Analytical methods for calculating Continuous Symmetry Measures and the Chirality Measure

Abstract

We provide analytical solutions of the Continuous Symmetry Measure (CSM) equation for several symmetry point-groups, and for the associated Continuous Chirality Measure (CCM), which are quantitative estimates of the degree of a symmetry-point group or chirality in a structure, respectively. We do it by solving analytically the problem of finding the minimal distance between the original structure and the result obtained by operating on it all of the operations of a specific G symmetry point group. Specifically, we provide solutions for the symmetry measures of all of the improper rotations point group symmetries, S(n), including the mirror (S(1), C(S)), inversion (S(2), C(i)) as well as the higher S(n)s (n > 2 is even) point group symmetries, for the rotational C(2) point group symmetry, for the higher rotational C(n) symmetries (n > 2), and finally for the C(nh) symmetry point group. The chirality measure is the minimal of all S(n) measures.

Chirality Measures for Vectors, Matrices, Operators and Functions

“…Over the years the CCM of vector coordinates was extensively used in that way to determine the chirality of many molecules and finding correlations between the chirality and various physical and chemical properties. [12][13][14][15] A simple example is provided by the chirality values of the twenty natural amino acids. [16] The fact that all the natural amino acids (except for glycine) are chiral is of course common knowledge, but which is, at the same time quite poor.…”

“…[9] Indeed, side by side with the traditional treatment of chirality, the notion that it is a measurable structural property was explored by various authors. [10][11] A methodology we have developed intensively in the past decade is the continuous chirality measure (CCM), [12] which is based on the continuous symmetry measures (CSM), [13] widely applied in many chemical, biochemical and physical studies. [14] It is a geometric measure that is based on the distance between a given structure to the closest achiral structure, and which offers several advantages: It is general enough to be used on different systems; the measure is universal and so different systems can be compared; chirality and symmetry measures for the same molecule can be derived and are on the same scale; it does not give false zero reading, and more.…”

Concurrent Symmetries: The Interplay Between Local and Global Molecular Symmetries

“…The procedure for obtaining WA C H T U N G T R E N N U N G (A,R i ) in each case has been described elsewhere and the reader interested in the mathematical details of symmetry operation measures is addressed to the more technical references. [4,10,11] Due to the different description of molecular shape, the actual values of SA C H T U N G T R E N N U N G (A,G) for a given molecule calculated by using its nuclear framework or its valence electron density are not the same. In general, the symmetry contents calculated for the electron density are lower, that is, SA C H T U N G T R E N N U N G (A,G) is larger than when it is calculated for the bare nuclear framework.…”

“…Only acyclic substitutions were allowed in the aromatic rings and structures with disorder were disregarded. The relevant fragments of the studied molecules, devoid of substituents, were analyzed by calculating the symmetry measures with the algorithms reported earlier, [4,10,13] implemented in the program SYMGROUP.…”

“…For that purpose we will study the structures of several families of organometallic, coordination, and organic compounds by means of the recently developed symmetry measures. [4,10] Summary of the methodology applied: Symmetry group measures (or, in short, symmetry measures) calibrate the proximity of a given structure to being invariant under all the operations of a symmetry group. In particular, we are interested in cyclic groups such as C n , S n , C i , or C s , since they gauge the presence of the corresponding symmetry elements, proper or improper axes, inversion center, or reflection planes.…”