\documentclass[a4paper,reqno,11pt,twoside]{book}

\input{packages.tex}
\input{macros.tex}

\begin{document}

$\bar v(13)$

$$ \chi^0(12) = L^0(11, 22) $$

$$ \hat\chi(12) = L(11, 22) $$

Dyson for L

\begin{equation}\nonumber
L = L^0 + L^0 K L 
\Longrightarrow
L = \bigl [ 1 - L^0  K]^{-1} L^0 
\end{equation}

Kernel

\begin{equation}\label{eq:BSE_kernel_LF}\nonumber
K(1234) = 
\underbrace{\delta(12)\delta(34)\bar v(13)}_{Exchange} 
- 
\underbrace{\delta(13)\delta(24)W(12)}_{Coulomb}
\end{equation}


\begin{equation}\nonumber
 \begin{cases}
   {\bar v(\qq)} = v(\qq) \quad {\text{if}}\; \qq \neq 0
   \\
   \\
   {\bar v(\qq=0)} = 0
 \end{cases}
\end{equation}


\begin{equation}\nonumber
\ee_M^{\LF}(\ww) = 1 - \lim_{\qq  \rightarrow 0}
 \vc(\qq)\,\hat\chi_{00}(\qq;\ww)
\end{equation}


Electron-hole expansion

\begin{equation}
F(1234) = 
\sum_{ \substack{(n_1 n_2) \\ (n_3 n_4)} }
\psi_{n_1}^\*(1) \psi_{n_2}(2)\,
F_{ (n_1 n_2) (n_3 n_4) }\,
\psi_{n_3}(3) \psi_{n_4}^\*(4) 
\end{equation}
%
\begin{equation}
F_{ (n_1 n_2) (n_3 n_4) } = 
\int 
F(1234)
\psi_{n_1}(1) \psi_{n_2}^\*(2) 
\psi_{n_3}^\*(3) \psi_{n_4}(4) 
\dd (1234)
\end{equation}


\begin{equation}
\chio_{ (n_1 n_2) (n_3 n_4) } (\ww) = 
\dfrac{ (f_{n_2} - f_{n_1}) } { (\ee_{n_2} - \ee_{n_1} - \ww)}
\delta_{n_1 n_3} \delta_{n_2 n_4} 
\end{equation}

\begin{equation}
\bar \chi_{ (n_1 n_2) (n_3 n_4) } (\ww) = 
\bigl [ H - \ww \bigr]^{-1}_{ (n_1 n_2) (n_3 n_4) } (f_{n_4} - f_{n_3}) 
\end{equation}

 L in terms of the Excitonic Hamilotian

\begin{equation}
L = \bigl [ H-\ww \bigr]^{-1}\,F
\end{equation}

$$ 
H =
\left( 
\begin{array}{c|cc} 
  &   |v'c'\kk'\> & |c'v'\kk'\>     \\ \hline 
\<vc\kk|  & R  &  C  \\ %\hline 
\<cv\kk|  &  -C^* & -R^*   \\
\end{array}
\right) 
$$

$$ 
F =
\left( 
\begin{array}{c|cc} 
  &   |v'c'\kk'\> & |c'v'\kk'\>     \\ \hline 
\<vc\kk|  & 1  &  0   \\ %\hline 
\<cv\kk|  & 0  & -1   \\
\end{array}
\right) 
$$

Resonant block

\begin{equation}
R_{vc\kk,v'c'\kk'} = 
( \ee_{c\kk} - \ee_{v\kk} )\delta_{vv'}\delta_{cc'}\delta_{\kk\kk'} + 
2 \bar v_{(vc\kk)(v'c'\kk')} - W_{(vc\kk)(v'c'\kk')}
\end{equation}

Matrix elements in real space


\begin{equation}
{\bar v}_{(n_1 n_2) (n_3 n_4)} =
\delta_{\sigma_1 \sigma_2}\, \delta_{\sigma_3 \sigma_4}
\iint \psi_{n_1}(\rr) \psi^*_{n_2}(\rr) {\bar v(\rr-\rr')} \psi_{n_3}^*(\rr') \psi_{n_4}(\rr') \dd \rr \dd \rr'
\end{equation}

\begin{equation}
W_{(n_1 n_2) (n_3 n_4)} =
\delta_{\sigma_1 \sigma_3}\, \delta_{\sigma_2 \sigma_4}
\iint \psi_{n_1}(\rr) \psi^*_{n_3}(\rr) {W(\rr,\rr',\ww=0)} \psi^*_{n_2}(\rr') \psi_{n_4}(\rr') \dd \rr \dd \rr'
\end{equation}


TDA approximation

\begin{equation}\nonumber
H^{\text{TDA}} =
\left( 
\begin{array}{c|cc} 
  &   |v'c'\kk'\> & |c'v'\kk'\>     \\ \hline 
\<vc\kk|  & R  &  0  \\ %\hline 
\<cv\kk|  &  0 & -R^*   \\
\end{array}
\right) 
\end{equation}


\begin{equation}\nonumber
\ee_M(\ww) = 1 - \lim_{\qq  \rightarrow 0}
 \vc(\qq)\,\tchi_{00}(\qq;\ww)
\end{equation}

Dipole operator

\begin{equation}\nonumber
P(\qq)_{n_1 n_2} = 
\<n_2|e^{i\qq\cdot\rr}|n_1\> 
\underset{\qq \rarr 0}{ \approx}
\delta_{n_1 n_2}  +
i \qq \cdot \<n_2|\rr|n_1\>
+ O(q^2)
\end{equation}

\begin{equation}\nonumber
\ee_M(\ww) = 
1 -  \lim_{\qq  \rightarrow 0}
v(\qq)\,\<P(\qq)|\bigl[ H - \ww \bigr]^{-1}\,F |P(\qq)\>
\end{equation}


%independent particle without local fields effects
%\begin{equation}\nonumber
%\Im\, \ee_M(\ww) \propto 
%| \<\psi_{c \kk+\qq} | e^{i\qq\cdot\rr}|\psi_{v\kk}\>|^2 \,
%\delta(\ww-\ee_{c\kk+\qq} + \ee_{v\kk})
%\end{equation}


It is possible to avoid the inversion of $\bigl[ H - \ww \bigr]$ by using the spectral representation

\begin{equation}\nonumber
\begin{cases}
H  |\lambda\> = \ee_\lambda |\lambda\>
\\
\\
O_{\lambda\lambda'} = \<\lambda|\lambda'\>
\\
\\ 
H = \sum_{\lambda \lambda'} \ee_\lambda |\lambda\> O_{\lambda \lambda'} \<\lambda'|
\end{cases}
\end{equation}

\begin{equation}
\bigl[ H -\ww \bigr]^{-1} = 
\sum_{\lambda \lambda'} |\lambda\> \dfrac{O_{\lambda \lambda'}^{-1}}{(\ee_\lambda - \ww)} \<\lambda'|
\end{equation}

Haydock


\begin{equation}
%\[
 R = R^\* =
\begin{pmatrix}
*  & *  & * & *  & * \\
*  & *  & * & *  & * \\
*  & *  & * & *  & * \\
*  & *  & * & *  & * \\
*  & *  & * & *  & *
\end{pmatrix}
%\]
\end{equation}


\begin{equation}\nonumber
%\[
\begin{pmatrix}
a_1 & b_2 &     &   &  \\
b_2 & a_2 & b_3 &   &  \\
    & b_3 & *   & * &  \\
    &     & *   & * & * \\
    &     &     & * & * 
\end{pmatrix}
%\]
\end{equation}



\begin{equation}
 R = R^\* =
\begin{pmatrix}
*  & *  & * & *  & * \\
*  & *  & * & *  & * \\
*  & *  & * & *  & * \\
*  & *  & * & *  & * \\
*  & *  & * & *  & *
\end{pmatrix}
\Longrightarrow
\begin{pmatrix}
a_1 & b_2 &     &   &  \\
b_2 & a_2 & b_3 &   &  \\
    & b_3 & *   & * &  \\
    &     & *   & * & * \\
    &     &     & * & * 
\end{pmatrix}
%\]
\end{equation}

Continued fraction

\begin{equation}
\ee_M(\ww) = 
1 - 
%\< u_0| (\ww-H)^{-1}|u_0\> =
% \cfrac{\norm{u_0}^2}
 \cfrac{F}
 {\ww - a_1 - \cfrac{b_2^2}{\ww - a_2 - \cfrac{b_3^2}{\cdots}}}
\end{equation}

Matrix elements in G-space


\begin{equation}
{\bar v}_{(vc\kk) (v'c'\kk')} = 
\dfrac{1}{V} \sum_{\GG \neq 0} {\bar v}(\GG) \;
\<c\kk  |e^{i\GG\cdot\rr} |v\kk\>
\<v'\kk'|e^{-i\GG\cdot\rr}|c'\kk'\>
\end{equation}

\begin{equation}
W_{(vc\kk) (v'c'\kk')} = 
\dfrac{1}{V} \sum_{\GG_1\GG_2} W_{\GG_1\GG_2}(\kk'-\kk,\ww=0)
\<v'\kk'|e^{i(\qq +\GG_1)\cdot \rr}|v\kk \> 
\<c \kk |e^{-i(\qq+\GG_2)\cdot\rr} |c'\kk'\> 
\end{equation}

equations for inclvkb 

\begin{equation}
\<b_1,\kmq|e^{-i\qq\cdot\rr}|b_2,\kk\> \underset{\qq \rarr 0}{\approx}
-i\,\qq\cdot \<b_1,\kk|\rr|b_2,\kk\> + \mcO(q^2)
\end{equation}

%\begin{equation}
%\label{eq:commutator_trick}
%\<\Psi_{\kk b_1}|\rr|\Psi_{\kk b_2}\> = 
% \dfrac {\<\Psi_{\kpq b_1}|\bigl[ \HH,\rr \bigr]|\Psi_{\kk b_2}\>}
%        {\ee_{\kk b_1} -\ee_{\kk b_2}}.
%\end{equation}


\begin{equation}
\<b_1,\kmq|e^{-i\qq\cdot\rr}|b_2,\kk\> \underset{\qq \rarr 0}{\approx}
\dfrac{
\<b_1,\kk|-i\qq\cdot\nabla + i\qq\cdot \bigl[\Vnl,\rr\bigr]|b_2,\kk\>
}
{
\varepsilon_{b_2\kk} - \varepsilon_{b_1\kk}
}
\end{equation}





\end{document}
