The phonon dispersion curves, and phonon density of states,  can be calculated with a standard ab initio and the direct method [1] codes. The procedure, originally applied to pure bulk crystals only, has been extended to study defected crystals, surfaces, adsorbed atoms on the surface, multilayers, etc.    The  classical phonon dispersion curves and  phonon density of state are now calculated routinely. Since the calculations provide not only the phonon frequencies, but also the polarization vectors, one can find the intensity of coherent and incoherent neutron and x-ray scattering  to carry on comparisons to the measurements.

Present computer clusters allow to treat quite complex systems. In recent years an interest to strong electron correlation systems has been increased. Along this line the ab initio codes were supplemented by modulus capable to treat such systems. Numerous calculations show that even simple approach which includes local Coulomb interaction energy U reproduces quite well the  phonon properties. The materials PuCoGa₅,  Fe₃O₄ are examples of such systems. Magnetic properties of  constituent atoms are another factor which frequently influences the phonon frequencies.

    Temperature dependence of many quantities can be described by the   quasiharmonic approximation. In particular, this approach allowed to establish the pressure-temperature phase diagram of AlN, Mg₂SiO₄, CaF₂. Ab initio phonon calculations can be performed on two-dimensional systems like graphene, multilayers and surfaces. Specially surfaces covered by monolayers of other compound are interesting to consider due to the appearance of stresses and their influences on the phonon spectra.

References

[1] http://wolf.ifj.edu.pl/phonon/

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1. First-principles study of phonon-based properties of magnetic CuFeS₂
2. Vibrational and thermodynamical properties of solids