Revisiting the definition of local hardness and hardness kernel
An analysis of the hardness kernel and the local hardness is performed to propose new definitions for these quantities that follow a similar pattern to the one that characterizes the quantities associated with softness, that is, we have derived new definitions for which the integral of the hardness kernel over the whole space of one of the variables leads to the local hardness, and the integral of the local hardness over the whole space leads to the global hardness. A basic aspect of the present approach is that the global hardness keeps its identity as the second derivative of the energy with respect to the number of electrons. The local hardness thus obtained depends on the first and second derivatives of the energy and the electron density with respect to the number of electrons. When these derivatives are approximated by a smooth quadratic interpolation of the energy, the expression for the local hardness reduces to the one intuitively proposed by Meneses, Tiznado, Contreras and Fuentealba. However, when one combines first directional derivatives with smooth second derivatives one finds additional terms that allow one to differentiate the local hardness for electrophilic attack from the one for nucleophilic attack. Numerical results related with electrophilic attacks on substituted pyridines, substituted benzenes and substituted ethenes are presented to show the overall performance of the new definition.