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23
Manuscript/rsdft-cipsi-qmc.bib
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@ -1057,6 +1057,18 @@
Year = {1996},
Bdsk-Url-1 = {https://doi.org/10.1016/S1380-7323(96)80091-4}}
@incollection{Sav-INC-96a,
author = {A. Savin},
title = {Beyond the Kohn-Sham Determinant},
booktitle = {Recent Advances in Density Functional Theory},
publisher = {World Scientific},
address = {},
editor = {D. P. Chong},
pages = {129-148},
year = {1996}
}
@article{Toulouse_2004,
Author = {Toulouse, Julien and Colonna, Fran{\c c}ois and Savin, Andreas},
Doi = {10.1103/PhysRevA.70.062505},
@ -1128,3 +1140,14 @@
Version = {2.1.2},
Year = 2020,
Bdsk-Url-1 = {https://doi.org/10.5281/zenodo.3677565}}
@article{TouSavFla-IJQC-04,
author = {J. Toulouse and A. Savin and H.-J. Flad},
title = {Short-range exchange-correlation energy of a uniform electron gas with modified electron-electron interaction},
journal = {Int. J. Quantum Chem.},
volume = {100},
pages = {1047},
year = {2004},
note = {}
}

71
Manuscript/rsdft-cipsi-qmc.tex
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@ -19,6 +19,7 @@
\definecolor{darkgreen}{HTML}{009900}
\usepackage[normalem]{ulem}
\newcommand{\toto}[1]{\textcolor{blue}{#1}}
\newcommand{\manu}[1]{\textcolor{green}{#1}}
\newcommand{\trashAS}[1]{\textcolor{blue}{\sout{#1}}}
\newcommand{\titou}[1]{\textcolor{red}{#1}}
\newcommand{\trashPFL}[1]{\textcolor{red}{\sout{#1}}}
@ -111,7 +112,7 @@ 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.
The polynomial scaling with the number of electrons and with the size
of the trial wave function makes the FN-DMC method particularly attractive.
of {\manu{in what sense is it polynomial?}the trial wave function makes the FN-DMC method particularly attractive.
In addition, the total energies obtained are usually far below
those obtained with the FCI method in computationally tractable basis
sets because the constraints imposed by the FN approximation
@ -154,14 +155,14 @@ FN-DMC.\cite{Petruzielo_2012}
Another approach consists in considering the FN-DMC method as a
\emph{post-FCI method}. The trial wave function is obtained by
approaching the FCI with a selected configuration interaction (SCI)
method such as CIPSI for instance.\cite{Giner_2013,Caffarel_2016_2}
method such as CIPSI for instance.\cite{Giner_2013,Giner_2015,Caffarel_2016_2}
\titou{When the basis set is increased, the trial wave function gets closer
to the exact wave function, so the nodal surface can be systematically
improved.\cite{Caffarel_2016} WRONG}
This technique has the advantage that using FCI nodes in a given basis
set is well defined, so the calculations are reproducible in a
This technique has the advantage \manu{of using the} FCI nodes in a given basis
set \manu{, which is perfectly well defined and therefore makes the calculations} reproducible in a
black-box way without needing any expertise in QMC.
But this technique cannot be applied to large systems because of the
\manu{Nevertheless,} this technique cannot be applied to large systems because of the
exponential scaling of the size of the trial wave function.
Extrapolation techniques have been used to estimate the FN-DMC energies
obtained with FCI wave functions,\cite{Scemama_2018} and other authors
@ -217,15 +218,15 @@ 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
If one wants to recover the 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 there exists several systematic
multi-determinant wave function one can obtain \manu{in a given basis set} is the FCI.
FCI is \manu{the ultimate goal of} \emph{post-Hartree-Fock} methods, 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
@ -262,8 +263,8 @@ accuracy so all the CIPSI selections were made such that $\abs{\EPT} <
\label{sec:rsdft}
%=================================
Following the seminal work of Savin,\cite{Savin_1996,Toulouse_2004}
the Coulomb operator entering the interelectronic repulsion is split into two pieces:
\manu{The range-separated DFT (RS-DFT)} was introduced in the seminal work of Savin,\cite{Sav-INC-96a,Toulouse_2004}
where the Coulomb operator entering the electron-electron repulsion is split into two pieces:
\begin{equation}
\frac{1}{r}
= w_{\text{ee}}^{\text{sr}, \mu}(r)
@ -278,7 +279,7 @@ where
are the singular short-range (sr) part and the non-singular long-range (lr) part, respectively, $\mu$ is the range-separation parameter which controls how rapidly the short-range part decays, $\erf(x)$ is the error function, and $\erfc(x) = 1 - \erf(x)$ is its complementary version.
The main idea behind RS-DFT is to treat the short-range part of the
interaction within KS-DFT, and the long-range part within a WFT method like FCI in the present case.
interaction \manu{using a density functional}, and the long-range part within a WFT method like FCI in the present case.
The parameter $\mu$ controls the range of the separation, and allows
to go continuously from the KS Hamiltonian ($\mu=0$) to
the FCI Hamiltonian ($\mu = \infty$).
@ -295,8 +296,8 @@ $\mathcal{F}^{\text{lr},\mu}$ is a long-range universal density
functional and $\bar{E}_{\text{Hxc}}^{\text{sr,}\mu}$ is the
complementary short-range Hartree-exchange-correlation (Hxc) density
functional. \cite{Savin_1996,Toulouse_2004}
One obtains the following expression for the ground-state
electronic energy
\manu{The exact ground state energy can be therefore obtained as a minimization
over a multi-determinant wave function as follows}:
\begin{equation}
\label{min_rsdft} E_0= \min_{\Psi} \qty{
\mel{\Psi}{\hat{T}+\hat{W}_\text{{ee}}^{\text{lr},\mu}+\hat{V}_{\text{ne}}}{\Psi}
@ -404,7 +405,7 @@ CCSD(T) and KS-DFT energies have been computed with
All the CIPSI calculations have been performed with \emph{Quantum
Package}.\cite{Garniron_2019,qp2_2020} We used the short-range version
of \titou{the local-density approximation (LDA)} and Perdew-Burke-Ernzerhof (PBE) \cite{PerBurErn-PRL-96} exchange
of the local-density approximation (LDA)\cite{Sav-INC-96a,TouSavFla-IJQC-04} and Perdew-Burke-Ernzerhof (PBE) \cite{PerBurErn-PRL-96} exchange
and correlation functionals defined in
Ref.~\onlinecite{GolWerStoLeiGorSav-CP-06} (see also
Refs.~\onlinecite{TouColSav-JCP-05,GolWerSto-PCCP-05}).
@ -478,13 +479,13 @@ For this purpose, we consider a weakly correlated molecular system, namely the w
molecule \titou{near its equilibrium geometry.} \cite{Caffarel_2016}
We then generate trial wave functions $\Psi^\mu$ for multiple values of
$\mu$, and compute the associated fixed-node energy keeping all the
parameters having an impact on the nodal surface fixed (\titou{such as ??}).
parameters having an impact on the nodal surface fixed \manu{such as CI coefficients and molecular orbitals}.
%======================================================
\subsection{Fixed-node energy of $\Psi^\mu$}
\label{sec:fndmc_mu}
%======================================================
From Table~\ref{tab:h2o-dmc} and Fig.~\ref{fig:h2o-dmc} where we report the fixed-node energy of \ce{H2O} as a function of $\mu$ for various short-range density functionals and basis sets,
From Table~\ref{tab:h2o-dmc} and Fig.~\ref{fig:h2o-dmc}, where we report the fixed-node energy of \ce{H2O} as a function of $\mu$ for various short-range density functionals and basis sets,
one can clearly observe that relying on FCI trial
wave functions ($\mu = \infty$) give FN-DMC energies lower
than the energies obtained with a single KS determinant ($\mu=0$):
@ -499,18 +500,20 @@ for an optimal value of $\mu$ (which is obviously basis set and functional depen
and then the FN-DMC error raises until it reaches the $\mu=\infty$ limit (\ie, the FCI wave function).
For instance, with respect to the FN-DMC/VDZ-BFD energy at $\mu=\infty$,
one can obtain a lowering of the FN-DMC energy of $2.6 \pm 0.7$~m\hartree{}
with an optimal value of $\mu=1.75$~bohr$^{-1}$.
with an optimal value of $\mu=1.75$~bohr$^{-1}$.
\manu{This lowering in FN-DMC energy is to be compared with the $3.2 \pm 0.7$~m\hartree{} of gain in FN-DMC energy between the KS wave function ($\mu=0$) and the FCI wave function ($\mu=\infty$)}.
When the basis set is increased, the gain in FN-DMC energy with
respect to the FCI trial wave function is reduced, and the optimal
value of $\mu$ is slightly shifted towards large $\mu$. Eventually, the nodes
of the wave functions $\Psi^\mu$ obtained with the srLDA
value of $\mu$ is slightly shifted towards large $\mu$.
Last but not least, the nodes of the wave functions $\Psi^\mu$ obtained with the srLDA
exchange-correlation functional give very similar FN-DMC energies with respect
to those obtained with the srPBE functional, even if the
RS-DFT energies obtained with these two functionals differ by several
tens of m\hartree{}.
tens of m\hartree{}.
\titou{The key fact here is that, at $\mu=1.75$~bohr$^{-1}$, $\Psi^{\mu}$ has \textit{only} $54\,540$ determinants at the srPBE/VDZ-BFD level, while the FCI wave function contains $200\,521$ determinants (see Table \ref{tab:h2o-dmc}). Even at the srPBE/VTZ-BFD level, we observe a reduction by a factor two in the number of determinants between the optimal $\mu$ value and $\mu = \infty$.
The take-home message here is that RS-DFT trial wave functions yield a lower fixed-node energy with more compact multideterminant expansion as compared to FCI.}
\manu{An other important aspect here regards the compactness of the trial wave functions $\Psi^\mu$:}
\titou{at $\mu=1.75$~bohr$^{-1}$, $\Psi^{\mu}$ has \textit{only} $54\,540$ determinants at the srPBE/VDZ-BFD level, while the FCI wave function contains $200\,521$ determinants (see Table \ref{tab:h2o-dmc}). Even at the srPBE/VTZ-BFD level, we observe a reduction by a factor two in the number of determinants between the optimal $\mu$ value and $\mu = \infty$.
The take-home message of this numerical study is that RS-DFT trial wave functions can yield a lower fixed-node energy with more compact multideterminant expansion as compared to FCI.}
%======================================================
\subsection{Link between RS-DFT and Jastrow factors }
@ -519,7 +522,7 @@ The take-home message here is that RS-DFT trial wave functions yield a lower fix
The data presented in Sec.~\ref{sec:fndmc_mu} evidence that, in a finite basis, RS-DFT can provide
trial wave functions with better nodes than FCI wave function.
Such behaviour can be directty compared to the common practice of
re-optimizing the multideterminant part of a trial wave function $\Psi$ (the so-called Slater part) in the presence of the exponentiated Jastrow factor $e^`J$. \cite{Umrigar_2005,Scemama_2006c,Umrigar_2007,Toulouse_2007,Toulouse_2008}
re-optimizing the multideterminant part of a trial wave function $\Psi$ (the so-called Slater part) in the presence of the exponentiated Jastrow factor $e^J$. \cite{Umrigar_2005,Scemama_2006c,Umrigar_2007,Toulouse_2007,Toulouse_2008}
Hence, in the present paragraph, we would like to elaborate further on the link between RS-DFT
and wave function optimization in the presence of a Jastrow factor.
\titou{T2: maybe we should mention that we only reoptimize the CI coefficients as it is of common practice to re-optimize more than this.}
@ -644,7 +647,7 @@ report in Table~\ref{table_on_top} the integrated on-top pair density
\begin{equation}
\expval{ n_2(\br,\br) } = \int d\br \,\,n_2(\br,\br)
\end{equation}
where $n_2(\br_1,\br_2$ is the two-body density [normalized to $N(N-1)$ where $N$ is the number of electrons]
where $n_2(\br_1,\br_2)$ is the two-body density [normalized to $N(N-1)$ where $N$ is the number of electrons]
obtained for both $\Psi^\mu$ and $\Psi^J$.
Then, in order to have a pictorial representation of both the on-top
pair density and the density, we report in Fig.~\ref{fig:n1} and Fig.~\ref{fig:n2}
@ -653,7 +656,7 @@ $n_2(\br,\br)$ along one \ce{O-H} axis of the water molecule.
From these data, one can clearly notice several trends.
First, from Table~\ref{table_on_top}, we can observe that the overall
on-top pair density decreases when $\mu$ increases. This is expected
on-top pair density decreases when $\mu$ increases, which is expected
as the two-electron interaction increases in $H^\mu[n]$.
Second, the relative variations of the on-top pair density with $\mu$
are much more important than that of the one-body density, the latter
@ -665,7 +668,7 @@ pair density obtained from $\Psi^J$ is superimposed with
$\Psi^{\mu=0.5}$, and at a large distance the on-top pair density is
the closest to $\mu=\infty$. The integrated on-top pair density
obtained with $\Psi^J$ lies between the values obtained with
$\mu=0.5$ and $\mu=1$~bohr$^{-1}$, constently with the FN-DMC energies
$\mu=0.5$ and $\mu=1$~bohr$^{-1}$, consistently with the FN-DMC energies
and the overlap curve.
These data suggest that the wave functions $\Psi^{0.5 \le \mu \le 1}$ and $\Psi^J$ are close,
@ -788,8 +791,7 @@ When pseudopotentials are used in a QMC calculation, it is common
practice to localize the non-local part of the pseudopotential on the
complete wave function (determinantal component and Jastrow).
If the wave function is not size-consistent,
so will be the locality approximation. Within, the determinant
localization approximation,\cite{Zen_2019} the Jastrow factor is
so will be the locality approximation. Within, the DLA,\cite{Zen_2019} the Jastrow factor is
removed from the wave function on which the pseudopotential is localized.
The great advantage of this approximation is that the FN-DMC energy
only depends on the parameters of the determinantal component. Using a
@ -826,7 +828,7 @@ Ref.~\onlinecite{Scemama_2015}).
In this section, we make a numerical verification that the produced
wave functions are size-consistent for a given range-separation
parameter.
We have computed the energy of the dissociated fluorine dimer, where
We have computed the \manu{FN-DMC} energy of the dissociated fluorine dimer, where
the two atoms are at a distance of 50~\AA. We expect that the energy
of this system is equal to twice the energy of the fluorine atom.
The data in table~\ref{tab:size-cons} shows that it is indeed the
@ -843,15 +845,14 @@ Closed-shell molecules often dissociate into open-shell
fragments. To get reliable atomization energies, it is important to
have a theory which is of comparable quality for open-shell and
closed-shell systems. A good test is to check that all the components
of a spin multiplet are degenerate.
FCI wave functions have this property and give degenrate energies with
of a spin multiplet are degenerate\manu{, as expected from exact solutions}.
FCI wave functions have this property and give degenerate energies with
respect to the spin quantum number $m_s$, but the multiplication by a
Jastrow factor introduces spin contamination if the parameters
for the same-spin electron pairs are different from those
for the opposite-spin pairs.\cite{Tenno_2004}
Again, when pseudopotentials are used this tiny error is transferred
in the FN-DMC energy unless the determinant localization approximation
is used.
in the FN-DMC energy unless the DLA is used.
Within DFT, the common density functionals make a difference for
same-spin and opposite-spin interactions. As DFT is a
@ -895,7 +896,7 @@ Although using $m_s=0$ the energy is higher than with $m_s=1$, the
bias is relatively small, more than one order of magnitude smaller
than the energy gained by reducing the fixed-node error going from the single
determinant to the FCI trial wave function. The highest bias, close to
2~m\hartree, is obtained for $\mu=0$, but the bias decreases quickly
2~m\hartree, is obtained for $\mu=0$, but the bias decreases rapidly
below 1~m\hartree{} when $\mu$ increases. As expected, with $\mu=\infty$
there is no bias (within the error bars), and the bias is not
noticeable with $\mu=5$~bohr$^{-1}$.
@ -1084,7 +1085,7 @@ solution would have been the PBE single determinant.
\section{Conclusion}
%%%%%%%%%%%%%%%%%%%%
We have seen that introducing short-range correation via
\manu{In the present work} we have shown that introducing short-range correation via
a range-separated Hamiltonian in a full CI expansion yields improved
nodal surfaces, especially with small basis sets. The effect of sr-DFT
on the determinant expansion is similar to the effect of re-optimizing