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\newcommand{\LCT}{Laboratoire de Chimie Théorique (UMR 7616), Sorbonne Université, CNRS, Paris, France}
\newcommand{\ANL}{Argonne Leadership Computing Facility, Argonne National Laboratory, Argonne, IL 60439, United States}
\newcommand{\LCPQ}{Laboratoire de Chimie et Physique Quantiques (UMR 5626), Université de Toulouse, CNRS, UPS, France}


\begin{document}	

\title{Enabling high accuracy diffusion Monte Carlo calculations with
  range-separated density functional theory and selected configuration interaction}

\author{Anthony Scemama}
\affiliation{\LCPQ}
\author{Emmanuel Giner}
\email{emmanuel.giner@lct.jussieu.fr}
\affiliation{\LCT}
\author{Anouar Benali}
\email{benali@anl.gov}
\affiliation{\ANL}
\author{Pierre-Fran\c{c}ois Loos}
\email{loos@irsamc.ups-tlse.fr}
\affiliation{\LCPQ}


\begin{abstract}
\end{abstract}

\maketitle


\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 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
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,
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}
and the
revival\cite{Abrams_2005,Bytautas_2009,Roth_2009,Giner_2013,Knowles_2015,Holmes_2016,Liu_2016}
of selected configuration interaction (sCI)
methods\cite{Bender_1969,Huron_1973,Buenker_1974} pushed the limits of
the sizes of the systems that could be computed at the FCI level, but
the scaling remains exponential unless some bias is introduced leading
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.
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
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.
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} 
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
parameters of the trial wave function.
The conventional approach consists in multiplying the trial wave
function by a positive function, the \emph{Jastrow factor}, taking
account of the electron-electron cusp and the short-range correlation
effects. The wave function is then re-optimized in the presence of the
Jastrow factor and the nodal surface is expected to be improved.
Using this technique, it has been shown that the chemical accuracy
could be reached within DMC.\cite{Petruzielo_2012}

Another approach consists in considering the DMC method as a
\emph{post-FCI method}. The trial wave function is obtained by
approaching the FCI with a selected configuration interaction
method such as CIPSI for instance.\cite{Giner_2013,Caffarel_2016_2}
When the basis set is increased, the trial wave function tends to the
exact wave function, so the nodal surface can be systematically
improved.\cite{Caffarel_2016}
This technique has the advantage that using FCI nodes in a given basis
set is well defined and has a unique solution. The optimization of the
wave function is deterministic, so the calculations are reproducible
and don't require the expertise of a QMC expert. However,
this technique can't be applied to large systems because of the
exponential scaling of the size of the wave function.  Extrapolation
techniques have been applied to estimate the DMC energy of a FCI
wave function in a large basis set,\cite{Scemama_2018} and other
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}



\section{Combining range-separated DFT with CIPSI}
\label{sec:rsdft-cipsi}

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
increasing the determinant expansion towards the FCI limit.
A third direction of improvement exists. It is the path which connects
the Hartree-Fock determinant to the Kohn-Sham determinant, which
usually has a better nodal surface. This third path is obtained by
the so-called \emph{range-separated} DFT framework which splits the
electron-electron interaction in a long-range part and a short-range
part. HELP MANU!



\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. Within 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}

\subsubsection{Exact equations}
\label{sec:exact}

The exact ground-state energy of a $N$-electron system with
nuclei-electron potential $v_\mathrm{ne}(\textbf{r})$ can be expressed
by the following minimization over $N$-representable densities
$n$~\cite{Lev-PNAS-79,Lie-IJQC-83}
\begin{equation}
  E_0 = \min_n \left\{ \mathcal{F}[n] + \int v_\mathrm{ne}(\textbf{r}) n(\textbf{r}) \mathrm{d} \textbf{r} \right\},
  \label{E0}
\end{equation}
with the standard constrained-search universal density functional
\begin{equation}
  \mathcal{F}[n] = \min_{\Psi\rightarrow n} \left \langle\Psi|\hat{T}+\hat{W}_\mathrm{ee}|\Psi \right \rangle,
  \label{Fn}
\end{equation}
where $\hat{T}$ and $\hat{W}_\mathrm{ee}$ are the kinetic-energy and
Coulomb electron-electron interaction operators, respectively. The
minimizing multideterminant wave function in Eq.~\eqref{Fn} will be
denoted by $\Psi[n]$.  In RS-DFT, the universal density functional is
decomposed as~\cite{Sav-INC-96,TouColSav-PRA-04}
\begin{equation}
  \mathcal{F}[n] = \mathcal{F}^{\mathrm{lr},\mu}[n] + \bar{E}_{\mathrm{Hxc}}^{\mathrm{sr,}\mu}[n],
  \label{Fdecomp}
\end{equation}
where $\mathcal{F}^{\mathrm{lr},\mu}[n]$ is a long-range (lr)
universal density functional
\begin{equation}
\label{lr_univ_fonc}
\mathcal{F}^{\mathrm{lr},\mu}[n]= \min_{\Psi\rightarrow n} \left
  \langle\Psi|\hat{T}+\hat{W}_\mathrm{ee}^{\mathrm{lr},\mu}|\Psi
\right \rangle,
\end{equation}
and $\bar{E}_{\mathrm{Hxc}}^{\,\mathrm{sr,}\mu}[n]$ is the
complementary short-range (sr) Hartree-exchange-correlation (Hxc)
density functional. In Eq.~(\ref{lr_univ_fonc}),
$\hat{W}_\mathrm{ee}^{\mathrm{lr}}$ is the long-range
electron-electron interaction defined as
\begin{equation}
  \hat{W}_\mathrm{ee}^{\mathrm{lr},\mu} = \frac{1}{2} \iint w_{\mathrm{ee}}^{\mathrm{lr},\mu}(r_{12}) \hat{n}_2(\textbf{r}_1,\textbf{r}_2)
  \mathrm{d} \textbf{r}_1 \mathrm{d} \textbf{r}_2,
\end{equation}
with the error-function potential
$w_{\mathrm{ee}}^{\mathrm{lr},\mu}(r_{12})=\mathrm{erf}(\mu\, r_{12}
)/r_{12}$ (expressed with the interelectronic distance $r_{12} =
||\textbf{r}_1-\textbf{r}_2||$) and the pair-density operator
$\hat{n}_2(\textbf{r}_1, \textbf{r}_2)=\hat{n}(\textbf{r}_1)
\hat{n}(\textbf{r}_2) - \delta(\textbf{r}_1-\textbf{r}_2)
\hat{n}(\textbf{r}_1)$ where $\hat{n}(\textbf{r})$ is the density
operator. The range-separation parameter $\mu$ corresponds to an
inverse distance controlling the range of the separation and the
strength of the interaction at $r_{12} = 0$. For a given density, we
will denote by $\Psi^\mu[n]$ the minimizing multideterminant wave
function in Eq.~\eqref{lr_univ_fonc}. Inserting the decomposition of
Eq.~\eqref{Fdecomp} into Eq.~\eqref{E0}, and recomposing the two-step
minimization into a single one, leads to the following expression for
the exact ground-state electronic energy
\begin{equation}
  \label{min_rsdft} E_0= \min_{\Psi} \left\{
\left
  \langle\Psi|\hat{T}+\hat{W}_\mathrm{{ee}}^{\mathrm{lr},\mu}+\hat{V}_{\mathrm{ne}}|\Psi\right
\rangle
+ \bar{E}^{\mathrm{sr},\mu}_{\mathrm{Hxc}}[n_\Psi]\right\},
\end{equation}
where the minimization is done over normalized $N$-electron
multideterminant wave functions, $\hat{V}_{\mathrm{ne}} = \int
v_{\mathrm{ne}} (\textbf{r}) \hat{n}(\textbf{r}) \mathrm{d}
\textbf{r}$, and $n_\Psi$ refers to the density of $\Psi$, i.e.
$n_\Psi(\textbf{r})=\left \langle\Psi|\hat{n}(\textbf{r})|\Psi\right
\rangle$.
The minimizing multideterminant wave function $\Psi^\mu$ in
Eq.~\eqref{min_rsdft} can be determined by the self-consistent
eigenvalue equation
\begin{equation}
  \label{rs-dft-eigen-equation}
  \hat{H}^\mu[n_{\Psi^{\mu}}] \ket{\Psi^{\mu}}= \mathcal{E}^{\mu} \ket{\Psi^{\mu}},
\end{equation}
with the long-range interacting Hamiltonian
\begin{equation}
  \label{H_mu}
  \hat{H}^\mu[n_{\Psi^{\mu}}] = \hat{T}+\hat{W}_{\mathrm{ee}}^{\mathrm{lr},\mu}+\hat{V}_{\mathrm{ne}}+ \hat{\bar{V}}_{\mathrm{Hxc}}^{\mathrm{sr},\mu}[n_{\Psi^{\mu}}],
\end{equation}
where
$\hat{\bar{V}}_{\mathrm{Hxc}}^{\mathrm{sr},\mu}[n]=\int \delta \bar{E}^{\mathrm{sr},\mu}_{\mathrm{Hxc}}[n]/\delta n(\textbf{r}) \, \hat{n}(\textbf{r}) \mathrm{d} \textbf{r}$
is the complementary short-range Hartree-exchange-correlation
potential operator. Note that $\Psi^{\mu}$ is not the exact physical
ground-state wave function but only an effective wave
function. However, its density $n_{\Psi^{\mu}}$ is the exact physical
ground-state density. Once $\Psi^{\mu}$ has been calculated, the exact
electronic ground-state energy is obtained by
\begin{equation}
  \label{E-rsdft}
  E_0=  \braket{\Psi^{\mu}|\hat{T}+\hat{W}_\mathrm{{ee}}^{\mathrm{lr},\mu}+\hat{V}_{\mathrm{ne}}|\Psi^{\mu}}+\bar{E}^{\mathrm{sr},\mu}_{\mathrm{Hxc}}[n_{\Psi^\mu}].
\end{equation}
Note that, for $\mu=0$, the long-range interaction vanishes,
$w_{\mathrm{ee}}^{\mathrm{lr},\mu=0}(r_{12}) = 0$, and thus RS-DFT
reduces to standard KS-DFT. For $\mu\to\infty$, the long-range
interaction becomes the standard Coulomb interaction,
$w_{\mathrm{ee}}^{\mathrm{lr},\mu\to\infty}(r_{12}) = 1/r_{12}$, and
thus RS-DFT reduces to standard wave-function theory (WFT).

\subsection{Approximations}
\label{sec:approx}

Provided that the exact short-range complementary functional is known
and that the wave function $\Psi^{\mu}$ is developed in a complete
basis set, the theory is exact and therefore provides the exact
density and ground state energy whatever value of $\mu$ is chosen to
split the universal Levy-Lieb functional (see Eqs. \eqref{Fdecomp} and
\eqref{lr_univ_fonc}).  Of course in practice, two kind of
approximations must be made: i) the approximation of the wave function
by using a finite basis set, ii) the approximation on the exact
complementary functionals.  As long as approximations are made, the
theory is not exact anymore and might depend on $\mu$.

Despite its $\mu$ dependence, approximated RS-DFT schemes provide
potentially appealing features as: i) the approximated wave function
$\Psi^{\mu}$ is an eigenvector of a non-divergent operator
$\hat{H}^\mu[n_{\Psi^{\mu}}]$ (see Eqs. \eqref{H_mu} and
\eqref{rs-dft-eigen-equation}) and therefore converges more rapidly
with respect to the basis set~\cite{FraMusLupTou-JCP-15}, and also
produces more compact wave function~\cite{GinPraFerAssSavTou-JCP-18}
as it will be illustrated here, ii) the semi-local approximations for
RS-DFT complementary functionals are usually better suited to describe
the remaining short-range part of the electron-electron correlation.

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}) which
takes the form
\begin{eqnarray}
  \bar{E}^{\mathrm{sr},\mu,\textsc{pbe}}_{\mathrm{x/c}}[n] = \int  \bar{e}_\mathrm{{x/c}}^\mathrm{sr,\mu,\textsc{pbe}}(n(\textbf{r}),\nabla n(\textbf{r})) \,            \mathrm{d}\textbf{r}.
\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} 

\begin{figure}[h]
  \centering
  \includegraphics[width=\columnwidth]{algorithm.pdf}  
  \caption{Algorithm showing the generation of the RSDFT-CIPSI wave
    function}
\end{figure}



\section{Influence of the range-separation parameter on the fixed-node
  error}
\label{sec:mu-dmc}
\begin{table}[h]
  \caption{Fixed-node energies of the water molecule.}
  \label{tab:h2o-dmc}
  \centering
  \begin{tabular}{crlrl}
    \hline
              &  \multicolumn{2}{c}{BFD-VDZ} &  \multicolumn{2}{c}{BFD-VTZ}\\
    $\mu$     &  $N_{\text{det}}$  &  E(DMC)      &  $N_{\text{det}}$  &  E(DMC)\\
    \hline
    $0.00$    &  $11$         &  $-17.253\,59(6)$  &  $23$           &  $-17.256\,74(7)$\\
    $0.20$    &  $23$         &  $-17.253\,73(7)$  &  $23$           &  $-17.256\,73(8)$\\
    $0.30$    &  $53$         &  $-17.253\,4(2)$   &  $219$          &  $-17.253\,7(5)$\\
    $0.50$    &  $1\,442$     &  $-17.253\,9(2)$   &  $16\,99$       &  $-17.257\,7(2)$\\
    $0.75$    &  $3\,213$     &  $-17.255\,1(2)$   &  $13\,362$      &  $-17.258\,4(3)$\\
    $1.00$    &  $6\,743$     &  $-17.256\,6(2)$   &  $256\,73$      &  $-17.261\,0(2)$\\
    $1.75$    &  $54\,540$    &  $-17.259\,5(3)$   &  $207\,475$     &  $-17.263\,5(2)$\\
    $2.50$    &  $51\,691$    &  $-17.259\,4(3)$   &  $858\,123$     &  $-17.264\,3(3)$\\
    $3.80$    &  $103\,059$   &  $-17.258\,7(3)$   &  $1\,621\,513$  &  $-17.263\,7(3)$\\
    $5.70$    &  $102\,599$   &  $-17.257\,7(3)$   &  $1\,629\,655$  &  $-17.263\,2(3)$\\
    $8.50$    &  $101\,803$   &  $-17.257\,3(3)$   &  $1\,643\,301$  &  $-17.263\,3(4)$\\
    $\infty$  &  $200\,521$   &  $-17.256\,8(6)$   &  $1\,631\,982$  &  $-17.263\,9(3)$\\
    \hline
  \end{tabular}
\end{table}

\begin{figure}[h]
  \centering
  \includegraphics[width=\columnwidth]{h2o-dmc.pdf}  
  \caption{Fixed-node energies of the water molecule with variable
    values of $\mu$.}
  \label{fig:h2o-dmc}
\end{figure}
The water molecule was taken at the equilibrium
geometry,\cite{Caffarel_2016} and RSDFT-CIPSI wave functions were
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. Then, these wave functions
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{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}

Atomization energies are challenging for post-Hartree-Fock methods
because their calculation requires a perfect balance in the
description of atoms and molecules. Basis sets used in molecular
calculations are atom-centered, so they are always better adapted to
atoms than molecules and atomization energies usually tend to be
underestimated.
In the context of FN-DMC calculations, the nodal surface is imposed by
the trial wavefunction which is expanded on an atom-centered basis
set. So we expect the fixed-node error to be also related to the basis
set incompleteness error.
Increasing the size of the basis set improves the description of
the density and of electron correlation, but also reduces the
imbalance in the quality of the description of the atoms and the
molecule, leading to more accurate atomization energies.

Another important feature required to get accurate atomization
energies is size-extensivity, since the numbers of correlated electrons
in the isolated atoms are different from the number of correlated
electrons in the molecule.
In the context of selected CI calculations, when the variational energy is
extrapolated to the FCI energy\cite{Holmes_2017} there is obviously no
size-consistence error. But when the selected wave function is used
for as a reference for post-Hartree-Fock methods or QMC calculations,
there is a residual size-consistence error originating from the
truncation of the determinant space.

QMC calculations can be made size-consistent by extrapolating the
FN-DMC energy to estimate the energy obtained with the FCI as a trial
wave function.\cite{Scemama_2018,Scemama_2018b} Alternatively, the
size-consistence error can be reduced by choosing the number of
selected determinants such that the sum of the PT2 corrections on the
atoms is equal to the PT2 correction of the molecule, enforcing that
the variational dissociation potential energy surface (PES) is
parallel to the perturbatively corrected PES, which is an accurate
estimate of the FCI PES.\cite{Giner_2015}

Another source of size-consistence error in QMC calculation may originate
from the Jastrow factor. Usually, the Jastrow factor contains
one-electron, two-electron and one-nucleus-two-electron terms.
The problematic part is the two-electron term, whose simplest form can
be expressed as
\begin{equation}
  J_\text{ee} = \sum_i \sum_{j<i} \frac{a r_{ij}}{1 + b r_{ij}}.
\end{equation}
$a$ is determined by cusp conditions, and $b$ is obtained by energy
or variance minimization.\cite{Coldwell_1977,Umrigar_2005}
One can easily see that this parameterization of the two-body
interation is not size-consistent. The dissociation of a
heteroatomic diatomic molecule $AB$ with a parameter $b_{AB}$
will lead to two different two-body Jastrow factors on each atom, each
with its own optimal value $b_A$ and $b_B$. To remove the
size-consistence error on a PES using this ansätz for $J_\text{ee}$,
one needs to impose that the parameters of $J_\text{ee}$ are fixed:
$b_A = b_B = b_{AB}$.

When pseudopotentials are used in a QMC calculation, it is common
practice to localize the pseudopotential on the complete wave
function. If the wave function is not size-consistent, so will be the
locality approximation. Recently, the determinant localization
approximation was introduced.\cite{Zen_2019} This approximation
consists in removing the Jastrow factor from the wave function on
which the pseudopotential is localized.
The great advantage of this approximation is that the FN-DMC energy
within this approximation only depends on the parameters of the
determinantal component. Using a size-inconsistent Jastrow factor, or
a non-optimal Jastrow factor will not introduce an additional
size-consistence error in FN-DMC calculations, although it will
reduce the statistical errors by reducing the variance of the local energy.


The energy computed within density functional theory is extensive, and
as it is a mean-field method the convergence to the complete basis set
(CBS) limit is relatively fast. Hence, DFT methods are very well adapted to
the calculation of atomization energies, especially with small basis
sets, but going to the CBS limit will converge to biased atomization
energies because of the use of approximate density functionals.

On the other hand, the convergence of the FCI energies to the CBS
limit will be slower because of the description of short-range electron
correlation with atom-centered functions, but ultimately the exact
energy will be reached.


The 55 molecules of the benchmark for the Gaussian-1
theory\cite{Pople_1989,Curtiss_1990} were chosen to test the quality
of the RSDFT-CIPSI trial wave functions for energy differences.

\begin{table}[h]
  \centering
  \begin{tabular}{lcrr}
    Method          &  Basis  & $\mu$    &  MAD (kcal/mol) \\
    \hline
    PBE             & VDZ-BFD &          &  5.02 \\
                    & VTZ-BFD &          &  4.57 \\
                    & VQZ-BFD &          &  5.31 \\
    BLYP            & VDZ-BFD &          &  9.53 \\
                    & VTZ-BFD &          &  5.58 \\
                    & VQZ-BFD &          &  5.86 \\
    PBE0            & VDZ-BFD &          & 11.20 \\
                    & VTZ-BFD &          &  6.40 \\
                    & VQZ-BFD &          &  6.28 \\
    B3LYP           & VDZ-BFD &          & 11.27 \\
                    & VTZ-BFD &          &  7.27 \\
                    & VQZ-BFD &          &  6.75 \\
    \hline
    CCSD(T)         & VDZ-BFD &          & 24.10 \\
                    & VTZ-BFD &          &  9.11 \\
                    & VQZ-BFD &          &  4.52 \\
    \hline
    RSDFT-CIPSI     & VDZ-BFD & 0        & 10.08 \\
                    &         & 1/4      &  5.55 \\
                    &         & 1/2      & 13.42 \\
                    &         & 1        & 17.07 \\
                    &         & $\infty$ & 23.62 \\
                    & VTZ-BFD & 0        &  6.31 \\
                    &         & 1/4      &  4.58 \\
                    &         & 1/2      &  6.77 \\
                    &         & 1        &  9.06 \\
                    & VQZ-BFD & 0        &  6.35 \\
                    &         & 1/4      &  5.48 \\
                    &         & 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 \\ 
                    & VTZ-BFD & 0        &  3.52 $\pm$ 0.19 \\
                    &         & 1/4      &  3.39 $\pm$ 0.77 \\
                    &         & 1/2      &  2.46 $\pm$ 0.18 \\
                    &         & 1        &  4.38 $\pm$ 0.94 \\
                    & VQZ-BFD & 0        &  3.16 $\pm$ 0.26 \\
                    &         & 1/4      &  2.90 $\pm$ 0.25 \\
                    &         & 1/2      &  2.06 $\pm$ 0.35 \\
    \hline
  \end{tabular}
  \caption{Mean absolute deviations obtained with the different
    methods and basis sets.}
  \label{tab:mad}
\end{table}

\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}




%%---------------------------------------
\begin{acknowledgments}
An award of computer time was provided by the Innovative and Novel
Computational Impact on Theory and Experiment (INCITE) program. This
research has used resources of the Argonne Leadership Computing
Facility, which is a DOE Office of Science User Facility supported
under Contract DE-AC02-06CH11357.  AB, was supported by the
U.S. Department of Energy, Office of Science, Basic Energy Sciences,
Materials Sciences and Engineering Division, as part of the
Computational Materials Sciences Program and Center for Predictive
Simulation of Functional Materials.
\end{acknowledgments}


\bibliography{rsdft-cipsi-qmc}

\end{document}
