Characters, character tables and symmetry species

So far, we have seen that individual molecules can possess different symmetry elements, and that if we collect this information together we can assign any given molecule to a particular point group. To ease this process it is common to use a decision tree, such as the one given in the inside back cover. 

We will need a systematic way to deduce whether a molecular property is symmetric or antisymmetric with respect to the symmetry operations for that molecule s point group, as things will quickly get more complicated. To this end, we define a number, Xp( ) called a character, which expresses the behavior of our property p when operated on by the symmetry operation, R. A collection of characters, one for each symmetry operation present in a point group, forms a representation, Fp. The property is technically referred to as the basis vector of the representation, Fp. We can define aU sorts of basis vectors, some of which have very little apparent connection to our original molecule, such as the non-symmetry operation translate along the z axis , often given the symbol z, or the non-symmetry operation rotate by an arbitrary amount about the X axis, often referred to as R. Strictly speaking, the characters are the trace of the transformation matrix for each symmetry operation, applied to the property, p. This is described in more detail in the on-line supplementary section for Chapter 2 on derivation of characters. 

Other possible basis vectors include true spatial vectors such as a dipole-moment component (e.g. or a displacement of an atom along a direction defined by a bond (i.e. a bond stretching motion). More elaborate basis vectors are also useful one very important one consists of the set of three Cartesian-axis displacements of each of the N atoms in a molecule, and has a dimensionality of 3N. We will see this basis vector in action in Section 8.5.3, where we use symmetry to characterize a set of molecular fundamental vibrational modes. We can also use the wavefunction of a molecular orbital as a basis vector, and so classify that orbital according to the effects of the various symmetry operations acting on that wave-function (Section 9.3). Even a tensor such as the molecular polarizability can be used as a basis vector. Throughout this book we use a number of subscript labels for the different basis vectors we discuss. The convention we use is as follows. 

Basis vector property, p Reducible representation subscript, Tp 

Collecting all the symmetry operations for a particular point group together generates its corresponding character table, which is an extremely useful tool that allows us to classify molecular properties according to their symmetry types. For example, the character table for the point group Cav is shown in Table 2.2. 

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