Improved parametrization schemes for electronic energy bands

In connection with semiempirical studies of electronic band structures we consider the development of parametrization schemes based mainly on the logarithmic derivatives $L_l$ rather than exclusively on the tangent of the phase shifts, $tan{\eta }_l$, as was done in an earlier study. Improvements over the earlier approach are sought in two different approaches considered here for the noble metals. In the first of these the logarithmic derivatives of the radial wave functions are exclusively treated as the fundamental quantities to be parametrized. For a comparable number of parameters, this scheme gives a small over-all improvement in the accuracy of fit to the results of several first-principles band calculations as compared to the fit obtained in the earlier $tan{\eta }_l$ parametrization scheme. The second scheme gives significantly better fits to the first-principles band-calculation results. This approach obtains these improved results by adopting a hybrid technique which uses $tan{\eta }_l$ parametrization to treat the resonant part of the $d$ scattering contribution, while the residual $d$ scattering and the complete $s$ and $p$ contributions are treated by parametrizing the logarithmic derivatives. The parametrization of the smoothly varying logarithmic derivatives is done in a uniform and systematic way in terms of square-well potentials. A simplified version of this scheme is also successful for a "simple" metal as demonstrated by an application to Al. An advantage of both schemes in an empirical application is that the goodness of fit is rather insensitive to the energy values used in evaluating the parameters. Finally, we discuss the relative advantages of the present scheme over others that have been proposed.