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\usepackage{listings}
\usepackage{calligra}
\usepackage{amsmath}
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\usepackage{tikz}
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\title{Teoria del funzionale densità.}
\author{\calligra Fulvio Ciriaco}
\subject{}
\begin{document}
  \pgfdeclareimage[width=1.5cm]{logo}{uniba.png}
  \logo{\pgfuseimage{logo}}
  \begin{frame}{\titlepage}
  \tableofcontents
  \end{frame}

\begin{frame} 
  \frametitle{Hohenberg e Kohn}
  \begin{align*}
    \Psi & \xrightleftharpoons[?]{}\rho \\
    \Psi_0 &\rightleftharpoons \rho_0
  \end{align*}
  Dipende dalla forma dell'hamiltoniana.
  \begin{equation}
    H=\underbrace{-\sum_i\frac {\nabla^2_i} 2 + 
    \sum_{i<j} \frac 1 {\vert r_i - r_j \vert }}_{U^N}
    +\sum_i V(r_i) 
  \end{equation} 

\end{frame}

\begin{frame} 
  \frametitle{per assurdo}
\begin{align*}
H^1&=U^N+\sum_i V^1(r_i) & H^1\Psi_0^1&=\epsilon_1\Psi_0^1 & 
   \langle \Psi_0^1|\hat d(r)| \Psi_0^1\rangle=\rho(r)\\
H^2&=U^N+\sum_i V^2(r_i) & H^2\Psi_0^2&=\epsilon_2\Psi_0^2 &
   \langle \Psi_0^2|\hat d(r)| \Psi_0^2\rangle=\rho(r)
\end{align*}
per ipotesi, o principio variazionale
\begin{align*}
  \langle \Psi_0^1 | H^1 | \Psi_0^1 \rangle &\le \langle\Psi_0^2 | H^1 | \Psi_0^2 \rangle\\ 
\langle \Psi_0^2 | U^N | \Psi_0^2 \rangle + \cancel{\int \rho V^1 dr}
&> \langle \Psi_0^1 | U^N | \Psi_0^1 \rangle + \cancel{\int \rho V^1 dr} \\
\langle \Psi_0^1 | U^N | \Psi_0^1 \rangle + \cancel{\int \rho V^2 dr}
&> \langle \Psi_0^2 | U^N | \Psi_0^2 \rangle + \cancel{\int \rho V^2 dr} 
\end{align*}
che assurdo!
\end{frame}

\begin{frame} 
  \frametitle{applicazione?}
  \begin{align*}
    E[\rho] &=F[\rho]+\int V(x) \rho(x) dx \\
    F[\rho] &= \min_\rho E_v[\rho]
  \end{align*}
\end{frame}

\begin{frame} 
  \frametitle{Levy-Lieb}
  \begin{align*}
    \langle \psi | U_N+V | \psi \rangle &\ge \langle \psi_0 | U_N+V| \psi_0 \rangle
    \qquad \text{(ancora)} \\[9pt]
    E&=\min_\rho\left[\min_{\psi\rightarrow \rho} \langle \psi | U| \psi \rangle + 
    \int V\rho\right]\\[9pt]
    F[\rho]&=\min_{\psi\rightarrow \rho} \langle \psi | U | \psi \rangle
  \end{align*}
    
\end{frame}

\begin{frame} 
  \frametitle{Kohn-Sham}
\begin{align*}
  F[\rho]&=T_S[\rho]+\frac 1 2 \iint \frac {\rho(r)\rho(r')}{|r-r'|} drdr'
  +E_{XC}[\rho] \\[0.5cm]
  T_S[\rho]&=\langle\psi_{\rho,\text{non interagente}}| T | \psi\rangle 
\end{align*}

\end{frame}

\begin{frame} 
  \frametitle{tassonomia}
  \begin{align*}
  F_L[\rho] &=\int f(\rho(x)) dx  &&\text{LDA, es. B97}\\
  F_G[\rho] &=\int g(\nabla\rho(x)) dx &&\text{GGA, es. PBE}\\
  F_H[\phi_k] &= w_{HF}Xc_{HF}(\phi_k)+ w_{DF}Xc(\rho[\phi_k]) &&\text{ibrido: Hybrid, es. B3LYP}\\
  F_M[\rho] &=\int g(\nabla^2\rho(x)) dx &&\text{meta, es. M05}\\
  F_N[\rho] &=\iint L(\rho(x_1),\rho(x_2)) dx_1dx_2 &&\text{non locale, es. CAM-B3LYP}
  \end{align*}
\end{frame}


\end{document}