Computer Generation of Nuclear Equivalence Classes Based on the Three-Dimensional Molecular Structure

The development of computational techniques for the vertex partitioning of graphs and for the machine generation of automorphism groups has received considerable attention in recent years.'-I4 The related problems pertinent to the molecular structure seek to generate nuclear partitioning under the three-dimensional molecular symmetry. In fact, it is envisaged that faster algorithms for the machine perception of molecular symmetry would require, as a first step, algorithms for the generation of nuclear equivalence classes. Compared to the graph automorphism and isomorphism problems the nuclear partitioning problem has received much less attention. The nuclear partitioning problem has several applications. The I3C and proton NMR depends on the nuclear equivalence ~1asses.I~ The number of equivalence classes yields the number of NMR signals, while the ratio of the number of elements in the various equivalence classes yields the intensity ratio in the case of proton NMR. Therefore an ideal starting point for such applications is the development of fast machine algorithms for the generation of nuclear equivalence classes. The hyperfine pattern in the ESR spectra of radicals is also dependent on the nuclear equivalence classes. l 5 9 l 6 The generation of symmetry-adapted linear combination'* (SALC) of atomic orbitals of large polyatomic systems requires, as a first step, the nuclear equivalence classes since the atomic orbitals belonging to nuclei that are in different equivalence classes would not mix in a SALC. Since fullerenes containing large numbers of nuclei exhibit several nuclear equivalence classes, computational techniques could be especially useful in the machine generation of SALCs for larger fullerenes and thus in quantum chemical computations that make use of SALCs. The objective of this study is to develop algorithms suitable for machine manipulation to generate nuclear equivalence classes. The algorithms developed up to now for this purpose are based on graph connectivity and not on the three-dimensional geometry of the molecule. We use the information contained in the Euclidian distance matrices for this purpose. These matrices have been used as molecular structural invariants el~ewhere. '~ Subsequent to the author's proposal to use the polynomials and other invariants derived from Euclidian matrices, Nikolic et a1.21 have shown that Euclidian matrices and their invariants can successfully