“Phylogenetic” Screening of External Potential Related Response Functions
Paweł Szarek · 2024
The energy functional, E [ N , v ( r )] defines mapping of electron number, N , and external potential, v ( r ) into energy field. The analysis of E variation upon changes in N is important tool for identifying the reactivity by characterizing transformations between electron-deficient, neutral, and hypervalent states. The responses to the second variable, the external potential, include sensitivities to electron-ion potential and any potential due to an applied electric field. The first energy derivative over the external potential defines electron density, which de facto is probability distribution of N in the chemical system. Considering that usually, the v ( r ) implicates potential solely from the nuclear charges, in the absence of other electric fields, the v -dependence is decomposed into the classes of response functions related to particular changes with respect to the nuclear charge, potential, and electric fields - in general, related through Maxwell's equations, Gauss's law, and Poisson's equation. This chapter focuses on E sensitivity to change in the magnitudes of nuclear charges. These superficial responses, tagged as “alchemical,” might be considered analogous to differentiability of energy functional with respect to N . However, while the global quantity N is characterized by its distribution in the system – electron density, ρ , and the changes in nuclear charges, Z i , are essentially point localized. The electron density-related responses might be loosely compared to nuclear potential derivatives producing “local” quantities, which change from point to point. Ultimately the electric field affects predominantly the electron density distribution and effects on nuclei can be ignored in general. Summary and interpretation of external potential-related energy derivatives split into nuclear charge, ∂ ∂ Z , external potential, ∂ ∂ v ( r ) , and electric field, ∂ ∂ F , thus sensitivities over “ Z ”, “ Z r ”, and “ Z / r 2 ” like series are superposed within one family tree.