The hierarchical equations of motion (HEOM) method was applied to calculate the emission spectra of molecular aggregates using the Frenkel exciton model. HEOM equations for the one-exciton excited state were first propagated until equilibration. The reduced density operator and auxiliary density operators (ADOs) were used to characterize the coupled system-bath equilibrium. The dipole-dipole correlation functions were then calculated to obtain the emission spectra of model dimers, and the B850 band of light-harvesting complex II (LH2) in purple bacteria. The effect of static disorder on equilibrium excited state and the emission spectra of LH2 was also explicitly considered. Several approximation schemes, including the high temperature approximation (HTA) of the HEOM, a modified version of the HTA, the stochastic Liouville equation approach, the perturbative time-local and time-nonlocal generalized quantum master equations, were assessed in the calculation of the equilibrium excited state and emission spectra.
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