The Characters of Representations

Since the Bravais cell contains lO atoms, it has 3x10 — 3 = 27 normal vibrations, excluding three translational motions of the cell as a whole.l These 27 vibrations can be classified into various symmetry species of the factor group Djd, using a procedure similar to that described in Sec. 1-7 for internal vibrations. First, we calculate the characters of representations corresponding to the entire freedom possessed by the Bravais primitive cell [a r( )] translational motions of the whole cell [ (T)], translatory lattice modes [ f(T )],... 

The traces of the representation matrices are called the characters of the representation, and (equation Al.4.36) shows that all equivalent representations have the same characters. Thus, the characters serve to distingiush inequivalent representations. 

The characters of the irreducible representations of a synnnetry group are collected together into a character table and the character table of the group 3 is given in table A1.4.3. The construction of character tables for finite groups is treated in section 4.4 of [2] and section 3-4 of [3]. 

In applications of group theory we often obtain a reducible representation, and we then need to reduce it to its irreducible components. The way that a given representation of a group is reduced to its irreducible components depends only on the characters of the matrices in the representation and on the characters of the matrices in the irreducible representations of the group. Suppose that the reducible representation is F and that the group involved... 

Whenever a fiinction can be written as a product of two or more fiinctions, each of which belongs to one of the synnnetry classes, the symmetry of the product fiinction is the direct product of the syimnetries of its constituents. This direct product is obtained in non-degenerate cases by taking the product of the characters for each symmetry operation. For example, the fiinction xy will have a symmetry given by the direct product of the syimnetries of v and ofy this direct product is obtained by taking the product of the characters for each synnnetry operation. In this example it may be seen that, for each operation, the product of the characters for Bj and B2 irreducible representations gives the character of the representation, so xy transfonns as A2. 

The symmetry argument actually goes beyond the above deterniination of the symmetries of Jahn-Teller active modes, the coefficients of the matrix element expansions in different coordinates are also symmetry determined. Consider, for simplicity, an electronic state of symmetiy in an even-electron molecule with a single threefold axis of symmetry, and choose a representation in which two complex electronic components, e ) = 1/v ( ca) i cb)), and two degenerate complex nuclear coordinate combinations Q = re " each have character T under the C3 operation, where x — The bras e have character x. Since the Hamiltonian operator is totally symmetric, the diagonal matrix elements e H e ) are totally symmetric, while the characters of the off-diagonal elements ezf H e ) are x. Since x = 1, it follows that an expansion of the complex Hamiltonian matrix to quadratic terms in Q. takes the form... 

Before considering other concepts and group-theoretical machinery, it should once again be stressed that these same tools can be used in symmetry analysis of the translational, vibrational and rotational motions of a molecule. The twelve motions of NH3 (three translations, three rotations, six vibrations) can be described in terms of combinations of displacements of each of the four atoms in each of three (x,y,z) directions. Hence, unit vectors placed on each atom directed in the x, y, and z directions form a basis for action by the operations S of the point group. In the case of NH3, the characters of the resultant 12x12 representation matrices form a reducible representation... 

In the E irreducible representation, the character of any symmetry operation corresponding to a rotation by an angle is given by... 

A planar molecule of point group 03b is shown in Fig. 5. The sigma orbitals i, <72 and (73 represented there will be taken as the basis set Application of the method developed in Section 8.9 yields the characters of the reducible representation given in Table 14. With the use of the magic formula (Eq. (37)] the structure of the reduced representation is of the form Ta — A, ... 

Consider the trans isomer of butadiene. Both the symmetry operations that define the group < 2h and the characters of the representation r are given in Table 3. The reduction of this representation leads to Tn =2Bg 2Aa. Thus, two linear combinations of the atomic orbitals can be constructed of symmetry Bg and two others of symmetry A. Their use will factor the secular determinant into two 2x2 blocks, as described in the following paragraph. 

Thus the sum of the diagonal elements, or trace, of a matrix D(R) is invariant under a transformation of the coordinate axes. When dealing with group representations the trace Dtl(R) is called the character of R in the... 

A matrix of order l has l2 elements. Each irreducible representation T, must therefore contribute If -dimensional vectors. The orthogonality theorem requires that the total set of Y f vectors must be mutually orthogonal. Since there can be no more than g orthogonal vectors in -dimensional space, the sum Y i cannot exceed g. For a complete set (19) is implied. Since the character of an identity matrix always equals the order of the representation it further follows that... 

That is, the character of the product representation is just the product of the characters. Hence the Clebsch-Gordon coefficients are... 

The dimension of a representation is the same as the order of the matrix. To reduce a representation it is necessary to reduce its order. It is noted that the dimension of a matrix representation corresponds to the character of the identity (E) matrix. 

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< >. 

To get the characters for the representation of subduced by a given representation of S, we just copy down the characters of that representation for the elements of which are also in . This is done for the irreducible representations of 4 subduced onto D2< in Table 3. Comparing Tables 2 and 3, and using the standard formula for finding the irreducible parts of a representation by means of the characters, we see that the representations subduced by. T<8> and /1<5) contain / ( )... 

Diagonalization of S produces states that carry representations of S6, the group of permutations of six objects, while diagonalization of the other operators produces states that transform according to the representations Alf , EUl, E2g, BXu of Dbh. This result can be verified by computing the characters of the representations carried by the eigenstates of 5 , as shown for example in Wilson, Decius, and Cross (1955). 

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