Permutation representation

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. 

A particular representation which we shall need is the regular representation of dimension g, which symbolically uses the group elements themselves as basis vectors of a permutation representation . 

Table 17.10. Basis functions (f. .. g, and their variables, for the IRs ofthe permutation representation of [1 1 1],...

Tm then follows from the character table for Oh and eq. (4.4.20). (The easiest way of determining (r, ) is to look at the figure representing the set of points to be permuted (in this case, a cube) and determine the number of points that are unshifted under one operation from each class. This number is the character for that class in the permutation representation Tm. For example, each of the dihedral planes contains four points which are invariant under reflections in that plane.)... 

Similarly, in mechanics, the extended Maxwell condition for rigidity of bar and joint assemblies can be used to generate a relation between the permutation representations of the bars and joints and those of the states of self-stress and mechanisms of the assembled framework [17a,b]. 

In the present case, the extension of the scalar counting rules for n(S) to symmetry theorems is straightforwardly achieved by replacing n(S), n(v), n(e) n( > ) by the permutation representations F,T(S), r,T(v), r,T(e) and ra(vj) [13]. A permutation representation /j/v), of a set of objects x has character y(R) under operation (R) of the symmetry group of the undistorted framework, where x(R) is equal to the number of objects wnshifted under operation R. The subscript a is often dropped if there is no danger of confusion. With these replacements, equation (4) becomes... 

The number of totally symmetric Heilbronner modes follows from counting of orbits. An orbit is a set of equivalent (structureless) objects, which are permuted amongst themselves by symmetry operations of the group every operation of the group either leaves a given member of the orbit in place and unchanged, or moves it to another location. Thus the six edges of benzene, the two face centres of pentalene and the pair of terminal vertices of an [w]-polyene chain, all form orbits. Each orbit has an associated permutation representation that contains the totally symmetric representation Jo exactly once. 

Consideration of the description of leapfrogging in terms of edge crossings suggests formulas for the permutational representations of the leapfrog polyhedron Lf in terms of structural representations of the parent, P. From... 

With the GT Calculator you can perform a variety of standard group theory calculations simply by entering the appropriate structure details for the molecular geometry. In addition, on the various worksheets of the calculator files, it is straightforward to determine more advanced group theoretical results, such as the numbers of isomers generated for a given structure by decoration, or to calculate and decompose the symmetric and antisymmetric powers of permutation representations. 

For any permutation representation the character is list of the traces of the permutation matrices with each trace equal to the number of labels unshifted under the corresponding symmetry operation of the groups. Thus, for the matrices in Table 2.1, the traces are 4, 1, 1, 2, 2, and 2 respectively, indicating that every operation leaves the nitrogen label unshifted... 

Figure 2.1 The actions of symmetry operations of the point group, C3V, in the structure of the ammonia molecule giving rise to the permutation representation based on the matrices in the second column of the figure. The principal rotational axis, C3, is normal to the plane of the paper.

Suppose that G is the group of symmetry operations of a polyhedron or polygon, with vertices corresponding to the atomic positions in a particular molecular structure. The division of the structure into orbits, as sets of vertices equivalent under the actions of the group symmetry operations and the calculation of associated permutation representations/characters were described in Chapter 2. In this chapter, the identity between the permutation representa-tion/character on the labels of the vertices of an orbit and the a representation/character on sets of local s-orbitals or a-oriented local functions is exploited to constmct the characters of the representations that follow from the transformation properties of higher order local functions. 

When the two substituents are distinguishable, the full square replaces its symmetric part in the permutation representation ... 

The difference between T(XY) and T(X2) is just the antisymmetric part of the square of Ter, with dimension XiaiCai - l)/2. In both cases, the number of isomers is immediately available once the reduction of the permutation representation of the vertices is known. 

To count the isomers with 2 substituents of one kind and 1 of another, consider first all possible placings of X2 and then add Y anywhere to each of them, excluding the case where Y falls on an X site. The permutation representation F(X2Y) follows as... 

For three distinct substituents X, Y and Z, consider all possible placings of XY and add Z anywhere except on an X site (where it would produce a heteropair XY) or a Y site (where it would produce a different heteropair XZ). The permutation representation F(XYZ) is... 

Theorem 1. A doubly transitive permutation representation of a group G over the complex field is the sum of the identical representation and an absolutely irreducible representation [13],... 

The (reducible) permutation representation (T ) is such that the character ( (R)) for each operation (R) of the point group is equal to the number of cluster vertices which remain fixed under that operation160. ... 

This permutational representation is also called the ground representation. It describes the transformation of the coset space. The dimension of this coset space is G / //. In the case of a cluster, where each coset corresponds to a site, it represents the permutation of the positions of the sites. For this reason, it is also called the positional representation. Indeed, Eq. (4.72) may equally well be written as... 

Thus if one chooses a set of AOs which are identical but on different centers then they carry a permutation representation of the operation, much like (O Eq. 2.94) but with appropriate transformation matrices in place of the Os and Is there. So if, for example, one had a set of identical s-orbitals with one on each center, the transformation matrices under any point group operation would just be 1, so that the matrix would remain just as in O Eq. 2.94. But if one had a set of three equivalent p-functions on each center the Os would become 3x3 null matrices and the Is would become those three-dimensional transformation matrices which represent the operation on a set of three equivalent p-functions at the origin and so on. 

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