results Ec G2

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Pierre-Francois Loos 2019-04-12 23:46:03 +02:00
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@ -189,7 +189,7 @@ Here, we only provide the main working equations.
We refer the interested reader to Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18} for a more formal derivation. We refer the interested reader to Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18} for a more formal derivation.
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 \alert{FCI} energy and density within $\Bas$, the exact ground state energy $\E{}{}$ may be written 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 written as
\begin{equation} \begin{equation}
\label{eq:e0basis} \label{eq:e0basis}
\E{}{} \E{}{}
@ -214,7 +214,7 @@ Importantly, in the limit of a complete basis set (which we refer to as $\Bas \t
\end{equation} \end{equation}
where $\E{\modY}{}$ is the energy associated with the method $\modY$ in the complete basis set. where $\E{\modY}{}$ is the energy associated with the method $\modY$ in the complete basis set.
In the case $\modY = \FCI$, we have a strict equality as $\E{\FCI}{} = \E{}{}$. In the case $\modY = \FCI$, we have a strict equality as $\E{\FCI}{} = \E{}{}$.
Provided that the functional $\bE{}{\Bas}[\n{}{}]$ is known exactly, the only source of error at this stage lies in the potential approximate nature of the methods $\modY$ and $\modZ$ for the \titou{FCI} energy and density within $\Bas$, respectively. Provided that the functional $\bE{}{\Bas}[\n{}{}]$ is known exactly, the only source of error at this stage lies in the potential approximate nature of the methods $\modY$ and $\modZ$ for the FCI energy and density within $\Bas$, respectively.
Rigorously speaking, the functional $\bE{}{\Bas}[\n{}{}]$ is obviously \textit{not} universal as it depends on $\Bas$. Rigorously speaking, the functional $\bE{}{\Bas}[\n{}{}]$ is obviously \textit{not} universal as it depends on $\Bas$.
Moreover, as $\bE{}{\Bas}[\n{}{}]$ aims at fixing the incompleteness of $\Bas$, its main role is to correct Moreover, as $\bE{}{\Bas}[\n{}{}]$ aims at fixing the incompleteness of $\Bas$, its main role is to correct
@ -223,15 +223,15 @@ Because the e-e cusp originates from the divergence of the Coulomb operator at $
Therefore, as we shall do later on, it feels natural to approximate $\bE{}{\Bas}[\n{}{}]$ with short-range density functionals which deal with a smooth long-range electron interaction. Therefore, as we shall do later on, it feels natural to approximate $\bE{}{\Bas}[\n{}{}]$ with short-range density functionals which deal with a smooth long-range electron interaction.
Contrary to the conventional RS-DFT scheme which requires a range-separated \textit{parameter} $\rsmu{}{}$, here we use a range-separated \textit{function} $\rsmu{\Bas}{}(\br{})$ which automatically adapts to quantify the incompleteness of $\Bas$ in $\mathbb{R}^3$. Contrary to the conventional RS-DFT scheme which requires a range-separated \textit{parameter} $\rsmu{}{}$, here we use a range-separated \textit{function} $\rsmu{\Bas}{}(\br{})$ which automatically adapts to quantify the incompleteness of $\Bas$ in $\mathbb{R}^3$.
The first step of the basis set correction consists in obtaining an effective two-electron interaction $\W{\Bas}{}(\br{1},\br{2})$ which represents the effect of the projection of the Coulomb operator in an incomplete basis set $\Bas$. The first step of the present basis set correction consists in obtaining an effective two-electron interaction $\W{\Bas}{}(\br{1},\br{2})$ which represents the effect of the projection of the Coulomb operator in an incomplete basis set $\Bas$.
The present definition ensures that $\W{\Bas}{}(\br{1},\br{2})$ is finite at the e-e coalescence point as long as an incomplete basis set is used, and tends to the genuine, unbounded $r_{12}^{-1}$ Coulomb interaction as $\Bas \to \infty$. %The present definition ensures that $\W{\Bas}{}(\br{1},\br{2})$ is finite at the e-e coalescence point as long as an incomplete basis set is used, and tends to the genuine, unbounded $r_{12}^{-1}$ Coulomb interaction as $\Bas \to \infty$.
In a second step, we shall link $\W{\Bas}{}(\br{1},\br{2})$ to $\rsmu{\Bas}{}(\br{})$. In a second step, we shall link $\W{\Bas}{}(\br{1},\br{2})$ to $\rsmu{\Bas}{}(\br{})$.
In the final step, we employ short-range density functionals \cite{TouGorSav-TCA-05} with $\rsmu{\Bas}{}(\br{})$ as range separation. In the final step, we employ short-range density functionals \cite{TouGorSav-TCA-05} with $\rsmu{\Bas}{}(\br{})$ as range separation.
%================================================================= %=================================================================
%\subsection{Effective Coulomb operator} %\subsection{Effective Coulomb operator}
%================================================================= %=================================================================
We define the effective operator as We define the effective operator as \cite{GinPraFerAssSavTou-JCP-18}
\begin{equation} \begin{equation}
\label{eq:def_weebasis} \label{eq:def_weebasis}
\W{\Bas}{}(\br{1},\br{2}) = \W{\Bas}{}(\br{1},\br{2}) =
@ -254,7 +254,7 @@ and $\Gam{pq}{rs} = \mel*{\wf{}{\Bas}}{ \aic{r}\aic{s}\ai{p}\ai{q} }{\wf{}{\Bas}
= \sum_{pqrstu \in \Bas} \SO{p}{1} \SO{q}{2} \V{pq}{rs} \Gam{rs}{tu} \SO{t}{1} \SO{u}{2}, = \sum_{pqrstu \in \Bas} \SO{p}{1} \SO{q}{2} \V{pq}{rs} \Gam{rs}{tu} \SO{t}{1} \SO{u}{2},
\end{equation} \end{equation}
and $\V{pq}{rs}$ are the usual two-electron Coulomb integrals. and $\V{pq}{rs}$ are the usual two-electron Coulomb integrals.
Note that the divergence condition of $\W{\Bas}{}(\br{1},\br{2})$ in Eq.~\eqref{eq:def_weebasis} ensures that one-electron systems do not have any basis set correction. Note that the divergence condition of $\W{\Bas}{}(\br{1},\br{2})$ in Eq.~\eqref{eq:def_weebasis} \titou{ensures} that one-electron systems do not have any basis set correction.
%\PFL{I don't agree with this. There must be a correction for one-electron system. %\PFL{I don't agree with this. There must be a correction for one-electron system.
%However, it does not come from the e-e cusp but from the e-n cusp.} %However, it does not come from the e-e cusp but from the e-n cusp.}
With such a definition, $\W{\Bas}{}(\br{1},\br{2})$ satisfies (see Appendix A of Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}) With such a definition, $\W{\Bas}{}(\br{1},\br{2})$ satisfies (see Appendix A of Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18})
@ -267,9 +267,9 @@ Because Eq.~\eqref{eq:int_eq_wee} can be rewritten as
\begin{equation} \begin{equation}
\iint r_{12}^{-1} \n{2}{}(\br{1},\br{2}) \dbr{1} \dbr{2} = \iint \W{\Bas}{}(\br{1},\br{2}) \n{2}{}(\br{1},\br{2}) \dbr{1} \dbr{2}, \iint r_{12}^{-1} \n{2}{}(\br{1},\br{2}) \dbr{1} \dbr{2} = \iint \W{\Bas}{}(\br{1},\br{2}) \n{2}{}(\br{1},\br{2}) \dbr{1} \dbr{2},
\end{equation} \end{equation}
it intuitively motivates $\W{\Bas}{}(\br{1},\br{2})$ as a potential candidate for an effective interaction. \alert{it intuitively motivates $\W{\Bas}{}(\br{1},\br{2})$ as a potential candidate for an effective interaction.}
As already discussed in Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, $\W{\Bas}{}(\br{1},\br{2})$ is symmetric, \textit{a priori} non translational, nor rotational invariant if $\Bas$ does not have such symmetries. As already discussed in Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, $\W{\Bas}{}(\br{1},\br{2})$ is symmetric, \textit{a priori} non translational, nor rotational invariant if $\Bas$ does not have such symmetries.
An important quantity to define in the present context is the value of the effective interaction at coalescence of opposite-spin electrons A key quantity is the value of the effective interaction at coalescence of opposite-spin electrons
\begin{equation} \begin{equation}
\label{eq:wcoal} \label{eq:wcoal}
\W{\Bas}{}(\br{}) = \W{\Bas}{}(\br{},{\br{}}), \W{\Bas}{}(\br{}) = \W{\Bas}{}(\br{},{\br{}}),
@ -289,7 +289,7 @@ for any $(\br{1},\br{2})$ such that $\n{2}{}(\br{1},\br{2}) \ne 0$ and for any $
%\subsection{Range-separation function} %\subsection{Range-separation function}
%================================================================= %=================================================================
Because $\W{\Bas}{}(\br{1},\br{2})$ is a non-divergent two-electron interaction, it can be straightforwardly linked to RS-DFT which employs smooth long-range operators. Because $\W{\Bas}{}(\br{1},\br{2})$ is a non-divergent two-electron interaction, it can be naturally linked to RS-DFT \titou{which employs smooth long-range operators.}
Although this choice is not unique, we choose here the range-separation function Although this choice is not unique, we choose here the range-separation function
\begin{equation} \begin{equation}
\label{eq:mu_of_r} \label{eq:mu_of_r}
@ -351,7 +351,7 @@ The ECMD functionals admit, for any density $\n{}{}(\br{})$, the following two l
\end{subequations} \end{subequations}
where $\Ec[\n{}{}(\br{})]$ is the usual universal correlation functional defined in KS-DFT. where $\Ec[\n{}{}(\br{})]$ is the usual universal correlation functional defined in KS-DFT.
The choice of ECMD in the present scheme is motivated by the analogy between the definition of $\bE{}{\Bas}[\n{}{}]$ [Eq.~\eqref{eq:E_funcbasis}] and the ECMD functionals [Eq.~\eqref{eq:ec_md_mu}]. The choice of ECMD in the present scheme is motivated by the analogy between the definition of $\bE{}{\Bas}[\n{}{}]$ [Eq.~\eqref{eq:E_funcbasis}] and the ECMD functionals [Eq.~\eqref{eq:ec_md_mu}].
Indeed, provided that $\w{}{\lr,\rsmu{\Bas}{}}(\br{1},\br{2}) \approx \W{\Bas}{}(\br{1},\br{2})$, then the wave function $\wf{}{\rsmu{\Bas}{}}$ coincides with $\wf{}{\Bas}$. Indeed, provided that $\w{}{\lr,\rsmu{\Bas}{}}(\br{1},\br{2}) \titou{=} \W{\Bas}{}(\br{1},\br{2})$, then $\wf{}{\rsmu{\Bas}{}}$ and $\wf{}{\Bas}$ coincide.
%The ECMD functionals differ from the standard RS-DFT correlation functional by the fact that the reference is not the KS Slater determinant but a multi-determinantal wave function. %The ECMD functionals differ from the standard RS-DFT correlation functional by the fact that the reference is not the KS Slater determinant but a multi-determinantal wave function.
%This makes them particularly well adapted to the present context where one aims at correcting a general WFT method. %This makes them particularly well adapted to the present context where one aims at correcting a general WFT method.
Therefore, following Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, we approximate $\bE{}{\Bas}[\n{}{}]$ by the ECMD functionals evaluated with the range separation function $\rsmu{\Bas}{}(\br{})$. Therefore, following Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, we approximate $\bE{}{\Bas}[\n{}{}]$ by the ECMD functionals evaluated with the range separation function $\rsmu{\Bas}{}(\br{})$.
@ -362,7 +362,7 @@ The LDA version of $\bE{}{\Bas}[\n{}{}]$ is defined as
\end{equation} \end{equation}
where $\be{\LDA}{\sr}(\n{}{},\rsmu{}{})$ is the short-range reduced (i.e.~per electron) correlation energy of the uniform electron gas (UEG) \cite{LooGil-WIRES-16} parametrized in Ref.~\onlinecite{PazMorGorBac-PRB-06}. where $\be{\LDA}{\sr}(\n{}{},\rsmu{}{})$ is the short-range reduced (i.e.~per electron) correlation energy of the uniform electron gas (UEG) \cite{LooGil-WIRES-16} parametrized in Ref.~\onlinecite{PazMorGorBac-PRB-06}.
The short-range LDA correlation functional defined in Eq.~\eqref{eq:def_lda_tot} 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 over correlate for small $\mu$.
In order to correct such a defect, we propose here a new ECMD functional inspired by the recent functional proposed by some of the authors \cite{FerGinTou-JCP-18} which interpolates between the usual PBE correlation functional $\e{\PBE}{}(\n{}{},\nabla \n{}{})$ for $\rsmu{}{}=0$ and the exact large-$\rsmu{}{}$ behavior, \cite{TouColSav-PRA-04, GoriSav-PRA-06, PazMorGorBac-PRB-06} yielding In order to correct such a defect, we propose here a new ECMD functional inspired by the recent functional proposed by some of the authors \cite{FerGinTou-JCP-18} which interpolates between the usual PBE correlation functional $\e{\PBE}{}(\n{}{},\nabla \n{}{})$ for $\rsmu{}{}=0$ and the exact large-$\rsmu{}{}$ behavior, \cite{TouColSav-PRA-04, GoriSav-PRA-06, PazMorGorBac-PRB-06} yielding
\begin{subequations} \begin{subequations}
\begin{gather} \begin{gather}
@ -373,7 +373,7 @@ In order to correct such a defect, we propose here a new ECMD functional inspire
\beta(n,\nabla n,\rsmu{}{}) = \frac{3}{2\sqrt{\pi} (1 - \sqrt{2} )} \frac{\e{\PBE}{}(\n{}{},\nabla \n{}{})}{\n{2}{\UEG}(\n{}{})}. \beta(n,\nabla n,\rsmu{}{}) = \frac{3}{2\sqrt{\pi} (1 - \sqrt{2} )} \frac{\e{\PBE}{}(\n{}{},\nabla \n{}{})}{\n{2}{\UEG}(\n{}{})}.
\end{gather} \end{gather}
\end{subequations} \end{subequations}
The difference between the ECMD PBE functional defined in Ref.~\onlinecite{FerGinTou-JCP-18} and the present expression \eqref{eq:epsilon_cmdpbe} is that we approximate here the \textit{exact} ground-state on-top pair density by its UEG version, i.e.~$\n{2}{}(\br{}) \approx \n{2}{\UEG}(\br{}) = \n{}{}(\br{})^2 g_0(\n{}{}(\br{}) )$, where $g_0(\n{}{})$ is the UEG correlation factor whose parametrization can be found in Eq.~(46) of Ref.~\onlinecite{GorSav-PRA-06}. The difference between the ECMD PBE functional defined in Ref.~\onlinecite{FerGinTou-JCP-18} and the present expression \eqref{eq:epsilon_cmdpbe} is that we approximate here the \textit{exact} ground-state on-top pair density by its UEG version, i.e.~$\n{2}{}(\br{}) \approx \n{2}{\UEG}(\n{}{}(\br{})) = \n{}{}(\br{})^2 g_0(\n{}{}(\br{}))$, where $g_0(\n{}{})$ is the UEG correlation factor whose parametrization can be found in Eq.~(46) of Ref.~\onlinecite{GorSav-PRA-06}.
This represents a major computational saving without loss of performance as we eschew the computation of $\n{2}{}(\br{})$. This represents a major computational saving without loss of performance as we eschew the computation of $\n{2}{}(\br{})$.
Therefore, the PBE complementary functional reads Therefore, the PBE complementary functional reads
\begin{equation} \begin{equation}
@ -494,7 +494,7 @@ iii) vanishes in the limit of a complete basis set, hence guaranteeing an unalte
%%% TABLE II %%% %%% TABLE II %%%
\begin{table} \begin{table}
\caption{ \caption{
Statistical analysis (in \kcal) of the G2 correlation energies depicted in Fig.~\ref{fig:G2_AE}. Statistical analysis (in \kcal) of the G2 correlation energies depicted in Fig.~\ref{fig:G2_Ec}.
Mean absolute deviation (MAD), root-mean-square deviation (RMSD), and maximum deviation (MAX) with respect to the CCSD(T)/CBS reference correlation energies. Mean absolute deviation (MAD), root-mean-square deviation (RMSD), and maximum deviation (MAX) with respect to the CCSD(T)/CBS reference correlation energies.
CA corresponds to the number of correlation energies (out of 55) obtained with chemical accuracy. CA corresponds to the number of correlation energies (out of 55) obtained with chemical accuracy.
\label{tab:stats}} \label{tab:stats}}
@ -526,7 +526,7 @@ iii) vanishes in the limit of a complete basis set, hence guaranteeing an unalte
\caption{ \caption{
Deviation (in \kcal) from CCSD(T)/CBS reference correlation energies obtained with various methods with the cc-pVDZ (top), cc-pVTZ (center) and cc-pVQZ (bottom) basis sets. Deviation (in \kcal) from CCSD(T)/CBS reference correlation energies 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_AE}} \label{fig:G2_Ec}}
\end{figure*} \end{figure*}
%\subsection{Comparison between the CIPSI and CCSD(T) models in the case of C$_2$, N$_2$, O$_2$, F$_2$} %\subsection{Comparison between the CIPSI and CCSD(T) models in the case of C$_2$, N$_2$, O$_2$, F$_2$}
@ -569,6 +569,18 @@ Importantly, the sensitivity with respect to the SR-DFT functional is quite larg
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}.
Such weak sensitivity to the approximated functionals in the DFT part when reaching large basis sets shows the robustness of the approach. Such weak sensitivity to the approximated functionals in the DFT part when reaching large basis sets shows the robustness of the approach.
We have computed the correlation energies of the G2-1 test sets with $\modY=\CCSDT$, $\modZ=\HF$ and the cc-pVXZ basis.
The plain CCSD(T) correlation energies as well as the corrected CCSD(T)+LDA and CCSD(T)+PBE values are reported.
A statistical analysis of these data is also provided in Table \ref{tab:stats}, where one can find the mean absolute deviation (MAD), the root-mean-square deviation (RMSD), and the maximum deviation (MAX) with respect to the CCSD(T)/CBS reference correlation energies.
From double-$\zeta$ to quintuple-$\zeta$ basis, the MAD associated with the CCSD(T) correlation energies goes down slowly from 14.29 to 1.28 {\kcal}.
For typical basis like cc-pVTZ, the MAD of CCSD(T) is still 6.06 {\kcal}.
Applying the basis set correction drastically reduces the basis set incompleteness error.
Already at the CCSD(T)+LDA/cc-pVDZ and CCSD(T)+PBE/cc-pVDZ levels, the MAD is reduced to 3.24 and 1.96 {\kcal}.
With the triple-$\zeta$ basis, the CCSD(T)+PBE/cc-pVTZ are already below 1 {\kcal} with 36 cases (out of 55) where we achieve chemical accuracy.
CCSD(T)+LDA/cc-pVQZ and CCSD(T)+PBE/cc-pVQZ return MAD of 0.33 and 0.31 kcal/mol (respectively) while CCSD(T)/cc-pVQZ still yields a fairly large MAD of 2.50 {\kcal}.
Therefore, similarly to F12 methods, \cite{TewKloNeiHat-PCCP-07} we can safely claim that the present basis set correction recover quintuple-$\zeta$ quality coupled-cluster correlation energies with triple-$\zeta$ basis sets for a much cheaper computational cost.
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\section*{Supporting information} \section*{Supporting information}
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