Analytic quadratic integration over the two-dimensional Brillouin zone

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Abstract
A new method is described to evaluate analytically integrals of quadratically interpolated functions over the two-dimensional Brillouin zone. The method is not geometric in nature like the quadratic method of Methfessel and co-workers, but is algebraic. It allows quadratic interpolation not only for the dispersion relation epsilon n(k), but for property functions fn(k) as well. Comparisons are made between the analytic quadratic integration and the commonly used analytic linear integration by calculating tight-binding Brillouin zone integrals with the same number of k-points for both methods. It is shown that convergence behaviour and convergence rate are far better for the analytic quadratic integration than for the analytic linear integration. Roughly, analytic quadratic integration can achieve the same accuracy as analytic linear integration with only about the square root of the total number of k-points needed.