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\begin{document}

%\preprint{APS/123-QED}

\title{Enabling high accuracy  DMC calculations with Range Separated DFT nodes}
%\\ breaks with \\
%\thanks{A footnote to the article title}%

\author{Anouar Benali*}%
\email{benali@anl.gov}
\affiliation{Computational Sciences Division, Argonne National Laboratory, Argonne, IL 60439, United States}
\author{Thomas Applencourt}
\affiliation{Argonne Leadership Computing Facility, Argonne National Laboratory, Argonne, IL 60439, United States}
\author{Pierre-Francois Loos}
\author{Anthony Scemama*}
\affiliation{Laboratoire de Chimie et Physique Quantiques, Universit\'e de Toulouse, CNRS, UPS, France}
\author{Emanuel Giner*}
\email{emmanuel.giner@lct.jussieu.fr}
\affiliation{Laboratoire de Chimie Th\'eorique, Sorbonne Universit\'e and CNRS, F-75005 Paris,
France}





\date{\today}% It is always \today, today,
             %  but any date may be explicitly specified

\begin{abstract}
%TODO
Fixed Node approximation sucks. Best way to solve it is with multideterminants. sCI approach great but only for small systems. Large number of determinants comes from high energy determinants describing electron electron cusp in the same manner as 2 body Jastrow used in Diffusion Monte Carlo. Range Separated Density Functional Theory removes the cusp by using a DFT function (erf function) to describe the cusp and CIPSI for long range. Determinants are then selected without the electron electron cusp. 
\end{abstract}

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%\tableofcontents

\section{\label{sec:level1}Introduction}

%TODO
Single determinant diffusion Monte Carlo (SD-DMC) has proven to be very accurate at describing properties of a wide range of molecules and solids (citation).
While closed-shell systems can reach accuracies of few kcal/mol for molecules and 0.05 to 0.1~eV for solids when compared to experimental data or coupled cluster calculations, a systematic accurate prediction of properties such as total energies and band/HOMO-LUMO gaps remains dependent on the nodes of the trial wavefunction; while DMC solves exactly the many-body Schr\"odinger equation, the fixed node approximation is the only practical way to maintain the fermionic behavior of the wavefunction and therefore is considered the \\

Using a CIPSI trial wavefunction in DMC allows a systematic improvement of the nodal surface for a large class of molecules (cite LCPQ papers) and solids(cite unpublished Diamond).
However, while CIPSI allows for a very compact determinant expansion, we are still limited in practice to a few hundreds of atomic orbitals per system.
Moreover, in many cases, even when it is possible to converge the CIPSI energy for a system, the size of determinant expansion is too large to be used in DMC as the wave function needs to be evaluated at every Monte Carlo step.
While the DMC energy associated with the FCI wave function can be estimated by extrapolation, it is still desirable to find a strategy to reduce the size of the determinant expansions.
%While the large number of determinants is often attributed to describing the electron-electron cusp, using a 2-body Jastrow often comes to achieving the same result at short range.   

\section{Short-range correlation}

\newcommand{\psidet}{\Psi_{\text{CI}}}
\newcommand{\psijas}{\exp(J)}

The common practice in QMC calculations is to first compute a trial wave function expressed as a configuration interaction (CI) expansion $\psidet$.
Then, the wave function is improved with a Jastrow factor $\psijas$ optimized in a variational Monte Carlo (VMC) framework.
The role of the Jastrow factor is to take account explicitly of electron correlation and 
the simplest Jastrow factor is expressed as
\begin{equation}
J = \sum_{i<j} \exp \qty( \frac{a r_12}{1+b r_{12}} ).
\end{equation}
The parameter $a$ can be set to impose the electron-electron cusp, and the parameter $b$ is related to the width of the Coulomb hole.
Conventional QMC calculations use more advanced forms of Jastrow factor, but the following can be applied to any form of Jastrow factor.
With a Jastrow factor, the variance of the local energy is significantly reduced because the wave function is now described beyond the limits of the one-electron basis set, similarly to F12 methods.
$\psidet$ is finally re-optimized in the presence of the Jastrow factor.(cite Claudia, Cyrus, Sandro, Julien) This last step improves the nodal surfaces, and also has the effect of shortening the CI expansion.

The short-range correlation effects described by the Jastrow factor are partly taken care of by the determinant expansion, but the Slater determinant basis is not well adapted to describe these effects. Moreover, the determinant expansion is responsible for the nodal surface, so it is desirable to express short-range correlations exclusively with a positive function and let the determinant expansion relax to treat exclusively static correlation effects.

In the ideal case, one would make the CIPSI determinant selection in the presence of the Jastrow factor to produce directly short CI expansions.

\section{Range-separated DFT in a nutshell}
\subsection{Exact equations}
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} \langle\Psi|\hat{T}+\hat{W}_\mathrm{ee}|\Psi \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
\beq
\label{lr_univ_fonc}
\mathcal{F}^{\mathrm{lr},\mu}[n]= \min_{\Psi\rightarrow n} \langle\Psi|\hat{T}+\hat{W}_\mathrm{ee}^{\mathrm{lr},\mu}|\Psi \rangle,
\eeq
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{eqnarray}
\label{min_rsdft}
E_0= \min_{\Psi} \Big\{ \langle\Psi|\hat{T}+\hat{W}_\mathrm{{ee}}^{\mathrm{lr},\mu}+\hat{V}_{\mathrm{ne}}|\Psi\rangle + \bar{E}^{\mathrm{sr},\mu}_{\mathrm{Hxc}}[n_\Psi]\Big\},
\nonumber\\
\end{eqnarray}                                                                                                                               
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.                                  $n_\Psi(\textbf{r})=\langle\Psi|\hat{n}(\textbf{r})|\Psi\rangle$.
The   minimizing multideterminant wave function $\Psi^\mu$ in Eq.~\eqref{min_rsdft} can be determined by the self-consistent eigenvalue equation
\beq
\label{rs-dft-eigen-equation}
\hat{H}^\mu[n_{\Psi^{\mu}}] \Ket{\Psi^{\mu}}= \mathcal{E}^{\mu} \Ket{\Psi^{\mu}},
\eeq
with the long-range interacting Hamiltonian
\beq
\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}}],
\eeq
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
\beq
\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}].
\eeq
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}
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{FerGinTou-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}



\section{\label{sec:Methods}Computational details}
All single determinant Hartree Fock (HF) and \textit{vanilla} density functional theory\cite{Hohenberg1964,kohn1965physrev} calculations were done using the PySCF code\cite{PYSCF}. To scan through multiple nodal surfaces, we used some of the most widely used XC functionals in the literature (LDA\cite{Kohn1965}, PBE\cite{PBE}, B3LYP\cite{B3LYP1,B3LYP2,B3LYP3,B3LYP4}, PBE0\cite{PBE,adamo1999jchemphys}, revPBE\cite{PhysRevLett.80.890}, and M06-L\cite{M06-L}).\\ 
Multideterminant wavefunctions were generated using selected Configuration Interaction\cite{Bender69,Whitten69,Huron73,Buenker78} within the Configuration Interaction Pertubatively Selected Iteratively (CIPSI)\cite{EVANGELISTI1983,Cimiraglia1985,Chirlian1987,Giner2013,Caffarel2014,Caffarel16} flavor as implemented in the Quantum Package code\cite{QP2}. Compared to other sCI methods, CIPSI has the advantage of generated a very compact trial wavefunction ideal for DMC. CIPSI calculations were converged when possible to a threshold $PT_2=0.0001$ or to a number of determinants explicitly specified for each system.  For the range-separated multideterminant density functional theory  calculations, we used a PBE short-range exchange and correlation functional as describe in ref[\onlinecite{2019JChPh.150h4103F}], also implemented in Quantum Package. The range separation value $\mu$ was optimized for each system in order to minimize the DMC energy. \\
All single and multideterminant DMC calculations used 1, 2 and three body Jastrows as implemented in the QMCPACK code\cite{Kim_2018}. The Jastrow parameters were optimized using VMC in presence of the fixed coefficients of the multideterminant expansions. While it has been shown that simultaneously reoptimizing the determinants coefficients, the Jastrow parameters while rotating the orbitals can lead to significantly small wavefunction with improved accuracies(cite Cyrus), such a procedure can only be applied to very small system as the number of parameters to optimize becomes too important. In our paper, the use of Jastrows has the benefit of reducing the variance and hence speeding up the calculations. All DMC calculation used a 0.001 time step. \\
All calculations used cc-pvNz basissets where N=D,T and Q unless when specified. To extrapolate to the CBS limit we used a mixed gaussian/exponential extrapolation scheme. 
\begin{equation}
    E_{cbs}(2,3,4)=\frac{(1+e^2)E_2-(e+e^3-e^5)E_3+e^6E_4}{(e-1)(e^5-e^2-1)}
\end{equation}
In the case of $C_2H_2-CH_2-O$ molecule, we also performed CCSD(T) calculations as a reference. 

\section{\label{sec:results}Results and Discussion}
\subsection{Water Molecule\label{subsec:Water}}
The water molecule have been extensively studied using CIPSI and DMC for both the ground state\cite{Caffarel2016} and excited states\cite{H2O-exci,H2O-exci-PP}. In their 2016 study, Caffarel \textit{et. al} were able to approximate the total energy of a water molecule to unprecedented accuracy (value here) using a about 1M determinant. In order to assess the accuracy using RS-MDDFT nodes we evaluate the 
\begin{table*}
\centering
\caption{Total energy (Ha) of H2O molecule using cc-pcvDz, cc-pcvTz, cc-pcvQz basis-sets. using RSDFT with a PBE functional using various ranges $\mu$. CIPSI(RSDFT) corresponds to the long range Wavefunction treated with selected CI. DMC(RSDFT) corresponds to the DMC with a RSDFT-PBE nodale surface containing 1 determinant. DMC(RSCI) corresponds to the DMC with a RSCI nodale surface. Nb dets is the number of determinants when sCI expansions are used. This number of determinants was also used in the DMC(RS-CI). In the case of RSDFT(PBE) and DMC(RSDFT) the number of determinants is 1.    }
\label{tab:RSDFT-H2O}
\begin{tabular}{cccccccccccccccc}
\hline \hline
                         Basis-set &  $\mu$    & Nb dets& RSDFT(PBE)& RS-CI  & DMC(RSDFT)& DMC(RS-CI)  \\ \hline
\multirow{9}{*}{cc-pcvDz}
%& 0 &  1 & -76.3355494055 &- & -76.410094 0.000624&  -\\
%& 0.0001&    2    &  -76.33560776488173 &  -76.3356077649   &   -76.409746 0.000239 &      -76.410145 0.000200\\
&  0.5  &   2396  &  -76.33604445222997 &  -76.3441442508   &   -76.410018 0.000198  &     -76.411258 0.000383\\     
&  1.0  &   40963 &  -76.27348956941330 &  -76.3336831179   &   -76.410771 0.000218  &     -76.415616 0.000479\\     
&  1.5  &  138253 &  -76.20172972611051 &  -76.3147073528   &   -76.411030 0.000143  &     -76.418327 0.000311\\
&  2.0  &  207380 &  -76.15446145849143 &  -76.3018300218   &   -76.410994 0.000168  &     -76.417751 0.000544\\
&  2.5  &  311071 &  -76.12743299891538 &  -76.2963616190   &   -76.410997 0.000138  &     -76.418830 0.000320\\
&  3.0  &  466607 &  -76.11231786534898 &  -76.2953530572   &   -76.410925 0.000175  &     -76.418333 0.000604\\
&  4.0  &  466607 &  -76.09737987601412 &  -76.2973907108  &    -76.410767 0.000183  &     -76.417842 0.000485\\
&  5.0  &  466607 &  -76.08892265698270 &  -76.299088319  &     -76.410719 0.000167  &     -76.417461 0.000481\\      
&  6.0  &  466607 &  -76.08222148947996 &  -76.2995315838 &     -76.410730 0.000172  &     -76.417162 0.000632\\
&  7.0  &  466607 &  -76.07645402100390 &  -76.2992732268  &    -76.410741 0.000192  &     -76.417398 0.000384\\
\hline
                         
\multirow{9}{*}{cc-pcvTz}
%&0.0001&    1    &  -76.37438352807571  &  -76.3743835281  &   -76.420977 0.000147  &           -\\
&  0.5&    8090  &   -76.37411359270587 &   -76.3853803202 &    -76.420407 0.000116 &      -76.422574 0.000253\\
&  1.0&   466607 &   -76.30808407357975 &   -76.3810129344 &    -76.420450 0.000117 &      -76.427268 0.000494\\
&  1.5&  2000000 &   -76.23450972568928 &   -76.3699240165 &    -76.420484 0.000129 &      -76.430965 0.000937\\
&  2.0&  2000000 &   -76.18647332232358 &   -76.3659759776 &    -76.420322 0.000123 &      -76.430956 0.000704\\
&  2.5&  2000000 &   -76.15905488759424 &   -76.3681700530 &    -76.420125 0.000170 &      -76.432259 0.000539\\
&  3.0&  2000000 &   -76.14366220610044 &   -76.3730744570 &    -76.420003 0.000149 &      -76.432267 0.000611\\
&  4.0&  2000000 &   -76.12827446549065 &   -76.3829546254 &    -76.420138 0.000139 &      -76.430869 0.000996\\
&  5.0&  2000000 &   -76.11950514107885 &   -76.3892344149 &    -76.419934 0.000119 &      -76.431917 0.000943\\
&  7.0&  2000000 &   -76.10674756885926 &   -76.3942237680 &    -76.419851 0.000161&       -76.430150 0.000752\\
  \hline
\multirow{9}{*}{cc-pcvQz}
&0.5 &12136   &-76.38318565911486      &-76.3952959489            &-76.422831 0.000207         &-76.425846  0.000254\\
&1.0 &2382775 &-76.31599767191094      &-76.3917462575            &-76.422337 0.000151         &-76.428929  0.000748\\
&1.5 &2408343 &-76.24211906926882      &-76.3816852280            &-76.422451 0.000188         &-76.432489  0.000531\\
&2.0 &3698407 &-76.19399206529708      &-76.3795304096            &-76.422567 0.000139         &-76.435818  0.000763\\
&2.5 &3847936 &-76.16655117814619      &-76.3834274230            &-76.422184 0.000103         &-76.435031  0.000546\\
&3.0 &2000000 &-76.15117119770653      &-76.3898019841            &-76.421932 0.000201         &-76.435098  0.000539\\
&4.0 &2000000 &-76.13583094033078      &-76.4016324447            &-76.422268 0.000097         &-76.433833  0.000530\\
&5.0 &2000000 &-76.12708615152579      &-76.4093388884            &-76.421901 0.000154         &-76.434136  0.000370\\
&7.0 &3468702 &-76.11433599757845      &-76.4159382168            &-76.421721 0.000183         &-76.434913  0.000523\\
 \hline
\end{tabular}
\end{table*}
\begin{figure*}
 \includegraphics[width=5 in]{DMC-RSDFT.pdf}
 \caption{SD-DMC energy (Ha) of $H_2O$ using an RSDFT trial wavefunction in cc-pcvNz (N=D,T,Q)   }  
 \label{fig:RSDFT(DMC)}
\end{figure*}

\begin{figure*}
 \includegraphics[width=5 in]{DMC-RS-CI.pdf}
 \caption{MD-DMC energy (Ha) of $H_2O$ using an RSDFT-CI trial wavefunction in cc-pcvNz (N=D,T,Q)   }  
 \label{fig:RSDFT(DMC)}
\end{figure*}


\begin{table*}
\centering
\caption{Total energy (Ha) of H2O molecule in cc-pcvDz, cc-pcvTz, cc-pcvQz basis-sets  using HF and DFT nodale surfaces with 1 determinant.     }
\label{tab:DFT-H2O}
\begin{tabular}{lcccc}
\hline 
Method&  cc-pcvDz &   cc-pcvTz &  cc-pcvQz &CBS  \\ \hline
Hartree Fock &-76.0271766506 &-76.0573172765 &-76.0649040131&-76.06898746\\ 
PBE & -76.3355494055&-76.3743286157 &-76.3840723976&-76.38931444\\ 
PBE0 & -76.3402905995& -76.3753524250&-76.3841796010&-76.38893093\\ 
B3LYP & -76.3849274149& -76.4237882022&-76.4332604748&-76.43831592\\ 
DMC(HF) & 76.409528 (501)& -76.419666 (143)&-76.421526 (360)&-76.42243128\\ 
DMC(PBE) &-76.410094 (624) & -76.420453 (434)&-76.422582 (361)&-76.42366162\\ 
DMC(PBE0) & -76.410533 (843)& -76.420505 (381)&-76.422983 (595)&-76.42431230\\ 
DMC(B3LYP) & -76.409494 (681)& -76.420054 (455)&-76.423077 (367)&-76.42475463\\ 
CIPSI$^\dagger$& -76.282136 & -76.388287 & -76.419324 &-76.43662728\\
DMC$^\dagger$&-76.41571(20)&-76.43182(19)&-76.43622(14)&-76.43863601\\

\end{tabular}
\begin{flushleft}

$^\dagger$Ref.\onlinecite{Caffarel2016}.\\

%$^1$Ref.~\onlinecite{stefanovich1994}.\\
%$^2$Ref.~\onlinecite{bouvier01}.\\
%$^3$Refs.~\onlinecite{fukuhara93,bouvier00,bouvier03}.\\
%$^4$Refs.~\onlinecite{aldebert85,howard88}.\\
%$^5$Ref.~\onlinecite{desgreniers99}.\\
%$^6$Ref.~\onlinecite{lynch74}.\\
%$^\parallel$Estimated in Ref.~\onlinecite{stefanovich1994} from enthalpy differences in Ref.~\onlinecite{ackermann1975}.\\
\end{flushleft}
\end{table*}
While a scan of the different values 

\begin{table*}[t]
\centering
\caption{Total energy (Ha) of H2O molecule using cc-pcvDz, cc-pcvTz, cc-pcvQz basis-sets. using RSDFT with a PBE functional using 4 various ranges $\mu$. CIPSI(RSDFT) corresponds to the long range Wavefunction treated with selected CI. DMC(RSDFT) corresponds to the DMC with a RSDFT-PBE nodale surface containing 1 determinant. DMC(RSCI) corresponds to the DMC with a RSCI nodale surface. Nb dets is the number of determinants when sCI expansions are used. In the case of RSDFT(PBE) and DMC(RSDFT) the number of determinants is 1.    }
\label{tab:RSDFT-H2O}
\begin{tabular}{ccccc}
\hline \hline
                         Basis-set &  Truncation   & Nb dets&   DMC(RS-CI)  \\ \hline
\multirow{6}{*}{cc-pcvDz}
& $10^{-4}$ &  1635  &   -76.414399 (526)  \\
& $10^{-5}$ &  13545 &  -76.416102 (459)  \\
& $10^{-6}$ &  50928 &  -76.417872 (480)  \\
& $10^{-10}$   &466607  &-76.418333 (604)\\
& sCI-CIPSI$^\dagger$ &172256& -76.415710 (200)\\
& extrapolated &  -  &   \\
\hline
                         
\multirow{6}{*}{cc-pcvTz}
& $10^{-4}$ & 2422&  -76.423138  (313)   \\
& $10^{-5}$ &  32128 &  -76.428206  (325)  \\
& $10^{-6}$ & 225188 &  -76.430162 (440)  \\
& $10^{-10}$ &2000000 & -76.432267 (611) \\
& sCI-CIPSI$^\dagger$ &640426& -76.431820 (190)\\
& extrapolated &  - &   \\
  \hline
\multirow{6}{*}{cc-pcvQz}
& $10^{-4}$ &  2771 & -76.424923 (304)    \\
& $10^{-5}$ &  44705&  -76.430322 (322)  \\
& $10^{-6}$ & 333724&  -76.432890 (311) \\
& $10^{-10}$ &2000000&-76.435098 (539)\\
&sCI-CIPSI$^\dagger$& 666927&-76.436220 (140)\\
& extrapolated & - &   \\
 \hline
\end{tabular}
\begin{flushleft}
$^\dagger$Ref.\onlinecite{Caffarel2016}.\\
\end{flushleft}
\end{table*}

The CBS extrapolated energy is -76.43652806 from DMC(RS-CI)
%\subsection{Heterocyclic rings}
%\label{subsec:rings}
%\subsubsection{symptomatic case}
%In this section, we describe a case where a closed shell molecule $(C_2H_2)(CH_2)_2O$(\ref{Fig:Amon}) (Weird molecule I don't know how to name) shows a large dependence of the nodal surface to the exchange and correlation functional used to generate the trial wavefunction. This molecule is part of the Amon5 data set (citation) and led to an error of 5 kcal/mol between nodal surface using B3LYP or PBE0.  As shown in Tab-\ref{tab:Amon}.


%\begin{table*}[t]
%\centering
%\caption{Total energy (Ha) of$(C_2H_2)(CH_2)_2O$ molecule using cc-pcvNz basis-sets, where N=D,T,Q. using RSDFT with a PBE functional using 4 various ranges $\mu$. CIPSI(RSDFT) corresponds to the long range Wavefunction treated with selected CI. DMC(RSDFT) corresponds to the DMC with a RSDFT-PBE nodale surface containing 1 determinant. DMC(RSCI) corresponds to the DMC with a RSCI nodale surface. Nb dets is the number of determinants when sCI expansions are used. In the case of RSDFT(PBE) and DMC(RSDFT) the number of determinants is 1.    }
%\label{tab:RSDFT-H2O}
%\begin{tabular}{ccccc}
%\hline \hline
%                         Basis-set &  $\mu$   & Nb dets& RS-CI&   DMC(RS-CI)  \\ \hline
%\multirow{6}{*}{cc-pcvDz}
%& 0.5 & 1071193   &-230.9072267556 &-231.171773 0.000919\\
%& 1.0 &   &  &\\
%& 1.5 &   &  &\\
%& 2.0 &   &  &\\
%& 3.0 &   &  &\\
%\hline
%                         
%\multirow{6}{*}{cc-pcvTz}
%& 0.5 &   &  &\\
%& 1.0 &   &  &\\
%& 1.5 &   &  &\\
%& 2.0 &   &  &\\
%& 3.0 &   &  &\\
%  \hline
%\end{tabular}
%\end{table*}




%\begin{table*}
%\centering
%\caption{CCSD(T), sCI and RS-CI total energies of $(C_2H_2)(CH_2)_2O$ with corresponding nodal surfaces.}
%\label{tab:Amon}
%\begin{tabular}{lcccc}
%\hline 
%Method & cc-pvDz & cc-pvTz & cc-pvQz & CBS\\
%\hline
%B3LYP        &-231.093329385307 &-231.170584346714 & -231.253886194304&-231.30756751\\
%LDA          &-226.788590559813 &-226.879468843886 & -226.897145319615&-226.90593931\\
%M06-L        &-231.203556502805 &-231.264239822355 & -231.289463622331&-231.30441811\\
%PBE          &-230.944478188641 &-231.016720117095 & -231.035891670978&-231.04634726\\
%PBE0         &-230.968714378770 &-231.035555580761 & -231.053142060619&-231.06271325\\
%REVPBE       &-231.167251334987 &-231.235909155231 & -231.255234805111&-231.26591964\\
%%TPSSH        &-231.253886194304 &-231.320766126893 & -231.338230427467 &-231.34771758\\
%DMC(B3LYP)   &-231.160238 +/- 0.000575 & -231.177485 +/- 0.000683 &-231.179359 +/- 0.000572  &-231.18002589 +/- 0.0011\\
%DMC(LDA)     &-231.158100 +/- 0.000889 & -231.176895 +/- 0.000951 &-231.178217 +/- 0.000577  &-231.17845634 +/- %0.0013\\
%DMC(M06-L)   &-231.157596 +/- 0.000634 & -231.172360 +/- 0.000821 &-231.176893 +/- 0.000653  &-231.17944595 +/- 0.0011\\
%DMC(PBE)     &-231.159047 +/- 0.000703 & -231.175740 +/- 0.000687 &-231.181002 +/- 0.000509  &-231.18398105 +/- 0.0011\\
%DMC(PBE0)    &-231.161049 +/- 0.000839 & -231.175554 +/- 0.000639 &-231.179289 +/- 0.000715  &-231.18131093 +/- 0.0013\\
%DMC(REVPBE)  &-231.159579 +/- 0.000627 & -231.173518 +/- 0.000690 &-231.178781 +/- 0.000718  &-231.18185675 +/- 0.0013\\
%DMC(TPSSH)   &   & -231.176844 +/- 0.000883  \\
%CIPSI & -230.573665595769 &  -230.830792783359 & \\
%RS-CI & & & \\
%DMC(CIPSI)&  -231.178695 0.000842& -231.179043 +/- 0.000795 \\
%DMC(RS-CI) & \\
%CCSD(T) &-230.6086906999973937& -230.9115831028316285& -231.051884751321992&-231.13627366\\
%\end{tabular}
%\end{table*}




%\begin{figure}
% \includegraphics[width=1.5in]{frag_00278.jpg}
% \caption{Cyclic $(C_2H_2)(CH_2)_2O$. Colored atoms represent Carbon (dark gray), Oxygen (red) and Hydrogen (light gray) }  
% \label{Fig:Amon}
%\end{figure}



\subsection{Reaction barrier heights for cycloreversion of heterocyclic rings \label{subsubsec:DMC}}
%TO MODIFY THIS TEXT!!! COPY PASTE FROM PAPER: http://dx.doi.org/10.1016/j.chemphys.2015.07.005 :\\
Cycloaddition reactions (or in the reverse direction; cycloreversion reactions) are one of the most important classes of organic reactions for converting simple unsaturated building blocks to cyclic structures and vice versa. Over the past decade a number of barrier height datasets of cycloaddition reactions have been constructed for the purpose of evaluating the performance of DFT and computationally economical ab initio methods [6–9]. It has been found that many DFT and ab initio methods perform poorly in computing the barrier heights of cycloaddition reactions.
END OF COPY PASTE\\

\subsubsection{All electrons \label{subsubsubsec:AE}}

\begin{figure}
 \includegraphics[width=3.5 in]{Reactant-Transition.jpg}
 \caption{Reaction path of the cyloreversion of $N-CH-O-CH_2O$. Left molecule represents the cyclic molecule $N-CH-O-CH_2O$ as a reactant phase, while the right molecules represent the 2 fragments after the cycloreversion $CH_2$=$O$ + $NCH$=$O$. Colored atoms represent Carbon (dark gray), Oxygen (red), Nitrogen (blue) and Hydrogen (light gray)}  
 \label{fig:Cycloreversion}
\end{figure}

In Single determinant DMC, All hell breaks loose. 
\begin{figure}
 \includegraphics[width=5 in]{Plot-Barrier.jpg}
 \caption{DMC energy (KCal/mol) of the cycloreversion barrier in function of multiple trial wavefunctions }  
 \label{fig:barrier}
\end{figure}

\begin{table*}[t]
\centering
\caption{Total energy (Ha) of Reactant $NCOHCH_2O$  molecule using cc-pcvTz basis-sets. using RSDFT with a PBE functional using 4 various ranges $\mu$. CIPSI(RSDFT) corresponds to the long range Wavefunction treated with selected CI. DMC(RSDFT) corresponds to the DMC with a RSDFT-PBE nodale surface containing 1 determinant. DMC(RSCI) corresponds to the DMC with a RSCI nodale surface. Nb dets is the number of determinants when sCI expansions are used. In the case of RSDFT(PBE) and DMC(RSDFT) the number of determinants is 1.    }
\label{tab:RSDFT-Reactant}
\begin{tabular}{cccccccccccccccc}
\hline \hline
  $\mu$    & Nb dets& RS-CI  & DMC(RS-CI)  \\ \hline
%\multirow{9}{*}{cc-pcvDz}
0.5&1157288&-282.9133058711&-283.094003 0.000927\\     
1.0&1904949&-282.8057502175&-283.103138 0.000927\\
1.5&1977527&-282.7101915981&-283.099866 0.000664\\ 
2.0&1013451&-282.6154071485&-283.099842 0.001062\\
3.0&3004675&-282.6828501511&-283.096674 0.001918\\
CIPSI (unconverged)&\\
\hline
\end{tabular}
\end{table*}

\begin{table*}[t]
\centering
\caption{Total energy (Ha) of Transition $NCOHCH_2O$  molecule using cc-pcvTz basis-sets. using RSDFT with a PBE functional using 4 various ranges $\mu$. CIPSI(RSDFT) corresponds to the long range Wavefunction treated with selected CI. DMC(RSDFT) corresponds to the DMC with a RSDFT-PBE nodale surface containing 1 determinant. DMC(RSCI) corresponds to the DMC with a RSCI nodale surface. Nb dets is the number of determinants when sCI expansions are used. In the case of RSDFT(PBE) and DMC(RSDFT) the number of determinants is 1.    }
\label{tab:RSDFT-Transition}
\begin{tabular}{cccccccccccccccc}
\hline \hline
  $\mu$    & Nb dets& RS-CI  & DMC(RS-CI)  \\ \hline
%\multirow{9}{*}{cc-pcvDz}
0.5&1107227&-282.8266993882&-283.016690 0.000992\\
1.0&1755071&-282.7189345822&-283.027787 0.000979\\
1.5&1685418&-282.6247889824&-283.022182 0.000827\\
2.0&1020333&-282.5500457103&-283.017502 0.000869\\
3.0&1034451&-282.5325749807&-283.013154 0.000668\\
CIPSI (unconverged)& \\
\hline
\end{tabular}
\end{table*}
\section{\label{sec:summary}Summary and Conclusions}



\begin{figure*}
 \includegraphics[width=7 in]{All-cyclo.pdf}
 \caption{All results for cyclo-reversion barrier with cc-pvNz and aug-ccpvNz where N=D,T,Q. ALL DATA AND BETTER CURVES TOMORROW }  
 \label{fig:Cycloreversion-ALL}
\end{figure*}



\begin{figure*}
 \includegraphics[width=7 in]{Raw-data.pdf}
 \caption{All results for cyclo-reversion barrier with cc-pvNz and aug-ccpvNz where N=D,T,Q.ALL DATA AND BETTER CURVES TOMORROW }  
 \label{fig:Cycloreversion-ALL}
\end{figure*}

\subsubsection{Using ECP \label{subsubsubsec:AE}}

\begin{table*}[t]
\centering
\caption{Total energy (Ha) of     }
\label{tab:RSDFT-H2O}
\begin{tabular}{cccccc}
\hline \hline \\
& & & Reactant& &Transition \\
$\mu$ &  Truncation   & Nb dets&   DMC(RS-CI) &  Nb dets&   DMC(RS-CI)\\ \hline
             
\multirow{6}{*}{0.5}
& $10^{-4}$ &    1680 &  -55.040788 +/- 0.000600 &    2193 & -54.965952 +/- 0.000564  \\
& $10^{-5}$ &  149014 &  -55.046736 +/- 0.000487 &  207757 & -54.974071 +/- 0.000438 \\
& $10^{-6}$ &  580538 &  -55.049582 +/- 0.000713 &  785845 & -54.975797 +/- 0.001028 \\
& $10^{-8}$ &  878696 &  -55.052633 +/- 0.000768 & 1208211 & -54.981078 +/- 0.001185\\
& Extrapolated & - & \\
  \hline
\multirow{6}{*}{1.0}
& $10^{-4}$ &   19096 & -55.037836 +/- 0.000937 &   19836 & -54.962879 +/- 0.000694  \\
& $10^{-5}$ &  421767 & -55.052299 +/- 0.000510 &  622751 & -54.979127 +/- 0.000759 \\
& $10^{-6}$ &  706987 & -55.055190 +/- 0.000866 & 1028113 & -54.982507 +/- 0.001025 \\
& $10^{-8}$ & 1028094 & -55.058087 +/- 0.001991 & 1456385 & -54.981470 +/- 0.001254 \\
& Extrapolated & - & \\
 \hline
 \multirow{6}{*}{1.5}
& $10^{-4}$ &   40490 & -55.037286 +/- 0.000549 &   37140 & -54.959592 +/- 0.000550  \\
& $10^{-5}$ &  684372 & -55.056664 +/- 0.000631 &  757980 & -54.979372 +/- 0.000675 \\
& $10^{-6}$ &  983169 & -55.057857 +/- 0.001030 & 1130973 & -54.981564 +/- 0.001080\\
& $10^{-8}$ & 1396355 & -55.058601 +/- 0.001201 &-1570198 & -54.980353 +/- 0.000821\\
& Extrapolated & - & \\
 \hline
 \multirow{6}{*}{1.8830139217}
& $10^{-4}$ &   51499 & -55.037119 +/- 0.000488 &   42355 & -54.959355 +/- 0.000739  \\
& $10^{-5}$ &  711366 & -55.056517 +/- 0.000673 &  688729 & -54.976469 +/- 0.000704\\
& $10^{-6}$ &  972113 & -55.054687 +/- 0.000715 &  974856 & -54.978992 +/- 0.000643 \\
& $10^{-8}$ & 1366649 & -55.054010 +/- 0.001094 & 1345970 & -54.978489 +/- 0.001518 \\
& Extrapolated & - & \\
 \hline
 \multirow{6}{*}{2.0}
& $10^{-4}$ &   52308 & -55.037627 +/- 0.000559 &   42913 & -54.958773 +/- 0.000654\\
& $10^{-5}$ &  727269 & -55.055610 +/- 0.000516 &  722231 & -54.978088 +/- 0.000736  \\
& $10^{-6}$ &  993584 & -55.053594 +/- 0.000896 & 1032111 & -54.978177 +/- 0.000855\\
& $10^{-8}$ & 1388907 & -55.053472 +/- 0.001130 & 1406982 & -54.979097 +/- 0.001066 \\
& Extrapolated & - & \\
 \hline
 \multirow{6}{*}{2.5}
& $10^{-4}$ &   62531 & -55.037055 +/- 0.000666 &   48428 & -54.955501 +/- 0.000494 \\
& $10^{-5}$ &  891316 & -55.055441 +/- 0.000816 &  696588 & -54.973473 +/- 0.000658 \\
& $10^{-6}$ & 1269257 & -55.056701 +/- 0.001269 &  971339 & -54.974249 +/- 0.001212 \\
& $10^{-8}$ & 1693057 & -55.053642 +/- 0.001261 & 1311271 & -54.975223 +/- 0.001043 \\
& Extrapolated & - & \\
 \hline
 \multirow{6}{*}{3.0}
& $10^{-4}$ &   64785 & -55.035541 +/- 0.000620 &   55062 & -54.956031 +/- 0.000838 \\
& $10^{-5}$ &  826390 & -55.053692 +/- 0.000548 &  715827 & -54.974286 +/- 0.000671 \\
& $10^{-6}$ & 1152612 & -55.053633 +/- 0.000814 &  994538 & -54.972605 +/- 0.001253 \\
& $10^{-8}$ & 1529509 & -55.049750 +/- 0.000949 & 1323590 & -54.972679 +/- 0.001963 \\
& Extrapolated & - & \\
 \hline
\end{tabular}
\begin{flushleft}

\end{flushleft}
\end{table*}




\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. 
% MDF citation - H. Shin 07/04
Input and output files for calculations are available on the Materials Data Facility, {\tt https://materialsdatafacility.org/}, DOI:To come.
\end{acknowledgments}
\bibliography{Bib,biblio}
%\bibliography{Bib}

\end{document}
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