  Conventional methods to calculate the thermodynamics of crystals evaluate the
harmonic phonon spectra and therefore do not work in frequent and important
situations where the crystal structure is unstable in the harmonic
approximation, such as the body-centered cubic (bcc) crystal structure when it
appears as a high-temperature phase of many metals. A method for calculating
temperature dependent phonon spectra self consistently from first principles
has been developed to address this issue. The method combines concepts from
Born's inter-atomic self-consistent phonon approach with first principles
calculations of accurate inter-atomic forces in a super-cell. The method has
been tested on the high temperature bcc phase of Ti, Zr and Hf, as
representative examples, and is found to reproduce the observed high
temperature phonon frequencies with good accuracy.
