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Manuscript/Ex-srDFT.tex
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@ -274,11 +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 as
\manu{
It is defined as follows
\begin{equation}
\label{eq:def_mu}
\rsmu{}{\Bas}(\br{}) = \frac{\sqrt{\pi}}{2} \W{}{\Bas}(\br{},\br{})
\end{equation}
\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}
@ -303,7 +304,7 @@ 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 $\Bas \to \CBS$ , which implies that $\lim_{\Bas \rightarrow \CBS} \rsmu{}{\Bas}(\br{}) = \infty$ and therefore 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 $\Bas \to \CBS$\manu{, 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.
%%%%%%%%%%%%%%%%%%%%%%%%
@ -318,8 +319,7 @@ 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}.
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.
\manu{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. \\
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}
@ -339,7 +339,6 @@ with
\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}
@ -352,8 +351,7 @@ 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}.
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 \textit{et. al} 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},
@ -372,6 +370,7 @@ Therefore, we propose the "PBE-ontop" (PBEot) functional which reads
\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}
@ -746,10 +745,20 @@ In other words, the UEG on-top density used in the LDA and PBE-UEG functionals (
It is interesting to study the behavior of the key quantities involved in the basis set correction for different states as the basis set incompleteness error is obviously state specific.
%\manu{To do so, we report the value of the range separation parameter in real space $\rsmu{}{\Bas}(\br{})$, the value of the energetic correction $\be{\text{c,md}}{\sr,\PBE}\qty(\n{}{}(\br{}),s(\br{}),\zeta(\br{}),\rsmu{}{\Bas}(\br{}) $ and the on-top pair density $\n{2}{\Bas}(\br{},\br{})$ computed with different basis sets for the ground state and second excited state of the carbon dimer which are both of $\Sigma_g^+$ symmetry, in Figures . }
We report $\rsmu{}{\Bas}(z)$, along the nuclear axis ($z$) for the two $^1 \Sigma_g^+$ electronic states of \ce{C2} computed with the AVDZ, AVTZ and AVQZ basis sets.
\manu{These figures illustrate several important things:
i) the maximal values of $\rsmu{}{\Bas}(\br{})$ are systematically close to the nuclei, a signature of the atom-centered basis set,
ii) the overall values of $\rsmu{}{\Bas}(\br{})$ increase with the basis set, which reflects the improvement of the description of the correlation effects when enlarging the basis set,
iii) the value of $\rsmu{}{\Bas}(\br{})$ are slightly larger near the oxygen atom, which traduces the fact that the inter-electronic distance is higher than close to the carbon atom due to a higher nuclear charge. }
\manu{These figures illustrate several general features regarding the present basis set correction:
\begin{itemize}
\item the maximal values of $\rsmu{}{\Bas}(\br{})$ are systematically close to the nuclei, a signature of the atom-centered basis set,
\item the overall values of $\rsmu{}{\Bas}(\br{})$ increase with the basis set, which reflects the improvement of the description of the correlation effects when enlarging the basis set,
\item the value of the energetic correction decreases, in absolute value, while increasing the basis set,
\item there is a clear correspondence between the value of the energetic correction and that of the on-top pair density.
\end{itemize}
Regarding now the differential effect of the basis set correction in the special case of the two states studied here, we can observe that:
\begin{itemize}
\item the value of $\rsmu{}{\Bas}(\br{})$ is overall the same between the two states, even if slightly more shaped in the case of the ground state. Such feature is coherent with the fact that the two states considered are both of $\Sigma$ symmetry and of valence character.
\item The on-top pair density is overall larger in the excited state, specially in the bonding region and the external parts. This is to be related to the fact that the excited state can be qualitatively described by a double excitation from $\pi$ orbitals to $\sigma$ orbitals, therefore increasing the overall population on the bond axis.
\item The energetic correction clearly stabilizes more the excited state than the ground state, illustrating that short-range correlation effects are more present in the former than the latter. This is to be linked with the value of the on-top pair density which is larger in the excited state.
\end{itemize}
}
%%% FIG 4 %%%
\begin{figure*}