Abstract
The `Green's function' method for the calculation of electronic band structure is investigated. The general T-matrix theory of Beeby and Edwards is used to transform the formulae of Korringa and of Kohn and Rostoker to a `reciprocal lattice representation' which is equivalent to a `nearly free electron' formalism. The matrix elements between plane waves then correspond to a `pseudo-potential', which depends mainly on the phase shifts of the atomic potentials but which is also a function of the wave vector and energy of the Bloch state being considered. For small phase shifts it tends to the scattering amplitude or `quasi-potential' of an atomic potential.
But when a phase shift passes through π/2 this pseudo-potential goes through infinity, i.e. at a virtual bound state there is a singularity in the reaction matrix (K matrix) of the atomic potential. Such a `resonance' can give rise to a narrow band, such as a d band in a transition metal. At energies below the zero of the muffin-tin potential (where the phase shifts become imaginary), the formalism is still valid, and one gets further bands from the singularities of the T matrix - the bound states - of each potential well.
The discussion is purely formal, but suggests that the Green's function method is more than a complicated scheme of calculation capable of giving good numerical results over a wide range of energies; it can be physically interpreted and used empirically in the same way as the orthogonalized plane-wave, tight-binding and other apparently more direct techniques.