2020-08-07 20:17:07 +0200 
+%% Created for Pierre-Francois Loos at 2020-08-08 08:15:47 +0200 
 
 
 %% Saved with string encoding Unicode (UTF-8) 
 
 
 
+@article{Pulay_1980,
+	Author = {Pulay, P{\'e}ter},
+	Date-Added = {2020-08-08 08:15:42 +0200},
+	Date-Modified = {2020-08-08 08:15:42 +0200},
+	Doi = {10.1016/0009-2614(80)80396-4},
+	Issn = {00092614},
+	Journal = {Chem. Phys. Lett.},
+	Language = {en},
+	Month = jul,
+	Number = {2},
+	Pages = {393--398},
+	Title = {Convergence Acceleration of Iterative Sequences. the Case of Scf Iteration},
+	Volume = {73},
+	Year = {1980},
+	Bdsk-Url-1 = {https://dx.doi.org/10.1016/0009-2614(80)80396-4}}
+
+@article{Pulay_1982,
+	Author = {Pulay, P.},
+	Date-Added = {2020-08-08 08:15:42 +0200},
+	Date-Modified = {2020-08-08 08:15:42 +0200},
+	Doi = {10.1002/jcc.540030413},
+	Issn = {0192-8651, 1096-987X},
+	Journal = {J. Comput. Chem.},
+	Language = {en},
+	Number = {4},
+	Pages = {556--560},
+	Title = {{{ImprovedSCF}} Convergence Acceleration},
+	Volume = {3},
+	Year = {1982},
+	Bdsk-Url-1 = {https://dx.doi.org/10.1002/jcc.540030413}}
+
 @article{Assaraf_2000,
 	Author = {Assaraf, Roland and Caffarel, Michel and Khelif, Anatole},
 	Date-Added = {2020-08-07 20:12:45 +0200},

Manuscript/rsdft-cipsi-qmc.tex

@@ -334,7 +334,10 @@ the FCI wave function.
   \centering
   \includegraphics[width=0.7\linewidth]{algorithm.pdf}
   \caption{Algorithm showing the generation of the RS-DFT wave
-    function $\Psi^{\mu}$ starting from the .}
+    function $\Psi^{\mu}$ starting from $\Psi^{(0)}$. 
+    The outer (macro-iteration) and inner (micro-iteration) loops are represented in red and blue, respectively.
+    The steps common to both loops are represented in purple.
+    DIIS extrapolations of the one-electron density are introduced in both the outer and inner loops in order to speed up convergence of the self-consistent process.}
   \label{fig:algo}
 \end{figure*}
 
@@ -347,23 +350,28 @@ connecting HF to FCI which consists in increasing the number of determinants.
 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 $\Psi^{(0)}$.
+single- or multi-determinant wave function $\Psi^{(0)}$ which can 
+be obtained in many different ways depending on the system that one considers.
 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 $\hat{H}^\mu$. \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
+use the CIPSI algorithm to perform approximate FCI calculations
+with the RS-DFT Hamiltonian $\hat{H}^\mu$. \cite{GinPraFerAssSavTou-JCP-18}
+This provides a multi-determinant trial wave function $\Psi^{\mu}$ that one can ``feed'' to DMC.
+In the outer (macro-iteration) loop (red), at the $k$th iteration, a CIPSI selection is performed 
+to obtain $\Psi^{\mu\,(k)}$ with the RS Hamiltonian $\hat{H}^{\mu\,(k)}$
+parameterized using the current one-electron density $n^{(k)}$.
+At each iteration, the number of determinants in $\Psi^{\mu\,(k)}$ increases.
+One exits the macro-iteration loop when the absolute energy difference between two successive iterations $\Delta E^{(k)}$ is below a threshold $\tau_1$ that has been set to \titou{???} in the present study.
+An inner (micro-iteration) 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
-RS-Hamiltonian is performed iteratively with the updated density.
+determinants in $\Psi^{\mu\,(k,l)}$ is kept fixed, and only the diagonalization of
+$\hat{H}^{\mu\,(k,l)}$ is performed iteratively with the updated density $n^{(k,l)}$.
+The micro-iteration loop is exited when the absolute energy difference between two successive iterations $\Delta E^{(k,l)}$ is below a threshold $\tau_2$ that has been set to \titou{???} in the present study.
 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.
+step to extrapolate the density both in the outer and inner loops. \cite{Pulay_1980,Pulay_1982}
 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 $\Psi^{(0)}$.