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@ -695,7 +695,7 @@ The performance of each of these functionals is tested in the following. Note th
\begin{table*}
{\color{red}
\caption{System-averaged on-top pair density $\langle n_2 \rangle$, extrapolated on-top pair density $\langle \mathring{n}_{2} \rangle$, and range-separation parameter $\langle \mu \rangle$ (all in atomic units) calculated with full-valence CASSCF and CIPSI wave functions (see text for details) for N$_2$ and N in the aug-cc-pVXZ basis sets (X=D,T,Q). All quantities were computed excluding all contributions from the 1s orbitals.}
\caption{System-averaged on-top pair density $\langle n_2 \rangle$, extrapolated on-top pair density $\langle \mathring{n}_{2} \rangle$, and range-separation parameter $\langle \mu \rangle$ (all in atomic units) calculated with full-valence CASSCF and CIPSI wave functions (see text for details) for \ce{N2} and \ce{N} in the aug-cc-pVXZ basis sets (X $=$ D, T, and Q). All quantities were computed within the frozen-core approximation, \ie, excluding all contributions from the 1s orbitals.}
\begin{ruledtabular}
\begin{tabular}{lrccccccc}
%\begin{tabular}{lrccccccc}
@ -708,8 +708,7 @@ The performance of each of these functionals is tested in the following. Note th
\ce{N} & aug-cc-pVDZ & 0.34464 & 0.19622 & 0.25484 & 0.14686 & 0.910 & 0.922 \\
& aug-cc-pVTZ & 0.34604 & 0.22630 & 0.22344 & 0.14828 & 1.263 & 1.299 \\
& aug-cc-pVQZ & 0.34614 & 0.24666 & 0.21224 & 0.15164 & 1.601 & 1.653 \\[0.1cm]
\hline
& aug-cc-pVQZ & 0.34614 & 0.24666 & 0.21224 & 0.15164 & 1.601 & 1.653 \\
\end{tabular}
\end{ruledtabular}
\label{tab:d1}
@ -745,7 +744,7 @@ For diatomics with the aug-cc-pVDZ and aug-cc-pVTZ basis sets,~\cite{KenDunHar-J
For the three diatomics, we performed an additional exFCI calculation with 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).
\alert{We note that, even though we use near-FCI energies in this work, the DFT-based basis-set correction could also be applied to any approximation to FCI such as multireference perturbation theory, similarly to what was done for weakly correlated systems for which the basis-set correction was applied to CCSD(T) calculations~\cite{LooPraSceTouGin-JCPL-19}.}
\alert{We note that, even though we use near-FCI energies in this work, the DFT-based basis-set correction could also be applied to any approximation to FCI such as multireference perturbation theory, similarly to what was done for weakly correlated systems for which the basis-set correction was applied to CCSD(T) calculations. \cite{LooPraSceTouGin-JCPL-19}}
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 $\mu(\br{})$ are calculated with this full-valence CASSCF wave function. The CASSCF calculations are 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}. We note that, instead of using CASSCF wave functions for $\psibasis$, one could of course use the same selected-CI wave functions used for calculating the energy but the calculations of $n_2(\br{})$ and $\mu(\br{})$ would then be more costly.
@ -830,7 +829,7 @@ Just as in \ce{H10}, the accuracy of the atomization energies is globally improv
It should be also noticed that when reaching the aug-cc-pVQZ basis set for \ce{N2}, the accuracy of the atomization energy slightly deteriorates for the $\pbeontXi$ and $\pbeontns$ functionals, but it remains nevertheless more accurate than the estimated FCI atomization energy and very close to chemical accuracy.
\alert{
The overestimation of the atomization energy with the basis-set correction seen for \ce{N2} in large basis sets reveals an unbalanced treatment between the molecule and the atom in favor of the molecular system. Since the integral over $\br{}$ of the on-top pair density $n_2(\br{})$ is proportional to the short-range correlation energy in the large-$\mu$ limit~\cite{PazMorGorBac-PRB-06,FerGinTou-JCP-18} [see Eq. \eqref{eq:lim_mularge}], the accuracy of a given approximation of the exact on-top pair density will have a direct influence on the accuracy of the related basis-set correction energy $\bar{E}^\Bas$. To quantify the quality of different flavors of on-top pair densities for a given system and a given basis set $\basis$, we define the system-averaged CASSCF on-top pair density and extrapolated on-top pair density
The overestimation of the basis-set-corrected atomization energy observed for \ce{N2} in large basis sets reveals an unbalanced treatment between the molecule and the atom in favor of the molecular system. Since the integral over $\br{}$ of the on-top pair density $\n{2}{}(\br{})$ is proportional to the short-range correlation energy in the large-$\mu$ limit \cite{PazMorGorBac-PRB-06,FerGinTou-JCP-18} [see Eq.~\eqref{eq:lim_mularge}], the accuracy of a given approximation of the exact on-top pair density will have a direct influence on the accuracy of the related basis-set correction energy $\bar{E}^\Bas$. To quantify the quality of different flavors of on-top pair densities for a given system and a given basis set $\basis$, we define the system-averaged CASSCF on-top pair density and extrapolated on-top pair density
\begin{subequations}
\begin{gather}
\ontopcas = \int \text{d}\br{}\, n_{2,\text{CASSCF}}(\br{}),
@ -840,41 +839,41 @@ The overestimation of the atomization energy with the basis-set correction seen
\label{eq:ontopextrap}
\end{gather}
\end{subequations}
where $\mathring{n}_{2,\text{CASSCF}}(\br{})=\ntwoextrap(n_{2,\text{CASSCF}}(\br{}),\murcas)$ [see Eq. \eqref{eq:def_n2extrap}] and $\murcas$ is the local range-separation function calculated with the CASSCF wave function, and similarly the system-averaged CIPSI on-top pair density and extrapolated on-top pair density
where $\mathring{n}_{2,\text{CASSCF}}(\br{})=\ntwoextrap(n_{2,\text{CASSCF}}(\br{}),\murcas)$ [see Eq.~\eqref{eq:def_n2extrap}] and $\murcas$ is the local range-separation function calculated with the CASSCF wave function, and similarly the system-averaged CIPSI on-top pair density and extrapolated on-top pair density
\begin{subequations}
\begin{gather}
\ontopcipsi = \int \text{d}\br{}\,n_{2,\text{CIPSI}}(\br{}),
\begin{align}
\ontopcipsi & = \int \text{d}\br{}\,n_{2,\text{CIPSI}}(\br{}),
\label{eq:ontopcipsi}
\\
\ontopextrapcipsi = \int \text{d}\br{}\, \mathring{n}_{2,\text{CIPSI}}(\br{}),
\ontopextrapcipsi & = \int \text{d}\br{}\, \mathring{n}_{2,\text{CIPSI}}(\br{}),
\label{eq:ontopextrapcipsi}
\end{gather}
\end{align}
\end{subequations}
where $\mathring{n}_{2,\text{CISPI}}(\br{})=\ntwoextrap(n_{2,\text{CIPSI}}(\br{}),\murcipsi)$ and $\murcipsi$ is the local range-separation function calculated with the CIPSI wave function. We also define the system-averaged range-separation parameters
\begin{subequations}
\begin{gather}
\muaverage = \frac{1}{N}\int \text{d}\br{}\,n_{\text{CASSCF}}(\br{}) \,\, \murcas,
\begin{align}
\muaverage & = \frac{1}{N}\int \text{d}\br{}\,n_{\text{CASSCF}}(\br{}) \,\, \murcas,
\label{eq:muaverage}
\\
\muaveragecipsi = \frac{1}{N}\int \text{d}\br{}\,n_{\text{CIPSI}}(\br{}) \,\, \murcipsi,
\muaveragecipsi & = \frac{1}{N}\int \text{d}\br{}\,n_{\text{CIPSI}}(\br{}) \,\, \murcipsi,
\label{eq:muaveragecipsi}
\end{gather}
\end{align}
\end{subequations}
where $n_{\text{CASSCF}}(\br{})$ and $n_{\text{CIPSI}}(\br{})$ are the CASSCF and CIPSI densities, respectively. All the CIPSI quantities have been calculated with the largest variational wave function computed in the CIPSI calculation with a given basis, which contains here at least $10^7$ Slater determinants. In particular, $\murcipsi$ has been calculated from Eqs. \eqref{eq:def_mur_val}-\eqref{eq:twordm_val} with the opposite-spin two-body density matrix $\Gam{pq}{rs}$ of the largest variational CIPSI wave function for a given basis. All quantities in Eqs. \eqref{eq:ontopcas}-\eqref{eq:muaverage} were computed excluding all contributions from the 1s orbitals, \ie, they are ``valence-only'' quantities.
where $n_{\text{CASSCF}}(\br{})$ and $n_{\text{CIPSI}}(\br{})$ are the CASSCF and CIPSI densities, respectively. All the CIPSI quantities have been calculated with the largest variational wave function computed in the CIPSI calculation with a given basis, which contains here at least $10^7$ Slater determinants. In particular, $\murcipsi$ has been calculated from Eqs.~\eqref{eq:def_mur_val}--\eqref{eq:twordm_val} with the opposite-spin two-body density matrix $\Gam{pq}{rs}$ of the largest variational CIPSI wave function for a given basis. All quantities in Eqs.~\eqref{eq:ontopcas}--\eqref{eq:muaverage} were computed excluding all contributions from the 1s orbitals, \ie, they are ``valence-only'' quantities.
}
\alert{
We report in Table \ref{tab:d1} these quantities for \ce{N2} and N in different basis sets. One notices that the system-averaged on-top pair density at the CIPSI level $\ontopcipsi$ is systematically lower than that at the CASSCF level $\ontopcas$, which is expected since short-range correlation, digging the correlation hole in a given basis set at near FCI level, is missing from the valence CASSCF wave function.
Also, $\ontopcipsi$ decreases in a monotonous way as the size of the basis set increases, leading to roughly a $20\%$ decrease from the aug-cc-pVDZ to the aug-cc-pVQZ basis sets, whereas $\ontopcas$ is almost constant with respect to the basis set. Regarding the extrapolated on-top pair densities, $\ontopextrap$ and $\ontopextrapcipsi$, it is interesting to notice that they are substantially lower than their non-extrapolated counterparts, $\ontopcas$ and $\ontopcipsi$. Nevertheless, the behaviors of $\ontopextrap$ and $\ontopextrapcipsi$ are qualitatively different: $\ontopextrap$ clearly increases when enlarging the basis set whereas $\ontopextrapcipsi$ remains almost constant. More precisely, in the case of \ce{N2}, the value of $\ontopextrap$ increases by about 30$\%$ from the aug-cc-pVDZ to the aug-cc-pVQZ basis sets, whereas the value of $\ontopextrapcipsi$ only fluctuates within 5$\%$ with the same basis sets. The behavior of $\ontopextrap$ can be understood by noticing that i) the value of $\murcas$ globally increases when increasing the size of the basis set (as evidenced by $\muaverage$), and ii) $\lim_{\mu \rightarrow \infty} \ntwoextrap(n_2,\mu) = n_2$ [see Eq. \eqref{eq:def_n2extrap}]. Therefore, in the CBS limit, $\murcas \rightarrow \infty$ and one obtains
We report in Table \ref{tab:d1} these quantities for \ce{N2} and \ce{N} for various basis sets. One notices that the system-averaged on-top pair density at the CIPSI level $\ontopcipsi$ is systematically lower than its CASSCF analogue $\ontopcas$, which is expected since short-range correlation, \ie, digging the correlation hole in a given basis set at near FCI level, is missing from the valence CASSCF wave function.
Also, $\ontopcipsi$ decreases in a monotonous way as the size of the basis set increases, leading to roughly a $20\%$ decrease from the aug-cc-pVDZ to the aug-cc-pVQZ basis sets, whereas $\ontopcas$ is almost constant with respect to the basis set. Regarding the extrapolated on-top pair densities, $\ontopextrap$ and $\ontopextrapcipsi$, it is interesting to notice that they are substantially lower than their non-extrapolated counterparts, $\ontopcas$ and $\ontopcipsi$. Nevertheless, the behaviors of $\ontopextrap$ and $\ontopextrapcipsi$ are qualitatively different: $\ontopextrap$ clearly increases when enlarging the basis set whereas $\ontopextrapcipsi$ remains almost constant. More precisely, in the case of \ce{N2}, the value of $\ontopextrap$ increases by about 30$\%$ from the aug-cc-pVDZ to the aug-cc-pVQZ basis sets, whereas the value of $\ontopextrapcipsi$ only fluctuates within 5$\%$ for the same basis sets. The behavior of $\ontopextrap$ can be understood by noticing that i) the value of $\murcas$ globally increases when enlarging the basis set (as evidenced by $\muaverage$), and ii) $\lim_{\mu \rightarrow \infty} \ntwoextrap(n_2,\mu) = n_2$ [see Eq.~\eqref{eq:def_n2extrap}]. Therefore, in the CBS limit, $\murcas \rightarrow \infty$ and one obtains
\begin{equation}
\lim_{\basis \rightarrow \text{CBS}} \ontopextrap = \lim_{\basis \rightarrow \text{CBS}} \ontopcas,
\end{equation}
\ie, $\ontopextrap$ must increase with the size of the basis set $\basis$ to eventually converge to $\lim_{\basis \rightarrow \text{CBS}} \ontopcas$, the latter limit being essentially reached with the present basis sets.
On the other hand, the stability of $\ontopextrapcipsi$ with respect to the basis set is quite remarkable and must come from the fact that i) $\ontopcipsi$ is a good approximation to the corresponding FCI value within the considered basis sets, and ii) the extrapolation formula in Eq. \eqref{eq:def_n2extrap} together with the choice of $\murcipsi$ are quantitatively correct. Therefore, we expect the calculated values of $\ontopextrapcipsi$ to be nearly converged with respect to the basis set, and we will take the value of $\ontopextrapcipsi$ in the aug-cc-pVQZ basis set as an estimate of the exact system-averaged on-top pair density.
On the other hand, the stability of $\ontopextrapcipsi$ with respect to the basis set is quite remarkable and must come from the fact that i) $\ontopcipsi$ is a good approximation to the corresponding FCI value within the considered basis sets, and ii) the extrapolation formula in Eq.~\eqref{eq:def_n2extrap} together with the choice of $\murcipsi$ are quantitatively correct. Therefore, we expect the calculated values of $\ontopextrapcipsi$ to be nearly converged with respect to the basis set, and we will take the value of $\ontopextrapcipsi$ in the aug-cc-pVQZ basis set as an estimate of the exact system-averaged on-top pair density.
}
\alert{
For the present work, it is important to keep in mind that it is $\ontopextrap$ which directly determines the basis-set correction in the large-$\mu$ limit, and more precisely the basis-set correction correlation energy (in absolute value) is an increasing function of $\ontopextrap$. Therefore, the error on $\ontopextrap$ with respect to the estimated exact system-averaged on-top pair density provides an indication of the error made on the basis-set correction for a given system and a given basis set. With the aug-cc-pVQZ basis set, we have the error $\ontopextrap - \ontopextrapcipsi = 0.120$ for the \ce{N2} molecule, whereas we have the error $2(\ontopextrap - \ontopextrapcipsi) = 0.095$ for the two isolated N atoms. We can then conclude that the overestimation of the system-averaged on-top pair density, and therefore of the basis-set correction, is more important for the \ce{N2} molecule at equilibrium distance than for the isolated N atoms, explaining probably the observed overestimation of the atomization energy. To confirm this statement, we computed the basis-set correction for the \ce{N2} molecule at equilibrium distance and for the isolated N atoms using $\murcipsi$ and $\mathring{n}_{2,\text{CIPSI}}(\br{})$ with the aug-cc-pVTZ and aug-cc-pVQZ basis sets, and obtained the following values for the atomization energies: 362.12 mH with aug-cc-pVTZ and 362.15 with aug-cc-pVQZ, which are indeed more accurate values than those obtained using $\murcas$ and $\mathring{n}_{2,\text{CASSCF}}(\br{})$.
For the present work, it is important to keep in mind that $\ontopextrap$ directly determines the basis-set correction in the large-$\mu$ limit. More precisely, the correlation energy contribution associated with the basis-set correction is (in absolute value) an increasing function of $\ontopextrap$. Therefore, the error on $\ontopextrap$ with respect to the estimated exact system-averaged on-top pair density provides an indication of the error made by the basis-set correction for a given system and basis set. With the aug-cc-pVQZ basis set, we have $\ontopextrap - \ontopextrapcipsi = 0.120$ for the \ce{N2} molecule, while $2(\ontopextrap - \ontopextrapcipsi) = 0.095$ for two isolated \ce{N} atoms. We can then conclude that the overestimation of the system-averaged on-top pair density, and therefore of the basis-set correction, is more important for the \ce{N2} molecule at equilibrium distance than for the isolated \ce{N} atoms. This probably explains the observed overestimation of the atomization energy. To confirm this statement, we computed the basis-set correction for both the \ce{N2} molecule at equilibrium distance and the isolated atoms using $\murcipsi$ and $\mathring{n}_{2,\text{CIPSI}}(\br{})$ with the aug-cc-pVTZ and aug-cc-pVQZ basis sets. We obtained the following values for the atomization energies: $362.12$ mH with aug-cc-pVTZ and $362.15$ mH with aug-cc-pVQZ, which are indeed more accurate values than those obtained using $\murcas$ and $\mathring{n}_{2,\text{CASSCF}}(\br{})$.
}
Finally, regarding now the performance of the basis-set correction along the whole potential energy curves reported in Figs.~\ref{fig:N2}, \ref{fig:O2}, and \ref{fig:F2}, it is interesting to notice that it fails to provide a noticeable improvement far from the equilibrium geometry. Acknowledging that the weak-correlation effects in these regions are dominated by dispersion interactions which are long-range effects, the failure of the present approximations for the complementary functional can be understood easily. Indeed, the whole scheme designed here is based on the physics of correlation near the electron-electron coalescence point: the local range-separation function $\mu(\br{})$ is based on the value of the effective electron-electron interaction at coalescence and the ECMD functionals are suited for short-range correlation effects. Therefore, the failure of the present basis-set correction to describe dispersion interactions is theoretically expected.
@ -939,7 +938,7 @@ where the left-hand-side quantities are for the supersystem and the right-hand-s
\ket*{\wf{\text{A}+\text{B}}{}} = \ket*{\wf{\text{A}}{}} \otimes \ket*{\wf{\text{B}}{}},
\label{PsiAB}
\end{equation}
where $\otimes$ is the antisymmetric tensor product. In this case, it is easy to shown that Eqs.~(\ref{nAB})-(\ref{sAB}) are valid, as well known, and it remains to show that Eqs.~(\ref{n2AB}) and~(\ref{muAB}) are also valid. Before showing this, we note that even though we do not explicitly consider the case of degeneracies, the lack of size consistency which could arise from spin-multiplet degeneracies can be avoided by the same strategy used for imposing the energy independence on $S_z$, \ie, by using the effective spin polarization $\tilde{\zeta}(n(\br{}),n_{2}(\br{}))$ or a zero spin polarization $\zeta(\br{}) = 0$. Moreover, \alert{for the systems treated in this work}, the lack of size consistency which could arise from spatial degeneracies (coming from atomic $p$ states) can also be avoided by selecting the \alert{same state} in the supersystem and in the isolated fragment. \alert{For example, for the F$_2$ molecule, the CASSCF wave function dissociates into the atomic configuration $\text{p}_\text{x}^2 \text{p}_\text{y}^2 \text{p}_\text{z}^1$ for each fragment, and we thus choose the same configuration for the calculation of the isolated atom. The same argument applies to the N$_2$ and O$_2$ molecules. For other systems, it may not be always possible to do so.}
where $\otimes$ is the antisymmetric tensor product. In this case, it is easy to shown that Eqs.~(\ref{nAB})-(\ref{sAB}) are valid, as well known, and it remains to show that Eqs.~(\ref{n2AB}) and~(\ref{muAB}) are also valid. Before showing this, we note that even though we do not explicitly consider the case of degeneracies, the lack of size consistency which could arise from spin-multiplet degeneracies can be avoided by the same strategy used for imposing the energy independence on $S_z$, \ie, by using the effective spin polarization $\tilde{\zeta}(n(\br{}),n_{2}(\br{}))$ or a zero spin polarization $\zeta(\br{}) = 0$. Moreover, \alert{for the systems treated in this work}, the lack of size consistency which could arise from spatial degeneracies (coming from atomic $p$ states) can also be avoided by selecting the \alert{same state} in the supersystem and in the isolated fragment. \alert{For example, for the \ce{F2} molecule, the CASSCF wave function dissociates into the atomic configuration $\text{p}_\text{x}^2 \text{p}_\text{y}^2 \text{p}_\text{z}^1$ for each fragment, and we thus choose the same configuration for the calculation of the isolated atom. The same argument applies to the \ce{N2} and \ce{O2} molecules. For other systems, it may not be always possible to do so.}
\subsection{Intensivity of the on-top pair density and the local range-separation function}