Application of the Augmented Plane Wave Method to Copper

An application of the augmented plane wave method of solving the periodic potential problem has been made to metallic copper. The aim is to investigate the value of the method and to learn details of the energy band structure of copper. The method has been found to converge very rapidly; eigenvalues of three- or four-figure accuracy can be obtained for any point in k-space without involving the solution of very large secular determinants. Throughout, good results can be obtained by considering only a single augmented plane wave; these appear to be good one-electron wave functions by themselves, and the possible use of these as a starting point in a self-consistent calculation is discussed. The method proves suitable ground for the application of a high-speed computer, by which the present computation has been carried out.The numerical results for copper are not in agreement with the results of a cellular calculation. Recalculation of certain states by the cellular method indicates that the dominant reason lies in the different potentials employed, the augmented plane wave method dealing with a potential constant in a region between the atoms. The sensitivity of the energy levels upon the exact crystal potential has become more apparent, though copper may be expected to show greater sensitivity than other materials. The present calculation of the width of the $3d$ band is 3.935 ev, compared with 3.46 ev by the cellular method and 2.7 ev by the tight-binding method. The appearance of a high density of states near the top of the $3d$ band is confirmed.