It is now possible to calculate accurately vibrational frequencies for transition metals entirely from first principles using the frozen-phonon approach. In addition, the interatomic forces arising from the electronic response to a particular phonon distortion can be calculated by making use of the Hellmann-Feynman theorem. Analysis of these forces helps in visualizing the complicated microscopic interactions responsible for phonon anomalies. The techniques and analysis are demonstrated for the important longitudinal ($\frac{2}{3}$,$\frac{2}{3}$,$\frac{2}{3}$) phonon in Mo, Nb, and bcc Zr. The stiffening of this mode as the electron-per-atom ratio increases from Nb to Mo is shown to arise from a development of directional bonding. The precipitous dip in this mode for the high-temperature bcc phase of Zr and the instability of bcc Zr towards the formation of the $\omega$ phase are related to the $d$-electron screening and details of the electronic structure.

• Received 5 July 1983

DOI:https://doi.org/10.1103/PhysRevB.28.6687