Nonequivalent representations

The only set of positive integers that satisfies (9.33) is 1, 1, 2. Thus, besides the irreducible nonequivalent representations (9.28) and (9.29), there is one further such representation, and its dimension is 1. This representation can be found using (9.31) in conjunction with the characters of (9.28) and (9.29). However, the fastest way to find it is to evaluate the determinants of the matrices (9.28). From (2.17), it follows that the determinants of the matrices of any representation multiply in the same way as the matrices multiply. Hence these determinants form a one-dimensional representation. Evaluating the determinants of the matrices (9.28), we get as the third nonequivalent irreducible representation of Qiv ... 

We see, therefore, that the characters of the matrices of the irreducible representations form sets of orthogonal vectors. Since the character is unchanged by a similarity transformation, we see that two nonequivalent representations have different character systems and that two irreducible representations with the same character system are equivalent. 

Before going on to consider applications of group theory in physical problems, it is necessary to discuss several general properties of irreducible representations. First, suppose that a given group is of order g and that the g operations have been collected into k different classes of mutually conjugate operations. It can be shown that the group Q possesses precisely k nonequivalent irreducible representations, T(1), r(2).r(t>, whose dimen-... 

Fig. 14 Experimental (a) and calculated (b) conductance values of Au-n-alkanedithiol-Au junctions vs number n of methylene units in a semilogarithmic representation. The three sets of conductance values - high (H), medium (M), and low (L) - are shown as squares, circles, and triangles. The straight lines were obtained from a linear regression analysis with decay constants (3n defined per methylene (CH2) unit. The conductances of many different, nonequivalent gauche isomers cover the window below the medium values in (b) [64]...

We are now in a position to show that two representations with a one-to-one correspondence in characters for each operation, are necessarily equivalent (see 7-3). If we consider two different nonequivalent irreducible representations then, since the characters are orthogonal (eqn (7-3.4)), there cannot be a one-to-one correspondence. If we consider two different reducible representations T° and Tb then, by eqn (7-4.2), if the characters are the same, the reduction will also be the same, that is the number of times occurs in P (a ) will, by the formula, be the same as the number of times T occurs in Fb. The reduced matrices can therefore be brought to the same form by reordering the basis functions of either Ta or Tb. The reduced matrices are therefore equivalent and necessarily Ta and Tb from whence the reduced matrices came (via a similarity transformation) must also be equivalent. Hence, we have proved our proposition. 

Since the order of C3v is 6, Theorem (5) tells us that (9.28) is irreducible. We saw in Section 9.2 that <3iv has three classes hence by Theorem (1), G3v has three nonequivalent irreducible representations. Calling the dimensions of these three representations /2, and we have from Theorem (2) ... 

The characters of the group symmetry operations for each nonequivalent irreducible representation are listed in the character table of the group. 

Note that to apply (9.39), we do not need the matrices of the nonequivalent irreducible representations only their traces (which are given in the character table) are needed. 

Before showing further applications of direct-product representations to quantum mechanics, we quote without proof a theorem we will need. Let rij a and rkip be the elements of the matrices corresponding to the symmetry operation R in the two different nonequivalent irreducible representations Ta and T it can be shown that... 

We shall see that it is the nonequivalent irreducible representations that are of quantum-mechanical significance. For these representations, the following theorems hold1 ... 

The number of nonequivalent irreducible representations of a group is equal to the number of classes in the group. 

Equation (9.31) implies that two nonequivalent irreducible representations cannot have the same set of characters. 

We have introduced certain labels in (9.36) for the three nonequivalent irreducible representations of 63v. The conventions (formulated by Mulliken) for such labels are the following ... 

Theorem (2) shows that there are only a finite number of nonequivalent irreducible representations of a group of finite order. Any reducible representation must either be the direct sum of two or more irreducible representations or be convertible to such a direct sum by the performance of a similarity transformation on its matrices. In the former case, it is easy to see by inspection what irreducible representations make up the reducible representation in the latter case, this is not obvious, since the matrices... 

This example has several features worthy of comment. The eight B atoms comprise two nonequivalent subsets, B — B4 and B5 — However, each subset gives the same representation (40002) and hence gives rise to SALCs of the same form which will require central AOs of the same symmetry types with which to interact. In the above results, we have placed each SALC opposite to the AO it seems likely to overlap with best when the shapes of the AOs are considered. For example, the E SALCs formed by the atoms of the flattened tetrahedron ( , - B4) will overlap better with the px and py orbitals than the E SALCs formed by the atoms of the elongated tetrahedron (Bs — B8). Of course, this is only a crude approximation and a real calculation would doubtless show that all orbitals of the same symmetry undergo some degree of mixing. 

It follows that the two seemingly different representations of the lone pairs of water, in terms of equivalent sp3 hybrids or nonequivalent canonical MOs, are both correct in that they correspond to the same unique polyelectronic wave functions. 

Figure 1 Schematic representation of the decrease in accuracy with increasing number of nonequivalent atomic sites that can be treated computationally by different levels of approximation. (Ref. 4. Reproduced by courtesy of D. Pettifor)...

Our objective is to ascertain how anomalous diffusion modihes the dielectric relaxation in a bistable potential with two nonequivalent wells, Eq. (195). The formal step-off transient solution of Eq. (172) for t > 0 is obtained from the Sturm-Liouville representation, Eq. (179), with the initial (equilibrium) distribution function... 

Here, we discuss an alternative scheme where the superposition state <1>) can be generated in two identical atoms driven in free space by a coherent laser field. This can happen when the atoms are in nonequivalent positions in the driving field, where the atoms experience different intensities and phases of the driving field. The populations of the collective states of the system can be found from the master equation (31). We use the set of the collective states (35) as an appropriate representation for the density operator... 

The two irreducible representations of the Csv group that we have obtained thus far are said to be nonequivalent, since they have different dimensions. There are several theorems governing irreducible representations for a particular group. ... 

If the symmetry of the system is broken, as in glycosyl derivatives, it is best to adopt the nonequivalent (p-type and n-type) representation of the lone pairs on oxygen. This is illustrated with a-D-xylopyranosyl fluoride in Fig. 2. With this representation it is important to note that the p-type lone pair is tilted at a dihedral angle of ca. 30% to the axial C—F bond. 

Figure 2. Representation of the chmr-to-chmr interconversion process in cyclohexane showing the possible intermediates (boat and twist-boat) and transition states (half-chair and half-boat). The magnetically nonequivalent protons, Hj, and Hb, undergo mutual exchange during ring inversion.

Suppose now that Ri and R are distinct (nonequivalent) irreducible representations. Then, by Theorem II, A = 0. Since B is completely arbitrary, it may be chosen in such a way that every element except, say, the a a "th, vanishes, the exceptional element being assigned the value of unity. Then, by (13), replacing the symbol T by R,... 

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