This metric is referred to as the "mean alignment-weighted frequency." It can be understood as a generalization of the phonon band center metric (eq 1): if all modes have the same alignment, the mean alignment-weighted frequency is directly proportional to the phonon band center. (Modes with alignments less than an arbitrary threshold are omitted from this sum in order to prevent numerical issues. A threshold of 10 -12 is used in the following analysis, although using any value between 10 -12 and 10 -4 yields identical results.)