Internal and external eigenvalue problems of Hermitian operators and their use in electronic structure theory

Within the fragment resolution of molecular systems the conceptual and interpretative advantages of using the separate eigenvalue problems for the internal and external part of the Hermitian matrix representing a physical quantity in quantum mechanics are examined. By definition, these two parts accordingly combine only the diagonal and off-diagonal subsystem-resolved blocks of matrix elements. These two partial eigenvalue problems bring about the matrix internal or external decouplings, respectively, which have recently been used in several interpretations of the molecular electronic structure. A character and structure of the external eigensolutions is examined in some detail and their recent applications in the Charge Sensitivity Analysis—to extract the most important electron-transfer effects between constituent atoms of model chemisorption systems, and in the Molecular-Orbital theory—to precisely identify the inter-orbital flows of electrons, are summarized and commented upon. The grouping relation, for combining the external/internal eigensolutions into those for the whole matrix, is derived in the context of the complementary “rotations” of the basis set vectors.