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Several improvements of the tetrahedron method for Brillouin-zone integrations are presented.(1) A translational grid of k points and tetrahedra is suggested that renders the results for insulators identical to those obtained with special-point methods with the same number of k points.(2) A simple correction formula goes beyond the linear approximation of matrix elements within the tetrahedra and also improves the results for metals significantly. For a required accuracy this reduces the number of k points by orders of magnitude.(3) Irreducible k points and tetrahedra are selected by a fully automated procedure, requiring as input only the space-group operations.(4) The integration is formulated as a weighted sum over irreducible k points with integration weights calculated using the tetrahedron method once for a given band structure. This allows an efficient use of the tetrahedron method also in plane-wave-based …