Average electronic energy is the central quantity in conceptual chemical reactivity theory

We show that, at both zero and nonzero temperature, the average electronic energy is the central quantity for the study of chemical reactivity using density functional theory. Specifically, we show that regardless of the temperature and the ensemble framework one chooses, the average electronic energy exhibits the functional dependence with respect to external potential and electron number of the zero-temperature case, but with the presence of an additional temperature-dependent term that vanishes in the zero-temperature limit. This reconciles the results from the finite-temperature chemical reactivity theory with those of the traditional zero-temperature approach, and resolves apparent discrepancies between the canonical and grand canonical ensembles. However, the grand canonical ensemble provides a more convenient mathematical framework for studying chemical reactivity because it regards chemical species as open quantum (sub)systems. The response functions of the average electronic energy reduce to the zero-temperature reactivity descriptors in the appropriate limit, helping to justify the use of idealized zero-temperature approximations for describing experimental results obtained at nonzero temperature. Extending this approach to the entropy, the responses of the entropy are shown to give a different set of interesting chemical reactivity response functions. The expressions for the heat capacity of an electronic system are especially elucidative.