saving work in spin part
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@ 761,180 +761,7 @@ 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 description of atoms and


molecule, leading to more accurate atomization energies.




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


\subsection{Size consistency}


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




An extremely important feature required to get accurate


atomization energies is sizeconsistency (or strict separability),


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. \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.}




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. \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


sizeconsistency error. But when the truncated SCI wave function is used


as a reference for postHF methods such as SCI+PT2


or for QMC calculations, there is a residual sizeconsistency error


originating from the truncation of the wave function.




QMC energies can be made sizeconsistent by extrapolating the


FNDMC energy to estimate the energy obtained with the FCI as a trial


wave function.\cite{Scemama_2018,Scemama_2018b} Alternatively, the


sizeconsistency error can be reduced by choosing the number of


selected determinants such that the sum of the PT2 corrections on the


fragments is equal to the PT2 correction of the molecule, enforcing that


the variational potential energy surface (PES) is


parallel to the perturbatively corrected PES, which is a relatively


accurate estimate of the FCI PES.\cite{Giner_2015}




Another source of sizeconsistency error in QMC calculations originates


from the Jastrow factor. Usually, the Jastrow factor contains


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 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


diatomic molecule \ce{AB} with a parameter $b_{\ce{AB}}$


will lead to two different twobody Jastrow factors, each


with its own optimal value $b_{\ce{A}}$ and $b_{\ce{B}}$. To remove the


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 of common


practice to localize the nonlocal part of the pseudopotential on the


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


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


nonsizeconsistent Jastrow factor, or a nonoptimal Jastrow factor will


not introduce an additional error in FNDMC calculations, although it


will reduce the statistical errors by reducing the variance of the


local energy. Moreover, the integrals involved in the pseudopotential


are computed analytically and the computational cost of the


pseudopotential is dramatically reduced (for more detail, see


Ref.~\onlinecite{Scemama_2015}).




%%% TABLE III %%%


\begin{table}


\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}


$\mu$ & \ce{F} & Dissociated \ce{F2} & Sizeconsistency error \\


\hline


0.00 & $24.188\,7(3)$ & $48.377\,7(3)$ & $0.000\,3(4)$ \\


0.25 & $24.188\,7(3)$ & $48.377\,2(4)$ & $+0.000\,2(5)$ \\


0.50 & $24.188\,8(1)$ & $48.376\,9(4)$ & $+0.000\,7(4)$ \\


1.00 & $24.189\,7(1)$ & $48.380\,2(4)$ & $0.000\,8(4)$ \\


2.00 & $24.194\,1(3)$ & $48.388\,4(4)$ & $0.000\,2(5)$ \\


5.00 & $24.194\,7(4)$ & $48.388\,5(7)$ & $+0.000\,9(8)$ \\


$\infty$ & $24.193\,5(2)$ & $48.386\,9(4)$ & $+0.000\,1(5)$ \\


\end{tabular}


\end{ruledtabular}


\end{table}


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




In this section, we make a numerical verification that the produced


wave functions are sizeconsistent for a given rangeseparation


parameter.


We have computed the FNDMC 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: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


twice the statistical error bars).






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


\subsection{Spin invariance}


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




Closedshell molecules often dissociate into openshell


fragments. To get reliable atomization energies, it is important to


have a theory which is of comparable quality for openshell and


closedshell systems. A good test is to check that all the components


of a spin multiplet are degenerate, 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 samespin electron pairs are different from those


for the oppositespin pairs.\cite{Tenno_2004}


Again, when pseudopotentials are used this tiny error is transferred


in the FNDMC energy unless the DLA is used.




Within DFT, the common density functionals make a difference for


samespin and oppositespin interactions. As DFT is a


singledeterminant theory, the density functionals are designed to be


used with the highest value of $m_s$, and therefore different values


of $m_s$ lead to different energies.


So in the context of RSDFT, the determinantal expansions will be


impacted by this spurious effect, as opposed to FCI.




%%% TABLE IV %%%


\begin{table}


\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}


$\mu$ & $m_s=1$ & $m_s=0$ & Spininvariance error \\


\hline


0.00 & $5.416\,8(1)$ & $5.414\,9(1)$ & $+0.001\,9(2)$ \\


0.25 & $5.417\,2(1)$ & $5.416\,5(1)$ & $+0.000\,7(1)$ \\


0.50 & $5.422\,3(1)$ & $5.421\,4(1)$ & $+0.000\,9(2)$ \\


1.00 & $5.429\,7(1)$ & $5.429\,2(1)$ & $+0.000\,5(2)$ \\


2.00 & $5.432\,1(1)$ & $5.431\,4(1)$ & $+0.000\,7(2)$ \\


5.00 & $5.431\,7(1)$ & $5.431\,4(1)$ & $+0.000\,3(2)$ \\


$\infty$ & $5.431\,6(1)$ & $5.431\,3(1)$ & $+0.000\,3(2)$ \\


\end{tabular}


\end{ruledtabular}


\end{table}


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




In this section, we investigate the impact of the spin contamination


due to the shortrange density functional on the FNDMC energy. We have


computed the energies of the carbon atom in its triplet state


with BFD pseudopotentials and the corresponding double$\zeta$ basis


set. The calculation was done with $m_s=1$ (3 spinup electrons


and 1 spindown electrons) and with $m_s=0$ (2 spinup and 2


spindown electrons).




The results are presented in Table~\ref{tab:spin}.


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 fixednode 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 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$^{1}$.




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


\subsection{Benchmark}


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


The sizeconsistency and the spininvariance of the present scheme are discussed in Appendices \ref{app:size} and \ref{app:spin}.




%%% FIG 6 %%%


\begin{squeezetable}


@ 1161,10 +988,186 @@ Challenge 2019gch0418) and from CALMIP (Toulouse) under allocation




The data that support the findings of this study are available within the article and its {\SI}, and are openly available in [repository name] at \url{http://doi.org/[doi]}, reference number [reference number].




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


\appendix


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




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


\section{Size consistency}


\label{app:size}


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




%%% TABLE III %%%


\begin{table}


\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}


$\mu$ & \ce{F} & Dissociated \ce{F2} & Sizeconsistency error \\


\hline


0.00 & $24.188\,7(3)$ & $48.377\,7(3)$ & $0.000\,3(4)$ \\


0.25 & $24.188\,7(3)$ & $48.377\,2(4)$ & $+0.000\,2(5)$ \\


0.50 & $24.188\,8(1)$ & $48.376\,9(4)$ & $+0.000\,7(4)$ \\


1.00 & $24.189\,7(1)$ & $48.380\,2(4)$ & $0.000\,8(4)$ \\


2.00 & $24.194\,1(3)$ & $48.388\,4(4)$ & $0.000\,2(5)$ \\


5.00 & $24.194\,7(4)$ & $48.388\,5(7)$ & $+0.000\,9(8)$ \\


$\infty$ & $24.193\,5(2)$ & $48.386\,9(4)$ & $+0.000\,1(5)$ \\


\end{tabular}


\end{ruledtabular}


\end{table}


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




An extremely important feature required to get accurate


atomization energies is sizeconsistency (or strict separability),


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. \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.}




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. \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


sizeconsistency error. But when the truncated SCI wave function is used


as a reference for postHF methods such as SCI+PT2


or for QMC calculations, there is a residual sizeconsistency error


originating from the truncation of the wave function.




QMC energies can be made sizeconsistent by extrapolating the


FNDMC energy to estimate the energy obtained with the FCI as a trial


wave function.\cite{Scemama_2018,Scemama_2018b} Alternatively, the


sizeconsistency error can be reduced by choosing the number of


selected determinants such that the sum of the PT2 corrections on the


fragments is equal to the PT2 correction of the molecule, enforcing that


the variational potential energy surface (PES) is


parallel to the perturbatively corrected PES, which is a relatively


accurate estimate of the FCI PES.\cite{Giner_2015}




Another source of sizeconsistency error in QMC calculations originates


from the Jastrow factor. Usually, the Jastrow factor contains


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 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


diatomic molecule \ce{AB} with a parameter $b_{\ce{AB}}$


will lead to two different twobody Jastrow factors, each


with its own optimal value $b_{\ce{A}}$ and $b_{\ce{B}}$. To remove the


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 of common


practice to localize the nonlocal part of the pseudopotential on the


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


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


nonsizeconsistent Jastrow factor, or a nonoptimal Jastrow factor will


not introduce an additional error in FNDMC calculations, although it


will reduce the statistical errors by reducing the variance of the


local energy. Moreover, the integrals involved in the pseudopotential


are computed analytically and the computational cost of the


pseudopotential is dramatically reduced (for more detail, see


Ref.~\onlinecite{Scemama_2015}).




In this section, we make a numerical verification that the produced


wave functions are sizeconsistent for a given rangeseparation


parameter.


We have computed the FNDMC 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: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


twice the statistical error bars).




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


\section{Spin invariance}


\label{app:spin}


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




%%% TABLE IV %%%


\begin{table}


\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}


$\mu$ & $m_s=1$ & $m_s=0$ & Spininvariance error \\


\hline


0.00 & $5.416\,8(1)$ & $5.414\,9(1)$ & $+0.001\,9(2)$ \\


0.25 & $5.417\,2(1)$ & $5.416\,5(1)$ & $+0.000\,7(1)$ \\


0.50 & $5.422\,3(1)$ & $5.421\,4(1)$ & $+0.000\,9(2)$ \\


1.00 & $5.429\,7(1)$ & $5.429\,2(1)$ & $+0.000\,5(2)$ \\


2.00 & $5.432\,1(1)$ & $5.431\,4(1)$ & $+0.000\,7(2)$ \\


5.00 & $5.431\,7(1)$ & $5.431\,4(1)$ & $+0.000\,3(2)$ \\


$\infty$ & $5.431\,6(1)$ & $5.431\,3(1)$ & $+0.000\,3(2)$ \\


\end{tabular}


\end{ruledtabular}


\end{table}


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




Closedshell molecules often dissociate into openshell


fragments. To get reliable atomization energies, it is important to


have a theory which is of comparable quality for open and


closedshell systems. A good test is to check that all the components


of a spin multiplet are degenerate, as expected from exact solutions.




FCI wave functions have this property and yield degenerate energies with


respect to the spin quantum number $m_s$.


However, multiplying the determinantal part of the trial wave function by a


Jastrow factor introduces spin contamination if the Jastrow parameters


for the samespin electron pairs are different from those


for the oppositespin pairs.\cite{Tenno_2004}


Again, when pseudopotentials are employed, this tiny error is transferred


to the FNDMC energy unless the DLA is enforced.




The context is rather different within DFT.


Indeed, mainstream density functionals have distinct functional forms to take


into account correlation effects of samespin and oppositespin electron pairs.


Therefore, KS determinants corresponding to different values of $m_s$ lead to different total energies.


Consequently, in the context of RSDFT, the determinant expansions is impacted by this spurious effect, as opposed to FCI.




In this Appendix, we investigate the impact of the spin contamination on the FNDMC energy


originating from the shortrange density functional. We have


computed the energies of the carbon atom in its triplet state


with the VDZBFD basis set and the srPBE functional.


The calculations are performed for $m_s=1$ (3 spinup


and 1 spindown electrons) and for $m_s=0$ (2 spinup and 2


spindown electrons).




The results are reported in Table~\ref{tab:spin}.


Although the energy obtained with $m_s=0$ is higher than the one obtained with $m_s=1$, the


bias is relatively small, \ie, more than one order of magnitude smaller


than the energy gained by reducing the fixednode error going from the single


determinant to the FCI trial wave function. The largest bias, close to


$2$ m\hartree, is obtained for $\mu=0$, but this bias decreases quickly


below $1$ m\hartree{} when $\mu$ increases. As expected, with $\mu=\infty$


we observe a perfect spininvariance of the energy (within the error bars), and the bias is not


noticeable for $\mu=5$~bohr$^{1}$.




\titou{T2: what do you conclude from this section? What value of $m_s$ do you use to compute the atoms?}




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


\bibliography{rsdftcipsiqmc}


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




\end{document}







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