Absorption matrix of multi-site systems calculated by a hybrid quantum-classical Liouville equation

The linear absorption spectrum of a multisite system can be written as a weighted accumulation of elements of an absorption matrix. In the framework of the quantum-classical Liouville equation (QCLE), a mean-field approximation is introduced to simplify the calculation of the absorption matrix. The classical bath oscillators are propagated under partially shifted potentials, which reflects an averaged behavior after considering quantum jumps of the system states. For a specified initial condition, the time-dependent shifting possibility of each bath potential is given by the time evolution of site population estimated by the Redfield equation or the noninteracting blip approximation. The two hybrid QCLE approaches are tested in various models, including biased and unbiased two-site models, a subnetwork and the whole monomer of Fenna-Matthews-Olson, and harmonic and anharmonic baths. With numerically excellent results, the numerical studies show reliability and flexibility of the hybrid QCLE in calculating the absorption matrix and spectrum.