Experiences with the Quadratic Korringa-Kohn-Rostoker Band Theory Method
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EXPERIENCES WITH THE QUADRATIC KORRINGA-KOHN-ROSTOKER BAND THEORY METHOD J. S. FAULKNER Alloy Research Center and Department of Physics, Florida Atlantic University, Boca Raton, Florida 33431.
ABSTRACT The Quadratic Korringa-Kohn-Rostoker method is a fast band theory method in the sense that all eigenvalues for a given k are obtained from one matrix diagonalization, but it differs from other fast band theory methods in that it is derived entirely from multiple-scattering theory, without the introduction of a Rayleigh-Ritz variational step. In this theory, the atomic potentials are shifted by Ao(r) with A equal to E-E0 and a(r) equal to one when r is inside the Wigner-Seitz cell and zero otherwise, and it turns out that the matrix of coefficients is an entire function of A. This matrix can be terminated to give a linear KKR, quadratic KKR, cubic KKR, ... , or not terminated at all to give the pivoted multiple-scattering equations. Full potentials are no harder to deal with than potentials with a shape approximation. INTRODUCTION The Quadratic Korringa-Kohn-Rostoker (QKKR) band theory method is, as its name implies, a modification of the Korringa-Kohn-Rostoker (KKR) band theory method [1-2]. The question of deriving a fast band-theory entirely on the basis of multiple-scattering theory [3] was addressed in 1979 [4], and the connection between this approach and earlier fast band-theories [5-7] was investigated. The possibility of a QKKR was discussed in this paper [4], but it was worked out in detail a few years later [8]. Some technical difficulties slowed the development of the theory, but these were overcome and self consistent band-theory calculations on muffin-tin and non-muffin-tin niobium, which included total energies, were published in 1989 [9]. Since that time, methods for further speeding up the QKKR have been developed [10], it has been applied to complex crystals [11], and used to treat the problem of an impurity in a host crystal [12]. A number of conclusions can be drawn from this experience with the QKKR. In the first place, all of the calculations since 1982 [8] have shown that the multiple-scattering theory will give useful eigenvalues and wave functions for general non-muffin tin potentials using angular momentum expansions that go to lmax = 4. It demonstrates that a fast band-theory method can be obtained from multiple-scattering theory without the introduction of a Rayleigh-Ritz variational step. Because of this, the resulting equations have characteristics that are unique. For example, the effect of the zeroes of the sine and cosine matrices on the eigenvalues obtained from linearized or quadraticized equations are easy to keep up with. The Rayleigh-Ritz approach leads naturally to a linearized theory, but the multiple scattering theory can produce one that is quadratic, cubic, or any higher power. Finally, this approach leads to a pivoted multiple-scattering formalism that is not a fast band theory, but it is useful in connection with the KKR-CPA theory of alloys or in dealing with impurity pro
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