MECHANICS OF MOLECULAR VIBRATIONS

M. A. Elyashevich

§ 1. THEORY OF VIBRATIONS OF POLYATOMIC MOLECULES AND ITS SIGNIFICANCE

The theory of vibrations of polyatomic molecules is one of the fundamental branches of molecular theory. In vibrations, both the geometrical structure of a molecule and the interactions of the atoms forming the molecule are vividly reflected. The vibrational energy of molecules plays a significant role in chemical kinetics. Vibrations manifest themselves most directly in vibrational spectra—infrared and Raman spectra. It is precisely these spectra that serve as the principal source of information about molecular vibrations. Therefore, the main part of the theory of vibrations of polyatomic molecules is the theory of vibrational spectra.

The theory of vibrational spectra makes it possible, on the basis of data from experimental investigation of Raman and infrared spectra, to find physical parameters characterizing the structure of molecules. The experimental study of vibrational spectra yields sets of numerical parameters of two types: first, values of vibrational frequencies and, second, values of intensities and polarizations for the observed spectral lines. The frequency values make it possible to calculate thermodynamic functions—heat capacity, entropy, and others—for an ensemble of molecules of a given kind. However, what is most essential is the determination, using the experimental values of vibrational frequencies, of the mechanical parameters of molecules—constants in the expression for the potential energy (“force constants”), which determine the properties of the molecule as a complex mechanical system consisting of atoms. Using experimental data on intensities and polarizations, one can find the electro-optical parameters of molecules—dipole moments and polarizabilities and their derivatives with respect to changes in interatomic distances in the molecule. Mechanical and electro-optical parameters are determined by the properties of the electronic shell of the molecule and provide an important characterization of the latter. Since their exact quantum-mechanical calculation for a polyatomic molecule is practically impossible, the determination of these parameters from experimental

data; on the basis of the application of the theory of vibrational spectra, is of very substantial importance for the theory of molecules in general. Having obtained for the molecule under study a set of mechanical and electro-optical parameters, one can draw a number of physical conclusions about its structure.

The theory of vibrational spectra also has substantial practical significance. Above all, it is important for structural analysis. Establishing the structure of particular complex molecules is one of the main tasks of modern chemistry. The full use of vibrational spectra for solving this problem is possible only on the basis of a well-developed theory. By interpreting vibrational spectra with the aid of such a theory, it is possible to determine the structural features of a given molecule as a whole and the properties of individual bonds. Let us note that the question of the presence in a molecule of particular bonds with definite properties is often decisive in assessing the properties of the molecule as a whole. Second, an especially important application of the theory of vibrational spectra is molecular spectral analysis, qualitative and quantitative. Here the theory is needed, first of all, for identifying the characteristic features of various groups of atoms and individual bonds, for example the groups $\mathrm{CH}_3$ and $\mathrm{CH}_2$, the bonds $\mathrm{C—J}$, $\mathrm{C—Br}$, $\mathrm{C—Cl}$, $\mathrm{C—F}$, and double bonds $\mathrm{C=C}$ in various organic molecules. No less essential is the possibility, by applying a theory worked out in detail, of predicting the spectra of various compounds. In principle, one can predict with sufficient accuracy the vibrational spectra of substances that have not yet been isolated chemically, for example, by measuring higher hydrocarbons—nonane, decane, etc.; the analysis of mixtures of these is one of the numerous tasks posed by practice. For quantitative molecular spectral analysis it is important to know the laws obeyed by intensities and polarizations in vibrational spectra.

At present, the development of the theory of vibrational spectra of polyatomic molecules represents a very urgent problem. The most essential part of this theory is the theory of the normal vibrations of polyatomic molecules. On the one hand, mechanical and electro-optical parameters of molecules are obtained most directly from experimental data on normal vibrations. On the other hand, in Raman spectra, which provide the richest and most readily interpretable experimental material, it is precisely the normal vibrations that appear with considerable intensity, while the overtones are weak.* In concrete calculations, first and foremost there must

* In infrared spectra the fundamental frequencies are also more intense than the overtones, but they lie in a more remote infrared region, which makes it difficult to obtain experimental data. It should be borne in mind, however, that for the interpretation of overtones it is necessary to know the properties of precisely the fundamental vibrations.

the mechanical problem of determining the frequencies of the fundamental vibrations must be solved. For the solution of the electro-optical problem—the finding of their intensities and polarizations—data on the form of the vibrations are needed, obtained as a result of solving the mechanical problem. It is precisely the problem of calculating the frequencies of the fundamental vibrations that is the central problem of the theory of vibrations of polyatomic molecules. Since the fundamental vibrations may be approximately regarded as harmonic, this is the problem of small vibrations of a system of particles—the atoms constituting the molecule.

Methods for calculating the properties of the fundamental vibrations of polyatomic molecules are set forth in detail in the author’s monograph, Solution of the Fundamental Problems of the Theory of Vibrational Spectra of Polyatomic Molecules[^1]. The present survey is devoted to the mechanical problem of calculating the frequencies of the fundamental vibrations of polyatomic molecules. Questions of the electro-optics of vibrations—the calculation of intensities and polarizations—are the subject of a survey by M. V. Vol’kenshtein1.

§ 2. GENERAL CONDITIONS FOR THE SOLVABILITY OF PROBLEMS IN THE THEORY OF VIBRATIONS OF POLYATOMIC MOLECULES

The problems of the mechanics of vibrations of polyatomic molecules, like problems of the theory of polyatomic molecules in general, are very complex. In principle, all problems in the theory of molecules can be solved by the methods of quantum mechanics. In this approach the molecule is regarded as a real physical system consisting of a definite number of nuclei and electrons. In the wave equation of the molecule one may approximately separate the variables describing electronic, vibrational, and rotational motion, and carry out the quantization of the system. Quantization gives the energy levels—the stationary states of the molecule. The vibrational levels that interest us are obtained by solving the problem of the motion of the nuclei, taking into account that the nuclei move slowly in comparison with the electrons. The role of the potential energy for the vibrations is played by the energy of the molecule with fixed nuclei, averaged over the motion of the electrons,

\[ \overline{E(\xi, r)} = V(r), \]

where \(\xi\) is the set of electronic coordinates, and \(r\) is the set of coordinates describing the relative positions of the nuclei. For the equilibrium configuration of the molecule the function \(V(r)\) has a minimum

\[ V_{\min} = V(r_0), \tag{2,1} \]

where by \(r_0\) are denoted the values of the relative coordinates of the nuclei for the equilibrium configuration.

For a diatomic molecule (\(r\) is the distance between the nuclei of the molecule) \(V(r)\) has the well-known form shown in Fig. 1.

(“potential energy curve”). Representing \(r\) in the form

\[ r=r_0+x, \tag{2,2} \]

where \(x=r-r_0\) is the vibrational coordinate, and expanding \(V(r)=V(r_0+x)\) in a series near \(r=r_0\), we have

\[ V(r_0+x)=V(r_0)+\frac{1}{2}kx^2+\frac{1}{6}lx^3+\frac{1}{24}mx^4+\cdots \tag{2,3} \]

Restricting ourselves only to quadratic terms, we obtain the potential energy of small harmonic vibrations in the form

\[ U(x)=\frac{1}{2}kx^2, \tag{2,4} \]

where the force constant is the constant of the quasi-elastic force

\[ k=\left(\frac{d^2V}{dr^2}\right)_{r=r_0}=\frac{d^2U}{dx^2}. \tag{2,5} \]

The subsequent terms in expansion (2,3) determine the anharmonicity of the vibrations.

Analogously, for the general case of a polyatomic molecule with \(n\) vibrational degrees of freedom, we obtain for small vibrations the potential energy in the form

\[ U(x)=U(x_1,x_2,\ldots,x_n) =\frac{1}{2}\sum k_{ij}x_i x_j, \tag{2,6} \]

where \(x_i(i=1,2,\ldots,n)\) are vibrational coordinates, characterizing the deviation of the instantaneous configuration of the nuclei from the equilibrium one and vanishing for the equilibrium configuration itself. The force constants

\[ k_{ij}=\left(\frac{\partial^2 V}{\partial r_i\,\partial r_j}\right) =\frac{\partial^2 U}{\partial x_i\,\partial x_j} \tag{2,7} \]

are determined by the form of the function \(V(r)=\bar E(\xi,r)\). However, even for the simplest hydrogen molecule \(\mathrm{H}_2\), an exact calculation of the potential energy curve is a very difficult mathematical problem, and for more complex cases it is in general practically impossible, i.e. it is practically impossible to compute, by applying successive quantum-mechanical methods, the force constants of molecules.

Fig. 1. Potential energy curve of a diatomic molecule.

We have an analogous situation also in calculating other physical and chemical properties of molecules. The chief significance of quantum theory for the doctrine of the structure of molecules consists in the fact that it gives

basic physical conceptions about molecules and makes it possible to understand the essence of the fundamental regularities that connect the various properties of molecules; in particular, it makes it possible to explain the origin of the mechanical and electro-optical properties of molecules. At the present time one cannot count on the existing gap between the foundations of the theory and the whole body of concrete experimental physical and chemical data on the properties of molecules being eliminated by the application of rigorous quantum-mechanical methods. At the same time, the task of the theory of molecules is not only the development of its general questions, but also bringing it to the point of obtaining concrete results for complex molecules, to the actual interpretation of experimental material and its full physical comprehension, which is of great importance both theoretically and practically. This task—the task of actual calculations of the properties of molecules—can be solved and must be solved on the basis of properly developed simplified methods. In constructing them one should take into account a number of very essential points.

First, it is rational to obtain from the theory only the relations between quantities characterizing the molecule, while the numerical values of the constants entering into the theoretical formulas should be determined empirically, from experiment. Thus, semi-empirical methods should be used. It is precisely such a semi-empirical formulation of the problem that underlies the very approach to the theory of molecular vibrations—the mechanical and electro-optical parameters are obtained from experimental data, on the basis of a theory giving the basic dependences between observable quantities and the parameters of the molecule (see § 1).

Second, the application of semi-empirical methods as the basic ones must include taking account of all those regularities which have been found by studying molecules by various physical and chemical methods as systems with their own specific features. Physical and chemical investigations have established a definite structure of molecules. Molecules contain certain structural elements, whose properties are preserved in passing from one complex system to another. Experience shows that, for a number of properties, the quantities characterizing them can be obtained approximately for the complete system as sums of the corresponding quantities for the structural elements. This idea of additivity is in general one of the guiding ideas in the study of the structure of matter. As the most important structural elements of molecules, modern chemistry singles out individual valence chemical bonds and individual groups of atoms in the molecule. The valence scheme of chemistry indicates the path for constructing methods of calculation based on the additivity of the properties of valence bonds and groups of atoms. Thus, one should apply methods based-

included in the valence scheme of chemistry. In the case of calculating the vibrational frequencies of molecules this leads above all to a natural choice of coordinates. Such natural “valence-force” coordinates for the majority of cases are changes in bond lengths and changes in the magnitudes of valence angles. Correspondingly, the force constants introduced into the theory (subsequently found by a semi-empirical method) determine the properties of individual bonds and the mutual influence of bonds—the forces arising when bond lengths and valence angles change.

Thirdly, a very essential feature of the structure of molecules, established in particular by extensive stereochemical investigations, is the presence in molecules of definite symmetry properties. Taking them into account makes it possible considerably to simplify the solution of problems in the theory of molecules, if symmetry properties are taken into account in the very construction of the methods of calculation. Thus, one should apply methods that take account of symmetry properties. This is especially essential in the theory of vibrations of polyatomic molecules. For molecules with a large number of vibrational degrees of freedom, taking symmetry properties into account at once considerably simplifies all calculations—the solution of one problem for a large number of degrees of freedom is reduced to the solution of a series of problems, each for a smaller number of degrees of freedom.

Fourthly, both in a theory proceeding from the idea of additivity and in taking account of symmetry properties, the correct choice of the approximations used is necessary. Calculations are always approximate in character. Only when we start from a physically reasonable approximation will the results obtained have meaning and, at the same time, will it be possible, by means of successive approximations, to obtain numerical characteristics of molecular properties with sufficient accuracy. In particular, the selection of valence bonds as structural elements determines, for an extensive range of problems, such a physically reasonable approximation, which is taken as the zeroth approximation. In these cases the interactions of bonds, which give deviations from additivity, are taken into account in the following approximations. In general, successful calculation of the properties of polyatomic molecules is possible only on the basis of constructing approximate methods, and doing so in the right way. Thus, one should apply approximate methods, correctly choosing the zeroth approximation and constructing, on its basis, the subsequent approximations. In considering molecular vibrations, the very separation of vibrational motion as independent of electronic (and also rotational) motion is already the result of a physically justified approximation. A further approximation consists in considering vibrations according to (2,6) as purely harmonic. In the very solution of the mechanical problem the use of methods proceeding from the additivity of bond properties leads to a quite definite zeroth approximation—to the valence-s…

force model; according to this model they are limited by force constants characterizing independent changes in the lengths of individual bonds and the magnitudes of individual valence angles. It is absolutely necessary to evaluate correctly the limits of applicability of such an approximation; as a rule, it is quite insufficient. The calculation method must be constructed so that transition to the next approximations can easily be carried out.

Finally, fifth, it is necessary especially to emphasize that, in view of the great complexity of the problems of molecular theory and the difficulty of bringing them to the point of obtaining concrete numerical results, for the practical solution of the problems posed it is very important to develop general calculation schemes and, if possible, to simplify and standardize the methodology employed and the very technique of the calculations. Only under these conditions will the theory be practically applicable not only in individual particular cases, but also for calculating the properties of extensive classes of chemical compounds. Thus, the development of calculation methods must be carried through to the acquisition of a quite concrete and standardized computational technique. For solving the problem of a possibly complete interpretation of the available experimental material on Raman and infrared spectra, this is of decisive importance. Just as with regard to symmetry properties, the especially important role of this condition precisely for the theory of vibrations of polyatomic molecules is connected with the large number of vibrational degrees of freedom. Even when symmetry properties are taken into account, the problem is still sufficiently complicated, and its further simplification is absolutely necessary for concrete calculations of the vibrational spectra of complex molecules. Only on the basis of applying such a computational technique were B. I. Stepanov and the author able to calculate the vibrations of such molecules as propane, butane, pentane, tetramethylmethane, and others.

§ 3. APPLICATION OF CLASSICAL METHODS FOR SOLVING THE PROBLEM OF THE MECHANICS OF VIBRATIONS

Along with the fulfillment of the general requirements for calculation methods listed in § 2, of extremely great importance for solving the problem of the mechanics of vibrations of polyatomic molecules is the possibility of calculating the frequencies of fundamental vibrations by classical methods. This possibility is connected with the large mass of the nuclei in comparison with the mass of the electrons (thanks to which the vibrational motion is separated from the electronic motion) and with the correspondence between the quantum and classical problems.

The wave equation for a harmonic oscillator with one degree of freedom, as is known, has the form

\[ [T(x)+U(x)]\psi_v(x)= \left[-\frac{h^2}{8\pi^2\mu}\frac{\partial^2}{\partial x^2} +\frac{1}{2}kx^2\right]\psi_v(x) =E_v\psi_v(x), \tag{3,1} \]

where the kinetic-energy operator \(T(x)=-\dfrac{h^2}{8\pi^2\mu}\dfrac{\partial^2}{\partial x^2}\) contains Planck’s constant \(h\) and the mass of the oscillator \(\mu\) (for a diatomic molecule—the reduced mass \(\mu=\dfrac{m_1m_2}{m_1+m_2}\), where \(m_1\) and \(m_2\) are the masses of the nuclei), while the potential energy \(U(x)=\dfrac{1}{2}kx^2\) according to (2,4). The solution of equation (3,1) gives the energy levels

\[ E_v=h\nu\left(v+\frac{1}{2}\right), \qquad (v=0,\ 1,\ 2,\ldots), \tag{3,2} \]

where the constant is

\[ \nu=\frac{1}{2\pi}\sqrt{\frac{k}{\mu}} . \tag{3,3} \]

Each level corresponding to the vibrational quantum number \(v\) is described by its own function \(\psi_v\). In emission and absorption, only transitions with a change of the quantum number \(v\) by \(\pm 1\) are possible (\(v=1 \rightleftarrows v=0,\ v=2 \rightleftarrows v=1\), etc.), giving the frequency \(\nu\) determined by formula (3,3). These frequencies coincide with the classical frequency of vibration of a harmonic oscillator with energy

\[ H=T(x)+U(x)=\frac{1}{2\mu}p^2+\frac{1}{2}kx^2, \tag{3,4} \]

where \(T(x)\) is the kinetic energy corresponding to the operator \(T(x)\) (3,1), and \(p=\mu\dot{x}\) is the momentum conjugate to the coordinate \(x\) and corresponding to the operator \(\dfrac{h}{2\pi i}\dfrac{\partial}{\partial x}\). Thus, there is a complete analogy between the quantum-mechanical problem (3,1), on the one hand, and the classical problem with energy (3,4), on the other hand, and the frequencies found by quantum-mechanical and classical calculations coincide. This analogy is preserved also for the case of small vibrations of a system with \(n\) degrees of freedom and potential energy (2,6).

The solution of the quantum-mechanical problem gives the same \(n\) frequencies of normal vibrations (fundamental frequencies) as the solution of the corresponding classical problem, since, upon introducing normal coordinates, for each degree of freedom one obtains a wave equation of type (3,1) and the corresponding classical problem with energy of type (3,4), where only instead of the coordinate \(x\) there enters the normal coordinate \(Q_s\) of the \(s\)-th normal vibration³.

It should be emphasized that the complete analogy of the equations and the coincidence of the solutions of the quantum-mechanical and classical problems of calculating vibrational frequencies holds only for the fundamental vibrations. The treatment of overtones already differs substantially. This

is connected with the difference between the equations. The classical equations of motion are ordinary differential equations, and the introduction of anharmonicity, i.e., the introduction, in addition to (2.4), of the following terms (cubic and higher) of the expansion (2.3), which also leads to the appearance of overtones, means a transition from linear equations to nonlinear ones. The wave equation is a partial differential equation which remains linear also when anharmonicity appears; the influence of the latter is reduced to a change in its eigenvalues (a displacement of levels) and its eigenfunctions (owing to which transitions appear that are associated with a change of the vibrational quantum number by more than unity, i.e., overtones)*).

Overtones should already be treated by quantum-mechanical methods. Semiclassical methods may also be applied. In what follows we restrict ourselves to considering only fundamental vibrations and therefore use, for calculating vibrational frequencies, exclusively purely classical methods.

§ 4. THE PRESENT STATE OF THE THEORY OF MOLECULAR VIBRATIONS

Let us consider, in general terms, the state of the theory of molecular vibrations from the point of view of solving its most important problem—the mechanical problem of calculating fundamental vibrations.

The problem of the mechanics of vibrations comprises a number of computational tasks: the construction of secular equations giving the relation between the frequencies of vibrations and the force constants; the derivation of numerical values of these constants from experimental data on vibrational frequencies and, conversely, the determination, with their aid, of vibrational frequencies; the finding of normal coordinates and of the corresponding form of the vibrations. Closely connected with the solution of the enumerated problems is also the question of the interpretation of the vibrational spectra of particular molecules and of the selection of characteristic frequencies that indicate the presence in molecules of definite structural elements.

A very large number of works has been devoted to the calculation of frequencies and the interpretation of the vibrational spectra, Raman and infrared, of specific polyatomic molecules; in particular, the works of Dennison, Rosenthal, Wilson, Sutherland, Mecke, Kohlrausch, Manneback, Ta-Yu Wu. A survey of the results obtained in the works of these authors and their collaborators, and in numerous other works, may be found in the books of Kohlrausch⁴ and Hibben⁵ and in the reviews of Dennison⁶ and Glockler⁷.

At the present time, a very large body of experimental material relating to various types of molecules has been systematized.

*) A more complete analogy is obtained between the solution of the quantum-mechanical problem of vibrations of a system of particles and the solution of the classical problem of vibrations of a plate. In both cases we have differential equations in partial derivatives with boundary conditions.

A considerable number of vibrational spectra have been interpreted, the characteristic frequencies of various structural groups and bonds in molecules have been established, certain regularities in the spectra of homologous series of compounds have been found. A number of empirical conclusions have been drawn about the properties of individual bonds; in a number of cases the relation between vibrational frequencies and the presence of electronic resonance has been investigated. In the main, however, the work on the theoretical interpretation of the experimental data has hitherto been reduced to systematizing the accumulated material, to empirical conclusions from it, and to rather crude calculations of vibrational frequencies, which did not give a correct idea of the actual interaction forces in molecules and often led to an erroneous interpretation of spectra. This was connected with very substantial shortcomings of the methods of calculation employed.

The methods for calculating vibrational frequencies are usually very cumbersome, and at the same time for each molecule its own special method is applied, developed for the given particular case. Vibrational coordinates \(x\) [see (2, 6)] are introduced in the most varied ways, often quite irrationally and artificially. Examples are the derivations of formulae for the vibrational frequencies of tetrahedral molecules of the types \(XY_4\), \(XY_3Z\), \(XY_2Z_2\) (examples: \(CH_4\), \(CH_3Cl\), \(CH_2Cl_2\)), carried out by Rosenthal\(^{8}\), and of a molecule of the type \(X(XY_3)_4\) (example tetramethylmethane \(C(CH_3)_4\)), carried out by Silver\(^{9}\). To solve the problem of normal vibrations, the standard method is often used: composing the expression for the kinetic energy in rectangular coordinates, with subsequent application of the condition that the moment of momentum of the motion during the vibrations be zero, in order to separate the vibration from the rotation of the molecule as a whole. In this case, even for a triatomic molecule, complicated calculations are necessary (Cross and Van Vleck\(^{10}\), Salant and Rosenthal\(^{11}\)). The symmetry properties are taken into account in different ways and at different stages of the calculation. For example, Rosenthal and Foge\(^{12}\), for tetrahedral molecules of the type \(XY_3Z\), immediately write the expression for the kinetic energy taking symmetry into account, which makes it difficult to compare the resulting relations with the relations for molecules of the types \(XY_4\) and \(XY_2Z_2\), which possess a different symmetry. Often, in order to take symmetry into account, symmetry coordinates are introduced, which are linear combinations of rectangular coordinates; as a result, complicated and nontransparent expressions are obtained (for example, in Silver’s papers\(^{9}\) for molecules of the type \(X(XY_3)_4\) and in Saxe’s\(^{13}\) for cyclopropane \(C_3H_6\)).

The complexity of the calculation methods used and their specificity for the molecules under consideration are also connected with the widespread use of simplified force models. For a polyatomic molecule the number of normal vibration frequencies is always smaller than the total number of force constants. For example, the methane molecule \(CH_4\) has 4 fundamental vibrational frequencies for 5 force constants; a molecule of the ethane type

$C_2H_4$—11 frequencies with 21 force constants*). Two paths are possible for determining the constants: either to consider only the given molecule and to confine oneself to only a portion of the constants, neglecting the others, or to bring in additional data. The overwhelming majority of investigators have followed the first path and have used simplified models—in particular, the valence-force model (see above, § 2) and the central-force model (only the quasi-elastic forces arising when distances between atoms change are taken into account), trying to find such values of the constants as would give the best agreement of the calculated frequencies with the observed ones. For methane, for example, according to the valence-force model only two constants are taken into account: the constant of the quasi-elastic force associated with a change in the length of the C—H bond, and the constant of the quasi-elastic force associated with a change in the H—C—H angle; according to the central-force model, likewise only two constants of the quasi-elastic forces are taken into account, for changes in the C—H and H—H distances. The constants obtained by such methods are usually not only very approximate, but in a very large number of cases do not at all reflect the actual interaction forces in the molecule. An example is furnished by the values of the force constants for the C—Cl bond in a number of molecules, according to the results of calculations by various authors (the constants are given in $10^6$ dyn/cm).

Molecule	Force constant
$\mathrm{CCl_4}$	$1.74$—$2.00^{14}$
$\mathrm{CH_2Cl_2}$	$2.22^{15}$
$\mathrm{CH_3Cl}$	$3.34^{16}$
$\mathrm{C_2H_5Cl}$	$2.2^{10}$
$\mathrm{ClCN}$	$5.15^{17}$

Even if one takes into account the possible difference between the C—Cl bond in the ClCN molecule and this bond in chlorinated hydrocarbons, such a large scatter of values of the force constants, which in essence refer to one and the same bond, is completely implausible and is in fact explained by the crudeness of the models used in the calculations.

The most important problem for the theory of molecules—the finding of sets of mechanical parameters, i.e. systems of force constants for different types of compounds, correctly reflecting their structure—has not been solved.

The main reason for the unsatisfactory solution of the basic problems of the mechanics of vibrations has been the absence of a sufficiently simple general method for solving the most important problem: the construction of the secu-

*) In the absence of symmetry, the number of independent constants for a molecule with $n$ degrees of freedom is $n(n+1)/2$. On the number of constants in the presence of symmetry, see the note on p. 534.

equations, giving the connection between vibrational frequencies and the force constants of molecules. The first steps in this direction were made by Born and Karman^18, who applied to a one-dimensional crystal (an infinite linear chain of atoms) the method of obtaining the secular equation by directly setting up the equations of motion, without explicitly using an expression for the kinetic energy. This method was successfully applied by Bartholomew and Teller^19, and subsequently by Kirkwood^20, in the case of a finite zigzag chain of atoms (a carbon chain), and by Lechner^21 for three-, four-, and five-atom molecules, starting from the valence-force model. In a number of works, when calculating vibrations in accordance with the valence scheme of chemistry (see § 2), changes in bond lengths and the magnitudes of valence angles were introduced as coordinates (Mecke^22, Manneback and co-workers^23, Lechner^21, and others). The question of introducing symmetry coordinates was developed—such linear combinations of vibrational coordinates in which one secular equation, common to all the fundamental vibrations of the molecule, decomposes into a series of equations of lower degree for vibrations of different symmetry (Wilson^24, Rosenthal and Murphy^25). Finally, the author succeeded in developing a general method^26–28,1 for setting up secular equations for the vibrations of polyatomic molecules; parallel work on developing general methods for solving problems in the mechanics of vibrations was also carried out by the American investigator Wilson and his co-workers^29–32; the method used by them is less transparent, although in essence equivalent to the author’s method.

In what follows, the foundations of the general method for solving problems in the mechanics of vibrations are set forth, primarily the method of setting up secular equations for finding vibrational frequencies^1. This method is constructed in accordance with the general conditions set out in § 2; decisive in its construction is the combination of the following factors.

1) The vibrational coordinates are chosen in a rational way: natural vibrational coordinates are introduced—changes in the distances between atoms and in the angles between bonds; these vibrational coordinates are expressed through vectors characterizing the equilibrium configuration of the molecule (§ 5). In natural vibrational coordinates the secular equation has the most transparent form (§ 6).

2) The classical equations of motion are set up by a vector method directly in natural coordinates; at the same time, the various types of interactions in the molecule are easily determined and classified (§ 7). Owing to the fact that the natural coordinates are relative, the conditions that the total momentum of the molecule be equal to zero are satisfied automatically.

3) The properties of symmetry are taken into account by introducing symmetry coordinates as linear combinations of natural coordinates (§ 8); the coefficients of the corresponding linear dependences—the symmetry coefficients—are calculated in advance for vibrations of various types.

symmetry and are applied according to definite rules in setting up the secular equations (§ 9).

By this method, the construction of the secular equations is carried out according to a general scheme and is reduced to the application of definite recipes, which can be well illustrated by the example of methane (§ 10). The secular equations are obtained in a form convenient for solution, which enabled B. I. Stepanov and the author to carry out a number of calculations of vibrations and to obtain real systems of force constants for various types of organic molecules \(^{33-38}\); an example is the calculation of the vibrations of methane and its deuterium-substituted species (§ 11). It is essential that force constants determined from experimental data on the frequencies of simpler molecules can be used to calculate the frequencies of more complex molecules (§ 12).

§ 5. INTRODUCTION OF NATURAL VIBRATIONAL COORDINATES

Vibrational coordinates determine changes in the relative positions of atoms in a molecule as compared with the equilibrium configuration; therefore, first of all, it is necessary to characterize the equilibrium configuration of the molecule.

Fig. 2. Equilibrium configuration of a molecule.

For a diatomic molecule the equilibrium configuration is characterized simply by specifying the equilibrium distance \(s=r_0\) between the nuclei [Fig. 1 and formula (2,1)]. For a polyatomic molecule it is necessary to specify the relative arrangement of the atoms for the equilibrium configuration. This is most rationally done in the following way. Let there be given a molecule of arbitrary form, consisting of atoms \(X, Y, Z,\ldots\), whose positions are determined by radius vectors \(\mathbf{a}_X, \mathbf{a}_Y, \mathbf{a}_Z\) (Fig. 2). Introducing the unit vector \(\mathbf{e}_{XY}\), directed from atom \(X\) to atom \(Y\), and denoting by \(s_{XY}\) the equilibrium distance between these atoms, we may write the relation

\[ \mathbf{a}_{XY}=\mathbf{a}_Y-\mathbf{a}_X=s_{XY}\,\mathbf{e}_{XY}. \tag{5,1} \]

The magnitude of the equilibrium angle \(\vartheta_{YZ}\) between the directions \(X—Y\) and \(X—Z\) is determined by the formula

\[ \mathbf{e}_{XY}\,\mathbf{e}_{XZ}=\cos \vartheta_{YZ}. \tag{5,2} \]

Numbering the distances \(X—Y, X—Z\) simply by the indices \(1,2,3,\ldots\), we

we can, for brevity, denote \(\mathbf e_{XY}=\mathbf e_1,\ \mathbf e_{XZ}=\mathbf e_2\), etc. Then (5.2) can be written in the form

\[ \mathbf e_1\mathbf e_2=\cos\vartheta_{12}. \tag{5.3} \]

In addition to the vectors \(\mathbf e\), it is also convenient to introduce auxiliary unit vectors \(\mathbf f\), lying in the plane \(1,2\) and perpendicular to the directions 1 and 2 (Fig. 3). The vector perpendicular to distance 1 and directed toward distance 2 will be denoted by \(\mathbf f_{12}\); the vector perpendicular to distance 2 and directed toward distance 1 will be denoted by \(\mathbf f_{21}\). It is easy to verify (see Fig. 4) that

Fig. 3. Vectors perpendicular to bonds.

Fig. 4. Relation of the vector \(\mathbf f_{12}\) to the vectors \(\mathbf e_1\) and \(\mathbf e_2\).

\[ \mathbf f_{12}=(\mathbf e_2-\mathbf e_1\cos\vartheta_{12})\frac{1}{\sin\vartheta_{12}} \tag{5.4} \]

and

\[ \mathbf f_{21}=(\mathbf e_1-\mathbf e_2\cos\vartheta_{12})\frac{1}{\sin\vartheta_{12}}. \tag{5.5} \]

In solving the problem of the frequencies of vibrations, we are not interested in the orientation of the equilibrium configuration of the molecule as a whole in space, and it is sufficient to specify, along with the distances \(s_1=s_{XY},\ s_2=s_{XZ}\), the relative directions of the vectors \(\mathbf e_1,\mathbf e_2\). These relative directions are determined by formulas of the type (5.3).

The normal configuration of the molecule is thus characterized, independently of its orientation, by the parameters \(s\) and \(\vartheta\). The number of independent parameters for an \(N\)-atomic molecule is equal to the number of internal degrees of freedom \(n=3N-6\). In the choice of independent parameters there is always a certain arbitrariness. For example, for a triatomic nonlinear molecule (Fig. 5, \(N=3,\ n=3\cdot3-6=3\)) one may choose as independent parameters two distances \(s_1, s_2\) and the angle \(\vartheta_{12}\) between them, or else three distances \(s_1, s_2, s_3\).

We shall now introduce vibrational coordinates. The natural vibrational coordinates are the changes \(q_i\) of the equilibrium distances

changes \(s_i\) and changes \(\gamma_{ik}\) of the equilibrium angles \(\vartheta_{ik}\). By specifying \(3N-6\) independent coordinates \(q\) and \(\gamma\)—changes of \(3N-6\) distances and angles—the change of the molecular configuration during vibrations is completely determined. Various choices of independent coordinates are possible, corresponding to various choices of the initial parameters \(s\) and \(\vartheta\). If one proceeds from valence bonds as the basic structural elements of the molecule, then it is natural to take as independent variables the changes in the lengths of the valence bonds and the changes in the angles between valence bonds. In such coordinates, the potential energy in the case of the valence-force model will contain only diagonal terms—the squares of the quantities \(q\) and \(\gamma\). Therefore we shall call such coordinates valence-force coordinates. In the case of the central-force model, the potential energy will contain only diagonal terms if, as vibrational coordinates, one takes \(3N-6\) independent changes of distances—central-force coordinates.

Fig. 5. Natural vibrational coordinates for the molecule XYZ.

For a nonlinear triatomic molecule \(XYZ\) (Fig. 5), the valence-force coordinates are the changes \(q_1\) and \(q_2\) in the bond lengths \(s_1\) and \(s_2\), and the change \(\gamma\) of the angle \(\vartheta_{12}\) between the bonds. The potential energy is written in the form

\[ U=\frac{1}{2}k_1q_1^2+\frac{1}{2}k_2q_2^2+\frac{1}{2}k_\gamma\gamma^2+hq_1q_2+a_1q_1\gamma+a_2q_2\gamma. \tag{5,6} \]

The valence-force model is obtained if the “interaction” terms are neglected and one retains only the purely quadratic terms*)

\[ U=\frac{1}{2}k_1q_1^2+\frac{1}{2}k_2q_2^2+\frac{1}{2}k_\gamma\gamma^2. \tag{5,7} \]

Here \(k_1\) and \(k_2\) determine the quasi-elastic forces associated with changes in the bond lengths, and \(k_\gamma\) the force associated with a change in the angle.

The central-force coordinates are the changes \(q_1\) and \(q_2\) in the lengths of the bonds \(X—Y\) and \(X—Z\), and the change \(q_3\) in the distance between atoms \(Y\) and \(Z\). The potential energy in these coordinates takes the form

\[ U=\frac{1}{2}k'_1q_1^2+\frac{1}{2}k'_2q_2^2+\frac{1}{2}k'q_3^2+h'q_1q_2+h'_1q_1q_3+h'_2q_2q_3 \tag{5,8} \]

*) At present, the valence-force model is often understood to mean a model in which certain basic interactions are also taken into account (when applying valence-force coordinates).

and for the central-force model we have

\[ U=\frac{1}{2}k'_1q_1^2+\frac{1}{2}k'_2q_2^2+\frac{1}{2}k'q_3^2. \tag{5,9} \]

Expressions (5,7) and (5,9) differ by the neglect of different terms, whereas the complete expressions (5,6) and (5,8) are equivalent*).

In calculating vibrations it is rational to use those coordinates in which the “off-diagonal” terms in the potential energy, determining the “interactions of the coordinates,” are as small as possible. For the water molecule, for example, such coordinates are the valence-force coordinates^39. It should be emphasized that what is at issue is the choice of the best initial zeroth approximation. If all force constants are taken into account, then the final result of solving the problem, of course, does not depend on whether valence-force or central-force coordinates are used [i.e., in the case of a triatomic molecule, on whether one starts from expression (5,6) or (5,8)]; the simplicity of the course of the solution, which is practically very essential, depends on a successful choice of coordinates and, consequently, on the choice of the zeroth approximation. In addition to valence-force and central-force coordinates, mixed coordinates may also be used, when, along with changes in the lengths of valence bonds, changes in certain other distances (between atoms not connected by chemical bonds) and certain valence angles are introduced.

Closely connected with the question of introducing natural vibrational coordinates is the question of the classification of molecular vibrations. Generally speaking, in each normal vibration occurring with a definite frequency all vibrational coordinates change. However, vibrations can be classified according to which coordinates change predominantly in the individual vibrations. A definite classification of vibrations also corresponds to a definite introduction of coordinates. The usual division of vibrations into valence and deformation vibrations^22 corresponds to the introduction of valence-force coordinates. In the valence vibration of the \(i\)-th bond the coordinate \(q_i\) changes; in the deformation vibration of the angle \(ij\), between the \(i\)-th and \(j\)-th bonds, the coordinate \(\gamma_{ij}\) changes. For organic molecules containing groups of the type

\[ >\mathrm{CH}_2,\ -\mathrm{CH}_3,\ =\mathrm{CH}_2,\ -\mathrm{NH}_2 \]

and so on, one should distinguish internal

*) We have the relation

\[ (s_3+q_3)^2=(s_1+q_1)^2+(s_2+q_2)^2-2(s_1+q_1)(s_2+q_2)\cos(\vartheta+\gamma), \tag{5,10} \]

whence, approximately, for small vibrations we obtain

\[ q_3=\frac{1}{s_3}\left[(s_1-s_2\cos\vartheta)q_1+(s_2-s_1\cos\vartheta)q_2+s_1s_2\sin\vartheta\cdot\gamma\right]. \tag{5,11} \]

By substituting (5,11) into (5,8) and comparing with (5,6), the relations between the force constants in (5,6) and (5,8) are readily determined.

and external deformation vibrations—in the former, the angles between bonds within the group change (internal angles, for example the angle H—C—H in the group \(>\mathrm{CH}_2\)); in the latter, the angles determining the rotation of the entire group as a whole change (external angles, for example the angles \(\mathrm{C}=\mathrm{C}-\mathrm{H}^{(1)}\) and \(\mathrm{C}=\mathrm{C}-\mathrm{H}^{(2)}\) for a \(-\mathrm{CH}_2\) group connected by a double bond to another carbon atom, as in ethylene).

Characteristic frequencies of particular bonds correspond to vibrations in which the lengths of these bonds change. In this connection, however, it must be borne in mind that in a number of cases the frequency may be quite characteristic (i.e. always appear when the given bond is present and retain its value), while at the same time, in the corresponding vibrations, the change of the given coordinate \(q_i\) will be accompanied by considerable changes of some other coordinates as well; for example, in valence vibrations of particular bonds, the adjacent angles often also change strongly. The form of the vibrations, determined by the ratios between the simultaneous changes of various bonds and angles, is not as characteristic as the frequency itself. In general, the question of the characteristic nature of the form of vibrations is sufficiently complex and requires detailed analysis in specific cases. In solving this question one should, in any event, use precisely natural coordinates.

Fig. 6. Molecule of the type \(XY_4\).

For the classification of vibrations, one may take the number of vibrations of each kind to be equal to the number of independent natural coordinates of the given kind. For a branched \(N\)-atomic molecule not containing closed rings of atoms, the number of bonds is equal to \(N-1\). We have \(N-1\) coordinates \(q\), to which correspond \(N-1\) valence vibrations, and \((3N-6)-(N-1)=2N-5\) angular coordinates \(\gamma\), to which correspond \(2N-5\) deformation vibrations*). For example, for a five-atomic molecule we have \(5-1=4\) valence vibrations and \(2\cdot 5-5=5\) deformation vibrations. For the case of a five-atomic molecule consisting of a central atom connected with four other atoms, like the methane-type molecule \(\mathrm{XY}_4\) (Fig. 6), the natural valence-force coordinates are the four changes of bond lengths \(q_1, q_2, q_3, q_4\) and the six changes of valence angles \(\gamma_{12}, \gamma_{13}, \gamma_{14}, \gamma_{23}, \gamma_{24}, \gamma_{34}\). The first correspond to four valence vibrations, the latter to five deformation vibrations. The angular coordinates in this case are not all independent. Between the six quantities \(\gamma\) there will be

*) And for complex molecules there also correspond rotational vibrations; to characterize these one must introduce angles of rotation \(\chi\) about the bonds joining parts of the molecule, as, for example, the angle of mutual rotation of the \(\mathrm{CH}_3\) groups for the ethane molecule\(^1\).

there is one relation. In calculating vibrations, generally speaking, it is not rational to take these additional relations into account and to eliminate the superfluous coordinates from the very beginning. It is much more convenient first to introduce such “superfluous” angular coordinates and to take the additional conditions into account at a later stage of the calculation. We have a similar situation for all organic molecules containing tetravalent carbon atoms forming single bonds. When all four bonds of the carbon atom are identical, the angles between the bonds are exactly tetrahedral; for this simplest case the additional relation between the six angular coordinates, as is easily shown (see the note on p. 501), will simply be

\[ \sum_{ij}\gamma_{ij}=\gamma_{12}+\gamma_{13}+\gamma_{14}+\gamma_{23}+\gamma_{24}+\gamma_{34}=0, \tag{5,12} \]

i.e. the sum of the changes of all six tetrahedral angles with vertex at the carbon atom is equal to zero. Condition (5,12) is easy to take into account at any stage of calculating the vibration frequencies.

Counting the number of vibrations of various kinds for specific molecules presents no difficulty when natural valence-force coordinates are used.

In conclusion to this paragraph, let us consider the connection of the natural coordinates with the displacements of atoms from their initial equilibrium positions, which must be known in order to write the equations of motion in natural coordinates. The natural coordinates can be expressed vectorially through the displacements

\[ \mathbf r_X,\ \mathbf r_Y,\ \mathbf r_Z \tag{5,13} \]

of atoms from their initial positions \(\mathbf a_X,\ \mathbf a_Y,\ \mathbf a_Z\) and through the vectors \(\mathbf e\) and \(\mathbf f\), characterizing the normal configuration \(^{26,27}\). We shall here consider the displacements to be small.

Fig. 7. Changes in the distance of atoms \(X\) and \(Y\).

The initial distance of atoms \(X\) and \(Y\), according to (5,1), is equal to \(s_{XY}=\mathbf e_{XY}(\mathbf a_Y-\mathbf a_X)\). The difference of the displaced positions is equal to

\[ (\mathbf a_Y+\mathbf r_Y)-(\mathbf a_X+\mathbf r_X), \tag{5,14} \]

and we find the changed distance by projecting this difference onto the direction \(X—Y\) (Fig. 7). We obtain

\[ \mathbf e_{XY}\left[(\mathbf a_Y+\mathbf r_Y)-(\mathbf a_X+\mathbf r_X)\right] =\mathbf e_{XY}(\mathbf a_Y-\mathbf a_X)+\mathbf e_{XY}(\mathbf r_Y-\mathbf r_X) = s_{XY}+\mathbf e_{XY}(\mathbf r_Y-\mathbf r_X). \tag{5,15} \]

Thus, the change of the coordinate \(q_{XY}\) of the distance \(s_{XY}\) has the form\(^*\)

\[ q_{XY}=\mathbf e_{XY}(\mathbf r_Y-\mathbf r_X). \tag{5,16} \]

The coordinate \(q_{XY}\) is therefore determined through the difference of the displacements of two atoms.

The change \(\gamma_{YZ}\) of the angle \(\vartheta_{YZ}=\vartheta_{12}\) is equal to the sum of the angles of rotation of the bonds \(X-Y\) and \(X-Z\) in the plane of this angle (Fig. 8). The angle of rotation \(\varphi_Y\) of the bond \(X-Y\) in the plane \(XYZ\) from \(Y\) toward \(Z\) will be approximately equal to

\[ \varphi_Y=\frac{(\mathbf r_Y-\mathbf r_X)\mathbf f_{YZ}}{s_{XY}}, \tag{5,17} \]

i.e., to the ratio of the projection of the difference of the displacements of atoms \(Y\) and \(X\) onto the direction perpendicular to the bond \(X-Y\), to the length of this bond \(s_{XY}\).

Fig. 8. Change of the angle between the bonds \(X-Y\) and \(X-Z\).

Similarly, the angle of rotation \(\varphi_Z\) of the bond \(X-Z\) in the plane \(XYZ\) from \(Z\) toward \(Y\) will be approximately equal to

\[ \varphi_Z=\frac{(\mathbf r_Z-\mathbf r_X)\mathbf f_{ZY}}{s_{XZ}}. \tag{5,18} \]

The total change (increase) of the angle \(Y-X-Z\) is equal to

\[ \gamma_{YZ}=-\varphi_Y-\varphi_Z= \frac{(\mathbf r_X-\mathbf r_Y)\mathbf f_{YZ}}{s_{XY}} + \frac{(\mathbf r_X-\mathbf r_Z)\mathbf f_{ZY}}{s_{XZ}}. \tag{5,19} \]

The coordinate \(\gamma_{YZ}\) is therefore determined through the differences of the displacements of three atoms.

\(^*\) In view of the smallness of the vibrations, we regard the length of the changed distance (5.14) as coinciding with the length of its projection (5.15).

The formulas (5.16) and (5.19)*) are the basic formulas expressing the natural coordinates \(q\) and \(Y\) in terms of the relative displacements of the atoms from their initial equilibrium positions.

§ 6. THE NATURAL SECULAR EQUATION

The frequencies of the normal vibrations of a system with \(n\) degrees of freedom are determined, as is known, by solving the secular equation of the \(n\)-th degree with respect to the square of the vibration frequency \(\omega^2\). The \(n\) roots \(\omega_s^2(s=1,2,\ldots,n)\) of this equation give the required vibration frequencies. What is essential is to obtain the secular equation in a form that is physically the most transparent and convenient for concrete calculations of the frequencies. We shall therefore consider the question of the form of the secular equation. For a system of points executing harmonic small vibrations about equilibrium positions, in generalized vibrational coordinates \(x_i\), where \(i=1,2,\ldots,n\) (see § 2), which determine the departure of the system from the equilibrium state (for which all \(x_i=0\)), the kinetic and potential energies have the form

\[ T=\frac{1}{2}\sum_{ij} T_{ij}\dot{x}_i\dot{x}_j \tag{6,1} \]

and [see (2.6)]

\[ U=\frac{1}{2}\sum_{ij} k_{ij}x_i x_j, \tag{6,2} \]

where \(T_{ij}\) are constants depending on the masses of the particles and on the equilibrium configuration, and \(k_{ij}\) are force constants.

Fig. 9. Relation between the vectors \(\mathbf f\).

The standard method of solving the problem consists in applying Lagrange’s equations

\[ \frac{d}{dt}\frac{\partial T}{\partial \dot{x}_i}+\frac{\partial U}{\partial x_i}=0. \tag{6,3} \]

*) Applying formula (5.19), one can obtain relation (5.12) for changes of tetrahedral angles. We have

\[ \begin{aligned} \gamma_{12}+\gamma_{13}+\gamma_{14}+\gamma_{23}+\gamma_{24}+\gamma_{34} &=\frac{(\mathbf r_X-\mathbf r_{Y1})}{s_1}\mathbf f_{12} +\frac{(\mathbf r_X-\mathbf r_{Y2})}{s_2}\mathbf f_{21}\\ &\quad+\frac{(\mathbf r_X-\mathbf r_{Y1})}{s_1}\mathbf f_{13} +\frac{(\mathbf r_X-\mathbf r_{Y3})}{s_3}\mathbf f_{31} +\frac{(\mathbf r_X-\mathbf r_{Y1})}{s_1}\mathbf f_{14} +\ldots\\ &=\frac{(\mathbf r_X-\mathbf r_{Y1})}{s_1} (\mathbf f_{12}+\mathbf f_{13}+\mathbf f_{14})+\ldots=0, \end{aligned} \tag{5,20} \]

since the sum of the unit vectors \(\mathbf f_{1l}\), directed from the bond \(X-Y_1\) to the bonds \(X-Y_i\) \((i=2,3,4)\) and forming angles of \(120^\circ\) with one another, vanishes (see Fig. 9).

We obtain, according to (6.1) and (6.2),

\[ \sum_j (T_{ij}\ddot{x}_j+k_{ij}x_j)=0. \tag{6.4} \]

Substitution of the solution in the form

\[ x_j=x_{j0}e^{i\omega t} \tag{6.5} \]

gives the system of equations

\[ \sum_j (k_{ij}-T_{ij}\omega^2)x_{j0}=0 \tag{6.6} \]

with respect to the amplitudes of the oscillations \(x_{j0}\), whose solvability condition is the equality to zero of the determinant

\[ \left| \begin{array}{cccc} k_{11}-T_{11}\omega^2 & k_{12}-T_{12}\omega^2 & k_{13}-T_{13}\omega^2 & \ldots\\ k_{21}-T_{21}\omega^2 & k_{22}-T_{22}\omega^2 & k_{23}-T_{23}\omega^2 & \ldots\\ k_{31}-T_{31}\omega^2 & k_{32}-T_{32}\omega^2 & k_{33}-T_{33}\omega^2 & \ldots\\ \ldots & \ldots & \ldots & \ldots \end{array} \right|=0 \tag{6.7} \]

or, in abbreviated notation,

\[ |k_{ij}-T_{ij}\omega^2|=0 \quad (i,j=1,2,\ldots,n). \tag{6.8} \]

The determinant (6.7) is an equation of degree \(n\) with respect to \(\omega^2\)—a secular equation. Solving it, we find \(n\) frequencies of oscillation \(\omega_s\).

The form (6.7) of the secular equation is inconvenient in that all elements of the determinant contain, generally speaking, the unknown \(\omega^2\). It can, however, be brought to a more convenient form, namely one in which \(\omega^2\) will enter only into the diagonal terms of the determinant. To this end we multiply the system of equations (6.4) by the coefficients \(A_{li}\), satisfying the conditions *)

\[ \sum_i A_{li}T_{ij}=\delta_{lj} \begin{cases} =1 & \text{for } l=j,\\ =0 & \text{for } l\ne j \end{cases} \tag{6.9} \]

and sum over \(i\):

\[ \sum_{ij} A_{li}(T_{ij}\ddot{x}_j+k_{ij}x_j)=0. \tag{6.10} \]

From (6.10), by virtue of (6.9), we obtain

\[ \ddot{x}_l=-\sum_{ij} A_{li}k_{ij}x_j =-\sum_i A_{li}\frac{\partial U}{\partial x_i}. \tag{6.11} \]

*) In other words, we introduce the matrix \(A\) with elements \(A_{li}\), inverse to the kinetic-energy matrix \(T\) with elements \(T_{ij}\).

Substitution of the solution (6.5) gives

\[ \omega^2 x_{l0}=\sum_{ij} A_{li} k_{ij} x_{j0}, \tag{6.12} \]

therefore,

\[ \sum_{ij}\left(A_{li}k_{ij}-\delta_{lj}\omega^2\right)x_{j0} =\sum_j\left[\left(\sum_i A_{li}k_{ij}\right)-\delta_{lj}\omega^2\right]x_{j0}=0. \tag{6.13} \]

Introducing the coefficients

\[ D_{lj}=\sum_i A_{li}k_{ij}=\sum_i A_{li}\frac{\partial^2 U}{\partial x_i \partial x_j}, \tag{6.14} \]

we can write this system in the form

\[ \sum_j \left(D_{lj}-\delta_{lj}\cdot\omega^2\right)x_{j0}=0. \tag{6.15} \]

From the condition for solvability of the system (6.15) we obtain the transformed secular equation in the form

\[ \left| \begin{array}{cccc} D_{11}-\omega^2 & D_{12} & D_{13} & \cdots\\ D_{21} & D_{22}-\omega^2 & D_{23} & \cdots\\ D_{31} & D_{32} & D_{33}-\omega^2 & \cdots\\ \cdots & \cdots & \cdots & \cdots \end{array} \right|=0 \tag{6.16} \]

or, in abbreviated notation,

\[ \left|D_{lj}-\delta_{lj}\omega^2\right|=0. \tag{6.17} \]

The unknown \(\omega^2\) enters only into the diagonal terms of the determinant (6.16). Now, however, in contrast to the form (6.7) of the secular equation, the elements \(D_{lj}\) are no longer symmetric with respect to the indices \(l\) and \(j\) \((D_{lj}\ne D_{jl})\). This drawback of the secular equation (6.16), however, is not particularly essential. The main advantage of the form (6.16) of the secular equation is that, when the nondiagonal elements \(D_{lj}\,(l\ne j)\) are small, the diagonal terms give approximate values of the squares of the vibration frequencies

\[ \omega_l^2 \cong \omega_{l0}^2 = D_{ll}, \tag{6.18} \]

and we obtain a natural zero approximation for the calculation of vibrations. Smallness of the nondiagonal terms, at least of a considerable part of them, can be achieved by a rational choice of coordinates. Natural vibrational coordinates are precisely such rationally introduced coordinates.

The determinant (6.16), when natural vibrational coordinates are introduced, constitutes the natural secular equation for determining the vibration frequencies of the molecule. We shall choose

as generalized coordinates \(x_j\) the natural vibrational coordinates \(q\) and \(\gamma\), introduced by us in § 5, and in what follows, by \(x_j\) in the formulas of the present paragraph given above, we shall mean precisely these coordinates.

Each diagonal element \(D_{ll}\) of the secular equation now gives the square of that frequency which would be obtained if only the corresponding natural coordinate \(x_j\) were varied, i.e., if the given distance or angle were varied independently of variations of all the other angles and distances. In this case the frequency \(\omega_{l0}=\sqrt{D_{ll}}\) would be characteristic of an oscillation consisting in the change of only the corresponding distance or angle.

Let us now consider in more detail the coefficients \(D_{lj}(l\ne j)\). Each such nondiagonal coefficient characterizes the “interaction” of two different natural coordinates \(x_l\) and \(x_j\). According to (6,11) and (6,14),

\[ \ddot{x}_l=-\sum_j D_{lj}x_j, \tag{6,19} \]

i.e., the coefficient \(D_{lj}\) determines the change in time of the coordinate \(x_l\) as a function of the value of the coordinate \(x_j\).

In the valence-force coordinates \(q\) and \(\gamma\) we can write (6,19) in the following form:

\[ \ddot{q}_r=-\sum_t D_{rt}q_t-\sum_\gamma D_{r\gamma}\gamma_\gamma, \tag{6,20} \]

\[ \ddot{\gamma}_\delta=-\sum_t D_{\delta t}q_t-\sum_\gamma D_{\delta\gamma}\gamma_\gamma. \tag{6,21} \]

Here the coefficients \(D_{rt}\) determine the interaction of bonds with one another, \(D_{\delta\gamma}\) that of angles with one another, and \(D_{r\gamma}\) and \(D_{\delta t}\) that of bonds with angles (namely, \(D_{r\gamma}\) characterizes the action of angles on bonds, and \(D_{\delta t}\) that of bonds on angles). Generally speaking, as was already noted above, \(D_{lj}\ne D_{jl}\) (for \(l\ne j\)), i.e., \(D_{rt}\ne D_{tr}\) \((r\ne t)\), \(D_{\delta\gamma}\ne D_{\gamma\delta}\) \((\delta\ne\gamma)\), \(D_{\delta t}\ne D_{t\delta}\). We shall call the coefficients \(D_{lj}\) the coefficients of total interaction, in contrast to the coefficients \(A_{lj}\) and \(k_{ij}\), through which the \(D_{lj}\) are expressed according to (6,14) and which also characterize the interaction of coordinates in a definite way.

Let us consider separately the coefficients \(A_{lj}\) and the coefficients \(k_{ij}\). The coefficients \(A_{lj}\), according to (6,12), determine the change of the coordinate \(x_j\) under the action of the generalized force \(\dfrac{\partial U}{\partial x_i}\) associated with the change of the coordinate \(x_i\). They are expressed in terms of the coefficients \(T_{ik}\) in the expression for the kinetic energy [as follows from their definition according to (6,9)] and depend on the masses of the atoms and on quantities characterizing the normal configuration—on the equilibrium distances \(s\) and equilibrium angles \(\vartheta\). The fact that the nondiagonal coefficients \(A_{li}(l\ne i)\) differ from zero is due to the fact that, in the motion of each atom,

participating in the change of a given distance or angle (see (5,16) and (5,19)), other distances and angles also change simultaneously. We shall return to this question below (see § 7).

We shall call the coefficients \(A_{li}\) coefficients of kinematic interaction, since they are not connected with the magnitude of the force of interaction between atoms and are determined only by the kinematics of the vibrations.

The coefficients \(k_{ij}\)—the coefficients in expression (6,2) for the potential energy—on the contrary, characterize the forces of interaction of atoms, and their magnitude is determined by the structure of the electron shell of the molecule. The diagonal coefficient \(k_{ii}\) is a force constant, giving the quasi-elastic force that arises when \(x_i\), the magnitude of the given distance or angle, is changed. The off-diagonal (“cross”) coefficients \(k_{ij}(i \ne j)\) characterize the mutual influence of changes in bonds and angles—the force acting on bond or angle \(i\) as a result of a change in bond or angle \(j\), and the equal force acting on bond or angle \(j\) as a result of a change in bond or angle \(i\) \((k_{ij}=k_{ji})\). The appearance of these forces is connected with the mutual influence of electrons localized on different bonds when bond lengths and angle magnitudes are changed.

The coefficients \(k_{ij}\)—the force constants—may be called, in accordance with the terminology introduced for \(D_{ij}\) and \(A_{li}\), coefficients of dynamical interaction. It is precisely these coefficients that are of greatest physical interest, since they characterize in a definite way the properties of the electron shell, and their values are closely connected with the properties of the chemical bond; in particular, they depend on the presence of electron resonance.

The basic formula (6,14) expresses the total interaction of the coordinates \(x_i\) and \(x_j\) through kinematic and dynamical interactions. The diagonal element \(D_{ii}\) determines the “interaction of the coordinate \(x_i\) with itself.” It should be emphasized that the magnitude of the “interaction” of different coordinates, i.e. the magnitude of the off-diagonal coefficients \((i \ne j)\) both of the total interaction and of the kinematic and dynamical interactions, depends on the choice of coordinates. It was already noted above (p. 32) that by introducing natural coordinates one achieves smallness of a significant part of the coefficients of total interaction. In § 5, when considering the question of valence-force and central-force natural coordinates, it was pointed out that it is rational to use those coordinates for which the off-diagonal terms in the expression for the potential energy are small, i.e. the coefficients of dynamical interaction \(k_{ij}(i \ne j)\). The smallness of the coefficients \(k_{ij}\) will contribute to a decrease, according to (6,14), also of the total coefficients \(D_{ij}\)*).

*) It should be kept in mind here that each coefficient \(D_{ij}\) also contains the diagonal dynamical coefficient \(k_{jj}\), multiplied by the off-diagonal kinematic coefficient \(A_{ij}\). With such a choice of natural ...

Let us also briefly examine the question of normal coordinates and the form of the vibrations. When the natural secular equation (6.16) has been solved, then from the system of equations (6.15) the amplitudes of the vibrations \(x_{j0s}\) are determined for each value of the vibration frequency \(\omega_s\). We obtain

\[ x_{js}=x_{j0s}e^{i\omega_s t}=a_{js}Q_{s0}e^{i\omega_s t}=a_{js}Q_s, \tag{6.22} \]

where \(Q_s=Q_{s0}e^{i\omega_s t}\) is the \(s\)-th normal coordinate, varying with frequency \(\omega_s\). Each normal vibration of the particles consists in the simultaneous variation, according to the law (6.22), of all natural coordinates, i.e. all coordinates vary simultaneously with the same frequency and with the same phases; the amplitudes of the natural coordinates for the given \(s\)-th normal vibration are expressed in terms of the amplitude of the normal coordinate \(Q_{s0}\) by the formula

\[ x_{j0s}=a_{js}Q_{s0}. \tag{6.23} \]

Substitution of the values \(\omega^2=\omega_s^2\) and \(x_{j0}=x_{j0s}\) into the system (6.15) gives, taking (6.23) into account and canceling by \(Q_{s0}\),

\[ \sum_{j=1}^{n}(D_{lj}-\delta_{lj}\omega_s^2)a_{js}=0. \tag{6.24} \]

Solving this system of equations, we find for the \(s\)-th normal vibration a set of coefficients \(a_{js}\), which, according to (6.23), determine the relative amplitudes of the natural coordinates \(x_{j0s}\)—the form of the given normal vibration, determining its character. For a purely valence vibration, the amplitudes of all angular coordinates \(\gamma\) are equal to zero; for a purely deformation vibration, the amplitudes of all coordinates \(q\) are equal to zero. In a real vibration, as a rule, both bond lengths and angles vary simultaneously. The predominant variation of coordinates of a definite type makes it possible to classify the corresponding vibration in a definite way (see § 5).

For each separate normal vibration its form is determined by formulas (6.22). If the values \(Q_1, Q_2, \ldots, Q_s, \ldots, Q_n\) of the normal coordinates are specified for all vibrations, then

\[ x_j=\sum_{s=1}^{n}x_{js}=\sum_s a_{js}Q_s. \tag{6.25} \]

Formulas (6.25) express the relation between the natural and normal coordinates.

coordinates for which the dynamic coefficients are small, the kinematic coefficients will not, generally speaking, be small. However, it is usually nevertheless rational to strive to reduce the full interaction coefficients precisely by reducing the dynamic interaction, with the proper introduction of natural coordinates.

The kinetic and potential energies (6.1) and (6.2), expressed in normal coordinates, as is easily shown, take the form

\[ T=\frac{1}{2}\sum_{s=1}^{n}\dot Q_s^{\,2}, \tag{6,26} \]

\[ U=\frac{1}{2}\sum_{s=1}^{n}\omega_s^2 Q_s^{\,2}, \tag{6,27} \]

i.e., they are sums of squares.

§ 7. DETERMINATION OF THE INTERACTION COEFFICIENTS

The solution of the basic problem of the mechanics of vibrations—the construction of the secular equation—reduces to the determination of the coefficients of the complete interaction \(D_{ij}\). For this, according to (6.14), it is necessary to find the dynamic coefficients \(k_{ij}\) and the kinematic coefficients \(A_{ij}\). The dynamic coefficients—the force constants, in accordance with the semiempirical character of the methods employed—are obtained from experimental values of vibrational frequencies (see §§ 1 and 2). Therefore they are either regarded in formulas (6.14) as unknowns to be determined, or, if their values have already been found by a semiempirical method (for example, from data for simpler molecules containing the same structural elements), are known numerical constants. The problem reduces to the determination of the kinematic coefficients. Finding them by explicitly forming expression (6.1) for the kinetic energy, with subsequent passage from the coefficients \(T_{ij}\) to the sought coefficients \(A_{ij}\) by formula (6.9), is in practice a rather complicated and laborious task. It is considerably simpler to determine the coefficients \(A_{ij}\) by directly forming the equation of motion (6.11) in natural coordinates, relating the accelerations \(\ddot x_i\) to the generalized forces \(\dfrac{dU}{dx_i}\). This method, developed by the author\(^{1,26,27,28}\), is a generalization of the Born and Karman method\(^{18}\). Its essence can be explained most clearly on the simplest example of a one-dimensional crystal—an infinite linear chain of atoms, considered in the original work of Born and Karman.

Fig. 10. Displacement of atoms of a linear chain.

Let \(y_i\) denote the displacement of the \(i\)-th atom of the chain (Fig. 10). Assuming that quasi-elastic forces act on the atom, depending on the relative displacements of neighboring atoms \(q_i=y_{i+1}-y_i\) and \(q_{i-1}=y_i-y_{i-1}\), we obtain the equations of motion in the form

\[ m\ddot y_i=k(y_{i+1}-y_i)-k(y_i-y_{i-1})=kq_i-kq_{i-1}. \tag{7,1} \]

For simplicity we shall assume the masses of all atoms and all quasi-elastic constants to be identical.

Similarly, for the \(i+1\)-st atom,

\[ m\ddot y_{i+1}=kq_{i+1}-kq_i . \tag{7,2} \]

Dividing both equations by \(m\) and subtracting (7,1) from (7,2), we find

\[ \ddot q_i=\ddot y_{i+1}-\ddot y_i=-\frac{k}{m}q_{i+1}-\frac{2k}{m}q_i+\frac{k}{m}q_{i-1}. \tag{7,3} \]

Thus, a system of equations of motion in the relative coordinates \(q_i\) is obtained. This system is of type (6,11), and from it follows the secular equation of type (6,16)

\[ \left| \begin{array}{cccccc} \cdot&\cdot&\cdot&\cdot&\cdot&\cdot\\ \cdots \dfrac{2k}{m}-\omega^2&-\dfrac{k}{m}&0&0\cdots\\ \cdots -\dfrac{k}{m}&\dfrac{2k}{m}-\omega^2&-\dfrac{k}{m}&0\cdots\\ \cdots 0&-\dfrac{k}{m}&\dfrac{2k}{m}-\omega^2&-\dfrac{k}{m}\cdots\\ \cdot&\cdot&\cdot&\cdot&\cdot&\cdot \end{array} \right|=0. \tag{7,4} \]

We shall now apply an analogous method to the vibrations of a branched molecule and shall formulate the equation of motion in natural coordinates and in vector form.

Fig. 11. Diagram of a branched molecule.

From the equation of motion we shall immediately find the required coefficients of the kinematic interaction. Let us consider in more detail the simplest case of purely valence vibrations.

Let there be given a branched molecule consisting of atoms \(X, Y, Z, U,\ldots\) with masses \(m_X, m_Y, m_Z, m_U\) (Fig. 11). We shall regard the potential energy as a function only of the changes \(q_{XY}, q_{XZ}, q_{YU}\), etc., of the distances between neighboring atoms:

\[ U=U(q)=U(q_{XY},q_{XZ},q_{YU},\ldots), \tag{7,5} \]

where, according to (5,16),

\[ q_{XY}=\mathbf e_{XY}(\mathbf r_Y-\mathbf r_X);\quad q_{XZ}=\mathbf e_{XZ}(\mathbf r_Z-\mathbf r_X);\quad \ldots, \tag{7,6} \]

i.e. they are expressed in terms of the displacements of the atoms \(\mathbf r_X, \mathbf r_Y, \mathbf r_Z\) and the unit vectors \(\mathbf e_{XY}, \mathbf e_{XZ},\ldots\).

The equation of motion of atom \(X\) will be

\[ m_X\ddot{\mathbf r}_X=-\nabla_X U(q_{XY},q_{XZ},q_{XY},\ldots). \tag{7,7} \]

On the right-hand side the gradient is taken with respect to the coordinates of atom \(X\). The potential energy depends on its displacement \(\mathbf r_X\) through the intermediary of

\(q_{XY}, q_{XZ}, \ldots\), i.e. through all changes of the distances from atom X to neighboring atoms. Therefore

\[ \nabla_X U=\frac{\partial U}{\partial q_{XY}}\nabla_X q_{XY}+\frac{\partial U}{\partial q_{XZ}}\nabla_X q_{XZ}+\ldots =\sum_R \frac{\partial U}{\partial q_{XR}}\nabla_X q_{XR}, \tag{7,8} \]

where the summation is over all neighboring atoms.

Taking into account that

\[ \nabla_X(\mathbf r_X \mathbf b)=\mathbf b \tag{7,9} \]

for any constant vector \(\mathbf b\), we obtain from (7,6)

\[ \nabla_X q_{XY}=-\mathbf e_{XY}. \tag{7,10} \]

Equation (7,7) will take the form

\[ m_Y \ddot{\mathbf r}_X=\sum_R \frac{\partial U}{\partial q_{XR}}\mathbf e_{XR}. \tag{7,11} \]

The right-hand side gives the force associated with the change of all distances from atom X to neighboring atoms. For atom Y we similarly have the equation of motion:

\[ m_Y \ddot{\mathbf r}_X=\sum_S \frac{\partial U}{\partial q_{YS}}\mathbf e_{YS}. \tag{7,12} \]

Equations (7,11) and (7,12) are analogous to equations (7,1) and (7,2). Dividing (7,11) by \(m_X\) and (7,12) by \(m_Y\) and subtracting the first from the second, we obtain

\[ \ddot{\mathbf r}_Y-\ddot{\mathbf r}_X =\frac{1}{m_Y}\sum_S \frac{\partial U}{\partial q_{YS}}\mathbf e_{YS} -\frac{1}{m_X}\sum_R \frac{\partial U}{\partial q_{XR}}\mathbf e_{XR}. \tag{7,13} \]

Taking into account that both the first and the second sums contain a term with \(\dfrac{\partial U}{\partial q_{XY}}\), we can rewrite (7,13) in the form (noting that \(\mathbf e_{XY}=-\mathbf e_{YX}\)):

\[ \ddot{\mathbf r}_Y-\ddot{\mathbf r}_X =\frac{d^2}{dt^2}(\mathbf r_Y-\mathbf r_X) =-\left(\frac{1}{m_X}+\frac{1}{m_Y}\right)\mathbf e_{XY}\frac{\partial U}{\partial q_{XY}}+ \]

\[ +\frac{1}{m_Y}\sum_{S\ne X}\mathbf e_{YS}\frac{\partial U}{\partial q_{YS}} -\frac{1}{m_X}\sum_{R\ne Y}\mathbf e_{XR}\frac{\partial U}{\partial q_{XR}}. \tag{7,14} \]

Multiplying (7,14) by \(\mathbf e_{XY}\), we obtain the required equation of motion in the natural coordinates \(q_{XY}\):

\[ \ddot q_{XY}=\mathbf e_{XY}(\ddot{\mathbf r}_Y-\ddot{\mathbf r}_X) =-\left(\frac{1}{m_X}+\frac{1}{m_Y}\right)\frac{\partial U}{\partial q_{XY}}- \]

\[ -\sum_{S\ne X}\frac{\mathbf e_{YX}\mathbf e_{YS}}{m_Y}\frac{\partial U}{\partial q_{YS}} -\sum_{R\ne Y}\frac{\mathbf e_{XY}\mathbf e_{XR}}{m_X}\frac{\partial U}{\partial q_{XR}}. \tag{7,15} \]

This equation is of type (6,11). The coefficients of \(-\dfrac{\partial U}{\partial q_{XR}}\) represent the required coefficients of the kinematic interaction

coordinate \(q_{XY}\) with coordinate \(q_{XX}\). In the present case we have two kinds of interaction coefficients—the interaction coefficient \((q_{XY}, q_{XY})\) of the coordinate \(q_{XY}\) “with itself,” equal to

\[ (q_{XY}, q_{XY}) \qquad \frac{1}{m_X}+\frac{1}{m_Y}, \tag{7,16} \]

and the interaction coefficient \((q_{XY}, q_{XZ})\) of the coordinates \(q_{XY}\) and \(q_{XZ}\), which have one common atom \(X\), equal to [cf. (5,2)]

\[ (q_{XY}, q_{XZ}) \qquad \frac{\mathbf e_{XY}\mathbf e_{XZ}}{m_X} = \frac{1}{m_X}\cos \vartheta_{YXZ}. \tag{7,17} \]

The coefficients of the kinematic interaction of bonds that do not have a common atom, for example \(Y—Z\) and \(X—U\) (see Fig. 11), are equal to zero. This is due to the fact that the acceleration of a given atom, for example \(X\) [cf. (7,7)], depends only on those coordinates which change under displacements of this atom (on \(q_{XY}, q_{XZ}\), etc.).

The interaction of the coordinates \(q_{XY}\) and \(q_{XY}\) is effected through the common atom, which participates simultaneously in two motions—the change of the bond \(X—Y\) and the change of the bond \(X—Z\)*). Accordingly, expression (7,17) contains the mass \(m_X\) of the common atom. The greater it is, the smaller the interaction will be. The interaction of the coordinate \(q_{XY}\) with itself is effected through both atoms \(X\) and \(Y\), and therefore (7,16) contains two masses—that of atom \(X\) as well as that of atom \(Y\).

By an entirely analogous method, but only with more complicated calculations, the kinematic interaction of angular coordinates \(\gamma\) with one another and with the coordinates \(q\) is determined. For this purpose the equations of motion of the three atoms \(X, Y\), and \(Z\), forming the angle \((XY, XZ)\), are written down, the potential energy being taken in the form of a function of both the coordinates \(q\) and the coordinates \(\gamma\). Similarly to (7,11) and (7,12), we obtain \(\ddot{\mathbf r}_X, \ddot{\mathbf r}_Y\), and \(\ddot{\mathbf r}_Z\), expressed in terms of the vectors \(\mathbf e\) and \(\mathbf f\). Composing, according to (5,19),

\[ \ddot{\gamma}_{YZ} = \frac{(\ddot{\mathbf r}_X-\ddot{\mathbf r}_Y)}{s_{XY}}\mathbf f_{YZ} + \frac{(\ddot{\mathbf r}_X-\ddot{\mathbf r}_Z)}{s_{XZ}}\mathbf f_{ZY}, \tag{7,18} \]

we obtain an equation of the type (6,11), from which the required coefficients of kinematic interaction are found. Bonds and angles having at least one common atom interact. In all, eleven types of kinematic interactions are possible [including the interactions (7,16) and (7,17)]; the corresponding coefficients are given in Table 1.

*) This circumstance was noted by Mekke^41, who introduced the corresponding term “Mitschwingen.”

Table 1

Coefficients of the kinematic interaction of the coordinates \(q\) and \(\gamma\)

Type of interaction	Number in order	Interacting coordinates	Explanatory drawing	Value of the interaction coefficient
Interaction of a bond with itself	1	\((q, q)\)	bond \(q\) between masses \(m_1\) and \(m_2\), with unit vector \(\vec e\)	\(\displaystyle \frac{1}{m_1}+\frac{1}{m_2}\)
Interaction of two bonds	2	\((q_1, q_2)\)	two bonds \(q_1,q_2\) meeting at \(m_0\), with unit vectors \(\vec e_1,\vec e_2\) and angle \(\vartheta_{12}\)	\(\displaystyle \frac{1}{m_0}\,\mathbf e_1\mathbf e_2=\frac{1}{m_0}\cos\vartheta_{12}\)
Interaction of a bond with the angle formed by this bond	3	\((q_1,\gamma_{12})\)	bonds \(q_1,q_2\) forming angle \(\gamma_{12}\) at \(m_0\); vectors \(\vec e_1,\vec e_2,\vec f_{21}\)	\(\displaystyle -\frac{1}{m_0s_2}\,\mathbf e_1\mathbf f_{21}= -\frac{1}{m_0s_2}\sin\vartheta_{12}\)
Interaction of a bond with an angle having one atom in common with the bond	4	\((q_1,\gamma_{23})\)	bond \(q_1\) adjacent to angle \(\gamma_{23}\); vectors \(\vec e_1,\vec e_2,\vec e_3,\vec f_{23},\vec f_{32}\)	\(\displaystyle -\frac{1}{m_0}\,\mathbf e_1\mathbf F_{23}\)
Interaction of a bond with an angle having one atom in common with the bond	5	\((q_1,\gamma_{23})\)	bond \(q_1\) adjacent to angle \(\gamma_{23}\); vectors \(\vec e_1,\vec e_2,\vec e_3,\vec f_{23}\)	\(\displaystyle \frac{1}{m_0s_2}\,\mathbf e_1\mathbf f_{23}\)
Interaction of an angle with itself	6	\((\gamma_{12},\gamma_{12})\)	angle \(\gamma_{12}\) at \(m_0\), with masses \(m_1,m_2\) and vectors \(\vec e_1,\vec e_2,\vec f_1,\vec f_{12},\vec f_{21}\)	\(\displaystyle \frac{1}{m_0}F_{12}^{2}+\frac{1}{m_1s_1^{2}}f_{12}^{2}+\frac{1}{m_2s_2^{2}}f_{21}^{2}=\frac{1}{m_0}\left(\frac{1}{s_1^{2}}+\frac{1}{s_2^{2}}-\frac{2}{s_1s_2}\cos\vartheta_{12}\right)+\frac{1}{m_1s_1^{2}}+\frac{1}{m_2s_2^{2}}\)

Continuation

Type of interaction	Number in order	Interacting coordinates	Explanatory drawing	Value of the interaction coefficient
Interaction of angles having two common atoms	7	\((\gamma_{12}, \gamma_{13})\)	Diagram with central atom \(m_0\), upper atom \(m_1\), bonds labeled \(\gamma_{12}\), \(\gamma_{13}\), vectors \(\mathbf e_2\), \(\mathbf e_3\), \(\mathbf f_{12}\), \(\mathbf f_{13}\).	\(\displaystyle \frac{1}{m_0}F_{12}F_{13}+\frac{1}{m_1s_1^{\,2}}f_{12}f_{13}\)
Interaction of angles having two common atoms	8	\((\gamma_{12}, \gamma_{13})\)	Diagram of two adjacent angles with common atoms \(m_2\), \(m_3\), bonds labeled \(\gamma_{12}\), \(\gamma_{13}\), and vectors \(\mathbf e_1\), \(\mathbf e_3\), \(\mathbf f_{12}\), \(\mathbf f_{13}\).	\(\displaystyle -\frac{1}{m_2s_1}F_{12}f_{13}-\frac{1}{m_3s_1}F_{13}f_{12}\)
Interaction of angles having one common atom	9	\((\gamma_{12}, \gamma_{34})\)	Diagram of two angles sharing central atom \(m_0\), bonds labeled \(\gamma_{12}\), \(\gamma_{34}\), vectors \(\mathbf e_1,\mathbf e_2,\mathbf e_3,\mathbf e_4\), \(\mathbf F_{12}\), \(\mathbf F_{34}\).	\(\displaystyle \frac{1}{m_0}F_{12}F_{34}\)
Interaction of angles having one common atom	10	\((\gamma_{12}, \gamma_{34})\)	Diagram with angles \(\gamma_{12}\), \(\gamma_{34}\), vectors \(\mathbf f_{12}\), \(\mathbf f_{34}\), and central atom \(m_0\).	\(\displaystyle -\frac{1}{m_0s_3}F_{12}f_{34}\)
Interaction of angles having one common atom	11	\((\gamma_{12}, \gamma_{34})\)	Diagram of a chain of four bonds with angles \(\gamma_{12}\), \(\gamma_{34}\), vectors \(\mathbf e_1,\mathbf e_2,\mathbf e_3,\mathbf e_4\), \(\mathbf f_{12}\), \(\mathbf f_{34}\), and atom \(m_0\).	\(\displaystyle \frac{1}{m_0s_1s_3}f_{12}f_{34}\)

Notation: \(\mathbf e_\beta\) is the unit vector along bond \(\beta\);

\[ \mathbf f_{\beta\gamma} = \frac{1}{\sin \vartheta_{\beta\gamma}} \left(\mathbf e_\gamma-\mathbf e_\beta\cos\vartheta_{\beta\gamma}\right) \]

is the unit vector in the plane \((\beta\gamma)\), perpendicular to bond \(\beta\);

\[ F_{\beta\gamma}=\frac{1}{s_\beta}f_{\beta\gamma}+\frac{1}{s_\gamma}f_{\gamma\beta}; \]

\(m_0, m_1, m_2, m_3\) are the masses of the atoms; \(s_\beta\) is the length of bond \(\beta\); \(\vartheta_{\beta\gamma}\) is the angle between bonds \(\beta\) and \(\gamma\).

They are expressed in terms of the vectors \(\mathbf e\), \(\mathbf f\), and

\[ F_{YX}=\frac{f_{YZ}}{s_{XY}}+\frac{f_{ZY}}{s_{XZ}}, \tag{7,19} \]

which, in the case of two identical bonds \((s_{XY}=s_{XZ}=s)\), are equal to

\[ F_{YZ}=\frac{1}{s}\left(f_{YZ}+f_{ZY}\right). \tag{7,20} \]

and are directed along the bisector of the angle \((XY, XZ)\) (Fig. 12). For each type of interaction an explanatory drawing is given. For brevity, bonds are numbered with a single digit. Atoms common to the interacting bonds and angles are shown in the drawings by solid circles; the others by open circles.

The table gives vector expressions for the interaction coefficients; from these expressions one can easily pass to explicit expressions through the angles \(\vartheta_{ik}\). For example, for the coefficient \((q_1, \gamma_{12})\) we have

\[ -\frac{1}{m_0}\frac{\mathbf e_1\mathbf f_{21}}{s_2} = -\frac{1}{m_0s_2}\sin\vartheta_{12}, \tag{7,21} \]

and for the coefficient \((\gamma_{12}, \gamma_{12})\), taking (7,19) into account,

\[ \frac{1}{m_0}F_{12}^{2} +\frac{1}{m_1s_1^{2}}f_{12}^{2} +\frac{1}{m_2s_2^{2}}f_{21}^{2} = \]

\[ = \frac{1}{m_0}\left(\frac{f_{12}}{s_1}+\frac{f_{21}}{s_2}\right)^2 +\frac{1}{m_1s_1^{2}} +\frac{1}{m_2s_2^{2}} = \]

\[ = \frac{1}{m_0}\left(\frac{1}{s_1^{2}}+\frac{1}{s_2^{2}}-\frac{2}{s_1s_2}\cos\vartheta_{12}\right) +\frac{1}{m_1s_1^{2}} +\frac{1}{m_2s_2^{2}}. \tag{7,22} \]

Table 1 makes it possible at once to write out the interaction coefficients for the various possible cases. Thus, the problem of determining them, and consequently also the problem of finding the coefficients of the complete interaction \(D\), is solved completely.

Fig. 12. Direction of the vector \(F\) for the case of two identical bonds.

Fig. 13. Triatomic molecule of type \(XYZ\).

Let us consider an example of setting up the secular equation for determining the vibrational frequencies of a triatomic molecule (Fig. 13). Introducing the valence-force coordinates \(q_1, q_2\) and \(\gamma\), and using Table 1 and formulas (7,17), (7,21), and (7,22), we immediately write, in the form

table of coefficients of kinematic interaction:

\[ \begin{array}{c|ccc} & q_1 & q_2 & \gamma \\[2mm] q_1 & \dfrac{1}{m_0}+\dfrac{1}{m_1} & \dfrac{1}{m_0}\cos\vartheta & -\dfrac{1}{m_0s_2}\sin\vartheta \\[3mm] q_2 & \dfrac{1}{m_0}\cos\vartheta & \dfrac{1}{m_0}+\dfrac{1}{m_2} & -\dfrac{1}{m_0s_1}\sin\vartheta \\[3mm] \gamma & -\dfrac{1}{m_0s_2}\sin\vartheta & -\dfrac{1}{m_0s_1}\sin\vartheta & \dfrac{1}{m_0}\left(\dfrac{1}{s_1^2}+\dfrac{1}{s_2^2}-\dfrac{2\cos\vartheta}{s_1s_2}\right) +\dfrac{1}{m_1s_1^2}+\dfrac{1}{m_2s_2^2} \end{array} \tag{7.23} \]

The coefficients of dynamical interaction \(k_{ij}\)—the force constants—we likewise write in the form of a table, using expression (5,6) for the potential energy in valence-force coordinates; this expression is a special case of the general formula (6,2) (where in the summation over \(i\) and \(j\) the terms with \(i\ne j\) enter twice):

\[ \begin{array}{c|ccc} & q_1 & q_2 & \gamma \\[1mm] q_1 & k_1 & h & a_1 \\ q_2 & h & k_2 & a_2 \\ \gamma & a_1 & a_2 & k_\gamma \end{array}. \tag{7.24} \]

The coefficients of total interaction are now determined by formula (6,14), which means that the \(j\)-th coefficient in the \(l\)-th row is obtained by multiplying the \(l\)-th row of table (7,23) by the \(j\)-th column of table (7,24), i.e. according to the rules of matrix multiplication. For example, the coefficient of total interaction \((q_1,\gamma)\) (the third coefficient of the first row) is obtained by multiplying the row \(q_1\) in (7,23) by the column \(\gamma\) in (7,24), i.e.

\[ D_{q_1\gamma} = \left(\dfrac{1}{m_0}+\dfrac{1}{m_1}\right)a_1 +\dfrac{1}{m_0}\cos\vartheta\cdot a_2 -\dfrac{1}{m_0s_2}\sin\vartheta\cdot k_\gamma . \tag{7.25} \]

Writing out the table of all coefficients \(D\), subtracting \(\omega^2\) from its diagonal elements, and setting the corresponding determinant equal to zero, we obtain the desired secular equation. For simplicity we give it for the model of valence forces, when in (7,24) the constants \(h=a_1=a_2=0\); it has the form

\[ \left| \begin{array}{ccc} \left(\dfrac{1}{m_0}+\dfrac{1}{m_1}\right)\cdot k_1-\omega^2 & \dfrac{1}{m_0}\cos\vartheta\cdot k_2 & -\dfrac{1}{m_0s_2}\sin\vartheta\cdot k_\gamma \\[3mm] \dfrac{1}{m_0}\cos\vartheta\cdot k_1 & \left(\dfrac{1}{m_0}+\dfrac{1}{m_2}\right)\cdot k_2-\omega^2 & -\dfrac{1}{m_0s_1}\sin\vartheta\cdot k_\gamma \\[3mm] -\dfrac{1}{m_0s_2}\sin\vartheta\cdot k_1 & -\dfrac{1}{m_0s_1}\sin\vartheta\cdot k_2 & \left[ \dfrac{1}{m_0}\left( \dfrac{1}{s_1^2}+\dfrac{1}{s_2^2} -\dfrac{2\cos\vartheta}{s_1s_2} \right) +\dfrac{1}{m_1s_1^2} +\dfrac{1}{m_2s_2^2} \right]k_\gamma-\omega^2 \end{array} \right|=0. \tag{7.26} \]

The roots of this cubic equation give the desired frequencies of vibration. It should be borne in mind that \(m_0, m_1, m_2, s_1, s_2\), and \(\vartheta\), for a particular molecule of known shape and dimensions, are assigned numerical coefficients.

In concluding this paragraph, let us note that from the equations of motion in the natural coordinates \(q\) and \(\gamma\) there automatically follows the equality to zero of the momentum and angular momentum when these coordinates are varied, i.e., the motion under consideration is purely vibrational. This can be verified directly by writing down the expressions for the momentum and angular momentum \(^{27,1}\), and is a consequence of the fact that the coordinates \(q\) and \(\gamma\) are relative ones, depending on differences of atomic displacements [cf. (5,16) and (5,19)]. The kinematic coefficients \(A_{li}\) found by us, related by the relation (6,9) to the coefficients in the expression for the kinetic energy (6,1), correspond to the case in which this expression (6,1) is already taken with account of the equality to zero of the momentum and angular momentum.

§ 8. SYMMETRY COORDINATES

If for a triatomic molecule a secular equation of the third degree is obtained, then already for a tetratomic molecule an equation of the sixth degree is obtained \((3N - 6 = 3 \cdot 4 - 6 = 6\) vibrational degrees of freedom), and for a pentatomic molecule, for example methane \(\mathrm{CH}_4\), already one of the ninth degree. Therefore it is extremely important to lower the degree of the secular equation and to simplify the method of constructing it, which becomes very cumbersome for equations of high degree. For molecules possessing symmetry, this is achieved by introducing symmetry coordinates.

Symmetry coordinates represent an intermediate link between the natural coordinates and the desired normal coordinates, in which the problem of vibrations is completely separated. In normal coordinates, for each degree of freedom we have, considering the vibrations to be small, a problem of harmonic vibration. According to (6,26) and (6,27), the total energy

\[ H = T + U = \frac{1}{2}\sum_{s=1}^{n}\dot Q_s^{\,2} + \frac{1}{2}\sum_{s=1}^{n}\omega_s^2 Q_s^2 = \]

\[ = \sum_{s=1}^{n}\left(\frac{1}{2}\dot Q_s^{\,2} + \frac{1}{2}\omega_s^2 Q_s^2\right) = \sum_{s=1}^{n} H_s, \tag{8,1} \]

where

\[ H_s = \frac{1}{2}\dot Q_s^{\,2} + \frac{1}{2}\omega_s^2 Q_s^2. \tag{8,2} \]

In the original natural coordinates \(x\), according to (6.1) and (6.2),

\[ H=T+U=\frac{1}{2}\sum_{i,j=1}^{n}\left(T_{ik}\dot{x}_{i}\dot{x}_{k}+k_{ij}x_i x_k\right). \tag{8.3} \]

Owing to the presence of terms containing \(T_{ij}\) and \(k_{ij}\) \((i\ne j)\), which characterize the interaction of the different coordinates \(x_i\) and \(x_j\), we have a single problem for all \(n\) degrees of freedom, whereas in normal coordinates we have \(n\) independent problems, each for one degree of freedom. In symmetry coordinates a partial separation of the problem is achieved—the problem for \(n\) degrees of freedom breaks up into several problems, each for a smaller number of degrees of freedom.

Fig. 14. Symmetry of a molecule of type \(XY_2\).

We shall consider the introduction of symmetry coordinates and the resulting simplification of the problem using the example of a symmetric nonlinear triatomic molecule \(XY_2\) (Fig. 14), possessing a plane of symmetry perpendicular to the plane of the molecule. Upon reflection in this plane) the normal configuration of the molecule is transformed into itself—the atom \(X\) remains in place, while the atoms \(Y\) interchange positions. The bonds \(X—Y^{(1)}\) and \(X—Y^{(2)}\), which thereby pass into one another, will be called equivalent; the changes \(q_1\) and \(q_2\) in their lengths are equivalent vibrational coordinates*\(^{42,43}\). In the valence-force coordinates \(q_1, q_2\), and \(\gamma\) (\(\gamma\) is the change of the angle \(Y^{(1)}XY^{(2)}\)), the kinetic and potential energies will have the form

\[ T=\frac{1}{2}T_q(\dot{q}_1^{\,2}+\dot{q}_2^{\,2})+\frac{1}{2}T_{\gamma}\dot{\gamma}^{\,2} +T_{qq}\dot{q}_1\dot{q}_2+T_{q\gamma}(\dot{q}_1+\dot{q}_2)\dot{\gamma} \tag{8.4} \]

and

\[ U=\frac{1}{2}k_q(q_1^2+q_2^2)+\frac{1}{2}k_{\gamma}\gamma^2 +hq_1q_2+a(q_1+q_2)\gamma . \tag{8.5} \]

The expression for \(U\) is a special case of (5.6): by virtue of symmetry,

\[ k_1=k_2=k \quad \text{and} \quad a_1=a_2=a. \]

Let us introduce, instead of \(q_1\) and \(q_2\), new coordinates—the symmetric

\[ q^{(A)}=\frac{1}{\sqrt{2}}(q_1+q_2) \tag{8.6} \]

*) And also upon rotation by \(180^\circ\) about the vertical axis of symmetry passing through the atom \(X\).

and antisymmetric

\[ q^{(B)}=\frac{1}{\sqrt{2}}(q_1-q_2). \tag{8.7} \]

The equivalent coordinates \(q_1\) and \(q_2\) are expressed, conversely, through \(q^{(A)}\) and \(q^{(B)}\) by the formulas

\[ \left. \begin{aligned} q_1&=\frac{1}{\sqrt{2}}\left(q^{(A)}+q^{(B)}\right),\\ q_2&=\frac{1}{\sqrt{2}}\left(q^{(A)}-q^{(B)}\right). \end{aligned} \right\} \tag{8.8} \]

When only \(q^{(A)}\) changes \((q^{(B)}=0)\), then

\[ q_1=q_2=\frac{1}{\sqrt{2}}q^{(A)}, \tag{8.9} \]

i.e., the displaced configuration of the molecule is not changed by reflection (Fig. 15). When only \(q^{(B)}\) changes \((q^{(A)}=0)\), then

\[ q_1=-q_2=\frac{1}{\sqrt{2}}q^{(B)} \tag{8.10} \]

and upon reflection the sign of the displacements of the atoms from their previous equilibrium positions changes (Fig. 16).

Fig. 15. Reflection for displacements of type \(A\) \((q_1=q_2)\).

Fig. 16. Reflection for displacements of type \(B\) \((q_1=-q_2)\).

The coordinate \(\gamma\) is symmetric: upon reflection the displaced configuration is transformed into itself (Fig. 17). Therefore one may write

\[ \gamma=\gamma^{(A)}. \tag{8.11} \]

Fig. 17. Symmetry of the angular coordinate \(\gamma\).

In the coordinates \(q^{(A)}, q^{(B)}\), and \(\gamma^{(A)}\), expressions (8.4) and (8.5) take the form

\[ T=\frac{1}{2}(T_q+T_{qq})\dot q^{(A)2} +\frac{1}{2}(T_q-T_{qq})\dot q^{(B)2} + \]

\[ +\frac{1}{2}T_\gamma\dot\gamma^{(A)2} +\sqrt{2}\,T_{q\gamma}\dot q^{(A)}\dot\gamma^{(A)}, \tag{8.12} \]

\[ U=\frac{1}{2}(k_q+h)q^{(A)2} +\frac{1}{2}(k_q-h)q^{(B)2} + \]

\[ +\frac{1}{2}k_\gamma\gamma^{(A)2} +\sqrt{2}\,\alpha q^{(A)}\gamma^{(A)}. \tag{8.13} \]

(8.12) and (8.13) contain no terms with “cross” products of symmetric coordinates by the antisymmetric one, and therefore the total energy can be written in the form

\[ H=T+U=H^{(A)}+H^{(B)}, \tag{8.14} \]

where

\[ \begin{aligned} H^{(A)}=T^{(A)}+U^{(A)} &=\frac12\left(T_q+T_{qq}\right)\dot q^{(A)2} +\frac12 T_\gamma \dot\gamma^{(A)2} \\ &\quad+\sqrt2\,T_{q\gamma}\dot q^{(A)}\dot\gamma^{(A)} +\frac12\left(k_q+h\right)q^{(A)2} +\frac12 k_\gamma \gamma^{(A)2} \\ &\quad+\sqrt2\,a q^{(A)}\gamma^{(A)}, \end{aligned} \tag{8.15} \]

\[ \begin{aligned} H^{(B)}=T^{(B)}+U^{(B)} =\frac12\left(T_q-T_{qq}\right)\dot q^{(B)2} +\frac12\left(k_q-h\right)q^{(B)2}. \end{aligned} \tag{8.16} \]

We have obtained two independent problems: for symmetric vibrations, a problem with two degrees of freedom, and for antisymmetric vibrations, a problem with one degree of freedom. The frequencies of the symmetric vibrations \(\omega_1^2\) and \(\omega_2^2\) are determined from a secular equation of the second degree, and the symmetric coordinates \(q^{(A)}\) and \(\gamma^{(A)}\) are expressed through the corresponding normal coordinates \(Q_1\) and \(Q_2\) by the formulas

\[ \left. \begin{aligned} q^{(A)}&=C_1Q_1+C_2Q_2,\\ \gamma^{(A)}&=D_1Q_1+D_2Q_2. \end{aligned} \right\} \tag{8.17} \]

The frequency of the antisymmetric vibration \(\omega_3^2\) will simply be

\[ \omega_3^2=\frac{k_q-h}{T_q-T_{qq}}, \tag{8.18} \]

and the coordinate \(q^{(B)}\) is proportional to the normal coordinate \(Q_3\),

\[ q^{(B)}=C_3Q_3. \tag{8.19} \]

The total energy in normal coordinates will be

\[ H=\frac12\left(\dot Q_1^2+\omega_1^2Q_1^2\right) +\frac12\left(\dot Q_2^2+\omega_2^2Q_2^2\right) +\frac12\left(\dot Q_3^2+\omega_3^2Q_3^2\right). \tag{8.20} \]

The coordinates \(q^{(A)}\), \(q^{(B)}\), \(\gamma^{(A)}\) and represent symmetry coordinates. They are linear combinations of the equivalent natural coordinates \(q_1\), \(q_2\), \(\gamma\) and are connected with them by formulas (8.8) and (8.11); through the normal coordinates they are expressed by formulas (8.17) and (8.19). The totality of these formulas gives the transition from natural coordinates

to normal ones by means of symmetry coordinates; this transition can be explained by the scheme

\[ \begin{array}{ccccc} \begin{array}{c} \text{equivalent}\\ \text{changes}\\ \text{of bond lengths} \end{array} & \left\{ \begin{array}{c} q_1\\ q_2 \end{array} \right\} &\longrightarrow& \left\{ \begin{array}{c} q^{(A)}\\ q^{(B)} \end{array} \right\} &\longrightarrow \left\{ \begin{array}{c} Q_1\\ Q_2 \end{array} \right\} \quad \begin{array}{c} \text{symmetric}\\ \text{normal}\\ \text{coordinates} \end{array} \\[1.5em] \begin{array}{c} \text{change of angle} \end{array} & \gamma &\longrightarrow& \gamma^{(A)} &\searrow\nearrow \quad \begin{array}{c} Q_3\ \text{antisymmetric}\\ \text{normal}\\ \text{coordinate} \end{array} \end{array} \]

Each symmetry coordinate is characterized, on the one hand, by what combination of equivalent natural coordinates it is, and, on the other hand, by its belonging to a definite type of vibration symmetry.

In general form, the transition from equivalent natural coordinates \(x_{\lambda a}\) (we denote them by two indices \(\lambda\) and \(a\); \(\lambda\) is the class of equivalent coordinates, \(a\) is the number of the equivalent coordinate of class \(\lambda\)) to normal coordinates \(Q_s^{(R)}\) of symmetry type \(R\) by means of symmetry coordinates \(z_\lambda^{(R)}\) is expressed by the formulas

\[ x_{\lambda a}=\sum_R c_{\lambda a}^{(R)} z_\lambda^{(R)} \tag{8,21} \]

(summation over the different types of vibration symmetry),

\[ z_\lambda^{(R)}=\sum_s g_{\lambda s}^{(R)} Q_s^{(R)} \tag{8,22} \]

(summation over all normal coordinates of the given symmetry type \(R\)).

For the example considered,

\[ \begin{array}{cc} \begin{array}{c} \text{equivalent}\\ \text{changes}\\ \text{of bond lengths} \end{array} & \left\{ \begin{array}{l} x_{q1}=q_1;\quad z_q^{(A)}=q^{(A)};\quad Q_1^{(A)}=Q_1\\ x_{q2}=q_2;\quad z_q^{(B)}=q^{(B)};\quad Q_2^{(A)}=Q_2 \end{array} \right\} \quad \begin{array}{c} \text{symmetric}\\ \text{normal}\\ \text{coordinates} \end{array} \\[2em] \begin{array}{c} \text{change of}\\ \text{angle} \end{array} & x_\gamma=\gamma;\quad z_\gamma^{(A)}=\gamma^{(A)};\quad Q_3^{(B)}=Q_3 \quad \begin{array}{c} \text{antisymmetric}\\ \text{normal}\\ \text{coordinate} \end{array} \end{array} \tag{8,23} \]

and (8,8), (8,11) represent a special case of formulas (8,21), while (8,17), (8,19) are a special case of formulas (8,22).

The coefficients \(c\) in (8,21) are determined exclusively by the symmetry properties of the molecule. We shall call them coefficients

symmetry. For the coordinates \(q_1\), \(q_2\), and \(\gamma\) they have the values, according to (8,8) and (8,11):

\[ \left. \begin{aligned} c_{q_1}^{(A)} &= \frac{1}{\sqrt{2}}, &\qquad c_{q_1}^{(B)} &= \frac{1}{\sqrt{2}},\\ c_{q_2}^{(A)} &= \frac{1}{\sqrt{2}}, &\qquad c_{q_2}^{(B)} &= -\,\frac{1}{\sqrt{2}} \end{aligned} \right\} \tag{8,24} \]

\[ c_{\gamma}^{(A)}=1. \tag{8,25} \]

For the various types of molecular symmetry, the symmetry coefficients were classified and calculated for the most important cases by the author\(^{43}\). For the simplest cases, such as the one considered above, they are obtained immediately from visual considerations; for more complicated cases their determination can be carried out by applying the mathematical apparatus of the doctrine of symmetry properties—the theory of groups\(^{42}\).

The coefficients \(g\) in (8,22) are determined by the specific properties of the molecule, primarily by the force constants, and are readily found when the problem of determining the vibrational frequencies of the given symmetry type has been solved.

The solution of the vibrational problem, with allowance for symmetry properties by introducing symmetry coordinates, may be represented by the following scheme:

\[ \begin{array}{cccccc} \text{energy} & \left\{ \begin{array}{c} \\[-0.4em] \end{array} \right. & H & \longrightarrow & \sum H^{(R)} & \longrightarrow \sum_{s=1}^{n} H_s \\[-0.2em] \text{of the system} && \begin{array}{c} \text{problem}\\ \text{for }3N-6=n\\ \text{degrees of freedom} \end{array} && \begin{array}{c} \text{a series of problems for}\\ n_R\text{ degrees}\\ \text{of freedom each} \end{array} & \begin{array}{c} n\text{ problems for}\\ \text{one degree}\\ \text{of freedom}\\ \text{each} \end{array} \\[1.2em] &&&& (8,21) & (8,22) \\[1.2em] \text{coordinates} & \left\{ \begin{array}{c} \\[-0.4em] \end{array} \right. & x_{\lambda\alpha} & \longrightarrow & z_{\lambda}^{(R)} & \longrightarrow Q_s \\[-0.2em] && \begin{array}{c} \text{natural}\\ \text{coordinates} \end{array} && \begin{array}{c} \text{symmetry}\\ \text{coordinates} \end{array} & \begin{array}{c} \text{normal}\\ \text{coordinates} \end{array} \\[1.2em] \text{path of} && \multicolumn{2}{c}{ \underbrace{\hspace{8em}}_{\begin{array}{c}\text{application of}\\ \text{symmetry properties}\end{array}} } & \multicolumn{2}{c}{ \underbrace{\hspace{8em}}_{\begin{array}{c}\text{solution of secular}\\ \text{equations of degree}\\ n_R\text{ each}\end{array}} } \\[-0.2em] \text{solution:} && &&& \end{array} \]

Thus it is necessary to set up the secular equations for each type \(R\) of vibrational symmetry separately. The number of such equations and their degrees \(n_R\), i.e., the numbers of vibrations of the different symmetry types, are obtained directly by counting the number of symmetry coordinates of the different types \(R\). In the case considered of the molecule \(XY_2\), we have one quadratic equation (coordinates \(q^{(A)}\), \(\gamma^{(A)}\)) and one equation of the first degree (coordinate \(q^{(B)}\)).

In general form, the energy of vibrations of symmetry type \(R\) will be

\[ H^{(R)}=\frac{1}{2}\sum_{\lambda,\mu=1}^{n_R} T_{\lambda\mu}^{(R)}\dot z_\lambda^{(R)}\dot z_\mu^{(R)} +\frac{1}{2}\sum_{\lambda,\mu=1}^{n_R} k_{\lambda\mu}^{(R)}z_\lambda^{(R)}z_\mu^{(R)}, \tag{8.26} \]

where the summation is carried out over the \(n_R\) vibrations of this type. In the case of formula (8.15), \(n_R=n_A=2\), and

\[ \begin{aligned} &T_{qq}^{(A)}=T_q+T_{qq};\qquad T_{\gamma\gamma}^{(A)}=T_\gamma;\qquad T_{q\gamma}^{(A)}=\sqrt{2}\,T_{q\gamma};\\ &k_{qq}^{(A)}=k_q+h;\qquad k_{\gamma\gamma}^{(A)}=k_\gamma;\qquad k_{q\gamma}^{(A)}=\sqrt{2}\,a. \end{aligned} \tag{8.27} \]

Instead of the constants \(T_{ij}\) and \(k_{ij}\) in (8.3), we have the constants \(T_{\lambda\mu}^{(R)}\) and \(k_{\lambda\mu}^{(R)}\) in (8.26). Similarly, instead of the interaction coefficients \(A_{\nu i}\) (6.9) and \(D_{ij}\) (6.14), we shall have the coefficients \(A_{\nu\lambda}^{(R)}\) and \(D_{\nu\mu}^{(R)}\), defined by the formulas

\[ \sum_{\lambda=1}^{n_R} A_{\nu\lambda}^{(R)} T_{\lambda\mu}^{R}=\delta_{\nu\mu}, \tag{8.28} \]

\[ D_{\nu\mu}^{(R)}=\sum_{\lambda=1}^{n_R} A_{\nu\lambda}^{(R)}k_{\lambda\mu}^{R}. \tag{8.29} \]

Analogously to (6.11), (6.17), and (6.24), we obtain the equations of motion

\[ \ddot z_\nu=-\sum_{\lambda\mu} A_{\nu\lambda}^{(R)}k_{\lambda\mu}^{(R)}z_\mu^{(R)} =-\sum_\lambda A_{\nu\lambda}^{(R)} \frac{\partial U^{(R)}}{\partial z_\lambda^{(R)}}, \tag{8.30} \]

the secular equation of degree \(n_R\):

\[ \left|D_{\nu\mu}^{(R)}-\delta_{\nu\mu}\omega^2\right|=0 \tag{8.31} \]

and a system of \(n_R\) equations for determining the \(n_R\) amplitudes:

\[ \sum_{\mu=1}^{n_R} \left(D_{\nu\mu}^{(R)}-\delta_{\nu\mu}\omega_s^{(R)2}\right) g_{\mu s}^{(R)}=0, \tag{8.32} \]

where \(g_{\mu s}^{(R)}\) are the coefficients relating the symmetry coordinates to the corresponding normal coordinates according to (8.22). The superscript \(R\) in all these formulas indicates that all quantities refer to vibrations of symmetry type \(R\). We shall call them “reduced” quantities: \(D_{\nu\mu}^{(R)}\), \(A_{\nu\lambda}^{(R)}\), and \(k_{\lambda\mu}^{(R)}\) are the reduced coefficients of the complete, kinematic, and dynamic interaction of the symmetry coordinates \(z_\lambda^{(R)}\).

§ 9. FORMULATION OF SECULAR EQUATIONS WITH SYMMETRY TAKEN INTO ACCOUNT

The formulation of the secular equations (8.31) for each type of vibration separately reduces to finding explicitly*) the reduced coefficients of the complete interaction \(D_{\mu}^{R}\). They can be found when the interaction coefficients in natural coordinates (§ 7) and the symmetry coefficients (§ 8) relating the natural coordinates to the symmetry coordinates are known. In doing so one may start either directly from the coefficients of the kinematic and dynamic interaction \(A_{ij}\) and \(k_{ij}\) in natural coordinates, or from the already obtained coefficients of the complete interaction \(D_{ij}\). We shall first consider the latter method for the example of a symmetric triatomic molecule \(XY_{2}\), restricting ourselves, for simplicity, to the valence-force model.

The coefficients of the complete interaction for the molecule \(XY_{2}\), as a particular case of the molecule \(XYZ\), are contained in the secular equation (7.26), which we write in the form

\[ \left| \begin{array}{ccc} D_{q_1,q_1}-\omega^2 & D_{q_1,q_2} & D_{q_1,\gamma}\\ D_{q_2,q_1} & D_{q_2,q_2}-\omega^2 & D_{q_2,\gamma}\\ D_{\gamma,q_1} & D_{\gamma,q_2} & D_{\gamma,\gamma}-\omega^2 \end{array} \right|=0, \tag{9.1} \]

where

\[ D_{q_1,q_1}=D_{q_2,q_2}=\left(\frac{1}{M}+\frac{1}{m}\right)k_q, \tag{9.2} \]

\[ D_{\gamma,\gamma}= \left[\frac{2}{Ms^2}(1-\cos\vartheta)+\frac{2}{ms^2}\right]k_\gamma, \tag{9.3} \]

\[ D_{q_1,q_2}=D_{q_2,q_1}=\frac{1}{M}\cos\vartheta\cdot k_q, \tag{9.4} \]

\[ D_{q_1,\gamma}=D_{q_2,\gamma}=-\frac{1}{Ms}\sin\vartheta\cdot k_\gamma, \tag{9.5} \]

\[ D_{\gamma,q_1}=D_{\gamma,q_2}=-\frac{1}{Ms}\sin\vartheta\cdot k_q. \tag{9.6} \]

Here \(M=m_0\) is the mass of atom \(X\), \(m=m_1=m_2\) is the mass of the atoms \(Y\), \(s\) is the equilibrium distance \(X-Y\), and \(k_q=k_1=k_2\) and \(k_\gamma\) are the force constants for the bond \(X-Y\) and the angle \(Y-X-Z\).

The equations of motion in the natural coordinates \(q_1, q_2, \gamma\) [i.e., equations of the type (6.11)], whose solvability condition gives the secular equation (9.1), have the form

\[ \left. \begin{aligned} \ddot q_1&=-D_{q_1,q_1}q_1-D_{q_1,q_2}q_2-D_{q_1,\gamma}\gamma,\\ \ddot q_2&=-D_{q_2,q_1}q_1-D_{q_2,q_2}q_2-D_{q_2,\gamma}\gamma,\\ \ddot\gamma&=-D_{\gamma,q_1}q_1-D_{\gamma,q_2}q_2-D_{\gamma,\gamma}\gamma. \end{aligned} \right\} \tag{9.7} \]

*) In the example of the molecule \(XY_{2}\), the formulas (8.15) and (8.16) contain coefficients \(T_{ij}\), which are not yet known to us explicitly.

Let us now take into account the symmetry of the vibrations. For symmetric vibrations we have, according to (8,9) and (8,11),

\[ q_1=q_2=\frac{1}{\sqrt{2}}\,q^{(A)},\qquad \gamma=\gamma^{(A)}. \tag{9,8} \]

Substituting these expressions into (9,7), we obtain

\[ \begin{aligned} \frac{1}{\sqrt{2}}\,\ddot q^{(A)} &=-\frac{1}{\sqrt{2}}\left(D_{q_1,q_1}+D_{q_1,q_2}\right)q^{(A)} -D_{q_1,\gamma}\gamma^{(A)},\\ \frac{1}{\sqrt{2}}\,\ddot q^{(A)} &=-\frac{1}{\sqrt{2}}\left(D_{q_2,q_1}+D_{q_2,q_2}\right)q^{(A)} -D_{q_2,\gamma}\gamma^{(A)},\\ \ddot\gamma^{(A)} &=-\frac{1}{\sqrt{2}}\left(D_{\gamma,q_1}+D_{\gamma,q_2}\right)q^{(A)} -D_{\gamma,\gamma}\gamma^{(A)}. \end{aligned} \tag{9,9} \]

The first two equations, according to (9,2), (9,4), and (9,5), coincide, and we obtain, taking also (9,6) into account:

\[ \begin{aligned} \ddot q^{(A)} &=-\left[\frac{1}{m}+\frac{1}{M}(1+\cos\vartheta)\right]k_q\,q^{(A)} +\frac{\sqrt{2}}{Ms}\sin\vartheta\, k_\gamma\,\gamma^{(A)} \\ &=-D^{(A)}_{q,q}q^{(A)}-D_{q,\gamma}\gamma^{(A)},\\[4pt] \ddot\gamma^{(A)} &=\frac{\sqrt{2}}{Ms}\sin\vartheta\, k_q\,q^{(A)} -\left[\frac{2}{Ms^2}(1-\cos\vartheta)+\frac{2}{ms^2}\right]k_\gamma\,\gamma^{(A)} \\ &=-D^{(A)}_{\gamma,q}q^{(A)}-D^{(A)}_{\gamma,\gamma}\gamma^{(A)}. \end{aligned} \tag{9,10} \]

The condition of solvability of these equations is the required secular equation of the second degree for symmetric vibrations

\[ \left| \begin{array}{cc} D^{(A)}_{q,q}-\omega^2 & D^{(A)}_{q,\gamma}\\ D^{(A)}_{\gamma,q} & D^{(A)}_{\gamma,\gamma}-\omega^2 \end{array} \right|=0. \tag{9,11} \]

Here

\[ D^{(A)}_{q,q} =\frac{1}{1/\sqrt{2}} \left(\frac{1}{\sqrt{2}}D_{q_1,q_1}+\frac{1}{\sqrt{2}}D_{q_1,q_1}\right) = \left[\frac{1}{m}+\frac{1}{M}(1+\cos\vartheta)\right]\cdot k_q, \tag{9,12} \]

\[ D^{(A)}_{q,\gamma} =\frac{1}{1/\sqrt{2}}D_{q_1,\gamma} =-\sqrt{2}\,\frac{1}{Ms}\sin\vartheta\, k_\gamma, \tag{9,13} \]

\[ D^{(A)}_{\gamma,q} =\frac{1}{\sqrt{2}}D_{\gamma,q_1} +\frac{1}{\sqrt{2}}D_{\gamma,q_2} =-\sqrt{2}\,\frac{1}{Ms}\sin\vartheta\, k_q, \tag{9,14} \]

\[ D^{(A)}_{\gamma,\gamma} =D_{\gamma,\gamma} =\frac{2}{s^2}\left[\frac{1}{m}+\frac{1}{M}(1-\cos\vartheta)\right]\cdot k_\gamma \tag{9,15} \]

— the reduced coefficients of the complete interaction for symmetric vibrations. The solution of the secular equation (9,11) gives the sought frequencies \(\omega_1\) and \(\omega_2\) of the symmetric vibrations.

Equations of type (8.32) for determining the amplitudes will be

\[ \left. \begin{aligned} \left(D^{(A)}_{q,q}-\omega_s^2\right)C_s-D^{(A)}_{q,\gamma}D_s&=0,\\ D^{(A)}_{\gamma,q}C_s-\left(D^{(A)}_{\gamma,\gamma}-\omega_s^2\right)D_s&=0 \end{aligned} \right\}, \tag{9.16} \]

where \(g_{qs}=C_s,\ g_{\gamma s}=D_s\) [see (8.17)]. It is easy to verify that both equations (9.16) give the same result for the ratio \(D_s/C_s\), which determines the relative amplitudes \(q^{(A)}\) and \(\gamma^{(A)}\) for the \(s\)-th \((s=1,2)\) normal vibration.

The case of the antisymmetric vibration is still simpler. According to (8.10) and by virtue of the symmetry of the angle \(\gamma\), for this vibration

\[ q_1=-q_2=\frac{1}{\sqrt{2}}q^{(B)}, \qquad \gamma=0, \tag{9.17} \]

and in system (9.7) the first and second equations give the same result

\[ \frac{1}{\sqrt{2}}\ddot q^{(B)} = -\frac{1}{\sqrt{2}}D_{q_1,q_1}q^{(B)} +\frac{1}{\sqrt{2}}D_{q_1,q_2}q^{(B)}, \tag{9.18} \]

while the third equation is identically reduced to zero, if (9.6) is taken into account:

\[ 0=-\frac{1}{\sqrt{2}}D_{\gamma,q_1}q^{(B)} +\frac{1}{\sqrt{2}}D_{\gamma,q_2}q^{(B)} +D_{\gamma,\gamma}\cdot 0 = \]

\[ =\frac{1}{\sqrt{2}}\left(-D_{\gamma,q_1}+D_{\gamma,q_2}\right)q^{B} +D_{\gamma,\gamma}\cdot 0 =0\cdot q^{(B)}+D_{\gamma,\gamma}\cdot 0=0. \tag{9.19} \]

Finally we obtain

\[ \ddot q^{(B)} = -\left[\frac{1}{m}+\frac{1}{M}(1-\cos\vartheta)\right]k_q\cdot q^{(B)} = D^{(B)}_{q,q}q^{(B)}, \tag{9.20} \]

where

\[ D^{(B)}_{q,q} = \frac{1}{1/\sqrt{2}} \left( \frac{1}{\sqrt{2}}D_{q_1,q_1} - \frac{1}{\sqrt{2}}D_{q_1,q_2} \right) = \left[\frac{1}{m}+\frac{1}{M}(1-\cos\vartheta)\right]\cdot k_q. \tag{9.21} \]

The frequency of the antisymmetric vibration is obtained in explicit form

\[ \omega_3^2=\left[\frac{1}{m}+\frac{1}{M}(1-\cos\vartheta)\right]\cdot k_q. \tag{9.22} \]

Formulas (9.12)—(9.15) and (9.21) determine the reduced coefficients of the complete interaction \(D^{(R)}_{\lambda\mu}\) through the coefficients of the complete interaction in natural coordinates \(D_{ij}\). The coefficients by which \(D_{q_1,q_1}\), \(D_{q_1,q_2}\), etc., are multiplied in these formulas are the symmetry coefficients (8.24), and we may write, for example, (9.12) in the form

\[ D^{(A)}_{qq} = \frac{1}{c^{(A)}_{q_1}} \left[ D_{q_1,q_1}c^{(A)}_{q_1} + D_{q_1,q_2}c^{(A)}_{q_2} \right], \tag{9.23} \]

(9.14) in the form

\[ D_{\gamma,q}^{(A)}=\frac{1}{c_\gamma^{(A)}}\left[D_{\gamma,q1}c_{q1}^{(A)}+D_{\gamma,q2}c_{q2}^{(A)}\right], \tag{9.24} \]

and (9.21) in the form

\[ D_{qq}^{(B)}=\frac{1}{c_{q1}^{(B)}}\left[D_{q1,q1}c_{q1}^{(B)}+D_{q1,q2}c_{q2}^{(B)}\right]. \tag{9.25} \]

Thus, in order to obtain the reduced coefficient of interaction of the symmetry coordinate \(z_\lambda^{(R)}\) (for example, \(\gamma^{(A)}\)) with the symmetry coordinate \(z_\mu^{(R)}\) (for example, \(q^{(A)}\)), it is necessary to multiply the coefficients \(D_{\lambda\alpha,\mu\beta}\) of interaction of two natural coordinates \(x_{\lambda\alpha}\) (in the present case \(\gamma\)) and \(x_{\mu\beta}\) (in the present case \(q_\beta,\ \beta=1,2\)) by the symmetry coefficients \(c_{\mu\beta}^{(R)}\) (in the present case \(c_{q\beta}^{(A)}\)), to sum over \(\beta\), i.e. over all coordinates equivalent to \(\mu\), and to divide by \(c_{\lambda\alpha}^{(R)}\) (in the present case \(c_\gamma^{(A)}=1\)),

\[ D_{\lambda\mu}^{(R)}=\frac{1}{c_{\lambda\alpha}^{(R)}}\sum_\beta D_{\lambda\alpha,\mu\beta}c_{\mu\beta}^{(R)}. \tag{9.26} \]

Formula (9.26) expresses the desired rule for determining the reduced coefficients of the complete interaction. We note that here \(\alpha\) is arbitrary, i.e. it is immaterial which one of the equivalent coordinates \(x_{\lambda\alpha}\) is taken, for example \(q_1\) or \(q_2\); indeed,

\[ D_{qq}=\frac{1}{c_{q1}^{(A)}}\left(D_{q1,q1}c_{q1}^{(A)}+D_{q1,q2}c_{q2}^{(A)}\right) =\frac{1}{c_{q2}^{(A)}}\left(D_{q2,q1}c_{q1}^{(A)}+ \right. \]

\[ \left. +D_{q2,q2}c_{q2}^{(A)}\right). \tag{9.27} \]

It can be shown\({}^{42}\) that the reduced coefficients of kinematic and dynamic interaction \(A_{\lambda\mu}^{(R)}\) and \(k_{\lambda\mu}^{(R)}\) are obtained from the corresponding coefficients \(A_{\lambda\alpha,\mu\beta}\) and \(k_{\lambda\alpha,\mu\beta}\) according to exactly the same rule, i.e.

\[ A_{\lambda\mu}^{(R)}=\frac{1}{c_{\lambda\alpha}^{(R)}}\sum_\beta A_{\lambda\alpha,\mu\beta}c_{\mu\beta}^{(R)} \quad(\alpha\text{—arbitrary}), \tag{9.28} \]

\[ k_{\lambda\mu}^{(R)}=\frac{1}{c_{\lambda\alpha}^{(R)}}\sum_\beta k_{\lambda\alpha,\mu\beta}c_{\mu\beta}^{(R)} \quad(\alpha\text{—arbitrary}). \tag{9.29} \]

Therefore, without forming the coefficients of the complete interaction \(D_{\lambda\alpha,\mu\beta}\) by formulas (6.14) (where now each natural coordinate is denoted by two indices \(\lambda\alpha\) or \(\mu\beta\)), one may directly determine the reduced kinematic and dynamic coefficients, and then find the coefficients \(D_{\nu\mu}^{(R)}\) by formula (8.29). In the case of a large number of vibrational degrees of freedom, this method is often very convenient, and we shall apply it in the next paragraph for the ca—

... for the methane molecule, here too we obtain the secular equations for the symmetric and antisymmetric vibrations of molecules \(XY_2\) by the same method. We start from the tables of kinematic and dynamic coefficients \((7,23)\) and \((7,24)\), again putting \(m_0=M,\ m_1=m_2=m,\ s_1=s_2=s,\ k_1=k_2=k_q,\ a_1=a_2=a\). We write out all the necessary coefficients in Table 2. Then we apply the rules \((9,28)\) and \((9,29)\). This reduces to multiplying the coefficients of the corresponding row \(q\) or \(\gamma\) by the symmetry coefficients standing below them, summing for coordinates of type \(q\) over the indices 1 and 2, and dividing by the corresponding coefficient \(c_{q1}=1/\sqrt{2}\) or \(c_\gamma=1\). For example,

Table 2

Interaction coefficients and symmetry coefficients for the molecule \(XY_2\)

Types of coefficients	Coordinates	Equivalent coordinates \(x_{\lambda\beta}\) of type \(q\): \(x_{q1}=q_1\)	Equivalent coordinates \(x_{\lambda\beta}\) of type \(q\): \(x_{q2}=q_2\)	Equivalent coordinates \(x_{\lambda\beta}\) of type \(q\): \(x_\gamma=\gamma\)
Coefficients of kinematic interaction	\(q_1\)	\(\dfrac{1}{m}+\dfrac{1}{M}\)	\(\dfrac{1}{M}\cos\vartheta\)	\(-\dfrac{1}{Ms}\sin\vartheta\)
Coefficients of kinematic interaction	\(\gamma\)	\(-\dfrac{1}{Ms}\sin\vartheta\)	\(-\dfrac{1}{Ms}\sin\vartheta\)	\(\dfrac{2}{s^2}\left[\dfrac{1}{m}+\dfrac{1}{M}(1-\cos\vartheta)\right]\)
Coefficients of dynamic interaction	\(q_1\)	\(k_q\)	\(h\)	\(a\)
Coefficients of dynamic interaction	\(\gamma\)	\(a\)	\(a\)	\(k_\gamma\)
Symmetry coefficients	symmetric vibrations	\(c_{q1}^{(A)}=\dfrac{1}{\sqrt{2}}\)	\(c_{q2}^{(A)}=\dfrac{1}{\sqrt{2}}\)	\(c_\gamma^{(A)}=1\)
Symmetry coefficients	antisymmetric vibrations	\(c_{q1}^{(B)}=\dfrac{1}{\sqrt{2}}\)	\(c_{q2}^{(B)}=-\dfrac{1}{\sqrt{2}}\)	\(c_\gamma^{(B)}=0\)
\(M\)—mass of atom \(X\), \(m\)—mass of atom \(Y\), \(s\)—length of the bond \(X—Y\).	\(M\)—mass of atom \(X\), \(m\)—mass of atom \(Y\), \(s\)—length of the bond \(X—Y\).	\(M\)—mass of atom \(X\), \(m\)—mass of atom \(Y\), \(s\)—length of the bond \(X—Y\).	\(M\)—mass of atom \(X\), \(m\)—mass of atom \(Y\), \(s\)—length of the bond \(X—Y\).	\(M\)—mass of atom \(X\), \(m\)—mass of atom \(Y\), \(s\)—length of the bond \(X—Y\).

\[ A_{q,q}^{(A)}= \frac{1}{c_{q1}^{(A)}}\left(A_{q1,q1}c_{q1}^{(A)}+A_{q1,q2}c_{q2}^{(A)}\right)= \]

\[ =-\sqrt{2}\left[\left(\frac{1}{m}+\frac{1}{M}\right)\frac{1}{\sqrt{2}}+ \frac{1}{M}\cos\vartheta\cdot\frac{1}{\sqrt{2}}\right]= \]

\[ =\frac{1}{m}+\frac{1}{M}(1+\cos\vartheta). \]

As a result, for symmetric vibrations we obtain:

\[ \begin{array}{c} \text{kinematic coefficients}\\[2mm] \begin{array}{c|cc} & q & \gamma\\ q & \dfrac{1}{m}+\dfrac{1}{M}(1+\cos\vartheta) & -\dfrac{\sqrt{2}}{Ms}\sin\vartheta\\[3mm] \gamma & -\dfrac{\sqrt{2}}{Ms}\sin\vartheta & \dfrac{2}{s^2}\left[\dfrac{1}{m}+\dfrac{1}{M}(1-\cos\vartheta)\right] \end{array} \end{array} \qquad \begin{array}{c} \text{dynamic}\\ \text{coefficients}\\[2mm] \begin{array}{c|cc} & q & \gamma\\ q & k_q+h & \sqrt{2}\,a\\[2mm] \gamma & \sqrt{2}\,a & k_\gamma \end{array} \end{array} \tag{9.30} \]

Multiplying the kinematic and dynamic coefficients according to (8.29), i.e., the rows of the first table by the columns of the second, and subtracting \(\omega^2\) from the diagonal elements, we obtain the secular equation for determining the frequencies of the symmetric vibrations:

\[ \left| \begin{array}{cc} \left[\dfrac{1}{m}+\dfrac{1}{M}(1+\cos\vartheta)\right](k_q+h) -\dfrac{2}{Ms}\sin\vartheta\,a-\omega^2 & \sqrt{2}\left[\dfrac{1}{m}+\dfrac{1}{M}(1+\cos\vartheta)\right]a -\dfrac{\sqrt{2}}{Ms}\sin\vartheta\,k_\gamma \\[3mm] -\dfrac{\sqrt{2}}{Ms}\sin\vartheta\,(k_q+h) +\dfrac{2\sqrt{2}}{s^2}\left[\dfrac{1}{m}+\dfrac{1}{M}(1-\cos\vartheta)\right]a & -\dfrac{2}{Ms}\sin\vartheta\,a +\dfrac{2}{s^2}\left[\dfrac{1}{m}+\dfrac{1}{M}(1-\cos\vartheta)\right]k_\gamma-\omega^2 \end{array} \right|=0 . \tag{9.31} \]

For \(h=a=0\) this equation coincides with equation (9.11), as it should.

For antisymmetric vibrations we have:

\[ \begin{array}{c} \text{kinematic}\\ \text{interaction}\\[1mm] \begin{array}{c|c} & q\\ q & \dfrac{1}{m}+\dfrac{1}{M}(1-\cos\vartheta) \end{array} \end{array} \qquad \begin{array}{c} \text{dynamic}\\ \text{interaction}\\[1mm] \begin{array}{c|c} & q\\ q & k_q-h \end{array} \end{array}; \tag{9.32} \]

whence

\[ \omega_3^2=\left[\dfrac{1}{m}+\dfrac{1}{M}(1-\cos\vartheta)\right](k_q-h), \]

in agreement with (9.22).

Thus, the construction of secular equations is reduced to writing out the coefficients of kinematic and dynamic interaction and the symmetry coefficients, and subsequently multiplying them according to a completely definite rule. In the next paragraph we shall consider a more complicated example of the construction of secular equations for the vibrations of the methane molecule.

§ 10. COORDINATES OF SYMMETRY AND CONSTRUCTION OF SECULAR EQUATIONS FOR A MOLECULE OF THE METHANE TYPE

Let us now consider a molecule \(XY_4\) of the methane type, consisting of a central atom \(X\) and four atoms \(Y\), situated at the vertices of a tetrahedron (Fig. 18). As natural coordinates we introduce four changes of bond lengths \(q_i\), \((i=1,2,3,4)\), and six changes \(\gamma_{ik}\), \((i,k=1,2,3,4)\), of the angles \(Y-X-Y\), of which five:

are independent (see p. 27). This molecule possesses high symmetry—the symmetry of a tetrahedron. It is transformed into itself under rotations by \(\frac{2\pi}{3}=120^\circ\) about any of the four bonds (i.e., it has 4 axes of symmetry of the third order), under rotations by \(\frac{2\pi}{2}=180^\circ\) about the axes \(x, y, z\) (i.e., it has three axes of symmetry of the second order), and under reflections in the planes \(Y^{(1)}-X-Y^{(2)}\), \(Y^{(1)}-X-Y^{(3)}\), etc. (i.e., it has six planes of symmetry*). To take this symmetry into account, we introduce symmetry coordinates. As can be shown\({}^{1,42,43}\), the natural coordinates will be expressed in terms of nine independent symmetry coordinates in the following way:

Fig. 18. Symmetry of a molecule of type \(XY_4\).

\[ \left. \begin{aligned} q_1&=\frac{1}{2}\left(q^{(A)}+q^{(FI)}+q^{(FII)}+q^{(FIII)}\right),\\ q_2&=\frac{1}{2}\left(q^{(A)}-q^{(FI)}-q^{(FII)}+q^{(FIII)}\right),\\ q_3&=\frac{1}{2}\left(q^{(A)}-q^{(FI)}+q^{(FII)}-q^{(FIII)}\right),\\ q_4&=\frac{1}{2}\left(q^{(A)}+q^{(FI)}-q^{(FII)}-q^{(FIII)}\right), \end{aligned} \right\} \tag{10,1} \]

\[ \left. \begin{aligned} \gamma_{12}&=\frac{1}{2}\gamma^{(EI)}-\frac{1}{2\sqrt{3}}\gamma^{(EII)} +\frac{1}{\sqrt{2}}\gamma^{(FIII)},\\ \gamma_{13}&=-\frac{1}{2}\gamma^{(EI)}-\frac{1}{2\sqrt{3}}\gamma^{(EII)} +\frac{1}{\sqrt{2}}\gamma^{(FII)},\\ \gamma_{14}&=-\frac{1}{\sqrt{3}}\gamma^{(EII)} +\frac{1}{\sqrt{2}}\gamma^{(FI)},\\ \gamma_{23}&=\frac{1}{\sqrt{3}}\gamma^{(EII)} -\frac{1}{\sqrt{2}}\gamma^{(FI)},\\ \gamma_{24}&=-\frac{1}{2}\gamma^{(EI)}-\frac{1}{2\sqrt{3}}\gamma^{(EII)} -\frac{1}{\sqrt{2}}\gamma^{(FII)},\\ \gamma_{34}&=\frac{1}{2}\gamma^{(EI)}-\frac{1}{2\sqrt{3}}\gamma^{(EII)} -\frac{1}{\sqrt{2}}\gamma^{(FIII)}. \end{aligned} \right\} \tag{10,2} \]

* And also under rotations by \(90^\circ\) about the axes \(x, y, z\) followed by reflection in the perpendicular plane, i.e., the axes of symmetry of the second order are rotary-reflection axes of the fourth order.

We have symmetry coordinates of three types—a symmetric coordinate \(q^{(A)}\), doubly degenerate coordinates \(\gamma^{(EI)}\) and \(\gamma^{(EII)}\), and triply degenerate coordinates \(q^{(FI)}\), \(q^{(FII)}\), \(q^{(FIII)}\) and \(\gamma^{(FI)}\), \(\gamma^{(FII)}\), \(\gamma^{(FIII)}\).

Coordinates of different types do not interact with one another, just as the symmetric and antisymmetric coordinates for the molecule \(XY_2\) do not.

We have one symmetric vibration with a certain frequency \(\omega_1\), for which we obtain, putting \(q^{(FI)}=q^{(FII)}=q^{(FIII)}=0\),

\[ q_1=q_2=q_3=q_4=\frac{1}{2}q^{(A)}. \tag{10,3} \]

In a symmetric vibration the angles do not change*).

The situation is more complicated for the remaining coordinates, which belong to the so-called degenerate coordinates. Let us examine their properties in more detail. Putting \(q^{(A)}=0\) and successively \(q^{(FII)}=q^{(FIII)}=0\), \(q^{(FI)}=q^{(FIII)}=0\), and \(q^{(FI)}=q^{(FII)}=0\) [cf. (8,9) and (8,10)], we obtain:

\[ \text{for a change of } q^{(FI)} \qquad q_1=-q_2=-q_3=q_4=\frac{1}{2}q^{(FI)}, \tag{10,4} \]

\[ \text{for a change of } q^{(FII)} \qquad q_1=-q_2=q_3=-q_4=\frac{1}{2}q^{(FII)}, \tag{10,5} \]

\[ \text{for a change of } q^{(FIII)} \qquad q_1=q_2=-q_3=-q_4=\frac{1}{2}q^{(FIII)}. \tag{10,6} \]

The displacements (10,4), (10,5), and (10,6) are symmetric with respect to the \(x\), \(y\), and \(z\) axes, respectively (Fig. 19, \(a\), \(b\), \(c\)). In exactly the same way, for the angular coordinates we have:

\[ \text{for a change of } \gamma^{(FI)} \qquad \gamma_{14}=-\gamma_{23}=\frac{1}{\sqrt{2}}\gamma^{(FI)}, \tag{10,7} \]

\[ \text{for a change of } \gamma^{(FII)} \qquad \gamma_{13}=-\gamma_{24}=\frac{1}{\sqrt{2}}\gamma^{(FII)}, \tag{10,8} \]

\[ \text{for a change of } \gamma^{(FIII)} \qquad \gamma_{12}=-\gamma_{34}=\frac{1}{\sqrt{2}}\gamma^{(FIII)}, \tag{10,9} \]

i.e., the displacements (10,7), (10,8), and (10,9) are symmetric with respect to the \(x\), \(y\), and \(z\) axes, respectively (Fig. 19, \(g\), \(d\), \(c\)). Owing to the different symmetry, coordinates of types I, II, and III do not interact with one another.

*) For the symmetric angular coordinate \(\gamma^{(A)}\) the relation would hold

\[ \gamma_{12}=\gamma_{13}=\gamma_{14}=\gamma_{23}=\gamma_{24}=\gamma_{34}=\frac{1}{\sqrt{6}}\gamma^{(A)}. \]

In view of (5,12) we obtain \(\gamma^{(A)}=0\).

and three independent oscillatory problems are obtained for the pairs of coordinates \(q^{(FI)}, \gamma^{(FI)};\ q^{(FII)}, \gamma^{(FII)};\ q^{(FIII)}, \gamma^{(FIII)}\). For example, for the coordinates \(q^{(FI)}\) and \(\gamma^{(FI)}\) we shall have the total energy

\[ H^{(FI)}=\frac{1}{2}T_q^{(F)}\dot q^{(FI)2} +\frac{1}{2}T_\gamma^{(F)}\dot\gamma^{(FI)2} +T_{q\gamma}^{(F)}\dot q^{(FI)}\dot\gamma^{(FI)} + \]

\[ +\frac{1}{2}k_q^{(F)}q^{(FI)2} +\frac{1}{2}k_\gamma^{(F)}\gamma^{(FI)2} +k_{q\gamma}^{(F)}q^{(FI)}\gamma^{(FI)}. \tag{10,10} \]

We obtain a secular equation of the second degree; solving it, we find two frequencies \(\omega_2^2\) and \(\omega_3^2\) of the \(F1\) vibrations, and, correspondingly, \(q^{(FI)}\) and \(\gamma^{(FI)}\) are expressed in terms of two normal coordinates

\[ \begin{aligned} q^{(FI)}&=C_2Q_2^{(I)}+C_3Q_3^{(I)},\\ \gamma^{(FI)}&=D_2Q_2^{(I)}+D_3Q_3^{(I)}. \end{aligned} \tag{10,11} \]

However, from the physical point of view all three pairs of coordinates do not differ in any way; the only difference between them consists in the fact that

Fig. 19. Symmetry of displacements for triply degenerate vibrations.

they are symmetric with respect to different, but equivalent, axes (cf. Fig. 19, \(a, г\); Fig. 19, \(б, д\); and Fig. 19, \(в, е\)). Therefore, both for the vibrations \(q^{(FII)}, \gamma^{(FII)}\) and for the vibrations \(q^{(FIII)}, \gamma^{(FIII)}\) the same vibration frequencies \(\omega_2, \omega_3\) are obtained as for \(q^{(FI)}, \gamma^{(FI)}\). As a result, each frequency is repeated three times—it will be triply degenerate. The six degrees of freedom correspond to only two different

MECHANICS OF MOLECULAR VIBRATIONS

frequencies. For vibrations \(FII\) and \(FIII\) we obtain

\[ \begin{aligned} q^{(FII)}&=C_2Q_2^{(II)}+C_3Q_3^{(II)},\\ \gamma^{(FII)}&=D_2Q_2^{(II)}+D_3Q_3^{(II)}, \end{aligned} \tag{10,12} \]

and

\[ \begin{aligned} q^{(FIII)}&=C_2Q_2^{(III)}+C_3Q_3^{(III)},\\ \gamma^{(FIII)}&=D_2Q_2^{(III)}+D_3Q_3^{(III)}, \end{aligned} \tag{10,13} \]

with the same coefficients as in (10,11). To the frequency \(\omega_2\) there correspond three normal coordinates \(Q_2^{(I)}, Q_2^{(II)}, Q_2^{(III)}\), and to the frequency \(\omega_3\)—three normal coordinates \(Q_3^{(I)}, Q_3^{(II)}, Q_3^{(III)}\).

The coordinates \(\gamma^{(EI)}\) and \(\gamma^{(EII)}\) are also degenerate, and moreover doubly so; they do not interact, and for each of these coordinates one obtains a problem for one degree of freedom with the same vibration frequency \(\omega_4\), which is thus doubly degenerate.

As a result, for the molecule \(XY_4\) we have not 9 different vibration frequencies, but only 4. They are determined by solving three problems—two problems for one degree of freedom each and one problem for two degrees of freedom—which constitutes an extraordinary simplification of the original problem for nine degrees of freedom.

The practical construction of the secular equations is carried out by the method described at the end of the preceding paragraph, with formulas (9,28), (9,29), and (9,26) being applied. The symmetry coefficients are taken from formulas (10,1), (10,2), which represent a special case of formulas (8,21). For the degenerate vibrations it is sufficient to take only one of the degenerate types (I or II or III for \(F\), and I or II for \(E\)). The coefficients of kinematic interaction are easily determined by using Table 1. We have interactions of types 1, 2, 3, 4, 6, 7, and 9. The scalar products of the vectors \(\mathbf e\), \(\mathbf f\), and \(\mathbf F\) are easily calculated by taking into account that all angles \(Y—X—Y\) are tetrahedral

\[ \left(\vartheta=109^\circ 28',\ \cos\vartheta=-\frac13,\ \sin\vartheta=\frac{\sqrt8}{3}\right). \]

The vectors \(\mathbf F\) are directed along the axes \(x,y,z\) and are equal in magnitude to

\[ \frac1s\,2\sin\frac{\vartheta}{2} = \frac2s\sqrt{\frac23} = \frac1s\sqrt{\frac83} \]

(see Fig. 20); therefore

\[ \mathbf F_{12}\mathbf F_{13}=0 \quad [\text{interaction }(\gamma_{12},\gamma_{13})], \]

and

\[ \mathbf e_1\mathbf F_{23} = -\frac1s\sqrt{\frac83}\cos\frac{\vartheta}{2} = -\frac1s\frac{\sqrt8}{3} \quad [\text{interaction }(q_1,\gamma_{23})]. \]

For the vectors \(\mathbf f\) we have \(f_{12} f_{13}=\cos 120^\circ=-\dfrac12\). The resulting coefficients of kinematic interaction are written in Table 3 for one of the equivalent changes \(q\) of the bond lengths, namely for \(q_1\), and for one of the equivalent changes \(\gamma\) of the valence angles, namely for \(\gamma_{12}\). Further, the corresponding coefficients of dynamic interaction are given and, finally, the coefficients of symmetry, according to (10,1) and (10,2), the types \(E1\) and \(F111\) being taken.

Fig. 20. Relations of the vectors \(\mathbf e\), \(\mathbf f\), and \(\mathbf F\) for a molecule of type \(XY_4\).

Multiplying, according to (9,28) and (9,29), for the coordinates \(q\) and the coordinates \(\gamma\) separately, the coefficients of interaction and the coefficients of symmetry, and summing over all equivalent coordinates (from 1 to 4 for \(q\) and from 1 to 6 for \(\gamma\)), we obtain the desired

Types of coefficients	Coordinates	Equivalent changes of bond lengths \(X—Y\): \(x_{q1}=q_1\)	Equivalent changes of bond lengths \(X—Y\): \(x_{q2}=q_2\)	Equivalent changes of bond lengths \(X—Y\): \(x_{q3}=q_3\)	Equivalent changes of bond lengths \(X—Y\): \(x_{q4}=q_4\)
Coefficients of kinematic interaction	\(q_1\)	\(\dfrac{1}{M}+\dfrac{1}{m}\)	\(-\dfrac{1}{3}\dfrac{1}{M}\)	\(-\dfrac{1}{3}\dfrac{1}{M}\)	\(-\dfrac{1}{3}\dfrac{1}{M}\)
Coefficients of kinematic interaction	\(\gamma_{12}\)	\(-\dfrac{\sqrt{8}}{3}\dfrac{1}{Ms}\)	\(-\dfrac{\sqrt{8}}{3}\dfrac{1}{Ms}\)	\(\dfrac{\sqrt{8}}{3}\dfrac{1}{Ms}\)	\(\dfrac{\sqrt{8}}{3}\dfrac{1}{Ms}\)
Coefficients of dynamic interaction	\(q_1\)	\(k_q\)	\(h\)	\(h\)	\(h\)
Coefficients of dynamic interaction	\(\gamma_{12}\)	\(a\)	\(a\)	\(b\)	\(b\)
Coefficients of symmetry	symmetric vibrations	\(c_{q_1}^{(A)}=\dfrac12\)	\(c_{q_2}^{(A)}=\dfrac12\)	\(c_{q_3}^{(A)}=\dfrac12\)	\(c_{q_4}^{(A)}=\dfrac12\)
Coefficients of symmetry	doubly degenerate vibrations	\(0\)	\(0\)	\(0\)	\(0\)
Coefficients of symmetry	triply degenerate vibrations	\(c_{q_1}^{(F)}=\dfrac12\)	\(c_{q_2}^{(F)}=\dfrac12\)	\(c_{q_3}^{(F)}=-\dfrac12\)	\(c_{q_4}^{(F)}=-\dfrac12\)

\(M\) is the mass of atom \(X\), \(m\) is the mass of atom \(Y\),

reduced interaction coefficients. For example, for triply degenerate vibrations

\[ D^{F}_{i,q}=\frac{1}{1/\sqrt{2}} \left[ -\frac{\sqrt{8}}{3}\frac{1}{Ms}\cdot\frac{1}{2} -\frac{\sqrt{8}}{3}\frac{1}{Ms}\cdot\frac{1}{2} -\frac{\sqrt{8}}{2}\frac{1}{Ms}\cdot\frac{1}{2} -\frac{\sqrt{8}}{3}\frac{1}{Ms}\cdot\frac{1}{2} \right] = \]

\[ =-\frac{8}{3}\frac{1}{Ms}. \]

Finally, for the symmetric vibrations \(A\) we have:

\[ \begin{array}{cc} \text{kinematic} & \text{dynamic}\\ \text{interaction} & \text{interaction}\\[2mm] q\left|\dfrac{1}{m}\right|q & q\left|\,k_q+3h\,\right|q, \end{array} \tag{10,14} \]

whence

\[ \omega_1^2=\frac{1}{m}(k_q+3h). \tag{10,15} \]

Table 3

and the symmetry coefficients for the molecule \(XY_4\)

coordinates	coordinates	coordinates	coordinates	coordinates	coordinates
changes of valence angles \(Y-X-Y\)	changes of valence angles \(Y-X-Y\)	changes of valence angles \(Y-X-Y\)	changes of valence angles \(Y-X-Y\)	changes of valence angles \(Y-X-Y\)	changes of valence angles \(Y-X-Y\)
\(x_{\gamma_1}=\gamma_{12}\)	\(x_{\gamma_2}=\gamma_{13}\)	\(x_{\gamma_3}=\gamma_{14}\)	\(x_{\gamma_4}=\gamma_{23}\)	\(x_{\gamma_5}=\gamma_{24}\)	\(x_{\gamma_6}=\gamma_{34}\)
\(-\dfrac{\sqrt{8}}{3}\dfrac{1}{Ms}\)	\(-\dfrac{\sqrt{8}}{3}\dfrac{1}{Ms}\)	\(-\dfrac{\sqrt{8}}{3}\dfrac{1}{Ms}\)	\(\dfrac{\sqrt{8}}{3}\dfrac{1}{Ms}\)	\(\dfrac{\sqrt{8}}{3}\dfrac{1}{Ms}\)	\(\dfrac{\sqrt{8}}{3}\dfrac{1}{Ms}\)
\(\dfrac{2}{s^2}\left(\dfrac{1}{m}+\dfrac{4}{3}\dfrac{1}{M}\right)\)	\(-\dfrac{1}{2}\dfrac{1}{Ms^2}\)	\(-\dfrac{1}{2}\dfrac{1}{Ms^2}\)	\(-\dfrac{1}{2}\dfrac{1}{Ms^2}\)	\(-\dfrac{1}{2}\dfrac{1}{Ms^2}\)	\(\dfrac{\sqrt{8}}{3}\dfrac{1}{Ms^2}\)
\(a\)	\(a\)	\(a\)	\(b\)	\(b\)	\(b\)
\(k_\gamma\)	\(l\)	\(l\)	\(l\)	\(l\)	\(v\)
\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(0\)
\(c^{(E)}_{\gamma_1}=\dfrac{1}{2}\)	\(c^{(E)}_{\gamma_2}=-\dfrac{1}{2}\)	\(c^{(E)}_{\gamma_3}=0\)	\(c^{(E)}_{\gamma_4}=0\)	\(c^{(E)}_{\gamma_5}=-\dfrac{1}{2}\)	\(c^{(E)}_{\gamma_6}=\dfrac{1}{2}\)
\(c^{(F)}_{\gamma_1}=\dfrac{1}{\sqrt{2}}\)	\(c^{(F)}_{\gamma_2}=0\)	\(c^{(F)}_{\gamma_3}=0\)	\(c^{(F)}_{\gamma_4}=0\)	\(c^{(F)}_{\gamma_5}=0\)	\(c^{(F)}_{\gamma_6}=-\dfrac{1}{\sqrt{2}}\)

\(s\) — length of the bond \(X-Y\).

For triply degenerate vibrations \(F\) we obtain:

\[ \begin{array}{cc} \text{kinematic} & \text{dynamic}\\ \text{coefficients} & \text{coefficients} \end{array} \]

\[ \begin{array}{c|cc} & q & \gamma\\ \hline q & \dfrac{1}{m}+\dfrac{4}{3}\dfrac{1}{M} & -\dfrac{8}{3}\dfrac{1}{Ms} \\[1.0em] \gamma & -\dfrac{8}{3}\dfrac{1}{Ms} & \dfrac{2}{s^{2}}\left(\dfrac{1}{m}+\dfrac{8}{3}\dfrac{1}{M}\right) \end{array} \qquad \begin{array}{c|cc} & q & \gamma\\ \hline q & k_q-h & \sqrt{2}\,(a'-b)\\[0.6em] \gamma & \sqrt{2}\,(a'-b) & k'_{\gamma}-v \end{array} \tag{10,16} \]

The corresponding secular equation for determining \(\omega_2^2\) and \(\omega_3^2\) will be

\[ \left| \begin{array}{cc} \left(\dfrac{1}{m}+\dfrac{4}{3}\dfrac{1}{M}\right)(k_q-h) -\dfrac{8\sqrt{2}}{3}\dfrac{1}{Ms}(a'-b)-\omega^2 & \sqrt{2}\left(\dfrac{1}{m}+\dfrac{4}{3}\dfrac{1}{M}\right)(a'-b) -\dfrac{8}{3}\dfrac{1}{Ms}(k'_{\gamma}-v) \\[1.0em] -\dfrac{8}{3}\dfrac{1}{Ms}(k_q-h) +\dfrac{2\sqrt{2}}{s^{2}}\left(\dfrac{1}{m}+\dfrac{8}{3}\dfrac{1}{M}\right)(a'-b) & -8\dfrac{\sqrt{2}}{3}\dfrac{1}{Ms}(a'-b) +\dfrac{2}{s^{2}}\left(\dfrac{1}{m}+\dfrac{8}{3}\dfrac{1}{M}\right)(k'_{\gamma}-v)-\omega^2 \end{array} \right|=0 \tag{10,17} \]

Finally, for the doubly degenerate vibrations \(E\):

\[ \begin{array}{cc} \text{kinematic} & \text{dynamic}\\ \text{interaction} & \text{interaction} \end{array} \]

\[ \gamma\left|3\cdot\dfrac{1}{m}\right| \qquad \gamma\left|k'_{\gamma}-2l'+v\right|. \tag{10,18} \]

whence

\[ \omega_4^2=3\dfrac{1}{m}\bigl[(k'_{\gamma}-v)-2(l'-v)\bigr]. \tag{10,19} \]

Thus, the problem of composing the secular equations is completely solved.

The frequencies are functions of five independent constants, namely: \(k_q, h, k_{\gamma}=k'_{\gamma}-v,\ l=l'-v\), and \(a=a'-b\) *). Since

*) The appearance of the differences \(k'_{\gamma}-v,\ l'-v\), and \(a'-b\) is connected with the presence of the additional condition (5,12) for the angular coordinates. The number of independent constants is the number of constants \(k^{(R)}_{\lambda\mu}\), when the potential energy is expressed in symmetry coordinates. If by \(n_R\) we denote the number of vibrations of type \(R\), then the number of constants for vibrations of this type is equal to \(n_R(n_R+1)/2\), and the total number of independent constants is

\[ \sum_R \frac{n_R(n_R+1)}{2}, \tag{10,20} \]

which gives for methane \(1+1+3=5\), and for ethane \(6+6+6+3=21\). This number is always greater than the number of frequencies, which is simply \(\sum_R n_R\).

if the number of frequencies is equal to four, then the constants cannot be determined completely from these frequencies alone.

The example considered of the molecule \(XY_4\) clearly shows the significance of taking the properties of symmetry into account. The higher the symmetry, the greater the effect of this accounting. At the same time, the presence of degeneracy of vibrations is very essential, owing to which the number of different vibrational frequencies is less than the number of vibrational degrees of freedom. Doubly degenerate vibrations occur for all molecules having one symmetry axis of order \(n\) not lower than the third, i.e. such an axis about which, upon rotation through an angle \(2\pi/n\) \((n \geq 3)\), the molecule is transformed into itself. Doubly degenerate vibrations are possessed, for example, by the pyramidal molecule \(XY_3\) of the ammonia type \(NH_3\) (a symmetry axis of the third order), the molecule \(X_2Y_6\) of the ethane type (also an axis of the third order), and the benzene molecule \(C_6H_6\) (a symmetry axis of the sixth order). In the presence of tetrahedral symmetry (the example considered of the molecule \(XY_4\)) and octahedral symmetry (ions of the type \(XY_6\), for example \(SiCl_6^{--}\)), which have several axes of order not lower than the third, one obtains not only doubly degenerate but also triply degenerate vibrations. For molecules possessing only planes of symmetry, a center of symmetry, and symmetry axes of the second order, all vibrations are nondegenerate, and the number of different vibrational frequencies is equal to the number of vibrational degrees of freedom.

§ 11. CALCULATION OF THE VIBRATIONAL FREQUENCIES OF METHANE AND ITS DEUTERO-SUBSTITUTED DERIVATIVES

The secular equations determine the relation of the vibrational frequencies of a molecule to its force constants. Applying the secular equations obtained by the method described, one can solve two types of problems:

I. From experimental data for vibrational frequencies, determine the force constants. Knowledge of the force constants makes it possible to draw important physical conclusions about the structure of molecules.

II. From known values of the force constants, calculate the vibrational frequencies. Knowledge of the vibrational frequencies makes it possible to interpret experimental data and to predict the frequencies of unknown vibrational spectra.

Concrete applications of the theory set forth to the solution of these two basic types of problems have shown the effectiveness of the methods developed for solving problems in the mechanics of vibrations.

The most important and primary problem is the determination of force constants from experimental data. Knowledge of these constants is necessary not only for calculating vibrational frequencies, but also for calculating intensities and polarizations, which depend on the particular form of the vibrations1. As we have already emphasized (§ 3, see also the end of the preceding paragraph), the number of force constants for an individual molecule always exceeds the number of fundamental frequencies

vibrations, while determining only part of the constants, by applying simplified force models, in most cases leads to completely incorrect results. Therefore, in order to determine the force constants, one should simultaneously use data for a series of molecules containing the same structural elements. It may be expected that identical structural elements in different molecules will be characterized by the same, or at least similar, constants. Thus the number of different constants is sharply reduced and, by rationally choosing a set of molecules of a definite type (for example, the simplest hydrocarbons and their halogen-substituted derivatives), we obtain for them a total number of observed frequencies considerably exceeding the number of constants, owing to which these constants can be determined in a sufficiently reliable manner. In this respect the most favorable case is that of isotopic molecules, above all deuterium-substituted compounds. The various deuterium-substituted molecules obtained by replacing different numbers of hydrogen atoms in the initial molecule by deuterium atoms possess differing frequencies with exactly identical force constants. Precisely such a case was considered by B. I. Stepanov^33. He calculated the vibrations of methane and of all its deuterium-substituted derivatives, namely: CH₃D, CH₂D₂, CHD₃, and CD₄, and determined the force constants of these molecules with great accuracy. Let us consider this case in more detail.

The potential energy of all five molecules under consideration is the same and contains, as we have seen (see § 10), five force constants: \(k_q\), \(k_\gamma\), \(h\) (bond–bond interaction), \(a\) (bond–angle interaction), and \(l\) (angle–angle interaction).

For the individual molecules we have the following number of frequencies:

CH₄ and CD₄. These are molecules of the type \(XY_4\), considered in § 10, and for them one obtains four frequencies each (1 symmetric \(A\), 1 doubly degenerate \(E\), and 2 triply degenerate \(F\)), which are determined by formulas (10,15) and (10,19) and by the secular equation (10,17). For the first molecule one must put \(m=m_{\mathrm H}\), for the second \(m=m_{\mathrm D}=2m_{\mathrm H}\).

CH₃D and CHD₃. These are molecules of the type \(XY_3Z\) and possess an axis of symmetry of the third order, i.e. the molecule transforms into itself under rotation by \(120^\circ\) about the bond \(X—Z\) (Fig. 21). The natural coordinates are:

\[ \begin{aligned} q_1,\ q_2,\ q_3 &\text{— changes in the lengths of the bonds } X—Y,\\ q_0 &\text{— change in the length of the bond } X—Z,\\ \gamma_{12},\ \gamma_{23},\ \gamma_{31} &\text{— changes in the angles } Y—X—Y,\\ \delta_1,\ \delta_2,\ \delta_3 &\text{— changes in the angles } Z—X—Y. \end{aligned} \]

The coordinate \(q_0\) is symmetric: \(q_0=q_0^{(A)}\). The three equivalent coordinates \(q_1,q_2,q_3\) are expressed in terms of the symmetric coordinate \(q^{(A)}\) and the doubly degenerate coordinates \(q^{(E I)}\) and \(q^{(E II)}\); the appearance of doubly degenerate coordinates is connected with the presence of a single axis of the third order (see § 10). Similarly, \(\gamma_{12},\ \gamma_{23},\ \gamma_{31}\) are expressed through

symmetry coordinates \(\gamma^{(A)}\), \(\gamma^{(EI)}\), \(\gamma^{(EII)}\), and \(\delta_1\), \(\delta_2\), \(\delta_3\)—in terms of the symmetry coordinates \(\delta^{(A)}\), \(\delta^{(EI)}\), \(\delta^{(EII)}\). As a result we have four interacting symmetry coordinates \(q_0^{(A)}\), \(q^{(A)}\), \(\gamma^{(A)}\), and \(\delta^{(A)}\), of which three are independent by virtue of the additional relation between the six angle changes. This gives one secular equation of the third degree for determining the three frequencies of the symmetric vibrations. For the interacting symmetry coordinates \(q^{(EI)}\), \(\gamma^{(EI)}\), and \(\delta^{(EI)}\) one obtains

Fig. 21. Symmetry of a molecule of type \(XY_3Z\).

Fig. 22. Symmetry of a molecule of type \(XY_2Z_2\).

likewise a secular equation of the third degree for determining the three vibration frequencies; the same equation is given by the coordinates \(q^{(EII)}\), \(\gamma^{(EII)}\), and \(\delta^{(EII)}\). We obtain three doubly expressed frequencies. In all, for each of the molecules \(\mathrm{CH_3D}\) and \(\mathrm{CHD_3}\) we have six vibration frequencies.

\(\mathrm{CH_2D_2}\). This molecule is of type \(XY_2Z_2\) and has two mutually perpendicular planes of symmetry (Fig. 22). The natural coordinates are:

\[ \begin{aligned} q_1,\ q_2&\text{— changes in the lengths of the } X-Y \text{ bonds,}\\ q_3,\ q_4&\text{— changes in the lengths of the } X-Z \text{ bonds,}\\ \gamma_{12}&\text{— change of the angle } Y-X-Y,\\ \gamma_{34}&\text{— change of the angle } Z-X-Z,\\ \gamma_{13},\ \gamma_{14},\ \gamma_{23},\ \gamma_{24}&\text{— changes of the angles } Y-X-Z. \end{aligned} \]

Four types of vibrations are possible:

\(A_1\)—symmetric with respect to both planes (fully symmetric),

\(A_2\)—antisymmetric with respect to both planes,

\(B_1\)—symmetric with respect to plane 1 and antisymmetric with respect to plane 2,

\(B_2\)—symmetric with respect to plane 2 and antisymmetric with respect to plane 1.

\(\gamma_{12}\) and \(\gamma_{34}\) are fully symmetric coordinates:

\[ \gamma_{12}=\gamma_Y^{(A_1)},\qquad \gamma_{34}=\gamma_Z^{(A_1)}, \]

\(q_1\) and \(q_2\) are expressed through symmetry coordinates:

\[ q_Y^{(A_1)}=\frac{1}{\sqrt{2}}(q_1+q_2),\qquad q_Y^{(B_2)}=\frac{1}{\sqrt{2}}(q_1-q_2), \]

\(q_3\) and \(q_4\)—through:

\[ q_Z^{(A_1)}=\frac{1}{\sqrt{2}}(q_3+q_4),\qquad q_Z^{(B_1)}=\frac{1}{\sqrt{2}}(q_3-q_4). \]

Finally, the four coordinates \(\gamma_{13}, \gamma_{14}, \gamma_{23}, \gamma_{24}\) are expressed through symmetry coordinates

\[ \gamma^{(A_1)}=\frac{1}{2}(\gamma_{13}+\gamma_{14}+\gamma_{23}+\gamma_{24}),\qquad \gamma^{(A_2)}=\frac{1}{2}(\gamma_{13}-\gamma_{14}-\gamma_{23}+\gamma_{24}), \]

\[ \gamma^{(B_1)}=\frac{1}{2}(\gamma_{13}-\gamma_{14}+\gamma_{23}-\gamma_{24}),\qquad \gamma^{(B_2)}=\frac{1}{2}(\gamma_{13}+\gamma_{14}-\gamma_{23}-\gamma_{24}). \]

As a result, we have one vibration of type \(A_2\) (coordinate \(\gamma^{(A_2)}\)), two vibrations of types \(B_1\) (interacting coordinates \(q_Z^{(B_1)}\) and \(\gamma^{(B_1)}\)) and \(B_2\) (interacting coordinates \(q_Y^{(B_2)}\) and \(\gamma^{(B_2)}\)) and, finally, four vibrations \(A_1\) (five interacting coordinates \(q_Y^{(A_1)}, q_Z^{(A_1)}, \gamma_{12}^{(A_1)}, \gamma_{34}^{(A_1)}\), and \(\gamma^{(A_1)}\), of which four are independent). Their frequencies are determined, respectively, from one secular equation of the first degree, two equations of the second degree, and one equation of the fourth degree.

For all five molecules we have 29 vibrational frequencies. To determine the constants, the values of 15 frequencies observed by MacWood and Urey\({}^{44}\) in Raman spectra were used. This number of frequencies exceeds threefold the number of constants, which made it possible to calculate the latter with sufficiently high accuracy, despite the presence of large errors in the experimental values (owing to the low intensity of the corresponding Raman lines in the experiments of MacWood and Urey). The following values of the constants best satisfy all 15 observed frequencies*):

\[ \begin{aligned} k_q&=(5.32\pm0.02)\cdot10^5\ \text{dyn/cm} &&\text{(quasi-elastic force constant of the C—H bond),}\\ k_\gamma&=(0.45\pm0.02)\cdot10^5\ \text{dyn/cm} &&\text{(quasi-elastic force constant of the H—C—H angle),}\\ h&=(0.032\pm0.004)\cdot10^5\ \text{dyn/cm} &&\text{(interaction of two C—H bonds),}\\ l&=(0.022\pm0.004)\cdot10^5\ \text{dyn/cm} &&\text{(interaction of two adjacent H—C—H angles),}\\ a&=(0.32\pm0.02)\cdot10^5\ \text{dyn/cm} &&\text{(interaction of the C—H bond with the adjacent H—C—H angle).} \end{aligned} \]

*) Here the angular coordinates are measured by an arc of radius \(s=1.09\ \text{Å}\).

Table 4

Results of calculating the frequencies of methane and its deuterium derivatives

Molecule	Symmetry type of vibrations	Bond type	Calculated frequency \(\omega_{\text{calc}}\) (in \(\mathrm{cm}^{-1}\))	Observed frequency \(\omega_{\text{obs}}\) (in \(\mathrm{cm}^{-1}\))	Difference \(\omega_{\text{calc}}-\omega_{\text{obs}}\) (in \(\mathrm{cm}^{-1}\))
\(\mathrm{CH}_4\)	\(A\) — symmetric	C—H	2914	2914	0
\(\mathrm{CH}_4\)	\(E\) — doubly degenerate	H—C—H	1536	—	—
\(\mathrm{CH}_4\)	\(F\) — triply degenerate	H—C—H C—H	1304 3022	1304 3022	0 0
\(\mathrm{CH}_3\mathrm{D}\)	\(A\) — symmetric	H—C—H C—D C—H	1309 2196 2962	1330 2200 2950 \((I)\)	−21 −4 12
\(\mathrm{CH}_3\mathrm{D}\)	\(E\) — doubly degenerate	H—C—D H—C—H C—H	1143 1486 3021	1156 \((I)\) 1474 \((I)\) 3031 \((I)\)	−13 12 −10
\(\mathrm{CH}_2\mathrm{D}_2\)	\(A_1\) — completely symmetric	D—C—D H—C—H C—D C—H	1032 1434 2164 2973	1033 1450 \((I)\) 2139 2974	−1 −16 25 −1
\(\mathrm{CH}_2\mathrm{D}_2\)	\(B_1\) — antisymmetric with respect to the plane D—C—D	H—C—D C—H	1091 3020	1091 \((I)\) 3020 \((I)\)	0 0
\(\mathrm{CH}_2\mathrm{D}_2\)	\(B_2\) — antisymmetric with respect to the plane H—C—H	H—C—D C—D	1260 2255	1286 2255 \((I)\)	−26 0
\(\mathrm{CH}_2\mathrm{D}_2\)	\(A_2\) — antisymmetric with respect to H—C—H and D—C—D	H—C—D	1336	1333	3
\(\mathrm{CHD}_3\)	\(A\) — symmetric	H—C—D C—D C—H	1010 2147 2990	938 \((I)\) 2141 3000 \((I)\)	22 6 −10
\(\mathrm{CHD}_3\)	\(E\) — doubly degenerate	D—C—D H—C—H C—D	1035 1300 2255	— 1299 2269	— 1 −14
\(\mathrm{CD}_4\)	\(A\) — symmetric	C—D	2085	2085	0
\(\mathrm{CD}_4\)	\(E\) — doubly degenerate	D—C—D	1098	—	—
\(\mathrm{CD}_4\)	\(F\) — triply degenerate	D—C—D C—D	1000 2255	988 \((I)\) 2258	12 −3

For each constant its error is indicated—the limits of change of the constant outside which the agreement between the observed and calculated frequencies is appreciably worsened. Table 4 gives the calculated frequencies of all 29 vibrations \(^{33,46}\) and the experimental values of the frequencies obtained from Raman spectra (those used to determine the constants are printed in boldface) and obtained from infrared spectra \(^{45}\) (marked by the letter \(I\)). For each vibration the type of symmetry is indicated, as well as the bonds or angles which chiefly change in this vibration *), and thus characterize the kind of vibration (valence or deformation vibrations of different bonds and angles). As the table shows, there is good agreement of the calculated frequencies not only with the frequencies observed in the Raman spectra, but also with the frequencies determined from infrared spectra, which were not used in finding the force constants; this serves as a good additional check on the correctness of the determination of the constants.

§ 12. CALCULATION OF VIBRATION FREQUENCIES OF COMPLEX MOLECULES

For very complex molecules, having tens of vibrational degrees of freedom, secular equations of high degrees are obtained, owing to which the determination of the unknown force constants from these algebraic equations becomes practically impossible. However, the corresponding constants can be determined for simpler molecules containing the same structural elements. Taking the numerical values of the force constants, we also obtain the secular equations in numerical form (see § 8), the only unknowns being the vibration frequencies; by applying a specially developed method \(^{34,1}\), one can solve secular equations with numerical coefficients even of very high degrees **).

Fig. 23. Configuration of the ethane molecule \(C_2H_6\).

The vibration frequencies of limiting hydrocarbons, both normal (\(n\)-propane, \(n\)-butane, \(n\)-pentane, etc.) and branched (isobutane, tetramethylmethane, etc.), can be calculated by applying the system of constants determined from the spectra of methane and ethane. For ethane (Fig. 23) B. I. Stepanov \(^{33}\) determined the force constants, starting from experi—

*) The form of all vibrations was calculated by M. V. Vol’kenshtein \(^{46}\).

**) In this case the secular equations to be solved are not expanded as power equations, but the method of gradual diagonalization of the interaction-coefficient matrix is used; a conclusion as to the approximate values can already be drawn from the general form of the secular equations and by approximate estimation of the off-diagonal terms.

experimental data for ordinary ethane \(C_2H_6\) and for heavy ethane \(C_2D_6\), each having 11 fundamental vibration frequencies. In the calculation the interaction constants \(h\), \(l\), and \(a\), found for methane (§ 11), were used; the constant \(k_\gamma\) turned out to have its previous value, while the quasi-elastic constant \(k_q\) of the \(C—H\) bonds in the \(CH_3\) group proved to be different from \(k_q\) for the \(C—H\) bonds in the \(CH_4\) group (5.15 instead of 5.32). The total number of constants for ethane is 21, and therefore, in the calculation, a number of simplifying assumptions were made; some of the constants corresponding to interactions of distant bonds and angles, for example of \(C—H\) bonds in different \(CH_3\) groups, were set equal to zero, while certain other constants were assumed to be identical, for example the interaction constants of adjacent \(H—C—H\) angles with one another, of \(H—C—C\) angles with one another, and of \(H—C—H\) angles with \(H—C—C\) angles. Thanks to this, the number of constants was reduced to 10; all 22 frequencies calculated with the aid of these constants agree very well (within \(10\ \mathrm{cm}^{-1}\)) with the observed frequencies. In the calculations it was assumed that the configuration of the ethane molecule is such that the \(CH_3\) groups are rotated relative to each other by \(60^\circ\) (Fig. 23).

Using the constants determined from the spectra of methane and ethane, one can already compute the frequencies of more complex molecules*). The vibrational spectra of propane and butane \(^{34}\), pentane \(^{35}\), isobutane and tetramethylmethane \(^{36}\) were calculated, and quite satisfactory agreement of the calculated frequencies with the observed ones was obtained; it proved possible to interpret completely the fundamental frequencies in the corresponding Raman and infrared spectra. As a result of the calculations it also became clear that the coefficients of complete interaction practically do not change in passing from one molecule to another, and a table of these coefficients \(^{35}\) can be compiled, which makes it possible to obtain the secular equations very quickly for any molecule of saturated hydrocarbons. On the basis of the calculations carried out, it is possible, with sufficient accuracy, to predict by analogy, without additional computations, the spectra of various isomers of hexanes, heptanes, and octanes, and, what is especially important, to establish characteristic signs of various types of branching, for example branchings of the type

\[ \mathrm{C—C—C} \begin{matrix} & \mathrm{C}\\[-0.4em] & |\\[-0.4em] & \mathrm{C} \end{matrix} \quad \text{and} \quad \mathrm{C—C—C—C} \begin{matrix} & \mathrm{C}\\[-0.4em] & |\\[-0.4em] & \mathrm{C}\\[-0.4em] & |\\[-0.4em] & \mathrm{C} \end{matrix} \]
\[ \text{[[unclear: structural formulas shown schematically on the page]]} \]

*) In doing this it is only necessary additionally to introduce the constants of the \(C—H\) bonds in the \(CH_2\) and \(CH\) groups and of the \(C—C—C\) angle; their values may be obtained by extrapolation and checked by comparing the calculated frequencies with the observed ones. In reality, of course, other constants besides \(h\), \(l\), \(k_q\), and \(k_\gamma\) change in passing from molecule to molecule; however, within the limits of accuracy of the calculations these changes will not be reflected.

The calculation of the vibration frequencies of halogen-substituted saturated hydrocarbons can be carried out by using the system of constants obtained by B. I. Stepanov from the spectra of simple halogen-substituted methanes[^37]. Experimental data were taken for all 12 molecules of this kind, namely:

$\mathrm{CH_3Br},\quad \mathrm{CH_3Cl},\quad \mathrm{CH_3F}$
(molecules of type $\mathrm{XY_3Z}$, having 6 vibration frequencies)

$\mathrm{CH_2Br_2},\quad \mathrm{CH_2Cl_2},\quad \mathrm{CH_2F_2}$
(molecules of type $\mathrm{XY_2Z_2}$, having 9 vibration frequencies)

$\mathrm{CHBr_3},\quad \mathrm{CHCl_3},\quad \mathrm{CHF_3}$
(molecules of type $\mathrm{XYZ_3}$, having 9 vibration frequencies)

$\mathrm{CBr_4},\quad \mathrm{CCl_4},\quad \mathrm{CF_4}$
(molecules of type $\mathrm{XZ_4}$, having 4 vibration frequencies)

These molecules are of the same symmetry types as those considered in §§ 10 and 11 (methane and its deuterium-substituted derivatives), and the secular equations for them are compiled without difficulty. A system of constants was found giving the best agreement between the calculated and observed frequencies. In this connection, considerable help is provided by the fact that the values of constants of a definite type change systematically in the substitution $\mathrm{Br \to Cl \to F}$ in a molecule containing a given number of atoms of the given halide, and also with a change in the number of atoms of the given halide. As an example, one may cite the data for the quasi-elastic constant of the $\mathrm{C—Z}$ bond, where $Z$ is a halogen atom (the constants are expressed in $10^5$ dyn/cm) (see table at left).

Molecule	$\mathrm{C—Br}$	$\mathrm{C—Cl}$	$\mathrm{C—F}$
$\mathrm{CH_3Z}$	3.00	3.70	6.25
$\mathrm{CH_2Z_2}$	3.05	3.76	6.80
$\mathrm{CHZ_3}$	3.12	3.88	6.95
$\mathrm{CZ_4}$	3.18	3.94	7.80

Comparison of the data for the $\mathrm{C—Cl}$ bond (middle column) with the data given in § 4 (p. 17) shows how essential the correct method of calculating the constants is. The interaction constants prove to be very significant, especially for fluorine-substituted compounds. Thus, in $\mathrm{CHF_3}$ the quasi-elastic constant of the angle $\mathrm{H—C—F}$ is equal to $1.12 \cdot 10^5$ dyn/cm, while the interaction constant of the $\mathrm{C—H}$ bond with the $\mathrm{H—C—F}$ angle is equal to $0.89 \cdot 10^5$ dyn/cm, i.e., only slightly smaller. In the valence-force model, however, this constant is assumed to be zero. In general, calculations have shown[^47] that both the valence-force model and the central-force model are completely unsuitable for calculating the frequencies of halogen-substituted methanes; without taking into account the interactions of different bonds and angles, it is impossible to obtain even approximately correct results.

The constants found from the spectra of simple halogen-substituted methanes were successfully applied to the calculation of mixed halogen-substituted methanes (containing atoms of different halogens, for example, $\mathrm{CCl_2Br_2}$, $\mathrm{CHClF_2}$, etc.) and of a series of halogen-substituted ethanes[^38].

A calculation of the vibrations of molecules of the so-called rotational isomers[^48], which differ in the mutual orientation of groups connected by a single bond, is of considerable interest. Such, for example, are the two rotational isomers of normal butane, in one of which all C—C bonds lie in one plane, while in the other one C—C bond is rotated relative to the plane of the two other C—C bonds (see Fig. 24), and the two rotational isomers of dichloroethane CH₂Cl·CH₂Cl (Fig. 25). The vibrational spectra of such molecules, chemically identical, differ in their frequencies. It has been possible to calculate these spectra both in the cases of butane[^35] and dichloroethane[^38], and in other cases as well. Taking into account the frequencies of the vibrations of rotational isomers is essential in calculating thermodynamic properties; for isomers the thermodynamic functions obtained are somewhat different[^49].

Fig. 24. Rotational isomers of butane:
a — planar form, b — rotated form.

Fig. 25. Rotational isomers of dichloroethane.

The calculations performed for the vibrations of complex molecules show that at present, for any type of complex molecule, one can fairly quickly and reliably determine, on the basis of experimental data for simpler molecules of an analogous type, the force constants, and interpret and predict the corresponding vibrational spectra.

The task of further investigations is[^50], on the one hand, the calculation of the fundamental vibrations of specific types of molecules and, on the other hand, the development of methods for taking account of the anharmonicity of vibrations both for fundamental frequencies and, in particular, for overtones.

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