Secular equation of Korringa, Kohn, and Rostoker for the case of non-muffin-tin, space-filling potential cells

Abstract

Proof is provided that the secular equation proposed by Korringa, Kohn, and Rostoker for determining the electronic structure of collections of mufFin-tin potentials, i.e. , potentials bounded by nonoverlapping spheres, remains valid in the completely general case of nonoverlapping, arbitrarily shaped, and particularly space-filling potential cells. Consequently, this secular equation leads to the proper coefBcients for determining the wave function of the system as a linear combination of cell wave functions.

The rigorous (nonvariational) solution of the Schrödinger equation for a molecular potential of arbitrary shape. I. General formulation

“…Define s, = Ron + t,,, n f 0, (26) sg = max(s1,s2 ,..., sN). (27) and let s o be a maximum value among s,: The value so corresponds to the radius of a sphere, circumscribing all the "molecule." Introduce a sphere '3Jbo, centered at the origin ( R o = 0) and having a radius po such as po > so.…”

“…Additionally, Keister [lo] has demonstrated that, in practice, the divergent series may not lead to some troubles, because of their asymptotic, semiconvergent nature. On the other hand, Gonis et al [13,24], Zhang et al [27,28], and Molenaar [19] have given reasons for the difficulties with the summations of divergent series virtually not to exist. To draw such a conclusion, they put forward a procedure by which the origins of cells should be shifted.…”

Reformulation of the Korringa - Kohn - Rostoker coherent potential approximation for the treatment of space-filling cell potentials and charge-transfer effects

“…Indeed, recently a great deal of work has been directed at the generalization of the KKR method to space-filling cells. Much of this work has established [25][26][27][28][29][30][31][32] that, as far as the band structure of a material is concerned, multiple-scattering theory (MST) assumes identical forms in both the case of MT and the case of space-filling cells. In fact, in all its manifestations, MST retains one of its most advantageous features, namely the separability of the structural aspects from the potential aspects of a scattering assembly.…”