It is important that any dynamics method approaches the correct population distribution at long times. In this paper, we derive a one-body reduced density matrix dynamics for electrons in energetic contact with a bath. We obtain a remarkable equation of motion which shows that in order to reach equilibrium properly, rates of electron transitions depend on the density matrix. Even though the bath drives the electrons towards a Boltzmann distribution, hole blocking factors in our equation of motion cause the electronic populations to relax to a Fermi-Dirac distribution. These factors are an old concept, but we show how they can be derived with a combination of time-dependent perturbation theory and the extended normal ordering of Mukherjee and Kutzelnigg for a general electronic state. The resulting non-equilibrium kinetic equations generalize the usual Redfield theory to many-electron systems, while ensuring that the orbital occupations remain between zero and one. In numerical applications of our equations, we show that relaxation rates of molecules are not constant because of the blocking effect. Other applications to model atomic chains are also presented which highlight the importance of treating both dephasing and relaxation. Finally, we show how the bath localizes the electron density matrix.
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Also note that an equation of motion for γ alone is not enough to exactly propagate an interacting many-fermion state which would require equations for higher RDMs. We assume non-interacting electrons, but our equations would be compatible with a time-dependent Hartree-Fock approximation.
See MK’s paper for details.
The two-body density does couple to the one-body density via the Coulomb interaction in an atomistic Hamiltonian.
Four Drude-Lorentz functions are chosen of the following form (width, amplitude, central frequency) (au): (1) (0.0001, 0.00001, 0.0001), (2) (0.001, 0.0017, 0.0017), (3) (0.013, 0.023, 0.027), and (4) (0.01, 0.017, 0.017).
