Spherically symmetrical properties of expansion coefficients for translation of spherical harmonics

A theorem regarding the angular dependence has been established for the expansion coefficients for translation of spherical harmonics (SH): If we add the products of all the translation coefficients with the same l values, but different m's, the result is independent of orientation. The spherically symmetrical properties of the translation coefficients K-lm,K-l'm' for SH obtained in the present paper and, the rotation coefficients T-lm,l'm'(lambda), the two-center overlap integrals over arbitrary atomic orbitals S-nlm,S-n'l'm' and the translation coefficients V-nlm,n'l'm'(N) for Slater-type orbitals (STO's) given recently by the author [I.I. Guseinov, J. Mel. Struct. (Theochem), 343 (1995) 173] are the same. The analytical formulas also have been derived for translation coefficients of SH in terms of binomial coefficients. The final results are especially useful for machine computations of arbitrary multi-electron molecular integrals for which the series expansion formulas have recently been derived by the author.