A Dynamical Density Field That Shows the Localizability of Electrons: The Exchange-Correlation Ehrenfest Force

A gradual but steady tide in theoretical chemistry is favoring the exploration of atomic and molecular interactions through the dynamical forces perceived and exerted by the particles of a system. By integrating the quantum mechanical force operator over all the spin and all but one of the spatial coordinates of the electrons, the Ehrenfest force density field reveals these forces directly and is separable into a classical term, related to the electric field, and a quantum mechanical correction, which we introduce and analyze for various atoms and molecules in this work. This exchange-correlation Ehrenfest force density field, F(r), excludes the dominant nuclear components that shape the full Ehrenfest field, revealing information about electron sharing, pairing, and delocalization. In a manner similar, though not equal, to the electron localization function, F(r) unveils covalent and core basins. Its divergence, ∇·F(r), indicates the presence of electron shells in atoms and recovers the positions of lone pairs and the shell structure of ionic, polar, and covalent interactions in molecules. It also exhibits a semiquantitative match with the Laplacian of the electron density that we also explore. In alignment with the established role of exchange-correlation as nature's glue, we demonstrate that a significant number of fundamental concepts in chemical bonding can be derived from the F(r) dynamical field.