Research: Science and Education
Modified Stereographic Projections of Point Groups and Diagrams of Their Irreducible Representations Sidney F. A. Kettle School of Chemical Sciences, University of East Anglia, Norwich NR4 7TJ, England
Introduction: Nodes Fundamental to a student’s understanding of quantum chemistry is the concept of nodes. It is in large measure their nodal patterns that distinguish s orbitals from p, p from d, and bonding orbitals from antibonding. A point seldom made sufficiently strongly is that the subject of group theory in chemistry is also concerned with nodal patterns. In particular, the different irreducible representations of a character table are distinguished by the different nodal patterns associated with them. The reason why there are sometimes irreducible representations that do not have basis functions listed for them at the right-hand side of a character table is that they require functions with so many nodal surfaces that no simple algebraic function describes them. This absence of basis functions can easily lead to the impression that there are irreducible representations which are more difficult than their more familiar counterparts. Fortunately, there is a way in which all irreducible representations can be treated equally in this respect, and as a bonus, the physical significance of each irreducible representation can be made more evident. This can be done by giving diagrammatic representations of each irreducible representation. Their nodal pattern differences are thus highlighted, all irreducible representations are treated equally, and it is a good exercise for the student to show that functions represented in the diagrams do, indeed, transform correctly. Although such diagrams can be drawn for the cubic groups, the presence of symmetry elements at inclined angles to each other makes them somewhat cumbersome; and it is doubtful if this, complicated by the triple degeneracies which have to be represented, makes them helpful to students. I shall therefore confine my discussion to the axial groups, for which the diagrams conform to a common pattern (a feature which makes them more readily understood), although I give the tetrahedral group as an example of the cubic. Stereographic Projections Stereographic projections of the crystallographic points are to be found in some books (1), but they seem currently to have gone out of fashion even in texts on group theory (although they are still to be found in crystallographic texts). Perhaps
this is because they indicate symmetry elements and symmetryequivalent points, whereas it is the symmetry operations which make the points equivalent that are the focus in contemporary chemical applications of group theory (2). The first step in the present discussion is to reintroduce stereographic projections, but in a slightly modified form. As an example of this approach, in Figure 1 I give a modified stereographic projection for the C2v point group alongside its traditional counterpart. The new form differs from the traditional in several ways. Most important is that, in the new, beside each point is indicated the unique operation that transforms the origin point (denoted by the identity element E) into it. Next, the new is based on a solid circle, whereas the traditional is based on a broken circle. In the traditional, a solid circle indicates a mirror plane perpendicular to the direction of observation. Unfortunately, this meant that symmetry-related points commonly eclipsed in the projection, necessitating a convention to cover this point. To make each point unique so that a unique symmetry operation may be associated with it, I will use a different convention to describe quantities such as a horizontal mirror plane reflection operation. Thus the use of a broken circle in the C2v projection to indicate the absence of such a mirror plane may be discarded; solid circles are easier to draw!
Figure 1. The (a) modified and (b) conventional stereographic projections of the C 2v group. The modified projection shows the effects of the operations of the C 2v group on the general point denoted by E. To remove any ambiguities (although these do not really exist because the labels in the figure remove them), the choice of orientations of the symmetry elements σv and σv ′ is shown.
JChemEd.chem.wisc.edu • Vol. 76 No. 5 May 1999 • Journal of Chemical Education
675
Research: Science and Education
Finally, not every symmetry element is diagrammatically represented (no C2 rotation axis is drawn in Figure 1). Again, in keeping with contemporary group theory, the primary concern is with the consequences of symmetry operations, not with the existence of symmetry elements. There is already an adequate supply of labels in the figure, sufficient to uniquely identify the omitted features, and so such labels will be introduced only when needed—or just helpful—to remove ambiguities. There is always enough information in the diagram to determine uniquely all symmetry elements and, more important, any additional labels such as (1), (2) which have been placed on them. For axial groups, the axis of highest symmetry will invariably be placed at the center of the diagram and will be made evident by the caption that always accompanies such figures.
C2v
E
C2
σv
A1
1
1
1
σ v′ 1
A2
1
1
{1
{1
B1
1
{1
1
{1
B2
1
{1
{1
1
Pictures of Irreducible Representations Using the stereographic projection in Figure 1 we can now give diagrams of the irreducible representations of the C2v point group. These are shown in Figure 2, where the pattern of Figure 1 is to be regarded as implicit. For convenience, the C2v character table is also given in Figure 2. It is a simple matter to show that the + and { signs in the diagrams in Figure 2 correspond to the entries 1 and {1, respectively, in the character table. As a second simple example, consider the point group C2h. This is a group for which the conventional stereographic projection contains eclipsing points, so it highlights the difference of the present diagrams. Figure 3 is an attempt to show the situation in three dimensions. The diagram includes the symmetry elements explicitly. Most important is the horizontal mirror plane (it leads to the eclipsing), which is shown as the large central circle (drawn in perspective as an ellipse). The smaller circles (ellipses) above and below it along the twofold axis contain the set of symmetry-related points that are the subject of this discussion. They correspond to the circles which appear in the subsequent diagrams (the large, central, mirror-plane circle does not appear in these diagrams). The new-type stereographic projection appropriate to the C2h group is shown in Figure 4. In it, points “above” the σh horizontal mirror plane in Figure 3 are indicated on the inner circle; points “below” the horizontal mirror plane are indicated on a second circle, concentric with the first and larger than it. Figure 5 shows diagrams of the irreducible representations of the C2h point group; again, the character table is included. As an aid to comprehension, Figure 6 shows these same irreducible representation diagrams using the perspective of Figure 3. However, perspectives akin to that of Figure 5 will be used in all the following diagrams. For the two groups so far exemplified, simple basis functions are available for all the irreducible representations. The final example is one in which this is not the case. It is the important group D4h, for which simple basis functions are not available for the A2g , A1u, B1u, and B2u irreducible representations. The last two are spanned by f functions; the first, by R z (but most students find rotations difficult to handle) and by a g orbital function; and A1u , by an h orbital function. In D4h there is not only a horizontal mirror plane reflection but also other symmetry operations for which an ambiguity arises. There are two sets of twofold axes perpendicular to the C4 axis and two sets of mirror planes containing this axis. Although it is not strictly necessary (given the final projec676
Figure 2. The nodal patterns associated with each of the irreducible representations of the C 2v point group. This diagram should be interpreted with reference to the modified projection of Fig. 1. For ease of reference, the C 2v character table is also given.
Figure 3. A three-dimensional diagram showing the symmetry elements of the C2h group and also the result of the corresponding operations acting on the point labeled E. The same labels are used in this diagram to represent both symmetry element and operation. In the new-type stereographic projections of the following figures, only the top and bottom circles (drawn here in perspective and so as ellipses), those containing the symmetr y-related points, are shown. Although in this diagram the top and bottom circles are the same size, in the following figures the bottom circle is drawn the larger, so that it is not eclipsed by the top circle. For the same reason, in the following figures the points here labeled i and σh are placed in the lower, larger, circle in the region outside that eclipsed by the top circle.
Figure 4. A modified stereographic projection for the C 2h group. E represents a general point. Two of the symmetry operations transform this into points below the horizontal mirror plane σh. In the diagram, these points are located between the “above the σh” circle and a larger, concentric, circle representing the “below the σh” region.
Journal of Chemical Education • Vol. 76 No. 5 May 1999 • JChemEd.chem.wisc.edu
Research: Science and Education σh
C2h
E
C2
i
Ag
1
1
1
1
Bg
1
{1
1
{1
Au
1
1
{1
{1
Bu
1
{1
{1
1
Figure 5. The nodal patterns associated with each of the irreducible representations of the C2h point group. This diagram should be interpreted with reference to the modified projection of Fig. 4 and the three-dimensional picture given in Fig. 3. For ease of reference, the C2h character table is also given.
Figure 7. A modified stereographic projection for the D4h group. E represents a general point. Points below the horizontal mirror plane σh are represented in a way similar to that adopted in Fig. 4.
tion one can always work backwards and determine the particular choice used in its construction), life is made much easier if the choice is indicated in the stereographic projection. I have adopted the convention that, for example, the members of one pair of σd mirror planes are separately denoted σd (1) and σd (2). The modified stereographic projection of the D4h group is given in Figure 7 and its irreducible representations are diagrammatically represented in Figure 8, along with its character table.
Figure 6. The irreducible representations of Fig. 5, this time drawn in the perspective of Fig. 3. Beware of one potentially misleading aspect of these diagrams. The phases shown do not just relate to the planes on which they are drawn but to the corresponding quadrant of space (quadrant, because there are four operations in the C2h group).
2C4 C2 2C2′ 2C 2′′
σ h 2σ d 2σ d′
D4h
E
i
2S4
A1g
1
1
1
1
1
1
1
1
1
1
A2g
1
1
1
{1
{1
1
1
1
{1
{1
B1g
1
{1
1
1
{1
1
{1
1
1
{1
B2g
1
{1
1
{1
1
1
{1
1
{1
1
Eg
2
0
{2
0
0
2
0
{2
0
0
A1u
1
1
1
1
1
{1
{1
{1
A2u
1
1
1
{1
{1
{1
{1
{1
1
1
B1u
1
{1
1
1
{1
{1
1
{1
{1
1
B2u
1
{1
1
{1
1
{1
1
{1
1
{1
Eu
2
0
{2
0
0
{2
0
2
0
0
{1 {1
Figure 8. The nodal patterns associated with each of the irreducible representations of the D4h point group. This diagram should be interpreted with reference to the modified projection of Fig. 7. For ease of reference, the D4h character table is also given.
A Cubic Group All the examples I have given are of axial groups, those for which there is a unique axis of symmetry, but this approach can be extended to the cubic groups. Inevitably, the resulting diagrams are somewhat complicated (many more symmetry operations have to be shown) and therefore less useful for teaching, although they are useful for research. As an example, Figure 9 gives the modified stereographic projection of the tet-
JChemEd.chem.wisc.edu • Vol. 76 No. 5 May 1999 • Journal of Chemical Education
677
Research: Science and Education Figure 9. A modified stereographic projection of the group Td . The four 3-fold rotation axes are denoted by the labels 1–4, shown at the center of the face through which they may be regarded as emerging. The stereographic projection consists of the six circles at the center of this figure; but for ease, the smallest circles (each of which may conveniently be associated with one of the faces of the tetrahedron), are shown enlarged along with the identification of each of the 24 symmetry-related general positions. Because the identity element is contained within the circle labeled 1, all symmetry operations are identified by a label that includes the 1 if possible. Clockwise rotations about the 3-fold axis emerging from a particular face are denoted by +; anticlockwise by {. Mirror-plane reflections are denoted by the two 3-fold axes that lie in the mirror plane; C 2 rotations by the two 3-fold axes the angle between which is bisected by the corresponding C2 axis. S 4 operations are similarly labeled, a rotation clockwise about an emerging axis being denoted +. The irreducible representations of the T d group could be shown if the four small circles at the center of the diagram (and/or their enlarged partners) were each subdivided into six and a phase was associated with each of the resulting 24 segments.
rahedral group Td , from which the interested reader can determine the details of the extension referred to above. The “tetrahedral” nature of the group is more evident in this representation than in the conventional. For convenience in denoting the effects of the symmetry operations, key parts of the diagram are reproduced alongside the main diagram. In this diagram, too, it is helpful to include the parentage of the labels used to denote the results of most of the symmetry operations (the numbers 1–4 suffice for this). For evident reasons, we do not represent the irreducible representations of the Td group. Conclusion It is inevitable when a line in a character table is replaced by a diagram (sometimes by several diagrams, in the case of degenerate irreducible representations) that the space occupied is increased considerably. This is the major disadvantage of the diagrams presented in this contribution. They are perhaps more suited for a lecture presentation, when they can be projected onto a screen and copies given to the students, than for inclusion in a textbook. However, they give a physical nodalplane meaning to irreducible representations, which can be very valuable when the student first encounters character tables. They have other advantages as well. First, when it is necessary to use projection operators in the construction of symmetry-adapted functions, the problem of correctly handling the operations arises. Even for some simple systems it is easy to become confused over whether a particular operation has already been considered. The modified stereographic projections presented here almost entirely eliminate this problem. The pictures of irreducible representations readily enable demonstration of the important fact that integration over a function
678
other than a totally symmetric one leads to an answer of zero. Simply draw tiny, symmetry-related boxes in each separate region of each of the diagrams; summing the contributions from these gives the answer zero for all but the totally symmetric irreducible representation. The same answer results for all choices of general positions of the small boxes, demonstrating the result. It can be helpful to superimpose two overhead transparencies. Symmetry-adapted functions can be obtained by inspection if one transparency depicts an individual irreducible representation as in Figures 2, 5, and 8 and the other shows the way a basis function (e.g., an orbital labeled “a”) is converted into b, c, d, etc. by the symmetry operations. Multiplication of the superimposed entries followed by their addition gives the required symmetry-adapted function. Next, the superposition of two transparencies depicting different irreducible representations (or two components of a degenerate irreducible representation) enables their orthogonality to be demonstrated easily. Finally, something for the more advanced user of group theory: these diagrams, both stereographic projections and diagrams of irreducible representations, are easily extended to cover the double groups, making them easier both to visualize and to handle (3). They can also be simply extended to triple and quadruple—and even higher—groups, should this become a valuable thing to do (3). Literature Cited 1. See, for example, Point Group Character Tables and Related Data; Salthouse, J. A.; Ware, M. J., Eds.; Cambridge University Press: New York, 1972. 2. See, for example, Symmetry and Structure; Kettle, S. F. A.; Wiley: Chichester and New York, 1995. 3. Kettle, S. F. A. Spectrochim. Acta A 1998, 54, 1633–1638.
Journal of Chemical Education • Vol. 76 No. 5 May 1999 • JChemEd.chem.wisc.edu
