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updated some minor corrections

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Emmanuel Giner 2020-04-06 11:31:48 +02:00
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Revised_Manuscript/srDFT_SC.tex
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@ -704,7 +704,7 @@ The performance of each of these functionals is tested in the following. Note th
\hline
\ce{N2} & aug-cc-pVDZ & 1.17542 & 0.65966 & 1.02792 & 0.58228 & 0.946 & 0.962 \\
& aug-cc-pVTZ & 1.18324 & 0.77012 & 0.92276 & 0.61074 & 1.328 & 1.364 \\
& aug-cc-pVQZ & 1.18484 & 0.84012 & 0.83866 & 0.59982 & 1.706 & \\[0.1cm]
& aug-cc-pVQZ & 1.18484 & 0.84012 & 0.83866 & 0.59982 & 1.706 & 1.746 \\[0.1cm]
\ce{N} & aug-cc-pVDZ & 0.34464 & 0.19622 & 0.25484 & 0.14686 & 0.910 & 0.922 \\
& aug-cc-pVTZ & 0.34604 & 0.22630 & 0.22344 & 0.14828 & 1.263 & 1.299 \\
@ -873,7 +873,7 @@ On the other hand, the stability of $\ontopextrapcipsi$ with respect to the basi
}
\alert{
For the present work, it is important to keep in mind that $\ontopextrap$ directly determines the basis-set correction in the large-$\mu$ limit. More precisely, the correlation energy contribution associated with the basis-set correction is (in absolute value) an increasing function of $\ontopextrap$. Therefore, the error on $\ontopextrap$ with respect to the estimated exact system-averaged on-top pair density provides an indication of the error made by the basis-set correction for a given system and basis set. With the aug-cc-pVQZ basis set, we have $\ontopextrap - \ontopextrapcipsi = 0.120$ for the \ce{N2} molecule, while $2(\ontopextrap - \ontopextrapcipsi) = 0.095$ for two isolated \ce{N} atoms. We can then conclude that the overestimation of the system-averaged on-top pair density, and therefore of the basis-set correction, is more important for the \ce{N2} molecule at equilibrium distance than for the isolated \ce{N} atoms. This probably explains the observed overestimation of the atomization energy. To confirm this statement, we computed the basis-set correction for both the \ce{N2} molecule at equilibrium distance and the isolated atoms using $\murcipsi$ and $\mathring{n}_{2,\text{CIPSI}}(\br{})$ with the aug-cc-pVTZ and aug-cc-pVQZ basis sets. We obtained the following values for the atomization energies: $362.12$ mH with aug-cc-pVTZ and $362.15$ mH with aug-cc-pVQZ, which are indeed more accurate values than those obtained using $\murcas$ and $\mathring{n}_{2,\text{CASSCF}}(\br{})$.
For the present work, it is important to keep in mind that $\ontopextrap$ directly determines the basis-set correction in the large-$\mu$ limit. More precisely, the correlation energy contribution associated with the basis-set correction is (in absolute value) an increasing function of $\ontopextrap$. Therefore, the error on $\ontopextrap$ with respect to the estimated exact system-averaged on-top pair density provides an indication of the error made by the basis-set correction for a given system and basis set. With the aug-cc-pVQZ basis set, we have $\ontopextrap - \ontopextrapcipsi = 0.240$ for the \ce{N2} molecule, while $2(\ontopextrap - \ontopextrapcipsi) = 0.190$ for two isolated \ce{N} atoms. We can then conclude that the overestimation of the system-averaged on-top pair density, and therefore of the basis-set correction, is more important for the \ce{N2} molecule at equilibrium distance than for the isolated \ce{N} atoms. This probably explains the observed overestimation of the atomization energy. To confirm this statement, we computed the basis-set correction for both the \ce{N2} molecule at equilibrium distance and the isolated atoms using $\murcipsi$ and $\mathring{n}_{2,\text{CIPSI}}(\br{})$ with the aug-cc-pVTZ and aug-cc-pVQZ basis sets. We obtained the following values for the atomization energies: $362.12$ mH with aug-cc-pVTZ and $362.15$ mH with aug-cc-pVQZ, which are indeed more accurate values than those obtained using $\murcas$ and $\mathring{n}_{2,\text{CASSCF}}(\br{})$.
}
Finally, regarding now the performance of the basis-set correction along the whole potential energy curves reported in Figs.~\ref{fig:N2}, \ref{fig:O2}, and \ref{fig:F2}, it is interesting to notice that it fails to provide a noticeable improvement far from the equilibrium geometry. Acknowledging that the weak-correlation effects in these regions are dominated by dispersion interactions which are long-range effects, the failure of the present approximations for the complementary functional can be understood easily. Indeed, the whole scheme designed here is based on the physics of correlation near the electron-electron coalescence point: the local range-separation function $\mu(\br{})$ is based on the value of the effective electron-electron interaction at coalescence and the ECMD functionals are suited for short-range correlation effects. Therefore, the failure of the present basis-set correction to describe dispersion interactions is theoretically expected.