RungeGross

RungeGross, also known as the Runge-Gross theorem, is a foundational result in time-dependent density functional theory (TDDFT) established by E. Runge and E.K.U. Gross in 1984. The theorem states that for a many-electron system evolving under a time-dependent external scalar potential v_ext(r,t), with a given initial N-electron state, there is a one-to-one correspondence between the time-dependent electron density n(r,t) and the external potential, up to an additive function of time.

Equivalently, the density uniquely determines the potential (modulo a purely time-dependent term) within a certain time interval, provided the potentials are analytic in time around the initial time t0 and the system is nonrelativistic and finite.

This result underpins TDDFT, validating the use of density-based functionals and leading to the time-dependent Kohn-Sham methodology, where a non-interacting reference system is constructed to reproduce the exact density.

Some limitations: the requirement of analyticity in time is mathematically strong and has motivated extensions and alternative formulations that relax assumptions; the theorem applies to scalar potentials and real physical systems; the existence of gauge freedom means potentials differing by a purely time-dependent function correspond to the same density.

The Runge-Gross theorem has become a standard reference in quantum chemistry and condensed-matter physics, providing the theoretical justification for density-based approaches to modeling the dynamics of many-electron systems.