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The low-frequency range of vibrational spectra is sensitive to collectivevibrations of the lattice. In molecular crystals, it can be decisive toidentify the structure of different polymorphs, and in addition, it plays animportant role on the magnitude of the temperature-dependent component ofvibrational free energy differences between these crystals. In this work westudy the vibrational Raman spectra and vibrational density of states ofdifferent polymorphs of the flexible Aspirin and Paracetamol crystals based ondispersion-corrected density-functional theory, density-functional perturbationtheory, and $ab\,initio$ molecular dynamics. We examine the effect ofquasi-harmonic lattice expansion and compare the results of harmonic theory andthe time correlation formalism for vibrational spectra. Lattice expansionstrongly affects the collective vibrations below 300 cm$^{-1}$, but it issignificantly less important at higher frequencies, while thermal nuclearmotion can be important in the full vibrational range. We also observe that theinclusion or neglect of many-body van der Waals dispersion interactions do notcause large differences in the low-frequency range of Raman spectra orvibrational density of states, provided the lattice constants are fixed. Weobtain quantitative agreement with experimental room-temperature Raman spectrabelow 300 cm$^{-1}$ for all polymorphs studied. Examining the two-dimensionalcorrelations between different vibrations, we find which modes show a largerdegree of anharmonic coupling to others, providing a possible route to assessthe accuracy of harmonic free energy evaluations in different cases.

Anharmonic Effects in the Low-Frequency Vibrational Modes of Aspirin and Paracetamol Crystals