Abstract
Pople has recently introduced the concept of a framework group to specify the full symmetry properties of a molecular structure. Furthermore, Pople has developed powerful algorithms for the use of framework groups to generate all distinguishable skeletons with a given number of sites. This paper studies the systematics of chirality arising from different framework groups. In this connection framework groups can be classified into four different types: linear, planar, achiral, and chiral. Chiral framework groups lead to chiral systems for any ligand partition including that with all ligands equivalent. Linear framework groups are never chiral even for the ligand partition with all ligands different. Planar framework groups are also never chiral since all sites are in the same plane, which therefore remains a symmetry plane for any ligand partition. However, the mirror symmetry of the molecular plane of a planar framework group can be destroyed by a process called polarization; this process can be viewed as the mathematical analogue of complexing a planar aromatic hydrocarbon to a transition metal. The chirality of four-, five-, and six-site framework groups is discussed in terms of the maximum symmetry ligand partitions resulting in removal of all of the symmetry elements corresponding to improper rotations S n (including reflections S 1 and inversions S 2) from achiral and polarized planar framework groups. The Ruch-Schönhofer group theoretical algorithms for the calculation of chiral ligand partitions and pseudoscalar polynomials of lowest degree (“chirality functions”) are adapted for use with these framework groups. Other properties of framework groups relevant to a study of their chirality are also discussed: these include their transitivity (i.e. whether all sites are equivalent or not), their normality (i.e. whether proper rotations correspond to even permutations and improper rotations correspond to odd permutations), and the number of sites in their symmetry planes.
Similar content being viewed by others
References
For part 12 of this series see King, R. B.: Inorg. Chim. Acta 57, 79 (1981)
Ruch, E.: Accts. Chem. Res. 5, 49 (1972)
Cotton, F. A.: Chemical applications of group theory, New York: Wiley-Interscience 1971
Ruch, E., Schönhofer, A.: Theoret. Chim. Acta (Berl.) 10, 91 (1968)
Ruch, E., Schönhofer, A. Theoret. Chim. Acta (Berl.) 19, 225 (1970)
Mead, C. A.: Top. Curr. Chem. 49, 1 (1974)
Keller, H., Langer, E., Lehner, H., Derflinger, G.: Theoret. Chim. Acta (Berl.) 49, 93 (1978)
Derflinger, G., Keller, H.: Theoret. Chim. Acta (Berl.) 49, 101 (1978)
Ruch, E.: Theoret. Chim. Acta (Berl.) 49, 107 (1978)
Mead, C. A.: Theoret. Chim. Acta (Berl.) 54, 165 (1980)
Dugundji, J., Marquarding, D., Ugi, I.: Chem. Scripta 9, 74 (1976)
Hasselbarth, W.: Chem. Scripta 10, 97 (1976)
Mead, C. A.: Chem. Scripta 10, 101 (1976)
Dugundji, J., Marquarding, D., Ugi, I.: Chem. Scripta 11, 17 (1977)
Hässelbarth, H.: Chem. Scripta 11, 148 (1977)
Mead, C. A.: Chem. Scripta 11, 145 (1977)
King, R. B.: J. Am. Chem. Soc. 91, 7211 (1969)
King, R. B.: Inorg. Chem. 20, 363 (1981)
Pople, J. A.: J. Am. Chem. Soc. 102, 4615 (1980)
Biggs, N. L.: Finite groups of automorphisms. London: Cambridge University Press 1971
Budden, F. J.: The fascination of groups. London: Cambridge University Press 1972
De Bruin, N. G.: in Applied combinatorial mathematics, E. F. Beckenbach, Ed., Chapter 5. New York: Wiley 1964
Chisholm, C. D. H.: Group theoretical techniques in quantum chemistry, Chapter 6. New York, New York: Academic Press 1976
Mumaghan, F. D.: The theory of group representations, Chapter 5. Baltimore, Maryland: Johns Hopkins 1938
Littlewood, D. E., Richardson, A. R.: Phil. Trans. R. Soc (London) Ser. A 233, 99–141 (1934)
Dehn, E.: Algebraic equations. New York, New York: Columbia University Press 1930
Keller, H., Krieger, C., Langer, E. H., Derflinger, G.: Liebigs Ann. Chem. 1296 (1977)
Keller, H., Krieger, C., Langer, E., Derflinger, G.: J. Mol. Struct. 40, 279 (1977)
Ruch, E., Runge, W., Kresze, G.: Angew. Chem. Intern. Ed. 12, 20 (1973)
Grünbaum, B.: Convex polytopes. New York, New York: Interscience Publishers 1967
Kuratowski, K.: Fundam. Math. 15, 271 (1930)
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
King, R.B. Chemical applications of topology and group theory 13. Chirality and framework groups [1]. Theoret. Chim. Acta 63, 103–132 (1983). https://doi.org/10.1007/BF00548015
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF00548015