An inquiry-based learning approach to the introduction of the improper rotation-reflection operation, S_n

Symmetry properties of molecules are generally introduced in second-year or third-year-level inorganic or physical chemistry courses. Students generally adapt readily to understanding and applying the operations of rotation (C_n), reflection (σ), and inversion (i). However, the two-step operation of improper rotation-reflection (S_n) often provides a greater challenge for students. S_n operations can be difficult to identify and visualize, and the reason their inclusion in the different types of symmetry operations is not always clear or explained. In this contribution, an inquiry-based learning exercise is used to introduce students to the S_n operation: The results of all symmetry operations are first listed by simple permutations of atoms, and then students search for operations to bring about these permutations.