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reply to reviewer 2

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Emmanuel Giner 2020-04-06 10:09:54 +02:00
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@ -82,7 +82,10 @@ However I still wonder if this is ok. For the spin I agree with the authors, but
\subsection*{Reply to reviewer 2} \subsection*{Reply to reviewer 2}
\begin{itemize} \begin{itemize}
\item[1] \item[1] As pointed out by reviewer 2, the atomization energies computed with the DFT-based basis set correction does not converge in a monotonic way in some cases, as for instance the error is larger for N$_2$ in aug-cc-pVQZ (-1.5 mH) than for the aug-cc-pVTZ basis set (-0.3 mH). The reviewer suggested that this could come from a non monotonic behaviour of the on-top pair density itself with respect to the basis set. The authors have investigated in that direction and reported several levels of approximations for the on-top pair density for N$_2$ and the Nitrogen atom in increasing basis sets in order to quantify that. It was found that average on-top pair density at near FCI level for a given system (\textit{i.e.} its integral in real-space) decays in a monotonic way as one enlarges the basis set, which is the signature that the depth of the coulomb hole is enlarged when one increases the basis set in a monotonic way.
Nevertheless, the on-top pair density used in the SU-PBE-OT functional is not the near FCI one but the on-top pair density at the CASSCF level extrapolated towards the exact on-top thanks to the known large $\mu$ behaviour and the $\mu_{\mathcal{B}}({\bf r })$ computed for a given point in real space in a given basis set. It was found by the authors that, unlike the near-FCI one, such extrapolated CASSCF on-top increases with the basis set as the CASSCF on-top is almost basis set independent and the global $\mu_{\mathcal{B}}({\bf r})$ increases with the basis set. On the other hand, the same extrapolation scheme based on the near FCI on-top is almost constant and can therefore be considered as a reliable estimate of the exact (\textit{i.e.} near CBS) on-top.
Comparing with estimated exact values, it was observed that extrapolated CASSCF on-top overestimates the on-top pair density, and that this overestimation is more pronounced in the molecular system than in the atomic limit. Acknowledging that the correlation energy is a growing function of the integral of the on-top pair density, this can explain the overestimation of the atomization energy.
To further confirm this statement, we perform a SU-PBE-OT calculation based on the largest variational CIPSI wave functions and using the extrapolated on-top pair density at such level. It was found that the atomization energies were clearly improved, which confirms that the unbalanced overestimation of the on-top pair density between the molecule and the atomic limit was the origin of the overestimation of the atomization energies.
\item[2] Our DFT-based basis set correction does not generally preserve spatial degeneracy for arbitrary states or ensembles. So, in that regard, it has the same problem than standard DFT. Specifically, for the example of C$_2$ and B$_2$, it will not give the same result for the spherically-averaged C or B atom as for the atoms with the occupied p orbital oritented along the bond axis. However, what we write in the Appendix is that for the systems treated in this work the CASSCF wave function dissociates into fragments in simple identified pure states and we can thus choose to perform the calculation on the isolated atom with the same pure state in order to preserve size-consistency. Of course, this may not be always possible for other more complicated systems. We have clarified this point in the Appendix. The new paragraph is now:\\ \item[2] Our DFT-based basis set correction does not generally preserve spatial degeneracy for arbitrary states or ensembles. So, in that regard, it has the same problem than standard DFT. Specifically, for the example of C$_2$ and B$_2$, it will not give the same result for the spherically-averaged C or B atom as for the atoms with the occupied p orbital oritented along the bond axis. However, what we write in the Appendix is that for the systems treated in this work the CASSCF wave function dissociates into fragments in simple identified pure states and we can thus choose to perform the calculation on the isolated atom with the same pure state in order to preserve size-consistency. Of course, this may not be always possible for other more complicated systems. We have clarified this point in the Appendix. The new paragraph is now:\\