Deriving the Redfield equation for electronically open molecules

We introduce a formalism to describe fractional charging of a molecule due to interactions with its environment. The interactions which induce fractional charging are contained in the Hamiltonian of the full system (molecule and environment). Such interactions can be singled out by expressing the Hamiltonian in a local spin orbital basis, and they are the main focus of this work. A reduced density operator for the molecule is derived starting from the Liouville-von Neumann equation for the full system by employing an explicitly constructed projection superoperator. By treating the molecule as an electronically open quantum system, we obtain a Redfield equation where the environment is included approximately. Phenomenological broadening of energy levels is included to mimic finite lifetimes of electronic states. The populations of the reduced density operator determine the mixture of different redox states and, hence, the fractional charging of the molecule. To illustrate the formalism, we use benzene physisorbed on a graphene sheet as a toy model. The work presented in this paper constitutes an initial step toward understanding molecules as electronically open quantum systems.