Lattice dynamics of transition metals in the resonance model

In this paper we reformulate the lattice dynamics of the transition metals in the framework of the recently proposed transition-metal model-potential (TMMP) method in order to study the nonlocal effects arising from the strong energy dependence of the $l=2\mathrm{}$ resonant term in the TMMP. Starting from the most general form of the electronic contribution to the dynamical matrix in the harmonic and self-consistent-field approximations, which involves the exact expression for the dielectric matrix $\epsilon (\overrightarrow{\mathrm{q}}+\overrightarrow{\mathrm{g}},\overrightarrow{\mathrm{q}}+{\overrightarrow{\mathrm{g}}}^{\prime })$ in the random-phase approximation, we apply a model-potential transformation to the true Bloch functions in the expression. The transformation shows that the modification of the free-electron (Lindhard) expression for the dielectric matrix arises from the depletion hole associated with the nonlocality ($E$ dependence) of the TMMP and the dominant contribution to the depletion hole comes from the strong energy dependence of the $l=2\mathrm{}$ resonant term. Finite-depletion hole is obtained by using the $T$ matrix for the $l=2\mathrm{}$ partial-wave electron-phonon scattering to handle the singularity in the TMMP well depth, $A_2\mathrm{}\left(E\right)\mathrm{}\propto {(E_d-E)}^{-1\mathrm{}}$ at the resonance energy $E_d$. Numerical results are obtained for the typical body-centered-cubic transition metal, vanadium, for which we have enough atomic spectroscopic data and for which no previous calculation of the phonon spectrum has been reported. The $l=2\mathrm{}$ resonance is found to determine a large percentage of the soft modes in the phonon spectrum, and the overall agreement with the experimental result of Colella and Batterman is good except for the lower [110] transverse branch.