Vibrational frequencies via total-energy calculations. Applications to transition metals

The important longitudinal ($\frac{2}{3},\frac{2}{3},\frac{2}{3}$) vibrational modes in Mo, Nb, and bcc Zr as well as the $H$- point modes in Mo and Nb have been studied using the frozen-phonon approach. These entirely first-principles calculations involve the precise evaluation of the total crystalline energy as a function of lattice displacement and yield calculated phonon frequencies to within a few percent of the experimental values. Anharmonic terms are readily obtained and are found to be very important for causing the tendency toward the $\omega$-phase instability in bcc Zr. The charge densities and single-particle energies obtained in the course of the calculations allow a detailed analysis of the electronic response to lattice distortions and the mechanisms causing phonon anomalies. The calculations also provide first-principles benchmarks at a few wave vectors where the validity of phenomenological models can be tested or their parameters determined.