[Paper Review] On Lunn-Senior's Mathematical Model of Isomerism in Organic Chemistry. Part II

Valentin Vankov Iliev|ArXiv.org|Jun 30, 2003

Chemistry and Stereochemistry Studies9 references3 citations

TL;DR

This paper formalizes Lunn-Senior's mathematical model of isomerism in organic chemistry by introducing a group-theoretic framework that models substitution reactions as order relations in partially ordered sets of isomers. It defines indistinguishability via substitution reactions through automorphisms preserving chiral pairs and orbit structures, with key results showing that indistinguishable isomers form specific orbit structures under automorphism groups, particularly in cyclopropane and ethene derivatives.

ABSTRACT

The second part of this paper is devoted to the following important question in organic chemistry: given two isomers of a molecule, how to identify them with their structural formulae using only type properties of that molecule? A classical answer of this question is given for benzene by the identification of its di-substituted (para, ortho, and meta), and tri-substituted (asymmetric, vicinal, and symmetric) derivatives via the Korner substitution reactions among them. Here we develop a machinery within the framework of the Lunn-Senior's mathematical model of isomerism in organic chemistry, which, in principle, answers this question. In particular, it is shown that the members of a chiral pair cannot be distinguished via substitution reactions. The examples of ethene, benzene, and cyclopropane are discussed.

Motivation & Objective

• To provide a rigorous mathematical foundation for Lunn-Senior’s ad hoc assertions about isomerism and substitution reactions in organic chemistry.
• To formalize the concept of indistinguishability of isomers via substitution reactions using automorphisms of partially ordered sets of isomer orbits.
• To introduce and characterize 'hidden symmetries' as automorphisms induced by renumberings of valences that preserve molecular symmetry groups.
• To extend the model to subsets of isomers (e.g., mono-, di-, tri-substituted derivatives) and define distinguishability within such restricted reaction networks.
• To apply the framework to concrete cases like cyclopropane and ethene, identifying orbit structures and chiral pairs under symmetry groups.

Proposed method

• Models isomers as orbits in $ T_{d;G} $, where $ G $ is the univalent substitution isomerism group acting on $ d $ unsatisfied valences.
• Defines distinguishability via substitution reactions using automorphisms $ \alpha \in \mathrm{Aut}_0(T_{D;G}) $ that preserve $ G $-orbits and chiral pairs.
• Introduces $ \mathrm{Aut}_0^\prime(T_{D;G}) $, the group of 'chiral automorphisms' that map chiral pairs onto chiral pairs.
• Uses the normalizer $ N $ of $ G $ in $ S_d $ to generate 'hidden symmetries' via induced automorphisms $ \hat{\nu} $, with $ \hat{\nu} \in \mathrm{Aut}_0^\prime(T_{D;G}) $ when $ \nu \in N' $.
• Applies the framework to specific compounds (e.g., cyclopropane) by computing orbit decompositions under $ G'' $, the structural isomerism group.
• Employs group isomorphisms such as $ \mathrm{Aut}_0(T_{D;G}) \simeq S_3 \times S_2 $ and $ \mathrm{Aut}_0^\prime(T_{D;G}) \simeq S_2 \times S_2 $ to classify indistinguishable isomer sets.

Experimental results

Research questions

• RQ1 When are two isomers of a compound indistinguishable via substitution reactions, according to the Lunn-Senior model?
• RQ2 How do automorphisms of the partially ordered set of isomers reflect chemical symmetries and chiral relationships?
• RQ3 What is the role of 'hidden symmetries' in preserving reaction indistinguishability beyond the standard substitution group?
• RQ4 How does restricting the set of observable isomers (e.g., mono- and di-substituted) affect the distinguishability of isomers?
• RQ5 What is the precise orbit structure of isomers under the structural isomerism group $ G'' $, and how does it classify indistinguishable derivatives?

Key findings

• The group $ \mathrm{Aut}_0(T_{D;G}) $ of automorphisms preserving $ G $-orbits is isomorphic to $ S_3 \times S_2 $, with $ S_3 $ permuting isomers labeled $ (4,2) $, $ (3^2) $, and $ (4,1^2) $, and $ S_2 $ permuting the chiral pair $ \{e_{(4,1^2)}, f_{(4,1^2)}\} $.
• The group $ \mathrm{Aut}_0^\prime(T_{D;G}) $ of chiral automorphisms is isomorphic to $ S_2 \times S_2 $, where one $ S_2 $ permutes the chiral pairs $ \{a_{(4,2)}, b_{(4,2)}\} $, $ \{a_{(3^2)}, b_{(3^2)}\} $, and $ \{a_{(4,1^2)}, b_{(4,1^2)}\} $, and the other permutes $ \{e_{(4,1^2)}, f_{(4,1^2)}\} $.
• Isomers within the same $ G'' $-orbit are structurally identical, and the paper identifies these orbits explicitly for cyclopropane: $ \{a_{(6)}\} $, $ \{a_{(5,1)}\} $, $ \{a_{(4,2)}, b_{(4,2)}, e_{(4,2)}\} $, $ \{c_{(4,2)}\} $, $ \{a_{(3^2)}, b_{(3^2)}\} $, $ \{c_{(3^2)}, e_{(3^2)}\} $, $ \{a_{(4,1^2)}, b_{(4,1^2)}, e_{(4,1^2)}, f_{(4,1^2)}\} $, $ \{c_{(4,1^2)}\} $.
• Products corresponding to isomers in the same set are indistinguishable via substitution reactions, while those in different sets are distinguishable, as confirmed by $ \mathrm{Aut}_0^\prime(T_{D;G}) $-stability.
• The homomorphism $ \mathrm{Aut}_0^\prime(T_{D';G}) \to \mathrm{Aut}_0^\prime(T_{D;G}) $ is not surjective in general, indicating that indistinguishability can be lost when extending the observable reaction network.
• The framework generalizes to any compound with a 6-valent skeleton and dihedral substitution group $ G $, with at least three homogeneous di-substitution derivatives, confirming broad applicability.