Analytic quadratic integration over the two-dimensional Brillouin zone

Brillouin-zone (BZ) integrations in systems with two-dimensional periodicity are discussed in the context of a method wherein the BZ is divided into simplices (triangles), and band energies and properties contributions are represented by quadratic interpolations based on six data points on the edges of each simplex. This paper describes a straightforward and easily implemented algorithm for identifying the paths bounding the occupied portion(s) of each simplex, and (in contrast to earlier work by others) provides completely analytic expressions, in closed form, for evaluating properties integrals over the occupied regions.