The eigenvalues and eigenvectors of the Localization-Delocalization Matrices (LDMs) are interpreted and computed for molecular examples. It is found that eigenvectors corresponding to the largest eigenvalues reflect the intuitive pictures of molecules, functional groups, and resonance structures. The eigenvectors can each be thought of as a collective direction of the flow of electrons between atoms. Localization modes and delocalization modes are introduced where the localization modes can be interpreted to define functional groups and delocalization modes can be interpreted as resonance structures and quantify the number of electrons netted from each flow. The eigenvalues and eigenvectors are found to be related to those of the negative of the inverse of the hardness matrix, the negative of the softness matrix, suggesting the eigenvectors associated with the largest eigenvalues are patterns of restricted electron flow and the suggesting the eigenvectors associated the smallest positive eigenvalues are patterns of free electron flow.
James S. M. Anderson (Tue,) studied this question.