We present a sampling method for Brillouin-zone integration in metals which converges exponentially with the number of sampling points, without the loss of precision of normal broadening techniques. The scheme is based on smooth approximants to the \ensuremath{\delta} and step functions which are constructed to give the exact result when integrating polynomials of a prescribed degree. In applications to the simple-cubic tight-binding band as well as to band structures of simple and transition metals, we demonstrate significant improvement over existing methods. The method promises general applicability in the fields of total-energy calculations and many-body physics.
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