The absorption line shapes of model molecular aggregates are investigated using the recently developed Liouville space hierarchical equations of motion (HEOM) method. The exact results are further exploited for the assessment of several approximation schemes, including the high temperature approximation of HEOM, the stochastic Liouville equation approach, and the perturbative time-local and time-nonlocal quantum master equations (QMEs). The calculations on dimers, larger ring-shaped aggregates, and a model of the B850 ring in the LH2 of purple bacteria show that while the other approximate methods can give reasonable absorption line shapes over a wide range of parameter regimes, the second-order time-nonlocal QME is generally inaccurate and may give spurious peaks in the absorption spectra.
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