By Geoffrey Wilkinson

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**Example text**

Similarly, arrows on the three coordinate axes may be used to represent the symmetry of the p x , p v , and p z orbitals. Changing the sign of the coordinate on an arrow inverts it. Probability density boundary surface pictures such as those in Figure 2-3 may be used to represent the symmetry of hydrogenlike orbitals. Using either arrows pointing along x, y, and z or the signed (plus and minus) Reducibl e R e p r e s e n t a t i o n s Table 3-3 51 Reducible representation for p orbitals in C 2v C2V E ft« rr 3 -1 σν(χζ) *v(yz) 1 1 boundary surfaces, we recognize that the symmetry operations may be applied to our pictorial representations of orbitals.

Exercise 3-1 Using polar coordinates, Figure 2-1, and the angular forms for the hydrogenlike wave functions (Table 2-2), show that in an octahedron, Cs[2p 2 ] = 2p y . * Odd functions have the quantum number 1 = 1, 3, 5; for even functions 1 = 0, 2, 4. f See Exercise 2-1, Chapter 2. Representation s of S y m m e t r y 45 Hint From Figure 2-1, z = r cos y = r sin sin φ, and x = r sin 0 cos φ. In a similar manner show that the octahedral C4 rotation takes Sdx2_v2 —» — 3dxi_y*. As the symmetry present in the molecule decreases, the number of orbitals which can be "identically" degenerate,* as opposed to being accidentally equally energetic, decreases also.

Similarly, under C' 2 and C" we have + 1 . Thus the reducible representation Td is given by the numbers 5, — 1, + 1 , + 1, and + 1 . For the three p functions the reducible representation, Γ ρ , is 3, 1, - 1 , - 1 , and - 1 (Table 3-5). (β) = Zx*(R)xj(R) = Stjh (3-1) ft ft (δ,-y = 0, i 7* j ; = 1, i = j) By definition, then, the reducible representation, ΓΓ, is a sum of irreducible representations. What is the result of the product of ΓΓ with an irreducible representation I \ for the point group in question [Eq.