Systematic Procedures for the Classification of Molecules into Point Groups: The Problem of the DndGroup Most introductions to group theory and symmetry in chemistry present a flow chart or similar systematic procedure far identifying the point group to which a molecule belongs.' Although various published procedures differ in detail, they are largely similar. In all such procedures, the student reaches s point where he must answer the question: "Are there any Cz axes perpendicular to the principal axis, C,?" If yes, the molecule belongs to one of the LJ point groups; if no, t o one of the C's. Such Cz axes are not difficult to spot if the paint group is Dnh, hut my experience has been that many students find the CP axes difficult to locate for the D,d molecules, even when told the molecule helongs to such a point group, and know they phould he there. In faet I often have difficulty locating such axes myself. The source of the difficulty lies perhaps in the fact that the Cz axes in most D M m~leculesdo not lie along any bands, or even bisect any obvious bond angles, and so are not along any of the lines that a chemist thinks to look in locating symmetry elements. Thus, a beginner, failing to see the Cz axes, often misidentifies a D,d molecule as C,, (the planes of symmetry, since they are natural planes of the molecule, are relatively easy to spot). Actually, it is easy to distinguish between C,, and Dad, prauided that the procedure does not depend on heing able to see the Cz axes. In Dad, the C, axis is always a colinear SP" axis: in C,. this is never the case. And recognizing a colinear S1, axis is relatively simple. Looking along the principal axis, the molecule divides into two identical halves along a horizontal plane (which is not a symmetry plane), with the "hack" half of the molecule twisted by an angle 3609/2n with respect to the "front" half. This feature is usually obvious, as considerstion of some typical D,d structures should make clear: D2d: allene; DM: staggered ethane; D u : the square antiprism, or B ~ H I O ~LJsd: - ; staggered ferrocene; Dgd: staggered dibenzene chromium, etc. If a molecule possesses a Sz, axis colinear with a C, axis and vertical planes of symmetry, but no horizontal plane, then it must he Dnd, whether or not the Ca axes are readily located. How this should he incorporated in the flow chart procedures is a question not easy to answer. Perhaps the question: "Is there a Su, axis colineer with C,?" should precede the question about Cz2s.Perhaps instead the beginner should alwavs check for a n Sg. axis whenever he has tentativelv identified the molecule as C.... This is a matter for the
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S u c h flow charts have appeared in this Journal; see Carter, R. L., J. CHEM. EDUC., 45.44 (1968); Donohue, J., J. CHEM. EDUC., 46.27 (1969). Of the most commonly used textbooks in symmetry and group theory, the only one which warns students of the difficulties in dientifying D,d molecules, and presents an adequate alternate procedure is Levine, I. N., "Quantum Chemistrv." and Soeetm." .~~~ , Orbitals..~ ,~ . ,. Allvn , and Bacon. Boston. 1970. Volume 1. . o. 335. Orchin.. M.. . and Jaffe. H. H... "Svmmetru. \Vilrs-Inrertcie~~~r. Nmv Y o r k . 1971. p. l i 2 . call* rhr nr1enl:qm of renders tu the d i t l ~ c,It" 1,ur only derrr~be;an allernnte procedure fctr identifymy this pomt croup for the qwrml rare or I).,
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East Texas State University Commerce, 75428
Flow Chart for Identification of Schoenfliess Point Groups Many flow charts have been published in various places to help students in identifying the point groups of molecules and solid bodies. Most of these charts have one assumption in common; namely, they rely on the student to infallibly identify the symmetry elements in the order indicated. In practice, this does not often happen. Indeed, the most common mistake made by students is to misidentify the Dnd groups as Gnu. The reason for this misidentification is quite understandable. The identification of the Cz axes perpendicular to the principal axes, which are found in all dicyclie groups, while it is quite easy in the D,h groups, it is very difficult in the D,d groups, even when a model is used. If a model is not used, the identification of these perpendicular CQ axes is nearly impossible in LJnd groups. The flow chart shown in the figure presumes that this mistake may he frequently made and takes advantage of the faet that all D,d groups will have, coincident with the principal axis, an improper axis SP,, which is of twice the order of the principal axis. Hence, if the student fails to identify the perpendicular Cz axes, the identification of the improper axis will tend to bring him back to the proper conclusion. All persons have free use of this chart for their classes provided they give proper acknowledgment of its source. Joseph H. Noggle University of Delaware Newark, Delaware 19711
190 / Journal of Chemical Education
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