Clarence H. Thomas University of Cincinnati Cincinnati, Ohio 45221
The Use of Group Theory to Determine ~olecularGeometry from R I Spectra
The importance of group theory for many branches of chemistry has been recognized for a long time, but only recently has this powerful tool for understanding many chemical problems been brought to the attention of the majority of chemists. Even at the undergraduate level the use of symmetry concepts can be very useful as an aid in classifying the properties of molecules. Since so large a number of physical and chemical properties depend upon the symmetries of a molecule a great amount of simplification is possible when these symmetries are taken into consideration. The introduction of symmetry concepts into the undergraduate curriculum has therefore become more and more common. A recent paper in thls Journal by Orchin and Jaffe ( I ) gives a presentation of group theory which is appropriate for the typical undergraduate chemistry major. Some of the more recent textbooks in physical chemistry, such as the 1972 edition of Moore's "Physical Chemistry'' ( 2 ) include introductory chapters on group theory. One of the fields of research in which symmetry properties have proved most useful is in the analysis of infrared spectra. The number and types of vibrations and the selection rules for vihrational transitions may all be determined solely by reference to the symmetry properties of the molecule. Conversely, by observing the infrared spectra we may, in many cases, deduce the point group to which a molecule belongs. This enables one to determine the overall shape of the molecule from low resolution ir spectra although the determination of exact hond lengths or hond angles requires high resolution spectra, either ir or microwave. In order to give the undergraduate chemistry majors some familiarity with this use of group theory, and especially to give them a chance to practice and deepen their knowledge of group theory we have provided an alternative to the usual experiments in infrared spectroscopy which are performed in the physical chemistry laboratory. The students take the spectra of several compounds for which they know only that the compound has a general formula, such as XY4, XYaZ, and/or XYz. From the low resolution infrared spectra of these compounds they are asked to determine the geometry of the molecule-whether it is linear or bent, planar or non-planar, etc. The background which they need before doing this analysis can be found in references ( I ) and ( 2 ) , as well as several other excellent references listed in the bibliography. The students should also be able, in most cases, to assign the observed hands to the proper vibrations of the molecule, needing only a minimum amount of information on the approximate frequencies at which stretches and bends might he thought to occur. We can best understand the steps involved in performing such an analysis by working out an example in detail. One of the best examples is methane, or a general XYa molecule. With its high degree of symmetry it provides
examples of most of the problems that are likely to be encountered in the analysis of any molecule. Although there is abundant chemical evidence that methane has tetrahedral symmetry, the infrared spectrum provides a striking confirmation of this fact. In addition to the tetrahedral shape, we will examine the expected spectrum for a square planar or a pyramidal shape. Other possibilities for a general XY4 molecule are also possible. For example, it was once suggested (incorrectly) that CCL is a distorted tetrahedron with C3" rather than Td symmetry. We want to first of all determine the number of vibrational degrees of freedom the molecule has. A molecule of N atoms will have 3N degrees of freedom (we are here excluding all consideration of electrons, of course). Of these, three will be translational, and three will he rotational (or two rotational for a linear molecule). The remainder 3N 6 (or 3N - 5) will be vihrational degrees of freedom. We may therefore expect methane to have (3.5) - 6 = 9 vibrational motions. This does not mean that there will be nine vihrational fundamentals observed in the ir. Some of the transitions will he forbidden in the ir bv the selection rules, others may be degenerate; that is, iwo or three of the motions may occur a t the same frequency so that only one fundamental is observed instead of two or three. Group theory enables us to find the degeneracies and the selection rules with the minimum amount of work; this is the great use of group theory in vihrational analysis. The first step in the analysis is to find the reducible re~resentationof the molecular vibrations. This may be d i n e by finding the reducible representation of a l i the molecular motions, includina translation and rotation, and then subtracting out the irreducible representations for translation and rotation, leaving only the representation for the vibrations. The steps involved in performing this analysis are described in Chapter 8 of "Introduction to Molecular Spectroscopy" by Barrow (3). Table 1 gives the characters for the irreducible representations of the tetrahedral group, as well as the reducible representation for the motions of the atoms in a tetrahedral XY4 molecule. (Row labeled r ( X L ,Y,, Z , ) ) . This reducible representation can he shown to reduce to
If the irreducible representations of the translations (T,, Table 1. Character Table for Td
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T,, T,) and rotations ( R , R,, R,) are subtracted out we are left with E 2F, Rvibration) = A,
+
+
Tetrahedral XY4 molecules will therefore have only four fundamental frequencies: one non-degenerate, one doubly degenerate, and two triply degenerate. A simple rule (which can easily be derived, but which may be assumed here) can be used to decide which fundamentals will occur in the ir spectrum. I t is: only those fundamentals having the same symmetry as T, T, or T, will be observed in the ir. For the XY4 molecule of tetrahedral symmetry this means that only the two FZfundamentals will be ir active. We can gain some further insight into the nature of the vibrations which the molecule is undergoing by another application of group theory to the problem. The coordinates which chemists ordinarily use when discussing molecular structure are the bond lengths and valence angles of the molecule, rather than the Cartesian coordinates of the molecule in some axis system. Likewise, when discussing the internal forces holding the molecule together and giving i t a certain shape it is most natural to think of these forces as acting primarily along chemical bonds and between valence angles. The bond stretches for the four X-Y bonds may be represented as arrows along the bonds, pointing in the direction which increases bond length, as shown in Figure 1.
sentation T(r) can be reduced to A1 + Fz. The A1 vibration is a totally symmetric stretch, with all of H's in CHI moving in and out together (sometimes called a "breathing" vibration); one of the FZ vibrations is a stretching mode. The other natural coordinates for vibration are the six angles. We can see that this presents a difficulty though; the four bond stretches plus six angle bends give us ten coordinates-and we need only nine altogether. We will go ahead and use all six aneles, and see how to reduce the tenmxdinates to the ninewe need at the end. We can represent the angle bending motions as arrows
Figure 2. Representation of angle at?
Figure 1. Stretching motion in methane.
Dividing the molecular motions into stretches and bends proves very useful when one is assigning the spectrum. Bending force constants are typically one tenth of the magnitude of stretching force constants. For example, the force constant for a C-H stretch is about 4.8 mdyn/ A, while an H-C-H bending force constant is near .3 mdyn/A in most molecules. From group theory it can be shown, as discussed in detail below, that only one stretching and one bending vibration should be infrared active in methane. Two infrared absorptions are observed, a t 3030.3 cm-' and 1306.2 cm-l. If one keeps in mind that the stretching force constants are always much larger than the bending force constants it is apparent that the 3020.3 em-' absorption should be assigned to the stretch and the 1306.2cm-' one to the bend. The characters for the representation of T, by the stretches may be found by considering the transformations of the arrows under the operations of Td.These are given in Table 1 in the row designated r(r). Under E the character is four, as all four stretches go into themselves. Under C3 one goes into itself, the others are interchanged; the character is therefore one. Under Cz and S4 none transform into themselves, so the character is zero. The reflection plane, ad, transforms two of the stretches into themselves, for a character of two. The reducible repre92
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on the atoms at the ends of the angle, pointing in the direction of maximum increase of the angle. The six angles may be labeled am als, am, aza, azr, and a34. Figure 2 shows the dis~lacementarrows to remesent aj2. The other angle displacements may he represented in ayimilar fashion. If we consider the transformations amone these aneles we obtain the characters given in the row m&ked r ( a j in Table 1. The characters for E and Ca are six and zero, respectively. Under Cz, the two bends for which the plane of the angle includes the CZ axis go into themselves for a character of two. Under Se all of the bends are interchanged, giving a character of zero. The reflection, od, transforms two bends into themselves; the one which is in the plane of reflection and the one a t right angles to it, for a character of two. r ( a ) may be reduced in the usual way t o give lYa) = A, E F,
+ +
The E and FZvibrations we may have expected, since Since there is only one A1 vibration, and we have decided this is the symmetric stretch, the A1 may be a bit unexpected. It occurs as a result of our starting with six angles instead of five and is a redundancy condition that is, if we increase all of the angles by the same amount, it is the same as doing nothing a t all. We may therefore discard the Al mode of vibration for the angles; i t will have an amplitude and frequency of zero. Another possible model for methane, or other XY4 molecule, is the square planar model seen in Figure 3. Convenient internal coordinates for this model are: four stretches, four in-plane angles al and two angles between opposite Y's: B13 and ,924. In Table 2 the character of the irreducible representations for D4h and for XI, YI, ZI), r ( r ) , r ( a ) and r(,9) are given.
Table 3. Character Table for C w
Table 2. Character Table for D4h
DM
E 2C.
C1
20' 2Cz"
i
ZSa
0 . 2
va
E
C4o
2oa
4
2C.
20.
2m
The allowed ir fundamentals will be those belonging to representations Az,, and EL A?.-an out-of-plane hend; degenerate stretch-doubly degenerate and E.,doubly bend. We may finally consider methane in the pyramidal model, Figure 5. In this case methane belongs to the group C4". Table 3 gives the characters for the reducihle and irreducible representations in this case. r(X,,Y,,Z,) = 3A, Translation: A,
Figure 3. Square planar XYI.
Rotation: A,
+ A, + 2B, + B, + 4E
+E
+E
Vibration: 2A,
+
28,
+ B, +
2E
+E + B, + E
r(r)= A, + B , r ( a ) = A,
r w = A, + B, Figure 4. The effect of the planar reflection on the out-of-plane bending vibrations.
The A1 in r ( n ) is again a redundancy condition. We are therefore left with ir active fundamentals 2241 + ZE,a total of four frequencies. In summary then, we may make a short list showing the expected spectra for different structures of methane. Group
T.9 D4e7
Figure 5. Pyramidal model of XY4 molecule.
The characters for the reducible representations may he obtained as before. Note that some of the characters in r (p) are -2; under these operations the bends p go into minus themselves, as illustrated in Figure 4. On breaking down these reducible representations we find
+
r ( x , , ~ . z , ) = ~ ,+#
+ B , ~ B , ~ +E, t
2A,"
+
Translation: A," Rotation: A,"
+
Vibration: A,
