  Finding a proper local measure of chemical hardness has been a long-standing
aim of density functional theory. The traditional approach to defining a local
hardness index, by the derivative of the chemical potential with respect to the
electron density subject to the constraint of a fixed external potential, has
raised several questions, and its chemical applicability has proved to be
limited. Here, we point out that the only actual possibility to obtain a local
hardness measure in the traditional approach emerges if the external potential
constraint is dropped; consequently, utilizing the ambiguity of a restricted
chemical potential derivative is not an option to gain alternative definitions
of local hardness. At the same time, however, the arising local hardness
concept turns out to be fatally undermined by its inherent connection with the
asymptotic value of the second derivative of the universal density functional.
The only other local hardness concept one may deduce from the traditional
definition is the one that gives a constant value, the global hardness itself,
throughout an electron system in its ground state. Consequently, the
traditional approach is in principle incapable of delivering a local hardness
indicator. The parallel case of defining a local version of the chemical
potential itself is also outlined, arriving at a similar conclusion. Namely,
the only local chemical potential concept that can be gained from a definition
dE[n]/dn(r)|v is the one that gives a constant, mu itself, for electron systems
in their ground state.
