The resultant measures of the entropy/information content in complex electronic states are discussed in the canonical position and momentum representations of molecular quantum mechanics. The nonclassical (phase/current) supplements of the classical (probability) descriptors of the overall entropy/information content in electronic states are identified and the associated entropy deficiency (information distance) quantities are introduced. The Shannon (global, logarithmic) and Fisher (local, gradient) information descriptors in both spaces are summarized, and the momentum continuity equation is used to establish the associated probability source. General relations between global and local information densities are examined and the etropic principles determining molecular phase equilibria are investigated.
The resultant measures of the entropy/information content in complex electronic states are discussed in the canonical position and momentum representations of molecular quantum mechanics. The nonclassical (phase/current) supplements of the classical (probability) descriptors of the overall entropy/information content in electronic states are identified and the associated entropy deficiency (information distance) quantities are introduced. The Shannon (global, logarithmic) and Fisher (local, gradient) information descriptors in both spaces are summarized, and the momentum continuity equation is used to establish the associated probability source. General relations between global and local information densities are examined and the etropic principles determining molecular phase equilibria are investigated.
Background: The resultant measures of the information content in molecular electronic states are revisited. In these overall measures the current-related terms complement the probability functionals of classical Information Theory. The nonclassical Shannon entropy reflects the average magnitude of the state phase distribution, while the related current term in the complementary Fisher measure accounts for the average phase-gradient. Method: These generalized Fisher and Shannon descriptors of the information content in complex electronic states are applied to determine the phase-equilibria in molecules. The vertical and horizontal equilibria in molecules, which mark the extrema of the nonclassical and resultant entropy/information descriptors, respectively, are explored. The requirement, that the resultant measures have common solutions of the associated extreme entropy/information principles, calls for the negative sign of the nonclassical gradient measure. Conclusion: These extrema imply the phase-transformation of molecular electronic states. A separation of the density (modulus) and current (phase) factors in many-electron states is effected using the Harriman-Zumbach-Maschke construction of complex antisymmetric states yielding the specified electron density. A combined description of molecular systems, which accounts for the density and current degrees-of-freedom of electronic states is summarized and continuity equations for the state phase and resultant entropy are discussed. Keywords: Continuity equations, current-information, information continuity, molecular equilibria, phase-entropy/information, entropy.
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.
Quantum-generalised descriptors of the information content of electronic states in molecules are proposed, in which non-classical (current) terms complement classical (probability) functionals of the ordinary information theory. The relation between densities of the familiar classical Fisher and Shannon information/entropy measures is applied to determine their non-classical complements. The quantum supplement of the classical Shannon entropy describes the average magnitude of the phase distribution, while the current term in the Fisher measure accounts for the gradient content of the state phase function. Illustrative applications of these quantum information concepts are presented and thermodynamical analogies are commented upon. The particle-density-constrained (vertical) and -unconstrained (horizontal) equilibria in molecules and their fragments are explored and the corresponding equilibrium ‘thermodynamic’ phases are determined. A separation of the density (modulus) and current (phase) factors of general many-electron states is effected using the Harriman–Zumbach–Maschke construction of antisymmetric states yielding the specified electron density. The phenomenological framework in spirit of the non-equilibrium thermodynamical description is proposed. It accounts for both the density and current degrees of freedom of molecular states. The associated entropy source in the information continuity equation is derived.
The entropic perspective on the molecular electronic structure is investigated. Information-theoretic description of electron probabilities is extended to cover the complex amplitudes (wave functions) of quantum mechanics. This analysis emphasizes the entropic concepts due to the phase part of electronic states, which generates the probability current density, thus allowing one to distinguish the information content of states generating the same electron density and differing in their current densities. The classical information measures of Fisher and Shannon, due to the probability/density distributions themselves, are supplemented by the nonclassical terms generated by the wave-function phase or the associated probability current. A complementary character of the Fisher and Shannon information measures is explored and the relationship between these classical information densities is derived. It is postulated to characterize also their nonclassical (phase/current-dependent) contributions. The continuity equations of the generalized information densities are examined and the associated nonclassical information sources are identified. The variational rules involving the quantum-generalized Shannon entropy, which generate the stationary and time-dependent Schrödinger equations from the relevant maximum entropy principles, are discussed and their implications for the system “thermodynamic” equilibrium states are examined. It is demonstrated that the lowest, stationary “thermodynamic” state differs from the true ground state of the system, by exhibiting the space-dependent phase, linked to the modulus part of the wave function, and hence also a nonvanishing probability current.
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.