Molecular Symmetry Groups

We collect syimnetry operations into various syimnetry groups , and this chapter is about the definition and use of such syimnetry operations and symmetry groups. Symmetry groups are used to label molecular states and this labelling makes the states, and their possible interactions, much easier to understand. One important syimnetry group that we describe is called the molecular symmetry group and the syimnetry operations it contains are pemuitations of identical nuclei with and without the inversion of the molecule at its centre of mass. One fascinating outcome is that indeed for... 

J. S. Griffith, The Irreducible Tensor Method for Molecular Symmetry Groups, Prentice-Hall, Englewood Cliffs, NJ, 1962, p. 20. 

Figure 16, Relation between symmetry in (a) the single space and (b) the double space of a system whose molecular symmetry group has a a plane in the single space, and is isomorphic with C2v in the double space.

The second major hurdle to be minimized is the evaluation of the PES. Obviously from Eq. (52) there are two tasks, calculating the weights and the Taylor series. The total number of data points in the PES may be very large because of symmetry. For example, a PES for CH5 with 2,500 symmetry-distinct data points has 2,500 x 5 = 300,000 data points in total. In some circumstances, the size of the symmetry group could be reduced without loss of accuracy by employing only the subset of feasible permutations, the molecular symmetry group,68 rather than the CNP group. 

Figure 4. Schematic of torsional energy levels in Si electronic state of phenylsilane and Do electronic state of phenylsilane+. The torsional state symmetry labels arise under the molecular symmetry group G12.

We therefore conclude that, for a combination of model, numerical and conceptual reasons the OHAO basis is well-adapted to a theory of valence. The hybrid orbital basis (for simple molecules) has a distinctive symmetry property it carries a permutation representation of the molecular symmetry group the equivalent orbitals are always sent into each other, never into linear combinations of each other. This simple fact enables the hybrid orbital basis to be studied in a way which is physically more transparent than the conventional AO basis. 

If there is a molecular symmetry group whose elements leave the hamiltonian 36 invariant, then the closed-shell wavefunction belongs to the totally symmetric representation of both the spin and symmetry groups.8 It is further true that under these symmetry operations the molecular orbitals transform among each other by means of an orthogonal transformation, such as mentioned in Eq. (5) 9) and, therefore, span a representation of the molecular symmetry group. In general, this representation is reducible. 

Finally, it must be mentioned that localized orbitals are not always simply related to symmetry. There are cases where the localized orbitals form neither a set of symmetry adapted orbitals, belonging to irreducible representations, nor a set of equivalent orbitals, permuting under symmetry operations, but a set of orbitals with little or no apparent relationship to the molecular symmetry group. This can occur, for example, when the symmetry is such that sev-... 

For the XY3 molecules considered here, we employ the Molecular Symmetry Group (MS group) D h(M) [3], given as the direct product [3],... 

These examples suggest the correct result The possible symmetry types, either for normal modes or electronic wave functions, that are compatible with an overall molecular symmetry, correspond to the full molecular symmetry group or its subgroups. Each normal mode, or electronic state, can be classified... 

Other than linear molecules. If molecules of symmetry other than axial are considered, it is not possible to describe their orientation by an azimuthal and polar angle, Euler angles, Q = a pi, y and Wigner rotation matrices are then needed as Eq. 4.8 suggests. In that case, besides the set of parameters X, X2, A, L that has been used for linear molecules, two new parameters, u, with i = 1,2, occur that enter through the rotation matrices. These must be chosen so that the dipole moment is invariant under any rotation belonging to the molecular symmetry group. The rotation matrix is expressed as a linear combination of such... 

The dipole moment p must be invariant under any rotation of the molecular symmetry group applied to any one of the two molecules [166, 374]. When one of these rotations is applied, the rotation matrix D V(Q) is transformed into a linear combination of the D v, Cl) matrices with different o. The proper linear combinations of the D%V(C1) are invariant under the rotational symmetry group. Such linear combinations are obtained from group-theoretical arguments. For example, for the case of methane pairs in the ground vibrational state, for Ai = 3, 4 and 6, we have the combinations... 

The original JT analysis examined the modification of central-field energy eigenvalues as a function of the representations of the important molecular symmetry groups. The results agree [65] with the heuristic analysis of o-a-m conservation. 

It is often convenient to use the symmetry coordinates that form the irreducible basis of the molecular symmetry group. This is because the potential-energy surface, being a consequence of the Born-Oppenheimer approximation and as such independent of the atomic masses, must be invariant with respect to the interchange of equivalent atoms inside the molecule. For example, the application of the projection operators for the irreducible representations of the symmetry point group D3h (whose subgroup... 

This is one of the reasons for the power and generality of group theoretical methods in discussing the properties of molecules for although the number of different imaginable molecules is unbounded, this is not true of their possible systems of axes and planes of symmetry. These are severely restricted by geometrical considerations and it is possible to write down a list of all the molecular symmetry groups that can exist and to discuss... 

Before going on to discuss the molecular symmetry groups in more detail we note one feature that they all possess. A symmetry operation which rotates or reflects a molecule into itself must leave the centre of mass (centre of gravity) of the molecule unmoved if the molecule has a plane or axis of symmetry, the centre of mass must lie on this plane or axis. It follows that all the axes and planes of symmetry of a molecule must intersect in at least one common point and that at least one point remains fixed under all the symmetry operations of the molecule. For this reason, the symmetry group of molecule is generally referred to as its point group and we shall use this name, which is taken over from crystallography, from now on. 

---