Added histograms

Manuscript/rsdft-cipsi-qmc.bib

@@ -309,3 +309,33 @@ density-functional theory: A systematically improvable approach},
     doi = {10.1103/PhysRevLett.94.150201}
 }
 
+@article{Huron_1973,
+    author = {Huron, B. and Malrieu, J. P. and Rancurel, P.},
+    title = {{Iterative perturbation calculations of ground and excited state energies
+    from multiconfigurational zeroth{-}order wavefunctions}},
+    journal = {J. Chem. Phys.},
+    volume = {58},
+    number = {12},
+    pages = {5745--5759},
+    year = {1973},
+    month = {Jun},
+    issn = {0021-9606},
+    publisher = {American Institute of Physics},
+    doi = {10.1063/1.1679199}
+}
+
+
+@article{Davidson_1975,
+    author = {Davidson, Ernest R.},
+    title = {{The iterative calculation of a few of the lowest eigenvalues and corresponding
+    eigenvectors of large real-symmetric matrices}},
+    journal = {J. Comput. Phys.},
+    volume = {17},
+    number = {1},
+    pages = {87--94},
+    year = {1975},
+    month = {Jan},
+    issn = {0021-9991},
+    publisher = {Academic Press},
+    doi = {10.1016/0021-9991(75)90065-0}
+}

Manuscript/rsdft-cipsi-qmc.tex

@@ -49,11 +49,36 @@
 \section{Introduction}
 \label{sec:intro}
 
+\section{Combining range-separated DFT with CIPSI}
+\label{sec:rsdft-cipsi}
 
-\section{Range-separated DFT in a nutshell}
+\subsection{CIPSI}
+
+\emph{Configuration interaction using a perturbative selection made
+iteratively} (CIPSI)\cite{Huron_1973} belongs to the class of selected
+CI methods, whose goal is to build compressed configuration interaction
+(CI) wave functions with a controlled accuracy, together with an estimate
+of the associated eigenvalue obtained with perturbation theory.
+
+Using a conventional iterative diagonalization method such as
+Davidson's,\cite{Davidson_1975} the size of the CI space is
+pre-defined, and at each iteration the length of the vectors is equal
+to the size of the CI space. With selected CI methods, at each
+iteration the size of the determinant space is increased such that
+one keeps only the most important determinants that will lead to an
+accurate description of the state of interest. The lowest eigenpairs
+are extracted from the CI matrix expressed in this determinant
+subspace, and the CI energy can be estimated by computing a
+second-order perturbative correction (PT2) to the variational energy
+computed in the complete CI space. The magnitude of the PT2 correction
+is a measure of the distance to the exact eigenvalue, and is an
+adjustable parameter to control the quality of the wave function.
+
+
+\subsection{Range-separated DFT in a nutshell}
 \label{sec:rsdft}
 
-\subsection{Exact equations}
+\subsubsection{Exact equations}
 \label{sec:exact}
 
 The exact ground-state energy of a $N$-electron system with
@@ -189,31 +214,13 @@ takes the form
 \end{eqnarray}
 
 
+\subsection{RSDFT-CIPSI}
 
+It is possible to use DFT for short-range interactions and CIPSI for
+the long-range. This scheme has been recently
+implemented,\cite{GinPraFerAssSavTou-JCP-18} 
 
 
-\section{Computational details}
-\label{sec:comp-details}
-
-All the calculations were made using BFD
-pseudopotentials\cite{Burkatzki_2008} with the associated double, triple
-and quadruple zeta basis sets.
-CCSD(T) and DFT calculations were made with
-\emph{Gaussian09},\cite{g09} using an unrestricted Hartree-Fock
-determinant as a reference for open-shell systems. All the CIPSI
-calculations and range-separated CIPSI calculations were made with
-\emph{Quantum Package}.\cite{Garniron_2019,qp2_2020}
-Quantum Monte Carlo calculations were made with QMC=Chem,\cite{scemama_2013}
-in the determinant localization approximation.\cite{Zen_2019}
-
-In the determinant localization approximation, only the determinantal
-component of the trial wave function is present in the expression of
-the wave function on which the pseudopotential is localized. Hence,
-the pseudopotential operator does not depend on the Jastrow factor, as
-it is the case in all-electron calculations. This improves the
-reproducibility of the results, as they depend only on parameters
-optimized in a deterministic framework.
-
 
 \section{Influence of the range-separation parameter on the fixed-node
   error}
@@ -260,6 +267,49 @@ were used as trial wave functions for FN-DMC calculations, and the
 corresponding energies are shown in table~\ref{tab:h2o-dmc} and
 figure~\ref{fig:h2o-dmc}.
 
+Using FCI trial wave functions gives FN-DMC energies which are lower
+than the energies obtained with a single Kohn-Sham determinant:
+3~m$E_h$ at the double-zeta level and 7~m$E_h$ at the triple-zeta
+level. Interestingly, with the double-zeta basis one can obtain a
+FN-DMC energy 2.5~m$E_h$ lower than the energy obtained with the FCI
+trial wave function, using the RSDFT-CIPSI with a range-separation
+parameter $\mu=1.75$. This can be explained by the inability of the
+basis set to properly describe short-range correlation, shifting
+the nodes from their optimal position. Using DFT to take account of
+short-range correlation frees the determinant expansion from describing
+short-range effects, and enables a better placement of the nodes.
+At the triple-zeta level, the short-range correlations can be better
+described, and the improvement due to DFT is insignificant. However,
+it is important to note that the same FN-DMC energy can be
+obtained with a CI expansion which is eight times smaller when sr-DFT
+is introduced. One can also remark that the minimum has been
+shifted towards the FCI, which is consistent with the fact that
+in the CBS limit we expect the minimum of the FN-DMC energy to be
+obtained for the FCI wave function, at $\mu=\infty$.
+
+
+\section{Computational details}
+\label{sec:comp-details}
+
+All the calculations were made using BFD
+pseudopotentials\cite{Burkatzki_2008} with the associated double, triple
+and quadruple zeta basis sets.
+CCSD(T) and DFT calculations were made with
+\emph{Gaussian09},\cite{g09} using an unrestricted Hartree-Fock
+determinant as a reference for open-shell systems. All the CIPSI
+calculations and range-separated CIPSI calculations were made with
+\emph{Quantum Package}.\cite{Garniron_2019,qp2_2020}
+Quantum Monte Carlo calculations were made with QMC=Chem,\cite{scemama_2013}
+in the determinant localization approximation.\cite{Zen_2019}
+
+In the determinant localization approximation, only the determinantal
+component of the trial wave function is present in the expression of
+the wave function on which the pseudopotential is localized. Hence,
+the pseudopotential operator does not depend on the Jastrow factor, as
+it is the case in all-electron calculations. This improves the
+reproducibility of the results, as they depend only on parameters
+optimized in a deterministic framework.
+
 
 \section{Atomization energy benchmarks}
 \label{sec:atomization}
@@ -354,6 +404,27 @@ of the RSDFT-CIPSI trial wave functions for energy differences.
 
 
 
+\begin{figure}[h]
+  \centering
+  \includegraphics[width=\columnwidth]{g2-dmc_dz.pdf}  
+  \caption{Histogram of the errors in atomization energies with the double-zeta basis set.}
+  \label{fig:g2-dmc_dz}
+\end{figure}
+
+\begin{figure}[h]
+  \centering
+  \includegraphics[width=\columnwidth]{g2-dmc_tz.pdf}  
+  \caption{Histogram of the errors in atomization energies with the triple-zeta basis set.}
+  \label{fig:g2-dmc_tz}
+\end{figure}
+
+\begin{figure}[h]
+  \centering
+  \includegraphics[width=\columnwidth]{g2-dmc_qz.pdf}  
+  \caption{Histogram of the errors in atomization energies with the quadruple-zeta basis set.}
+  \label{fig:g2-dmc_qz}
+\end{figure}
+