By Philip Bunker (Auth.)
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Additional info for Molecular Symmetry and Spectroscopy
Representations of Groups and a 3 = £ [ 7 x 2 + 1 x 0 + 1 x 0 + 1 x 0 + 1 x ( - 1 ) + 1 x (-1)] - 2. (4-52) Thus Γ = 2Γ1ΘΓ202Γ3. (4-53) We now test to see if this is correct by using Eq. (4-42b). For the operation E Eq. (4-42b) gives χ Γ = 7= 2x1 + 1x1 + 2x2 (4-54) which is correct. For the operations (12), (23), or (13) Eq. (4-42b) gives x r = l = 2 x l + lx(-l) + 2x0 (4-55) which is correct, and for the operations (123) and (132) Eq. (4-42b) gives χ Γ = 1 = 2 χ 1 + 1χ1 + 2χ(-1) (4-56) which is also correct.
2-15) Problem 2-4. Draw the effect of separately applying (12)(34)* and (13)(24)(56) to the ethylene molecule in Fig. 1-2. Is there any difference between the effects of these operations? Answer. Yes. If the nuclei of the ethylene molecule are in the equilibrium configuration (as they are in Fig. 1-2) then (12)(34)* and (13)(24)(56) have the same effect on the nuclear coordinates, but (12)(34)* inverts the electron coordinates whereas (13)(24)(56) does not. If the nuclei are not in the equilibrium configuration then the effects of these operations on the nuclear coordinates will usually be different, and this is discussed in more detail in the next section.
4-35) each matrix consists of an upper left 2 x 2 block and a lower right l x l block, with zeros in the off diagonal positions that connect the elements in one block with those of the other. Block diagonal matrices can consist of more than two blocks along the diagonal, for example the matrix "1 0 7 0 0 0 0 2 5 4 0 0 0 0 3 6 8 0 0 0 0 0 0 0 6 4 0 0 0 0 0 5 3 0 0 0 0 0 0 0 8 1 0 0 0 0 0 9 2 is also block diagonal and consists of a 3 x 3 block and two 2 x 2 blocks. The reader can use Eq. (4-3) to prove that the product of two block diagonal matrices having the same block structure produces a third block diagonal matrix that has the same block structure.