Quadratic integration: Theory and application to the electronic structure of platinum

In this paper we present a general computational technique for deriving fine histogram representations of singular k-dependent integrals for any crystalline solid. Detailed consideration is given to the errors involved in such integrals, and in their representations. Comparisons are made to previous work and the special advantage of each technique is emphasized. Applications of the quadratic (QUAD) technique are made to platinum using the combined interpolation scheme showing the total, band and basis function (S-part) density of states. These densities of states are based on histograms of width 0.001 Ry, 1 000 000 Monte Carlo points in 1/48th of the Brillouin zone, and have an intrinsic accuracy of better than 1%. The extrinsic accuracy of the platinum band structure used here is somewhat poorer than 1%, and the special problems of materials with a steep derivative in the density of states at the Fermi energy is considered. Comparisons with the temperature and magnetic field dependent susceptibility and the specific heat data suggest that enhancement effects in platinum are small.