© Division of Chemical Education •  www.JCE.DivCHED.org •  Vol. 86 No. 2 February 2009  •  Journal of Chemical Education 251 Research: Science and Education Group theory has been extensively used to study various chemical problems, such as the determination of Raman- and infrared-active vibrations or the construction of symmetry adapted linear combinations (SALC) in the linear combination of atomic orbitals (LCAO) theory. For a given molecular point group, the irst step is to construct a reducible representation that will be subsequently reduced to a direct sum of the irreduc- ible representations belonging to the relevant group. In a number of favorable cases, a reducible representation can be reduced quickly just by searching for each irreducible representation in a row of the character table of the molecular point group for the correct combination matching the character of the reduc- ible representation. However, for more complicated cases, the following equation is used (1, 2) 1 a h i i  ( ) * ( ) χ χ R R R ･ (1) where a i is the number of times the i th irreducible representation appears in the reducible representation where h is the order of the group, χ(R) is the character of the matrix corresponding to operation R in the reducible representation, and χ i *(R) is the character of the matrix corresponding to operation R in the i th irreducible rpresentation. For groups having many similar operations in distinct class- es, the use of this equation is oten tedious. Hence, computer programs have been developed to make this laborious task easier (3, 4). In this article, we propose a simple and straightforward procedure for decomposing a given reducible representation directly using a spreadsheet template. It is based on the inversion of the central part of the character table, which is considered as a matrix. To illustrate this topic, two basic examples are developed: (i) the SALC of the Γ σ representation for the O h symmetry and (ii) molecular orbitals (MOs) obtained from symmetry orbitals (SOs) for the benzene molecule. For a given point group, the character table has the general form presented in Table 1 and a reducible representation can be written as:  ⊕ ⊕ ⊕ Γ Γ Γ Γ Γ red a b c z z 1 2 3 | | (2) where a, b, c, … , z indicate the number of times the ith irre- ducible representation occurs in the reducible representation Γ red . For a given class C n , each character χ n of the reducible representation Γ red can be expressed as a linear combination of the characters of the appropriate irreducible representations belonging to the point group under study:    |  a b c z i 1 11 21 31 1 χ χ χ χ χ (3)   a b 2 12 χ χ χ χ χ 22 32 2  |  c z i (4) Ȩ χ χ χ χ χ 1 2 3    |  a b c z n n n n in (5) his set of linear combinations can be summarized using the following matrix notation: χ χ χ 1 2  ab n | | z n n n χ χ χ χ χ χ χ χ χ χ χ χ 11 12 13 1 21 22 23 2 31 32 33 3 | | | Ȩ | || | | χ χ χ χ i i i in 1 2 3 (6) Note that the matrix corresponds to the central part of Table 1 and will be denoted T. Hence, the a, b, c, … , z coeicients are easily obtained using the inverse matrix T ‒1 : | ab z | χ χ χ n 1 2  T 1 (7) If one is able to determine T ‒1 , the reduction of Γ red is obtained directly without needing to use eq 1. At irst sight, the inversion of T appears to be complicated but it can be performed easily by means of the matrix inversion function (called Miniverse in Ex- cel) and also matrix product function of a spreadsheet sotware (e.g., Excel or OpenOice). Note that spreadsheets have already been used in group theory calculations, for example, to compute the number of IR- and Raman-active bands (5). he irst example deals with a [ML 6 ] transition metal in octahedral symmetry. It is of general importance in inorganic chemistry to determine the SALC of atomic orbitals that can be mixed to give rise to the σ molecular orbitals. he irst task is to obtain the reducible representation Γ σ . his is easily performed by considering that each combination is represented by a vector Towards “Inverse” Character Tables? A One-Step Method for Decomposing Reducible Representations J.-Y. Piquemal,* R. Losno, and B. Ancian Department of Chemistry, Université Paris 7, Bâtiment Lavoisier, 15 rue Jean de Baïf, 75205 Paris Cedex 13, France; *jean-yves.piquemal@univ-paris-diderot.fr Table 1. General Form of a Character Table Group C 1 C 2 … C n Γ 1 χ 11 χ 12 … χ 1n Γ 2 χ 21 χ 22 … χ 2n … … … … … Γ i χ i1 χ i1 … χ in Γ red χ 1 χ 2 … χ n NOTE: C n are the classes; Γ i are the irreducible representations for a given molecular point group; Γ red are the reducible representations; and χ in are the character of the matrix corresponding to an operation of the i th irreducible representation for the nth class C n . Advanced Chemistry Classroom and Laboratory edited by Joseph J. BelBruno Dartmouth College Hanover, NH 03755