VIBRATIONS OF POLYATOMIC MOLECULES AND THEIR STUDY USING GROUP THEORY

M. V. Vol'kenshtein

Submitted 1936 | SovietRxiv: ru-193601.05934 | Translated from Russian

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VIBRATIONS OF POLYATOMIC MOLECULES AND THEIR STUDY USING GROUP THEORY

M. V. Volʹkenshtein, Moscow

§ 1. Introduction. Normal vibrations.
§ 2. Vibrational spectra of molecules.
§ 3. Basic concepts of group theory.
§ 4. Classification of the normal vibrations of a molecule on the basis of group theory.
§ 5. Irreducible representations of symmetry groups and their characters.
§ 6. Expression of normal vibrations.
§ 7. Selection rules.
§ 8. Some examples. Associated molecules.
§ 9. Range of applicability.
§ 10. Experimental material.
Literature.

§ 1. Introduction. Normal Vibrations

The theory of molecular spectra is, naturally, considerably more complex than the theory of atomic spectra. In essence it may be considered complete only for diatomic molecules; here the combined efforts of numerous investigators have made it possible to give a detailed qualitative and quantitative characterization of the spectrum in relation to the structure of the molecule. It is true that even here, for a number of questions, one has to confine oneself to approximations. For polyatomic molecules the problem is considerably more complicated, and their theory is at present in the initial stage of its development. In this review we disregard the electronic and rotational structure of the spectrum and confine ourselves to the vibrations of the molecule in its fundamental, electronically unexcited state. These vibrations find their expression in infrared spectra and in Raman spectra. The theory of both would achieve its aim if, on the basis of other physical and chemical properties of a substance, we were able to predict the number and numerical values of the frequencies of the spectrum, their intensities, their polarization properties, and so on. Of greatest interest, of course, is the inverse transition: the determination of the structure of a molecule from its spectrum.

However, even for diatomic molecules the theoretical determination

properties of a single vibration encounters great difficulties. We can treat such a molecule approximately as a harmonic oscillator. Then it is quantized according to the corresponding law, and the value of the single frequency is found from elementary considerations. But the assumption of the elastic character of the bond contradicts the ability of a diatomic molecule to dissociate. Thus one has to seek a potential function different from a quadratic one; it has been obtained only in a certain approximation1. Here, as in the case of a polyatomic molecule, we are faced with the need to know the character of the bond between the atoms. Up to now quantum chemistry has not been able to give us the information needed for this. Therefore, for polyatomic molecules one must make do with still cruder approximations. Usually, in a first approximation, the forces are regarded as elastic. In doing so one uses one of three methods of treatment[^2]: either one adopts the so-called system of valence forces, acting along the directions of the valence strokes, or a system of central forces, acting between all atoms, or (more rarely) a system of atomic forces—the atoms vibrate in potential wells with cross sections analogous to the potential curves for diatomic molecules, while the action of the remaining atoms is regarded as a perturbation. By specifying one or another simplified mechanism of the action of the forces for the simplest molecules, it is possible to obtain visual models of the vibrations and to calculate the values of the frequencies. The results are often in good agreement with experiment. Here it will be appropriate to mention the work of certain American scientists who constructed mechanical models of molecules from metal balls connected by springs and studied oscillograms of the vibrations of such models[^3]. The vibration spectra obtained in this way sometimes agreed well with the Raman and infrared spectra of real molecules. However, it is in principle impossible to go further along this path. The approximate mechanism of the forces is knowingly incorrect, the complexity of the calculation grows rapidly with an increase in the number of atoms in the molecule, and in essence the calculation is possible only for the simplest cases. The reliability of the results is always open to doubt.

Another path, which from our point of view is more fruitful, consists in renouncing the finding of numerical values of the frequencies, and in renouncing approximate mechanical representations. A theory of this kind proceeds exclusively from the symmetry of the molecule as a whole. It proves capable of characterizing all those properties of the vibrations that depend on the symmetry of the molecule—the selection rules in the Raman spectrum and in the infrared spectrum, the polarization relations in the Raman spectrum, the degrees of degeneracy of the normal vibrations. Conversely, knowing the number of lines in the spectra and the other properties just enumerated, with the aid of such a structural-symmetry theory one can give a qualitative characterization of the structure of the molecule, find the symmetry of the molecule, which is an unquestionable achievement. The present review will be devoted to this theory.

VIBRATIONS OF POLYATOMIC MOLECULES

A vibrating polyatomic molecule is at every instant in a state of very complex vibrational motion. Indeed, the vibrations of the individual degrees of freedom interact with one another; we are dealing with a problem of coupled vibrations, the simplest example of which is the vibration of two pendulums connected by a spring. We shall regard the forces, in first approximation, as linear functions of the displacements of the particles from the equilibrium position, in other words, as elastic forces obeying Hooke’s law. Then the potential energy of the vibrating molecule is represented by a quadratic function of the displacements, containing, in addition to the squares of the displacements, also their products, corresponding to the presence of coupling between the vibrations of the individual degrees of freedom of the molecule. But every complex coupled vibration can be represented as the result of a superposition of vibrations that do not act upon one another, in which all particles vibrate in phase and with the same frequency—the so-called normal vibrations$^{4,5}$. In normal vibrations the particles simultaneously pass through their equilibrium positions, whence it follows that the ratios of the accelerations to the displacements for the individual atoms are equal. Such displacements are rectilinear. The total number $s$ of internal degrees of freedom for an $N$-atomic molecule, if the atoms are not arranged along a single straight line, is equal to $3N - 6$ ($3N$ minus the degrees of freedom of motion of the molecule as a whole—3 rotations and 3 translations). The number of normal vibrations will be the same. The individual normal vibrations are orthogonal to one another: this means that the forms of motion in these vibrations are such that if one atom, in two normal vibrations, vibrates in the same direction, then another atom, in the two vibrations, vibrates in opposite directions. The work done by the forces of the first normal vibration upon the displacements of the second normal vibration is equal to zero.

\[ \sum_{x,i} m_i x_\lambda^i x_\mu^i = 0 \tag{1, 1} \]

$\lambda, \mu$—indices corresponding to two normal vibrations, $i$—the index of the atom, $m$—mass, $x$—displacement from the equilibrium position.

If the system of displacements

\[ x_\lambda^1,\ y_\lambda^1,\ z_\lambda^1,\ x_\lambda^2,\ y_\lambda^2,\ z_\lambda^2,\ldots,\ x_\lambda^{3N-6},\ y_\lambda^{3N-6},\ z_\lambda^{3N-6} \tag{1, 2} \]

is denoted by $s^\lambda$, where $\lambda = 1,2,\ldots,3N - 6$, then any displacement of the particles in the complex vibrational motion of the molecule will be represented as a linear combination,

\[ q_1 s^{(1)} + q_2 s^{(2)} + \ldots + q_{3N-6} s^{(3N-6)} \tag{1, 3} \]

$q_\lambda$ are normal coordinates obtained from real displacements by a linear transformation whose coefficients are determined by the scheme [1, 2]. It is obvious that, for an individual normal vibration $\lambda$, only one normal coordinate $q_\lambda$ is present. Having carried out such a transformation, we have eliminated from the expression for the potential energy the terms having the form of products of coordinates, and, on the basis of the orthogonality relations [1, 1], have brought it to the form

\[ U=\frac{\beta_1}{2}q_1^2+\frac{\beta_2}{2}q_2^2+\ldots+\frac{\beta_{3N-6}}{2}q_{3N-6}^2 . \tag{1, 4} \]

The kinetic energy is represented simply by the sum of squares

\[ T=\frac{\dot q_1^2}{2}+\frac{\dot q_2^2}{2}+\ldots+\frac{\dot q_{3N-6}^2}{2}. \tag{1, 5} \]

Quantum-mechanically, this means the representation of the vibrational function of the molecule in the form of a product of functions of the individual normal coordinates

\[ \Psi_{\mathrm{osc}}=\psi(q_1)\psi(q_2)\ldots\psi(q_{3N-6}). \tag{1, 6} \]

The eigenvalue of the energy for an individual normal vibration is equal to

\[ E_\lambda=\left(n_\lambda+\frac{1}{2}\right)h\nu_\lambda . \tag{1, 7} \]

In view of this, the spectrum will contain, as separate frequency lines, precisely the noninteracting normal vibrations. Such is the property of all spectral instruments in general, whether a spectrograph for optical vibrations, a wavemeter for radio waves, or a set of Helmholtz resonators for acoustic vibrations: they decompose complex vibrations into a sum of normal ones.

The translational and rotational motions of the molecule are also its normal vibrations, with frequency equal to zero. The symmetry properties of normal vibrations are characterized by the laws of their transformation under one or another operation of self-superposition in the molecule—rotations, mirror reflections, and rotations followed by reflection. In a number of cases the frequencies of several normal vibrations coincide with one another. Such vibrations are called degenerate. As we shall see, degeneracy is determined by the symmetry of the molecule. Knowing the number and degree of degeneracy of the normal vibrations, we shall be able to say how many lines the spectrum will consist of. Considering the behavior of the electric-moment vector and of the polarizability tensor under symmetry operations, we shall be able to say which of these vibrations will be active in the infrared and, correspondingly, in the Raman spectrum. Finally, the polarization properties of Raman lines are likewise—though, it is true, not always—determined by the symmetry properties of the normal vibrations.

The method of calculation may consist in solving the $3N$ equations of analytical mechanics for the $3N$ vibrations of an $N$-atomic molecule (for generality, rotations and translations are included here) with respect to the amplitudes. The equations have the form

\[ \nu^{2} m_k a_{kx}+\sum_{k'=1}^{N}\sum_{y=x,y,z} A_{kk'xy}a_{k'y}=0. \tag{1, 8} \]

Here $\nu$ is the frequency, $m$ the mass, $a$ the amplitude, and $A$ a constant. The indices $k, k'$ refer to the number of the atom, and $x, y, z$ are coordinates in space. Using the simplifications introduced by symmetry, one can answer the question of the number and multiplicity of the frequencies and of the laws of transformation of the amplitudes for all admissible types of symmetry, even without knowing the constants $A_{kk'xy}$, whose values are determined by the character of the bond. Expressing the electric moment through the amplitudes, we find the selection rules for the infrared spectrum. This was the procedure of Brester$^6$, who carried out his calculations for all 32 point crystallographic symmetry groups and succeeded in characterizing the infrared spectra of crystals obtained by the residual-ray method. Placzek$^7,8$ supplemented Brester’s work with calculations for the polarizability and created a detailed theory of infrared and, chiefly, Raman spectra. However, Brester’s method is very cumbersome and complicated. The most natural approach is consideration on the basis of group theory, relying on the group properties of symmetry operations. Here knowledge of the behavior of the amplitudes will not be needed—we shall proceed only from the most general symmetry properties of the molecule. The results obtained are, in principle, the same as in the application of Brester’s theory, since group theory is merely a translation of the theory of linear transformations into another, simpler mathematical language. The further exposition will be devoted to the method of applying group theory to the vibrations of polyatomic molecules.

Before turning to group theory, let us briefly discuss the principal properties of infrared and, chiefly, Raman spectra.

§ 2. Vibrational Spectra of Molecules

On interacting with a material medium, light undergoes more or less profound changes. These changes are always characteristic of the given medium and may concern the frequency, the state of polarization, the amplitude, the phase, the direction of propagation—in short, the entire set of properties of the light wave. Whereas the changes in infrared absorption concern only the spectral composition and intensity, the spectrum of scattered light gives us more detailed information about the structure of the particles of the medium, since here the changes also concern the state of polarization, and the laws

these changes—the polarization relations—are closely connected with the structure of the substance.

The oscillation of a system of charged particles, in which the dipole moment of the molecule changes, is associated with the emission of infrared waves of length from 3 to 20 μ. Conversely, if such an oscillation is excited by an external light wave, then the molecule, on the one hand, absorbs light in the same interval of wavelengths and, on the other hand, scatters it; moreover, along with light scattered without a change in wavelength, a definite part of the total intensity also falls on shifted wavelengths—on combination or Raman scattering. The intensity of the light emitted in all directions is proportional to the fourth power of the frequency and to the square of the second time derivative of the electric moment. For the infrared spectrum, only the change of the electric moment during the oscillation is essential—normal oscillations not accompanied by such a change are forbidden in the infrared spectrum. For the spectrum of scattered light, the behavior of the electron shell during the oscillations of the molecule is essential: its ability to be deformed, its polarizability. We can construct a theory of scattering on the basis of Bohr’s correspondence principle, treating radiation classically, but starting from the Schrödinger equation for the scattering medium[^8]. In this case the matrix elements of the electric moment $\mathbf{M}$ are constructed with the aid of the eigenfunctions modified by the external light wave. The intensity of the shifted Raman line associated with the transition from state $n$ to state $k$ is given by the expression

\[ J_{nk}=\frac{64\pi^4}{3c^3}(\nu+\nu_{nk})^4|\mathbf{E}_{nk}|^2, \tag{2, 1} \]

where

\[ \mathbf{E}_{nk}=\frac{1}{h}\sum_r\left\{\frac{(\mathbf{A}\mathbf{M}_{nr})\mathbf{M}_{rk}}{\nu_{rn}-\nu}+\frac{\mathbf{M}_{nr}(\mathbf{A}\mathbf{M}_{rk})}{\nu_{rk}+\nu}\right\}, \tag{2, 2} \]

the vector $\mathbf{A}$ is the amplitude of the incident wave, and $\nu$ is its frequency.

The calculation of the matrix elements of the electric moment requires knowledge of the $\psi$-functions of the state. However, for polyatomic molecules the $\psi$-functions are unknown. Therefore here we are essentially deprived of the possibility of using formula [2, 1]. But this circumstance should not embarrass us in analyzing the problems under consideration. Following the propositions set forth in the preceding paragraph, we can take the classical path, abandoning knowledge of the character of the bond between the particles and, consequently, knowledge of the $\psi$-functions. If classical mechanics is inapplicable for interpreting atomic spectra and the electronic spectra of molecules, which are connected with the motions of particles possessing small mass and high velocities—electrons—then, in the case of molecular oscillations, we should expect great success for the classical theory. In fact, the oscillation of a molecule is the oscillation of its nuclei, i.e., of particles possessing masses at least 1860 times greater than the mass of the electron,

and moving much more slowly than the latter. The vibrations of the nuclei are so slow that at each instant we may regard the electronic configuration as the same as if the nuclei were not moving. In other words, the “vibrations” of the electrons are so rapid that during one vibration of the nuclei the electrons manage to return many times to their former position, and we may calmly operate with the average state of the electronic configuration.

The preceding quantum-mechanical considerations apply equally to any scattering system. For vibrations of the nuclei in a molecule we may confine ourselves to simplified ideas. Here one may consider the Raman effect as the result of the action of the vibrations of the nuclei on the state of the electronic shell—the periodic deformation of the latter. The external electric field of the light wave excites in the molecule a certain electric moment:

\[ \mathbf{M}=\alpha \mathbf{E}. \tag{2, 3} \]

Moreover, in the general case $\mathbf{M}$ is not parallel to $\mathbf{E}$, since the polarizability $\alpha$ is a tensor, which may be represented in the form of an ellipsoid. This tensor must be symmetric (Hermitian in the case of the presence of complex elements—the absorption of light), which follows from the requirement that the law of conservation of energy be observed*.

The polarizability $\alpha$ characterizes the electronic shell of the molecule. But the vibrations of the nuclei periodically deform it. Therefore $\alpha$ will be some function of the normal coordinates. Expanding it in a series in the normal coordinates $q_\lambda$ gives:

\[ \alpha(q)=\alpha_0+\sum_{\lambda=1}^{3N-6}\left(\frac{\partial \alpha}{\partial q_\lambda}\right)_0 q_\lambda +\frac{1}{2}\sum_{\lambda,\mu}^{3N-6}\left(\frac{\partial^2 \alpha}{\partial q_\lambda \partial q_\mu}\right)q_\lambda q_\mu+\ldots \tag{2, 10} \]

* Expression (2,3) may be written as follows:

\[ \begin{aligned} M_x&=\alpha_{xx}E_x+\alpha_{xy}E_y+\alpha_{xz}E_z,\\ M_y&=\alpha_{yx}E_x+\alpha_{yy}E_y+\alpha_{yz}E_z,\\ M_z&=\alpha_{zx}E_x+\alpha_{zy}E_y+\alpha_{zz}E_z. \end{aligned} \tag{2, 4} \]

$x, y$, and $z$ are coordinates in the molecule.

Let us calculate the work that is obtained when a force $E_x$, increasing from 0 to 1, acts on a molecule at rest:

\[ A_1=\int_0^1 M_x\,dE_x=\int_0^1 \alpha_{xx}E_x\,dE_x=\frac{1}{2}\alpha_{xx}. \tag{2, 5} \]

Now let there act a force $E_y$ from 0 to 1; $(E_x=1)$

\[ A_2=\int_0^1 M_y\,dE_y=\int_0^1(\alpha_{yx}+\alpha_{yy}E_y)\,dE_y =\alpha_{yx}+\frac{1}{2}\alpha_{yy}. \tag{2, 6} \]

M. V. VOLKENSTEIN

Constructing, with the aid of an oscillatory function of the form (1,6), the matrix elements for the polarizability $\alpha$ (2, 10),

\[ \alpha_{nn'}=\int \widetilde{\psi}_{\mathrm{osc}\, n\lambda}(q)\psi_{\mathrm{osc}\, n'}\,d\tau, \tag{2, 11} \]

we obtain, for Rayleigh unshifted scattering, the polarizability element $(\alpha)^{\lambda}_{n}$, while the elements $(\alpha)^{\lambda\mu\ldots}_{\lambda\mu\ldots}$ will give us Raman scattering. The equality of a given matrix element to zero means the prohibition of the corresponding vibration in the spectrum.

The appearance of new Raman frequencies is easy to understand if one bears in mind the periodicity of the deformations of the shell and of the external field of the light wave. Indeed, if in [2, 10], instead of $q_\lambda$, one substitutes periodic functions

\[ q_\lambda=q_\lambda^{(0)}\sin(2\pi\nu_\lambda t+\theta) \tag{2, 12} \]

and substitutes instead of $\mathbf{E}$ in (2,3)

\[ \mathbf{E}=\mathbf{E}_0\cos(2\pi\nu t+\theta_0), \tag{2, 13} \]

then in the expression for $M$, alongside terms varying with frequency $\nu$, there will also appear terms with frequencies $\nu\pm\nu_\lambda$, $\nu\pm2\nu_\lambda,\ldots,\nu\pm\nu_\lambda\pm\nu_\mu$, etc. Moreover, if in the expansion (2, 10) we restrict ourselves to first-order terms, assuming a harmonic change of the polarizability under the action of vibrations, then only the frequencies $\nu\pm\nu_\lambda$, $\nu\pm\nu_\mu$, etc., will be observed, but not overtones and combination frequencies. Overtones and combination frequencies can appear only in the presence of an anharmonic term in the expression for the potential energy of the molecule or in the presence of second-order derivatives in the expansion (2,10). Since the corresponding terms in most molecules are comparatively very small, overtones and combinations have very low intensity in experiment and are rarely observed.$^9$ In the present survey we shall hardly touch upon them.

The most important polarization relations in the Raman spectrum are also connected with the properties of the polarizability tensor. If $\alpha$ were a scalar quantity, then light scattered at a right angle to the direction of incidence would be completely polarized, as is explained by the following drawing (Fig. 1).

$^9$ The total work is equal to

\[ E_1=A_1+A_2=\frac{1}{2}\alpha_{xx}+\frac{1}{2}\alpha_{yy}+\alpha_{yx}. \tag{2, 7} \]

Carrying out the calculation in the other order, we obtain

\[ E_2=\frac{1}{2}\alpha_{xx}+\frac{1}{2}\alpha_{yy}+\alpha_{xy} \tag{2, 8} \]

and by the law of conservation of energy $E_1=E_2$ and

\[ \alpha_{xy}=\alpha_{yx} \]

and analogously

\[ \alpha_{xz}=\alpha_{zx}, \qquad \alpha_{yz}=\alpha_{zy}. \tag{2, 9} \]

If $M_z$ is parallel to $E$, then only the component $M_{zz}$ is observed. If $\alpha$ is not scalar, then $M_z$ is not parallel to $E$, and therefore the component $M_{zx}$ is observed. The light is partially depolarized. The degree of depolarization $\rho$ for a given Raman line (the ratio of the intensity of the depolarized part to the total intensity) characterizes the behavior of the polarizability under various vibrations and is determined by the ratio between the scalar and anisotropic parts of the tensor $\alpha$.

Fig. 1.

Every symmetric tensor consists of a set of 6 quantities

\[ \alpha_{xx},\ \alpha_{yy},\ \alpha_{zz},\ \alpha_{xy},\ \alpha_{yz},\ \alpha_{zx}, \]

which transform as products of vectors. Indeed, in passing from the system of axes $x, y, z$ to the system $x', y', z'$, we have

\[ \begin{aligned} \alpha_{x'x'} &= \sum_{xy} \alpha_{xy}\cos(xx')\cos(yx');\\ \alpha_{x'y'} &= \sum_{xy} \alpha_{xy}\cos(xx')\cos(yy'). \end{aligned} \tag{2, 14} \]

Such a tensor can be decomposed into two parts—the scalar $\alpha_I$ (mean polarizability), represented by a sphere, and the anisotropic $\alpha_{II}$, namely:

\[ \begin{aligned} \alpha &= \alpha_I+\alpha_{II};\\ \alpha_I &= \frac{\alpha_x+\alpha_y+\alpha_z}{3};\\ \alpha_{II} &= \sqrt{\frac{1}{2}\left[(\alpha_y-\alpha_z)^2+(\alpha_z-\alpha_x)^2+(\alpha_x-\alpha_y)^2\right]}. \end{aligned} \tag{2, 15} \]

$\alpha_{11}$, $\alpha_{12}$, $\alpha_1$ are different tensor components of the polarizability. Both $\alpha_1$ and $\alpha_{11}$ participate additively in the scattering. The theory gives for the degree of depolarization of light scattered without a shift (Rayleigh),${}^{8}$ under illumination by natural light,

\[ \rho_r=\frac{6\alpha_{11}^{2}}{45\alpha_1^{2}+7\alpha_{11}^{2}} . \tag{2, 16} \]

For the Raman effect, what is essential is not the polarization itself, but, as follows from the expansion (2, 10), its variation as a function of the normal coordinates $\left(\dfrac{d\alpha}{dq_i}\right)_0$. Denoting respectively $\left(\dfrac{d\alpha_I}{dq_i}\right)_0$ and $\left(\dfrac{d\alpha_{11}}{dq_i}\right)_0$ by $\alpha'_1$ and $\alpha'_{11}$, we obtain

\[ \rho_R(\nu_i)=\frac{6\alpha_{11}^{\prime\,2}}{45\alpha_1^{\prime\,2}+7\alpha_{11}^{\prime\,2}} . \tag{2, 17} \]

For totally symmetric vibrations of molecules possessing cubic symmetry, the $\alpha$-scalar $\alpha'_{11}=0$, $\left(\dfrac{d\alpha}{dq_i}\right)_0=\alpha_1$, and $\rho_R=0$.

In order to pass to the group-theoretical analysis of vibrations, it is necessary to pause over several basic concepts of group theory.

§ 3. Basic concepts of group theory${}^{10,11,12,13}$

We encounter the concept of a group every time we have to do with some set of elements united by a common property, and moreover to any pair of elements there may be associated, according to a known rule, a third element belonging to the same set. This element is called the product of the two elements of the pair,

\[ AB=C, \tag{3, 1} \]

where the order of multiplication is not always immaterial—there are groups for which

\[ BA=D\ne C. \tag{3, 2} \]

But $D$ nevertheless belongs to the same set.

In order that our set actually constitute a group, it must also possess the following properties.

Associativity of the law of multiplication:

\[ A(BC)=(AB)C. \tag{3, 3} \]

There must exist a “unit” of the group, i.e. an element $E$ such that multiplication by it leaves the element being multiplied unchanged

\[ AE=EA=A. \tag{3,4} \]

For each element of the group there must exist an inverse element

\[ AB=BA=E. \tag{3,5} \]

We say that $B=A^{-1}$, and conversely.

It should be remembered that throughout we are speaking of symbolic multiplication, i.e. only of some operation, different for different groups, which associates to two elements a third.

The most important physical phenomena and mathematical operations satisfy the conditions for a group. Thus, for example, all motions of a rigid body in a given coordinate system form a group. Condition $(3,1)$ is expressed by the fact that passing from position $A$ to $B$, and then from $B$ to $C$, is equivalent to the direct passage from $A$ to $C$, which forms the symbolic product of the two successive passages. It is obvious that the identity element will be the state of rest of the body, and the inverse will be the reverse displacement of it.

All motions of a rigid body decompose into translations and rotations. Any rotations also form a group. Thus, for example, all rotations in the plane about the $z$-axis form a group, each member of which can be represented by the usual transformation of coordinates

\[ \left. \begin{aligned} x'&=x\cos\varphi-y\sin\varphi,\\ y'&=x\sin\varphi+y\cos\varphi,\\ z'&=z. \end{aligned} \right\} \tag{3,6} \]

Any element of this group may be represented by the matrix composed of the coefficients of the transformation

\[ \begin{pmatrix} \cos\varphi & -\sin\varphi & 0\\ \sin\varphi & \cos\varphi & 0\\ 0 & 0 & 1 \end{pmatrix}. \tag{3,7} \]

The symbolic product of two elements of the group is replaced by ordinary multiplication of matrices. Condition $(3,1)$ is expressed by the relation:

\[ \begin{pmatrix} \cos\varphi_1 & -\sin\varphi_1 & 0\\ \sin\varphi_1 & \cos\varphi_1 & 0\\ 0 & 0 & 1 \end{pmatrix} \begin{pmatrix} \cos\varphi_2 & -\sin\varphi_2 & 0\\ \sin\varphi_2 & \cos\varphi_2 & 0\\ 0 & 0 & 1 \end{pmatrix} = \]

\[ = \begin{pmatrix} \cos(\varphi_1+\varphi_2) & -\sin(\varphi_1+\varphi_2) & 0\\ \sin(\varphi_1+\varphi_2) & \cos(\varphi_1+\varphi_2) & 0\\ 0 & 0 & 1 \end{pmatrix} \tag{3,8} \]

\[ \varphi_1+\varphi_2=\varphi_3. \]

The rotation group (3, 7) is continuous; to each value of the angle of rotation $\varphi$ from a continuous series there corresponds an element of the group. But in what follows we shall have to deal chiefly with finite, discrete groups.

Let us give a few more definitions. Two elements $A$ and $B$ of a group $G$ are called conjugate if in $G$ there is an element $C$ such that

\[ A=C^{-1}BC, \tag{3, 9} \]

or, what is the same thing,

\[ CA=BC. \tag{3, 9'} \]

Conjugate elements are also called similar; moreover, equation (3, 9) means that $B$, by means of a similarity transformation, has been transformed into $A$. Thus, by means of a similarity transformation one can bring the matrix (3, 7) to the simpler diagonal form:

\[ \begin{pmatrix} \dfrac{i}{\sqrt{2}} & \dfrac{1}{\sqrt{2}} & 0\\ \dfrac{1}{\sqrt{2}} & -\dfrac{i}{\sqrt{2}} & 0\\ 0 & 0 & 1 \end{pmatrix} \times \begin{pmatrix} \cos\varphi & -\sin\varphi & 0\\ \sin\varphi & \cos\varphi & 0\\ 0 & 0 & 1 \end{pmatrix} \times \begin{pmatrix} -\dfrac{i}{\sqrt{2}} & \dfrac{i}{\sqrt{2}} & 0\\ \dfrac{1}{\sqrt{2}} & \dfrac{1}{\sqrt{2}} & 0\\ 0 & 0 & 1 \end{pmatrix} = \begin{pmatrix} e^{i\varphi} & 0 & 0\\ 0 & e^{-i\varphi} & 0\\ 0 & 0 & 1 \end{pmatrix}. \tag{3, 10} \]

The matrices by which the matrix (3, 7) is multiplied on the left and on the right are indeed inverses of one another. To verify this, it is enough to multiply them—the result is the identity matrix.

It is easy to prove that two elements similar to a third are similar to one another. Every group, on the basis of this property, can be divided into classes of elements conjugate to one another and only to one another. The order of a class is the number of elements contained in it. Analogously, the order of a finite group is the number of its elements.

Group theory and all possibilities of its application are based on the idea of isomorphism,$^{14}$ which is of exceptionally profound significance in natural science. If to some system of objects $G$, standing in certain relations to one another, there is put into one-to-one correspondence another system of objects $G^1$, and moreover the same correspondence exists between all relations within the system $G$ and the relations within the system $G^1$, then we say that the systems $G$ and $G^1$ are isomorphic. It is not difficult to narrow

this definition, passing to the isomorphism of two groups. In the most general form, isomorphism is the basis of any application of mathematics to physics—the system of physical objects $G$ (phenomena, laws) proves to be isomorphic to a system of mathematical objects $G^1$ (operations, symbols). In our case we shall be concerned with the isomorphism of the symmetry group of a molecule with a certain group of matrices. As we shall see, this correspondence makes it possible to solve, in the simplest way, the difficult problem of the vibrations of a molecule.

In connection with the idea of isomorphism, the concept of a representation of a group is introduced. A representation of a group is a group of matrices isomorphic to it. Thus, a representation of the group of rotations in a plane about the $z$-axis is the group of matrices (3, 7). The simplifications introduced by representations are connected with the fact that we may pass from symbolic multiplication to the simple multiplication of matrices. Through the properties of representations, group theory is connected with the theory of linear transformations, which is of great importance for theoretical physics, since each matrix is essentially only a symbolic, operator expression of a linear transformation—the matrix (3, 7) corresponds to the transformation (3, 6). Let us show, by the example of the classification of atomic terms, the method of applying the theory of representations, and thereby also group theory, to a problem of theoretical physics $^{12,13,15}$.

A problem of eigenvalues is posed—the differential Schrödinger equation. Without solving it, we of course cannot find the numerical eigenvalues of energy and momentum; but on the basis of group-theoretical considerations we can indicate the number and quantum numbers of the eigenvalues, and hence also the degrees of degeneracy of the corresponding eigenfunctions. In this circumstance is reflected the similarity of the problem of the proper vibrations of a molecule to the problem of atomic terms. The discussion is carried out as follows.

The Schrödinger equation possesses invariance with respect to certain symmetry operations $R_i$—with respect to rotations and reflections of the coordinates of the electrons and with respect to permutations of the electrons. If the function $\psi(x_1, x_2, \ldots, x_n)$ is a solution of the equation corresponding to the eigenvalue $\varepsilon$, then the result of applying the linear transformation $R_i$ to this function, $R_i \psi(x_1, x_2, \ldots, x_n)$, will also be a solution of the equation corresponding to the same energy $\varepsilon$. By virtue of the linearity of the equation, any linear combination of such functions will be its solution. The number $l$ of linearly independent eigenfunctions corresponding to one and the same $\varepsilon$ is called the degree of degeneracy of the function. Any function $R_i \psi$ is expressed in terms of the $l$ functions of the given $\varepsilon$:

\[ R_i \psi=\sum_{\lambda=1}^{l} a_{\lambda x}^{R_j}\psi_{\lambda x}. \tag{3, 11} \]

The matrices $(a_{\lambda\chi}^{R_i})$ (the upper index corresponds to one or another element of the group of symmetry operations) form an $l$-dimensional representation of the group of elements $R_i$. Indeed, to each element of the group there corresponds its own $l$-dimensional matrix; for example, the product of the elements $R_iR_j$

\[ R_iR_j\psi_\lambda = \sum_{\chi=1}^{l} a_{\lambda\chi}^{R_i} R_j\psi_\chi = \sum_{\chi=1}^{l}\sum_{\mu=1}^{l} a_{\lambda\chi}^{R_i} a_{\chi\mu}^{R_j}\psi_\mu \tag{3, 12} \]

corresponds to the product of the matrices

\[ a_{\lambda\mu}^{R_iR_j} = \sum_{\chi=1}^{l} a_{\lambda\chi}^{R_i} a_{\chi\mu}^{R_j}. \tag{3, 13} \]

We have obtained a representation of the term $\varepsilon$. Since the $l$ functions $\psi_\chi$ are linearly dependent, this representation is irreducible. In other words, there exists no similarity transformation, carried out simultaneously for all elements of the group, that would bring the matrices $(a_{\lambda\chi}^{R_i})$ to diagonal, or at least to stepped, form (Fig. 2).

Fig. 2.

Fig. 2.

In the matrix of Fig. 2, only those functions are linearly dependent which are connected by the coefficients of the shaded squares—the stepped form means the absence of linear dependence between the functions of separate squares.

Conversely, if we consider the entire set of proper functions of the atom corresponding to all values of $\varepsilon$, then we obtain a reducible representation, which by a certain similarity transformation can be brought to the form of Fig. 2. In this case the irreducible parts of the matrices will form irreducible representations of the group corresponding to different values of $\varepsilon$, different terms. The degree of degeneracy of the proper functions is given simply by the order of the irreducible representation. This is the so-called necessary degeneracy, which can be removed only by breaking the symmetry, by changing the group $R_i$—for example, by introducing an external field. In contrast to this, accidental degeneracy is the coincidence of eigenvalues $\varepsilon$ for functions corresponding to different irreducible representations. Accidental degeneracy can be removed by changing the energy conditions in the atom even while preserving symmetry, i.e. in the absence of an external field.

VIBRATIONS OF POLYATOMIC MOLECULES

The eigenfunctions of an atom with spherical symmetry are spherical functions[^15]. They have the form:

\[ \psi = P_m^l(\vartheta \varphi)=e^{-im\varphi}P_m^l(\vartheta), \tag{3,14} \]

where for $m \geq 0$

\[ P_m^l(\vartheta)=P_{-m}^l(\vartheta) = \frac{\sin^m \vartheta}{2^l l!}\, \frac{d^{\,l+m}\sin^{2l}\vartheta}{(d\cos\vartheta)^{l+m}} . \tag{3,15} \]

Applying to such a function one of the operations of rotation in space, characterized by the Euler angles $\alpha,\ \beta,\ \gamma$, we, because of the $2l+1$-fold degeneracy of the spherical functions, obtain a $(2l+1)$-dimensional representation of the rotation group

\[ R_{\{\alpha\beta\gamma\}}P_m^l(\vartheta,\varphi) = \sum_{m'=-l}^{l} D^{(l)}(\{\alpha\beta\gamma\})_{m'm} P_{m'}^l(\vartheta,\varphi). \tag{3,16} \]

If, as the element $R$, we choose a rotation about some axis $z$ through an angle $a=\varphi$, then, by virtue of (3,14), we obtain the representation in the form

\[ D^{(l)}(\{\varphi 00\}) = \begin{bmatrix} e^{-il\varphi} & 0 & \cdots & 0 & 0\\ 0 & e^{-i(l-1)\varphi} & \cdots & 0 & 0\\ \cdot & \cdot & & \cdot & \cdot\\ \cdot & \cdot & & \cdot & \cdot\\ 0 & 0 & \cdots & e^{i(l-1)\varphi} & 0\\ 0 & 0 & \cdots & 0 & e^{il\varphi} \end{bmatrix}. \tag{3,17} \]

Thus, if we were to introduce a field along the $z$-axis, then the removal of degeneracy would take place: instead of a single $2l+1$-dimensional representation, instead of a single term, we would obtain $2l+1$ one-dimensional representations, $2l+1$ nondegenerate terms. For the special case $l=1$ we had the matrix (3,10).

Thus, the problem of finding the number of terms and their corresponding degrees of degeneracy reduces to finding all the irreducible representations of the given group. This is carried out on the basis of the very important orthogonality relations in the theory of linear representations, and of the orthogonality relations following from them between irreducible representations and between classes of similar elements of the group. The same relations also hold between the so-called characters of representations—the sums of their diagonal elements. The significance of these quantities is determined by the fact that they are invariant with respect to similarity transformations. Indeed,

Let the elements of some matrix $R$ be $r_{kl}$, and the elements of the matrices $C^{-1}$ and $C$ be $c_{uk}$ and $c_{lw}$. The general element of the matrix $D=C^{-1}RC$ will be

\[ d_{uw}=\sum_l\sum_k c_{uk}r_{kl}c_{lw}. \tag{3,18} \]

The character is the sum of the diagonal terms

\[ \chi_D=\sum_u d_{uu}=\sum_u\sum_l\sum_k \bar c_{uk} r_{kl} c_{lu}. \tag{3,19} \]

By the definition of the inverse matrix,

\[ \sum_u \bar c_{uk}c_{lu}=\delta_{lk}= \begin{cases} 1 & k=l\\ 0 & k\ne l . \end{cases} \tag{3,20} \]

Consequently,

\[ \chi_D=\sum_u d_{uu}=\sum_k r_{kk}=\chi_R, \tag{3,21} \]

Q.E.D.

The character of the 3-dimensional representation of the rotation group (3, 7) is equal to

\[ \chi_3=1+2\cos\varphi. \tag{3,22} \]

The character of the transformed matrix (3, 10) is equal to

\[ \chi_3=1+e^{i\varphi}+e^{-i\varphi}, \tag{3,23} \]

which is the same thing.

The character of the $(2l+1)$-dimensional representation of the rotation group (3, 17) is equal to

\[ \chi_{2l+1}=\sum_{m=-l}^{l} e^{im\varphi} =1+2\cos\varphi+2\cos2\varphi+\ldots+2\cos l\varphi = \frac{\sin\left(l+\frac12\right)\varphi}{\sin\frac12\varphi}. \tag{3.24} \]

By virtue of the invariance of the character with respect to similarity transformations, the characters of all elements of one and the same class are equal. The orthogonality relations between characters make it possible to decompose any representation of a group into its irreducible parts. Indeed, the character of the entire representation in Fig. 2 is, evidently, equal to the sum of the characters of the irreducible, shaded parts. Let us represent the character in the form of such a sum

\[ \chi=a_1\chi_1+a_2\chi_2+\ldots+a_p\chi_p. \tag{3,25} \]

The coefficients $a_1,a_2,\ldots,a_p$ give us the number of repetitions in the decomposition into irreducible parts of one or another irreducible part. On the basis of the orthogonality relations these coefficients are calculated analogously to Fourier coefficients,

\[ a_m=\frac1n\sum_R \chi(R)\chi_m(R). \tag{3,26} \]

The summation extends over all elements of the group $R$; $n$ is the order of the group.

The expansion into the sum (3, 25) is a kind of Fourier series.

Thus, in order to find the number and degrees of irreducible representations, one must know their characters.

We shall dwell on other relations in the theory of groups and representations in the further exposition.

§ 4. Classification of the Natural Vibrations of a Molecule on the Basis of Group Theory

A polyatomic molecule always possesses a definite symmetry, i.e., there is a series of operations—rotations, reflections, and rotations followed by reflection—which leave the equilibrium state of the molecule unchanged. The symmetry operations form a group, as follows from the fact that the result of the successive application of two such operations can be represented as the result of applying a third operation belonging to the same set of symmetry elements. One can classify the normal vibrations of a molecule on the basis of the laws of their transformations under the performance of one or another symmetry operation. We distinguish vibrations that are symmetric with respect to a given operation, i.e., those that do not change when it is performed; antisymmetric vibrations, for which all displacements change sign when the operation is performed; and degenerate vibrations, which, when the symmetry operation is performed, pass into new vibrations with the same frequency. Knowing the degrees of degeneracy and the symmetry properties of the vibrations, we shall be able to indicate how many frequencies there will be in the infrared and in the Raman spectrum of the molecule.

Fig. 3.

The problem of finding the degrees of degeneracy of vibrations is essentially analogous to the problem, considered in the preceding paragraph, of finding atomic terms. In place of eigenfunctions we now have normal coordinates, and hence vibrations; and in place of the eigenvalues of energy, momentum, etc., frequencies appear. The group-theoretical solution of the problem was carried out by Wigner[^17]. It is the simplest.

Let there be placed, in some spatial coordinate system, $N$ material points forming the nuclear skeleton of the molecule (Fig. 3). The nuclei of the molecule execute a certain vibrational motion, determined by a set of vectors $s_k$ (where $k = 1, 2, \ldots, N$) of the displacements of the individual particles from the equilibrium position. For different normal vibrations these sets of vectors are different. We denote the set of vectors corresponding to the $p$-th normal vibration $(p = 1, 2, \ldots, 3N)$ by $s_p$.

Any displacement of the molecule will be represented by a linear combination of these sets (1, 3).

Since we are dealing with normal vibrations, the quantities $s^{(1)}, s^{(2)}$ are orthogonal and normalized. The coefficients $q_1, q_2 \ldots q_{3N}$ are normal coordinates.

The molecule possesses a symmetry forming a group $G$. Let us perform on the molecule, vibrating with frequency $\nu_p$, one of the transformations of the group $R$. We obtain the displacements $R s^{(p)}$. In this case, in place of the $k$-th particle there is the particle with number $l$. We shall write this in the form $l=R(k)$, or $k=R^{-1}(l)$,

\[ R s_k = R s_{R^{-1}(l)}. \tag{4,1} \]

The frequency of the vibration has obviously not changed. However, $R s^{(p)}$ still does not represent a new normal vibration with the same frequency $\nu_p$. Here we have merely rotated the molecule. But

Fig. 4.

if we renumber the particles anew, placing the $l$-th particle in the position of the $k$-th, then, owing to the equivalence of the particles, the configuration will not change (Fig. 4c). We shall denote the operation $R$ with the subsequent renumbering by $\overline{R}$. Then

\[ R s_k = R s_{R^{-1}(l)} = \overline{R}s_l . \tag{4,2} \]

The displacements $\overline{R}s_l$ form a system of amplitudes of a certain new normal vibration $R s^{(p)}$, possessing the same frequency $\nu_p$ as $s^{(p)}$, i.e. degenerate with it. If there are altogether $f$ such vibrations (the number $f$ is the degree of degeneracy following from the symmetry properties*), then

\[ \overline{R}s^{(x)}=\sum_{\lambda=1}^{f} D(R)_{\lambda x}s^{(\lambda)};\qquad x=1,2,\ldots,f, \tag{4,3} \]

* In Fig. 4, $f=3$, corresponding to the three rotations about particle No. 4 through $\frac{2\pi}{3}$. Meanwhile, as will be shown in § 5, the maximum degree

i.e. any normal vibration, being $f$-fold degenerate, is accordingly expressed in terms of a linear combination of $f$ components. Expression (4,3) is quite analogous to expression (3,11). It is clear that the $f$-row square matrices $D(R)_{\lambda\mu}$ form an irreducible representation of the group $G$. Indeed, if $T$ is an operation of the same group, then

\[ \overline{TR}s^{(\mu)} = \sum_{\lambda=1}^{f} \overline{T}D(R)_{\lambda\mu}s^{(\lambda)} = \sum_{\lambda=1}^{f}\sum_{\nu=1}^{f} D(T)_{\nu\lambda}D(R)_{\lambda\mu}s^{(\nu)} = \]

\[ = \sum_{\nu=1}^{f}D(TR)_{\nu\mu}s^{(\nu)}. \tag{4,4} \]

[Compare with (3,12)],

i.e. the product of two matrices corresponding to elements of the group again gives a matrix corresponding to an element of the same group. The remaining group properties are easily proved. The fact that the representation is irreducible follows, as in § 3, from the very definition of degeneracy.

Thus, to each $f$-fold degenerate normal vibration of a molecule there corresponds an irreducible $f$-dimensional representation of the group $G$ of symmetry operations. But the converse is also true: to each irreducible representation of the group $G$ there corresponds a definite type of vibration, and the degree of degeneracy is given by the order of the corresponding representation. We have as many types of normal vibrations as the group $G$ has irreducible representations.

If we consider not the $f$ vibrations with frequency $\nu_p$, but in general all $3N$ normal vibrations, then

\[ \overline{R}s^{(\alpha)} = \sum_{\lambda=1}^{3N} \Delta(R)_{\lambda\alpha}s^{(\lambda)}; \qquad \alpha=1,2\ldots 3N. \tag{4,5} \]

In view of what has just been said, the representation $\Delta(R)$ is reducible and is reduced to the form

\[ \Delta(R)= \begin{bmatrix} D^{(1)}(R)&0&\ldots\\ 0&D^{(2)}(R)&\ldots\\ \cdot&\cdot&\cdot&\cdot\\ \cdot&\cdot&\cdot&\cdot \end{bmatrix}. \tag{4,6} \]

As was indicated in § 2, in order to determine which and how many irreducible representations a group representation contains, one must compute its character and use the orthogonality relations existing between them. These characters are, of course, different for different symmetry groups, but the general form of the character $\Delta(R)$ can be obtained without further work.

\[ \chi'(R)=\sum_{\xi=1}^{3N}\Delta(R)_{\xi\xi}. \tag{4,7} \]

We shall carry out this calculation following Wigner. Let us denote the displacement in which only the $k$-th particle is displaced by one unit in the direction $\alpha$ $(\alpha=x,y,z)$ by

\[ s_{k\alpha}. \]

It is clear that

\[ (s_{k\alpha})_{l\beta}=\delta_{kl}\delta_{\alpha\beta} = \begin{cases} 0, & \text{if } k\ne l \text{ or } \alpha\ne\beta,\\ 1, & \text{if } k=l \text{ and } \alpha=\beta . \end{cases} \tag{4,8} \]

Any aggregate vector $s^{(i)}$ can be represented in the form

\[ s^{(i)}=\sum_{k=1}^{N}\sum_{\alpha=x,y,z}s^{(i)}_{k\alpha}s_{k\alpha}. \tag{4,9} \]

Conversely, by virtue of the orthogonality and normalization of the vectors,

\[ s_{k\alpha}=\sum_{i=1}^{3N}s^{(i)}_{k\alpha}s^{(i)}. \tag{4,10} \]

Applying the operation $\overline{R}$ to $s_{k\alpha}$, we obtain

\[ \overline{R}s_{k\alpha} = \sum_{l=1}^{N}\sum_{\beta=x,y,z} \overline{\Delta}(R)_{l\beta;k\alpha}s_{\beta l}. \tag{4,11} \]

Here $\overline{\Delta}(R)$ is again a representation of the group $G$. It is obtained by a transformation of similarity from $\Delta(R)$:

\[ \overline{\Delta}(R)=A^{-1}\Delta(R)A, \tag{4,12} \]

where $A$ is the matrix $\bigl(s^{(i)}_{k\alpha}\bigr)$. Consequently (cf. p. 344), the character $\overline{\Delta}(R)$ is equal to the character $\Delta(R)$:

\[ \chi'(R)=\sum_{l\beta}\overline{\Delta}(R)_{l\beta;l\beta}. \tag{4,13} \]

In order to compute the character in explicit form, it is necessary to specify what, in essence, happens to the displacement images when the operation $\overline{R}$ is applied to it (4,13). After the operation $R$ [[unclear: continuation cut off at bottom of page]]

the number $k$ acquires displacement components $R_{\alpha x}, R_{\alpha y}, R_{\alpha z}$ along the axes $x,y,z$, and

\[ \overline{R}_{s k\alpha} = \sum_{\beta=x,y,z} R_{\alpha\beta}s_{R^{-1}(k)\beta}. \tag{4,14} \]

At the same time, (4,13) holds. Hence a separate diagonal term is

\[ \overline{\Delta}(R)_{k\alpha;k\alpha} = \begin{cases} 0 & \text{for } R^{-1}(k)\ne k,\\ R_{\alpha\alpha} & \text{for } R^{-1}(k)=k. \end{cases} \tag{4,15} \]

And the sum of the diagonal terms of the matrix $R_{\alpha\beta}$, which we shall denote by $H(R)$, is

\[ H(R)=\sum_{\alpha=x,y,z}R_{\alpha\alpha} = \sum_{\alpha=x,y,z}\overline{\Delta}(R)_{k\alpha;k\alpha}, \quad \text{if } R^{-1}(k)=k. \tag{4,16} \]

We have passed from calculating the character of a $3N$-dimensional matrix to a three-dimensional one.

Finally, if $u_R$ is the number of equilibrium positions of individual particles that $R$ leaves unchanged—the so-called proper symmetry of the particles—then

\[ \chi'(R)= \sum_{k=1}^{N}\sum_{\alpha} \overline{\Delta}(R)_{k\alpha;k\alpha} = u_R H(R), \tag{4,17} \]

since $u_R$ gives that fraction of the number $N$ for which the condition $R^{-1}k=k$ is satisfied.

The question arises what $H(R)$ is equal to. $R$ is either a rotation, or a reflection, or a rotation followed by a reflection. As we saw in § 3 (3,22), the character of the three-dimensional representation of the rotation group is

\[ H(R)=1+2\cos\varphi_R. \tag{4,18} \]

If, in addition, there is a reflection, then the sign changes. Consequently

\[ \chi'(R)=\pm u_R(1+2\cos\varphi_R). \tag{4,19} \]

The plus sign corresponds to a pure rotation, the minus sign to a rotation followed by a reflection.

It should not be forgotten that, for generality, we included among the normal vibrations of the molecule also translation and rotation. In reality we are interested not in $3N$, but in $3N-6$ internal vibrations of the molecule. Therefore one may subtract from $\chi'(R)$ the characters of the irreducible representations of translation and rotation. These are, respectively, the representations of the polar and axial vector. Both vectors transform in the same way under rotation, but under mirror reflection the polar vector changes sign, whereas the axial vector retains it. The characters of their irreducible representations will therefore be

\[ \begin{aligned} &\text{for translation} \quad \pm(1+2\cos\varphi_R),\\ &\text{for rotation} \quad 1+2\cos\varphi_R. \end{aligned} \]

Subtracting from $\chi'(R)$, we have

\[ \chi(R)= \begin{cases} (u_R-2)(1+2\cos\varphi_R), & \text{if } R \text{ is a pure rotation},\\ -u_R(1+2\cos\varphi_R), & \text{if } R \text{ is a rotation with reflection}. \end{cases} \]

Thus the character $\Delta(R)$ has been calculated. To determine what vibrations can occur here, it is necessary to decompose $\Delta(R)$ into the irreducible representations of the symmetry group $G$. As we saw in § 3 (3.25), this is done by expanding $\chi(R)$ in a Fourier series in the characters of the irreducible representations $\chi_m(R)$. To obtain numerical results, therefore, it is necessary to know the quantities $u_R$, $\varphi_R$, $\chi_m(R)$, and the number of elements of the group. How these quantities are found for various symmetry groups will be shown in the next paragraph.

§ 5. Irreducible representations of symmetry groups and their characters

The symmetry of a molecule is point symmetry. By carrying out on a molecule a symmetric operation of rotation, rotation with reflection, reflection in a point or in a plane, we leave fixed at least one point—the center of gravity of the molecule.

In other words, we exclude the symmetry of translation, screw motion, glide reflection, and in this lies the essential difference between the symmetry of a molecule and the symmetry of a crystal. In fact, a crystal is a spatial lattice and, as Schönflies[^18] showed, there exist 230 kinds of such lattices. However, if one excludes the symmetry of translation, etc., and considers the symmetry of the unit cell of a crystal, then the number of (point) symmetry groups is reduced to 32; correspondingly, the same will be the number of independent classes into which the 230 spatial symmetries of crystal lattices can be divided. The point symmetry of a molecule in principle coincides with the point symmetry of the unit cell of a crystal. However, the coincidence is not complete: the fact that a crystal is a spatial lattice restricts the orders of the rotation axes to the numbers 2, 3, 4, 6 (self-coincidence under rotations respectively by $180$, $120$, $90$, and $60^\circ$) and the angles of intersection of axes to the values $0^\circ$, $60^\circ$, $70^\circ 31' 44''$, $90^\circ$, $109^\circ 28' 16''$, and $180^\circ$.

For molecules, in addition, axes of the icosahedral group $J$ of order 5 (rotation by $72^\circ$) are also possible, but the number of molecular symmetry groups is again, of course, determined by the maximum possible number of symmetrically constructed polyhedra.

A complete enumeration of the molecular symmetry groups is given by Placzek.

We give a summary of the basic notations of symmetry elements according to Schönflies.

Axes of symmetry—axes of rotation, denoted by the letter $C$ with indices: $C_p$—rotations through $\dfrac{2\pi}{p}$, $C_\infty$—continuous rotation. Two mutually perpendicular axes form a dihedral group $D$.
Planes of reflection—$C_s$ or $\sigma, \sigma_d$. A plane perpendicular to the axis of rotation is $\sigma_h$, one parallel to it is $\sigma_v$.
Center of reflection—inversion $C_i$ or $i$.
Axis of rotation followed by reflection—$S_p$.

Let us explain the meaning of rotation followed by reflection by means of a drawing borrowed from Ewald$^ {18}$ (Fig. 5).

As a result of a rotation about an axis lying in the plane of the drawing by $90^\circ$, followed by reflection in a plane perpendicular to the plane of the drawing, the directions 1, 2 pass into directions 3, 4 equivalent to them. This cannot be achieved by simple reflection or by simple rotation.

Fig. 5.

Fig. 5.

As we saw in the preceding paragraph, the classification of the vibrations of a symmetric molecule can be carried out provided that the characters of the irreducible representations of the corresponding symmetry group are found. Obviously, by exactly the same method we can find which vibrations are characteristic of an elementary crystal cell of a given symmetry. The calculation of the characters of irreducible representations for point crystal groups was carried out by Bethe$^ {19}$. The problem before him was that of determining the laws of splitting of atomic terms in a crystal. In fact, it is clear that, by placing an atom in an electric field of specified symmetry, we obtain a splitting of terms depending on the magnitude of the angular momentum in the atom and on the symmetry of the external field (Stark effect). As was indicated in § 3, the Schrödinger equation is invariant with respect to certain symmetry operations in the atom, in particular with respect to continuous rotation, represented by the matrices (3, 17). The expression of the atomic eigenfunctions has degree $2l+1$, where $l$ is the quantum number of the orbital moment. However, when a field of symmetry lower than spherical is introduced, the degeneracy is lowered, disappearing altogether in the limiting case of an axial field. Finding the number and degrees of expression of the crystal terms into which the $(2l+1)$-fold expressed eigenvalue of the free atom is split is naturally carried out by decomposing the character (3, 17), equal according to (3, 24) to

\[ \chi=\frac{\sin\left(l+\frac{1}{2}\right)\varphi}{\sin \frac{1}{2}\varphi}. \tag{5,1} \]

in Fourier series, by the characters of the irreducible representations of the various symmetry groups. The number and degree of these representations again give us the number and degrees of degeneracy of the split terms. Here we again see an analogy with the methods for solving problems on atomic terms and on molecular vibrations. We may therefore, for our purposes, make use of the characters of the irreducible representations of the crystallographic symmetry groups computed by Bethe.

Let us consider the method of calculation by the example of one of the cubic groups—the tetrahedral group, supplemented by rotations followed by reflection $T_d$. This is the symmetry of tetrahedral molecules of the type $XY_4$ (methane, carbon tetrachloride, etc.)

1) $E$—identity—1 element.
2) $C_2$—rotations by $\pi$, such axes 3—3 elements,
3) $\sigma$—planes of reflection—there are 6 of them.
4) $C_3$—rotations by $\dfrac{2\pi}{3}$, such axes 4—8 elements

and

5) $S_4$—rotations with subsequent reflections about three axes $C_p$ by $\dfrac{\pi}{2}$—6 elements.

Let us clarify this enumeration with a drawing (Fig. 6). In the figure one axis each of $C_2$, $C_3$, $S_4$ is shown, and one of the planes $\sigma$ is shaded.

Fig. 6.

According to the rules of group theory, a group consisting of a finite number $h$ (in the present case 24) of elements must possess as many $l$ irreducible representations as it contains classes. Indeed, we saw (p. 344) that a group has as many different characters of irreducible representations as it contains classes of similar or conjugate elements. The question is: how large is this number $l$?

In group theory it is proved that to any finite group with $h$ elements (in our case—24) one can associate an isomorphic group of permutations of $n$ elements; moreover, since the full group of permutations of $n$ numbers contains $n!$ permutations, then

\[ h = n! \tag{5,2} \]

and in our case

\[ 24 = 4! \]

i.e., the elements of our group are isomorphic to the permutations of four numbers, for example

\[ \begin{pmatrix} 1 & 2 & 3 & 4\\ 2 & 3 & 4 & 1 \end{pmatrix} \quad \text{etc.—24 in all.} \tag{5,3} \]

Every permutation can be represented as a product of cyclic permutations of the form

$$ (a\ b\ c\ d). \tag{5,4} $$

Expression (5,4) denotes the permutation that sends $a$ to $b$, $b$ to $c$, $c$ to $d$, $d$ to $a$. It is clear that the permutation (5,3) is represented by the cycle

$$ (1\ 2\ 3\ 4) \tag{5,5} $$

and, for example, the permutation

$$ \begin{pmatrix} 1&2&3&4\\ 2&1&3&4 \end{pmatrix} \tag{5,6} $$

by the product of cycles

$$ (1\ 2)\ (3)\ (4). \tag{5,7} $$

It can be shown that under a similarity transformation the number of members of a cycle does not change. Therefore the number of classes in our group of 24 elements is given by the number of different ways of representing permutations of the type (5,3), (5,6), etc., in the form of cycles, namely:

$$ \begin{gathered} 1)\ (a\ b\ c\ d),\quad 2)\ (a\ b\ c)\ (d),\quad 3)\ (a\ b)\ (c\ d),\quad 4)\ (a\ b)\ (c)\ (d),\\ 5)\ (a)\ (b)\ (c)\ (d) \end{gathered} $$

or, what is the same thing, the number of classes $l$ is equal to the number of ways of partitioning the number $n$ into a sum of positive integers${}^{20}$

$$ \begin{aligned} &1)\quad 4=4; &&4)\quad 4=2+1+1;\\ &2)\quad 4=3+1; &&5)\quad 4=1+1+1+1.\\ &3)\quad 4=2+2; \end{aligned} $$

Thus, in our case we have 5 classes, which corresponds to the 5 kinds of symmetry elements listed on p. 352. The number of irreducible representations will also be $l$. Their orders we shall obtain, according to the rules of group theory, by decomposing $h$ into the sum of $l$ squares of integers—in our case

$$ 24=3^2+3^2+2^2+1^2+1^2 \tag{5,8} $$

the orders of the irreducible representations will be $3, 3, 2, 1, 1$.

The total number of characters will obviously be 25, since to each of the five representations of each of the five classes of the group there corresponds one character. The characters are computed on the basis of the orthogonality relations between classes and representations. Bethe uses the relations${}^{21}$

$$ h_i h_k \chi_j^{\,i}\chi_j^{\,k} = \chi_1 \sum_{j=1}^{l} c_{ikj} h_j^{\,i}\chi_j, \tag{5,9} $$

where $h_i$ is the number of elements of the given class; $\chi_i$ is its character; $i,k,j$ range from 1 to $l$; $\chi_1$ is the character of the class $E$, consisting of one

element—the identity substitution (1) (2) (3) (4), represented by the unit matrix. It is obvious that $\chi_1$ for each of the representations is equal to the order of the given representation, i.e. $3, 3, 2, 1$, and $1$. $c_{lkj}$ are integers characterizing the relations between the classes.

As a result, the following character table is obtained for the group $T_d$ (Table 1).

TABLE 1

representations	$E(1)$	$C_3(8)$	$C_2(3)$	$\sigma(6)$	$S_4(6)$	In parentheses, the numbers of elements in the given class
$A_1$	1	1	1	1	1	$\chi_1$
$A_2$	1	1	1	$-1$	$-1$	$\chi_2$
$K$	2	$-1$	$-2$	0	0	$\chi_3$
$F_1$	3	0	$-1$	$-1$	1	$\chi_4$
$F_2$	3	0	$-1$	1	$-1$	$\chi_5$

$A$—denotes a one-dimensional representation, $K$—two-, $F$—three-dimensional.

Thus we have obtained all the characters of the irreducible representations of the symmetry group $T_d$. In order to determine from this what vibrations, for example, the methane molecule $\mathrm{CH}_4$ is capable of performing, it remains to write down also the quantities $\varphi_R$—the angles of rotation for one or another operation—and $u_R$, the number of particles that remain fixed under these operations. Looking at Fig. 6, we can find these numbers without difficulty.

From this it is already easy to calculate the characters of the reducible representations $\chi'(R)$, and then also $\chi(R)$ (see § 4). We collect all the data in Table 2.

TABLE 2

$T_d$	$E(1)$	$C_3(8)$	$C_2(3)$	$\sigma(6)$	$S_4(6)$
$U_R$	5	2	1	3	1
$\varphi_R$	0	$\dfrac{2\pi}{3}$	$\pi$	$\pi$	$\dfrac{\pi}{2}$
$\pm(1+2\cos\varphi_R)$	3	0	$-1$	1	$-1$
$\chi'(R)$	15	0	$-1$	3	$-1$
$\chi(R)$	9	0	1	3	$-1$

The number and degrees of degeneracy of the vibrations we shall find by expanding $\chi(R)$ in a Fourier series in the characters taken from Table 1—see (3.25), where

\[ \chi(R)=\sum_{i=1}^{5} a_i \chi_i(R), \tag{5.10} \]

according to (3.26)

\[ a_i=\frac{1}{24}\sum_{j=1}^{5}\chi_i(R)\chi(R) \tag{5.11} \]

or, since the characters of all elements of a given class are equal to one another,

\[ a_i=\frac{1}{24}\sum_{i=1}^{5} h_i \chi'_i(R)\chi(R), \tag{5.12} \]

where $h_i$ is the number of elements of the class. We carry out the calculation for our case

\[ \left. \begin{aligned} a_1&=\frac{1}{24}(1\cdot1\cdot9+8\cdot1\cdot0+3\cdot1\cdot1+6\cdot1\cdot3+6\cdot1\cdot-1)=1,\\ a_2&=\frac{1}{24}(1\cdot1\cdot9+8\cdot1\cdot0+3\cdot1\cdot1+6\cdot-1\cdot-3+6\cdot-1\cdot-1)=0,\\ a_3&=\frac{1}{24}(1\cdot2\cdot9+8\cdot-1\cdot0+3\cdot-2\cdot1+6\cdot0\cdot3+6\cdot0\cdot-1)=1,\\ a_4&=\frac{1}{24}(1\cdot3\cdot9+8\cdot0\cdot0+3\cdot-1\cdot1+6\cdot-1\cdot3+{}\\ &\qquad\qquad\qquad\qquad\qquad\qquad {}+6\cdot1\cdot-1)=0,\\ a_5&=\frac{1}{24}(1\cdot3\cdot9+8\cdot0\cdot0+3\cdot-1\cdot1+6\cdot1\cdot3+{}\\ &\qquad\qquad\qquad\qquad\qquad\qquad {}+6\cdot-1\cdot-1)=2 \end{aligned} \right\} \tag{5.13} \]

Thus the character of the matrix representing the symmetry properties of the vibrations of the molecule is expanded in the series

\[ \chi(R)=\chi_1(R)+\chi_3(R)+2\chi_5(R), \tag{5.14} \]

and the matrix itself is reduced to a step-like form (Fig. 7).

Fig. 7.

We obtain for tetrahedral molecules of the type $XY_4$ in all 4 vibrations, of which 2 are triply degenerate, one is doubly degenerate, and one is nondegenerate.

§ 6. Degeneracy of normal vibrations

We have seen that, as a result of the existence of the specific symmetry of the molecule, some of its normal vibrations turn out to be degenerate. In the example considered of an $XY_4$ molecule, we obtained two- and threefold degeneracy of vibrations.

As already indicated, those normal vibrations are called degenerate which have identical frequencies and transform into one another under the performance of one or another symmetry operation. Since we have assumed the harmonic character of the vibrations near the equilibrium position (for deviations see below in § 9), any superposition of degenerate normal vibrations will again give us a normal vibration with the same frequency. However, the degree of degeneracy will not thereby become equal to infinity, since the number of linearly independent, jointly degenerate vibrations is strictly limited. A molecule $XY_2$, constructed in the form of a rod (for example $CO_2$), has the following normal vibrations,

Fig. 8.

The vibration in Fig. 8a is fully symmetric: it transforms into itself (the direction of all displacements is preserved) under the performance of all symmetry operations proper to such a molecule—continuous rotation about the $YXY$ axis and reflection in the point $X$. The vibration in Fig. 8b is antisymmetric with respect to reflection in the point $X$: the directions of the displacements change sign under reflection. Finally, by rotating Fig. 8c about the $YXY$ axis, we obtain an infinite number of jointly degenerate vibrations, all of which, however, can be obtained by the superposition of two vibrations having, obviously, the same frequency—the vibration shown in Fig. 8c and the same vibration rotated by $90^\circ$. Thus the vibration in Fig. 8c is doubly degenerate.

We must modify the very definition of a normal vibration for the case of degenerate vibrations. We defined a normal vibration (§ 1) as one in which all degrees of freedom simultaneously perform harmonic vibrations with the same frequency and phase. The new definition, correct for degenerate vibrations, no longer requires equality of phase, since any vibrations composed of degenerate ones, even with a difference of phase, are also normal.

The necessary degeneracy of normal vibrations connected with the symmetry of the problem is characteristic of any physical oscillatory systems. Thus the vibration of a conical pendulum is a superposition of two jointly degenerate vibrations of the same frequency in mutually perpendicular planes. In an analogous way, coupled electrical oscillations in complex circuits may also have identical frequencies. Here, instead of linear functions of displacements, linear functions of electric charges are considered as normal coordinates. The presence of a definite symmetry must, as before, lead to degen-

tion of normal vibrations. It should be supposed that the application of group theory must also, in these cases, give a substantial simplification in the solution of these difficult problems.

The degree of degeneracy of vibrations increases as the symmetry of the molecule increases. If the symmetry group consists of only two operations (the classes $E$ and $\sigma, i, C_2$), then there exist only two types of normal vibrations—antisymmetric or symmetric with respect to reflection $\sigma, i$, or rotation through $\pi$ $(C_2)$. Correspondingly, the normal coordinates are transformed with a change or with preservation of sign. Degeneracy first appears in the presence of symmetry elements $C_p$, where $p > 2$, but if there are no other symmetry elements, i.e. the symmetry is plane, then the degeneracy is at most twofold[^22]. Indeed, the irreducible representation of the rotation group in the plane is two-dimensional and has the form

\[ D= \begin{pmatrix} \cos \dfrac{2\pi l}{p} & -\sin \dfrac{2\pi l}{p} \\ \sin \dfrac{2\pi l}{p} & \cos \dfrac{2\pi l}{p} \end{pmatrix}. \tag{6,1} \]

\[ l=0,1,2,\ldots,p. \]

Fig. 9.

Physically, such a degeneracy is expressed in the fact that to each vibration invariant with respect to a rotation through $\dfrac{2\pi l_1}{p}$ there corresponds a vibration, invariant with respect to a rotation through $\dfrac{2\pi l_2}{p}$, with $l_2=p-l_1$. In other words, it is immaterial whether the system rotates clockwise or counterclockwise. A vibration invariant only for the value $l=0$ is nondegenerate.

Together with the degenerate forms of the normal vibrations in Fig. 4, the vibrations are not those shown in the figure, but those obtained by the superposition $4a+4d+4c$ and $4a+4c+4d$ of the vibrations in Fig. 9. The points in Fig. 9 vibrate along circles.

It is interesting that we obtain these same degenerate vibrations simply by changing the direction in which time is reckoned. The frequency under this change will remain the same. If, for example, the vibrations of three groups rotated relative to one another by $120^\circ$ occur with

by a phase difference of $\frac{2\pi}{3}$, then upon changing the direction of time reckoning we obtain another picture with a phase difference $-\frac{2\pi}{3} \equiv \frac{4\pi}{3}$.

The types of vibrations for the group $C_p$ are described by an integer defined modulo $p$, i.e., for example, for $p=3$ (as in Fig. 9) $l=1 \equiv 4 \pmod p$, since $4-1=p$.

To simplify the discussion we may combine the jointly expressed normal coordinates $q_1$ and $q_2$ into the complex coordinates $q_1+iq_2$ and $q_1-iq_2$. In this representation (6,1) is reduced to the form

\[ D= \begin{pmatrix} e^{i\frac{2\pi l}{p}} & 0\\ 0 & e^{-i\frac{2\pi l}{p}} \end{pmatrix} = \begin{pmatrix} e^{i\frac{2\pi l}{p}} & 0\\ 0 & e^{i\frac{2\pi(p-l)}{p}} \end{pmatrix}. \tag{6,2} \]

The expression $q_1+iq_2$, when the rotation $C_p^{(l)}$ is carried out, is simply multiplied by $e^{-i\frac{2\pi l}{p}}$

\[ C_p^{(l)}(q_1+iq_2)=(q_1+iq_2)e^{-i\frac{2\pi l}{p}}. \tag{6,3} \]

It is obvious that for $l=\frac{p}{2}$ ($p$ even)

\[ C_p^{\left(\frac{p}{2}\right)}(q_1+q_2)=(q_1+iq_2)e^{-i\pi}=-(q_1+iq_2). \tag{6,4} \]

We cannot call such a vibration degenerate, since here there is only a simple change of sign of the normal coordinate. This is a vibration antisymmetric with respect to $C_p$. The vibration symmetric with respect to $C_p$ is obtained, as already stated, by putting $l=0$.

The choice of the sign of $q_2$ is arbitrary, and the sign of $l$, according to (6,3), depends on it—the types with $l$ and with $-l$ are identical. For even $p$ we therefore obtain $\frac{p}{2}$, and for odd $p$, $\frac{p}{2}-1$ types of vibrations.

Passing to the higher symmetry of the tetrahedron and octahedron, we encounter, as follows from the calculations of § 5, threefold degeneracy. Here the normal coordinates transform as three-dimensional vectors, according to representations of the form (3,8). Finally, molecules of the icosahedron group have 4- and 5-fold degeneracy. No higher necessary degeneracy occurs in molecules. The situation is somewhat different in crystals. The vibrations of crystals have the closest relation to the questions we have been discussing; therefore it will be appropriate to dwell on them$^{23}$.

A crystal may be regarded as a very large molecule. From the point of view of symmetry relations, the distinction between molecules and crystals reduces to the fact that, in addition to point symmetry, a crystal also has translational symmetry along the spatial lattice, which, in combination with ordinary rotations and reflections, can give the symmetry of a screw motion—

… and mirror sliding. We may subdivide the vibrations of a crystal into external and internal ones. By the latter we shall understand the vibrations of crystalline groups upon which the other unit structural cells have little influence, for example the vibrations of the ions $CO_3^{\prime\prime}$, $NO_3^\prime$, $SO_4^{\prime\prime}$, etc., in crystals of carbonates, nitrates, and sulfates. It is obvious that these vibrations, as well as their degeneracy, are analogous to the vibrations of a molecule, with the difference that the maximum degree of degeneracy of internal crystalline vibrations is equal to three, since icosahedral symmetry is impossible for crystals.

Fig. 10.

On the contrary, external crystalline vibrations have no analogue in polyatomic molecules, unless one counts the simple translational motion of a molecule with frequency $\nu = 0$. The most simplified model of a crystal will be an infinite linear chain of identical atoms (Fig. 10). The atoms can be displaced only along the straight line; the normal vibration will be a longitudinal wave, changing its phase by a certain amount $\varphi$ in traversing the distance $l$ between neighboring atoms. This vibration is doubly degenerate, since the waves—direct and reverse—propagate with the same frequency.

Fig. 11.

\[ x_i = a \sin (2\pi \nu t \pm l\varphi). \tag{6,5} \]

The superposition of two degenerate vibrations gives a standing wave. There will be no degeneracy only when $\varphi = 0$ or $\varphi = \pi$. The first value corresponds to translation of the crystal as a whole, the second to the vibration of neighboring particles in mutually opposite directions.

In passing to a spatial model of the same type (Fig. 11) we encounter an eightfold degeneracy of external vibrations, corresponding to the eight possible directions of longitudinal waves.

	1	2	3	4	5	6	7	8
$x$	+	−	+	+	−	−	+	−
$y$	+	+	−	+	−	+	−	−
$z$	+	+	+	−	+	−	−	−

Up to this point we have been speaking of necessary degeneracy, i.e., of degeneracy connected with the symmetry of the problem. In the language of group theory this means that to all jointly degenerate coordinates there corresponds one irreducible representation, whose order is equal to the degree of degeneracy of the vibration. But cases are also possible in which, owing to the special character of the force function, some of the frequencies of such vibrations coincide although they are not necessarily degenerate. Such degeneracy is called accidental.^24 In practice one more often has to deal with such cases where either there is an integral relation between frequencies (the frequencies $\nu_1=667.5\ \mathrm{cm}^{-1}$ and $\nu_2\simeq 2\nu_1=1330\ \mathrm{cm}^{-1}$ for $\mathrm{CO}_2$), or the sum of two frequencies is equal to a third (the frequencies $\nu_1=313\ \mathrm{cm}^{-1}$, $\nu_2=459\ \mathrm{cm}^{-1}$, and $\nu_3\simeq \nu_1+\nu_2=775\ \mathrm{cm}^{-1}$ for $\mathrm{CCl}_4$), etc. Such coincidences give rise to resonance and lead to substantial complications of the spectrum. For example, in the case of $\mathrm{CO}_2$,^9,25 because of resonance, the accidental degeneracy is removed and the frequency $1330\ \mathrm{cm}^{-1}$ splits into two frequencies: $1285\ \mathrm{cm}^{-1}$ and $1388\ \mathrm{cm}^{-1}$. This occurs because accidental degeneracy is not characteristic of the symmetry of the molecule. The introduction of some perturbation that does not change the symmetry of the system and therefore preserves the normal degeneracy—namely, the always-present anharmonic term in the expression for the potential energy—removes the accidental degeneracy. L. I. Mandelstam gives a beautiful interpretation of the relations in the spectrum of $\mathrm{CO}_2$ by means of a classical analogy with parametric resonance.^26

§. 7. Selection Rules

In order to pass from theory to experiment and thereby test it, it is necessary to find out which of the normal vibrations possible for a given molecule are allowed and which are forbidden in spectra by the selection rules.

Infrared spectra are determined by the behavior of the electric-moment vector during vibrations. Raman spectra are determined by the behavior of the polarizability tensor. The difference in the character of these physical quantities creates a difference in the selection rules for the one and the other case. What is common to the electric moment and to the polarizability is that they depend additively on the individual normal displacements; therefore in the infrared and in the Raman spectrum there appear separate normal frequencies that do not influence one another.

Quantum-mechanically, the selection rules are found by constructing the matrix elements of the transition between the corresponding energy levels.^27 Thus, for example, the matrix element of the transition between the levels $E_n$ and $E_k$, associated with the emission or absorption of the frequency $\nu_{nk}$, for the infrared spectrum will be

\[ \mathbf{M}_{nk}=\int \tilde{\psi}(q)\,\mathbf{M}(q)\,\psi_k(q)\,dq. \tag{7,1} \]

If $M_{nk}=0$, then the frequency $\nu_{nk}$ is forbidden in the infrared spectrum.

The dependence of the selection rules on the symmetry of the molecule has been studied by many authors. However, the simplest classification can be carried out only with the aid of group theory.

The eigenfunctions $\widetilde{\psi}_n(q)$ and $\psi_k(q)$ for the Schrödinger equation admitting the symmetry of the group $G$ transform like the corresponding normal coordinates $q$. The introduction of an anharmonic term does not change the situation, since the symmetry of the molecule is not thereby disturbed—the perturbed eigenfunctions transform according to the same representations. It is on this circumstance that the possibility of applying group theory to the question of the activity of vibrations is based.

The rule of group theory for determining the activity of vibrations may be most conveniently formulated as follows$^{24,45}$. One must construct the direct product $D_n \cdot D_k$ of the irreducible representations $D_n$ and $D_k$, corresponding to the eigenfunctions $\widetilde{\psi}_n(q)$ and $\psi_k(q)$ (cf. p. 342). In group theory, the direct product is the group constructed from all possible products of elements of two groups that have no common elements, with the exception of the identity element, each element of one of the groups being paired with each element of the other. The same definition also applies to representations of a single group $G$. The direct product can again be decomposed into irreducible components,

\[ D_n \cdot D_k = D_1 + D_2 + \ldots + D_t . \tag{7,2} \]

Since for a vibrational transition from the ground state $0 \to 1$ the function $\widetilde{\psi}_n(q)$ transforms according to the identity representation, $D_n=1$, and instead of the series (7,2) we write the series

\[ D_k = D_1 + D_2 + \ldots + D_t . \tag{7,3} \]

Here on the right stand the representations of the symmetry group with characters computed by Bethe (§ 5). It is also necessary to decompose into irreducible representations of the symmetry group $G$ the representation of the vector $\mathbf{M}$ of the electric moment, $D_M$. This representation of the three-dimensional polar vector (3,7) has the character

\[ \chi_M = \pm(1 + 2\cos\varphi). \tag{7,3'} \]

The decomposition into irreducible representations is carried out, as was shown, by expanding the character in a Fourier series. Thus,

\[ D_M = D'_1 + D'_2 + \ldots + D'_{t'} . \tag{7,4} \]

If at least one of the representations in the series (7,2) coincides with at least one of the representations in the series (7,4), then the transition $n \to k$ is allowed, and the frequency $\nu_{nk}$ is active in the infrared spectrum, since only in this case is expression (7,1) nonzero.*

* This rule follows from the orthogonality relations between irreducible representations and can be derived as follows: Van der Waerden proves a theorem$^{18}$, according to which, upon decompo-

This is the abstract-mathematical formulation of physically quite obvious relations—namely, that in the infrared spectrum only those vibrations are active which are associated with a change in the electric moment of the molecule.

The selection rules for Raman spectra are found in an analogous way, by constructing the corresponding matrix elements

\[ \alpha_{n k}=\int \widetilde{\psi}_{n}(q)\,\alpha(q)\,\psi_{k}(q)\,d\tau . \tag{7,5} \]

Here the essential quantity is the polarizability, $\alpha$, and its changes as a function of changes in the normal coordinates $q$. We saw in § 2 that $\alpha$ can be expanded in a power series in $q$. If the vibrations are regarded as harmonic and, consequently, one restricts oneself to the fundamental tone, then

\[ \alpha=\alpha_{0}+\left(\frac{\partial\alpha}{\partial q}\right)_{q=q_{0}} q . \tag{7,9} \]

The Raman effect is associated precisely with the value of $\left(\dfrac{\partial\alpha}{\partial q}\right)_{0}$. The equality of the first derivative to zero—the presence of an extremum (minimum or maximum) of the polarizability—means the prohibition of the corresponding vibration in the Raman spectrum. Thus, for example, it is easy to understand the inactivity in the Raman spectrum of the fundamental vibrations of an ionic bond. Indeed, here $\alpha$ does not depend on $q$ at all, since the entire valence, and hence optically active, electron shell of the molecule is drawn to only one nucleus and therefore is not deformed during the vibration.

Group theory finds the selection rules for the Raman effect in the same way as for infrared spectra, with the difference that instead of the representation $D_{M}$ of the vector of the electric moment one must decompose

the product of some system of functions corresponding to the reducible representation $D$ in a complete orthogonal system of functions to which irreducible representations $D_{\lambda}$ correspond—there will be nonzero coefficients only for those fundamental functions whose representations are contained in $D$ as component parts. In our case:

\[ \widetilde{\psi}_{n}M=\sum_{i}\psi_{i}a_{n i}. \tag{7,5'} \]

and by virtue of the orthogonality of the functions $\psi_i$

\[ M_{n k}=a_{n k}. \tag{7,6} \]

and $a_{n k}\ne 0$ only if the direct product

\[ D_{n}\cdot D_{M}=D_{1}^{\prime\prime}+D_{2}^{\prime\prime}+\ldots+D_{\nu}^{\prime\prime} \tag{7,7} \]

contains $D_{k}$ as a component part. From this it is not difficult to pass to the formulation given above, having proved on the basis of relations between characters that if $D_{n}\cdot D_{M}$ contains $D_{k}$, then $D_{n}\cdot D_{k}$ contains $D_{k}$, and $D_{M}$ contains $D_{n}$.

to the irreducible parts, the representation of the polarizability $D_\alpha$. In § 2 it was shown that $\alpha$ transforms as a symmetric tensor. The representation of a symmetric tensor is six-dimensional; it decomposes into two irreducible parts—one-dimensional and five-dimensional—corresponding to the scalar and anisotropic parts of the polarizability tensor (Fig. 12).

The character of the representation $D_\alpha$ shown in the figure will, obviously, be equal to the sum of the characters of the one-dimensional and five-dimensional representations of the rotation group (cf. p. 344). Namely,

\[ \chi(\varphi)=2+2\cos\varphi+2\cos 2\varphi =4\cos^2\varphi+2\cos\varphi. \tag{7,10} \]

Decomposing $\chi(\varphi)$ according to the characters of the symmetry group $C$, we shall find the selection rules for the Raman effect just as for the infrared spectrum. That these rules must differ is evident even without calculations. Consider, for example, a system with a center of symmetry. Upon reflection in the center the sign of $M$ changes, whereas the sign of $\alpha$ does not change, since the polarizability ellipsoid itself has a center. Hence it follows that for a symmetric vibration

Fig. 12.

Fig. 12.

\[ \left. \begin{aligned} M_{nk}&=\int \tilde{\psi}_n(q)M(q)\psi_k(q)\,dq \\ &=-\int \tilde{\psi}_n(q)M(q)\psi_k(q)\,dq=0;\\ \alpha_{nk}&=\int \tilde{\psi}_n(q)\alpha(q)\psi_k(q)\,dq\ne 0, \end{aligned} \right\} \tag{7,11} \]

i.e. it is allowed in the Raman spectrum, but forbidden in the infrared one; for an antisymmetric vibration the function $\psi_k(q)$ changes sign under reflection, and therefore the opposite condition holds—the antisymmetric vibration is allowed in the infrared spectrum and forbidden in the Raman spectrum. A more general expression of this so-called alternative prohibition states that in the Raman spectrum only terms of the same species combine—either only symmetric or only antisymmetric ones. For the infrared spectrum, on the contrary, terms only of different species combine—the well-known Laporte rule.

From the relations between the scalar and anisotropic parts of the polarizability tensor $\alpha_\perp$ and $\alpha_{\parallel}$ follow the polarization properties of the lines. As was indicated in § 2, the degree of depolarization $\rho_R$ for Raman scattering is equal to

\[ \rho_R=\frac{6{\alpha'_{\parallel}}^{\,2}}{45{\alpha'_{\perp}}^{\,2}+7{\alpha'_{\parallel}}^{\,2}}, \tag{7,12} \]

where $\alpha'_{\perp}$ and $\alpha'_{\parallel}$ are the derivatives of the corresponding parts of the polarizability with respect to the normal coordinates. The maximum value of $\rho_R$ is attain—

is achieved when the scalar part $\alpha_i'$ is equal to zero. It is equal to $\frac{6}{7}=0.86$. Conversely, if the anisotropic part is equal to zero, then $\rho_R=0$, and the line is completely polarized. To find the polarization ratios one must seek the selection rules separately for $\alpha_{\perp}$, which transforms according to the unit representation, and for $\alpha_{\parallel}$, which transforms according to a five-dimensional representation with character $1+2\cos\varphi+2\cos2\varphi$. The calculation shows that for all Raman lines belonging to non-totally symmetric vibrations, $\rho_R=\frac{6}{7}$, i.e. the scalar part is equal to zero. For totally symmetric vibrations of molecules belonging to cubic symmetry, the ellipsoid of polarizability turns into a sphere and $\alpha_{\parallel}=0,\rho_R=0$; the corresponding spectral line is not depolarized. For totally symmetric vibrations of molecules with symmetry lower than cubic, $\alpha_i'\ne0$ and $\alpha_{\parallel}'\ne0$, and therefore $0\leq \rho_R\leq \frac{6}{7}$. The exact value is not determined by symmetry, but in practice it is for the most part less than 0.5.

Up to now we have spoken only about fundamental tones in spectra. However, real vibrations are never strictly harmonic, as a result of which, in addition to the fundamental tones, overtones $n\nu$ and combination vibrations with frequencies possessing considerably lower intensity are observed in spectra,

\[ \nu_\lambda \pm \nu_\mu . \tag{7,13} \]

They appear precisely as a consequence of the violation of harmonicity and of the interaction of normal vibrations arising thereby. The selection rules and polarization ratios for combination frequencies are also connected with the symmetry of the molecule, but differ from the selection rules and polarization ratios for the fundamental tone. For example, if there is a center of symmetry and there exist two vibrations $\nu_s$ and $\nu_a$, symmetric and antisymmetric with respect to the center, then $\nu_s$ is allowed in the Raman spectrum, and $\nu_a$ in the infrared. As for combination frequencies, in the Raman effect all frequencies of the form

\[ n\nu_s + 2m\nu_a, \tag{7,14} \]

where $n,m=0,1,2\ldots$,

are allowed, and in the infrared all frequencies of the form

\[ n\nu_s + (2m+1)\nu_a, \tag{7,15} \]

where $n,m=0,1,2\ldots$.

Thus an inactive vibration may also become active in combination with an active one. The selection rules in these cases too are derived by means of group theory, which was first carried out by Tisza${}^{24}$.

§ 8. Some Examples. Associated Molecules

Let us consider the results of such calculations for some of the simplest molecules.

The molecule $XY_2$. If the atoms $Y$ are equivalent, then two models are possible—a linear one (Fig. 13), discussed on p. 356, and an angular one. The symmetry group for the linear model consists of the identity $E$ of reflection in the center $X$ and of an infinite series of rotations about the axis $XYX$. This is the group $D_{\infty}$. The number of normal vibrations is $3N-5$, i.e. four. As has already been shown, two of them are nondegenerate—symmetric and antisymmetric—and the third is doubly degenerate. Fig. 8 depicts these vibrations, if the system of valence forces is adopted. The vibrations of Figs. 8a and 8b are called, following Mecke$^2$, valence vibrations, respectively symmetric and antisymmetric, and are denoted $\nu(s)$ and $\nu(a)$. The degenerate vibration of Fig. 8c takes place in a direction perpendicular to the valence strokes; this is the deformation vibration $\delta$.

$Fig. 13 and Fig. 14: schematic vibrations of $XY_2$$

Fig. 13. Fig. 14.

The vibrations $\nu(a)$ and $\delta$ are antisymmetric with respect to the center of symmetry and are therefore forbidden in the Raman effect. In the infrared, the vibration $\nu(s)$ is forbidden.

For the angular model the number of normal vibrations is $3N-6=3$. The vibrations are the same, but with the disappearance of the rotation axis $YXY$ the degeneracy also disappears (Fig. 14).

Mecke$^2$ introduces the symbols $\pi$ and $\sigma$ to denote vibrations occurring along the vector of the electric moment and perpendicular to it, in the case when the molecule possesses a moment.

All vibrations are resolved both in the Raman spectrum and in the infrared spectrum; moreover $\rho_{\nu(\pi)} \leqslant 0.86$, $\rho_{\delta(\pi)} \leqslant 0.86$, while $\rho_{\nu(\sigma)} = 0.86$. We see that the measurement of the degree of depolarization, and even the very number of lines in the spectrum, tells us how the molecule is constructed. Such molecules as $CO_2$, $CS_2$, etc., turn out to be linear; angular ones are $H_2O$, $SO_2$, and others.

The molecule $XY_3$. General symmetry considerations permit both a planar model of an equilateral triangle and a pyramid. We shall carry out the calculation in full for both cases.

The symmetry of the planar model is the symmetry of the dihedral group with a plane of symmetry perpendicular to the axis—the group $D_3$. It consists of six elements—the identity, two rotations through $\frac{2\pi}{3}$, and three rotations through $\pi$. The table of characters and of all quantities needed for the calculations has the following form (Table 3).

TABLE 3

Group $D_3$	$E(1)$	$C_3(2)$	$C_2(3)$
Representation $A_1$	1	1	1	$\chi_1$
$A_2$	1	1	$-1$	$\chi_2$
$K$	2	$-1$	0	$\chi_3$
$u_R$	4	1	2
$\varphi_R$	0	$\dfrac{2\pi}{3}$	$\pi$
$\pm(1+2\cos\varphi_R)$	3	0	1
$\chi(R)=(u_R-2)(1+2\cos\varphi_R)$ (for pure rotation)	6	0	0
$\chi_M=+(1+2\cos\varphi_R)$	3	0	$-1$
$\chi_\sigma=4\cos^2\varphi_R+4\cos\varphi_R$	6	0	2
$\chi_{\sigma I}=1$	1	1	1
$\chi_{\sigma II}=4\cos^2\varphi_R+2\cos\varphi_R-1$	5	$-1$	1

First of all we find which vibrations are possible in general. For this we carry out the decomposition of $\chi(R)$ into $\chi_1,\chi_2$, and $\chi_3$ (cf. p. 355). The result is

\[ \chi(R)=\chi_1(R)+\chi_2(R)+2\chi_3(R). \tag{8,1} \]

The number of frequencies is equal to 4, of which two belong to nondegenerate and two to doubly degenerate vibrations. In all there are $3N-6=6$ vibrations. The representation $A_1$ corresponds to symmetric, $A_2$ to antisymmetric, and $K$ to degenerate vibrations.

Let us carry out the decomposition of $\chi_M$

\[ \chi_M=a_1\chi_1+a_2\chi_2+a_3\chi_3 \tag{8,2} \]

\[ \left. \begin{aligned} a_1&=\frac{1}{6}(1\cdot1\cdot3+2\cdot1\cdot0+3\cdot1\cdot-1)=0,\\ a_2&=\frac{1}{6}(1\cdot1\cdot3+2\cdot1\cdot0+3\cdot-1\cdot-1)=1,\\ a_3&=\frac{1}{6}(1\cdot2\cdot3+2\cdot-1\cdot0+3\cdot0\cdot-1)=1, \end{aligned} \right\} \tag{8,3} \]

i.e.

\[ \chi_M=\chi_2+\chi_3. \tag{8,4} \]

The representation of the electric moment decomposes into representations with characters $\chi_{a_1}$ and $\chi_{\alpha_{\mathrm{II}}}$, contained also in the decomposition $\chi(R)$ (of these, $\chi_{a_1}$ occurs twice). Thus, of the 4 frequencies permitted in the infrared spectrum, one nondegenerate frequency is forbidden.

To find the selection rules and polarization relations, we carry out the decomposition into irreducible representations of the representations $a_1$—the scalar part of the polarizability tensor—and $a_{\mathrm{II}}$—its anisotropic part.

The calculation gives:

\[ \chi_{a_1}=\chi_1, \tag{8,5} \]

\[ \chi_{a_{\mathrm{II}}}=\chi_1+2\chi_3, \tag{8,6} \]

i.e., the nondegenerate vibration with representation $A_1$ is allowed in the Raman spectrum and has $\rho_R<0.86$, since both $a_1$ and $a_{\mathrm{II}}$ are nonzero for it. The vibration $A_2$ is forbidden in the Raman spectrum. Finally, both degenerate vibrations $K$ are allowed in the Raman spectrum and have degrees of depolarization equal to 0.86, since here the scalar part is forbidden. The vibration $A_1$, allowed in the Raman spectrum and forbidden in the infrared, is evidently a totally symmetric vibration (Fig. 15).

Fig. 15.

For the pyramidal model we have the symmetry group $C_{3v}$, isomorphic to the group $D_3$. It contains as many elements and their classes as the group $D_3$, and has the same characters. The difference consists in the fact that instead of rotations $C_2$, reflections now appear in three planes passing through the vertex and one of the angles—$\sigma_v$. The table has the form (the characters $\chi_1,\chi_2,\chi_3$ are as before):

TABLE 4

Group $C_{3v}$	$E(1)$	$C_3(2)$	$\sigma_v(3)$
$u_R$	4	1	2
$\varphi_R$	0	$\dfrac{2\pi}{3}$	$\pi$
$\pm(1+2\cos\varphi_R)$	3	0	1
$\chi(R)=\begin{cases}(u_R-2)(1+2\cos\varphi_R)\ \text{rot.}\\ -u_R(1+2\cos\varphi_R)\ \text{refl.}\end{cases}$	6	0	2
$\chi_M=\pm(1+2\cos R)$	3	0	1
$\chi_{\alpha_1}$	1	1	1
$\chi_{\alpha_{\mathrm{II}}}$	5	$-1$	1

Analytical calculation gives:

\[ \chi(R)=2\chi_1(R)+2\chi_6(R). \tag{8,7} \]

Four frequencies are possible—2 symmetric, nondegenerate, and 2 doubly degenerate:

\[ \chi_M=\chi_1+\chi_3, \tag{8,8} \]

\[ \chi_{a1}=\chi_1, \tag{8,9} \]

\[ \chi_{a11}=\chi_1+2\chi_3. \tag{8,10} \]

All vibrations are allowed both in the infrared and in the Raman spectrum. The degrees of depolarization are

\[ <0.86;\quad <0.86;\quad 0.86;\quad 0.86. \]

If one starts from the system of valence forces, then graphically these vibrations are represented as follows (the vibrations are projected onto a plane passing through the axis of symmetry) (Fig. 16):

\[ \nu(\pi)\quad \rho<0.86 \qquad \nu(\sigma)\quad \rho=0.86 \qquad \delta(\pi)\quad \rho<0.86 \qquad \delta(\sigma)\quad \rho=0.86 \]

Fig. 16.

The doubly degenerate vibrations, $\nu(\sigma)$ and $\delta(\sigma)$, are such because here the displacements occur perpendicular to the axis of symmetry, and there are two such independent directions.

We see that the spectra of a planar and a pyramidal molecule differ. The ions $NO_3'$ and $CO_3''$ are planar; ammonia $NH_3$ and many halide compounds of trivalent metals and metalloids are pyramidal. For example, the Raman spectrum of $PCl_3$ consists of the following frequencies:

TABLE 5

$\nu\ \mathrm{cm}^{-1}$	$\rho_R$
190	0.86
258	0.28
484	0.86
511	0.16

Molecule XV. We have already found what vibrations are possible for the tetrahedral model $XY_4$ (p. 356). We obtained

\[ \chi_l(R)=\chi_1(R)+\chi_3(R)+2\chi_5(R). \tag{8,11} \]

The same decomposition for $M$ and $a_1, a_{11}$ gives

\[ \chi_M+\chi_5, \tag{8,12} \]

\[ \chi_{a1}=\chi_1, \tag{8,13} \]

\[ \chi_{a11}=\chi_3+\chi_5. \tag{8,14} \]

In the infrared spectrum only two triply degenerate vibrations are allowed. In the Raman spectrum all three vibra-

—of them, for the first, nondegenerate one there is only a scalar part, while for the doubly and triply degenerate vibrations only an anisotropic one. Consequently, the degrees of depolarization will be
$\rho = 0;\ 0.86;\ 0.86;\ 0.86.$

Adopting the system of central forces, we may, following Teller$^{29}$, depict these vibrations as follows (Fig. 17).

Here Fig. 17a depicts the fully symmetric nondegenerate vibration $\nu_1$; Fig. 17b, three jointly degenerate vibrations $\nu_2$, symmetric with respect to the plane passing through the axis $X$ and the edge $(1,2)$, and unchanged under rotation followed by reflection about the axis $X$. These vibrations are represented, respectively, by solid, broken, and dotted arrows. However, the degree of degeneracy is not 3 but 2, since there is a linear dependence among the three vibrations drawn (cf. Fig. 4 and the note on p. 346). In Fig. 17c the vibration $\nu_3$ is shown, corresponding to an equal displacement of the rigid tetrahedron with respect to the center. Since the displacement may be carried out in any of the three coordinate directions, this vibration is triply degenerate. Finally, Fig. 17d likewise gives a triply degenerate vibration.

Fig. 17.

Fig. 17.

If the system of valence forces is adopted, the graphical form of the vibrations of $XY_4$ will be different. We give it, as well as a representation of those vibrations which are obtained upon successive replacement of the atoms $Y$ by atoms $Z$ (for example $CCl_4$, $CHCl_3$, $CH_2Cl_2$, $CH_3Cl$, $CH_4$) (Fig. 18$^{30}$).

The central atom remains at rest for $\delta_{1,2}$ and $\nu_1$, while for $\delta_{3,4,5}$ and $\nu_{2,3,4}$ it describes spatial trajectories that cannot be shown in the drawing; likewise, the unrepresented phase differences of the linear paths of the angular atoms are denoted by a double arrow. We see that under successive substitution, connected with a lowering of symmetry, the degeneracy is removed, disappearing completely for $XY_2Z_2$. This same circumstance of the removal of degeneracy is encountered in the association of molecules$^{31}$.

In a number of cases, as a result of a van der Waals interaction not accompanied by the appearance of new bonds, the spectra may chan—

Fig. 18.

Fig. 18.

…be altered owing to slight changes in symmetry. For example, the linear molecule $XY_2$, under the action of van der Waals forces, may bend, which will evidently lead to the appearance of two new lines in the Raman spectrum. It can be shown that the symmetry as a result of such an interaction cannot increase, but may either be preserved, pass into an isomorphic group, or decrease. This is connected with the fact that van der Waals forces—dispersion, electrostatic, and induction forces—are determined by the magnitudes of the electric moment and the polarizability, which themselves possess the same symmetry. Namely, the principal axes of the polarizability ellipsoid are directed along the symmetry axes. The same applies to the vector of the resultant moment. Thus displacement of the particle can occur only along an axis of symmetry. The degeneracy as a result of associative interaction may decrease, but cannot increase.

The application of group theory to the vibrations of all possible molecules is the subject of Wilson’s work.^46 Using this same method, he has analyzed in particularly great detail the vibrations of the benzene molecule $C_6H_6$.^47

§ 9. Range of applicability

The range of applicability of group theory to molecular spectra is very broad, since it is based on assumptions about symmetry that are of the most general character. A significant limitation is introduced by the requirement that the vibrations be harmonic; however, since the perturbation introduced by an anharmonic term does not violate the symmetry of the molecule, we may operate with harmonic vibrations. Anharmonicity will manifest itself only in the appearance of combination bands and overtones, the selection rules for which can again be obtained with the aid of group theory.

The theory is of greatest importance for the Raman effect. It is precisely here, owing to the existence of polarization relations, that it can be fully tested experimentally. And here the domain of its application coincides with the domain of application of Placzek’s classical theory of polarizability[^32]. The basic condition for its application is that the frequency of the incident light be sufficiently far removed from all absorption frequencies. This separation must be large in comparison with the splitting of the electronic states caused by the motion of the nuclei.

\[ \nu \gg \nu_k,\qquad \nu_e-\nu \gg \nu_k . \tag{9.1} \]

In the absorption region the polarizability tensor may contain an antisymmetric part, which leads to a change in the selection rules and in the polarization relations.

All the relations derived by us referred to the principal, unexcited electronic state. For an excited state all the relations may be strongly distorted. The theory of polarizability neglects the degeneracy of the principal electronic state. Thus we completely abstract from the character of the structure and excitation of the electronic shell, leaving to it only the possibility of following the vibrations of the nuclei.

On the contrary, the presence of rotational levels introduces no restrictions. Using the same theory of polarizability, we can derive all the most important relations, fully confirmed by experiment[^33]. Thus, for example, an alternative prohibition is obtained for rotational levels in the infrared spectrum and in the Raman spectrum.

It is of interest to what extent, by means of the theory set forth, of the influence of symmetry on the properties of spectra, it is possible to arrive at numerical values of the frequencies, and at the calculation of the intramolecular distribution of forces. This problem, simple for diatomic molecules, becomes very complicated for polyatomic molecules. Here we can no longer unambiguously calculate the coefficient of elasticity of a bond, knowing the frequency of vibration. An $N$-atomic molecule has $s=3N-6$ $(3N-5)$ normal frequencies. Meanwhile, the forces acting during vibration are specified by

\[ \frac{s(s+1)}{2} \]

constants; consequently, frequencies alone are insufficient for determining these constants. It is necessary, in addition, to characterize the relative magnitudes of the displacements and the directions of the forces acting on the particles in one or another vibration—to specify the form of this vibration. To a certain extent this problem is solved with the aid of group theory, and, proceeding from definite experimental material, one can, by comparing it with theory, determine the character of the bonds in the molecule. However, this fundamental possibility is greatly diminished by the fact that the actual conditions are not as we have set them forth. In fact, we have throughout operated with a certain idealization, which proves already unsuitable for the calculation of forces. The molecule not only vibrates; it is also capable of rotating, like an asymmetric top, and this rotation interacts

with the vibration, producing Coriolis forces, etc.^48 This greatly complicates the picture. The problem becomes practically insoluble. Moreover, in speaking of $s(s+1)/2$ constants, we are starting from a harmonic law of vibrations. Anharmonicity, which has little effect on general structural, symmetry relations, is very strongly reflected in the values of the frequencies. Thus, for the time being, the question of whether the structural-symmetry theory can be applied to determine the forces acting inside a molecule must be answered in the negative. This applies equally to the question of the intensity of vibrations. It is true that in some cases we can give a qualitative law of intensity. Namely, considering the selection rules, one may say that totally symmetric vibrations give the most intense lines in the Raman spectrum, since for them both the scalar and the anisotropic parts of the polarizability tensor are represented. The same applies to the totally symmetric vibrations of molecules of the cubic system, for which only the scalar part is allowed, because the scalar part is the mean of the values of the principal axes of the polarizability ellipsoid, while the anisotropic part is determined by the differences between these principal values and is therefore much smaller than the scalar part. These rules are, of course, only qualitative. Buckingham^34 attempts to estimate the intensity of a Raman line in comparison with the known intensity of a Rayleigh line, proceeding from the same classical theory of polarizability. Namely, since

\[ \alpha = \alpha_0 + \left(\frac{\partial \alpha}{\partial q}\right)_0 q_0, \tag{9.2} \]

the intensities of the Raman and Rayleigh lines are related as

\[ \frac{I_{\nu-\nu_k}}{I_\nu} \sim \frac{\left(\frac{\partial \alpha}{\partial q}\right)_0^2 q_0}{\alpha^2}. \tag{9.3} \]

Experimentally this ratio is about 0.1. From this Buckingham obtains for $\left(\frac{\partial \alpha}{\partial q}\right)_0$ a value equal to $\sim 10^{-10}\,\text{cm}^2$ ($\alpha \sim 10^{-24}\,\text{cm}^3,\ q_0 \sim \sim 10^{-9}\,\text{cm}$) and for $\left(\frac{\partial^2 \alpha}{\partial q^2}\right)_0$ a value equal to $\sim 10^{-8}\,\text{cm}$. Great significance should not be attached to these calculations, if only because the classical theory is in principle inapplicable to intensities: it gives neither the value of $q_0$ nor, for example, the correct relation between the Stokes and anti-Stokes satellites in the Raman spectrum.

Structural-symmetry theory can characterize for us the structure of a molecule as a whole, its symmetry, but it cannot give us the values of the forces acting on the particles or of their amplitudes.

Other phenomena may be adduced—for example, isotopy. Here, by replacing some atoms by others as indicators (which has recently become possible owing to the fact that heavy hydrogen has become available to physical chemists), we shall be able to simplify the deciphering of the spectrum and arrive, ultimately, at definite results concerning the distribution of forces in the molecule. By comparing the results of a spectroscopic investigation of a substance with its Kerr effect, with refraction, with the values of the dipole moment, etc., one can construct the ellipsoid of polarizability of the molecule and indicate the magnitudes of its principal axes. Rayleigh scattering is also important here. However, consideration of these interesting questions would lead us too far afield; we shall deliberately confine ourselves to one area. In the following, concluding paragraph, some examples will be given of the application of the theory to experimental material, illustrating the propositions set forth.

§ 10. Experimental Material

The theory, as has already been indicated, can be tested with particular completeness in the study of Raman spectra. In the short time of its existence, Raman spectroscopy has surpassed infrared spectroscopy in the amount of material. In a review printed about two years ago, Kohlrausch^35 indicates that 760 works had been done on the Raman effect, in which 1,880 spectra of 1,000 molecules were investigated. In the past two years, another hundred, if not more, have probably been added to these works. However, unfortunately, in the overwhelming majority of the works no polarization measurements were made, which considerably reduced the value of these works. It was reduced still more by the fact that the principal aim of the investigators was the identification of substances and the determination of bond energies. At the same time, general structural determinations receded into the background. The theoretical interpretation of the spectra obtained was carried out on the basis of the crudest considerations concerning the harmonic character of bonds. For calculation, either a system of valence forces or a system of central forces was introduced. Below we shall show, by an example, to what errors such a method of treatment can lead if structural-symmetry relations are not taken into account. Of course, comparison of the spectra of molecules with one another can give much. Establishing the invariance of frequencies inherent in particular groups and bonds is quite a real achievement. However, only a comparison of the usual data with the data of the polarization experiment and of structural-symmetry theory makes it possible to characterize most fully the structure and properties of a molecule. Certain successes can also be achieved without polarization relations, in the study of intermolecular interaction. However, here too, as has already been pointed out and will be shown again below, structural-symmetry theory is very essential.

The limited and controversial nature of the data of the system of central,

force field is illustrated by the work of Brodsky and Zakaz[^36]. These authors studied the Raman spectra of arsenic trichloride in various solvents. They undertook a calculation of the constants characterizing the AsCl₃ molecule, proceeding from the correct assumption that this molecule is pyramidal. The calculation according to the central-force system, carried out by Dennison[^37], was based on assigning the frequencies $\nu_1 = 410\ \mathrm{cm}^{-1}$ and $\nu_2 = 159\ \mathrm{cm}^{-1}$ to nondegenerate vibrations along the axis of symmetry, and $\nu_3 = 372\ \mathrm{cm}^{-1}$ and $\nu_4 = 195\ \mathrm{cm}^{-1}$ to doubly degenerate vibrations perpendicular to this axis. The result of the calculation showed good agreement with electron-diffraction data; moreover, the inverse recalculation from these data gave the frequencies $\nu_1 = 436\ \mathrm{cm}^{-1}$, $\nu_2 = 150\ \mathrm{cm}^{-1}$, $\nu_3 = 387\ \mathrm{cm}^{-1}$, $\nu_4 = 188\ \mathrm{cm}^{-1}$. But in fact this agreement is purely accidental. The authors did not take into account the polarization ratios in the AsCl₃ spectrum, which give a completely different distribution of the frequencies. Specifically for AsCl₃ there are the following data[^38],[^39].

TABLE 6

$\nu\ \mathrm{cm}^{-1}$	$\rho_R$	Vibration
410	0.08	$\nu(\pi)$ nondegenerate
195	0.42	$\delta(\pi)$ nondegenerate
372	0.86	degenerate doubly
159	0.86	degenerate doubly

Thus a central-force system alone, without taking symmetry relations into account, cannot lead to the correct result[^48].

The fact that we can say almost nothing about the numerical values of the frequencies on the basis of molecular structure can again be illustrated by the example of the trihalides. Consider the following table, relating to the pyramidal molecules PBr₃ and PCl₃.

TABLE 7

| \multicolumn{2}{c}{PBr₃} | \multicolumn{2}{c}{PCl₃} |
|---:|---:|---:|---:|
| $\nu\ \mathrm{cm}^{-1}$ | $\rho_R$ | $\nu\ \mathrm{cm}^{-1}$ | $\rho_R$ |
| 116 | 0.86 | 190 | 0.86 |
| 162 | 0.185 | 258 | 0.28 |
| 380 | 0.28 | 484 | 0.86 |
| 400 | 0.86 | 511 | 0.16 |

We see that the sequence of frequencies changes even for molecules very close in their structure and composition; the largest

for $\mathrm{PCl}_3$ the frequency $511\ \mathrm{cm}^{-1}$ belongs to the totally symmetric vibration $\nu(\pi)$ and has the smallest degree of depolarization, $\rho_R = 0.16$. But if, without carrying out polarization measurements, we were to regard the largest frequency for $\mathrm{PBr}_3$, $400\ \mathrm{cm}^{-1}$, likewise as the frequency of a totally symmetric vibration, we would make a gross error, since $\rho_R$ of this frequency is equal to $0.86$. The frequency $511\ \mathrm{cm}^{-1}$ of $\mathrm{PCl}_3$ corresponds to the frequency $380\ \mathrm{cm}^{-1}$ of $\mathrm{PBr}_3$.

Errors connected with underestimating symmetry theory are also often encountered in the identification of combination tones.

Let us show by an example how even knowledge of the simplest structural-symmetry relations makes it possible to determine the structure of a molecule. For dihalogen-substituted ethylenes—$\mathrm{C}_2\mathrm{H}_2\mathrm{Cl}_2$ and $\mathrm{C}_2\mathrm{H}_2\mathrm{Br}_2$—two isomers are possible—Cis and Trans. Trumpy[^41], who investigated the Raman spectra of both isomers, correctly indicates that the spectrum of the Cis isomer must contain a greater number of lines and a greater number of lines with a high degree of depolarization than the Trans isomer. Indeed, the latter has a center of symmetry, and vibrations antisymmetric with respect to the center are forbidden in the Raman spectrum (see p. 363). Experiment confirms this conclusion (see Table 8, on p. 376).

$Fig. 19. Cis-dichloroethylene. Group $C_{2v}$. Trans-dichloroethylene. Group $C_{2h}$.$

Fig. 19.

Common to the isomers are the high frequencies of the $\mathrm{C—H}$ and $\mathrm{C=C}$ bonds.

We have already indicated in § 8 the significance of structural theory in studying the association of molecules. We proceeded from the preservation of the individuality of the molecule upon association and showed that changes in the spectrum may be caused exclusively by a change in the symmetry of the molecule. However, even in the formation of a molecular complex, symmetry relations are very important for determining the structure of the associated compound. Unfortunately, the experimental material here is rather scanty. Let us consider one example[^31]. Leightman and Uholin[^42] found that the frequency $622\ \mathrm{cm}^{-1}$ in the Raman spectrum of acetic acid is weakened more strongly upon dilution than the other lines. The authors assigned this frequency to the mutual vibration of two $\mathrm{CH}_3\mathrm{COOH}$ molecules in the binary associated compound, already previously known to chemists. This frequency

TABLE 8

\[ \mathrm{C_2H_2Cl_2} \]

| | \multicolumn{3}{c}{Cis} | | \multicolumn{3}{c}{Trans} |
|---:|---:|---:|---:|---:|---:|---:|---:|
| | $\nu\ \mathrm{cm}^{-1}$ | intensity | $\rho_R$ | | $\nu\ \mathrm{cm}^{-1}$ | intensity | $\rho_R$ |
| 1 | 171 | 18 | 0.50 | 1 | 350 | 20 | 0.29 |
| 2 | 407 | 10 | 0.82 | 2 | 752 | 5 | 0.7 |
| 3 | 561 | 5 | 0.86 | 3 | 840 | 6 | 0.08 |
| 4 | 711 | 15 | 0.05 | 4 | 1271 | 15 | 0.2 |
| 5 | 806 | 0.5 | — | 5 | 1575 | 10 | 0.07 |
| 6 | 880 | 1 | about 1 | 6 | 1625 | 0.5 | about 0 |
| 7 | 1180 | 10 | 0.7 | 7 | 1690 | 1 | about 0 |
| 8 | 1586 | 15 | 0.08 | 8 | 3072 | 10 | 0.2 |
| 9 | 1688 | 2 | about 1 (?) | 9 | 3140 | 1 | about 1 (?) |
| 10 | 3078 | 20 | 0.31 | | | | |
| 11 | 3158 | 1 | about 1 (?) | | | | |

is absent from the infrared spectrum of the molecule, which indicates the symmetric character of the vibration. However, the degree of depolarization of the line is close to 0.86 (according to some data $\rho_R=0.89$, according to others $\rho_R=0.70$), to the value corresponding to an antisymmetric vibration. Following Tsang$^{43}$, one may imagine two kinds of association of acids of the acetic-acid series (Fig. 20).

Fig. 20.

Fig. 20.

In case a the four oxygens form a rectangle, the plane of which is perpendicular to the two other planes. Such a model is asymmetric. In the second model the center of symmetry and the normal vibrations antisymmetric with respect to it must be forbidden in the Raman spectrum. In the infrared spectrum, on the contrary, symmetric vibrations are forbidden. If, therefore, the frequency $622\ \mathrm{cm}^{-1}$ belongs to an antisymmetric vibration, then we must prefer model a; if Leitman and Ukholin are right, then model b must be adopted. Further experimental material is required to resolve the question.

The number of works taking structural theory, and in particular Placzek’s theory, into account has recently increased. The recent works of Kohlrausch compare favorably in this respect with all preceding ones$^{29,44}$.

Until quantum chemistry can teach us how to interpret polyatomic molecules, we shall have to confine ourselves to general structural judgments. And here Wigner’s rigorous theory—the application of group theory to these questions—plays an essential role.

In conclusion I should like to express my sincere gratitude to Prof. Yu. B. Rumer, who supervised my diploma work, from which the present survey grew, and to Prof. M. A. Leontovich for a number of valuable comments.

LITERATURE

R. Morse, Phys. Rev., 34, 57, 1929; E. Teller u. Pöschl, Z. Physik, 83, 143, 1933.
R. Mecke, Hand- u. Jahrbuch d. Chem. Phys., 9, II, 334, 1935; R. Mecke, Advances in the Physical Sciences, 14, 326, 1934.
Kettering, Shutts & Andrews, Phys. Rev., 35, 1422, 1930; Phys. Rev., 36, 531, 1930, D. Andrews a. J. Murray, Journ. chem. Phys, 2, 634, 1934, and others.
E. Teller, Hand- u. Jahrbuch d. chem. Phys. 9, II, 90, 1935.
R. de L. Kronig, The Optical basis of the Theory of Valency, Cambridge, p. 143, 1935.
C. Brester, Kristallsymmetrie und Reststrahlen, Utrecht, 1923.
G. Placzek, Leipziger Vorträge, S. 71, 1931.
G. Placzek, Handb. d. Rad VI/2, 1934; G. Placzek, Rayleigh scattering and the Raman effect, ONTI, Ukraine X.—K. 1935.
F. Rasetti, Leipziger Vorträge, S. 59, 1931.
O. Yu. Schmidt, Abstract Group Theory, GTTI, 1933.
A. Speiser, Theorie der Gruppen endlichen Ordnung, Berlin, Springer, 1927.
E. Wigner, Gruppentheorie und ihre Anwendung auf die Quantenmechanik der Atomspektren, Braunschweig, Vieweg, 1931.
Van-der-Waerden, Die Gruppentheoretische Methode in der Quantenmechanik, Berlin, Springer 1932.
G. Weyl, On the Philosophy of Mathematics, GTTI, p. 54, 1934.
E. Wigner, Z. Physik, 43, 624, 1927.
See, for example, A. Sommerfeld, Wave Mechanics, GTTI, p. 14, 1933.
E. Wigner, Göttinger Nachrichten, 133, 1930.
See R. Ewald, Handb. d. Physik, 24, 199, 1927.
H. Bethe, Ann. d. Phys., 3, 133, 1929.
K. Nikol’skii, Quantum Mechanics of Molecules, GTTI, p. 105, 1934.
A. Speiser, l. c. p. 171.
G. Placzek, l. c. p. 83.
F. Teller, Hand- und Jahrbuch d. Chem. Phys. 9, II, 162, 1935.
L. Tisza, Z. Physik, 82, 48, 1932.
E. Fermi, Z. Physik, 71, 250, 1931.
G. Landsberg, Advances in Chemistry, 1, 491, 1932.
Weizel, Bandenspektren, Leipzig, S. 133, 1931.
Van-der-Waerden, l. c. S. 71.
M. Born, Optik, Berlin, Springer, S. 553, 1933.
K. W. F. Kohlrausch, Z. physik. Chem. (B) 28, 340, 1935.
M. Wolkenstein, Acta physicochimica URSS, 4, 357, 1936. “Der Raman-Effekt der Ammoniaklösungen,” being printed there.
G. Placzek, l. c. p. 64 ff., p. 164 ff.
G. Placzek u. E. Teller, Z. Physik, 81, 209, 1932.
Buchheim, Phys. Z., 36, 694, 1935.

K. W. F. Kohlrausch, Uspekhi khimii, 3, 1001, 1934.
A. Brodskii & A. Sack, Journ. chem. Phys., 3, 449, 1935.
D. Dennison, Phil. Mag., 1, 195, 1926.
J. Cabannes et A. Rousset, C. R. 194, 79, 1931.
D. Jost & T. Anderson, Journ. chem. Phys., 3, 754, 1935.
J. Cabannes et A. Rousset, C. R. 194, 707, 1931.
B. Trumpy, Z. Physik, 88, 226, 1934; 90, 1934.
S. Leitumann u. Uchida, C. R. Acad. Sci. URSS, 4, 12, 1934; Journ. chem. Phys., 2, 825, 1934.
Zahn, Trans. Farad. Soc., 30, 809, 1934.
K. W. F. Kohlrausch u. F. Köpl, Z. physik. Chem. (B), 26, 209, 1934.
R. S. Mulliken, Phys. Rev., 43, 279, 1933.
E. B. Wilson Jr., Journ. chem. Phys., 2, 432, 1934.
E. B. Wilson Jr., Phys. Rev., 45, 705, 1934.
J. Howard & E. B. Wilson Jr., Journ. chem. Phys., 2, 630, 1934.

The origin of the vibrations for molecules with axial symmetry [[unclear: footnote text damaged]]; in such cases Fig. 4 is not linearly independent. This was [[unclear]] in § 6. Fig. 4 has, therefore, only [[unclear: plane/planar]] [[unclear: continuation damaged]]. ↩

Submission history

[v1] 1936

representations	\(E(1)\)	\(C_3(8)\)	\(C_2(3)\)	\(\sigma(6)\)	\(S_4(6)\)	In parentheses, the numbers of elements in the given class
\(A_1\)	1	1	1	1	1	\(\chi_1\)
\(A_2\)	1	1	1	\(-1\)	\(-1\)	\(\chi_2\)
\(K\)	2	\(-1\)	\(-2\)	0	0	\(\chi_3\)
\(F_1\)	3	0	\(-1\)	\(-1\)	1	\(\chi_4\)
\(F_2\)	3	0	\(-1\)	1	\(-1\)	\(\chi_5\)

\(T_d\)	\(E(1)\)	\(C_3(8)\)	\(C_2(3)\)	\(\sigma(6)\)	\(S_4(6)\)
\(U_R\)	5	2	1	3	1
\(\varphi_R\)	0	\(\dfrac{2\pi}{3}\)	\(\pi\)	\(\pi\)	\(\dfrac{\pi}{2}\)
\(\pm(1+2\cos\varphi_R)\)	3	0	\(-1\)	1	\(-1\)
\(\chi'(R)\)	15	0	\(-1\)	3	\(-1\)
\(\chi(R)\)	9	0	1	3	\(-1\)

Group \(D_3\)	\(E(1)\)	\(C_3(2)\)	\(C_2(3)\)
Representation \(A_1\)	1	1	1	\(\chi_1\)
\(A_2\)	1	1	\(-1\)	\(\chi_2\)
\(K\)	2	\(-1\)	0	\(\chi_3\)
\(u_R\)	4	1	2
\(\varphi_R\)	0	\(\dfrac{2\pi}{3}\)	\(\pi\)
\(\pm(1+2\cos\varphi_R)\)	3	0	1
\(\chi(R)=(u_R-2)(1+2\cos\varphi_R)\) (for pure rotation)	6	0	0
\(\chi_M=+(1+2\cos\varphi_R)\)	3	0	\(-1\)
\(\chi_\sigma=4\cos^2\varphi_R+4\cos\varphi_R\)	6	0	2
\(\chi_{\sigma I}=1\)	1	1	1
\(\chi_{\sigma II}=4\cos^2\varphi_R+2\cos\varphi_R-1\)	5	\(-1\)	1

Group \(C_{3v}\)	\(E(1)\)	\(C_3(2)\)	\(\sigma_v(3)\)
\(u_R\)	4	1	2
\(\varphi_R\)	0	\(\dfrac{2\pi}{3}\)	\(\pi\)
\(\pm(1+2\cos\varphi_R)\)	3	0	1
\(\chi(R)=\begin{cases}(u_R-2)(1+2\cos\varphi_R)\ \text{rot.}\\ -u_R(1+2\cos\varphi_R)\ \text{refl.}\end{cases}\)	6	0	2
\(\chi_M=\pm(1+2\cos R)\)	3	0	1
\(\chi_{\alpha_1}\)	1	1	1
\(\chi_{\alpha_{\mathrm{II}}}\)	5	\(-1\)	1

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VIBRATIONS OF POLYATOMIC MOLECULES AND THEIR STUDY USING GROUP THEORY

Contents

§ 1. Introduction. Normal Vibrations

VIBRATIONS OF POLYATOMIC MOLECULES

§ 2. Vibrational Spectra of Molecules

§ 3. Basic concepts of group theory\({}^{10,11,12,13}\)

§ 4. Classification of the Natural Vibrations of a Molecule on the Basis of Group Theory

§ 5. Irreducible representations of symmetry groups and their characters

§ 6. Degeneracy of normal vibrations

§ 8. Some Examples. Associated Molecules

§ 9. Range of applicability

§ 10. Experimental Material

LITERATURE

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VIBRATIONS OF POLYATOMIC MOLECULES AND THEIR STUDY USING GROUP THEORY

	1	2	3	4	5	6	7	8
\(x\)	+	−	+	+	−	−	+	−
\(y\)	+	+	−	+	−	+	−	−
\(z\)	+	+	+	−	+	−	−	−

\(\nu\ \mathrm{cm}^{-1}\)	\(\rho_R\)	Vibration
410	0.08	\(\nu(\pi)\) nondegenerate
195	0.42	\(\delta(\pi)\) nondegenerate
372	0.86	degenerate doubly
159	0.86	degenerate doubly