A Dynamical Density\nField That Shows the Localizability\nof Electrons: The Exchange-Correlation Ehrenfest Force

A gradual but steady tide in theoretical chemistry is\nfavoring\nthe exploration of atomic and molecular interactions through the dynamical\nforces perceived and exerted by the particles of a system. By integrating\nthe quantum mechanical force operator over all the spin and all but\none of the spatial coordinates of the electrons, the Ehrenfest force\ndensity field reveals these forces directly and is separable into\na classical term, related to the electric field, and a quantum mechanical\ncorrection, which we introduce and analyze for various atoms and molecules\nin this work. This exchange-correlation Ehrenfest force density field, <b>F</b><sub><b>xc</b></sub>(<b>r</b>), excludes the\ndominant nuclear components that shape the full Ehrenfest field, revealing\ninformation about electron sharing, pairing, and delocalization. In\na manner similar, though not equal, to the electron localization function, <b>F</b><sub><b>xc</b></sub>(<b>r</b>) unveils covalent\nand core basins. Its divergence, <b>∇·F</b><sub><b>xc</b></sub>(<b>r</b>), indicates the presence of\nelectron shells in atoms and recovers the positions of lone pairs\nand the shell structure of ionic, polar, and covalent interactions\nin molecules. It also exhibits a semiquantitative match with the Laplacian\nof the electron density that we also explore. In alignment with the\nestablished role of exchange-correlation as nature’s glue,\nwe demonstrate that a significant number of fundamental concepts in\nchemical bonding can be derived from the <b>F</b><sub><b>xc</b></sub>(<b>r</b>) dynamical field.