John P. Fackler, Jr. Case Western Reserve University Cleveland, Ohio 44106
Symmetry in Molecular Structure -Facts, Fiction and Fun
The advances we make in understanding molecular shapes and structures rest heavily on our ability to describe the average ("near sighted") structure of the species in question. This "near sighted" description of nature leads us to a "fiction:' cen tered on our attempt to model the nature we observe. Our understanding of molecular shapes and structures has advanced enormously over the past decade, due largely to the strides made in the development of computing hardware and software for X-ray crystallography. Paralleling this advancement has been the utilization of mathematical conceots originating in abstract group theory to summarize the properties of these molerules. Moleculnr crysttllloflaphy developed out ul'an understandingofcr~stulsymmetrieandthe realization that finite limitations exist for the number of different ways substances may crystallize. Crystallization itself is a process in which randomness (in liquid or aas phases) gives way to ordering in the solid state. The fact thnt moleculeionly a few angstroms on edgei arrange themselves in a highly ordered manner as they crystallize into solids permits is t i explore this order and thereby deduce the shapes of the molecules themselves. The orderliness of the arrangement of molecules in q s t a l s is never prrfert. Indeed, relief from perfect orddiness is essential toadequately study and nmplity the molecular structure in crvstals using X-rav techniaurs. Similnrlv, the svmmetry in nature that we visially observe is never rezly perfect. Our mind, howevrr, uses the apparent s p m e t r y to reduce the work required to rememher all the dt:tail. We remember differences from perfect symmetry. The fashion of creating beauty marks is an ancient example of man's attempt to dey stroy the monotony associated with apparent symmetry. These beauty marks amplify our perception of the object itself. (I am reminded of a conversation I had with a former student about svmmetrv. We remarked about the fact that some people perceivesymmetry in ot,j(!ct.i much more readily than do others. We are both badlv near-sighted and have h e n so since childhood. Our inahilitito distinguish imperfections in objects clearly distinguishable upon closer inspection may have been a factor in our ability to readily grasp symmetry relationships.) The existence of symmetry relationships, no matter how imperfect, dictates that these same symmetry relationships underlie our description of matter. Thus a molecule such as methane, CH4, with an average molecular shape described as a tetrahedron, has electronic structural properties which conform to this tetrahedral symmetry. If we descrihe one C-H bond in methane. the other three bonds are also described by virtue of thk symmetry accorded this species. However. if we could studv methane molecules individuallv a t some instantaneous moment, the symmetry would not aiways be there. Yet the advances we make in understandine molecular shapes and structures rest heavily on ourabilit). i; describe the average ("neur-siahttnl? structure of the s~ecies in question. It is this "near-sighted" description of nature that leads us to a "fiction."This fiction is centered on o w attempt to model the nature we observe. As scientists we develop generalized 'The Angstrom unit, 1 X 10-8cm,is being superseded by the pirn. Thus, a chemical bond is 100-300pm corneter, pm; 1 pm = in length.
descriptions of nature in order to better evaluate and predict its behavior. These models never are perfect but they tend to get better as more facts are uncovered. This, then, is the fun part of scientific investigation, the discovery of facts which destrov the fiction created earlier and the development of a new fiction to "understand" the& facts. The Facts Upon crystallization, molecules of similar shape and form arrange themselves in ordered arrays "infinite" in length compared with molecular dimensions. In a crystal 0.1 X 0.1 X 0.1 cm there are about 1020such molecules. Yet svmmetrv dictates that with only seven crystal systems, all crystals are describable in terms of the aneles and leueths of their repeating units. In two-dimensionk patterns such as are found in mosaic tiles or wallvaver, onlv four general svmmetrv " svs" tems exist (ignoring cbGr differknces)rone of these four systems (hexagonal) is seen in the Chinese pattern. F i m e 1.The parameters associated with repeating units i n t w o - and
Table 1. Parameters Associated with Repeating Units in Two-Dimensional Patterns
n t b
oblique rhombic
o+b
y=90° o = b y=90° o = b y = 120'
square hexagonal
All two-dimensional patterns are describable in termr o f the coordinateso, b, and theangle y in the reDeat unit.
Table
2. The Seven Crystal Synemr
Unir ceil. NO. inde-
-
crvvtat wstem
Laffice svmmew
ucnoent ~ilratneters
Triciinic
. . .. .. .-. 6 a
Tetragonal Rhombohedra1 Hexagonal Cubic
2 2 2 1
Parameters
-..- . . . -.
tbtc:u+p+y a+b+c;o=n=9o":p+90" a c; = = = e = b # c ; o = p =?=90D a = b = c;u = 0 = 7 + 90' a = b f c;,u = p = 90': y = 120°
E;z:;$bic 4
.
(I
+
+
@
.
= b = e:u = 3' = , = go9
I 21m mmm 41mmm 31m 61mmm m3m
All t,,ree.dimenriona~ are describable in termr of the coordinates a, b, and c and the angler o. p and 7 in the repeat unit (unit cell).
c;la*
Figure 2. A representation of a square planar complex such as FiCir2- with a Cfold rotation axis, C4,and four 2fold rotation axes. C2'(x).Cs'(y). C2"(&y). G r ( x . - y ) , perpendicular to C4.
Fiqne 1. Chinese panern, an example ofa panem confwmingtoa2dimensioMI space group in the hexagonal system. Table 3. Symbols for Symmetry Operations and Thelr Deflnnlons Symbola
Operation
C... In) ..
Rotation bv 2rlnradians Inversion through a symmetry canter Reflectomthrabgh a plane Rotat on oy 2 n l n followed oy rel8ection through a plane perpendicular to S, Rotation by ~dnfo~lowed by Inversion
;(TI n (m) Sn
(a
aSpeclroxoplc(SEhOBnfliesI SymbDllSm wlm notafion I" psrenmesis.
crystallographic (Hem*vln+&uguinl
three-dimensional patterns are listed in Tables 1and 2. X - R a y have wavelengths on the same order as molecular dimensions. 1-3 A.'l'hese X-ravs can be diffracted from planes of electron density associated-with atoms' repeating units in crystals. The conditions for the diffraction discovered by Laue in 1912 were developed by W. L. Bragg in 1913. The symmetry of the diffraction pattern is limited to one of only 230 possibilities. By mapping out the various possibilities resulting from repeating molecular units, structures can be determined. Since 3N narameters are essential to describe the ~ o s i t i o n s of N atoms in three-dimensional space, a t least 3N reflection intensities are needed. I n practice. it is desirable to overdetermine each required para&er. ~urihennore,since electron densities are affected hv thermal vibrations of the atoms and are not generally spherical, more parameters are needed. For a molecule having- 20 svmmetrv independent atom positions, 80-180 reflections are required and-more like 800-2000 reflections are desirable to deduce the structure with a high probability. Fortunately, this number of independent reflections usually can be obtained. Before we delve into the fiction of orhital relationships used to help classify electronic properties of molecules, I am going to present briefly the language used to describe molecular structure^.^ This language depends on the fact that we classify svmmetrv in a molecule bv assumine -Dronerties .. - the molecule to he fixed a t some point in space. It assumes that we have a precise description of the equilibrium nuclear framework. Five kinds i f symmetry operations are conventionally used, althoueh onlv four are necessary to describe point symmetry. ~istor~cally,~~ectrosco~~ists and crystallogriphen have used different but related definitions. Spectroscopic nomenclature generally will be used throughout this article, nlthough Table 3 contains both. 1) Pmper rotation. Rotation about an axis by lln of a circle ( 2 r h radians) may cause a molecule to return to a position indistinguishable from the first. For example, in PtCb2-, rotation by 90°, 2~14,Figure 2, about an axis perpendicular to the plane of the molecule leads to a structure apparently no different from the original. Rotation by 270°, C2, also pmduces a similar
result. A C4 rotation axis is said to be present. In general, rotations are designated by C In a square there are also five Cz axes. One is coincidental with Cr; the others are perpendicular to the C1 axis. 80 / Journal of ChemicalEducation
Flgue 3. T t e symmeby in a square. Dm. (a) Points poduced by C4.(b)New palm found on addition of a C2 perpendicular to C4.(c) New points generated by Ca. (dl Addition of ch. This prDduces elght new DOints, giving a total of 16 for D,,. Symmetry elements are included in the drawing.
2) Reflection (mirror symmetry, m). A planar molecule obviously has mirror symmetry since reflection in the plane of the molecule produces an identical structure. The planar PtCL2- anion also contains four mirror planes perpendicular to the plane of' the molecule itself. The plane of the molecule is labeled oh (horizontal)since it is perpendicular3to C4,the principal C, axis. The four planeswhich contain both the Cnand the C z ( l ) axes are labeled s, (vertical). 3) Inversion (center of symmetry).Reflection of all atoms in the molecule through a point (the center of symmetry) leads to inversion. A general position in space (x,y,z) becomes (-x,-y,-z) upon carrying out the inversion operation, symbolized by i. A square contains a center of symmetry. 4) Rototion-Reflection. A C4 rotation followed by reflection in the lane of the molecule.. m. a structure indistin... oroduces . guizhnhle imm the origlnnl one. This oper.~tionis Inheld S4 and can be thuught of as n nmhinnriun nf C, with on An S, operation may exist independent of the presence of C, and r h . The staggered form of ethane illustrates this. Along the carbon-carbon Ca axis it is posaihle to carry out a Sg symmetry operation,although there is no Cg axis or oh plane present. 4a) Rotatron-Inuersion (crystallographic).The crystallographic ooeration which is used in dace of rotation-reflection is rotation-inversion-rotation hv 2rln followed bv, inversion. It nrromplishrs the snm* ultimarc god hut i~ nor identical u i t h S,. Therymhnl R destgnates nn n-tikl n,tntim-invcr+n a & 6 is identical withSccarried out iwe times. tar -(isi.
An illustrative description of the effect symmetry operations can have on an arbitrary point is presented in Figure 3. In each case the general point (r,y,z) produces every other point when all the symmetry operations designated are performed. Such a figure4 can he helpful to use to discuss point The reader who is uninitiated into this topic should read an introductory cha~terin one of the hooks listed in the bibliopraphy. ~ryst&gr~phicnotation Labels such a plane as 4lm. A stereoeraohic oroiection. .. . , . F o r a detailed examination of the electron pair repulswn rhrory see (Xlespie. R. J., "Molecular Geometry," Van Nuatrand Rheinhuld, London. 1972.
'
group symmetry. Note that the number of points generated equals the number of operations. While we generally describe symmetry with four specific types of operations, it is possible to classify three together, S,, o, and i, into the general category improper rotations. Groups contain only proper, C,, and improper rotations. In fact, all groups are either entirely proper (no improper rotations) or one-half Droner and one-half imoroner. This classification is derived dam' the observation that S; (rotation hy 2a followed bv reflection) is indeed a and S3 is i. O ~ t i c aactivity l in molecules requires the absence of improper rotations. Classillcatlon of Molecular Structures The operations we have heen discussing are the operations required to precisely describe a mathematical group. We must understand what it means to combine these o~erations.In other words, what is the meaning of C p ( l ) X c4? The combination or multinlication of C d L ) X CA(molecule~implies that afteropera;ing with C4on the molicule the operation ('2( - ) is carried out. With C4 the molecule is rorated 18). 90° ahout oneaxisand then with C n ( l ) the rotated molrculr is nttated b~ 180° ahout another axis. A table which systematically lists each product of every symmetry operation in a group is called a group multiplication table, Table 4. The specific relationships required in order for symmetry operations to constitute a mathematical group are the following 1) Closure. The combination of operations can lead to no new
ooerations.
.~
2) There is one ooeration.. E.. a trivial one. which com-. Montitv. ~, ~~
~
~
~
~
mtrrps with all i,ther, and leaver them unchanged. 3, Ac>winttue1.0" The romhrnation 14operation4 is associative. Flrr example, the triple product A X ( R X C, equals \ A X Bl X
".
I~
4) Reciprocal. For every operation there is another (or the same)
operation, its inverse, which combines with it to produce the identity, E. In Table 4, note that each of these four relationships holds. A molecule which disnlavs . none of the svmmetrv we have heen discussing has only one symmetry operation present, E, the trivial one. This eroun - . is desienated CI. There are three groups which contain only two operations, C,, CZand Ci. C, contains a mirror plane, a,along with E. C2 implies the existence of the rotation Cz (and E), while Cj has inversion, i, along with E. Few simple molecules belong to Ci. Molecules with C, and Cz symmetry occur quite commonly. Groups containing only operations caused by a proper rotation axis are designated by the symbol C,. The operations found are C,, Cn2,Cn3,. . . ,Cnn(E).If the group contains n mirror nlanes in addition. it is labeled C.,.. The oresence of an Sznbperation, coincidental with C,, &hout'additional svmmetrv. bv. ST,. There are ".leads to the "eroun. svmholized . n proper rotations and n improper rotations present in this
-
.PIOUD. , .
C-type point groups are commonly found for small molecules. HzO, for example, belongs to CzU,NH3 to C3", and trans-dichloroethylene to C2h. Molecules belonging to Sz, are seldom found. Symmetry point groups belong to the D classification if they have C2 axes per~endicularto a ~ r i n c i ~C, a l axis. For D,. there are n C; axes perpendicul& to 6,. here are no improper rotational operations. There are two mirror image related structures for this type of molecule. The molecule is said to be chiral or dissymmetric. All molecules in C, and D, groups are dissymmetric. There are no improper rotations n m-. r. m rLr"" ".t
The presence of a mirror plane perpendicular to C, in the D classification produces D f i . Mirrors containing C,, without the presence of ah lead to the D,d groups. A line diagram summarizing the important steps that must be taken to deduce the point group symbolism for a given molecule is presented in Figure 4. A linear molecule such as HCI contains a symmetry axis about which rotation by any number of degrees leads to an indistinguishable structure. This axis is labeled an infinitefold rotation axis, C,. There are an infinite number of vertical planes which contain this axis. The point -a o . u svmbol ~. used for this group is C,,. A linear molecule such as Con has, in addition to C,,, a ah and an infinite number of Cp axes perpendicular to C,. Its point group symhol is D-h. A tetrahedron and an octahedron are related to a cube in that all three structures contain four C3 axes positioned 109.47 degrees (the tetrahedral anele) from each other. Molecules with this symmetry are speciall.y described to he cubic. In full tetrahedral svmmetw such as for the molecule methane.. CHa. .. there are 24symmetry operations present of the C2, CQ,S4, and the ad type. The symbol used for this group is T d . Removing all the improper rotations from the group leaves us with the 12 proper rotations present in the point group T . Addition of a symmetry center to T produces Th. An octahedron displays 48 symmetry operations. Its symhol is Oh. Removal of the 24 improper rotations leaves the point group 0 ,rarely found for real molecules. However, often the
Figure 4. A line cham used to identity point group symmehy
The presence of a mirror plane perpendicular to C, produces the group Cnh. The existence of this oh plane requires the presence of n improper rotation operations of the type S, (including ah and i). Note that C,h appears to be formed by adding o h to the group C,.
Table 4. Group Munlplkatlon Table for C, C,
E(Ca4)
C,
C4Wd
Ca3
L
~ i g u r e 5The . icosahedral Bt2H&anion
with
4, symmetry.
Volume 55, Number 2 Februaty 1978 1 81
energetically equivalent or very nearly so, Figure 6. These possibilities should not he forgotten when attempts are made to correlate physical properties such as the vibrational or electronic soectra with structure. A particularly simple approach that helps us remember molecular structures is one popularized by R. Nyholm and R. Gille~pie.~ I t assumes that valence electron pairs surrounding atoms take uv vositions consistent with electrostatic revulsive atom with only two electron pairs such as forces. T~US-A BeHz is linear, D-j,, since the electron pair distance is maximized by this arrangement. With BH3, the three electron pairs also lie in a plane which contains the B atom, the molecule having D3h symmetry. In situations where there is only a single electron pair bond between the central atom and the atoms hound to it, this electrostatic electron pair repulsion theory works quite well. Boron trifluoride is planar, while SnC12 is bent, Czu.as is the carhene CC12. Water, with its four electron pairs, also is bent, not linear. However, we must not assume that those non-bonding electron pairs produce an electron density tetrahedrally related to the two H-0 0 honds, Figure 7. I will return to this point later. Table 5 lists a number of examples of structures conforming to ti these electrostatic structural rules.
point group 0 is used by spectroscopists instead of the complete Oh group. The group 0 contains all of the spectroscopically useful symmetry information about the molecules exceptthe inversion operation. The pentagonal dodecahedron and the icosahedron are two regular polyhedra which display I h symmetry. There are 120 operations in this group with 60 in its proper counterpart I. The anion B,7H172.- .- is known.. Fiaure - 5. to have the full icosahedral symmetry. Some examples of this classificationare as follows: D-h: la-, C02, U0z2+; C,,: COS, HgClBr, CO; Cz,: SOz, Hz0, BrF3, CHzClz; D3h: BF3, PF5, Ni(CN)s3-, Ni(CN)s3-, crystalline MoSz, [ReHgI2-; C3": HCC12 '23": NH3, cis[CoBr?CI~l~-. ~is[CoBr~C13]~-, HCC1;; Dw: Dad: Co(OzCH)3; D4h: trans[CoBr~C trans[CoBr2Cla]3-, PtC1d2-; Td: CC14, NiC142-; Oh: NiF&. The Fiction and the Fun
Recognizing that we can describe properties of molecules in terms of their molecular structures, we realize immediately that these vroperties cannot contradict the correct structural results. of co&se, structures sometimes are incorrectly determined even using X-ray crystallography. Furthermore, some molecules may have two or more structures which are
I '''"
Figure 7. "Rabbit ear" model for the non-banding electrons In H.0
Figure 8. The trigonal bipyramid-square pyramid rearrangement according to the Bern mechanism. me structure lb) has . . has C..svmmetrv. .. . . It orssumablv . a somewnal dilteronl energy man m e groLna slate str~nurewlm L % rymmeny. However. the Energy dinerenee may be small a1 owing reanangements m occur at room temperature
Figure 6. Stereochemical rearrangement or pseudorotation of a tetragonally distovted Octahedral complex between the three equaly energetic structural minima, all of Din symmetry.
Table 5. Electron Pair Arrangements Using Glllesple-Nyholm Rules Number of Electron Pairs in Valency Shell
Number of Pairs
Number of Lone Pairs
Bonding
Arrangement
Shape of Molecule
Examples
2
Linear
2
0
Linear
HgCi,, As(CN)Z-
3
Equilateral triangle
3
0
BFa
2
1
Equilateral triangle V-shape
4
Tenahedron
4
0
Tetrahedral
3
1
Trigonal pyramid V-shape
BF4-, TiC14. Zn(NH3)2C12. CU(CN),~NHs. PCla. PFs
5
6
Trigonal bipyramid
Octahedron
2
2
5
0
4
1
Triwnal bipyamid SeFlb
3 2
2 3
T-shape Linear
6
0
Octahedron
5
1
4
2
Square pyramid Square
Non-nanssonelements and "anslim elemems wim c 9 and doconfigurations. 1s in an equatorial position.
*me lon*palr
82 1 Journal of ChemicalEducation
SnCh (vapor)
H20. F20. H C PClr, NbCI5, Tack SeF,, TeCI,, (CH3)~TeCIt. PbO. SF4 CIFJ. BrFa. C&, lCln IC12-. 13MOFs, P C K , SnCls2TiFsZ-. VFsIFs. SbClr21CI.-.
XeF,
....
Figure 9. Electron density of water in the plane of the lone p a i s of H20. Taken by Permission from Streihveiser, A.. Jr. an6 Owens. P. H.. "Orbital and Electron Density Diagrams." MacMillan and Ca.. New Yo*, 1973.
An interesting structural problem existed until recently with regard to the five electron pair structure associated with PFs. Electrostatics would predict the trigonal bipyramid, Fieure 8a. to be the observed confieuration. with its Dlr s G m e t r y . ' ~ nexamination of this str;cture suggests that the three eauatorial bonds are identical bv svmmetrv but different from thk two axial bonds. Several str;ctures studied by X-ray crystallography show this geometry. PF5 vibrational spectroscopy displayed the expected result, namely two different P-F stretching energies. Yet, the verv-i m ~.o r t a nstructural t tool in solution,nucle& magnetic resonance, which probes the electronic environment of atoms. showed onlv one tvDe of fluorine atom. The answer to the nuzzle was realized bv recoenizine that the PFs molecules insolution undergo a rapid (
