Introduction

This document explains a tetrahedron method library libtetrabz. libtetrabz is a library to calculate the total energy, the charge density, partial density of states, response functions, etc. in a solid by using the optimized tetrahedron method [1]. Subroutines in this library receive the orbital (Kohn-Sham) energies as an input and calculate weights \(w_{n n' k}\) for integration such as

\[\begin{align} \sum_{n n' k} F(\varepsilon_{n k}, \varepsilon_{n' k+q})X_{n n' k} = \sum_{n n' k} w_{n n' k} X_{n n' k} \end{align}\]

libtetrabz supports following Brillouin-zone integrations

\[\begin{align} \sum_{n k} \theta(\varepsilon_{\rm F} - \varepsilon_{n k}) X_{n k} \end{align}\]
\[\begin{align} \sum_{n k} \delta(\omega - \varepsilon_{n k}) X_{n k}(\omega) \end{align}\]
\[\begin{align} \sum_{n n' k} \delta(\varepsilon_{\rm F} - \varepsilon_{n k}) \delta(\varepsilon_{\rm F} - \varepsilon'_{n' k}) X_{n n' k} \end{align}\]
\[\begin{align} \sum_{n n' k} \theta(\varepsilon_{\rm F} - \varepsilon_{n k}) \theta(\varepsilon_{n k} - \varepsilon'_{n' k}) X_{n n' k} \end{align}\]
\[\begin{align} \sum_{n n' k} \frac{ \theta(\varepsilon_{\rm F} - \varepsilon_{n k}) \theta(\varepsilon'_{n' k} - \varepsilon_{\rm F})} {\varepsilon'_{n' k} - \varepsilon_{n k}} X_{n n' k} \end{align}\]
\[\begin{align} \sum_{n n' k} \theta(\varepsilon_{\rm F} - \varepsilon_{n k}) \theta(\varepsilon'_{n' k} - \varepsilon_{\rm F}) \delta(\varepsilon'_{n' k} - \varepsilon_{n k} - \omega) X_{n n' k}(\omega) \end{align}\]
\[\begin{align} \sum_{n n' k} \frac{ \theta(\varepsilon_{\rm F} - \varepsilon_{n k}) \theta(\varepsilon'_{n' k} - \varepsilon_{\rm F})} {\varepsilon'_{n' k} - \varepsilon_{n k} + i \omega} X_{n n' k}(\omega) \end{align}\]

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