Screens for Displaying Chirality Changing Mechanisms of a Series of Peroxides and Persulfides from Conformational Structures Computed by Quantum Chemistry

Abstract

A great variety of data on molecular structure and changes, accumulated both experimentally and theoretically, need be compacted and classified to extract the information arguably relevant to understand the basic mechanisms of chemical transformations. Here a screen for displaying four-center processes is developed and as an illustration applied to conformations involving torsions around O – O and S – S bonds, extending the structural properties previously calculated in this laboratory. The construction of the screen follows from connections recently established between the classical kinematic mechanism – the four-bar linkage – and the basic ingredient of quantum angular momentum theory – the 6j symbol.

Appendix. Quadrilaterals, Quadrangles, Tetrahedra: The 6-Distance Representation

We extend here geometrical considerations alluded to in Sect. 2.1. The six distances that define the stereogenic unit of peroxides and persulfides, i.e. the sequence of bonds R₁ – O₁ – O₂ – R₂ (for the sake of simplicity we refer only to the peroxide case, but it can be simply applied to persulfides), individuate a tetrahedron, whose edges are the distances O₁R₁, O₁R₂, O₂R₂, O₂R₁, O₁O₂, and R₁R₂. The planar projection of the tetrahedron permits to define a quadrilateral, whose diagonals can be chosen among the pairs of opposite distances R₁O₁ and R₂O₂; R₁O₂ and R₂O₁; O₁O₂ and R₁R₂. The tetrahedron and quadrilateral coincide when the stereogenic unit has a planar geometry, e.g. for the cis and trans configurations. The length of the diagonals vary within a range, keeping the four sides of the quadrilateral fixed. In these terms, we represent the variation of the diagonals in a two-dimensional diagram, the screen. We are here inspired by the Wigner-Racah-Regge approach to theory of the most basic ingredients of quantum angular momentum and of spin networks, the 6j symbol. In a similar fashion, let’s arrange the six distances as follows,

$$ \left\{ {\begin{array}{*{20}c} {{\text{O}}_{1} {\text{O}}_{2} } & {{\text{O}}_{2} {\text{R}}_{2} } & {{\text{O}}_{1} {\text{R}}_{2} } \\ {{\text{R}}_{1} {\text{R}}_{2} } & {{\text{O}}_{1} {\text{R}}_{1} } & {{\text{O}}_{2} {\text{R}}_{1} } \\ \end{array} } \right\} , $$

(A.1)

the 6d symbols. In the application to peroxides, in the text the diagonals correspond to the first column, since in the specific case of peroxides and persulfides, the variation of R1 – R2 distance is the most suitable to monitor the chirality change transition. Two other choices of diagonals is possible, since by taking into account that the symbol is invariant under permutation of the three columns:

$$ \left\{ {\begin{array}{*{20}c} {{\text{O}}_{1} {\text{O}}_{2} } & {{\text{O}}_{2} {\text{R}}_{2} } & {{\text{O}}_{1} {\text{R}}_{2} } \\ {{\text{R}}_{1} {\text{R}}_{2} } & {{\text{O}}_{1} {\text{R}}_{1} } & {{\text{O}}_{2} {\text{R}}_{1} } \\ \end{array} } \right\} = \left\{ {\begin{array}{*{20}c} {{\text{O}}_{2} {\text{R}}_{2} } & {{\text{O}}_{1} {\text{O}}_{2} } & {{\text{O}}_{1} {\text{R}}_{2} } \\ {{\text{O}}_{1} {\text{R}}_{1} } & {{\text{R}}_{1} {\text{R}}_{2} } & {{\text{O}}_{2} {\text{R}}_{1} } \\ \end{array} } \right\} = \cdots $$

(A.2)

(there are 3! = 6 ways identical symbols).

The first row represents a triad (a triangular face of the tetrahedron into a vertex), while the second row exhibits the convergence of three edges of the tetrahedron. One can have four triads, $\left\{ {{\text{O}}_{1} {\text{R}}_{1} , {\text{O}}_{1} {\text{R}}_{2} , {\text{O}}_{1} {\text{O}}_{2} } \right\}$, $\left\{ {{\text{O}}_{2} {\text{R}}_{2} , {\text{O}}_{2} {\text{R}}_{1} , {\text{O}}_{1} {\text{O}}_{2} } \right\}, \left\{ {{\text{O}}_{1} {\text{R}}_{1} , {\text{O}}_{2} {\text{R}}_{1} , {\text{R}}_{1} {\text{R}}_{1} } \right\}, \left\{ {{\text{O}}_{2} {\text{R}}_{2} , {\text{O}}_{1} {\text{R}}_{2} , {\text{R}}_{1} {\text{R}}_{2} } \right\}$. The invariance with respect to permutations of the four triads and the related triangles generate the invariance of the 6j under the interchange of upper and lower arguments of any two columns, for example

$$ \left\{ {\begin{array}{*{20}c} {{\text{O}}_{1} {\text{O}}_{2} } & {{\text{O}}_{2} {\text{R}}_{2} } & {{\text{O}}_{1} {\text{R}}_{2} } \\ {{\text{R}}_{1} {\text{R}}_{2} } & {{\text{O}}_{1} {\text{R}}_{1} } & {{\text{O}}_{2} {\text{R}}_{1} } \\ \end{array} } \right\} = \left\{ {\begin{array}{*{20}c} {{\text{R}}_{1} {\text{R}}_{2} } & {{\text{O}}_{1} {\text{R}}_{1} } & {{\text{O}}_{1} {\text{R}}_{2} } \\ {{\text{O}}_{1} {\text{O}}_{2} } & {{\text{O}}_{2} {\text{R}}_{2} } & {{\text{O}}_{2} {\text{R}}_{1} } \\ \end{array} } \right\} = \cdots $$

(A.3)

(four ways). A total of 24 symmetries can be enumerated in this way. Each symbol has in addition the six Regge symmetries replicas, for a total of 144 symmetries. The important relationship of these quantum mechanically discovered (1959) symmetries were later found. Surprisingly they apply to properties of Euclidean (Ponzano and Regge, 1968) and non-Euclidean tetrahedra and to the operating rules of the most venerable of the kinematic mechanisms. For further discussions, see companion papers in this volume [38, 39].

This “distance only” formulation may be suitable to describe peroxides and similar classes of molecules for which often mapping based on pairs of dihedral angle (e.g. Ramachandran plot) is not applicable. In turn, as a viewpoint in the same spirit of the displays on screen examined in the mean text, it can be arguably extendible to other systems, such as molecules characterized by the asymmetric carbon covalently bound to four different ligands, or to describe folding involving the peptidic bonds.