  A general computational scheme for the (non-relativistic) Bethe logarithm is
developed opening the route to `routine' evaluation of the leading-order
quantum electrodynamics correction (QED) relevant for spectroscopic
applications for small polyatomic and polyelectronic molecular systems. The
implementation relies on Schwartz' method and minimization of a Hylleraas
functional. In relation with electronically excited states, a projection
technique is considered, which ensures positive definiteness of the functional
over the entire parameter (photon momentum) range. Using this implementation,
the Bethe logarithm is converged to a relative precision better than 1:10$^3$
for selected electronic states of the two-electron H$_2$ and H$_3^+$, and the
three-electron He$_2^+$ and H+H$_2$ molecular systems. The present work focuses
at nuclear configurations near the local minimum of the potential energy
surface, but the computations can be repeated also for other structures.
