INTRAMOLECULAR INTERACTIONS AND OPTICS

M. V. Vol’kenshtein

1. INTRODUCTION

A large part of the modern physical methods for investigating the structure of molecules is based, to one degree or another, on the application of optical phenomena. These are, first of all, spectroscopy and the methods of molecular optics: the study of refraction and scattering of light, the study of optical activity and of the optical behavior of a substance in an external force field. In the broad sense of the word, X-ray and electron diffraction also belong to optics. These methods have already played a considerable role in the development of chemistry; in the future their significance cannot fail to become still more substantial.

We do not propose, of course, in this short article to consider with any completeness the whole vast field of applications of optical methods to the solution of chemical problems, to the investigation of molecular structure. We shall confine ourselves to several examples of the optical study of a question that is very topical in chemistry—the study of nonadditive properties of molecules, properties determined by nonvalent interactions, by the mutual influence of bonds.

Up to now one often encounters the erroneous opinion that the valence scheme used in chemistry and A. M. Butlerov’s theory of the structure of organic compounds are identical. By a valence scheme we mean the method of representing structural formulas of molecules by means of chemical symbols for atoms and little lines—valence strokes. Having no other means of expression, the valence scheme, taken by itself, presupposes additivity of molecular properties. Indeed, if one confines oneself, for example, to the notation C—Cl for the bond of carbon with chlorine in any compound, the valence scheme does not convey the interactions of this bond with the remaining parts of the molecule. It is thereby assumed that the C—Cl bonds are everywhere one and the same. In reality this, of course, is not so. A. M. Butlerov’s theory is, in content, incomparably richer than the valence scheme.

schemes; A. M. Butlerov constantly emphasized that all the constituent parts of a molecule are in interaction and that valence strokes convey only the strongest of these interactions. Thus, the circumstance, for example, that benzene does not fit into a valence scheme is, from the point of view of structural theory, nothing mysterious. Guided by the ideas of A. M. Butlerov’s theory, we can and must abandon a number of conclusions following from a formally understood valence scheme. Thus the classical notion of the complete freedom of rotation of parts of a molecule about single bonds falls away: the interaction of atoms not connected by valence strokes can obviously hinder this internal rotation (see below).

The most important area of modern theoretical chemistry is the area of compounds in which there is not even approximate additivity. The properties of such compounds cannot be conveyed by a valence scheme. These include molecules with conjugated bonds, aromatic compounds, etc. It was precisely to explain the peculiarities of such substances that the theory of electronic resonance was created. However, as is known, this theory did not justify the hopes placed in it, which is connected with the absence of a physical foundation for the approximate methods of calculation and structural formulas employed. The real physical fact of the delocalization of the electrons of chemical bonds in nonadditive molecules determines their specific properties.

It is necessary to define precisely the physical meaning of the concept of “delocalization” of electrons. In the case of ordinary chemical bonds in additive compounds, the electrons are completely localized on the bond: they are mainly under the action of the fields of the two bonded atoms and interact comparatively weakly with the electrons of other bonds. By contrast, in the case, for example, of the benzene molecule, a portion of the electrons proves to be completely delocalized, i.e., belonging equally to all six force centers. In other words, the interaction of these electrons is especially great. Thus, what is involved is a large interaction of the electrons of different bonds, their socialization, collectivization in the molecule. It is evident that delocalization of electrons by no means signifies delocalization of bonds, as is sometimes expressed. Only some portion of the electrons is socialized, and this leads to a change in the properties of the bonds, which, however, are still characterized by definite values of energy, interatomic distance, etc.

At the present time, only a detailed and systematic study of nonadditive compounds by physical methods can ensure further penetration into this area. The most valuable information on the nature of nonadditive substances can be obtained as a result of the study of their electronic spectra.

2. DELOCALIZATION OF ELECTRONS IN NON-ADDITIVE COMPOUNDS

The properties of organic compounds with conjugated bonds are so specific that one may speak of their interpretation as a special physical problem. The simplest example of such substances is furnished by hydrocarbons with conjugated double bonds—the homologous series of polyenes. The study of the electronic properties of polyenes, beginning with ethylene and butadiene and ending with chains containing 11 double bonds—carotene:

\[ \begin{aligned} &\text{(carotene structural formula)}\\[-2mm] &\mathrm{ \begin{array}{c} \text{cyclohexene ring with } \mathrm{H_3C,\ CH_3,\ CH_3} \end{array} \!-\!CH{=}CH{-}C(CH_3){=}CH{-}CH{=}CH{-}C(CH_3){=}CH{-}CH{=} }\\ &\mathrm{ {=}CH{-}CH{=}C(CH_3){-}CH{=}CH{-}CH{=}C(CH_3){-}CH{=}CH\!-\! \begin{array}{c} \text{cyclohexene ring with } \mathrm{CH_3,\ CH_3,\ H_3C} \end{array} } \end{aligned} \]

has shown that the frequencies and intensities of the absorption bands in these spectra vary regularly with the increase in the number of double bonds. The absorption bands shift into the long-wavelength region of the spectrum, and their intensities increase at the same time.

A number of works have been devoted to this phenomenon. There are numerous attempts to calculate the spectra of polyenes on the basis of various classical models[^1]. Apparently, the greatest fundamental interest is presented by the quantum-mechanical calculation in which the polyene chain is regarded as a potential box of width equal to the length of the chain[^2]. Taking this quantity to be

\[ L = Nl, \]

where \(l\) is the length of the \(C=C-C\) unit, and \(N\) is the number of such units in the chain, we obtain the following eigenvalues of the energy of an electron placed in the box,

\[ \varepsilon_n=\frac{h^2}{8m}\frac{n^2}{L^2},\qquad n=1,\ 2,\ldots, \tag{1} \]

with the corresponding eigenfunctions

\[ \psi_n=\left(\frac{2}{L}\right)^{1/2}\sin\frac{\pi n x}{L}. \tag{2} \]

Thus it is assumed that the \(\pi\)-electrons of the double bonds in the polyene

completely delocalized are the same “free” electrons in the chain as electrons in a metal. Indeed, a calculation made on this basis gives satisfactory results. A chain consisting of \(N\) links contains \(N\) pairs of mobile electrons. According to the Pauli principle, the first \(N\) levels of the box are occupied by these electrons. The longest-wavelength absorption band, responsible for the color of the substance, is associated with the transition

\[ \varepsilon_N \to \varepsilon_{N+1}. \]

We have:

\[ \nu_N=\frac{1}{h}\left(\varepsilon_{N+1}-\varepsilon_N\right) =\frac{h}{8mN^2l^2}(2N+1)=\operatorname{const}(2N+1) \]

and

\[ \lambda_N=\frac{\operatorname{const}}{2N+1}. \tag{3} \]

Let us calculate the corresponding intensities. The matrix element of the transition is

\[ p_{n'n}=\frac{2L}{\pi^2}\int_0^\pi \sin n\theta \cdot \theta \cdot \sin n'\theta\,d\theta = \]

\[ = \begin{cases} 0 \ldots\ldots & n'-n\ \text{even},\\[6pt] \dfrac{2L}{\pi^2}\left\{\dfrac{1}{(n'+n)^2}-\dfrac{1}{(n'-n)^2}\right\}\ldots & n'-n\ \text{odd}. \end{cases} \]

Here

\[ \theta=\frac{\pi x}{L}. \]

For the transition \(N\to N+1\) we have:

\[ p_{N+1,N}=-\frac{2Nl}{\pi^2}\left\{\frac{1}{(2N+1)^2}-1\right\}. \]

The corresponding oscillator strength is equal to

\[ f_{N+1,N}=1.08\cdot 10^{11}\nu_N(p_{N+1,N})^2\simeq 0.134(2N+1). \tag{4} \]

The hyperbolic course of the wavelengths and the linear course of the oscillator strengths (for \(2N\gg 1\)) are confirmed by experiment to an accuracy of 10–20%. It is obvious that the absorption band under consideration must be polarized along the \(x\)-axis, i.e., along the length of the chain.

The model of the potential box is, of course, very crude. It becomes completely inapplicable for a small number of conjugated bonds (ethylene, butadiene). It should be noted that in this case the value \(\varepsilon_1\), calculated by formula (1), proves to be comparable with the ionization potential of the molecule. It is clear that in this case the model of a potential box with infinitely high walls is unsuitable. Despite these circumstances,

the calculations described are instructive, since they show that the electrons in polyenes are in fact delocalized. From the model presented there follow some consequences not devoid of interest. Polyene chains can exist in the trans form

\[ \diagup \mathrm{C}=\mathrm{C}\diagup \mathrm{C}=\mathrm{C}\diagup \]

and in the cis form

\[ \diagup \mathrm{C}=\mathrm{C}\backslash \mathrm{C}=\mathrm{C}\diagup \]

In the second case the total length of the chain must be smaller and, consequently, the values of \(\nu\) must increase and the values of \(f\) decrease. This is, in general, confirmed by data concerning the separation of the forms of carotene (rotational isomers) by the method of chromatographic adsorption[^3].

Of course, the assertion of the delocalization of electrons in a system of conjugated bonds, the assertion of the “metal-like” character of such a system, would sound insufficiently convincing if it were based only on the facts cited. However, we also know other kinds of data which speak of the same thing. This is, first of all, the electrical conductivity of graphite. In the hexagonal lattice of graphite the crystal planes are built in the form of “benzene parquet”—as a continuous aggregate of C—C bonds forming regular hexagons with an interatomic distance close to the corresponding distance in benzene. In view of the fact that all C—C bonds in benzene and in the “benzene parquet” are identical, it may be asserted that the \(\pi\)-electrons in them are completely delocalized. An optical study of very thin flakes of graphite shows that in such flakes light whose direction of vibration lies in the plane of the “parquet” is completely absorbed; for light with a perpendicular direction of vibration the thin flakes of graphite are transparent. The electrical conductivity of graphite is very large and has considerable anisotropy, being greatest in directions that also lie in the plane of the “parquet.” Thus, the properties of graphite show that delocalized \(\pi\)-electrons can play the role of conduction electrons.

Despite a number of attempts, it has not been possible to detect directly electrical conductivity in polyene chains—in carotene. However, carotene possesses an anomalously high polarizability; its refractive exaltation is of the same order of magnitude as the refraction calculated according to the additive scheme. It is obvious that the anomalies in refraction, always observed in the case of conjugated systems, are explained by the peculiarities of their electronic spectra. At the same time it proves possible to cite one more important

a fact that may be regarded as indirect evidence of the metal-like character of compounds with conjugated bonds. This concerns the anisotropy of the diamagnetic susceptibility of aromatic compounds.

As experiment shows, the diamagnetic susceptibility of benzene, naphthalene, etc., is anomalously large in the direction perpendicular to the plane of the aromatic rings. The values of the mean susceptibilities in the plane of the rings and in the direction perpendicular to it are given in Table I.

Table I

P. Ehrenfest had already proposed an explanation of this fact by means of closed electric currents flowing along the perimeters of the rings[^4]. Further calculations confirmed the correctness of this point of view: the benzene ring can indeed be regarded as a superconductor. In this case the strength of the current induced by a magnetic field in a closed contour must be proportional to the field strength and to the area enclosed by the superconductor, and inversely proportional to the length of the superconductor:

\[ i \sim \frac{SH}{l}. \tag{5} \]

This equation is a consequence of the fundamental equation for a superconductor, which replaces Ohm’s law for ordinary conductors,

\[ \operatorname{rot}\Lambda \mathbf{i}=-\frac{1}{c}\mathbf{H}, \tag{6} \]

where

\[ \Lambda=\frac{m}{ne^2} \]

and $n$ is the number of electrons per unit length of the superconductor. In integral form

\[ \Lambda c \oint i_s\,ds=-\iint H_n\,dS, \tag{6′} \]

whence (5) follows. Let us calculate the diamagnetic susceptibility. The induced magnetic moment of the current $i$ is equal to

\[ M=\frac{1}{c}Si\sim \frac{S^2H}{l} \]

and, consequently,

\[ \chi=\frac{S^2}{lc\Lambda}. \tag{7} \]

Assuming that the anisotropy of the diamagnetic susceptibility of benzene \(\chi_{\perp}-\chi_{\parallel}\) is entirely determined by its superconductivity, we find:

\[ c\Lambda=\frac{S^2}{l(\chi_{\perp}-\chi_{\parallel})}=1.12\cdot 10^{-6}\ \mathrm{sec}\cdot\mathrm{cm}^{-1}. \]

If we take the 6 \(\pi\)-electrons of benzene as the electrons of superconductivity, we find,

\[ n=\frac{6}{6\cdot 1.39\cdot 10^{-8}}, \]

whence

\[ c\Lambda=\frac{mc}{ne^2}=1.66\cdot 10^{-6}\ \mathrm{sec}\cdot\mathrm{cm}^{-1}. \]

The agreement is satisfactory. Thus it indeed follows that delocalized \(\pi\)-electrons play the role of conduction electrons. Calculations of \(\Delta\chi\) for condensed aromatic compounds carried out on this basis (here, along with “Ohm’s law” (6), one has to use Kirchhoff’s law for branched currents) give good results. The magnitude of the anisotropy of the diamagnetic susceptibility turns out to be approximately equal to this magnitude for benzene, multiplied by the number of benzene rings in the molecule.

The possibility of connecting such heterogeneous facts as the features of electromagnetic spectra and anomalies in molecular refraction—on the one hand—and the anisotropy of diamagnetic susceptibility—on the other, on the basis of the idea of the “metal-like character” of the corresponding compounds, appears very attractive. Recently a number of attempts have been made to apply the potential-box model to aromatic compounds. L. A. Borovinskii and M. V. Vol’kenshtein used this model to calculate the spectra of benzene, naphthalene, and anthracene. In the calculation they succeeded in explaining the peculiarity of these spectra, which consists in the fact that the longest-wavelength band is polarized along the transverse axis of the molecule[^7]. This fact was established by I. V. Obreimov, A. F. Prikhot’ko, and their collaborators[^6]. However, the theoretical problem of establishing the connection between spectra and anisotropy of diamagnetic susceptibility remains unresolved to this day. The importance of this problem is obvious.

The delocalization of electrons in a chain of conjugated bonds finds its expression in all chemical and physicochemical properties of the corresponding substances. Apparently, these phenomena may lead to a special character of the forces of intermolecular interaction. If an electron is capable of “moving” in a molecule over a large distance, the forces of interaction may

have a unipolar character (London^7) and, consequently, decrease with distance much more slowly than the van der Waals forces between ordinary molecules. Indeed, at a large distance between the centers of gravity of the positive and negative charges in a molecule, the interaction energy may be calculated as the energy of attraction of unlike charges. Consequently, the interaction energy is inversely proportional to the first power of the intermolecular distance. It is possible that these unipolar forces play an essential role in the properties of rubber.

Some delocalization of electrons apparently exists in any long chains. In this case the unipolar forces bind individual molecules at points remote from one another, preserving a certain freedom of orientation for a large number of chain links situated between these centers of mutual attraction.

Delocalization of electrons occurs in a very extensive set of organic compounds. Thus, the optical properties of all dyes are determined precisely by this circumstance. Here especially great importance attaches to the investigations carried out by the school of Academician S. I. Vavilov (P. P. Feofilov). The study of polarization in the fluorescence spectra of dyes makes it possible, by comparatively simple means, to study the nature of electronic transitions in these substances. The classical theory developed by S. I. Vavilov, which makes it possible to determine the character of the “oscillators” responsible for absorption and emission in the act of luminescence, provides an opportunity to penetrate especially deeply into the essence of the processes occurring in the electron shell of a molecule with delocalized electrons during its interaction with light.

The physical facts set forth by us, in general, prove the existence of electron delocalization with sufficient certainty. This is of great importance for chemistry. It is obvious that further study of the optical properties of chemical compounds will give chemical science incomparably more than any quantum-chemical calculations based on approximate methods.

3. ANOMALOUS INTENSITIES IN SPECTRA OF COMBINATION SCATTERING

In 1941 M. V. Vol’kenshtein first observed anomalously high intensities in the spectra of combination scattering of substances with delocalized electrons.^8 It was found that the lines corresponding to the characteristic vibration of the carbonyl bond sharply increase their intensity

upon transition from substances with additive properties to nonadditive substances. We present a table of the data obtained (Table II).

Table II

Subsequently, P. P. Shorygin investigated this effect deeply and comprehensively\(^9\). It turned out that the intensities in combination-scattering spectra are extraordinarily sensitive to changes in the structure of the molecule, considerably surpassing in this respect any other molecular constants. In Table III we give data obtained by P. P. Shorygin for the intensities of the characteristic line of the totally symmetric vibration of the nitro group. In the last column of the table are indicated the degrees of depolarization of the corresponding lines, measured by Ya. S. Bobovich and M. V. Wolkenstein\(^ {10}\).

Table III

From the very large number of other similar data obtained by P. P. Shorygin and his collaborators, we shall also cite (Table IV) the intensities of carbonyl lines\(^ {11}\); these data supplement Table II.

Table IV

The investigations of P. P. Shorygin showed that any electronic interaction in a molecule, connected even with an insignificant degree of electron delocalization, finds a clear-cut...

expression in terms of the intensities of combination lines. This effect has already found, in the works of P. P. Shorygin and his collaborators, a number of important chemical applications, making it possible to determine the structure of very complex chemical substances. Recently A. N. Nesmeyanov established very interesting general regularities in the chemical properties of various organometallic compounds. It turned out that conjugation of the carbon–halogen and carbon–metal bonds is possible if a double bond C=C is contained between them (for example, in Cl—CH=CH—Hg)^{12}. This conjugation, and consequently also the delocalization of the electrons, manifests itself in the chemical behavior of the substance and, to a certain extent, in the value of the molecular refraction. One may think that A. N. Nesmeyanov’s data can be directly confirmed, and that knowledge of the properties of organometallic compounds can be substantially deepened, if one investigates the intensities in the combination spectra of these compounds.

The first attempts to explain the effect of anomalous intensities proceeded from the theory of “electronic resonance.” However, the qualitative considerations obtained on this basis did not lead, and indeed could not lead, to any sufficiently convincing results.

The intensities in combination-scattering spectra are determined by the most important constants of the molecule: the positions of the energy levels of the electron shell and the transition probabilities between them. Quantum mechanics leads to the following formula for the matrix element of a component of the polarizability tensor

\[ (\alpha_{\rho\sigma})_{mn} = \frac{1}{h}\sum_k \left\{ \frac{(p_\sigma)_{nk}(p_\rho)_{km}}{\nu_{kn}-\nu} - \frac{(p_\sigma)^*_{mk}(p_\rho)^*_{kn}}{\nu_{mk}-\nu} \right\}. \tag{8} \]

Here \((p_\sigma)_{nk}\), etc., are the matrix elements of the components of the dipole moment. The intensity of a combination line associated with a vibrational (or rotational) transition \(n \to m\) is equal to

\[ I=\mathrm{const}\,(\nu-\nu_{nm})^4\left(5b_{nm}^2+13g_{nm}^2\right), \tag{9} \]

where

\[ b_{nm}=\sum_\sigma(\alpha_{\sigma\sigma})_{nm}, \]

\[ g_{nm}^2 = \frac{3}{2}\sum_{\sigma,\rho}(\alpha_{\sigma\rho})_{nm}^2 - \frac{1}{3} \left(\sum_\sigma(\alpha_{\sigma\sigma})_{nm}\right)^2 . \]

It can be shown^{13} that for the vibrational transitions \(0 \rightleftarrows 1\), predominantly observed in scattering, to good accuracy

the condition is satisfied

\[ (\alpha_{\sigma\rho})^{01}_{00} = \left\{ \frac{\partial(\alpha_{\sigma\rho})_{00}}{\partial Q} \right\}_{Q=0} Q^{01}. \tag{10} \]

Here the lower indices refer to electronic states. Formula (10) makes it possible to replace the quantum-mechanical calculation by a classical one, expressing the intensities through the derivatives of the components of the polarizability tensor in the ground electronic state of the molecule with respect to the normal coordinates \(Q\). On this basis, A. V. Vol’kenshtein proposed a general theory of intensities and polarizations in vibrational spectra, applicable to molecules with approximately additive properties\(^{14}\). In this case it proves possible to represent the polarizability tensor of the molecule as the sum of the polarizability tensors of the individual groups and bonds in the molecule. Denoting the natural coordinates—the changes in bond lengths and valence angles—by the letters \(q\), we have:

\[ \frac{\partial \alpha_{\sigma\rho}}{\partial Q_l} = \sum_n \frac{\partial \alpha^{(n)}_{\sigma\rho}}{\partial Q_l} = \sum_{n,m} \frac{\partial \alpha^{(n)}_{\sigma\rho}}{\partial q_m} \frac{\partial q_m}{\partial Q_l}, \tag{11} \]

where \(n\) is the number of the group or bond in the molecule, and \(m\) is the number of the natural coordinate. The intensities and polarizations in the combination spectrum are ultimately expressed through the quantities \(\alpha_{nm}\) and

\[ \frac{\partial \alpha_{nm}}{\partial q}, \]

which refer to individual bonds in an additive molecule.

However, we are now considering the case of manifest nonadditivity, associated with electron delocalization. In this case, the question is evidently one of anomalously high values of the derivatives themselves,

\[ \frac{\partial \alpha}{\partial q}. \]

It can be shown that these anomalies are determined by specific features in the electronic spectra.

If the frequency of the incident light is sufficiently far removed from the frequencies of all electronic transitions, then from equation (8) we obtain:

\[ (\alpha_{\sigma\rho})_{nn} = \frac{2}{h} \sum_k \frac{\nu_{nk}}{\nu_{nk}^{2}-\nu^{2}} (p_\sigma)_{nk}(p_\rho)_{nk}. \tag{12} \]

We find:

\[ \frac{\partial(\alpha_{\sigma\sigma})_{00}}{\partial Q} = \frac{2}{h} \sum_k \left\{ \frac{2\nu_{0k}}{\nu_{0k}^{2}-\nu^{2}} (p_\sigma)_{0k} \frac{\partial(p_\sigma)_{0k}}{\partial Q} - \frac{2(\nu_{0k}^{2}+\nu^{2})}{(\nu_{0k}^{2}-\nu^{2})^{2}} \left|(p_\sigma)_{0k}\right|^{2} \frac{\partial\nu_{0k}}{\partial Q} \right\}. \tag{13} \]

where \(k\) numbers the electronic levels. It can be shown approximately,

that the second term in (13) is considerably larger than the first. Therefore[^15]

\[ \frac{\partial \alpha_\sigma}{\partial Q} \cong -\frac{2}{h}\sum_k \frac{\nu_{0k}^{2}+\nu^{2}}{\left(\nu_{0k}^{2}-\nu^{2}\right)^{2}} \left|(p_\sigma)_{0k}\right|^{2} \frac{\partial \nu_{0k}}{\partial Q}. \tag{13a} \]

The intensity of a combination line is determined by the squares of quantities of the type (13a). The lower the frequency of the electronic band and the greater its intensity, proportional to \((p_\sigma)_{0k}^{2}\), the greater the role played by the corresponding term in the sum (13a). On the basis of a study of the electronic spectra of typical nonadditive compounds (polyenes), we may roughly assume that in their spectra there is one intense long-wave band \(\nu_e\). In this case

\[ \frac{\partial \alpha_\sigma}{\partial Q} \cong -\frac{2}{h} \frac{\nu_e^{2}+\nu^{2}}{\left(\nu_e^{2}-\nu^{2}\right)^{2}} (p_\sigma)_e^{2} \frac{\partial \nu_e}{\partial Q}. \tag{13b} \]

The anomalies in the intensities of combination lines are thus explained by the special closeness of \(\nu\) to \(\nu_e\) and by the increased values of the oscillator strengths, proportional to \((p_\sigma)^2\) (M. V. Vol’kenshtein[^15], P. P. Shorygin[^16]). The dependence of the intensities on \(\nu_e\) and \(p_\sigma^2\) should be much sharper, and the anomalies much more strongly expressed, than in refraction. Indeed, the intensity of a combination line is proportional to

\[ \frac{\left(\nu_e^{2}+\nu^{2}\right)^{2}} {\left(\nu_e^{2}-\nu^{2}\right)^{4}} (p_\sigma)^4, \]

whereas the refraction is proportional to

\[ \frac{\nu_e}{\nu_e^{2}-\nu^{2}}(p_\sigma)^2. \]

Recently P. P. Shorygin and A. I. Finkel’shtein carried out a detailed verification of the above propositions[^17]. For this purpose they studied the dependence of the intensities of combination lines on the frequency of the incident light. It is clear that, according to (9), (11), (13b), the intensities obtained upon excitation by light with frequencies \(\nu_1\) and \(\nu_2\) are related as

\[ \frac{I_1}{I_2} \cong \frac{\nu_1^{4}}{\nu_2^{4}} \frac{\left(\nu_e^{2}+\nu_1^{2}\right)^{2}} {\left(\nu_e^{2}-\nu_1^{2}\right)^{4}} \frac{\left(\nu_e^{2}-\nu_2^{2}\right)^{4}} {\left(\nu_e^{2}+\nu_2^{2}\right)^{2}} . \]

Knowing \(I_1\), \(I_2\), \(\nu_1\), and \(\nu_2\), one can determine \(\nu_e\) and compare the values obtained with those measured in electronic spectra. We present the data obtained by A. I. Finkel’shtein for nitro compounds (Table V).

Table V

Taking into account the crudeness of the assumptions made, the agreement must be regarded as sufficiently good. We see that the key to understanding the phenomenon under consideration should indeed be sought in the electronic spectra.

The same applies to the peculiarities in the values of the degrees of depolarization of anomalously intense combination lines. The degree of depolarization is expressed by the relation (cf. (9)):

\[ \rho = \frac{6g^2}{5b^2 + 7g^2}. \tag{14} \]

As was shown by M. V. Vol’kenshtein, the limiting possible value of the quantity $\rho$ for characteristic valence vibrations is equal to $0.5^{18}$. This occurs if the tensor $\dfrac{\partial \alpha}{\partial Q}$ is maximally anisotropic: only one of its principal values is different from zero. The increase of $\rho$ from 0.30 to 0.50, proceeding in general parallel with the growth of intensities in the series of nitro compounds (Table III), shows that the electronic transition determining the phenomenon is polarized in one direction—evidently along the bisector of the $\mathrm{NO_2}$ triangle. The increase in the intensity and degree of depolarization for characteristic vibrations in polyenes indicates the same thing. According to A. I. Finkel’shtein, $\rho$ for triene (allocymene) is equal to 0.42, for dienes 0.18–0.33. At the same time, for olefins (one $\mathrm{C{=}C}$ bond) $\rho = 0.10$–$0.16$. Polarization measurements in spectra of combination scattering thus constitute a source of information about the polarization of electronic transitions.

There is no doubt that the prospects for applying the phenomena indicated to the solution of chemical problems, to the determination of the structure of molecules and of the electronic interactions in them, are very great. Work in this direction, begun comparatively recently by Soviet scientists, must lead to a number of substantial discoveries.

4. “Intramolecular Interferometry”

The delocalization of electrons finds its direct expression in electronic spectra and, in particular, in the intensities of lines in combination-scattering spectra. The material considered above relates mainly to nonadditive compounds, whose structure cannot be conveyed by classical valence formulas.* However, in molecules with electrons localized in bonds there also occur interactions that cannot be described by means of a valence scheme. Nonadditivity in these cases, too, is successfully investigated by optical methods. Here especially great importance attaches to the study of optical activity—“intramolecular interferometry,” in I. V. Obreimov’s apt expression. According to the classical theory of this phenomenon, natural optical activity occurs if a molecule lacks a plane and a center of symmetry, and if the electrons responsible for its optical behavior interact while being at a finite distance from one another. In the theory of optical activity we can no longer, as is done in other problems of molecular optics, regard the dimensions of molecules and the interatomic distances as vanishingly small in comparison with the wavelength. In other words, it is necessary to take into account the difference of the phases of the light wave at different points of the molecule. The rotation of the plane of polarization is ultimately explained precisely by these small differences in phase, which are very sensitive to changes in the nature of the structural units of the molecule and to changes in their mutual arrangement. Therefore one can indeed say that an optically active molecule contains a kind of internal “interferometer,” distinguished by exceptional sensitivity.

We can approach a quantitative calculation of the magnitude of optical activity starting from the same ideas on the basis of which it proved possible to construct a theory of the intensities in vibrational spectra, on the basis of the so-called valence-optical scheme[^19]. Let us represent a molecule as an aggregate of parts arranged in a definite way (groups of atoms, chemical bonds), possessing anisotropic polarizabilities \(\alpha_{ks}\). Here \(k\) is the number of the part and \(s = 1, 2, 3\), corresponding respectively to the three principal directions of its polarizability ellipsoid. In the zero approximation (used in the theory of vibrational spectra) we may disregard the interactions of the individual parts of the molecule. In doing so, we obviously regard the molecule as additive. However, an explanation of optical activity can be obtained only by taking interaction into account. Under the action of the incident light wave, in each constituent part of the molecule a dipole moment is induced, determined by the values of \(\alpha_{ks}\) and by the position of the part. To calculate the optical activity it is necessary to take into account the fields created in neighboring parts by these induced-

MOLECULAR INTERACTIONS AND OPTICS

dipoles. Considering only pair interactions, we arrive, by a purely classical route, at the following expression for the mean value of the gyration tensor \(g\) (the rotatory power of the substance is proportional to this quantity):

\[ g=\frac{2\pi}{3\lambda}\sum_{k,l}'\sum_{s,t}\frac{1}{2R_{kl}^{3}}\, \alpha_{ks}\alpha_{lt}(\mathbf R_{kl}[\mathbf{lt},\mathbf{ks}])\times \]

\[ \times\left\{\frac{3}{R_{kl}^{2}}(\mathbf R_{kl},\mathbf{ks})(\mathbf R_{kl},\mathbf{lt})-(\mathbf{ks},\mathbf{lt})\right\}. \tag{15} \]

Here \(\mathbf R_{kl}\) is the distance between the \(k\)-th and \(l\)-th particles, while \(\mathbf{lt}\) and \(\mathbf{ks}\) are unit vectors of the principal directions of the corresponding polarizability ellipsoids. If the latter possess axial symmetry, i.e. \(\alpha_{k2}=\alpha_{k3}\), then

\[ g=\frac{\pi}{3\lambda}\sum_{k,l}'\frac{1}{R_{kl}^{3}}\, (\alpha_{k1}-\alpha_{k2})(\alpha_{l1}-\alpha_{l2})(\mathbf R_{kl}[\mathbf{ll},\mathbf{kl}])\times \]

\[ \times\left\{\frac{3}{R_{kl}^{2}}(\mathbf R_{kl},\mathbf{kl})(\mathbf R_{kl},\mathbf{ll})-(\mathbf{kl},\mathbf{ll})\right\}. \tag{15a} \]

The magnitude of optical activity is expressed through the anisotropies of the polarizabilities of the constituent parts of the molecule, through their mutual orientations, and through the distances between them. These distances should evidently be referred to the centers of gravity of the electron shells of groups and bonds.

Formulas (15) and (15a) are analogous to those obtained by Kirkwood, who solved the same problem by a quantum-mechanical method.

We see that natural optical activity is, by its very nature, a nonadditive phenomenon. The quantity \(g\) represents a sum of products of quantities referring to the constituent parts of the molecule. Obviously, in this approximation we shall obtain no activity if the interacting groups are isotropic, or if the vectors \(\mathbf R_{kl}\), \(\mathbf{kl}\), \(\mathbf{ll}\) are coplanar. In these cases the optical activity may differ from zero in subsequent approximations. If one takes into account the interaction of two groups through the medium of a third group, then we obtain an additional term in \(g\), differing in order of magnitude from (15) by the factor \(\alpha/R^{3}\). This term may fail to vanish if at least one of the three interacting groups is anisotropic. Finally, when the interaction of four groups is taken into account, optical activity proves possible even for isotropic groups, since through four points—the centers of the corresponding polarizability spheres—in the general case no plane of symmetry can be drawn. In this case we find, denoting the polarizabilities

abilities \(\alpha\),

\[ g^{(3)}=-\frac{\pi}{27\lambda}\sum_{k,n,m,l}' \frac{\alpha_k\alpha_n\alpha_m\alpha_l}{R_{kn}^3R_{nm}^3R_{ml}^3} \left([{\bf R}_{kn},{\bf R}_{nm}]\mid{\bf R}_{kl}\right)\times \]

\[ \times\left\{3({\bf R}_{kn},{\bf R}_{nm})({\bf R}_{nm},{\bf R}_{ml})+ R_{kn}^2({\bf R}_{nm},{\bf R}_{nl})+ R_{nl}^2({\bf R}_{kn},{\bf R}_{nm})- R_{nm}^2({\bf R}_{kn},{\bf R}_{ml})\right\}. \tag{16} \]

The theory of optical activity based on the valence-optical scheme proceeds, in essence, from the dipole, van der Waals interaction of bonds and, consequently, is applicable only to not very compact molecules, in which the quantities \(R^3\) are considerably greater than \(\alpha\). There are sufficiently many such molecules. The theory shows that the share of participation of a given group in the magnitude of the rotatory power rapidly decreases with increasing distance of this group from the asymmetric part of the molecule. This conclusion is in complete agreement with the so-called rule of distance, found experimentally by L. A. Chugaev. “The closer an inactive substituent is to the asymmetric complex, the more significant its optical effect,” wrote L. A. Chugaev \(^{20}\). This rule can be illustrated by the data obtained by L. A. Chugaev (Table VI) on the optical activity of menthol esters of fatty acids (menthol—optically active alcohol \(C_{10}H_{19}OH\)).

Table VI

The rule of distance is a physical confirmation of the general proposition of the theory of structure, which V. V. Markovnikov formulated as follows: “... the influence of any element upon another weakens as they are removed from one another in the common chain of chemical action that holds all the elements in the particle...” \(^{21}\). The theory based on the valence-optical scheme makes it possible to carry out the corresponding calculations.

It is easy to see that formulas (15), (16) imply a sharp change in optical activity upon the introduction of a new group into the asymmetric complex or upon any change in the geometrical configuration of the molecule. This is indeed the case in experiment. The value of the rotatory power of a substance is an important molecular constant that directly expresses the structural features of the molecule and is very sensitive to intermolecular interaction. It is characteristic that in the case of a sharp

non-additive substances containing, for example, a large number of conjugated bonds, optical activity assumes very great values. For example, for

![[structural formula of a camphor residue–containing conjugated compound]]

\[ [M]^{35^\circ}_{D}=12460^\circ \]

(\(\mathrm{C_8H_{14}CO}\) — the camphor residue).

The optical activity of a substance is closely connected with its electronic absorption spectrum. Just as ordinary dispersion of light is determined by the frequencies and intensities of the absorption bands, optical activity—circular double refraction—is determined by the frequencies of the absorption bands and by the difference in absorption intensities for the right and left waves—circular dichroism. The theory of the phenomenon shows that the most important role in the magnitude of optical activity is played by comparatively weak long-wave absorption bands; in many substances their occurrence is caused by delocalization of electrons.

Optical activity has repeatedly been used in solving delicate chemical problems. Of especially great importance here are the works of one of the greatest Russian chemists, L. A. Chugaev \(^{20}\). However, until recently the wider application of “intramolecular interferometry” was hindered by the absence of a theory allowing quantitative calculation. Despite all its limitations, the “valence-optical” theory set forth above makes it possible, at least semi-quantitatively, to estimate the optical activity of a substance of known structure and, conversely, to draw conclusions about the structure of a molecule from the magnitude of the rotatory power of the substance. It is evident that in the future optical activity will find ever broader chemical applications.

5. ROTATIONAL ISOMERISM AND OPTICAL METHODS FOR ITS STUDY

We shall now turn to a special case of intramolecular interaction that is not described by the valence scheme. The question is the hindrance of internal rotation of parts of a molecule about single bonds. Classical stereochemistry, proceeding from the valence scheme and the direct results of chemical experience, regards this internal rotation as completely free. Indeed, in contrast to the case of compounds with a double bond, it is impossible by chemical means to isolate stereoisomers of the cis and trans type for compounds with single bonds, for example 1,2-dichloroethane. Since rotation of one of the \(\mathrm{CH_2Cl}\) groups relative to the other about the \(\mathrm{C—C}\) axis

does not change the arrangement of the valence strokes, classical stereochemistry, taking into account only valence interactions, considers any configurations of the molecule corresponding to different angles of rotation of \(\mathrm{CH_2Cl}\) about the \(\mathrm{C—C}\) axis to be energetically equivalent. Subsequent thermodynamic studies showed that internal rotation about single bonds is always hindered to one degree or another. Only on the basis of this assumption is it possible to obtain agreement between experimental values of thermodynamic constants and the values calculated statistically with the aid of the third law of thermodynamics. Complete agreement for ethane, for example, is achieved at a height of the retarding potential barrier \(V_0\) equal to \(2750\ \mathrm{cal/mol}\). From symmetry considerations it is obvious that the potential barrier in ethane must be described by a function of the type

\[ V_t = V_0 \frac{1}{2}(1 - \cos 3\varphi). \tag{17} \]

It may be thought that the presence of potential energy of internal rotation is determined by electrostatic dipole–dipole, dipole–quadrupole, and quadrupole–quadrupole interaction of atoms not joined by valence bonds. The empirical dependence on interatomic distances indeed has the form\(^{23}\)

\[ V = \sum_{i,k} \frac{C_{ik}}{r_{ik}^{5}} . \tag{18} \]

Thus, the minima of the potential curve (17) correspond to the greatest distance between the hydrogens of the methyl groups of ethane in the trans configuration. The preferential trans arrangement with respect to single bonds is also confirmed by X-ray and electron-diffraction investigations of a number of organic compounds and may be considered no less general a principle of stereochemistry than, say, the planar arrangement of atoms attached to a double bond. However, from the chemical point of view it is not so essential whether internal rotation about the \(\mathrm{C—C}\) bond in ethane is completely free or is hindered by a periodic potential of the type (17). In either case there is no positional isomerism—all three minima of the curve (17) are equivalent. The situation is different in the case of a molecule lacking axial symmetry \(\mathrm{C—C}\), for example 1,2-dichloroethane. In this case the potential is not expressed by a simple periodic function; the minima corresponding to different configurations are characterized by different energy values. Consequently, rotational isomerism occurs. It was precisely the application of optical methods of investigation that made it possible to discover and study this new type of isomerism\(^{24}\). Rotational isomerism was discovered by the method of vibrational spectra of combinational scattering—

…tions. Each rotational isomer is characterized by its own set of vibrational frequencies, which appear in the spectrum, despite the fact that the rate of rotational isomerization is very high. It can be shown that the mutual transformation of isomers of ordinary aliphatic compounds takes place in a time of the order of \(10^{-10}\) sec. Thus the impossibility of separating rotational isomers by chemical methods becomes understandable: a substance for which rotational isomerism is possible is in a state of dynamic equilibrium, whose constant depends on the temperature and on the differences in the free energies of the rotational isomers—a quantity of the order of several large calories per mole. However, optically, rotational isomers can be detected; during the lifetime of a rotational isomer—\(10^{-10}\) sec.—a molecule vibrating with a frequency of the order of \(10^{12}\)–\(10^{13}\ \text{sec.}^{-1}\) has time to emit or absorb a sufficiently long wave train. Therefore each rotational isomer gives its own spectrum; the ratio of the intensities of the spectral lines corresponds to the equilibrium constant of the mixture. At present there is a large number of works devoted to the spectroscopic study of rotational isomerism. The presence of rotational isomerism can be judged on the basis of an anomalous number of lines in the spectrum. For example, the molecule \(\mathrm{CF_2Cl\cdot CF_2Cl}\), if it has the trans configuration \(C_{2h}\), can possess nine active vibrations in the Raman spectrum. For the folded configuration \(C_2\) the number of such vibrations increases to 18. Since 23 different frequencies are observed in the spectrum, one may think that the substance is a mixture of two rotational isomers. Similar conclusions can also be drawn on the basis of an increased number of characteristic frequencies. Thus, chlorine derivatives of paraffins \(\mathrm{RCl}\) should have one characteristic frequency \(\mathrm{C—Cl}\) (for \(\mathrm{CH_3Cl}\), \(712\ \text{cm}^{-1}\)). Beginning with \(\mathrm{C_3H_7Cl}\), there are two such frequencies (\(\sim 650\) and \(\sim 720\ \text{cm}^{-1}\)). The vibrational spectra of a number of substances are greatly simplified when the substance is frozen—the lines of the less stable isomer are “frozen out.” Spectroscopic studies show that the equilibrium constant of rotational isomers depends strongly on the solvent: the magnitudes of the differences in free energies \(\Delta F\) change substantially under the influence of the environment. In this sense one may say that these quantities characterize, rather, the properties of the medium as a whole than the properties of individual molecules. This is easy to understand if one takes into account that the energetics of rotational isomerism, of hindered internal rotation, is determined by dipole and quadrupole van der Waals interaction of nonbonded atoms. But the interaction with atoms of neighboring molecules has the same order of magnitude. Therefore the energetics of internal rotation changes especially strongly under the influence of intermolecular forces.

We have already indicated that among optical phenomena the greatest sensitivity to external influences and to any internal changes…

...changes in the structure of the molecule possesses optical activity. The application of this phenomenon to the study of rotational isomerism is especially expedient. We have seen that the value of \(g\) depends, on the one hand, on the anisotropic polarizabilities of the parts of the molecule, and, on the other, on their geometrical arrangement and, consequently, on rotational isomerism. The values of \(g\) for rotational isomers may differ by hundreds of percent and even in sign. It is obvious that rotational isomerism may be expressed in the temperature dependence of optical activity. The study of the dependence of \([M]\) on \(T\) makes it possible to determine the equilibrium constant of the mixture and the values of \(\Delta F\). This was done in two works devoted to esters of tartaric acid and to secondary butyl alcohol.

The existence of rotational isomerism has been firmly established not only by optical, but also by other physical methods (measurement of dipole moments, etc.). However, chemistry has until now practically ignored this very important and interesting phenomenon. Meanwhile, there is no doubt that taking it into account is necessary not only for understanding a number of stereochemical regularities, but also when considering the general chemical properties of organic compounds. Rotational isomerism is especially significant in the case of a number of high-molecular substances, the most important properties of which are apparently determined by this phenomenon.

6. THE “ROTATIONAL-ISOMERIC” THEORY OF LINEAR POLYMERS

A characteristic feature of rubber-like linear polymers is their great extensibility. At the same time, elastic deformation of the polymer is not accompanied by a change in its internal energy, and the elastic modulus proves to be a linear function of temperature. The so-called kinetic theory of rubber, proposed to explain its mechanical and thermodynamic properties, proceeds from consideration of internal rotation in the polymer chain. The elasticity of rubber has a configurational, entropic origin. Under deformation, the internal energy remains unchanged, but the entropy changes (decreases) and, consequently, the free energy increases

\[ F = U - TS. \]

The decrease in entropy is determined by the fact that, under the action of an external force, the polymer chain passes from the most probable state into a less probable one. The different states, the different configurations of the chain, are possible precisely because internal rotation exists in it. Thanks to the internal rotation of the links of the chain, its most probable state corresponds to a coiled ball (Kuhn\(^{25}\)). If the internal rotation in the chain is regarded as completely free, we obtain the following expression

for its length—the mean-square distance from the first to the last atom of the chain:

\[ \overline{r^2}=Nl^2\frac{1+\cos\alpha}{1-\cos\alpha}. \tag{19} \]

Here \(N\) is the number of links, \(l\) is the length of a link, and \(\pi-\alpha\) is the fixed valence angle. This formula is valid for \(N\gg 1\).

However, as we have seen, internal rotation is always hindered. The first to introduce this fact into the theory of polymers were S. E. Bresler and Ya. I. Frenkel\(^{26}\), who assumed that each link has one definite value of the azimuth of internal rotation (trans chain); around this azimuth the links execute only torsional vibrations. If certain computational errors made by S. E. Bresler and Ya. I. Frenkel are corrected, one obtains the formula

\[ \overline{r^2}=Nl^2\frac{1+\cos\alpha}{1-\cos\alpha}\frac{2}{1-\eta}, \tag{20} \]

where \(\eta\) is the mean cosine of the azimuth of internal rotation, calculated by the formula

\[ \eta= \frac{ \displaystyle\int_{-\pi}^{\pi}\cos\varphi\, e^{-\frac{V(\varphi)}{kT}}\,d\varphi }{ \displaystyle\int_{-\pi}^{\pi} e^{-\frac{V(\varphi)}{kT}}\,d\varphi }. \tag{21} \]

\(V(\varphi)\) is the potential of internal rotation, which, in accordance with the initial assumptions of the theory of Bresler and Frenkel, should be written in the form

\[ V=V_0\sin^2\frac{\varphi}{2}. \tag{22} \]

The energy minimum (the trans-configuration of the links) corresponds to \(\varphi=0\). Formula (20) is valid for very long chains (large \(N\)) and values of \(\eta\) close to 1 (strong hindrance—a rigid chain). In the general case of an arbitrary hindering potential, for sufficiently large \(N\) we obtain:

\[ \overline{r^2}=Nl^2\frac{1+\cos\alpha}{1-\cos\alpha}\frac{1+\eta}{1-\eta}. \tag{23} \]

It is evident that (23) goes over into (20) as \(\eta\to 1\). All the expressions presented above have been obtained under the assumption that rotation about a given bond does not depend on rotation about the remaining bonds. This assumption is, of course, admissible only as a crude approximation. However, already at this stage the theory encounters great difficulties. The quantity \(\eta\) must be calculated by formula (21). Meanwhile, the form of the function \(V(\varphi)\) is unknown. The usual point of view, accepted in the literature, on the properties of polymers reduces to

this, that the chain is the more flexible, the smaller the inhibiting potential \(V_0\). The study of low-molecular compounds has made it possible to establish these potentials for a number of compounds. We give some data (Table VII).

Table VII

In accordance with expression (18), the barrier is especially high in substances with lateral methyl groups. Conversely, rotation about single bonds adjacent to a double bond is substantially facilitated. It is customary to explain in this way the flexibility of the chains of rubber and of a number of synthetic rubbers containing double bonds. However, this explanation is contradicted by the flexibility of the chains of polyisobutylene

\[ \begin{array}{ccccccccc} & & \mathrm{H} & & \mathrm{H} & \mathrm{H} & & \mathrm{H} & \\ & & \diagdown & \diagup & & \diagdown & \diagup & & \\ \mathrm{C} & & \mathrm{C} & & & \mathrm{C} & & & \mathrm{C} \\ & \diagdown & & \diagdown & \diagup & & \diagdown & \diagup & \\ & \mathrm{C} & & & \mathrm{C} & & & \mathrm{C} & \\ & \diagdown & \diagup & & \diagdown & \diagup & & \diagdown & \diagup \\ \mathrm{CH_3} & & \mathrm{CH_3} & \mathrm{CH_3} & & \mathrm{CH_3} & \mathrm{CH_3} & & \mathrm{CH_3} \end{array} \]

which possesses rubber-like properties, despite the fact that the barriers here, owing to the presence of lateral methyl groups, should be high.

In the light of the ideas set forth in Section 5, one can formulate a new point of view on the properties of linear polymers. A linear polymer may be treated, in the zero approximation, as an equilibrium mixture of rotational isomers. Indeed, if, for example, in polyisobutylene we single out an individual C—C bond, we shall write

its formula in the form

\[ \ldots \mathrm{H_2C(H_3C)_2C{-}C{-}C{-}CH_2C(CH_3)_2}\ldots \]

with the substituents on the two central carbon atoms

\[ \begin{array}{cc} \mathrm{H} & \mathrm{CH_3}\\[-2mm] | & |\\[-1mm] \mathrm{C} & \mathrm{C}\\[-1mm] | & |\\[-2mm] \mathrm{H} & \mathrm{CH_3} \end{array} \]

Upon rotations about the indicated \(C{-}C\) bond, energetically nonequivalent positions arise. The deepest minimum corresponds to the trans configuration; two equivalent positions, rotated from the trans form by \(120^\circ\) and \(240^\circ\), correspond to a less deep minimum. Statistical weights may be assigned to the isomers:

\[ c_i = e^{-\frac{\Delta U_i}{kT}}, \tag{24} \]

where \(\Delta U_i\) are the relative internal energies of the isomers (for the trans form \(\Delta U_i=0\)).

Thus, in the zero approximation we assume that the bonds are in strictly defined positions, and that internal rotation as such is absent. Thereby we can calculate \(\eta\) not from formula (21), but from the formula

\[ \eta = \frac{\displaystyle \sum_{i=1}^{n} \cos \varphi_i e^{-\frac{\Delta U_i}{kT}}} {\displaystyle \sum_{i=1}^{n} e^{-\frac{\Delta U_i}{kT}}}. \tag{25} \]

In the case of polyisobutylene, for which only three isomers are possible, corresponding to

\[ \varphi_1=0,\ \Delta U_1=0,\qquad \varphi_2 \simeq 120^\circ,\ \Delta U_2=\Delta U, \]

\[ \varphi_3 \simeq 240^\circ,\ \Delta U_3=\Delta U, \]

we obtain:

\[ \eta = \frac{1-e^{-\frac{\Delta U}{kT}}} {1+2e^{-\frac{\Delta U}{kT}}}. \tag{25a} \]

Thus, for the calculation of \(\eta\) it is necessary to know only a single constant \(\Delta U\). We see that \(\eta\) in the zero approximation depends only on differences in the energies of the isomers, and not on the heights of the torsional barriers. If all \(\Delta U=0\), as in the case of a periodic

of the potential (17), the quantity \(\eta=0\). Indeed, in this case

\[ \eta=\frac{1}{n}\sum_{i=1}^{n}\cos\frac{2\pi}{n}i=0, \tag{26} \]

i.e., in the presence of a periodic potential the behavior of the chain is the same as for completely free internal rotation. Consequently, the flexibility of the chain is determined not by the potential barriers \(V_0\), but by the differences in the energies of the isomers \(\Delta U\). The greater flexibility of the polyisobutylene chain becomes understandable; it is not difficult to see that, because of the small difference in the properties of the groups \(\mathrm{CH}_2\) and \(\mathrm{CH}_3\), the quantity \(\Delta U\) must be sufficiently small here. The greater flexibility of polybenzyl chains than of polystyrene chains, etc., is explained by the same considerations. The physical picture of the changes in chain configurations upon its stretching is as follows. In the free state the rotations are distributed along the chain in the most probable way. The mean-square length of the chain, according to (23) and (25), is then equal to

\[ \overline{r^2} = Nl^2\frac{1+\cos\alpha}{1-\cos\alpha} \frac{\displaystyle\sum_{i=1}^{n}c_i(1+\cos\varphi_i)} {\displaystyle\sum_{i=1}^{n}c_i(1-\cos\varphi_i)}. \tag{27} \]

Under isoenergetic deformation of the chain the values of \(c_i\) and \(\varphi_i\) do not change, but the rotations are distributed along the chain in a less probable way, so that its length increases. It is obvious that under isoenergetic stretching the chain cannot reach the maximum possible length corresponding to the purely trans configuration:

\[ r^2_{\max}=N^2l^2\cos^2\frac{\alpha}{2}. \]

In the next approximation it is necessary to take into account that, in reality, torsional vibrations occur about each equilibrium position \(\varphi_i\). We assign to each rotational isomer a potential of the type (22):

\[ V_i=V_{i0}\sin^2\frac{\varphi}{2}, \]

where \(\varphi\) is measured from \(\varphi_i\). Instead of \(\cos\varphi_i\), in formulas (25) and (27) one must substitute the expression \(\overline{\cos(\varphi_i+\varphi)}\). At temperature \(T\), \(\varphi\) varies within limits corresponding to the condition

\[ kT=V_{i0}\sin^2\frac{\varphi}{2}, \]

i.e.,

\[ \overline{\cos(\varphi_i+\varphi)} = \frac{ \displaystyle \int\limits_{-\,2\arcsin\sqrt{\frac{kT}{V_{i0}}}}^{\,2\arcsin\sqrt{\frac{kT}{V_{i0}}}} (\cos\varphi_i\cos\varphi-\sin\varphi_i\sin\varphi)\,d\varphi }{ \displaystyle \int\limits_{-\,2\arcsin\sqrt{\frac{kT}{V_{i0}}}}^{\,2\arcsin\sqrt{\frac{kT}{V_{i0}}}} d\varphi } = \cos\varphi_i\, \frac{ 2\sin 2\arcsin\sqrt{\frac{kT}{V_{i0}}} }{ 4\arcsin\sqrt{\frac{kT}{V_{i0}}} }. \tag{28} \]

For \(V_{i0}\gg kT\) we obtain:

\[ \overline{\cos(\varphi_i+\varphi)} \simeq \cos\varphi_i \left(1-\frac{2kT}{3V_{i0}}\right). \tag{29} \]

The question may arise as to how legitimate it is to substitute into formula (23), derived under the assumption of continuous rotation, the value \(\eta\) (25), computed on the basis of the discrete distribution of rotations. However, it turns out that a direct calculation, which from the very beginning takes account of rotational isomerism, gives the same result. Such a calculation corresponds to the stochastic problem of the “mean free path” in the diamond crystal lattice, the probabilities of displacement in different directions being unequal and depending on the preceding step. Thus, this problem could be solved by the method of Markov chains.

Thus, thermodynamics and, in the final analysis, all the most important properties of linear polymers in the zeroth approximation are determined by the differences of the energies of the rotational isomers \(\Delta U_i\). By contrast, the kinetics and the relaxation phenomena are governed by the heights of the barriers \(V_0\). Both quantities, as we have already indicated, depend substantially on intermolecular interaction.

The further development of polymer theory must evidently be based on the considerations set forth here. We see that the importance of rotational isomerism in polymer physics is very great.

In this article we have considered certain types of intramolecular interactions and their optical manifestations. When the interaction of electrons of different bonds is strong, they prove to be delocalized—collectivized in the molecule. This phenomenon

finds its expression in the electronic spectra of molecules, in the anisotropy of the diamagnetic susceptibility of aromatic compounds, but the effect of anomalous intensities in combination scattering proves to be the most sensitive to interactions of this kind. The weaker induction interaction of bonds in a molecule is clearly manifested in the phenomenon of optical activity. Finally, the dipole and quadrupole interaction of non-bonded atoms finds its expression in the phenomenon of rotational isomerism, discovered and successfully studied by optical methods. This phenomenon is extremely important for understanding the properties of linear polymers.

We see that there exists a whole series of interactions in a molecule that are not described by the valence scheme. It is clear that the existence of these interactions follows from A. M. Butlerov’s theory of the structure of organic compounds. Only by forgetting the true content of the works of this great scholar can one consider contradictions with the valence scheme impossible. Unfortunately, this erroneous point of view is widespread. Thus, in the physical facts of electron delocalization, rotational isomerism, etc., some see a kind of “collapse of the foundations.” In the present article we have attempted to demonstrate how optical methods make it possible to investigate these facts, and thereby to show that the importance of optical methods in solving current problems of chemistry is very great.

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