The complete and original calculation scheme beyond the finite difference approximation, for the atomic (and orbital) Fukui function (FF) indices is proposed. The method explores an expansion for derivatives of LCAO coefficients, partial derivative C/partial derivative N = CU. The separation scheme for the U matrix has been elaborated at the ab initio level. Nucleophilic and electrophilic FF indices, as well as atomic softness, have been derived from the standard result of SCF HF ab initio calculations. The indices reproduce two effects; the change in orbital occupancy and the relaxation of the electronic system. The molecular hardness (softness) provided by this scheme explicitly includes these two effects.