  Radiative corrections to electronic structure are characterized by
perturbative expansions in $\alpha$ and $Z\alpha$, where $\alpha$ is the
fine-structure constant and $Z$ is the nuclear charge. A formulation of the
leading-order $\alpha(Z\alpha)$ Uehling contribution to the renormalized vacuum
polarization is reported in a form that is convenient for implementation in
computational studies of the relativistic electronic structures of molecules.
Benchmark calculations based on these methods are reported that include the
leading-order vacuum polarization effects within relativistic mean-field
methods for the E119$^+$ ion and the closed-shell diatomic species E119F.
