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.     

What is an atom in a molecule?

The derivation of the Hirshfeld atoms in molecules from information theory is clarified. The importance for chemistry of the concept of atoms in molecules (AIM) is stressed, and it is argued that this concept, while highly useful, constitutes a noumenon in the sense of Kant.     

Normal (decoupled) representation of electronegativity equalization equations in a molecule

The decoupled (normal) representation of the electronegativity (chemical potential) equalization equations, in which the hardness tensor {η_ij}={?μ_i/?N_j} becomes diagonal, is examined in the atoms-in-a-molecule (AIM ) approximation; μ_i=?E/?N_i is the chemical potential of the i-th AIM , N_i is its electron population, and E is the system energy. All relevant chemical potential, hardness, softness, and Fukui function quantities corresponding to the normal electron redistribution channels, Q_y, are discussed and expressed in terms of the canonical AIM parameters. The normal chemical potentials, μ _γ=?E/?Q_γ, provide a natural classification of the normal modes into three groups: (a) acceptor normal modes (μ _a < 0, positive mode Fukui function, hardness, and softness), (b) donor normal modes (μ _d > 0, negative mode Fukui function, hardness, and softness), and (c) polarization normal modes (μ _p=0, zero mode Fukui function, hardness, and softness). The implications of the normal mode analysis for the theory of chemical reactivity are briefly summarized.     

Use of charge sensitivity analysis in diagnosing chemisorption clusters: Minimum-energy coordinate and Fukui function study of model toluene–[V2O5] systems

Charge sensitivity analysis (CSA ) is carried out for model toluene–vanadium pentoxide chemisorption complexes involving the two-pyramidal model of the active site on the (010)—V₂O₅ surface. Maps of the electrostatic potential around the adsorbate and the substrate cluster are used to rationalize energetical preferences of alternative perpendicular and parallel arrangements of the toluene ring relative to the pyramid bases, known from previous SCF MO studies. The minimum-energy coordinates (MEC ) in the electron population space are determined from the CSA semiempirical, finite difference atomic hardness matrix for the actual SCF MO charges in the chemisorption clusters. They represent collective charge displacements which minimize the system energy per unit change in the oxidation state of a specified atom, thus providing a convenient diagnostic tool for testing the alternative charge rearrangements and range of perturbations due to the chemisorption bond or changes in the cluster environment. The MEC relaxed hardnesses diagnose mode stabilities and together with the MEC topologies identify the most probable locations of the adsorbate activation. Finally, the atomic Fukui function indices are used to explore trends in the distribution of the external charge transfer due to the system environment. © 1995 John Wiley & Sons, Inc.