Auxiliary Functions for STO Molecular Integrals: Si, Ci and Ei

Abstract

The renewed interest in molecular integrals for Slater-type orbitals (STO’s)^[1] is increasing the urgency of the development of more efficient methods for calculating sine, cosine, and exponential integrals. The sine integral arises in expansions of the Fourier transform of a two-center STO product, and the exponential integral occurs in analytic multi-center integral formulations.^[2] The standard analytical methods for evaluating these integrals include the power series expansions for small arguments, and continued-fraction or asymptotic expansions for large arguments.^[3] None of these methods are fast for a large intermediate argument range, and the integrals are often evaluated by finite numerically fitted Chebyshev^[4] or rational-fraction^[5] expansions.