diff --git a/Manuscript/Ex-srDFT.tex b/Manuscript/Ex-srDFT.tex index 40994a3..1b2a975 100644 --- a/Manuscript/Ex-srDFT.tex +++ b/Manuscript/Ex-srDFT.tex @@ -150,6 +150,8 @@ \newcommand{\si}{\sigma} \newcommand{\sis}{\sigma^\star} +\newcommand{\extrfunc}[0]{\epsilon} + \newcommand{\LCPQ}{Laboratoire de Chimie et Physique Quantiques (UMR 5626), Universit\'e de Toulouse, CNRS, UPS, France} \newcommand{\LCT}{Laboratoire de Chimie Th\'eorique (UMR 7616), Universit\'e Pierre et Marie Curie, Sorbonne Universit\'e, CNRS, Paris, France} @@ -272,7 +274,12 @@ which correspond to the WFT limit ($\mu = \infty$) and the DFT limit ($\mu = 0$) In Eq.~\eqref{eq:large_mu_ecmd}, $\Ec[\n{}{}]$ is the usual universal correlation density functional defined in Kohn-Sham DFT. \cite{HohKoh-PR-64, KohSha-PR-65} The key ingredient --- the range-separated function $\rsmu{}{\Bas}(\br{})$ --- automatically adapts to the spatial non-homogeneity of the basis set incompleteness error. -It is defined such that the long-range interaction of RS-DFT, $\w{}{\lr,\mu}(r_{12}) = \erf( \mu r_{12})/r_{12}$, coincides, at coalescence, with an effective two-electron interaction $\W{}{\Bas}(\br{1},\br{2})$ ``mimicking'' the Coulomb operator in an incomplete basis $\Bas$, i.e.~$\w{}{\lr,\rsmu{}{\Bas}(\br{})}(0) = \W{}{\Bas}(\br{},\br{})$ at any $\br{}$. \cite{GinPraFerAssSavTou-JCP-18} +It is defined as +\begin{equation} + \label{eq:def_mu} + \rsmu{}{\Bas}(\br{}) = \frac{\sqrt{\pi}}{2} \W{}{\Bas}(\br{},\br{}) +\end{equation} +such that the long-range interaction of RS-DFT, $\w{}{\lr,\mu}(r_{12}) = \erf( \mu r_{12})/r_{12}$, coincides, at coalescence, with an effective two-electron interaction $\W{}{\Bas}(\br{1},\br{2})$ ``mimicking'' the Coulomb operator in an incomplete basis $\Bas$, i.e.~$\w{}{\lr,\rsmu{}{\Bas}(\br{})}(0) = \W{}{\Bas}(\br{},\br{})$ at any $\br{}$. \cite{GinPraFerAssSavTou-JCP-18} The explicit expression of $\W{}{\Bas}(\br{1},\br{2})$ is given by \begin{equation} \label{eq:def_weebasis} @@ -296,8 +303,7 @@ and $\Gam{pq}{rs} = 2 \mel*{\wf{}{\Bas}}{ \aic{r_\downarrow}\aic{s_\uparrow}\ai{ = \sum_{pqrstu \in \Bas} \SO{p}{1} \SO{q}{2} \V{pq}{rs} \Gam{rs}{tu} \SO{t}{1} \SO{u}{2}, \end{equation} and $\V{pq}{rs}= \braket{pq}{rs}$ are two-electron Coulomb integrals. -An important feature of $\W{}{\Bas}(\br{1},\br{2})$ is that it tends to the regular Coulomb operator $r_{12}^{-1}$ as $\Bas \to \CBS$ , and -ensures that $\bE{}{\Bas}[\n{}{}]$ vanishes when $\Bas$ is complete. +An important feature of $\W{}{\Bas}(\br{1},\br{2})$ is that it tends to the regular Coulomb operator $r_{12}^{-1}$ as $\Bas \to \CBS$ , which implies that $\lim_{\Bas \rightarrow \CBS} \rsmu{}{\Bas}(\br{}) = \infty$ and therefore ensures that $\bE{}{\Bas}[\n{}{}]$ vanishes when $\Bas$ is complete. We refer the interested readers to Refs.~\onlinecite{GinPraFerAssSavTou-JCP-18,LooPraSceTouGin-JPCL-19} for additional details. %%%%%%%%%%%%%%%%%%%%%%%% @@ -312,50 +318,60 @@ The local-density approximation (LDA) of the ECMD complementary functional is de \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.~\citenum{PazMorGorBac-PRB-06}. -To go beyond the LDA and cure its over correlation at small $\mu$, some of the authors recently proposed a Perdew-Burke-Ernzerhof (PBE)-based ECMD functional\cite{LooPraSceTouGin-JPCL-19}, -\begin{equation} - \label{eq:def_pbe_tot} - \bE{\PBE}{\Bas}[\n{}{},\rsmu{}{}] = - \int \n{}{}(\br{}) \be{\text{c,md}}{\sr,\PBE}\qty(\n{}{}(\br{}),s(\br{}),\zeta(\br{}),\rsmu{}{}(\br{})) \dbr{}, -\end{equation} -where $s = \abs{\nabla \n{}{}}/\n{}{4/3}$ is the reduced density gradient. -$\be{\text{c,md}}{\sr,\PBE}\qty(\n{}{},s,\zeta,\rsmu{}{})$ interpolates between the usual PBE correlation functional, \cite{PerBurErn-PRL-96} $\e{\text{c}}{\PBE}(\n{}{},s,\zeta)$, at $\rsmu{}{} = 0$ and the exact large-$\rsmu{}{}$ behavior where the on-top pair density naturally appears. \cite{TouColSav-PRA-04, GorSav-PRA-06, PazMorGorBac-PRB-06} -In order to avoid the computation of the exact on-top pair density, we proposed in Ref.~\onlinecite{LooPraSceTouGin-JPCL-19} to use an approximated alternative which yields -\begin{subequations} -\begin{gather} - \label{eq:epsilon_cmdpbe} - \be{\text{c,md}}{\sr,\PBE}(\n{}{},s,\zeta,\rsmu{}{}) = \frac{\e{\text{c}}{\PBE}(\n{}{},s,\zeta)}{1 + \beta^\PBE(\n{}{},s,\zeta) \rsmu{}{3} }, - \\ - \label{eq:beta_cmdpbe} - \beta^\PBE(\n{}{},s,\zeta) = \frac{3}{2\sqrt{\pi} (1 - \sqrt{2} )} \frac{\e{\text{c}}{\PBE}(\n{}{},s,\zeta)}{\n{2}{\UEG}(\n{}{},\zeta)/\n{}{}}. -\end{gather} -\end{subequations} -where $\n{2}{\UEG}(\n{}{},\zeta) \approx \n{}{2} (1-\zeta^2) g_0(n)$ is the parametrization of the UEG on-top pair-distribution function $g_0(n)$ given in Eq.~(46) of Ref.~\citenum{GorSav-PRA-06}. -We will refer to this functional as the PBE ECMD functional. +The functional purely based on the LDA presents two different defects: i) at small $\mu$ it tends to overestimate the correlation energy, and ii) the quantities based on the UEG are hardly transferable when the system becomes strongly-correlated or multi-configurational. An attempt to solve these problems has been proposed by some of the authors~\cite{FerGinTou-JCP-18} in the context of the RS-DFT where they proposed to connect between the exact behaviour at large $\mu$ and the Perdew-Burke-Ernzerhof (PBE) functional at $\mu=0$. The use of the PBE correlation functional has clearly shown to improve the results for small $\mu$, and the exact behaviour at large $\mu$ naturally introduces the \textit{exact} on-top pair density $n_2(\br{})$ which contains information about the level of strong correlation of the system. -As the UEG on-top pair density might not be suitable to treat strongly correlated systems, \titou{we propose here to use an extrapolation of the exact on-top pair density based on the on-top pair density} -\begin{equation} - \tn{2}{}(\br{},\n{2}{},\rsmu{}{}) = \n{2}{}(\br{},\br{}) \qty(1+ \frac{2}{\sqrt{\pi}\rsmu{}{}(\br{})})^{-1}. -\end{equation} -Therefore, we introduce the ``on top'' PBE (PBEot) ECMD functional which reads +Obviously, the exact on-top pair density cannot be accessed in practice and must be approximated by a function referred here as $\n{2}{X}$, $X$ standing for the label of a given approximation. +Therefore, based on the propositions of ~Ref\cite{FerGinTou-JCP-18}, we introduce the general functional form for the PBE linked complementary functional: \begin{multline} \label{eq:def_pbe_tot} - \bE{\PBEot}{\Bas}[\n{}{},\n{2}{},\rsmu{}{}] = + \bE{\PBE,X}{\Bas}[\n{}{},\n{2}{X},\rsmu{}{}] = \int \n{}{}(\br{}) \\ - \times \be{\text{c,md}}{\sr,\PBEot}\qty(\n{}{}(\br{}),\n{2}{}(\br{},\br{}),s(\br{}),\zeta(\br{}),\rsmu{}{}(\br{})) \dbr{}, + \times \be{\text{c,md}}{\sr,\PBE}\qty(\n{}{}(\br{}),\n{2}{X}(\br{}),s(\br{}),\zeta(\br{}),\rsmu{}{\Bas}(\br{})) \dbr{}, \end{multline} with \begin{subequations} \begin{gather} \label{eq:epsilon_cmdpbe} - \be{\text{c,md}}{\sr,\PBEot}(\n{}{},\n{2}{},s,\zeta,\rsmu{}{}) = \frac{\e{\text{c}}{\PBE}(\n{}{},s,\zeta)}{1 + \beta^\PBEot(\n{}{},\n{2}{},s,\zeta) \rsmu{}{3} }, + \be{\text{c,md}}{\sr,\PBE}(\n{}{},\n{2}{X},s,\zeta,\rsmu{}{}) = \frac{\e{\text{c}}{\PBE}(\n{}{},s,\zeta)}{1 + \beta^\PBE(\n{}{},\n{2}{X},s,\zeta) \rsmu{}{3} }, \\ \label{eq:beta_cmdpbe} - \beta^\PBEot(\n{}{},\n{2}{},s,\zeta) = \frac{3}{2\sqrt{\pi} (1 - \sqrt{2})} \frac{\e{\text{c}}{\PBE}(\n{}{},s,\zeta)}{\tn{2}{}(\n{2}{},\rsmu{}{})/\n{}{}}. + \beta^\PBE(\n{}{},\n{2}{X},s,\zeta) = \frac{3}{2\sqrt{\pi} (1 - \sqrt{2})} \frac{\e{\text{c}}{\PBE}(\n{}{},s,\zeta)}{\n{2}{X}/\n{}{}}. \end{gather} \end{subequations} +In Ref~\cite{LooPraSceTouGin-JPCL-19}, some of the present authors introduced a version, here-referred as "PBE-UEG", where the exact on-top pair density $\n{2}{}(\br{})$ was approximated by that of the UEG. In practice, within the present notation, the PBE-UEG functional reads: +\begin{multline} + \label{eq:def_pbe_tot} + \bE{\PBE,\UEG}{\Bas}[\n{}{},\n{2}{\UEG},\rsmu{}{}] = + \int \n{}{}(\br{}) + \\ + \times \be{\text{c,md}}{\sr,\PBE}\qty(\n{}{}(\br{}),\n{2}{\UEG}(\br{}),s(\br{}),\zeta(\br{}),\rsmu{}{\Bas}(\br{})) \dbr{}, +\end{multline} +with +\begin{equation} + \n{2}{\UEG}(\br{}) = n(\br{})^2 (1-\zeta(\br{})^2) g_0(n(\br{})), +\end{equation} +where the UEG on-top pair-distribution function $g_0(n)$ given in Eq.~(46) of Ref.~\citenum{GorSav-PRA-06}. + +As shown in Ref~\cite{LooPraSceTouGin-JPCL-19}, the link with the usual PBE functional has shown to improve the results over LDA for weakly correlated systems, but the remaining on-top pair density obtained from the UEG might not be suited for the treatment of excited states and/or strongly-correlated systems. Therefore, we propose here a variant inspired by the work of ~Ref\cite{FerGinTou-JCP-18} where we obtain an approximation of the exact on-top pair density based on an extrapolation proposed by Gori-Giorgi in Ref.~\onlinecite{GorSav-PRA-06}. Introducing the extrapolation function $\extrfunc(\n{2}{},\mu)$ +\begin{equation} + \extrfunc(\n{2}{},\mu) = \n{2}{} \qty(1+ \frac{2}{\sqrt{\pi}\mu})^{-1}, +\end{equation} +we propose the following approximation for the on-top pair density: +\begin{equation} + \tn{2}{\Bas}(\br{}) = \extrfunc(\n{2}{\Bas}(\br{}),\rsmu{}{\Bas}(\br{})). +\end{equation} +Such formula relies on the association between $\n{2}{\Bas}(\br{ },\br{ })$, which is the on-top pair density computed in the basis set $\Bas$ in a given point, and the local value of the range-separation parameter $\rsmu{}{\Bas}$ at the same point. +Therefore, we propose the "PBE-ontop" (PBEot) functional which reads +\begin{multline} + \label{eq:def_pbe_tot} + \bE{\PBEot}{\Bas}[\n{}{},\tn{2}{\Bas},\rsmu{}{}] = + \int \n{}{}(\br{}) + \\ + \times \be{\text{c,md}}{\sr,\PBE}\qty(\n{}{}(\br{}),\tn{2}{\Bas}(\br{}),s(\br{}),\zeta(\br{}),\rsmu{}{\Bas}(\br{})) \dbr{}. +\end{multline} +Therefore, the unique difference between the PBE-UEG and PBEot functionals are the approximation for the exact on-top pair density. %%%%%%%%%%%%%%%%%%%%%%%% \section{Computational details} @@ -402,9 +418,9 @@ Total energies for each state can be found in the {\SI}. Figure \ref{fig:CH2} clearly shows that, for the double-$\zeta$ basis, the exFCI adiabatic energies are far from being chemically accurate with errors as high as 0.015 eV. From the triple-$\zeta$ basis onward, the exFCI excitation energies are chemically-accurate though, and converge steadily to the CBS limit when one increases the size of the basis set. Concerning the basis set correction, already at the double-$\zeta$ level, the PBEot correction returns chemically accurate excitation energies. -The performance of the PBE and LDA functionals (which does not require the computation of the on-top density of each state) is less impressive. +The performance of the PBE-UEG and LDA functionals (which does not require the computation of the on-top density of each state) is less impressive. Yet, they still yield significant reductions of the basis set incompleteness error, hence representing a good compromise between computational cost and accuracy. -Note that the results for the PBE functional are not represented in Fig.~\ref{fig:CH2} as they are very similar to the LDA ones (similar considerations apply to the other systems studied below). +Note that the results for the PBE-UEG functional are not represented in Fig.~\ref{fig:CH2} as they are very similar to the LDA ones (similar considerations apply to the other systems studied below). It is also quite evident that, the basis set correction has the tendency of over-correcting the excitation energies via an over-stabilization of the excited states compared to the ground state. This trend is quite systematic as we shall see below. @@ -458,7 +474,7 @@ This trend is quite systematic as we shall see below. & $1.378$ [$-0.011$] & $2.489$ [$-0.016$] \\ \\ - exFCI+PBE & AVDZ + exFCI+PBE-UEG & AVDZ & $0.308$ [$-0.080$] & $1.388$ [$-0.002$] & $2.560$ [$+0.056$] \\ @@ -555,7 +571,7 @@ However, these results also clearly evidence that special care has to be taken f \cline{5-16} & & & & \mc{3}{c}{exFCI} & \mc{3}{c}{exFCI+PBEot} - & \mc{3}{c}{exFCI+PBE} + & \mc{3}{c}{exFCI+PBE-UEG} & \mc{3}{c}{exFCI+LDA} \\ \cline{5-7} \cline{8-10} \cline{11-13} \cline{14-16} @@ -715,7 +731,7 @@ They have been recently studied with state-of-the-art methods, and have been sho The vertical excitation energies associated with these transitions are reported in Table \ref{tab:Mol} and represented in Fig.~\ref{fig:C2}. An interesting point here is that one really needs to consider the PBEot functional to get chemically-accurate absorption energies with the AVDZ atomic basis set. We believe that the present result is a direct consequence of the multireference character of the \ce{C2} molecule. -In other words, the UEG on-top density used in the LDA and PBE functionals (see Sec.~\ref{sec:func}) is a particularly bad approximation of the true on-top density for the present system. +In other words, the UEG on-top density used in the LDA and PBE-UEG functionals (see Sec.~\ref{sec:func}) is a particularly bad approximation of the true on-top density for the present system. %%% FIG 5 %%% \begin{figure} @@ -758,7 +774,7 @@ In the present context, ethylene is a particularly interesting system as it cont Our basis set corrected vertical excitation energies are gathered in Table \ref{tab:Mol} and depicted in Fig.~\ref{fig:C2H4}. Except for one particular excitation (the lowest singlet-triplet excitation $1\,^{1}A_{1g} \ra 1\,^{3}B_{1u}$), the exFCI+PBEot/AVDZ excitation energies are chemically accurate and the errors drop further when one goes to the triple-$\zeta$ basis. %(Note that one cannot afford exFCI/AVQZ calculations for ethylene.) -Consistently with the previous examples, the LDA and PBE functionals are slightly less accurate, although they still correct the excitation energies in the right direction. +Consistently with the previous examples, the LDA and PBE-UEG functionals are slightly less accurate, although they still correct the excitation energies in the right direction. %%% FIG 6 %%% \begin{figure} @@ -777,7 +793,7 @@ Consistently with the previous examples, the LDA and PBE functionals are slightl We have shown that, by employing the recently proposed density-based basis set correction developed by some of the authors, \cite{GinPraFerAssSavTou-JCP-18} one can obtain, using sCI methods, chemically-accurate excitation energies with typically augmented double-$\zeta$ basis sets. This nicely complements our recent investigation on ground-state properties, \cite{LooPraSceTouGin-JPCL-19} which has evidenced that one recovers quintuple-$\zeta$ quality atomization and correlation energies with triple-$\zeta$ basis sets. The present study clearly shows that, for very diffuse excited states, the present correction relying on short-range correlation functionals from RS-DFT might not be enough to catch the radial incompleteness of the one-electron basis set. -Also, in the case of multireference systems, we have evidenced that the PBEot functional is more appropriate than the LDA and PBE functionals relying on the UEG on-top density. +Also, in the case of multireference systems, we have evidenced that the PBEot functional is more appropriate than the LDA and PBE-UEG functionals relying on the UEG on-top density. We are currently investigating the performance of the present basis set correction for strongly correlated systems and we hope to report on this in the near future. %%%%%%%%%%%%%%%%%%%%%%%%