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The Theory of Infrared and Raman Vibrational Spectra
By E. Bright Wilson Jr., J. C. Decius, Paul C. Cross
Dover Publications, Inc.Copyright © 1955 E. Bright Wilson, Jr., J. C. Decius and Mrs. Paul C. Cross
All rights reserved.
An immense amount of experimental data has been accumulated from investigations of the infrared absorption spectra and of the Raman effect in polyatomic molecules. Only an extremely small fraction of this material has been subjected to analysis, although the theoretical tools for such an analysis are quite well developed and the results which could be obtained are of considerable interest. One reason for this situation is the amount of labor required to unravel the spectrum of a complex molecule, but an additional deterrent has been the unfamiliar mathematics, such as group theory, in terms of which the most powerful forms of the theory of molecular dynamics have been couched. When only the necessary parts of these mathematical techniques are considered, the difficulty of understanding the theory of the vibrational and rotational spectra of polyatomic molecules is greatly reduced.
In this first chapter, a short general survey of the background of the subject will be given, to serve as an introduction to the more mathematical treatment which follows.
1-1. Infrared Spectra
The absorption or emission spectrum arising from the rotational and vibrational motions of a molecule which is not electronically excited is mostly in the infrared region. A small molecule having an electric moment emits and absorbs light of frequency below about 250 wave numbers because of its rotational motion. Molecules which are absorbing or emitting 1 quantum of vibrational energy show bands in the region from about 200 to 3,500 cm-1, while bands due to several vibrational quantum jumps are detected all the way from a few hundred to many thousand wave numbers, sometimes being observable in the visible portion of the spectrum (13,000 to 26,000 cm-1). See Fig. 1-1.
Infrared spectra may be observed either in emission or in absorption, although the latter method is by far the more common. In absorption experiments light from a suitable source is passed through a tube containing the gas to be studied and thence into the spectrograph. If the spectrograph is of low resolving power, a series of wide bands is observed which correspond to the vibrational transitions, but if a spectrograph of higher resolution is used, these bands may break up into lines which can be correlated with the energy levels of rotation. In practice, only a few light molecules (H2O, NH3, CH4, CO2, etc.) have been observed in the infrared region with high enough resolving power to resolve the rotational structure. Figure 1-1 shows some observed spectra.
Liquids and solids are also studied, and they yield interesting results, but except in so far as they give vibrational spectra in close agreement with those found for the corresponding gases they will not be discussed in this book, in which interactions between separate molecules will be neglected.
1-2. Raman Spectra
If the substance being studied (as a gas, liquid, or solid) is strongly illuminated by monochromatic light in the visible or ultraviolet region and the scattered light observed in a spectrograph, a spectrum is obtained (see Fig. 1-2) which consists of a strong line (the exciting line) of the same frequency as the incident illumination together with weaker lines on either side shifted from the strong line by frequencies ranging from a few to about 3,500 wave numbers. The pattern of lines is symmetrical about the exciting line except with regard to intensities, the lines on the high-frequency side being considerably weaker than the others. In fact, they are frequently too weak to be observed. The lines of frequency less than the exciting lines are called Stokes lines, the others anti-Stokes lines.
These lines differing in frequency from the exciting line, the Raman lines, have their origin in an interchange of energy between the light quanta and the molecules of the substance scattering the light. The lines which appear very near the exciting line are correlated with changes in the rotational energy states of the molecules without changes in the vibrational energy states and form the pure rotation Raman spectrum. The lines farther from the exciting line are really bands of unresolved lines and are associated with simultaneous changes in the vibrational and rotational energy states.
The frequency shifts, that is, the differences between the frequencies of the Raman lines and the exciting line, are independent of the frequency of the exciting line. A mercury arc is usually used for illumination, and there are a number of the strong mercury lines which are used to excite Raman spectra. The frequencies for a given molecule found in infrared absorption frequently agree with the frequency shifts found in the Raman effect, but this is not always true and depends on the symmetry of the molecule in a way which is now well understood.
By polarizing the incident light or in other ways, it is possible to find the degree of depolarization of each frequency shift in the Raman spectrum, a quantity which will be important in the interpretation of the experimental results. This quantity is the ratio, for the scattered light, of the intensities of the components polarized perpendicular and parallel, respectively, to the direction of polarization of the incident illumination.
1-3. The Molecular Model
In attempting to account for the observed infrared and Raman spectra of real molecules, a certain simplified model for such molecules is adopted, and then the spectra which this model would exhibit are calculated. The specification of the model involves certain parameters such as size, stiffness of valence bonds, etc., which can be varied within limits set by other types of experimental evidence until the best agreement with experiment is obtained. An attempt is usually made to select a number of such parameters which is much smaller than the number of experimental quantities so that the success of the theory can be tested by the agreement which it provides with experiment.
The model which will be used in this book consists of particles held together by certain forces. The particles, which are to be endowed with mass and certain electrical properties, represent the atoms and are to be treated as if all the mass were concentrated at a point. It is assumed that the atoms may be electrically polarized by an external electrical field, such as that of a beam of light, and that they may or may not be permanently polarized by their mutual interactions in such a manner that the whole molecule has a resultant electric moment. Both the polarizability and the electric moment of the model may vary as the particles (hereafter called atoms) change their relative positions. Finally, the atoms may possess an internal degree of freedom or nuclear spin which introduces certain symmetry restrictions.
The forces between the particles may be crudely thought of as weightless springs which only approximately obey Hooke's law and which hold the atoms in the neighborhood of certain configurations relative to one another. This picture of the forces as springs is useful for visualization, but is not sufficiently general for all cases. For example, it does not cover cases of restricted rotation about single bonds such as may occur in ethane. The nature of these interatomic forces is one of the chief problems still being studied and will be discussed in Chap. 8. The search for a potential function which involves a small number of parameters and which at the same time permits good agreement with experiment is by no means ended.
The statement that the model obeys the laws of quantum mechanics is an essential part of its specification. However, since atoms are fairly heavy particles (compared to electrons), it will sometimes be true that classical mechanics when properly used gives results which are good approximations to those of quantum mechanics.
Since the atoms of this model have been regarded as point masses with certain electrical properties, there is an apparent disagreement with the fact that many experiments require that atoms be made of electrons and nuclei. It is possible to reconcile these two points of view. If the wave equation for a molecule made up of electrons and nuclei is set up, a procedure exists whereby this equation may be separated into two equations, one of which governs the electronic motions and yields the forces between the atoms, whereas the other is the equation for the rotation and vibration of the atoms and is identical with the equation for the model adopted here. In principle, therefore, the forces between the atoms can be calculated a priori from the electronic wave equation, but in practice this is not mathematically feasible (except for H2), and it is necessary to postulate the forces in such a manner as to obtain agreement with experiment. Therefore, although it is theoretically possible to start with a model consisting of electrons and nuclei interacting coulombically and obeying the laws of quantum mechanics, in practice it is necessary to assume the nature of the equilibrium configuration and of the forces between the atoms, so that it seems more desirable to start with the model in which the atoms are the units.
This separation of the electronic motion and the nuclear motions is only an approximation which may break down in certain cases, especially for high electronic states. If there were no interaction between the two types of motions, there would be no Raman effect of any importance. However, the coupling is small for the lowest electronic state.
1-4. Classical Theory of Vibrational and Rotational Spectra
Classical electromagnetic theory requires that an accelerated charged particle emit radiant energy. On this basis a rotating molecule with an electric moment should emit light of the same frequency as the frequency of rotation. Because of the Maxwellian distribution of rotational velocities, a collection of gas molecules should emit a band of frequencies possessing an intensity maximum which is related to the most probable frequency of rotation. In practice, this prediction of classical theory is quite closely verified experimentally for heavy molecules and low resolution.
The molecular model of the previous section can move as a whole, rotate about its center of mass, and vibrate. The translational motion does not ordinarily give rise to radiation. Classically, this follows because acceleration of charges is required for radiation. The rotational motion causes practically observable radiation if, and only if, the molecule has an electric (dipole) moment. The vibrational motions of the atoms within the molecule may also be associated with radiation if these motions alter the electric moment. A diatomic molecule has only one fundamental frequency of vibration so that if it has an electric moment its infrared emission spectrum will consist of a series of bands, the lowest of which in frequency corresponds to the distribution of rotational frequencies for nonvibrating molecules. The other bands arise from combined rotation and vibration; their centers correspond to the fundamental vibration frequency and its overtones. A polyatomic molecule has more than one fundamental frequency of vibration so that its spectrum is correspondingly richer.
The Raman effect can also be explained classically. The electric vector of the incident illumination induces in the molecule an oscillating electric moment which emits radiation. If the molecule is at rest, the induced moment, and therefore the scattered light, has the same frequency as the incident light. If, however, the molecule is rotating or vibrating, this is not necessarily the case, because the amplitude of the induced electric moment may depend on the orientation of the molecule and the relative positions of its atoms. Since the configuration changes periodically because of rotation and vibration, the scattered radiation is "modulated" by the rotational and vibrational frequencies so that it consists of light of frequencies equal to the sum and to the difference of the incident frequency and the frequencies of the molecular motions, in addition to the incident frequency.
Thus, the classical theory of radiation and classical mechanics provides an explanation of the general features of both infrared and Raman spectra. It cannot, however, account for the details and is to be regarded as only a rough approximate method of treatment.
1-5. The Quantum Viewpoint
When the molecular spectra of a few light molecules are observed with spectrographs of high resolving power, the bands previously discussed are resolved into a series of closely spaced lines. Classical theory is unable to explain this phenomenon. The explanation, of course, depends on the use of quantum theory, in which the molecule is restricted to definite, discrete energy levels of rotation and vibration. Radiation occurs only when a molecule undergoes a transition from one stationary state to another of different energy. The Bohr frequency rule gives the frequency v of the light radiated or absorbed on transitions between states of energies Wn" and Wn' It is
vn"n' = vn'n" = Wn' - Wn"/h (1)
where h is Planck's constant.
Not every transition can occur with the emission or absorption of radiation. The rules which tell which transitions may occur are called selection rules.
Although the classical theory is not correct in predicting that the observed radiation will consist of frequencies occurring in the motion of the system, there is an asymptotic relationship between the frequencies predicted by the classical and by the quantum theory, known as Bohr's correspondence theorem for frequencies. According to this theorem the frequencies emitted and absorbed by a quantum system approach asymptotically the classical frequencies of the system as the quantum numbers of the initial and final states are increased. The intensities of the quantum transitions will likewise asymptotically approach the intensities calculated classically, as the quantum numbers increase.
From the quantum viewpoint, the band of lowest frequency in the infrared spectrum (it may extend into the microwave region) of a molecule with an electric moment consists of discrete lines, each of which corresponds to a transition between two different rotational energy levels of the nonvibrating molecule (or rather, of the molecule in its lowest vibrational energy level). The other bands with higher frequencies correspond to transitions involving simultaneous changes of rotational and vibrational energies. The spacing between adjacent vibrational levels is considerably greater than that between adjacent rotational levels so that, although the various vibrational bands are usually fairly widely spaced, it requires a spectrograph of very high resolving power to separate the rotational lines.
The fundamental frequency of the classical explanation corresponds to a quantum transition from one vibrational state to the next, while the overtone frequencies correspond to transitions to other than adjacent levels. Since the vibrational levels are nearly but not quite evenly spaced, the vibrational bands will fall into series with frequencies which are almost but not quite multiples of the fundamental frequencies.
The quantum picture of the Raman effect is that a photon of energy hv0 (v0 being the frequency of the incident light) comes up to a molecule in a given stationary state, causing a transition to another higher (or lower) energy level different in energy by an amount hvn"n'. This amount of energy is subtracted from (or added to) the incident photon so that the emitted or "scattered" photon then has the energy hv0 [??] hvn"n' and therefore has the frequency v0 [??] vn"n'. Since in general more molecules are in the lower than in the higher energy states, there will be more cases in which the photon gives up some of its energy than vice versa, so that the Stokes lines will be stronger than the anti-Stokes lines.
In calculating the energy levels and selection rules, the principles of quantum mechanics must be used. This is usually done through the medium of the Schrödinger equation and wave mechanics, but the equivalent mathematical techniques of matrix mechanics and the operator calculus are frequently useful.
There are three main applications of the interpreted results of infrared and Raman studies. These are the study of the nature of the forces acting between the atoms of a molecule, the determination of molecular structure, and the calculation of thermodynamic quantities.
Excerpted from Molecular Vibrations by E. Bright Wilson Jr., J. C. Decius, Paul C. Cross. Copyright © 1955 E. Bright Wilson, Jr., J. C. Decius and Mrs. Paul C. Cross. Excerpted by permission of Dover Publications, Inc..
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