Symmetry operators group theory

Atoms have complete spherical synnnetry, and the angidar momentum states can be considered as different synnnetry classes of that spherical symmetry. The nuclear framework of a molecule has a much lower synnnetry. Synnnetry operations for the molecule are transfonnations such as rotations about an axis, reflection in a plane, or inversion tlnough a point at the centre of the molecule, which leave the molecule in an equivalent configuration. Every molecule has one such operation, the identity operation, which just leaves the molecule alone. Many molecules have one or more additional operations. The set of operations for a molecule fonn a mathematical group, and the methods of group theory provide a way to classify electronic and vibrational states according to whatever symmetry does exist. That classification leads to selection rules for transitions between those states. A complete discussion of the methods is beyond the scope of this chapter, but we will consider a few illustrative examples. Additional details will also be found in section A 1.4 on molecular symmetry. 

We now turn to electronic selection rules for syimnetrical nonlinear molecules. The procedure here is to examme the structure of a molecule to detennine what synnnetry operations exist which will leave the molecular framework in an equivalent configuration. Then one looks at the various possible point groups to see what group would consist of those particular operations. The character table for that group will then pennit one to classify electronic states by symmetry and to work out the selection rules. Character tables for all relevant groups can be found in many books on spectroscopy or group theory. Ftere we will only pick one very sunple point group called 2 and look at some simple examples to illustrate the method. 

It is recommended that the reader become familiar with the point-group symmetry tools developed in Appendix E before proceeding with this section. In particular, it is important to know how to label atomic orbitals as well as the various hybrids that can be formed from them according to the irreducible representations of the molecule s point group and how to construct symmetry adapted combinations of atomic, hybrid, and molecular orbitals using projection operator methods. If additional material on group theory is needed. Cotton s book on this subject is very good and provides many excellent chemical applications. 

We have found three distinet irredueible representations for the C3V symmetry group two different one-dimensional and one two dimensional representations. Are there any more An important theorem of group theory shows that the number of irredueible representations of a group is equal to the number of elasses. Sinee there are three elasses of operation, we have found all the irredueible representations of the C3V point group. There are no more. 

All molecules possess the identity element of symmetry, for which the symbol is / (some authors use E, but this may cause confusion with the E symmetry species see Section 4.3.2). The symmetry operation / consists of doing nothing to the molecule, so that it may seem too trivial to be of importance but it is a necessary element required by the mles of group theory. Since the C operation is a rotation by 2n radians, Ci = I and the symbol is not used. 

In the examples presented in the previous section, the vectors % of displace meat coordinates [Eqs. (12) and (19)] were used as a basis. It should not be surprising that the matrices employed to represent the symmetry operations have different forms depending on the basis coordinates. In effect, there is an infinite number of matrices that can serve as representations of a given symmetry operation. Nevertheless, there is one quantity that is characteristic of the operation - the trace of the matrix - as it is invariant under a change of basis coordinates. In group theory it is known as the character. 

Although not listed among the symmetry elements for a structure, there is also the identity operation, E. This operation leaves the orientation of the molecule unchanged from the original. This operation is essential when considering the properties that are associated with group theory. When a Cn operation is carried out n times, it returns the structure to its original orientation. Therefore, we can write... 

The mathematical apparatus for treating combinations of symmetry operations lies in the branch of mathematics known as group theory. A mathematical group behaves according to the following set of rules. A group is a set of elements and the operations that obey these rules. 

Each set of four numbers ( 1) constitutes an irreducible representation (i.r.) of the symmetry group, on the basis of either a coordinate axis or an axial rotation. According to a well-known theorem of group theory [2.7.4(v)], the number of i.r. s is equal to the number of classes of that group. The four different i.r. s obtained above therefore cover all possibilities for C2V. The theorem thus implies that any representation of the symmetry operators of the group, on whatever basis, can be reduced to one of these four. In summary, the i.r. s of C2 are given by Table 1. 

In addition, group theory can be used to assess when transition dipole moments must be zero. The product of the irreducible representations of the two wave functions and the dipole moment operator within the molecular point group symmetry must contain the totally symmetric representation for the matrix element to be non-zero (note that, if the molecule does not contain an inversion center, the operator r does not belong to any single irrep, except for the trivial case of Ci symmetry see Appendix B for more details). A consequence of this consideration is that, for instance, electronic transitions between states of the same symmetry are forbidden in molecules possessing inversion centers. 

As already evident from the previous section, symmetry properties of a molecule are of utmost importance in understanding its chemical and physical behaviour in general, and spectroscopy and photochemistry in particular. The selection rules which govern the transition between the energy states of atoms and molecules can be established from considerations of the behaviour of atoms or molecules under certain symmetry operations. For each type of symmetry, there is a group of operations and, therefore, they can be treated by group theory, a branch of mathematics. 

In the molecular orbital theory and electronic spectroscopy we are interested in the electronic wave functions of the molecules. Since each of the symmetry operations of the point group carries the molecule into a physically equivalent configuration, any physically observable property of the molecule must remain unchanged by the symmetry operation. Energy of the molecule is one such property and the Hamiltonian must be unchanged by any symmetry operation of the point group. This is only possible if the symmetry operator has values 1. Hence, the only possible wave functions of the molecules are those which are either symmetric or antisymmetric towards the symmetry operations of the... 

Since each MO belongs to some irreducible representation of the molecular point group, we must find linear combinations of the AOs that transform according to the irreducible representations group theory enables us to do this. A symmetry operation sends each nucleus either into itself or into an atom of the same type a symmetry operation will thus transform each AO into some linear combination of the AOs (9.63). Therefore (Section 9.6), the AOs form a basis for some representation TAO of the point group of the molecule. This representation (as any reducible representation) will be the direct sum of certain of the irreducible representations r r2,...,r (not necessarily all different) of the molecular point group ... 

A group is a collection of elements that are interrelated according to certain rules. We need not specify what the elements are or attribute any physical significance to them in order to discuss the group which they constitute. In this book, of course, we shall be concerned with the groups formed by the sets of symmetry operations that may be carried out on molecules or crystals, but the basic definitions and theorems of group theory are far more general. 

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