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JPCL_revision/G2-srDFT.tex
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@ -10,7 +10,7 @@
\definecolor{darkgreen}{HTML}{009900} \definecolor{darkgreen}{HTML}{009900}
\usepackage[normalem]{ulem} \usepackage[normalem]{ulem}
\newcommand{\titou}[1]{\textcolor{red}{#1}} \newcommand{\titou}[1]{\textcolor{red}{#1}}
\newcommand{\juju}[1]{\textcolor{purple}{#1}} \newcommand{\jt}[1]{\textcolor{purple}{#1}}
\newcommand{\manu}[1]{\textcolor{darkgreen}{#1}} \newcommand{\manu}[1]{\textcolor{darkgreen}{#1}}
\newcommand{\toto}[1]{\textcolor{brown}{#1}} \newcommand{\toto}[1]{\textcolor{brown}{#1}}
\newcommand{\trashPFL}[1]{\textcolor{red}{\sout{#1}}} \newcommand{\trashPFL}[1]{\textcolor{red}{\sout{#1}}}
@ -197,8 +197,11 @@ We refer the interested reader to Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18} fo
%Let us assume we have both the energy $\E{\modY}{\Bas}$ and density $\n{\modZ}{\Bas}$ of a $\Ne$-electron system described by two methods $\modY$ and $\modZ$ (potentially identical) in an incomplete basis set $\Bas$. %Let us assume we have both the energy $\E{\modY}{\Bas}$ and density $\n{\modZ}{\Bas}$ of a $\Ne$-electron system described by two methods $\modY$ and $\modZ$ (potentially identical) in an incomplete basis set $\Bas$.
%According to Eq.~(15) of Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, assuming that $\E{\modY}{\Bas}$ and $\n{\modZ}{\Bas}$ are reasonable approximations of the FCI energy and density within $\Bas$, the exact ground state energy $\E{}{}$ may be approximated as %According to Eq.~(15) of Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, assuming that $\E{\modY}{\Bas}$ and $\n{\modZ}{\Bas}$ are reasonable approximations of the FCI energy and density within $\Bas$, the exact ground state energy $\E{}{}$ may be approximated as
Let us assume we have both the energy \titou{$\E{\CCSDT}{\Bas}$ and density $\n{\HF}{\Bas}$ of a $\Ne$-electron system in an incomplete basis set $\Bas$.} Let us assume
According to Eq.~(15) of Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, assuming that \titou{$\E{\CCSDT}{\Bas}$ and $\n{\HF}{\Bas}$} are reasonable approximations of the FCI energy and density within $\Bas$, the exact ground state energy $\E{}{}$ may be approximated as \titou{that we have reasonable approximations of the FCI energy and density of a $\Ne$-electron system in an incomplete basis set $\Bas$, say the CCSD(T) energy $\E{\CCSDT}{\Bas}$ and the Hartree-Fock (HF) density $\n{\HF}{\Bas}$. According to Eq.~(15) of Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, the exact ground-state energy $\E{}{}$ may be approximated as
}
%we have both the energy \titou{$\E{\CCSDT}{\Bas}$ and density $\n{\HF}{\Bas}$ of a $\Ne$-electron system in an incomplete basis set $\Bas$.}
%According to Eq.~(15) of Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, assuming that \titou{$\E{\CCSDT}{\Bas}$ and $\n{\HF}{\Bas}$} are reasonable approximations of the FCI energy and density within $\Bas$, the exact ground state energy $\E{}{}$ may be approximated as
\begin{equation} \begin{equation}
\label{eq:e0basis} \label{eq:e0basis}
\titou{\E{}{} \titou{\E{}{}
@ -215,17 +218,17 @@ where
is the basis-dependent complementary density functional, $\hT$ is the kinetic operator and $\hWee{} = \sum_{i<j} r_{ij}^{-1}$ is the interelectronic repulsion operator. is the basis-dependent complementary density functional, $\hT$ is the kinetic operator and $\hWee{} = \sum_{i<j} r_{ij}^{-1}$ is the interelectronic repulsion operator.
In Eq.~\eqref{eq:E_funcbasis}, $\wf{}{\Bas}$ and $\wf{}{}$ are two general $\Ne$-electron normalized wave functions belonging to the Hilbert space spanned by $\Bas$ and the complete basis set (CBS), respectively. In Eq.~\eqref{eq:E_funcbasis}, $\wf{}{\Bas}$ and $\wf{}{}$ are two general $\Ne$-electron normalized wave functions belonging to the Hilbert space spanned by $\Bas$ and the complete basis set (CBS), respectively.
Both wave functions yield the same target density $\n{}{}$ (assumed to be representable in $\Bas$). Both wave functions yield the same target density $\n{}{}$ (assumed to be representable in $\Bas$).
Importantly, in the CBS limit (which we refer to as $\Bas \to \infty$), we have, for any density $\n{}{}$, $\lim_{\Bas \to \infty} \bE{}{\Bas}[\n{}{}] = 0$. Importantly, in the CBS limit (which we refer to as $\Bas \to \CBS$), we have, for any density $\n{}{}$, $\lim_{\Bas \to \CBS} \bE{}{\Bas}[\n{}{}] = 0$.
This implies that This implies that
\begin{equation} \begin{equation}
\label{eq:limitfunc} \label{eq:limitfunc}
\titou{\lim_{\Bas \to \infty} \qty( \E{\CCSDT}{\Bas} + \bE{}{\Bas}[\n{\HF}{\Bas}] ) = \E{\CCSDT}{} \approx \E{}{},} \titou{\lim_{\Bas \to \CBS} \qty( \E{\CCSDT}{\Bas} + \bE{}{\Bas}[\n{\HF}{\Bas}] ) = \E{\CCSDT}{\CBS} \approx \E{}{},}
\end{equation} \end{equation}
%where $\E{\modY}{}$ is the energy associated with the method $\modY$ in the CBS limit. %where $\E{\modY}{}$ is the energy associated with the method $\modY$ in the CBS limit.
where \titou{$\E{\CCSDT}{}$ is the $\CCSDT$ energy} in the CBS limit. where \titou{$\E{\CCSDT}{\CBS}$ is the $\CCSDT$ energy} in the CBS limit.
\titou{Of course, the above holds true for any method that provides a good approximation to the energy and density, not just CCSD(T) and HF.} \titou{Of course, the above holds true for any method that provides a good approximation to the energy and density, not just CCSD(T) and HF.}
%In the case where $\modY = \FCI$ in Eq.~\eqref{eq:limitfunc}, we have a strict equality as $\E{\FCI}{} = \E{}{}$. %In the case where $\modY = \FCI$ in Eq.~\eqref{eq:limitfunc}, we have a strict equality as $\E{\FCI}{} = \E{}{}$.
In the case where \titou{$\CCSDT$ is replaced by $\FCI$} in Eq.~\eqref{eq:limitfunc}, we have a strict equality as $\E{\FCI}{} = \E{}{}$. In the case where \titou{$\CCSDT$ is replaced by $\FCI$} in Eq.~\eqref{eq:limitfunc}, we have a strict equality as $\E{\FCI}{\CBS} = \E{}{}$.
%Provided that the functional $\bE{}{\Bas}[\n{}{}]$ is known exactly, the only sources of error at this stage lie in the potential approximate nature of the methods $\modY$ and $\modZ$, and the lack of self-consistency in the present scheme. %Provided that the functional $\bE{}{\Bas}[\n{}{}]$ is known exactly, the only sources of error at this stage lie in the potential approximate nature of the methods $\modY$ and $\modZ$, and the lack of self-consistency in the present scheme.
Provided that the functional $\bE{}{\Bas}[\n{}{}]$ is known exactly, the only sources of error at this stage lie in the approximate nature of the \titou{$\CCSDT$ and $\HF$ methods}, and the lack of self-consistency of the present scheme. Provided that the functional $\bE{}{\Bas}[\n{}{}]$ is known exactly, the only sources of error at this stage lie in the approximate nature of the \titou{$\CCSDT$ and $\HF$ methods}, and the lack of self-consistency of the present scheme.
@ -278,7 +281,7 @@ As already discussed in Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, $\W{}{\Bas}
Thanks to its definition one can show that (see Appendix B of Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}) Thanks to its definition one can show that (see Appendix B of Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18})
\begin{equation} \begin{equation}
\label{eq:lim_W} \label{eq:lim_W}
\lim_{\Bas \to \infty}\W{}{\Bas}(\br{1},\br{2}) = r_{12}^{-1}, \lim_{\Bas \to \CBS}\W{}{\Bas}(\br{1},\br{2}) = \frac{1}{r_{12}},
\end{equation} \end{equation}
for any $(\br{1},\br{2})$ such that $\n{2}{\Bas}(\br{1},\br{2}) \ne 0$. for any $(\br{1},\br{2})$ such that $\n{2}{\Bas}(\br{1},\br{2}) \ne 0$.
@ -305,15 +308,15 @@ As in Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, we consider here a specific c
\label{eq:ec_md_mu} \label{eq:ec_md_mu}
\bE{\text{c,md}}{\sr}[\n{}{},\rsmu{}{}] \bE{\text{c,md}}{\sr}[\n{}{},\rsmu{}{}]
= \min_{\wf{}{} \to \n{}{}} \mel*{\Psi}{\hT + \hWee{}}{\wf{}{}} = \min_{\wf{}{} \to \n{}{}} \mel*{\Psi}{\hT + \hWee{}}{\wf{}{}}
- \mel*{\wf{}{\rsmu{}{}}}{\hT + \hWee{}}{\wf{}{\rsmu{}{}}}, - \mel*{\wf{}{\rsmu{}{}}[n]}{\hT + \hWee{}}{\wf{}{\rsmu{}{}}[n]},
\end{equation} \end{equation}
where $\wf{}{\rsmu{}{}}$ is defined by the constrained minimization where $\wf{}{\rsmu{}{}}[n]$ is defined by the constrained minimization
\begin{equation} \begin{equation}
\label{eq:argmin} \label{eq:argmin}
\wf{}{\rsmu{}{}} = \arg \min_{\wf{}{} \to \n{}{}} \mel*{\wf{}{}}{\hT + \hWee{\lr,\rsmu{}{}}}{\wf{}{}}, \wf{}{\rsmu{}{}}[n] = \arg \min_{\wf{}{} \to \n{}{}} \mel*{\wf{}{}}{\hT + \hWee{\lr,\rsmu{}{}}}{\wf{}{}},
\end{equation} \end{equation}
with $\hWee{\lr,\rsmu{}{}} = \sum_{i<j} \w{}{\lr,\rsmu{}{}}(r_{ij})$. with $\hWee{\lr,\rsmu{}{}} = \sum_{i<j} \w{}{\lr,\rsmu{}{}}(r_{ij})$.
The ECMD functionals admit, for any $\n{}{}$, the following two limiting forms The ECMD functionals admit, for any $\n{}{}$, the following two limits
\begin{align} \begin{align}
\label{eq:large_mu_ecmd} \label{eq:large_mu_ecmd}
\lim_{\mu \to \infty} \bE{\text{c,md}}{\sr}[\n{}{},\rsmu{}{}] & = 0, \lim_{\mu \to \infty} \bE{\text{c,md}}{\sr}[\n{}{},\rsmu{}{}] & = 0,
@ -354,7 +357,7 @@ The difference between the ECMD functional defined in Ref.~\onlinecite{FerGinTou
This represents a major computational saving without loss of accuracy for weakly correlated systems as we eschew the computation of $\n{2}{\Bas}(\br{},\br{})$. This represents a major computational saving without loss of accuracy for weakly correlated systems as we eschew the computation of $\n{2}{\Bas}(\br{},\br{})$.
%Depending on the functional choice, the complementary functional $\bE{}{\Bas}[\n{\modZ}{\Bas}]$ is approximated by $\bE{\LDA}{\Bas}[\n{\modZ}{\Bas},\rsmu{}{\Bas}]$ or $\bE{\PBE}{\Bas}[\n{\modZ}{\Bas},\rsmu{}{\Bas}]$ where $\rsmu{}{\Bas}(\br{})$ is given by Eq.~\eqref{eq:mu_of_r}. %Depending on the functional choice, the complementary functional $\bE{}{\Bas}[\n{\modZ}{\Bas}]$ is approximated by $\bE{\LDA}{\Bas}[\n{\modZ}{\Bas},\rsmu{}{\Bas}]$ or $\bE{\PBE}{\Bas}[\n{\modZ}{\Bas},\rsmu{}{\Bas}]$ where $\rsmu{}{\Bas}(\br{})$ is given by Eq.~\eqref{eq:mu_of_r}.
\titou{The complementary functional $\bE{}{\Bas}[\n{\HF}{\Bas}]$ is approximated by $\bE{\PBE}{\Bas}[\n{\HF}{\Bas},\rsmu{}{\Bas}]$ where $\rsmu{}{\Bas}(\br{})$ is given by Eq.~\eqref{eq:mu_of_r}.} \titou{The complementary functional $\bE{}{\Bas}[\n{\HF}{\Bas}]$ is approximated by $\bE{\PBE}{\Bas}[\n{\HF}{\Bas},\rsmu{}{\Bas}]$ where $\rsmu{}{\Bas}(\br{})$ is given by Eq.~\eqref{eq:mu_of_r}.}
\titou{The local-density approximation (LDA) version of the ECMD functional is discussed in the {\SI}.} \titou{The slightly simpler local-density approximation (LDA) version of the ECMD functional is discussed in the {\SI}.}
%================================================================= %=================================================================
%\subsection{Frozen-core approximation} %\subsection{Frozen-core approximation}
%================================================================= %=================================================================
@ -382,7 +385,7 @@ with
\end{gather} \end{gather}
\end{subequations} \end{subequations}
and the corresponding FC range-separation function $\rsmuFC{}{\Bas}(\br{}) = (\sqrt{\pi}/2) \WFC{}{\Bas}(\br{},\br{})$. and the corresponding FC range-separation function $\rsmuFC{}{\Bas}(\br{}) = (\sqrt{\pi}/2) \WFC{}{\Bas}(\br{},\br{})$.
It is noteworthy that, within the present definition, $\WFC{}{\Bas}(\br{1},\br{2})$ still tends to the regular Coulomb interaction as $\Bas \to \infty$. It is noteworthy that, within the present definition, $\WFC{}{\Bas}(\br{1},\br{2})$ still tends to the regular Coulomb interaction as $\Bas \to \CBS$.
%Defining $\nFC{\modZ}{\Bas}$ as the FC (i.e.~valence-only) one-electron density obtained with a method $\modZ$ in $\Bas$, the FC contribution of the complementary functional is then approximated by $\bE{\LDA}{\Bas}[\nFC{\modZ}{\Bas},\rsmuFC{}{\Bas}]$ or $\bE{\PBE}{\Bas}[\nFC{\modZ}{\Bas},\rsmuFC{}{\Bas}]$. %Defining $\nFC{\modZ}{\Bas}$ as the FC (i.e.~valence-only) one-electron density obtained with a method $\modZ$ in $\Bas$, the FC contribution of the complementary functional is then approximated by $\bE{\LDA}{\Bas}[\nFC{\modZ}{\Bas},\rsmuFC{}{\Bas}]$ or $\bE{\PBE}{\Bas}[\nFC{\modZ}{\Bas},\rsmuFC{}{\Bas}]$.
\titou{Defining $\nFC{\HF}{\Bas}$ as the FC (i.e.~valence-only) $\HF$ one-electron density in $\Bas$, the FC contribution of the complementary functional is then approximated by $\bE{\PBE}{\Bas}[\nFC{\HF}{\Bas},\rsmuFC{}{\Bas}]$}. \titou{Defining $\nFC{\HF}{\Bas}$ as the FC (i.e.~valence-only) $\HF$ one-electron density in $\Bas$, the FC contribution of the complementary functional is then approximated by $\bE{\PBE}{\Bas}[\nFC{\HF}{\Bas},\rsmuFC{}{\Bas}]$}.
@ -463,22 +466,22 @@ In most cases, the basis-set corrected triple-$\zeta$ atomization energies are o
\begin{figure} \begin{figure}
\includegraphics[width=0.5\linewidth]{fig2} \includegraphics[width=0.5\linewidth]{fig2}
\caption{ \caption{
\titou{$\rsmu{}{\Bas}$ (top) and $\n{}{} \times \be{\text{c,md}}{\sr,\PBE}$ (bottom) along the molecular axis ($z$) for \ce{N2} in various basis sets. \titou{$\rsmu{}{\Bas}$ (top) and $\n{}{} \times \be{\text{c,md}}{\sr,\PBE}$ (bottom) along the molecular axis ($z$) for \ce{N2} for various basis sets.
The two nitrogen atoms are located at $z=0$ and $z=2.076$ bohr.} The two nitrogen nuclei are located at $z=0$ and $z=2.076$ bohr.}
\label{fig:N2}} \label{fig:N2}}
\end{figure} \end{figure}
\titou{The fundamental quantity of the present basis set correction is $\rsmu{}{\Bas}(\br{})$. \titou{The fundamental quantity of the present basis-set correction is $\rsmu{}{\Bas}(\br{})$.
As it grows when one gets closer to the CBS limit, the value of $\rsmu{}{\Bas}(\br{})$ quantifies the quality of a given basis set at a given $\br{}$. As it grows when one gets closer to the CBS limit, the value of $\rsmu{}{\Bas}(\br{})$ quantifies the quality of a given basis set at a given $\br{}$.
Another important quantity closely related to $\rsmu{}{\Bas}(\br{})$ is the local energetic correction, $\n{}{}(\br{}) \be{\text{c,md}}{\sr,\PBE}\qty(\n{}{}(\br{}),s(\br{}),\zeta(\br{}),\rsmu{}{\Bas}(\br{}))$, which integrates to the total basis set correction $\bE{\PBE}{\Bas}[\n{}{},\rsmu{}{\Bas}]$ [see Eq.~\eqref{eq:def_pbe_tot}]. Another important quantity closely related to $\rsmu{}{\Bas}(\br{})$ is the local energetic correction, $\n{}{}(\br{}) \be{\text{c,md}}{\sr,\PBE}\qty(\n{}{}(\br{}),s(\br{}),\zeta(\br{}),\rsmu{}{\Bas}(\br{}))$, which integrates to the total basis set correction $\bE{\PBE}{\Bas}[\n{}{},\rsmu{}{\Bas}]$ [see Eq.~\eqref{eq:def_pbe_tot}].
Such quantity essentially depends on the local values of both $\rsmu{}{\Bas}(\br{})$ and $\n{}{}(\br{})$. Such a quantity essentially depends on the local values of both $\rsmu{}{\Bas}(\br{})$ and $\n{}{}(\br{})$.
In order to qualitatively illustrate how the basis set correction operates, we report, in Figure \ref{fig:N2}, $\rsmu{}{\Bas}$ and $\n{}{} \times \be{\text{c,md}}{\sr,\PBE}$ along the molecular axis ($z$) of \ce{N2} for $\Bas=\{\text{cc-pVDZ, cc-pVTZ, cc-pVQZ}\}$. In order to qualitatively illustrate how the basis-set correction operates, we report, in Figure \ref{fig:N2}, $\rsmu{}{\Bas}$ and $\n{}{} \times \be{\text{c,md}}{\sr,\PBE}$ along the molecular axis ($z$) of \ce{N2} for $\Bas=\text{cc-pVDZ, cc-pVTZ, cc-pVQZ}$.
This figure illustrates several general trends: This figure illustrates several general trends:
i) the global value of $\rsmu{}{\Bas}(z)$ is much larger than 0.5 which is the standard value used in RS-DFT, i) the value of $\rsmu{}{\Bas}(z)$ tends to be much larger than 0.5 bohr$^{-1}$ which is the common value used in RS-DFT,
ii) $\rsmu{}{\Bas}(z)$ is highly non uniform in space, illustrating the non-homogeneity of basis set quality in quantum chemistry, ii) $\rsmu{}{\Bas}(z)$ is highly non-uniform in space, illustrating the non-homogeneity of basis-set quality in quantum chemistry,
iii) $\rsmu{}{\Bas}(z)$ is significantly larger close to the nuclei, a signature that atom-centered basis sets better describe these high-density regions than the bonding regions, iii) $\rsmu{}{\Bas}(z)$ is significantly larger close to the nuclei, a signature that nucleus-centered basis sets better describe these high-density regions than the bonding regions,
v) the global value of the energy correction get smaller as one improves the basis set quality, and the reduction is spectacular close to the nuclei, and v) the value of the energy correction gets smaller as one improves the basis-set quality, the reduction being spectacular close to the nuclei, and
iv) a large energetic contribution comes from the bonding regions highlighting the poor description of correlation effects in these region with Gaussian basis sets.} iv) a large energetic contribution comes from the bonding regions, highlighting the imperfect description of correlation effects in these regions with Gaussian basis sets.}
%%% TABLE II %%% %%% TABLE II %%%
\begin{table} \begin{table}
@ -517,7 +520,7 @@ iv) a large energetic contribution comes from the bonding regions highlighting t
Deviation (in \kcal) from the CCSD(T)/CBS atomization energy obtained with \titou{various basis sets for CCSD(T) (top) and CCSD(T)+PBE (bottom).} Deviation (in \kcal) from the CCSD(T)/CBS atomization energy obtained with \titou{various basis sets for CCSD(T) (top) and CCSD(T)+PBE (bottom).}
% Deviation (in \kcal) from the CCSD(T)/CBS atomization energy obtained with various methods with the cc-pVDZ (top), cc-pVTZ (center) and cc-pVQZ (bottom) basis sets. % Deviation (in \kcal) from the CCSD(T)/CBS atomization energy obtained with various methods with the cc-pVDZ (top), cc-pVTZ (center) and cc-pVQZ (bottom) basis sets.
The green region corresponds to chemical accuracy (i.e.~error below 1 {\kcal}). The green region corresponds to chemical accuracy (i.e.~error below 1 {\kcal}).
\titou{Note the difference in scaling of the vertical axes.} \titou{Note the different scales of the vertical axes.}
See {\SI} for raw data \titou{and the corresponding LDA results}. See {\SI} for raw data \titou{and the corresponding LDA results}.
\label{fig:G2_Ec}} \label{fig:G2_Ec}}
\end{figure*} \end{figure*}
@ -545,7 +548,7 @@ Encouraged by these promising results, we are currently pursuing various avenues
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section*{Supporting Information Available} \section*{Supporting Information Available}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
See {\SI} for raw data associated with the atomization energies of the four diatomic molecules and the G2 set \titou{as well as the definition of the LDA ECMD functionals (and the corresponding numerical results).} See {\SI} for raw data associated with the atomization energies of the four diatomic molecules and the G2 set \titou{as well as the definition of the LDA ECMD functional (and the corresponding numerical results).}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{acknowledgements} \begin{acknowledgements}

10
JPCL_revision/SI/G2_srDFT-SI.tex
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@ -139,7 +139,7 @@
\maketitle \maketitle
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Local-density approximation} %\section{Local-density approximation}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The local-density approximation (LDA) of the ECMD complementary functional is defined as The local-density approximation (LDA) of the ECMD complementary functional is defined as
\begin{equation} \begin{equation}
@ -147,7 +147,7 @@ The local-density approximation (LDA) of the ECMD complementary functional is de
\bE{\LDA}{\Bas}[\n{}{},\rsmu{}{\Bas}] = \int \n{}{}(\br{}) \be{\text{c,md}}{\sr,\LDA}\qty(\n{}{}(\br{}),\zeta(\br{}),\rsmu{}{\Bas}(\br{})) \dbr{}, \bE{\LDA}{\Bas}[\n{}{},\rsmu{}{\Bas}] = \int \n{}{}(\br{}) \be{\text{c,md}}{\sr,\LDA}\qty(\n{}{}(\br{}),\zeta(\br{}),\rsmu{}{\Bas}(\br{})) \dbr{},
\end{equation} \end{equation}
where $\zeta = (\n{\uparrow}{} - \n{\downarrow}{})/\n{}{}$ is the spin polarization and $\be{\text{c,md}}{\sr,\LDA}(\n{}{},\zeta,\rsmu{}{})$ is the ECMD short-range correlation energy per electron of the uniform electron gas (UEG) \cite{LooGil-WIRES-16} parameterized in Ref.~\onlinecite{PazMorGorBac-PRB-06}. where $\zeta = (\n{\uparrow}{} - \n{\downarrow}{})/\n{}{}$ is the spin polarization and $\be{\text{c,md}}{\sr,\LDA}(\n{}{},\zeta,\rsmu{}{})$ is the ECMD short-range correlation energy per electron of the uniform electron gas (UEG) \cite{LooGil-WIRES-16} parameterized in Ref.~\onlinecite{PazMorGorBac-PRB-06}.
The short-range LDA correlation functional relies on the transferability of the physics of the UEG which is certainly valid for large $\mu$ but is known to over correlate for small $\mu$. The short-range LDA correlation functional relies on the transferability of the physics of the UEG which is certainly valid for large $\mu$ but is known to overcorrelate for small $\mu$.
The sensitivity with respect to the RS-DFT functional is quite large for the double- and triple-$\zeta$ basis sets, where clearly the PBE functional performs better. The sensitivity with respect to the RS-DFT functional is quite large for the double- and triple-$\zeta$ basis sets, where clearly the PBE functional performs better.
However, from the quadruple-$\zeta$ basis, the LDA and PBE functionals agree within a few tenths of a {\kcal}. However, from the quadruple-$\zeta$ basis, the LDA and PBE functionals agree within a few tenths of a {\kcal}.
@ -163,7 +163,7 @@ Such weak sensitivity to the density-functional approximation when reaching larg
\hspace{1cm} \hspace{1cm}
\includegraphics[width=0.30\linewidth]{fig1d} \includegraphics[width=0.30\linewidth]{fig1d}
\caption{ \caption{
Deviation (in \kcal) from CBS atomization energies of \ce{C2} (top left), \ce{O2} (top right), \ce{N2} (bottom left) and \ce{F2} (bottom right) obtained with various methods and basis sets. Deviation (in \kcal) from CBS atomization energies of \ce{C2} (top left), \ce{O2} (top right), \ce{N2} (bottom left), and \ce{F2} (bottom right) obtained with various methods and basis sets.
The green region corresponds to chemical accuracy (i.e.~error below 1 {\kcal}). The green region corresponds to chemical accuracy (i.e.~error below 1 {\kcal}).
\label{fig:diatomics}} \label{fig:diatomics}}
\end{figure*} \end{figure*}
@ -201,7 +201,7 @@ Such weak sensitivity to the density-functional approximation when reaching larg
\includegraphics[width=\linewidth]{fig2b} \includegraphics[width=\linewidth]{fig2b}
\includegraphics[width=\linewidth]{fig2c} \includegraphics[width=\linewidth]{fig2c}
\caption{ \caption{
Deviation (in \kcal) from the CCSD(T)/CBS atomization energy obtained with various methods with the cc-pVDZ (top), cc-pVTZ (center) and cc-pVQZ (bottom) basis sets. Deviation (in \kcal) from the CCSD(T)/CBS atomization energy obtained with various methods with the cc-pVDZ (top), cc-pVTZ (center), and cc-pVQZ (bottom) basis sets.
The green region corresponds to chemical accuracy (i.e.~error below 1 {\kcal}). The green region corresponds to chemical accuracy (i.e.~error below 1 {\kcal}).
\label{fig:G2_Ec}} \label{fig:G2_Ec}}
\end{figure*} \end{figure*}
@ -210,7 +210,7 @@ Such weak sensitivity to the density-functional approximation when reaching larg
\begin{table*} \begin{table*}
\caption{ \caption{
\label{tab:diatomics} \label{tab:diatomics}
Frozen-core atomization energies (in {\kcal}) of \ce{C2}, \ce{O2}, \ce{N2} and \ce{F2} computed with various methods and basis sets. Frozen-core atomization energies (in {\kcal}) of \ce{C2}, \ce{O2}, \ce{N2}, and \ce{F2} computed with various methods and basis sets.
The deviations with respect to the corresponding CBS values are reported in parenthesis. The deviations with respect to the corresponding CBS values are reported in parenthesis.
See main text for more details. See main text for more details.
} }