Pure rotation subgroup

Thus the group Td has a pure rotational subgroup, T, of order 12. It consists of the following classes ... 

A3.1 What are the highest-order pure rotational subgroups of Cm, Djj, C5,. ... 

To solve this problem, we first recognize that we need not employ all of the 48 operations of Oh instead, we can deal with the 7, representation of the pure rotational subgroup O, which has only one half as many operations. Let us label axes and basis functions as shown in the sketch below ... 

The second expression is simply a list of the six /s, in numerical order, each multiplied by the character for one of the six operations, in the conventional order , CA, Cl,..., C. This must be true for each and every representation. Hence, the sets of characters of the group are the coefficients of the LCAO-MOs. The argument is obviously a general one and applies to all cyclic (CH) systems belonging to the point groups Dnh, each of which has a uniaxial pure rotation subgroup, C . 

Although the full symmetry of the octahedron is O, we can gain all required information about the d orbitals by using only the pure rotational subgroup O because Oh may be obtained from O by adding the inversion, i. However, we already know that d orbitals are even to inversion, so that it is only the pure rotational operations of O which bring us new information. 

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. 

For simplicity and brevity, we consider the pure rotational subgroup T of the tetrahedral point group T4. The extension of the analysis to Td is straightforward. We want to And the complete set of symmetry operators Z for which... 

The group R(3) comprises the infinite number of possible proper rotations about a point in three-dimensional space. It is the pure rotation subgroup of a sphere. The character x/ ) of the irreducible representation under pure rotation through an angle is... 

Looking along the row of the multiplication table for the inversion centre, it is also clear that all the other members of the full point group occur as a product of one of the simple rotational operations and the inversion centre. In the Dsd character table there are three different possibilities for the characters under the pure rotation subgroup labels Ai, A2 and E. For each of these there are then two possibilities for the behaviour of an object under the inversion operation ... 

In groups without an inversion centre but containing a horizontal mirror plane, a similar argument can be put forward based on the combination of the pure rotational subgroup with the CTi, plane. 

If the crystal and sample symmetries are higher than triclinic, the multiple physically equivalent choices of the coordinate systems (x ) and (y ) impose constraints on the ODF. The symmetry group of the ODF must be the product of the proper subgroups (only pure rotations) of the crystal and sample point groups. 

For example, for the group are identified two subgroups the one formed only from the unit operation E and the subgroups of pure rotations ofC... 

Moreover, thanks to the symmetry operations cyclicality from the pure subgroups of the rotations Cy that will be called cyclic group. Any cyclic group is abelian, natural property derived from the of the group objects nature only rotations, by their nature commutative. 

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