Microscopic analysis of interatomic forces in transition metals with lattice distortions

Abstract

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.