Spatial-Order Hierarchy of Time-Dependent Exchange-Correlation Potential

Exact time-dependent density-functional theory (DFT) separates the exchange-correlation potential into interaction and kinetic-correlation components, but the structural relation between them remains unclear. We establish a representability constraint based on the off-diagonal expansion of the one-electron reduced density matrix (1RDM) relative to a chosen reference dynamics. A non-zero linear off-diagonal term of the 1RDM generates a current density beyond the representable space of the reference, while a nontrivial quadratic off-diagonal structure activates the kinetic-correlation component. The resulting hierarchy provides a criterion for assessing, from the spatial locality of electronic correlation, whether a given exchange-correlation functional or a more general electron-correlation approximation remains adequate. In a one-dimensional two-electron correlation quench, retaining the exact interaction component while setting the kinetic component to zero suppresses the density broadening of the exact evolution. These results establish a spatial-order hierarchy among the force components governing the density dynamics and provide an exact constraint for the construction and assessment of nonadiabatic approximations.