diff --git a/Manuscript/G2-srDFT.tex b/Manuscript/G2-srDFT.tex index ab2b81f..0e6e2d9 100644 --- a/Manuscript/G2-srDFT.tex +++ b/Manuscript/G2-srDFT.tex @@ -136,11 +136,10 @@ \begin{abstract} We report a universal density-based basis set incompleteness correction that can be applied to any wave function method. -The present correction 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 and appropriately vanishes in the complete basis set (CBS) limit. -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) atomization energies near the CBS limit for the G2-1 set of molecules with compact Gaussian basis sets. -For example, while CCSD(T)/cc-pVTZ yields a mean absolute deviation (MAD) of 7.79 kcal/mol compared to CCSD(T)/CBS atomization energies, the CCSD(T)+LDA and CCSD(T)+PBE corrected methods return MAD of 2.89 and 2.46 kcal/mol (respectively) with the same basis. -These values drop below 1 {\kcal} with the cc-pVQZ basis set. +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. +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} \maketitle @@ -256,8 +255,8 @@ and $\Gam{pq}{rs} = \mel*{\wf{}{\Bas}}{ \aic{r}\aic{s}\ai{p}\ai{q} }{\wf{}{\Bas} \end{equation} 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. -\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.} +%\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.} With such a definition, $\W{\Bas}{}(\br{1},\br{2})$ satisfies (see Appendix A of Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}) \begin{equation} \label{eq:int_eq_wee} @@ -275,9 +274,9 @@ An important quantity to define in the present context is the value of the effec \label{eq:wcoal} \W{\Bas}{}(\br{}) = \W{\Bas}{}(\br{},{\br{}}), \end{equation} -which is necessarily \textit{finite} for an incomplete basis set as long as the on-top pair density $\n{2}{}(\br{},\br{})$ is non vanishing. - -Of course, there exists \textit{a priori} an infinite set of functions in $\mathbb{R}^6$ satisfying \eqref{eq:int_eq_wee}, but thanks to its definition one can show that (see Appendix B of Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}) +which is necessarily \textit{finite} for an incomplete basis set as long as the on-top pair density $\n{2}{}(\br{})$ is non vanishing. +%Of course, there exists \textit{a priori} an infinite set of functions in $\mathbb{R}^6$ satisfying \eqref{eq:int_eq_wee}, but +Thanks to its definition one can show that (see Appendix B of Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}) \begin{equation} \label{eq:lim_W} \lim_{\Bas \to \infty}\W{\Bas}{}(\br{1},\br{2}) = r_{12}^{-1}\ @@ -290,7 +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} %================================================================= -\alert{Because the Coulomb operator within a basis set $\Bas$ is a non divergent two-electron interaction, we can straightforwardly link the present theory with the RS-DFT which uses the so-called long-range interaction which are smooth bounded two-electron operators.} +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. Although this choice is not unique, we choose here the range-separation function \begin{equation} \label{eq:mu_of_r} @@ -300,9 +299,9 @@ such that the long-range interaction \begin{equation} \w{}{\lr,\rsmu{\Bas}{}}(\br{1},\br{2}) = \frac{1}{2} \qty{ \frac{\erf[ \rsmu{\Bas}{}(\br{1}) r_{12}]}{r_{12}} + \frac{\erf[ \rsmu{\Bas}{}(\br{2}) r_{12}]}{ r_{12}} } \end{equation} -\PFL{This expression looks like a cheap spherical average. -What about $\rsmu{\Bas}{}(\br{1},\br{2}) = \sqrt{\rsmu{\Bas}{}(\br{1}) \rsmu{\Bas}{}(\br{2})}$ and a proper spherical average to get $\rsmu{\Bas}{}(r_{12})$?} coincides with the effective interaction at coalescence, i.e.~$\w{}{\lr,\rsmu{\Bas}{}}(\br{},\br{}) = \W{\Bas}{}(\br{})$. +%\PFL{This expression looks like a cheap spherical average. +%What about $\rsmu{\Bas}{}(\br{1},\br{2}) = \sqrt{\rsmu{\Bas}{}(\br{1}) \rsmu{\Bas}{}(\br{2})}$ and a proper spherical average to get $\rsmu{\Bas}{}(r_{12})$?} %================================================================= %\subsection{Short-range correlation functionals} @@ -327,7 +326,7 @@ with $\hWee{\lr,\rsmu{}{}} = \sum_{i