  Time-dependent density functional theory (TDDFT) is widely used for
understanding and predicting properties and behaviors of matter. As one of the
fundamental theorems in TDDFT, van Leeuwen's theorem [Phys. Rev. Lett. 82, 3863
(1999)] guarantees how to construct a unique potential with the same one-body
density evolution. Here we extend van Leeuwen's theorem by exploring truncation
criteria in BBGKY-hierarchy. Our generalized theorem demonstrates the existence
of a unique non-local potential to accurately reconstruct the multi-body
density evolution in binary interacting systems. Under non-stringent
conditions, truncation of the BBGKY-hierarchy equations aligns with the
behavior of multi-body density evolution, and maintains consistency in the
reduced equations. As one of applications within the extended TDDFT supported
by our theorem, multiple excitation energy can be typically solved as the
eigenvalue of a generalized Casida's equation. The extended TDDFT provides an
accurate and first-principle framework capable of describing the kinetic
processes of correlated system, including strongly coupled particle transport,
multiple excitation and ionization processes.
