Solid-harmonic derivatives of generalized Gaussian functions-exponential functions of a scalar argument that has no third derivatives with respect to any nuclear coordinate-are evaluated for three, four, and five centers without coupling any of the original angular momenta. Generalized Gaunt coefficients arise in this approach. They represent scalar coupling of all angular momenta lost from cross differentiation. All formulas are independent of all original angular momenta, which aids the evaluation of all integrals involving n centers at one time. Recurrence relations are given for the 3-j generalized Gaunt coefficient. The methods of Racah are used to obtain the coefficients that transform the generalized Gaunt coefficients into a representation in which the angular momentum lost due to cross differentiation are arbitrarily coupled, and thus show directly that the generalized Gaunt coefficients always represent scalar coupling. More intermediate information can be reused if the coupled generalized Gaunt coefficients are used to evaluate all the integrals involving a given set of centers. (C) 2003 American Institute of Physics.