  We report first principles LDA calculations of the electronic structure and
thermoelectric properties of $\beta $-Zn$_{4}$Sb$_{3}$. The material is found
to be a low carrier density metal with a complex Fermi surface topology and
non-trivial dependence of Hall concentration on band filling. The band
structure is rather covalent, consistent with experimental observations of good
carrier mobility. Calculations of the variation with band filling are used to
extract the doping level (band filling) from the experimental Hall number. At
this band filling, which actually corresponds to 0.1 electrons per 22 atom unit
cell, the calculated thermopower and its temperature dependence are in good
agreement with experiment. The high Seebeck coefficient in a metallic material
is remarkable, and arises in part from the strong energy dependence of the
Fermiology near the experimental band filling. Improved thermoelectric
performance is predicted for lower doping levels which corresponds to higher Zn
concentrations.
