saving work in Sec V
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@article{Kutzelnigg_1985,


Author = {W. Kutzelnigg},


DateAdded = {20200818 15:05:16 +0200},


DateModified = {20200818 15:05:51 +0200},


Journal = {Theor. Chim. Acta},


Pages = {445},


Title = {r12Dependent terms in the wave function as closed sums of partial wave amplitudes for large l},


Volume = {68},


Year = {1985}}




@article{Kutzelnigg_1991,


Author = {W. Kutzelnigg and W. Klopper},


DateAdded = {20200818 15:05:16 +0200},


DateModified = {20200818 15:05:24 +0200},


Journal = {J. Chem. Phys.},


Pages = {1985},


Title = {Wave functions with terms linear in the interelectronic coordinates to take care of the correlation cusp. I. General theory},


Volume = {94},


Year = {1991}}




@article{Pack_1966,


Author = {{R. T. Pack and W. ByersBrown}},


DateAdded = {20200818 09:51:18 +0200},


@ 943,8 +963,9 @@


Year = {2015},


BdskUrl1 = {https://doi.org/10.1063/1.4907920}}




@article{GinPraFerAssSavTouJCP18,


@article{Giner_2018,


Author = {Emmanuel Giner and Barth\'elemy Pradines and Anthony Fert\'e and Roland Assaraf and Andreas Savin and Julien Toulouse},


DateModified = {20200818 15:08:15 +0200},


Journal = {J. Chem. Phys.},


Pages = {194301},


Title = {Curing basisset convergence of wavefunction theory using densityfunctional theory: A systematically improvable approach},



@ 120,7 +120,7 @@ Another route to solve the Schr\"odinger equation is densityfunctional theory (


Presentday DFT calculations are almost exclusively done within the socalled KohnSham (KS) formalism, \cite{Kohn_1965} which


transfers the complexity of the manybody problem to the universal and yet unknown exchangecorrelation (xc) functional thanks to a judicious mapping between a noninteracting reference system and its interacting analog which both have exactly the same oneelectron density.


KSDFT \cite{Hohenberg_1964,Kohn_1965} is now the workhorse of electronic structure calculations for atoms, molecules and solids thanks to its very favorable accuracy/cost ratio. \cite{ParrBook}


As compared to WFT, DFT has the indisputable advantage of converging much faster with respect to the size of the basis set. \cite{FraMusLupTouJCP15,Loos_2019d,Giner_2020}


As compared to WFT, DFT has the indisputable advantage of converging much faster with respect to the size of the basis set. \cite{FraMusLupTouJCP15,Giner_2018,Loos_2019d,Giner_2020}


However, unlike WFT where manybody perturbation theory provides a precious tool to go toward the exact wave function, there is no systematic way to improve approximate xc functionals toward the exact functional.


Therefore, one faces, in practice, the unsettling choice of the \emph{approximate} xc functional. \cite{Becke_2014}


Moreover, because of the approximate nature of the xc functional, although the resolution of the KS equations is variational, the resulting KS energy does not have such property.


@ 397,7 +397,7 @@ single or multideterminant wave function $\Psi^{(0)}$ which can


be obtained in many different ways depending on the system that one considers.


One of the particularity of the present work is that we


use the CIPSI algorithm to perform approximate FCI calculations


with the RSDFT Hamiltonian $\hat{H}^\mu$. \cite{GinPraFerAssSavTouJCP18}


with the RSDFT Hamiltonian $\hat{H}^\mu$. \cite{Giner_2018}


This provides a multideterminant trial wave function $\Psi^{\mu}$ that one can ``feed'' to DMC.


In the outer (macroiteration) loop (red), at the $k$th iteration, a CIPSI selection is performed


to obtain $\Psi^{\mu\,(k)}$ with the RSDFT Hamiltonian $\hat{H}^{\mu\,(k)}$


@ 656,14 +656,14 @@ This is yet another key result of the present study.


\begin{figure*}


\includegraphics[width=\columnwidth]{densitymu.pdf}


\includegraphics[width=\columnwidth]{ontopmu.pdf}


\caption{\toto{Oneelectron density $n(\br)$ (left) and ontop pair


\caption{Oneelectron density $n(\br)$ (left) and ontop pair


density $n_2(\br,\br)$ (right) along the \ce{OH} axis of \ce{H2O}


as a function of $\mu$ for $\Psi^\mu$, and $\Psi^J$ (dashed


curve).


The integrated ontop pair density $\expval{P}$ is


given in the legend.


For all trial wave functions, the CI expansion consists of the 200 most important


determinants of the FCI expansion obtained with the VDZBFD basis (see Sec.~\ref{sec:rsdftj} for more details).}}


determinants of the FCI expansion obtained with the VDZBFD basis (see Sec.~\ref{sec:rsdftj} for more details).}


\label{fig:densities}


\end{figure*}


%%% %%% %%% %%%


@ 671,9 +671,9 @@ This is yet another key result of the present study.


In order to refine the comparison between $\Psi^\mu$ and $\Psi^J$, we


report several quantities related to the one and twobody densities of


$\Psi^J$ and $\Psi^\mu$ with different values of $\mu$. First, we


report in the legend of Fig~\ref{fig:densities} the integrated ontop pair density


report in the legend of the right panel of Fig~\ref{fig:densities} the integrated ontop pair density


\begin{equation}


\expval{ P } = \int d\br \,\,n_2(\br,\br)


\expval{ P } = \int d\br \,\,n_2(\br,\br),


\end{equation}


where $n_2(\br_1,\br_2)$ is the twobody density [normalized to $\Nelec(\Nelec1)$]


obtained for both $\Psi^\mu$ and $\Psi^J$.


@ 691,13 +691,13 @@ are much more important than that of the onebody density, the latter


being essentially unchanged between $\mu=0$ and $\mu=\infty$ while the


former can vary by about 10$\%$ in some regions.


%TODO TOTO


\toto{In the highdensity region of the \ce{OH} bond, the value of the ontop


In the highdensity region of the \ce{OH} bond, the value of the ontop


pair density obtained from $\Psi^J$ is superimposed with


$\Psi^{\mu=0.5}$, and at a large distance the ontop pair density of $\Psi^J$ is


the closest to $\mu=\infty$. The integrated ontop pair density


obtained with $\Psi^J$ is $\expval{P}=1.404$, between the values obtained with


obtained with $\Psi^J$ is $\expval{P}=1.404$, which nestles between the values obtained at


$\mu=0.5$ and $\mu=1$~bohr$^{1}$, consistently with the FNDMC energies


and the overlap curve depicted in Fig.~\ref{fig:overlap}.}


and the overlap curve depicted in Fig.~\ref{fig:overlap}.




These data suggest that the wave functions $\Psi^{0.5 \le \mu \le 1}$ and $\Psi^J$ are close,


and therefore that the operators that produced these wave functions (\ie, $H^\mu[n]$ and $e^{J}He^J$) contain similar physics.


@ 748,18 +748,18 @@ As a conclusion of the first part of this study, we can highlight the following


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%




Atomization energies are challenging for postHF methods


because their calculation requires a perfect balance in the


because their calculation requires a subtle balance in the


description of atoms and molecules. The mainstream oneelectron basis sets employed in molecular


calculations are atomcentered, so they are, by construction, better adapted to


atoms than molecules and atomization energies usually tend to be


underestimated by variational methods.


In the context of FNDMC calculations, the nodal surface is imposed by


the trial wavefunction which is expanded in an atomcentered basis


set, so we expect the fixednode error to be also intimately related to


the trial wavefunction which is expanded in the very same atomcentered basis


set. Thus, we expect the fixednode error to be also intimately related to


the basis set incompleteness error.


Increasing the size of the basis set improves the description of


the density and of the electron correlation, but also reduces the


imbalance in the quality of the description of the atoms and the


imbalance in the description of atoms and


molecule, leading to more accurate atomization energies.




%============================


@ 768,19 +768,21 @@ molecule, leading to more accurate atomization energies.




An extremely important feature required to get accurate


atomization energies is sizeconsistency (or strict separability),


since the numbers of correlated electron pairs in the isolated atoms


are different from those of the molecules.


The energy computed within DFT is sizeconsistent, and


as it is a meanfield method the convergence to the CBS limit


is relatively fast. Hence, DFT methods are very well adapted to


since the numbers of correlated electron pairs in the molecule and its isolated atoms


are different.


The energies computed within DFT are sizeconsistent, and


\titou{because it is a meanfield method the convergence to the CBS limit


is relatively fast}. \cite{FraMusLupTouJCP15}


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.


sets. \cite{Giner_2018,Loos_2019d,Giner_2020}


\titou{But going to the CBS limit will converge to biased atomization


energies because of the use of approximate density functionals.}




On the other hand, FCI is also sizeconsistent, but the convergence of


the FCI energies to the CBS limit is much slower because of the


Likewise, FCI is also sizeconsistent, but the convergence of


the FCI energies towards the CBS limit is much slower because of the


description of shortrange electron correlation using atomcentered


functions. But ultimately the exact energy will be reached.


functions. \cite{Kutzelnigg_1985,Kutzelnigg_1991} But ultimately the exact energy will be reached.




In the context of SCI calculations, when the variational energy is


extrapolated to the FCI energy \cite{Holmes_2017} there is no


@ 805,7 +807,7 @@ oneelectron, twoelectron and onenucleustwoelectron terms.


The problematic part is the twoelectron term, whose simplest form can


be expressed as in Eq.~\eqref{eq:jastee}.


The parameter


$a$ is determined by cusp conditions, and $b$ is obtained by energy


$a$ is determined by the electronelectron cusp condition, \cite{Kato_1957,Pack_1966} and $b$ is obtained by energy


or variance minimization.\cite{Coldwell_1977,Umrigar_2005}


One can easily see that this parameterization of the twobody


interaction is not sizeconsistent: the dissociation of a


@ 816,11 +818,11 @@ sizeconsistency error on a PES using this ans\"atz for $J_\text{ee}$,


one needs to impose that the parameters of $J_\text{ee}$ are fixed:


$b_A = b_B = b_{\ce{AB}}$.




When pseudopotentials are used in a QMC calculation, it is common


When pseudopotentials are used in a QMC calculation, it is of common


practice to localize the nonlocal part of the pseudopotential on the


complete wave function (determinantal component and Jastrow).


complete trial wave function $\Phi$.


If the wave function is not sizeconsistent,


so will be the locality approximation. Within, the DLA,\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 FNDMC energy


only depends on the parameters of the determinantal component. Using a


@ 834,9 +836,9 @@ Ref.~\onlinecite{Scemama_2015}).




%%% TABLE III %%%


\begin{table}


\caption{FNDMC energies (in hartree) using the VDZBFD basis set


and pseudopotential of the fluorine atom and the dissociated fluorine


dimer, and sizeconsistency error. }


\caption{FNDMC energy (in hartree) using the VDZBFD basis set and the srPBE functional


of the fluorine atom and the dissociated \ce{F2} molecule for various $\mu$ values.


The sizeconsistency error is also reported.}


\label{tab:sizecons}


\begin{ruledtabular}


\begin{tabular}{cccc}


@ 863,7 +865,7 @@ of this system is equal to twice the energy of the fluorine atom.


The data in Table~\ref{tab:sizecons} shows that it is indeed the


case, so we can conclude that the proposed scheme provides


sizeconsistent FNDMC energies for all values of $\mu$ (within


$2\times$ statistical error bars).


twice the statistical error bars).






%============================


@ 893,8 +895,8 @@ impacted by this spurious effect, as opposed to FCI.




%%% TABLE IV %%%


\begin{table}


\caption{FNDMC energies (in hartree) of the triplet carbon atom (VDZBFD) with


different values of $m_s$.}


\caption{FNDMC energy (in hartree) for various $\mu$ values of the triplet carbon atom with


different values of $m_s$ computed with the VDZBFD basis set and the srPBE functional.}


\label{tab:spin}


\begin{ruledtabular}


\begin{tabular}{cccc}


@ 937,15 +939,15 @@ noticeable with $\mu=5$~bohr$^{1}$.


%%% FIG 6 %%%


\begin{squeezetable}


\begin{table*}


\caption{Mean absolute errors (MAE), mean signed errors (MSE) and


standard deviations (RMSD) obtained with various methods and


\caption{Mean absolute errors (MAE), mean signed errors (MSE), and


root mean square errors (RMSE) obtained with various methods and


basis sets.}


\label{tab:mad}


\begin{ruledtabular}


\begin{tabular}{ll ddd ddd ddd}


& & \mc{3}{c}{VDZBFD} & \mc{3}{c}{VTZBFD} & \mc{3}{c}{VQZBFD} \\


\cline{35} \cline{68} \cline{911}


Method & $\mu$ & \tabc{MAE} & \tabc{MSE} & \tabc{RMSD} & \tabc{MAE} & \tabc{MSE} & \tabc{RMSD} & \tabc{MAE} & \tabc{MSE} & \tabc{RMSD} \\


Method & $\mu$ & \tabc{MAE} & \tabc{MSE} & \tabc{RMSE} & \tabc{MAE} & \tabc{MSE} & \tabc{RMSE} & \tabc{MAE} & \tabc{MSE} & \tabc{RMSE} \\


\hline


PBE & 0 & 5.02 & 3.70 & 6.04 & 4.57 & 1.00 & 5.32 & 5.31 & 0.79 & 6.27 \\


BLYP & 0 & 9.53 & 9.21 & 7.91 & 5.58 & 4.44 & 5.80 & 5.86 & 4.47 & 6.43 \\


@ 976,20 +978,21 @@ DMC@ & 0 & 4.61(34) & 3.62(34) & 5.30(09) & 3.52(19) & 


\end{squeezetable}


%%% %%% %%% %%%




The 55 molecules of the benchmark for the Gaussian1


theory\cite{Pople_1989,Curtiss_1990} were chosen to test the


The atomization energies of the 55 molecules of the Gaussian1


theory\cite{Pople_1989,Curtiss_1990} were chosen as a benchmark set to test the


performance of the RSDFTCIPSI trial wave functions in the context of


energy differences. Calculations were made in the double, triple


energy differences. \titou{REFERENCES??}


Calculations were made in the double, triple


and quadruple$\zeta$ basis sets with different values of $\mu$, and using


NOs from a preliminary CIPSI calculation \titou{as a starting point}.


For comparison, we have computed the energies of all atoms and


NOs from a preliminary CIPSI calculation as a starting point (see Fig.~\ref{fig:algo}).


For comparison, we have computed the energies of all the atoms and


molecules at the KSDFT level with various semilocal and hybrid density functionals [PBE, BLYP, PBE0, and B3LYP], and at


the CCSD(T) level. Table~\ref{tab:mad} gives the corresponding mean


absolute errors (MAE), mean signed errors (MSE) and \titou{standard


deviations (RMSE)}. For FCI (RSDFTCIPSI, $\mu=\infty$) we have


absolute errors (MAE), mean signed errors (MSE), and root mean square errors (RMSE).


For FCI (RSDFTCIPSI, $\mu=\infty$) we have


provided the extrapolated values at $\EPT \to 0$, and the error bars


correspond to the difference between the energies \titou{computed with a


twopoint and with a threepoint linear extrapolation}. \cite{Loos_2018a,Loos_2019,Loos_2020b,Loos_2020c}


correspond to the difference between the extrapolated energies computed with a


twopoint and a threepoint linear extrapolation. \cite{Loos_2018a,Loos_2019,Loos_2020b,Loos_2020c}




In this benchmark, the great majority of the systems are weakly correlated and are then well


described by a single determinant. Therefore, the atomization energies


@ 998,7 +1001,7 @@ the basis set is small. The introduction of exact exchange (B3LYP and


PBE0) make the results more sensitive to the basis set, and reduce the


accuracy. Thanks to the singlereference character of these systems,


the CCSD(T) energy is an excellent estimate of the FCI energy, as


shown by the very good agreement of the MAE, MSE and RMSD of CCSDT(T)


shown by the very good agreement of the MAE, MSE and RMSE of CCSDT(T)


and FCI energies.


The imbalance of the quality of description of molecules compared


to atoms is exhibited by a very negative value of the MSE for


@ 1022,9 +1025,9 @@ trial wave function was estimated for each system by searching for the


minimum of the spline interpolation curve of the FNDMC energy as a


function of $\mu$.


This corresponds the line of Table~\ref{tab:mad} labelled as ``Opt.''


\titou{The optimal $\mu$ value for each system is reported in the \SI.}


The optimal $\mu$ value for each system is reported in the \SI.


Using the optimal value of $\mu$ clearly improves the


MAE, the MSE an the RMSD compared to the FCI wave function. This


MAE, the MSE an the RMSE compared to the FCI wave function. This


result is in line with the common knowledge that reoptimizing


the determinantal component of the trial wave function in the presence


of electron correlation reduces the errors due to the basis set incompleteness.


@ 1059,11 +1062,11 @@ $\mu=0.5$~bohr$^{1}$ with the quadruple$\zeta$ basis set.


%%% %%% %%% %%%




Searching for the optimal value of $\mu$ may be too costly and time consuming, so we have


computed the MAD, MSE and RMSD for fixed values of $\mu$.


computed the MAD, MSE and RMSE for fixed values of $\mu$.


As illustrated in Fig.~\ref{fig:g2dmc} and Table \ref{tab:mad},


the best choice for a fixed value of $\mu$ is


0.5~bohr$^{1}$ for all three basis sets. It is the value for which


the MAE [$3.74(35)$, $2.46(18)$, and $2.06(35)$ kcal/mol] and RMSD [$4.03(23)$,


the MAE [$3.74(35)$, $2.46(18)$, and $2.06(35)$ kcal/mol] and RMSE [$4.03(23)$,


$3.02(06)$, and $2.74(13)$ kcal/mol] are minimal. Note that these values


are even lower than those obtained with the optimal value of


$\mu$. Although the FNDMC energies are higher, the numbers show that



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