On Quantum Groups and Their Potential Use in Mathematical Chemistry
Maurice Kibler, Tidjani Négadi
备注：
16 pages, Tex file. Paper written from a lecture presented (by M.K.) at the ``IV International Conference on Mathematical and Computational Chemistry'', Bled (Yugoslavia), 3 - 7 June 1991
The quantum algebra suq(2) is introduced as a deformation of the ordinary Lie algebra su(2). This is achieved in a simple way by making use of $q$-bosons. In connection with the quantum algebra suq(2), we discuss the q-analogues of the harmonic oscillator and the angular momentum. We also introduce q-analogues of the hydrogen atom by means of a q-deformation of the Pauli equations and of the so-called Kustaanheimo-Stiefel transformation.
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, wh…
Thanks to an algebraic duality property of reduced states, the Schmidt best approximation theorems have important corollaries in the rigorous theory of two-electron moleculae. In turn, the "harmonium mode" or "Moshinsky atom" constitutes a non-trivial laboratory bench for energy functionals proposed over the years (1964--today), purporting to recover the full ground state of the system from knowledge of the reduced 1-body matrix. That model is usually regarded as solvable, but some important aspects of it, in particular the exact energy and full state functionals ---unraveling the "phase dilem…
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The aim of this paper is to present a generalization of Lunn-Senior's mathematical model of isomerism in organic chemistry. The main idea of Lunn and Senior is that if the type of isomerism is fixed, a molecule with a fixed skeleton and d univalent substituents has a symmetry group $W\leq S_d$ which is generally not the molecule's 3-dimensional symmetry group. The unit character of W induces a representation of the symmetric group $S_d$ which governs the combinatorics of the isomers of the given molecule. Lunn-Senior's thesis is that certain non-negative integers established by this representa…
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