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Manuscript/rsdft-cipsi-qmc.tex
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@ -14,6 +14,7 @@
\urlstyle{same}
\newcommand{\alert}[1]{\textcolor{red}{#1}}
\newcommand{\eg}[1]{\textcolor{blue}{#1}}
\definecolor{darkgreen}{HTML}{009900}
\usepackage[normalem]{ulem}
@ -49,9 +50,9 @@
\section{Introduction}
\label{sec:intro}
The full configuration interaction (FCI) method leads to the exact
solution of the Schrödinger equation with an approximate Hamiltonian
expressed in a finite basis of Slater determinants.
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
true Hamiltonian obtained with the additional constraint that it
can only span the space provided by the basis. At the complete
@ -59,7 +60,7 @@ basis set (CBS) limit, the constraint vanishes and the exact solution
is obtained.
Hence the FCI method enables a systematic improvement of the
calculations by increasing the size of the basis set. Nevertheless,
calculations by \sout{increasing the size} \eg{improving the quality} of the basis set. Nevertheless,
its exponential scaling with the number of electrons and with the size
of the basis is prohibitive to treat large systems.
In recent years, the introduction of new algorithms\cite{Booth_2009}
@ -74,27 +75,28 @@ to a loss of size extensivity.
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 is known as the
\emph{fixed-node} approximation. When the nodes of the trial wave
function coincide with the nodes of the exact wave function, the exact
energy and wave function are obtained.
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 }
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
function. Moreover, the total energies obtained are usually below
those obtained with FCI because the fixed-node approximation imposes
those obtained with FCI \eg{in computationally tractable basis sets} because the fixed-node approximation imposes
less constraints on the solution than the finite-basis approximation.
In many cases, the systems under study are well described by a single
Slater determinant. Single-determinant DMC can be used as a post-Hatree-Fock
single-reference method with an accuracy comparable to coupled cluster.
\eg{The qualitative picture of the electronic structure of weakly correlated systems, such as organic molecule near their equilibrium geometry, }\sout{In many cases, the systems under study} are well described by a single
Slater determinant. \sout{Single-determinant} DMC \eg{with a single-determinant trial wave function} can be used as a post-Hatree-Fock
single-reference method with an accuracy comparable to coupled cluster\eg{mettre la ref}.
The favorable scaling of QMC and its adequation with massively
parallel architectures makes it an attractive alternative for large
systems.
It has been shown that the nodal surfaces obtained with
Kohn-Sham determinants are in general better than those obtained with
the Hartree-Fock determinant,\cite{Per_2012} and of comparable quality
to those obtained with natural orbitals of single-determinant
correlated calculations.\cite{Wang_2019}
\eg{The choice of the Slater determinant entirely depends on the type of orbitals used to build it, for which three main options are available: the Kohn-Sham\ref{} (KS) scheme, the HF scheme or the natural orbitals (NO) of a correlated wave function. }
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}
However, the fixed-node approximation is much more difficult to
control than the finite-basis approximation, as it is not possible
to minimize directly the DMC energy with respect to the variational
@ -126,6 +128,35 @@ authors have used a combination of the two approaches where CIPSI
trial wave functions are re-optimized under the presence of a Jastrow
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 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 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 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}
\section{Combining range-separated DFT with CIPSI}