last changes

Revised_Manuscript/srDFT_SC.tex

@@ -151,7 +151,7 @@
 \newcommand{\ntwo}[0]{n_{2}}
 \newcommand{\ntwohf}[0]{n_2^{\text{HF}}}
 \newcommand{\ntwophi}[0]{n_2^{{\phi}}}
-\newcommand{\ntwoextrap}[0]{\mathring{n}_{2}^{\text{}}}
+\newcommand{\ntwoextrap}[0]{\mathring{n}_{2}}
 \newcommand{\ntwoextrapcas}[0]{\mathring{n}_2^{\text{}}}
 \newcommand{\mur}[0]{\mu({\bf r})}
 \newcommand{\murr}[1]{\mu({\bf r}_{#1})}
@@ -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 $\langle \mu \rangle$ 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 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.}
 \begin{ruledtabular}
 \begin{tabular}{lrccccccc}
 %\begin{tabular}{lrccccccc}
@@ -748,7 +748,7 @@ to investigate the performance of the basis-set correction in regimes of both we
 The considered systems are the \ce{H10} linear chain with equally-spaced atoms, and the \ce{N2}, \ce{O2}, and \ce{F2} diatomics.
 
 The computation of the ground-state energy in Eq.~\eqref{eq:e0approx} in a given basis set requires approximations to the FCI energy $\efci$ and to the basis-set correction $\efuncbasisFCI$. 
-For diatomics with the aug-cc-pVDZ and aug-cc-pVTZ basis sets,~\cite{KenDunHar-JCP-92} 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 detail). All these calculations are performed with the latest version of \textsc{Quantum Package}, \cite{QP2} and will be labelled 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 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 diatomics with the aug-cc-pVDZ and aug-cc-pVTZ basis sets,~\cite{KenDunHar-JCP-92} 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 detail). 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 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 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). 
 
@@ -837,27 +837,27 @@ 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 flavor 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 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
 \begin{subequations}
 \begin{gather}
  \ontopcas = \int \text{d}\br{}\, n_{2,\text{CASSCF}}(\br{}),
  \label{eq:ontopcas}
 \\
- \ontopextrap = \int \text{d}\br{}\, \ntwoextrap(n_{2,\text{CASSCF}}(\br{}),\murcas), 
+ \ontopextrap = \int \text{d}\br{}\, \mathring{n}_{2,\text{CASSCF}}(\br{}),
  \label{eq:ontopextrap}
 \end{gather}
 \end{subequations}
-where $\murcas$ 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{}), 
  \label{eq:ontopcipsi}
 \\
- \ontopextrapcipsi = \int \text{d}\br{}\, \ntwoextrap(n_{2,\text{CIPSI}}(\br{}),\murcipsi), 
+ \ontopextrapcipsi = \int \text{d}\br{}\, \mathring{n}_{2,\text{CIPSI}}(\br{}), 
  \label{eq:ontopextrapcipsi}
 \end{gather}
 \end{subequations}
-where $\murcipsi$ is the local range-separation function calculated with the CIPSI wave function. We also define the system-averaged range-separation parameters
+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,
@@ -871,7 +871,7 @@ where $n_{\text{CASSCF}}(\br{})$ and $n_{\text{CIPSI}}(\br{})$ are the CASSCF an
 }
 
 \alert{
-We report in Table \ref{tab:d1} these quantities for \ce{N2} and N in different basis sets. One can notice 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 Coulomb hole in a given basis set at near FCI level, is missing from the valence CASSCF wave function. 
+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 
 \begin{equation}
  \lim_{\basis \rightarrow \text{CBS}} \ontopextrap =  \lim_{\basis \rightarrow \text{CBS}} \ontopcas,
@@ -880,13 +880,11 @@ Also, $\ontopcipsi$ decreases in a monotonous way as the size of the basis set i
 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.
 }
 
-\manu{
-In the case of the present work, it is important to keep in mind that $\ontopextrap$ is directly linked to the basis set correction in the large $\mu$ limit, and more precisely the correlation energy (in absolute value) is a growing function of $\ontopextrap$. Therefore, the error on $\ontopextrap$ with respect to the estimated exact (here taken as $\ontopextrapcipsi$ in the aug- cc-pVQZ basis set) provides an indication on the magnitude of the error on the basis set correction for a given system and a given basis set. 
-In the aug-cc-pVQZ, for \ce{N2} $\ontopextrap - \ontopextrapcipsi = 0.120$ whereas $2\times(\ontopextrap - \ontopextrapcipsi) = 0.095$. We can then conclude that the overestimation of the on-top pair density and therefore of the basis set correction is more important on the \ce{N2} molecule at equilibrium distance than on the dissociated molecule, explaining probably the overestimation of the atomization energy. 
-To confirm such statement, we computed the basis set correction at the equilibrium geometry of \ce{N2} and the isolated N atoms using $\murcipsi$ and $\ntwoextrap(n_2^{\text{CIPSI}}(\br{},\br{}),\murcipsi)$ in the aug-cc-pVTZ and aug-cc-PVQZ basis sets, and obtained the following values for the atomization energies: 362.12 mH in aug-cc-pVTZ and ????? in the aug-cc-pVQZ, which are more accurate values than those obtained using $\murcas$ and $\ntwoextrap(n_2^{\text{CASSCF}}(\br{},\br{}),\murcas)$.  
+\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 ????? with aug-cc-pVQZ, which are indeed more accurate values than those obtained using $\murcas$ and $\mathring{n}_{2,\text{CASSCF}}(\br{})$.
 }
 
-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 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.
+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.
 We hope to report further on this in the near future.
 
 \section{Conclusion}