[...]... make up a point group form such a group e.g for C the group members are Ε {= C/), Q (i.e C / ) and two vertical planes a(xy), oiyz) Groups can be formed in many different ways, of which a collection of symmetry operations is only one There are four mathematical rules which any group must obey, and are: 2v • the group must be closed i.e any combination of two or more members of the group must be equivalent... descriptors of individual symmetry operations - will be introduced, as will numerical representations for the point group symmetry operations These will allow group theory to be applied in a more quantitative way to the analysis of vibrational spectra and molecular orbital theory {Parts II and III) 2.1 GROUPS A "group" can be thought of as an exclusive club to which only a number of members belong These members... irreducible representations for the C point group, this is not the case for all such groups In the case of C the translation / rotation vectors either transform onto themselves or their reverse under each of the symmetry operations; in general this is not true Consider the case of NH ( C ) : 2 v 2 v 2v 3 3Y V Fig 2.1 Vectors representing R„ T^T, and T for the C point group For clarity, the translational... along, and rotation about, ζ (T , R ) generate irreducible representations, as follows: z Ε 1 1 ci 1 1 σ (1) 1 -1 J Cs 1 1 ν σ/2) 1 -1 z 1 -1 τ, However, the vectors for T, and T (Fig 2.1 focuses solely on the vectors on nitrogen for simplicity) move to completely new positions under, for example C ' In y 3 Groups and Representations Ch.2] 23 this case, T and T need to be treated as a pair and the... 0.866 etc) Similar transformation matrices can be derived for the (T„T ) pair of vectors for all the other operations of the C point group, as can matrices for the pair of rotations (R„ R ) The complete table of irreducible representations for the C point group is: y 3v y 3v 24 Groups and Representations [Ch 2 Table 2.2 Representations of the point group C j y Ε Cs' 1 1 1 1 Γ ι ol [ο lj Η σ/1) S A] \-λ... elements of the O point group h 1.2.2 Assigning Point Groups From the classification of point groups given above, it should be apparent that (0 not all symmetry elements need to be located in order to assign a point group and (it) some symmetry elements take precedence over others For example, [PtCL,] " has C (and C? co-incident), four further C axes perpendicular to C , σ*, two σ„ two σ S and i although only... members of the group must be equivalent to another member of the group For example: ζ oiyz) Η σ(χζ) χ a(yz) = C 2 Ch.2] Groups and Representations 17 In this context the multiplication sign (x) should be read as "followed by" SAQ 2.1 : Complete the following table and hence show that all operations of the C& point group form a closed group c Ε oixz) 2 Ε C oixz) a(yz) oiyz) 2 C 2 Entries in the Table... = σ(χζ) 2 Therefore: σ(χζ) χ [oiyz) χ C ] = σ(χζ) χ σ(χζ) = Ε 2 2 2 Groups and Representations 18 [Ch.2 2.2 TRANSFORMATION MATRICES The mathematical tool for describing the effect of a symmetry operation is a transformation matrix Before explaining how these matrices work, it is necessary to give a brief explanation of matrix multiplication A matrix is a grid of numbers with χ rows and y columns, in... the main axis) or a (containing the main axis and bisecting bond angles), though σ and a can be grouped together rotations and improper axes can generate several operations (C„ , 5„") while only one operation is associated with either /' or σ a point group is a collection of symmetry elements (operations) and is identified by a point group symbol a point group can be derived without identifying every... hydrogen atoms; all faces of the cube are equivalent) 4 2* Identify the point groups of each of the following pairs of molecules: C0 and S0 ferrocene (staggered) and ruthenocene (eclipsed) cis- and /ra«y-Mo(CO) Cl [IF ] and[ IF ]SnCl(F)andXeCl(F) mer- and yoc-WCl F 2 2 4 2 + 6 6 3 3 3 Link the following species with the correct point group: [AuCL,]C [C1F ] 2 T 2v d + BrF 5 D S0 3 [C1F ]2 C 3h 3h Ni(CO) C . undergraduate students. This book Group Theory for Chemists is largely based on that award-winning lecture course. Group Theory for Chemists Fundamental Theory and Applications Second Edition. Symmetry and Group Theory in Chemistry, Horwood Chemical Science Series, 1998. R. L. Carter, Molecular Symmetry and Group Theory, Wiley and Sons, 1998. G Davidson, Group Theory for Chemists, . Molecular Symmetry and Group Theory: A Programmed Introduction to Chemical Applications, 2 nd Edition, John Wiley and Sons, 2000. Ρ Η Walton, Beginning Group Theory for Chemistry, OUP,