Total Energy ofd-Band Metals: Alkaline-Earth and Noble Metals

We consider the sum of the one-electron energies of the occupied bands for a metal which has $d$ bands interacting with nearly-free-electron bands. Using Hubbard's hybrid form of the Korringa-Kohn-Rostoker band-structure method, we consistently retain terms in the energy to first order in the width of the $d$ scattering resonance. For the alkaline-earth and noble metals (the metals at each extreme of the transition series), it is shown that the lowest-order effect of the interaction between the $d$ bands and the free-electron bands, and, in the case of a noble metal, also the effect of the finite width of the full $d$ bands, results in a simple net contribution to the total energy given by $U_d=\left(\frac{10\Gamma }{\pi }\right)\ln \left(\frac{|{\epsilon }_0-{\epsilon }_F|}{{\epsilon }_0}\right)$, where ${\epsilon }_0$ and $\Gamma$ are the energy and width of the $d$ scattering resonance, and ${\epsilon }_F$ is the Fermi energy for the unperturbed free-electron band. It is shown that this term is to be added to the usual pseudopotential contribution to the total energy, where, if the pseudopotential is expressed in Ziman's phase-shift formulation, the $d$ phase shift is to be replaced by the residual phase shift which remains after the resonance part is extracted. It is also shown that the term $U_d$ gives a negligibly small contribution to the cohesive energy and to the compressibility of Cu, indicating that the $d$-band contributions to these properties are to be found either in the volume dependence of the energy of the $d$ resonance, or in contributions to the total energy of the metal other than that of the total band-structure energy.