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

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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.