Simple refinements of Brillouin zone integration

Calculations of thermodynamic properties of crystals by means of quasi-harmonic lattice dynamics require numerical integrations over the Brillouin zone, using successively finer grids to achieve convergence ro the required precision; but for complex crystals convergence may be uneconomically slow. A model for orthorhombic polyethylene is used to show how convergence may be improved (1) at low temperatures by taking successively finer grids close to the origin of reciprocal space, and (2) at all temperatures by using a three-dimensional Simpson's rule.