Roman F. Nalewajski

Two‐electron valence indices from the Kohn‐Sham orbitals

The recent Hartree-Fock (HF) difference approach to the chemical valence indices (ionic and covalent), formulated in the framework of the pair-density matrix, is implemented within the Kohn-Sham (KS) density functional theory (DFT). The valence numbers are quadratic in terms of displacements of the molecular spin-resolved charge-and-bond-order (CBO) matrix elements, relative to values in the separated atoms limit (SAL). It is shown that the global valence represents a generalized “distance” quantity measuring a degree of similarity between the two CBO matrices: the molecular and SAL. Numerical values for typical molecules exhibiting single and multiple bonds demonstrate that the KS orbitals give rise to these new bond valences in good agreement with both chemical and HF predictions. This KS bond multiplicity analysis is applied to the chemisorption system including the allyl radical and a model surface cluster of molybdenum oxide. It is concluded that the quadratic valence analysis represents a valuable procedure for extracting useful chemical information from standard DFT calculations.

Hirshfeld analysis of molecular densities: subsystem probabilities and charge sensitivities

The Hirshfeld (“stockholder”) partitioning of molecular one-electron densities (or probabilities) is generalized to many-electron distributions using the minimum entropy deficiency principle of the Information Theory, and the ensemble interpretation of this division scheme is given. A distinction is made between changes of the ground-state density (“horizontal” displacements) and alternative divisions of the fixed molecular density (“vertical” displacements) and the relevant equilibrium criteria are discussed. Within the density functional theory the subsystem densities are interpreted as the ground-state densities for the effective external potentials of such molecular fragments. The charge sensitivities for the global equilibrium (mutually open subsystems) and constrained equilibria (mutually closed subsystems) are briefly summarized and the derivative properties of the quadratic Taylor expansion of the system energy in terms of the stockholder subsystem densities are examined. In the subsystem resolution the additive and nonadditive components of the hardness and softness kernels are identified and the relation between the linear response and softness kernels of molecular fragments is derived.

REACTIVITY CRITERIA IN CHARGE SENSITIVITY ANALYSIS

The charge sensitivity analysis, based upon the hardness/softness concepts and the chemical potential (electronegativity) equalization principle established within the density functional theory, is used as a diagnostic tool for probing trends in the chemical reactivity of large molecular systems. The new criteria are reviewed with special emphasis on two-reactant reactivity concepts, which explicitly take into account the interaction between reactants in a general donor-acceptor system, and the collective charge displacement coordinate systems that give the most compact description of the charge reorganization accompanying chemical reactions. The global collective populational reference frames discussed include populational normal modes, minimum energy coordinates, and the relaxational modes; the reactant reference frames include internal modes of reactants as well as externally decoupled and inter-reactant-coupling modes. The charge-coupling information in the atoms-in-molecules resolution is modelled by the hardness matrix, which provides the canonical input data for determining a series of chemically interesting probes for diagnosing reactivity trends. A survey of these concepts includes the global treatment of molecular systems and the reactant-resolved description of general reactive systems. The molecular-fragment development emphasizes the relaxational influence of one reactant upon another, reflected by the off-diagonal charge sensitivities that measure the responses of one reactant to charge displacements in the other. Both open (exchanging electrons with the reservoir) and closed (preserving the number of electrons) reactive systems are investigated. Stability criteria for the equilibrium charge distribution in such systems are summarized and their implications discussed. The ground-state mapping relations between geometrical (nuclear position) and electronic (atomic charge) degrees of freedom are related to the Gutmann rules of structural chemistry. Illustrative examples are given of the application of these reactivity concepts to model catalytic clusters of transition metal oxide surfaces and large adsorbates (toluene).

Electronic Structure and Chemical Reactivity: Density Functional and Information-Theoretic Perspectives

A general theme of this survey is the electronic density as a source and carrier of the information about molecular structure and reactivity. First, the classical rules of the electronic structure and chemical reactivity are reexamined, stressing the continuity of ideas in chemistry and exposing the interrelations between their original, mostly intuitive basis, and more rigorous foundations within the Density Functional Theory (DFT). The conceptual advantages of the DFT approach are stressed and a distinction is made between changes in the ground-state density (‘horizontal’ displacements) and in a division of the fixed molecular density (‘vertical’ displacements). The molecular charge sensitivities for the global equilibrium (of the mutually open subsystems) and the constrained equilibrium (of the mutually closed subsystems) are summarized and expressed in terms of the principal second derivatives of the system energy. The DFT treatment of a wide variety of molecular responses, to be eventually used as reactivity indices, is briefly surveyed. A wealth of reactivity concepts originating from the DFT rooted charge sensitivity analysis is reviewed, including both the electronic and external potential related quantities. This overview also covers general criteria for the equilibrium distribution of electrons in molecular systems and their fragments, explicit relations between alternative sets of the conjugate state-parameters, reflecting the transformations between displacements (perturbations) of the controlled state variables into the equilibrium linear responses of their respective energy conjugates, and the DFT characterization of the embedded molecular fragments, e.g., atoms-in-molecules, reactants, etc. Recent developments in a ‘thermodynamic’-like approach to molecules and their constituent fragments are then summarized, with the entropic representation description provided by the information theory, and the elements of the ‘communication’ system approach to the chemical bond multiplicities. In particular, the Hirshfeld partitioning of molecular one-electron densities (probabilities) is summarized and its minimum entropy deficiency basis is stressed. The same entropic principle can be used to derive the ‘stockholder’ rule of dividing the joint many-electron probabilities and the local softness (Fukui function) characteristics of the electronic gas in the molecular/reactive systems and their constituent fragments. For example, the proportionality relations between the charge responses of subsystems in the molecular/reactive system and their ‘promolecular’ reference, respectively, directly follow from the relevant minimum entropy deficiency principles formulated in terms of the local softness or Fukui function descriptors of molecular fragments. The charge sensitivities of the Hirshfeld fragments are examined and the surprisal analysis of molecular densities is advocated as an attractive interpretative tool, supplementary to the familiar density difference diagrams. The local equalization rules of the subsystem information-distance densities at the corresponding global value, which are satisfied by the Hirshfeld molecular fragments, are then emphasized. These local measures of the information distance with respect to the separated subsystems of the relevant promolecular reference are semi-quantitatively related to the molecular density difference function. The entropy representation ‘forces’ driving the charge transfer (CT) between reactants in the donor–acceptor systems are defined. These CT affinities combine the familiar Fukui functions of the subsystems with the information-distance densities, i.e., the entropy representation ‘intensive’ conjugates of the subsystem electron densities (‘extensive’ local state variables), and exactly vanish for the ‘stockholder’ densities of reactants.

Partial communication channels of molecular fragments and their entropy/information indices

Within the ‘communication’ theory of the chemical bond the additive decomposition of the global entropy/information indices into corresponding quantities characterizing the partial communication channels of the embedded molecular fragments is examined. The so called row and column channels of molecular subsystems are introduced. They represent alternative communication networks of the constituent fragments in the molecule, which fully take into account the internal (intra-subsystem) and external (between the molecular fragment and its molecular environment) communications (bonds). These concepts are illustrated using the two-orbital model of a chemical bond. The three-orbital model of the triatomic, symmetric transition-state complex is then examined to explore the influence of the system spin polarization on the entropy/information bond descriptors of the internal and external chemical interactions of the model diatomic fragments. Applications to the π-bond systems (butadiene and benzene) in the Hückel theory approximation are also reported and the results are compared with predictions from molecular orbital (MO) theory. These results indicate that the information-theoretic indices of the partial channels of molecular fragments emphasize the information equilibrium of the molecular subsystems, which is manifested by a remarkable equalization of various atomic and diatomic indices, irrespective of the fragment identity. This is in contrast to the complementary MO bond indices, which strongly differentiate between different subsystems, in accordance with the molecule structural formula.Within the ‘communication’ theory of the chemical bond the additive decomposition of the global entropy/information indices into corresponding quantities characterizing the partial communication channels of the embedded molecular fragments is examined. The so called row and column channels of molecular subsystems are introduced. They represent alternative communication networks of the constituent fragments in the molecule, which fully take into account the internal (intra-subsystem) and external (between the molecular fragment and its molecular environment) communications (bonds). These concepts are illustrated using the two-orbital model of a chemical bond. The three-orbital model of the triatomic, symmetric transition-state complex is then examined to explore the influence of the system spin polarization on the entropy/information bond descriptors of the internal and external chemical interactions of the model diatomic fragments. Applications to the π-bond systems (butadiene and benzene) in the Huckel t...

Information theoretic approach to molecular and reactive systems

The information-theoretic basis of the “stockholder” partitioning of the molecular electron density into fragment densities is reexamined in terms of the variational principles for alternative measures of the minimum information distance between the subsystem densities and promolecule electron distribution, for which the Hirshfeld scheme represents the optimum division. The local equalization of the subsystem information distance densities is discussed and illustrated for selected diatomics and triatomics. Approximate relations between the information content diagrams and familiar density difference plots of quantum chemistry are explored and the surprisal analysis of the molecular electron density is advocated as the entropic complement of to the familiar density difference diagrams. The generalized forces (affinities) and the Fukui function quantities of the Hirshfeld reactants in the donor-acceptor reactive system are examined. These affinities, combining both the information entropy and the Fukui function information, drive the charge transfer processes between the subsystems. Various Fukui function descriptors of acidic and basic reactants are defined and the associated minimum entropy deficiency rule for the Fukui function “distributions” is established.

Basic concepts and illustrative applications of the sensitivity analysis of molecular charge distribution

Abstract Basic concepts of molecular charge sensitivity analysis, including the global and fragment (rigid and relaxed) hardness and softness (Fukui function) parameters, are summarized. The resultant and regional parameters are defined, corresponding to the global and constrained (regional) equilibrium cases, respectively. Possible ways of modeling the canonical hardness and softness tensors of atoms-in-a-molecule (AIM) are discussed and the alternative representations of the chemical potential (electronegativity) equalization equations are examined. Of great interest in the chemical reactivity and catalysis are the decoupled normal representation, corresponding to the principal axes (diagonal) hardness tensor, and the partially decoupled Lanczos representation, generating a tridiagonal (Jacobi) hardness tensor. The illustrative numerical results for pyrrole and cyclopentadiene are reported. They include the resultant hardness and Fukui function parameters for both isolated molecules and protonation reactive systems. The effect of the molecular structural changes (bending of the CH bonds at α and β positions) on the resultant Fukui function indices is examined in some detail. Finally, possible applications in chemisorption and catalysis are identified and briefly commented upon.

Chapter 5 Multiple, Localized, and Delocalized/Conjugated Bonds in the Orbital Communication Theory of Molecular Systems

Information theory (IT) probe of the molecular electronic structure, within the communication theory of chemical bonds (CTCB), uses the standard entropy/information descriptors of the Shannon theory of communication to characterize a scattering of the electronic probabilities and their information content throughout the system chemical bonds generated by the occupied molecular orbitals (MO). These “communications” between the basis-set orbitals are determined by the two-orbital conditional probabilities: one- and two-electron in character. They define the molecular information system, in which the electron-allocation “signals” are transmitted between various orbital “inputs” and “outputs”. It is argued, using the quantum mechanical superposition principle, that the one-electron conditional probabilities are proportional to the squares of corresponding elements of the charge and bond-order (CBO) matrix of the standard LCAO MO theory. Therefore, the probability of the interorbital connections in the molecular communication system is directly related to Wiberg’s quadratic covalency indices of chemical bonds. The conditional-entropy (communication “noise”) and mutual-information (information capacity) descriptors of these molecular channels generate the IT-covalent and IT-ionic bond components, respectively. The former reflects the electron delocalization (indeterminacy) due to the orbital mixing, throughout all chemical bonds in the system under consideration. The latter characterizes the localization (determinacy) in the probability scattering in the molecule. These two IT indices, respectively, indicate a fraction of the input information lost in the channel output, due to the communication noise, and its surviving part, due to deterministic elements in probability scattering in the molecular network. Together, these two components generate the system overall bond index. By a straightforward output reduction (condensation) of the molecular channel, the IT indices of molecular fragments, for example, localized bonds, functional groups, and forward and back donations accompanying the bond formation, and so on, can be extracted. The flow of information in such molecular communication networks is investigated in several prototype molecules. These illustrative (model) applications of the orbital communication theory of chemical bonds (CTCB) deal with several classical issues in the electronic structure theory: atom hybridization/promotion, single and multiple chemical bonds, bond conjugation, and so on. The localized bonds in hydrides and delocalized π-bonds in simple hydrocarbons, as well as the multiple bonds in CO and CO2, are diagnosed using the entropy/information descriptors of CTCB. The atom promotion in hydrides and bond conjugation in π-electron systems are investigated in more detail. A major drawback of the previous two-electron approach to molecular channels, namely, two weak bond differentiation in aromatic systems, has been shown to be remedied in the one-electron approach.

Entropy descriptors of the chemical bond in information theory. I. Basic concepts and relations

The elements of the ‘communication theory’ of the chemical bond are outlined. The average uncertainties of the information theory characterizing the communication channels are summarized. The molecular system in atomic resolution is interpreted as the ‘communication’ channel, in which signals of the electron allocations to constituent atoms are propagated from the molecular input (‘source’), determined by the atomic ‘promolecule’, to the molecular output (‘receiver’), via the system chemicals bonds. This transmission network, connecting the free atoms of the promolecule reference and the bonded atoms-in-a-molecule, is determined by the molecular conditional two-electron probabilities in atomic resolution. Owing to the electron delocalization, the molecular information channels exhibit a chemical ‘noise’, which affects the transmission of the atom-assignment signals and the associated flow of information through the communication system. Several entropy/information concepts, including the conditional entropy, mutual information and information distance (cross-entropy, entropy deficiency), between the input and output probability distributions, are used to characterize the chemical bond and its covalent and ionic components. The conditional entropy and mutual information are identified as overall measures of the covalent and ionic bond components, respectively. Several reference input probabilities for probing different aspects of the chemical bond are examined. A distinction between the electron-sharing and the pair-sharing (coordination) bonds is explored and the entropic indices of the localized bonds are proposed. The average entropies of the local ‘stockholder’ communication channel, generated by the Hirshfeld partition of molecular electron densities, are proposed as complementary information descriptors of the chemical bond.

Information distance approach to Hammond postulate

Abstract The entropy deficiencies of the Information Theory are used to measure the information ‘distance’ between the electron densities of the transition-state complex in collinear atom exchange reactions and the corresponding ‘promolecular’ densities obtained from electron distributions of the reaction substrates and products. These quantities are used to probe similarities between electronic structures of the transition-state complex and those of the dissociation subsystems for the forward (products) and reverse (substrates) reactions, thus providing entropic criteria of a degree of closeness of the transition-state configuration relative to substrates and products. The electronic and geometric components of such information-theoretic distances are separated and discussed. They both satisfy the qualitative implications of the Hammond postulate.