Products of Irreducible Representations

As shown in Sec. 1.10, the characters for direct products can be obtained by multiplying the corresponding characters of two representations and resolving the result into those of the irreducible representations by using Eq. 1.74. This procedure, however, can be greatly simplified if we use the following rules (Ref. 3 of Chapter 1). 

For groups in the lists above that have symbols A, B, or without subscripts, read Ai = A2 = A, and so on. 

APPENDIX V. NUMBER OF INFRARED- AND RAMAN-ACTIVE STRETCHING VIBRATIONS FOR MX Y -TYPE MOLECULES 

Compound Structure Point Group IRor Raman M-X Stretching M-Y Stretching 

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The calculated state energies, the transition moments, and the symmetry classification are given in Table 3. The symmetry species of the triplet functions is obtained by taking the direct product of irreducible representation of the space and the spin functions Fx, Fy, Fz, which transform as the rotations Rx, Ry, and Rz-... 

Like in the ordinary point groups, the direct product of irreducible representations is a reducible representation and the characters for individual symmetry operations obey the relationship... 

The direct product of irreducible representations follows the rules compiled in Table 1.20. For degenerate representations the rules are more complex and specific for the given group (Tables 1.21 and 1.22). 

We are approaching the application of group theory in optical transitions in spectroscopy. The most important issue will be a decision whether an integral is zero or nonzero. If the integral is zero, then the transition is forbidden, while if it is nonzero, then it is allowed. To make such a decision, we have to use what is known as the direct product of irreducible representations. Imagine basis functions and that correspond to irreducible representations a and f of... 

X-6. Symmetry Species of Combinations Direct Products of Irreducible Representations... 

Table 3. Symmetrized direct products of irreducible representations and vibronic activity...

The underlined modes are contained in symmetrized as well as in antisymmetrized direct products of irreducible representations of electronic states. There must be at least two sets of coordinates belonging to these modes and at least one of them is vibronic active. 

To have a nonvanishing matrix element (4.53), it is necessary for the direct product of irreducible representations... 

But the functions we want to work with can be extremely complicated. How can we be sure that any function can be classified according to the representations of the point group In each point group, it is possible to express the symmetry properties of any function in terms of a finite set of irreducible representations of the group. In the same way that any whole number can be broken down into a unique product of prime numbers, we can break down the symmetry properties of any function into a unique product of irreducible representations. Each irreducible representation F, stands for a particular combination of symmetry properties that make it distinct from every other irreducible representation Fj, such that functions given by any two different irreducible representations are always orthogonal ... 

Determine the resulting representations for the following products of irreducible representations. 

Here we find a new concept, the direct product between irreducible representations of a symmetry group. This direct product is related to the product of their corresponding space functions. For our purposes, we will only mention that the direct product between two, Pj and A, (or more) irreducible representations of a group is a new... 

EXAMPLE 7.4 The direct product between irreducible representations of group O. 

Similarly, we could perform other direct products between irreducible representations of the 0 group, and then decompose them into irreducible representations of this group. We would obtain ... 

An electric dipole transition will be allowed with x, y, or z polarization if the direct product of the representations of the two states concerned is or contains the irreducible representation to which Jt y, or z, respectively, belongs. 

The direct products of representations of double groups can be taken in the usual way and reduced to sums of irreducible representations. 

Table 14.6. Clebsch-Gordan coefficients for the inner direct products of irreducible co-representations.

Table 66 Decomposition of the direct product F F of irreducible representations in some double groups... 

In general, 71/ is a reducible representation, i.e., a combination of irreducible representations. The number of times the irreducible representation 71 occurs in the direct product 71/ can be determined by ... 

Irreducible representations are orthogonal to each other. The sum of the products of the characters (multiplied together for each class) for any pair of irreducible representations is 0. 

Taking any pair of irreducible representations, multiplying together the characters for each class and multiplying by the number of cper-ations in the class (see Table 4-8 for an exam-pe), and adding the products gives zero. 

The restrictions listed until now are of the type of necessary conditions. In order to obtain sufficient conditions for the observation of the effect of magnetic optical activity of molecules in T states, it is necessary to pass under the sign of the spur in the expression Sp pLK-K Cl J]Clf. J for allowed values of the pairs (JJ ) to appropriate products of irreducible tensor operators for the point group of the molecule. Using the decomposition of the representation DJ of the full spherical group into irreducible representations of the molecular point group, one can present the result in the form of a sum of following terms ... 

When the direct product of two irreducible matrix representations of a group is reducible, it can be reduced to a direct sum of irreducible representations by cin equivalent transformation with a constant matrix, i.e. the same matrix for all the matrix representatives of the symmetry operators of the group (2). We shall assume the irreducible representations in unitary form then the constant matrix can be chosen as the real orthogonal matrix whose elements are the coupling coefficients occuring in Eq. (5). The orthogonality properties can be expressed as... 

The basic operation in tensor theory is the unitary transformation that reduces the direct product of two irreducible unitary representations and transforms it into a direct sum of irreducible representations. In the three-dimensional rotation group a term of a certain symmetry type (irreducible representation) occurs at most once (23) in the direct sum. Thus, the effect of the transformation on the bases for the initial representations to 5deld an element of one of the direct product sets may be expressed according to Wigner s formula... 

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