Theoretical studies of the optical and electronic properties of V, Nb, and Ta

Band structures of V, Nb, and Ta have been calculated by the nonlocal empirical pseudopotential method (EPM). The bands have the same ordering and about the same shapes as previous first-principles results, but show important differences in the locations of the $N_{1^{\prime }}$ and $H_{12}$ states. It is shown that these differences suggest there is considerably less $s-p$ and $d-p$ hybridization of the augmented-plane-wave valence bands compared to the EPM bands. It is further shown that charge distributions with charge maxima located between second nearest neighbors occur when the $H_{12}$ ($d_{x^{2-}{y}^2}, d_{z^2}$) states are more than 2.1 eV below the $P_4$ ($d_{\mathrm{xy}}, d_{\mathrm{yz}}, d_{\mathrm{zx}}$) states. The present charge distributions provide a picture of partially metallic $s-p$ bonding together with additional bonding associated with a "bilocal covalency" wherein a nearly nonoverlapping pair of tightly bound atomic $d$-like charge maxima are aligned between nearest-neighbor atoms. The theoretical reflectivity spectra, based on the direct-transition model (with $\overrightarrow{\mathrm{k}}$-dependent matrix elements), are in reasonable agreement with the data by Weaver et al. The main structure in the optical reflectivity arises from transitions between hybridized $s-d$ bands and hybridized $s-p$ bands near the $N$ symmetry point (e.g., $N_1\to N_{1^{\prime }}$). Reflectivities calculated with the nondirect-transition model are in unequivocally poorer agreement with the data than are the direct-transition reflectivities. The calculated densities of states and Fermi surfaces are shown to be in general agreement with available data. The partial densities of $s$, $p$, and $d$ states are also presented.