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fix problem with Eq (7)

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Pierre-Francois Loos 2019-11-23 21:40:25 +01:00
parent 4118afd1bb
commit 8b70a52134
1 changed files with 7 additions and 5 deletions
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12
Manuscript/FarDFT.tex
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@ -49,6 +49,7 @@
\newcommand{\F}[2]{F_{#1}^{#2}} \newcommand{\F}[2]{F_{#1}^{#2}}
\newcommand{\Ts}[1]{T_\text{s}^{#1}} \newcommand{\Ts}[1]{T_\text{s}^{#1}}
\newcommand{\eps}[2]{\varepsilon_{#1}^{#2}} \newcommand{\eps}[2]{\varepsilon_{#1}^{#2}}
\newcommand{\Eps}[2]{\mathcal{E}_{#1}^{#2}}
\newcommand{\e}[2]{\epsilon_{#1}^{#2}} \newcommand{\e}[2]{\epsilon_{#1}^{#2}}
\newcommand{\kin}[2]{t_\text{#1}^{#2}} \newcommand{\kin}[2]{t_\text{#1}^{#2}}
\newcommand{\E}[2]{E_{#1}^{#2}} \newcommand{\E}[2]{E_{#1}^{#2}}
@ -91,11 +92,12 @@
\newcommand{\hGamma}[2]{\Hat{\Gamma}_{#1}^{#2}} \newcommand{\hGamma}[2]{\Hat{\Gamma}_{#1}^{#2}}
\newcommand{\eHc}[1]{h_{#1}} \newcommand{\eHc}[1]{h_{#1}}
\newcommand{\eF}[2]{F_{#1}^{#2}} \newcommand{\eF}[2]{F_{#1}^{#2}}
\newcommand{\ON}[2]{f_{#1}^{#2}}
% Numbers % Numbers
\newcommand{\Nens}{M} \newcommand{\Nens}{M}
\newcommand{\Nel}{N} \newcommand{\Nel}{N}
\newcommand{\Nbas}{K} \newcommand{\Norb}{K}
% Ao and MO basis % Ao and MO basis
\newcommand{\MO}[2]{\phi_{#1}^{#2}} \newcommand{\MO}[2]{\phi_{#1}^{#2}}
@ -228,13 +230,13 @@ From the GOK-DFT ensemble energy expression in Eq.~\eqref{eq:Ew-GOK}, we obtain
\label{eq:dEdw} \label{eq:dEdw}
\pdv{\E{}{\bw}}{\ew{I}} \pdv{\E{}{\bw}}{\ew{I}}
= \E{}{(I)} - \E{}{(0)} = \E{}{(I)} - \E{}{(0)}
= \eps{I}{\bw} - \eps{0}{\bw} + \left. \pdv{\E{\xc}{\bw}[\n{}{}]}{\ew{I}} \right|_{\n{}{} = \n{}{\bw}}, = \Eps{I}{\bw} - \Eps{0}{\bw} + \left. \pdv{\E{\xc}{\bw}[\n{}{}]}{\ew{I}} \right|_{\n{}{} = \n{}{\bw}},
\end{equation} \end{equation}
\titou{where $\eps{I}{\bw}$ is the $I$th KS orbital energy (T2: wrong!)} given by the ensemble KS equation where $\Eps{I}{\bw} = \sum_{p}^{\Norb} \ON{p}{(I)} \eps{p}{\bw}$, $\eps{p}{\bw}$ is the $p$th KS orbital energy given by the ensemble KS equation
\begin{equation} \begin{equation}
\qty( -\frac{\nabla^2}{2} + \vext(\br{}) + \fdv{\E{\Hxc}{\bw}[\n{}{}]}{\n{}{}(\br{})}) \MO{p}{\bw}(\br{}) = \eps{p}{\bw} \MO{p}{\bw}(\br{}), \qty( -\frac{\nabla^2}{2} + \vext(\br{}) + \fdv{\E{\Hxc}{\bw}[\n{}{}]}{\n{}{}(\br{})}) \MO{p}{\bw}(\br{}) = \eps{p}{\bw} \MO{p}{\bw}(\br{}),
\end{equation} \end{equation}
(where $\MO{p}{\bw}(\br{})$ is a KS orbital) and $\n{}{\bw} = \sum_{I=0}^{\Nens-1} \ew{I} \n{}{(I)}$ is the ensemble density. (where $\MO{p}{\bw}(\br{})$ is a KS orbital), $\ON{p}{(I)}$ its occupancy for the state $I$, and $\n{}{\bw} = \sum_{I=0}^{\Nens-1} \ew{I} \n{}{(I)}$ is the ensemble density.
Equation \eqref{eq:dEdw} is our working equation for computing excitation energies. Equation \eqref{eq:dEdw} is our working equation for computing excitation energies.
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@ -476,8 +478,8 @@ As concluding remarks, we would like to say that, what we have done is awesome.
%%% ACKNOWLEDGEMENTS %%% %%% ACKNOWLEDGEMENTS %%%
%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%
\begin{acknowledgements} \begin{acknowledgements}
PFL acknowledges funding from the \textit{Centre National de la Recherche Scientifique}.
CM thanks the \textit{Universit\'e Paul Sabatier} (Toulouse, France) for a PhD scholarship. CM thanks the \textit{Universit\'e Paul Sabatier} (Toulouse, France) for a PhD scholarship.
PFL acknowledges funding from the \textit{Centre National de la Recherche Scientifique}.
\end{acknowledgements} \end{acknowledgements}
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