The quantum information‐theoretic description of electron probabilities and currents in molecules is extended. The Harriman‐Zumbach‐Maschke framework of equidensity orbitals is reexamined and the nonclassical Fisher information contribution it generates is used to determine the system equilibrium states for the fixed (ground‐state) electron density/energy. The lowest of such variational “thermodynamic” states can in general exhibit the space‐dependent phase and hence also nonvanishing probability current. The phase/current feature of electronic states in Harriman's representation is emphasized throughout, the probability interpretation of its key constructs is given, and the phase shifts accompanying interactions between the equidensity orbitals are examined. The phase‐“temperature” concept is introduced as the information‐theoretic descriptor of probability currents in molecules and their fragments.
The quantum information‐theoretic description of electron probabilities and currents in molecules is extended. The Harriman‐Zumbach‐Maschke framework of equidensity orbitals is reexamined and the nonclassical Fisher information contribution it generates is used to determine the system equilibrium states for the fixed (ground‐state) electron density/energy. The lowest of such variational “thermodynamic” states can in general exhibit the space‐dependent phase and hence also nonvanishing probability current. The phase/current feature of electronic states in Harriman's representation is emphasized throughout, the probability interpretation of its key constructs is given, and the phase shifts accompanying interactions between the equidensity orbitals are examined. The phase‐“temperature” concept is introduced as the information‐theoretic descriptor of probability currents in molecules and their fragments.
Information-theoretic description of the electron probabilities and currents in molecules is extended to cover the complex amplitudes (wave functions) of quantum mechanics. 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. The previous one-electron development in such an entropic perspective on the molecular electronic structure is extended to cover N-electron states by adopting the Harriman-type framework of equidensity orbitals. This analysis emphasizes the phase part of electronic states, which generates the probability-current density and the associated non-classical entropy contributions, which allow one to distinguish the information content of states generating the same electron density and differing in their current composition. A complementary character of the Fisher and Shannon information measures is explored in the associated vertical (density-constrained) information principles, for determining the equilibrium state corresponding to the fixed ground-state electron density. It is argued that the lowest “thermodynamic” state generally differs from the true ground state of the system, by exhibiting the space-dependent phase and hence also the non-vanishing probability current, linked to the system electron distribution.
The equidensity orbitals (EO) are used in the resultant entropic description of molecular states which combines the probability and current contributions in the overall information content. Continuities of the modulus and phase components of electronic wavefunctions are examined, and the Harriman–Zumbach–Maschke (HZM) construction of Slater determinants yielding the prescribed electron density is explored. The conditional probability interpretation of (complex) HZM wavefunctions is formulated, the entropy/information contributions due to the state phase component are summarized, and a nonclassical origin of quantum dynamics of the resultant gradient information, related to average kinetic energy of electrons, is emphasized. The phase equilibria maximizing the resultant-entropy measures are explored, and “thermodynamic” phase minimizing the overall gradient information is determined. It generates finite orbital currents giving rise to the vanishing resultant flow of electrons in the system as a whole. Potential use of atomic and molecular EO bases in electronic structure calculations and interpretations in chemistry is discussed, and illustrative example of Gaussian probability distribution is examined in some detail.
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.
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.