From Molecular Point Group Symmetry to Space Group Symmetry
Brian Hathaway University College cork, Ireland
An undergraduate experiment in model building One of the more difficult conceots that a universitv student in chemistry is rrquired tu grasp is that of the threr dimen. sionsl nattrn! of the sternrheniistrv of even simnle mulerulc.i. Very early in the course the student is introduied to the tetrahedral stereochemistrv of the carbon atom in methane (taught ideally through the \'alrnce Electnm Pair Hepulsiun Theory) and wch quickly has tu f a r r the prohlem nf reprrsenting a three dimensional stereochemistry on a t w di~ mrnsional sheet of paper. The uvailahility of siml~lrmolecular mcdels undoubtedly hrlps to redun: this pruhlun, hut thr uie of tal1ert.d h o n d or the drawing (ria tetr:lhedron uithin the framework of a cuhe (Fie. l J wls seems to sidetrack the t u dent from the mure impurtant question, namely, an appreciation of the thret: diniensiunal nature of the sterenchemistrv of simple molecules. In manv undergraduate courses molecular symmetry is taught early in thrcourse as a suhject in its own right andcan he used to "describe" thr shape o i a mnlecule \in the element.; of symmetry present, and c& ultimately he used to determine the point group symmetry of a three dimensional molecule. Prooerlv illustrated molecular svmmetrv makes a substantial contrib;tion to the students feeiing for the three dimensional asoect of molecular structure. The next step in the suhject of "molecular symmetry" is usually a discussion of character tables, includina svmmetrv represmtations, and their use in deducing selectibn-rules foi the various branches of molecular spectroscopy and to the calculations of the quantum merhanicd propciiies w simple mdecules. Unfortunately, while thrse are all clearly lnudirhle ublrctives. thrv all tend to il~rertthe student irom the three dimensional basis of molecular symmetry, and this important aspect is not developed further. This stunted growth may he prevented by the use of space erouo models as a set exoeriment in the undergraduate Lavhing laboratory, prefrr;~t,lyrestricted to a s i m a e Sp:lc(. Group, such a i 1'2,ir, which is sufficienr to dernonstratt. the eitlvt of the two trnnslational elements of symmetry ( n a m ~ l y a 21 screwaxli a t ~ da t-glide plane) m d the rfftvt uf a rrnter of symmetry in three dikensonal space. The set experiment may involve
. .
1) building a space group model (see later), or
2) using prebuilt space group models to identify the effect of the three elements of symmetry present on the asymmetric unit of the structure. The table sets out the undergraduate timetable of the two parallel streams that develop from the initial introduction of symmetry a t University College, Cork, in the second year of a four-year fbl
undereraduate course. In the third vear. . . on the left-hand side are the more mi~thematicallyorientated ropivs such as character tablrs, selection rules, iluckel r a l c d i t i m etc. while rhe right-hand side follows a second year course on crystal chemistry (including X-ray powder photography) with a third-year course on space group symmetry. In the fourth year the left-hand course of the table leads naturally to the applications of symmetry in theoretical chemistry and in various physical techniques such as single-crystal magnetism, esr and electronic spectroscopy, while the right-hand course leads from space groups into the more advanced crystallography course. includina.. crvstal . structure determination. It is realized that most univtnities already give a Iwture nmrse on crvstallwranhy (includinr. sl~acr . -rruup symmetry), but rhe atnhor has found that there are advantapes he obtained in splitting the course into two unequal sections in the two successive yean. l i a n l e ~ y ~ ~ ~ a ~ : e ~ r o uin~ the ~)m third m t year . t ~ f~llow~~d hy~. crvstallorraohy . . . in the fturth sear. These are 1) An introductory course on space group symmetry, restricted to one simple space group, P21/c, acts as a bridge between point group .~ symmetry and a &&advanced erystallogra~hycourse 2) The opportunity to build, or examine in detail, a three dimensional model of a simule . soace . erauodaes - . enable the student to ohtain a feeling fur the three dimensional nature of a space group before embarkingon themuch moredifficult question ofhowastructureis actually determined 3) The student obtains from the building, or examining,of a simple space group model, a better understanding of the relationship between general and special positions in a space group and the related question of the point group symmetry of a molecule in ageneral position and the site svmmetrv. .. when the same molecule is mesent in a suecial position 4) The student obtains a better idea of how the local molecular axes ~
ficult to convey in a lectureor on paper and which is crucial in interpreting the physical properties of compounds involving oriented single-crystal techniques especially, electrun spin resonance, electronic, infrared and Raman spectra and magnetic anisotropy. In the author's experience the advantage of a student having a preliminary stab a t space group symmetry, separate from the more difficult tooic of crvstal structure determination. clearly overweighs tde disadvantages of splitting the tradi: tional crystallography course into two (unequal) sections.
Molecular Svmmetrv Timetable First Year
Nil
Second Year
Elements of Symmetry and Point Group Determination.
Third Year
Character Table 3b Seiedion Rules 3c Huckel M.O.
J
Fou~thYear
Figure 1. The tetrahedral stereochemistryrepresentation (a) by tapered bonds and ( b )within a cube.
166 1 Journal of Chemical Education
3a
4a 4b
Crystal Field Calc. Single-crystal phy~icaitechniques
I
3.9
Translational Elements of Symmetry
3b
Space Group Symmetry
4a Crystallography Crystal Structure Determination
4b
proleeted Figure 3. A Plut-plot of the [Cu(dien)2]zfcation of Cu(dienl2Cl2HS0 down Me a' axis, with the elements of symmetry added.
Figure 2. A Pluto-plot of Me [Cydienb12+cation ot Cu(dienWirHz0prdeeted down M e &xis with the elements of symmetry added. Space Group Model (Building) Experiment Each student is provided with 1) A copy of the original paper of the X-ray structure determination (differentfor each student); 2) A corkboard, cork spheres, and brass wire for model huilding (or B space group model).
The student is asked by reference to the original paper to either build a model of one asymmetric unit in the unit cell and to indicate the positions of the symmetryrelatedunits by putting in key atoms, or to identify on the prebuilt model
provided the asynmrtric unit and the symmetry related units. Each student then enters, \,ia thv computer terminal (availablr in the lal~oratury)the coordinates of the main atoms in the as)mmerric unit and penwttes, via the plortw facility on the comnuter. of the unit cell \,ieu,ed alunrr the a-, . .a .nmiection . b-, or c-axis. The student is asked to write up an account of his model, to answer a list of general questions, and to indicate on the relevant "view" of his unit cell the positions of the elements of symmetry (center, screw, and glide-Figs. 2 and 3) relevant to his structure. Time Inuolvements-approximately 10 hr of practical time, is required by the Space Group model building experiment which can be reduced to 5 hr by the use of prehuilt models, hut the problems that students have in calculating for the first time the atomic coordinates of the symmetry related atoms in a unit cell should not be underestimated. Should this approach become too time consuming, however, the student can be transferred to a set of prebuilt Space Groups models available in the laboratory. The author wishes to acknowledge the help of Dr. S. Motherwell (Cambridge University) for providing a copy of his X-ray crystallography plotting program PLUTO and to the Computer Bureau UCC for computing facilities.
Volume 56, Number 3, March 1979 1 167
