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Data/RSDFT-CIPSI.org
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Manuscript/rsdft-cipsi-qmc.tex
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@ -23,7 +23,7 @@
\newcommand{\LCPQ}{Laboratoire de Chimie et Physique Quantiques (UMR 5626), Université de Toulouse, CNRS, UPS, France}
\begin{document}
\begin{document}
\title{Enabling high accuracy diffusion Monte Carlo calculations with
range-separated density functional theory and selected configuration interaction}
@ -50,7 +50,7 @@
\section{Introduction}
\label{sec:intro}
The full configuration interaction (FCI) method \eg{within an incomplete basis set}
The full configuration interaction (FCI) method \eg{within an incomplete basis set}
leads to the exact solution of the Schrödinger equation with an approximate Hamiltonian
\eg{which consists in the exact one projected onto } \sout{expressed in} a finite basis of Slater determinants.
The FCI wave function can be interpreted as the exact solution of the
@ -76,7 +76,7 @@ The Diffusion Monte Carlo (DMC) method is a numerical scheme to obtain
the exact solution of the Schrödinger equation with an additional
constraint, imposing the solution to have the same nodal hypersurface
as a given trial wave function. This approximation, known as the
\emph{fixed-node} approximation, \eg{is variational with respect to the nodes of the trial wave function: the DMC energy obtained with a given trial wave function is an upper bound to the exact energy, and the latter is recovered only }
\emph{fixed-node} approximation, \eg{is variational with respect to the nodes of the trial wave function: the DMC energy obtained with a given trial wave function is an upper bound to the exact energy, and the latter is recovered only }
when the nodes of the trial wave function coincide with the nodes of the exact wave function\sout{, the exact energy and wave function are obtained}.
The DMC method is attractive because its scaling is polynomial with
the number of electrons and with the size of the trial wave
@ -95,7 +95,7 @@ systems.
As it has been shown by many studies\cite{Per_2012}, the nodal surfaces obtained with the
KS determinant are in general better than those obtained with
the HF determinant, and of comparable quality
to those obtained with a Slater determinant built with NO.\cite{Wang_2019}
to those obtained with a Slater determinant built with NO.\cite{Wang_2019}
However, the fixed-node approximation is much more difficult to
control than the finite-basis approximation, as it is not possible
@ -131,25 +131,25 @@ factor.\cite{Giner_2016,Dash_2018,Dash_2019}
\begin{enumerate}
\item Total energies and nodal quality:
\begin{itemize}
\item Factual stuffs: KS occupied orbitals closer to NOs than HF
\item Factual stuffs: 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 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 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 RSDFT we cannot optimize energy with $\mu$ , but in FNDMC
\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
\item $\Rightarrow$ reproducible
\end{itemize}
\item with the optimal $\mu$:
\begin{itemize}
\item Direct optimization of FNDMC with one parameter
\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 $\mathcal{B}\rightarrow \text{CBS}$
@ -162,7 +162,7 @@ factor.\cite{Giner_2016,Dash_2018,Dash_2019}
\section{Combining range-separated DFT with CIPSI}
\label{sec:rsdft-cipsi}
Starting from a Hartree-Fock determinant in a small basis set,
Starting from a Hartree-Fock determinant in a small basis set,
we have seen that we can systematically improve the trial wave
function in two directions. The first one is by increasing the
size of the atomic basis set, and the second one is by
@ -342,11 +342,11 @@ takes the form
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}
implemented.\cite{GinPraFerAssSavTou-JCP-18}
\begin{figure}[h]
\centering
\includegraphics[width=\columnwidth]{algorithm.pdf}
\includegraphics[width=\columnwidth]{algorithm.pdf}
\caption{Algorithm showing the generation of the RSDFT-CIPSI wave
function}
\end{figure}
@ -386,7 +386,7 @@ implemented.\cite{GinPraFerAssSavTou-JCP-18}
\begin{figure}[h]
\centering
\includegraphics[width=\columnwidth]{h2o-dmc.pdf}
\includegraphics[width=\columnwidth]{h2o-dmc.pdf}
\caption{Fixed-node energies of the water molecule with variable
values of $\mu$.}
\label{fig:h2o-dmc}
@ -563,6 +563,8 @@ of the RSDFT-CIPSI trial wave functions for energy differences.
& & 1/4 & 5.55 \\
& & 1/2 & 13.42 \\
& & 1 & 17.07 \\
& & 2 & 19.20 \\
& & 5 & 22.93 \\
& & $\infty$ & 23.62 \\
& VTZ-BFD & 0 & 6.31 \\
& & 1/4 & 4.58 \\
@ -573,10 +575,12 @@ of the RSDFT-CIPSI trial wave functions for energy differences.
& & 1/2 & 6.35 \\
\hline
DMC/RSDFT-CIPSI & VDZ-BFD & 0 & 5.07 $\pm$ 0.44 \\
& & 1/4 & 4.04 $\pm$ 0.37 \\
& & 1/2 & 3.74 $\pm$ 0.35 \\
& & 1 & 5.42 $\pm$ 0.29 \\
& & $\infty$ & 7.38 $\pm$ 1.08 \\
& & 1/4 & 4.04 $\pm$ 0.37 \\
& & 1/2 & 3.74 $\pm$ 0.35 \\
& & 1 & 5.42 $\pm$ 0.29 \\
& & 2 & 5.98 $\pm$ 0.83 \\
& & 5 & 6.68 $\pm$ 1.07 \\
& & $\infty$ & 7.38 $\pm$ 1.08 \\
& VTZ-BFD & 0 & 3.52 $\pm$ 0.19 \\
& & 1/4 & 3.39 $\pm$ 0.77 \\
& & 1/2 & 2.46 $\pm$ 0.18 \\
@ -593,21 +597,21 @@ of the RSDFT-CIPSI trial wave functions for energy differences.
\begin{figure}[h]
\centering
\includegraphics[width=\columnwidth]{g2-dmc-dz.pdf}
\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}
\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}
\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}