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@article{Karolewski_2013,


Author = {Andreas Karolewski and Leeor Kronik and Stephan Kummel},


DateAdded = {20200819 14:04:10 +0200},


DateModified = {20200819 14:05:08 +0200},


Doi = {10.1063/1.4807325},


Journal = {J. Chem. Phys.},


Pages = {204115},


Title = {Using optimally tuned range separated hybrid functionals in groundstate calculations: Consequences and caveats},


Volume = {138},


Year = {2013}}




@article{Kronik_2012,


Author = {Leeor Kronik and Tamar Stein and Sivan {RefaelyAbramson} and Roi Baer},


DateAdded = {20200819 14:01:54 +0200},


DateModified = {20200819 14:01:54 +0200},


Doi = {10.1021/ct2009363},


Journal = {J. Chem. Theory Comput.},


Pages = {15151531},


Title = {Excitation Gaps of FiniteSized Systems from Optimally Tuned RangeSeparated Hybrid Functionals},


Volume = {8},


Year = {2012},


BdskUrl1 = {https://doi.org/10.1021/ct2009363}}




@article{Stein_2009,


Author = {Stein, Tamar and Kronik, Leeor and Baer, Roi},


DateAdded = {20200819 14:01:19 +0200},


DateModified = {20200819 14:01:19 +0200},


Doi = {10.1021/ja8087482},


Eprint = {https://doi.org/10.1021/ja8087482},


Journal = {J. Am. Chem. Soc.},


Note = {PMID: 19239266},


Number = {8},


Pages = {28182820},


Title = {Reliable Prediction of Charge Transfer Excitations in Molecular Complexes Using TimeDependent Density Functional Theory},


Url = {https://doi.org/10.1021/ja8087482},


Volume = {131},


Year = {2009},


BdskUrl1 = {https://doi.org/10.1021/ja8087482}}




@article{Hattig_2012,


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


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


@ 1679,15 +1718,14 @@


Year = {2019},


BdskUrl1 = {https://doi.org/10.1063/1.5116024}}








@misc{nist,


doi = {10.18434/T47C7Z},


url = {http://cccbdb.nist.gov/},


Note = {\url{http://cccbdb.nist.gov/}},


author = {Johnson, RD},


title = {Computational Chemistry Comparison and Benchmark Database, NIST Standard Reference Database 101},


publisher = {National Institute of Standards and Technology},


year = {2002},


copyright = {License Information for NIST data}


}


Author = {Johnson, RD},


Copyright = {License Information for NIST data},


Doi = {10.18434/T47C7Z},


Note = {\url{http://cccbdb.nist.gov/}},


Publisher = {National Institute of Standards and Technology},


Title = {Computational Chemistry Comparison and Benchmark Database, NIST Standard Reference Database 101},


Url = {http://cccbdb.nist.gov/},


Year = {2002},


BdskUrl1 = {http://cccbdb.nist.gov/},


BdskUrl2 = {https://doi.org/10.18434/T47C7Z}}



@ 424,17 +424,19 @@ the final trial wave function $\Psi^{\mu}$ is independent of the starting wave f


\label{sec:compdetails}


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




For all the systems considered here, experimental geometries have been used and they have been extracted from the NIST website \titou{[REF]}.


Geometries for each system are reported in the {\SI}.


All reference data (geometries, atomization


energies, zeropoint energy corrections, etc) were taken from the NIST


computational chemistry comparison and benchmark database


(CCCBDB).\cite{nist}




All the calculations have been performed using BurkatzkiFilippiDolg (BFD)


All calculations have been performed using BurkatzkiFilippiDolg (BFD)


pseudopotentials \cite{Burkatzki_2007,Burkatzki_2008} with the associated double,


triple, and quadruple$\zeta$ basis sets (VXZBFD).


The smallcore BFD pseudopotentials include scalar relativistic effects.


Coupled cluster with singles, doubles, and perturbative triples [CCSD(T)] and KSDFT energies have been computed with


\emph{Gaussian09},\cite{g16} using the unrestricted formalism for openshell systems.




All the CIPSI calculations have been performed with \emph{Quantum


The CIPSI calculations have been performed with \emph{Quantum


Package}.\cite{Garniron_2019,qp2_2020} We consider the shortrange version


of the localdensity approximation (LDA)\cite{SavINC96a,TouSavFlaIJQC04} and PerdewBurkeErnzerhof (PBE) \cite{PerBurErnPRL96}


xc functionals defined in


@ 728,7 +730,7 @@ produce a reasonable onebody density.


\subsection{Intermediate conclusions}


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




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


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


\begin{itemize}


\item With respect to the nodes of a KS determinant or a FCI wave function,


one can obtain a multideterminant trial wave function $\Psi^\mu$ with a smaller


@ 763,14 +765,17 @@ 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


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}.


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}, respectively.




%%% FIG 6 %%%


\begin{squeezetable}


\begin{table*}


\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.}


root mean square errors (RMSEs) with respect to the NIST reference values obtained with various methods and


basis sets.


All quantities are given in kcal/mol.}


\label{tab:mad}


\begin{ruledtabular}


\begin{tabular}{ll ddd ddd ddd}


@ 810,47 +815,55 @@ DMC@ & 0 & 4.61(34) & 3.62(34) & 5.30(09) & 3.52(19) & 


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. \toto{All reference data (geometries, atomization


energies, zeropoint energy corrections) were taken from the NIST


computational chemistry comparison and benchmark database


(CCCBDB).}\cite{nist}


energy differences.


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 as a starting point (see Fig.~\ref{fig:algo}).


\footnote{At $\mu=0$, the number of determinants is not equal to one because


we have used the natural orbitals of a preliminary CIPSI calculation, and


not the srPBE orbitals.


So the KohnSham determinant is expressed as a linear combination of


determinants built with NOs. It is possible to add


an extra step to the algorithm to compute the NOs from the


RSDFTCIPSI wave function, and redo the RSDFTCIPSI calculation with


these orbitals to get an even more compact expansion. In that case, we would


have converged to the KS orbitals with $\mu=0$, and the


solution would have been the PBE single determinant.}


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


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


with respect to the NIST reference values as explained in Sec.~\ref{sec:compdetails}.


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


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


correspond to the difference between the extrapolated energies computed with a


provided the extrapolated values (\ie, when $\EPT \to 0$), and, although one cannot provide theoretically sound error bars, they


correspond here 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


calculated at the DFT level are relatively accurate, even when


calculated at the KSDFT level are relatively accurate, even when


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. Note that, due to the approximate nature of the xc functionals,


the MAEs associated with KSDFT atomization energies do not converge towards zero and remain altered even in the CBS limit.


the statistical quantities associated with KSDFT atomization energies do not converge towards zero and remain altered even in the CBS limit.


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.


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


and FCI energies for each basis set.


The imbalance in the description of molecules compared


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


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


CCSD(T)/VDZBFD ($23.96$ kcal/mol) and FCI/VDZBFD ($23.49\pm0.04$ kcal/mol), 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 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 \titou{singledeterminant} MAE ($4.61\pm0.34$ kcal/mol).


trial wave functions are much lower than the FNDMC


energies at $\mu = 0$, the MAE obtained with FCI ($7.38\pm1.08$ kcal/mol) is


larger than the MAE at $\mu = 0$ ($4.61\pm0.34$ kcal/mol).


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


negative MAE which confirms that the atomization energies are systematically


underestimated. This confirms that some of the basis set


underestimated. This corroborates 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


@ 861,40 +874,29 @@ function of $\mu$.


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 as compared to the FCI wave function. This


MAEs, MSEs, and RMSEs 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 contains more than a few million


determinants.


determinants. \cite{Scemama_2016}


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].


%TOTO TODO


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 (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.}


FCI ($8.43\pm0.39$ kcal/mol).


The same comment applies to $\mu=0.5$~bohr$^{1}$ with the quadruple$\zeta$ basis.




%%% FIG 5 %%%


\begin{figure*}


\centering


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


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


\caption{Errors 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.


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


with a cross.}


\label{fig:g2dmc}


\end{figure*}


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


@ 910,7 +912,8 @@ 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{}}


This is another key result of the present study, and it can be explained by the lack of sizeconsistency when one considers different $\mu$ values for the molecule and the isolated atoms.


This observation was also mentioned in the context of optimallytune rangeseparated hybrids. \cite{Stein_2009,Karolewski_2013,Kronik_2012}




%%% FIG 6 %%%


\begin{figure*}


@ 920,8 +923,7 @@ 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.


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


with a cross.}


\label{fig:g2ndet}


\end{figure*}


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


@ 934,7 +936,7 @@ 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 \titou{two orders of magnitude}.


reduced by more than 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


@ 942,17 +944,14 @@ MAE. Hence, RSDFTCIPSI trial wave functions with small values of


$\mu$ could be very useful for large systems to go beyond the


singledeterminant approximation at a very low computational cost


while ensuring sizeconsistency.




Note that when $\mu=0$ the number of determinants is not equal to one because


we have used the natural orbitals of a preliminary CIPSI calculation, and


not the srPBE orbitals.


So the KohnSham determinant is expressed as a linear combination of


determinants built with NOs. It is possible to add


an extra step to the algorithm to compute the NOs from the


RSDFTCIPSI wave function, and redo the RSDFTCIPSI calculation with


these orbitals to get an even more compact expansion. In that case, we would


have converged to the KS orbitals with $\mu=0$, and the


solution would have been the PBE single determinant.


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 (see Appendix \ref{app:size}).




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


\section{Conclusion}


@ 991,7 +990,9 @@ cancellations of errors.


\begin{acknowledgments}


This work was performed using HPC resources from GENCITGCC (Grand


Challenge 2019gch0418) and from CALMIP (Toulouse) under allocation


20190510.


202018005.


Funding from \textit{``Projet International de Coop\'eration Scientifique''} (PICS08310) and from the \textit{``Centre National de la Recherche Scientifique''} is acknowledged.


This study has been (partially) supported through the EUR grant NanoX No.~ANR17EURE0009 in the framework of the \textit{``Programme des Investissements d'Avenir''}.


\end{acknowledgments}


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




@ 1015,9 +1016,9 @@ atomization energies is sizeconsistency (or strict separability),


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


are different.




KSDFT energies are sizeconsistent, and


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


is relatively fast}. \cite{FraMusLupTouJCP15}


KSDFT energies are sizeconsistent, and because xc functionals are


directly constructed in complete basis, their convergence with respect


to the size of the basis set is relatively fast. \cite{FraMusLupTouJCP15,Giner_2018,Loos_2019d,Giner_2020}


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}



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