The irreducible representations of SO

The irreducible representations of SO(n) are also characterized by a set of integers, but corresponding to the partition... 

The results of the current section, both the lowering operators and the classification, will come in handy in Section 8.4, where we classify the irreducible representations of so(4). One can apply the classification of the irreducible representations of the Lie algebra sm(2) to the study of intrinsic spin, as an alternative to our analysis of spin in Section 10.4. More generally, raising and lowering operators are widely useful in the study of Lie algebra representations. 

The representation of so(2r + l,IR) is irreducible, while the representations of so(2r, IR) and u r) are reducible. The irreducible representations of so(2r, IR) are obtained by restricting to the subspace of even or odd number states in LAp, while the irreducible representations of u r) are obtained by restricting to subspaces of states of fixed integer particle number. 

Having done this we solve the Scln-ddinger equation for the molecule by diagonalizing the Hamiltonian matrix in a complete set of known basis fiinctions. We choose the basis functions so that they transfonn according to the irreducible representations of the synnnetry group. 

We can summarize our work above by writing that the natural representation of SO (3) on C is irreducible. In contrast, we have seen in Section 5.1 that the representation of the circle group defined by Formula 5.1 is not irreducible. 

In this section we classify the finite-dimensional irreducible representations of 50(3). Compared to the work we did classifying the irreducible representations of SU(2) in Section 6.5, the calculation in this section is a piece of cake. However, the reader should note that we use the 51/ (2) classification in this section. So our classification for 50(3) is not inherently easier. Our trick is to use the group homomorphism 4> 51/(2) —> 50(3) (defined in Section 4.3) to show that any representation of 5 O (3) is just a representation of 51/ (2) in disguise. At the end of the section we show how to use weights to identify irreducible representations. 

The Qn > are essentially the only finite-dimensional irreducible representations of SO(.3). 

Proposition 7.2 Suppose f is a nonnegative integer. Then the natural representation of SO (3) on is irreducible. 

The results of this section are another confirmation of the philosophy spelled out in Section 6.2. We expect that the irreducible representations of the symmetry group determined by equivalent observers should correspond to the elementary systems. In fact, the experimentally observed spin properties of elementary particles correspond to irreducible projective unitary representations of the Lie group SO(3). Once again, we see that representation theory makes a testable physical prediction. 

Note first that y(E) = 6 while all other characters are zero. The reason is that the operation E transforms each into itself while every rotation operation necessarily shifts every 0, to a different place. Clearly this kind of result will be obtained for any /z-membered ring in a pure rotation group C . Second, note that the only way to add up characters of irreducible representations so as to obtain y = 6 for E and y - 0 for every operation other than E is to sum each column of the character table. From the basic properties of the irreducible representations of the uniaxial pure rotation groups (see Section 4.5), this is a general property for all C groups. Thus, the results just obtained for the benzene molecule merely illustrate the following general rule ... 

The conceptual simplicity of the CASSCF model lies in the fact that once the inactive and active orbitals are chosen, the wave function is completely specified. In addition such a model leads to certain simplifications in the computational procedures used to obtain optimized orbitals and Cl coefficients, as was illustrated in the preceding chapters. The major technical difficulty inherent to the CASSCF method is the size of the complete Cl expansion, NCAS. It is given by the so-called Weyl formula, which gives the dimension of the irreducible representation of the unitary group U(n) associated with n active orbitals, N active electrons, and a total spin S ... 

So the eight pairs of electrons of this molecule occupy delocalized molecular orbitals lag to 1 3U, while the first vacant orbital is l g- Note that the names of these orbitals are simply the symmetry species of theZ)2h point group. In other words, molecular orbitals are labeled by the irreducible representations of the point group to which the molecule belongs. So for ethylene there are three filled orbitals with Ag symmetry the one with the lowest energy is called lag, the next one is 2ag, etc. Similarly, there are two orbitals with Z iu symmetry and they are called lb u and 2bi . All the molecular orbitals listed above, except the first two, are illustrated pictorially in Fig. 6.4.2. By checking the >2h character table with reference to the chosen coordinate system shown in Fig. 6.4.2, it can be readily confirmed that these orbitals do have the labeled symmetry. In passing, it is noted that the two filled molecular orbitals of ethylene not displayed in Fig. 6.4.2, lag and l iu, are simply the sum and difference, respectively, of the two carbon Is orbitals. 

Thus it may produce the transitional densities of the alg, egc, and tluz symmetries. At this point selection rules pertinent to the frontier orbitals approximation enter for the 12-electron complexes the symmetries of the frontier orbitals are Th = eg and Tl = ai3, the tensor product Th <8> TL = eg aig = eg contains only the irreducible representation eg so that the selection rules allow only the density component of the egc symmetry to appear. In its turn this density induces additional deformation of the same symmetry. That means that in the frontier orbitals approximation, only the elastic constant for the vibration modes of the symmetry eg is renormalized. This result is to be understood in terms of individual nuclear shifts of the ligands in the trans- and cis-positions relative to the apical one. They, respectively, are ... 

Fig. 1. Regions in the (k, q) plane where the -eigenvalue spectrum corresponds to unitary irreducible representations of so(3) (horizontal shading) and so(2, 1) (vertical shading). See text for details. 

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