abinit/doc/theory/mbt_assets/equations.tex

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

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

\begin{document}

\begin{equation}
\ee(1, 2) = \delta(1, 2) - \int v(1, 3)\tchi(3, 2)\dd(3)
\end{equation}

\begin{equation}
\ee_{\GG_1\GG_2}(\qq;\ww) = \delta_{\GG_1,\GG_2} - v(\qq, \GG_1) \tchi_{\GG_1\GG_2}(\qq;\ww)
\end{equation}

\begin{equation}
W = v + (\ee^{-1} - 1) v
\end{equation}


\begin{equation}
\label{eq:GW_space_frequency}
\Sigma(\rr_1,\rr_2;\ww) = 
%\lim_{\delta \rarr 0^+}
\dfrac{i}{\twopi} \int e^{i\ww'\delta^+} G(\rr_1,\rr_2;\ww+\ww') W(\rr_1,\rr_2;\ww')\dd\ww'
\end{equation}

 Macro epsilon with and without LFE

$\ee_M^{\LF}(\ww)$

\begin{equation}
\label{eq:abs_LFE}
\ee_M^{\LF}(\ww) = \lim_{\qq \rarr 0} \dfrac{1}{\ee^{-1}_{0 0}(\qq,\ww)}
\end{equation}

$\ee_M^\NLF(\ww)$

\begin{equation}
\label{eq:abs_NLFE}
\ee_M^\NLF(\ww) = \lim_{\qq \rarr 0} {\ee_{0 0}(\qq,\ww)}
\end{equation}

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

 Oscillators

\begin{equation}
\label{eq:def_oscillator}
M^{b_1b_2}_\GG (\kk,\qq) \df
\<\kmq,b_1|e^{-i(\qpG)\cdot\rr}|\kk,b_2\> = 
 \sum_{\GG'=1} 
 u_{\kmq b_1}^\*(\GG')u_{\kk b_2}(\GG+\GG')
%= 
% \\
% = \int u_{\overline{\kmq} b_1}^\*(\rr) e^{-i(\GG-\GGo)\cdot \rr}\, u_{\kk b_2}(\rr)\dd\rr
% = \hat\mcF \bigl[ u_{\overline{\kmq} b_1} u_{\kk b_2}^\* \bigr] (\GG-\GGo).
\end{equation}

 Spectral method

$\tchi^\mcS(\ww')$ 

$\ww'$


\begin{multline}
\label{eq:im_part_chi0}
 \tchi^\mcS_{\GG_1\GG_2}(\qq,\ww') = \\
 = \dfrac{2}{V} \sum_{\substack{\kk \\ b_1 b_2}} \,
 \,\Bigl[ f(\varepsilon_{b_1\kmq})-f(\varepsilon_{b_2\kk}) \Bigr]
 \times 
 %\\
 M^{b_1 b_2}_{\GG_1}(\kk,\qq) \Bigl[{M^{b_1 b_2}_{\GG_2}}(\kk,\qq)\Bigr]^\*\,
 {\sign(\ww')\,
 \delta(\ww'-\varepsilon_{b_1\kmq}+\varepsilon_{b_2\kk})}
\end{multline}


\begin{equation}
\label{eq:chi0_Hilbert_transform}
 {\tchio}_{\GG_1\GG_2} (\qq,\ww) = 
 \int_0^{+\infty} \tchi^\mcS_{\GG_1\GG_2} (\qq,\ww') 
 \times \biggl(\frac{1}{\ww-\ww'+i\delta}-\frac{1}{\ww+\ww'-i\delta}\biggr)\dd\ww'
\end{equation}


Sigma


\begin{equation}
W = v + (\ee^{-1}-1) v
\end{equation}

\begin{equation}
\Sigma(\rr_1,\rr_2;\ww) \df \Sigma_x(\rr_1,\rr_2) + \Sigma_c(\rr_1,\rr_2;\ww)
\end{equation}

\begin{equation}\label{eq:Sigma_x}
\Sigma_x(\rr_1,\rr_2)=  
-\sum_\kk^\BZ \sum_\nu^\text{occ} \Psi_{n\kk}(\rr_1){\Psi^\*_{n\kk}}(\rr_2)\,\vc(\rr_1,\rr_2)
\end{equation}


\begin{equation}
\label{eq:diag_mat_el_Sigma_x}
 \<b_1\kkt|\oSigma_x|b_1\kkt\> =
 -\dfrac{\fourpi}{V} \sum_{\nu}^\occ \sum_\qq^\BZ \sum_{\GG}
  \dfrac{\abs{M_\GG^{b_1b_1}(\kkt,\qq)}^2}{\abs{\qpG}^2}
\end{equation}

\begin{equation}
\label{eq:mat_el_Sigma_c}
\<b_1\kkt|\oSigma_c(\ww)|b_2\kkt\> = 
\dfrac{i}{\twopi V}
\sum_\qq^\BZ \sum_{\GG_1\GG_2}\sum_{n=1}^\pinf
\Bigl[M_{\GG_1}^{nb_1}(\kkt,\qq)\Bigr]^\* M_{\GG_2}^{nb_2}(\kkt,\qq)\,\tilde v_{\GG_1 \GG_2}(\qq)\,
J_{\GG_1\GG_2}^{\,n \kkt-\qq}(\qq,\ww)
\end{equation}


 ppmodel

\begin{equation}
\label{eq:ppmodel_imag}
\Im\,\ee^{-1}_{\GG_1\GG_2}(\qq,\ww) = 
A_{\GG_1\GG_2}(\qq)\,
\bigl[
\delta(\ww-\ww_{\GG_1\GG_2}(\qq)) - 
\delta(\ww+\ww_{\GG_1\GG_2}(\qq))
\bigr]
\end{equation}

\begin{equation}
\label{eq:ppmodel_real}
\Re\,\ee^{-1}_{\GG_1 \GG_2} (\qq,\ww) = 
\delta_{\GG_1 \GG_2} + \dfrac{\Omega_{\GG_1\GG_2}^2(\qq)}{\ww^2-\wwt^2_{\GG_1\GG_2}(\qq)}
\end{equation}


 CD method

\begin{multline}
\label{eq:GW_CD}
\Sigma_c(\ww) = \dfrac{i}{\twopi} \Bigl\{ 
 \twopi\,i \sum_{z_p}^\mcC \lim_{z\rarr z_p} G(z)\,\Wc(z)\,(z-z_p) -\int_\minf^\pinf G(\ww+i\ww')\,\Wc(i\ww') \dd(i\ww')
 \Bigr\}.
\end{multline}

\begin{equation}
\Wc(\qq,\ww)_{\GG_1\GG_2} \df 
%\tac_{\GG_1\GG_2}(\qq,\ww)
\bigl(\eet^{-1}_{\GG_1\GG_2}(\qq,\ww) -\delta_{\GG_1,\GG_2}\bigr)
\,\tvc_{\GG_1\GG_2}(\qq)
\end{equation}


Notations

$(1) \df (\rr_1,t_1)$ 

$\vc(\rr_1,\rr_2)$ 

\begin{equation}
\delta(12) = \delta(\rr_1-\rr_2)\,\delta(t_1-t_2)
\end{equation}
%
\begin{equation}
\int \dd1 = \int \ddrr_1 \int_{-\infty}^{+\infty} \dd t_1
\end{equation}
%
\begin{equation}
v(12)= \vc(\rr_1,\rr_2)\,\delta(t_1-t_2)
\end{equation}
%
\begin{equation}
1^+ = (\rr_1,t_1 + \eta)_{\eta  \rarr 0^+}
\end{equation}

\begin{alignat}{2}\label{eq:FT_1point_convention}
 u(\rr)= \sum_\GG u(\GG)e^{i\GG\cdot\rr}, &\quad
 u(\GG) = \frac{1}{\Omega} \int_\Omega u(\rr)e^{-i\GG\cdot\rr}\dd\rr
\end{alignat}


\begin{equation}\label{eq:IFT_2points_convention}
 f(\rr_1,\rr_2)= \frac{1}{V} \sum_{\substack{\qq \\ \GG_1 \GG_2}}  
 e^{i (\qq +\GG_1) \cdot \rr_1}\,f_{\GG_1 \GG_2}(\qq)\,e^{-i (\qq+\GG_2) \cdot \rr_2} 
\end{equation}

\begin{equation}\label{eq:FT_2points_convention}
 f_{\GG_1\GG_2}(\qq) = \frac{1}{V} \iint_V                                                     
 e^{-i(\qq+\GG_1) \cdot \rr_1}\,f(\rr_1, \rr_2)\,e^{i (\qq+\GG_2) \cdot \rr_2}\dd\rr_1\dd\rr_2 
\end{equation}

The volume of the unit cell is denoted with $\Omega$, while $V$ is the total volume of the crystal 

\begin{equation}\label{eq:Poles of G and W}
 G(\ww-\ww')  
 W(\ww')
\end{equation}


\end{document}