The conventional approach to the electronegativity equalization principle suffers from a serious conceptual drawback that stems from discontinuities in the first derivative of the electronic energy with respect to the total charge of a molecule. A formalism that avoids the resulting ill-defined atomic hardness matrices employs a simple definition of the atomic softness matrix sigma. The matrix sigma, which is a generalization of the Huckel atom-atom polarizability matrix, can be easily and rigorously computed to arbitrary accuracy with any of the contemporary electronic structure methods. Properties of the atomic softness matrix are reviewed and illustrated with several numerical examples involving the first-row hydrides and a linear polyyne.