By the use of ellipsoidal coordinates, the two-center Coulomb and hybrid integrals over complete orthonormal sets of Psi(alpha)- type orbitals arising in ab initio calculations of molecules are evaluated, where alpha - 1, 0, -1, -2,...,. These integrals are expressed through the auxiliary functions Q(ns)(q) and G-(ns)(q). The comparison is made with some values of integrals for Slater type orbitals the computation results of which are in good agreement with those obtained in the literature. The relationships obtained are valid for the arbitrary quantum numbers, screening constants and location of orbitals. Closed form expressions for two-center Coulomb and hybrid integrals for 1s and 2s orbitals with alpha = 1 are also presented. As an example of application, the Hartree-Fock-Roothaan calculations for the ground state of H-2 molecule are carried out with alpha = 1 and alpha = 0.