Molecular crystals possess a highly complex crystallographic landscape, which in many cases results in the experimental observation of multiple crystal structures for the same compound. Accurate results can often be obtained for such systems by employing periodic density functional theory using hybrid functionals; however, this is not always computationally feasible. One possibility to circumvent these expensive periodic calculations is the utilization of multimer embedding methods. Therein, the fully periodic crystal is described at a lower level of theory, and subsequently monomer energies, dimer interaction energies, etc., are corrected via high-level calculations. In this paper, we further extend such a multimer embedding approach by one multimer order for all investigated properties, allowing us to compute lattice energies up to the tetramer embedding level, and atomic forces, the stress tensor, and harmonic phonons up to the trimer level. We test the significance of including these higher-order multimers by embedding PBE0+MBD multimers into periodic PBE+MBD calculations utilizing the X23 benchmark set of molecular crystals and comparing the results to explicit periodic PBE0+MBD calculations. We show that tetramer interactions systematically improve the lattice energy approximation and explore multiple possibilities for multimer selection. Furthermore, we confirm that trimer interactions are crucial for the description of the stress tensor, yielding cell volumes, on average, within 0.3% of those of PBE0+MBD. Subsequently, this also results in an improvement in the description of vibrational properties, giving on average Gamma-point frequencies within 1.3 cm-1 and vibrational free energies within 0.3 kJ/mol of the PBE0+MBD results.
List et al. (Mon,) studied this question.