Communication pubs.acs.org/jchemeduc
Teaching Molecular Symmetry of Dihedral Point Groups by Drawing Useful 2D Projections Lan Chen, Hongwei Sun,* and Chengming Lai Department of Chemistry, Nankai University, Tianjin, 300071, P. R. China ABSTRACT: There are two main difficulties in studying molecular symmetry of dihedral point groups. One is locating the C2 axes perpendicular to the Cn axis, while the other is finding the σd planes which pass through the Cn axis and bisect the angles formed by adjacent C2 axes. In this paper, a method of learning to draw 2D projections has been proposed to identify symmetry elements. Molecules with similar projective polygons are also compared to discuss the differences in point groups. KEYWORDS: Second-Year Undergraduate, Physical Chemistry, Hands-On Learning/Manipulatives, Group Theory/Symmetry, Learning Theories, Molecular Properties/Structure, Molecular Symmetry, Dihedral Point Groups, Projection
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INTRODUCTION In molecular point groups, the dihedral groups include Dn, Dnh, and Dnd. The group Dn has an n-fold axis of symmetry Cn and n C2 axes perpendicular to the Cn axis. In addition to the symmetry elements of Dn, the group Dnh has a horizontal mirror plane perpendicular to the Cn axis, n vertical mirror planes containing the Cn axis, and a center of symmetry if n is even; the group Dnd has n vertical mirror planes which pass through the Cn axis and bisect the angles between adjacent C2 axes, and a center of symmetry if n is odd.1,2 It is a little bewildering for students to find all the symmetry elements mentioned above. Several good 3D interactive visualizations of symmetry operations and elements using known molecular structures that have helped students learning molecular symmetry are available on the Internet,3−8 but we may still take advantage of planar demonstration. From the following approach, it can be seen that the symmetry elements of a molecule can be easily shown if the molecule is projected onto a suitable plane.
1a, just like a chair. In order to look for symmetry elements, tilt the chair backward and rotate to view down the front and back face (Figure 1b); it can be seen that the six carbons are evenly distributed in an alternating pattern above and below an imaginary horizontal plane bisecting the molecule. Now rotate to make this imaginary horizontal plane the plane of the paper (Figure 1c and alternatively as color coded: Figure 1d), then choose the paper as the projection plane, and project the six carbons onto the paper; a hexagon is obtained (Figure 1d). The six carbons are divided into two groups; the red ones were above the paper before projection, while the blue ones were below. From Figure 1d, the line passing through the center of the hexagon and perpendicular to the paper is a C3 axis (also an S6 axis), since rotation by 120° about it only interchanges carbons of the same color. The line connecting the midpoints of two opposite sides is a C2 axis, since rotation by 180° about it just sends red carbons to blue ones, and vice versa. There are three C2 axes all lying on the paper and perpendicular to the C3 axis. The plane passing through two opposite vertices of the hexagon and perpendicular to the paper is a σd plane; it bisects the angle formed by adjacent C2 axes as shown in Figure 1d. There are three σd planes too. All these symmetry elements intersect at the center of the hexagon, which is also the center of symmetry of the molecule, since the inversion through the center takes red carbons (above the paper at a distance of d before projection) to blue ones (below the paper at the same distance of d before projection), and vice versa. Now all symmetry elements of the molecule are clearly presented in Figure 1d, and the cyclohexane molecule in the chair conformation belongs to the D3d point group. Now turn to the [Co(NH2CH2CH2NH2)3]3+ complex built around an octahedral metal scaffolding (Figure 2a). If we view the molecule down the top front triangular face, we will get Figure 2b. Choose the paper as the projection plane; Figure 2c is the projection. Compared with Figure 1d, there are no changes in the axes of symmetry, but the three ethylenediamines destroy the σd planes and the center of symmetry, and thus [Co(en)3]3+ belongs to the D3 point group.
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MOLECULES WITH HEXAGONAL PROJECTIONS We begin with the cyclohexane molecule in its chair conformation. In textbooks, it is usually illustrated as Figure
Figure 1. Cyclohexane in the chair conformation and its projection.
Figure 2. [Co(NH2CH2CH2NH2)3]3+ and its projection. © XXXX American Chemical Society and Division of Chemical Education, Inc.
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DOI: 10.1021/ed500898p J. Chem. Educ. XXXX, XXX, XXX−XXX
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MOLECULES WITH OCTAGONAL PROJECTIONS Next we discuss molecules with eight-membered rings. Figure 3a is sulfur crown S8, and the view from above is Figure 3b.
paper is an 8-fold rotation−reflection axis S8 (also a C4 axis), since rotation by 45° around it sends red sulfurs to blue ones; then reflection through the paper changes blue to red again. The line connecting the midpoints of two opposite sides is a C2 axis; there are four. The plane passing through two opposite vertices of the octagon and perpendicular to the paper is a σd plane, and there are four of them. All these symmetry elements intersect at the center of the octagon, but sulfur crown has no center of symmetry, since the colors of two sulfurs at opposite vertices are just the same, so they were both above the paper or both below the paper before projection and, thus, cannot be interchanged with each other through an inversion operation. Now we can conclude that sulfur crown belongs to the D4d point group. Next we consider [8]annulene (Figure 4a,b). From the projection (Figure 4c), the line passing through the center of the octagon and perpendicular to the paper is a 4-fold rotation−reflection axis S4 (also a C2 axis). Two lines lying on the paper and passing through the midpoints of two opposite single bonds are C2 axes. Two planes perpendicular to the paper and passing through the midpoints of two opposite double bonds are σd planes. Like sulfur crown, [8]annulene has no center of symmetry; the colors of the two carbons at opposite vertices are just the same, so it belongs to the D2d point group. Now look at 1,3,5,7-tetra-methyl-[8]annulene (Figure 5a); its projection is Figure 5b. Compared with Figure 4c, the only symmetry element left is the 4-fold rotation−reflection axis S4, and this molecule belongs to the S4 point group.
Figure 3. Sulfur crown S8 and its projection.
Figure 4. [8]Annulene and its projection.
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Figure 5. 1,3,5,7-Tetramethyl-[8]annulene and its projection.
OTHER TYPICAL MOLECULES OF DIHEDRAL GROUPS Another typical molecule of dihedral groups is allene. We will not discuss it in detail and just give results in Table 1. For
Choose the paper as the projection plane; the projection of S8 is an octagon (Figure 3c). From Figure 3c, the line passing through the center of the octagon and perpendicular to the Table 1. Allene and Its Derivatives
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Table 2. Some Typical Molecules of Dihedral Groups
a If the atoms sit in the plane of the paper, they will be black. bFor Dnh groups, the σh plane is the plane of the paper. cFor eclipsed ferrocene, the pentagon in the paper represents dihedral planes contained within the plane of the paper and perpendicular to the principal rotation axis.
perpendicular to the polygon, the lines connecting the midpoints of two opposite sides, the lines connecting two opposite vertices, and the lines connecting a vertex and the midpoint of the opposite side. As for planes of symmetry, it may be the plane of the polygon itself, and planes perpendicular to the polygon and passing through the midpoints of two opposite sides or two opposite vertices, or passing through a vertex and the midpoint of the opposite side. If the molecule has a center of symmetry, it must be the center of the polygon. Apart from dihedral groups, the method of learning to draw 2D projections may also be useful for some other point groups and convenient to compare a molecule with its derivatives. As a complement to computer aided visualizations, this method is simple, effective, and easily carried out with colored pens or pencils and a piece of paper. This method also develops student’s skills to draw three-dimensional molecules in a twodimensional orientation and to find views of molecules that
comparison, several derivatives of allene are also contained. Table 2 gives other typical molecules of dihedral groups. In teaching the Dnd point groups, we found that the 2D projections of molecules are especially helpful to demonstrate the rotation−reflection symmetry operation of an S2n axis. It is not difficult for students to identify the Cn axis about which a rotation by 360°/n sends equivalent atoms into one another; then we make students think about a rotation by 360°/2n, which takes red atoms to blue ones, and ask them how to change blue to red againmake a reflection of course, so the Cn rotation axis is also an S2n rotation−reflection axis.
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CONCLUSIONS In summary, projected onto a suitable plane, the projection of a molecule of dihedral groups is usually a polygon or two staggered polygons. To look for axes of symmetry, pay attention to the line passing through the center and C
DOI: 10.1021/ed500898p J. Chem. Educ. XXXX, XXX, XXX−XXX
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illustrate the symmetry elements. Feedback from students indicates that 2D projections of molecules can clearly show the locations of C2 axes and how a σd plane bisects the angle formed by adjacent C2 axes.
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AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS This work has been supported by NFFTBS (No. J1103306). The authors sincerely thank the reviewers for their valuable comments and suggestions on this manuscript.
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REFERENCES
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