Ab initio calculations of the phonon spectra and the thermal expansion coefficients of the 4d metals

We have performed first principles calculations of the phonon spectra of the 4d transition metals with the so-called supercell method and found good agreement with observations. Furthermore, the electron and phonon contributions to the free energy for the 4d metals have been calculated from a first principles method. From the free energy, the thermal expansion of the metals is calculated. The calculated thermal expansion coefficients for the cubic elements are in overall good agreement with the experiment when the local density approximation is employed, while the generalized gradient approximation calculations yield a discrepancy as big as similar to 30%. For the hexagonal elements, reasonable agreement is found between calculations and experiment for the volume thermal expansion at temperatures 200 K less than or similar to T less than or similar to 300 K.