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We extend the Runge-Gross theorem for a very general class of Markovian andnon-Markovian open quantum systems under weak assumptions about the nature ofthe bath and its coupling to the system. We show that for Kohn-Sham (KS)Time-Dependent Density Functional Theory, it is possible to rigorously includethe effects of the environment within a bath functional in the KS potential,thus placing the interactions between the particles of the system and thecoupling to the environment on the same footing. A Markovian bath functionalinspired by the theory of nonlinear Schrodinger equations is suggested, whichcan be readily implemented in currently existing real-time codes. Finally,calculations on a helium model system are presented.

Time-Dependent Density Functional Theory for Open Quantum Systems with Unitary Propagation