Quadratic integration over the three-dimensional Brillouin zone

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Abstract
A new method is described to evaluate integrals of quadratically interpolated functions over the three-dimensional Brillouin zone. The method is based on the method of the authors for analytic quadratic integration over the two-dimensional Brillouin zone. It uses quadratic interpolation not only for the dispersion relation epsilon (k), but for property functions f(k) as well. The method allows a 'machine accuracy' evaluation of the integrals and may therefore be regarded as equivalent to a truly analytic evaluation of the integrals. It is compared to other methods of integral approximation by calculating tight-binding Brillouin zone integrals using the same number of k-points for all methods. Also shown are cohesive energy calculations for a number of elements. When the quadratic method is compared to the commonly used linear method, it is found that far fewer k-points are needed to obtain a desired accuracy.