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In this paper, which constitutes the first part of the series, we considercalculation of two-centre Coulomb and hybrid integrals over Slater-typeorbitals (STOs). General formulae for these integrals are derived with norestrictions on the values of the quantum numbers and nonlinear parameters.Direct integration over the coordinates of one of the electrons leaves us withthe set of overlap-like integrals which are evaluated by using two distinctmethods. The first one is based on the transformation to the ellipsoidalcoordinates system and the second utilises a recursive scheme for consecutiveincrease of the angular momenta in the integrand. In both methods simpleone-dimensional numerical integrations are used in order to avoid severedigital erosion connected with the straightforward use of the alternativeanalytical formulae. It is discussed that the numerical integration does notintroduce a large computational overhead since the integrands are well-behavedfunctions, calculated recursively with decent speed. Special attention is paidto the numerical stability of the algorithms. Applicability of the resultingscheme over a large range of the nonlinear parameters is tested on examples ofthe most difficult integrals appearing in the actual calculations including atmost 7i-type functions (l=6).

Calculation of two-centre two-electron integrals over Slater-type orbitals revisited. I. Coulomb and hybrid integrals