Rotational symmetry point groups

The point groups Td,Oh, and / are the respective symmetry group of Tetrahedron, Cube, and Icosahedron the point groups T, O, and I are their respective normal subgroup of rotations. The point group 7 is generated by T and the central symmetry inversion of the centre of the Isobarycenter of the Tetrahedron. 

The wavefunctions V /, may be classified according to their symmetry properties. If we take the symmetry point group of this system to be C3V, there are three symmetry species in this group A (symmetric with respect to all operations of this group), A2 (symmetric with respect to the threefold rotations but antisymmetric with respect to the vertical symmetry planes), and E (a two-dimensional representation). 

S,S)-Tartaric acid has a single symmetry element, a C2 axis, and therefore belongs to the symmetry point group C2. On rotation through 180° the pairs of carbon centres 1 and 4, and 2 and 3 are transformed into each other. This is also the case if the molecule adopts another conformation as illustrated by the two examples shown below. 

Some decades ago, Swalen and Costain(1959), Myers and Wilson (1960) have extended the use of the symmetry point groups for studying the double internal rotation in acetone [1-2]. Dreizler generalized their considerations to two Csv rotor molecules with frames of a lower symmetry than C2U, and deduced their character tables [3]. 

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. 

Notice that, since the external rotation is relatively slower than the internal motions, the external rotation symmetry group may be expected to be isomorphic with the symmetry point group of the molecule in its most symmetric configuration [4]. As a result, the local full NRG, defined by operator (19), may be expected to be isomorphic to the direct product of the restricted NRG by the symmetry point group of the molecule ... 

As to be expected, the restricted NRG is seen to be isomorphic to the full NRG, and the external symmetry rotation group is found to be isomorphic to the symmetry point group of the molecule, C, in its most synunetric configuration. Similar expressions may be deduced for benzaldehyde, pyrocatechin and acetone. 

The local full NRG appears then as a direct product of two subgroups the restricted NRG corresponding to the internal motions, and a single switch subgroup, Uf, which corresponds to the external rotation. The restricted NRG is found to be smaller than the full NRG and the external rotation one isomor-pic to the symmetry point group of the molecule, Cs, in its most symmetric configuration. As a result, it may be written as in the case of phenol ... 

To say that the x,y, and z axes are equivalent in the Oh point group (which denotes octahedral synunetry see Symmetry Point Groups) is to assert that there exist symmetry operations that interchange these axes. Thus, if we rotate the octahedron in Figme 1 and 2 by 120° about the axis that joins the centroids of the triangular faces defined by ligands 125 and 346, we are performing the symmetry operation C3 which transforms x - z, z - y and y - x. Thus, the function (x — y ) becomes (z — x ) and (2z — x — y )... 

Linear objects are the only objects having infinite rather than finite symmetry point groups since the linear axis corresponds to an infinite order rotation axis, namely C. If there is a reflection plane perpendicular to the infinite order rotation axis (dividing the object into two equivalent halves), then the symmetry point group is Doo, if not, the symmetry point group is Coo-... 

Does the object has an even-order improper rotation axis S2 but no planes of symmetry or any proper rotation axis other than one collinear with the improper rotation axis The presence of an improper rotation axis of even order S2 without any noncollinear proper rotation axes or any reflection planes indicates the symmetry point group S2 with 2n operations. 

The high-symmetry point groups Ik, Of, and 7 are well known in chemistry and are represented by such classic molecules as C6o Sp6, and CH4. For each of these point groups, there is also a purely rotational subgroup (/, O, and T, respectively) in which the only symmetry operations other than the identity operation are proper axes of rotation. The symmetry operations for these point groups are in Table 4-5. 

The vibronic functions ne) belong to the representation A], A2, and E of the symmetry point group. Therefore in the group Td the matrix elements of the operator of the dipole moment (belonging to the T2 representation) calculated by the functions in (36) are identically equal zero. Hence the dipole transitions between the states (36) are forbidden as pure rotational or vibronic-rotational ones. 

The simplest way to treat it qualitatively is to consider the way the hydrogenic wave functions transform when reducing the symmetry from f +(3) (the 3D rotation group) to the T symmetry point group of the acceptor in a cubic crystal. 

As already noted, the crystal field potential should be invariant under all the transformations of the symmetry point group. This means that it is transformed according to the totally symmetric representation (A ) of the group G. The spherical tensor component Tk m appears within the crystal field potential when the decomposition of the irreducible representations Dof the rotation group R3 in terms of irreducible representations rt of the point group G contains the totally symmetric representation A ... 

---