Raman Effect and Intermolecular Interaction

M. V. Volkenstein, Moscow

Contents

§ 1. Raman effect. § 2. Applications to chemical problems. § 3. Structural-symmetry determinations. § 4. Van der Waals forces. § 5. Raman effect and van der Waals forces. 1. Electric field. § 6. Raman effect and van der Waals forces. 2. Successive calculation. § 7. Molecular compounds and the Raman effect. § 8. Raman effect and the structure of liquids. § 9. Symmetry relations. Literature.

§ 1. Raman Effect

Modern physics has developed a number of methods for studying the structure of matter, and our conceptions of the structure of atoms, molecules, crystals, liquids, etc., are based on the results obtained by applying them. For the most part these methods reduce to studying the character of the interaction of matter with radiation. Among the physical methods of investigation, an important place belongs to the Raman effect. The Raman effect, or combination scattering of light, was discovered simultaneously and independently in the spring of 1928 by Raman and Krishnan in Calcutta and by Mandelstam and Landsberg in Moscow. They observed that in the spectrum of light scattered by a substance (by the Indians, liquid benzene; by Mandelstam and Landsberg, crystalline quartz), alongside the undisplaced lines of the spectrum of the light source (a mercury lamp), there appear weaker new lines characteristic of the scattering substance. It turned out that the differences between the frequencies of the shifted Raman lines and the unshifted lines of classical Rayleigh scattering correspond to the frequencies of the natural vibrations of atoms in a molecule relative to one another—that is, to those frequencies which are observed in the infrared spectrum of the substance.

Thus, the Raman effect makes it possible to obtain directly the values of the most important molecular constants, namely the frequencies of vibrations of atoms in molecules and of molecules relative to one another in molecular compounds and in solids.

It must be said that long before the discovery of the Raman effect, the study of scattered light—chiefly of its polarization properties—was directed toward solving questions of molecular structure. But, surprisingly, it occurred to no one to investigate the spectrum of scattered light, although this is considerably easier than studying its polarization properties. In one of his articles Smekal emphasizes the extraordinary simplicity of observing the Raman effect, pointing out that the phenomenon could have been discovered by a second- or third-year student. However, it seems beyond doubt to us that this discovery could have been made only on the basis of far-reaching theoretical ideas, which matured only around 1928; this is confirmed by the fact of the simultaneous discovery of the effect by different scientists (besides the authors already mentioned, the Frenchmen Cabannes and Rocard were on the approaches to the discovery in the spring of 1928). On the other hand, the theory of the phenomenon had been developed several years before its first observation, namely in 1923 and 1925.

The nature of the effect is essentially quantum. As early as 1923, Smekal[^1] pointed to the possibility of inelastic and superelastic collisions of light quanta with a molecule, leading in the first case to a decrease in the frequency of the scattered light, and in the second to its increase. At the same time, within the molecule itself a transition takes place from one vibrational state to another, and the difference between the frequencies of the incident and scattered light directly gives us the frequency of the molecule’s own vibrations. This possibility was quantitatively studied by Kramers and Heisenberg[^2] (1925), who considered the question of the interaction of light with matter on the basis of Bohr’s correspondence principle; the radiation was treated by them classically, while quantum mechanics was applied to the scattering medium. Thus, the following expression was obtained for the intensity of the shifted frequency associated with the transition of the scattering particle from state \(n\) to state \(k\):

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

where

\[ E_{nk}=\frac{1}{h}\sum_r\left\{\frac{(AM_{nr})M_{rk}}{\nu_{rn}-\nu}+\frac{M_{nr}(AM_{rk})}{\nu_{rk}+\nu}\right\} \tag{1,2} \]

\(A\) is the amplitude of the incident wave, \(M_{ll}\) is the matrix element of the electric moment, \(\nu\) is the frequency of the incident wave, and \(c\) is the speed of light. A similar result can be obtained if the calculation is carried out using Dirac’s more consistent theory of radiation[^3][^4], which considers the interaction of quantized matter with quantized radiation. However, for the subsequent exposition we shall be able to abandon the quantum picture of the phenomenon and restrict ourselves to its more intuitive classical interpretation. The Raman effect permits such an application of classical ideas,

Of course, with certain limitations. The point is that when we speak of the Raman effect of molecules, the phenomenon can be defined as the result of the action of nuclear vibrations on the state of the electron shell, with changes in which the radiation is also connected. If classical physics is inapplicable for interpreting atomic spectra or the electronic spectra of molecules associated with the motions of particles possessing very small mass and very high velocities, then in the case of molecular vibrations we should expect greater success for the classical theory. Indeed, the vibration of a molecule is the vibration of its nuclei, i.e. of particles possessing a mass at least 1840 times greater than the mass of the electron, and moving much more slowly than the latter. The vibrations of the nuclei are so slow that we may regard the electronic configuration at each instant as the same as if the nuclei were not moving. In other words, the “vibrations” of the electrons are so rapid that during one vibration of the nuclei the electrons manage to return many times to their former position, and we may safely operate with the mean state of the electronic configuration.

What happens when light falls on such a classical model? The external electric field of the light wave induces in the molecule a certain electric moment

\[ \mathbf{M}=\alpha \mathbf{E}. \tag{1,3} \]

\(\alpha\) is the polarizability of the molecule. This quantity is, in the general case, tensorial in character; moreover, in the absence of absorption of light or optical activity of the molecule this tensor, as can easily be shown, is real and symmetric. The quantity \(\alpha\) characterizes the displaceability of the electron shell of the molecule and is measured in cubic centimeters. But vibrations of the nuclei periodically deform the shell. Each molecule, built of \(N\) nuclei, \(N\) atoms, is capable of performing \(3N-6\) normal vibrations \((3N-5\), if the molecule is linear)—such vibrations in which all particles vibrate in phase and with the same frequency. \(\alpha\) will be some function of the displacements from the equilibrium position that are associated with such vibrations of the normal coordinates. Expanding \(\alpha\) in a series in the normal coordinates, and restricting ourselves to the linear term, we have

\[ \alpha(q)=\alpha_0+\sum_{\lambda=1}^{3N-6} \left(\frac{\partial \alpha}{\partial q_\lambda}\right)_0 q_\lambda . \tag{1,4} \]

Substituting here the value of \(q_\lambda\)

\[ q_\lambda=q_\lambda^0 \sin(\omega_\lambda t+\delta) \tag{1,5} \]

and substituting in (1,3) the value of \(\mathbf{E}\)

\[ \mathbf{E}=\mathbf{E}_0 \cos(\omega_0 t+\theta_0), \tag{1,6} \]

we obtain in the expression for \(\mathbf{M}\), along with terms periodically varying with frequency \(\omega_0\), also two terms with frequencies \(\omega_0 \pm \omega_\lambda\). If we were to carry the expansion further—to the quadratic term

or if we adopted for \(q_\lambda\) an anharmonic law, then we would obtain, in addition, terms oscillating with frequencies \(\omega_0 \pm 2\omega_\lambda\), etc.—overtones and combination frequencies in the Raman spectrum. But their intensity is very small, and for the questions of interest to us they are of no great significance.

Thus we have obtained the Raman effect as the result of a modulation of the oscillations of the external electromagnetic field of a light wave by the vibrations of the atoms in a molecule.

Classical theory cannot, naturally, give us either the values of the quantities \(q_\lambda\) or the intensities of the lines. But a whole series of the most important properties of molecular vibrations—their number, selection rules, and polarization relations connected with the symmetry of the vibrations—are correctly illuminated by classical theory \(^{5,6}\).

The value of the Raman effect lies in the possibility of transferring the vibrational spectrum of a molecule from the infrared region, which is difficult for experiment, into the region of visible or ultraviolet light. The vibrational frequencies that we obtain from the Raman spectrum coincide with the infrared ones, with the exception of frequencies forbidden by alternative selection rules. Thus, symmetric vibrations, not accompanied by a change in the electric moment of the molecule, are forbidden in the infrared spectrum and, on the contrary, are especially intense in the Raman spectrum. The reverse relation holds for vibrations antisymmetric with respect to the center of symmetry of the molecule. But the greater part of vibrations is allowed in both cases.

§ 2. Applications to Chemical Problems

After this brief outline of the essence of the phenomenon, let us proceed to an equally brief consideration of the usual applications of the Raman effect in chemistry. We do not consider it necessary to dwell on these questions in detail, since, first, our article treats an entirely different and more particular circle of phenomena, and, second, there exists a considerable number of monographs and review articles devoted to the indicated problems, and there is no need to repeat them \(^{7,8,9,10}\).

The first type of application of the Raman effect to chemistry is the determination of the proper frequencies of molecular vibrations and the discovery, by means of the Raman spectrum, of which groups and bonds enter into the molecule under investigation. The proper vibrational frequency of a given group of atoms is a very characteristic constant, changing little upon replacement of other groups entering into the molecule. Such are the frequencies of the methyl group, the benzene nucleus, the groups \(\mathrm{NH_2}\), \(\mathrm{NO_2}\), and many others. On the other hand, individual bonds in molecules also preserve their definite frequencies. Such are the bonds \(\mathrm{C—C}\), \(\mathrm{C=C}\), \(\mathrm{C\equiv C}\), \(\mathrm{C—Cl}\), \(\mathrm{C—Br}\), etc. in the aliphatic series of compounds, the aromatic bond, the frequencies \(\mathrm{O—H}\), \(\mathrm{N—H}\), etc. An experienced Raman spectroscopist, already at first glance at a spectrum, can say, for example, whether the given hydrocarbon belongs to the aliphatic or aromatic group, etc. The constancy of the frequencies of individual groups and bonds determines the possibility of analyzing the structure of compounds forming

homologous series, the study of those changes which are introduced into the Raman spectrum in passing from one isomer of a given compound to another, etc., and thereby also the general possibility of identifying the structure of a molecule, and with it the possibility of analytical applications of the Raman effect. Thus, for example, keto-enol tautomerism and cis-trans isomerism were studied. In both cases it proved possible to observe very characteristic differences in the spectra of the isomers. Here the essential circumstance is that the Raman spectra of noninteracting substances are additive when we are dealing with their mixture. The intensity of the Raman lines is proportional to the concentration of the substance. This creates the possibility of technical applications of Raman spectroscopy—the possibility of quantitative analysis of mixtures that are difficult to analyze by ordinary methods, such as, for example, mixtures of hydrocarbons in petroleum fractions.

However, the determination of frequencies characteristic of individual substances is not reduced to pure empiricism. The determination of the vibrational frequencies for one or another bond also has the significance that it makes it possible to calculate the coefficient of a quasi-elastic bond and to determine its character. The cyclic frequency of oscillation of any harmonic oscillator consisting of two particles is expressed by the following formula:

\[ \omega = \sqrt{\frac{k}{m}}, \tag{2,1} \]

where

\[ m=\frac{m_1 m_2}{m_1 + m_2} \]

is the reduced mass, and \(k\) is the coefficient of the elastic bond, characterizing the force with which the displaced particle tends to return to the equilibrium position. Proceeding from this formula, it proved possible, for example, to show that the coefficients of the bonds \(C—C\), \(C=C\), and \(C\equiv C\) are related as \(1:2:3\), and that, for example, in the CO molecule the bond is triple.

As is known, the heat capacity of molecules, associated with the degrees of freedom corresponding to their vibrations, is expressed through the natural frequencies of vibration. Using the data for these frequencies obtained from Raman spectra, it was possible to calculate the specific heats of a number of compounds in very good agreement with experiment (Table 1)\(^{9}\).

TABLE 1

	Heat capacity	Heat capacity
	calculated	observed
CO\(_2\) gas . . .	0.159	0.154
SO\(_2\) gas . . .	0.117	0.117
PCl\(_3\) 117°C . .	0.118	0.119
AsCl\(_3\) 217°C .	0.095	0.098
SiCl\(_4\) 97°C .	0.123	0.117
TiCl\(_4\) 167°C .	0.108	0.125
SnCl\(_4\) 157°C .	0.084	0.087

The second application of the Raman effect to chemical and physicochemical problems is the study of electrolytic disso-

ciation, by means of the Raman spectrum. The ionic bond is not active in a first-order Raman spectrum—in vibrations of such a molecule as NaCl, the polarizability changes hardly at all and the quantity \(\dfrac{\partial \alpha}{\partial q}\) is close to zero. Therefore, in the spectrum of a purely ionic compound we observe only the frequency of complex ions and, in general, observe nothing if the ions are monatomic. Let us give an example taken from one of our works[^11]. In the spectrum of \([(C_2H_5)_4N^+]J'\) we observe only the frequencies of the \((C_2H_5)_4N^+\) ion, but not the frequencies \(N — J\) or \(C — J\), although in the spectrum of \(C_2H_5J\), from which the substance under investigation is obtained by combination with \((C_2H_5)_3N\), there was a very intense \(C — J\) line.

The Raman effect shows that many compounds to which a purely ionic structure is usually ascribed in fact are not such. For example, the frequencies of the \(NO_3\) ion are observed in the Raman spectrum of \(HNO_3\) only beginning with sufficient dilution, whereas at higher concentration the frequencies of the undissociated molecule are observed. Conversely, in the Raman spectra of nitrates we observe only the frequencies of \(NO_3\). The Raman spectra of \(H_2SO_4\), \(HSO_4\), and \(SO_4''\) ions are different, and from the relative intensity of the lines one can judge the degree of dissociation of the acid in a solution of one concentration or another.

A third, very important and topical kind of application of the Raman effect to problems of physical chemistry is the study of phenomena connected with intermolecular interaction—solvation, changes in the state of aggregation, and so forth. These questions will be considered in detail below; the present article is mainly devoted to them. Here we shall only point out that changes in the state of a substance—its dissolution, a change in the state of aggregation, a change in temperature and pressure—are in a number of cases accompanied by characteristic changes in the Raman spectrum, making it possible to draw far-reaching conclusions about the nature of intermolecular interaction.

A fourth kind of application is the use of the Raman effect in questions of chemical kinetics. These applications have acquired special interest recently in connection with the development of experimental technique that permits the use of sufficiently short exposures when photographing Raman spectra. However, even now the application of the Raman effect is limited to the study of homogeneous reactions that proceed comparatively slowly. The application is based on the fact that the Raman spectrum of the reaction products differs from the Raman spectra of the reacting substances. By obtaining a consecutive series of photographs at definite intervals of time, one can record the course of the reaction from the relative intensity of the Raman lines of the reacting substances and products. In studying photochemical reactions, the source of the spectrum itself can be used as the source of light energy. This ingenious method was demonstrated in the work of Bezhold and Ornstein[^12], who studied the kinetics of the photochlorination of chloroform with formation of carbon tetrachloride, judging the course of the reaction from the relative intensity of the Raman lines. Partasarathi[^13] studied, under ...

with the aid of the Raman effect, the kinetics of the reaction between acetic anhydride and water and between acetic anhydride and alcohol. Finally, Kohlrausch and co-workers \(^{14}\) studied the reaction of the isomeric cis–trans conversion of dibromoethylene. In a number of cases the Raman effect makes it possible to detect intermediate products which cannot be isolated and detected by ordinary chemical methods.

Of considerable interest in the Raman effect is the phenomenon of isotopy. The replacement of one isotope by another, i.e. a change in the mass of the vibrating nucleus, should be reflected in the value of the frequency, as follows from formula (1,7). This effect should be especially large when hydrogen is replaced by deuterium. Thus, the frequency of the OH group should decrease by a factor of \(\sqrt{\frac{17}{9}}\) on passing to OD. This indeed occurs in the transition from \(\mathrm{H_2O}\) to \(\mathrm{D_2O}\). The Raman effect can, on the one hand, serve for the discovery of new isotopes and, on the other, for determining the structure of a molecule from those changes in the spectrum which are introduced when an element is replaced by its isotope—in particular, hydrogen by deuterium. Thus the Raman-effect method may prove to be very useful and convenient in reactions with isotopes as indicators.

§ 3. Structural-Symmetry Determinations

In addition to determining the frequencies of vibrations of molecules and of their constituent parts, the Raman effect gives us the possibility of characterizing the molecule as a whole: of determining the geometry of the molecule, its symmetry. If a molecule possesses a definite symmetry, then not all of the possible \(3N - 6\) normal vibrations will be represented in the spectrum by separate frequencies. Some of these vibrations may be jointly degenerate and their frequencies coincide. Thus, for a tetrahedral structure of the methane type, \(3 \cdot 5 - 6 = 9\) normal vibrations are possible. However, owing to the high symmetry of the structure, all vibrations except one—the totally symmetric one—are degenerate, and only 4 frequencies are observed in the spectrum. One of these frequencies is doubly degenerate, 2 are triply degenerate, and one is nondegenerate. On passing to a lower symmetry we have a smaller degree of degeneracy—not more than two. Thus, when one of the hydrogen atoms in methane is replaced by a halogen atom, we obtain a spectrum already consisting of six frequencies; of these, three are doubly degenerate. Finally, when two hydrogen atoms in \(\mathrm{CH_4}\) are replaced by halogen atoms, the degeneracy is removed completely, and the spectrum will consist of 9 frequencies. The character of the symmetry of one or another vibration is substantially reflected in the selection rules in the Raman spectrum and in the infrared spectrum. We have already indicated that vibrations symmetric with respect to the center of symmetry are forbidden in the infrared spectrum and allowed in the Raman spectrum. The opposite holds for antisymmetric vibrations. For example, the Raman spectrum of cis-dichloroethylene, a molecule having no center of symmetry, is considerably richer in lines than the spectrum of the trans isomer, owing to the absence of the indicated prohibition. The quantum-mechanical selection rules for the infrared

spectra are found when calculating the matrix element of the electric moment.

\[ \mathbf{M}_{nk}=\int \widetilde{\psi}_n(q)\,\mathbf{M}(q)\,\psi_k(q)\,d\tau . \tag{3,1} \]

The equality to zero of the element \(\mathbf{M}_{nk}\) means the prohibition of the corresponding vibrational transition. For a Raman spectrum we must calculate the matrix element of the polarizability

\[ \alpha_{nk}=\int \widetilde{\psi}_n(q)\,\alpha(q)\,\psi_k(q)\,d\tau . \tag{3,2} \]

Naturally, owing to the vector character of \(\mathbf{M}(q)\) and the tensor character of \(\alpha(q)\), we must obtain different selection rules in these two cases.

Thus we see that already a mere count of the number of observed lines in the spectrum makes it possible to draw conclusions about the symmetry of the molecule. In addition, however, there is still another important characteristic of the spectrum, one which considerably increases the accuracy of determinations of this kind. This concerns the polarization properties of the spectral lines.

[Diagram with labels: \(E\), \(E_y\), “Medium,” \(M_{22}\), \(M_{12}\), \(M_1\), \(M_2\), \(M_{2x}\), \(x\), \(y\), \(z\), “Observer,” \(90^\circ\).]

Fig. 1.

If the molecule were perfectly symmetric and, consequently, the polarizability \(\alpha\) were a scalar quantity, then the light scattered by the molecule would be completely linearly polarized, as is explained by the following drawing (Fig. 1). If \(\mathbf{M}_1\) is parallel to \(\mathbf{E}\), then the observer sees only the component \(\mathbf{M}_{1x}\). If, on the contrary, \(\alpha\) is not scalar, then \(\mathbf{M}_2\) is not parallel to \(\mathbf{E}\), and therefore the component \(\mathbf{M}_{2x}\) is observed. The light is partially depolarized. The degree of depolarization \(\rho\) for a given Raman line (the ratio of the intensity of the depolarized part to the total intensity) characterizes the behavior of the polarizability in one or another vibration. As calculation shows, for totally symmetric vibrations of molecules belonging to the cubic symmetry group, \(\rho=0\). For nonsymmetric vibrations \(\rho \leq \dfrac{6}{7}\). For symmetric vibrations of molecules possessing lower symmetry, \(\rho>0\), but in practice for the most part small,

less than 0.5. Thus, by counting the number of lines and measuring their degrees of depolarization, we can, even without knowing the numerical values of the frequencies, say how the molecule is constructed.

The structural-symmetry relations in the Raman spectrum that do not concern the numerical values of the frequencies and their intensities can be obtained within the framework of a purely classical theory. Such a theory was developed by Placzek ^5,15. Wigner ^16 applied to it the mathematical apparatus of group theory, which made it possible to obtain especially simply all the most important structural properties of vibrations: the degrees of degeneracy of frequencies, selection rules, and polarization relations for a molecule of any symmetry. This method is considered in the work of Tisza ^17 and in more detail in our article ^18, as well as in the article by Rosenthal and Murphy ^19 that appeared after it. Some special structural-symmetry relations connected with intermolecular interaction will be analyzed in § 9.

§ 4. Van der Waals Forces

The most important problem of theoretical chemistry is the problem of the nature of the forces of chemical bonding, forces usually and thoughtlessly depicted by chemists by means of valence strokes. At present it may be asserted that quantum chemistry, as well as the less powerful methods that preceded it (for example, Lewis’s theory), if they have not succeeded in giving quantitative answers to the questions of interest to the chemist, are in any case capable of characterizing the nature of the chemical bond in general terms. Along with the necessity of interpreting the nature of the bond in an individual molecule, physical chemistry was faced with the task of explaining the ability of molecules with already saturated valences to form more or less stable compounds with their neighbors. This ability for a long time seemed mysterious, although the exceptionally broad and important significance of the phenomena of intermolecular interaction had long been known to science. Deviations of the behavior of a real gas from the ideal, the properties of liquids and solids, the phenomena of capillarity and adsorption, etc., led to the necessity of introducing certain forces acting between molecules. The existence of these forces found its expression, in particular, in the correction for internal pressure in the van der Waals equation. However, up to the last two decades, this correction was introduced in a purely phenomenological way, and the mechanism of intermolecular forces remained obscure. Modern physics in principle explains completely the nature of these forces; however, exact calculations are possible by no means in all cases.

In our exposition we shall for the present restrict ourselves to those intermolecular forces which, like the correction for internal pressure, are called van der Waals forces. It should immediately be emphasized that, alongside these forces, intermolecular forces of a much greater order of magnitude are possible, close to the order of magnitude of chemical forces—the forces of complex formation. See below concerning them.

Modern physics distinguishes three types of van der Waals forces. Let us consider their essence ^20.

Each molecule is electrically neutral. However, if the centers of gravity of the positive and negative charges do not coincide, then such a molecule will possess a dipole moment. The presence of dipoles may be the cause of intermolecular interaction. The interaction energy will depend on the mutual arrangement of the dipoles. If two molecules possess dipole moments \(\mu_1\) and \(\mu_2\), then the dipoles tend to occupy a position with minimum potential energy: they turn toward each other with unlike poles and draw together. The interaction energy is

\[ U_1=\frac{\mu_1\mu_2}{R^3}\,[2\cos\theta_1\cos\theta_2-\sin\theta_1\sin\theta_2\cos(\varphi_1-\varphi_2)], \tag{4,1} \]

where \(R\) is the distance between the dipoles, and \(\theta\) and \(\varphi\) are the angles determining their mutual arrangement. This formula applies to two dipoles oriented in a definite manner. In the real case the molecules will be in a state of disordered thermal motion, and for the calculation it is necessary to average \(U_1\) over all possible positions of the dipoles, following Boltzmann statistics:

\[ \overline{U}_1= \frac{\displaystyle\int U_1 e^{-\frac{U_1}{kT}}\,d\tau} {\displaystyle\int e^{-\frac{U}{kT}}\,d\tau}. \tag{4,2} \]

The calculation gives

\[ \overline{U}_1=-\frac{2}{3R^6}\,\frac{\mu_1^2\mu_2^2}{kT}. \tag{4,3} \]

It is obvious that this effect—the Keesom orientation effect—must depend on temperature. In the limit, as \(T\to\infty\), complete disorder should set in, and the averaged interaction proves to be equal to zero. On the contrary, at absolute zero the interaction will be especially strong: the dipoles will be arranged parallel to one another \((\theta_1=\theta_2=0)\).

Up to now the discussion has concerned dipolar molecules. But a molecule that does not possess a permanent dipole moment is capable of acquiring one under the action of an external field—in particular, the field of a dipolar molecule—owing to the presence of polarizability \(\alpha\). Assuming that the force of the external field produces an elastic stretching of the molecule, we obtain for the energy \(U_2\) the following expression:

\[ U_2=-\frac{\alpha}{2}E^2. \tag{4,4} \]

If the field \(E\) is created by a neighboring molecule with moment \(\mu\), then

\[ E=\frac{\mu}{R^3}\sqrt{1+3\cos^2\theta} \tag{4,5} \]

and

\[ U_2=-\frac{\alpha}{2}\frac{\mu^2}{R^6}(1+3\cos^2\theta). \tag{4,6} \]

Averaging over all possible values of \(\theta\), we have the mean value of the interaction energy of molecule No. 1 with moment \(\mu_1\) and molecule No. 2 with polarizability \(\alpha_2\):

\[ (\bar U_2)_{1-2}=-\alpha_2\frac{\mu_1^2}{R^6} \tag{4,7} \]

and conversely

\[ (\bar U_2)_{2-1}=-\alpha_1\frac{\mu_2^2}{R^6}. \tag{4,8} \]

In the case of the induction interaction of two identical molecules with moment \(\mu\) and polarizability \(\alpha\), we obtain:

\[ \bar U_2=-2\frac{\alpha\mu^2}{R^6}. \tag{4,9} \]

This effect—the Debye induction effect—depends on temperature insofar as the induced moments in turn interact with one another. However, calculation shows that the influence of temperature is very small.

If a molecule has no dipole moment but has a quadrupole moment, etc., then such higher moments can also interact; however, the energy of their interaction decreases much more rapidly with distance. For quadrupole moments it is proportional to \(R^{-8}\) in the case of induction forces and to \(R^{-10}\) in the case of orientation forces.

However, experiment shows that even in the absence of both dipole and multipole moments, a definite interaction occurs between molecules with spherical symmetry of the electron cloud, for example between molecules of a noble gas. This compels us to think that, in addition to electrostatic forces, there exist forces acting also between spherically isotropic molecules. From the point of view of classical physics, the existence of such forces is completely incomprehensible. Indeed, as London showed, their nature is purely quantum.

Two noble-gas molecules located at a finite distance at absolute zero, from the point of view of classical physics, are absolutely motionless and do not affect one another in any way. However, quantum mechanics recognizes motion that does not cease even at \(T=0^\circ\mathrm{K}\). This is the so-called zero-point motion. Every particle capable of oscillating, be it an electron in an atom or a nucleus in a molecule, possesses the so-called zero-point energy, equal to

\[ U_0=\frac{h\nu}{2}, \tag{4,10} \]

where \(\nu\) is the frequency of oscillation and \(h\) is Planck’s constant. The existence of zero-point energy follows from Heisenberg’s uncertainty relation, according to which, when the coordinate is determined exactly—when particles are fixed in a definite region of space—we inevitably make an error in determining the momentum of the particles—...

turns out to possess a certain zero-point impulse, and hence also a zero-point energy. Any oscillator composed of electrons and nuclei turns out to be oscillating even at absolute zero.

Such “zero-point oscillations” of two particles may be regarded as coupled. At absolute zero the total energy of two particles, say, two atoms of a noble gas, is expressed by the sum of the zero-point oscillation energies of their electrons, diminished by a certain interaction energy, equal for two spherical isotropic oscillators to the quantity

\[ U_3=-\frac{3}{4}\frac{h\nu_0\alpha^2}{R^6}, \tag{4,11} \]

where \(\alpha\) is the polarizability of the atom. It is obvious that this effect will be independent of temperature, so long as the latter does not become so high as to cause a change in the electronic state of the molecule. The existence of this effect is determined by the dependence of the polarizability on the oscillations of the electrons, expressed by the dispersion formula:

\[ \alpha_k(\nu)=\frac{2}{3h}\sum_l \frac{\mu_{kl}^{\,2}\nu_{kl}}{\nu_{kl}^{\,2}-\nu^2}. \tag{4,12} \]

Thus depends the polarizability of a molecule (or atom) in state \(k\) on the external frequency \(\nu\). The summation extends over all quantum transitions \(k \to l\); \(\nu_{kl}\) are the corresponding frequencies, \(\mu_{kl}\) the “oscillator strengths”—the matrix elements of the corresponding moments. The final expression for the energy of the dispersion interaction of two molecules, one in state \(p\), and the other in state \(k\), has the form:

\[ (U_3)_{p-k}=\frac{2}{3hR^6}\sum_{la}\frac{\mu_{kl}^{\,2}\mu_{p\sigma}^{\,2}}{\nu_{kl}+\nu_{p\sigma}}. \tag{4,13} \]

A numerical estimate of the energy of the dispersive London interaction can be obtained with the aid of the formula

\[ U_3=-\frac{3}{2R^6}\frac{I_1I_2}{I_1+I_2}\alpha_1\alpha_2, \tag{4,14} \]

where \(I_1\)—\(I_2\)—are the ionization potentials of the outer electron. This is London’s estimate. Kirkwood \(^{21}\) estimates \(I\) as

\[ I=10.4\sqrt{\frac{N}{10^{24}\alpha}}\ \mathrm{eV}, \tag{4,15} \]

where \(N\) is the total number of electrons. Kirkwood and Slater \(^{22}\) take for \(N\) not the total number of electrons, but the number of electrons in the outer shell. Different estimates lead to different numerical results, which may differ from one another by several times, but are nevertheless of the same order of magnitude. The Kirkwood–Slater estimate пред-

is the best of those given. Hellmann\(^ {23}\) gives a more exact formula for \(U_3\):

\[ U_3=-\frac{1}{R^6}\frac{3}{2}\alpha_1\alpha_2\frac{I_1I_2}{I_1+I_2}\times \left[ 1+ \frac{ \xi_1 I_2\left(\frac{3}{2}+\frac{I_2}{I_1}\right) + \xi_2 I_1\left(\frac{3}{2}+\frac{I_1}{I_2}\right) }{ I_1+I_2 } \right], \tag{4,16} \]

where \(\xi=\frac{\alpha''}{\alpha}\), \(\alpha''\) is the part of the polarizability due to the inner electrons of the core (Rumpf); \(I\) is estimated according to Kirkwood—Slater.

In what follows, however, we shall use the simple formula (4,14), since we shall be interested only in the order of magnitude of the effect.

The additivity of dispersion forces is essential—in the case of many molecules they do not mutually cancel, but add up.

In most cases dispersion forces play a more significant role than the other two kinds of van der Waals forces. Only for molecules possessing large dipole moments do the orientation forces play the principal role. The induction effect is in most cases especially small. London gives the following data (Table 2).

TABLE 2

Gas	\(\mu\cdot10^{18}\)	\(\alpha\cdot10^{24}\)	\(h\nu_0(\mathrm{V})\)	Orientation effect \(\dfrac{2}{3}\dfrac{\mu^4}{k\,293}\cdot10^{60}\) \((\mathrm{erg}\cdot\mathrm{cm}^6)\)	Induction effect \(2\mu^2\alpha\cdot10^{60}\) \((\mathrm{erg}\cdot\mathrm{cm}^6)\)	Dispersion effect \(\dfrac{3}{4}\alpha^2h\nu_0\cdot10^{60}\) \((\mathrm{erg}\cdot\mathrm{cm}^6)\)	Sum
CO	0.12	1.99	14.3	0.0034	0.057	67.5	67.56
HJ	0.38	5.4	12	0.35	1.68	382	384.03
HBr	0.78	3.58	13.3	6.2	4.05	176	186.25
HCl	1.03	2.63	13.7	18.6	5.4	105	129
NH\(_3\)	1.5	2.21	16	84	10	93	187
H\(_2\)O	1.84	1.48	18	190	10	47	247

We see that only in the case of water are the dispersion forces smaller than the orientation forces.

§ 6. Raman Effect and van der Waals Forces

1. Electric Field

As we have seen, Raman frequencies are determined by the existence of definite valence bonds in the molecule. In the intermolecular interaction of two or a larger number of molecules, in the case where the energy of such interaction is, in order of magnitude, close to the usual chemical energy, new bonds arise, the molecules lose their individuality, and significant changes in the spectrum may be expected. On the one hand, new lines may appear; on the other—

former lines may change their position and intensity. If the interaction energy is less than that of an ordinary chemical one, but still large enough to create definite directed bonds, as occurs in crystals with a molecular lattice, then the spectrum of the individual molecule remains unchanged, and in addition there appear frequencies of intermolecular vibrations with low numerical values, of the order of the rotational frequencies in the spectrum of the molecule. Below we shall examine in detail the most important cases of this kind of interaction. But first of all let us consider what happens in the Raman spectrum of a molecule in the presence of comparatively weak van der Waals forces of intermolecular interaction, which do not create new bonds and in general do not alter the individuality of the molecule. In other words, we are interested in the question of what changes, for example, take place in the Raman spectrum upon the transition of a substance from the gaseous state to the liquid state, upon its dissolution, etc. Experience shows that even in those cases where no new lines appear characteristic of new vibrational possibilities, i.e., in those cases where such possibilities are not created, nevertheless changes occur in the frequencies, intensities, and widths of the Raman lines of a substance when its state of aggregation is changed, etc. In this and the following paragraph we shall attempt to give a qualitative theory of these phenomena.

Van der Waals forces are, as we have seen, forces of electrical origin. By placing a molecule in a liquid, we as it were introduce a certain electrostatic field with a potential equal to the potential of the van der Waals forces. This is so, at least, for the orientational and induction effects. Thus one may approach the solution of the problem of the influence of van der Waals forces on the Raman spectrum by considering the question of the influence of an electrostatic field on the Raman spectrum.

Attempts to detect in the Raman spectrum a phenomenon analogous to the Stark effect in atomic spectra have been undertaken more than once. By placing a substance in a strong electrostatic field, investigators looked for the appearance of new frequencies, the splitting of old frequencies, and other changes. In no case was such an effect observed, despite the large magnitude of the applied fields. Sirkar^24 managed to note only small changes in the degree of depolarization of certain lines in cyclohexane, benzene, and chlorobenzene upon the application of fields of 15,000–25,000 V/cm. However, these results are not yet entirely reliable.

Buchheim^25 made an attempt at a quantitative estimate of the influence of an electric field on the Raman spectrum of a molecule. His work is based on very simple premises and constitutes an extremely simplified classical treatment of the question.

Let us imagine a diatomic molecule vibrating with its fundamental frequency

$$ \omega_0=\sqrt{\frac{k}{m}}, $$

with values of the polarizability along the axis of the molecule equal to $\alpha_1$, and perpendicular to it equal to $\alpha_2$. The polarizability tensor may in this case be represented by an ellipsoid of rota-

RAMAN EFFECT AND INTERMOLECULAR INTERACTION

tion. Let us impose on this molecule an electric field \(\mathbf{E}\) with components: \(E_1\)—along the axis of the molecule and \(E_2\)—perpendicular to it. The external field will, on the one hand, tend to rotate the molecule in its own direction, if the molecule possesses a permanent moment, \(\mu\)—the orientational effect. On the other hand, it will induce in the molecule a certain moment—the induction effect. The additional energy is equal to

\[ V=-\mu E_1-\frac{1}{2}\alpha_1 E_1^2+\alpha_2 E_2^2; \tag{5.1} \]

The total energy of the molecule in the field \(\mathbf{E}\)

\[ H=\frac{1}{2}m\dot q^{\,2}+\frac{1}{2}kq^2+V, \tag{5.2} \]

where \(q\) is the relative displacement of the constituent atoms from the equilibrium position. We assume that \(q\) does not change—the changes of frequency due to the change in the bond force are much smaller than those obtained owing to the dependence of \(V\) on \(q\). Restricting ourselves in the expansion of \(\mu\) and \(\alpha\) in a series in \(q\) to terms of the second order, we have:

\[ V=-E_1\left(\mu_0+\frac{\partial\mu}{\partial q}q+\frac{1}{2}\frac{\partial^2\mu}{\partial q^2}q^2\right)- \]

\[ -\frac{E_1^2}{2}\left(\alpha_1^{(0)}+\frac{\partial\alpha_1}{\partial q}q+\frac{\partial^2\alpha_1}{\partial q^2}q^2\right)- \]

\[ -\frac{E_2^2}{2}\left(\alpha_2^{(0)}+\frac{\partial\alpha_2}{\partial q}q+\frac{\partial^2\alpha_2}{\partial q^2}q^2\right). \tag{5.3} \]

The equation of motion

\[ m\ddot q=-\frac{\partial}{\partial q}(U+V) \tag{5.4} \]

will have the form

\[ m\ddot q+\left(m\omega_0^2-\frac{\partial^2\mu}{\partial q^2}E_1-\frac{1}{2}\frac{\partial^2\alpha_1}{\partial q^2}E_1^2-\frac{1}{2}\frac{\partial^2\alpha_2}{\partial q^2}E_2^2\right)q= \]

\[ =E_1\frac{\partial\mu}{\partial q}+\frac{E_1^2}{2}\frac{\partial\alpha_1}{\partial q}+\frac{E_2^2}{2}\frac{\partial\alpha_2}{\partial q}. \tag{5.5} \]

We seek the solution in the form

\[ q=\bar q+q_0\cos(\omega t+\varphi). \tag{5.6} \]

Substitution gives

\[ \bar q= \frac{ E_1\dfrac{\partial\mu}{\partial q} +\dfrac{E_1^2}{2}\dfrac{\partial\alpha_1}{\partial q} +\dfrac{E_2^2}{2}\dfrac{\partial\alpha_2}{\partial q} }{ m\omega_0^2 -E_1\dfrac{\partial^2\mu}{\partial q^2} -\dfrac{E_1^2}{2}\dfrac{\partial^2\alpha_1}{\partial q^2} -\dfrac{E_2^2}{2}\dfrac{\partial^2\alpha_2}{\partial q^2} } \tag{5.7} \]

and

\[ \omega= \sqrt{ \omega_0^2 -\frac{E_1}{m}\frac{\partial^2\mu}{\partial q^2} -\frac{E_1^2}{2m}\frac{\partial^2\alpha_1}{\partial q^2} -\frac{E_2^2}{2m}\frac{\partial^2\alpha_2}{\partial q^2} }. \tag{5.8} \]

As a result of the action of the external electric field, the equilibrium distance changes by \(\overline q\). In addition, the Raman line undergoes, on the one hand, a broadening depending on \(E\) and on the orientation of the molecule relative to the field—such will be the effect, averaged over all molecules, of the term \(\dfrac{E_1}{m}\dfrac{\partial^2\mu}{\partial q^2}\); on the other hand, a shift of the center of the line associated with terms quadratic in \(E_1\) and \(E_2\). Taking the terms with \(E_1\), \(E_1^2\), \(E_2^2\) as small in comparison with \(\omega_0^2\) and expanding the root in a series, we have for the broadening of the line

\[ \Delta \omega=\frac{E}{2m}\frac{1}{\omega_0}\frac{\partial^2\mu}{\partial q^2}. \tag{5,9} \]

Putting \(\dfrac{\partial^2\alpha_1}{\partial q^2}=\dfrac{\partial^2\alpha_2}{\partial q^2}\), we have for the shift of the center of the line

\[ \delta \omega=-\frac{E^2}{4m}\frac{1}{\omega^2}\frac{\partial^2\alpha_1}{\partial q^2}. \tag{5,10} \]

For a numerical estimate of the quantities \(\Delta\omega\) and \(\delta\omega\), it is necessary to estimate \(\dfrac{\partial^2\mu}{\partial q^2}\) and \(\dfrac{\partial^2\alpha}{\partial q^2}\).

Buchheim obtains the order of magnitude of these derivatives from the ratios between the intensities of the fundamental tone and the overtone in the Raman spectrum (for \(\alpha\)) and in the infrared absorption spectrum (for \(\mu\)), measured respectively by Weiler \(^{26}\) and Imes \(^{27}\). Indeed, according to the classical approximate theory of the Raman effect [see equation (1, 4)] we have for the ratio of the intensities of the Rayleigh line and the fundamental tone in the Raman spectrum

\[ \frac{I_{\nu\to\nu_0}}{I_\nu}\simeq \frac{\left(\dfrac{\partial\alpha}{\partial q}\right)^2 q_0^2}{\alpha^2} \tag{5,11} \]

and for the ratio of the intensities of the overtone and the fundamental tone

\[ \frac{I_{\nu\to 2\nu_0}}{I_{\nu\to\nu_0}}\simeq \frac{\left(\dfrac{\partial^2\alpha}{\partial q^2}\right)^2 q_0^4} {\left(\dfrac{\partial\alpha}{\partial q}\right)^2 q_0^2}. \tag{5,12} \]

Similarly, for the absorption coefficients of the fundamental tone and overtone in the infrared spectrum we have

\[ \frac{\beta_{\nu_0}}{\beta_{2\nu_0}}\simeq \frac{\left(\dfrac{\partial\mu}{\partial q}\right)^2 q_0^2} {\left(\dfrac{\partial^2\mu}{\partial q^2}\right)^2 q_0^4}. \tag{5,13} \]

Since all these ratios are known from experiment, then, proceeding from the orders of magnitude

\[ \alpha\sim 10^{-24}\,\text{cm}^3;\quad \mu\sim 10^{-18}\,CGSE;\quad q_0\sim 10^{-9}\,\text{cm}, \]

we obtain, first,

\[ \frac{\partial \chi}{\partial q}\sim 10^{-16}\ \mathrm{cm}^2 \quad\text{and}\quad \frac{\partial^2 \alpha}{\partial q^2}\sim 10^{-8}\ \mathrm{cm}. \]

To determine the quantity \(\dfrac{\partial^2\mu}{\partial q^2}\), one must know \(\dfrac{\partial\mu}{\partial q}\). This quantity is the so-called effective charge. For molecules of the type X—X, \(\dfrac{\partial\mu}{\partial q}\) is, obviously, equal to zero. For polar molecules and for optically active vibrations of polyatomic nonpolar molecules, \(\dfrac{\partial\mu}{\partial q}\ne 0\), and we can obtain information about its numerical value in several ways. Let us consider the following equations, expressing the value of the polarization of a molecule:

\[ P=\frac{\varepsilon-1}{\varepsilon+2}\frac{M}{d}=P_E+P_\mu+P_A, \tag{5,14} \]

where \(\varepsilon\) is the dielectric constant, \(M\) the molecular weight, \(d\) the density, and \(P_E\), \(P_\mu\), and \(P_A\) are the electronic, dipole, and atomic polarizations—quantities having the following values:

\[ P_E+P_\mu=\frac{4\pi N}{3}\left(\alpha+\frac{\mu^2}{3kT}\right) \tag{5,15} \]

(for an external frequency \(\nu=0\); \(\alpha\) is the electronic polarization)

and

\[ P_A=\frac{\left(\dfrac{\partial\mu}{\partial q}\right)^2N}{3\pi m\nu_0^2}, \tag{5,16} \]

where \(m\) is the reduced mass, and \(\nu_0\) is the natural frequency of vibration of the molecule. For the refractive index in the visible region we have

\[ \frac{n^2-1}{n^2+2}\frac{M}{d}=P_L. \tag{5,17} \]

Finally, for the overwhelming majority of substances in the solid state, the entire polarization consists of electronic and atomic polarizations:

\[ P_{Tb}=P_L+P_A. \tag{5,18} \]

Comparing equations (5,14) and (5,17) and (5,18), we can determine \(\dfrac{\partial\mu}{\partial q}\), which proves to be of the order of magnitude \(1\cdot10^{-9}\) CGSE.

However, these determinations are very inaccurate, and an error even by a factor of 10 is possible.

A second method of determining \(\dfrac{\partial\mu}{\partial q}\) is the measurement of the absolute intensity of absorption in the infrared region. The measurements of Burgin\({}^{28}\) for HCl give the value \(\dfrac{\partial\mu}{\partial q}\sim 0.86\cdot10^{-10}\) CGSE. Barto-

Lome[^29] gives an approximately two times smaller value. However, these studies also cannot be relied upon because of the great difficulties that accompany measurements in the infrared region. The most accurate, according to Van Vleck[^30], should be considered the third method of determining \(\dfrac{\partial \mu}{\partial q}\) from the dispersion in the infrared region. The atomic refraction for an infrared frequency is equal to

\[ P-\frac{\left(\dfrac{\partial \mu}{\partial q}\right)^2 N}{3\pi m(\nu_0^2-\nu^2)}. \tag{5,19} \]

The values of \(\dfrac{\partial \mu}{\partial q}\) for the active vibrations of \(\mathrm{CO}_2\), accompanied by changes of the dipole moment, turned out to be of the order of \(1\cdot 10^{-9}\) CGSE and \(3\cdot 10^{-10}\) CGSE[^31].

Thus, at present we may say that \(\dfrac{\partial \mu}{\partial q}\) has the order of magnitude \(10^{-9}\)—\(10^{-10}\) CGSE. Taking the value \(1\cdot 10^{-9}\), exaggerated by approximately a factor of two, Bouguer obtains for \(\dfrac{\partial^2 \mu}{\partial q^2}\) the value \(10^{-10}\) CGSE. Using the indicated values of the derivatives for the HCl molecule, which has a fundamental frequency \(\nu \sim 3000\ \mathrm{cm}^{-1}\), i.e. \(\omega_0 = 5.4\cdot 10^{-14}\ \mathrm{sec}^{-1}\), at a field strength of \(10^5\ \mathrm{V/cm}\), we have:

\[ \delta\omega \sim 10^5\ \mathrm{sec}^{-1} \]

\[ \Delta\omega \sim 10^9—10^{10}\ \mathrm{sec}^{-1}, \]

whence

\[ \Delta\lambda \sim 10^{-2}\ \text{\AA} \]

\[ \delta\lambda \sim 10^{-6}\ \text{\AA} \]

\[ q \sim 10^{-4}\ \text{\AA}. \]

We see that the changes caused even by the strongest fields obtainable in laboratory practice lie below the limit of observation. In addition to changes in the frequency and width of the lines, one may expect changes in the intensity of the lines and in the degree of their depolarization. The intensity of a Raman line can be expressed classically by the formula

\[ I=\mathrm{const}\left(15A^2+\frac{7}{6}B\right), \tag{5,20} \]

where

\[ A=\frac{\alpha_1' + \alpha_2' + \alpha_3'}{3} \tag{5,21} \]

and

\[ B=(\alpha_1'-\alpha_2')^2+(\alpha_2'-\alpha_3')^2+(\alpha_3'-\alpha_1')^2 \tag{5,22} \]

\[ \alpha_i=\frac{\partial x_i}{\partial q}, \]
the components of the polarizability tensor referred to the principal axes. In the absence of a field we had the tensor

\[ \left| \begin{array}{ccc} \dfrac{\partial \alpha_1}{\partial q} & 0 & 0\\[6pt] 0 & \dfrac{\partial \alpha_2}{\partial q} & 0\\[6pt] 0 & 0 & \dfrac{\partial \alpha_3}{\partial q} \end{array} \right| \tag{5.23} \]

In the presence of a field,

\[ \left| \begin{array}{ccc} \dfrac{\partial \alpha_1}{\partial q}+\dfrac{\partial^2\alpha_1}{\partial q^2}\,\bar q & 0 & 0\\[8pt] 0 & \dfrac{\partial \alpha_2}{\partial q}+\dfrac{\partial^2\alpha_2}{\partial q^2}\,\bar q & 0\\[8pt] 0 & 0 & \dfrac{\partial \alpha_3}{\partial q}+\dfrac{\partial^2\alpha_3}{\partial q^2}\,\bar q \end{array} \right| \tag{5.24} \]

In connection with this, the intensity and the degree of depolarization will change, the latter being determined by the ratio between the isotropic and anisotropic parts, \(\alpha'\)—between \(A\) and \(B\).

Up to now we have been speaking of an external electric field. But what interests us is intermolecular interaction. This problem can be treated analogously, with the difference that the fields acting between molecules are 100–1000 times greater than those accessible to experiment. Then

\[ \delta x \sim 1\ \text{\AA}, \]

\[ \Delta\lambda \sim 1\text{—}10\ \text{\AA}, \]

i.e., quantities quite accessible to observation. The intensity, too, can vary noticeably, up to several percent.

§ 6. Raman Effect and van der Waals Forces

2. Sequential Calculation

Buchheim does not, however, give a comparative estimate of the influence of the various kinds of van der Waals forces—orientational, induction, and dispersion—on the Raman spectrum of a molecule. Moreover, the method proposed by him is in principle inexact for the further reason that one cannot consider an isolated molecule and take into account the influence of its neighbors as an external action. It should be taken into account that no molecule is free, and its vibrations are coupled with the vibrations of the surrounding molecules. In Buchheim’s work there is an indication of the importance of molecular resonance, but he does not carry out a sequential calculation. Meanwhile, it is very easy to do this, proceeding from the same approximate representations.

Let us restrict our consideration to two identical molecules. We shall regard them as two identical pendulums connected by a spring (Fig. 2). The role of the spring will be played here by the van der Waals forces. Their existence produces a transfer of vibrational energy from one pendulum to the other and a splitting of the natural frequencies of the free pendulums. Let us consider this phenomenon mathematically, solving the problem by the general method for solving the problem of coupled vibrations. For this it is first necessary to write the equations of motion of the system.

Fig. 2.

The kinetic energy of two such oscillators is expressed by the equation

\[ T=\frac{1}{2}m(\dot q_1^{\,2}+\dot q_2^{\,2}). \tag{6,1} \]

The potential energy contains, in addition to the elastic energy of the vibrations, a coupling term

\[ U=\frac{1}{2}m\omega_0^{2}(q_1^{2}+q_2^{2})+V, \tag{6,2} \]

where \(V\) is the energy of the van der Waals interaction, equal to

\[ V=-\frac{1}{R^{6}}\left(\frac{2}{3kT}\mu_1^{2}\mu_2^{2}+\mu_1^{2}\alpha_2+\mu_2^{2}\alpha_1+\frac{3}{4}I\alpha_1\alpha_2\right). \tag{6,3} \]

The equations of motion have the form

\[ \left\{ \begin{aligned} m\ddot q_1+m\omega_0^{2}q_1+\frac{\partial V}{\partial q_1}&=0,\\ m\ddot q_2+m\omega_0^{2}q_2+\frac{\partial V}{\partial q_2}&=0. \end{aligned} \right. \tag{6,4} \]

They must be solved simultaneously. First of all we shall calculate \(\frac{\partial V}{\partial q}\). For this purpose, as before, in expression (6, 3) we expand \(\mu_1\) and \(\alpha_1\) in powers of \(q_1\) and \(\mu_2\) and \(\alpha_2\) in powers of \(q_2\), retaining terms up to the second order. Such an expansion without simultaneous expansion of \(R\) is permissible for \(R \gg q\), which indeed holds under real conditions \((R\sim 10^{-8}\ \text{cm},\ q\sim 10^{-9}\ \text{cm})\). We take the derivatives and retain in their expressions the terms linear with respect to \(q\). As a result, equations (6, 4) are transformed into the form

\[ \left\{ \begin{aligned} m\ddot q_1+(m\omega_0^{2}-B)q_1-Aq_2&=C,\\ m\ddot q_2+(m\omega_0^{2}-B)q_2-Aq_1&=C, \end{aligned} \right. \tag{6,5} \]

where \(A\), \(B\), and \(C\) are composed of terms characteristic of the orien-

tational, induction, and dispersion interactions (indices \(o, i, d\)); moreover

\[ \left\{ \begin{aligned} A_o&=\frac{8}{3kTR^6}\mu^2\left(\frac{\partial\mu}{\partial q}\right)^2;\quad B_o=\frac{4}{3kTR^6}\left\{\mu^2\left(\frac{\partial\mu}{\partial q}\right)^2+\mu^3\frac{\partial^2\mu}{\partial q^2}\right\}\\[4pt] A_i&=\frac{4}{R^6}\mu\frac{\partial\mu}{\partial q}\frac{\partial\alpha}{\partial q};\quad B_i=\frac{2}{R^6}\left\{\alpha\left(\frac{\partial\mu}{\partial q}\right)^2+\alpha\mu\frac{\partial^2\mu}{\partial q^2}+\mu^2\frac{\partial^2\alpha}{\partial q^2}\right\}\\[4pt] A_d&=\frac{3I}{4R^6}\left(\frac{\partial\alpha}{\partial q}\right)^2;\quad B_d=\frac{3I}{4R^6}\alpha\frac{\partial^2\alpha}{\partial q^2}\\[6pt] C_o&=\frac{4}{3kTR^6}\mu^3\frac{\partial\mu}{\partial q}\\[4pt] C_i&=\frac{1}{R^6}\left\{2\alpha\mu\frac{\partial\mu}{\partial q}+\mu^2\frac{\partial\alpha}{\partial q}\right\}\\[4pt] C_d&=\frac{3I}{4R^6}\alpha\frac{\partial\alpha}{\partial q} \end{aligned} \right. \tag{6,6} \]

We seek solutions in the form

\[ \left\{ \begin{aligned} q_1&=\bar q_1+q_{10}\cos(\omega t+\varphi)\\ q_2&=\bar q_2+q_{20}\cos(\omega t+\varphi) \end{aligned} \right\} \tag{6,7} \]

\(\omega\) are the normal frequencies of the coupled system. Substituting and equating separately to zero the constant terms and the terms depending on time, we obtain two systems of equations:

\[ \left\{ \begin{aligned} (m\omega_0^2-B)\bar q_1-A\bar q_2&=C\\ -A\bar q_1+(m\omega_0^2-B)\bar q_2&=C \end{aligned} \right\} \tag{6,8} \]

and

\[ \left\{ \begin{aligned} [m(\omega_0^2-\omega^2)-B]q_{10}-Aq_{20}&=0\\ -Aq_{10}+[m(\omega_0^2-\omega^2)-B]q_{20}&=0 \end{aligned} \right\} \tag{6,9} \]

From (6,8) we find

\[ \bar q_1=\bar q_2=\frac{C}{m\omega_0^2-A-B}=q. \tag{6,10} \]

Such is the displacement of the equilibrium distance caused by the van der Waals forces. For the solvability of the second system (homogeneous equations) it is necessary that the determinant of the system be equal to zero

\[ \left| \begin{array}{cc} m(\omega_0^2-\omega^2)-B & -A\\ -A & m(\omega_0^2-\omega^2)-B \end{array} \right|=0 \tag{6,11} \]

or

\[ m^2\omega^4-2(m^2\omega_0^2-mB)\omega^2-A^2+m^2\omega_0^4-2mB\omega_0^2=0 \tag{6,12} \]

and

\[ \omega^2=\omega_0^2-\frac{B}{m}\pm\frac{\sqrt{A^2+B^2}}{m}. \tag{6,13} \]

Denoting \(\omega_0-\dfrac{B}{m}=\omega'^2\) and \(A^2+B^2=D^2\), we have

\[ \omega^2=\omega'^2-\frac{D}{m};\quad \omega=\sqrt{\omega'^2 \pm \frac{D}{m}} \tag{6,14} \]

and, taking \(\dfrac{D}{m}\ll\omega'^2\), approximately

\[ \omega=\omega' \pm \frac{1}{2}\frac{D}{m\omega'}; \tag{6,15} \]

taking \(\dfrac{B}{m}\ll\omega_0^2\), we have

\[ \omega'=\omega_0-\frac{1}{2}\frac{B}{m\omega_0}. \tag{6,16} \]

Finally

\[ \omega=\omega_0-\frac{1}{2}\frac{B}{m\omega_0}\pm \frac{1}{2}\frac{D}{m\omega'}, \tag{6,17} \]

and

\[ q=\frac{C}{m\omega_0^2-A-B}. \tag{6,10'} \]

For many molecules we again obtain the same three effects as in Buckingham’s calculations: 1. A shift of the center of the band, determined by

\[ \delta\omega=\frac{1}{2}\frac{B}{m}\omega_0, \tag{6,18} \]

1. Broadening of the band

\[ \Delta\omega=\frac{D}{m\omega'} \tag{6,19} \]

and 3. A change of the equilibrium distance, determined by (6,10)a. Let us estimate these effects numerically. Both \(A\), and \(B\), and \(C\) are additively composed of terms characteristic of the three types of van der Waals interaction, which we shall denote by the indices \(o, i, d\). Let us carry out the calculation again for the HCl molecule with \(\omega_0\simeq 5.4\cdot 10^{14}\ \mathrm{sec}^{-1}\) and \(m\simeq 1.7\cdot 10^{-24}\ \mathrm{g}\). The distance between molecules in a liquid \(R\), for the densest packing, is \(\simeq 4.2\cdot 10^{-8}\ \mathrm{cm}\), \(I\simeq 10\ \mathrm{eV}\simeq 1.5\cdot 10^{-12}\ \mathrm{erg}\), \(T=300^\circ\). The values of \(\delta\omega\), \(\Delta\omega\), and \(\bar q\) will be determined by the quantities \(\dfrac{\partial\mu}{\partial q}\), \(\dfrac{\partial^2\mu}{\partial q^2}\), \(\dfrac{\partial\alpha}{\partial q}\), \(\dfrac{\partial^2\alpha}{\partial q^2}\). For these derivatives we take the former orders of magnitude. The calculation gives the same orders of magnitude both for \(\delta\omega\) and for \(\Delta\omega\), namely:

For the orientational interaction

\[ \frac{\Delta\omega}{\omega_0}\sim 0.04\% \text{ for } \frac{\partial\mu}{\partial q}=1\cdot 10^{-10}; \quad \frac{\Delta\omega}{\omega_0}\sim 1\% \text{ for } \frac{\partial\mu}{\partial q}=5\cdot 10^{-10}; \]

\[ \frac{\Delta\omega}{\omega_0}\sim 4\% \text{ for } \frac{\partial\mu}{\partial q}=1\cdot 10^{-9}. \]

Raman Effect and Intermolecular Interaction

The same orders of magnitude of $\omega_0$ obtain for $\dfrac{\delta\omega}{\omega_0}$.

For induction interaction

\[ \frac{\Delta\omega}{\omega_0}\sim 0.020\% \quad \text{even for} \quad \frac{\partial\mu}{\partial q}=1\cdot 10^{-9} \]

and for dispersion interaction

\[ \frac{\Delta\omega}{\omega_0}\sim 0.005\% \quad \text{for} \quad \frac{\partial\mu}{\partial q}=1\cdot 10^{-9}. \]

As for the magnitude $\bar q$, its values are very small, namely

\[ q_0\sim 10^{-11}\ \text{cm}; \qquad q_i\sim 10^{-13}\ \text{cm}; \qquad q_d\sim 10^{-13}\ \text{cm}, \]

$\dfrac{\partial\mu}{\partial q}$ is probably greater than $1\cdot 10^{-10}$ and undoubtedly less than $1\cdot 10^{-9}$. We see that a noticeable effect, reaching the order of a percent of the frequency value, is obtained only for orientational interaction, despite the fact that the energy of the dispersion interaction is considerably higher. This is explained by the fact that the dipole moment changes more strongly with $q$ than does the polarizability, and it is precisely the values of the derivatives that determine the magnitudes $\Delta\omega$ and $\delta\omega$. Experiment shows—

TABLE 3

Substance	$\omega_0$ (cm$^{-1}$), gas	$\omega_0$ (cm$^{-1}$), liquid	$\delta\omega$	%	$\mu\cdot 10^{18}$	Note
H$_2$	4162	4149	13	0.3	0
N$_2$	2331	2326	5	0.2	0
O$_2$	1555	1550	5	0.3	0
N$_2$O	1286	1282	4	0.3
N$_2$O	2229	2223	6	0.3	0.14
SnCl$_4$	367	368	−1	0	0	in the gas only 2 frequencies were observed
SnCl$_4$	400	403	−3	0	0	in the gas only 2 frequencies were observed
PCl$_3$	523	511	12	2.3	1.1	frequencies of completely symmetric vibrations
AsCl$_3$	422	405	17	4.0	2.1	frequencies of completely symmetric vibrations
AsBr$_3$	287	276	11	4.0	1.63	frequencies of completely symmetric vibrations
SbCl$_3$	382	355	27	7.0	3.1—4.6	frequencies of completely symmetric vibrations
SF$_6$	772	776	−4	0	0	frequencies of completely symmetric vibrations
H$_2$S	2615	2576	39	1.5	0.93	frequencies of completely symmetric vibrations
HCl	2886	2770	116	4.0	1.04	frequencies of completely symmetric vibrations
HBr	2558	2466	92	3.6	0.79	frequencies of completely symmetric vibrations
HJ	2233	2165	68	3.1	0.38	frequencies of completely symmetric vibrations
SO$_2$	546	526	20	4.0
SO$_2$	1152	1146	6	0.5	1.61
SO$_2$	1361	1340	21	1.4

shows that Raman lines of dipolar liquids are indeed broader than those of nonpolar ones. The frequency shift in the transition from gas to liquid is likewise more considerable for dipolar substances. It will be appropriate here to give a table of Raman frequencies in the liquid and gaseous states (Table 3). We see that the shifts are especially large for dipolar substances, and they are the larger the higher the dipole moment. For the HCl molecule the effect reaches 4% of the value of the frequency, which is several times greater than the effect calculated by us $\left(\dfrac{\partial \lambda}{\partial q}\right)$ for HCl, undoubtedly less than $5 \cdot 10^{-10}$), although their orders of magnitude coincide. Such a discrepancy is understandable. Both Buchheim’s calculation and our calculation are very rough and approximate. First, the method itself is in principle inaccurate, since it is based on a crude classical theory of the phenomenon. Second, reducing the interaction to a coupling between the vibrations of only two molecules is likewise certainly incorrect. An error is also introduced by restricting the expression for the derivative to linear terms. Finally, the numerical estimates of the derivatives $\alpha$ and $\mu$ are very inaccurate. Thus these calculations can give only the order of magnitude of the effect. But a number of conclusions of a physical nature remain correct. Such are the general results of the consideration, which give us the causes of the appearance of the shift, the broadening of the lines, and the change of the equilibrium distance. The form of the dependence of $\Delta \omega$, $\delta \omega$, and $\bar q$ on the derivatives $\mu$ and $\alpha$, on $\omega_0$ and on the reduced mass of the molecule is physically correct. In addition, the fact that only orientational interaction can be the cause of the observed changes in the spectrum, while induction and dispersion interactions do not give an appreciable effect, is essential and reliable. Thus, the considerations set forth have a certain scientific value, making it possible to draw a number of conclusions about the influence of van der Waals forces on the Raman spectrum of a molecule.

In the experimental part of his work Buchheim investigated changes in the intensities of the Raman lines of benzene in binary mixtures with cyclohexane and chloroform, $\mathrm{CCl}_4$ with acetone and benzene, and $\mathrm{CHCl}_3$ with benzene and cyclohexane. He was able to observe changes in the relative intensities by several percent, as well as broadening of the lines. The latter have often been observed in binary mixtures by other authors as well.

The existence of intermolecular forces should be associated with the shift and blurring of Raman frequencies in binary mixtures. This effect is essentially analogous to the effect of a change in the aggregate state. Such shifts have by no means always been recorded—mostly additivity of the spectra has occurred. Unfortunately, measurements of the intensities and polarization of the lines were made only in a few cases. Let us list some of the most interesting works.

Dadieu and Kohlrausch31 were the first to undertake an extensive investigation of binary mixtures. They studied Raman spectra of mixtures of benzene with chloroform, acetic and benzoic acids, and of chloroform with eth—

romide, methyl acetate and acetone, ethanol with hexachloroethane, acetic and benzoic acids, ether with the same substances, acetic acid with water, nitrobenzene with carbon tetrachloride, and nitromethane with methanol. In almost all cases additivity of the spectra occurred within the limits of experimental error (up to \(5\ \mathrm{cm}^{-1}\)). However, in mixtures with carboxylic acids weak but noticeable changes were observed, especially in the frequencies belonging to the \(C=O\) group—the constituent part of the molecule possessing the largest dipole moment. Here both a shift and, in some cases, a splitting of the frequency occurred. For example, the frequency \(1669\ \mathrm{cm}^{-1}\) of pure acetic acid takes, in an equimolecular mixture with \(\mathrm{C_2H_5OH}\), the value \(1706\ \mathrm{cm}^{-1}\), and with \(\mathrm{C_6H_6}\)—\(1656\ \mathrm{cm}^{-1}\), falling on raising the temperature to \(16^\circ\) and to \(1641\ \mathrm{cm}^{-1}\). In a mixture with ether two frequencies appear: \(1664\) and \(1750\ \mathrm{cm}^{-1}\). However, in a mixture of acetic acid with such a dipolar substance as pyridine, complete additivity takes place[^32]. We shall return again to acetic acid in § 7.

Brodsky, Zak, and Bezugly studied the spectra of \(\mathrm{AsCl_3}\) in \(\mathrm{C_6H_6}\), \(\mathrm{CCl_4}\), and alcohols[^33]. The changes found are too small for any definite significance to be ascribed to them. This is also confirmed by the work of Kurnosova and Ashkinazi[^34], [^35] on the spectra of \(\mathrm{AsCl_3}\) and \(\mathrm{AsBr_3}\) in \(\mathrm{C_6H_6}\) and \((\mathrm{C_2H_5})_2\mathrm{O}\) and of \(\mathrm{SbCl_3}\) in \((\mathrm{C_2H_5})_2\mathrm{O}\). On the contrary, \(\mathrm{SbCl_3}\) in \(\mathrm{C_6H_6}\) gives new lines, about which see below. Of interest are the changes undergone by the frequency of vibration of the HCl molecule in various solvents. Let us compare the data in Table 4.

TABLE 4

HCl

State / solution	Frequency	Width
Gas \(1\ atm\)[^36]	\(2885\ \mathrm{cm}^{-1}\)	Width \(35\ \mathrm{cm}^{-1}\)
Liquid[^37]	\(2776\ \mathrm{cm}^{-1}\)	\(45\ \mathrm{cm}^{-1}\)
Solutions \(0.2\ m\) in[^38]: \(\mathrm{SiCl_4}\)	\(2860\ \mathrm{cm}^{-1}\)
Solutions \(0.2\ m\) in[^38]: \(\mathrm{PCl_3}\)	\(2824\ \mathrm{cm}^{-1}\)
Solutions \(0.2\ m\) in[^38]: \(\mathrm{CHCl_3}\)	\(2826\ \mathrm{cm}^{-1}\)
Solutions \(0.2\ m\) in[^38]: \(\mathrm{CH_3COCl}\)	\(2804\ \mathrm{cm}^{-1}\)
Solutions \(0.2\ m\) in[^38]: \(\mathrm{SO_2}\)	\(2798\ \mathrm{cm}^{-1}\)
Solutions \(0.2\ m\) in[^38]: \(\mathrm{C_2H_5Br}\)	\(2797\ \mathrm{cm}^{-1}\)
Solid[^39]	\(2763\ \mathrm{cm}^{-1}\)	\(40\ \mathrm{cm}^{-1}\)

Simon and Feher[^40] investigated the Raman spectra of dioxane in mixtures with \(\mathrm{H_2O}\), \(\mathrm{H_2O_2}\), \(\mathrm{CCl_4}\), \(\mathrm{CH_3CN}\), \(\mathrm{NH_2CH_3}\), \(\mathrm{HCN}\), \(\mathrm{NH_3}\), \(\mathrm{HCl}\), \(\mathrm{PCl_3}\), and \(\mathrm{AsCl_3}\). They succeeded in observing appreciable shifts only in the most dipolar solvents, \(\mathrm{H_2O}\) and \(\mathrm{H_2O_2}\). A number of works were devoted to solutions in alcohols. Krishnamurti[^41] found that, in a mixture with water, the \(\nu(C-O)\) frequency of methanol decreases from \(1034\) to \(1018\ \mathrm{cm}^{-1}\) (\(\sim 1.5\%\)). Similar effects were observed by Gubo[^42] when dissolving \(\mathrm{LiClO_4}\) in alcohol and by Bore[^43] when dissolving \(\mathrm{LiBr}\). With ethanol the effect is considerably weaker. Gibben[^44] found that \(\mathrm{ZnCl_2}\) in methanolic solution lowers the \(\nu C-O\) frequency from \(1034\) to \(1012\ \mathrm{cm}^{-1}\) (by \(2\%\)) and raises the frequencies of the \(\mathrm{CH_3}\) group from \(2835\) to \(2844\ \mathrm{cm}^{-1}\), which is explained by facil-

of the vibrations of this group. Gibben’s results are in good agreement with our data, which show that the addition of HCl to methanol leads to a considerable lowering of the frequency \(\nu(\mathrm{C—O})\) (from 1034 to \(1000\ \mathrm{cm}^{-1}\), \(\sim 3.3\%\)) and to its great broadening. The frequencies of the \(\mathrm{CH_3}\)-group increase by \(15\)—\(20\ \mathrm{cm}^{-1}\). With ethanol there is no such effect, but a large broadening of the lines occurs in this case as well. We investigated a series of binary mixtures with \(\mathrm{NH_3}\), \(\mathrm{SO_2}\), and pyridine. The spectra of solutions in ammonia of such substances as \(\mathrm{NH_4NO_3}\), \(\mathrm{NH_4CNS}\) \(^{45}\), \(\mathrm{Hg(CN)_2}\) \(^{46}\), showed that a considerable intermolecular interaction occurs here, accompanied both by shifts of the frequencies of \(\mathrm{NH_3}\) and by changes in the frequencies of \(\mathrm{Hg(CN)_2}\) and of the \(\mathrm{CNS'}\)-ion. Comparing our data and those of other investigators for \(\mathrm{Hg(CN)_2}\) in various solvents, we obtain the following Table 5.

TABLE 5

\(\mathrm{Hg(CN)_2}\)

	Frequencies \((\mathrm{cm}^{-1})\)	Frequencies \((\mathrm{cm}^{-1})\)
	\(\mathrm{Hg—(CN)_2}\)	\(\mathrm{C—N}\)
Crystal \(^{47}\)	275	2192
Saturated aqueous solution \(^{48}\)	260	2194
Solution in \(\mathrm{CH_3OH}\) \(^{49}\)	—	2204
Solution in \(\mathrm{NH_3}\) \(\mathrm{Hg(CN)_2\cdot 6NH_3}\)	—	2164
Solution in \(\mathrm{NH_3}\) \(\mathrm{Hg(CN)_2\cdot 23NH_3}\)	—	2163
Solution in pyridine	260	2180

We see that the frequency of the CN group depends very strongly on the solvent. As for \(\mathrm{SO_2}\), it was investigated by us in the liquid state in mixtures with \(\mathrm{CCl_4}\) and \(\mathrm{CHCl_3}\) (additivity), with \(\mathrm{BCl_3}\) (chemical reaction), and with \((\mathrm{CH_3})_2\mathrm{O}\) \(^{50}\). In the last case the interaction was expressed in a lowering of the frequency \(\nu(\mathrm{C—O})\) from 920 to \(908\ \mathrm{cm}^{-1}\) and in an increase of the frequencies of the \(\mathrm{CH_3}\)-group. The interaction is undoubtedly orientational (see below).

The list of works could also be continued, but this does not seem necessary to us.

In this paragraph we have spoken of those cases of intermolecular interaction in which definite molecular compounds and new bonds are not formed. In other words, we have limited ourselves to considering possible changes in an already existing spectrum without the appearance of new lines. Below we shall analyze the possibility of the formation of molecular compounds, the phenomenon of association, and related questions.

§ 7. Molecular compounds and the Raman effect

Up to now the discussion has concerned intermolecular interaction that does not lead to the formation of stable products. Meanwhile,

Both van der Waals forces and other kinds of bonding—exchange, Coulomb, hydrogen—can lead to the formation of stable interaction products of already saturated molecules. Such molecular compounds prove to have a quite definite composition—they obey stoichiometric ratios[^51][^52]. Formally, compounds of this kind are interpreted in chemistry in several ways.

Let us consider in parallel the products of the addition of HCl to ammonia and to dimethyl ether—ammonium chloride and the oxonium compound of dimethyl ether

\[ \mathrm{NH_4Cl} \qquad (\mathrm{CH_3})_2\mathrm{OHCl}. \]

The most primitive view is the idea of an increase in the valence of one or another atom upon interaction with saturated molecules. The indicated compounds were formerly represented in this way:

\[ \begin{array}{cc} \begin{array}{c} \mathrm{H}\ \ \ \ \mathrm{Cl}\\[-2mm] \diagdown\ \ \diagup\\[-1mm] \mathrm{N}^{\mathrm{V}}\\[-1mm] \diagup\ |\ \diagdown\\[-2mm] \mathrm{H}\ \mathrm{H}\\[-1mm] \ \ \mathrm{H} \end{array} & \begin{array}{c} \mathrm{CH_3}\ \ \ \ \mathrm{H}\\[-2mm] \diagdown\ \ \diagup\\[-1mm] \mathrm{O}^{\mathrm{IV}}\\[-1mm] \diagup\ \diagdown\\[-2mm] \mathrm{CH_3}\ \ \ \ \mathrm{Cl} \end{array} \end{array} \]

Nitrogen increases its valence from three to five, and oxygen from two to four. All valences are regarded as equivalent; intermolecular interaction in this picture is associated with the appearance of new chemical bonds with an energy of the order of hundreds of calories. However, quite apart from other facts contradicting such a view (see below), it should already be noted here that a whole series of molecular compounds cannot be fitted into such a scheme; such, for example, are all anomalous oxonium compounds of the type

\[ (\mathrm{C_2H_5})_2\mathrm{O}\cdot 5\mathrm{HCl}. \]

A second method is the extension to molecular compounds of Werner’s ideas on coordination bonds, successfully applied by him to complex compounds[^53]. In this case the bond differs from an ordinary chemical bond precisely in that the central atom, while increasing its valence, acquires a charge. The bond energy here is of the same order as in ordinary compounds, and the bond is formed by the same forces—exchange, Coulomb, and polarization. The formulas of our compounds are then written as follows:

\[ \left[ \begin{array}{c} \mathrm{H}\\ \mathrm{H}\ \mathrm{N}\ \mathrm{H}\\ \mathrm{H} \end{array} \right]^+ \mathrm{Cl}^- \quad\text{and}\quad \left[ \begin{array}{c} \mathrm{CH_3}\backslash\\ \mathrm{O}\!-\!\mathrm{H}\\ \mathrm{CH_3}/ \end{array} \right]^+ \mathrm{Cl}^- \]

with tetravalent nitrogen and trivalent oxygen. Such substances should behave as ionic compounds, in particular conducting current well in the molten state. It must be said that the possibility of formation of such compounds follows directly from quantum-mechanical considerations, according to which

nitrogen can be homeopolar tetravalent and at the same time heteropolar monovalent. Oxygen can be homeopolar trivalent and heteropolar monovalent[^53].

We know that for $\mathrm{NHCl_4}$ the coordination formula corresponds more closely to the facts than the formula with pentavalent nitrogen. However, the representation of every molecular compound as a complex compound is incorrect. In particular, as will be shown below, such a formula is unsuitable for $(\mathrm{CH_3})_2\mathrm{O}\cdot\mathrm{HCl}$.

The third way of representing molecular compounds is the introduction of the concept of a new kind of force of intermolecular interaction, weaker than ordinary valence forces. Depicting these forces by a dotted line, we shall write the formulas of ammonium chloride and of dimethyl ether hydrochloride as follows:

\[ \mathrm{NH_3 \ldots HCl}; \qquad (\mathrm{CH_3})_2 \ \mathrm{O \ldots HCl}. \]

In these cases the bond is either purely van der Waals or hydrogen. We shall briefly characterize the latter. It has long been known that an especially strong association occurs in hydrogen-containing liquids—in water, hydrofluoric acid, carboxylic acids, and alcohols. At the present time, the explanation of the association of hydrogen-containing substances by means of the hydrogen bond is gaining ever greater reliability. The concept of the hydrogen bond was introduced by Latimer and Rodebush[^54] in 1920. In view of the special position of hydrogen in the Mendeleev system—because it lacks inner electrons and has a small atomic radius—hydrogen is capable of being placed between two atoms, such as oxygen, nitrogen, fluorine, and of producing a new type of bond. Within the framework of Lewis’s octet theory, the formulas of associated molecules of water and hydrofluoric acid are represented in the following way:

\[ \mathrm{H : O : \boxed{H} : O : \quad and \quad H : F : \boxed{H} : F :} \]

Rodebush[^55] points out that the hydrogen bond may be formally regarded as coordination with a proton. The coordination number of the proton is then equal to two. The hydrogen bond occurs only between the atoms $\mathrm{O}$, $\mathrm{N}$, and $\mathrm{F}$, and its energy does not exceed 6–8 cal. The distance between two oxygens connected by $\mathrm{H}$ is less than their atomic diameter and is equal to $2.6\ \mathrm{\AA}$. Calculations carried out by Gillette and Sherman[^56] for the interaction energy of two $\mathrm{HCOOH}$ molecules giving an association binary compound (the experimental value of the energy is 14.125 cal[^57]; for acetic acid, 13.790[^58]) lead to the conception of the hydrogen bond as a superposition of three kinds of interaction:

\[ \mathrm{X^- \ H^+ \ X^-}; \qquad \mathrm{X{-}H \ X} \quad \text{and} \quad \mathrm{X \ H{-}X}. \]

Consequently, the hydrogen bond is composed of polar and nonpolar interaction. The last two states can formally be

describe as an exchange of hydrogen. In addition to the hydrogen bond effected by a single hydrogen atom placed between atoms O, N, F, Bernal and Megaw\(^{58a}\) consider a bond effected between two hydroxyls. In the case of a hydroxyl bond, the distance between the two oxygens is of the order of 2.7–2.9 Å, and the bond energy is about 5 cal. Obviously, here too we encounter a hydrogen bond, but one weakened owing to the electrostatic repulsion of the two hydrogen atoms. The hydroxyl bond explains, in particular, the phenomenon of association of alcohols.

The task of physical methods of investigation—such as measurement of dipole moments, study of the Kerr effect, absorption spectra, and, finally, the Raman effect—is to determine the nature of the intermolecular bond in molecular compounds—to solve the question of what the energy and character of the intermolecular bond are. We have already said that there are grounds for thinking that secondary (neither chemical nor coordination) bonds arise at the expense of van der Waals interaction—orientational, induction, and dispersion\(^{59}\)—or at the expense of the hydrogen bond. Thus, for molecular compounds of trinitrobenzene with aromatic hydrocarbons, Briegleb and co-workers\(^{60,61,62}\) assume an induction interaction—the dipolar nitro group polarizes the cloud of \(p\)-electrons of the hydrocarbon. Briegleb and Shakhovskoy,\(^{60}\) studying the deviation of the absorption coefficient from additivity and its variation with temperature, find for such compounds a heat of bonding of about 1–2 cal and only in isolated cases 3–4 cal. Interaction between dipolar molecules may proceed with a decrease of the mean dipole moment when the dipoles are oriented antiparallel, or with an increase of the moment when the dipoles become arranged tail to tail. Anomalously high values of the moments, up to 9 Debye units, were found by Ulich and co-workers\(^{63}\) for molecular compounds of \(AlCl_3\), \(SnCl_4\), and \(BeCl_2\) with organic molecules. These values are anomalous and can be explained only by a strong deformation of the complex that is formed. Spectroscopically, the formation of molecular compounds is in a number of cases detected directly by the appearance of coloration in a compound of two colorless molecules. Such a displacement of the absorption band into the longer-wavelength part of the spectrum is called the phenomenon of halochromy. Thus, for example, chinhydrone, obtained from colorless quinone and hydroquinone, is colored dark green. The phenomenon of halochromy also occurs for numerous inorganic–organic compounds, such as compounds of \(SnCl_4\), etc.

Molecular compounds can often be isolated in crystalline form; many of them are very stable, retaining their individuality also in the vapor. Thus, for example, the double molecules of acetic acid dissociate only at a fairly high temperature, while the reaction of formation of the simplest oxonium compound—the already mentioned compound of dimethyl ether with hydrogen chloride—proceeds in the vapor practically instantaneously, as

this was shown by Syrkin and Gladyshev. These circumstances usually lead chemists to ascribe an increased valence to the binding atoms in such compounds and to speak of such compounds as of ordinary molecules. Meanwhile, the fact of the relative stability of a molecular compound says nothing as yet about the bond energy and the magnitude of the elastic bond constant. The true nature of such compounds can be studied only with the aid of direct physical methods of investigation, among which Raman spectroscopy is very effective.

Up to the present time only isolated works on the Raman effect devoted to molecular compounds have been published. Raman-spectroscopic investigations of oxonium compounds were undertaken simultaneously and independently by us and by Syrkin, and by Brigleb and Lappe. In their first article \(^{64}\) the latter give data on the spectra of the compounds \((\mathrm{C}_2\mathrm{H}_5)_2\mathrm{O}\cdot\mathrm{HBr}\) and \((\mathrm{C}_2\mathrm{H}_5)_2\mathrm{O}\cdot\mathrm{SnCl}_4\). At room temperature the bromination reaction proceeds rapidly, and it was not possible to observe the spectrum of the molecular compound. At a temperature of \(-40^\circ\) the spectrum of the ether changes strongly (Table 6). Despite a large excess of HBr, only a weak band at \(2100\text{--}2200\ \mathrm{cm}^{-1}\) was observed, evidently belonging to HBr bound to the molecular compound \((\mathrm{C}_2\mathrm{H}_5)_2\mathrm{O}\cdot\mathrm{HBr}\) by forces of subsidiary valence. The HBr molecule under these conditions is evidently strongly deformed, since the frequency of gaseous HBr is \(2558\ \mathrm{cm}^{-1}\), and of liquid HBr \(2487\ \mathrm{cm}^{-1}\) \(^{37}\). As our calculation shows (see above), such a shift and broadening of the band may be caused by orientational forces. The data of Brigleb and Lappe concerning the band \(3394 \pm 32\), characteristic of the OH bond, are, by their own admission, doubtful.

In the spectrum of the compound \((\mathrm{C}_2\mathrm{H}_5)_2\mathrm{O}\cdot\mathrm{SnCl}_4\), which these authors studied in the molten state at \(100^\circ\mathrm{C}\), they failed to observe any changes. On the contrary, such changes occurred in a benzene solution of the complex. These data were obtained unambiguously only in the second part of the work of Brigleb and Lappe \(^{65}\) and will be set forth by us in § 9. In the same article they give data for the spectrum of molten \((\mathrm{C}_2\mathrm{H}_5)_2\mathrm{O}\cdot\mathrm{AlCl}_3\) (Table 6). Here, too, significant changes in the spectrum of the ether take place. The frequencies 310(8) and 534(6) evidently belong to the \(\mathrm{AlCl}_3\) molecule \(^{10}\). As Brigleb and Lappe emphasize, the study of molecular compounds of \(\mathrm{AlCl}_3\) may shed light on the nature of the catalytic activity of \(\mathrm{AlCl}_3\) and its role in cracking.

For the compound \((\mathrm{C}_2\mathrm{H}_5)_2\mathrm{O}\cdot\mathrm{HCl}\) Brigleb and Lappe give no data. Evidently, they were unable to observe deviations from additivity. This is also confirmed by our work. The author of this article and Ya. K. Syrkin investigated the Raman spectra of mixtures of dimethyl ether with HCl and with HBr and of diethyl ether with HCl \(^{66}\). It turned out that the spectrum of \((\mathrm{CH}_3)_2\mathrm{O}\cdot\mathrm{HCl}\) contains no new lines in comparison with the spectrum of the pure ether, and that only small shifts of frequencies take place. The greatest shift has

there is a place for the frequency \(\nu(\pi)\) of the symmetric vibration of oxygen relative to the methyl groups from \(920\ \mathrm{cm}^{-1}\) to \(890\ \mathrm{cm}^{-1}\). This becomes understandable if one takes into account that the attachment of HCl to oxygen, however it may occur, must slow the vibrations of oxygen and at the same time facilitate the vibrations in the methyl groups, which in fact does take place.

TABLE 6*
Frequencies in \(\mathrm{cm}^{-1}\)

\((\mathrm{C}_2\mathrm{H}_5)_2\mathrm{O}\)	\((\mathrm{C}_2\mathrm{H}_5)_2\mathrm{O}\cdot 2\mathrm{HBr}\)	\((\mathrm{C}_2\mathrm{H}_5)_2\mathrm{O}\cdot \mathrm{AlCl}_3\)
435 (5)	406 (5)	310 (8)
500 (1)	573 (1)	406 (10)
	678 (1 s.b.)	534 (6)
		625 (4)
		690 (2 b.)
841 (5)		823 (3)
934 (0)	920 (0)	891 (4)
1026 (1)	996 (4 b.)	1054 (4)
1144 (1)	1137 (0)	1206 (1)
1286 (9)	1270 (1)	1262 (2)
1459 (6b.)	1459 (6 b.)	1450 (5 b.)
2866 (8)	2866 (0)
2933 (10)	2940 (10)	2901 (10)
2978 (10)	2984 (10)	2976 (10)
	3394 ± 32? (0)
	2100—2200 (0)

Three of the 5 frequencies of the \(\mathrm{CH}_3\)-group prove to be somewhat elevated in the mixture with HCl. The complete absence of new frequencies and of significant changes in the spectrum contradicts the idea of an increase in the valence of oxygen in the oxonium compound. These ideas are also contradicted by the data of Syrkin and Gladyshev, as well as of Maas and Morrison\(^{67}\), who found the heat of the reaction \((\mathrm{CH}_3)_2\mathrm{O} + \mathrm{HCl}\) to be equal to 6 cal. A rough calculation shows that the heat of formation of the complex

TABLE 7**
Frequencies in \(\mathrm{cm}^{-1}\)

Frequency values	\((\mathrm{CH}_3)_2\mathrm{O}\)	\((\mathrm{CH}_3)_2\mathrm{O}\cdot \mathrm{HCl}\)	\((\mathrm{CH}_3)_2\mathrm{O}\cdot \mathrm{SO}_2\)
\(\nu(\mathrm{CH}_3)\)	2988 s. st.	3003 st. b.	2996 st.
\(\nu(\mathrm{CH}_3)\)	2947 st.	2953 m. b.	2952 m.
\(\nu(\mathrm{CH}_3)\)	2908 st.	2923 m.	2919 m.
\(\nu(\mathrm{CH}_3)\)	2868 st. b.	2875 m.	2868 m.
\(\nu(\mathrm{CH}_3)\)	2813 s. st.	2829 s. st.	2821 st.
\(\delta(\mathrm{CH}_3)\)	1448 m. b.	1449 m. b.	1453 st. b.
\((\mathrm{CH}_3)_2\mathrm{O}\ \nu(\sigma)\)	1095 s. s.	1081 s. s.	1088 s.
\((\mathrm{CH}_3)_2\mathrm{O}\ \nu(\pi)\)	920 st.	891 st. b.	908 st.
\((\mathrm{CH}_3)_2\mathrm{O}\ \delta(\pi)\)	408 s. s.	418 s.	411 s.

* In parentheses are intensities: \(b.\)—broad, \(s.\ b.\)—very broad.
** \(s.\ st.\)—very intense, \(st.\)—intense, \(m.\)—medium, \(s.\)—weak, \(s.\ s.\)—very weak, \(b.\)—broad line.

with tetravalent oxygen, equal to the sum of the heats of formation of the OH and OCl bonds minus the heat of formation of HCl, has an essentially different order of magnitude: it is approximately 10 times larger. The notion of trivalent oxygen should likewise be rejected, since, on the one hand, an increase in valence should have been accompanied by the appearance of new permitted lines in the Raman spectrum, while on the other, the low value of the electrical conductivity obtained by us for \((\mathrm{CH}_3)_2\mathrm{O}\cdot\mathrm{HCl}\) shows that this substance cannot be regarded as a molten salt. Both the absence of new lines and the value of the heat of formation suggest that the interaction in this case is effected through a hydrogen bond. It is interesting that frequency shifts in the same directions, although somewhat smaller, occur, according to our measurements, in the molecular compound of \((\mathrm{CH}_3)_2\mathrm{O}\) with \(\mathrm{SO}_2\), formed, evidently, by orientational forces \(^{60}\). Let us compare all these data in Table 7.

Of considerable interest would be the spectroscopic study of oxonium compounds of the pyron type, for example, the compound of dimethylpyrone with HCl

\[ \begin{array}{c} \mathrm{CH}_3-\mathrm{C}\!\left(\begin{array}{c} \diagup\mathrm{O}\diagdown \end{array}\right)\mathrm{C}-\mathrm{CH}_3 \;+\;\mathrm{HCl}\\[-2mm] \quad\;\;\Vert \qquad\qquad \Vert\\[-1mm] \quad\;\mathrm{HC}\qquad\quad \mathrm{CH}\\[-1mm] \qquad\diagdown\mathrm{C}\diagup\\[-1mm] \qquad\;\;\Vert\\[-1mm] \qquad\;\;\mathrm{O} \end{array} \]

This would make it possible to determine to which oxygen HCl is preferentially added—to the ether oxygen or to the carbonyl oxygen.

Ashkinazi and Kurnosova succeeded in observing the formation of the molecular compound \(\mathrm{C}_6\mathrm{H}_6\cdot 2\mathrm{SbCl}_3\) \(^{35}\). In addition to the unchanged frequencies of benzene and antimony trichloride, they find two frequencies, 477 and \(1236\ \mathrm{cm}^{-1}\). If the molecular compound is assigned the structure

\[ \begin{array}{ccc} m & M & m\\ \mathrm{SbCl}_3 & \mathrm{C}_6\mathrm{H}_6 & \mathrm{SbCl}_3 \end{array} \]

and one of the frequencies is calculated by the Dale and Kohlrausch formulas \(^{68}\), taking the other \((1236\ \mathrm{cm}^{-1})\) as known, then good agreement of the numerical values is obtained (470 instead of \(477\ \mathrm{cm}^{-1}\)). However, the results of this work should be treated with caution, since the high value of the frequency \(1236\ \mathrm{cm}^{-1}\), with such a large magnitude of the vibrating masses, indicates a high value of the elastic constant of the bond, which is improbable for a molecular compound. Moreover, the absence of any changes in the spectra of \(\mathrm{C}_6\mathrm{H}_6\) and \(\mathrm{SbCl}_3\) in the presence of high frequencies of new bonds also does not confirm the conclusions of the authors of this work.

Another example of a Raman-spectroscopic study of molecular compounds is provided by the work of Leitman and Ukholin \(^{69}\) on acetic-

acid. These authors found that the intensity of the frequency 622 cm\(^{-1}\) in the spectrum of acetic acid decreases, when the concentration of CH\(_3\)CO\(_2\)H in aqueous solution is lowered, more strongly than for the other frequencies. This leads one to suppose that here we are dealing with the frequency of intermolecular vibrations of two CH\(_3\)CO\(_2\)H molecules relative to one another in their associated double compound. According to all the data, such a compound is formed by two hydrogen bonds (interaction energy 13.79 kcal), and it has either a symmetrical configuration (Fig. 3), or a configuration in which the four oxygens form a tetrahedron. One might have expected that upon dilution of the acid the frequencies of the C=O and C—O bonds would change owing to the destruction of the hydrogen bond. However, the choice of solvent is unfortunate—water itself contains hydrogen (hydroxyl) bonds, and the liberated acetic-acid molecule can enter into interaction with water. The results obtained upon raising the temperature would be more unambiguous. Gillette and Daniels

TABLE 8

CH\(_3\)COOH

	25°	172°
\(\nu_1\)	2985	2985
\(\nu_2\)	1740	1786
\(\nu_3\)	1435	1398
\(\nu_4\)	1296	1288
\(\nu_5\)	1190	1185

Fig. 3.

studied the infrared spectrum of acetic acid at temperatures of 25 and 72°C. They found the following changes (Table 8)\(^{70}\). The vibrations of binary and individual molecules may be represented graphically as follows (Fig. 4). The vibrations of the dimer \(\omega'_2\), \(\omega'_3\), and \(\omega'_5\), active in the infrared spectrum, correspond to the vibrations of the monomer \(\omega_2\), \(\omega_3\), and \(\omega_5\). Calculation shows that \(\omega'_5 \simeq \omega_5\), as is also the case in experiment. The unchanged frequency \(\nu_1\) belongs to the CH bond. As for assigning the frequency 622 cm\(^{-1}\) to intermolecular vibrations, this is contradicted by data obtained by K. P. Godina in our laboratory. The frequency 622 cm\(^{-1}\) does not disappear and is not weakened in a mixture of CH\(_3\)CO\(_2\)H with HCl of composition 1:1, although here one might have expected the formation of \((\mathrm{CH}_3\mathrm{CO}_2\mathrm{H}_2)^+\) and Cl′ ions with rupture of the hydrogen bond. A final solution of the question can be obtained only on the basis of more extensive experimental material.

Questions concerning the nature of compounds of molecules with molecules are closely connected with questions concerning the combination of molecules with ions. The simplest example is the solvated proton—the hydroxonium ion H\(_3\)O\(^+\) in water, the ammonium ion NH\(_4^+\) in ammonia, and a proton to which is attached

more than one molecule of solvent \([{\rm H}({\rm H}_2{\rm O})_n]^+\), \([{\rm H}({\rm NH}_3)_n]^+\) ^45,96. In particular, in our work^45 on the Raman spectra of solutions of \({\rm NH}_4{\rm NO}_3\) and \({\rm NH}_4{\rm CNS}\) in ammonia, it was shown that the ammonium ion in ammonia loses—

Vibrations of \(({\rm CH}_3{\rm COOH})_2\)
“inactive”

active

\(\omega'_3\)  \(\omega'_2\)  \(\omega'_3\)

Vibrations of \({\rm CH}_3{\rm COOH}\)

\(\omega_5\)  \(\omega_2\)  \(\omega_3\)

Fig. 4.

—its individuality, adding one or two more molecules of ammonia. Alongside the molecular compounds formed—

TABLE 9
Cyanides

Substance	Frequency of coord. bond	CN frequencies
\({\rm K}[{\rm Cu}({\rm CN})_2]\)		2110
\({\rm K}[{\rm Ag}({\rm CN})_2]\)	239 st. 855 s.	2134 st.
\({\rm K}_2[{\rm Ni}({\rm CN})_3]\)		2142 st.
\({\rm K}_2[{\rm Zn}({\rm CN})_4]\)		2055 s. 2149 st.
\({\rm K}_2[{\rm Cd}({\rm CN})_4]\)		2140 s. b.
\({\rm K}_2[{\rm Hg}({\rm CN})_4]\)		2149 s. b.
\({\rm K}_2[{\rm Ni}({\rm CN})_4]\)		2144 st.
\({\rm K}_3[{\rm Cu}({\rm CN})_4]\)		2090 sl. 2176 s. s.
\({\rm K}_3[{\rm Cr}({\rm CN})_6]\)	782 st.	2137 s.
\({\rm K}_3[{\rm Co}({\rm CN})_6]\)	340 st.	2070 st. 2144 st.
\({\rm K}_3[{\rm Rh}({\rm CN})_6]\)	593 s.	2149 st.
\({\rm K}_4[{\rm Cr}({\rm CN})_6]\)	619 s.	2130 st.
\({\rm K}_4[{\rm Fe}({\rm CN})_6]\)		2051 st. 2092 st. 2153 s. 2195 s.
\({\rm K}_4[{\rm Ru}({\rm CN})_6]\)	281 s. s.	2068 st. 2107 m.

nymi van der Waals and semi-chemical forces (hydrogen bond), let us stop, for the sake of completeness, at complex compounds. Here, as was already indicated above, we are dealing with forces which, in magnitude and character, are analogous to the forces in ordinary molecules—exchange, Coulomb, and dispersion forces. A large number of works have been devoted to the Raman effect of complex compounds. Of interest

TABLE 10

Halogen-containing complexes

Substance	Frequencies of the coordination bond
\(K_2HgCl_4\)	266 st.
\((NH_4)_2HgCl_4\)	273 st.
\(K_2HgBr_4\)	166 st.
\(K_2HgJ_4\)	126 s.
\(K_2CdBr_4\)	160
\(Na_2CdJ_4\)	109
\((NH_4)_4ZnCl_6\)	274
\(H_2SnCl_6\)	154 st.; 235 m.; 313 s. st.
\(Li_2SnCl_6\)	159 m.; 234 m.; 314 st.
\(MgSnCl_6\)	157 m.; 237 m.; 320 st.
\(H_2SbCl_6\)	172 st.; 277 st.; 337 s. st.; 419 s.; 605 s. s.
\(H_2SiF_6\)	—; —; —; —; 649 st.

are both the changes in the frequencies of the molecules located in the coordination sphere and the frequency of the coordination bond itself. Let us compare the data obtained for a series of complex compounds by various authors (Damashun 71 et al., Tables 9, 10, and 11).

TABLE 11

Ammine compounds

Substance	Frequencies of the coordination bond	Frequencies of the anion	Frequencies of \(NH_3\)
\(Ag(NH_3)_2Cl\)	—		3228 m.; 3308 st.; 3396 st.
\(Cu(NH_3)_4Cl_2\)	419
\(Cu(NH_3)_4SO_4\)	410 s.	977 st.	3173; (3309)
\(Zn(NH_3)_4Cl_2\)	418 st.		3183; 3274, 3307
\(Zn(NH_3)_6SO_4\)	428 st.	979 st.; 1110 s.	3193 st.; 3268 m.
\(Cd(NH_3)_6Cl_2\)	340 st.		3287, 3327
\(Ni(NH_3)_6Cl_2\)	—		3386, 3374
\(Co(NH_3)_6Cl_3\)	482,570

In the first half of the table the single frequency for each compound belongs to the totally symmetric vibration. We

we see that the frequencies of the coordination bond of the anion depend almost not at all on the cation. We obtained data for \(H_2SiF_6\) \(^{72}\) (Table 10) and for \(Ag(NH_3)_2Cl\) \(^{45}\) (Table 11).

Mathieu \(^{73}\) studied platinum and rhodium complexes with ammonia and ethylenediamine. The complex frequencies were obtained with such intensity that it proved possible to study even their polarization properties. In full agreement with Placzek’s theory, Mathieu found the most intense frequencies of the totally symmetric vibrations to be the least depolarized. For the last complex, given in Table 12,

TABLE 12

Substance	Symmetry group	Frequency in \(cm^{-1}\)	Properties
\([Pt(NH_3)_4]Cl_2\)	\(D_{4h}\)	505	Slight depolariz., totally symm.
\([Pt\, en_2]Cl_2\)	\(V_h\)	525 225	Intense. polar. weak
\([Rh(NH_3)_6]Cl_3\)	\(O_h\)	475 390 560	Intense.
\([Pt\, en_3]Cl_4\)	\(D_3\)	250 439 550 960	Symm.

he succeeded in obtaining all 4 frequencies required by the theory for the symmetry of the pyramid \(D_3\). These facts once again confirm the directional character of coordination bonds.

In addition, Damashun and other authors investigated complex molybdates, tungstates, etc. To this same category should be assigned our work on the Raman spectrum of tetraethylammonium iodide \(^{11}\). For the totally symmetric vibration of the tetrahedral ion \([N(C_2H_5)_4]^+\) we found the frequency \(667\ cm^{-1}\).

§ 8. The Raman Effect and the Structure of Liquids

In recent years the scientific point of view on the nature of the liquid state has changed substantially. If earlier the liquid was brought closer to the gas, proceeding from the possibility of a continuous transition between these two states, then now, despite the princi-

the real impossibility of a continuous transition from a liquid to a solid (owing to the absence of anisotropy in a liquid), theory is developing in the direction of bringing a liquid closer to a crystal. It is precisely the results of investigations undertaken by modern physical methods, such as radiography and the Raman effect, that have led to the emergence of a new view of the liquid. According to these conceptions, most liquids may be brought closer to crystals in the sense that the molecules in a liquid are not arranged with respect to one another in a wholly disorderly fashion. As Debye showed^74, the X-ray scattering curve of liquid mercury contains maxima similar to those observed in scattering by a solid. From this curve one may infer that the mercury atoms (mercury—an atomic liquid) are located predominantly at distances of 3.3 Å, and avoid distances of 4.4 Å. Thus liquid mercury represents a certain likeness of a crystal—its atoms are arranged more or less regularly. As Bernal writes^75, the scattering curve from a liquid is a blurred copy of the crystal curve. He believes that the molecular structure of a simple liquid can always be represented by a statistical distribution function determined by only three variables: the mean smallest distance between two molecules, the number of nearest neighbors of a molecule, and a parameter characterizing the irregularity of the distribution. The true configuration of a liquid can be obtained from the conditions of the minimum of potential energy with respect to changes in intermolecular distances and the minimum of free energy with respect to changes in the other two variables.

Proceeding from these conceptions, Bernal proposes the following classification of liquids (Table 13).

TABLE 13

Types of liquid structures

Type of intermolecular forces	Type of intermolecular forces	Type of molecules	Type of molecules
Type of intermolecular forces	Type of intermolecular forces	spherical	nonspherical
Nondirectional	Decrease slowly with distance	Liquid metals	—
Nondirectional	Decrease rapidly with distance	Liquid noble gases $\mathrm{CH_4}$, $\mathrm{CCl_4}$, $\mathrm{C_6H_{12}}$	Liquid paraffins, benzenes, etc.
Directional		Water, $\mathrm{NH_3}$, glycerin, etc.	Alcohols with long chains and acids

If one may put it this way, the unit cell of a liquid structure is the entire liquid.

A particularly interesting liquid is water. Its anomalies have long attracted attention. We can dwell only briefly on the nature of water, although the number of works devoted to it, and especially works on the Raman effect, is extremely large.

Bernal and Fowler^76 constructed a theory of the structure of water which proved suitable for interpreting both the electrical and optical properties of water and, in particular, its Raman spectrum. In essence this theory amounts to the following. Water is a quasi-crystalline structure, and the character of the relative arrangement of the water molecules is different at different temperatures.

Fig. 5.

Specifically, three modifications are possible: 1) of the ice type—tridymite \((\mathrm{SiO}_2)\), with tetrahedral coordination at temperatures below \(4^\circ\mathrm{C}\); 2) of the quartz type, with tetrahedral coordination, in the range \(4\text{—}200^\circ\mathrm{C}\); 3) of the close-packed liquid type—between \(200^\circ\mathrm{C}\) and the critical temperature. Of course, the transition from one structure to another occurs not at once, by a jump, but continuously. Thus at room temperature one may, for example, expect the simultaneous coexistence of all three structures, but predominantly the quartz-like one. Water molecules bound by hydroxyl bonds are grouped tetrahedrally (Fig. 5). The ability of water to form this kind of structure is connected with its geometrical structure, in particular with the value of the angle \(\mathrm{H—O—H}\), which is close to the tetrahedral one. The structure is infinitely repeating; the entire volume of the liquid represents, as it were, a single whole—in this sense Langmuir’s expression is apt, that the whole ocean is one molecule, and catching fish in it is a process of dissociation^77. Such a three-dimensional, infinitely repeating structure is impossible for ammonia, which is isoelectronic with water and forms only short chains or rings. By contrast, hydrofluoric acid, likewise isoelectronic with water, can polymerize indefinitely precisely by means of the hydrogen bond.

The Raman spectrum of water is interesting in its difference from the Raman spectrum of the vapor. The latter contains, in the OH frequency region, one characteristic narrow line with a frequency of about \(3700\ \mathrm{cm}^{-1}\). In liquid water, in this region there is observed an extremely broad band (up to \(600\ \mathrm{cm}^{-1}\) wide) with three maxima at \(3220,\ 3440,\ 3580\ \mathrm{cm}^{-1}\). Raising the temperature weakens the first maximum, at first strengthens the second, and then the third, and greatly narrows the band. The same act—

Fig. 6. Scattering spectra of crystals near the Rayleigh line Hg 4047 Å (after Vuks, Acta Physicochimica URSS, 6, No. 1, 1937),
a — p-dibromobenzene;
b — p-bromochlorobenzene;
c — α-modification of p-dichlorobenzene (stable at \(t < 32^\circ\text{C}\));
d — β-modification of dichlorobenzene (stable at \(t > 32^\circ\text{C}\)).

causes the addition of strong electrolytes, as is shown by the study of the spectrum of CaCl₂·6H₂O, for example, undertaken by Gerlach^78 and by us. The predominant maximum for ice is 3200 cm⁻¹, for cold water—3400 cm⁻¹, and for hot water—3600 cm⁻¹. These facts can be interpreted, proceeding from the theory of Bernal and Fowler, by regarding the corresponding frequencies as characteristic of the three structures of water listed above. Naturally, the addition of an electrolyte disturbs the quasi-crystalline structure and must act in the same direction as an increase of temperature.

In addition to the broadening of the vapor band, intermolecular frequencies should be observed in the liquid. This possibility was excluded by the classical theory of a close-packed liquid, which left no room for definite complexes of molecules with a lifetime sufficiently long to yield a narrow band. In place of discrete intermolecular lines, the classical theory predicted a continuum. The intermolecular frequencies actually observed once more refute the old theory.

The aim of investigation by means of the Raman effect is to determine which of the two hypotheses about liquids is correct—the micro- or the quasi-crystalline one^79. According to the first of them, intermolecular frequencies in a liquid owe their existence to a large number of microcrystals floating in the liquid. According to the second hypothesis, already set forth above, the entire liquid represents a certain likeness of a crystal. If the first hypothesis is correct, the position and width of the intermolecular bands should be the same in the liquid and in the solid. According to the quasi-crystalline hypothesis, these bands should be broadened in the liquid and shifted toward lower frequencies. This is indeed confirmed by the work of Gross and Vuks^80.

Gross and Vuks studied the nature of the “wings” of the Rayleigh line—the broadened bands accompanying the unshifted line in the spectrum of light scattered by certain liquids. The existence of “wings” had been known for quite a long time, and this phenomenon had been studied by many authors, who gave it various interpretations. Thus Bhagavantam^81 considered that the “wings” owe their existence to the rotation of molecules retarded in the quasi-crystalline groups present in the liquid. The nature of the phenomenon was revealed by Gross and Vuks, who showed that when a liquid is frozen, in place of the wings there appear discrete frequencies characteristic of the crystal (Fig. 6). Gross and Vuks ascribed these discrete frequencies to intermolecular vibrations in the crystal lattice. However, Bhagavantam^82 continued to insist on his interpretation, believing that the possibility of hindered rotation exists also in the solid. On the other hand, Sirkar^83 believes that these frequencies characterize not a crystal lattice, but the intermolecular vibrations of polymerized groups of molecules. Sirkar assumes that the presence of wings in a liquid is an independent fact indicating the presence of molecular rotation in the liquid. Nevertheless, the interpretation

Gross and Vuks, who find in their discovery confirmation of the quasi-crystalline hypothesis, appears to us the only correct one. This is once again proved by the subsequent works of Gross and Vuks, Vuks and Gross and Komarov. Thus Vuks \(^{84,85}\) was able to discover that the position of the low intermolecular frequencies depends on the crystalline modification of the substance, as, for example, in \(\alpha\)- and \(\beta\)-\(p\)-dichlorobenzene (Table 14). At the same time the spectra of substances of different chemical composition, but crystallizing isomorphously, in the region of low intermolecular frequencies proved to be very similar (Table 14—\(p\)-dichlorobenzene, dibromobenzene, bromochlorobenzene). Thus the indicated frequencies undoubtedly belong precisely to the crystal lattice. In turn, the circumstance that the wings of the Rayleigh line in a liquid characterize precisely intermolecular vibrations, and not molecular rotation, is illustrated with sufficient clarity by the work of Gross and Komarov \(^{86}\), who recorded the Raman spectrum of gaseous carbon disulfide at a pressure of 0.5 atm. Despite the freedom of molecular rotation that exists here, the wings disappeared completely. The entire body of work in this direction gives us substantial and direct proof of the quasi-crystalline structure of the liquid, confirmed, on the other hand, by X-ray structural analysis.

As for the intermolecular frequencies of water, they were studied by Mara \(^{87}\) and Boll \(^{88}\). Mara also succeeded in constructing a theory \(^{89}\) of the intermolecular vibrations of water, starting from the ideas of Bernal and Fowler. Mara considers that electrical and dispersion forces, as well as Born repulsion, act between the water molecules. Mara finds the energetically most favorable configuration of the water molecules and calculates the vibrational frequencies of the tetrahedral grouping in good agreement with experiment. Namely, instead of the frequencies 60, 175, 500, and 740 \(\text{cm}^{-1}\), he finds the frequencies 60, 166, 670, and 700 \(\text{cm}^{-1}\). The frequency 570 \(\text{cm}^{-1}\), the frequency of torsional vibrations, libration, was obtained from somewhat different considerations by Bernal and Tamm \(^{90}\), which reduce to the difference in its value from the difference in the properties of \(\mathrm{H_2O}\) and \(\mathrm{D_2O}\). In good agreement with Mara’s theory is also the fact that the frequency 175 \(\text{cm}^{-1}\) practically does not change in passing from \(\mathrm{H_2O}\) to \(\mathrm{D_2O}\), whereas the remaining frequencies change approximately by a factor of \(\sqrt{2}\), as should follow from Mara’s calculations. We shall compare certain intermolecular frequencies in the following table (Table 14).

§ 9. Symmetry relations

As was set forth above, ordinary van der Waals forces do not create new bonds, and under their action we may expect not the appearance of new Raman lines, but only shifts and broadenings of the old ones. The number of atoms and bonds in each molecule does not change; the molecule retains its individuality, and with it the individuality of the spectrum is preserved. However, up to now we have left out

toward the possibility of the appearance of new lines even under the action of comparatively weak forces, as a consequence of a distortion of the symmetry of the molecule. However, such a possibility exists. It was pointed out independently by Brillouin and Laue ^64 and by us ^94.

TABLE 14

Frequencies of intermolecular vibrations in cm\(^{-1}\)

Substance	Frequencies
C\(_6\)H\(_6\) ^80	62      104 (p.)
p—C\(_6\)H\(_4\)Br\(_2\) ^80	20 (p.)    38 (p.)    93 (p.)
p—C\(_6\)H\(_4\)Br\(_2\) ^91 (32°)	36      93
p—C\(_6\)H\(_4\)Br\(_2\) ^91 (−40°)	37.5      95
p—C\(_6\)H\(_4\)Br\(_2\) ^91 (−180°)	42      104
p—C\(_6\)H\(_4\)BrCl ^85	22.4    42.5    94
p—C\(_6\)H\(_4\)Cl\(_2\) α ^85	27.5    46.5    54.0    93
p—C\(_6\)H\(_4\)Cl\(_2\) β ^85	43.3    54.5    82
p—C\(_6\)H\(_4\)Cl\(_2\) ^91 (45°)	40 (2)    50 (2)    82 (2b.)
p—C\(_6\)H\(_4\)Cl\(_2\) ^91 (32°)	40 (2)    50 (2)    82 (2b.)
p—C\(_6\)H\(_4\)Cl\(_2\) ^91 (−180°)	55 (2)    60 (2)    105 (2b.)
C\(_{10}\)H\(_8\) ^80	45 (p)    73 (p)    109 (p)    124 (p)
(C\(_6\)H\(_5\))\(_2\)O ^80	22 (st. p.)    38 (s. p)    67 (s. p.)    104 (st. p)
S (rhomb.) ^92	88 (st)
S (liquid) ^92	80 (s. d.)
CS\(_2\) ^93	70 (s. st.)    81 (s)
P ^92	36 (st.)
H\(_2\)O ^79	60 (s)    176 (m. d.)    500 (s. p.)    700 (s. p.)
D\(_2\)O ^79	170 (m.)    350 (s.)    500 (s. s.)

p — polarized, d — depolarized lines.

Van der Waals forces, insufficient for the creation of new bonds, are capable of changing the symmetry of a molecule by changing the geometrical arrangement of the nuclei or the distribution of forces. Every change of symmetry entails changes in the selection rules of polarization relations and in the degeneracy of frequencies. Let us first consider three examples.

1) The molecule XY\(_2\). If both atoms Y are completely identical, then the molecule can have two kinds of symmetry—it may be either linear or bent. The CS\(_2\) molecule, for example, is linear and belongs to the symmetry group \(D_{\infty h}\). Calculation gives in this case three normal vibrations, one of which is doubly degenerate. Two frequencies are active in the infrared spectrum, and in the Raman spectrum only one is. However, it may happen that the molecule XY\(_2\) is bent by association forces. The symmetry will then pass into the group \(C_{2v}\). Here all three normal vibrations are nondegenerate and allowed both in the infrared and in the Raman spectrum. Intermolecular interaction has as its consequence the removal of degeneracy and an enrichment of the spectrum.

2) The molecule XY\(_3\). General symmetry considerations admit a planar—triangular—or pyramidal—model of such a mole-

cules. A planar molecule (for example, \( \mathrm{BCl}_3 \)) belongs to the symmetry group \(D_{3h}\) and has four frequencies. Of these, two are doubly degenerate. Three frequencies are allowed in the Raman spectrum, and the same number in the infrared, but not all of these frequencies coincide in both spectra, owing to the alternative prohibition (center of symmetry).

As a result of interaction with external molecules, the atom \(X\) may be brought out of the plane of the triangle \(Y_3\). If the symmetry becomes pyramidal, then we obtain the group \(C_{3v}\) of the trigonal pyramid, isomorphic with \(D_{3h}\). The number of frequencies here is the same, but the selection rules change—all 4 frequencies are allowed both in the infrared and in the Raman spectrum.

3) The molecule \(XY_4\). Let us consider a tetrahedral molecule (for example, \(\mathrm{SnCl}_4\), symmetry group \(T_d\)). As was indicated above, it has four frequencies, of which one is doubly degenerate and two are triply degenerate. In the infrared spectrum only these latter two are allowed, while in the Raman spectrum all four are allowed. As a result of associative interaction, the central atom \(X\) may be displaced in the direction of one of the \(XY\) axes. In this case the symmetry \(T_d\) is lowered to \(C_{3v}\), and instead of four we obtain six frequencies. Three of them are doubly degenerate, and three are nondegenerate. All frequencies are allowed both in the infrared and in the Raman spectrum. Here, besides the change in the selection rules, the removal of degeneracy owing to the lowering of symmetry also plays an important role. The same effect occurred in the first example; however, there the total number of frequencies did not increase, since the transition from a linear molecule to a bent one was accompanied by the loss of one degree of freedom (\(3N-6\) instead of \(3N-5\)).

We might expect the following changes of symmetry as a result of association:

\[ \begin{aligned} a.\;& \text{Lowering of symmetry}\\ b.\;& \text{Transition to an isomorphic group}\\ c.\;& \text{Preservation of the group}\\ d.\;& \text{Increase of symmetry.} \end{aligned} \]

It follows from simple considerations that case \(d\) must be excluded if the number of atoms and bonds in the molecule is regarded as unchanged. Indeed, the van der Waals forces consist of the three parts considered in § 1 of this chapter—dispersion, orientation, and induction. The quantities determining the directionality of the van der Waals forces are the polarizability \(\alpha\) and the dipole moment \(\mu\). Both quantities are closely connected with the symmetry of the molecule. The principal values of the polarizability ellipsoid are directed along the axes of symmetry. The same applies to the vector of the resultant moment. If the individuality of the molecule, adopted by us as a prerequisite, is preserved, the symmetry may either be preserved (case \(c\)), or the displacement of the particle along one of the axes of symmetry will lead to cases \(a\) and \(b\). Case \(d\)—an increase of symmetry—requires a complete regrouping of the atoms in the molecule, which must be accompanied by the rupture of old and the formation of new valence

bonds. The forces needed for this are of an order of magnitude greater than van der Waals forces.

In the most interesting case a.—lowered symmetry—the spectrum is enriched as a result of the removal of degeneracy. Not only does the number of fundamental frequencies increase, but also the number of combination tones and overtones. The possible increase in the number of forbidden lines is always outweighed by the general increase in the number of frequencies. In case b.—transition to an isomorphic group—the number of lines changes, but only as a consequence of a change in the selection rules. Finally, in case c.—preservation of symmetry—only the numerical values of the frequencies and their intensities can change, but not the number of active frequencies and their degrees of depolarization.

The order of magnitude of the possible changes in symmetry is connected, on the one hand, with changes in the equilibrium distances of the nuclei $\bar q$, which, as we have shown, in the case of orientational interaction may be of the order of at most $10^{-10}\ \mathrm{cm}$, i.e., very small. However, distortion of symmetry may also occur with preservation of the positions of the nuclei, owing to a change in the symmetry of the bonds, since the electron cloud can be displaced considerably as a result of polarization. The effect may be very significant and quite observable. Indeed, this effect has in some cases been observed. Brigleb and Lauppe$^{65}$, studying the spectra of $\mathrm{SnCl}_4$ in its molecular compounds with methyl alcohol and diethyl ether, found an increase in the number of frequencies from 4 to 6, as considered in the third example. We give their results (Table 15).

We proposed interpreting changes in the spectrum of ammonia upon its transition from gas to liquid, solution, and complex compound as the result of distortion of symmetry$^{45}$. The spectrum of gaseous ammonia consists of the following frequencies (Table 16), characteristic of a pyramidal structure (ammonia is a pyramidal molecule with a dipole moment of 1.5). The frequency $\delta(\pi)$ is split into two because of the tunnel effect$^{97}$. Owing to the existence of two equilibrium positions for the atom, on both sides of the triangle $\mathrm{H}_3$, nitrogen is able to oscillate between these two positions, passing through the potential barrier, and these vibrations modulate the $\delta(\pi)$ vibration and split its frequency. This splitting also occurs for other frequencies, but there it is smaller. The frequency $\nu(\sigma)$ is not observed in the Raman spectrum of gaseous $\mathrm{NH}_3$; nevertheless, it evidently exists$^{98}$, since such a $\sigma$ band is observed in the infrared

TABLE 15

$\mathrm{SnCl}_4$

	Pure	In $\mathrm{CH_3OH}$	With $(\mathrm{C_2H_5})_2\mathrm{O}$ in $\mathrm{C_6H_6}$
$\delta_s$	104 (8)	161 (3)	171 (5)
$\delta_a$	131 (5)	218 (1)	253 (4)
$\nu_s$	371 (10)	296 (1)	320 (10)
$\nu_a$	407 (7)	334 (10)	395 (6)
		404 (2)
		497 (2)

of the hydrogen atoms of one ammonia molecule with neighboring molecules. In other words, the hydrogen bond which here undoubtedly exists may, in a somewhat weakened form (perhaps it should be called an ammonia bond, by analogy with a hydroxyl bond), be effected by one or two, but not all three hydrogens of the given molecule. Such packing of molecules, in which all three H atoms are the carriers of the bond, cannot occur because of the mutual repulsion of the hydrogens and the geometrical structure of NH\(_3\). Thus, in each NH\(_3\) molecule one of the hydrogens becomes inequivalent to the other two, which is reflected in the transition from \(C_{3v}\) symmetry to \(C_s\). At the same time, as a result of the intermolecular bond that has arisen, the splitting of the frequency \(\delta(\pi)\), associated with the tunnel effect and actually not observed in liquid NH\(_3\), must disappear.

Analogous distortions of symmetry must occur in concentrated aqueous solutions of NH\(_3\). Here, along with a small percentage of NH\(_4\)OH (the frequencies of the NH\(_4^+\) ion could not be observed in these cases), complexes of the type NH\(_3\)H\(_2\)O are formed. It may be thought that the intermolecular bond is a hydrogen bond, and that again one or two, but not three, hydrogens take part in it—most likely one,

\[ \begin{array}{c} \mathrm{H}\backslash \\ \mathrm{H}/\mathrm{O}\,\ldots\,\mathrm{H}-\mathrm{N}\!\begin{array}{l} /\mathrm{H}\\ \backslash \mathrm{H} \end{array} \end{array} \]

which again leads to \(C_3\) symmetry and the disappearance of the tunnel effect.

It is more difficult to interpret the splitting of the ammonia frequencies in complex compounds\(^{36,45}\). We suppose that in those cases where the number of ammonias in the complex ion is large, for example in \([\mathrm{Cd}(\mathrm{NH}_3)_6]^{\bullet\bullet}\) or in \([\mathrm{Zn}(\mathrm{NH}_3)_6]^{\bullet\bullet}\), the distortion of symmetry is effected through interaction between ammonias located in the coordination sphere, and is fundamentally analogous to the interaction in liquid NH\(_3\). On the other hand, in those cases where the number of ammonias is small, for example in \([\mathrm{Ag}(\mathrm{NH}_3)_2]^{\bullet}\), such interaction is hardly present, since the ammonias are probably spatially separated by the silver ion

\[ \mathrm{NH}_3-\mathrm{Ag}^{\bullet}-\mathrm{NH}_3. \]

Here one may suppose that, in aqueous solutions, water molecules penetrate into the coordination sphere and interact with NH\(_3\) according to the type of interaction in aqueous solutions of ammonia. It is very characteristic that the NH\(_3\) frequencies in \([\mathrm{Ni}(\mathrm{NH}_3)_6]^{\bullet\bullet}\), \([\mathrm{Zn}(\mathrm{NH}_3)_6]^{\bullet\bullet}\), \([\mathrm{Cd}(\mathrm{NH}_3)_6]^{\bullet\bullet}\) are lowered, whereas in \([\mathrm{Ag}(\mathrm{NH}_3)_2]^{\bullet}\) they are, as our observations have shown, within the limits of experimental error the same as in aqueous solution. In the complex ions, moreover, there may take place a rotation of the plane of H\(_3\) relative to the connecting line metal—nitrogen, again leading to the removal of the interaction.

Thus, one may point to two types of associative bonding of ammonia with neighboring molecules: 1) the ammonia—ammonia type, occurring in liquid ammonia and in saturated complex ions with large coordination numbers; 2) the ammonia—water type, occurring in aqueous solutions of ammonia and in complexes with small coordination numbers, as well as in alcoholic and similar solutions of ammonia.

The associative character of the splitting of the frequencies of NH$_3$ is confirmed by the character of the change of the frequencies with temperature$^{104}$. With increasing temperature the bands of liquid NH$_3$ shift toward shorter waves and become narrower. The same occurs in aqueous solution, and upon dilution the effect increases. The water band in the presence of NH$_3$ weakens and disappears.

The data relating to ordinary ammonia are confirmed by data relating to deuteroammonia. Indeed, the corresponding splitting occurs in this case as well. We give the table (cf. Table 18). We believe that the splitting resulting from distortion of the symmetry proceeds here according to the same scheme.

TABLE 18

ND$_3$

	Infrared frequencies of the gas$^{105}$	Observed Raman frequencies of the gas$^{106}$	Observed Raman frequencies of the liquid$^{107}$
$\delta(\pi)$	748	786
$\delta(\sigma)$	1191	—	1588 (0)?
$\nu(\pi)$	2420	2420	2399 (5)
$\nu(\sigma)$	2556	—	{ 2341 (5)
$\nu(\sigma)$	2556	—	{ 2500 (3)

Let us sum up in the form of a brief résumé, although this is not customary in review articles. We do not, of course, consider the present review exhaustive. In particular, such important and topical questions as those relating to the Raman spectra of water have been treated comparatively little. In this connection we have only briefly dwelt on the question of the hydrogen and hydroxyl bond. The focus of the review lay in a detailed consideration of the influence of van der Waals forces on the Raman spectrum both in the absence of symmetry distortions caused by these forces and in their presence. On the other hand, we have considered a number of works concerning the Raman spectra of molecular compounds and complex compounds. Finally, works on the Raman effect relating to the structure of liquids have been analyzed.

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