Use of Harriman’s construction in determining molecular equilibrium states

Roman F Nalewajski

doi:10.1007/s10910-012-0088-5

Abstract

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.

Highlights

The classical Fisher [1] or Shannon [2] information descriptors of the particle spatial distribution are determined solely by the probability aspect of the complex wave function
Alternative constructions and extensions have been suggested, e.g., [20,21,22]. In this analysis we focus on the use of this type of constructing the molecular wave functions in probing the physical implications of the lowest equilibrium (“thermodynamic”) state [8] in the molecular quantum mechanics
In what follows we extend the previous one-electron analysis to the entropy/information equilibria in general N -electron systems by adopting the equidensity orbital framework of the Harriman– Zumbach–Maschke construction

Summary

Introduction

The classical Fisher (local) [1] or Shannon (global) [2] information descriptors of the particle spatial distribution are determined solely by the probability aspect of the complex wave function. This phase side of the molecular electronic structure reflects its “entropic” aspect, which still remains largely unexplored It has been recently demonstrated [8] for the one-electron system that the lowest equilibrium state corresponding to the maximum (quantum) entropy, which gives rise to the ground-state particle distribution, is characterized by the space-dependent (local) phase linked to the ground-state probability density itself. In the one-electron case the molecular (maximum-entropy) equilibrium exhibits the same energy as the true ground state, it involves the local, real-valued phase function related to the probability distribution, and the nonvanishing probability current This is contrary to the non-degenerate eigenstate of the Hamiltonian, for which the current identically vanishes. In what follows we extend the previous one-electron analysis to the entropy/information equilibria in general N -electron systems by adopting the equidensity orbital framework of the Harriman– Zumbach–Maschke construction

Harriman construction

Information principles

Conclusion