\documentclass{beamer}
\usepackage[english,italian]{babel}
\usepackage[utf8]{inputenc}
\usepackage[T1]{fontenc}
\usepackage{lmodern}
\usepackage{listings}
\usepackage{calligra}
\usepackage{amsmath}
\usepackage{graphicx}
\usepackage{cancel}
\graphicspath{ {./figure/} }
\usetheme{default}
\usecolortheme{beaver}

\title{equazioni HF}
\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{attraverso il principio variazionale}
  \begin{align*}
    \cal{L} &= \left\langle \Psi \middle| H \middle|\Psi \right\rangle - 
      \mu (\left\langle \Psi \middle | \Psi \right\rangle -1) \\
    \left\langle \Psi \middle | H \middle | \Psi \right\rangle &= 
    \sum_m \langle \phi_m | h |\phi_m \rangle + 
    \sum_{m<n} \langle \phi_m(x_1) \phi_n(x_2) | g | \phi_m(x_1) \phi_n(x_2)  \rangle\\
                                                               &-\sum_{m<n} \langle \phi_m(x_1) \phi_n(x_2) | g | \phi_n(x_1) \phi_m(x_2) \rangle \\
    h &= - \frac {\nabla^2} 2 + V \\
    g &= \frac 1 {|r_1-r_2|}
    \end{align*}
\end{frame}

\begin{frame} 
  \frametitle{attraverso il principio variazionale}
  \begin{align*}
    {\cal L} &= \sum_m \langle \phi_m | h |\phi_m \rangle + \\ 
             &+\sum_{m<n} \langle \phi_m \phi_n | g | \phi_m \phi_n \rangle 
    -\sum_{m<n} \langle \phi_m \phi_n | g | \phi_n \phi_m \rangle \\
    &+ \sum_m \lambda_m (\langle \phi_m | \phi_m \rangle -1)
  \end{align*}
\end{frame}

\begin{frame} 
  \frametitle{se solo non ci fossero interazioni fra gli elettroni!}
  \begin{align*}
    {\cal L} &= \sum_m \langle \phi_m | h |\phi_m \rangle + 
    \sum_m \lambda_m (\langle \phi_m | \phi_m \rangle -1) \\
    \frac {\partial \cal L} {\partial \phi^*_n} &=
    h |\phi_n\rangle + \lambda_n |\phi_n\rangle = 0
  \end{align*}
\end{frame}

\begin{frame} 
  \frametitle{coniugare o non coniugare, non è un problema}
  una variabile complessa o due variabili?
  \begin{align*}
    \frac {\partial f(|x|^2)}{\partial x^*}&=\frac {\partial f(|x|^2}{\partial |x|^2} =
    \frac {\partial f(|x|^2)}{\partial |x|^2} x \\
    \frac {\partial |x|^2} {\partial x^*} &= \frac {\partial x^* x} {\partial x^*} = x
  \end{align*}
  possiamo vedere $x^*,x$ come una trasformazione lineare delle variabili cartesiane del piano complesso:
  \begin{equation*}
    \left\{ \begin{matrix}
        x &=& a+ib\\
        x^* &=& a-ib
    \end{matrix}\right.
    \implies
    \left\{ \begin{matrix}
        a &=& \frac {x+x^*} 2\\
        b^* &=& \frac {x-x^*} 2
    \end{matrix}\right.
  \end{equation*}
  un test:
  \begin{equation*}
    \frac {\partial |x|^2} {\partial a}=\frac {\partial (a^2+b^2)} {\partial a} =
    2a =
    \frac {\partial |x|^2} {\partial x^*} \frac {\partial x^*} {\partial a} + 
    \frac {\partial |x|^2} {\partial x} \frac {\partial x} {\partial a}  
  \end{equation*}
\end{frame}

\begin{frame} 
  \frametitle{se solo non ci fossero interazioni fra gli elettroni!}
  \begin{align*}
    {\cal L} &= \sum_m \langle \phi_m | h |\phi_m \rangle + 
    \sum_m \lambda_m (\langle \phi_m | \phi_m \rangle -1) \\
    \frac {\partial \cal L} {\partial \phi^*_n} &=
    h |\phi_n\rangle + \lambda_n |\phi_n\rangle = 0
  \end{align*}
  invece,
  \begin{align*}
    {\cal L} &= \sum_m \langle \phi_m | h |\phi_m \rangle + \\ 
             &+\frac 1 2 \sum_{m,n} (\langle \phi_m \phi_n | g | \phi_m \phi_n \rangle 
             - \langle \phi_m \phi_n | g | \phi_n \phi_m \rangle) \\
    &+ \sum_m \lambda_m (\langle \phi_m | \phi_m \rangle -1)
  \end{align*}

\end{frame}

\begin{frame} 
  \frametitle{iterazione}
  \begin{align*}
    f(y) &= ay \\
    f(y) &=a y + y^2
  \end{align*}
  \pause
  \begin{align*}
    y_0 &\\
    f(y_1) &= a y_1 + y_1 y_0 \\
    f(y_2) &= a y_2 + y_2 y_1 \\
    ... &
  \end{align*}
\end{frame}

\begin{frame} 
  \frametitle{hartree-fock}
  \begin{align*}
    {\cal L} &= \sum_m \langle \phi_m | h |\phi_m \rangle + \\ 
             &+\frac 1 2 \sum_{m,n} (\langle \phi_m \phi^{(i)}_n | g | \phi_m \phi^{(i)}_n \rangle 
             - \langle \phi_m \phi^{(i)}_n | g | \phi^{(i)}_n \phi_m \rangle) \\
    &+ \sum_m \lambda_m (\langle \phi_m | \phi_m \rangle -1)
  \end{align*}
  \begin{equation*}
    (h+K+G)\phi_n = \lambda_n \phi_n
  \end{equation*}
\end{frame}

\end{document}