Communication pubs.acs.org/jchemeduc
An Inquiry-Based Learning Approach to the Introduction of the Improper Rotation−Reflection Operation, Sn John P. Graham* Department of Chemistry, United Arab Emirates University, Al Ain, United Arab Emirates ABSTRACT: 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 (Cn), reflection (σ), and inversion (i). However, the two-step operation of improper rotation−reflection (Sn) often provides a greater challenge for students. Sn 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 Sn 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. KEYWORDS: Second-Year Undergraduate, Upper-Division Undergraduate, Inorganic Chemistry, Inquiry-Based/Discovery Learning, Problem Solving/Decision Making, Group Theory/Symmetry, Molecular Properties/Structure
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This is a fact at the heart of molecular symmetry, although rarely stated in this way. Next we assign the student a molecule which contains an Sn axis and ask the student to write out all possible permutations of atoms that might arise from symmetry operations, without explicit consideration of the symmetry elements of the molecule. In this example, where all of the equivalent atoms are bound to one central carbon atom, this can be stated more simply: “All possible unique permutations of atoms without consideration of the symmetry elements of the molecule”. This step could be completed by students outside of the classroom and requires no knowledge of symmetry operations at all. The example used here is CH4, but any molecule with an Sn operation could be used for this purpose (however it is preferable to choose a molecule that does not give rise to too many permutations and one in which Sn has a visibly unique effect on atomic positions). It is suggested that students work in groups from here on as the exercise is lengthy and well suited to collaborative work. It may be helpful to work out a subset of the unique permutations in class to illustrate a systematic approach, for example, the first six permutations given in Figure 1, all of which have H1 “on top”. All of the permutations of H atoms in methane are given in Figure 1. With all of the possible permutations drawn, students are then asked to determine which symmetry operation will take the initial arrangement of atoms to each of the new permutations. It is helpful during this process for students to note the general effects of each operation. In the case of CH4, students can readily conclude the following: • C2 rotations result in all H atoms moving to new positions, and the atoms are exchanged in pairs. • C3 (and C32) operations result in three H atoms moving to new positions and one atom remaining unmoved. • σ operations result in two H atoms remaining unmoved and two H atoms exchanging positions.
n understanding of group theory and the basic symmetry properties of molecules is essential background for chemistry students who wish to advance in understanding of the bonding and spectra of molecules. Most undergraduate inorganic textbooks begin discussion of symmetry through introduction of the different classes of symmetry operation: Proper rotation (Cn), reflection through a plane of symmetry (σ), inversion (i), improper rotation−reflection (Sn), and the identity operator (E). Of these different types of symmetry operation, Sn usually provides the greatest challenge for students. There are two reasons for difficulty in understanding the use of Sn operations: first, the ability to visualize the effects of Sn and locate Sn axes/planes, and second, the lack of justification as to why such a two-step operation is necessary. It seems that the latter may be the most significant problem: why do we need this odd looking two-step operation when the other operations Cn, σ, and i are relatively clear and completed in one step? And why is this particular combination of steps necessary at all? A survey of modern undergraduate textbooks shows that it is common practice to introduce the different types of symmetry elements all at once and to illustrate the effect of each type of operation with appropriate examples.1−5 Usually the most difficult, Sn, is left until last. However, no clear explanation as to why Sn is necessary is given. In this contribution, an alternative approach to introduction of the Sn operation is presented: Students discover the necessity of this operation through investigation of all indistinguishable permutations of H atoms in methane. We begin by introducing the students to E, i, Cn, and σ operations with illustrative examples. For most students these operations are readily accepted and understood. To help students visualize these symmetry elements and operations, the “Unique Atom Rule”6 and 3D computer programs7−9 may prove very helpful. Students are then informed that symmetry operations should exist such that all possible interchanges of equivalent atoms, without changing bonding connections between atoms, can be carried out through their application. © 2014 American Chemical Society and Division of Chemical Education, Inc.
Published: August 14, 2014 2213
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Figure 1. All possible permutations of H atoms in methane.
With the above generalizations in mind students can readily assign operations to take the initial arrangement to each of the new permutations. Students should group the permutations according to type of operation as in Figure 2. Students should find that six of the permutations from Figure 1 cannot arise from the symmetry operations E, C2, C3, C32, or σ, given in Figure 3. Figure 3. Six permutations of H atoms that cannot be derived from Cn or σ.
It is noted that, in the 6 remaining arrangements, all H atoms have moved, but they have not exchanged in a pairwise fashion. The question then arises, how can we get from the original arrangement to each or any of these? We can now ask the students to devise methods to get from the initial arrangement to one of those in Figure 3. A hint to the class that this involves two steps using a rotation axis and reflection plane, neither of which is actually a symmetry element on its own, may be sufficient for some students to discover the concept of Sn (It should also be noted that some students may discover Sn as the product of Cn−1 and i instead of Cn and σ). Even if students fail to discover any Sn symmetry element, they will be far more receptive to its introduction immediately after completing the above exercise. Sn now arises out of necessity, and not an arbitrary looking list of operation types. If students have not yet discovered Sn on their own, this is an ideal time to introduce Sn and show how it can be used to reach the 6 arrangements of Figure 3. Use of interactive Internet 3D resources are also very useful at this stage.7−9 Finally, once students have discovered and accepted the need for the Sn type operation, it is enlightening to reveal that the operations σ and i correspond to the improper rotation−reflections S1 and S2, respectively. Although it may be argued that the procedures suggested above are time-consuming, students can benefit from the exercise in several ways: • Students learn to work together in groups to solve a complex problem. • Students gain considerable practice in the identification of and visualization of the effects of the symmetry operations Cn and σ.
Figure 2. Assignment of H atom permutations to symmetry operations. 2214
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• Students gain a greater appreciation for the relationship between possible indistinguishable arrangements of atoms and symmetry operations. • Students actively discover the common effects of related symmetry operations. • Students see a clear justification for the consideration of the Sn operation and its unique effects.
AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. Notes
The authors declare no competing financial interest.
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REFERENCES
(1) Housecroft, C. E.; Sharpe, A. G. Inorganic Chemistry, 3rd ed.; Prentice Hall: Upper Saddle River, NJ, 2008. (2) Overton, T.; Rourke, J.; Weller, M.; Armstrong, F. Inorganic Chemistry, 6th ed.; Oxford University Press: Oxford, U.K., 2014. (3) Canham, G. R.; Overton, T. Descriptive Inorganic Chemistry, 5th ed.; W. H. Freeman: New York, 2010. (4) House, J. E. Inorganic Chemistry, 2nd ed.; Elsevier/Academic Press: Waltham, MA, 2013. (5) Miessler, G. L.; Tarr, D. A. Inorganic Chemistry, 3rd ed.; Prentice Hall: Upper Saddle River, NJ, 2010. (6) Graham, J. P. Unique Atoms and the Identification of the Symmetry Elements of Molecules. J. Chem. Educ. 2011, 88, 1010. (7) Charistos, N. D.; Tsipis, C. A.; Sigalas, M. P. 3D Molecular Symmetry Shockwave: A Web Application for Interactive Visualization and Three-Dimensional Perception of Molecular Symmetry. J. Chem. Educ. 2005, 82, 1741. (8) Cass, M. E.; Rzepa, H. S.; Rzepa, D. R.; Williams, C. K. An Animated Interactive Overview of Molecular Symmetry. J. Chem. Educ. 2005, 82, 1742. (9) Johnston, D. H. Symmetry Resources at Otterbein University. http://symmetry.otterbein.edu/ (accessed Aug 2014).
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