In the Laboratory
Dynamic Paper Constructions for Easier Visualization of Molecular Symmetry Lawrence T. Sein, Jr. Department of Chemical and Physical Sciences, Cedar Crest College, Allentown, Pennsylvania 18104
[email protected] Though symmetry is a powerful tool for solving problems in chemistry (1, 2), undergraduate students must grapple with their inability to easily visualize symmetry operations. To classify molecules into the appropriate point group, the student needs to make an inventory of the symmetry operations present in the molecule, apply a scheme to determine which symmetry operations permit a definitive assignment, and proceed quickly to the next molecule of interest. A number of articles have appeared that have introduced innovative methods for assisting the student in this task (3-6). In spite of the conventional wisdom that today's students are particularly visual as a result of experience with video-game technology (7), they are perhaps less capable of visualizing complex geometric or chemical shapes without assistance. Their difficulties can be minimized by use of a textbook that includes graphics that demonstrate the symmetry operations. But even the most excellent textbook is limited in space. Excellent computerbased models for assisting students with symmetry have been developed (8, 9), but these fail to show the interrelationships among groups and their subgroups. Memory and understanding are enhanced by involving as many of the student's senses as possible; static images in a textbook can at most stimulate only the visual sense. Therefore, I have designed an exercise for the students to improve their ability to visualize symmetry operations by adding the tactile features of touch and motion. The students construct a series of paper models during a 2-3 h laboratory period or as a homework assignment. The laboratory period would be a module in the laboratory course that accompanies the third- or fourth-year undergraduate-level (advanced) inorganic chemistry course. These models, unlike the standard ball and stick wooden models, allow instant identification of symmetry operations for a large number of the most commonly encountered point groups in inorganic chemistry. Furthermore, the models demonstrate the operations dynamically. Discussion The fundamental symmetry operations of point groups are reflections (σ), proper rotations (Cn), improper rotations (Sn), and inversions (i). Improper rotations and inversions, which frequently pose the greatest difficulties for students, can be clearly illustrated with this method. The student needs construction paper (multiple colors), poster board or cardboard (optional), scissors, a hole-punch, glue, metal ruler, pencil, pipe cleaners, and brass paper fasteners (brads). The students print patterns onto construction paper or poster board and then cut them out. To simulate rotations, two pieces of paper or poster board are attached with a brad that forms the rotational axis. The upper piece of paper is arranged so that it lines up with the lower piece
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Figure 1. Model of square (D4h), showing (A) C2, C4 axes and (B) showing σd.
of paper. Rotation of the upper piece by the proper angle will return the upper piece to a configuration where it lines up with the lower piece (Figure 1A). Reflections are simulated by careful scoring and prefolding along lines on the upper piece of paper that represent the relevant mirror planes (Figure 1B). If the portions of the upper piece of paper are lifted so that the piece fold along the line representing the mirror plane, the portion on each side of the line will perfectly align with the other. Inversion is simulated by using a strip of paper or a pipe cleaner cut to an appropriate length, which is dictated by the symmetry in question. The strip is slid between the two pieces of paper with its midpoint at the brad and is thus projecting from the center of inversion. When the strip lines up with one atom of the molecule, the other end of the strip (representing the inversion of a point) will exactly align with another atom of the same element. Improper rotation is simulated by a combined rotation and reflection. Two different techniques are employed to demonstrate this operation. The simpler method involves the assembly and disassembly of the model. For the second, more complicated technique, the rotation portion is simulated by a twist within a tube by the required angle. The reflection portion is simulated by movement of the upper portion backward. The student can build several copies of each model. Changing the colors of the “atoms” attached at different points will lower the group symmetry and remove one or more symmetry operations. Symmetry lines and planes are labeled on the models, which will help the student associate the operations with the model. Hazards The edge of poster board or cardboard is capable of cutting through the skin. Scissors are sharp and can cause either puncture or slashing wounds. Models should be kept away from small children, as brass paper fasteners (“brads”) could be swallowed. Proper ventilation should be used with the glue.
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r 2010 American Chemical Society and Division of Chemical Education, Inc. pubs.acs.org/jchemeduc Vol. 87 No. 8 August 2010 10.1021/ed100210h Published on Web 06/03/2010
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Journal of Chemical Education
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In the Laboratory
Conclusions The construction of paper models to assist the student's visualization of symmetry operations is easy, useful, and fun. Though the students are initially puzzled by the intrusion of lowtech into the science curriculum, they quickly master the identification of symmetry operations, and their use in classifying molecules of chemical interest. The 2-3 h exercise, undertaken during the first laboratory period (which is often unutilized), allows the efficient introduction to an otherwise mathematically daunting curriculum.
2. Kim, S. K. Group Theoretical Methods and Applications to Molecules and Crystals; Cambridge University Press: Cambridge, 1999. 3. Vining, W. J.; Grosso, R. P., Jr. J. Chem. Educ. 2003, 80, 110. 4. McKay, S. E.; Boone, S. R. J. Chem. Educ. 2001, 78, 1487. 5. Potillo, L. A.; Kantardjieff, K. A. J. Chem. Educ. 1995, 72, 399. 6. Jackson, W. G. J. Chem. Educ. 1992, 69, 624. 7. Johnson, S. Everything Bad Is Good for You: How Today's Popular Culture Is Actually Making Us Smarter; Riverhead Publishing: New York, 2005. 8. Chen, F. M. C. J. Chem. Educ. 2005, 82, 174. 9. Kijewski, L. J. Chem. Educ. 2005, 82, 174.
Literature Cited
Supporting Information Available
1. Cotton, F. A. Chemical Applications of Group Theory; 3rd ed.; John Wiley and Sons Inc.: Hoboken, NJ, 1990.
Student handout with 11 figures in color, and 23 patterns. This material is available via the Internet at http://pubs.acs.org.
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Vol. 87 No. 8 August 2010
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r 2010 American Chemical Society and Division of Chemical Education, Inc.
