Group Theory in Advanced Inorganic Chemistry An Introductory Exercise Robert A. Faltynek Philadelphia College of Pharmacy and Science, Philadelphia. PA 19104 Group theoretical principles are important in constructing molecular orbitals, interpreting spectra, and simplifying quantum mechanical calculations. Recently published inoreanic textbooks eenerallv introduce a o u p theom a s a prel;de to discussion of covalent bonding i n d spe&oscopv: a welcome inclusion that facilitates in-depth study of thkse topics (1-5). Regardless of the lucid tekt coverage now available, many students struggle during their first encounter with group theory for two reasons: inability to connect abstract mathematical formalism with concrete application, and inability to visualize or mentally manipulate three-dimensional objects. The exercise described here demands active class oartici~ationin a Drocess that pictorially develops some key group theoretical principles. It. therefore. addresses both areas of difficultv mentioned above and provides a basis for a deeper look a t molecular symmetry and its consequences. W o somewhat unrelated pedagogical goals stimulated development of the exercise. The p r h a y goal was to differentiate symmetry operations from symmetry elements in an emphatic and unambiguous manner. Inorganic textbooks often define symmetry operations and symmetry elements on the same page ( I d ) , and usually in the same paragraph ( I 4).Althoughit is bothlogical and reasonable for experts in the field to wnsider operations and elements as inextricably r e lated concepts(6),there is a tendency for uninitiated students to confuse the more familiar idea of element with the more pertinent idea of operation when both definitions are offered a t the same time. The secondary goal was to demystify the origin of improper rotations S,". An improper rotation is a binary operation involving wnsecutive application of the unitary operations of rotation and reflection through a perpenWhy is this particular dicular plane (Cnma h = a&""' = Snm). binary operation given its own special name and operator symbol? Why do we not invent symbols for all binary or higher order operations? Answers to these questions are not provided in recently published inorganic texts (161,or in Presented in part at the 204th National Meeting of the American Chemical Society. Division of Chemical Education; Washington, DC; 23 August 1992.
20
Journal of Chemical Education
other commonly available sources on chemical applications of gmup theory (G11). Looking beyond the original two goals, the exercise is a n outline for assembling concepts prerequisite to the study of character tables, which in turn are the basis for applying group theory to chemical problems. I n special topics courses or other situations where more time is allotted to molecular symmetry, questions raised during the exercise can be elaborated into detailed discussions of group theory a t the level of Cotton's classic presentation (6). Evaluation data from students in one-semester advanced inorganic courses offered over a five-year period endorse the exercise a s a means of both introducing pointgroup symmetry arguments and sharpening spatial visualization skills. The data also indicate favorable reaction to the participatory, active mode in which the exercise unfolds a s described below. Classroom Methodology Begin by drawing the four parent squares illustrated in Figure 1.The labels CY, Y,Z) correspond to Cartesian coordinate axes t h a t m u s t remain mutually orthogonal throughout the exercise. Introduce the convention of righthanded versus left-handed coordinates and explain that the squares in Figure 1are spatially indistinguishable except for the (X, Y, Z) labels, which are included only to define different axis permutations describing the same geometrical figure. Note that the four parent squares are equivalent but not identical when the coordinate labels (X, Y, Z) are specified. Show that each of the four squares in Figure 1can be rotated sequentially counterclockwise by 9Ooaboutthe Z axis to generate three more squares with permuted axes a s inFigure 2. Point out that under the rule requiringmutual perpendicularity of CY, Y, Z), there are 16 possible ways of labeling a square with permuted (X, Y, Z) axes, because Z can be directed either into or out of the plane of the figure (2 degrees of locational freedom), leaving 4 degrees of freedom for the speciiieation of X and 2 degrees of freedom for the specification of Y.Then the overall number of permutations is [(2 x 4 x 2) = 161. Accordingly, Figure 2 illustrates all possible ways
tions C,m, cu,cd,ah, and i to answer this question, being sure to emphasize the action component of rotation, reflection, and inversion. The appropriate symmetry operations are listed below each equivalent square in Figure 3. Verify that XorY XorY Ca4 yields square 1 unpermuted, (+) (+I establishing the identity or doX,Y axes in nothing operation. Lead the class figure plane to realize that 14 of the 16 squares Lefthandedaxes Right handed axes in Figure 3 obtain from applying a Z axis out of figure plane (3) one-step or unitary symmetry opZ axis out of figure plane (1) eration to square (I), but that a two-step, or binary operation is required to generate the second and fourth equivalent squares highlighted in the second row of Figure 3. Formally define the improper rotation operation S,m = ohCnm = Cam a h and assert that S,m is the only binary symmetry operation necessary to complete Figure 3. Inquire a s to why no other binary opRight handed axes Left handed axes erations appear in Figure 3, and Z axis into figure plane (4) 2 axis into figure plane (2) promise to account empirically for their absence with the definition of Figure 1. Cartesian coordinate axes definingfourparent squares. closure.
A -
of drawing equivalent squares with labels (X, Y,Z) in different ~ositionsif axis ortho~onalityis maintained. Challenge the Elass to draw coordinatkabefed squares other than thoie in Figure 2 and show that any proposed additions to the figure are, in fact, redundant. Now inquire what must be done to the square in the upper left-hand comer of Figure 2 [labeled square (111 in order to permute X, Y, and Z to their new positions in each of the other squares. Define and discuss symmetry opera-
Figure 3. Symmetry operations applied to square (1) in Figure 1, generating 16 equivalent squares. Point group = D4h. Figure 3 Conventions
Figure 2. Counterclockwise rotation of four parent squares.
Operations applied to square (1) in Figure 1. Zaxis perpendicular to the figure. C,,m(Z)= m counterclockwise (36O0/n)rotations about thezaxis. a,(* or aJY), and C2(4or C2(Y)= reflection through XZ or M pLane, and 180" rotation about Xor Yaxis, respectively. odXV and C,'(XY) = reflectionthrough plane definedby theZaxis and the line y = -x, and 180" rotation about the line y = - X respec and G(XY)= reflection through plane defined by the Zaxis and the line y = x, and 180" rotation about the line y = x, respectively. Volume 72 Number 1 January 1995
21
Figure 4. Binary operations and group closure. Operations applied to square 1 in Figure 1 in the order right-to-left. All rows illustrate closure. Rows 3 and 4 illustrate non-commutivity. Row 5 illustrates the identity operation.
Figure 5: Group operations obey the associative law. Row 1: Ternary operation odXY) oh C&Z) = Cz'(XY). Rows 2 and 3: Subset binary operations oh C&Z) and oflXY) oh = i and Czf(XY),respectively. ROWS4 and 5: Reconstitution of temaly operations odXV [oh c&z)] and [odXY)oh] C&Z) = W(XY).
Figure 4 Conventions Operations are executed around the configuration of (X, Y, Z) indicated in square (1)in Figure I ; that is, right-handed coordinates with the Zaxis outward. Operations applied in the order right-to-left in the eauations.
Because Fipure 3 elaborates the svmmetm onerations necessary to generate all possible pe-utations bf the (X, Y,Z) labeled square, it is logical to postulate that the ODeratious taken together comprise a kosed collection of discriptors unique to this particular geometrical figure. 11lustrate, a s partially done i n Figure 4, t h a t binary combinations of operations other than the apparently nec.. essary S,m are identical to unitary operations already cousidered, thus defining closure. Observe that Figure 4 is the start of a multinliea~ontable of svmmetrv on&ations ,~ ~ - as-~ sociated with t6e square and tha
