Results done

Manuscript/srDFT_SC.bib

@@ -1,13 +1,23 @@
 %% This BibTeX bibliography file was created using BibDesk.
 %% http://bibdesk.sourceforge.net/
 
-%% Created for Pierre-Francois Loos at 2019-12-12 03:48:35 +0100 
+%% Created for Pierre-Francois Loos at 2019-12-13 10:13:42 +0100 
 
 
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+@article{BooCleThoAla-JCP-11,
+	Author = {G. H. Booth and D. Cleland and A. J. W. Thom and A. Alavi},
+	Date-Added = {2019-12-13 10:13:33 +0100},
+	Date-Modified = {2019-12-13 10:13:33 +0100},
+	Journal = {J. Chem. Phys.},
+	Pages = {084104},
+	Title = {Breaking the carbon dimer: The challenges of multiple bond dissociation with full configuration interaction quantum Monte Carlo methods},
+	Volume = {135},
+	Year = {2011}}
+
 @article{StaDav-CPL-01,
 	Author = {Viktor N. Staroverov and Ernest R. Davidson},
 	Date-Added = {2019-12-12 03:48:34 +0100},
@@ -1456,13 +1466,6 @@
 	Volume = {80},
 	Year = {1984}}
 
-@article{BooCleThoAla-JCP-11,
-	Author = {G. H. Booth and D. Cleland and A. J. W. Thom and A. Alavi},
-	Journal = {J. Chem. Phys.},
-	Pages = {084104},
-	Volume = {135},
-	Year = {2011}}
-
 @article{BooGruKreAla-Nat-13,
 	Author = {G. H. Booth and A. Gr\"uneis and G. Kresse and A. Alavi},
 	Journal = {Nature},

Manuscript/srDFT_SC.tex

@@ -709,14 +709,15 @@ Also, as the frozen-core approximation is used in all our selected CI calculatio
 
 \subsection{H$_{10}$ chain}
 
+The study of the \ce{H10} chain with equally distant atoms is a good prototype of strongly-correlated systems as it consists in the simultaneous breaking of 10 covalent $\sigma$ bonds which all interact with each other. Also, being a relatively small system, benchmark calculations at near-CBS values can be obtained (see Ref.~\onlinecite{h10_prx} for a detailed study of this problem). 
 
-The study of the \ce{H10} chain with equally distant atoms is a good prototype of strongly-correlated systems as it consists in the simultaneous breaking of 10 covalent $\sigma$ bonds which all interact with each other. Also, being a relatively small system, benchmark calculations at near CBS values can be obtained (see Ref.~\onlinecite{h10_prx} for a detailed study of this problem). 
+We report in Fig.~\ref{fig:H10} the potential energy curves computed using the cc-pVXZ (X $=$ D, T, and Q) basis sets for different levels of approximation. The computation of the atomization energies $D_0$ for each level of theory is reported in Table \ref{tab:d0}. A general trend that can be observed from these data is that, in a given basis set, the quality of the potential energy curves are globally improved by adding the basis-set correction, independently of the approximation level of \titou{$\efuncbasis$}. Also, no erratic behavior is found when stretching the bonds, which shows that the present procedure and their functionals are robust when reaching the strong-correlation regime.  
+In other words, smooth potential energy surfaces are obtained with the present basis-set correction.
+More quantitatively, the values of $D_0$ are within chemical accuracy (\ie, an error below $1.4$ mHa) from the cc-pVTZ basis set when using the $\pbeontXi$ and $\pbeontns$ functionals, whereas such an accuracy is not even reached at the standard MRCI+Q/cc-pVQZ level of theory. 
 
-We report in Figure \ref{fig:H10} the potential energy curves computed using the cc-pVXZ (X=D,T,Q) basis sets for different levels of approximations. The computation of the atomization energies $D_0$ at each level of theory used here is reported in Table \ref{tab:d0}. A general trend that can be observed from these data is that, in a given basis set, the quality of the potential energy curves are globally improved by adding the basis-set correction, whatever the level of approximation used for the functional $\efuncbasisFCI$. Also, no bizarre behaviors are found when stretching the bonds, which shows that the functionals are robust when reaching the strong-correlation regime.  
-
-More quantitatively, the values of $D_0$ are within chemical accuracy (\ie, an error below 1.4 mHa) from the cc-pVTZ basis set when using the PBE-ot-$\tilde{\zeta}$ and PBE-ot-$0{\zeta}$ functionals, whereas such an accuracy is not reached at the cc-pVQZ basis set using standard MRCI+Q. 
-
-Regarding in more details the performance of the different types of approximate functionals, the results show that PBE-ot-$\tilde{\zeta}$ and PBE-ot-$0{\zeta}$ are very similar (the maximal difference on $D_0$ being 0.3 mHa), and they give slightly more accurate results than PBE-UEG-$\tilde{\zeta}$. These findings bring two important clues on the role of the different physical ingredients used in the functionals: i) the explicit use of the on-top pair density coming from the CASSCF wave function (see Eq.~\eqref{eq:def_n2extrap}) is preferable to the use of the on-top pair density based on the UEG (see Eq.~\eqref{eq:def_n2ueg}); ii) removing the dependency on any kind of spin polarization does not lead to significant loss of accuracy provided that one uses a qualitatively correct on-top pair density. Point ii) is important as it shows that spin polarization in density-functional approximations essentially plays the same role as that of the on-top pair density.  
+Analyzing more carefully the performance of the different types of approximate density functionals, the results show that $\pbeontXi$ and $\pbeontns$ are very similar (the maximal difference on $D_0$ being 0.3 mHa), and that they give slightly more accurate results than $\pbeuegXi$. These findings provides two important clues on the role of the different physical ingredients used in these functionals: i) the explicit use of the on-top pair density coming from the \titou{CAS} wave function [see Eq.~\eqref{eq:def_n2extrap}] is preferable over the use of the UEG on-top pair density [see Eq.~\eqref{eq:def_n2ueg}] which is somehow understandable, and ii) removing the dependency on any kind of spin polarization does not lead to significant loss of accuracy providing that one employs a qualitatively correct on-top pair density. The latter point is crucial as it shows that the spin polarization in density-functional approximations essentially plays the same role as the on-top pair density.
+\titou{This could have significant implications for the construction of more robust families of density-functional approximations within DFT.}
+\PFL{Why can't we see the effect of dispersion in that system?}
 
 \subsection{Dissociation of diatomics}
 
@@ -768,13 +769,14 @@ Regarding in more details the performance of the different types of approximate
 \end{figure*}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
-The \ce{C2}, \ce{N2}, \ce{O2} and \ce{F2} molecules are complementary to the \ce{H10} system for the present study as the level of strong correlation increases while stretching the bond similarly to the case of \ce{H10}, but in addition these molecules exhibit more important and versatile types of weak correlations due to the larger number of electrons. Indeed, the short-range correlation effects are known to play a strong differential effect on the computation of $D_0$, while the shape of the curve far from the equilibrium geometry is governed by dispersion interactions which are medium to long-range weak-correlation effects. Also, \ce{O2} has a triplet ground state and is therefore a good check for the performance of the dependence on the spin polarization of the different types of functionals proposed here.
+The \ce{C2}, \ce{N2}, \ce{O2} and \ce{F2} molecules are complementary to the \ce{H10} system for the present study as the level of strong correlation in these diatomics also increases while stretching the bond similarly to the case of \ce{H10}. In addition, these molecules exhibit more important and versatile types of weak correlations due to the larger number of electrons. Indeed, the short-range correlation effects are known to play a strong differential effect on the computation of $D_0$, while the shape of the curve far from the equilibrium geometry is governed by dispersion interactions which are medium to long-range weak-correlation effects. Also, \ce{O2} has a triplet ground state and is therefore a good candidate for checking the spin-polarization dependence of the various functionals proposed here.
 
-We report in Figure \ref{fig:C2}, \ref{fig:N2}, \ref{fig:O2}, and \ref{fig:F2} the potential energy curves computed using the aug-cc-pVDZ and aug-cc-pVTZ basis sets of C$_2$, N$_2$, O$_2$, and N$_2$, respectively, for different approximation levels. The computation of the atomization energies $D_0$ at each level of theory used here is reported in Table \ref{tab:d0}. 
+We report in Figs.~\ref{fig:C2}, \ref{fig:N2}, \ref{fig:O2}, and \ref{fig:F2} the potential energy curves of \ce{C2}, \ce{N2}, \ce{O2}, and \ce{N2} computed at various approximation levels using the aug-cc-pVDZ and aug-cc-pVTZ basis sets. The computation of the atomization energies $D_0$ at each level of theory is reported in Table \ref{tab:d0}. 
 
-Just as in the case of \ce{H10}, the quality of $D_0$ are globally improved by adding the basis-set correction and it is remarkable that PBE-ot-$\tilde{\zeta}$ and PBE-ot-$0{\zeta}$ give very similar results. The latter observation confirms that the dependency on the on-top pair density allows one to remove the dependency of any kind of spin polarization for a quite wide range of electron density and also for a high-spin system like \ce{O2}. More quantitatively, an error below 1.0 mHa on the estimated exact valence-only $D_0$ is found for \ce{N2}, \ce{O2}, and \ce{F2} with the aug-cc-pVTZ basis set using the PBE-ot-$0{\zeta}$ functional, whereas such a result is far from reach within the same basis set at the near-FCI level. In the case of \ce{C2} with the aug-cc-pVTZ basis set, an error of about 5.5 mHa is found with respect to the estimated exact $D_0$. Such an error is remarkably large with respect to the other diatomic molecules studied here and might be associated to the level of strong correlation in the \ce{C2} molecule. 
+Just as in \ce{H10}, the quality of $D_0$ is globally improved by adding the basis-set correction and it is remarkable that $\pbeontXi$ and $\pbeontns$ provide again very similar results. The latter observation confirms that the dependency on the on-top pair density allows one to remove the dependency of any kind of spin polarization for a quite wide range of electron density and also for \titou{open-shell} systems like \ce{O2}. More quantitatively, an error below 1.0 mHa on the estimated exact valence-only $D_0$ is found for \ce{N2}, \ce{O2}, and \ce{F2} with the aug-cc-pVTZ basis set using the $\pbeontns$ functional, whereas such a feat is far from being reached within the same basis set at the near-FCI level. In the case of \ce{C2} with the aug-cc-pVTZ basis set, an error of about 5.5 mHa is found with respect to the estimated exact $D_0$. \titou{Such an error is remarkably large compared to the other diatomic molecules studied here and might be associated to the significant level of strong correlation in the \ce{C2} molecule. \cite{BooCleThoAla-JCP-11}}
 
-Regarding now the performance of the basis-set correction along the whole potential energy curve, 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 \titou{complementary basis functionals} can be understood easily. Indeed, the whole scheme designed here is based on the physics of correlation near the electron-electron cusp: the local range-separation function $\mu(\br{})$ is designed by looking at the electron-electron coalescence point 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 expected.
+Regarding now the performance of the basis-set correction along the whole potential energy curve, 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 \titou{complementary basis functionals} 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 designed thanks to the universal condition provided by the electron-electron cusp 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 and predictable.
+\titou{We hope to report further on this in the near future.}
 
 \section{Conclusion}
 \label{sec:conclusion}