Representation of symmetry operations

Is it possible that consideration of other examples might expose a case in which (C2 followed by av) does not give the same result as additional tests, it will prove more effective to treat the geometric manipulations as independent entities, without reference to illustrative objects on which they operate. The development of numerical representations of symmetry operations will then provide straightforward arithmetic tests for equality. 

Because of the restrictions imposed on the values of the rotation angles (see Table 1.4), sincp and cos(p in Cartesian basis are 0, 1 or -1 for one, two and four-fold rotations, and they are 1/2 or Vs/2 for three and six-fold rotations. However, when the same rotational transformations are considered in the appropriate crystallographic coordinate system, all matrix elements become equal to 0, -1 or 1. This simplicity (and undeniably, beauty) of the matrix representation of symmetry operations is the result of restrictions imposed by the three-dimensional periodicity of crystal lattice. The presence of rotational symmetry of any other order (e.g. five-fold rotation) will result in the non-integer values of the elements of corresponding matrices in three dimensions. 

In this particular case, the point group can be described as a set of permutations of the labels 0,1,2,3 of the comers of the tetrahedron or as symmetry operations of this polyhedron. However, we should carefully note that the representation of symmetry operations as permutations of comers, or atoms, is not generally unique. For example, in the case of a planar molecule both the reflection in the plane and the identity mapping are represented by the identity permutation, although they are different symmetry operations. consists of all 24 permutations of the corners of the tetrahedron,... 

So, if the results of operations on sets of basis vectors are considered collectively, we gain more information than when they are considered in isolation. This approach is formalized in the matrix representation of symmetry operations. The matrices are mathematical objects that contain enough information to carry out the symmetry operations on a whole set of basis vectors simultaneously. 

So, as we found for the case of the simpler basis in H2O, the matrix representation of symmetry operations allows products of operations to be considered algebraically. This matrix product approach can be extended to matrices of any size and so for any size of basis. 

It is important to distinguish between mmetiy properties of wave functions on one hand and those of density matrices and densities on the other. The symmetry properties of wave functions are derived from those of the Hamiltonian. The "normal" situation is that the Hamiltonian commutes with a set of symmetry operations which form a group. The eigenfunctions of that Hamiltonian must then transform according to the irreducible representations of the group. Approximate wave functions with the same symmetry properties can be constructed, and they make it possible to simplify the calculations. 

The tables of characters have the general form shown in Table 5. Each colipua represents a class of symmetry operation, while the rows designate the different irreducible representations. The entries in the table are simply the characters (traces) of the corresponding matrices. Two specific properties of the character tables will now be considered. 

To illustrate the application of Eq. (37), consider the ammonia molecule with the system of 12 Cartesian displacement coordinates given by Eq. (19) as the basis. The reducible representation for the identity operation then corresponds to the unit matrix of order 12, whose character is obviously equal to 12. The symmetry operation A = Cj of Eq. (18) is represented by the matrix of Eq. (20) whore character is equal to zero. Hie same result is of course obtained for die operation , as it belongs to the same class. For the class 3av the character is equal to two, as exemplified by the matrices given by Eqs. (21) and (22) for the operations C and Z), respectively. The representation of the operation F is analogous to D (problem 12). 

Under the action of the electric field of the 0 environment, the state of the ( ) 3d configuration will split up into two states. The orbital degeneracy of a D state is 2 X 2 -f- 1 = 5. From the above discussion each of the resulting states must belong to one of the irreducible representations of Oh given in Table I. The state W corresponds to an irreducible representation of the group of symmetry operations of a sphere, i.e., the full rotation group R(3). 

The actual linear combinations whi.ch reduce P, P, and P could be found by the projection operator technique of 7-6 but because of the large number of symmetry operations contained in 0h and because of the problems connected with the multi-dimensional representations (similar to those encountered in 11-6), to do so would require much time and even more patience. Fortunately, we can find the correct combinations by inspection. 

Since symmetry operations in the same class have the same character in a given representation and are grouped together in the character table, it is convenient to rewrite (9.38) in a form where the sum goes over the various classes, rather than over the individual symmetry operations. If ht is the number of symmetry operations in class /, then (9.38) becomes... 

The column headings are the classes of symmetry operations for the group, and each row depicts one irreducible representation. The +1 and —1 numbers, which... 

Now the trace of the new representation should correspond to a character of the irreducible representation of the double group. By inspecting all classes of symmetry operations, the given z-th energy level is unambiguously classified according to the character table of the double group. 

A group-theoretical treatment of this symmetry contraint leads to the requirement that an MO must belong to an irreducible representation of the point group. A representation is a set of matrices - one for each symmetry operation - which constitutes a group isomorphous with the group of symmetry operations and can be used to represent the symmetry group. When we say that a function belongs to (or transforms as , or forms a basis for ) a particular representation, we mean that the matrices which constitute the representation act as operators which transform the function in the same way as the symmetry operations of the molecule. (The reader who knows little about matrices and their application as transformation operators can skip over such remarks.) An irreducible representation is one whose matrices cannot be simplified to sets of lower order. 

Table 4-4 shows a preliminary character table for the C3v point group. The complete set of symmetry operations is listed in the upper row. Clearly, some of them must belong to the same class since the number of irreducible representations is 3 and the number of symmetry operations is 6. A closer look at this table reveals that the characters of all irreducible representations are equal in C3 and Cf and also in oy, o+ and a", respectively. Thus, according to rule 4 C3 and Cl form one class, and ay, ct and a" together form another class. 

A complete character table is given in Table 4-5 for the C3v point group. The classes of symmetry operations are listed in the upper row, together with the number of operations in each class. Thus, it is clear from looking at this character table that there are two operations in the class of threefold rotations and three in the class of vertical reflections. The identity operation, E, always forms a class by itself, and the same is true for the inversion operation, i (which is, however, not present in the C3v point group). The number of classes in C3v is 3 this is also the number of irreducible representations, satisfying rule 5 as well. 

The character tables usually consist of four main areas (sometimes three if the last two are merged), as is seen in Table 4-5 for the C3v and in Table 4-7 for the C2h point group. The first area contains the symbol of the group (in the upper left corner) and the Mulliken symbols referring to the dimensionality of the representations and their relationship to various symmetry operations. The second area contains the classes of symmetry operations (in the upper row) and the characters of the irreducible representations of the group. 

Figure 6-16a). Thus, they must be treated together in this way they form a basis for a representation. All symmetry operations are indicated in Figure 6-16b. The D h character table is given in Table 5-3. The characters of this representation will be...

If ra is an irreducible representation of dimension k, and if F, F2, is a set of degenerate eigenfunctions that form the basis for theyth irreducible representation of the group of symmetry operations, these eigenfunctions transform according to the relation... 

For a symmetrical atomic or molecular system, these considerations place a severe restriction on the possible eigenfunctions of the system. All possible eigenfunctions must form bases for some irreducible representation of the group of symmetry operations. The form of the possible eigenfunctions is also determined to a large extent since they must transform in a quite definite way under the operations of the group. 

The number of inequivalent irreducible representations is equal to the number of classes in the group of symmetry operators. 

Figure 2.7-6 A Assignment of the Cartesian coordinate axes and the symmetry operations of a planar molecule of point group C2,.. B Character table, 1 symbol of the point group after Schoen-flies 2 international notation of the point group 3 symmetry species (irreducible representations) 4 symmetry operations 5 characters of the symmetry operations in the symmetry species +1 means symmetric, -1 antisymmetric 6 x, y, z assignment of the normal coordinates of the translations, direction of the change of the dipole moment by the infrared active vibrations, R, Ry, and R stand for rotations about the axes specified in the subscript 7 x, xy,. .. assign the Raman active species by the change of the components of the tensor of polarizability, aw, (Xxy,. ...

The A 2 representation of the C2v group can now be explained. The character table has four columns it has four classes of symmetry operations (Property 2 in Table 4-7). It must therefore have four irreducible representations (Property 3). The sum of the products of the characters of any two representations must equal zero (orthogonality. Property 6). Therefore, a product of A and the unknown representation must have 1 for two of the characters and — 1 for the other two. The character for the identity operation of this new representation must be 1 [x(i ) = 1 ] in order to have the sum of the squares... 

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. 

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