On quantifying chirality — Obstacles and problems towards unification

Recently there has been an ever growing interest and activity in the attempts on quantifying chirality which is causing this concept to become a diverse and uncorrelated entity. Possible reasons for this complication are presently discussed. It is shown that it becomes necessary to distinguish between geometric and physical chiralities. For geometrical chiral sets it is necessary to distinguish between equi- and sub-dimensional sets where the metrization of their chirality can be generalized and unified only for equi-dimensional sets. This is accomplished by the method of overlap. For sub-dimensional sets there exists no general and unique mode of quantifying chiralities, except for discrete and finite sets of points such as the comers of polyhedron, for which the approach of Hausdorff distances proves to be an efficient method of quantifying the chirality presented by their distribution. The domain of physical chiralities, although being of natural significance, is still in a premature state of development. Each physical property may have a different chiral measure so that there is no sense in a claim of unification. Equi- and sub-dimensionality exist also for physical chiralities and they can be quantified by the overlap method for equi-dimensional sets.     

Entropy in dissimilarity and chirality measures

Within the prospect of quantifying the geometrical dissimilarity of molecular models on the basis of a thermodynamical formalism, the algebra of stereogenic pairing equilibria is reviewed and applied to molecular geometry: developing Rassat's proposition, an interaction energy of two figures F and F is taken as proportional tod _H ^Emphasis>/2 (F, F), whered _H denotes the Hausdorff distance. IfG is a group of rotations in E _n the geometrical version of the general equation (E) of the chemical algebra defines a distance extensionD _p(F,F) ofd _H(F,F), which is independent of the orientations of F and F, and where the coefficientp is interpreted as the reciprocal of a temperature-like parameter:p 1/T. At K (p = ), no formal entropy contributes to the definition of the uniform distanceD . At K (p = 0), the discrimination between homo- and hetero-pairing of figures by the harmonic distance Do is averaged over orientation states. Temperature-dependent chirality measuresc _p are derived fromD _p, andc is analogous to Mislow's chirality measure. If T and oT are normalized enantiomorphic triangles with coincident centroids inE ₂,c _p(T) =D _p (T, T) is calculated forp = 0 andp = , and discussed for 0 <p < . Finally, the Hausdorff interaction model is putatively related to energy profiles versus dihedral angle inmeso- anddl-molecules.     

Distance functions as generators of chirality measures

A non-numerical analysis is presented of chirality measures associated with a set of topologically equivalent distance functions. A chirality measure is defined as the minimum distance that separates a chiral and an achiral object (first kind) or two enantiomorphs (second kind). On the basis of this analysis, as applied to triangles in the Euclidean plane, results of an earlier computational study of the Hausdorff chirality measure are now fully understood. Analytical proof has been provided for an earlier conjecture, based on a numerical analysis, that the union of enantiomorphous triangles is achiral under conditions of maximal overlap. Geometric parameters for the most chiral triangle, as determined by a family of three measures of the first kind, are found to differ substantially from those determined by the corresponding measures of the second kind; none of these extremal triangles is degenerate.     

Continuous chirality measures in transition metal chemistry

The definition of the continuous chirality measure(CCM) is provided and its applications are summarized in this tutorial review, with special emphasis on the field of transition metal complexes. The CCM approach, developed in recent years, provides a quantitative parameter that evaluates the degree of chirality of a given molecule. Many quantitative structural correlations with chirality have been identified for most of the important families of metal complexes. Our recent research has shown that one can associate the chirality measures with, e.g., enantioselectivity in asymmetric catalysis. We also explore a fragment approach to chirality in which we investigate which part of a molecule is responsible for the chirality-associated properties of a given family of compounds.     

About second kind continuous chirality measures. 1. Planar sets

The chirality index of a d-dimensional set of n points is defined as the sum of the n squared distances between the vertices of the set and those of its inverted image, normalized to 4T/d,T being the inertia of the set. The index is computed after minimization of the sum of the squared distances with respect to all rotations and translations and all permutations between equivalent vertices. The properties of the chiral index are examined for planar sets. The most achiral triangles are obtained analytically for all equivalence situations: one, two, and three equivalent vertices. These triangles are different from those obtained by Weinberg and Mislow with distance functions. This revised version was published online in July 2006 with corrections to the Cover Date.     

A topological account of chirality.

Graph invariants may differentiate structural isomers but are inappropriate to account for stereoisomerism and to distinguish between chiral structures. This work is an attempt to address this problem. A chiral function F satisfying condition F(D) = -F(L), where D and L denote enantiomers of the same structure, has been applied in combination with Randic's index (1)chi(v). The resulting index chi(c) was used to explain the variance in thin-layer chromatographic retention indices.(1)     

The temperature-dependent optical activity of quartz: from Le Châtelier to chirality measures

Quartz, the most abundant mineral in Earth’s crust, is a chiral crystal. It was the first material for which the phenomenon of the optical rotation was observed. In the late 19th century/beginning of the 20th, several researchers, the most famous of which is Le Châtelier, investigated how this optical rotation changes with temperature. By employing a modern analytical/computational tool for evaluating the degree of chirality on a continuous scale, we were able to show a remarkable agreement between the original optical rotation/temperature curve, and the chirality/temperature curve. We thus provide a direct interpretation of the early observations, as reflected in the dependence of the optical rotation of the degree of chirality, linking these two properties quantitatively.     

A note on chirality in NMR spectroscopy

Using simple symmetry arguments we give proofs of the derivations of the manifestation of chirality in the chemical shift and spin-spin coupling constant in nuclear magnetic resonance and relate our proofs to earlier discussions.     

Dynamic chirality, chirality transfer and aggregation behaviour of dithienylethene switches

The synthesis and characterisation of a series of chiral and achiral low molecular weight organogelators (LMWGs) based on bis-amide substituted dithienylethene photochromic switches is reported. The LMWGs gelate a range of solvents depending on the specific functionalisation of the hydrogen bonding amide groups. In mixtures of chiral and achiral LMWGs the stereochemical outcome of the chiral aggregation is determined by the chiral LMWG molecules in most cases. However, for the first time we demonstrate that the stereochemical outcome of the aggregation can be influenced by the achiral LWMG molecules in some cases. Furthermore specific π-π (and/or van der Waals) interactions of chiral LMWGs 1-3o with the solvent allow the solvent to influence the control of chirality of aggregation. This influence of the solvent has a dramatic effect on whether four- or two-gel states are available.     

Quantitative chirality of helicenes

Quantitative evaluation of geometrical chirality through the use of the Continuous Chirality Measure is employed to unveil correlations between chirality and several properties of helicenes. These include the efficiency of their chiral-chromatographic racemate separation, their optical rotation, their racemization energy and the melting points of the enantiomer-crystals. The non-monotonic behavior of the chirality of helicenes is shown to reflect the behavior of a theoretical helix.     

Geometric chirality products

Developed from Guye's produit d'asymétrie and formally similar to Ruch's chirality products, geometric chirality products are functions purely of molecular shape, without reference to chemical characteristics. In their normalized versions, geometric chirality products have all the attributes' of a chirality measure, i.e. they are similarity invariant and dimensionless in the interval [–1, 1]. An application to Boys' model of the tetrahedron is presented, and a detailed study of the results for triangular domains in E² is reported. According to this measure, the most chiral triangle is infinitely flat and infinitely skewed. The analysis leads to the paradoxical conclusion that the most chiral triangle is infinitesimally close to an achiral one, The results are compared with those obtained for an overlap measure of chirality, and the relationship between molecular models and measures of chirality is briefly discussed.On leave from the Department of Chemistry, University of the Western Cape, Bellville 7530, South Africa.     

Dipeptide-induced chirality organization

This review describes an outline of dipeptide-induced chirality organization by using molecular scaffolds. A variety of ferrocene-dipeptide conjugates as bioorganometallics are designed to induce chirality-organized structures of peptides. The ferrocene serves as a reliable organometallic scaffold with a central reverse-turn unit for the construction of protein secondary structures via intramolecular hydrogen bondings, wherein the attached dipeptide strands are constrained within the appropriate dimensions. Another interesting feature of ferrocene-dipeptide conjugates is their strong tendency to self-assemble through contribution of available hydrogen bonding sites for helical architectures in solid states. Symmetrical introduction of two dipeptide chains into a urea molecular scaffold is performed to induce the formation of the chiral hydrogen-bonded duplex, wherein each hydrogen-bonded duplex is connected by continuous intermolecular hydrogen bonds to form a double helix-like arrangement.     

A geometrical approach to the degree of chirality and asymmetry

A new measure of the degree of chirality and asymmetry of a finite number of particles is proposed. To this end a space of configurations of identical particles is defined as the orbit space of the group of all permutations of particles embedded in an Euclidean space. This space is shown to be a metric space and the action of the translation and orthogonal groups is also defined. The results are applied to the study of an algebra of polynomials on the configuration space and its equivalence to the algebra of symmetric Cartesian tensors is demonstrated. An illustrative example is presented. Some general features of chirality are also briefly discussed.     

On the chirality of paper

On a macroscopic level, a paper sheet has a fiber structure similar to the chains of a molecular chiral membrane. The continuum model of membranes based on chiral elasticity can be applied to calculate the spontaneous curl observed in paper sheets. Twist of the paper sheet is shown to occur in the presence of a tilt relative to the sheet geometry of the constituent cellulose fibers. The dependence of the twist angle on the width of the sheet, on external mechanical fields and on the elastic constants of the paper is given.     

Desymmetrization and degree of chirality

Chiral objects, viewed as distorted derivatives of achiral ones, may be represented by points in a configuration space that is spanned by a set of symmetry coordinates derived for the symmetry group of the achiral object of highest symmetry. We propose a measure (d) that quantifies the displacement of the representative point for a chiral object away from thenearest point representing an achiral object in such a multi-dimensional configuration space. If the symmetry coordinates are chosen so as to yield a similarity invariant measure, then the valuesd; obtained for a series ofi chiral objects can serve as a basis for comparing the degrees of chirality of these objects. The chirality of triangles inE ² is studied by this method, and it is shown that the most chiral triangle in terms of this measure corresponds to one that is infinitely flat, and that may be approached but is never attained. This result is compared to others obtained previously for the same system by the use of different measures of chirality.On leave from the Department of Chemistry, University of the Western Cape, Bellville 7530, South Africa.