mirror of https://github.com/abinit/abinit.git
304 lines
6.0 KiB
TeX
304 lines
6.0 KiB
TeX
\documentclass[a4paper,reqno,11pt,twoside]{book}
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\input{packages.tex}
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\input{macros.tex}
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\begin{document}
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$\bar v(13)$
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$$ \chi^0(12) = L^0(11, 22) $$
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$$ \hat\chi(12) = L(11, 22) $$
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Dyson for L
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\begin{equation}\nonumber
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L = L^0 + L^0 K L
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\Longrightarrow
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L = \bigl [ 1 - L^0 K]^{-1} L^0
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\end{equation}
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Kernel
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\begin{equation}\label{eq:BSE_kernel_LF}\nonumber
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K(1234) =
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\underbrace{\delta(12)\delta(34)\bar v(13)}_{Exchange}
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-
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\underbrace{\delta(13)\delta(24)W(12)}_{Coulomb}
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\end{equation}
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\begin{equation}\nonumber
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\begin{cases}
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{\bar v(\qq)} = v(\qq) \quad {\text{if}}\; \qq \neq 0
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\\
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\\
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{\bar v(\qq=0)} = 0
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\end{cases}
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\end{equation}
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\begin{equation}\nonumber
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\ee_M^{\LF}(\ww) = 1 - \lim_{\qq \rightarrow 0}
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\vc(\qq)\,\hat\chi_{00}(\qq;\ww)
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\end{equation}
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Electron-hole expansion
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\begin{equation}
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F(1234) =
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\sum_{ \substack{(n_1 n_2) \\ (n_3 n_4)} }
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\psi_{n_1}^\*(1) \psi_{n_2}(2)\,
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F_{ (n_1 n_2) (n_3 n_4) }\,
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\psi_{n_3}(3) \psi_{n_4}^\*(4)
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\end{equation}
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%
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\begin{equation}
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F_{ (n_1 n_2) (n_3 n_4) } =
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\int
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F(1234)
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\psi_{n_1}(1) \psi_{n_2}^\*(2)
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\psi_{n_3}^\*(3) \psi_{n_4}(4)
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\dd (1234)
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\end{equation}
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\begin{equation}
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\chio_{ (n_1 n_2) (n_3 n_4) } (\ww) =
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\dfrac{ (f_{n_2} - f_{n_1}) } { (\ee_{n_2} - \ee_{n_1} - \ww)}
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\delta_{n_1 n_3} \delta_{n_2 n_4}
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\end{equation}
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\begin{equation}
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\bar \chi_{ (n_1 n_2) (n_3 n_4) } (\ww) =
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\bigl [ H - \ww \bigr]^{-1}_{ (n_1 n_2) (n_3 n_4) } (f_{n_4} - f_{n_3})
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\end{equation}
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L in terms of the Excitonic Hamilotian
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\begin{equation}
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L = \bigl [ H-\ww \bigr]^{-1}\,F
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\end{equation}
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$$
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H =
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\left(
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\begin{array}{c|cc}
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& |v'c'\kk'\> & |c'v'\kk'\> \\ \hline
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\<vc\kk| & R & C \\ %\hline
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\<cv\kk| & -C^* & -R^* \\
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\end{array}
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\right)
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$$
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$$
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F =
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\left(
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\begin{array}{c|cc}
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& |v'c'\kk'\> & |c'v'\kk'\> \\ \hline
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\<vc\kk| & 1 & 0 \\ %\hline
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\<cv\kk| & 0 & -1 \\
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\end{array}
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\right)
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$$
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Resonant block
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\begin{equation}
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R_{vc\kk,v'c'\kk'} =
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( \ee_{c\kk} - \ee_{v\kk} )\delta_{vv'}\delta_{cc'}\delta_{\kk\kk'} +
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2 \bar v_{(vc\kk)(v'c'\kk')} - W_{(vc\kk)(v'c'\kk')}
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\end{equation}
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Matrix elements in real space
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\begin{equation}
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{\bar v}_{(n_1 n_2) (n_3 n_4)} =
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\delta_{\sigma_1 \sigma_2}\, \delta_{\sigma_3 \sigma_4}
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\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'
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\end{equation}
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\begin{equation}
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W_{(n_1 n_2) (n_3 n_4)} =
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\delta_{\sigma_1 \sigma_3}\, \delta_{\sigma_2 \sigma_4}
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\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'
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\end{equation}
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TDA approximation
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\begin{equation}\nonumber
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H^{\text{TDA}} =
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\left(
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\begin{array}{c|cc}
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& |v'c'\kk'\> & |c'v'\kk'\> \\ \hline
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\<vc\kk| & R & 0 \\ %\hline
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\<cv\kk| & 0 & -R^* \\
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\end{array}
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\right)
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\end{equation}
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\begin{equation}\nonumber
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\ee_M(\ww) = 1 - \lim_{\qq \rightarrow 0}
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\vc(\qq)\,\tchi_{00}(\qq;\ww)
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\end{equation}
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Dipole operator
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\begin{equation}\nonumber
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P(\qq)_{n_1 n_2} =
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\<n_2|e^{i\qq\cdot\rr}|n_1\>
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\underset{\qq \rarr 0}{ \approx}
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\delta_{n_1 n_2} +
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i \qq \cdot \<n_2|\rr|n_1\>
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+ O(q^2)
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\end{equation}
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\begin{equation}\nonumber
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\ee_M(\ww) =
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1 - \lim_{\qq \rightarrow 0}
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v(\qq)\,\<P(\qq)|\bigl[ H - \ww \bigr]^{-1}\,F |P(\qq)\>
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\end{equation}
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%independent particle without local fields effects
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%\begin{equation}\nonumber
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%\Im\, \ee_M(\ww) \propto
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%| \<\psi_{c \kk+\qq} | e^{i\qq\cdot\rr}|\psi_{v\kk}\>|^2 \,
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%\delta(\ww-\ee_{c\kk+\qq} + \ee_{v\kk})
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%\end{equation}
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It is possible to avoid the inversion of $\bigl[ H - \ww \bigr]$ by using the spectral representation
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\begin{equation}\nonumber
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\begin{cases}
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H |\lambda\> = \ee_\lambda |\lambda\>
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\\
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\\
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O_{\lambda\lambda'} = \<\lambda|\lambda'\>
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\\
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\\
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H = \sum_{\lambda \lambda'} \ee_\lambda |\lambda\> O_{\lambda \lambda'} \<\lambda'|
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\end{cases}
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\end{equation}
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\begin{equation}
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\bigl[ H -\ww \bigr]^{-1} =
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\sum_{\lambda \lambda'} |\lambda\> \dfrac{O_{\lambda \lambda'}^{-1}}{(\ee_\lambda - \ww)} \<\lambda'|
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\end{equation}
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Haydock
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\begin{equation}
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%\[
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R = R^\* =
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\begin{pmatrix}
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* & * & * & * & * \\
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* & * & * & * & * \\
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* & * & * & * & * \\
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* & * & * & * & * \\
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* & * & * & * & *
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\end{pmatrix}
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%\]
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\end{equation}
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\begin{equation}\nonumber
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%\[
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\begin{pmatrix}
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a_1 & b_2 & & & \\
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b_2 & a_2 & b_3 & & \\
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& b_3 & * & * & \\
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& & * & * & * \\
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& & & * & *
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\end{pmatrix}
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%\]
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\end{equation}
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\begin{equation}
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R = R^\* =
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\begin{pmatrix}
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* & * & * & * & * \\
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* & * & * & * & * \\
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* & * & * & * & * \\
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* & * & * & * & * \\
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* & * & * & * & *
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\end{pmatrix}
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\Longrightarrow
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\begin{pmatrix}
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a_1 & b_2 & & & \\
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b_2 & a_2 & b_3 & & \\
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& b_3 & * & * & \\
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& & * & * & * \\
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& & & * & *
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\end{pmatrix}
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%\]
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\end{equation}
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Continued fraction
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\begin{equation}
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\ee_M(\ww) =
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1 -
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%\< u_0| (\ww-H)^{-1}|u_0\> =
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% \cfrac{\norm{u_0}^2}
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\cfrac{F}
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{\ww - a_1 - \cfrac{b_2^2}{\ww - a_2 - \cfrac{b_3^2}{\cdots}}}
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\end{equation}
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Matrix elements in G-space
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\begin{equation}
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{\bar v}_{(vc\kk) (v'c'\kk')} =
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\dfrac{1}{V} \sum_{\GG \neq 0} {\bar v}(\GG) \;
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\<c\kk |e^{i\GG\cdot\rr} |v\kk\>
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\<v'\kk'|e^{-i\GG\cdot\rr}|c'\kk'\>
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\end{equation}
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\begin{equation}
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W_{(vc\kk) (v'c'\kk')} =
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\dfrac{1}{V} \sum_{\GG_1\GG_2} W_{\GG_1\GG_2}(\kk'-\kk,\ww=0)
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\<v'\kk'|e^{i(\qq +\GG_1)\cdot \rr}|v\kk \>
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\<c \kk |e^{-i(\qq+\GG_2)\cdot\rr} |c'\kk'\>
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\end{equation}
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equations for inclvkb
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\begin{equation}
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\<b_1,\kmq|e^{-i\qq\cdot\rr}|b_2,\kk\> \underset{\qq \rarr 0}{\approx}
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-i\,\qq\cdot \<b_1,\kk|\rr|b_2,\kk\> + \mcO(q^2)
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\end{equation}
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%\begin{equation}
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%\label{eq:commutator_trick}
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%\<\Psi_{\kk b_1}|\rr|\Psi_{\kk b_2}\> =
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% \dfrac {\<\Psi_{\kpq b_1}|\bigl[ \HH,\rr \bigr]|\Psi_{\kk b_2}\>}
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% {\ee_{\kk b_1} -\ee_{\kk b_2}}.
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%\end{equation}
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\begin{equation}
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\<b_1,\kmq|e^{-i\qq\cdot\rr}|b_2,\kk\> \underset{\qq \rarr 0}{\approx}
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\dfrac{
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\<b_1,\kk|-i\qq\cdot\nabla + i\qq\cdot \bigl[\Vnl,\rr\bigr]|b_2,\kk\>
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}
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{
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\varepsilon_{b_2\kk} - \varepsilon_{b_1\kk}
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}
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\end{equation}
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\end{document}
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