Resultant Information Descriptors, Equilibrium States and Ensemble Entropy †.

In this article, sources of information in electronic states are reexamined and a need for the resultant measures of the entropy/information content, combining contributions due to probability and phase/current densities, is emphasized. Probability distribution reflects the wavefunction modulus and generates classical contributions to Shannon's global entropy and Fisher's gradient information. The phase component of molecular states similarly determines their nonclassical supplements, due to probability "convection". The local-energy concept is used to examine the phase equalization in the equilibrium, phase-transformed states. Continuity relations for the wavefunction modulus and phase components are reexamined, the convectional character of the local source of the resultant gradient information is stressed, and latent probability currents in the equilibrium (stationary) quantum states are related to the horizontal ("thermodynamic") phase. The equivalence of the energy and resultant gradient information (kinetic energy) descriptors of chemical processes is stressed. In the grand-ensemble description, the reactivity criteria are defined by the populational derivatives of the system average electronic energy. Their entropic analogs, given by the associated derivatives of the overall gradient information, are shown to provide an equivalent set of reactivity indices for describing the charge transfer phenomena.

Information-Theoretic Descriptors of Molecular States and Electronic Communications between Reactants.

The classical (modulus/probability) and nonclassical (phase/current) components of molecular states are reexamined and their information contributions are summarized. The state and information continuity relations are discussed and a nonclassical character of the resultant gradient information source is emphasized. The states of noninteracting and interacting subsystems in the model donor-acceptor reactive system are compared and configurations of the mutually-closed and -open equidensity orbitals are tackled. The density matrices for subsystems in reactive complexes are used to describe the entangled molecular fragments and electron communications in donor-acceptor systems which determine the entropic multiplicity and composition of chemical bonds between reactants.

Resultant Information Description of Electronic States and Chemical Processes †.

General variations of electronic states are discussed, and the state "vertical" and "horizontal" displacements are explored. Quantum dynamics of the wave function modulus and phase components is examined, and the associated continuity relations are summarized. The probability and current contributions to overall information-theoretic (IT) descriptors of the state global and gradient entropy/information content are identified. These resultant measures are used to determine the phase equilibria in molecular systems. The currents corresponding to the entropy- and information-optimum phases are explored, the classical (probability) and nonclassical (current) flows in molecular information systems are identified, and probability interpretation of equidensity orbitals yielding the prescribed electron density is given. The physical equivalence of variational principles for the electronic energy and resultant gradient-information in open systems is stressed. It implies that their populational derivatives have the same capacity in describing charge transfer phenomena in molecules and their fragments. Illustrative application in determining thermodynamic equilibria and optimum charge transfer in molecular systems is discussed, and implications of the molecular virial theorem for bond-formation process and chemical reactions are investigated. The crucial role of the resultant gradient-information in chemical bonding is emphasized, and the Hammond postulate of reactivity theory is shown to be indexed by the reaction-coordinate derivative of electronic kinetic energy at the transition-state complex.

Role of electronic kinetic energy and resultant gradient information in chemical reactivity.

The role of resultant gradient-information concept, reflecting the kinetic energy of electrons, in shaping the molecular electronic structure and reactivity preferences of open reactants is examined. This quantum-information descriptor combines contributions due to both the modulus (probability) and phase (current) components of electronic wavefunctions. The importance of resultant entropy/information concepts for distinguishing the bonded (entangled) and nonbonded (disentangled) states of molecular fragments is emphasized and variational principle for the minimum of ensemble-average electronic energy is interpreted as a physically equivalent rule for the minimum of resultant gradient-information, and the information descriptors of charge-transfer (CT) phenomena are introduced. The in situ reactivity criteria, represented by the populational CT derivatives of the ensemble-average values of electronic energy or resultant information, are mutually related, giving rise to identical predictions of electron flows in the acid(A) - base(B), reactive systems. The virial theorem decomposition of electronic energy is used to reveal changes in the resultant information content due to the chemical bond formation, and to rationalize the Hammond postulate of reactivity theory. The complementarity principle of structural chemistry is confronted with the regional hard (soft) acid and bases (HSAB) rule by examining the polarizational and relaxational flows in such acceptor-donor reactive systems, responses to the external potential and CT displacements, respectively. The frontier-electron basis of the HSAB principle is reexamined and the intra- and inter-reactant communications in A-B systems are explored.

Information equilibria, subsystem entanglement, and dynamics of the overall entropic descriptors of molecular electronic structure.

Overall descriptors of the information (determinicity) and entropy (uncertainty) content of complex molecular states are reexamined. These resultant concepts combine the classical (probability) contributions of Fisher and Shannon, and the relevant nonclassical supplements due to the state phase/current. The information-theoretic principles determining equilibria in molecules and their fragments are explored and the nonadditive part of the global entropy is advocated as a descriptor of the classical index of the quantum entanglement of molecular subsystems. Affinities associated with the probability and phase fluxes are identified and the criterion of vanishing overall information-source is shown to identify the system stationary electronic states. The production of resultant density of the gradient-information is expressed in terms of the conjugate affinities (forces, perturbations) and fluxes (currents, responses). The Schrödinger dynamics of probability and phase components of molecular electronic states is used to determine the temporal evolution of the overall gradient information and complex entropy. The global sources of the resultant information/entropy descriptors are shown to be of purely nonclassical origin, thus identically vanishing in real electronic states, e.g., the nondegenerate ground state of a molecule.

Information-scattering perspective on orbital hybridization.

Within the communication theory of the chemical bond, the transformation of atomic orbitals (AO) into molecular orbitals (MO) generates the information system for the associated electronic promotion of AOs in a molecule. It consists of the two orbital-mixing stages involving AOs and MOs, and one MO-occupation subchannel. The conditional-entropy and mutual-information descriptors of this resultant "communication" system, which measure the average "noise" and the amount of information in the molecular channel, provide novel information-theoretic measures of the system bond covalency and ionicity, respectively. This information-theoretic approach to the many-center probability-scattering in AO resolution is now applied to the one-center orbital transformations to examine the entropic indices of an effective promotion of the canonical AO in alternative valence states of an atom, identified by different occupations of hybrid orbitals (HO). This phenomenon is first illustrated and tested using the simplest scheme of mixing two atomic orbitals in a generalized sp hybridization. The conditions for the maximum of the AO-promotion covalency are examined, and the shape independence of the orbital channels and their entropy/information descriptors for the equalized probability weights of HO in the specified atomic valence state is commented upon. Entropic indices are then generated for selected valence states of the carbon atom, resulting from different hybridization schemes, in order to characterize their complementary aspects of the system electron polarization (one-center ionicity) and electron delocalization (one-center covalency). The interpretation of these components as measures of a degree of the acquired "order" and surviving "disorder" (electron uncertainty) in the valence state is also given. These results are found to generally agree with intuitive expectations. The exact HO-occupation subchannel is derived, which reproduces the average AO occupations in the valence state. This approach is also proposed for the multicenter probability scattering in molecules, via the system occupied MO, in probing the system chemical bonds.

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.

Electron localization function as information measure.

The conditional two-electron probability function, which defines the electron localization function (ELF) of Becke and Edgecombe in the Kohn-Sham theory, is interpreted as the nonadditive (interorbital) Fisher information contained in the electron distribution. The probability normalization considerations suggest a use of the related information measure defined in terms of the unity-normalized probability distributions (shape factors of the electron densities), as the key ingredient of the modified information-theoretic ELF. This modified Fisher information density is validated by a comparison with the original two-electron probability function. Illustrative applications to typical molecular systems demonstrate the adequacy of the modified information-theoretic ELF in extracting the key features of the electron distributions in molecules. The overall Fisher information itself and the associated information-distance quantities are also proposed as complementary localization functions.