Rotational subgroups

Here /, are the three moments of inertia. The symmetry index a is the order of the rotational subgroup in the molecular point group (i.e. the number of proper symmetry operations), for H2O it is 2, for NH3 it is 3, for benzene it is 12 etc. The rotational partition function requires only information about the atomic masses and positions (eq. (12.14)), i.e. the molecular geometry. 

For readers unfamiliar with these techniques, it might be helpful at this point to work out an example in some detail. We choose that of the allene skeleton, already discussed somewhat in this section, and at first we limit ourselves to achiral ligands, so that G = S4. The character table for S4 is shown in Table 1. In this case, the subgroup is just D2a, and its rotational subgroup is D2. Table 2 shows the classes of T>za, the number of elements in each, the class of S4 and of S4 to which each belongs, and the character of each for the representation, T< >. 

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.)... 

For nonlinear molecules, the symmetry number is simply the order of the rotational subgroup (these terms will be defined in Chapter 9). 

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 ... 

As pointed out in Section 6.3, for the (CH)3 case, all the essential symmetry properties of the LCAOs we seek are determined by the operations of the uniaxial rotational subgroup, Cft. When the set of six pn orbitals is used as the basis for a representation of the group C6, the following results are obtained ... 

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. 

A2u symmetry orbital this corresponds to the totally symmetric representation in the rotational subgroup Cy, so, even without using the projection operator, its form can be given by ... 

K Lagrange multiplier K Transmission coefficient K Compressibility constant fcg Boltzmann constant k, k Force constant (for atoms A, B,...) K Anharmonic constants (third derivative) K F.xchange operator r, 9, (f) Polar coordinates (T Order of rotational subgroup (T Charge density Pauli 2x2 spin matrices s Electron spin operator S Entropy... 

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 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. 

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 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 4.5 Symmetry Operations for High-Symmetry Point Groups and Their Rotational Subgroups... 

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