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4
Manuscript/SI/srDFT_SC-SI.tex
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@ -137,6 +137,7 @@
% effective interaction % effective interaction
\newcommand{\twodm}[4]{\elemm{\Psi}{\psixc{#4}\psixc{#3} \psix{#2}\psix{#1}}{\Psi}} \newcommand{\twodm}[4]{\elemm{\Psi}{\psixc{#4}\psixc{#3} \psix{#2}\psix{#1}}{\Psi}}
\newcommand{\murpsibas}[0]{\mu_{\wf{}{\Bas}}({\bf r})}
\newcommand{\murpsi}[0]{\mu({\bf r};\wf{}{A+B})} \newcommand{\murpsi}[0]{\mu({\bf r};\wf{}{A+B})}
\newcommand{\murpsia}[0]{\mu({\bf r};\wf{}{A})} \newcommand{\murpsia}[0]{\mu({\bf r};\wf{}{A})}
\newcommand{\murpsib}[0]{\mu({\bf r};\wf{}{B})} \newcommand{\murpsib}[0]{\mu({\bf r};\wf{}{B})}
@ -406,7 +407,8 @@ which, in the case of a multiplicative wave function is nothing but
As $\murpsia = 0 \text{ if }\br{} \in B$ (and equivalently for $\murpsib $ on $B$), $\murpsi$ is an intensive quantity. The conclusion of this paragraph is that, provided that the wave function for the system $A+B$ is multiplicative in the limit of the dissociated fragments, all quantities used for the basis set correction are intensive and therefore the basis set correction is size consistent. As $\murpsia = 0 \text{ if }\br{} \in B$ (and equivalently for $\murpsib $ on $B$), $\murpsi$ is an intensive quantity. The conclusion of this paragraph is that, provided that the wave function for the system $A+B$ is multiplicative in the limit of the dissociated fragments, all quantities used for the basis set correction are intensive and therefore the basis set correction is size consistent.
\section{Computational considerations} \section{Computational considerations}
The computational cost of the present approach is driven by two quantities: the computation of the on-top pair density and the $\murpsibas$ in real-space on the DFT grid. Within a blind approach, for each grid point the computational cost is of order $n_{\Bas}^4$ and $n_{\Bas}^6$ for the on-top pair density and $\murpsibas$, respectively.
Nevertheless, using CASSCF wave functions to compute these quantities leads to significant simplifications
\bibliography{../srDFT_SC} \bibliography{../srDFT_SC}
\end{document} \end{document}

277
Manuscript/srDFT_SC.bbl
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@ -6,7 +6,7 @@
%Control: page (0) single %Control: page (0) single
%Control: year (1) truncated %Control: year (1) truncated
%Control: production of eprint (0) enabled %Control: production of eprint (0) enabled
\begin{thebibliography}{66}% \begin{thebibliography}{84}%
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@ -124,11 +124,11 @@
{journal} {Chem. Phys. Lett.}\ }\textbf {\bibinfo {volume} {286}},\ \bibinfo {journal} {Chem. Phys. Lett.}\ }\textbf {\bibinfo {volume} {286}},\ \bibinfo
{pages} {243} (\bibinfo {year} {1998})}\BibitemShut {NoStop}% {pages} {243} (\bibinfo {year} {1998})}\BibitemShut {NoStop}%
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(\bibinfo {year} {2019}),\ \Eprint {http://arxiv.org/abs/1903.05458} (\bibinfo {year} {2019}),\ \Eprint {http://arxiv.org/abs/1903.05458}
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{\bibfield {journal} {\bibinfo {journal} {Phys. Chem. Chem. Phys.}\ {\bibfield {journal} {\bibinfo {journal} {Phys. Chem. Chem. Phys.}\
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@ -247,31 +251,25 @@
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{pages} {6062} (\bibinfo {year} {1979})}\BibitemShut {NoStop}% {76}},\ \bibinfo {pages} {6062} (\bibinfo {year} {1979})}\BibitemShut
{NoStop}%
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@ -576,6 +584,14 @@
{\doibase 10.1103/PhysRevA.98.062510} {\bibfield {journal} {\bibinfo {\doibase 10.1103/PhysRevA.98.062510} {\bibfield {journal} {\bibinfo
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{pages} {062510} (\bibinfo {year} {2018})}\BibitemShut {NoStop}% {pages} {062510} (\bibinfo {year} {2018})}\BibitemShut {NoStop}%
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{Phys. Rev. A}\ }\textbf {\bibinfo {volume} {44}},\ \bibinfo {pages} {1549}
(\bibinfo {year} {1991})}\BibitemShut {NoStop}%
\bibitem [{\citenamefont {Becke}, \citenamefont {Savin},\ and\ \citenamefont \bibitem [{\citenamefont {Becke}, \citenamefont {Savin},\ and\ \citenamefont
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@ -585,6 +601,116 @@
{\bibfield {journal} {\bibinfo {journal} {Theoret. Chim. Acta}\ }\textbf {\bibfield {journal} {\bibinfo {journal} {Theoret. Chim. Acta}\ }\textbf
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{publisher} {World Scientific},\ \bibinfo {year} {1996})\ pp.\ \bibinfo
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@ -594,6 +720,14 @@
}\href@noop {} {\bibfield {journal} {\bibinfo {journal} {Phys. Rev. A}\ }\href@noop {} {\bibfield {journal} {\bibinfo {journal} {Phys. Rev. A}\
}\textbf {\bibinfo {volume} {51}},\ \bibinfo {pages} {4531} (\bibinfo {year} }\textbf {\bibinfo {volume} {51}},\ \bibinfo {pages} {4531} (\bibinfo {year}
{1995})}\BibitemShut {NoStop}% {1995})}\BibitemShut {NoStop}%
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{142} (\bibinfo {year} {2001})}\BibitemShut {NoStop}%
\bibitem [{\citenamefont {Holmes}, \citenamefont {Umrigar},\ and\ \citenamefont \bibitem [{\citenamefont {Holmes}, \citenamefont {Umrigar},\ and\ \citenamefont
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@ -747,8 +881,17 @@
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{\bibinfo {journal} {The Journal of Chemical Physics}\ }\textbf {\bibinfo {\bibinfo {journal} {J. Chem. Phys.}\ }\textbf {\bibinfo {volume} {122}},\
{volume} {122}},\ \bibinfo {pages} {154110} (\bibinfo {year} {2005})},\ \bibinfo {pages} {154110} (\bibinfo {year} {2005})}\BibitemShut {NoStop}%
\Eprint {http://arxiv.org/abs/https://doi.org/10.1063/1.1869493} \bibitem [{\citenamefont {Booth}\ \emph {et~al.}(2011)\citenamefont {Booth},
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\end{thebibliography}% \end{thebibliography}%

49
Manuscript/srDFT_SC.tex
View File

@ -550,7 +550,7 @@ Second, the fact that $\efuncdenpbe{\argebasis}$ vanishes for systems with vanis
\subsection{Requirements for strong correlation} \subsection{Requirements for strong correlation}
\label{sec:requirements} \label{sec:requirements}
An important requirement for any electronic-structure method is size-consistency, \ie, the additivity of the energies of non-interacting fragments, which is mandatory to avoid any ambiguity in computing interaction energies. When two subsystems \ce{A} and \ce{B} dissociate in closed-shell systems, as in the case of weak intermolecular interactions for instance, spin-restricted Hartree-Fock (RHF) is size-consistent. When the two subsystems dissociate in open-shell systems, such as in covalent bond breaking, it is well known that the RHF approach fails and an alternative is to use a complete-active-space (CAS) wave function which, provided that the active space has been properly chosen, leads to additive energies. An important requirement for any electronic-structure method is size-consistency, \ie, the additivity of the energies of non-interacting fragments, which is mandatory to avoid any ambiguity in computing interaction energies. When two subsystems \ce{A} and \ce{B} dissociate in closed-shell systems, as in the case of weak intermolecular interactions for instance, spin-restricted Hartree-Fock (RHF) is size-consistent. When the two subsystems dissociate in open-shell systems, such as in covalent bond breaking, it is well known that the RHF approach fails and an alternative is to use a complete-active-space-self-consistent-field (CASSCF) wave function which, provided that the active space has been properly chosen, leads to additive energies.
Another important requirement is spin-multiplet degeneracy, \ie, the independence of the energy with respect to the $S_z$ component of a given spin state, which is also a property of any exact wave function. Such a property is also important in the context of covalent bond breaking where the ground state of the supersystem $\ce{A + B}$ is generally of lower spin than the corresponding ground states of the fragments (\ce{A} and \ce{B}) which can have multiple $S_z$ components. Another important requirement is spin-multiplet degeneracy, \ie, the independence of the energy with respect to the $S_z$ component of a given spin state, which is also a property of any exact wave function. Such a property is also important in the context of covalent bond breaking where the ground state of the supersystem $\ce{A + B}$ is generally of lower spin than the corresponding ground states of the fragments (\ce{A} and \ce{B}) which can have multiple $S_z$ components.
@ -578,19 +578,19 @@ An alternative way to eliminate the $S_z$ dependency is to simply set $\zeta=0$,
\subsubsection{Size consistency} \subsubsection{Size consistency}
Since $\efuncdenpbe{\argebasis}$ is computed via a single integral over $\mathbb{R}^3$ [see Eq.~\eqref{eq:def_ecmdpbebasis}] which involves only local quantities [$n(\br{})$, $\zeta(\br{})$, $s(\br{})$, $n_2(\br{})$, and $\mu(\br{})$], in the case of non-overlapping fragments \ce{A\bond{...}B}, it can be written as the sum of two local contributions: one coming from the integration over the region of subsystem \ce{A} and the other one from the region of subsystem \ce{B}. Therefore, a sufficient condition for size consistency is that these local quantities coincide in the isolated systems and in the subsystems of the supersystem \ce{A\bond{...}B}. Since these local quantities are calculated from the wave function $\psibasis$, a sufficient condition is that the wave function is multiplicatively separable in the limit of non-interacting fragments, \ie, $\Psi_{\ce{A\bond{...}B}}^{\basis} = \Psi_{\ce{A}}^{\basis} \Psi_{\ce{B}}^{\basis}$ \manu{(see SI for more detailed demonstration of that statement)}. In the case where the two subsystems \ce{A} and \ce{B} dissociate in closed-shell systems, a simple RHF wave function ensures this property, but when one or several covalent bonds are broken, a properly chosen CAS wave function is sufficient to recover this property. \titou{The underlying active space must however be chosen in such a way that it leads to size-consistent energies in the limit of dissociated fragments.} Since $\efuncdenpbe{\argebasis}$ is computed via a single integral over $\mathbb{R}^3$ [see Eq.~\eqref{eq:def_ecmdpbebasis}] which involves only local quantities [$n(\br{})$, $\zeta(\br{})$, $s(\br{})$, $n_2(\br{})$, and $\mu(\br{})$], in the case of non-overlapping fragments \ce{A\bond{...}B}, it can be written as the sum of two local contributions: one coming from the integration over the region of subsystem \ce{A} and the other one from the region of subsystem \ce{B}. Therefore, a sufficient condition for size consistency is that these local quantities coincide in the isolated systems and in the subsystems of the supersystem \ce{A\bond{...}B}. Since these local quantities are calculated from the wave function $\psibasis$, a sufficient condition is that the wave function is multiplicatively separable in the limit of non-interacting fragments, \ie, $\Psi_{\ce{A\bond{...}B}}^{\basis} = \Psi_{\ce{A}}^{\basis} \Psi_{\ce{B}}^{\basis}$ \manu{(see SI for more detailed demonstration of that statement)}. In the case where the two subsystems \ce{A} and \ce{B} dissociate in closed-shell systems, a simple RHF wave function ensures this property, but when one or several covalent bonds are broken, a properly chosen CASSCF wave function is sufficient to recover this property. \titou{The underlying active space must however be chosen in such a way that it leads to size-consistent energies in the limit of dissociated fragments.}
\subsection{\titou{Complementary density functional approximations}} \subsection{\titou{Complementary density functional approximations}}
\label{sec:def_func} \label{sec:def_func}
%\subsubsection{Definition of the protocol to design functionals} %\subsubsection{Definition of the protocol to design functionals}
As the present work focuses on the strong-correlation regime, we propose here to investigate only approximate functionals which are $S_z$ independent and size-consistent in the case of covalent bond breaking. Therefore, the wave functions $\psibasis$ used throughout this paper are CAS wave functions in order to ensure size consistency of all local quantities. The difference between two flavors of functionals are only due to the type of i) spin polarization, and ii) on-top pair density. As the present work focuses on the strong-correlation regime, we propose here to investigate only approximate functionals which are $S_z$ independent and size-consistent in the case of covalent bond breaking. Therefore, the wave functions $\psibasis$ used throughout this paper are CASSCF wave functions in order to ensure size consistency of all local quantities. The difference between two flavors of functionals are only due to the type of i) spin polarization, and ii) on-top pair density.
<<<<<<< HEAD <<<<<<< HEAD
Regarding the approximation to the \textit{exact} on-top pair density entering in eq. \eqref{eq:def_beta}, we use two different approximations. The first one is based on the uniform electron gas (UEG) and reads Regarding the approximation to the \textit{exact} on-top pair density entering in eq. \eqref{eq:def_beta}, we use two different approximations. The first one is based on the uniform electron gas (UEG) and reads
======= =======
Regarding the spin polarization that enters into $\varepsilon_{\text{c}}^{\text{PBE}}(\argepbe)$, two different types of $S_z$-independent formulations are considered: i) the \textit{effective} spin polarization $\tilde{\zeta}$ defined in Eq.~\eqref{eq:def_effspin} and calculated from the CAS wave function, and ii) a \textit{zero} spin polarization. Regarding the spin polarization that enters into $\varepsilon_{\text{c}}^{\text{PBE}}(\argepbe)$, two different types of $S_z$-independent formulations are considered: i) the \textit{effective} spin polarization $\tilde{\zeta}$ defined in Eq.~\eqref{eq:def_effspin} and calculated from the CASSCF wave function, and ii) a \textit{zero} spin polarization.
Regarding the on-top pair density entering in Eq.~\eqref{eq:def_beta}, we use two different approximations. The first one is based on the uniform electron gas (UEG) and reads Regarding the on-top pair density entering in Eq.~\eqref{eq:def_beta}, we use two different approximations. The first one is based on the uniform electron gas (UEG) and reads
>>>>>>> 7d17cec3065dbb2c36d90f24f286b41985ef6529 >>>>>>> 7d17cec3065dbb2c36d90f24f286b41985ef6529
@ -598,9 +598,9 @@ Regarding the on-top pair density entering in Eq.~\eqref{eq:def_beta}, we use tw
\label{eq:def_n2ueg} \label{eq:def_n2ueg}
\ntwo^{\text{UEG}}(n,\zeta) \approx n^2\big(1-\zeta^2\big)g_0(n), \ntwo^{\text{UEG}}(n,\zeta) \approx n^2\big(1-\zeta^2\big)g_0(n),
\end{equation} \end{equation}
where the pair-distribution function $g_0(n)$ is taken from Eq.~(46) of Ref.~\onlinecite{GorSav-PRA-06}. As the spin polarization appears in Eq.~\eqref{eq:def_n2ueg}, we use the effective spin polarization $\tilde{\zeta}$ of Eq.~\eqref{eq:def_effspin} in order to ensure $S_z$ independence. Thus, $\ntwo^{\text{UEG}}$ will depend indirectly on the on-top pair density of the CAS wave function through $\tilde{\zeta}$. where the pair-distribution function $g_0(n)$ is taken from Eq.~(46) of Ref.~\onlinecite{GorSav-PRA-06}. As the spin polarization appears in Eq.~\eqref{eq:def_n2ueg}, we use the effective spin polarization $\tilde{\zeta}$ of Eq.~\eqref{eq:def_effspin} in order to ensure $S_z$ independence. Thus, $\ntwo^{\text{UEG}}$ will depend indirectly on the on-top pair density of the CASSCF wave function through $\tilde{\zeta}$.
Another approach to approximate the exact on-top pair density consists in using directly the on-top pair density of the CAS wave function. Following the work of some of the previous authors, \cite{FerGinTou-JCP-18,GinSceTouLoo-JCP-19} we introduce the extrapolated on-top pair density Another approach to approximate the exact on-top pair density consists in using directly the on-top pair density of the CASSCF wave function. Following the work of some of the previous authors, \cite{FerGinTou-JCP-18,GinSceTouLoo-JCP-19} we introduce the extrapolated on-top pair density
\begin{equation} \begin{equation}
\label{eq:def_n2extrap} \label{eq:def_n2extrap}
\ntwoextrap(\ntwo,\mu) = \bigg( 1 + \frac{2}{\sqrt{\pi}\mu} \bigg)^{-1} \; \ntwo, \ntwoextrap(\ntwo,\mu) = \bigg( 1 + \frac{2}{\sqrt{\pi}\mu} \bigg)^{-1} \; \ntwo,
@ -713,6 +713,8 @@ Regarding the \titou{complementary basis functional}, we first perform full-vale
Also, as the frozen-core approximation is used in all our selected CI calculations, we use the corresponding valence-only \titou{complementary functionals}. Therefore, all density-like quantities exclude any contribution from the $1s$ core orbitals, and the range-separation function is taken as the one defined in Eq.~\eqref{eq:def_mur_val}. Also, as the frozen-core approximation is used in all our selected CI calculations, we use the corresponding valence-only \titou{complementary functionals}. Therefore, all density-like quantities exclude any contribution from the $1s$ core orbitals, and the range-separation function is taken as the one defined in Eq.~\eqref{eq:def_mur_val}.
Regarding the computational cost of the present approach, it should be stressed (see supplementary information) that the basis set correction represents, for all systems and basis sets studied here, a much smaller computational cost than any WFT-based calculations (CIPSI. We thus believe that this approach is a significant step towards the routine calculation of near-CBS energetic quantities in strongly correlated systems.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{table*} \begin{table*}
\label{tab:d0} \label{tab:d0}
@ -728,26 +730,26 @@ Also, as the frozen-core approximation is used in all our selected CI calculatio
%\cline{2-6} %\cline{2-6}
& &\multicolumn{4}{c}{Estimated exact:\fnm[1] 665.4} \\[0.2cm] & &\multicolumn{4}{c}{Estimated exact:\fnm[1] 665.4} \\[0.2cm]
\hline \hline
& & \tabc{FCI} & \tabc{FCI+$\pbeuegXi$} & \tabc{FCI+$\pbeontXi$} & \tabc{FCI+$\pbeontns$}\\ & & \tabc{exFCI} & \tabc{exFCI+$\pbeuegXi$} & \tabc{exFCI+$\pbeontXi$} & \tabc{exFCI+$\pbeontns$}\\
\hline \hline
\ce{C2} & aug-cc-pVDZ & 204.6 [29.5] & 218.0 [16.1] & 217.4 [16.7] & 217.0 [17.1] \\ \ce{C2} & aug-cc-pVDZ & 204.6 [29.5] & 218.0 [16.1] & 217.4 [16.7] & 217.0 [17.1] \\
& aug-cc-pVTZ & 223.4 [10.9] & 228.1 [6.0] & 228.6 [5.5] & 226.5 [5.6] \\[0.1cm] & aug-cc-pVTZ & 223.4 [10.9] & 228.1 [6.0] & 228.6 [5.5] & 226.5 [5.6] \\[0.1cm]
%\cline{2-6} %\cline{2-6}
& & \multicolumn{4}{c}{Estimated exact:\fnm[2] 234.1} \\[0.2cm] & & \multicolumn{4}{c}{Estimated exact:\fnm[2] 234.1} \\[0.2cm]
\hline \hline
& & \tabc{FCI} & \tabc{FCI+$\pbeuegXi$} & \tabc{FCI+$\pbeontXi$} & \tabc{FCI+$\pbeontns$}\\ & & \tabc{exFCI} & \tabc{exFCI+$\pbeuegXi$} & \tabc{exFCI+$\pbeontXi$} & \tabc{exFCI+$\pbeontns$}\\
\hline \hline
\ce{N2} & aug-cc-pVDZ & 321.9 [40.8] & 356.2 [6.5] & 355.5 [7.2] & 354.6 [8.1] \\ \ce{N2} & aug-cc-pVDZ & 321.9 [40.8] & 356.2 [6.5] & 355.5 [7.2] & 354.6 [8.1] \\
& aug-cc-pVTZ & 348.5 [14.2] & 361.5 [1.2] & 363.5 [-0.5] & 363.2 [-0.3] \\[0.1cm] & aug-cc-pVTZ & 348.5 [14.2] & 361.5 [1.2] & 363.5 [-0.5] & 363.2 [-0.3] \\[0.1cm]
& & \multicolumn{4}{c}{Estimated exact:\fnm[2] 362.7} \\[0.2cm] & & \multicolumn{4}{c}{Estimated exact:\fnm[2] 362.7} \\[0.2cm]
\hline \hline
& & \tabc{FCI} & \tabc{FCI+$\pbeuegXi$} & \tabc{FCI+$\pbeontXi$} & \tabc{FCI+$\pbeontns$}\\ & & \tabc{exFCI} & \tabc{exFCI+$\pbeuegXi$} & \tabc{exFCI+$\pbeontXi$} & \tabc{exFCI+$\pbeontns$}\\
\hline \hline
\ce{O2} & aug-cc-pVDZ & 171.4 [20.5] & 187.6 [4.3] & 187.6 [4.3] & 187.1 [4.8] \\ \ce{O2} & aug-cc-pVDZ & 171.4 [20.5] & 187.6 [4.3] & 187.6 [4.3] & 187.1 [4.8] \\
& aug-cc-pVTZ & 184.5 [7.4] & 190.3 [1.6] & 191.2 [0.7] & 191.0 [0.9] \\[0.1cm] & aug-cc-pVTZ & 184.5 [7.4] & 190.3 [1.6] & 191.2 [0.7] & 191.0 [0.9] \\[0.1cm]
& & \multicolumn{4}{c}{Estimated exact:\fnm[2] 191.9} \\[0.2cm] & & \multicolumn{4}{c}{Estimated exact:\fnm[2] 191.9} \\[0.2cm]
\hline \hline
& & \tabc{FCI} & \tabc{FCI+$\pbeuegXi$} & \tabc{FCI+$\pbeontXi$} & \tabc{FCI+$\pbeontns$}\\ & & \tabc{exFCI} & \tabc{exFCI+$\pbeuegXi$} & \tabc{exFCI+$\pbeontXi$} & \tabc{exFCI+$\pbeontns$}\\
\hline \hline
\ce{F2} & aug-cc-pVDZ & 49.6 [12.6] & 54.8 [7.4] & 54.9 [7.3] & 54.8 [7.4] \\ \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-pVTZ & 59.3 [2.9] & 61.2 [1.0] & 61.5 [0.7] & 61.5 [0.7] \\[0.1cm]
@ -769,9 +771,9 @@ We report in Fig.~\ref{fig:H10} the potential energy curves computed using the c
In other words, smooth potential energy surfaces are obtained with the present basis-set correction. 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. 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.
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. 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{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.
\titou{This could have significant implications for the construction of more robust families of density-functional approximations within DFT.} 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{Why can't we see the effect of dispersion in that system?}
\subsection{Dissociation of diatomics} \subsection{Dissociation of diatomics}
@ -782,7 +784,7 @@ Analyzing more carefully the performance of the different types of approximate d
\includegraphics[width=0.45\linewidth]{data/C2/DFT_avtzE_relat.eps} \includegraphics[width=0.45\linewidth]{data/C2/DFT_avtzE_relat.eps}
\includegraphics[width=0.45\linewidth]{data/C2/DFT_avtzE_relat_zoom.eps} \includegraphics[width=0.45\linewidth]{data/C2/DFT_avtzE_relat_zoom.eps}
\caption{ \caption{
Potential energy curves of the \ce{C2} molecule calculated with near-FCI and basis-set corrected near-FCI using the aug-cc-pVDZ (top) and aug-cc-pVTZ (bottom) basis sets. The estimated exact energies are from Ref.~\onlinecite{LieCle-JCP-74a}. Potential energy curves of the \ce{C2} molecule calculated with exFCI and basis-set corrected exFCI using the aug-cc-pVDZ (top) and aug-cc-pVTZ (bottom) basis sets. The estimated exact energies are based on fit of experimental data and obtained from Ref.~\onlinecite{LieCle-JCP-74a}.
\label{fig:C2}} \label{fig:C2}}
\end{figure*} \end{figure*}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@ -794,7 +796,7 @@ Analyzing more carefully the performance of the different types of approximate d
\includegraphics[width=0.45\linewidth]{data/N2/DFT_avtzE_relat.eps} \includegraphics[width=0.45\linewidth]{data/N2/DFT_avtzE_relat.eps}
\includegraphics[width=0.45\linewidth]{data/N2/DFT_avtzE_relat_zoom.eps} \includegraphics[width=0.45\linewidth]{data/N2/DFT_avtzE_relat_zoom.eps}
\caption{ \caption{
Potential energy curves of the \ce{N2} molecule calculated with near-FCI and basis-set corrected near-FCI using the aug-cc-pVDZ (top) and aug-cc-pVTZ (bottom) basis sets. The estimated exact energies are from Ref.~\onlinecite{LieCle-JCP-74a}. Potential energy curves of the \ce{N2} molecule calculated with exFCI and basis-set corrected exFCI using the aug-cc-pVDZ (top) and aug-cc-pVTZ (bottom) basis sets. The estimated exact energies are based on fit of experimental data and obtained from Ref.~\onlinecite{LieCle-JCP-74a}.
\label{fig:N2}} \label{fig:N2}}
\end{figure*} \end{figure*}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@ -806,7 +808,7 @@ Analyzing more carefully the performance of the different types of approximate d
% \includegraphics[width=0.45\linewidth]{data/O2/DFT_avtzE_relat.eps} % \includegraphics[width=0.45\linewidth]{data/O2/DFT_avtzE_relat.eps}
% \includegraphics[width=0.45\linewidth]{data/O2/DFT_avtzE_relat_zoom.eps} % \includegraphics[width=0.45\linewidth]{data/O2/DFT_avtzE_relat_zoom.eps}
\caption{ \caption{
Potential energy curves of the \ce{O2} molecule calculated with near-FCI and basis-set corrected near-FCI using the aug-cc-pVDZ (top) and \titou{aug-cc-pVTZ (bottom)} basis sets. The estimated exact energies are from Ref.~\onlinecite{LieCle-JCP-74a}. Potential energy curves of the \ce{O2} molecule calculated with exFCI and basis-set corrected exFCI using the aug-cc-pVDZ (top) and \titou{aug-cc-pVTZ (bottom)} basis sets. The estimated exact energies are based on fit of experimental data and obtained from Ref.~\onlinecite{LieCle-JCP-74a}.
\label{fig:O2}} \label{fig:O2}}
\end{figure*} \end{figure*}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@ -818,12 +820,12 @@ Analyzing more carefully the performance of the different types of approximate d
\includegraphics[width=0.45\linewidth]{data/F2/DFT_avtzE_relat.eps} \includegraphics[width=0.45\linewidth]{data/F2/DFT_avtzE_relat.eps}
\includegraphics[width=0.45\linewidth]{data/F2/DFT_avtzE_relat_zoom.eps} \includegraphics[width=0.45\linewidth]{data/F2/DFT_avtzE_relat_zoom.eps}
\caption{ \caption{
Potential energy curves of the \ce{F2} molecule calculated with near-FCI and basis-set corrected near-FCI using the aug-cc-pVDZ (top) and aug-cc-pVTZ (bottom) basis sets. The estimated exact energies are from Ref.~\onlinecite{LieCle-JCP-74a}. Potential energy curves of the \ce{F2} molecule calculated with exFCI and basis-set corrected exFCI using the aug-cc-pVDZ (top) and aug-cc-pVTZ (bottom) basis sets. The estimated exact energies are based on fit of experimental data and obtained from Ref.~\onlinecite{LieCle-JCP-74a}.
\label{fig:F2}} \label{fig:F2}}
\end{figure*} \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 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. 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. 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{C2}, \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: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}. 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}.
@ -837,7 +839,7 @@ Regarding now the performance of the basis-set correction along the whole potent
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{C2}, \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. 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{C2}, \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 \titou{CAS} wave functions leading to size-consistent energies, and to develop functionals using only $S_z$-independent density-like quantities. 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 \titou{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 \trashPFL{calculated from a multideterminant wave function} 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. 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 \trashPFL{calculated from a multideterminant wave function} 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.
\titou{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.} \titou{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.}
@ -846,13 +848,12 @@ Regarding the results of the present approach, the basis-set correction systemat
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 has a long-range correlation nature and the present approach is designed to recover only short-range correlation effects.
\PFL{I think the paragraph below should be placed WAY earlier.} %\PFL{I think the paragraph below should be placed WAY earlier.}
Finally, regarding the computational cost of the present approach, it should be stressed (see supplementary information) that it represents, for all systems and basis sets studied here, a minor computational overhead. We thus believe that this approach is a significant step towards the routine calculation of near-CBS energetic quantities in strongly correlated systems.
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