  The traditional approach to establishing a local measure of chemical
hardness, by defining a local hardness concept through the derivative of the
chemical potential with respect to the electron density, has been found to have
limited chemical applicability, and has proved to be an unfeasible approach in
principle. Here, we propose a new approach via a unique local energy concept.
This local energy is shown to emerge from the Hamilton-Jacobi kind of
construction of Schrodinger's quantum mechanics. It then leads to the concepts
of a local chemical potential, i.e. negative of local electronegativity, and a
local hardness just as the chemical potential and hardness are obtained from
the energy, namely via differentiations with respect to the number of
electrons. The emerging local hardness adds corrections to a recently proposed
local hardness expression that has been found to be a good local measure of
hardness for a series of atomic and molecular systems. These corrections become
relevant for molecules with a large number of electrons. It is pointed out
further that the definition of local softness that yields it as the Fukui
function times the softness is not well-established, explaining recent
observations of failure of this local softness concept as a proper local
reactivity index for hard systems.
