Some Properties of the Sharma–Mittal Statistical Distribution

Bakiev, T. N.; Nakashidze, D. V.; Savchenko, A. M.; Semenov, K. M.

doi:10.3103/s0027134923040033

Cited by 1 publication

(1 citation statement)

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“…It is interesting to mention that the two-parameter Sharma-Mittal (S-M) uncertainty measure represents the most general approach to attack the Stockholder partitioning scheme, since it represents a generalization of the Gibbs-Shannon, Renyi, and Tsallis information measures. [77,78] Applications of the Sharma-Mittal entropy in chemistry are still relatively limited [75] compared to other areas, [78,79] its ability to capture nonextensive behavior and long-range interactions makes it a valuable tool for studying complex chemical systems and processes where traditional entropy measures may not be sufficient. Although it is not the goal of this study to perform any numerical calculations to test the S-M uncertainty measure, other than demonstrating its utility to provide a theoretical justification for employing the stockholder partitioning scheme without resource of any arbitrary reference atoms.…”

Section: A Stockholder Partitioning Aim Schemementioning

confidence: 99%

The Separability Problem in Molecular Quantum Systems: Information‐Theoretic Framework for Atoms in Molecules

Esquivel,

Carrera

2024

ChemPhysChem

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Even though molecules are fundamentally quantum entities, the concept of a molecule retains certain classical attributes concerning its constituents. This includes the empirical separability of a molecule into its three‐dimensional, rigid structure in Euclidean space, a framework often obtained through experimental methods like X‐Ray crystallography. In this work, we delve into the mathematical implications of partitioning a molecule into its constituent parts using the widely recognized Atoms‐In‐Molecules (AIM) schemes, aiming to establish their validity within the framework of Information Theory concepts. We have uncovered information‐theoretical justifications for employing some of the most prevalent AIM schemes in the field of Chemistry, including Hirshfeld (stockholder partitioning), Bader's (topological dissection), and the quantum approach (Hilbert's space definition). In the first approach we have applied the generalized principle of minimum relative entropy derived from Sharma‐Mittal relative entropy, avoiding the need for an arbitrary selection of reference promolecular atoms. Within the realm of topological‐information partitioning, we have demonstrated that the Fisher information of Bader's atoms conform to a comprehensive theory based on the Principle of Extreme Physical Information. For the quantum approach we have presented information‐theoretic justifications for conducting Löwdin symmetric transformations on the density matrix to form atomic Hilbert spaces.

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