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minor modifs on t2's modifs on the effective operator

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JCTC_revision/GW-srDFT.tex
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@ -436,14 +436,20 @@ 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 which equals its Coulomb counterpart in an incomplete basis, \ie,
\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{
\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}
Because the value of $W^{\Bas}(\br{},\br{}')$ at coalescence, $W^{\Bas}(\br{},\br{})$, is necessarily finite in a finite basis $\Bas$, one can map this operator to a non-divergent, long-range interaction of the form
}
\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
\begin{equation}
\label{eq:Wapprox}
W^{\Bas}(\br{},\br{}')
@ -459,32 +465,42 @@ Knowing that $\lim_{r \to 0} \erf(\mu r)/r = 2\mu/\sqrt{\pi}$, this yields
\end{equation}
}
\titou{A convenient choice for the on-top value of the two-electron effective operator is, for example,
\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{
\begin{equation}
\label{eq:W}
W^{\Bas}(\br{},\br{}) =
W^{\Bas}(\br{},\br{}') =
\begin{cases}
f^{\Bas}(\br{})/\n{2}{\Bas}(\br{}), & \n{2}{\Bas}(\br{}) \neq 0, \\
f^{\Bas}(\br{},\br{}')/\n{2}{\Bas}(\br{},\br{}'), & \n{2}{\Bas}(\br{},\br{}') \neq 0, \\
\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{}) = \sum_{pq}^{\Nbas} \sum_{ij}^{\Nocc} \MO{p}(\br{}) \MO{q}(\br{}) (pi|qj) \MO{i}(\br{}) \MO{j}(\br{}),
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}
and
\begin{equation}
\n{2}{\Bas}(\br{}) = \frac{1}{4} [\n{}{\Bas}(\br{})]^2
\n{2}{\Bas}(\br{}) = n_{\uparrow}^{\Bas}(\br{}) n_{\downarrow}^{\Bas}(\br{}')
\end{equation}
is the opposite-spin on-top pair density in a closed-shell system.
}
is the opposite-spin pair density in a closed-shell system.
\titou{The effective operator $W^{\Bas}(\br{},\br{}')$ has some interesting properties.
\manu{Because of this definition, }
\titou{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 operator, and that the magnitude of the energetic correction tends to zero [see Eq.~\eqref{eq:limSig}].
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{
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.
}