To check the dynamic stability of the predicted structures, finite displacement phonon calculations have been carried out. The predicted Li 2 FeSO structure was found to have no imaginary frequencies across the first Brouline zone. The Li 2 Fe 2 S 2 O structure obtained by the search (Fig. ) has a highly symmetric perovskite layer as shown in Fig. . However, when we calculate the phonon band structure, it was found to have imaginary frequencies (Fig. ). Pushing along the imaginary mode (at (-0.5, 0.5, 0) of the conventional cell) allows us to obtain a structure with tilted OLi 4 Fe 2 octahedra in a a 0 b + c 0 tilt system with space group P2 1 (Fig. ). Recomputing the phonon dispersion reveals that the imaginary mode has disappeared, which can be seen in Fig. . The energy per atom is reduced by about 1 meV for this distorted structure. For Li 4 Fe 3 S 3 O 2 , the lowest energy structure generated by the enumeration of symmetrically inequivalent structures (Fig. ) was found to be dynamically stable. Imaginary frequencies have been also found in Li 2 Fe 3 S 2 O 2 and Li 2 Fe 4 S 3 O 2 , which are subsequently eliminated by following the corresponding modes. Their final dynamically stable structures have lower energies by 2.6 meV and 0.2 meV compared with the non-distorted structures respectively, giving negligible effects for their thermodynamic stabilities. Since the structure searching was carried out with small unit cells containing symmetry operations, distortions that require increased cell sizes is not possible to be found. Nevertheless, since typically energy reduction as result of mode-pushing is small, the structures obtained directly from the search are already sufficiently accurate models for evaluating the thermodynamic stabilities. Phonon band structure for the phases mentioned above can be be found in the electronic supporting information (ESI) Fig. .