Molecular Integrals for Exponential-Type Orbitals Using Hyperspherical Harmonics

Exponential-type orbitals are better suited to calculations of molecular electronic structure than are Gaussians, since ETO's can accurately represent the behavior of molecular orbitals near to atomic nuclei, as well as their long-distance exponential decay. Orbitals based on Gaussians fail in both these respects. Nevertheless, Gaussian technology continues to dominate computational quantum chemistry, because of the ease with which difficult molecular integrals may be evaluated when Gaussians are used as a basis. In the present chapter, we hope to contribute to a new movement in quantum chemistry, in which ETO's will not only be able to produce more accurate results than could be obtained using Gaussians, but also will compete with Gaussian technology in the speed of integral evaluation. The method presented here makes use of V. Fock's projection of three-dimensional momentum-space onto a four-dimensional hypersphere. Using this projection, Fock was able to show that the Fourier transforms of Coulomb Sturmian basis functions are very simply related to four-dimensional hyperspherical harmonics. With the help of Fock's relationships and the theory of hyperspherical harmonics we are able to evaluate molecular integrals based on Coulomb Sturmians both rapidly and accurately. The method is then extended to Slater-Type Orbitals by using a closed-form expression for expanding STO's in terms of Coulomb Sturmians. A general theorem is presented for the rapid evaluation of the necessary angular and hyperangular integrals. The general methods are illustrated by a few examples.