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@ -526,7 +526,6 @@ First, thanks to the properties in Eqs.~\eqref{eq:cbs_mu} and~\eqref{eq:lim_muin
Second, $\efuncdenpbe{\argebasis}$ correctly vanishes for systems with uniformly vanishing on-top pair density, such as one-electron systems and for the stretched H$_2$ molecule, Second, $\efuncdenpbe{\argebasis}$ correctly vanishes for systems with uniformly vanishing on-top pair density, such as one-electron systems and for the stretched H$_2$ molecule,
\begin{equation} \begin{equation}
\label{eq:lim_ebasis}
\lim_{n_2 \to 0} \efuncdenpbe{\argebasis} = 0. \lim_{n_2 \to 0} \efuncdenpbe{\argebasis} = 0.
\end{equation} \end{equation}
This property is doubly guaranteed by i) the choice of setting $\wbasis = +\infty$ for a vanishing pair density [see Eq.~\eqref{eq:wbasis}], which leads to $\mu(\br{}) \to \infty$ and thus a vanishing $\ecmd(\argecmd)$ [see Eq.~\eqref{eq:lim_muinf}], and ii) the fact that $\ecmd(\argecmd)$ vanishes anyway when the on-top pair density vanishes [see Eq.~\eqref{eq:lim_n2}]. This property is doubly guaranteed by i) the choice of setting $\wbasis = +\infty$ for a vanishing pair density [see Eq.~\eqref{eq:wbasis}], which leads to $\mu(\br{}) \to \infty$ and thus a vanishing $\ecmd(\argecmd)$ [see Eq.~\eqref{eq:lim_muinf}], and ii) the fact that $\ecmd(\argecmd)$ vanishes anyway when the on-top pair density vanishes [see Eq.~\eqref{eq:lim_n2}].
@ -628,7 +627,7 @@ The performance of each of these functionals is tested in the following. Note th
\includegraphics[width=0.45\linewidth]{data/H10/DFT_vqzE_relat.pdf} \includegraphics[width=0.45\linewidth]{data/H10/DFT_vqzE_relat.pdf}
\includegraphics[width=0.45\linewidth]{data/H10/DFT_vqzE_relat_zoom.pdf} \includegraphics[width=0.45\linewidth]{data/H10/DFT_vqzE_relat_zoom.pdf}
\caption{ \caption{
Potential energy curves of the H$_{10}$ chain with equally-spaced atoms calculated with MRCI+Q and basis-set corrected MRCI+Q using the cc-pVDZ (top, labelled vdz), cc-pVTZ (middle, labelled vtz), and cc-pVQZ (bottom, labelled vqz) basis sets. Potential energy curves of the H$_{10}$ chain with equally-spaced atoms calculated with MRCI+Q and basis-set corrected MRCI+Q using the cc-pVDZ (top, labelled vdz), cc-pVTZ (middle, labelled as ``vtz''), and cc-pVQZ (bottom, labelled as ``vqz'') basis sets.
The MRCI+Q energies and the estimated exact energies have been extracted from Ref.~\onlinecite{h10_prx}. The MRCI+Q energies and the estimated exact energies have been extracted from Ref.~\onlinecite{h10_prx}.
\label{fig:H10}} \label{fig:H10}}
\end{figure*} \end{figure*}
@ -638,7 +637,7 @@ The performance of each of these functionals is tested in the following. Note th
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{table*} \begin{table*}
\caption{Atomization energies $D_0$ (in mHa) and associated errors (in square brackets) with respect to the estimated exact values computed at different approximation levels with various basis sets.} \caption{Atomization energies $D_0$ (in mHa) and associated errors (in square brackets) with respect to the estimated exact values computed at different levels of theory with various basis sets.}
\begin{ruledtabular} \begin{ruledtabular}
\begin{tabular}{lrdddd} \begin{tabular}{lrdddd}
@ -696,7 +695,7 @@ The performance of each of these functionals is tested in the following. Note th
The purpose of the present paper being to investigate the performance of the density-based basis-set correction in regimes of both weak and strong correlation, we study the potential energy curves up to the dissociation limit of a \ce{H10} chain with equally-spaced atoms and the \ce{N2}, \ce{O2}, and \ce{F2} diatomics. For a given basis set, in order to compute the ground-state energy in Eq.~\eqref{eq:e0approx}, one needs an approximation to both the FCI energy $\efci$ and the basis-set correction $\efuncbasisFCI$. The purpose of the present paper being to investigate the performance of the density-based basis-set correction in regimes of both weak and strong correlation, we study the potential energy curves up to the dissociation limit of a \ce{H10} chain with equally-spaced atoms and the \ce{N2}, \ce{O2}, and \ce{F2} diatomics. For a given basis set, in order to compute the ground-state energy in Eq.~\eqref{eq:e0approx}, one needs an approximation to both the FCI energy $\efci$ and the basis-set correction $\efuncbasisFCI$.
In the case of the \ce{N2}, \ce{O2}, and \ce{F2} molecules for the aug-cc-pVDZ and aug-cc-pVTZ basis sets,~\cite{KenDunHar-JCP-92} approximations to the FCI energies are obtained using frozen-core selected CI calculations (using the CIPSI algorithm) followed by the extrapolation scheme proposed by Holmes \textit{et al.} (see Refs.~\onlinecite{HolUmrSha-JCP-17, SceGarCafLoo-JCTC-18, LooSceBloGarCafJac-JCTC-18, SceBenJacCafLoo-JCP-18, LooBogSceCafJac-JCTC-19, QP2} for more details). All these calculations are performed with the latest version of \textsc{QUANTUM PACKAGE}, \cite{QP2} and will be labeled as exFCI in the following. In the case of \ce{F2}, we also use the correlation energy extrapolated by intrinsic scaling (CEEIS) \cite{BytNagGorRue-JCP-07} as an estimation of the FCI correlation energy with the cc-pVXZ (X $=$ D, T, and Q) basis sets.~\cite{Dun-JCP-89} The estimated exact potential energy curves are obtained from experimental data \cite{LieCle-JCP-74a} for the \ce{N2} and \ce{O2} molecules, and from CEEIS calculations in the case of \ce{F2}. For all geometries and basis sets, the error with respect to the exact FCI energies are estimated to be of the order of $0.5$ mHa. For the \ce{N2}, \ce{O2}, and \ce{F2} molecules, we also performed single-point exFCI calculations in the aug-cc-pVQZ basis set at the equilibrium geometry to obtain reliable estimates of the FCI/CBS dissociation energy. In the case of the \ce{N2}, \ce{O2}, and \ce{F2} molecules for the aug-cc-pVDZ and aug-cc-pVTZ basis sets,~\cite{KenDunHar-JCP-92} approximations to the FCI energies are obtained using frozen-core selected CI calculations (using the CIPSI algorithm) followed by the extrapolation scheme proposed by Holmes \textit{et al.} (see Refs.~\onlinecite{HolUmrSha-JCP-17, SceGarCafLoo-JCTC-18, LooSceBloGarCafJac-JCTC-18, SceBenJacCafLoo-JCP-18, LooBogSceCafJac-JCTC-19, QP2} for more details). All these calculations are performed with the latest version of \textsc{QUANTUM PACKAGE}, \cite{QP2} and will be labeled as exFCI in the following. In the case of \ce{F2}, we also use the correlation energy extrapolated by intrinsic scaling (CEEIS) \cite{BytNagGorRue-JCP-07} method as an estimate of the FCI correlation energy with the cc-pVXZ (X $=$ D, T, and Q) basis sets.~\cite{Dun-JCP-89} The estimated exact potential energy curves are obtained from experimental data \cite{LieCle-JCP-74a} for the \ce{N2} and \ce{O2} molecules, and from CEEIS calculations in the case of \ce{F2}. For all geometries and basis sets, the error with respect to the exact FCI energies are estimated to be of the order of $0.5$ mHa. For the \ce{N2}, \ce{O2}, and \ce{F2} molecules, we also performed single-point exFCI calculations in the aug-cc-pVQZ basis set at the equilibrium geometry to obtain reliable estimates of the FCI/CBS dissociation energy.
In the case of the \ce{H10} chain, the approximation to the FCI energies together with the estimated exact potential energy curves are obtained from the data of Ref.~\onlinecite{h10_prx} where the authors performed MRCI+Q calculations with a minimal valence active space as reference (see below for the description of the active space). In the case of the \ce{H10} chain, the approximation to the FCI energies together with the estimated exact potential energy curves are obtained from the data of Ref.~\onlinecite{h10_prx} where the authors performed MRCI+Q calculations with a minimal valence active space as reference (see below for the description of the active space).
Regarding the complementary functional, we first perform full-valence CASSCF calculations with the GAMESS-US software~\cite{gamess} to obtain the wave function $\psibasis$. Then, all density-related quantities involved in the functional [density $n(\br{})$, effective spin polarization $\tilde{\zeta}(\br{})$, reduced density gradient $s(\br{})$, and on-top pair density $n_2(\br{})$] together with the local range-separation function $\mu(\br{})$ of Eq.~\eqref{eq:def_mur} are calculated with this full-valence CASSCF wave function. The CASSCF calculations have been performed with the following active spaces: (10e,10o) for \ce{H10}, (10e,8o) for \ce{N2}, (12e,8o) for \ce{O2}, and (14e,8o) for \ce{F2}. Regarding the complementary functional, we first perform full-valence CASSCF calculations with the GAMESS-US software~\cite{gamess} to obtain the wave function $\psibasis$. Then, all density-related quantities involved in the functional [density $n(\br{})$, effective spin polarization $\tilde{\zeta}(\br{})$, reduced density gradient $s(\br{})$, and on-top pair density $n_2(\br{})$] together with the local range-separation function $\mu(\br{})$ of Eq.~\eqref{eq:def_mur} are calculated with this full-valence CASSCF wave function. The CASSCF calculations have been performed with the following active spaces: (10e,10o) for \ce{H10}, (10e,8o) for \ce{N2}, (12e,8o) for \ce{O2}, and (14e,8o) for \ce{F2}.
@ -713,20 +712,20 @@ Regarding the computational cost of the present approach, it should be stressed
\includegraphics[width=0.45\linewidth]{data/N2/DFT_avtzE_relat.pdf} \includegraphics[width=0.45\linewidth]{data/N2/DFT_avtzE_relat.pdf}
\includegraphics[width=0.45\linewidth]{data/N2/DFT_avtzE_relat_zoom.pdf} \includegraphics[width=0.45\linewidth]{data/N2/DFT_avtzE_relat_zoom.pdf}
\caption{ \caption{
Potential energy curves of the \ce{N2} molecule calculated with exFCI and basis-set corrected exFCI using the aug-cc-pVDZ (top, labelled avdz) and aug-cc-pVTZ (bottom, labelled avtz) basis sets. The estimated exact energies are based on a fit of experimental data and obtained 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, labelled as ``avdz'') and aug-cc-pVTZ (bottom, labelled as ``avtz'') basis sets. The estimated exact energies are based on a fit of experimental data and obtained from Ref.~\onlinecite{LieCle-JCP-74a}.
\label{fig:N2}} \label{fig:N2}}
\end{figure*} \end{figure*}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{H$_{10}$ chain} \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 system). The \ce{H10} chain with equally-spaced atoms is a 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 system).
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 $\efuncbasis$. Also, no erratic behavior is found when stretching the bonds, which shows that the present procedure (\ie, the determination of the range-separation function and the definition of the functionals) is robust when reaching the strong-correlation regime. 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 $\efuncbasis$. Also, no erratic behavior is found when stretching the bonds, which shows that the present procedure (\ie, the determination of the range-separation function and the definition of the functionals) is robust when reaching the strong-correlation regime.
In other words, smooth potential energy curves are obtained with the present basis-set correction. In other words, smooth potential energy curves 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 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 [Eq.~\eqref{eq:def_n2extrap}] is preferable over the use of the UEG on-top pair density [Eq.~\eqref{eq:def_n2ueg}] which is somewhat understandable, and ii) removing the dependence 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 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 included in these functionals: i) the explicit use of the on-top pair density originating 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 somewhat understandable, and ii) removing the dependence 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. This could have significant implications for the construction of more robust families of density-functional approximations within DFT.
%Finally, the reader would have noticed that we did not report the performance of the $\pbeuegns$ functional as its performance are significantly inferior than the three other functionals. The main reason behind this comes from the fact that $\pbeuegns$ has no direct or indirect knowledge of the on-top pair density of the system. Therefore, it yields a non-zero correlation energy for the totally dissociated \ce{H10} chain even if the on-top pair density is vanishingly small. This necessary lowers the value of $D_0$. Therefore, from hereon, we simply discard the $\pbeuegns$ functional. %Finally, the reader would have noticed that we did not report the performance of the $\pbeuegns$ functional as its performance are significantly inferior than the three other functionals. The main reason behind this comes from the fact that $\pbeuegns$ has no direct or indirect knowledge of the on-top pair density of the system. Therefore, it yields a non-zero correlation energy for the totally dissociated \ce{H10} chain even if the on-top pair density is vanishingly small. This necessary lowers the value of $D_0$. Therefore, from hereon, we simply discard the $\pbeuegns$ functional.
@ -752,7 +751,7 @@ This could have significant implications for the construction of more robust fam
\includegraphics[width=0.45\linewidth]{data/O2/DFT_avtzE_relat.pdf} \includegraphics[width=0.45\linewidth]{data/O2/DFT_avtzE_relat.pdf}
\includegraphics[width=0.45\linewidth]{data/O2/DFT_avtzE_relat_zoom.pdf} \includegraphics[width=0.45\linewidth]{data/O2/DFT_avtzE_relat_zoom.pdf}
\caption{ \caption{
Potential energy curves of the \ce{O2} molecule calculated with exFCI and basis-set corrected exFCI using the aug-cc-pVDZ (top, labelled avdz) and aug-cc-pVTZ (bottom, labelled avtz) basis sets. The estimated exact energies are based on a fit of experimental data and obtained 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, labelled as ``avdz'') and aug-cc-pVTZ (bottom, labelled as ``avtz'') basis sets. The estimated exact energies are based on a fit of experimental data and obtained from Ref.~\onlinecite{LieCle-JCP-74a}.
\label{fig:O2}} \label{fig:O2}}
\end{figure*} \end{figure*}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@ -778,7 +777,7 @@ This could have significant implications for the construction of more robust fam
\includegraphics[width=0.45\linewidth]{data/F2/DFT_avtzE_relat.pdf} \includegraphics[width=0.45\linewidth]{data/F2/DFT_avtzE_relat.pdf}
\includegraphics[width=0.45\linewidth]{data/F2/DFT_avtzE_relat_zoom.pdf} \includegraphics[width=0.45\linewidth]{data/F2/DFT_avtzE_relat_zoom.pdf}
\caption{ \caption{
Potential energy curves of the \ce{F2} molecule calculated with exFCI and basis-set corrected exFCI using the aug-cc-pVDZ (top, labelled avdz), aug-cc-pVTZ (bottom, labelled avtz) basis sets. Potential energy curves of the \ce{F2} molecule calculated with exFCI and basis-set corrected exFCI using the aug-cc-pVDZ (top, labelled as ``avdz''), aug-cc-pVTZ (bottom, labelled as ``avtz'') basis sets.
The estimated exact energies are based on a fit of the non-relativistic valence-only CEEIS data extracted from Ref.~\onlinecite{BytNagGorRue-JCP-07}. The estimated exact energies are based on a fit of the non-relativistic valence-only CEEIS data extracted from Ref.~\onlinecite{BytNagGorRue-JCP-07}.
\label{fig:F2}} \label{fig:F2}}
\end{figure*} \end{figure*}
@ -786,7 +785,7 @@ This could have significant implications for the construction of more robust fam
\subsection{Dissociation of diatomics} \subsection{Dissociation of diatomics}
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 interactions in \ce{H10} play a minor role on the potential energy curve 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 bonds 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. \cite{AngDobJanGou-BOOK-20} The dispersion interactions in \ce{H10} play a minor role on the potential energy curve 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} 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 curve of \ce{F2} using the cc-pVXZ (X $=$ D, T, and Q) basis sets. The value of $D_0$ for 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 curve of \ce{F2} using the cc-pVXZ (X $=$ D, T, and Q) basis sets. The value of $D_0$ for each level of theory is reported in Table \ref{tab:d0}.
@ -866,12 +865,10 @@ The on-top pair density can be written in an orthonormal spatial orbital basis $
\end{equation} \end{equation}
with $\Gam{pq}{rs} = 2 \mel*{\wf{}{}}{ \aic{r_\downarrow}\aic{s_\uparrow}\ai{q_\uparrow}\ai{p_\downarrow}}{\wf{}{}}$. As the summations run over all orbitals in the basis set $\Bas$, $n_{2{}}(\br{})$ is invariant to orbital rotations and can thus be expressed in terms of localized orbitals. For two non-overlapping fragments $\text{A}+\text{B}$, the basis set can then be partitioned into orbitals localized on the fragment A and orbitals localized on B, \ie, $\Bas=\Bas_\text{A}\cup \Bas_\text{B}$. Therefore, we see that the on-top pair density of the supersystem $\text{A}+\text{B}$ is additively separable with $\Gam{pq}{rs} = 2 \mel*{\wf{}{}}{ \aic{r_\downarrow}\aic{s_\uparrow}\ai{q_\uparrow}\ai{p_\downarrow}}{\wf{}{}}$. As the summations run over all orbitals in the basis set $\Bas$, $n_{2{}}(\br{})$ is invariant to orbital rotations and can thus be expressed in terms of localized orbitals. For two non-overlapping fragments $\text{A}+\text{B}$, the basis set can then be partitioned into orbitals localized on the fragment A and orbitals localized on B, \ie, $\Bas=\Bas_\text{A}\cup \Bas_\text{B}$. Therefore, we see that the on-top pair density of the supersystem $\text{A}+\text{B}$ is additively separable
\begin{equation} \begin{equation}
\label{eq:def_n2}
n_{2,\text{A}+\text{B}}(\br{}) = n_{2,\text{A}}(\br{}) + n_{2,\text{B}}(\br{}), n_{2,\text{A}+\text{B}}(\br{}) = n_{2,\text{A}}(\br{}) + n_{2,\text{B}}(\br{}),
\end{equation} \end{equation}
where $n_{2,\text{X}}(\br{})$ is the on-top pair density of the fragment X where $n_{2,\text{X}}(\br{})$ is the on-top pair density of the fragment X
\begin{equation} \begin{equation}
\label{eq:def_n2}
n_{2,\text{X}}(\br{}) = \sum_{pqrs \in \Bas_\text{X}} \SO{p}{} \SO{q}{} \Gam{pq}{rs} \SO{r}{} \SO{s}{}, n_{2,\text{X}}(\br{}) = \sum_{pqrs \in \Bas_\text{X}} \SO{p}{} \SO{q}{} \Gam{pq}{rs} \SO{r}{} \SO{s}{},
\end{equation} \end{equation}
in which the elements $\Gam{pq}{rs}$ with orbital indices restricted to the fragment X are $\Gam{pq}{rs} = 2 \mel*{\wf{\text{A}+\text{B}}{}}{ \aic{r_\downarrow}\aic{s_\uparrow}\ai{q_\uparrow}\ai{p_\downarrow}}{\wf{\text{A}+\text{B}}{}} = 2 \mel*{\wf{\text{X}}{}}{ \aic{r_\downarrow}\aic{s_\uparrow}\ai{q_\uparrow}\ai{p_\downarrow}}{\wf{\text{X}}{}}$, owing to the multiplicative structure of the wave function [see Eq.~\eqref{PsiAB}]. This shows that the on-top pair density is a local intensive quantity. in which the elements $\Gam{pq}{rs}$ with orbital indices restricted to the fragment X are $\Gam{pq}{rs} = 2 \mel*{\wf{\text{A}+\text{B}}{}}{ \aic{r_\downarrow}\aic{s_\uparrow}\ai{q_\uparrow}\ai{p_\downarrow}}{\wf{\text{A}+\text{B}}{}} = 2 \mel*{\wf{\text{X}}{}}{ \aic{r_\downarrow}\aic{s_\uparrow}\ai{q_\uparrow}\ai{p_\downarrow}}{\wf{\text{X}}{}}$, owing to the multiplicative structure of the wave function [see Eq.~\eqref{PsiAB}]. This shows that the on-top pair density is a local intensive quantity.
@ -923,20 +920,19 @@ For a CASSCF wave function $\Psi$, the occupied orbitals can be partitioned into
\label{def_n2_good} \label{def_n2_good}
n_{2}(\br{}) = n_{2,\mathcal{A}}(\br{}) + n_{\mathcal{A}}(\br{}) n_{\mathcal{I}}(\br{}) + \frac{n_{\mathcal{I}}(\br{})^2}{2}, n_{2}(\br{}) = n_{2,\mathcal{A}}(\br{}) + n_{\mathcal{A}}(\br{}) n_{\mathcal{I}}(\br{}) + \frac{n_{\mathcal{I}}(\br{})^2}{2},
\end{equation} \end{equation}
where $n_{2,\mathcal{A}}(\br{})$ is the purely active part of the on-top pair density where
\begin{equation} \begin{subequations}
\begin{align}
\label{def_n2_act} \label{def_n2_act}
n_{2,\mathcal{A}}(\br{}) = \sum_{pqrs \in \mathcal{A}} \SO{p}{} \SO{q}{} \Gam{pq}{rs} \SO{r}{} \SO{s}{}, n_{2,\mathcal{A}}(\br{}) & = \sum_{pqrs \in \mathcal{A}} \SO{p}{} \SO{q}{} \Gam{pq}{rs} \SO{r}{} \SO{s}{},
\end{equation} \\
$n_{\mathcal{A}}(\br{})$ is the active part of the density n_{\mathcal{A}}(\br{}) & = \sum_{pq\, \in \mathcal{A}} \phi_p (\br{}) \phi_q (\br{})
\begin{equation}
n_{\mathcal{A}}(\br{}) = \sum_{pq\, \in \mathcal{A}} \phi_p (\br{}) \phi_q (\br{})
\mel*{\wf{}{}}{ \aic{p_\uparrow}\ai{q_\uparrow} + \aic{p_\downarrow}\ai{q_\downarrow} }{\wf{}{}}, \mel*{\wf{}{}}{ \aic{p_\uparrow}\ai{q_\uparrow} + \aic{p_\downarrow}\ai{q_\downarrow} }{\wf{}{}},
\end{equation} \\
and $n_{\mathcal{I}}(\br{})$ is the inactive part of the density n_{\mathcal{I}}(\br{}) & = 2 \sum_{p\, \in \mathcal{I}} \phi_p (\br{})^2
\begin{equation} \end{align}
n_{\mathcal{I}}(\br{}) = 2 \sum_{p\, \in \mathcal{I}} \phi_p (\br{})^2. \end{subequations}
\end{equation} are the purely active part of the on-top pair density, the active part of the density, and the inactive part of the density, respectively.
The leading computational cost is the evaluation of $n_{2,\mathcal{A}}(\br{})$ on the grid which, according to Eq.~\eqref{def_n2_act}, scales as $O(N_\text{grid} N_\mathcal{A}^4)$ where $N_{\mathcal{A}}$ is the number of active orbitals which is much smaller than the number of basis functions $N_{\Bas}$. The leading computational cost is the evaluation of $n_{2,\mathcal{A}}(\br{})$ on the grid which, according to Eq.~\eqref{def_n2_act}, scales as $O(N_\text{grid} N_\mathcal{A}^4)$ where $N_{\mathcal{A}}$ is the number of active orbitals which is much smaller than the number of basis functions $N_{\Bas}$.
\subsection{Computation of the local range-separation function} \subsection{Computation of the local range-separation function}
@ -947,16 +943,16 @@ In addition to the on-top pair density, the computation of $\mur$ needs the comp
f(\bfr{},\bfr{}) = \sum_{rs \in \Bas} V^{rs}(\bfr{}) \, \Gamma_{rs}(\bfr{}), f(\bfr{},\bfr{}) = \sum_{rs \in \Bas} V^{rs}(\bfr{}) \, \Gamma_{rs}(\bfr{}),
\end{equation} \end{equation}
where where
\begin{equation} \begin{subequations}
V^{rs}(\bfr{}) = \sum_{pq \in \Bas} V_{pq}^{rs} \phi_p(\br{}) \phi_q(\br{}), \begin{align}
\end{equation} V^{rs}(\bfr{}) & = \sum_{pq \in \Bas} V_{pq}^{rs} \phi_p(\br{}) \phi_q(\br{}),
and \\
\begin{equation} \Gamma_{rs}(\bfr{}) & = \sum_{pq \in \Bas} \Gam{rs}{pq} \phi_p(\br{}) \phi_q(\br{}) .
\Gamma_{rs}(\bfr{}) = \sum_{pq \in \Bas} \Gam{rs}{pq} \phi_p(\br{}) \phi_q(\br{}) . \end{align}
\end{equation} \end{subequations}
For a general multideterminant wave function, the computational cost of $f(\bfr{},\bfr{})$ thus scales as $O(N_\text{grid}N_{\Bas}^4)$. For a general multideterminant wave function, the computational cost of $f(\bfr{},\bfr{})$ thus scales as $O(N_\text{grid}N_{\Bas}^4)$.
In the case of a CASSCF wave function, $\Gam{rs}{pq}$ vanishes if one index $p,q,r,s$ does not belong to the set of inactive or active occupied orbitals $\mathcal{I}\cup \mathcal{A}$. Therefore, at a given grid point, the number of non-zero elements $\Gamma_{rs}(\bfr{})$ is only at most $(N_{\mathcal{I}}+N_{\mathcal{A}})^2$, which is usually much smaller than $N_{\Bas}^2$. As a consequence, one can also restrict the sum in the calculation of $f(\bfr{},\bfr{})$ In the case of a CASSCF wave function, $\Gam{rs}{pq}$ vanishes if one index $p,q,r,s$ does not belong to the set of inactive or active occupied orbitals $\mathcal{I}\cup \mathcal{A}$. Therefore, at a given grid point, the number of non-zero elements $\Gamma_{rs}(\bfr{})$ is only at most $(N_{\mathcal{I}}+N_{\mathcal{A}})^2$, which is usually much smaller than $N_{\Bas}^2$. As a consequence, one can also restrict the sum in the calculation of
\begin{equation} \begin{equation}
f(\bfr{},\bfr{}) = \sum_{rs \in \mathcal{I}\cup\mathcal{A}} V^{rs}(\bfr{}) \, \Gamma_{rs}(\bfr{}). f(\bfr{},\bfr{}) = \sum_{rs \in \mathcal{I}\cup\mathcal{A}} V^{rs}(\bfr{}) \, \Gamma_{rs}(\bfr{}).
\end{equation} \end{equation}