poster

beamercolorthemegemini.sty

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+% Gemini theme
+% https://github.com/anishathalye/gemini
+
+% ====================
+% Definitions
+% ====================
+
+\definecolor{lightgray}{RGB}{245, 246, 250}
+\definecolor{blue}{RGB}{64, 115, 158}
+\definecolor{darkblue}{RGB}{39, 60, 117}
+\definecolor{lightblue}{RGB}{232, 244, 255}
+
+% ====================
+% Theme
+% ====================
+
+% Basic colors
+\setbeamercolor{palette primary}{fg=black,bg=white}
+\setbeamercolor{palette secondary}{fg=black,bg=white}
+\setbeamercolor{palette tertiary}{bg=black,fg=white}
+\setbeamercolor{palette quaternary}{fg=black,bg=white}
+\setbeamercolor{structure}{fg=darkblue}
+
+% Headline
+\setbeamercolor{headline}{fg=lightgray,bg=blue}
+\setbeamercolor{headline rule}{bg=darkblue}
+
+% Block
+\setbeamercolor{block title}{fg=blue,bg=white}
+\setbeamercolor{block separator}{bg=black}
+\setbeamercolor{block body}{fg=black,bg=white}
+
+% Alert Block
+\setbeamercolor{block alerted title}{fg=blue,bg=lightblue}
+\setbeamercolor{block alerted separator}{bg=black}
+\setbeamercolor{block alerted body}{fg=black,bg=lightblue}
+
+% Example Block
+\setbeamercolor{block example title}{fg=blue,bg=lightgray}
+\setbeamercolor{block example separator}{bg=black}
+\setbeamercolor{block example body}{fg=black,bg=lightgray}
+
+% Heading
+\setbeamercolor{heading}{fg=black}
+
+% Itemize
+\setbeamercolor{item}{fg=darkblue}
+
+% Bibliography
+\setbeamercolor{bibliography item}{fg=black}
+\setbeamercolor{bibliography entry author}{fg=black}
+\setbeamercolor{bibliography entry title}{fg=black}
+\setbeamercolor{bibliography entry location}{fg=black}
+\setbeamercolor{bibliography entry note}{fg=black}
+\setbeamertemplate{bibliography entry article}{}
+\setbeamertemplate{bibliography entry title}{}
+\setbeamertemplate{bibliography entry location}{}
+\setbeamertemplate{bibliography entry note}{}

beamercolorthemelabsix.sty

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+% Gemini theme
+% https://github.com/anishathalye/gemini
+
+% ====================
+% Definitions
+% ====================
+
+\definecolor{labsixorange}{RGB}{243, 111, 33}
+
+% Extra colors
+\definecolor{lightgray}{RGB}{240, 240, 240}
+\definecolor{lightorange}{RGB}{255, 240, 230}
+
+% ====================
+% Theme
+% ====================
+
+% Basic colors
+\setbeamercolor{palette primary}{fg=black,bg=white}
+\setbeamercolor{palette secondary}{fg=black,bg=white}
+\setbeamercolor{palette tertiary}{bg=black,fg=white}
+\setbeamercolor{palette quaternary}{fg=black,bg=white}
+\setbeamercolor{structure}{fg=labsixorange}
+
+% Headline
+\setbeamercolor{headline}{fg=white,bg=labsixorange}
+\setbeamercolor{headline rule}{bg=black}
+
+% Block
+\setbeamercolor{block title}{fg=labsixorange,bg=white}
+\setbeamercolor{block separator}{bg=black}
+\setbeamercolor{block body}{fg=black,bg=white}
+
+% Alert Block
+\setbeamercolor{block alerted title}{fg=labsixorange,bg=lightorange}
+\setbeamercolor{block alerted separator}{bg=black}
+\setbeamercolor{block alerted body}{fg=black,bg=lightorange}
+
+% Example Block
+\setbeamercolor{block example title}{fg=labsixorange,bg=lightgray}
+\setbeamercolor{block example separator}{bg=black}
+\setbeamercolor{block example body}{fg=black,bg=lightgray}
+
+% Heading
+\setbeamercolor{heading}{fg=black}
+
+% Itemize
+\setbeamercolor{item}{fg=labsixorange}
+
+% Bibliography
+\setbeamercolor{bibliography item}{fg=black}
+\setbeamercolor{bibliography entry author}{fg=black}
+\setbeamercolor{bibliography entry title}{fg=black}
+\setbeamercolor{bibliography entry location}{fg=black}
+\setbeamercolor{bibliography entry note}{fg=black}

beamercolorthememit.sty

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+% Gemini theme
+% https://github.com/anishathalye/gemini
+
+% ====================
+% Definitions
+% ====================
+
+% Colors from http://web.mit.edu/graphicidentity/colors.html
+\definecolor{mitred}{cmyk}{0.24, 1.0, 0.78, 0.17}
+\definecolor{mitdarkgray}{cmyk}{0.48, 0.39, 0.39, 0.04}
+\definecolor{mitlightgray}{cmyk}{0.24, 0.20, 0.20, 0.0}
+
+% Extra colors
+\definecolor{lightgray}{RGB}{240, 240, 240}
+\definecolor{lightorange}{RGB}{255, 245, 242}
+
+% ====================
+% Theme
+% ====================
+
+% Basic colors
+\setbeamercolor{palette primary}{fg=black,bg=white}
+\setbeamercolor{palette secondary}{fg=black,bg=white}
+\setbeamercolor{palette tertiary}{bg=black,fg=white}
+\setbeamercolor{palette quaternary}{fg=black,bg=white}
+\setbeamercolor{structure}{fg=mitred}
+
+% Headline
+\setbeamercolor{headline}{fg=black,bg=lightgray}
+
+% Block
+\setbeamercolor{block title}{fg=mitred,bg=white}
+\setbeamercolor{block separator}{bg=black}
+\setbeamercolor{block body}{fg=black,bg=white}
+
+% Alert Block
+\setbeamercolor{block alerted title}{fg=mitred,bg=lightorange}
+\setbeamercolor{block alerted separator}{bg=black}
+\setbeamercolor{block alerted body}{fg=black,bg=lightorange}
+
+% Example Block
+\setbeamercolor{block example title}{fg=mitred,bg=lightgray}
+\setbeamercolor{block example separator}{bg=black}
+\setbeamercolor{block example body}{fg=black,bg=lightgray}
+
+% Heading
+\setbeamercolor{heading}{fg=black}
+
+% Itemize
+\setbeamercolor{item}{fg=mitred}
+
+% Bibliography
+\setbeamercolor{bibliography item}{fg=black}
+\setbeamercolor{bibliography entry author}{fg=black}
+\setbeamercolor{bibliography entry title}{fg=black}
+\setbeamercolor{bibliography entry location}{fg=black}
+\setbeamercolor{bibliography entry note}{fg=black}

beamerthemegemini.sty

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+% Gemini theme
+% https://github.com/anishathalye/gemini
+
+% ====================
+% Dependencies
+% ====================
+
+\RequirePackage{raleway}
+\RequirePackage{exscale}
+\RequirePackage{ragged2e}
+\RequirePackage{changepage}
+\RequirePackage{fontspec}
+\RequirePackage{calc}
+
+% ====================
+% Fonts
+% ====================
+
+\newfontfamily\Raleway[Ligatures=TeX]{Raleway}
+\newfontfamily\Lato[Ligatures=TeX]{Lato}
+
+\usefonttheme{professionalfonts}
+
+\setsansfont{Lato}[
+  UprightFont=*-Light,
+  ItalicFont=*-LightItalic,
+  BoldFont=*-Regular,
+  BoldItalicFont=*-Italic
+]
+
+\setbeamerfont{headline}{family=\Raleway}
+\setbeamerfont{headline title}{size=\Huge,series=\bfseries}
+\setbeamerfont{headline author}{size=\Large}
+\setbeamerfont{headline institute}{size=\normalsize}
+\setbeamerfont{block title}{family=\Raleway,size=\large,series=\bfseries}
+\setbeamerfont{heading}{family=\Lato,series=\bfseries}
+\setbeamerfont{caption}{size=\small}
+\setbeamerfont{footline}{family=\Raleway,size=\normalsize}
+
+% ====================
+% Macros
+% ====================
+
+\newcommand{\samelineand}{\qquad}
+
+% ====================
+% Elements
+% ====================
+
+% List
+\def\@listi{\leftmargin\leftmarginii
+\topsep 1ex % spacing before
+\parsep 0\p@ \@plus\p@
+\itemsep 0.5ex} % spacing between
+
+% Itemize
+
+\setbeamertemplate{itemize item}{\raise0.5ex \hbox{\vrule width 0.5ex height 0.5ex}}
+\setbeamertemplate{itemize subitem}{\raise0.3ex \hbox{\vrule width 0.5ex height 0.5ex}}
+\setbeamertemplate{itemize subsubitem}{\raise0.2ex \hbox{\vrule width 0.5ex height 0.5ex}}
+
+% Enumerate
+
+\setbeamertemplate{enumerate item}{\insertenumlabel.}
+\setbeamertemplate{enumerate subitem}{\insertsubenumlabel.}
+\setbeamertemplate{enumerate subsubitem}{\insertsubsubenumlabel.}
+
+% Equation
+\setlength\belowdisplayshortskip{2ex}
+
+% Caption
+\setbeamertemplate{caption}[numbered]
+\setbeamertemplate{caption label separator}[period]
+\setlength{\abovecaptionskip}{2ex}
+\setlength{\belowcaptionskip}{1ex}
+
+% Bibliography
+\setbeamertemplate{bibliography item}[text]
+
+% Navigation
+\beamertemplatenavigationsymbolsempty
+
+% ====================
+% Components
+% ====================
+
+% Heading
+\newcommand\heading[1]
+{%
+  \par\bigskip
+  {\usebeamerfont{heading}\usebeamercolor[fg]{heading}#1}\par\smallskip
+}
+
+% logo
+\newlength{\logoleftwidth}
+\setlength{\logoleftwidth}{0cm}
+\newlength{\logorightwidth}
+\setlength{\logorightwidth}{0cm}
+\newlength{\maxlogowidth}  % space on both sides set to maxlogowidth to keep title centered
+\setlength{\maxlogowidth}{0cm}
+
+\newcommand{\logoright}[1]{
+  \newcommand{\insertlogoright}{#1}
+  \settowidth{\logorightwidth}{\insertlogoright}
+  \addtolength{\logorightwidth}{10ex}
+  \setlength{\maxlogowidth}{\maxof{\logoleftwidth}{\logorightwidth}}
+}
+\newcommand{\logoleft}[1]{
+  \newcommand{\insertlogoleft}{#1}
+  \settowidth{\logoleftwidth}{\insertlogoleft}
+  \addtolength{\logoleftwidth}{10ex}
+  \setlength{\maxlogowidth}{\maxof{\logoleftwidth}{\logorightwidth}}
+}
+
+% Headline
+\setbeamertemplate{headline}
+{
+  \begin{beamercolorbox}{headline}
+    \begin{columns}
+      \begin{column}{\maxlogowidth}
+        \vskip5ex
+        \ifdefined\insertlogoleft
+        \vspace*{\fill}
+        \hspace{10ex}
+        \raggedright
+        \insertlogoleft
+        \vspace*{\fill}
+        \else\fi
+      \end{column}
+      \begin{column}{\dimexpr\paperwidth-\maxlogowidth-\maxlogowidth}
+        \usebeamerfont{headline}
+        \vskip3ex
+        \centering
+        \ifx \inserttitle \empty \else
+        {\usebeamerfont{headline title}\usebeamercolor[fg]{headline title}\inserttitle\\[0.5ex]}
+        \fi
+        \ifx \beamer@shortauthor \empty \else
+        {\usebeamerfont{headline author}\usebeamercolor[fg]{headline author}\insertauthor\\[1ex]}
+        \fi
+        \ifx \insertinstitute \empty \else
+        {\usebeamerfont{headline institute}\usebeamercolor[fg]{headline institute}\insertinstitute\\[1ex]}
+        \fi
+      \end{column}
+      \begin{column}{\maxlogowidth}
+        \vskip5ex
+        \ifdefined\insertlogoright
+        \vspace*{\fill}
+        \raggedleft
+        \insertlogoright
+        \hspace{10ex}
+        \vspace*{\fill}
+        \else\fi
+      \end{column}
+    \end{columns}
+    \vspace{5ex}
+    \ifbeamercolorempty[bg]{headline rule}{}{
+      \begin{beamercolorbox}[wd=\paperwidth,colsep=0.5ex]{headline rule}\end{beamercolorbox}
+    }
+  \end{beamercolorbox}
+}
+
+% Block
+\setbeamertemplate{block begin}
+{
+  \begin{beamercolorbox}[colsep*=0ex,dp=2ex,center]{block title}
+    \vskip0pt
+    \usebeamerfont{block title}\insertblocktitle
+    \vskip-1.25ex
+    \begin{beamercolorbox}[colsep=0.025ex]{block separator}\end{beamercolorbox}
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+  {\parskip0pt\par}
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+  \justifying
+  \setlength{\parskip}{1ex}
+  \vskip-2ex
+}
+\setbeamertemplate{block end}
+{
+  \end{beamercolorbox}
+  \vskip0pt
+  \vspace*{2ex}
+}
+
+% Alert Block
+\setbeamertemplate{block alerted begin}
+{
+  \begin{beamercolorbox}[colsep*=0ex,dp=2ex,center]{block alerted title}
+    \vskip0pt
+    \usebeamerfont{block title}\insertblocktitle
+    \vskip-1.25ex
+    \begin{beamercolorbox}[colsep=0.025ex]{block alerted separator}\end{beamercolorbox}
+  \end{beamercolorbox}
+  {\parskip0pt\par}
+  \usebeamerfont{block body}
+  \vskip-0.5ex
+  \begin{beamercolorbox}[colsep*=0ex]{block alerted body}
+  \justifying
+  \begin{adjustwidth}{1ex}{1ex}
+  \setlength{\parskip}{1ex}
+  \vskip-2ex
+}
+\setbeamertemplate{block alerted end}
+{
+  \end{adjustwidth}
+  \vskip1ex
+  \end{beamercolorbox}
+  \vskip0pt
+  \vspace*{2ex}
+}
+
+% Example Block
+\setbeamertemplate{block example begin}
+{
+  \begin{beamercolorbox}[colsep*=0ex,dp=2ex,center]{block example title}
+    \vskip0pt
+    \usebeamerfont{block title}\insertblocktitle
+    \vskip-1.25ex
+    \begin{beamercolorbox}[colsep=0.025ex]{block example separator}\end{beamercolorbox}
+  \end{beamercolorbox}
+  {\parskip0pt\par}
+  \usebeamerfont{block body}
+  \vskip-0.5ex
+  \begin{beamercolorbox}[colsep*=0ex]{block example body}
+  \justifying
+  \begin{adjustwidth}{1ex}{1ex}
+  \setlength{\parskip}{1ex}
+  \vskip-2ex
+}
+\setbeamertemplate{block example end}
+{
+  \end{adjustwidth}
+  \vskip1ex
+  \end{beamercolorbox}
+  \vskip0pt
+  \vspace*{2ex}
+}
+
+% Footer
+\newcommand{\footercontent}[1]{\newcommand{\insertfootercontent}{#1}}
+
+\setbeamertemplate{footline}{
+  \ifdefined\insertfootercontent
+  \begin{beamercolorbox}[vmode]{headline}
+    \ifbeamercolorempty[bg]{headline rule}{}{
+      \begin{beamercolorbox}[wd=\paperwidth,colsep=0.25ex]{headline rule}\end{beamercolorbox}
+    }
+    \vspace{1.5ex}
+    \hspace{\sepwidth}
+    \usebeamerfont{footline}
+    \centering
+    \insertfootercontent
+    \hspace{\sepwidth}
+    \vspace{1.5ex}
+  \end{beamercolorbox}
+  \else\fi
+}

eesws_poster.tex

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+\documentclass[final]{beamer}
+%% Possible paper sizes: a0, a0b, a1, a2, a3, a4.
+%% Possible orientations: portrait, landscape
+%% Font sizes can be changed using the scale option.
+\usepackage[size=a0,orientation=portrait,scale=1.2]{beamerposter}
+\usepackage[backend=biber,style=numeric,sorting=none]{biblatex}
+\addbibresource{HKvsLieb.bib}
+\usepackage{mathtools}
+
+\newcommand{\red}[1]{\textcolor{red}{#1}}
+\newcommand{\blue}[1]{\textcolor{blue}{#1}}
+\newcommand{\green}[1]{\textcolor{green}{#1}}
+\newcommand{\purple}[1]{\textcolor{purple}{#1}}
+\newcommand{\hH}{\Hat{H}} 
+\newcommand{\hT}{\Hat{T}} 
+\newcommand{\hV}{\Hat{V}} 
+\newcommand{\hn}{\Hat{n}}
+\newcommand{\cre}[1]{a_{#1}^\dagger}
+\newcommand{\ani}[1]{a_{#1}} 
+
+\newcommand{\br}{\boldsymbol{r}} 
+
+\newcommand{\up}{\uparrow} 
+\newcommand{\dw}{\downarrow} 
+\newcommand{\Dv}{\Delta v} 
+\newcommand{\Dn}{\Delta n} 
+\newcommand{\ie}{\textit{i.e.}\xspace}
+
+\AtEveryBibitem{%
+\ifentrytype{article}{
+    \clearfield{url}%
+    \clearfield{urlyear}%
+    \clearfield{doi}%
+    \clearfield{issn}%
+    \clearfield{month}%
+    \clearfield{day}%
+    \clearfield{title}%
+}{}
+}
+
+\DeclareFieldFormat[article]{pages}{#1}
+
+\renewbibmacro{in:}{%
+  \ifentrytype{article}{}{\printtext{\bibstring{in}\intitlepunct}}}
+
+\usetheme{gemini}
+\usecolortheme{gemini}
+\useinnertheme{rectangles}
+
+% ====================
+% Packages
+% ====================
+
+\usepackage[utf8]{inputenc}
+\usepackage{graphicx,dcolumn,xcolor,microtype,multirow,amscd,amsmath,amssymb,amsfonts,physics,wrapfig,tikz,siunitx,bm}
+\usepackage{booktabs}
+\usepackage{tikz}
+\usepackage{pgfplots}
+
+\newfontfamily\qp{Times New Roman}[
+  % the font has no small caps, so we use another one for them
+  SmallCapsFont=TeX Gyre Termes,
+  SmallCapsFeatures={Letters=SmallCaps}
+]
+
+
+% ====================
+% Lengths
+% ====================
+
+% If you have N columns, choose \sepwidth and \colwidth such that
+% (N+1)*\sepwidth + N*\colwidth = \paperwidth
+\newlength{\sepwidth}
+\newlength{\colwidth}
+\setlength{\sepwidth}{0.03\paperwidth}
+\setlength{\colwidth}{0.45\paperwidth}
+
+\newcommand{\separatorcolumn}{\begin{column}{\sepwidth}\end{column}}
+
+
+% ====================
+% Logo (optional)
+% ====================
+
+% LaTeX logo taken from https://commons.wikimedia.org/wiki/File:LaTeX_logo.svg
+% use this to include logos on the left and/or right side of the header:
+\logoright{\vspace{6cm} \includegraphics[width=10cm]{fig/Fermi.jpg}}
+\logoleft{\vspace{-6cm} \includegraphics[width=10cm]{fig/ERC.jpg}}% \hspace{1cm} \includegraphics[height=6cm]{fig/CNRS.png} }
+\addtobeamertemplate{headline}{} 
+{
+\begin{tikzpicture}[remember picture,overlay] 
+\node [shift={(7.5 cm,-3.5cm)}] at (current page.north west) {\includegraphics[height=6cm]{fig/CNRS.png}}; 
+\node [shift={(-7.25 cm,-4.5cm)}] at (current page.north east) {\includegraphics[width=10cm]{fig/LCPQ.jpg}}; 
+\end{tikzpicture} 
+}
+
+% ====================
+% Footer (optional)
+% ====================
+
+\footercontent{
+	\href{mailto:sgiarrusso@irsamc.ups-tlse.fr}{\texttt{sgiarrusso@irsamc.ups-tlse.fr}}
+	\hfill
+	Sara Giarrusso and Pierre-Fran\c cois Loos, arXiv preprint arXiv:2303.15084
+	\hfill
+	\href{mailto:loos@irsamc.ups-tlse.fr}{\texttt{loos@irsamc.ups-tlse.fr}
+	}
+}
+% (can be left out to remove footer)
+
+% ====================
+% My own customization
+% - BibLaTeX
+% - Boxes with tcolorbox
+% - User-defined commands
+% ====================
+%\input{custom-defs.tex}
+
+%% Reference Sources
+%\addbibresource{refs.bib}
+%\renewcommand{\pgfuseimage}[1]{\includegraphics[scale=2.0]{#1}}
+
+\newcommand{\LCPQ}{Laboratoire de Chimie et Physique Quantiques (UMR 5626), Universit\'e de Toulouse, CNRS, UPS, France}
+
+\title{Exact Excited-State Functionals of the Asymmetric Hubbard Dimer}
+
+\author{\underline{Sara Giarrusso},\inst{1}, Pierre-Fran\c cois Loos,\inst{1}}
+\institute[shortinst]{\inst{1} \LCPQ}
+
+\begin{document}
+
+\begin{frame}[t]
+
+%%%%%%%%%%%%%%%%
+%%% Abstract %%%
+%%%%%%%%%%%%%%%%
+\begin{alertblock}{Abstract}
+We derive, for the case of the asymmetric Hubbard dimer at half-filling, the exact functional associated with each singlet ground and excited state, using both Levy's constrained search and Lieb's convex formulation.
+While the ground-state functional is, as commonly known, a convex function with respect to the density (or, more precisely, the site occupation), the functional associated with the (highest) doubly-excited state is found to be concave.
+Also, we find that, because the density of the first excited state is non-invertible, its ``functional'' is a partial, multi-valued function composed of one concave and one convex branch that correspond to two separate sets of values of the external potential. 
+These findings offer insight into the challenges of developing state-specific excited-state density functionals for general applications in electronic structure theory.
+\end{alertblock}
+%%%%%%%%%%%%%%%%
+
+\vspace{-1.3cm}
+\begin{columns}[t]
+\separatorcolumn
+
+\begin{column}{\colwidth}
+
+    %\vspace{-2cm}
+     %%%%%%%%%%%%%%%%%%%%%%
+    \begin{block}{Theoretical background}
+The variational principle can be reformulated in terms of the Hohenberg-Kohn functional as
+\begin{equation}
+	\label{eq:HKvar}
+	E_0[v] = \min_{\rho}\qty{ F[\rho] + \int v(\br) \rho(\br) d\br } \Leftrightarrow  \fdv{F[\rho(\br)]}{\rho(\br)} = - v(\br) 
+\end{equation}
+$ F[\rho]$ is the ``Fenchel conjugate" of $E_0[v]$ and can be obtained from the Lieb variational principle (or convex formulation) as: 
+\begin{equation}
+	\label{eq:Lvar}
+	F[\rho] = \max_{v}\qty{ E[v] - \int v(\br) \rho(\br) d\br } \Leftrightarrow \fdv{E[v(\br)]}{v(\br)} = \rho(\br) 
+\end{equation}
+Levy's constrained search requires $\rho$ to be only $N$-representable rather than $v$-representable, defining the Levy functional as
+\begin{equation}
+\label{eq:FLL}
+	F[\rho] 
+	= \min_{\Psi \leadsto \rho} \mel{\Psi}{\hT + \hV_{ee}}{\Psi}
+\end{equation}
+    \end{block}
+    %%%%%%%%%%%%%%%%%%%%%%
+
+    %%%%%%%%%%%%%%%%%%%%%%
+    %\vspace{-1.cm}
+    \begin{block}{The model}
+The Hamiltonian of the asymmetric Hubbard dimer is
+\begin{equation}
+    \hH = 
+    - t \sum_{\sigma=\up,\dw} \qty( \cre{0\sigma} \ani{1\sigma} + \text{h.c.} )
+    + U \sum_{i=0}^{1} \hat{n}_{i\up} \hat{n}_{i\dw} 
+	+  \Dv \frac{\hn_{1} - \hn_{0}}{2}
+\end{equation}
+A generic singlet wave function can be written as 
+\begin{equation}
+	\ket{\Psi} = x \ket{0_\up0_\dw} + y \frac{\ket{0_\up1_\dw} - \ket{0_\dw1_\up}}{\sqrt{2}} + z \ket{1_\up1_\dw}
+\end{equation}
+with $-1 \le x,y,z \le 1$ and the normalization condition 
+\begin{equation}
+\label{eq:normalization}
+	x^2 + y^2 + z^2 = 1
+\end{equation}
+    \end{block}
+    %%%%%%%%%%%%%%%%%%%%%%
+
+    %%%%%%%%%%%%%%%%%%%%%%
+      %  \vspace{-0.5cm}
+    \begin{block}{Ground and excited-state energies and densities}
+   
+    The total energy, $E$, is given by the sum of
+    \begin{subequations}
+\begin{align}
+	\label{eq:T}
+	&T  = - 2\sqrt{2} t y \qty(x + z)
+	\\
+	\label{eq:Vee}
+	&V_{ee}  = U \qty(x^2 + z^2)
+	\\
+	\label{eq:V}
+	&V  = \rho \cdot \Dv \quad \text{with} \quad \rho=\mel{\Psi}{\frac{\hn_{1} - \hn_{0}}{2}}{\Psi} =  (z^2 - x^2)
+\end{align}
+\end{subequations}
+        \includegraphics[scale=1.7]{fig/fig1} 
+        
+        \begin{columns}[T]
+    \begin{column}{.45\textwidth}
+        \includegraphics[scale=1.4]{fig/fig2} 
+        \end{column}
+         \begin{column}{.45\textwidth}
+         In agreement with Eq.~\eqref{eq:Lvar}, one finds
+         \begin{equation}
+         \dv{E_0(\Dv)}{\Dv} =2\, \rho_0(\Dv)
+         \end{equation}
+         However, the same holds for the excited states	
+       \begin{subequations}
+\begin{align}
+        & \dv{E_1(\Dv)}{\Dv} =2\, \rho_1(\Dv)\\
+        & \dv{E_2(\Dv)}{\Dv} =2\, \rho_2(\Dv)
+         \end{align}
+         \end{subequations}
+        \end{column}
+         \begin{column}{.1\textwidth}
+         
+          \end{column}
+         \end{columns}
+    \end{block}
+    %%%%%%%%%%%%%%%%%%%%%%
+
+    %%%%%%%%%%%%%%%%%%%%%%
+        %\vspace{-0.5cm}
+    \begin{block}{Levy's constrained search}
+    \begin{columns}[T]
+         \begin{column}{.35\textwidth}
+         \hspace{-3.5cm}
+        $\phantom{0\,\,}$\includegraphics[scale=1.2]{fig/fig5} 
+        \end{column}
+         \begin{column}{.5\textwidth}
+         \vspace{1cm}
+        Substituting $x$ and $z$, we obtain the four-branch function
+        \hspace{-3cm}
+{\small{\begin{align}
+	\label{eq:fpm}
+	f_{\pm\pm}(\rho,y) 
+	= & - 2 t y \qty (\pm\sqrt{1 - y^2 - \rho} 
+	\pm \sqrt{1 - y^2 + \rho}) \notag
+	\\
+	&+ U \qty(1 - y^2)
+\end{align}} }
+
+Rather than only minimizing $f_{\pm\pm}(\rho,y) $
+for a given $\rho$, we seek \textit{all}  its stationary points with respect to $y$:
+\begin{equation}
+	\pdv{f_{\pm\pm}(\rho,y)}{y} = 0
+\end{equation}
+        \end{column}
+         \begin{column}{.1\textwidth}
+          \end{column}
+         \end{columns}
+    \end{block}
+    %%%%%%%%%%%%%%%%%%%%%%
+
+\end{column}
+
+\separatorcolumn
+
+\begin{column}{\colwidth}
+
+    \vspace{3cm}
+    %               
+    %%%%%%%%%%%%%%%%%%%%%%
+%    \begin{block}{}
+    
+    \includegraphics[scale=1.35]{fig/fig3} 
+    $f_{--}(\rho,y)$ (red) and $f_{++}(\rho,y)$ (green) have each one stationary point for $y \geq 0$, while $f_{+-}(\rho,y)$ (yellow) has two stationary points that disappear for $\rho > \rho_c$.
+ %   \end{block}
+\vspace{0.5cm}
+    %%%%%%%%%%%%%%%%%%%%%%
+    \begin{block}{Ground and excited-state ``universal" function(al)s}
+     %
+     \begin{columns}[T]
+         \begin{column}{.45\textwidth}
+         \includegraphics[scale=1.5]{fig/fig4} \\
+         \includegraphics[scale=1.5]{fig/Vvsrho} 
+         \end{column}
+         \begin{column}{.45\textwidth}
+         \vspace{0.5cm}
+             \begin{subequations}
+         \begin{align}
+        & F_0(\rho)=f_{++}(\rho,y_0)\\
+        & F_2(\rho)=f_{--}(\rho,y_2)\\
+        &  \begin{rcases}    F_1^{\cup}(\rho) = f_{+-}(\rho,y_1^{\cup}))\\
+          F_1^{\cap}(\rho) = f_{+-}(\rho,y_1^{\cap}
+      \end{rcases} \quad \rho < \rho_\text{c}
+         \end{align}
+         \end{subequations}
+ 
+\vspace{1.5cm}
+%\dv{F_2(\rho)}{\rho} = -\Dv_2 (\rho)
+ In agreement with Eq.~\eqref{eq:HKvar} and beyond the ground state, one finds
+    \begin{subequations}
+\begin{align}
+        & \dv{F_0(\rho)}{\rho} = -\Dv_0 (\rho)\\
+        & \dv{F_2(\rho)}{\rho} = -\Dv_2 (\rho)\\
+        &  \begin{rcases}   \dv{F_1^{\cup}(\rho)}{\rho} = - \Dv_1^{\cup}(\rho)\\
+        \dv{F_1^{\cap}(\rho)}{\rho} = -\Dv_1^{\cap}(\rho) \end{rcases} \quad \rho < \rho_\text{c}
+         \end{align}
+         \end{subequations}
+         \end{column}
+         \end{columns}
+    \end{block}
+    %%%%%%%%%%%%%%%%%%%%%%
+
+    %%%%%%%%%%%%%%%%%%%%%%
+    \begin{block}{Lieb's convex formulation}
+    \vspace{0.5cm}
+      \begin{columns}
+\begin{column}{.45\textwidth}
+         \includegraphics[scale=1.5]{fig/fig6} 
+          \end{column}
+          \begin{column}{.5\textwidth}
+\begin{equation}
+	f_{n}(\rho,\Dv) = E_n - \Dv \rho
+\end{equation}
+Rather than only maximizing $ f_{n}(\rho,\Dv) $ for a given $\rho$, we seek \textit{all} its stationary points with respect to $\Dv$ for each $n$ value, 
+\begin{equation}
+	\pdv{f_{n}(\rho,\Dv)}{\Dv} = 0
+\end{equation}         
+           \end{column}
+         \end{columns}
+    \end{block}
+    \vspace{-1.3cm}
+    %%%%%%%%%%%%%%%%%%%%%%
+        \begin{block}{Conclusions}
+        While the ground-state functional is well-known to be a convex function with respect to the difference in site occupations, the functional associated with the highest doubly-excited state is found to be concave. 
+Additionally, and more importantly, we discovered that the ``functional'' for the first excited state is a partial, multi-valued function of the density that is constructed from one concave and one convex branch associated with two separate sets of values of the external potential. 
+Finally, we find that Levy's constrained search and Lieb's convex formulation are entirely consistent for all the states of the model, producing the same landscape of state-specific functionals.
+  \end{block}
+    %%%%%%%%%%%%%%%%%%%%%%%
+      \vspace{-0.4cm}
+    %%% Acknowledgement %%%
+    %%%%%%%%%%%%%%%%%%%%%%%
+    \begin{block}{Acknowledgement}
+        \textit{This project has received funding from the European Research Council (ERC) 
+        under the European Union's Horizon 2020 research and innovation programme,
+        grant agreement No.~863481.}
+    \end{block}
+    %%%%%%%%%%%%%%%%%%%%%%%
+ \vspace{-1.cm}
+    %%%%%%%%%%%%%%%%%%
+    %%% References %%%
+    %%%%%%%%%%%%%%%%%%
+    \begin{block}{References}
+
+        %\printbibliography
+        {\footnotesize{
+        \parbox[b]{\hsize}{[1] P. Hohenberg, and W. Kohn, \textit{Inhomogeneous electron gas}, Phys. rev. 136.3B (1964): B864.}\\
+
+\parbox[b]{\hsize}{[2] M. Levy, \textit{Universal variational functionals of electron densities, first-order density matrices, and natural spin-orbitals and solution of the v-representability problem} Proc. Nat. Ac. Sci 76.12 (1979): 6062-6065.}\\
+        
+ \parbox[b]{\hsize}{[3] E. H. Lieb, \textit{Density functionals for Coulomb systems}  Int. J. Quantum Chem. \textbf{24}, 243 (1983).}\\
+        
+        \parbox[b]{\hsize}{[4]  D. J. Carrascal, J. Ferrer, J. C. Smith, and K. Burke, \textit{The Hubbard dimer: a density functional case study of a many-body problem}, J.Phys. Condens. Matter \textbf{27}, 393001 (2015).}\\
+        
+       \parbox[b]{\hsize}{[5]     J. P. Perdew, and M. Levy, \textit{Extrema of the density functional for the energy: Excited states from the ground-state theory}, Phys. Rev. B 31.10 (1985): 6264. \\
+     }
+      }}
+
+    \end{block}
+    %%%%%%%%%%%%%%%%%%
+
+\end{column}
+
+\separatorcolumn
+\end{columns}
+
+\end{frame}
+
+\end{document}