saving work in Sec V and appendix A
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@article{Hattig_2012,


Author = {C. Hattig and W. Klopper and A. Kohn and D. P. Tew},


DateAdded = {20200819 09:28:48 +0200},


DateModified = {20200819 09:28:59 +0200},


Journal = {Chem. Rev.},


Pages = {4},


Title = {Explicitly Correlated Electrons in Molecules},


Volume = {112},


Year = {2012}}




@article{Kutzelnigg_1985,


Author = {W. Kutzelnigg},


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



@ 458,6 +458,8 @@ The FNDMC simulations are performed with allelectron moves using the


stochastic reconfiguration algorithm developed by Assaraf \textit{et al.},


\cite{Assaraf_2000} with a time step of $5 \times 10^{4}$ a.u.




All the data related to the present study (geometries, basis sets, total energies, etc) can be found in the {\SI}.




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


\section{Influence of the rangeseparation parameter on the fixednode error}


\label{sec:mudmc}


@ 723,7 +725,7 @@ they both deal with an effective nondivergent interaction but still


produce a reasonable onebody density.




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


\subsection{Intermediate conclusion}


\subsection{Intermediate conclusions}


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




As a conclusion of the first part of this study, we can highlight the following observations:


@ 749,25 +751,25 @@ 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 subtle balance in the


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


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


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


atoms than molecules and atomization energies usually tend to be


atoms than molecules. Thus, 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 the very same atomcentered basis


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


the determinantal part of the trial wavefunction which is expanded in the very same atomcentered basis


set. Thus, we expect the fixednode error to be also intimately connected 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 description of atoms and


molecule, leading to more accurate atomization energies.


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


molecules, leading to more accurate atomization energies.


The sizeconsistency and the spininvariance of the present scheme, two key properties to obtain accurate atomization energies, are discussed in Appendices \ref{app:size} and \ref{app:spin}.




%%% FIG 6 %%%


\begin{squeezetable}


\begin{table*}


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


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


\caption{Mean absolute errors (MAEs), mean signed errors (MSEs), and


root mean square errors (RMSEs) \titou{with respect to ??? (in kcal/mol)} obtained with various methods and


basis sets.}


\label{tab:mad}


\begin{ruledtabular}


@ 815,9 +817,9 @@ NOs from a preliminary CIPSI calculation as a starting point (see Fig.~\ref{fig:


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 root mean square errors (RMSE).


absolute errors (MAEs), mean signed errors (MSEs), and root mean square errors (RMSEs).


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


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


provided the extrapolated values (\ie, when $\EPT \to 0$), and the error bars


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}




@ 830,20 +832,20 @@ 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 RMSE of CCSDT(T)


and FCI energies.


The imbalance of the quality of description of molecules compared


The imbalance in the description of molecules compared


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


CCSD(T) and FCI/VDZBFD, which is reduced by a factor of two


CCSD(T)/VDZBFD and FCI/VDZBFD, which is reduced by a factor of two


when going to the triple$\zeta$ basis, and again by a factor of two when


going to the quadruple$\zeta$ basis.




This large imbalance at the VDZBFD level affects the nodal


This significant imbalance at the VDZBFD level affects the nodal


surfaces, because although the FNDMC energies obtained with nearFCI


trial wave functions are much lower than the singledeterminant FNDMC


energies, the MAE obtained with FCI ($7.38\pm1.08$ kcal/mol) is


larger than the singledeterminant MAE ($4.61\pm 0.34$ kcal/mol).


larger than the \titou{singledeterminant} MAE ($4.61\pm 0.34$ kcal/mol).


Using the FCI trial wave function the MSE is equal to the


negative MAE which confirms that all the atomization energies are


underestimated. This confirms that some of the basisset


negative MAE which confirms that the atomization energies are systematically


underestimated. This confirms that some of the basis set


incompleteness error is transferred in the fixednode error.




Within the double$\zeta$ basis set, the calculations could be performed for the


@ 851,45 +853,46 @@ whole range of values of $\mu$, and the optimal value of $\mu$ for the


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.''


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


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 RMSE compared to the FCI wave function. This


MAE, the MSE an the RMSE as 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.


These calculations were done only for the smallest basis set


because of the expensive computational cost of the QMC calculations


when the trial wave function is expanded on more than a few million


when the trial wave function contains more than a few million


determinants.


At the RSDFTCIPSI level, one can see that with the VTZBFD


basis the MAEs are larger for $\mu=1$~bohr$^{1}$ than for the


FCI. For the largest systems, as shown in Fig.~\ref{fig:g2ndet}


there are many systems which did not reach the threshold


$\EPT<1$~m\hartree{}, and the number of determinants exceeded


10~million so the calculation stopped. In this regime, there is a


At the RSDFTCIPSI/VTZBFD level, one can see that


the MAEs are larger for $\mu=1$~bohr$^{1}$ ($9.06$ kcal/mol) than for


FCI [$8.43(39)$ kcal/mol]. For the largest systems, as shown in Fig.~\ref{fig:g2ndet},


there are many systems for which we could not reach the threshold


$\EPT<1$~m\hartree{} as the number of determinants exceeded


10~million before this threshold was reached. For these cases, there is then a


small sizeconsistency error originating from the imbalanced


truncation of the wave functions, which is not present in the


extrapolated FCI energies. The same comment applies to


$\mu=0.5$~bohr$^{1}$ with the quadruple$\zeta$ basis set.




extrapolated FCI energies (see Appendix \ref{app:size}). The same comment applies to


$\mu=0.5$~bohr$^{1}$ with the quadruple$\zeta$ basis.


\titou{T2: Fig. 6 is mentioned before Fig. 5.}




%%% FIG 5 %%%


\begin{figure*}


\centering


\includegraphics[width=\textwidth]{g2dmc.pdf}


\caption{Errors in the FNDMC atomization energies with various


\caption{Errors \titou{(with respect to ???)} in the FNDMC atomization energies (in kcal/mol) with various


trial wave functions. Each dot corresponds to an atomization


energy.


The boxes contain the data between first and third quartiles, and


the line in the box represents the median. The outliers are shown


with a cross.}


with a cross.


\titou{T2: change basis set labels.}}


\label{fig:g2dmc}


\end{figure*}


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




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


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


computed the MAEs, MSEs and RMSEs 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


@ 899,6 +902,7 @@ are even lower than those obtained with the optimal value of


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


they are more consistent from one system to another, giving improved


cancellations of errors.


\titou{This is another key result of the present study and it can be explained by the lack of sizeconsistentcy when one uses different $\mu$ values for different systems like in optimallytune rangeseparated hybrids. \cite{}}




%%% FIG 6 %%%


\begin{figure*}


@ 908,7 +912,8 @@ cancellations of errors.


functions. Each dot corresponds to an atomization energy.


The boxes contain the data between first and third quartiles, and


the line in the box represents the median. The outliers are shown


with a cross.}


with a cross.


\titou{T2: change basis set labels.}}


\label{fig:g2ndet}


\end{figure*}


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


@ 916,12 +921,12 @@ cancellations of errors.


The number of determinants in the trial wave functions are shown in


Fig.~\ref{fig:g2ndet}. As expected, the number of determinants


is smaller when $\mu$ is small and larger when $\mu$ is large.


It is important to remark that the median of the number of


It is important to note that the median of the number of


determinants when $\mu=0.5$~bohr$^{1}$ is below $100\,000$ determinants


with the VQZBFD basis, making these calculations feasible


with such a large basis set. At the double$\zeta$ level, compared to the


FCI trial wave functions the median of the number of determinants is


reduced by more than two orders of magnitude.


FCI trial wave functions, the median of the number of determinants is


reduced by more than \titou{two orders of magnitude}.


Moreover, going to $\mu=0.25$~bohr$^{1}$ gives a median close to 100


determinants at the VDZBFD level, and close to $1\,000$ determinants


at the quadruple$\zeta$ level for only a slight increase of the


@ 997,28 +1002,6 @@ The data that support the findings of this study are available within the articl


\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


@ 1030,13 +1013,13 @@ 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.}


However, in the CBS, KSDFT atomization energies do not match the exact values due to the approximate nature of the xc 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.


functions. \cite{Kutzelnigg_1985,Kutzelnigg_1991,Hattig_2012}


Eventually though, the exact atomization energies 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


@ 1069,8 +1052,8 @@ 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:


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


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


$b_{\ce{A}} = b_{\ce{B}} = b_{\ce{AB}}$.




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


practice to localize the nonlocal part of the pseudopotential on the


@ 1085,46 +1068,47 @@ 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


pseudopotential is dramatically reduced (for more details, 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


the two atoms are separated by 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


The data in Table~\ref{tab:sizecons} shows that this 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).




%%% 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}


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




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


\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


@ 1159,11 +1143,33 @@ Although the energy obtained with $m_s=0$ is higher than the one obtained with $


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


$2$ 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$^{1}$.




%%% 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.


The spininvariance error is also reported.}


\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}


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




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




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



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