  The dissipative dynamics of many-electron systems interacting with a thermal
environment has remained a long-standing challenge within time-dependent
density functional theory (TDDFT). Recently, the formal foundations of open
quantum systems time-dependent density functional theory (OQS-TDDFT) within the
master equation approach were established. It was proven that the exact
time-dependent density of a many-electron open quantum system evolving under a
master equation can be reproduced with a closed (unitarily evolving) and
non-interacting Kohn-Sham system. This potentially offers a great advantage
over previous approaches to OQS-TDDFT, since with suitable functionals one
could obtain the dissipative open-systems dynamics by simply propagating a set
of Kohn-Sham orbitals as in usual TDDFT. However, the properties and exact
conditions of such open-systems functionals are largely unknown. In the present
article, we examine a simple and exactly-solvable model open quantum system:
one electron in a harmonic well evolving under the Lindblad master equation. We
examine two different representitive limits of the Lindblad equation
(relaxation and pure dephasing) and are able to deduce a number of properties
of the exact OQS-TDDFT functional. Challenges associated with developing
approximate functionals for many-electron open quantum systems are also
discussed.
