Resonances and Transition-Metal Pseudopotentials

Two other views of the d-state electronic structure should be introduced. They are related mathematical formulations but, as with the use of both LCAOand pseudopotential descriptions of eovalent solids, the different approaches are relevant for answering different questions. 

Imagine again that spheres are constructed around each atom of a transition metal. A muffin-tin potential, constant between spheres and spherically symmetric within, is assumed. In the context of this section, it will be best to let the spheres be nonoverlapping. 

The solution of the corresponding radial equation is a familiar problem -we follow Schiff (1968) outside the atomic sphere, there arc two independent solutions, the spherical Bessel function and Neumann function, 7i(fcr) and ni kr), both corresponding to energy For conceptual purposes, they maybe thought 

The two arbitrary constants have been written A cos and —A sin (5,. The asymptotic form at large r (following Schiff, 1968) is 

We fit to a vanishing boundary condition at R (it is Ri R) = 0) and must fit also at the atomic sphere. Only the ), is regular at the origin, so if the potential had the same constant value within the sphere, the eorrcct solution would be simply /, that is, the phase shift 5, would be zero. The effect of a scattering potential is simply to introduce a nonzero phase shift, which indeed simply shifts the phase of the sinusoidal oscillations in the asymptotic form given in Eq. (20-21). 

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The electronic structure is reformulated in terms of free electrons and a d resonance in order to relate the band width Wj to the resonance width T, and is then reformulated again in terms of transition-metal pseudopotential theory, in which the hybridization between the free-electron states and the d state is treated in perturbation theory. The pseudopotential theory provides both a definition of the d-state radius and a derivation of all interatomic matrix elements and the free-electron effective mass in terms of it. Thus it provides all of the parameters for the LCAO theory, as well as a means of direct calculation of many properties, as was possible in the simple metals. ... 

The expansion in F/( , - tj.) would suggest the possibility of incorporating the effect of the resonance in perturbation theory and thus extending the pseudopotential perturbation theory of simple metals to transition metals. This has in fact been done (Harrison, 1969) and the pseudopotential approach has been extensively developed by Moriarty (1970, I972a,b,c), but the application.s have been largely restricted to the ends of the transition scries where the expansion is clearest. 

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