.k& PACIFIC SOUTHWEST ASSOCIATION O F CHEMISTRY TEACHERS
ON THE ORIGIN OF CHARACTERISTIC GROUP FREQUENCIES IN INFRARED SPECTRA DAVID A. DOWS University of Southern California, Los Angeles 7
T H E identification of organic compouuds by means of infrared spectra is now nearly as common as the use of melting points or boiling points. While the latter physical properties will always he important for analytical purposes, the infrared spectrum has two important advantages. (1) In the usual infrared spectral trace, many absorptions occur. Thus one obtains several experimental data (frequencies) for each single con~pound. These absorptions are characterized not only by their frequencies, but also by their intensities. (2) The spectra of molecules containiug similar functional groups (e.g., C = 0, CH,, C = C, etc.) show absorptions in narrow frequency ranges characteristic of the groups; the presence of an absorption in a characteristic frequency region is often evidence for the mesence of the corresponding functional groups. I t is these common frequencies, characteristic of functional groups, which are the subject of this article. Because of the power of the infrared method for structure determination, a text in organic analysis is not considered up-to-date if it does not contain a section devoted to infrared analysis, and more and more undergraduate laboratories are including use of this tool as part of the training of a chemist. But this recent interest is basedon the availability of convenient spectrometers, not on any recent discovery of group frequencies. In fact, Coblentz, in his classic work on infrared spectra in 1905 (1)pointed out the existence of group frequencies and the inferences which could be drawn from their occurrence. More recently, Colthnp published his well-known chart of Spectra-Structure Correlations (2); and in the past few years a t least two important books have treated the subject. The first is by Bellamy (3); it is the most comprehensive collection and critical evaluation now available of group frequencies, largely of organic molecules, but confined to the sodium chloride region of the spectrum (2-15 microns). The second is the volume on "Chemical Applications of Spectroscopy," in Weissberger's series (4), which contains excellent chapters on infrared methods and applications. Any use of empirical data such as given in the literature or in the book by Bellamy should be made with nnderstanding of the principles discussed in (4). We shan, in this article, he concerned primarily with the principles underlying the existence of group frequencies. It is well known that infrared spectra are directly concerned with the vibrations of the atoms in
VOLUME 35, NO. 12, DECEMBER, 1958
a molecule with respect to one another (as differentiated, say, from rotations of the molecule as a whole, or from electronic motions such as give rise to visible and ultraviolet absorption spectra). Thus a study of infrared characteristic absorptions is a study of group vibrations. To gain insight into group vibrations we shall first consider the vibrations of small molecules and the usual methods of treating such systems. VIBRATIONS IN SMALL MOLECULES
The calculation of vibrational frequencies of molecules is, in principle, straightforward. I n practice, however, it becomes complicated and tedious very rapidly as the number of atoms increases-it is only rarely that molecules of more than, say, fifteen atoms are treated fully. Complete treatments of the theory of molecular vibrations and of infrared spectra are t o be found in the books by Herzberg (6) aud by Wilson, Decius, and Cross (6). Somewhat shorter and simpler treatments are available by West in (4) and by Barnes and Bonner in THIS JOURNAL (7), among many. UIIsupported statements in this article will in general be verifiable in these sources. The infrared spectra of solutions and liquids (and usually those of gases and solids) arise simply from transitions between the vibrational energy levels of the molecules under study. For any molecule with N atoms, there are 3N-6 (3N-5 for a linear molecule) vibrational frequencies, although they may not all he different, and may not all appear in the infrared spectrum. (For discussion of degeneracy of frequencies and selection rules one is referred t o references 4-6.) It is profitable, therefore, to look briefly a t the simplest vibrating system, a diatomic molecule, as a model for the 3N-6 vibrational motions of a general molecule. The diatomic molecule, a mass suspended by a spring, and a simple pendulum are analogous in their physical motion and their mathematical treatment. Each is characterized by a mass whose motion is resisted by a force proportional t o the distance from some equilibrium position (Hooke's law force). An equivalent statement is t o say that all the cases are subject t o a certain potential energy ( V )which is given by V = ('/*)k(r
-T
~
)
~
(1)
where (r - ro)is the displacement from the equilibrium position and k is the "force constaut." Actually the potential energy for a diatomic molecule is similar t o
energy level (v = 0) has significant population, and infrared spectra have primarily to do with transitions involving the absorption of light of energy corresponding to the jump from v = 0 to v = 1; A E = he vo, absorption a t vo cm.-'. Occasionally "overtone" transitions, or "harmonics," may be observed, corresponding to v = 0 v = 2. The frequency is usually a little less than 2vuhecause of the anharmonic effects mentioned above. The mathematical treatment also allows a computation of the relative magnitudes of the motion of the two atoms in a diatomic molecule (since the center of mass must remain stationary). Figure 2 shows the relative magnitudes of motion of the atoms in two molecules, CO and CH. The point of interest is in the case of two widely separated masses (CH) where only one mass undergoes significant motion. When we consider a polyatomic molecule, it is logical to look on it as a combination of several (3N - 6) simple vibrators comparable to diatomic molecules. Of course some of the 3N - 6 frequencies involve angle bending motions instead of bond stretching, hut it is easily shown that the two types of motion are completely analogous. There is, however, an important modification which must be added to the simple sum-
-
the solid curve of Figure 1. At large separations of the atoms the potential goes to a constant energy D, the dissociation energy. The dashed curve of Figure 1 gives the parabola (equation 1) which most closely matches the actual curve a t the equilibrium internuclear distance ro. Since most vibrations a t usual temperatures are confined to the vicinity of the equilibrium position, this parabola will be a good approximation in that region. The deviation of the true potential from the parabola gives rise to the anharmonic effects found spectroscopically for higher energy states. When the system of two masses ml and mz, moving in the potential field of equation (I), is treated either classically or quantum mechanically (8) the resulting vibration is found to have the frequency (in wavenumbers)
where c is the velocity of light, and m is the "reduced mass," in the case of a diatomic molecule m = mlmJml mz. The units "wavenumbers" are common among infrared spectroscopists, since they are proportional to the energy differences directly and have convenient magnitudes. They refer to the number of waves per centimeter (cm.-I) of the absorbed light (cm.-I = wavenumber = 104/wavelength in microns = 108/wave length in Angstroms). Normal vibration frequencies of molecules range from about 4000 to 100 cm.-I (2.5 t o 100 microns); the common range available in commercial instruments is 4000 t o 650 em.? (2.5 to 15 microns). The quantum mechanical treatment (8) results in a series of energy levels in which the molecule may find itself, given by 0 = 0, 1,2, . . . . . E = ~ C U ~+ ( Y (3)
+
where h is Planck's constant and v is the vibrational quantum number, taking on only the integral values indicated. At normal temperatures only the lowest 630
Pigun, 2. R e l a t h Magnitudes of Motion of Atoms in Diatomic Mole. cu1e Vibrations
mation of a number of simple vibrators, since the vibrators are not isolated, but will couple with one another in the polyatomic molecule. This gives rise t o new frequencies not identical to those of the diatomic molecules involved. It is the magnitude of this coupling which ultimately determines the isolation or lack thereof of a particular vibration, and therefore the constancy of its frequency in going from one molecule t o another. Coupling can arise from two sources: first, the presence of a common atom (or a common linkage of atoms) in the two primary vibrators, and second, electrical coupling due to the changes in electronic configuration in one bond which accompany motion within another bond. These effects will be illustrated for the COz molecule. Carbon dioxide may be thought of as two CO molecules connected by a common C atom. Since we are interested in coupling and changes of frequencies we shall only consider the two "CO stretching" frequencies of COz and shall ignore the bending frequency (this procedure is allowable because of the symmetry of the molecule; see ref. 5 or 6). We shall first assume that the stretching of the CO bond obeys a simple potential function such as (I), and that its frequency is given by (2). The potential function for two CO bonds is, in analogy V = '/&(r, - ro)' + '/&(n - rO)¶ (4) JOURNAL OF CHEMICAL EDUCATION
where rl and rz are the two CO bond lengths. It is easily shown (see ref. 5, pages 153,172) that the two CO stretching frequencies based on this potential are given by
TABLE 1 C 4 C 1 Frequencies 2966 cm.-' 1355 732 3042
1455 1015
Figure 3 shows the magnitudes and directions of the motions of the atoms in CO and COI on the basis of this calculation. Also shown in Figure 3 are the frequencies of the vibrations, v,, vl, and vz. I t is seen that the effect of a common atom is to split the frequencies widely (the split is very large because of the exact likeness of the two diatomic molecules involved, and would be mnch smaller for, say, OCS). In the calculations the value k = 15.5 X lo5dyne per centimeter has been used for both CO and CO,. The frequencies found are not exactly those of COz which cannot be matched by such a simple potential function. It is necessary t o assume that there is further coupling between the vibrators, of the electrical kind, which is represented as a further (parabolic) term in the potential function connecting (ref. 5) are
TI
and rz. The result for the frequencies
CH stretch CH. bend non-deeenerate
CCI stretch
\
C H stretch
CH, rorkine
Although it is not difficult to go through the calculations for this molecule, analogous t o those for COz, we can observe the effect of coupling without going to that labor. We notice that the two frequencies labeled "CH stretch" differ by 76 cm.-', or 2.5y0, while in the isolated CH molecule there is only one such frequency. Likewise the two CH, bending frequencies differ by 7.5%. This clearly indicates coupling between these vibrations. It is of interest to inquire why the splitting of the coupled CH stretches in the CHJ group is so mnch less (3y0 compared to 50%) than the splitting of CO frequencies in COz. Two reasons are apparent, even ignoring the question of the amount of electrical coupling. First, the C in CO has a share in the motion of somewhat more than 50% (Figure 2) while in CH it is only about 6%. This means that the common atom is less important in the vibrational motion and cannot transmit or couple the motion between the vibrators efficiently. Second, in COz the direction of motion of the C is the same in both CO bonds, while in the CHa group each CH bond is more closely a t right angles (actually about 109') to its neighbors, and the motion in one bond is inefficient in perturbing the other (the perturbation increases proportionally with the cosine of the angle between the bonds).
and these frequencies are also shown in Figure 3. The VIBRATIONS OF LARGE MOLECULES values k = 15.5 X lo5and k' = 1.3 X 105.arechosen to reproduce in equation (7) the frequencies of GOz. We have seen that if we consider a polyatomic moleIt is seen that in the case of COz the close coupling of cule as made up of a group of basic vibrators (similar to the two halves of the molecule leads to two "CO diatomic molecules) we must consider the coupling stretching" frequencies which are quite different from between the vibrators to arrive a t the frequencies of the that in an isolated CO molecule of the same bond polyatomic molecule. These coupling factors can vary strength. from a negligible change to as much as a 50% shift in As a second example of the vibrations of a small frequency. This treatment is analogous to that of molecule we shall examine methyl chloride. The 5 atoms will produce 3 X 5 - 6 = 9 vibrational frequencies; however there will be only 6 discrete frequencies, there being three pairs of doubly degenerate (i.e., the same) vibrations. Looking a t the molecule as a collection of diatomic vibrators we recognize immediately that there should be one vibration arising from C-CI and three from C-H (of which two are degenerate, leaving two discrete CH frequencies). The other three discrete frequencies must then be of the bending type. The observed frequencies (9) for CHaCl are given WWVENUP70ER (CM-') in Table 1, with approximate ri- 3. Fr.qu.nci.. and Charect.~i.tis Motion. of COl and of a n ~ p o t h e t i c a lCO Mdelecule with the characterizations. Same Bond Strength VOLUME 35, NO. 12, DECEMBER, 1958
631
assuming the additivity of bond energies to compute the heat of formation of a molecule; the results are indicative but often only approximate. We shall now take the next step, or approximation, which is t o assume the constancy of frequencies within a group of atoms such as -CHa, -NH2, -CO, -CeHs, etc. Some groups are found t o have frequencies which are characteristic regardless of the remainder of the molecule in which the group finds itself, but others are found t o have frequencies which depend in a sensitive way on the molecular environment. Of course, when two or more groups are joined to form a molecule there are always additional vibrations not attributable to any of the constituent groups' vibrations, since, for example, if two groups of N atoms are joined together there are 3(2N) - 6 = 6N - 6 frequencies, while in the two groups originally there were 2(3N - 6) = 6 N - 12 frequencies, a gain of six frequencies in the complete molecule. These correspond to losses of rotational and translational degrees of freedom. As an example we take the case of ethane, composed of two CH, groups. Each group has 6 frequenciesthree CH stretches and three CHa bending motionswith degeneracy as in CHCl. The frequencies 2966, 1355, 3042, and 1455 cm.-' (Table 1) serve as our model of a CHI group. The observed ethane frequencies (5) (half are not seen in the infrared because of selection rules, but may be obtained from the Raman effect,except for the 275 cm.-I value) are given in Table 2. A double asterisk indicates a doubly degenerate frequency. TABLE 2 Frequencies of Ethane
Infrared 2993*' 2954 1486** 1379 821**
CHaCl 3042** CH stretch 2966 CH stretch 1455** CHa bend 1355 CHI bend .. bend . . CC stretch . . bend . . torsion
The results for ethane are a t least close to what we suspected. Each CH, frequency yields two in ethane (since there are two coupled CH3 groups), and the additional frequency shifts are not large. I n addition there are other frequencies, not fourd in the isolated groups, due to the C-C stretch, two types of bendiug motion, analogous t o the 1015 frequency of CHCI, and the frequency of torsion of the two CH, groups about the C-C bond against the 3 kcal. barrier to internal rotation about that bond. The procedure for predicting the spectrum of a large molecule is thus clear, a t least in principle. One chooses a set of characteristic groups and predicts that each group will contribute its set of characteristic.frequencies. If there are two or more groups of a given kind (e.g., CHagroups) there will be two or more sets of frequencies which will be separated t o some extent depending on the magnitude of the coupling between the groups. If the two CHa groups (for instance) are at the ends of a long molecule, i t is clear that little coupling is expected, and there wiU be only one set of the CH3 frequencies apparent, though their intensity wiU be greater than for a single such group. This effect of lack of coupling,
and coincidence of frequencies, is called "accidental degeneracy," since the frequencies are degenerate (the same) only due to lack of coupling. The "necessary degeneracy" within the CH3 group is a result of the symmetry of the group. I n addition to the combined groups frequencies, there will, of course, be the frequencies attributable to the connection between the groups. There is no way to predict these except by analogy to other molecules (essentially the use of larger groups). The practical problem in using group frequencies in this way is to find which groups transfer frequencies from one molecule to another without appreciable change. Only three examples will be mentioned here, since other works (3-4) are comprehensive on this suhject. As an example of good frequency transferability Bellamy (3) lists, for the CH stretching mode of -CH3, 2962 and 2872 ern.-' 10 em.-'; and for -CHI, 2926 and 2853 cm.-' 5 10 cm.-I It is clearly possible, then, given sufficient resolution and accuracy in the spectrometer, to analyze for these groups with little uncertainty. The CH, rocking frequencies in a paraffin hydrocarbon are examples of non-transferable frequencies. Snyder ( l o ) , most recently, has studied these vibrations, which lie in the region 720-1060 cm.? and are strongly dependent on the chain length. Third, Bellamy gives data on the -CH3 bending frequencies for -C-CHa, C(CH&, and -C(CH& groupings, which show that it should be possible to decide between these three structures on the basis of the spectrum. These three examples show, respectively, good isolation (negligible coupling) of a vibration in the C H bond, large coupling (poor isolation) of the CH2rocking motion, as might be expected, and small coupling between adjacent CH3 groups giving rise to analytically useful, small frequency shifts.
*
CONCLUSION
We have seen how it is possible to approach the vibrations of a large molecule through the constituent group frequencies. The magnitude of the coupling between groups in the molecule determines the extent of the changes in frequency in going from one molecule to another, and therefore the usefulness of the idea of constant or characteristic group frequencies. Sometimes a small amount of coupling yields small, analytically useful shifts. It is often possible to make reasonable guesses as t o the magnitude of the coupling, based on the isolation of motion within the group and the closeness of 'pproach of the groups. The problem of the skeletal vibration frequencies of large molecules (the ones not properly assignable t o simpler functional groups) can be solved, essentially, only by considering larger and larger groups as structural units, to include more of these vibrations. When sufficient correlative work is done on the spectra of large molecules, these frequencies will become useful in analysis of unknown structures. LITERATURE CITED (1) COBLENTZ, W. W., Publ. Cnrnegie Inst. of Wash., D. C., No. 35, Part I(1905). N. B., J. Opt. Sac. Am., 40, 397 (1950). (2) COLTAUP, L. J.. ('The Infrared S~ectraof Comdex Mole(3) BELLAMY. eules,"'~ohniVileiley & Sons, 1nc.t New York, 1954.
JOURNAL OF CHEMICAL EDUCATION
(4) WEISSBERGER, A,, "Technique of Organic Chemistry," Vol. IX, Interscience Publishers, Inc., New York, 1956.
(5) HERZBERG, G., "Infrared and Raman Spectra," D. Van Nostrend Co., Inc., New York, 1945. (6) WILSON,E. B., J. C. DECIUS,AND P. C. CROSS,"Molecular Vibrat.ions," McGraw-Hill Book Co., New York, 1955.
VOLUME 35, NO. 12, DECEMBER, 1958
(7) BARNES, R. B., A N D L. G. BONNER, J. CAEM.EDUC.,14,564 (1937); 15, 25 (1938). (8) PAULING, L.,AND E. B. WILSON,"Introduction to Quantum Mechanics," McGraw-Hill Book Co., 1935. (9) KING, W. T., J. M. MILLS,AND B. L. CRAWFORD, JR., J . Chem. Phys., 27, 455 (1957). (10) SNYDER, R. G., J . Chem. Phys., 27, 969 (1957).
