Quantum information approach to electronic equilibria: molecular fragments and non-equilibrium thermodynamic description

Roman F Nalewajski

doi:10.1007/s10910-014-0357-6

Abstract

The quantum-generalized Information Theory is applied to explore molecular equilibrium states by using the resultant information content of electronic states, determind by the classical (probability based) measures and their non-classical (phase/current related) complements, in the extremum entropy/information principles. The “vertical” (probability-constrained) entropic rules are investigated within the familiar Levy and Harriman–Zumbach–Maschke constructions of Density Functional Theory. A close parallelism between the vertical maximum-entropy and minimum-energy principles in quantum mechanics and their thermodynamic analogs is emphasized and a relation between the probability and phase distributions in the “horizontal” (probability-unconstrained) phase-equilibria is examined. These solutions are shown to involve the spatial phase contribution related to the system electron density.The complete specification of the equilibrium states of molecular/promolecular fragments, including the subsystem density and the equilibrium phase of the system as a whole, is advocated and illustrated for bonded hydrogens in $$\hbox {H}_{2}$$ . Elements of the non-equilibrium thermodynamic description of molecular systems are formulated. They recognize the independent probability and phase state parameters, the associated currents, and their contributions to the quantum entropy density and its current. The phase and entropy continuity equations are explored and the local sources of these quantities are identified.

Highlights

The Information Theory (IT) [1,2,3,4,5,6,7,8] has been successfully applied to explore the electron distributions and chemical bonds in molecules, e.g., [9,10,11,12,13,14,15]
The non-additive Fisher information in the Atomic Orbital (AO) resolution [26] has been used as the Contra–Gradience (CG) criterion for localizing bonding regions in molecules [11,12,13,14,15], while the related information density in the Molecular Orbital (MO) resolution has been shown [11,21,22,23] to determine the vital ingredient of the Electron-Localization Function (ELF) [27,28,29]
In the present work we focus on quantum measures of the entropy/information content of the molecular electronic states, and on the non-classical contributions due to the wave-function phase, or its gradient – the probability current density

Summary

Introduction

The Information Theory (IT) [1,2,3,4,5,6,7,8] has been successfully applied to explore the electron distributions and chemical bonds in molecules, e.g., [9,10,11,12,13,14,15]. The non-classical density-per-electron Sφ(r) is proportional to the local magnitude of the phase function, |φ| = [φ2]1/2, the square root of the phase-density π = φ2, with the particle probability p providing the local “weighting” factor in the associated average (global) functional, while the classical density Sclass.(r) is seen to be determined by the negative logarithm of the system probability density Together these two components generate the overall, resultant Shannon measure of the quantum indeterminicity content due to both the probability and current distributions in the complex quantum state φ [16,17,18,19]:. Together these two contributions allow one to monitor the full information content of the non-equilibrium or variational quantum states, providing the complete IT description of their evolution towards the final equilibrium

Vertical solutions and horizontal phase-equilibria

Molecular equilibria in N-electron systems

Phase descriptors of molecular fragments

Elements of non-equilibrium thermodynamic description

Conclusion