loduction to Molecular Symmetry
It has become apparent from the recent literature that the successfuluse of modern spectroscopic methods for the determination of strncture and bonding properties of molecular species requires a familiarity with molecular symmetry. The purpose of this article is to introduce symmetry terminology as well ss to provide a methodical scheme for deducing symmetry point groups by inspection of molecular geometry. The following procedure has been developed and tested successfully on graduate and advanced undergraduate students. The determination of point groups is based on the student's ability to identify certain symmetry elements and to perform symmetry operations. All molecules can be effectivelydescribed by specifying a maximum of four symmetry elements. Before discussing these elements and operations in detail, a clear distinction het,ween the two must be made since they are closely related and often confused. A synzinetry elemant may be defined as a line, point, or plane with respect to which one or more symmetry ~perat~ions may be applied. A symmetry operation, however, is the movement of a geometrical figure (or molcmle) relative t o some symmetry element such that all the points on the figure coincide with equivalent points in the original orientation. I t might be noted that the center of gravity is usually, hut not always, maintained. As an illustration, let us consider the linear molecule, carbon dioxide:
Proper Rotation Axis, C.
A molecule possesses a proper rotation axis of order n if rotation about the axis by 2 n / n or 360°/n produces a configuration indistinguishable from the original. For example, there are a multitude of rotation axes for the benzene molecule, which can be considered as a regular hexagon (Fig. 1). One readily recognizes a Ceaxis
Figure 1.
Proper rotetion axes for the benzene molecule.
lying perpendicular t o the molecular plane and passing through the geometric center. Clockwise or counter6 1into 2, 2 into 3, etc., clockwise rotation by 2 ~ / brings and results in an identical orientation. I n addition, one also observes six C2 axes at right angles to the Ca axis. Three of them bisect the sides and three bisect the angles of the molecule. The representative rotation axes for the regular Table I.
Symmetry Elements, Notations and Operations
Element 1. Proper rotation axis
There exists an axis (dotted line) through the carbon atom and perpendicular to the bond axis such that a clockwise or counterclockwise rotation of 180" will bring Ol into 02. Since the oxygen atoms are indistinguishable, the final orientation is indistinguishable from the original. I t is readily evident from the above definitions that the axis is the symmetry element, and the process of rotat,ion by 180" is the appropriate symmetry operation. The four symmetry elements and their operations are listed in Table 1 along with the Schoenflies symbols.' Each system will he discussed along with illustrative examples.
Notation"
Operation
C,(n)
Rotation about axis by
i (7)
27/"
Inversion of aU points through the center c (m) Reflection in the plane S,(S,) Rotation about the axis by 2r/n and reflection in the plane perpendicular to the axis Parentheses indicate the Herrnann-Mmquin terms.
2. Center of symmetry (inversion center) 3. Plane of symmetry 4. Improper rotation axis
1 Unfortuniltely, two notation systems exist. Spectroscopists and chemists use the older Sohoenliies notation despite the restriction to point group symmetry. More recently crystallographers have used the Hermann-Marquin terms, since they are applicable to both point and space group symmetries. The latter will not be discussed.
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octahedron are given in Figure 2. Practice with molecular models should enable the student to identify readily all four C2 axes located perpendicular to the Cp axis. These will be depicted later as nC21C, and will be significant in determining symmetry point groups.
here is that the two methyl groups are uot freely rotating but occupy the preferred orientation as indicated. The center of symmetry, therefore, lies at point x since each hydrogen and carbon would find its analogue on passing through a. Plane of Symmetry, u
Figure 2.
Representative rotation oxes for the regular octahedron.
A plane of symmetry is one in which reflection through the plane transforms the molecule into its mirror image. There are two types of symmetry planes that must be characterized. A horizontal plane, denoted by u,, is defined as the symmetry plane which lies perpendicular to the highest order rotation axis. Any other plane of of symmetry can be designated as a,, a vertical plane. Both uh and a , occur in the planar BCla nlolecule and are illustrated in Figure 5 . It is apparent that the highest proper rotation axis in this species is the C3 axis which passes through the boron atom perpendicular to the plane of the molecule. Consequently, uh must lie coincident with the n~olecularplane. On careful inspection, one observes a maximum of three vertical planes at right angles to uh and bisecting the C1-B-C1 angles. Only one such u , is drawn.
Cenkr of Symmetry, i
If there is a point on a molecule having coordinates 0, 0, 0 such that changing the coordinates (x, y, z) of all the atoms to (-x, -y, - 2 ) brings the molecule into an equivalent configuration, that molecule is said to have a center of symmetry or inversion center. I n other words, if we can draw a line to a particular point from any element in a molecule, direct extension of that line in the same direction for an equivalent distance will find the identical element.
I Figure 5.
Planer of symmetry for the trichlorobutone molecule.
Improper Rotation Axis,
Figure 3.
Inversion center for N i I C N P .
To illustrate, consider the Ni(CN)&- ion where the nickel(II1) has bonding orbitals directed toward the corners of a square (Fig. 3). It is clear that the species has a center of symmetry at the center of the nickel ion. Applying the correct symmetry operations for i would transform CN(1) through the central metal ion to CN(3). The same is true for the extension of CN(2) to CN(4). Such operations will leave the ion unchanged. A less obvious example occurs in the cyclohutane derivative drawn in Figure 4. The assumption made
{7L
/
S.
A molecule has an improper axis of rotation of ordern if it can be rotated by 2 ~ / nand subsequently brought into an indistinguishable configuration by reflection in a plane perpendicular to that axis. A tetrahedral molecule, say methane, represents an excellent example. For convenience and ease of perception, consider the tetrahedron inscribed in a cube (Fig. 6). The Sa axis (Fig. 6a) is the symmetry element.. The sylnmetry operation entails first a rotation about S4by 90' (Fig. 6b) followed by reflechn in
'
/H/;-H
'X'
H H-C
--"
'/
H H
H Figure 4.
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Representative inversion center for 1.3-dimethylcyclobutone.
Journal of Chemical Education
Figure 6.
lC1
A reprerentotive improper rotation oxis for methane.
the plane, rI (Fig. 6c). The indistinguishability of 6a and 6c serves as a criteria for assigning the S4axis. One should note that rI is not a plane of symmetry and need not be to perform the S, operations. Symmetry Point Groups
Of the almost infinite number of molecules that can exist, it has been found that only a relatively few combinations of symmetry elements can occur. These combinations of elements are called point groups. The term point group is used in contrast to space group in that the above discussed symmetry operations must leave a specific point of the molecule unchanged. Space groups are associated with symmetry operations applied to unit cells and will not be discussed further. A detailed analysis of the mathematics and applications of group theory and the derivation of point groups is beyond the scope of this article. Several excellent treatises have been published in this area and are referred to in the bibliography. It is adequate to state that there are a total of 32 point groups. The designation of the point group to which a molecule belongs is essentially a summary of the symmetry elements of that molecule and, thus, a statement of its symmetry. Reference should now be made to Figure 7, which is an easily learned scheme for the deduction of the proper point group from a knowledge of the symmetry elements of any molecule. It should be noted that the italic terms are point group designations and generally represent the principal symmetv element of that
Linear 3Iolecules
i
group. The subscripts are introduced to further associate the symbol with the group elements. The directions for using the diagram are as follows: (1) Linear molecules are considered first (extreme left) and are designated either C,. or D,n. In order to distinguish between them one looks for the presence of a C, axis and mC, 1 C,. If these conditions are verified, the molecule belongs to the point group, Doh; if not, the point group is C,* (2) Molecules with especially high symmetry are placed toward the middle of the scheme. These are the easily recognized tetrahedron, Ta, octahedron, Oh, and icosahedron, l a , symmetries. (3) If neither of the above are found, we Look to the extreme right of the diagram and search for the proper rotation axk of highest order (C, where n = m a ~ i m u m ) . ~If there is no such axis, then the molecule is of very low symmetry, i.e., C,, C*,or C,. Any plane of symmetry or the presence of an inversion centcr m b v uwd t c s di-ti\.pui-h bctwm rhcnt I f rherc3rc111,~or mwc (1. .~xc=,HC s r l m tllc mlr 101h i r h c * ~ 4 ordrr. I f . in ndditiw. 1hr ~r~ilrculr~or?ro~rsm,lunr~.S:, nrlz. w:th or without an inversibn center, then the point &up designation is Sn(n = even). (5) If (4) is not applicable, then we inspect the molecule for nCa axes perpendicular to the major C , axis (nCe IC.). Lack of such a set implies a Cnh, Cno,or Cn species. Distinction can be made by looking jimf for a cj, (indicating Cnh symmetry) or next for na,'s (indicating C,, symmetry). If no plane of symmetry is observed, the point group is C,. (6) If uie do find the condition of nC2 1 Cn,the molecule bslongs to one of the point groups: Dm, D.s, or D,. These are confirmed by detecting either on, or nook or no a,respectively.
a If there is more than one C , axis (n = maximum), it is convenient to select the axis whirhlies perpendinlar to the molecular plane.
Special Symmetries
I
Proper Axis, C, n = maximum
1
yes
TdOJh
1
I
S2" (alone or with Z]
Yes I
--
D.h Figure 7.
D"d
D,
I',
Cn* Cna Cn
Scheme for point group selection.
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I, January 1966
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Examples
Several examples using Figure 7 will serve to illustrate the practicability of the scheme. Methyl Chloride. Although methane is a T d molecule, the presence of a substituted halogen lowers the symmetry considerably (Fig. 8). We must, therefore, look to the extreme right of the scheme. There exists only one rotation axis, a C3axis collinear with the C1-C bond.
ngure 8. Methyl chloride.
Since there is no Sz. axis, the molecule must belong to the C or D class. D symmetries can easily he eliminated from our selection since there are no C2axes. The only planes of symmetry are the 3r,'s lying in the three C1-GH planes confirming the point group designation, Caw Ethylene. Inspection of the geometry of ethylene excludes any of the special high symmetry point groups (Fig. 9). We do notice that there are three mutually perpendicular Cz axes. The major axis may be arbitrarily chosen perpendicular to the molecular plane. The remaining C2 axes can represent 2 C z l C 2 and establishes a D type point group. Final specification occurs with the recognition of a a, lying collinear with the molecular plane indicating D2*symmetry.
n u n s [Co(NH3)C12]+.The symmetry of this molecule is less than 0, due to the presence of halide substituents. Following the scheme, one notes a Cp axis passing through the apical halogens. Again Sz, symmetry is eliminated due to the presence of several symmetry elements in addition to i . 4 C z l C a are observed bisecting the equatorial sides and angles, thereby confirming a D symmetry. Since there is a r l C 4 ,the point group is Dm. Cis [Co(NH3)C12]+.This ion does not possess a Cp axis as above. The highest Ch is to a twofold rotation axis bisecting the side containing the two halogens. S2. is eliminated as above. There are 2~:s both collinear with CSand perpendicular to each other. Thus, the point group is Cz,. The preceding discussion is designed to enable the student to quickly identify symmetry elements, and from this to identify symmetry point groups. Point groups, in themselves, are unimportant; however, it must be emphasized that the determination of point groups is the essential primary step in the elucidation of more interesting molecular properties such as bond hydridization, molecular orbital wave functions, transition probabilities, and the number of infrared or Raman fundamentals.
rranr
Figure 10.
cis
[Co(NHshC121f.
Acknowledgment
The author wishes to acknowledge with gratitude the useful suggestions of Dr. T. Wartik, Dr. C. D. Schmulbaclr, and Dr. N. Matwiyoff during the preparation of this manuscript. Figure
9. Ethylene.
Benzene. We observed before (Figure 1) that the highest order rotation axis for benzene is Co. Since there are many more symmetry elements (C3, CZ,Sg, . . .) in addition to i,the point group cannot be S2.. I.'ollowing our scheme, we can conclude that the molecule has D symmetry due to the 6CzlCe. The existence of a ah, in the molecular plane takes precedence over the 6r,'s, implying Dea symmetry for the benzene molecule.
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Bibliography BARROW,GORDONM., "Introduction t o Maleculm Spectroscopy," McGraw-Hill Book Ca., New York, 1962. BELLAMY, L. J., "The Infra-Red Spectra of Complex Molecules," John Wiley & Sons, Inc., New York, 1954. COTTON, F. ALBERT,"Chernirsl Applications of Group Theory," Inteecience Publishers (division of John Wiley & Sons), New York, 1963. WHEATLEY, P. J., "The Determination of Molecular Structure," Clarendon Press, London, 1959. WIGNER,E. P., "Group Theory," Academic Press Inc., New York, 1959.
