Local vibrational mode theory meets graph theory: Complete and non-redundant local mode sets
Mateus Quintano, Renaldo T. Moura, Elfi Kraka
Abstract
This Frontiers Article introduces a unique perspective on the well-established concept of completeness in a chemically meaningful set of non-redundant local vibrational modes. By utilizing graph theory, we demonstrate how this concept naturally arises when Euler’s theorem is fulfilled in molecular graphs of tree, cycle, and polyhedral types. This significantly advances our understanding of topology, leading to a new interpretation for deriving such a set. A key aspect of the local vibrational mode theory is the decomposition of normal modes into local mode contributions, which provides a powerful approach for analyzing vibrational spectra. This however requires a complete and meaningful set of non-redundant local vibrational modes, as demonstrated for the IR spectra of both non-zwitterionic and zwitterionic forms of glycine, the cubane and perfluorocubane pair, and the Ar–benzene dimer. The mathematical concept is put to the test by applying our counting formulas for complete and non-redundant local mode sets to a series of organic molecules with increasing complexity.