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Julien Toulouse 2020-03-27 17:51:46 +01:00
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Revised_Manuscript/srDFT_SC.tex
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@ -144,7 +144,9 @@
\newcommand{\twodm}[4]{\mel{\Psi}{\psixc{#4}\psixc{#3} \psix{#2}\psix{#1}}{\Psi}} \newcommand{\twodm}[4]{\mel{\Psi}{\psixc{#4}\psixc{#3} \psix{#2}\psix{#1}}{\Psi}}
\newcommand{\murpsi}[0]{\mu_{\wf{}{\Bas}}({\bf r})} \newcommand{\murpsi}[0]{\mu_{\wf{}{\Bas}}({\bf r})}
\newcommand{\murcas}[0]{\mu_{\text{CASSCF}}({\bf r})} \newcommand{\murcas}[0]{\mu_{\text{CASSCF}}({\bf r})}
\newcommand{\mucas}[0]{\mu_{\text{CASSCF}}}
\newcommand{\murcipsi}[0]{\mu_{\text{CIPSI}}({\bf r})} \newcommand{\murcipsi}[0]{\mu_{\text{CIPSI}}({\bf r})}
\newcommand{\mucipsi}[0]{\mu_{\text{CIPSI}}}
\newcommand{\murpsibas}[0]{\mu_{\wf{}{\Bas}}({\bf r})} \newcommand{\murpsibas}[0]{\mu_{\wf{}{\Bas}}({\bf r})}
\newcommand{\ntwo}[0]{n_{2}} \newcommand{\ntwo}[0]{n_{2}}
\newcommand{\ntwohf}[0]{n_2^{\text{HF}}} \newcommand{\ntwohf}[0]{n_2^{\text{HF}}}
@ -269,12 +271,12 @@
\newcommand{\dbr}[1]{d\br{#1}} \newcommand{\dbr}[1]{d\br{#1}}
\newcommand{\PBEspin}{PBEspin} \newcommand{\PBEspin}{PBEspin}
\newcommand{\PBEueg}{PBE-UEG-{$\tilde{\zeta}$}} \newcommand{\PBEueg}{PBE-UEG-{$\tilde{\zeta}$}}
\newcommand{\ontopcas}{\langle n_2^{\text{CAS}}(\br{},\br{}) \rangle} \newcommand{\ontopcas}{\langle n_{2,\text{CASSCF}} \rangle}
\newcommand{\ontopextrap}{\langle \mathring{n}_{2}^{\text{CAS}}(\br{},\br{}) \rangle} \newcommand{\ontopextrap}{\langle \mathring{n}_{2,\text{CASSCF}} \rangle}
\newcommand{\ontopextrapcipsi}{\langle \mathring{n}_{2}^{\text{CIPSI}}(\br{},\br{}) \rangle} \newcommand{\ontopextrapcipsi}{\langle \mathring{n}_{2,\text{CIPSI}} \rangle}
\newcommand{\ontopcipsi}{\langle n_2^{\text{CIPSI}}(\br{},\br{}) \rangle} \newcommand{\ontopcipsi}{\langle n_{2,\text{CIPSI}} \rangle}
\newcommand{\muaverage}{\langle \murcas \rangle} \newcommand{\muaverage}{\langle \mucas \rangle}
\newcommand{\muaveragecipsi}{\langle \murcipsi \rangle} \newcommand{\muaveragecipsi}{\langle \mucipsi \rangle}
\newcommand{\largemu}{E_{c,md}^{\mu \rightarrow \infty}} \newcommand{\largemu}{E_{c,md}^{\mu \rightarrow \infty}}
\newcommand{\LCT}{Laboratoire de Chimie Th\'eorique (UMR 7616), Sorbonne Universit\'e, CNRS, Paris, France} \newcommand{\LCT}{Laboratoire de Chimie Th\'eorique (UMR 7616), Sorbonne Universit\'e, CNRS, Paris, France}
@ -468,7 +470,7 @@ is the valence-only effective interaction and
= \sum_{pqrs \in \BasFC} \SO{p}{1} \SO{q}{2} \Gam{pq}{rs} \SO{r}{1} \SO{s}{2}. = \sum_{pqrs \in \BasFC} \SO{p}{1} \SO{q}{2} \Gam{pq}{rs} \SO{r}{1} \SO{s}{2}.
\end{gather} \end{gather}
One would note the restrictions of the sums to the set $\BasFC$ in Eqs.~\eqref{eq:fbasis_val} and \eqref{eq:twordm_val}. One would note the restrictions of the sums to the set $\BasFC$ in Eqs.~\eqref{eq:fbasis_val} and \eqref{eq:twordm_val}.
It is also noteworthy that, with the present definition, $\wbasisval$ still tends to the usual Coulomb interaction as $\Bas \to \CBS$. It is also noteworthy that, with the present definition, $\wbasisval$ still tends to the usual Coulomb interaction as $\Bas \to \CBS$. \alert{For simplicity, we will drop the indication ``val'' in the notation for the rest of the paper.}
\subsection{General form of the complementary functional} \subsection{General form of the complementary functional}
\label{sec:functional} \label{sec:functional}
@ -692,6 +694,7 @@ The performance of each of these functionals is tested in the following. Note th
\begin{table*} \begin{table*}
{\color{red}
\caption{Integral of the on-top pair density in real space at various levels of theory (see text for details) for N$_2$ and N in the aug-cc-pVXZ basis sets (X=D,T,Q).} \caption{Integral of the on-top pair density in real space at various levels of theory (see text for details) for N$_2$ and N in the aug-cc-pVXZ basis sets (X=D,T,Q).}
\begin{ruledtabular} \begin{ruledtabular}
\begin{tabular}{lrccccccc} \begin{tabular}{lrccccccc}
@ -714,10 +717,10 @@ The performance of each of these functionals is tested in the following. Note th
%\ce{O} & aug-cc-pVDZ & 0.51391 & 0.31497 & 0.41604 & & 1.080 & 1.080 \\ %\ce{O} & aug-cc-pVDZ & 0.51391 & 0.31497 & 0.41604 & & 1.080 & 1.080 \\
% & aug-cc-pVTZ & 0.51607 & 0.36905 & 0.37689 & & 1.499 & 1.499 \\ % & aug-cc-pVTZ & 0.51607 & 0.36905 & 0.37689 & & 1.499 & 1.499 \\
% & aug-cc-pVQZ & 0.51628 & 0.38335 & 0.36249 & & 1.924 & 1.924 \\[0.1cm] % & aug-cc-pVQZ & 0.51628 & 0.38335 & 0.36249 & & 1.924 & 1.924 \\[0.1cm]
\hline
\end{tabular} \end{tabular}
\end{ruledtabular} \end{ruledtabular}
\label{tab:d1} \label{tab:d1}
}
\end{table*} \end{table*}
%& \tabc{$\largemu$} %& \tabc{$\largemu$}
@ -833,38 +836,39 @@ We report in Figs.~\ref{fig:N2}, \ref{fig:O2}, and \ref{fig:F2} the potential en
Just as in \ce{H10}, the accuracy of the atomization energies is globally improved by adding the basis-set correction and it is remarkable that $\pbeontXi$ and $\pbeontns$ provide again very similar results. The latter observation confirms that the dependence on the on-top pair density allows one to remove the dependence of any kind of spin polarization for a quite wide range of covalent bonds and also for an open-shell system like \ce{O2}. More quantitatively, an error below 1.0 mHa compared to the estimated exact valence-only atomization energy is found for \ce{N2}, \ce{O2}, and \ce{F2} with the aug-cc-pVTZ basis set using the $\pbeontns$ functional, whereas such a feat is far from being reached within the same basis set at the near-FCI level. In the case of \ce{F2} it is clear that the addition of diffuse functions in the double- and triple-$\zeta$ basis sets strongly improves the accuracy of the results, which could have been anticipated due to the strong breathing-orbital effect induced by the ionic valence-bond forms in this molecule. \cite{HibHumByrLen-JCP-94} Just as in \ce{H10}, the accuracy of the atomization energies is globally improved by adding the basis-set correction and it is remarkable that $\pbeontXi$ and $\pbeontns$ provide again very similar results. The latter observation confirms that the dependence on the on-top pair density allows one to remove the dependence of any kind of spin polarization for a quite wide range of covalent bonds and also for an open-shell system like \ce{O2}. More quantitatively, an error below 1.0 mHa compared to the estimated exact valence-only atomization energy is found for \ce{N2}, \ce{O2}, and \ce{F2} with the aug-cc-pVTZ basis set using the $\pbeontns$ functional, whereas such a feat is far from being reached within the same basis set at the near-FCI level. In the case of \ce{F2} it is clear that the addition of diffuse functions in the double- and triple-$\zeta$ basis sets strongly improves the accuracy of the results, which could have been anticipated due to the strong breathing-orbital effect induced by the ionic valence-bond forms in this molecule. \cite{HibHumByrLen-JCP-94}
It should be also noticed that when reaching the aug-cc-pVQZ basis set for \ce{N2}, the 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. 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.
\manu{ \alert{
The overestimation of the atomization energy appearing for \ce{N2} in large basis sets reveals a kind of unbalanced treatment between the molecule and atoms in favour of the molecular system. 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
As the integral of the exact on-top pair density is proportional to the 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 to the exact on-top pair density will have a direct influence on the accuracy of the related correlation energy. \begin{subequations}
To quantify the quality of various flavour of on-top pair densities for a given system and a given basis set $\basis$, we define the following quantities \begin{gather}
\begin{equation} \ontopcas = \int \text{d}\br{}\, n_{2,\text{CASSCF}}(\br{}),
\label{eq:ontopcas} \label{eq:ontopcas}
\ontopcas = \int \text{d}\br{}\, n_2^{\text{CASSCF}}(\br{},\br{}), \\
\end{equation} \ontopextrap = \int \text{d}\br{}\, \ntwoextrap(n_{2,\text{CASSCF}}(\br{}),\murcas),
\begin{equation}
\label{eq:ontopextrap} \label{eq:ontopextrap}
\ontopextrap = \int \text{d}\br{}\, \ntwoextrap(n_2^{\text{CASSCF}}(\br{},\br{}),\murcas), \end{gather}
\end{equation} \end{subequations}
\begin{equation} 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
\begin{subequations}
\begin{gather}
\ontopcipsi = \int \text{d}\br{}\,n_{2,\text{CIPSI}}(\br{}),
\label{eq:ontopcipsi} \label{eq:ontopcipsi}
\ontopcipsi = \int \text{d}\br{}\,n_2^{\text{CIPSI}}(\br{},\br{}), \\
\end{equation} \ontopextrapcipsi = \int \text{d}\br{}\, \ntwoextrap(n_{2,\text{CIPSI}}(\br{}),\murcipsi),
\begin{equation}
\label{eq:ontopextrapcipsi} \label{eq:ontopextrapcipsi}
\ontopextrapcipsi = \int \text{d}\br{}\, \ntwoextrap(n_2^{\text{CIPSI}}(\br{},\br{}),\murcipsi), \end{gather}
\end{equation} \end{subequations}
\begin{equation} where $\murcipsi$ 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,
\label{eq:muaverage} \label{eq:muaverage}
\muaverage = \frac{1}{N_{e}}\int \text{d}\br{}\,n^{\text{CASSCF}}(\br{}) \,\, \murcas \\
\end{equation} \muaveragecipsi = \frac{1}{N}\int \text{d}\br{}\,n_{\text{CIPSI}}(\br{}) \,\, \murcipsi,
\begin{equation}
\label{eq:muaveragecipsi} \label{eq:muaveragecipsi}
\muaveragecipsi = \frac{1}{N_{e}}\int \text{d}\br{}\,n^{\text{CIPSI}}(\br{}) \,\, \murcipsi \end{gather}
\end{equation} \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.
} }
\manu{
The quantity $n_2^{\text{CIPSI}}(\br{},\br{})$ is the on-top pair density of the largest variational wave function for a given CIPSI calculation in a given basis, which contains here at leas $10^7$ Slater determinants. The quantity $\murcipsi$ is the $\murpsi$ obtained with the definition of Eq. \eqref{eq:def_mur} with the two-body tensor $\Gam{pq}{rs}$ and on-top pair density $\twodmrdiagpsi$ associated with the largest variational CIPSI wave function for a given basis.
All quantities from Eqs. \eqref{eq:ontopcas} to \eqref{eq:muaverage} were computed excluding all contributions from the $1s$ orbitals. }
\manu{ \manu{
We report in Table \ref{tab:d1} these quantities for N and \ce{N2} in different basis sets. We report in Table \ref{tab:d1} these quantities for N and \ce{N2} in different basis sets.