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
\[\begin{align}
\sum_{n n'}
\int_{\rm BZ} \frac{d^3 k}{V_{\rm BZ}}
F(\varepsilon_{n k}, \varepsilon_{n' k+q})X_{n n' k}
= \sum_{n n'} \sum_{k}^{N_k} w_{n n' k} X_{n n' k}
\end{align}\]
libtetrabz supports following Brillouin-zone integrations
\[\begin{align}
\sum_{n}
\int_{\rm BZ} \frac{d^3 k}{V_{\rm BZ}}
\theta(\varepsilon_{\rm F} - \varepsilon_{n k})
X_{n k}
\end{align}\]
\[\begin{align}
\sum_{n}
\int_{\rm BZ} \frac{d^3 k}{V_{\rm BZ}}
\delta(\omega - \varepsilon_{n k})
X_{n k}(\omega)
\end{align}\]
\[\begin{align}
\sum_{n n'}
\int_{\rm BZ} \frac{d^3 k}{V_{\rm BZ}}
\delta(\varepsilon_{\rm F} - \varepsilon_{n k})
\delta(\varepsilon_{\rm F} - \varepsilon'_{n' k})
X_{n n' k}
\end{align}\]
\[\begin{align}
\sum_{n n'}
\int_{\rm BZ} \frac{d^3 k}{V_{\rm BZ}}
\theta(\varepsilon_{\rm F} - \varepsilon_{n k})
\theta(\varepsilon_{n k} - \varepsilon'_{n' k})
X_{n n' k}
\end{align}\]
\[\begin{align}
\sum_{n n'}
\int_{\rm BZ} \frac{d^3 k}{V_{\rm BZ}}
\frac{
\theta(\varepsilon_{\rm F} - \varepsilon_{n k})
\theta(\varepsilon'_{n' k} - \varepsilon_{\rm F})}
{\varepsilon'_{n' k} - \varepsilon_{n k}}
X_{n n' k}
\end{align}\]
\[\begin{align}
\sum_{n n'}
\int_{\rm BZ} \frac{d^3 k}{V_{\rm BZ}}
\theta(\varepsilon_{\rm F} - \varepsilon_{n k})
\theta(\varepsilon'_{n' k} - \varepsilon_{\rm F})
\delta(\varepsilon'_{n' k} - \varepsilon_{n k} - \omega)
X_{n n' k}(\omega)
\end{align}\]
\[\begin{align}
\sum_{n n'}
\int_{\rm BZ} \frac{d^3 k}{V_{\rm BZ}}
\frac{
\theta(\varepsilon_{\rm F} - \varepsilon_{n k})
\theta(\varepsilon'_{n' k} - \varepsilon_{\rm F})}
{\varepsilon'_{n' k} - \varepsilon_{n k} + i \omega}
X_{n n' k}(\omega)
\end{align}\]
