Performance of numerical approximation on the calculation of two-center   two-electron integrals over non-integer Slater-type orbitals using elliptical   coordinates
  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. 
Ali Ba\u{g}c{\i} and Philip E. Hoggan
quant-ph