Transition-Metal Pseudopotentials

The pseudopotential method is reformulated and directly generalized to transition metals. The Schrödinger equation, in the self-consistent field approximation, is transformed without approximation into a transition-metal pseudopotential equation. The usual simple-metal pseudopotential appears in a modified form, but is nevertheless weak. An additional term appears which plays a role akin to $s-d$ hybridization. Eigenstates and eigenvalues are then sought in a perturbation expansion in the pseudopotential and hybridization terms. It is possible, as with simple-metal pseudopotentials, to sum over states to obtain both the screening field and the total energy, though this summation is simple only when we neglect the partial filling or partial emptying of $d$ bands. This is appropriate for the noble metals, the alkaline earths, and perhaps some transition metals between. For such cases there exists a pseudopotential form factor, just as in the simple metals, and an energy-wave-number characteristic from which a two-body central-force interaction may be derived. The form factor is evaluated for copper (using a semilocal approximation); it yields a good estimate of the $s-p$ gap at $L$ and of the resistivity of liquid copper.