Electronic chemical response indexes at finite temperature in the canonical ensemble

Assuming that the electronic energy is given by a smooth function of the number of electrons and within the extension of density functional theory to finite temperature, the first and second order chemical reactivity response functions of the Helmholtz free energy with respect to the temperature, the number of electrons, and the external potential are derived. It is found that in all cases related to the first or second derivatives with respect to the number of electrons or the external potential, there is a term given by the average of the corresponding derivative of the electronic energy of each state (ground and excited). For the second derivatives, including those related with the temperature, there is a thermal fluctuation contribution that is zero at zero temperature. Thus, all expressions reduce correctly to their corresponding chemical reactivity expressions at zero temperature and show that, at room temperature, the corrections are very small. When the assumption that the electronic energy is given by a smooth function of the number of electrons is replaced by the straight lines behavior connecting integer values, as required by the ensemble theorem, one needs to introduce directional derivatives in most cases, so that the temperature dependent expressions reduce correctly to their zero temperature counterparts. However, the main result holds, namely, at finite temperature the thermal corrections to the chemical reactivity response functions are very small. Consequently, the present work validates the usage of reactivity indexes calculated at zero temperature to infer chemical behavior at room and even higher temperatures.