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manu's comments : that is a brilliant paper

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28
Manuscript/G2-srDFT.bib
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@ -12276,18 +12276,16 @@
Year = {2017},
Bdsk-Url-1 = {https://doi.org/10.1002/jcc.24761}}
@article{rs_dft_toul_colo_savin,
Author = {J. Toulouse and F. Colonna and A. Savin},
Doi = {10.1103/PhysRevA.70.062505},
Issue = {6},
Journal = {Phys. Rev. A},
Month = {Dec},
Numpages = {16},
Pages = {062505},
Publisher = {American Physical Society},
Title = {Long-range--short-range separation of the electron-electron interaction in density-functional theory},
Url = {https://link.aps.org/doi/10.1103/PhysRevA.70.062505},
Volume = {70},
Year = {2004},
Bdsk-Url-1 = {https://link.aps.org/doi/10.1103/PhysRevA.70.062505},
Bdsk-Url-2 = {https://doi.org/10.1103/PhysRevA.70.062505}}
@article{Tenno-CPL-04,
title = "Initiation of explicitly correlated Slater-type geminal theory",
journal = "Chemical Physics Letters",
volume = "398",
number = "1",
pages = "56 - 61",
year = "2004",
issn = "0009-2614",
doi = "https://doi.org/10.1016/j.cplett.2004.09.041",
url = "http://www.sciencedirect.com/science/article/pii/S000926140401379X",
author = "Seiichiro Ten-no",
abstract = "We employ the Slater-type function as a geminal basis function to incorporate the inter-electron distance in explicitly correlated theory. It is shown that the use of the Slater-type geminals confers numerical and computational advantages over the previous explicitly correlated methods. The performance of the resulting method is examined in some benchmark calculations at the second order Møller–Plesset perturbation theory. The results reveal that the Slater-type function is promising compared to the ordinary Gaussian-type geminals and linear r12 function."
}

26
Manuscript/G2-srDFT.tex
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@ -138,7 +138,7 @@
We report a universal density-based basis set incompleteness correction that can be applied to any wave function method.
The present correction, which appropriately vanishes in the complete basis set (CBS) limit, relies on short-range correlation density functionals (with multi-determinant reference) from range-separated density-functional theory (RS-DFT) to estimate the basis set incompleteness error.
Contrary to conventional RS-DFT schemes which require an \textit{ad hoc} range-separation \textit{parameter} $\mu$, the key ingredient here is a range-separation \textit{function} $\mu(\bf{r})$ which automatically adapts to the spatial non-homogeneity of the basis set incompleteness error.
As illustrative examples, we show how this density-based correction allows us to obtain CCSD(T) correlation energies near the CBS limit for the G2-1 set of molecules with compact Gaussian basis sets.
As illustrative examples, we show how this density-based correction allows us to obtain coupled-cluster with single and double substitutions and triple CCSD(T) correlation energies near the CBS limit for the G2-1 set of molecules with compact Gaussian basis sets.
\titou{For example, while CCSD(T)/cc-pVTZ yields a mean absolute deviation (MAD) of 6.06 kcal/mol compared to CCSD(T)/CBS atomization energies, the CCSD(T)+LDA and CCSD(T)+PBE corrected methods return MAD of 1.19 and 0.85 kcal/mol (respectively) with the same basis.}
\end{abstract}
@ -213,17 +213,17 @@ Importantly, in the limit of a complete basis set (which we refer to as $\Bas \t
\lim_{\Bas \to \infty} \qty( \E{\modY}{\Bas} + \bE{}{\Bas}[\n{\modZ}{\Bas}] ) = \E{\modY}{} \approx E,
\end{equation}
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 $\modY = \FCI$ \manu{in \eqref{eq:limitfunc}}, 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 FCI energy and density within $\Bas$, respectively.
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
for the lack of cusp in $\wf{}{\Bas}$ (i.e.~a discontinuous derivative) at the e-e coalescence points, a universal condition of exact wave functions.
Because the e-e cusp originates from the divergence of the Coulomb operator at $r_{12} = 0$, a cuspless wave function could equivalently originate from a Hamiltonian with a non-divergent Coulomb interaction at $r_{12} = 0$.
Because the e-e cusp originates from the divergence of the Coulomb operator at $r_{12} = 0$, a cuspless wave function could equivalently originate from a Hamiltonian with a non-divergent \trashMG{Coulomb} \manu{two-electron} interaction at $r_{12} = 0$.
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 $\mathbb{R}^3$.
The first step of the present basis set correction consists of obtaining an effective two-electron interaction $\W{\Bas}{}(\br{1},\br{2})$ ``mimicking'' the Coulomb operator in a finite basis $\Bas$.
The first step of the present basis set correction consists of obtaining an effective two-electron interaction $\W{\Bas}{}(\br{1},\br{2})$ ``mimicking'' the \manu{effect of the basis set incompleteness on the }Coulomb operator \trashMG{in a finite basis $\Bas$}.
%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 $\Bas \to \infty$.
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.
@ -247,7 +247,7 @@ where
\n{2}{}(\br{1},\br{2})
= \sum_{pqrs \in \Bas} \SO{p}{1} \SO{q}{2} \Gam{pq}{rs} \SO{r}{1} \SO{s}{2},
\end{equation}
and $\Gam{pq}{rs} = \mel*{\wf{}{\Bas}}{ \aic{r}\aic{s}\ai{p}\ai{q} }{\wf{}{\Bas}}$ are the opposite-spin pair density associated with $\wf{}{\Bas}$ and its corresponding tensor (respectively), $\SO{p}{}$ is a spinorbital,
and $\Gam{pq}{rs} = \manu{1/2\mel*{\wf{}{\Bas}}{ \aic{r\downarrow}\aic{s\uparrow}\ai{p\uparrow}\ai{q\downarrow} + \aic{r\uparrow}\aic{s\downarrow}\ai{p\downarrow}\ai{q\uparrow} }{\wf{}{\Bas}}}$ are the opposite-spin pair density associated with $\wf{}{\Bas}$ and its corresponding tensor (respectively), $\SO{p}{}$ is a \trashMG{spinorbital}\manu{spatial orbital},
\begin{equation}
\label{eq:fbasis}
\f{\Bas}{}(\br{1},\br{2})
@ -306,8 +306,9 @@ coincides with the effective interaction at coalescence, i.e.~$\w{}{\lr,\rsmu{\B
%=================================================================
%\subsection{Short-range correlation functionals}
%=================================================================
%Once defined, $\rsmu{\Bas}{}(\br{})$ can be used in RS-DFT functionals to approximate $\bE{}{\Bas}[\n{}{}]$.
As in Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, we consider here a specific class of short-range correlation functionals known as ECMD whose general definition reads \cite{TouGorSav-TCA-05}
\manu{Once defined $\rsmu{\Bas}{}(\br{})$, we can use RS-DFT functionals to approximate $\bE{}{\Bas}[\n{}{}]$, and
}
as in Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, we consider here a specific class of short-range correlation functionals known as ECMD whose general definition reads \cite{TouGorSav-TCA-05}
\begin{multline}
\label{eq:ec_md_mu}
\bE{}{\sr}[\n{}{}(\br{}),\rsmu{}{}]
@ -360,7 +361,7 @@ The LDA version of the ECMD complementary functional is defined as
\label{eq:def_lda_tot}
\bE{\LDA}{\sr}[\n{}{}(\br{}),\rsmu{}{}(\br{})] = \int \be{\LDA}{\sr}\big(\n{}{}(\br{}),\rsmu{}{}(\br{})\big) \n{}{}(\br{}) \dbr{},
\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 \trashMG{short-range} reduced (i.e.~per electron) ECMD of the uniform electron gas (UEG) \cite{LooGil-WIRES-16} parametrized 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 $\mu$.
In order to correct such a defect, we propose here a new PBE ECMD functional
@ -413,7 +414,7 @@ and the corresponding FC range-separation function
\label{eq:muval}
\rsmu{\Bas}{\FC}(\br{}) = \frac{\sqrt{\pi}}{2} \W{\Bas}{\FC}(\br{},\br{}).
\end{equation}
It is worth not\manu{ic}ing that, within the present definition, $\W{\Bas}{\FC}(\br{1},\br{2})$ still satisfies Eq.~\eqref{eq:lim_W}.
It is worth not\manu{ic}ing that, within the present definition, $\W{\Bas}{\FC}(\br{1},\br{2})$ still \trashMG{satisfies Eq.~\eqref{eq:lim_W}} \manu{tends to the regular Coulomb interaction when $\Bas \to \infty$}.
Defining $\n{\modZ}{\FC}$ as the FC (i.e.~valence-only) one-electron density obtained with a model $\modZ$, the FC contribution of the complementary functional is then evaluated as $\bE{\LDA}{\sr}[\n{\modZ}{\FC}(\br{}),\rsmu{\Bas}{\FC}(\br{})]$ or $\bE{\PBE}{\sr}[\n{\modZ}{\FC}(\br{}),\rsmu{\Bas}{\FC}(\br{})]$.
@ -421,7 +422,7 @@ Defining $\n{\modZ}{\FC}$ as the FC (i.e.~valence-only) one-electron density obt
%\subsection{Computational considerations}
%=================================================================
One of the most computationally intensive task of the present approach is the evaluation of $\W{\Bas}{}(\br{})$ [see Eqs.~\eqref{eq:wcoal}] at each quadrature grid point.
Yet embarrassingly parallel, this step scales, in the general (multi-determinantal) case, as $\Ng \Nb^4$ (where $\Nb$ is the number of basis functions in $\Bas$) but is reduced to $\order*{ \Ng \Ne^2 \Nb^2}$ in the case of a single Slater determinant.
Yet embarrassingly parallel, this step scales, in the general (multi-determinantal) case, as \manu{$\Ng \Nb^6$} (where $\Nb$ is the number of basis functions in $\Bas$) but is reduced to $\order*{ \Ng \Ne^2 \Nb^2}$ in the case of a single Slater determinant.
%\begin{equation}
% \label{eq:fcoal}
% \f{\Bas}{\HF}(\br{}) = \sum_{pq \in \Bas} \sum_{ij}^{\occ} \SO{p}{} \SO{q}{} \V{pq}{ij} \SO{i}{} \SO{j}{},
@ -537,7 +538,8 @@ 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}.
Such weak sensitivity when reaching large basis sets shows the robustness of the approach.
As a second set of numerical examples, we compute the correlation energy contribution to the atomization energies of the G2-1 test sets with $\modY=\CCSDT$, $\modZ=\HF$ and the cc-pVXZ basis sets.
As a second set of numerical examples, we compute the \manu{error with respect to the CBS values} of the correlation energy contribution to the atomization energies of the G2-1 test sets with $\modY=\CCSDT$, $\modZ=\HF$ and the cc-pVXZ basis sets.
\manu{Investigating the convergence of correlation energies or difference of such quantities is usually done to appreciate the performance of basis set corrections aiming at correcting two-electron effects\cite{Tenno-CPL-04,TewKloNeiHat-PCCP-07,IrmGru-arXiv-2019}, as these quantities do not contain the HF energy component whose rate of convergence is very different depending on the molecular system. }
The ``plain'' CCSD(T) correlation energies as well as the corrected CCSD(T)+LDA and CCSD(T)+PBE values are depicted in Fig.~\ref{fig:G2_Ec}.
The raw data 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 correlation energies.
@ -549,6 +551,8 @@ With the triple-$\zeta$ basis, the MAD of CCSD(T)+PBE/cc-pVTZ is already below 1
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}.
\titou{Therefore, similar to F12 methods, \cite{TewKloNeiHat-PCCP-07} we can safely claim that the present basis set correction recovers quintuple-$\zeta$ quality correlation energies with triple-$\zeta$ basis sets for a much cheaper computational cost.}
\manu{Encouraged by these results for weakly correlated ground states molecules, ongoing development based on the same strategy point towards the correction of the basis set error for strongly correlated systems, excited states and the treatment of the one-electron error in the basis set incompleteness. }
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section*{Supporting information}