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Pierre-Francois Loos 2019-12-06 22:12:11 +01:00
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JCTC_revision/GW-srDFT.tex
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@ -436,20 +436,14 @@ where $\eGOWO{\HOMO}$ and $\eGOWO{\LUMO}$ are the HOMO and LUMO orbital energies
\subsection{Basis set correction}
\label{sec:BSC}
%%%%%%%%%%%%%%%%%%%%%%%%
\titou{The fundamental idea behind the present basis set correction is to recognize that the singular two-electron Coulomb interaction $\abs*{\br{} - \br{}'}^{-1}$ projected in a finite basis $\Bas$ is a finite, non-divergent quantity at $\abs*{\br{} - \br{}'} = 0$, which ``resembles'' the long-range interaction operator $\abs*{\br{} - \br{}'}^{-1} \erf(\rsmu{}{} \abs*{\br{} - \br{}'})$ used within RS-DFT. \cite{Giner_2018}}
%We can therefore define an effective two-electron operator $W^{\Bas}(\br{},\br{}')$ which equals its Coulomb counterpart in an incomplete basis, \ie,
\manu{
We can therefore define an effective non divergent two-electron interaction $W^{\Bas}(\br{},\br{}')$ which reproduces the expectation value of the Coulomb interaction over a given two-body density $\n{2}{\Bas}(\br{},\br{}')$, \ie,}
\titou{
\titou{The fundamental idea behind the present basis set correction is to recognize that the singular two-electron Coulomb interaction $\abs*{\br{} - \br{}'}^{-1}$ projected in a finite basis $\Bas$ is a finite, non-divergent quantity at $\abs*{\br{} - \br{}'} = 0$, which ``resembles'' the long-range interaction operator $\abs*{\br{} - \br{}'}^{-1} \erf(\rsmu{}{} \abs*{\br{} - \br{}'})$ used within RS-DFT. \cite{Giner_2018}
We can therefore define an effective non-divergent two-electron interaction $W^{\Bas}(\br{},\br{}')$ which reproduces the expectation value of the Coulomb interaction over a given pair density $\n{2}{\Bas}(\br{},\br{}')$, \ie,
\begin{equation}
\iint \frac{\n{2}{\Bas}(\br{},\br{}')}{\abs*{\br{} - \br{}'}} d\br{} d\br{}'
=
\iint \n{2}{\Bas}(\br{},\br{}') W^{\Bas}(\br{},\br{}') d\br{} d\br{}'.
\end{equation}
}
\manu{The derivation and properties of $W^{\Bas}(\br{},\br{}')$ can be found in details in Ref. \cite{Giner_2018}, but a key aspect is that }
\titou{
because the value of $W^{\Bas}(\br{},\br{}')$ at coalescence, $W^{\Bas}(\br{},\br{})$, is necessarily finite in a finite basis $\Bas$, one can} \manu{fit} \titou{$W^{\Bas}(\br{},\br{}')$ by a non-divergent, long-range interaction of the form
The derivation and properties of $W^{\Bas}(\br{},\br{}')$ are detailed in Ref.~\onlinecite{Giner_2018}, but a key aspect that is worth mentioning here is that because the value of $W^{\Bas}(\br{},\br{}')$ at coalescence, $W^{\Bas}(\br{},\br{})$, is necessarily finite in a finite basis $\Bas$, one can fit $W^{\Bas}(\br{},\br{}')$ by a non-divergent, long-range interaction of the form
\begin{equation}
\label{eq:Wapprox}
W^{\Bas}(\br{},\br{}')
@ -458,17 +452,14 @@ because the value of $W^{\Bas}(\br{},\br{}')$ at coalescence, $W^{\Bas}(\br{},\b
+ \frac{\erf[\rsmu{}{\Bas}(\br{}') \abs*{\br{} - \br{}'}]}{\abs*{\br{} - \br{}'}}
}.
\end{equation}
The information about the finiteness of the basis set has been transferred to the range-separation \textit{function} $\rsmu{}{\Bas}(\br{})$, and its value can be set by ensuring that the two sides of Eq.~\eqref{eq:Wapprox} are strictly equal at $\abs*{\br{} - \br{}'} = 0$.
The information about the finiteness of the basis set is then transferred to the range-separation \textit{function} $\rsmu{}{\Bas}(\br{})$, and its value can be set by ensuring that the two sides of Eq.~\eqref{eq:Wapprox} are strictly equal at $\abs*{\br{} - \br{}'} = 0$.
Knowing that $\lim_{r \to 0} \erf(\mu r)/r = 2\mu/\sqrt{\pi}$, this yields
\begin{equation}
\rsmu{}{\Bas}(\br{}) = \frac{\sqrt{\pi}}{2} W^{\Bas}(\br{},\br{}).
\end{equation}
}
\manu{In Refs. \cite{Giner_2018,Loos_2019} some of the present authors proposed the following definition for the $W^{\Bas}(\br{},\br{}')$}
%\titou{A convenient choice for the on-top value of the two-electron effective operator is, for example,
\titou{
\titou{Following Refs.~\onlinecite{Giner_2018,Loos_2019,Giner_2019}, we adopt the following definition:
\begin{equation}
\label{eq:W}
W^{\Bas}(\br{},\br{}') =
@ -477,7 +468,7 @@ Knowing that $\lim_{r \to 0} \erf(\mu r)/r = 2\mu/\sqrt{\pi}$, this yields
\infty, & \text{otherwise}, \\
\end{cases}
\end{equation}
where, in the case of a single-determinant method (such as HF and KS-DFT),}
where, in the case of a single-determinant method (such as HF and KS-DFT),
\begin{equation}
f^{\Bas}(\br{},\br{}') = \sum_{pq}^{\Nbas} \sum_{ij}^{\Nocc} \MO{p}(\br{})\MO{i}(\br{}) (pi|qj)\MO{q}(\br{}') \MO{j}(\br{}'),
\end{equation}
@ -485,22 +476,17 @@ and
\begin{equation}
\n{2}{\Bas}(\br{},\br{}') = n_{\uparrow}^{\Bas}(\br{}) n_{\downarrow}^{\Bas}(\br{}')
\end{equation}
is the opposite-spin pair density in a closed-shell system.
is the opposite-spin pair density in a closed-shell system, and $n_{\sigma}^{\Bas}(\br{})$ is the spin-$\sigma$ one-electron density.}
\manu{Because of this definition, }
\titou{the effective interaction $W^{\Bas}(\br{},\br{}')$ has some interesting properties.
\titou{Because of this definition, the effective interaction $W^{\Bas}(\br{},\br{}')$ has some interesting properties.
For example, we have
\begin{equation}
\lim_{\Bas \to \CBS} W^{\Bas}(\br{},\br{}') = \abs*{\br{} - \br{}'}^{-1},
\end{equation}
which means that, in the limit of a complete basis, one recovers the genuine (divergent) Coulomb interaction}.
\manu{
Therefore, at the CBS limit the coalescence value $W^{\Bas}(\br{},\br{})$ goes to infinity and so does $\rsmu{}{\Bas}(\br{})$.
As the present basis set correction uses complementary short-range functionals derivatives from RS-DFT which go to zero when $\mu$ goes to infinity,
the present basis set correction vanishes at the CBS limit.
}
\titou{
which means that, in the limit of a complete basis, one recovers the genuine (divergent) Coulomb interaction.
Therefore, in the CBS limit, the coalescence value $W^{\Bas}(\br{},\br{})$ goes to infinity, and so does $\rsmu{}{\Bas}(\br{})$.
As the present basis set correction employs complementary short-range potentials from RS-DFT which have the property of going to zero when $\mu$ goes to infinity,
the present basis set correction vanishes in the CBS limit.
Note also that the divergence condition of $W^{\Bas}(\br{},\br{}')$ in Eq.~\eqref{eq:W} ensures that one-electron systems are free of correction.
}