From e1c56ef7687fb294f82dd8d93db55aacb3469dcf Mon Sep 17 00:00:00 2001 From: Pierre-Francois Loos Date: Fri, 6 Dec 2019 22:12:11 +0100 Subject: [PATCH] Done with theory section --- JCTC_revision/GW-srDFT.tex | 38 ++++++++++++-------------------------- 1 file changed, 12 insertions(+), 26 deletions(-) diff --git a/JCTC_revision/GW-srDFT.tex b/JCTC_revision/GW-srDFT.tex index 763f8cb..a0b6f0e 100644 --- a/JCTC_revision/GW-srDFT.tex +++ b/JCTC_revision/GW-srDFT.tex @@ -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. }