From 10b38145f99d095760f2caeadf6c566a28e01786 Mon Sep 17 00:00:00 2001 From: Pierre-Francois Loos Date: Mon, 6 Jan 2020 16:07:15 +0100 Subject: [PATCH] results --- Manuscript/srDFT_SC.tex | 21 ++++++++++----------- 1 file changed, 10 insertions(+), 11 deletions(-) diff --git a/Manuscript/srDFT_SC.tex b/Manuscript/srDFT_SC.tex index 6ed8039..25404a8 100644 --- a/Manuscript/srDFT_SC.tex +++ b/Manuscript/srDFT_SC.tex @@ -305,7 +305,6 @@ In the general context of multiconfigurational DFT, this finding shows that one The general goal of quantum chemistry is to provide reliable theoretical tools to explore the rich area of chemistry. More specifically, developments in quantum chemistry primarily aim at accurately computing the electronic structure of molecular systems. Despite intense developments, no definitive solution to this problem has been found. The theoretical challenge to tackle belongs to the quantum many-body problem, due the intrinsic quantum nature of the electrons and the Coulomb repulsion between them. This so-called electronic correlation problem corresponds to finding a solution to the Schr\"odinger equation for a $N$-electron system, and two main roads have emerged to approximate this solution: wave-function theory (WFT) \cite{Pop-RMP-99} and density-functional theory (DFT). \cite{Koh-RMP-99} Although both WFT and DFT spring from the same Schr\"odinger equation, they rely on very different formalisms, as the former deals with the complicated $N$-electron wave function whereas the latter focuses on the much simpler one-electron density. In its Kohn-Sham (KS) formulation, \cite{KohSha-PR-65} the computational cost of DFT is very appealing since it is a simple mean-field procedure. Therefore, although continued efforts have been done to reduce the computational cost of WFT, DFT still remains the workhorse of quantum computational chemistry. The difficulty of obtaining a reliable theoretical description of a given chemical system can be roughly categorized by the strength of the electronic correlation. The so-called weakly correlated systems, such as closed-shell organic molecules near their equilibrium geometry, are typically dominated by correlation effects which do not affect the qualitative mean-field picture of the system. These weak-correlation effects can be either short range (near the electron-electron coalescence points) or long range (London dispersion interactions). The theoretical description of weakly correlated systems is one of the most concrete achievement of quantum chemistry, and the main remaining issue for these systems is to push the limit of the chemical system size that can be treated. The case of the so-called strongly correlated systems, which are ubiquitous in chemistry, is more problematic as they exhibit a much more complex electronic structure. For example, transition metal complexes, low-spin open-shell systems, covalent bond breaking situations have all in common that they cannot be even qualitatively described by a single electronic configuration. It is now clear that the usual semilocal density-functional approximations of KS DFT fail to accurately describe these situations and WFT is king for the treatment of strongly correlated systems. -%\PFL{I think we should add some references in the paragraph above.} In practice, WFT uses a finite one-particle basis set (here denoted as $\basis$). The exact solution of the Schr\"odinger equation within this basis set is then provided by full configuration interaction (FCI) which consists in a linear-algebra eigenvalue problem with a dimension scaling exponentially with the system size. Due to this exponential growth of the FCI computational cost, introducing approximations is necessary, with at least two difficulties for strongly correlated systems: i) the qualitative description of the wave function is determined by a primary set of electronic configurations (whose size can scale exponentially in many cases) among which near degeneracies and/or strong interactions appear in the Hamiltonian matrix; ii) the quantitative description of the system requires also to account for weak correlation effects which involve many other electronic configurations with typically much smaller weights in the wave function. Addressing these two issues is a rather complicated task for a given approximate WFT method, especially if one adds the requirement of satisfying formal properties, such as spin-multiplet degeneracy (\ie, invariance with respect to the spin operator $S_z$) and size consistency. @@ -689,7 +688,7 @@ Regarding the computational cost of the present approach, it should be stressed \hline \ce{F2} & aug-cc-pVDZ & 49.6 [12.6] & 54.8 [7.4] & 54.9 [7.3] & 54.8 [7.4] \\ & aug-cc-pVTZ & 59.3 [2.9] & 61.2 [1.0] & 61.5 [0.7] & 61.5 [0.7] \\[0.1cm] - & aug-cc-pVQZ & 60.1 [ ] & 61.0 [1.2] & 61.3 [0.9] & 61.3 [0.9] \\[0.1cm] + & aug-cc-pVQZ & 60.1 [\titou{???}] & 61.0 [1.2] & 61.3 [0.9] & 61.3 [0.9] \\[0.1cm] \hline & & \tabc{CEEIS\fnm[3]} & \tabc{CEEIS\fnm[3]+$\pbeuegXi$} & \tabc{CEEIS\fnm[3]+$\pbeontXi$} & \tabc{CEEIS\fnm[3]+$\pbeontns$}\\ \hline @@ -717,7 +716,7 @@ More quantitatively, the values of $D_0$ are within chemical accuracy (\ie, an e 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 provide 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 CASSCF 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. 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?} +\PFL{The functional $\pbeuegns$ is not mentioned and should be discarded at this stage (and explain why).} \subsection{Dissociation of diatomics} @@ -766,18 +765,18 @@ This could have significant implications for the construction of more robust fam \includegraphics[width=0.45\linewidth]{data/F2/DFT_vqzE_relat.eps} \includegraphics[width=0.45\linewidth]{data/F2/DFT_vqzE_relat_zoom.eps} \caption{ - Potential energy curves of the \ce{F2} molecule calculated with CEEIS$^1$ and basis-set corrected CEEIS$^1$ using the cc-pVDZ (top), cc-pVTZ (middle) and cc-pVQZ (bottom) basis sets. The estimated exact energies are based on fit on valence-only extrapolated CEEIS data obtained from Ref.~\onlinecite{BytNagGorRue-JCP-07}. -The non-relativistic CEEIS energies have been extracted from Ref.~\onlinecite{BytNagGorRue-JCP-07}. + Potential energy curves of the \ce{F2} molecule calculated with CEEIS$^1$ and basis-set corrected CEEIS$^1$ using the cc-pVDZ (top), cc-pVTZ (middle) and cc-pVQZ (bottom) basis sets. + The estimated exact energies are based on a fit of the valence-only extrapolated CEEIS data extracted from Ref.~\onlinecite{BytNagGorRue-JCP-07}. \label{fig:F2}} \end{figure*} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -The \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. The dispersion forces in \ce{H10} play a much minor role in the PES due to the much smaller number of near-neighboring electron pairs compared to \ce{N2}, \ce{O2} or \ce{F2}. 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. +The \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. The dispersion forces in \ce{H10} play a minor role in the PES due to the much smaller number of near-neighboring electron pairs compared to \ce{N2}, \ce{O2} or \ce{F2}. 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 Figs.~\ref{fig:N2} and \ref{fig:O2} the potential energy curves of \ce{N2}, \ce{O2}, and computed at various approximation levels using the aug-cc-pVDZ and aug-cc-pVTZ basis sets, and in Fig~\ref{fig:F2} the potential energy surface of \ce{F2} using the cc-pVXZ (X=D,T,Q) basis set. The computation of the atomization energies $D_0$ at each level of theory is reported in Table \ref{tab:d0}. +We report in Figs.~\ref{fig:N2} and \ref{fig:O2} the potential energy curves of \ce{N2} and \ce{O2} computed at various approximation levels using the aug-cc-pVDZ and aug-cc-pVTZ basis sets. Figure \ref{fig:F2} reports the potential energy surface of \ce{F2} using the cc-pVXZ (X $=$ D, T, and Q) basis set. The value of $D_0$ for each level of theory is reported in Table \ref{tab:d0}. -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 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{F2} it is clear that the presence of diffuse functions in for double- and triple-zeta types basis sets strongly improves the results, which is somehow understandable due to the strong breathing-orbital effect in this molecule induced by the ionic valence bond forms. \cite{HibHumByrLen-JCP-94} -It should be also noticed that when reaching the aug-cc-pVQZ basis set for \ce{N2}, the quality of $D_0$ slightly deteriorates for the $\pbeontXi$ and $\pbeontns$ functionals, but it remains nevertheless more accurate than the estimated FCI $D_0$ and very close the to chemical accuracy. +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 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{F2} it is clear that the addition of diffuse functions in the double- and triple-$\zeta$ basis sets strongly improves the results, a result that can be anticipated due to the strong breathing-orbital effect induced by the ionic valence bond forms in this molecule. \cite{HibHumByrLen-JCP-94} +It should be also noticed that when reaching the aug-cc-pVQZ basis set for \ce{N2}, the quality of $D_0$ slightly deteriorates for the $\pbeontXi$ and $\pbeontns$ functionals, but it remains nevertheless more accurate than the estimated FCI $D_0$ and very close to chemical accuracy. 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 complementary density 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. We hope to report further on this in the near future. @@ -787,14 +786,14 @@ We hope to report further on this in the near future. In the present paper we have extended the recently proposed DFT-based basis-set correction to strongly correlated systems. We studied the \ce{H10}, \ce{N2}, \ce{O2}, and \ce{F2} molecules up to the dissociation limit at near-FCI level in increasing-large basis sets, and investigated how the basis-set correction affects the convergence toward the CBS limit of the potential energy curves of these molecular systems. -The DFT-based basis-set correction relies on three aspects: i) the definition of an effective non-divergent electron-electron interaction obtained from the expectation value over a wave function $\psibasis$ of the Coulomb interaction projected into an incomplete basis set $\basis$, ii) the fit of this effective interaction with the long-range interaction used in RS-DFT, and iii) the use of a short-range, complementary correlation functional from RS-DFT. In the present paper, we investigated i) and iii) in the context of strong correlation and focused on potential energy curves and atomization energies. More precisely, we proposed a new scheme to design functionals fulfilling i) spin-multiplet degeneracy, and ii) size consistency. To fulfil such requirements we proposed to use CASSCF wave functions leading to size-consistent energies, and to develop functionals using only $S_z$-independent density-like quantities. +The density-based basis-set correction relies on three aspects: i) the definition of an effective non-divergent electron-electron interaction obtained from the expectation value over a wave function $\psibasis$ of the Coulomb interaction projected into an incomplete basis set $\basis$, ii) the fit of this effective interaction with the long-range interaction used in RS-DFT, and iii) the use of a short-range, complementary density functional borrowed from RS-DFT. In the present paper, we investigated i) and iii) in the context of strong correlation and focused on potential energy curves and atomization energies. More precisely, we proposed a new scheme to design functionals fulfilling spin-multiplet degeneracy and size consistency. To fulfil such requirements we proposed to use CASSCF wave functions leading to size-consistent energies, and to develop functionals using only $S_z$-independent density-like quantities. The development of new $S_z$-independent and size-consistent functionals has lead us to investigate the role of two related quantities: the spin polarization and the on-top pair density. One important result of the present study is that by using functionals \textit{explicitly} depending on the on-top pair density, one can eschew its spin polarization dependency without loss of accuracy. This avoids the commonly used effective spin polarization originally proposed in Ref.~\onlinecite{BecSavSto-TCA-95} which has the disadvantage of possibly becoming complex-valued in the multideterminant case. From a more fundamental aspect, this shows that, in a DFT framework, the spin polarization mimics the role of the on-top pair density. Consequently, we believe that one could potentially develop new families of density functional approximations where the spin polarization is abondonned and replaced by the on-top pair density. Regarding the results of the present approach, the basis-set correction systematically improves the near-FCI calculations in a given basis set. More quantitatively, it is shown that with only triple-$\zeta$ quality basis sets chemically accurate atomization energies, $D_0$, are obtained for all systems whereas the uncorrected near-FCI results are far from this accuracy within the same basis set. -Also, it is shown that the basis-set correction gives substantial differential contribution to potential energy curves close to the equilibrium geometries, but at long internuclear distances it cannot recover the dispersion energy missing because of the basis set incompleteness. This behaviour is actually expected as dispersion has a long-range correlation nature and the present approach is designed to recover only short-range correlation effects. +Also, it is shown that the basis-set correction gives substantial differential contribution to potential energy curves close to the equilibrium geometries, but at long internuclear distances it cannot recover the dispersion energy missing because of the basis set incompleteness. This behaviour is actually expected as dispersion is of long-range nature and the present approach is designed to recover only short-range correlation effects. \bibliography{srDFT_SC}