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monir corrections from titou

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Pierre-Francois Loos 2019-05-10 15:01:53 +02:00
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JPCL_revision/G2-srDFT.tex
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@ -227,7 +227,7 @@ where \titou{$\E{\CCSDT}{}$ is the $\CCSDT$ energy} in the CBS limit.
%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 $\E{\FCI}{} = \E{}{}$.
%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 potential approximate nature of the \titou{$\CCSDT$ and $\HF$ methods}, 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.
The functional $\bE{}{\Bas}[\n{}{}]$ is obviously \textit{not} universal as it depends on $\Bas$. 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
@ -429,7 +429,7 @@ We begin our investigation of the performance of the basis-set correction by com
In a second time, we compute the atomization energies of the entire G2 set \cite{CurRagTruPop-JCP-91} composed by 55 molecules with the cc-pVXZ basis set family. In a second time, we compute the atomization energies of the entire G2 set \cite{CurRagTruPop-JCP-91} composed by 55 molecules with the cc-pVXZ basis set family.
This molecular set has been intensively studied in the last 20 years (see, for example, Refs.~\onlinecite{FelPetDix-JCP-08, Gro-JCP-09, FelPet-JCP-09, NemTowNee-JCP-10, FelPetHil-JCP-11, HauKlo-JCP-12, PetTouUmr-JCP-12, FelPet-JCP-13, KesSylKohTewMar-JCP-18}) and can be considered as a representative set of small organic and inorganic molecules. This molecular set has been intensively studied in the last 20 years (see, for example, Refs.~\onlinecite{FelPetDix-JCP-08, Gro-JCP-09, FelPet-JCP-09, NemTowNee-JCP-10, FelPetHil-JCP-11, HauKlo-JCP-12, PetTouUmr-JCP-12, FelPet-JCP-13, KesSylKohTewMar-JCP-18}) and can be considered as a representative set of small organic and inorganic molecules.
%As a method $\modY$ we employ either CCSD(T) or exFCI. %As a method $\modY$ we employ either CCSD(T) or exFCI.
As \titou{a ``reference'' method}, we employ either CCSD(T) or exFCI. \titou{We employ either CCSD(T) or exFCI to compute the energy of these systems.}
Here, exFCI stands for extrapolated FCI energies computed with the CIPSI algorithm. \cite{HurMalRan-JCP-73, GinSceCaf-CJC-13, GinSceCaf-JCP-15} Here, exFCI stands for extrapolated FCI energies computed with the CIPSI algorithm. \cite{HurMalRan-JCP-73, GinSceCaf-CJC-13, GinSceCaf-JCP-15}
We refer the interested reader to Refs.~\onlinecite{HolUmrSha-JCP-17, SceGarCafLoo-JCTC-18, LooSceBloGarCafJac-JCTC-18, SceBenJacCafLoo-JCP-18, LooBogSceCafJAc-JCTC-19} for more details. We refer the interested reader to Refs.~\onlinecite{HolUmrSha-JCP-17, SceGarCafLoo-JCTC-18, LooSceBloGarCafJac-JCTC-18, SceBenJacCafLoo-JCP-18, LooBogSceCafJAc-JCTC-19} for more details.
%In the case of the CCSD(T) calculations, we have $\modZ = \ROHF$ as we use the restricted open-shell HF (ROHF) one-electron density to compute the complementary basis-set correction energy. %In the case of the CCSD(T) calculations, we have $\modZ = \ROHF$ as we use the restricted open-shell HF (ROHF) one-electron density to compute the complementary basis-set correction energy.
@ -463,8 +463,8 @@ 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{
$\rsmu{}{\Bas}$ (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} in various basis sets.
The two nitrogen atoms are located at $z=0$ and $z=2.076$ bohr. The two nitrogen atoms are located at $z=0$ and $z=2.076$ bohr.}
\label{fig:N2}} \label{fig:N2}}
\end{figure} \end{figure}
@ -524,7 +524,7 @@ The ``plain'' CCSD(T) atomization energies as well as the corrected \titou{CCSD(
The raw data \titou{(as well as the corresponding LDA results)} can be found in the {\SI}. The raw data \titou{(as well as the corresponding LDA results)} can be found in the {\SI}.
A statistical analysis of these data is also provided in Table \ref{tab:stats}, where we report the mean absolute deviation (MAD), root-mean-square deviation (RMSD), and maximum deviation (MAX) with respect to the CCSD(T)/CBS atomization energies. A statistical analysis of these data is also provided in Table \ref{tab:stats}, where we report the mean absolute deviation (MAD), root-mean-square deviation (RMSD), and maximum deviation (MAX) with respect to the CCSD(T)/CBS atomization energies.
Note that the MAD of our CCSD(T)/CBS atomization energies is only 0.37 {\kcal} compared to the values extracted from Ref.~\onlinecite{HauKlo-JCP-12} which corresponds to frozen-core non-relativistic atomization energies obtained at the CCSD(T)(F12)/cc-pVQZ-F12 level of theory corrected for higher-excitation contributions ($E_\text{CCSDT(Q)/cc-pV(D+d)Z} - E_\text{CCSD(T)/cc-pV(D+d)Z})$. Note that the MAD of our CCSD(T)/CBS atomization energies is only 0.37 {\kcal} compared to the values extracted from Ref.~\onlinecite{HauKlo-JCP-12} which corresponds to frozen-core non-relativistic atomization energies obtained at the CCSD(T)(F12)/cc-pVQZ-F12 level of theory corrected for higher-excitation contributions ($E_\text{CCSDT(Q)/cc-pV(D+d)Z} - E_\text{CCSD(T)/cc-pV(D+d)Z})$.
From double-$\zeta$ to quintuple-$\zeta$ basis, the MAD associated with the CCSD(T) atomization energies goes down slowly from 14.29 to 1.28 {\kcal}. From the double- to the quintuple-$\zeta$ basis, the MAD associated with the CCSD(T) atomization energies goes down slowly from 14.29 to 1.28 {\kcal}.
For a commonly used basis like cc-pVTZ, the MAD of CCSD(T) is still 6.06 {\kcal}. For a commonly used 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. 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 level, the MAD is reduced to 3.24 and 1.96 {\kcal}. %Already at the CCSD(T)+LDA/cc-pVDZ and CCSD(T)+PBE/cc-pVDZ level, the MAD is reduced to 3.24 and 1.96 {\kcal}.