that enables calculations for an arbitrary number of molecules and will prove to be convenient for our purposes. Notice that the mode âðkÞ D is highly localized at âk but has a long tail for â1 i kÀ1 (for a visualization, see Supplementary Fig. ); furthermore, it is fully characterized by the index k and thus does not depend explicitly on N. In terms of these dark modes, the reaction in Eq. ( ) can be drastically simplified from an N þ 1 to a three-body process,