Interionic interactions in transition metals

An approximate calculation of the total energy of a transition metal as a function of volume and ionic configuration at constant volume obtained by extending the nearly-free-electron theory of the simple metals to include the effects of transition-metal $d$ bands provides a qualitative first-principles prediction of the elastic and bonding properties of the transition metals. The description becomes quantitative if one allows the adjustment of two parameters. The $s$ electrons, treated with an empty-core pseudopotential and Thomas-Fermi theory, contribute volume-dependent terms and an effective two-body repulsive potential between ions at constant total volume. The Harrison-Froyen formulation of the $d-d$ matrix elements is combined with a similar treatment of the overlap between $d$ states on different ions and the Friedel model of the density of $d$ states to include the effects of the $d$ bands in the total energy, resulting in a bonding term proportional to the bandwidth, varying as $d^{-5\mathrm{}}$, and a shift in the center of gravity of the $d$ band, varying as $d^{-8\mathrm{}}$, both of which can be expressed as effective two-body interactions between ions to be combined with the repulsion from the $s$ electrons in describing properties at constant total volume. The effect of $s-d$ hybridization is shown to be approximately accounted for by a shift in relative band occupations. The volume dependence is tested by prediction of the equilibrium volume, the bulk modulus, and a Grüneisen constant for all the transition metals, and the effective interionic potential is used to predict the elastic constants of the cubic metals. All properties can be calculated by hand; the agreement between predicted and observed values is as good as that obtained with the corresponding theory in the simple metals.