Energy Bands in Periodic Lattices—Green's Function Method

The mathematical basis of calculations of energy bands in periodic lattices using the Green's function method is presented and the method's usefulness discussed. The original formulation of the method by Kohn and Rostoker is modified to achieve more efficient and accurate evaluation of "structure constants" using symmetry considerations and the full Ewald summation procedure. Formulas are derived giving the wave function both inside and outside the sphere inscribed in the unit cell. The method is demonstrated with the 3-dimensional Mathieu potential. Convergence is found to be very rapid both in this test case and in practical calculations on metals, and accurate energies and wave functions can be obtained without elaborate calculation even at points of low symmetry within the Brillouin zone.