Formal theory of equivalent potentials in solids: II. Scattering theory approach for muffin-tin potentials

Different equivalent potentials variously proposed for muffin-tin ionic potentials in solids are described in a unified manner as obeying a conservation condition for the scattering amplitudes. This condition is the basis of a reformulation of the problem in which a projector operator method is used to generate different families of equivalent potentials. When the projectors are constructed with bound and scattering states, one obtains the orthogonalized plane wave pseudopotentials. When these are in and out projectors in real space, with respect to the muffintin spheres, one obtains the augmented plane wave effective potentials. It is shown that the plane wave representation of the generator of this family yields the Slater effective potential. A special choice consisting of eliminating the potential inside, while leaving the right surface value to ensure matching, yields the Ziman pseudopotential, whose precise relationship with Slater's is thus established. A quantitative theory of phase shift cancellation is proposed and applied to perform a phase shift analysis of the orthogonalized plane wave pseudopotentials. It is shown that the number of multiples of π deducted from the trace of the phase shift operator is equal to the number of pseudo-bound energy levels (of the pseudo-Hamiltonian) which lie above the energy level under study.