\documentclass{article}

\def\w{\omega}
\def\wp{\omega^\prime}
\def\wc{{\omega_{\rm C}}}
\def\ket{\rangle}
\def\bra{\langle}
\def\H{\hat{H}}
\def\P{\hat{P}_{\rm occ}}
\def\E{\varepsilon}
\def\vp{{v^\prime}}
\def\q{{\bf q}}
\def\s{\sigma}
\def\k{{\bf k}}
\def\qp{{\bf q^\prime}}
\def\G{{\bf G}}
\def\Gp{{\bf G^\prime}}
\def\rt{\tilde{r}}
\def\pt{\tilde{p}}
\def\rb{{\bf r}}
\def\rp{{\bf r^\prime}}
\def\rpp{{\bf r^{\prime\prime}}}
\def\rppp{{\bf r^{\prime\prime\prime}}}
\def\mo{$\overline{1}$}
\def\mt{$\overline{2}$}
\def\S{\mathcal{S}}
\def\Gs{\mathcal{G}}
\def\v{\mathbf{v}}
\def\symm{\left\{\mathcal{S}|\mathbf{v}\right\}}


\begin{document}
\section{Theory}
  For MgB$_{2}$ we have calculated the isoptropic and anisotropic Coulomb interaction using RPA screening
and RPA+Vxc.


In the isotropic case I have used the standard definitions:
%
\begin{equation}
\mu = N(0)\bra\bra W_{n\k,n'\k'} \ket\ket_{\rm FS}
\end{equation}
%
\begin{equation}
\mu = N(0) \frac{\sum_{n\k}\sum_{n'\k'}W_{n\k,n'\k'}\delta(\epsilon_{n\k}) \delta(\epsilon_{n'\k'})}{\sum_{n\k}\sum_{n'\k'}\delta(\epsilon_{n\k})\delta(\epsilon_{n'\k'})}
\end{equation}
%
For the Anisotropic tensors \emph{I am still performing the average over the whole Fermi Surface} but with a truncation in $\k$-space for the delta functions I am using 
Mazin's definition:
%
\begin{equation}
\mu_{ij} = <<W_{n\k,n'\k'}>>_{ij} N_{j}
\end{equation}


\begin{equation}
\mu_{mm'} = N(0) \frac{\sum_{n\k}\sum_{n'\k'}W_{n\k,n'\k'}\delta(\epsilon_{m}) \delta(\epsilon_{m'})}{\sum_{n\k}\sum_{n'\k'}\delta(\epsilon_{n\k})\delta(\epsilon_{n'\k'})}
\end{equation}

\section{Results}
\begin{table}
\begin{center}
\begin{tabular}{l c c}
$\sigma$  & 0.0719  & 0.02593   \\
$\pi$     & 0.0239  & 0.09823   \\
\end{tabular}
\caption{Anisotropic $\mu$ for MgB$_2$ using RPA. \label{tab:mgb2aniso}}
\end{center}
\end{table}


\end{document}
