DOI: 10.1002/chem.201002128 Concurrent Symmetries: The Interplay Between Local and Global Molecular Symmetries Jorge Echeverría, [a, b] Abel Carreras, [b, c] David Casanova, [b] Pere Alemany,* [b, c] and Santiago Alvarez* [a, b] Introduction Symmetry is a central concept in chemistry, usually de- scribed through the formalism of symmetry point groups and crystallographic space groups. Évariste Galois (1811– 1832) developed group theory and Camille Jordan general- ized it in 1870 with the introduction of representation theory. Crystallographers have been dealing with symmetry groups since 1913, but it was only in 1931 that Eugene P. Wigner and Hermann Weyl applied the group representa- tion theory to quantum mechanics, thus setting the founda- tions for its use in chemistry. [1] However, the popularization of the use of group theory for chemical applications came in the 1960s, with the pedagogical books of Cotton and McWeeney, followed by a plethora of other authors, of which we cite here only a representative sample. [2] There are, however, two aspects of the commonly accept- ed application of symmetry concepts to chemistry that are biased by the limitations of group theory. One is the dichot- omic character of symmetry: an object either has some sym- metry or it does not, with no halftones. Chemists have always been aware that there are different degrees of sym- metry, and one molecule may be closer to having a certain symmetry than another one. An implementation of such a consideration of symmetry as a quantifiable parameter has been developed in recent years by Avnir and co-workers through the definition of continuous symmetry measures [3] and symmetry operation measures. [4] The other limitation associated with the use of group theory in chemistry is the consideration of the symmetry of a molecule as a whole, thus disregarding the symmetry of its components when it is not encompassed by that of the full molecule. To the best of our knowledge, there is presently no systematic way of handling different, unrelated symme- tries of molecular assemblies. Let us try to clarify our point by way of simple geometric two-dimensional designs. Con- sider objects 1 and 2, which clearly have both threefold rota- tional symmetry, as well as three mirror planes perpendicu- lar to the paper and passing through each of the vertices of Keywords: electronic structure · density functional calculations · molecular modeling · sandwich complexes · symmetry Abstract: We analyze in this article the degree to which different groups of atoms retain local symmetries when assembled in a molecule. This study is carried out by applying continuous symmetry measures to several families of mixed sandwiches, a variety of piano-stool molecules, and several organic groups. An analysis of the local symmetry of the electron density shows that, sandwiched between two re- gions of different symmetry that correspond to the ligand sets, its symmetry is cy- lindrical at the central metal atom. [a] Dr. J. Echeverría, Prof. Dr. S. Alvarez Departament de Química Inorgànica Universitat de Barcelona, Martí i Franqus 1–11 08028 Barcelona (Spain) Fax: (+ 34)-93-490-7725 E-mail: santiago@ub.qi.es [b] Dr. J. Echeverría, A. Carreras, Dr.D. Casanova, Dr. P. Alemany, Prof. Dr. S. Alvarez Institut de Química Teòrica i Computacional (IQTCUB) Universitat de Barcelona, Martí i Franqus 1–11 08028 Barcelona (Spain) [c] A. Carreras, Dr. P. Alemany Departament de Química Física Universitat de Barcelona, Martí i Franqus 1–11 08028 Barcelona (Spain) Fax: (+ 34)-93 402 1231 E-mail: p.alemany@ub.edu Chem. Eur. J. 2011, 17, 359 – 367  2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim 359 FULL PAPER