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Manuscript/rsdft-cipsi-qmc.tex
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@ -175,20 +175,24 @@ Nevertheless, as the orbitals are one-electron functions,
the procedure of orbital optimization in the presence of the
Jastrow factor can be interpreted as a self-consistent field procedure
with an effective Hamiltonian,\cite{Filippi_2000} similarly to DFT.
So \eg{KS-}DFT can be viewed as a very cheap way of introducing the effect of
correlation in the \eg{orbital }parameters determining the nodal surface \eg{of a single Slater determinant}.
\sout{But in the general case, even} \eg{Nevertheless, even using the exact exchange correlation potential at } the CBS limit, a fixed-node error \sout{will} \eg{necessary} remains because the single-determinant ans\"atz does not
have enough \sout{freedom} \eg{flexibility} to describe the \sout{exact} nodal surface \eg{of the exact correlated wave function of a generic $N$-electron system}.
If one wants to have to exact CBS limit, a multi-determinant parameterization of the wave functions is required.
So KS-DFT can be viewed as a very cheap way of introducing the effect of
correlation in the orbital parameters determining the nodal surface
of a single Slater determinant.
Nevertheless, even when using the exact exchange correlation potential at the
CBS limit, a fixed-node error necessarily remains because the
single-determinant ansätz does not have enough flexibility to describe the
nodal surface of the exact correlated wave function of a generic $N$-electron
system.
If one wants to have to exact CBS limit, a multi-determinant parameterization
of the wave functions is required.
\subsection{CIPSI}
Beyond the single-determinant representation, the best
multi-determinant wave function one can obtain is the FCI. FCI is
a \emph{post-Hartree-Fock} method, and \sout{there is a continuous} \eg{there exists several systematic improvements}
\sout{connections} between the Hartree-Fock and FCI wave functions:
\sout{Multiple paths exist: one can for example use}
increasing the maximum degree of excitation of CI methods (CISD, CISDT,
a \emph{post-Hartree-Fock} method, and there exists several systematic
improvements between the Hartree-Fock and FCI wave functions:
increasing the maximum degree of excitation of CI methods (CISD, CISDT,
CISDTQ, \emph{etc}), or increasing the complete active space
(CAS) wave functions until all the orbitals are in the active space.
Selected CI methods take a shorter path between the Hartree-Fock
@ -206,7 +210,8 @@ Within the \emph{Configuration interaction using a perturbative
selection made iteratively} (CIPSI)\cite{Huron_1973} method, the PT2
correction is computed along with the determinant selection. So the
magnitude of $\EPT$ can be made the only parameter of the algorithm,
and we choose this parameter as the convergence criterion.
and we choose this parameter as the convergence criterion of the CIPSI
algorithm.
Considering that the perturbatively corrected energy is a reliable
estimate of the FCI energy, using a fixed value of the PT2 correction
@ -237,7 +242,9 @@ The parameter $\mu$ controls the range of the separation, and allows
to go continuously from the Kohn-Sham Hamiltonian ($\mu=0$) to
the FCI Hamiltinoan ($\mu = \infty$).
\eg{To rigorously connect wave function theory and DFT,} the universal \eg{Levy-Lieb} density functional\cite{Lev-PNAS-79,Lie-IJQC-83} is decomposed as
To rigorously connect wave function theory and DFT, the universal
Levy-Lieb density functional\cite{Lev-PNAS-79,Lie-IJQC-83} is
decomposed as
\begin{equation}
\mathcal{F}[n] = \mathcal{F}^{\mathrm{lr},\mu}[n] + \bar{E}_{\mathrm{Hxc}}^{\mathrm{sr,}\mu}[n],
\label{Fdecomp}
@ -314,63 +321,44 @@ with the RS-DFT Hamiltonians,\cite{GinPraFerAssSavTou-JCP-18} as shown
in figure~\ref{fig:algo}. In the outer loop (red), a CIPSI selection
is performed with a RS-Hamiltonian parameterized using the current
density. An inner loop (blue) is introduced to accelerate the
\eg{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.
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.
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.
\sout{As always, } \eg{As mentioned above,} the convergence criterion for CIPSI was set to $\EPT <
1$~m$E_h$.
As mentioned above, the convergence criterion for CIPSI was set to
$\EPT < 1$~m$E_h$.
\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{g16} using an unrestricted Hartree-Fock
determinant as a reference for open-shell systems.
\subsection{Approximations}
In this work, we use the short-range version of the
Perdew-Burke-Ernzerhof (PBE)~\cite{PerBurErn-PRL-96} exchange and
correlation functionals of Ref.~\onlinecite{GolWerStoLeiGorSav-CP-06}
(see also Refs.~\onlinecite{TouColSav-JCP-05,GolWerSto-PCCP-05}).
All the CIPSI calculations were made with \emph{Quantum
Package}.\cite{Garniron_2019,qp2_2020} We used the short-range version
of the Perdew-Burke-Ernzerhof (PBE)~\cite{PerBurErn-PRL-96} exchange
and correlation functionals of
Ref.~\onlinecite{GolWerStoLeiGorSav-CP-06} (see also
Refs.~\onlinecite{TouColSav-JCP-05,GolWerSto-PCCP-05}).
The convergence criterion for stopping the CIPSI calculations
was $\EPT < 1$~m$E_h \vee \Ndet > 10^7$.
Quantum Monte Carlo calculations were made with QMC=Chem,\cite{scemama_2013}
in the determinant localization approximation (DLA),\cite{Zen_2019}
where 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, in the DLA the fixed-node
energy is independent of the Jatrow factor, as in all-electron
calculations.
\subsection{RSDFT-CIPSI}
\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 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}
\begin{itemize}
\item plot $N_{det}$ en fonction de $\mu$
\end{itemize}
\end{itemize}
\end{enumerate}
% Overlap with reoptimized
% Plot Ndets as a function of mu
\section{Influence of the range-separation parameter on the fixed-node
@ -416,26 +404,24 @@ correlation functionals of Ref.~\onlinecite{GolWerStoLeiGorSav-CP-06}
values of $\mu$.}
\label{fig:f2-dmc}
\end{figure}
\eg{The first question we would like to address is the quality of the nodes of the wave functions $\Psi^{\mu}$ obtained
with an intermediate range separation parameter $\mu$ (\textit{i.e.} $\mu > 0$ and $\mu < + \infty$).
Therefore, we computed the fixed node energy obtained with $\Psi^{\mu}$
without re optimizing any parameters having an impact on the nodes
(such as Slater determinant coefficients or orbitals),
and this for several values of $\mu$. We considered two weakly correlated molecular systems:
the water molecule and fluorine dimer, both studied near their equilibrium geometry\cite{Caffarel_2016}.
All RSDFT-CIPSI wave functions were obtained calculations using BFD pseudopotentials
and the corresponding double- and triple-zeta basis sets for the water molecule,
and double-zeta quality for the fluorine dimer.}
\sout{The water molecule was taken at the equilibrium
The first question we would like to address is the quality of the
nodes of the wave functions $\Psi^{\mu}$ obtained with an intermediate
range separation parameter $\mu$ (\textit{i.e.} $0 < \mu < +\infty$).
We generated trial wave functions $\Psi^\mu$ with multiple
values of $\mu$, and computed the associated fixed node energy
keeping fixed all the parameters having an impact on the nodal surface.
We considered two weakly correlated molecular systems: the water
molecule and fluorine dimer, near their equilibrium
geometry\cite{Caffarel_2016}.
geometry,\cite{Caffarel_2016} and
generated with BFD pseudopotentials and the corresponding double-zeta
basis set using multiple values of the range-separation parameter
$\mu$. The convergence criterion for stopping the CIPSI calculation
was set to 1~m$E_h$ on the PT2 correction.
$\mu$.
Then, these wave functions
were used as trial wave functions for FN-DMC calculations,}
\eg{We report the values of the FN-DMC energies of the water molecule in table~\ref{tab:h2o-dmc} and
figure~\ref{fig:h2o-dmc}.}
We report the values of the FN-DMC energies of the water molecule in
table~\ref{tab:h2o-dmc} and figure~\ref{fig:h2o-dmc}.
\eg{From table~\ref{tab:h2o-dmc} and figure~\ref{fig:h2o-dmc} one can clearly observe that }using FCI trial wave functions gives FN-DMC energies which are lower
than the energies obtained with a single Kohn-Sham determinant:
@ -473,28 +459,6 @@ Nevertheless, one can notice that the value of such optimal $\mu$ is sensibly la
wave function reoptimized in the presence of a Jastrow factor.}
\label{fig:overlap}
\end{figure}
\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{g16} 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}
@ -668,7 +632,6 @@ been the KS single determinant.
We could have obtained
single-determinant wave functions by using the natural orbitals of a first
% TODO : Get FCI calculations on Irene and redo figure with Ndet -> 10^7
@ -691,4 +654,39 @@ Simulation of Functional Materials.
\bibliography{rsdft-cipsi-qmc}
\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 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}
\begin{itemize}
\item plot $N_{det}$ en fonction de $\mu$
\end{itemize}
\end{itemize}
\end{enumerate}
\end{document}