Parametrization of electronic band structure using the Green's-function method: Empirical application to Cu and Ag

A new empirical energy-band parametrization scheme based on the Green's-function method has been developed and was applied to Cu and Ag. The scheme utilizes the logarithmic derivatives associated with an ab initio muffin-tin potential $V^{\mathrm{}\left(0\right)\mathrm{}}\mathrm{}\left(r\right)\mathrm{}$. The scheme can be understood in terms of the addition to $V^{\mathrm{}\left(0\right)\mathrm{}}$ of $E$- and $l$ -dependent square-well potentials the depths of which ${\nu }_l\mathrm{}\left(E\right)\mathrm{}$ are adjusted to yield the correct (empirical) energy bands. The ${\nu }_l\mathrm{}\left(E\right)\mathrm{}$ are found to be smooth functions of $E$ which can be accurately approximated by low-order polynomials. An accurate fit for $d$ -band metals over a roughly 1-Ry range requires only seven adjustable parameters— a number smaller than required by other schemes. Extensive tests of the approach using results of first-principles calculations were carried out in precisely the same manner as proposed for the empirical application, and the results indicate that this scheme is more accurate than other approaches using more parameters. The seven pieces of data used in the empirical parametrization for Cu and Ag were the $s$, $p$, and $d$ phase shifts required to fit the Fermi-surface geometry and four firmly identified vertical energy gaps: $E_F-X_5$, $X_4^{}{}_{}{}^{\prime }-X_5$, $X_5-X_3$, and $L_1^u-L_2^{}{}_{}{}^{\prime }$. The empirical $E_n\left(\overrightarrow{\mathrm{k}}\right)$ were obtained for a large range of $E_F$ values (relative to the constant part of the potential). Except for the rather high energy levels (e.g., the upper $X$, $W$, and $K$ states) the relative band structures prove to be rather insensitive to the $E_F$ value. The presently available and firmly established data do not narrow the permissible range of $E_F$. Comparisons with several recent Cu and Ag calculations show that the present bands are in better accord with experiments. The empirical $E_n\left(\overrightarrow{\mathrm{k}}\right)$, which are required to fit the input data, are also found to agree within experimental uncertainties with all additional data related to level positions. Cu, for which more data are available, is particularly well checked. To obtain some information about the effective interactions $V_l\mathrm{}(E,r)\mathrm{}$ and the associated wave functions, a set of coupled integro-differential equations is derived which relates these quantities to the ${\nu }_l\mathrm{}\left(E\right)\mathrm{}$. However, it appears that the solutions to these equations are not unique unless some constraints are imposed on the correction $V_l\mathrm{}(E,r)-V^{\mathrm{}\left(0\right)\mathrm{}}\mathrm{}\left(r\right)\mathrm{}$. A suggestion is made for obtaining approximate useful wave functions prior to the resolution of the nonuniqueness problem. From our experiences and a consideration of the merits of the new scheme, it is evident that it should be very useful in the study of the electronic structure of various solids.