From 80bc17a9daab1b95b4489aff903f45a959e9a6e9 Mon Sep 17 00:00:00 2001 From: Emmanuel Fromager Date: Thu, 23 Apr 2020 14:05:36 +0200 Subject: [PATCH] Manu: saving work --- Manuscript/FarDFT.tex | 8 +++++--- 1 file changed, 5 insertions(+), 3 deletions(-) diff --git a/Manuscript/FarDFT.tex b/Manuscript/FarDFT.tex index 4686f5f..2dd3f8d 100644 --- a/Manuscript/FarDFT.tex +++ b/Manuscript/FarDFT.tex @@ -361,7 +361,7 @@ where $\hHc(\br{}) = -\nabla^2/2 + \vne(\br{})$, and \end{equation} The ensemble density can be obtained directly (and exactly, if no approximation is made) from those orbitals: -\beq +\beq\label{eq:ens_KS_dens} \n{}{\bw}(\br{})=\sum_{I=0}^{\nEns-1} \ew{I}\left(\sum_{p}^{\nOrb} \ON{p}{(I)} [\MO{p}{\bw}(\br{})]^2\right), \eeq @@ -482,7 +482,8 @@ is the Hxc potential, with \section{Computational details} \label{sec:compdet} -The self-consistent GOK-DFT calculations have been performed in a restricted formalism with the \texttt{QuAcK} software, \cite{QuAcK} which is freely available on \texttt{github}, and where the present weight-dependent functionals have been implemented. +The self-consistent GOK-DFT calculations \manuf{[see Eqs.~(\ref{eq:eKS}) +and (\ref{eq:ens_KS_dens})]} have been performed in a restricted formalism with the \texttt{QuAcK} software, \cite{QuAcK} which is freely available on \texttt{github}, and where the present weight-dependent functionals have been implemented. For more details about the self-consistent implementation of GOK-DFT, we refer the interested reader to Ref.~\onlinecite{Loos_2020} where additional technical details can be found. For all calculations, we use the aug-cc-pVXZ (X = D, T, Q, and 5) Dunning family of atomic basis sets. \cite{Dunning_1989,Kendall_1992,Woon_1994} Numerical quadratures are performed with the \texttt{numgrid} library \cite{numgrid} using 194 angular points (Lebedev grid) and a radial precision of $10^{-6}$. \cite{Becke_1988b,Lindh_2001} @@ -490,7 +491,8 @@ This study deals only with spin-unpolarised systems, \ie, $\n{\uparrow}{} = \n{\ Moreover, we restrict our study to the case of a two-state ensemble (\ie, $\nEns = 2$) where both the ground state ($I=0$ with weight $1 - \ew{}$) and the first doubly-excited state ($I=1$ with weight $\ew{}$) are considered. Although one should have $0 \le \ew{} \le 1/2$ to ensure the GOK variational principle, we will sometimes ``violate'' this variational constraint. Indeed, the limit $\ew{} = 1$ is of particular interest as it corresponds to a genuine saddle point of the KS equations, and match perfectly the results obtained with the maximum overlap method (MOM) developed by Gilbert, Gill and coworkers. \cite{Gilbert_2008,Barca_2018a,Barca_2018b} - +\manu{Maybe we should be more clear about what we mean with $\ew{} = 1$. +In the range $1/2\leq \ew{}\leq 1$, } %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Results} \label{sec:res}