Rotational Subgroup of

Some of the information pertaining to a group is stored in property lists. Table I exemplifies how this looks for the simple case of the cyclic group of order three. (This would be isomorphic to the rotational subgroup of a molecule such as methyl fluoride. The operators (1 2 3) and (1 3 2) would correspond to the permutations of the three hydrogen nucleii numbered 1, 2 and 3. NIL, the language s symbol for the empty list, serves as the identity.)... 

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

The main effect is already taken into account if symmetry numbers are included in the densities of states. The symmetry number is a correction to the density of states that allows for the fact that indistinguishable atoms occupy symmetry-related positions and these atoms have to obey the constraints of the Pauli principle (i.e. the wave function must have a definite symmetry with respect to any permutation), whereas the classical density of states contains no such constraint. The density of states is reduced by a factor that is equal to the dimension of the rotational subgroup of the molecule. When a molecule is distorted, its symmetry is reduced, and so its symmetry number changes by a proportion that is equivalent to the number of indistinguishable ways in which the distortion may be produced. For example, the rotational subgroup of the methane molecule is T, whose dimension is 12, whereas the rotational subgroup of a distorted molecule in which one bond is stretched is C3, whose dimension is 3. The ratio of these symmetry numbers, 4, is the number of ways in which the distortion can occur, i.e. the reaction path degeneracy. 

The use of symmetry—at least the translational subgroup—is essential to modem first-principles calculations on crystalline solids. Group theory is simplest for Abelian groups such as the translational subgroup of a crystal or the six-fold-rotational subgroup of the benzene molecule. For such simple cyclic groups, the irreducible representations are characterized by a phase, exp(ifc), associated with each step in a direction of periodicity. For one-dimensional (or cyclic) periodicity,... 

E17.4(b) The symmetry number is the order of the rotational subgroup of the group to which a molecule belongs (except for linear molecules, for which a = 2 if the molecule has inversion symmetry and 1 otherwise). 

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... 

Table 7.6 The projection operator method for finding the SALCs for H(1s) orbitals in BH, using the Dj rotational subgroup of Djh. The assignment of the horizontal Cj axes follows the convention that 2, C2 and are through B—H, B—H2 and B—Hj respectively, as shown in Figure 7.13.

For the big representation we can use the projection operator method along with the >4 rotational subgroup of Ah as laid out in Table 7.15. The rotational subgroup does not contain the inversion centre, and so the gerade (g) and ungerade (u) labels do not appear. 

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