  In this paper we consider calculation of two-centre exchange integrals over
Slater-type orbitals (STOs). We apply the Neumann expansion of the Coulomb
interaction potential and consider calculation of all basic quantities which
appear in the resulting expression. Analytical closed-form equations for all
auxiliary quantities have already been known but they suffer from large digital
erosion when some of the parameters are large or small. We derive two
differential equations which are obeyed by the most difficult basic integrals.
Taking them as a starting point, useful series expansions for small parameter
values or asymptotic expansions for large parameter values are systematically
derived. The resulting novel expansions replace the corresponding analytical
expressions when the latter introduce significant cancellations. Additionally,
we reconsider numerical integration of some necessary quantities and present a
new way to calculate the integrand with a controlled precision. All proposed
methods are combined to lead to a general, stable algorithm. We perform
extensive numerical tests of the introduced expressions to verify their
validity and usefulness. Advances reported here provide methodology to compute
two-electron exchange integrals over STOs for a broad range of the nonlinear
parameters and large angular momenta.
