  In this paper we consider the evaluation of the Araki-Sucher correction for
arbitrary many-electron atomic and molecular systems. This contribution appears
in the leading order quantum electrodynamics corrections to the energy of a
bound state. The conventional one-electron basis set of Gaussian-type orbitals
(GTOs) is adopted; this leads to two-electron matrix elements which are
evaluated with help of generalised the McMurchie-Davidson scheme. We also
consider the convergence of the results towards the complete basis set. A
rigorous analytic result for the convergence rate is obtained and verified by
comparing with independent numerical values for the helium atom. Finally, we
present a selection of numerical examples and compare our results with the
available reference data for small systems. In contrast with other methods used
for the evaluation of the Araki-Sucher correction, our method is not restricted
to few-electron atoms or molecules. This is illustrated by calculations for
several many-electron atoms and molecules.
