  For general two-electron two-centre integrals over Slater-type orbitals
(STOs), the use of the Neumann expansion for the Coulomb interaction potential
yields infinite series in terms of few basic functions. In many important cases
the number of terms necessary to achieve convergence by a straightforward
summation is large and one is forced to calculate the basic integrals of high
order. We present a systematic approach to calculation of the higher-order
terms in the Neumann series by large-order expansions of the basic integrals.
The final expressions are shown to be transparent and straightforward to
implement, and all auxiliary quantities can be calculated analytically.
Moreover, numerical stability and computational efficiency are also discussed.
Results of the present work can be used to speed up calculations of the STOs
integral files, but also to study convergence of the Neumann expansion and
develop appropriate convergence accelerators.
