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@ 692,3 +692,28 @@ note={Gaussian Inc. Wallingford CT}


publisher = {American Physical Society},


doi = {10.1103/PhysRevA.70.062505}


}




@misc{Scemama_2015,


author = {Scemama, Anthony and Giner, Emmanuel and


Applencourt, Thomas and Caffarel, Michel},


title = {{QMC using very large configuration interactiontype expansions}},


howpublished = {Pacifichem, Advances in Quantum Monte Carlo},


year = {2015},


month = {Dec},


doi = {10.13140/RG.2.1.3187.9766}


}




@article{Tenno_2004,


author = {Tenno, Seiichiro},


title = {{Explicitly correlated second order perturbation theory: Introduction of a


rational generator and numerical quadratures}},


journal = {J. Chem. Phys.},


volume = {121},


number = {1},


pages = {117129},


year = {2004},


month = {Jul},


issn = {00219606},


publisher = {American Institute of Physics},


doi = {10.1063/1.1757439}


}

@ 465,7 +465,7 @@ $\mu=\infty$.


\label{fig:overlap}


\end{figure}




This data confirms that RSDFT/CIPSI can give improved CI coefficients


This data confirms that RSDFTCIPSI can give improved CI coefficients


with small basis sets, similarly to the common practice of


reoptimizing the wave function in the presence of the Jastrow


factor. To confirm that the introduction of RSDFT has the same impact


@ 490,8 +490,7 @@ $1$. Nevertheless, a slight maximum is obtained for


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








\section{Atomization energy benchmarks}


\section{Atomization energies}


\label{sec:atomization}




Atomization energies are challenging for postHartreeFock methods


@ 499,40 +498,40 @@ because their calculation requires a perfect balance in the


description of atoms and molecules. Basis sets used in molecular


calculations are atomcentered, so they are always better adapted to


atoms than molecules and atomization energies usually tend to be


underestimated.


underestimated with variational methods.


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


the trial wavefunction which is expanded on an atomcentered basis


set. So we expect the fixednode error to be also related to the basis


set incompleteness error.


set. So we expect the fixednode error to be also tightly related to


the basis set incompleteness error.


Increasing the size of the basis set improves the description of


the density and of electron correlation, but also reduces the


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


molecule, leading to more accurate atomization energies.




Another important feature required to get accurate atomization


energies is sizeextensivity, since the numbers of correlated electrons


in the isolated atoms are different from the number of correlated


electrons in the molecule.


\subsection{Sizeconsistence}




An extremely important feature required to get accurate


atomization energies is sizeextensivity, since the numbers of


correlated electron pairs in the isolated atoms are different from


those of the molecules.


In the context of selected CI calculations, when the variational energy is


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


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


sizeconsistence error. But when the selected wave function is used


for as a reference for postHartreeFock methods or QMC calculations,


there is a residual sizeconsistence error originating from the


truncation of the determinant space.


as a reference for postHartreeFock methods (sCI+PT2 for instance)


or for QMC calculations, there is a residual sizeconsistence error


originating from the truncation of the determinant space.




% Invariance with m_s




QMC calculations can be made sizeconsistent by extrapolating the


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


sizeconsistence error can be reduced by choosing the number of


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


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


the variational dissociation potential energy surface (PES) is


the variational potential energy surface (PES) is


parallel to the perturbatively corrected PES, which is an accurate


estimate of the FCI PES.\cite{Giner_2015}




Another source of sizeconsistence error in QMC calculation may originate


Another source of sizeconsistence 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


@ 540,11 +539,12 @@ be expressed as


\begin{equation}


J_\text{ee} = \sum_i \sum_{j<i} \frac{a r_{ij}}{1 + b r_{ij}}.


\end{equation}


The parameter


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


or variance minimization.\cite{Coldwell_1977,Umrigar_2005}


One can easily see that this parameterization of the twobody


interation is not sizeconsistent. The dissociation of a


heteroatomic diatomic molecule $AB$ with a parameter $b_{AB}$


interation is not sizeconsistent: the dissociation of a


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


will lead to two different twobody Jastrow factors on each atom, each


with its own optimal value $b_A$ and $b_B$. To remove the


sizeconsistence error on a PES using this ansätz for $J_\text{ee}$,


@ 552,18 +552,21 @@ one needs to impose that the parameters of $J_\text{ee}$ are fixed:


$b_A = b_B = b_{AB}$.




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


practice to localize the pseudopotential on the complete wave


function. If the wave function is not sizeconsistent, so will be the


locality approximation. Recently, the determinant localization


approximation was introduced.\cite{Zen_2019} This approximation


consists in removing the Jastrow factor from the wave function on


which the pseudopotential is localized.


practice to localize the nonlocal part of the pseudopotential on the


complete wave function (determinantal component and Jastrow).


If the wave function is not sizeconsistent,


so will be the locality approximation. Within, the determinant


localization approximation,\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


within this approximation only depends on the parameters of the


determinantal component. Using a sizeinconsistent Jastrow factor, or


a nonoptimal Jastrow factor will not introduce an additional


sizeconsistence error in FNDMC calculations, although it will


reduce the statistical errors by reducing the variance of the local energy.


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


sizeinconsistent Jastrow factor, or a nonoptimal Jastrow factor will


not introduce an additional sizeconsistence 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}).






The energy computed within density functional theory is extensive, and


@ 572,18 +575,40 @@ as it is a meanfield method the convergence to the complete basis set


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.




On the other hand, the convergence of the FCI energies to the CBS


limit will be slower because of the description of shortrange electron


correlation with atomcentered functions, but ultimately the exact


energy will be reached.






\subsection{Spininvariance}




Closedshell molecules usually 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.


FCI wave functions are invariant with respect to the spin quantum


number $m_s$, but the introduction of a


Jastrow factor breaks this spininvariance if the parameters


for the samespin electron pairs are different from those


for the oppositespin pairs.\cite{Tenno_2004}


Again, using pseudopotentials this error is transferred in the DMC


calculation unless the determinant localization approximation is used.




To check that the RSDFTCIPSI are spininvariant, we compute the


FNDMC energies of the ?? dimer with different values of the spin


quantum number $m_s$.








\subsection{Benchmark}




The 55 molecules of the benchmark for the Gaussian1


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


of the RSDFTCIPSI trial wave functions for energy differences.








%\begin{squeezetable}


\begin{table*}


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


@ 665,7 +690,28 @@ We could have obtained


singledeterminant wave functions by using the natural orbitals of a first






\section{Conclusion}




We have seen that introducing shortrange correation via


a rangeseparated Hamiltonian in a full CI expansion yields improved


nodes, especially with small basis sets. The effect is similar to the


effect of reoptimizing the CI coefficients in the presence of a


Jastrow factor, but without the burden of performing a stochastic


optimization.


The proposed procedure provides a method to optimize the


FNDMC energy via a single parameter, namely the rangeseparation


parameter $\mu$. The sizeconsistence error is controlled, as well as the


invariance with respect to the spin projection $m_s$.


Finding the optimal value of $\mu$ gives the lowest FNDMC energies


within basis set. However, if one wants to compute an energy


difference, one should not minimize the


FNDMC energies of the reactants independently. It is preferable to


choose a value of $\mu$ for which the fixednode errors are well


balanced, leading to a good cncellation of errors. We found that a


value of $\mu=0.5$~bohr${^1}$ is the value where the errors are the


smallest. Moreover, such a small value of $\mu$ gives extermely


compact wave functions, making this recipe a good candidate for


accurate calcultions of large systems with a multireference character.








@ 712,9 +758,8 @@ Simulation of Functional Materials.


\item basis set dependent: $\mu \rightarrow \infty$ when $\mathcal{B}\rightarrow \text{CBS}$


\item large wave functions


\end{itemize}


\begin{itemize}


\item plot $N_{det}$ en fonction de $\mu$


\end{itemize}


\item Invariance with m_s




\end{itemize}


\end{enumerate}





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