Information Theoretical Study of Chirality: Enantiomers with One and Two Asymmetric Centra

Abstract

In this work, the Kullback-Leibler information deficiency is probed as a chirality measure. It is argued that the information deficiency, calculated using the shape functions of the R and S enantiomers, considering one as reference for the other, gives an information theory based expression useful for quantifying chirality. The measure is evaluated for five chiral halomethanes possessing one asymmetric carbon atom with hydrogen, fluorine, chlorine, bromine, and iodine as substituents. To demonstrate the general applicability, a study of two halogen-substituted ethanes possessing two asymmetric carbon atoms has been included as well. The basic expression of the sum of the local information deficiency over all atoms can be decomposed into separate summations over coinciding and noncoinciding atoms, or into a global and a mixing entropy term, or into a local entropy contribution for each atom individually based on the Hirshfeld partitioning. Avnir's continuous chirality measure (CCM) has been computed and confronted with the information deficiency. Finally, the relationship between chirality and optical rotation is used to study the proposed measure. The results illustrate Mezey's holographic electron density theorem with an intuitively appealing division of the strength of propagation of the atomic chirality from an asymmetric carbon atom throughout the molecule. The local information deficiency of the carbon atom is proposed as a measure of chirality; more precisely, the difference in information between the R and the S enantiomer turns out to be a quantitative measure of the chirality of the system. It may be evaluated as the arithmetic mean of the different alignments, or considering only the alignment resulting in the highest similarity value, or using the QSSA alignment.

Alternative Kullback−Leibler Information Entropy for Enantiomers

“…Thus a small value of the information deficiency leads to the most similar alignment for the enantiomer and the racemate. With our previous information deficiency measure 33 the same trend was found for the two enantiomers.…”

“…In the author's previous publication 33 information theory was used as a quantum chirality indicator. It was evaluated for chiral halomethanes possessing one asymmetric carbon atom and H, F, Cl, Br and I as substituents, resulting in five different enantiomeric pairs, and also for two halogen substituted ethanes.…”

Fisher and Jensen–Shannon divergences: Quantitative comparisons among distributions. Application to position and momentum atomic densities

“…[29][30][31] Very recently a new application of the KL measure to the study of molecular chirality has been carried out. 32 Let us also point out that the concept of Shannon entropy provides additional ways of defining functionals, displaying adequate properties ͑namely non-negativity and saturation for identical distributions, as occurs with QD, KLS, and QSI͒, with the aim of comparing distribution functions, as, e.g., the so-called JSD, among others. It is defined as 9,11,12…”

“…Properties and inequalities among these global information measures have also been obtained, and more recently a new generalization including weighted measures has been provided. 11,12,37,38 However, it would be also of great interest to dispose of information measures displaying a deeper local character, namely their values being more sensitive to the aforementioned strong local changes. The main quantity studied and employed in the literature with such a characteristic ͑in what concerns measures on a single distribution͒ is the so-called Fisher information I, [39][40][41][42][43] …”

Symmetry measures of the electron density

“…In the search for a criterion on how the two molecular densities may be superimposed, several methods and algorithms have been proposed. Some examples are alignments according to common physicochemical features, 35 the backbone (BB) alignment method, [20][21][22]25 the topo-geometrical superposition algorithm (TGSA), 36 and algo-rithms designed to maximize the molecular similarity (QSSA).…”