Identity for Kullback-Leibler divergence in density functional reactivity theory

Density functional reactivity theory (DFRT) makes use of simple electron density functionals to appreciate chemical reactivity. The Kullback-Leibler divergence, also called information gain or relative Shannon entropy, as a key component of the DFRT framework has entertained a few fascinating applications in the recent literature. For instance, it was employed to rationalize the Hirshfeld stockholder partition of atoms in molecules. It has also been applied to both determine regioselectivity and simultaneously quantify molecular electrophilicity and nucleophilicity, the capability of atoms in molecules to accept and donate electrons. In this work, we examine the local behavior of the Kullback-Leibler divergence and present a novel identity for it. The validity of the identity is subsequently verified by the numerical results of neutral atoms from He to Kr with different choices of the reference density. Analytical properties and local behaviors of the three new functions introduced in the identity are discussed. These new local functions, together with the identity unveiled in this work, should provide new perspectives for us to further develop the DFRT framework with added insights about the local behavior of molecular physiochemical properties.