The vibrational frequencies of selected normal modes can be obtained entirely from first principles with use of frozen phonon calculations which involve the precise evaluation of crystal total energy as a function of lattice displacement. The calculations allow a detailed analysis of the microscopic mechanisms causing phonon anomalies and softmode phase transitions. Successful calculations for Zr, Nb, and Mo have been made with use of both tight-binding and pseudopotential methods.

• Received 1 June 1982

DOI:https://doi.org/10.1103/PhysRevLett.49.673