442 Chapter 13 Introduction to Symmetry in Quantum Mechanics EXAMPLE 13.3 Determ

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442 Chapter 13 Introduction to Symmetry in Quantum Mechanics EXAMPLE 13.3 Determine the point groups that define the symmetry of the following compounds, whose structures are illustrated in Figure 13.15. a. Hydrogen sulfide, H,S b. Sulfur hexafluoride, SF c. Acetylene, C,H2 d. Benzene, CH e. The nitrate ion, NO, (assume resonance averages the structure into a flat, trian- (a) planar (like H2o) gular species). SOLUTION (all S-F bonds (b) octahedral equivalent) Using Figure 13.14, satisfy yourself that the following point groups are indeed correct by identifying the individual symmetry elements, if any exist other than E. a. H,S: C, just like H,0 b. SF: O, This molecule has the shape of an octahedron. c. C,H: D, because it is linear and symmetric (if it had no center of inversion, (c) linear it would be Cwy). (d) planar (Planar, all N-O bonds equivalent) O The applicability of symmetry to molecules is deeper than just the shape of the molecule. Mathematical equations also have symmetry properties. We have already discussed the concept of odd and even functions. This is a symmetry property. An even function implies that a plane of symmetry exists, typically a plane that inter- sects the y-axis. You can verify this by looking at plots of cosine, an even function, (e) planar FIGURE 13.15 What are the point and sine, an odd function. groups of these five molecules? See Example 13.3. As mathematical functions, quantum-mechanical wavefunctions can also have certain symmetry properties. But what symmetry properties does a wavefunction have? Because a wavefunction determines the distribution of electron probability in

Explanation / Answer

Point group for Benzene is D6h

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