Electro-optics of Molecular Vibrations

M. V. Vol'kenshtein

§ 1. Introduction

Molecular vibrations are of the greatest importance in a number of physical phenomena. Vibrational spectra—infrared and Raman—constitute one of the principal sources of our knowledge about the structure and properties of molecules. At the present time vibrational spectra are finding ever broader application for the purposes of structural analysis—determining the details of the structure of a molecule, the arrangement of its constituent atoms, and the nature of the bonds between the atoms. In the infrared and Raman spectra of a molecule the following quantities can be observed and measured quantitatively:

1. The number of lines (bands).
2. The frequencies (wavelengths) of the intensity maxima of the lines (bands).
3. The intensity of the lines (bands)—differential and integral.
4. The form of the lines (bands)—their half-width and contour.
5. The polarizations of the lines—in the case of Raman spectra.

We shall confine ourselves here to questions relating to purely vibrational spectra, and therefore leave aside rotational structure, which is also very important for characterizing the spectrum.

Until recently, in the science of the structure of matter and in the practice of structural analysis, spectral parameters not connected with intensities were used chiefly—the number of lines and their frequencies. In the present review we shall set forth the modern theory of intensities and polarizations in vibrational spectra. As will be said below, the study of intensities and polarizations gives very valuable and, in some respects, irreplaceable information about molecular structure. There is no doubt that investigations in this direction—still very few in number—will develop strongly and find wide application.

In considering molecular vibrations and the quantitative parameters of vibrational spectra, we have the right to divide the properties of vibrations into three kinds: geometrical, mechanical, and electro-optical. Such a division is expedient; only on its basis has it been possible to construct a reasonable theory of vibrational spectra, making it possible to interpret spectra and apply them in practice. Strictly speaking, all properties

vibrations may be called electro-optical, since molecules are built of nuclei and electrons, and their vibrations manifest themselves in optics—in spectra.

However, introducing terminology that is to some extent conventional, we may speak of the geometrical properties of vibrations—those properties which depend on the number and arrangement of atoms and ions in a molecule, on the symmetry of this arrangement, independently of the nature of the forces acting between the atoms. By bringing these forces into consideration as characterized by experimentally determined values of the elastic constants, we may abstract from their ultimate cause—from the structure and properties of the electron shell of the molecule—and solve the problem of the vibrations of the molecule as a problem of classical mechanics. In this sense the number of observed lines—the degree of degeneracy of the vibrations—is a geometrical property, while the values of the frequencies are mechanical, for they are found from the solution of the mechanical problem. Finally, the intensities and polarizations are directly determined by the properties of the electron shell; it is precisely these parameters that we shall have in mind when speaking of the electro-optics of vibrations.

The indicated division of properties follows in its very essence from the fundamental features of the modern theory—its classical character and semi-empiricism. As applied to the mechanical problem, the theory of molecular vibrations has been set forth in the survey by M. A. El'yashevich¹ (see also the last chapter of our monograph²).

Classical methods in this case are quite natural. Less obvious is the possibility of applying them to the electro-optical problem. However, as we shall show below, such a possibility is real. The semi-empiricism of the theory—both in mechanics and in electro-optics—means finding characteristic constants of some aggregate of spectroscopic data and calculating, by means of these constants, a considerably larger amount of data.

The problem of the electro-optics of vibrations consists, in particular, in establishing the connection between vibrational spectra and other electro-optical properties of molecules—the Kerr effect, the magnitude and temperature dependence of the dielectric constant, etc. In the course of solving this problem, new quantitative characteristics of molecules are found: effective charges of bonds and the derivatives of their polarizabilities with respect to bond length. These characteristics are inseparable from the chemical properties of the molecule. The study of the electro-optics of vibrations is a new source of information on the structure of matter, one hitherto little used. However, the few data already obtained testify to the exceptional sensitivity of intensities in vibrational spectra to one or another change in chemical properties. Thus, intensities (as well as polarizations) acquire great importance for structural molecular spectral analysis.

At the present time, in the field of vibrational spectra, the history of atomic spectra is being repeated. After the establishment of the basic laws-

regularities in frequencies—series laws—and the creation of the corresponding theory—the Bohr theory, physics proceeded to the study of the intensities of spectral lines, to the theory of transition probability, and to the measurement of “oscillator strengths.” In the field of molecular spectra we have likewise entered the “era of intensities.” There can be no doubt that the study of the electro-optics of vibrational spectra will give much to the science of the structure of matter.

§ 2. CLASSICAL AND QUANTUM THEORIES

The experimental material relating to our problem has been obtained chiefly by the method of Raman spectra. Quantum mechanics, in principle, makes it possible to determine the intensity of a Raman line. Kramers and Heisenberg, using the correspondence principle (quantization of the electronic system and a classical treatment of radiation), obtained the following formula3, 3a:

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

where

\[ E_{kn}=\frac{1}{h}\sum_{r}\left\{ \frac{(AM_{kr})M_{rn}}{\nu_{rk}-\nu} + \frac{M_{kr}(AM_{rn})}{\nu_{rn}+\nu} \right\}, \]

\(\nu\) is the frequency of the incident light, \(\nu_{kn}\) is the frequency of the virtual transition, \(M_{kr}\) is the matrix element of the electric moment of the system, and \(A\) is the amplitude of the incident light wave, written in the form

\[ E=A^{*}e^{2\pi i\nu t}+Ae^{-2\pi i\nu t}. \tag{2,2} \]

Using Dirac’s consistent quantum theory of radiation, we arrive at the same formula. However, despite its great fundamental significance, it is difficult to apply it to concrete calculations of intensities in the Raman spectra of molecules. To use formula (2,1) it is necessary to know the whole set of electronic and vibrational states of a polyatomic molecule. At present this is inaccessible to us. The problem must be simplified.

G. Placzek proceeded along the path of considering separately the motion of electrons and nuclei in the molecule and only subsequently establishing the connection between them, which is manifested in the Raman effect4. The eigenfunction characterizing the vibrational and electronic state of the molecule is written in the form

\[ \Psi_{nv}(\xi,Q)=\varphi_{n}(\xi,Q)u_{v}(Q); \tag{2,3} \]

\(\xi\) are the coordinates of the electrons, \(Q\) are the normal coordinates of the nuclei; \(\varphi_{n}(\xi,Q)\) is the eigenfunction of the system with fixed nuclei, \(Q\) being here a parameter. The matrix element of the electric moment corresponding to the vibrational transition \(v\to v'\) has the form

\[ M_{vv'}=E\int \varphi_{n}(\xi,Q)u_{v}(Q)b(\xi,Q)\varphi_{n}(\xi,Q)u_{v'}(Q)d\tau =E[b_{n}(Q)]_{vv'}. \tag{2,4} \]

Obviously, \([b_n(Q)]_{vv'}\) is the matrix element of the polarizability in the \(n\)-th electronic state, corresponding to the transition \(v \to v'\). In fact, in the Raman effect we always deal with the ground, unexcited electronic state. Therefore we shall omit the index \(n\). The polarizability is a function of the normal coordinates of the molecule. In the first approximation (regarding the vibrations as harmonic) we have

\[ u_v(Q)=u_{v_1}(Q_1)u_{v_2}(Q_2)\ldots u_{v_{3N-6}}(Q_{3N-6}), \tag{2,5} \]

i.e. all normal coordinates separate. Here \(u_{v_j}\) are the proper functions of harmonic oscillators. Expanding the polarizability in a series in the values of the \(3N-6\) normal coordinates (\(3N-5\) in the case of a linear molecule) about the equilibrium position, we write

\[ b(Q)=b^0+\sum_j \left(\frac{\partial b}{\partial Q_j}\right)_0 Q_j +\frac{1}{2}\sum_{jk}\left(\frac{\partial^2 b}{\partial Q_j\partial Q_k}\right)_0 Q_jQ_k+\ldots \tag{2,6} \]

Restricting ourselves in (2,6) to the linear term (the case of the so-called electrical anharmonicity will be considered below), we obtain, with the aid of the known functions \(u_{v_j}\), the matrix element of the polarizability

\[ b_{v_j,\;v_j\pm1}=\sqrt{v_j+1}\left(\frac{\partial b}{\partial Q_j}\right)_0 Q_{j0}; \tag{2,7} \]

\(Q_{j0}\) is the zero amplitude of the normal vibration; from the magnitude of the matrix element of the coordinate of the harmonic oscillator corresponding to the transition \(j \to j\pm1\), it follows that

\[ Q_{j0}=h^{1/2}/2\sqrt{2\pi\nu_j}^{\,1/2}, \tag{2,8} \]

i.e. \(Q_{j0}\) is the matrix element \(Q\) for the transition \(0\to1\). In Planck’s theory the expression (2,7) determines the intensities and polarizations of the fundamentals in the Raman spectrum of the molecule. Like the polarizability \(b\) itself, the quantity \(\dfrac{\partial b}{\partial Q_j}\) is a symmetric tensor. We denote its trace—the sum of the diagonal terms—by

\[ A_j=\sum_{i=\xi,\eta,\zeta}\frac{\partial b_{ii}}{\partial Q_j} \tag{2,9} \]

and the anisotropy by:

\[ B_j=\sqrt{\frac{3}{2}\sum_{i,k}\left(\frac{\partial b_{ik}}{\partial Q_j}\right)^2 -\frac{1}{2}\sum_i\left(\frac{\partial b_{ii}}{\partial Q_j}\right)^2}. \tag{2,10} \]

The degree of depolarization of a Raman line under illumination by natural light is equal to

\[ \rho=\frac{6B_j^2}{5A_j^2+7B_j^2}. \tag{2,11} \]

The intensity of the Stokes line, taking into account the dependence on the temperature and the frequency of the incident light,

\[ J_j\sim J_0(\nu-\nu_j)^4(5A_j^2+13B_j^2)\, \frac{Q_{j0}^2}{1-\exp(-h\nu_j/kT)}. \tag{2,12} \]

Thus, in Placzek’s semiclassical theory, the problem of determining the intensities and polarizations of Raman lines reduces to the calculation of the tensor \(\left(\dfrac{\partial b}{\partial Q_j}\right)_0\). G. Placzek himself confines himself to conclusions that can be obtained independently of the numerical values of the tensor. These conclusions, based exclusively on symmetry properties,\(^{1,4}\) are the following:

1. For totally symmetric vibrations of molecules belonging to the cubic symmetry system, \(B_j=0\) and, consequently, \(\rho=0\).

2. For non-totally symmetric vibrations one always has \(A_j=0\) and, consequently, \(\rho=\dfrac{6}{7}\).

3. For totally symmetric vibrations of molecules belonging to a symmetry system lower than cubic, \(A_j\ne0;\ B_j\ne0,\)

\[ 0<\rho<\frac{6}{7}. \]

1. The most intense Raman lines generally belong to totally symmetric vibrations (qualitative rule).

It is obvious that, for all their value, the conclusions presented do not make it possible to construct a theory of intensities and polarizations. The question of the true values of \(\left(\dfrac{\partial b}{\partial Q_j}\right)_0\), of their dependence on the structure of the molecule, the nature of the chemical bonds, and the form of the vibrations remains open.

In the theory of intensities of infrared vibrational spectra the situation is analogous. The absorption intensity in the transition \(v_j\to v'_j\) is proportional to the Einstein coefficient \(B^{v'_j}_{v_j}\), determined by the expression

\[ B^{v'_j}_{v_j}=\frac{2\pi^3}{3h^2}\, \frac{g_{v'_j}}{g_{v_j}}\, C_{v_jv'_j}^2, \tag{2,13} \]

Here \(g\) is the statistical weight of the corresponding state, and the quantity \(C_{v_jv'_j}\) is the matrix element of the electric moment \(\mathbf{p}\). According to the prin-

chain of correspondence, we may write, in the same approximation as before,

\[ C_{v_j,v'_j}^{\,2}=\sum_{i=x,y,z}\left(\frac{\partial p_i}{\partial Q_j}\right)_0^2 Q_{j0}^{\,2}. \tag{2,14} \]

The symmetry properties make it possible in this case also to determine the selection rules\(^{1,4,5}\), but the complete solution of the problem requires knowledge of the vector \(\left(\dfrac{dp}{dQ_j}\right)_0\).

As we shall show below, purely theoretical calculation of the required quantities is impossible even for diatomic molecules. The theory of the electro-optical properties of vibrations must be constructed as a semiempirical one. The classical character of those methods which so far have proved to be the only ones leading to the goal is connected with the impossibility of a consistent quantum-mechanical calculation. These two features—semiempiricism and classicality, as we have already indicated, are characteristic of the modern theory of the electro-optics of vibrational spectra.

§ 3. POLARIZABILITY AND SILBERSTEIN’S THEORY

The behavior of the electron shell of a molecule is determined, in a number of phenomena, by its polarizability. This most important molecular constant, along with the dipole electric moment, magnetic susceptibility, and magnetic moment, characterizes the molecule in all phenomena of molecular optics. In particular, refraction, the Kerr effect, Rayleigh scattering, optical activity, and, finally, combination scattering of light (the Raman effect) are connected with polarizability. The dependence of polarizability on the frequency of oscillations of the strength of the applied field is given by the well-known dispersion formula. The polarizability at a given frequency of the incident light is called the optical polarizability; at an infinitely large period—static. If we disregard the phenomenon of optical activity and work outside absorption regions, the polarizability is represented by a real symmetric tensor of the second rank. The trace and anisotropy of this tensor [cf. (2,6), (2,9), (2,10)] appear in various formulas of molecular optics. We shall use the expression of the tensor reduced to its principal axes:

\[ \|b_{\xi\eta}\| = \begin{pmatrix} b_1 & 0 & 0\\ 0 & b_2 & 0\\ 0 & 0 & b_3 \end{pmatrix}. \tag{3,1} \]

The trace of the tensor is given by the expression for molecular refraction (the Lorentz–Lorenz formula)

\[ a=b_1+b_2+b_3=\frac{n^2-1}{n^2+2}\frac{M}{d}\frac{9}{4\pi N_A}. \tag{3,2} \]

Here \(n\) is the refractive index, \(M\) the molecular weight, \(d\) the density, \(N_A\) Avogadro’s number.

Similarly, the mean value of the static polarizability \(a^0\) can be obtained from the value of the temperature-independent part of the dielectric constant.

The anisotropy of the tensors \(b\) and \(b^0\) determines the value of the Kerr constant,

\[ K=\frac{2\pi}{9}\frac{N_A}{45kT} \left\{ \sum_{l,k=1,2,3}(b_l-b_k)(b_l^0-b_k^0)+ \right. \]

\[ \left. +\frac{1}{kT}\sum_{l,k}(b_l-b_k)(p_l^2-p_k^2) \right\}, \tag{3,3} \]

where \(p_l\) is the component of the dipole moment in a coordinate system fixed in the molecule.

The degree of depolarization of Rayleigh scattering is determined by the relation between the anisotropies \(g\) and the trace \(a\) of the tensor \(b_{\xi\eta}\), analogously to (2,11),

\[ \rho_0=\frac{6g^2}{5a^2+7g^2}. \tag{3,4} \]

By comparing (3,2), (3,3), and (3,4), it is often possible to determine \(b_1, b_2, b_3\)—the polarizability ellipsoid of the molecule\({}^{7}\). To study the electro-optical properties of Raman spectra it is necessary to know not only the polarizability tensor \(b_{\xi\eta}\), but, chiefly, the tensor \(b'_{\xi\eta}\), the derivative of the polarizability with respect to the normal coordinate. In other words, it is necessary to know the dependence of the polarizability on the instantaneous configuration of the vibrating molecule—on changes in interatomic distances and the mutual arrangement of the atoms.

The first attempt to calculate the polarizability ellipsoid of a molecule as a function of the interatomic distances was the well-known theory of Silberstein\({}^{8}\). This theory proceeds from the mutual induction of electric moments induced in isotropically polarizable atoms located at definite distances from one another. In the case of a diatomic molecule composed of atoms with polarizabilities \(\alpha_1\) and \(\alpha_2\), situated at a distance \(R\), we have

\[ \left. \begin{aligned} b_{\parallel}=b_3&=\frac{\alpha_1+\alpha_2+4\alpha_1\alpha_2/R^3}{1-4\alpha_1\alpha_2/R^6},\\[4pt] b_{\perp}=b_1=b_2&=\frac{\alpha_1+\alpha_2-2\alpha_1\alpha_2/R^3}{1-2\alpha_1\alpha_2/R^6}. \end{aligned} \right\} \tag{3,5} \]

A correct qualitative result is obtained: a larger polarizability along the axis of the molecule and a smaller one perpendicular to it.

Analogous calculations were carried out for a symmetric triatomic and for a tetrahedral molecule\(^{9,10}\). In the latter case we obtain

\[ b=b_1=b_2=b_3= \frac{ 216(4+2\alpha_1/R^3)\alpha_1+ \{216+108\alpha_1/R^3-(33011-2304\sqrt{6})^6\,4\alpha_1^2/R^6\}\alpha_2 }{ 27(8+4\alpha_1/R^3-36\alpha_1^2/R^6)^2-8192\cdot 4\alpha_1\alpha_2/R^6 }; \tag{3,6} \]

\(R\) is the distance between the atoms situated at the vertices of the tetrahedron, \(\alpha_1\) is their polarizability, and \(\alpha_2\) is the polarizability of the central atom. If this latter can be neglected, then

\[ b=\frac{4\alpha_1+3\sqrt{6}\,\alpha_1^2/r^3} {1+3\sqrt{6}\,\alpha_1/64r^3-243\alpha_1^2/1024r^6}, \tag{3,6a} \]

where \(r\) is the distance from the center to a vertex. A calculation of the polarizability of \(\mathrm{CCl}_4\), performed by this formula, gives \(10.2\cdot 10^{24}\ \mathrm{cm}^3\) instead of the experimental value \(10.4\cdot 10^{24}\ \mathrm{cm}^3\). However, such good agreement is accidental. In general, Silberstein’s theory gives incorrect quantitative results. The point is that it is built on an incorrect physical basis. The electron cloud in a homeopolar molecule is an exchange cloud, and, in essence, the electron shells of the atoms cannot be regarded as separate. But even for ionic molecules, where such a separation has greater grounds, Silberstein’s theory is incorrect, since it assumes the field of the induced dipoles to be uniform over interatomic distances. These circumstances were rightly pointed out by Plachek\(^{4}\). At the same time, the induction effect considered by Silberstein must indeed take place, though not as the sole factor determining the polarizability of the molecule, but as a certain additional factor. Bidermann\(^{11}\) showed that, in the quantum-mechanical calculation of polarizability by the method of successive approximations, the second-order terms are responsible for the Silberstein interaction (see also the work of Neugebauer\(^{12}\)). Nevertheless, in the literature there are isolated attempts to apply Silberstein’s theory to the calculation of electro-optical parameters in the Raman spectra of simple molecules. Cabannes and Rousset\(^{13}\) calculated the degrees of depolarization of the symmetric vibrations of the linear symmetric triatomic molecules \(\mathrm{CO}_2\) and \(\mathrm{CS}_2\), of the triangle \(\mathrm{CO}_3\), and of the hexagon \(\mathrm{C}_6\mathrm{H}_6\), using the corresponding expressions of Silberstein’s theory, which give the polarizability as a function of the interatomic distances, with which the symmetric normal coordinates are connected by simple linear relations. In all cases \(\rho\) comes out too large. We give the table (Table 1).

P. Rao^14 calculates the intensity of the Raman line corresponding to the totally symmetric vibration of \( \mathrm{CCl}_4 \), starting from expression (3.6a). He neglects the polarizability of the carbon atom, regarding it as \( \mathrm{C}^{4+} \). Taking the derivative \(db/dr\), one can calculate the ratio of the intensity of the Raman line to that of Rayleigh scattering by the formula

\[ \frac{I_{\mathrm{Raman}}}{I_{\mathrm{Rayleigh}}} \sim \frac{h}{8\pi^2\nu}\, \frac{\left(\dfrac{db}{dr}\right)^2}{b^2}. \tag{3.7} \]

P. Rao obtains \(db/dr = 1.63 \cdot 10^{-16}\ \mathrm{cm}^2\) and a value of the ratio (3.7) equal to \(3.6 \cdot 10^{-4}\), instead of the experimental \(2.0 \cdot 10^{-4}\). The order of magnitude is thereby given correctly. Nevertheless, both because of the above-mentioned fundamental shortcomings and because of the great mathematical difficulties and the unreliability of the results obtained, Silberstein’s theory is unsuitable for solving problems of the electro-optics of vibrational spectra.

Table 1

Degree of depolarization of Raman lines

molecule	frequency, \(\mathrm{cm}^{-1}\)	\(\rho\) calculated	\(\rho\) measured
\(\mathrm{O}_2\)	1556	0.66	0.3
\(\mathrm{CO}_2\)	1286	0.51	0.18
\(\mathrm{CO}_2\)	1389	0.51	0.14
\(\mathrm{CS}_2\)	654	0.46	0.25
\(\mathrm{CS}_2\)	799	0.46	0.25
\(\mathrm{CO}_3^{\prime\prime}\)	1067	0.78	0.2
\(\mathrm{C}_6\mathrm{H}_6\)	992	0.37	0.07

A consistent quantum-mechanical theory, in principle, makes it possible to calculate the polarizability of a molecule as a function of the interatomic distances. Using the variational method, Kirkwood^15 derived the approximate formula

\[ b_i=\frac{4\left[(q_i^2)_{00}\right]^2}{N}, \tag{3.8} \]

where \((q_i^2)_{00}\) is the value of the square of the \(i\)-th coordinate of an electron in state 0, and \(N\) is the number of electrons in the molecule. Calculation by this formula is possible only when the proper functions of the system are known, in other words, in the simplest cases. Formula (3.8) gives good results for atoms. Using various proper functions, Hirschfelder^16 calculated the polarizability of the hydrogen molecule. Similar calculations were carried out by Stenhsolt^17, Icksol^18, and Mrowka^19. Mrowka’s initial results were erroneous and were later corrected by him. We give the table:

MOLECULAR VIBRATIONAL ELECTRO-OPTICS

Table 2

Polarizability of H₂ × 10²⁴ cm⁻³

Author	Eigenfunctions	$b_3$	$b_1 = b_2$	$a/3$	$b_3 - b_1$
Hirschfelder¹⁶	Rosen with 2 parameters	0.75	0.74	0.74	0.01
Hirschfelder	Rosen with 1 parameter	0.71	0.67	0.68	0.04
Hirschfelder	Wang with 2 parameters	0.75	0.73	0.74	0.02
Hirschfelder	Wang with 1 parameter	0.71	0.66	0.68	0.05
Stenschol¹⁷	Wang with 1 parameter	0.77	0.52	0.60	0.25
Isthol¹⁸		0.72	0.59	0.63	0.13
Mrovka¹⁹		0.61	0.85	0.77	−0.24
Mrovka²⁰		0.82	0.77	0.79	0.05
Experiment		—	—	0.80	0.36

In more complicated cases, calculations are practically impossible. It is evident that quantum mechanics is at present little applicable to the solution of our problems. The situation with the theory of dipole moments and effective charges—quantities necessary for calculating intensities in infrared spectra—is approximately the same.

§ 4. ADDITIVE SCHEME OF MOLECULAR OPTICS

Modern science of the structure of matter makes broad use of the idea of additivity—essentially one of the guiding ideas in natural science. Additivity means that the properties of a certain system can be represented as sums of the properties of definite structural units that are preserved in the transition from one complex system to another. It is precisely with the mental decomposition of a complex system into such structural units that analytical investigation must, in a number of cases, begin, treating the interaction of the units as a deviation from additivity. The task of the physicist at the first stage of work consists in such an identification of real structural units as has genuine physical meaning, and in finding the limits of applicability of the additivity scheme. At the next stage, the interaction of the structural units, the deviations from the adopted scheme, are considered. Naturally, the scheme of additivity has the greater content the less significant the deviations from it are in comparison with the additive quantities. These ideas underlie modern atomism.

Molecular optics as a whole and, in particular, the electro-optics of vibrations deal with two basic physical quantities: the polarizability tensor and the vector of electric moment. The question arises as to what structural units a complex molecule can be divided into so that its polarizability and dipole moment would be expressed additively—at least in the zeroth approximation. It is obvious that finding such units substantially simplifies the problem.

It suggests itself to single out individual atoms and ions as structural units. Indeed, it is known that, for example, molecular refraction—in other words, the trace of the polarizability tensor—obeys with high accuracy the law of additivity of the refractions of individual atoms. However, this is rather the exception than the rule; we saw above that Silberstein’s theory, operating with individual atoms, has a very limited significance.

Modern chemistry singles out the valence bonds of atoms as the basic structural units. Individual bonds are characterized by definite values of physical quantities—lengths, energies, dipole moments, etc. In a number of cases good additivity is indeed observed. However, deviations from additivity, and large interactions of bonds, often occur. These deviations are interpreted in the modern valence scheme of chemistry as the so-called electronic resonance, as a first approximation starting from the zero approximation of the additive valence scheme. The valence scheme of chemistry in the broad sense of the word, including deviations from additivity—electronic resonance—finds its physical justification in quantum mechanics, in particular in the theory of directed valences. The singling out of individual valence bonds apparently leads to reasonable results. A substantial simplification of problems in the electro-optics of vibrations is achieved by applying the idea of additivity of the valence scheme[^22]. This is a natural development of the additive scheme of molecular optics, which reduces to the following. The electric dipole moment and the polarizability of a molecule (as well as the magnetic moment and the diamagnetic susceptibility) can be obtained additively from the corresponding constants of individual bonds.

The additivity of dipole moments is widely used in the theory of the dielectric constant[^23]. A quantum-mechanical justification of such additivity can be found, for example, in Allara’s work[^24]. We write the moment of the molecule $\mathbf{p}$ as the vector sum of the moments of the individual bonds

$$ \mathbf{p}=\sum_{n=1}^{N}\mathbf{p}^{(n)}. \tag{4,1} $$

The summation is carried out over all bonds in the molecule, whose number is $N$.

The scalar additivity of the traces of the polarizability tensor

$$ a=\sum_{n=1}^{N} a^{(n)} \tag{4,2} $$

is used in calculating molecular refraction.

Tensor additivity of polarizabilities was first applied by Otterbein[^25], Zakse[^26], Dalaporta and Daskol[^27], for calculating po-

ELECTRO-OPTICS OF MOLECULAR VIBRATIONS

Kerr constant. They postulated additivity of the energy of a molecule in an electric field as the sum of the energies of the individual bonds. In this case the static and optical polarizabilities were identified, which can be done in a region sufficiently far removed from absorption bands. The potential energy of a molecule in the field is

\[ U=-\frac{1}{2}\sum_{ik} b_{ik}E_iE_k, \tag{4,3} \]

where \(i,k=\xi,\eta,\zeta\) are axes fixed in the molecule. On the other hand,

\[ U=-\frac{1}{2}\sum_{ik}E_iE_k\sum_{nm}\alpha_{nm}\cos(nmi)\cos(nmk), \tag{4,3a} \]

\(\alpha_{nm}\) is the polarizability of an individual bond, \(n\) is the bond index, \(m=1,2,3\) correspond respectively to the three principal directions of the polarizability ellipsoid of the individual bond. It is assumed that the direction of the bond itself is one of these principal directions and that the other two are perpendicular to it. For an isolated bond, and for a diatomic molecule, this is always true. \(\cos(nmi)\) is the direction cosine of the \(m\)-th direction of the \(n\)-th bond.

Consequently, one may write

\[ b_{ik}=\sum_{n=1}^{N} b_{ik}^{(n)}, \tag{4,4} \]

where

\[ b_{ik}^{(n)}=\sum_{m=1}^{3}\alpha_{nm}\cos(nmi)\cos(nmk), \tag{4,5} \]

and analogously

\[ p_i^{(n)}=\mu_n\cos(ni), \tag{4,6} \]

\(\mu_n\) is the electric moment of the bond.

The values \(\alpha_{nm}\), characterizing the \(n\)-th bond, can be obtained by comparing the values of the Kerr constants of several compounds containing identical bonds. Thus, comparing these quantities for \(\mathrm{CH_3Cl}\), \(\mathrm{CH_2Cl_2}\), and \(\mathrm{CHCl_3}\), Zaks found \(\alpha_{nm}\) for the \(\mathrm{C-H}\) and \(\mathrm{C-Cl}\) bonds. Refractivities were used as auxiliary quantities. The reverse calculation of the Kerr constants gives rather good agreement with experiment (see Table 3).

Table 3

Kerr constant \(\cdot 10^{12}\)

Molecule	\(K\) measured	\(K\) calculated
\(\mathrm{CH_3Cl}\)	41	42
\(\mathrm{CH_2Cl_2}\)	13.7	12.0
\(\mathrm{CHCl_3}\)	\(-28\)	\(-23\)

Later, Sheng-Nien Wang\(^{28}\) and Denbigh\(^{29}\) calculated the polarizability ellipsoids of a number of homeopolar bonds by comparing the Kerr constants, the degrees of depolarization of re-

Rayleigh scattering and the magnitude of molecular refraction. We give a table of comparative data:

Table 4

Polarizabilities of bonds

Bond	\(\alpha_m \cdot 10^{24}\ \mathrm{cm}^{-3}\)	Zakse	Wang	Denbigh
C—H	\(\alpha_1\)	0,81	0,72	0,79
C—H	\(\alpha_2=\alpha_3\)	0,57	0,62	0,58
C—C	\(\alpha_1\)	0,23	1,82	1,82
C—C	\(\alpha_2=\alpha_3\)	0,54	0,02	0,02
C=C	\(\alpha_1\)	3,92	3,02	2,86
C=C	\(\alpha_2\)	0,94	0,96	1,06
C=C	\(\alpha_3\)	0,35	0,96	1,06
C—Cl	\(\alpha_1\)	3,37	3,53	3,67
C—Cl	\(\alpha_2=\alpha_3\)	2,21	2,15	2,08
C—Br	\(\alpha_1\)	—	—	5,04
C—Br	\(\alpha_2=\alpha_3\)	—	—	2,88
H—Cl	\(\alpha_1\)	—	3,13	3,13
H—Cl	\(\alpha_2=\alpha_3\)	—	2,39	2,39
H—Br	\(\alpha_1\)	—	—	4,23
H—Br	\(\alpha_2=\alpha_3\)	—	—	3,32
H—J	\(\alpha_1\)	—	—	6,58
H—J	\(\alpha_2=\alpha_3\)	—	—	4,89

In Zakse, the smaller polarizability of the C—C bond along the bond than perpendicular to it is scarcely plausible. At the same time, the anisotropy of the C—C bond in Wang and Denbigh is evidently too high. Wang and Denbigh erroneously regard the polarizability ellipsoid of the C=C bond as an ellipsoid of revolution; \(\pi\)-bonds in complex nonlinear molecules do not have cylindrical symmetry.

Additivity of the polarizability ellipsoids of individual bonds is substantiated, as we have indicated, by the valence scheme. One confirmation of it is the constancy of the electronic frequencies for a given bond, independently of other bonds in the molecule. The values of polarizability are determined by the electronic spectrum of the molecule, according to the dispersion formula. We give some data:

Table 5

Electronic bands of various bonds

Bond	Molecule	\(\lambda\) in Å	Bond	Molecule	\(\lambda\) in Å
\(\mathrm{C{=}C}\)	Butene-1	1875, 1819 1750—1730	\(\mathrm{C-Br}\)	\(\mathrm{CH_3Br}\)	2880—1920
\(\mathrm{C{=}C}\)	Pentene-1	1884, 1830 1750—1730	\(\mathrm{C-Br}\)	\(\mathrm{C_2H_5Br}\)	2850—1900
\(\mathrm{C{=}C}\)	Heptene-1	1886, 1831 1750—1730	\(\mathrm{C-Br}\)	Other saturated bromides	2850—1900
\(\mathrm{C-Cl}\)	\(\mathrm{CH_3Cl}\)	1610—1540	\(\mathrm{C-J}\)	\(\mathrm{CH_3J}\)	3600—2270
\(\mathrm{C-Cl}\)	\(\mathrm{C_2H_5Cl}\)	1600—1540	\(\mathrm{C-J}\)	\(\mathrm{C_2H_5J}\)	3600—2100
\(\mathrm{C-Cl}\)	Other saturated chlorides	1600—1500	\(\mathrm{C-J}\)

Stuart’s criticism of these notions of the additivity of polarizabilities[^31] is unfounded. He pointed out that interactions must be taken into account. However, in the absence of electronic resonance, the additive scheme of molecular optics gives a quite reasonable zero approximation. The presence of resonance will require new empirical characteristics of the bond. Thus, the bond of carbon atoms in benzene undoubtedly has \(\alpha_m\) values different from those for \(\mathrm{C-C}\) and \(\mathrm{C{=}C}\) bonds in aliphatic and ethylenic compounds.

It is known that such quantities as the Kerr constant and the degree of depolarization of Rayleigh scattering depend strongly on intermolecular interaction. In particular, large changes in these quantities in liquids, as compared with gases, are characteristic. These phenomena are explained by the joint orientation of groups of molecules in a liquid and are not directly connected with questions concerning the isolated molecules with which we are concerned here (see, however, § 11).

In application to problems of the theory of vibrational spectra, the additive scheme is somewhat modified. Here the essential quantities are not the values of the dipole moments and polarizabilities of molecules themselves, but their derivatives with respect to the normal vibrational coordinates. The assumption of additivity means in this case that the changes \(\partial\mu/\partial q\) and \(\partial\alpha/\partial q\) for a given bond depend only on the change in the length of that bond. This more far-reaching assumption was made by M. V. Vol’kenshtein\(^{22}\). The additivity of the derivatives \(\mu'\) and \(\alpha'\) is apparently observed within narrower limits than the additivity of the quantities \(\mu\) and \(\alpha\) themselves. This is indicated, in particular, by the very great role of interaction (cross terms in the potential energy) in molecular vibrations, when the vibrations of some bonds strongly affect the vibrations of other bonds\(^{1,2,32}\). In the following, first approximation, it is of course necessary to take into account the dependence of \(\mu\) and \(\alpha\) of a given bond on changes in the lengths of the neighboring bonds interacting with it, and on changes in the adjacent angles.

§ 5. THEORY OF INTENSITIES AND POLARIZATIONS

In the works of M. V. Vol’kenshtein\(^{22,33}\) and later of M. V. Vol’kenshtein and M. A. Elyashevich\(^{32,34,35,36}\), a theory of electro-optical parameters—intensities and polarizations in vibrational spectra—was developed, based on the additive scheme. By analogy with the valence-force scheme for calculating vibrational frequencies\(^{1,2,32}\), we shall speak of the valence-optical scheme of the theory of intensities.

Let us first set forth the basic propositions of the theory in the zeroth approximation. Extending the principle of additivity to the derivatives of the polarizability and of the dipole moment of a molecule with respect to the normal coordinates, we write

\[ \frac{\partial b_{ik}}{\partial Q_j} = \sum \frac{\partial b_{ik}^{(n)}}{\partial Q_j} \quad\text{and}\quad \frac{\partial p_i}{\partial Q_j} = \sum \frac{\partial p_i^{(n)}}{\partial Q_j}. \tag{5,1} \]

As M. A. Elyashevich\(^{1,32}\) has shown, a rational method for solving problems relating to molecular vibrations is connected with the introduction of the so-called natural valence-force vibrational coordinates—changes in bond lengths \(q_n\) and changes in valence angles \(\gamma_s\). Obviously, the coordinates \(q_n\) are precisely suitable for the valence-optical scheme. We may write\(^{22,34}\)

\[ \frac{\partial b_{ik}}{\partial Q_j} = \sum_n \sum_t \frac{\partial b_{ik}^{(n)}}{\partial q_t} \frac{\partial q_t}{\partial Q_j} + \sum_n \sum_s \frac{\partial b_{ik}^{(n)}}{\partial \gamma_s} \frac{\partial \gamma_s}{\partial Q_j} \tag{5,2} \]

and an analogous expression for the dipole moment.

Finding the derivatives of the natural coordinates with respect to the normal coordinates is a problem of mechanics, not of the electro-optics of vibrations. In approxima-

for the study of harmonic vibrations—the fundamental frequencies in the spectrum, the natural and normal coordinates are related by linear relations. The derivatives \(\partial q_i/\partial Q_j\) and \(\partial \gamma_s/\partial Q_j\) thus give us certain numerical coefficients. By contrast, the derivatives of the polarizabilities and dipole moments of the bonds with respect to the natural coordinates require special consideration.

Rewrite (5.2) in the form of the sum of “valence” and “deformation” terms

\[ \frac{\partial b_{ik}}{\partial Q_j} = \left(\frac{\partial b_{ik}}{\partial Q_j}\right)_{\gamma} + \left(\frac{\partial b_{ik}}{\partial Q_j}\right)_{q}. \tag{5.2a} \]

Using expressions (4.5) and (4.6), we obtain

\[ \left(\frac{\partial b_{ik}}{\partial Q_j}\right)_{\gamma} = \sum_{nm} \frac{\partial a_{nm}}{\partial q_n} \cos(nmi)\cos(nmk)\, \frac{\partial q_n}{\partial Q_j}, \tag{5.3} \]

\[ \left(\frac{\partial b_{ik}}{\partial Q_j}\right)_{q} = \sum_{nms} a_{nm} \frac{\partial}{\partial \gamma_s} \left[\cos(nmi)\cos(nmk)\right] \frac{\partial \gamma_s}{\partial Q_j}. \tag{5.4} \]

These expressions are valid only in the zero approximation of the valence-optical scheme, according to which the polarizability and dipole moment of the \(n\)-th bond depend only on the change in the length of this bond,

\[ \frac{\partial a_{nm}}{\partial q_t} = \frac{\partial a_{nm}}{\partial q_n}\delta_{nt} \quad \text{and} \quad \frac{\partial a_{nm}}{\partial \gamma_s}=0, \tag{5.5} \]

where

\[ \delta_{nt}= \begin{cases} 1, & n=t,\\ 0, & n\ne t. \end{cases} \]

In a rectangular coordinate system, expressions (5.3) and (5.4) can be substantially simplified. Since the three principal directions of the polarizability of a bond, corresponding to \(m=1,2,3\), are perpendicular to one another, we obtain

\[ \frac{\partial}{\partial \gamma_s} \sum_m \cos(nmi)\cos(nmk) = \frac{\partial}{\partial \gamma_s}\delta_{ik} = 0. \]

Whence

\[ \frac{\partial}{\partial \gamma_s} [\cos(n1i)\cos(n1k)] = - \frac{\partial}{\partial \gamma_s} [\cos(n2i)\cos(n2k)+ \cos(n3i)\cos(n3k)] \]

and therefore

\[ \left(\frac{\partial b_{ik}}{\partial Q_j}\right)_{q} = -\sum \left\{ (a_{n1}-a_{n2}) \frac{\partial}{\partial \gamma_s} [\cos(n2i)\cos(n2k)] + (a_{n1}-a_{n3}) \frac{\partial}{\partial \gamma_s} [\cos(n3i)\cos(n3k)] \right\}. \tag{5.4a} \]

A practically especially important case is that of single, so-called \(\sigma\)-bonds possessing rotational symmetry. For them \(a_{n2}=a_{n3}\), and expression (5,4a) assumes a quite simple form, containing only the direction cosines of the bond itself:

\[ \left(\frac{\partial b_{ik}}{\partial Q_j}\right)_q = \sum_{ns}(a_{n1}-a_{n2}) \frac{\partial}{\partial r_s} \{\cos(n1i)\cos(n1k)\} \frac{\partial r_s}{\partial Q_j}. \tag{5,6} \]

Analogous transformations bring, in the case of \(\sigma\)-bonds, the valence term (5,3) to the form

\[ \left(\frac{\partial b_{ik}}{\partial Q_j}\right)_r = \sum_n \left\{ \left( \frac{\partial a_{n1}}{\partial q_n} - \frac{\partial a_{n2}}{\partial q_n} \right) \cos(n1i)\cos(n1k) + \frac{\partial a_{n2}}{\partial q_n}\delta_{ik} \right\} \frac{\partial q_n}{\partial Q_j}. \tag{5,7} \]

Still simpler expressions are obtained for the derivatives of the dipole moment.

Thus, the problem of calculating the intensity (and polarization) of a vibrational line is reduced, in the case of \(\sigma\)-bonds, to formulas (5,6) and (5,7). First of all, of course, it is necessary to know the exact form of the molecule; then—the form of the vibrations, the vibrational coordinates and, consequently, the derivatives \(\partial q_n/\partial Q_j\) and \(\partial r_s/\partial Q_j\). The methods developed by M. A. Eliashevich\(^{1,32}\) and B. I. Stepanov\(^{2,32}\) make it possible to find these derivatives without great difficulty. After this there remains the determination of the optical parameters proper. In the zero approximation, for each \(\sigma\)-bond it is necessary to know three quantities in the case of Raman spectra: \(a_1-a_2\), \(\partial a_1/\partial q\) and \(\partial a_2/\partial q\), and two in the case of infrared spectra: \(\mu\) and \(\partial\mu/\partial q\). For a \(\pi\)-bond, in the case of Raman spectra, \(a_1-a_3\) and \(\partial a_3/\partial q\) are added as well.

The basic idea of the valence-optical scheme consists in the fact that one and the same bonds are characterized in different molecules by identical values of \(a_m\), \(\partial a_m/\partial q\), \(\mu\), and \(\partial\mu/\partial q\). Thus, for example, to calculate the intensities and polarizations in the Raman spectrum of a molecule containing two kinds of \(\sigma\)-bonds, say \(\mathrm{CH_2Cl_2}\), knowledge of six parameters is required. The very same parameters are suitable for calculations in the cases of other molecules containing the previous bonds: \(\mathrm{CHCl_3}\), \(\mathrm{CH_3Cl}\), \(\mathrm{CH_4}\), \(\mathrm{CCl_4}\). In this case the quantities \(\partial a_m/\partial q\) and \(\partial\mu/\partial q\) play the role of new physical constants characterizing the bond, along with \(a_m\) and \(\mu\).

We have already seen that a direct theoretical calculation of \(a_m\) and \(\mu\), and hence also of \(\partial a_m/\partial q\) and \(\partial\mu/\partial q\), is in fact impossible at the present time. Natural, therefore, is a semiempirical method for finding these quantities\(^{22,32,37}\). The values \(a_m-a_{m'}\) may be found, as we have seen, from the Kerr constant and the degree of depolarization of Rayleigh scattering. The values \(\mu\) are found from the magnitudes of the dipole moments of molecules. To determine \(\partial a_m/\partial q\) one has to make use of:

data on intensities and polarizations in Raman spectra*). However, by using part of the experimental spectroscopic material relating to particular molecules, and by determining the necessary parameters, we shall be able to predict the intensities of vibrational lines for a number of other molecules containing the same bonds.

In the case of infrared spectra, a valuable source for determining the effective bond charges \(\partial\mu/\partial q\) is provided by the values of atomic polarization and dispersion of the refractive index (see § 9).

§ 6. ELECTRO-OPTICS OF VALENCE AND DEFORMATION VIBRATIONS

The formulas of the preceding section, even before their application to effective calculations, make it possible to draw a number of general conclusions about the electro-optical properties of molecular vibrations.

The generalized valence-force model used in calculating the mechanical properties—the frequencies and forms of vibrations of molecules \(^{1,2,32}\)—naturally leads to a division of vibrations into valence ones, i.e. those in which the lengths of valence bonds change and the coordinates \(\gamma\) are equal to zero, and deformation ones, in which valence angles change and the coordinates \(q\) are zero. In fact, in the majority of cases both \(q\) and \(\gamma\) are nonzero simultaneously. However, the shares of participation of bond lengths and valence angles in a given vibration are unequal, and the division of vibrations into valence and deformation ones has meaning as a certain rough approximation. With full justification, within the framework of the valence scheme, we may speak of the valence and deformation shares of a given vibration. This is what we did in the preceding section, separating the tensor \(\partial b_{ik}/\partial Q_j\) and the vector \(\partial p_i/\partial Q_j\) into valence and deformation parts.

Let us consider separately the deformation part \((\partial b_{ik}/\partial Q_j)_q\). We shall calculate the trace of this tensor, \(A_j\). On the basis of (5,4)

\[ A_j=\sum_i\left(\frac{\partial b_{ii}}{\partial Q_j}\right)_q =\sum_i\sum_n\sum_{ms} a_{nm}\frac{\partial}{\partial\gamma_s}\left[\cos^2(nmi)\right]\frac{\partial\gamma_s}{\partial Q_j} \]

\[ =\sum_n\sum_{ms} a_{nm}\frac{\partial}{\partial\gamma_s}\left[\sum_i\cos^2(nmi)\right]\frac{\partial\gamma_s}{\partial Q_j}=0. \tag{6,1} \]

This has the following clear meaning. Deformation vibrations of polyatomic molecules reduce to rotations of individual bonds from their equilibrium positions. In this process the spherical part of the polarizability ellipsoid—its trace:

\[ a=\sum_i b_{ii}=\sum_n\sum_i b_{ii}^{(n)}=\sum_{nm} a_{nm}\left[\sum_i\cos^2(nmi)\right]=\sum_{nm} a_{nm} \tag{6,2} \]

* An idea of the order of magnitude of the trace \(\partial a/\partial q\) in the case of simple hydrogen-containing molecules can be obtained by comparing the refractions of hydrogen and deuterium compounds \(^{38}\).

invariant, since, according to the assumption of additivity, the quantities \(a_{,m}\) themselves do not change upon rotations of the bonds. Therefore the change \(A_\gamma\) of the trace \(a\) is equal to zero. According to (2,11), it follows from this that the degree of depolarization of a Raman line belonging to a deformation vibration is equal to \(6/7\), independently of its symmetry[^36]. This proposition, valid, of course, only in the zero approximation, supplements Placzek’s rule cited in § 2, which states that \(\rho = 6/7\) for nonsymmetric vibrations. At the same time, for valence fully symmetric vibrations \(\rho\) is always different from zero. Consequently, other conditions being equal, the value of \(\rho\) may serve as an approximate measure of the degree of participation of angle deformation in a given fully symmetric vibration, varying from small values for purely valence vibrations to values close to \(6/7\) for purely deformation vibrations. Characteristic valence vibrations should give polarized Raman lines, and to a greater degree the more anisotropic the tensor \(\partial \alpha / \partial q\) of the bond is. The degree of depolarization of symmetric vibrations of one and the same type should be greater for those vibrational frequencies in which deformation is represented to a greater extent.

Table 6

Degrees of depolarization of Raman lines

Molecule	Bond	\(\nu\ \mathrm{cm}^{-1}\)	Character of vibration*)	\(\rho\)
\(\mathrm{CH_3Cl}\)	\(\mathrm{C—Cl}\)	712	\(\nu(s)\)	0.2
\(\mathrm{C_2H_5Cl}\)	\(\mathrm{C—Cl}\)	656	\(\nu(s)\)	0.15
\(\mathrm{CH_3Br}\)	\(\mathrm{C—Br}\)	594	\(\nu(s)\)	0.2
\(\mathrm{C_2H_5Br}\)	\(\mathrm{C—Br}\)	560	\(\nu(s)\)	0.19
\(\mathrm{C_2H_5J}\)	\(\mathrm{C—J}\)	500	\(\nu(s)\)	0.22
\(\mathrm{CH_3SH}\)	\(\mathrm{C—S}\)	704	\(\nu(s)\)	0.3
\(\mathrm{XYCO}\)	\(\mathrm{C=O}\)	1663—1735	\(\nu\)	0.34—0.42
\(\mathrm{CH_2Cl_2}\)	\(\mathrm{C—Cl}\)	283	\(\delta(s)\)	0.43
\(\mathrm{CH_2Cl_2}\)	\(\mathrm{C—Cl}\)	700	\(\nu(s)\)	0.09
\(\mathrm{CH_2Br_2}\)	\(\mathrm{C—Br}\)	174	\(\delta(s)\)	0.35
\(\mathrm{CH_2Br_2}\)	\(\mathrm{C—Br}\)	577	\(\nu(s)\)	0.11
\(\mathrm{CH_2J_2}\)	\(\mathrm{C—J}\)	121	\(\delta(s)\)	0.42
\(\mathrm{CH_2J_2}\)	\(\mathrm{C—J}\)	483	\(\nu(s)\)	0.19
\((\mathrm{C_2H_5})_2\mathrm{O}\)	\(\mathrm{C—O}\)	438	\(\delta(s)\)	0.48
\((\mathrm{C_2H_5})_2\mathrm{O}\)	\(\mathrm{C—O}\)	840	\(\nu(s)\)	0.29
\(\mathrm{SO_2}\)	\(\mathrm{S=O}\)	525	\(\delta(s)\)	0.50
\(\mathrm{SO_2}\)	\(\mathrm{S=O}\)	1145	\(\nu(s)\)	0.18
\(\mathrm{>CH_2}\)	\(\mathrm{C—H}\)	\(\sim 1450\)	\(\delta(s)\)	\(\sim 0.86\)
\(\mathrm{>CH_2}\)	\(\mathrm{C—H}\)	\(\sim 2850\)	\(\nu(s)\)	\(\sim 0.10\)

*) \(\nu\) — predominantly valence, \(\delta\) — deformation, \(s\) — symmetric vibration.

These propositions are confirmed by experiment. We shall give a table of values of \(\rho\) for characteristic vibrations of molecules of the type \(XY\) and symmetric vibrations of triangular molecules of the type \(XY_2\) (Table 6).

Especially characteristic is the case of the \(CH_2\) group, for which the symmetric deformation vibration gives a line with \(\rho=6.7^*)\).

In general, according to (5,4a) and (5,6), the intensities and polarizations of Raman lines of deformation vibrations depend only on the anisotropies \(a_{mn}-a_{mn'}\) of the polarizability tensors themselves of the individual bonds. On the contrary, it can readily be shown, starting from (5,7), that in the case of valence vibrations the trace \(A_q\) depends only on the traces of the tensors \(\partial a/\partial q\) of the individual bonds, and the anisotropy \(B_q\) on their anisotropies.

Of particular practical interest for structural molecular analysis is the case of valence vibrations of a group of \(N\) identical \(\sigma\)-bonds forming equal angles \(\vartheta\) with one another (for example, the groups \(CH\), \(CH_2\), \(CH_3\), which frequently occur in molecules of organic compounds). Let us determine how, in this special case, the electro-optical parameters of purely valence symmetric vibrations depend on the number of bonds \(N\) and on the angle \(\vartheta^{41}\).

For the Raman spectrum, according to (5,7), the trace is

\[ A_N=\sum_i \left(\frac{\partial b_{ii}}{\partial Q}\right)_\gamma =\frac{\partial q}{\partial Q}N\left(\frac{d\alpha_1}{dq}+2\frac{d\alpha_2}{dq}\right) =\frac{\partial q}{\partial Q}NA_1 \tag{6,3} \]

the square of the anisotropy is

\[ B_N^2=\frac{3}{2}\sum_{ik}\left(\frac{\partial b_{ik}}{\partial Q}\right)_\gamma -\frac{1}{2}\left[\sum_i\left(\frac{\partial b_{ii}}{\partial Q}\right)_\gamma\right]^2= \]

\[ =\left(\frac{\partial q}{\partial Q}\right)^2 N^2\left(\frac{d\alpha_1}{dq}-\frac{d\alpha_2}{dq}\right)^2 \left(1-\frac{N-1}{2N}3\sin^2\vartheta\right)= \]

\[ =\left(\frac{\partial q}{\partial Q}\right)^2 N^2B_1^2 \left(1-\frac{N-1}{2N}3\sin^2\vartheta\right). \tag{6,4} \]

Here \(A_1\) and \(B_1\) are the trace and anisotropy of the tensor \(\partial a/\partial q\) for an individual bond. From (6,3) and (6,4) we obtain an expression for the degree of depolarization as a function of \(N\) and \(\vartheta\),

\[ \rho_N= \frac{ 6B_1^2\left(1-\frac{N-1}{2N}3\sin^2\vartheta\right) }{ 5A_1^2+7B_1^2\left(1-\frac{N-1}{2N}3\sin^2\vartheta\right) }, \tag{6,5} \]

\(\rho_N\) decreases rapidly with increasing \(N\). For the tetrahedron \(XY_4\), with \(N=4\) and \(\vartheta=109^\circ 28'\), \(\rho_4=0\) in accordance with Placzek’s rule (p. 58).

We shall give calculations of \(\rho\) relating to symmetric valence \(C—H\) vibrations in bromomethanes (Table 7). In this case, for the semiempir—

\(^*)\) Ignorance of the rules set forth led Ta-Yu\({}^{39}\) to the erroneous assignment of some lines with \(\rho=6/7\) to nonsymmetric vibrations (cf. 40).

rical determination of \(A_1\) and \(B_1\), the Raman spectra of the chloromethanes were used. The values in the third column were obtained by rigorous calculations taking into account the true form of the vibrations\(^{38}\), and those in the fourth column by formula (6.5).

Table 7

Degrees of depolarization of Raman lines

Molecule	Frequency, \(\mathrm{cm}^{-1}\)	\(\rho_N\) \(^{35}\)	\(\rho_N\) \(^{5}\)	\(\rho_N\), experiment
\(\mathrm{CH_4}\)	2914	0	0	(0)
\(\mathrm{CH_3Br}\)	2973	0.05	0.05	0.10
\(\mathrm{CH_2Br_2}\)	2988	0.13	0.11	0.10
\(\mathrm{CHBr_3}\)	3021	0.31	0.22	0.25

Formula (6.5) can be successfully applied for orientational calculations of the degree of depolarization. Let us give one more example—the S—H bond. In methyl mercaptan \(\mathrm{H_3C—SH}\) the \(2573\ \mathrm{cm}^{-1}\) line of the characteristic vibration of the S—H bond has \(\rho_1 = 0.30\). Using the relation

\[ \rho_1=\frac{6B_1^2}{5A_1^2+7B_1^2} =\frac{6(\alpha'_1-\alpha'_2)^2}{5(\alpha'_1-2\alpha'_2)^2+7(\alpha'_1-\alpha'_2)^2} =0.30, \tag{6.6} \]

we find the ratio \(\alpha'_2:\alpha'_1=0.17\). Substituting into (6.5), for \(N=2\) and \(\vartheta=92^\circ20'\) (the experimental value of the angle), we obtain for hydrogen sulfide \(\mathrm{H—S—H}\) \(\rho_2=0.10\) instead of the experimental value \(\rho=0.15^*)\).

To calculate intensities it is necessary to know \(\partial q/\partial Q\). In the case of purely valence vibrations of a system of identical bonds,

\[ \partial q/\partial Q=\frac{2\pi\nu}{\sqrt{Nk}}, \tag{6.7} \]

where \(k\) is the bond force constant. This relation is obtained from the expression for the potential energy of a fully symmetric valence vibration:

\[ U=\sum_{n=1}^{N}\frac{kq_n^2}{2} =\frac{Nk}{2}q^2 =2\pi^2\nu^2 Q. \]

In the case of small changes in \(\nu\) with change in \(N\) (cf. the second column of Table 7), equation (6.7) assumes the form

\[ \partial q/\partial Q \cong \frac{\mathrm{const}}{\sqrt{N}}. \tag{6.7a} \]

\[ \rule{4cm}{0.4pt} \]

\(^*)\) Let us note that the accuracy of experimental determinations of \(\rho\) is usually low; in most works the errors reach tens of percent.

and the intensity of the Raman line of the totally symmetric vibration, according to (2.12),

\[ I_N^R \sim Nf\left\{5A_1^2+13B_1^2\left(1-\frac{N-1}{2N}3\sin^2\vartheta\right)\right\}, \tag{6.8} \]

where

\[ f \sim \frac{(\nu-\nu_0)^4}{\nu}\,\frac{1}{1-\exp(-h\nu/kT)}. \]

The intensity \(I_N^R\) of the Raman line of the totally symmetric vibration increases with increasing number of bonds \(N\). This circumstance is also confirmed by experiment. On the contrary, in the case of infrared spectra the intensity decreases with increasing \(N^{41,42}\). We have, according to (2.13) and (2.14),

\[ I_N^{\mathrm{ir}} \sim \sum_N\left(\frac{\partial p_i}{\partial Q}\right)_\gamma^2 = \sum_i\left(\sum_{n=1}^{N}\frac{\partial \mu_n}{\partial q}\cos(ni)\frac{\partial q}{\partial Q}\right)^2 = \]

\[ = \left(\frac{\partial q}{\partial Q}\right)^2 \left(\frac{\partial \mu}{\partial q}\right)^2 N\{1+(N-1)\cos\vartheta\} \tag{6.9} \]

and, by virtue of (6.7a),

\[ I_N^{\mathrm{ir}} \cong I_1^{\mathrm{ir}}\{1+(N-1)\cos\vartheta\}. \tag{6.9a} \]

If \(\vartheta\) is the tetrahedral angle,

\[ I_2 \cong \frac{2}{3}I_1,\qquad I_3 \cong \frac{1}{3}I_1,\qquad I_4=0. \]

The most intense and characteristic in infrared spectra, however, are not the symmetric but the antisymmetric vibrations. Consideration of the problem in general form does not lead here to a simple formula, since we are dealing with degenerate vibrations.

§ 7. METHOD OF CALCULATING INTENSITIES AND POLARIZATIONS \(^{22,32,33,34,35}\)

Effective calculations of intensities and polarizations, according to the theory set forth, despite its fundamental simplicity, require a number of special devices, without which they become extremely difficult. Especially important is the consistent application of symmetry properties, which considerably simplifies the problem.

The basic expressions (5.3), (5.4), and the analogous ones for the dipole moment contain cosines of the angles formed by the valence bonds and the perpendiculars to them (the principal directions of the ellipsoids of bond polarizability) with the rectangular coordinates fixed in the molecule. In the case of \(\sigma\)-bonds, according to (5.6), (5.7), only the direction cosines of the bonds themselves are needed. Therefore, in order to solve the electro-optical problem, instead of valence-force coordinates \(Y\)—changes in the valence angles between bonds—one has to introduce the so-called valence-optical

coordinates \(\beta\)—changes of the angles formed by the bonds with the coordinate axes. We have

\[ d\cos(n1i)=-\sin(n1i)\beta_{ni}. \tag{7,1} \]

By virtue of the orthogonality condition

\[ \sum_i \cos(n1i)\sin(n1i)\beta_{ni}=0 \tag{7,2} \]

each triple \(\beta_{n1}, \beta_{n2}, \beta_{n3}\) is expressed in terms of two independent angular coordinates \(\varphi_n\) and \(\psi_n\), which determine the rotation of the \(n\)-th bond (for example, changes in latitude and longitude). The valence-optical coordinates for each bond will thus be \(q_i, \varphi_n\), and \(\psi_i\). For a total number of bonds \(N\), we have obtained altogether \(3N\) valence-optical coordinates, of which 3 correspond to rotational degrees of freedom. The number of natural valence-force coordinates \((q_i,\gamma_s)\) is \(3N-3\). The three missing conditions needed for passing from valence-force to valence-optical coordinates are the conditions that the components of the angular momentum be equal to zero. Taking these conditions into account, we separate vibrations from rotations and compute the tensors \(\dfrac{\partial b}{\partial Q}\) or the vectors \(\dfrac{\partial p}{\partial Q}\) for purely vibrational motions. These conditions are readily obtained in explicit form with the aid of Jacobi coordinates \(^{22,43,44}\), which describe the position of each subsequent mass relative to the center of gravity of all the preceding ones.

Fig. 1.

The symmetry properties are taken into account as follows. As in the problem of vibrational mechanics, the electro-optical problem is solved separately for each type of vibration. For this purpose symmetry coordinates are introduced—linear combinations of valence-optical coordinates satisfying the requirements of symmetry, i.e. transforming among themselves under symmetry operations in irreducible fashion \(^{1,5,32,34,35}\). Each symmetry coordinate is a linear combination of equivalent coordinates—changes of bonds of the same type, or of angles formed by bonds of the same type and arranged in the same way.

The expression for the derivative of the polarizability (or dipole moment) with respect to a normal coordinate belonging to the given symmetry type “\(\chi\)” is brought to the form

\[ \frac{\partial b_{ik}}{\partial Q_j^{(\chi)}}= \sum_{\lambda} \frac{\partial b_{ik}}{\partial q_{\lambda}^{(\chi)}}C_{\lambda j}^{(\chi)} + \frac{\partial b_{ik}}{\partial \varphi_{\lambda}^{(\chi)}}E_{\lambda j}^{(\chi)} + \frac{\partial b_{ik}}{\partial \psi_{\lambda}^{(\chi)}}F_{\lambda j}^{(\chi)}, \tag{7,3} \]

where the coefficients \(C, E, F\) denote derivatives of the corresponding valence-optical coordinates with respect to the normals. The summation is carried out over all equivalent coordinates of the given type.

Let us consider the molecule \(XY_2\). First of all we choose rectangular coordinate axes fixed with respect to the equilibrium configuration of the molecule. From simple symmetry considerations there follows the choice of axes shown in Fig. 1: the axis passes along the bisector of the angle \(YXY\), and the axis lies in the plane of the drawing. The expressions for the direction cosines and sines of the equilibrium angles between the bonds and the chosen axes are given in Table 8.

The equilibrium values are: bond length \(XY—s\), angle \(YXY—2\vartheta\). The valence-force coordinates are \(q_1, q_2, \gamma\). The valence-optical coordinates are: \(q_1, q_2, \beta_{1\xi}, \beta_{1\eta}, \beta_{1\zeta}, \beta_{2\xi}, \beta_{2\eta}, \beta_{2\zeta}\). Of these, \(\beta_{1\eta}=\beta_{2\eta}=0\), since the vibrations occur in the plane \(\xi\zeta\). Denoting by \(\varphi_1\) and \(\varphi_2\) the changes of the angles between the bonds and the negative direction of the \(\zeta\) axis (Fig. 2), we obtain

\[ \beta_{1\xi}=\varphi_1,\qquad \beta_{1\zeta}=-\varphi_1,\qquad \beta_{2\xi}=-\varphi_2,\qquad \beta_{2\zeta}=-\varphi_2. \tag{7,4} \]

Fig. 2.

The number of coordinates \(\varphi_1, \varphi_2\) is greater by one than the number of valence-force

Table 8

Direction cosines and sines

Bond numbers \(n\)	1	1	1	2	2	2
Coordinate axes \(i\)	\(\xi\)	\(\eta\)	\(\zeta\)	\(\xi\)	\(\eta\)	\(\zeta\)
\(\cos(ni)\)	\(-\sin\vartheta\)	\(0\)	\(-\cos\vartheta\)	\(\sin\vartheta\)	\(0\)	\(\cos\vartheta\)
\(\sin(ni)\)	\(\cos\vartheta\)	\(1\)	\(\sin\vartheta\)	\(\cos\vartheta\)	\(1\)	\(\sin\vartheta\)

angular coordinates \((\gamma)\), since one rotational degree of freedom corresponds to them (rotation about the \(\eta\) axis).

The vibrations of the molecule \(XY_2\) belong to two types of symmetry: they may be symmetric and antisymmetric with respect to the plane \(\xi\eta\) (cf. Fig. 3). We introduce the symmetry coordinates \(q^{(s)}, q^{(as)}, \varphi^{(s)}, \varphi^{(as)}\). In the case of symmetric vibrations we have

Fig. 3.

\[ \varphi_1=\frac{1}{\sqrt{2}}\varphi^{(s)},\qquad \varphi_2=\frac{1}{\sqrt{2}}\varphi^{(s)}, \tag{7,5} \]

in the case of an antisymmetric vibration

\[ \varphi_1=\frac{1}{\sqrt{2}}\varphi^{(as)},\quad \varphi_2=-\frac{1}{\sqrt{2}}\varphi^{(as)} \tag{7,5a} \]

and analogous expressions for \(q_1\) and \(q_2\).

The factor \(1/\sqrt{2}\) has been introduced for normalization. Expressing the symmetry coordinates through the normal coordinates \(Q_1,\ Q_2,\ Q_3\), we have:

\[ \left. \begin{aligned} q^{(s)}&=C_1Q_1+C_2Q_2, &\quad q^{(as)}&=C_3Q_3,\\ \gamma^{(s)}&=D_1Q_1+D_2Q_2, &\quad \gamma^{(as)}&=0,\\ \varphi^{(s)}&=E_1Q_1+E_2Q_2, &\quad \varphi^{(as)}&=E_3Q_3. \end{aligned} \right\} \tag{7,6} \]

The type \(as\) includes rotation about the axis \(\eta\). \(C,\ D,\ E\) are numerical coefficients. We find the relation between them (see Fig. 2)

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

Whence, according to (7,5),

\[ \gamma^{(s)}=\sqrt{2}\varphi^{(s)} \]

and hence:

\[ \left. \begin{aligned} D_{1,2}&=\sqrt{2}\,E_{1,2},\\ E_{1,2}&=\frac{1}{\sqrt{2}}D_{1,2}. \end{aligned} \right\} \tag{7,7} \]

Fig. 4.

The coefficient \(E_3\) is found from the condition that the component of the angular momentum about the axis \(\eta\) be equal to zero (Fig. 4).

\[ M_{\eta}=\mu_a\cdot 2a\cdot \dot{\xi}_a+\mu_b\cdot h\cdot \dot{\xi}_b=0. \tag{7,8} \]

Here

\[ \mu_a=\frac{m_y}{2},\quad \mu_b=\frac{2m_xm_y}{m_x+2m_y}, \]

\(a\) is half the base, \(h\) is the height of the triangle YXY. Expressing the Jacobi coordinates \(\xi_a\) and \(\xi_b\) through the valence-optical coordinates, we obtain

\[ \left. \begin{aligned} \xi_a&=(q_1-q_2)\cos\vartheta-s(\varphi_1-\varphi_2)\sin\vartheta,\\ \xi_b&=\frac{1}{2}(q_1-q_2)\sin\vartheta+\frac{1}{2}s(\varphi_1-\varphi_2)\cos\vartheta \end{aligned} \right\} \tag{7,9} \]

and, substituting into (7,8), we have

\[ s(\dot{\varphi}_2-\dot{\varphi}_1)= \frac{\left(1-\dfrac{m_x}{m_x+2m_y}\right)\operatorname{ctg}\vartheta} {1+\dfrac{m_x}{m_x+2m_y}\operatorname{ctg}^2\vartheta} (\dot{q}_2-\dot{q}_1). \tag{7,8a} \]

Or

\[ E_3=\frac{1}{s}\frac{\left(1-\dfrac{m_x}{m_x+2m_y}\right)\operatorname{ctg}\vartheta} {1+\dfrac{m_x}{m_x+2m_y}\operatorname{ctg}^2\vartheta}\,C_3. \tag{7,10} \]

Thus, the coefficients \(E\) are expressed in terms of \(C\) and \(D\). The latter are found from the normalization condition. The potential energy of vibrations of the molecule \(XY_2\) \(^{1,2,32,45}\)

\[ U=\frac{k_q}{2}(q_1^2+q_2^2)+hq_1q_2+a(q_1+q_2)\gamma+\frac{k_\gamma\gamma^2}{2}, \tag{7,11} \]

on the other hand,

\[ U=\frac{\omega_1^2Q_1^2}{2}+\frac{\omega_2^2Q_2^2}{2}+\frac{\omega_3^2Q_3^2}{2}. \tag{7,12} \]

Substituting (7,6) into (7,11) and comparing with (7,12), we find \(C\) and \(D\) as functions of \(\omega_i\) and of the dynamical coefficients \(k_q, k_\gamma, h, a\). Thus, we find the numerical coefficients \(C, E\) needed for expression (7,3).

Let us pass to the specifically electro-optical part of the problem. We form the derivatives \(b_{ik}\) and \(p_i\) with respect to the symmetry coordinates and sum over all equivalent coordinates, according to (7,3). On the basis of (5,6) and (5,7) we have in our case (\(\sigma\)-bonds)

\[ \left. \begin{aligned} \frac{\partial b_{ik}}{\partial q_n} &=\left(\frac{\partial\alpha_1}{\partial q_n}-\frac{\partial\alpha_2}{\partial q_n}\right)\cos(ni)\cos(nk) +\frac{\partial\alpha_2}{\partial q_n}\delta_{ik},\\ \frac{\partial p_i}{\partial q_n} &=\frac{\partial\mu}{\partial q_n}\cos(ni),\\ \frac{\partial b_{ik}}{\partial \varphi_n} &=(\alpha_1-\alpha_2)\frac{\partial}{\partial\varphi_n}\{\cos(ni)\cos(nk)\},\\ \frac{\partial p_i}{\partial\varphi_n} &=\mu\frac{\partial}{\partial\varphi_n}\cos(ni) \end{aligned} \right\} \tag{7,13} \]

and similar expressions for derivatives with respect to the symmetry coordinates. Using Table 8, we obtain the following table of derivatives (Table 9).

Summing over \(n=1,2\), according to (7,3), we obtain the final expressions for the tensors \(\partial b_i/\partial Q_j\) and the vectors \(\partial p_i/\partial Q_j\).

Type \(s,\ j=1,2\).

\[ \left. \begin{aligned} \frac{\partial b_{\xi\xi}}{\partial Q_j} &=C_j\sqrt{2}\,[(\alpha_1'-\alpha_2')\sin^2\vartheta+\alpha_2'] +E_j\,2\sqrt{2}\,(\alpha_1-\alpha_2)\sin\vartheta\cos\vartheta,\\ \frac{\partial b_{\eta\eta}}{\partial Q_j} &=C_j\sqrt{2}\,\alpha_2',\\ \frac{\partial b_{\zeta\zeta}}{\partial Q_j} &=C_j\sqrt{2}\,[(\alpha_1'-\alpha_2')\sin^2\vartheta+\alpha_2'] -E_j\,2\sqrt{2}\,(\alpha_1-\alpha_2)\sin\vartheta\cos\vartheta. \end{aligned} \right\} \tag{7,14} \]

Composition of the derivatives of polarizability and dipole

Coordinates	Expressions through symmetry coordinates, type \(s\)	Expressions through symmetry coordinates, type \(as\)	Components \(b_{\xi\xi}\)	Components \(b_{\eta\eta}\)
\(q_1\)	\(\dfrac{1}{\sqrt{2}}\,q^{(s)}\)	\(\dfrac{1}{\sqrt{2}}\,q^{(as)}\)	\((a_1'-a_2')\sin^2\vartheta+a_2'\)	\(a_2'\)
\(q_2\)	\(\dfrac{1}{\sqrt{2}}\,q^{(s)}\)	\(-\dfrac{1}{\sqrt{2}}\,q^{(as)}\)	\((a_1'-a_2')\sin^2\vartheta+a_2'\)	\(a_2'\)
\(\beta_{1\xi}=\varphi_1\)	\(\dfrac{1}{\sqrt{2}}\,\varphi^{(s)}\)	\(\dfrac{1}{\sqrt{2}}\,\varphi^{(as)}\)	\(2(a_1-a_2)\sin\vartheta\cos\vartheta\)	\(0\)
\(\beta_{1\xi}=-\varphi_1\)	\(\dfrac{1}{\sqrt{2}}\,\varphi^{(s)}\)	\(-\dfrac{1}{\sqrt{2}}\,\varphi^{(as)}\)	\(0\)	\(0\)
\(\beta_{2\xi}=-\varphi_2\)	\(-\dfrac{1}{\sqrt{2}}\,\varphi^{(s)}\)	\(\dfrac{1}{\sqrt{2}}\,\varphi^{(as)}\)	\(-2(a_1-a_2)\sin\vartheta\cos\vartheta\)	\(0\)
\(\beta_{2\xi}=-\varphi_2\)	\(-\dfrac{1}{\sqrt{2}}\,\varphi^{(s)}\)	\(\dfrac{1}{\sqrt{2}}\,\varphi^{(as)}\)	\(0\)	\(0\)
Derivatives with respect to symmetry coordinates	with respect to \(q^{(s)}\)	—	\(\sqrt{2}\big[(a_1'-a_2')\sin^2\vartheta+a_2'\big]\)	\(\sqrt{2}\,a_2'\)
Derivatives with respect to symmetry coordinates	with respect to \(\varphi^{(s)}\)	—	\(2\sqrt{2}(a_1-a_2)\sin\vartheta\cos\vartheta\)	\(0\)
Derivatives with respect to symmetry coordinates	—	with respect to \(q^{(as)}\)	\(0\)	\(0\)
Derivatives with respect to symmetry coordinates	—	with respect to \(\varphi^{(as)}\)	\(0\)	\(0\)

The off-diagonal terms of the tensor are zero.

\[ \frac{\partial p_\xi}{\partial Q_j} = \frac{\partial p_\eta}{\partial Q_j} =0,\qquad \frac{\partial p_\zeta}{\partial Q_j} = -C_j\sqrt{2}\,\mu'\cos\vartheta + E_j\sqrt{2}\,\mu\sin\vartheta . \tag{7,14a} \]

Type \(as,\ j=3\).

All components of the tensor are zero, with the exception of

\[ \frac{\partial b_{\xi\eta}}{\partial Q_3} = C_3\sqrt{2}(a_1'-a_2')\sin\vartheta\cos\vartheta + E_3\sqrt{2}(a_1-a_2)(\cos^2\vartheta-\sin^2\vartheta), \tag{7,15} \]

\[ \frac{\partial p_\xi}{\partial Q_3} = -C_3\sqrt{2}\,\mu'\sin\vartheta - E_3\sqrt{2}\,\mu\cos\vartheta,\qquad \frac{\partial p_\eta}{\partial Q_3} = \frac{\partial p_\zeta}{\partial Q_3} =0. \tag{7,15a} \]

moments with respect to symmetry coordinates for \(XY_2\)

Table 9

\(b_{zz}\)	\(b_{\eta\eta}\)	\(b_{\xi\xi}\)	\(b_{\eta\xi}\)	\(p_z\)	\(p_\eta\)	\(p_\xi\)
\((a'_1-a'_2)\cos^2\vartheta+a'_2\)	\(0\)	\((a'_1-a'_2)\sin\vartheta\cos\vartheta\)	\(0\)	\(-\mu'\sin\vartheta\)	\(0\)	\(-\mu'\cos\vartheta\)
\((a'_1-a'_2)\cos^2\vartheta+a'_2\)	\(0\)	\(-\,(a'_1-a'_2)\sin\vartheta\cos\vartheta\)	\(0\)	\(\mu'\sin\vartheta\)	\(0\)	\(-\mu'\cos\vartheta\)
\(0\)	\(0\)	\((a_1-a_2)\cos^2\vartheta\)	\(0\)	\(-\mu\cos\vartheta\)	\(0\)	\(0\)
\(2(a_1-a_2)\sin\vartheta\cos\vartheta\)	\(0\)	\((a_1-a_2)\sin^2\vartheta\)	\(0\)	\(0\)	\(0\)	\(-\mu\sin\vartheta\)
\(0\)	\(0\)	\((a_1-a_2)\cos^2\vartheta\)	\(0\)	\(-\mu\cos\vartheta\)	\(0\)	\(0\)
\(2(a_1-a_2)\sin\vartheta\cos\vartheta\)	\(0\)	\((a_1-a_2)\sin^2\vartheta\)	\(0\)	\(0\)	\(0\)	\(-\mu\sin\vartheta\)
\(\sqrt{2}\big[(a'_1-a'_2)\cos^2\vartheta+a'_2\big]\)	\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(-\sqrt{2}\,\mu\cos\vartheta\)
\(-2\sqrt{2}(a_1-a_2)\sin\vartheta\cos\vartheta\)	\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(\sqrt{2}\,\mu\sin\vartheta\)
\(0\)	\(0\)	\(\sqrt{2}(a'_1-a'_2)\sin\vartheta\cos\vartheta\)	\(0\)	\(-\sqrt{2}\,\mu'\sin\vartheta\)	\(0\)	\(0\)
\(0\)	\(0\)	\(\sqrt{2}(a_1-a_2)\big[\cos^2\vartheta-\sin^2\vartheta\big]\)	\(0\)	\(-\sqrt{2}\,\mu\cos\vartheta\)	\(0\)	\(0\)

Thus the problem is solved in the zero approximation of the additive scheme. At the same time, of course, knowledge of the exact form of the vibrations is necessary in order to find the coefficients \(C\) and \(E\). Modern methods provide us with this possibility\(^{1,2,32}\). The parameters \(a'_1\), \(a'_2\), \(\mu'\) are found, as was already indicated, semiempirically, with the aid of the values \(a_1-a_2\), \(\mu\), obtained by other methods. In the works of M. Wolkenstein and M. Elyashevich\(^{32,34,35}\) the tensors \(\dfrac{\partial b_{ik}}{\partial Q}\) and the vectors \(\dfrac{\partial p_i}{\partial Q}\) are also given for the molecules \(XY_2Z_2\), \(XY_3Z\), \(XY_4\).

In the theory of the first approximation the number of parameters increases, since, alongside changes in the polarizability and dipole moment,

of a given bond when its length changes, it is necessary, because of the departure from additivity, to take into account the derivatives \(\dfrac{\partial \alpha_{nm}}{\partial q_t}\) and \(\dfrac{\partial \mu_n}{\partial q_t}\) for \(t \ne n\), as well as derivatives with respect to the valence angles. In the case \(XY_2\) the following parameters are added:

\[ \frac{\partial \alpha_{11}}{\partial q_2} = \frac{\partial \alpha_{21}}{\partial q_1}, \quad \frac{\partial \alpha_{12}}{\partial q_2} = \frac{\partial \alpha_{22}}{\partial q_1}, \quad \frac{\partial \mu_1}{\partial q_2} = \frac{\partial \mu_2}{\partial q_1} \quad \text{and} \quad \frac{\partial \alpha_1}{\partial \gamma}, \frac{\partial \alpha_2}{\partial \gamma}, \frac{\partial \mu_1}{\partial \gamma}, \frac{\partial \mu_2}{\partial \gamma}. \]

The corresponding method of calculation was developed by M. A. Elyashevich\(^{32,46}\).

The selection rules for \(A\) and \(B\), derived by Pláček (p. 58), are observed automatically in the theory being presented. This is a natural result. In the case of highly symmetric molecules, the tensors \(\dfrac{\partial b_{ik}}{\partial Q}\) and the vectors \(\dfrac{\partial p_i}{\partial Q}\) for jointly degenerate vibrations are equal in absolute value. Therefore the corresponding intensity must simply be multiplied by the degree of degeneracy.

§ 8. RESULTS OF THEORETICAL CALCULATIONS

The method described was applied to the calculation of intensities and polarizations in the Raman spectra of chloro- and bromo-derivatives of methane\(^{35}\). For these compounds there are comparatively reliable experimental data, which cannot be said of the overwhelming majority of other molecules. Experimental work in this field is only beginning. The group of molecules \( \mathrm{CCl_4}, \mathrm{CHCl_3}, \mathrm{CH_2Cl_2}, \mathrm{CH_3Cl}, \mathrm{CH_4}, \mathrm{CH_3Br}, \mathrm{CH_2Br_2}, \mathrm{CHBr_3}, \mathrm{CBr_4} \), according to the additive scheme, is constructed from three types of bonds: C—Cl, C—Br, C—H. These are \(\sigma\)-bonds. Altogether, therefore, six parameters \(\dfrac{\partial \alpha_m}{\partial q}\)—two for each bond—and three quantities \(a_1 — a_2\) are needed. The latter quantities were obtained by Denbigh (§ 4). To find the six derivatives, use is made of the six most reliable values of the degrees of depolarization of certain valence vibrations, marked in Table 10 in bold type. The amplitudes and frequencies of the vibrations of the indicated molecules were accurately determined by B. Stepanov\(^{47}\). The results are given in Table 10.

Using the values \(a_1 — a_2\) obtained by Denbigh, and the values \(\rho\) underlined in the table, we obtain:

For C—H
\[ a'_1 = 1.28 \cdot 10^{-16}\ \mathrm{cm}^2, \qquad a'_2 = 0.31 \cdot 10^{-16}\ \mathrm{cm}^2 \]

For C—Cl
\[ a'_1 = 2.82 \cdot 10^{-16}\ \mathrm{cm}^2, \qquad a'_2 = 0.68 \cdot 10^{-16}\ \mathrm{cm}^2 \]

For C—Br
\[ a'_1 = 3.23 \cdot 10^{-16}\ \mathrm{cm}^2, \qquad a'_2 = 0.83 \cdot 10^{-16}\ \mathrm{cm}^2 \]

Table 10

Calculated and measured intensities and depolarizations
in the Raman spectra of halomethanes

Molecule	Frequency in cm\(^{-1}\)	Form of vibration	Intensity calc.	Intensity meas.	Depolarization calc.	Depolarization meas.
CCl\(_4\)	459	\(s\nu\ (C—Cl)\)	(10)	(10)	0	—
CCl\(_4\)	217	\(as_2\delta\ (Cl—C—Cl)\)	13,2	9	6/7	—
CCl\(_4\)	775	\(as_3\nu\ (C—Cl)\)	2,3	5	6/7	—
CCl\(_4\)	313	\(as_3\delta\ (Cl—C—Cl)\)	9,2	10	6/7	—
CHCl\(_3\)	667	\(s\nu\ (C—Cl)\)	(10)	(10)	0,04	0,10
CHCl\(_3\)	3020	\(s\nu\ (C—H)\)	2,1	3	0,22	0,22
CHCl\(_3\)	366	\(s\delta\ (Cl—C—Cl)\)	5,6	5	0,41	0,21
CHCl\(_3\)	761	\(as_2\nu\ (C—Cl)\)	18,8?	4	(6/7)	0,78
CHCl\(_3\)	1215	\(as_2\delta\ (H—C—Cl)\)	1,4	0	(6/7)	0,85
CHCl\(_3\)	262	\(as_2\delta\ (Cl—C—Cl)\)	7,4	5	(6/7)	0,77
CH\(_2\)Cl\(_2\)	700	\(s\nu\ (C—Cl)\)	(10)	(10)	0,09	0,09
CH\(_2\)Cl\(_2\)	2986	\(s\nu\ (C—H)\)	2,5	3,1	0,11	0,27
CH\(_2\)Cl\(_2\)	283	\(s\delta\ (Cl—C—Cl)\)	6,3	6,3	0,77	0,43
CH\(_2\)Cl\(_2\)	1418	\(s\delta\ (H—C—H)\)	0,8	0,9	0,75	0,88
CH\(_2\)Cl\(_2\)	3046	\(as\nu\ (C—H)\)	1,0	—	(6/7)	\(D\)
CH\(_2\)Cl\(_2\)	896	\(as\delta\ (H—C—Cl)\)	0,4	—	(6/7)	—
CH\(_2\)Cl\(_2\)	736	\(as\nu\ (C—Cl)\)	2,3	2,0	(6/7)	0,79
CH\(_2\)Cl\(_2\)	1266	\(as\delta\ (H—C—Cl)\)	0,1	—	(6/7)	—
CH\(_2\)Cl\(_2\)	1149	\(as\delta\ (H—C—Cl)\)	0,6	0,5	(6/7)	0,89
CH\(_3\)Cl	712	\(s\nu\ (C—Cl)\)	(10)	(10)	0,20	0,20
CH\(_3\)Cl	2967	\(s\nu\ (C—H)\)	4,8	10	0,05	0,05
CH\(_3\)Cl	1355	\(s\delta\ (H—C—H)\)	0,3	0	0,52	\(P\)
CH\(_3\)Cl	3038	\(as_2\nu\ (C—H)\)	2,4	2	(6/7)	0,86
CH\(_3\)Cl	1020	\(as_2\delta\ (H—C—Cl)\)	1,6	0	(6/7)	—
CH\(_3\)Cl	1460	\(as_2\delta\ (H—C—H)\)	2,2	2	(6/7)	\(\sim 1\)
CH\(_4\)	2914	\(s\nu\ (C—H)\)	(10)	(10)	(0)	—
CH\(_4\)	1520	\(as_2\delta\ (H—C—H)\)	3,1	0	(6/7)	—
CH\(_4\)	3022	\(as_3\nu\ (C—H)\)	9,0	2	(6/7)	—
CH\(_4\)	1304	\(as_3\delta\ (H—C—H)\)	2,1	0	(6/7)	—
CH\(_3\)Br	610	\(s\nu\ (C—Br)\)	(10)	(10)	0,20	0,20
CH\(_3\)Br	2973	\(s\nu\ (C—H)\)	3,9	7,5	0,05	0,10
CH\(_3\)Br	1306	\(s\delta\ (H—C—H)\)	0,1	0,9	0,24	\(P\)
CH\(_3\)Br	3061	\(as_2\nu\ (C—H)\)	2,0	1,2	(6/7)	0,86
CH\(_3\)Br	957	\(as_2\delta\ (H—C—Br)\)	1,6	0	(6/7)	—
CH\(_3\)Br	1450	\(as_2\delta\ (H—C—H)\)	0,9	0	(6/7)	\(D\)

Table 10 (continued)

Molecule	Frequency in cm\(^{-1}\)	Form of vibration	Intensity, calc.	Intensity, meas.	Depolarization, calc.	Depolarization, meas.
CH\(_2\)Br\(_2\)	577	\(s\nu(\mathrm{C—Br})\)	(10)	(10)	0,11	0,11
CH\(_2\)Br\(_2\)	2988	\(s\nu(\mathrm{C—H})\)	2,0	1,6	0,11	0,10
CH\(_2\)Br\(_2\)	174	\(s\delta(\mathrm{Br—C—Br})\)	12,1	7,6	0,77	0,37
CH\(_2\)Br\(_2\)	1387	\(s\delta(\mathrm{H—C—H})\)	0,6	0,8	0,44	0,53
CH\(_2\)Br\(_2\)	3054	\(as\nu(\mathrm{C—H})\)	0,8	0,1	(6/7)	0,85
CH\(_2\)Br\(_2\)	806	\(as\delta(\mathrm{H—C—Br})\)	0,7	—	(6/7)	—
CH\(_2\)Br\(_2\)	638	\(as\nu(\mathrm{C—Br})\)	2,8	2,4	(6/7)	0,87
CH\(_2\)Br\(_2\)	1181	\(as\delta(\mathrm{H—C—Br})\)	0,1	0,1	(6/7)	0,82
CH\(_2\)Br\(_2\)	1090	\(as\delta(\mathrm{H—C—Br})\)	0,5	0,2	(6/7)	0,92
CHBr\(_3\)	539	\(s\nu(\mathrm{C—Br})\)	(10)	(10)	(0,10)	0,09
CHBr\(_3\)	3021	\(s\nu(\mathrm{C—H})\)	1,6	5,7	0,22	0,25
CHBr\(_3\)	222	\(s\delta(\mathrm{Br—C—Br})\)	7,5	11,4	0,39	0,18
CHBr\(_3\)	655	\(as\nu(\mathrm{C—Br})\)	24,0?	7,2	(6/7)	0,58
CHBr\(_3\)	1144	\(as\delta^{2}(\mathrm{H—C—Br})\)	0,8	2,9	(6/7)	0,81
CHBr\(_3\)	154	\(as\delta^{2}(\mathrm{Br—C—Br})\)	9,8	8,6	(6/7)	0,79
CBr\(_4\)	265	\(s\nu(\mathrm{C—Br})\)	(10)	(10)	(0)	—
CBr\(_4\)	125	\(as\delta(\mathrm{Br—C—Br})\)	20,3	10	(6/7)	—
CBr\(_4\)	667	\(as^{3}\nu(\mathrm{C—Br})\)	0,8	2	(6/7)	—
CBr\(_4\)	183	\(as^{3}\delta(\mathrm{Br—C—Br})\)	14,6	8	(6/7)	—

\(s\) — symmetric, \(as\) — antisymmetric, \(\nu\) — predominantly stretching, \(\delta\) — predominantly deformation vibrations. The indices 2 and 3 are the degrees of expression.

On the physical meaning of these quantities, see below, § 10. With the aid of these values the remaining intensities and polarizations were calculated.

Table 10 shows that, already in the zeroth approximation, the theory set forth makes it possible to establish the correct distribution of intensities and polarizations in the spectrum, i.e., to find their dependence on the form of the vibrations. The table gives relative intensities separately for each molecule; the theory makes it possible to calculate them on a single scale\(^ {35}\), but we do not have the corresponding experimental data for verification. The discrepancies between the measured and calculated values of \(I\) and \(\rho\) are explained, for the most part, by the inadequacy of the zeroth approximation of the additive scheme. Thus, according to the rule of § 6, in the zeroth approximation for fully symmetric deformation vibrations we have \(\rho = 0{,}86\) \((= {}^{6}/_{7})\). In reality \(\rho < 0{,}86\). Hence one obtains theoretical values of \(\rho\) that are too high for the frequencies CHCl\(_3\) 366, CH\(_2\)Cl\(_2\) 283, CHBr\(_3\) 222, CH\(_2\)Br\(_2\) 174. Probably the anomalously large ...

the theoretical intensities of the antisymmetric valence vibrations: CHCl\(_3\) 761 and CHBr\(_3\) 655. For further details see the original paper \(^{35}\).

It should be noted that the greater part of the experimental data on intensities was obtained by means of a rough visual estimate. In addition, the calculations refer to integral intensities, whereas the measurements usually refer to differential intensities. Therefore it is difficult to speak of exact quantitative agreement.

In the tables, the values of \(\rho\) that are obtained already from the symmetry properties alone, according to Placzek’s rules (§ 2), independently of the proposed theory, are enclosed in parentheses.

With the aid of the valence-optical scheme, the intensities and polarizations in the Raman and infrared spectra of the deutero-methanes CH\(_3\)D, CH\(_2\)D\(_2\), CHD\(_3\), CD\(_4\) \(^{48}\) were also calculated; however, these calculations cannot yet be fully verified because of the insufficiency of the experimental data.

Recently Gertrude Nordheim and Gerta Spöner \(^{49}\) published a paper in which the valence-optical scheme is applied to the characterization of C—Cl vibrations in dichlorobenzenes. The problem set by the authors did not amount to a rigorous calculation, but only to an approximate estimate useful in the analysis of the spectrum. Therefore the question of the form of the vibrations is not considered, and the molecules of \(P\)-, \(M\)-, and \(O\)-dichlorobenzene are treated as two C—Cl bonds situated, respectively, at angles \(2\vartheta = 180\), 120, and \(60^\circ\). To characterize the ellipsoid \(\dfrac{d a_m}{d q}\) of the C—Cl bond, the value \(\rho = 0.13\) for the C—Cl frequency in monochlorobenzene is used.

Nordheim and Spöner write the tensors \(a'\) for each of the C—Cl bonds in a form not reduced to the principal axes, as \(a'_{ik}\), where \(i, k = x, y, z\) (the axes are referred to the bond). In the molecule, fixed axes \(\xi, \eta, \zeta\) are established; the \(\xi\)-axis is drawn along the bisector of the angle formed by the two C—Cl bonds, while the \(\eta\)-axis lies in the plane of the molecule (Fig. 5). The polarization components contributed by each of the bonds are considered additive. We have

Fig. 5.

\[ a'_{\xi \eta}=\sum_{xy} a'_{xy}\cos(x,\xi)\cos(y,\eta). \tag{8,1} \]

The angle of rotation for the first bond is \(\vartheta\), and for the second, \(-\vartheta\). The tensors \(b'\) for purely valence symmetric and antisymmetric vibrations

have the form

\[ \begin{aligned} b'(s)&=a_{\xi\eta}^{(1)}(\vartheta)+a_{\xi\eta}^{(2)}(-\vartheta),\\ b'(as)&=a_{\xi\eta}^{(1)}(\vartheta)-a_{\xi\eta}^{(2)}(-\vartheta). \end{aligned} \tag{8,2} \]

Here the indices (1) and (2) number the bonds. We obtain

\[ \left. \begin{aligned} b'_{\xi\xi}(s)&=2\left(a'_{xx}\cos^2\vartheta+a'_{yy}\sin^2\vartheta\right),\\ b'_{\eta\eta}(s)&=2\left(a'_{xx}\sin^2\vartheta+a'_{yy}\cos^2\vartheta\right),\\ b'_{zz}(s)&=2a'_{zz}, \end{aligned} \right\} \tag{8,3} \]

the remaining components are zeros, and

\[ \left. \begin{aligned} b'_{\xi\xi}(as)&=b'_{yy}(as)=b'_{\xi z}(as)=b'_{zz}(as)=b'_{z\eta}(as)=0,\\ b'_{\xi\eta}(as)&=-2\left(a'_{xx}-a'_{yy}\right)\sin\vartheta\cos\vartheta . \end{aligned} \right\} \tag{8,4} \]

These expressions should be compared with (7,14) and (7,15). Obviously, in our former notation

\[ a'_{xx}=a'_1 \quad \text{and} \quad a'_{yy}=a'_2 . \]

We give a table of the calculated intensities and polarizations and of the measured polarizations:

Table 11

Valence vibrations of dichlorobenzenes

Molecule	\(I^{(s)}\)	\(I^{(as)}\)	\(\rho^{(s)}\)	\(\rho^{(as)}\)	\(\rho^{(s)}\) experiment
Dichlorobenzene
Para: \(\vartheta=90^\circ\)	\(4\times 40\)	0	0.13	—	0.06
Meta: \(\vartheta=60^\circ\)	\(4\times 34.4\)	\(4\times 5.65\)	0.06	\(6/7\)	0.25
Ortho: \(\vartheta=30^\circ\)	\(4\times 34.4\)	\(4\times 5.65\)	0.06	\(6/7\)	0.35
Monochlorobenzene	40	—	0.13	—	0.13

The intensity for \(\mathrm{C_6H_5Cl}\) has conventionally been taken as 40. The remaining quantities are referred to it. Unfortunately, there are no experimental data on the intensities of these four molecules expressed on a single scale.

With the aid of similar simple methods, the intensities of planar and non-planar deformation vibrations have been calculated. Their \(\rho=6/7\), in accordance with the rule of § 6, since it is obvious that Nordheim and Shponer work in the same zero approximation, disregarding the true form of the vibrations of the molecule. In this connection it is interesting to note that the results of Table 11 can be obtained directly with the aid of formulas (6,3)—(6,5). Indeed, from \(\rho=0.13\) for monochlorobenzene we have \(A_1^2=7.83\,B^2\). Substituting in (6,5) \(N=2\) and \(\vartheta\) correspond-

respectively equal to 180, 120, and 60°, we obtain for the \(P\)-, \(M\)-, and \(O\)-molecules \(\rho^{(s)}=0.13,\ 0.06,\) and \(0.06\). The intensities obtained by Nordheim and Shponer are calculated directly from the formula (cf. (6,3) and (6,4))

\[ I \sim N^2\left\{54A_1^2+13B_1^2\left(1-\frac{N-1}{2N}\,3\sin^2\vartheta\right)\right\}. \tag{8,5} \]

Thus, the simplified method of Nordheim and Shponer gives inaccurate results, not going beyond what can be obtained with the aid of the considerations developed in § 5. The applicability of the method is limited, since the fundamental question of the relation to the form of the vibrations remains unresolved, and in any somewhat complicated cases the method of Nordheim and Shponer is altogether unsuitable.

Finally, let us give one more example of a calculation of the electro-optical properties of a spectrum, applicable only to one special case, but distinguished by very high accuracy.

Lord and Teller[^50] calculated the intensity ratios of the Raman lines of certain vibrations of benzene and hexadeuterobenzene. Not having a general method for solving the problem, these authors restricted themselves to comparing intensities for such vibrations whose forms admit a distinct separation into motions of the C and H (respectively D) atoms. Such is the doubly degenerate deformation vibration \(E_g^{-}\), in which the carbon and hydrogen six-membered rings rotate about a common axis lying in the plane of the ring, and the totally symmetric purely valence vibrations \(A_{1g}\)—pulsations of the carbon and hydrogen rings. Starting from the known intensities and degrees of depolarization of these vibrations in benzene, Lord and Teller calculated with great accuracy the intensities and polarizations for the same vibrations of hexadeuterobenzene. These results are very interesting, but, as has already been said, the method has a very narrow range of application; the general question of the dependence of electro-optical properties on the form of the vibrations remains open.

Up to now we have spoken exclusively about Raman spectra. Calculations in the region of infrared spectra are hampered by the lack of experimental material. Indeed, reliable measurements of the intensities of infrared bands are available only for a small number of simple molecules: hydrogen halides, CO, CO\(_2\). Meanwhile, calculations of the quantities \(\dfrac{\partial\mu}{\partial q}\)—the effective charges of the bonds—are very interesting for the study of the electrical properties of molecules. Attempts to determine \(\mu'\) from absolute intensity measurements have been made only in the cases just mentioned (see the following paragraph). The valence-optical scheme evidently makes it possible to calculate \(\mu'\) not from absolute, but from relative determinations of intensity, which are much simpler. Here, of course, the values of the bond moments \(\mu\) themselves are necessary. Starting from the value of \(\mu\) for the C—H bond, equal to

\(0.4 \cdot 10^{-18}\) CGSE. M. V. Vol'kenshtein\({}^{48}\) calculated, using rough data on the intensities of the infrared spectra of deuteriomethanes \(\mu'\) for the C—H bond, and found the value \(\mu' = \pm 0.3 \cdot 10^{-10}\) CGSE (the question of the sign remains open, since the effective charge enters into the expressions for intensities squared). Along with intensities in infrared spectra, one may use, for finding effective charges, the value of the atomic polarization\({}^{50}\), which in the case of diatomic molecules is equal to

\[ P_A = \frac{\left(\dfrac{\partial \mu}{\partial a}\right)_0^2 N'} {3\pi m \nu^2}, \tag{8,6} \]

as well as the dispersion in the infrared region. The latter method is applied in the cited work\({}^{48}\). We can write for the refractive index of a gas

\[ n^2 - 1 = \sum_i \frac{K_i}{\nu_i^2 - \nu^2}, \tag{8,7} \]

where \(\nu_i\) are the natural frequencies of oscillations. Let us express the coefficients \(K_i\) through \(\dfrac{\partial p}{\partial Q_i}\). In the simplest case of oscillation of one bond, the classical equation of motion of an oscillator in a field has the form

\[ m\ddot q + kq = \varepsilon E_0 e^{2\pi i\nu t}. \tag{8,8} \]

Passing to the normal coordinate

\[ Q = \frac{1}{C} q = Q_0 e^{2\pi i\nu_0 t}, \tag{8,9} \]

we obtain

\[ - C4\pi^2 \nu^2 Q_0 + C4\pi^2 \nu_0^2 Q_0 = \frac{\varepsilon}{m} E_0. \tag{8,10} \]

But

\[ \varepsilon = \frac{\partial p}{\partial q} = \frac{1}{C}\frac{\partial p}{\partial Q}, \tag{8,11} \]

whence the induced moment \(\mu_i\)

\[ \mu_i = \frac{\partial p}{\partial Q} Q_0 = \frac{1}{4\pi^2 m C^2} \frac{(\partial p/\partial Q)^2}{\nu_0^2 - \nu^2} E_0, \tag{8,12} \]

and, according to the Lorentz–Lorenz formula and (8,12), the trace of the polarizability is

\[ \alpha = \frac{M}{4\pi N_A d}(n^2 - 1) = \frac{1}{4\pi^2 m C^2} \frac{\left(\dfrac{\partial p}{\partial Q}\right)^2}{\nu_0^2 - \nu^2}. \tag{8,13} \]

Consequently,

\[ K = \frac{N_A d}{\pi m M C^2} \left(\frac{\partial p}{\partial Q}\right)^2, \tag{8,14} \]

\(d\) is the density, \(N_A\) is Avogadro’s number, \(M\) is the molecular weight, \(m\) is the reduced mass.

Rollefson and Havens\(^{51}\) carried out precise measurements of the dispersion of gaseous methane in the infrared region. Using the value of \(K\) obtained by these authors, we again find\(^{48}\) for the C—H bond

\[ \frac{\partial \mu}{\partial q}=\pm 0.3\cdot 10^{-10}CGSE=\pm 0.06e \tag{8,15} \]

in complete agreement with the results of the other method of determination; \(e\) is the charge of the electron.

§ 9. VALENCE-OPTICAL SCHEME, ANHARMONICITY AND FERMI RESONANCE

We have considered questions relating to the fundamental frequencies in vibrational spectra. Let us now turn to the electro-optical properties of overtones and combination frequencies. They are especially important in infrared spectra, where the region of the fundamental frequencies is difficult for experimental investigation, and also in Raman spectra in the case of the so-called Fermi resonance. As is known, the appearance of combination frequencies and overtones is possible only owing to anharmonicity. It is useful to distinguish mechanical and electrical anharmonicity\(^{53}\). Mechanical anharmonicity means a deviation of the force law from the linear one and, correspondingly, of the law of potential energy from the quadratic one. Electrical anharmonicity means the nonlinearity of the dependence of the electric dipole moment or polarizability on the normal coordinates—the appearance in the corresponding series expansions of terms

\[ \left(\frac{\partial^2 p_l}{\partial Q^2}\right)_0 \quad \text{and} \quad \left(\frac{\partial^2 b_{ik}}{\partial Q^2}\right)_0, \]

as well as mixed and higher derivatives [cf. (2,6)]. The numerical values of overtone frequencies are determined by mechanical anharmonicity, whereas electro-optical quantities are determined by mechanical and electrical anharmonicity.

Strictly speaking, in the presence of mechanical anharmonicity the concepts of normal coordinates and normal vibrations lose their meaning. Nevertheless, for small deviations from the harmonic law it is possible, for purposes of calculation, to use normal coordinates in the zeroth approximation, introducing anharmonic corrections as a perturbation. Let us consider the intensity of the first overtone of a certain normal vibration in the Raman spectrum. We have the approximate classical equation of motion:

\[ \ddot Q+\omega^2Q=xQ^2, \tag{9,1} \]

where \(x\) is the coefficient of mechanical anharmonicity.

The solution of (9.1) may be found in the form

\[ Q=Q_0\cos\omega t+y\cos2\omega t. \tag{9.2} \]

Restricting ourselves, in the expansion of the polarizability \(b\) in a series in \(Q\), to the first three terms, we have

\[ b_{ik}=b^0_{ik}+b'_{ik}Q+\frac{1}{2}b''_{ik}Q^2, \tag{9.3} \]

where \(b'_{ik}=\left(\dfrac{\partial b_{ik}}{\partial Q}\right)_0\), and \(b''_{ik}=\left(\dfrac{\partial^2 b_{ik}}{\partial Q^2}\right)_0\) is the electrical anharmonicity.

Substituting (9.2) into (9.3), we obtain

\[ \begin{aligned} b_{ik}={}& b^0_{ik}+\frac{1}{4}b''_{ik}Q_0^2+\frac{1}{4}b''_{ik}y^2+{}\\ &+\left(b'_{ik}Q_0+\frac{1}{2}b''_{ik}yQ_0\right)\cos\omega t+{}\\ &+\left(b'_{ik}y+\frac{1}{2}b''_{ik}Q_0^2\right)\cos2\omega t+{}\\ &+\frac{1}{4}b''_{ik}yQ_0\cos3\omega t+{}\\ &+\frac{1}{4}b''_{ik}y^2\cos4\omega t. \end{aligned} \tag{9.4} \]

The intensity of the overtone with frequency \(2\omega\) is determined by the sum of the terms: \(b'_{ik}y\), which depends on the mechanical anharmonicity, and \(\dfrac{1}{2}b''_{ik}Q_0^2\), which depends on the electrical anharmonicity. Let us derive an expression for \(b''_{ik}\) by means of the valence-optical scheme \(^{54}\). According to (6.3) and (6.4)

\[ \begin{aligned} b''_{ik}={}&\left(\frac{\partial^2 b_{ik}}{\partial Q^2}\right)_0 =\left(\frac{\partial^2 b_{ik}}{\partial Q^2}\right)_{\gamma\gamma} +2\left(\frac{\partial^2 b_{ik}}{\partial Q^2}\right)_{\gamma,q} +\left(\frac{\partial^2 b_{ik}}{\partial Q^2}\right)_{qq} ={}\\ ={}&\sum_{nm}\frac{\partial^2 a_{nm}}{\partial q_n^2}\cos(nmi)\cos(nmk) \left(\frac{\partial q_n}{\partial Q}\right)^2 +\sum_{nm}\frac{\partial a_{nm}}{\partial q_n}\cos(nmi)\cos(nmk)\frac{\partial^2 q_n}{\partial Q^2}+{}\\ &+2\sum_{nms}\frac{\partial a_{nm}}{\partial q_n} \frac{\partial}{\partial\gamma_s}\,[\cos(nmi)\cos(nmk)] \frac{\partial q_n}{\partial Q}\frac{\partial\gamma_s}{\partial Q}+{}\\ &+\sum_{nms}a_{nm}\frac{\partial^2}{\partial\gamma_s^2}\,[\cos(nmi)\cos(nmk)] \left(\frac{\partial\gamma_s}{\partial Q}\right)^2+{}\\ &+\sum_{nms}a_{nm}\frac{\partial}{\partial\gamma_s}\,[\cos(nmi)\cos(nmk)] \frac{\partial^2\gamma_s}{\partial Q^2}. \end{aligned} \tag{9.5} \]

In the case of small vibrations the natural coordinates \(\gamma_s\) and \(q_n\) are linear functions of \(Q\), and, consequently, the second and fifth terms on the right-hand side of expression (9.5) are zero. However, for vibrations with large amplitudes—in the case of overtones—this approximation is insufficient, and in the general case \(q\) and \(\gamma\) may be related to \(Q\) nonlinearly.

The electrical anharmonicity of the molecule is characterized, in addition, by the term containing \(\dfrac{\partial^2 a_{nm}}{\partial q_n^2}\) or \(\dfrac{\partial^2 \mu_n}{\partial q_n^2}\) (only this term is available

in the case of a diatomic molecule). Finally, as is seen from expression (9.5), deformation vibrations of polyatomic molecules occupy a special position. Here, for the appearance of overtones, neither electrical anharmonicity of individual bonds nor a nonlinear dependence of \(q\) and \(\gamma\) on \(Q\) is required. Overtones always arise when they are permitted by the selection rules, since in these cases the fourth term on the right-hand side of (9.5) is nonzero.

The determination of the coefficients of mechanical anharmonicity \(x\) is possible only with the aid of quantum-mechanical calculations, since the usual classical prescription for the calculation gives overtone frequencies exactly equal to multiples of the frequencies of the fundamental vibrations. In reality they differ from the latter, and only on these differences can the calculation of \(x\) be based. It is clear that only by knowing the coefficients \(x\) can one proceed to the determination of the coefficients of electrical anharmonicity. Effective calculations are at present possible only in cases of Fermi resonance, which, however, are of especially great practical significance.

As Fermi showed\(^{55}\), when the frequency of a combination tone or of an overtone of one vibration accidentally coincides with the fundamental frequency of another vibration, owing to mechanical anharmonicity an interaction of the vibrations and a splitting of the corresponding levels may occur. Examples of Fermi-resonance cases are numerous; among them the most striking are \(CO_2^{55,56}\), \(CS_2^{57}\), \(CCl_4^{58}\), dimethylacetylene\(^{59}\), molecules containing the groups \(CH_2\), \(CH_3^{60}\), etc. Placzek\(^{4}\) derived formulas for the intensities in cases of triatomic linear symmetric molecules \((CO_2, CS_2)\), in which the frequency of the valence totally symmetric vibration is close to twice the frequency of the deformation vibration. The intensities of the components of the resulting doublets are determined, according to formula (2.12), by the tensors

\[ \left(b^{00}_{10}\right)_{ik} = \frac{1}{\sqrt{2a}} \left\{ \sqrt{a+|\Delta|}\,\frac{\partial b_{ik}}{\partial Q_1}Q_{10} \pm \sqrt{a-|\Delta|}\,\frac{\sqrt{2}}{2}\, \frac{\partial^2 b_{ik}}{\partial Q_2^2}Q_{20}^{2} \right\}, \]

\[ \left(b^{00}_{02}\right)_{ik} = \frac{1}{\sqrt{2a}} \left\{ \mp \sqrt{a-|\Delta|}\,\frac{\partial b_{ik}}{\partial Q_1}Q_{10} + \sqrt{a+|\Delta|}\,\frac{\sqrt{2}}{2}\, \frac{\partial^2 b_{ik}}{\partial Q_2^2}Q_{20}^{2} \right\}. \tag{9,6} \]

Here \(\Delta=\nu_1-2\nu_2\), \(a=\sqrt{\Delta^2+16x^2}\), where \(x\) is a quantity proportional to the coefficient of mechanical anharmonicity appearing in the expression for the potential energy in the term \(Q_1Q_2^2\). The upper signs correspond to the case of identical signs \((CO_2)\), the lower—to different signs of \(x\) and \(\Delta\) \((CS_2)\). For doublet components associated with transitions not from the zero but from the first excited vibrational level \(Q_2\), we have:

\[ \left(b^{01}_{11}\right)_{ik} = \frac{1}{\sqrt{2a_1}} \left\{ \sqrt{a_1+|\Delta|}\,\frac{\partial b_{ik}}{\partial Q_1}Q_{10} \pm \sqrt{a_1-|\Delta|}\,\frac{\sqrt{6}}{2}\, \frac{\partial^2 b_{ik}}{\partial Q_2^2}Q_2^{2} \right\}, \]

\[ \left(b^{01}_{03}\right)_{ik} = \frac{1}{\sqrt{2a_1}} \left\{ \mp \sqrt{a_1-|\Delta|}\,\frac{\partial b_{ik}}{\partial Q_1}Q_{10} + \sqrt{a_1+|\Delta|}\,\frac{\sqrt{6}}{2}\, \frac{\partial^2 b_{ik}}{\partial Q_2^2}Q_2^{2} \right\}. \tag{9,7} \]

Here \(a_1=\sqrt{\Delta^2+32x^2}\).

Calculations on the basis of the valence-optical scheme were carried out for CO$_2$ by Nath and Chulam$^{61}$, and for CO$_2$ and CS$_2$ by M. V. Vol’kenshtein$^{54}$. Nath and Chulam applied an additivity scheme specifically to this case; M. Vol’kenshtein proceeded from the general theory set forth above, in particular from expressions (9,5) and (6,3)—(6,4). In the case of CO$_2$ and CS$_2$, the most important point is to take account of the nonlinearity in the expressions for $q$ and $\gamma$ through $Q$. In the calculation, values of the quantities $a_1$ and $a_2$ for the bonds C=O and C=S, known from other sources, were used. We give the results of the calculations$^{54}$.

Table 12

Molecule CO$_2$

Frequency	Transition	Measured I: Dickinson	Measured I: Langseth	Measured I: Hanson 50°	Measured I: Hanson 200°	Calculated I 50°	Calculated I 200°
1264	(01)→(11) └→(03)	–	–	–	–	0,9	3,7
1289	(00)→(10) └→(02)	6,7	6,1	5,7	6,1	7,7	7,7
1389	(00)→(10) └→(02)	(10)	(10)	(10)	(10)	(10)	(10)
1409	(01)→(11) └→(03)	–	–	2,3	5,2	1,0	4,0

Table 13

Molecule CS$_2$

Frequency	Transition	Measured I: Langseth and coauthors	Measured I: Calderola and D’Iulot	Calculated I
641*		–	0,9	0,9
648	(01)→(11) └→(03)	2,2	2,9	2,3
656,5	(00)→(10) └→(02)	(10)	(10)	(10)
787,7*		0,05	0,17	0,14
796	(00)→(10) └→(02)	0,79	1,75	1,6
804,9	(01)→(11) └→(03)	0,26	0,9	1,3

Frequencies marked with an asterisk are connected with the isotope effect—with CS$^{32}$S$^{34}$ molecules, present in an amount of $\sim 8\%$ in the isotopic mixture.

In these calculations the following values of $\alpha$ were used:

\[ \begin{aligned} &\text{for the } \mathrm{C}= \mathrm{O}\text{ bond:}\\ &\alpha_1 = 2.05 \cdot 10^{-24}\ \mathrm{cm}^3,\qquad \alpha_2 = 0.96 \cdot 10^{-24}\ \mathrm{cm}^3,\\ &\text{for the } \mathrm{C}= \mathrm{S}\text{ bond:}\\ &\alpha_1 = 7.57 \cdot 10^{-24}\ \mathrm{cm}^3,\qquad \alpha_2 = 2.78 \cdot 10^{-24}\ \mathrm{cm}^3. \end{aligned} \]

The derivatives $\alpha'$ obtained are:

\[ \begin{aligned} &\text{for the } \mathrm{C}= \mathrm{O}\text{ bond:}\\ &\alpha'_1 = 2.56 \cdot 10^{-16}\ \mathrm{cm}^2,\qquad \alpha'_2 = 0.77 \cdot 10^{-16}\ \mathrm{cm}^2,\\ &\text{for the } \mathrm{C}= \mathrm{S}\text{ bond:}\\ &\alpha'_1 = 1.80 \cdot 10^{-16}\ \mathrm{cm}^2,\qquad \alpha'_2 = 0.60 \cdot 10^{-16}\ \mathrm{cm}^2. \end{aligned} \]

Thus, the valence-optical scheme makes it possible to calculate the intensities of the components of the splitting in the case of Fermi resonance in good agreement with experiment.

§ 10. ELECTRO-OPTICS OF VIBRATIONS AND THE NATURE OF THE CHEMICAL BOND

The study of questions of the electro-optics of vibrational spectra, despite their great physical interest, is not an end in itself. As we indicated at the beginning, the application of the electro-optical properties of a molecule to problems of the theory of the structure of matter, and to molecular structural analysis, is essential. From what follows it is clear that the possibility of a meaningful analysis of a spectrum on the basis not only of frequencies, but also of polarizations and intensities, greatly simplifies the spectral analysis of a molecule. In this sense the approximate estimates and rules set forth above in § 6 are very useful. Of greatest interest, however, is the elucidation of the relationship between the intensities and polarizations of a spectrum and the parameters determining them, on the one hand, and the nature of the chemical bonds in the molecules under consideration, on the other.

Table 14

Effective charges of bonds

Molecule	Author	$\dfrac{\partial \mu}{\partial q}$	$\mu \cdot 10^{18}$	$r_0,\ \mathring{\mathrm{A}}$	$\dfrac{\mu}{r_0}$
HCl	Burgen $^{62}$	$0.173\,e$	1.03	1.28	$0.184\,e$
HCl	Bartolomé $^{63}$	$0.186\,e$	1.03	1.28	$0.184\,e$
HBr	»	$0.075\,e$	0.78	1.42	$0.115\,e$
HJ	»	$0.033\,e$	0.38	1.62	$0.049\,e$
$> \mathrm{CH}$	Wolkenstein $^{48}$	$0.062\,e$	0.4	1.09	$0.077\,e$
$> \mathrm{CO}$	Mathison $^{54}$	$0.82\,e$	0.12	1.15	$0.022\,e$

Let us begin with the infrared spectra. One may state the general proposition (as yet insufficiently confirmed by experiment): the intensities

the valence vibrations of the bonds the larger their dipole moments. In other words, the behavior of \(\mu\) and \(\dfrac{\partial \mu}{\partial q}\) is symbate. We give (Table 14) the available results of the determination of the effective charges \(\dfrac{\partial \mu}{\partial q}\) from the absolute intensities of infrared absorption bands \(^{62-64}\) and by the method set forth above (§ 8).

We see that the behavior of \(\mu'\) is symbate to \(\mu\) and, moreover, the values of \(\mu\) for HCl, HBr, HJ, and CH are close to the quotient obtained by dividing the dipole moment by the internuclear distance. Mathison’s result for CO falls outside this regularity; the explanation for this should probably be sought not only in the inaccuracy of a difficult experiment, but also in the somewhat different character of the chemical bond in CO than in the hydrogen halides or in CH. In the valence scheme of modern chemistry the concept of the so-called electronic resonance \(^{21}\) is widely used, according to which the real valence state of a molecule or chemical bond, described by a certain wave function \(\psi\), may be represented as the result of the superposition of two or more “pure” states, described by functions \(\psi_k\), with which definite valence formulas may be associated. In mathematical form we write*)

\[ \psi=\sum c_k\psi_k \tag{10,1} \]

In particular, according to Pauling \(^{21}\), an ordinary \(\sigma\)-bond may be described as the result of the superposition of a homeopolar \(\psi_h\)- and an ionic \(\psi_i\)-state:

\[ \psi=\psi_h+c\psi_i . \tag{10,2} \]

Woll \(^{65}\), and also Dyatkina \(^{66}\), postulated that the dipole moment of a molecule or bond is determined entirely by the function \(\psi_i\), and for a purely homeopolar bond \(\mu=0\); these propositions are supported by approximate calculations. Another treatment of the question by Coulson \(^{67}\) is, apparently, less convincing. Thus we have

\[ \mu=-\frac{e\int \psi \sum x_i\psi^{*}\,d\tau}{\int \psi\psi^{*}\,d\tau}, \tag{10,3} \]

where \(e\) is the charge of the electron, and \(x_i\) are the coordinates of all electrons in the system.

*) The value of the coefficients \(c_k\)—the “weights of the states”—is to a large extent conventional, since they depend on the method of approximate quantum-mechanical calculation. The concept of electronic resonance, arising from the localized-pair method, is, in essence, a very visual representation—in the customary images of the valence scheme—of the approximation given by the Heitler–London–Slater–Pauling method. In using this concept for qualitative conclusions we undoubtedly approach the truth. But detailed quantitative estimates—for example, the sometimes performed calculations of the coefficients \(c_k\) to fractions of a percent—are obviously devoid of meaning.

According to (10.2),

\[ \mu=\frac{\mu_h+2c\mu_{hi}+c^2\mu_i}{2+2s^2+c^2+4cs}, \tag{10.4} \]

where \(s\) is the nonorthogonality, \(\mu_h\), according to the adopted postulate, is zero, and \(\mu_{hi}\) is approximately equal to

\[ \mu_{hi}=\frac{\mu_i s}{2}. \tag{10.5} \]

At the same time,

\[ \mu_i=e\cdot r, \tag{10.6} \]

where \(r\) is the length of the dipole. Finally we have

\[ \mu=\frac{cs+c^2}{2+2s^2+4cs+c^2}\,er. \tag{10.7} \]

It is evident that the ratio \(\mu:er_0\) is a good measure of the ionic character of the bond.

The dependence of \(\mu\) on \(r\) in the case of a purely ionic bond is linear (Fig. 6). For real molecules, application of Frank’s criterion gives the following. In the case of ionic molecules the curve \(\mu(r)\) will asymptotically approach the straight line \(er\), while in the case of atomic molecules, after passing through a certain maximum, it will approach the abscissa axis[^65]. Such a course of the curve in both cases is explained by the fact that \(s\) and \(c\), in turn, depend on \(r\), which makes the direct theoretical determination of the derivative \(\partial\mu/\partial q=\partial\mu/\partial r\) very difficult. The effective charges \(\mu'\) given in Table 14 characterize the slope of the curve \(\mu(r)\) at the point corresponding to the equilibrium value of \(r\), equal to \(r_0\) (Fig. 6). In one way or another, it is evident that the values of \(\partial\mu/\partial q\) serve in some manner as a measure of the ionicity of the bond, which is also confirmed by Table 14. The CH bond here occupies a position intermediate between HBr and HJ. As for CO, the state of this molecule can be described with the aid of the principal structures

\[ \mathrm{C}=\mathrm{O}\qquad \mathrm{C}^{-}\equiv \mathrm{O}^{+} \]

with oppositely directed dipole moments, almost compensating one another at the equilibrium interatomic distance (the magnitude of these moments is of the order of \(2.7\cdot 10^{-18}\)). Thus, two ionic structures are involved. During vibration the compensation can be sharply disturbed, which possibly determines the large value of \(\partial\mu/\partial q\), many times exceeding the values of \(\mu/r_0\).

Fig. 6.

Large values of $\mu'$ and, consequently, large intensities of the infrared bands of valence vibrations of polar bonds may be of substantial importance in molecular structural analysis. As H. M. Thompson has recently shown$^{68}$, these circumstances can be successfully used for the identification of characteristic bonds. The bonds C—O, C—N, C—F in the presence of C—C bonds are not spectrally distinguishable from them, since their vibrations interact and mix. The Raman spectrum of a system with C—O, C—N, C—C bonds is essentially indistinguishable from the spectrum of a similar system consisting only of C—C bonds. But in the infrared spectrum there is a sharp difference in intensities, for C—C bonds are nonpolar, whereas C—O, C—N bonds contain large fractions of ionic states.

It is also evident that the presence in a given molecule of intramolecular ionized structures with large moments must manifest itself in the high intensity of the corresponding infrared bands.

Table 15

Intensities of Raman lines C—Hal

Molecule	Frequency	Intensity	Half-width	Author
$\mathrm{C_2H_5Cl}$	655	37	1.1	73
$\mathrm{C_2H_5Br}$	557	85	1.1	73
$\mathrm{C_2H_5J}$	497	171	1.1	73
$\mathrm{C_3H_7Cl}$	651	71	1.0	73
$\mathrm{C_3H_7Cl}$	725	33	0.7	73
$\mathrm{C_3H_7Br}$	565	138	1.0	73
$\mathrm{C_3H_7Br}$	648	61	1.0	73
$\mathrm{C_3H_7J}$	503	191	1.5	73
$\mathrm{C_3H_7J}$	590	139	1.1	73
$\mathrm{C_4H_9Cl}$	651	24	13.9	72
$\mathrm{C_4H_9Cl}$	722	5.5	—	72
$\mathrm{C_4H_9Br}$	559	45.5	16.8	72
$\mathrm{C_4H_9Br}$	640	28	16.8	72
$\mathrm{C_4H_9J}$	505	98	14.9	72
$\mathrm{C_6H_5Cl}$	592	93	12.3	73
$\mathrm{C_6H_5Cl}$	704	23	0.5	73
$\mathrm{C_6H_5Br}$	607	35	0.5	73
$\mathrm{C_6H_5J}$	555	84	0.6	73

The dependence of the intensities of Raman lines, and thereby of the quantities $\dfrac{\partial \alpha}{\partial q}$, on the degree of ionicity of the molecule is opposite to the dependence for infrared bands. Plachek had already pointed out$^{69}$ that for a purely ionic bond, if mutual polarization of the ions is neglected, $\dfrac{\partial \alpha}{\partial q}$ is close to zero, since in this case the outer electron shell is located entirely at the negative ion and is not deformed during vibration. The value $\partial \alpha / \partial q$ is thus a certain measure of homeopolarity. This can be illustrated by calculation according to Kirkwood’s formula (3.8) for the simplest systems$^{33}$. Indeed, Callihan and Salant$^{70}$ also found that the intensity for HBr is higher than for HCl. We shall also mention the work of Hansen-Damaschin$^{71}$, comparing—

governing the intensity of the Raman lines of the ions \(\mathrm{CO}_3''\), \(\mathrm{NO}_3'\), \(\mathrm{SO}_4''\), \(\mathrm{ClO}_4'\), and of certain complex compounds, with the intensities of benzene lines. Other conditions being equal, the behavior of \(\dfrac{d\alpha}{dq}\) is similar to the behavior of \(a\). For these reasons the intensities in the series \(\mathrm{C—Cl}\), \(\mathrm{C—Br}\), \(\mathrm{C—J}\) increase (for infrared bands the opposite behavior should be expected). Let us give a table (Table 15).

The intensities and half-widths are expressed here in conventional units, different in the cited works \(^{72}\) and \(^{73}\).

Let us consider the quantities \(\dfrac{d\alpha}{dq}\) determined above (§ 8). We emphasize that, owing to the insufficiency of experimental data, these quantities are effective in character and require refinement (Table 16).

Table 16

Polarizabilities of bonds and their derivatives

Bond	\(\alpha_1,\ \text{Å}^3\)	\(\alpha_2,\ \text{Å}^3\)	\(\alpha_1',\ \text{Å}^2\)	\(\alpha_2',\ \text{Å}^2\)	\(\dfrac{\alpha_1}{r_0},\ \text{Å}^2\)	\(\dfrac{\alpha_2}{r_0},\ \text{Å}^2\)	Note
\(\mathrm{C—H}\)	0.79	0.58	1.28	0.31	0.73	0.53	In the molecules \(\mathrm{CH_3Cl}\), \(\mathrm{CH_3Br}\), \(\mathrm{CH_2Cl_2}\), \(\mathrm{CH_2Br_2}\),
\(\mathrm{C—Cl}\)	3.67	2.08	2.82	0.68	2.04	1.15	\(\mathrm{CHCl_3}\), \(\mathrm{CHBr_3}\), \(\mathrm{CCl_4}\),
\(\mathrm{C—Br}\)	5.04	2.88	3.23	0.83	2.52	1.44	\(\mathrm{CBr_4}\), \(\mathrm{CH_4}^{35}\)
\(\mathrm{C{=}O}\)	2.05	0.96	2.56	0.77	1.87	0.87	\(\mathrm{CO_2}^{54}\)
\(\mathrm{C{=}S}\)	7.57	2.78	1.80	0.60	4.73	1.74	\(\mathrm{CS_2}^{54}\)
\(\mathrm{H—H}\)	0.75	0.74	1.10	0.95	1.00	0.99	Quantum-mechanical calculation \(^{16}\)
\(\mathrm{H—H}\)	0.73	0.73	1.61	1.61	0.97	0.97	Mean values, Bell \(^{38}\)
\(\mathrm{H—Cl}\)	2.49	2.49	1.23	1.23	1.94	1.94	» »
\(\mathrm{H—Br}\)	3.39	3.39	1.07	1.07	2.40	2.40	» »

The polarizability of the \(\mathrm{C—H}\) bond is small, in accordance with the small volume of the electron cloud. But \(\alpha'\) is large, and its ratios to \(a\) considerably exceed the corresponding quantities for the \(\mathrm{C—Cl}\) and \(\mathrm{C—Br}\) bonds. This is evidently determined by the great homeopolarity of the \(\mathrm{C—H}\) bond. The values of \(\dfrac{d\alpha}{dq}\) for \(\mathrm{C—Br}\) are naturally higher than for \(\mathrm{C—Cl}\). The reasons why \(\dfrac{d\alpha}{dq}\) for \(\mathrm{C{=}S}\) is lower than for \(\mathrm{C{=}O}\) are as yet unclear.

Of great interest are the still rather few results concerning the dependence of the intensities of Raman lines on characteristic-

...of valence vibrations from the constitutional features of the molecules. The intensities prove to be extremely sensitive to one or another change in the structure of the molecule; they constitute a kind of “indicator of electronic resonance.” At the same time, the polarizations are little sensitive to such changes. Let us consider several examples (Tables 17 and 18):

Table 17

Intensities of the Raman lines of vibrations of the CO group $^{73}$

Molecule	Frequency	Intensities	Half-width, $cm^{-1}$
$\mathrm{CH_3COCl}$	1798	11.2	16.9
$\mathrm{CH_3COCH_3}$	1708	14.3	18
$\mathrm{CH_3COC_6H_5}$	1678	14[[unclear: decimal part]]	11.8
$\mathrm{C_6H_5COC_6H_5}$	1653	262	13.5

The values of $\rho$ for $\mathrm{C=O}$ vibrations in all known cases lie within the limits $0.34—0.42$.

Table 18*)

Intensities of Raman lines

Molecule	Benzene ring, $\sim 1000\ cm^{-1}$	Benzene ring, $\sim 1600\ cm^{-1}$	$\mathrm{C=C}$	$\mathrm{C\equiv N}$
$\mathrm{C_6H_6}$	150	6	—	—
$\mathrm{C_6H_5—CH=CH_2}$	100	100	140	—
$\mathrm{C_6H_5—O—CH=CH_2}$	120	40	80	—
$\mathrm{C_6H_5—CN}$	220	150	—	120
$\mathrm{C_6H_5—CO—CH_3}$	190	190	—	—
$\mathrm{CH_3—CN}$	—	—	—	12
$\mathrm{H_2C=CH—CN}$	—	—	50	50
$\mathrm{H_2C=CH—CH=CH_2}$	—	—	40	—

These striking results (changes of intensity by hundreds of percent) find their qualitative explanation in the concept of electronic resonance $^{73}$. In the cases given in Table 17, there exist—

*) In Table 18 and further in the text, preliminary data are cited from the latest (as yet unpublished) work of P. P. Shorygin. I take this opportunity to express my gratitude to him for his kind permission to use these data.

the following internally ionized structures are possible:

\[ \begin{array}{cccc} \mathrm{H_3C} & \mathrm{H_3C} & \mathrm{H_3C} & \mathrm{C_6H_5} \\ \backslash & \backslash & \backslash & \backslash \\ \mathrm{C}\equiv\mathrm{O}^{+} & \mathrm{C}=\mathrm{O} & \mathrm{C}-\mathrm{O}^{-} & \mathrm{C}-\mathrm{O}^{-} \\ / & / & \Vert\ \ \ \ & \Vert\ \ \ \\ \mathrm{Cl} & \mathrm{H_3C} & \mathrm{H_5C_6}^{+} & \mathrm{C_6H_5}^{+} \end{array} \]

\[ \begin{array}{cc} & \text{3 structures} \qquad\qquad \text{6 structures} \end{array} \]

In the first molecule the electron cloud of the \(\mathrm{C}=\mathrm{O}\) bond is in the field of an additional positive charge, which should give a decrease in polarizability in comparison with acetone, which has no internally ionized structures. Conversely, an additional negative charge means an increase in polarizability. Since the course of \(a\) and \(\dfrac{da}{dq}\) is in general similar, the above distribution of intensities is explained. The results relating to the \(\mathrm{C}\equiv\mathrm{N}\)-bond line are explained analogously:

\[ \mathrm{H_3C}-\mathrm{C}\equiv\mathrm{N},\quad \mathrm{H_2C}^{+}-\mathrm{CH}=\mathrm{CN}^{-},\quad \mathrm{C_6H_5}^{+}=\mathrm{C}=\mathrm{N}^{-}. \]

\[ \text{3 structures} \]

However, this explanation is far from always sufficient. It is unsuitable, for example, for the intensities of \(\mathrm{C}-\mathrm{C}\) bonds, for which, in particular, the following data have been obtained:

\[ \begin{array}{lr} \mathrm{H_3C}-\mathrm{CO}-\mathrm{O}-\mathrm{CH}=\mathrm{CH_2} & 55 \\ \mathrm{Br}-\mathrm{CH_2}-\mathrm{CH}=\mathrm{CH_2} & 85 \end{array} \]

Meanwhile, structural considerations similar to those set forth above compel one to expect the reverse order of intensities.

P. P. Shorygin\(^{74}\) explains these phenomena by the fact that, during molecular vibrations, the relative weights of one or another structure change. Such an assumption is plausible. There is no doubt that the question requires further experimental study—it has only just begun. However, it is already clear that the significance of intensities for molecular structural analysis can hardly be overestimated. Let us note, incidentally, that from all the considerations set forth in this article it follows that, in cases of characteristic vibrations of identical bonds, the intensities of Raman lines and infrared bands are also characteristic—they do not change from compound to compound.

§ 11. INTERMOLECULAR INTERACTION

The question of the influence of intermolecular forces on the values of frequencies in vibrational (Raman) spectra was in its time the subject of our review in this same journal\(^{75}\). Here we shall set forth the little that can presently be said about the influence of these forces on electro-optical properties and parameters.

That circumstance, namely that the valence-optical scheme proves applicable to the vibrational spectra of substances in the liquid state, shows that the intensities and polarizations of Raman lines are, in general, little sensitive to van der Waals forces, in contrast to the Kerr constant and the degree of depolarization of Rayleigh scattering. In the latter cases we encounter the collective behavior of whole groups of molecules, different from the behavior of an individual molecule, which determines its spectrum. The low sensitivity of the electro-optical parameters in spectra to weak intermolecular interaction is also evidenced by the well-known possibility of quantitative analysis of mixtures of certain substances, for example hydrocarbons, by the method of vibrational spectra, on the basis of the proportionality of intensities to the relative content of the substance. However, this position cannot be regarded as a general rule. In many cases van der Waals forces noticeably affect the spectrum.

As was shown earlier \(^{75}\), appreciable frequency shifts under the action of van der Waals forces arise only because of the orientation effect.

Induction and dispersion forces do not act on the mechanical parameters of vibrations. However, they can affect the electro-optical parameters. The point is that the polarizability of the molecule itself depends on dispersion forces \(^{33,76}\). In the dispersion interaction of two three-dimensional isotropic particles with polarizability \(b_0\), the polarizability of each of the particles increases along the line joining them and decreases perpendicular to it:

\[ b_1 \lessgtr b_0 \left( 1 + \frac{2b_0}{R^3} \right), \qquad b_2 \lessgtr b_0 \left( 1 - \frac{b_0}{R^3} \right), \tag{11,1} \]

where \(R\) is the distance between the particles. Thus, the anisotropy of the tensor \(b\) changes from 0 to a value close to \(\frac{2b_0^2}{R^3}\), while the mean value (the trace) remains practically unchanged. This latter circumstance is confirmed by the practical independence of molecular refraction from the aggregate state. In liquids consisting of anisotropic, though nonpolar, molecules, a preferential orientation of them under the influence of dispersion forces is possible. It may be thought that the most important properties of such liquids—their viscosity, etc.—are to a significant extent determined by this effect. As a result of the change in polarizability anisotropy, the Kerr constants and the degree of depolarization of Rayleigh scattering change. In accordance with what has been set forth, one may expect that the intensities of Raman lines belonging to nonsymmetric and deformation vibrations, which depend on \(\alpha'_1-\alpha'_2\), \(\alpha_1-\alpha_2\), etc., should change more strongly under the action of dispersion forces than the intensities of valence fully symmetric ...

oscillations, depending on \(a_1' + 2a_2'\). Experimental investigation of these questions is only just beginning.

Undoubtedly, in cases of stronger intermolecular interaction of a semichemical character, not reducible to van der Waals forces, the electro-optical properties must change very sharply, since in such cases (hydrogen bonding, for example) what is involved is the formation of intermolecular compounds, changes in the properties of bonds—their ionicity, etc. These questions have also as yet been little investigated.

Of particular interest is the case of a change in the spectrum as a result of changes in the selection rules caused by intermolecular interaction \({}^{77}\). Thus, for example, in the Raman spectrum of liquid \(\mathrm{CS}_2\) there are observed the frequencies \(395\) and \(1523\ \mathrm{cm}^{-1}\) of the deformation and antisymmetric valence vibrations, which are forbidden for a linear symmetric triatomic molecule. Evidently, the \(\mathrm{CS}_2\) molecules are deformed by van der Waals forces. Let us estimate the possible magnitude of the change in the valence angle under the action of an electric field \(F\), directed along its bisector. Denote the dipole moment of each of the two bonds by \(\mu\). Let the valence angle in the field be \(\varphi\), and in the absence of the field \(\varphi_0\). The potential energy in the field is

\[ U=\frac{C_\delta}{2}(\varphi-\varphi_0)^2-2\mu F\cos\frac{\varphi}{2}. \tag{11,2} \]

The equilibrium condition is

\[ \frac{\partial U}{\partial \varphi} = C_\delta(\varphi-\varphi_0)+\mu F\sin\frac{\varphi}{2} =0, \tag{11,3} \]

\(\varphi-\varphi_0\) is a small quantity and, for \(\varphi_0=\pi\), \(\sin \frac{\varphi}{2}\simeq 1\).

We have

\[ \varphi-\pi \simeq -\frac{\mu F}{C_\delta}. \tag{11,4} \]

The order of magnitude is \(\mu \simeq 10^{-18} CGSE\), \(C_\delta\)—the elastic constant of the valence angle—\(\sim 10^{-11} CGSE\).

For strong intermolecular fields \(F\sim 10^8—10^9\ V/\mathrm{cm}\). We have \(\varphi-\pi\sim 0.1\). The intensity of the antisymmetric valence vibration is determined by the tensor (7,15). We see that, in the case of strong intermolecular interaction, the appearance of previously forbidden lines with intensities accessible to observation is indeed possible.

Some interesting data on the influence of intermolecular forces on the intensity of Raman lines are contained in the work of Buchheim \({}^{78}\).

We have set forth in the present review the current state of the theory of intensities and polarizations in the vibrational spectra of molecules. This is a new field, and the results presented belong

at the initial stage of investigations, which undoubtedly must and will expand in the future.

The main part of the material presented in the review was obtained in the work of a group of staff members of the Laboratory of Spectral Analysis of the State Optical Institute.

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