Characterizing vibrational motion beyond internal coordinates

We present a procedure for the decomposition of the normal modes of a composite system, including its rotations and translations, into those of fragments. The method permits—by the cross-contraction of dyads of mass-weighted displacement vectors, without recourse to valence coordinates—the direct comparison of nuclear motions of structurally similar but otherwise arbitrary fragments of molecules, and it leads to a quantitative definition of the similarity and the overlap of nuclear motions. We illustrate its usefulness by the quantification of the mixing of the normal modes of formic acid monomers upon the formation of a dimer, by the comparison of the overlap of the intermolecular normal vibrations of the water dimer computed with different ab initio schemes, and by the comparison of similarity and overlap of vibrations of (4S,7R)-galaxolide and (4S)-4-methylisochromane. The approach is expected to become a standard tool in vibrational analysis.