Computation of auxiliary functions in STO molecular integrals up to arbitrary accuracy. I. evaluation of incomplete gamma function En(x) by forward recursion

A new inequality and an approximate relation of incomplete Gamma function E_n(x) are derived in this paper. Applying both to simplify the error analysis on the recursive algorithm of E_n(x), we analytically derived a criterion about relative error. The criterion decides whether the forward recursive relation can be used to compute {E_n(x), n = 0, 1, ?, N} with an acceptable loss of significant figures. Further analytical exploration leads to another criterion that decides whether the group of E_n(x) can be evaluated with an acceptable absolute error by the same methods. Both criteria are not only superior to those given in previous work by other authors, but also strongly supported by our numerical experiments. A strategy is illustrated to explain how to apply both criteria to evaluate molecular integral over Slater-type orbitals. © 1993 John Wiley & Sons, Inc.