Accurate and fast algorithm of the molecular incomplete gamma function with a complex argument

Several efficient algorithms for the accurate and fast calculation of the molecular incomplete gamma function Fm(z) with a complex argument z are developed. The complex incomplete gamma function is arising in molecular integrals over the gauge-including atomic orbitals. Two kinds of algorithms are recommended: (1) a high-precision version and (2) a fast version. The high-precision version is able to guarantee 15 significant figures (10(-15) in the relative error) and the fast version is able to guarantee 12 significant figures (10(-12) in the relative error), at worst, within the double-precision arithmetic. The fast version is about 5-20 times faster than the high-precision version. For most molecular calculations, the fast version will give a satisfied precision.