Local chemical potential, local hardness, and dual descriptors in temperature dependent chemical reactivity theory

Abstract

In this work we establish a new temperature dependent procedure within the grand canonical ensemble, to avoid the Dirac delta function exhibited by some of the second order chemical reactivity descriptors based on density functional theory, at a temperature of 0 K. Through the definition of a local chemical potential designed to integrate to the global temperature dependent electronic chemical potential, the local chemical hardness is expressed in terms of the derivative of this local chemical potential with respect to the average number of electrons. For the three-ground-states ensemble model, this local hardness contains a term that is equal to the one intuitively proposed by Meneses, Tiznado, Contreras and Fuentealba, which integrates to the global hardness given by the difference in the first ionization potential, I, and the electron affinity, A, at any temperature. However, in the present approach one finds an additional temperature-dependent term that introduces changes at the local level and integrates to zero. Additionally, a τ-hard dual descriptor and a τ-soft dual descriptor given in terms of the product of the global hardness and the global softness multiplied by the dual descriptor, respectively, are derived. Since all these reactivity indices are given by expressions composed of terms that correspond to products of the global properties multiplied by the electrophilic or nucleophilic Fukui functions, they may be useful for studying and comparing equivalent sites in different chemical environments.

Grand-canonical approach to density functional theory of electrocatalytic systems: Thermodynamics of solid-liquid interfaces at constant ion and electrode potentials

“…A position dependent chemical potential also allows defining regions with different electrophilicility or ionophilicility. 51 Accordingly, the position dependent chemical potential density is…”

Temperature‐dependent approach to chemical reactivity concepts in density functional theory

“…By this procedure a local chemical potential has been defined replacing the number of electrons by the electronic density in the numerator of Equation , that is, which implies that, …”

“…Now, recalling that hardness and softness are inverses of each other only at T = 0, and as the integral of the local softness over all space is equal to the global softness, we decided to define a new local hardness, that instead of being the inverse of the local softness, its integral over all space should lead to the global hardness . At the same time, we expected that through this approach we could eliminate the Dirac delta function that appears in Equation , and also the ambiguity associated with the definition given by Equation .…”

“…An interesting aspect of Equation is related to the fact that if one considers the derivative of this equation with respect to 〈 N 〉 one is led to a quantity that is proportional to the dual descriptor given in Equation , that is, …”

“…An alternative approach is to consider the derivative of the Fukui function with respect to the electronic chemical potential. Using the chain rule in this case one finds that, where we have used Equation for the dual descriptor, and Equation for the hardness. As the function P ( ω ) cancels, one is led to a quantity that is well‐behaved and which we called the τ ‐soft dual descriptor, because it is proportional to the inverse of the Parr and Pearson hardness, that is, the Parr and Pearson softness.…”