  The two-center two-electron Coulomb and hybrid integrals arising in
relativistic and nonrelativistic ab-initio calculations of molecules are
evaluated over the non-integer Slater-type orbitals via ellipsoidal
coordinates. These integrals are expressed through new molecular auxiliary
functions and calculated with numerical Global-adaptive method according to
parameters of non-integer Slatertype orbitals. The convergence properties of
new molecular auxiliary functions are investigated and the results obtained are
compared with results found in the literature. The comparison for two-center
twoelectron integrals is made with results obtained from one-center expansions
by translation of wavefunction to same center with integer principal quantum
number and results obtained from the Cuba numerical integration algorithm,
respectively. The procedures discussed in this work are capable of yielding
highly accurate two-center two-electron integrals for all ranges of orbital
parameters.
