The Molecular Symmetry Group

So far, the symmetry of the Hamiltonian was defined as the set of all operations that leave the Hamiltonian invariant. This invariance group was assumed to coincide with the point group of the nuclear frame of the molecule, but it is now time to provide a clear explanation of this connection. This section relies on the definition of the 

The T operators are the kinetic energy operators for electrons and nuclei  

The Hamiltonian may contain additional terms describing coupling between orbital and spin momenta. Longuet-Higgins stated that this full Hamiltonian must be invariant under the following types of transformations  

Any rotation of the positions and spins of all particles (electrons and nuclei) about any axis through the center of mass. 

The simultaneous inversion of the positions of all particles in the center of mass. 

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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... 

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],... 

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... 

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. 

Stone applied the theory of Longuet-Higgins to deduce the character tables for the multiple internal rotation in neopentane and in octahedral hexa-ammonium metallic complexes [6]. Dalton examined the use of the permutation-inversion groups for determining statistical weights and selection rules for radiative processes in non-rigid systems [7]. Many applications of the Molecular Symmetry Groups have been reviewed later by Bunker [8,9]. 

The abstract character of the Molecular Symmetry Group of Longuet- Higgins does not make easy the physical interpretation of the interconversion... 

It was sometimes believed, in the scientific literature, that the Molecular Symmetry Group of Longuet-Higgins and the Schrodinger Supergroup of Altmann were isomorphic, i.e., both theories were equivalent [19]. We shall see, however, that is not generally true, especially when some symmetry is retained in the molecule. 

Let P be any permutation of positions and spins of identical nuclei, or any product of sudi permutations. Let E be the identity, E the inversion of all particle positions with respect to the center of mass, and P" the product PE = E P. Then the Molecular Symmetry Group is the set of ... 

In order to illustrate the Molecular symmetry Group Theory, let us consider the methyl boron difluoride molecule CH3 — BF2), which contains a nearly free rotating methyl group. We shall see next that the torsional levels of this molecule can be classified according to the irreducible representations of the symmetry point group Csv, although this molecule does not possess, in a random configuration, any symmetry at all. 

These twelve operations form a group, called Gu. Its multiplication table, as well as, its character table are easily deduced. We present this in Table 1. The Molecular Symmetry Group of CH3 — BF2 is then seen to be isomorphic to the symmetry point group Ce . 

Table 1 Character Table for the Molecular Symmetry Group G12 corresponding to the BF2 - CH3 molecule.

Inspection of Fig. 2, shows that the equivalence between the Schrodinger Supergroup and the Molecular Symmetry Group operations, for the CH3—BF2 molecule, is easily verified. In Table 2, the two groups appear to be identical. 

I ble 2 Equivalence between the Molecular Symmetry Group and Schrodinger Supergroup operations in the CH3 — BFj molecule. 

Let us now consider Longuet-Higgins point of view. Following the example of CH3 — BH2, the Molecular Symmetry Group is such a system and is easily written down. It will contain all the feasible operations ... 

This result was foreseen earlier by using the Longuet-Higgins formalism. The Molecular Symmetry Group for PFs may be indeed written as the permutation group of five identical nuclei multiplied by the inversion subgroup [20] ... 

The concepts of full and restricted NRG s are defined. The full NRG s, which consider the overall rotations, are seen to be entirely equivalent to the Molecular Symmetry Groups of Longuet-Higgins [5], whereas the restricted NRG s, limited to the interconversion motions, may be compared with the Isodynamic groups of Altmann, and the isometric groups of Gunthard [10,27]. 

Bunker, P. R. Practically everything you ought to know about the molecular symmetry group. New York M. Dekker 1976. 

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