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@ 249,7 +249,7 @@ still an active field of research. The present paper falls


within this context.




The central idea of the present work, and the launchpad for the remainder of this study, is that one can combine the various strengths of WFT, DFT, and QMC in order to create a new hybrid method with more attractive features and higher accuracy.


In particular, we show here that one can combine CIPSI and KSDFT via the range separation (RS) of the interelectronic Coulomb operator \cite{SavINC96a,Toulouse_2004} to obtain accurate FNDMC energies with compact multideterminant trial wave functions.


In particular, we show here that one can combine CIPSI and KSDFT via the range separation (RS) of the interelectronic Coulomb operator \cite{SavINC96a,Toulouse_2004}  a scheme that we label RSDFTCIPSI in the following  to obtain accurate FNDMC energies with compact multideterminant trial wave functions.






The present manuscript is organized as follows.


@ 1029,21 +1029,22 @@ going to the quadruple$\zeta$ basis.


This large 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~$\pm$ 1.08~kcal/mol) is


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


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).


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


incompleteness error is transferred in the fixednode error.




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


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


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 the line of the table labelled by the \emph{Opt}


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


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


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.}


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


MAE, the MSE an the RMSD 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.


@ 1051,8 +1052,8 @@ 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


determinants.


At the RSDFTCIPSI level, we can remark that with the triple$\zeta$


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


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


@ 1077,13 +1078,13 @@ $\mu=0.5$~bohr$^{1}$ with the quadruple$\zeta$ basis set.


\end{figure*}


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




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


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


are illustrated in Fig.~\ref{fig:g2dmc}. As seen on the figure and


in Table~\ref{tab:mad}, the best choice for a fixed value of $\mu$ is


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 $\mu$.


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),


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


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


$3.02(06)$, 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


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


@ 1106,28 +1107,28 @@ 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


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


with the quadruple$\zeta$ basis set, making these calculations feasilble


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.


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


determinants at the double$\zeta$ level, and close to 1~000 determinants


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


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


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 keeping the sizeconsistency.


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 first CIPSI calculation, and


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 natural orbitals. It is possible to add


an extra step to the algorithm to compute the natural orbitals from the


RSDFT/CIPSI wave function, and redo the RSDFT/CIPSI calculation with


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 KohnSham orbitals with $\mu=0$, and the


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


solution would have been the PBE single determinant.




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