Tight-binding calculations of the elastic constants and phonons of hcp Zr: Complications due to anisotropic stress and long-range forces

We have calculated the phonons and elastic constants of zirconium in the hexagonal-close-packed (hcp) crystal structure using the Naval Research Laboratory (NRL) empirical tight-binding (TB) approach; the tight-binding parameters are obtained by fitting to ab initio density-functional theory-generalized gradient approximation energy bands and total energies for many different structures and volumes. We address difficulties involved with the fitting procedure and give results for elastic constants, force constants, quasiharmonic phonons, and specific heat. Because the predicted TB lattice constants at the zero-temperature energy minimum are slightly different from those experimentally observed at room temperature, our TB model has an anisotropic stress at the experimental lattice constants. We correct for these stresses in our calculations of the elastic constants and sound speeds. Such techniques are also useful for calculating such properties for arbitrary c/a. Our phonon calculations were done by the direct-force method in real space using calculated force constants; these fall off quite slowly with distance, which causes problems with the calculated phonon spectrum due to the slow convergence with increasing supercell size. This long-range behavior could play a large role in determining the unusually anharmonic and anomalous physical properties of Zr. We show that similar, although less severe, problems should arise for other metals. These considerations suggest that the direct-force method for calculating phonons may be problematic for many metals.