Practical criterion for the determination of translation factors. III. A common translation factor with optimized asymptotic form

As a continuation of parts I and II of this series [A. Riera, Phys. Rev. A 30, 2304 (1984) and L. F. Errea, J. M. Gómez-Llorente, L. Méndez, and A. Riera, Phys. Rev. A 32, 2158 (1985)], and for ${\mathrm{He}}^{2+}$-H(1s) collisions as a benchmark case, we use the criterion based on the minimization of the Euclidean (N) and weighted ($N_w$) norms, to obtain a common translation factor with optimal asymptotic form. Mathematical and intuitive conditions for this translation factor are developed, and pitfalls of the use of these conditions without an objective criterion, such as provided by N or $N_w$, are pointed out.