From 9e2144718c8d2d2e549f88b8e403356d23297c2f Mon Sep 17 00:00:00 2001 From: eginer Date: Fri, 6 Dec 2019 16:28:02 +0100 Subject: [PATCH] minor modifs on t2's modifs on the effective operator --- JCTC_revision/GW-srDFT.tex | 42 ++++++++++++++++++++++++++------------ 1 file changed, 29 insertions(+), 13 deletions(-) diff --git a/JCTC_revision/GW-srDFT.tex b/JCTC_revision/GW-srDFT.tex index aab3193..f08e038 100644 --- a/JCTC_revision/GW-srDFT.tex +++ b/JCTC_revision/GW-srDFT.tex @@ -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. }