Experimental Data on the Spectroscopy of Polyatomic Molecules*

R. Mecke, Heidelberg

In the present report we shall deal with experimental data on the spectroscopy of polyatomic molecules, but this in no way means that research in this field may be regarded as in any degree completed. The study of the spectra of polyatomic molecules—in contrast to the study of the spectra of diatomic molecules—is still in its initial stage of investigation, and only in the coming years can major advances in this field be expected. Thus I am, unfortunately, obliged first to illuminate the negative aspects, pointing to the many difficulties confronting the spectroscopist when the number of atoms in a molecule increases from two to three and more. At the same time, however, we shall become acquainted with the paths along which the spectroscopy of polyatomic molecules will develop in the near future, and I hope to show that already at the present time the application of combined methods of study—namely, thanks to investigation in the infrared region (of a thermal and photographic character), while simultaneously using Raman spectra and molecular-interferometric measurements—makes it possible to say much concerning the structure of polyatomic molecules. In speaking of these positive data, I shall have to specialize somewhat the subject of my report, and therefore in conclusion I shall deal chiefly with one problem, which I consider exceptionally important in the present state of research—namely, the calculation and interpretation of the natural frequencies of polyatomic molecules.

I consider the development of this question so important because it will make it possible successfully to use, in the solution of physical and especially chemical molecular problems, the enormous factual material of Raman-spectrum spectroscopy that has accumulated in an avalanche in recent years.

1. The spectrum of a diatomic molecule (an electronic-vibrational spectrum) is analyzed, as is known, in the following manner: first, by “tone—

* Leipziger Vortrag 1931. The present article scarcely touches on the theory of the question. One of the best reviews of a theoretical character is by Dennison (Rev. of Mod. Phys. 3, 280, 1931).

“structure” of bands, by means of the so-called “combination relations,” one determines the moment of inertia of the molecule and thereby the distance between the atomic nuclei; then, in analogous fashion, the “coarse structure” of its system of bands is arranged into a “quantum scheme,” which at once gives the frequency of the nuclear vibrations and the degree of anharmonicity of the bond. Further, from the type of construction of the bands one attempts to determine the character of the electronic terms and, finally, one seeks the products of decomposition of the molecule under adiabatic separation of the nuclei and determines the amount of energy required for this. Subsequent, more detailed and subtle investigations aim at clarifying other questions of the electronic structure of the term. Thus, over a period of more than 10 years, more than 100 different molecules have been studied. Although a great deal of minor experimental work will still be required in order to supplement the numerical material and give a complete survey of molecular data for simple compounds, all the fundamental questions concerning diatomic molecules have now been fully resolved. Unfortunately, the study of polyatomic molecules has by no means yet attained such a fortunate state. Quite the contrary: I should like especially to emphasize that up to now it has not once been possible to analyze in such detail the electronic-vibrational spectrum of a polyatomic molecule. To be sure, in this field there are already highly promising beginnings (for example, the works of Henri, Herzberg, Mekke, and others), but nevertheless these are no more than beginnings. The reasons why triatomic and polyatomic molecules have hitherto yielded with such difficulty to spectroscopic investigation, despite numerous attempts, are very diverse.

First of all, it is necessary to overcome considerable experimental difficulties. Even for a diatomic molecule it is not easy to obtain a sufficiently intense emission spectrum, and this applies to an even greater degree to polyatomic molecules, since, owing to the increased number of possibilities for rotation and vibration of the molecule, its tendency to decompose rises extraordinarily. In this case one can speak only of very weak exciters, such as, for example, fluorescence or a weak glow discharge. But precisely in the investigation of polyatomic molecules we cannot dispense with the use of a spectral apparatus possessing large dispersion and high resolving power; therefore, proceeding precisely from considerations of luminosity and molecular stability, we are compelled almost exclusively to confine ourselves to the study of absorption spectra. Unfortunately, the greater part of molecules absorb only in the far ultraviolet, and the difficulty of creating in this region good light sources giving a continuous spectrum has by no means contributed to the success of the investigation.

Alongside these experimental difficulties, which must be taken into account already when obtaining the spectra, there exist numerous difficulties of a technical order which we have to encounter in the analysis of spectra. We do not

as concerns the requirements imposed by the resolving power of the spectrum; but for a diatomic molecule we had to determine from the fine structure and the band scheme only one moment of inertia and one nuclear frequency for each electronic state; for a polyatomic molecule with \(n\) atoms we must determine three principal moments of inertia and \(3n-6\)* frequencies; the number of lines and bands in the spectrum increases to the same degree. Therefore its analysis is often almost impossible because of the large number of bands with indistinct edges and unresolved fine structure. If certain successes have been achieved in this field, they must be evaluated accordingly. As an example I give part of the absorption spectrum of \(\mathrm{NO}_2\) in the visible region (Fig. 1, \(a\)). In addition, I may refer to the excellent photographs made by Henri.

Fig. 1. \(a\)—\(\mathrm{NO}_2\), \(b\)—\(\mathrm{H}_2\mathrm{O}\), \(\lambda 8230\), \(c\)—\(\mathrm{NH}_3\), \(\lambda 7920\), \(d\)—\(\mathrm{C}_2\mathrm{H}_2\), \(\lambda 7886\).

In conclusion it is necessary to speak of the difficulties of the theory which have not yet been overcome. In doing so I shall not touch upon the theory of electronic states; likewise I shall leave aside possible dissociation processes, which can be followed spectroscopically. Below I shall dwell in detail on the calculation and interpretation of the \(3n-6\) natural frequencies of such a molecule, while now I shall first touch only on certain questions of fine structure, i.e. of rotational states. As is known, the quantization of the simple rotator (Schwarzschild) led to the discovery that the mutual relation between the lines in a band of a diatomic molecule must be very simple, and this discovery was fully confirmed experimentally. Thus, if

* In elongated rod-shaped molecules there are \(3n-5\) natural frequencies, i.e. \(2n-3\) distinct frequencies, since \(n-2\) frequencies are so-called double vibrations.

once the experimental difficulties of obtaining the spectrum and resolving its bands have been overcome, then the assignment of all lines of this spectrum is only a question of the investigator’s diligence and perseverance. The greatest difficulties were presented, strictly speaking, only by the analysis of the many-line spectrum of hydrogen; but this case is an exception and, moreover, at the present time the questions relating to it have been successfully resolved thanks to the work of Richardson and his collaborators, Finkelnburg, Mecke, Weizel, and others. Matters stand otherwise with polyatomic molecules, not to speak of several fortunate cases where, owing to the arrangement of the atoms in one straight line, the rotational character of a diatomic molecule is obtained. In the case of a symmetric top (a molecule with two equal principal moments of inertia), the mathematical treatment of the problem presents no difficulties; in the case of a serial arrangement, the only real hindrance is the excessive abundance of lines in the band. But as soon as even small symmetry disturbances appear in the molecule, very considerable difficulties immediately arise: the quantization of an “asymmetric top” (Kramers, Hund, Dennison, and others) cannot be carried out in finite form. The approximate methods applied in this case show that here one must not expect a regular sequence in the distances between lines. Unfortunately, experience confirms this conclusion: at the present time we still do not have a reliable criterion for determining how, in an asymmetric molecule (three different principal moments of inertia), the lines of one band are combined into series. With small symmetry disturbances one may use approximate relations obtained in measurements of the spectrum, where, as a rule, only the frequency curves (Häufigkeitskurven) of the lines are measured, and not the lines themselves. From them one can approximately calculate the moments of inertia, but no more. As a typical example I shall mention investigations of water, ammonia, and the halogen derivatives of methane, about which I shall speak later. It is also necessary to draw attention to the following circumstance. The relative coordinates of the position of an \(n\)-atomic molecule are determined uniquely by \(3n - 6\) data, but, in addition, a permutation (Vertauschung) of the atomic masses is possible. Therefore the calculation of the moments of inertia is in no case sufficient for an unambiguous determination of the structure of a polyatomic molecule. In this case it is always necessary to make certain assumptions a priori. Thus, for example, if for ammonia and for water one assumes that the distances of H from the central atom are indeed equal to one another, then for determining the internuclear distances and angles it is sufficient to know two moments of inertia \((\mathrm{NH}_3 : I_1 = I_2,\ \mathrm{H}_2\mathrm{O} : I_1 = I_2 + I_3)\). But nevertheless there are always two solutions: a “flat” and a “pointed” (spitzes) model of the molecule, between which a choice cannot be made on the basis of the available data. Fortunately, however, the laws of constancy of angles and distances between nuclei

apparently, are not exactly observed, so that these difficulties are not of great significance here.

1. I have enumerated so many difficulties and pointed out so many negative aspects that the impression might easily be created that analysis of the spectrum of a polyatomic molecule is, in general, a hopeless matter. It goes without saying that this is not so. It is enough to recall that only a few years ago we were faced with entirely analogous difficulties in investigating the spectra of diatomic molecules. At that time the analysis of bands was begun with consideration of rotational and rotational-vibrational bands (Bjerrum, Kratzer), and in this way the desired goal was reached. We too shall confine ourselves only to this class of spectra, and therefore shall take into account only the ground electronic state of the molecule. This restriction substantially simplifies the investigation. But as long as we use only spectroscopy in the infrared region, we shall not succeed in obtaining any very significant results. The methods of investigation themselves are the reason why, as a rule, the resolution of the bands is still insufficient for determining their fine structure, or for reliably establishing the vibrational character of individual bands (the fundamental frequency, an overtone, combination vibrations), in the presence of a large number of possible combinations of natural frequencies. Moreover, the short-wave (less than \(10\mu\)) infrared region usually investigated by no means embraces all the frequencies of the molecule. However, in the very last few years three new, very important methods of investigation have come to the aid of spectroscopy in the infrared region. The most significant of these is Raman spectroscopy, on whose successes I need not dwell in the present article*. According to this method, with the aid of simple spectroscopic auxiliary means, on both sides of the “exciting” line there is produced a greatly simplified rotational and vibrational-rotational spectrum of the molecule, containing for the most part only fundamental frequencies. But this spectrum by no means corresponds to a simplified infrared spectrum. Thus, for example, in simple molecules \(H_2\), \(N_2\), \(O_2\), etc., on the basis of the laws of symmetry, infrared absorption bands cannot arise, but in the Raman spectrum the rotational and vibrational frequencies are clearly visible; they are “not active in the infrared spectrum, but active in the Raman spectrum.” On the other hand, Raman spectroscopy cannot dispense with the results of measurements of infrared absorption bands. Thus both methods constantly supplement one another. A complete analysis of the Raman spectrum becomes possible only thanks to infrared measurements, and conversely. Below I shall speak further of the results achieved thanks to this point of view.

* For review literature see Kohlrausch, Smekal. Raman Effekt, Berlin, Springer 1931; also the articles by Placzek and Rasetti in the same volume of Leipziger Vorträge, where the present article by Mecke (“Molekülstruktur”) is printed.

Next, a very important method for studying the structure of molecules is the method of molecular interferometry proposed by Debye1. This method makes it possible to measure directly the distances between nuclei by means of the diffraction of X-rays (Debye) and of electrons (Wierl). I cannot dwell here on the technique and results of this method; I should like only to emphasize that this method has rendered invaluable service to molecular spectroscopy, since thanks to it we can calculate moments of inertia independently of spectroscopic investigation and then apply the results obtained to the solution of various questions connected with the fine structure of bands. Even for diatomic molecules it is not easy to resolve the fine structure of bands if heavy atoms are present; this applies to an even greater extent to polyatomic molecules, and thus interferometry is in this case a great aid.

A further supplement to the methods of investigation is photography in the infrared region. With the aid of the newest sensitizers of the Agfa and Kodak firms, absorption spectra with maximum dispersion can easily be photographed in the region up to \(\lambda = 8800\ \text{\AA}\), and, if desired, even up to \(1\ \mu\). I shall show below that rotation-vibration bands also occur in this region, although they have lower intensity there and high overtones. Nevertheless, the advantages of this method are very evident, since the possibility of using a large concave Rowland diffraction grating increases the accuracy of measurements and the resolving power (in comparison with the old measurements in the infrared region) by several tens of times. Since this is the region in which I myself work, I shall allow myself to dwell on it in somewhat more detail and to report on the results achieved by my collaborators and me. In Table 1 all the rotation-vibration bands photographed up to the present time are compared; moreover, the matter here concerns chiefly such vibrations as can be attributed to the bonds C—H, N—H, and O—H. The method of designating the bands and their meanings will be discussed in greater detail below.

The first molecule successfully investigated was the ammonia molecule2; three bands were photographed: \(\lambda = 8800\ \text{\AA}\) \((3\nu + \delta)\), \(\lambda = 7920\ \text{\AA}\) \((4\nu)\) (Fig. 1, c), and \(\lambda = 6470\ \text{\AA}\) \((5\nu)\). In Fig. 2 I give a comparison of the fundamental band \(3\mu\) \((\nu)\), investigated with all possible care by Barker and Stinchcomb3, and of its third overtone \((4\nu)\), photographed by us in the region \(\lambda = 7920\ \text{\AA}\). This overtone should have had, and in fact did have, the same structure as the fundamental band. Both bands are drawn on the same frequency scale. From this comparison it is clearly seen that even with the most careful measurements by means of a diffraction grating in the infrared region, nevertheless in

R. Mekke

TABLE 1

Photographed rotational-vibrational bands

as a result, only statistical frequency curves (Häufigkeitskurven) are obtained; moreover, we see that the fine structure of the ammonia bands is far more complex than could have been supposed on the basis of the investigations carried out up to now in the infrared

Fig. 2. Ammonia bands at $3\mu$ and [[unclear: near/at]] $20\mu$.

region. The theory of such a molecule, however, necessarily leads one to expect precisely such a structure. The ammonia pyramid was considered a symmetric molecule, but in the structure of the photographed band noticeable unpleasant violations of symmetry are apparent. Therefore only an approximate determination of the moments of inertia could be made, which gave, for $\mathrm{N-H}$, $0{,}977\ \text{Å}$, for $\mathrm{H-H}$, $1{,}43\ \text{Å}$, and for $h$ (the height of the pyramid), $0{,}517\ \text{Å}$. In this re-

set in motion new investigations, which will be reported in the near future. Contrary to expectation, good results were obtained by studies of the absorption spectrum of acetylene^4. Both bands, photographed at \(\lambda = 7885\ \text{Å}\) (Fig. 1, \(d\)) and \(\lambda = 8620\ \text{Å}\), showed a remarkably simple structure—namely the structure of a diatomic molecule, from which it followed that the four atoms lie on one straight line. But already at \(\lambda = 7956\ \text{Å}\) a very slight bending of the rod \(\mathrm{H—C—C—H}\) became noticeable, characterized by the appearance of weak side bands to the right and to the left of the principal one. Nevertheless, it proved possible to determine the moments of inertia with complete accuracy, which gave for the distances \(\mathrm{C—H}\) \(1.08\ \text{Å}\), and for \(\mathrm{C \equiv C}\) \(1.19\ \text{Å}\). In addition, an alternation of intensities was clearly visible in the bands, which I had already discovered several years earlier in symmetric diatomic molecules. We were even able to make precise measurements of the intensities and determine the ratio of the intensities (along with a simultaneous check of the formulas for the intensities of a “diatomic” molecule) with an accuracy up to \(1:3.0 \pm 0.1\) (Fig. 3)^5.

Fig. 3. Alternating intensities in \(\mathrm{C_2H_2}\), \(\lambda 7886\).

Thanks to this, the spin of the hydrogen nucleus could also be unambiguously determined spectroscopically, and for it the value \(1/2\) was obtained.* Similar investigations were successfully carried out by Badger and Binder^6 for the molecule of hydrocyanic acid (HCN), which is also linear, the distances here being \(\mathrm{C—H}\) \(1.08\ \text{Å}\) and \(\mathrm{C \equiv N}\) \(1.15\ \text{Å}\). It goes without saying that in this case the bands (\(\lambda = 7912\ \text{Å}\) and \(\lambda = 8563\ \text{Å}\)) revealed no alternating intensities. These two molecules, however, exhaust the group of “isosteric” compounds \(\mathrm{N_2}\), \(\mathrm{CO}\), \(\mathrm{C_2H_2}\), and \(\mathrm{HCN}\) (14 electrons), and it turns out^7 that isosterism also causes a considerable similarity of these molecules in spectroscopic respects, as I shall discuss in more detail below (Table 5).

* Until now this quantity could be determined only from the change in the specific heat of hydrogen. The many-line spectrum of hydrogen was not suitable for this investigation because of its short series, and also because the temperature at which it was obtained was not known. Moreover, it was doubtful whether the intensity ratio is in fact an integral number and equal to \(1:3\).

The largest number of bands could be photographed8 for water vapor (Fig. 1, b)—namely at least 5 bands, and if the atmospheric lines of the solar spectrum are added, then several more bands are added. At the same time, as a pleasant by-product, it was found that all atmospheric lines in the solar spectrum must be assigned to pairs of water and oxygen8. Since the molecule is asymmetric, the analysis of the fine structure on the basis of the considerations set forth above presents great difficulties. But the investigations gave, for the moments of inertia, approximate values \(O—H\ 0.94\ \text{Å}\) and \(H—H\ 1.43\ \text{Å}\)9. Further, with the aid of photography it proved possible to determine the third proper frequency, which until now had not yet been found in the infrared region.

The spectrum of methane is being investigated at the present time, and I shall not now discuss the results of this investigation. In the spectrum of methane there is a remarkably very strong violation of symmetry in the band \(\lambda = 8860\ \text{Å}\), which considerably complicates the analysis. Further, for all these molecules, with the aid of photography and measurements of the infrared spectra and Raman spectra, the proper frequencies and their combination possibilities could be determined, of which more will be said below. Although the field of application of photography in the infrared region for the investigation of rotational-vibrational spectra is limited to \(X—H\) vibrations, I nevertheless look very optimistically upon the further development of this method of investigation, whose advantage consists mainly in the use of instruments of large dispersion; I believe that this method will play a major role in solving problems of fine structure.

1. In what follows I should like to specialize the question somewhat and, above all, to abandon entirely the consideration of those problems of fine structure which we have not yet mastered to a sufficient degree either practically or theoretically. We shall also not touch upon such properties of molecular symmetry, important both for theory and for practice, i.e. questions connected with these properties, as, for example: under what conditions a band disappears in the infrared absorption spectrum as a so-called “inactive vibration,” when one should expect the appearance of combination frequencies, and what frequencies may arise in the Raman spectrum. On these questions a paper by Placzek is expected in the near future.* The formulation of the problem should be as follows: is it possible, on the basis of the experimental data at our disposal, to estimate with sufficient reliability the order of magnitude of the proper frequencies of a polyatomic molecule and at the same time to form a clear representation of the vibrations of such a molecule. This question is very important not only for the deciphering of infrared bands and Raman frequencies, but, in particular, for application to the solution of chemical problems.

* See Leipziger Vorträge, 1931 (“Molekülstruktur”).

Only when we learn to determine and characterize natural frequencies with complete reliability will it become possible to create structural chemistry purely spectroscopically and thereby help the chemist who establishes structural formulas on the basis of the data of reactive chemistry, and sometimes also to improve these formulas. In the solution of this problem I see the principal aim of the spectroscopic study of polyatomic molecules, and I hope that this aim is fully attainable.

In order to outline the path toward attaining this aim, let us recall some data from the spectroscopy of diatomic molecules. We shall see that in diatomic compounds there are molecular quantities which remain practically unchanged in complex compounds; that is, we shall construct a polyatomic molecule—just as chemists do—from diatomic, “standardized” parts.

Fig. 4. Curve of internuclear distances in diatomic hydrides.

Such an additive quantity is, first of all, the distance between nuclei. In Fig. 4 one can see how the distance between nuclei changes regularly in diatomic hydride compounds and how from it one can determine the structure of the atomic shell, with breaks in the curve for the noble gases. I have no opportunity here to dwell on details, but the distance between nuclei remains to a considerable extent unchanged when we pass from unsaturated radicals to saturated compounds, for example, from CH, NH, OH to CH\(_4\), NH\(_3\), and H\(_2\)O. Only a slight decrease in the distances, approximately by 5–10%, indicates some strengthening of the molecule, which was to be expected from the chemical point of view (Table 2). Everything said here about hydrogen compounds also applies to other molecules. Thus, for example, in carbon–nitrogen–oxygen compounds we have, for different types of bonds, the following internuclear distances: for a single bond 1.5 Å; for a double bond 1.23 Å, and for a triple bond 1.12 Å, with a very small range of deviations, contrary to expectation (Table 3). This result

TABLE 2

Distances between nuclei and nuclear frequencies of saturated and unsaturated hydrides

TABLE 3

Characteristic values of $\nu$, $r$, and $k$ for various carbon–oxygen-like compounds
(average values)

very important, since it gives us the possibility, using the moments of inertia found spectroscopically and by interferometric measurements (Debye), to calculate the unknown distances between nuclei and the angles between the valence strokes connecting atoms in a polyatomic molecule; moreover, this result is important in the calculation and estimation of natural frequencies$^{1,12}$.

Further, a characteristic quantity is the nuclear frequency of a diatomic molecule, since from it one can unambiguously determine the character of the valence bond. I give the frequency curve for diatomic oxide compounds (Fig. 5). This curve clearly shows that, as one moves through the periodic system, each newly added electron at first increases the frequency of vibration of the nucleus, and then acts by increasing the bond until, at CO (respectively SiO), a maximum is reached. The electron in the compounds N (respectively P) also strengthens the bond, but not to the same degree as in the preceding oxides in the ground state; such an unusually great bond strength is possessed rather by the next higher electronically excited state. In the molecule in the ground state the new electron, on the contrary, loosens the bond, and the removal of the second electron from the valence shell takes place. This loosening of the bond continues from

oxygen up to the noble gas, where a minimum is reached—zero; here, in general, not a single electron can form a valence shell with oxygen. This example is intended mainly to show how, from the magnitude of the nuclear vibrational frequencies, one can unambiguously determine the character of the bond. Thus, for example, it can readily be shown that in different electronic states of a diatomic molecule the character of the bond is different, i.e. the number of bonding electrons changes from level to level. This phenomenon finds clear expression in the abrupt change of nuclear frequencies. In Fig. 5 the dotted curves depict some such excitation stages corresponding to simple bonds. Further, an example may be the ultraviolet absorption of nitrogen and oxygen, when, owing to the absorption of light, the bond changes from a triple

Fig. 5. Curve of nuclear frequencies of diatomic oxides.

\((\mathrm{N}\equiv\mathrm{N})\) to a double \((\dot{\mathrm{N}}=\dot{\mathrm{N}})\), and from a double \((\mathrm{O}=\mathrm{O})\) to a single \((\dot{\mathrm{O}}-\dot{\mathrm{O}})\) with “straightened” (“aufgerichteten”) valences, this phenomenon being characterized by an abrupt change of the nuclear frequencies from 2360 and 1576 to 1680 and 708. In the already mentioned diatomic carbon–nitrogen–oxygen compounds, a statistical investigation of the various excited states shows that the obtained quantities group themselves very closely about the mean values \(X\equiv X\ 2200\), \(X=X\ 1650\), and \(X-X\ 1025\) (Table 3). All these considerations, set forth here only in part, inevitably lead to an expansion or even a change of our old concept of valence; we are now compelled to distinguish strictly between the “valency” (Wertigkeit), or valence of an atom, and the character of the bond of a molecule. Thus, for example, the spectroscopic similarity of CO and nitrogen, extending even to details (in contrast to carbonyl \(=\mathrm{C}=\mathrm{O}\)), compels us to assume a triple bond \(\mathrm{C}\equiv\mathrm{O}\) for carbon monoxide; this requirement is fully justified by other analogies, both of a physical kind (isosterism) and chemical (for example, parachor, dissociation energy). In order to

to reveal a similar character of the bond also in the chemical structural formula, I propose that between the symbols of the elements, instead of valence strokes, which express only the valence of the atom, one should always put the number of bonding electrons; thus, for example, for carbon monoxide the formula will be written in the form C(4 + 2)O, so that here four valence electrons of the carbon atom and two valence electrons of oxygen give a triple bond. Thanks to such a notation, the difficulties connected with the presence of diatomic C in a stable molecular compound are also removed. For the three electronic levels NO with nuclear frequencies 2345, 1891, and 1030 one may analogously write: N(5 + 2)O; N(3 + 2)O and N(1 + 2)O. The task of further investigations is to determine the character of the electronic terms ($\pi$- and $\sigma$-electrons).

Fig. 6. Magnitude of the bond $k$ in hydrides.

1. After these remarks, which have diverted us somewhat to the side, but which are nevertheless very important, let us return to the question of the additive magnitude of the bond, from which it is possible to calculate the natural frequencies of polyatomic molecules. The natural frequency of the bond between two atoms is not suitable for this investigation; by analogy with the mechanical modulus of elasticity, we introduce as the bond constant that work which is necessary in order to double the distance between the nuclei of the molecule, with strict observance of Hooke’s law. This constant is obtained directly from the formula for the potential of the so-called harmonic bond:

\[ P = k \left( \frac{\Delta r}{r} \right)^2 . \tag{1} \]

Its magnitude can easily be calculated for a diatomic molecule from the frequency of vibration of the nucleus $\omega$ and the moment of inertia $\mu r^2$, namely:

\[ 2\pi\omega = \sqrt{\frac{2k}{\mu r^2}} . \tag{2} \]

I shall show the significance of this bond constant $k$ on several examples. In Fig. 6 a graphical representation of this quantity is given for diatomic hydrides. Starting from zero at the noble gases, the curve rises regularly for each period of the periodic system. The curvature at each first element (H, Li, Na) shows that here the bond is produced by $s$-electrons; in the case when

there are only \(p\)-electrons, the curve is almost rectilinear, so that the constant \(k\) may be approximately (and, of course, purely empirically) expressed by the formula \(k = 2.4\,(Z' + 2)\,\mathrm{V}\), or \(55.5\,(Z' + 2)\,6\ \mathrm{cal/mol}\), where \(Z'\) denotes the total number of external electrons. Thus all the external electrons act here upon the bond in a strengthening manner, i.e. the character of the simple bond \(X—H\) is expressed in the fact that the hydrogen electron is accepted into the family of electrons of the other element as a fully equal member. We cannot now dwell on other, very interesting details. In carbon–nitrogen–oxygen compounds the values for the single, double, and triple bond (as I shall show below) also group very closely around the values 27, 54, 77 V (Table 3); their ratio may with sufficient approximation be taken as \(1 : 2 : 3\), i.e. it is considerably higher than in hydride compounds. But for what follows the fact is especially important that this value of the bond energy is to a considerable degree constant for chemically similar elements, i.e. for those elements which stand in the separate groups of the periodic system one below another and possess the same number of valence electrons. As an example I shall first of all point to the hydrogen halides (Table 4), for which it is clearly seen that the value \(k\) remains constant (24.0 V), although the nuclear frequency and the distance between nuclei change greatly. But owing to the greater affinity for the electron this value is much higher than for the molecule \(\mathrm{H_2}\) (9.3 V), where the bond is formed by only two K-electrons.

TABLE 4

Hydrogen halides

The following Table 5 contains bond values for the triple, double, and single bonds of molecules containing 10, 12, and 14 external electrons. Here it is also obvious that these values are constant. In Table 6 are given data for the halo derivatives of methane, denoted \((\mathrm{H_3C})—X\), namely those frequencies which are the only ones characteristic of this bond and from which, as we shall see below, the values of \(k\) can be calculated. In this case the constancy of this quantity is also quite sufficient. Thus, for example, for \((\mathrm{H_3C})—J\) the distance between nuclei was not

TABLE 5

Bond constants \(X \equiv X\), \(X = X\), and \(X—X\)

10 valence electrons

12 valence electrons

14 valence electrons

TABLE 6

Halogen derivatives of methane

* Measurements of Raman spectra.
** Interferometric.

determined neither spectroscopically nor by interferometric means, and it proved possible to determine it by the method indicated above, whereby it turned out to be equal to 2.50 Å. The great electron affinity of the halogens is again revealed in comparison with methane \((\mathrm{H}_3\mathrm{C})-\mathrm{C}\), which has only one force constant, equal to 18.4 V. Analogous conclusions may be drawn if we gradually chlorinate methane until it is converted into carbon tetrachloride (Table 7). In this case the constancy of the value \(k\) is not preserved so well; one may rather say that \(k\) gradually increases as the number of chlorine atoms increases; this is explained by the fact that chlorine atoms, owing to their great electron affinity, exert upon one another a slight additional attractive action. Moreover, there arises here an increase, not provided for above, of the angle between the valence strokes (Spreizung).

TABLE 7

Chlorinated methane*

If we now turn to various tetrachlorides, in which the structure of the molecule remains unchanged (Table 8), then here we shall see a remarkable constancy of the quantity \(k\). Although the value of this constant here is greater than for the monochlorides and considerably higher than for methane, the influence of the central atom and of the distance between the nuclei is evidently very small. These examples should show us with sufficient conviction that, in the “rigidity” (Starrheit) of the chemical valence bond thus defined, we have indeed found a good measure of the character of the bond, and that this measure to a considerable degree possesses additive properties, i.e., in practice preserves its value also in more complex compounds. The results thus obtained allow us, from the distance between nuclei and from the rigidity of the valence bond, to determine with sufficient reliability a large number of natural frequencies of a polyatomic molecule.

1. In order to express the vibrational character of a natural frequency, more than a year ago I proposed introducing a division into so-called valence vibrations and deformation vibrations \(^{10}\). Below I shall briefly show that this division always

* Raman and interferometric measurements.

TABLE 8*

Tetrachlorides (inactive vibration)

possible. We begin first with the simplest conception of the valence bond as a thin elastic rod, without losing sight of the existence of the binding electron shells, which in reality determine the relations of the atoms in the molecule. Such a rod can above all be stretched, and we shall try to express this elastic extensibility through our bond constant \(k\). But, in addition, the rod is subject to bending, and its resistance to bending, generally speaking, must be expressed through two bending constants: the constant \(b\), corresponding to a bending force acting in the direction of the least stress (Zwang), and the constant \(b'\)—in the direction of the greatest stress. This simply means that the vibration in bending of the rod in the general case represents a Lissajous figure, which can always be resolved into two mutually perpendicular natural frequencies—in the direction of the greatest and the least stress. If we now abstract from cyclic bonds (which present no difficulty in principle), then we immediately obtain that such a “rod molecule” (“Stabmolekül”) with \(n\) atoms must have \(n-1\) “rods” and therefore also \(n-1\) vibrations caused by stretching (valence frequencies) and \(2n-5\) vibrations caused by bending (deformation frequencies). It can further be seen how the appearance of each new valence stroke in the molecule, i.e. the addition of each new atom, creates 3 new frequencies, namely one valence vibration and two deformation vibrations.

Up to now we have been dealing with the simplest mechanical conception of the molecule.** But I should like to warn against the desire

* Raman and interferometric measurements.

** An analogous conception also underlies the work of Andrews,¹¹ who constructs, on the basis of the same considerations, models of molecules. But he does not make a strict distinction between valence and deformation vibrations. The essential point in my considerations—the expression of this distinction in vibration schemes and in the calculation—is absent in Andrews. In addition, it must be noted, in order to avoid misunderstandings, that his bond constant \(k\) is not identical with mine. In Andrews it has the usual dimension of force, dyn/cm, and in no way characterizes the type of bond, since here the distance between nuclei is not taken into account. See also K. W. F. Kohlrausch,¹² who gives yet another definition of bond strength.

specialize this representation by further deepening it in detail. The calculation of natural frequencies is carried out by the well-known method of small amplitudes of vibration, by establishing the corresponding normal coordinates; in doing so, one must by no means forget that here we are dealing only with an approximate method. Then, in accordance with the considerations set forth above, we compose a simple formula for the potential, containing \(3n - 3\) molecular constants:

\[ P=\sum_i^{n-1}\left[k_i\left(\frac{\Delta r}{r}\right)^2+b_i\Delta\varphi^2+b'_i\Delta\psi^2\right]. \tag{3} \]

But since in vibration neither the translational nor the rotational resultant momenta change, the number of independent constants is reduced to \(3n-6\), and the solution gives \(3n-6\) natural frequencies. It goes without saying that, in the general case, they are functions of all the masses present in the molecule, the distances between nuclei, and the quantities \(k_i\) and \(b_i\) (\(b'_i\)):

\[ \omega_k=F(m_i,r_i,k_i,b_i,b'_i). \]

But experience has shown that in practice the strength of a bond in stretching (rigidity) is approximately ten times greater than its strength in bending. We directly determine the possibility of the existence and the essence of a valence stroke by the feature that the ratio of the axes of its quasi-elastic stress ellipsoid is \(1:10\). If this relation does not hold, then the possibility of drawing valence strokes between two atoms is certainly excluded. On the other hand, the determination of the values of \(k\) and \(b\) has often given us a spectroscopic criterion for establishing the correctness of a chemical structural formula. In this respect molecular spectroscopy acquires exceptional interest for stereochemistry and, in particular, for the stereochemistry of cyclic bonds (the benzene ring).

On the basis of these assumptions, we can always expand the solution for the natural frequencies into well-convergent series in powers of \(\frac{b_i}{k_i}\). Therefore, in most cases we remain within the limits of small amplitudes of vibration if we neglect the first term \(\frac{b}{k}\), i.e. in this first approximation we obtain a certain number of natural frequencies, namely always \(n-1\), which are functions only of \(k_i\), and \(2n-5\) frequencies, which in the first approximation depend only on \(b_i\) (\(b'_i\)). Therefore the former are called, according to the definition, valence vibrations, and the latter—deformation vibrations. This approximate method considerably simplifies calculations, which were formerly very complicated; moreover, it makes possible a simple, graphical interpretation of valence and deformation vibrations. Expansion into series and their truncation already at the first term means only that,

that, for atoms in terminal positions, having only one valence bond, the vibrational motions of the valence frequencies occur in the direction of the valence stroke, while the vibrations of the deformation frequencies occur in the direction perpendicular to it. This reasoning is, of course, not applicable to central atoms. But if the molecule (radical) possesses an axis of symmetry, as, for example, when several different atoms are bound to the central one ($\mathrm{NH}_3$, $\mathrm{CH}_3$, $\mathrm{OH}_2$), then we may speak of $\pi$- or $\sigma$-vibrations, depending on whether the central atom vibrates parallel or perpendicular to the axis of symmetry. In other cases of symmetry we shall have to distinguish between symmetric and antisymmetric vibrations. Therefore, I should like to propose the following mode of notation: if one abandons the specification of the natural frequency itself, it may be denoted by $\omega_i$, in accordance with the terminology adopted for diatomic molecules. But when subdividing into valence and deformation vibrations, as was shown above, one should choose the designations $\nu_i$ and $\delta_i$, which are then further subdivided into $\pi$- and $\sigma$- (respectively $s$- and $a$-) vibrations. In most cases such a system of notation is quite sufficient.

Further, a substantial simplification of the calculation of natural frequencies can be achieved if one does not seek the complete solution and determines only the most interesting natural frequencies by combining atoms into groups (radicals). In doing so one can make wide use of the advantage of large differences in masses. Thus, for example, the radical —($\mathrm{CH}_n$) in $X—C$ vibrations can always be regarded as a whole, since the joint vibration of the C and H atoms has no appreciable influence on the frequency of the $X—C$ vibration. On the other hand, in the so-called $C—H$ vibration, for example in chloroform $\mathrm{HCCl}_3$, the group ($\mathrm{Cl}_3\mathrm{C}$) may be regarded as a whole, since the C atom takes only a slight part in the vibration, and the Cl atoms practically do not participate in it at all. In this way we easily arrive at a “diatomic” molecule (Tables 9—11). But such molecules, in contrast to “true” diatomic molecules, also possess deformation vibrations. It is characteristic that these bending vibrations degenerate into a double vibration in the case when the axis of symmetry of the molecule coincides with the valence stroke; in a molecule that does not possess symmetry, however, they split into two natural frequencies ($\mathrm{H}_2\mathrm{CO}$, $\mathrm{C}_6\mathrm{H}_5\mathrm{CH}_3$). Table 10 gives several typical examples, and the valence frequencies are also given. The values of $k$ and $b$ determined from them are, of course, only approximate, since other values of $k$ and $b$ also exert some influence on the frequencies. In Tables 5, 9, and 11 there are other examples of valence vibrations of diatomic and triatomic molecules (CN compounds).

In exactly the same way, several identical atoms bound to a central one can be combined into a single group (radicals).

TABLE 9

Stretching vibrations of “diatomic” halogen derivatives
(Raman lines)

TABLE 10

Stretching and deformation vibrations of “diatomic” molecules

TABLE 11

“Triatomic” cyanogen compounds

Let us denote the angle of such an “\(n\)-fold split” valence with the axis of symmetry by the letter \(\alpha\); a molecule of the type \(Z_m—X—Y_n\), with its two \(\sqrt{k}\)-frequencies, vibrates like the linear molecule \(Z—X—Y\); only instead of \(k_z\) and \(k_y\), there are the bond constant \(k_y n \cos^2\alpha\) and the apparent mass \(Y n \cos^2\alpha\). The same applies to \(Z_m\). Then both frequencies are calculated by a formula having the somewhat complicated form:

\[ (2\pi \nu_\mu)^2 = \left(\frac{k}{\mu r^2}\right)_z + \left(\frac{k}{\mu r^2}\right)_y \pm \sqrt{ \left[ \left(\frac{k}{\mu r^2}\right)_z - \left(\frac{k}{\mu r^2}\right)_y \right]^2 + \frac{4k_z k_y \cos^2\gamma_z \cos^2\alpha_y mn}{r_z^2 r_y^2 X^2} }, \tag{4} \]

where both reduced masses \(\mu_z\) and \(\mu_y\) are determined by the formulas:

\[ \frac{1}{\mu_z}=\frac{1}{Z}+\frac{m\cos^2\alpha_z}{X} \quad \text{and} \quad \frac{1}{\mu_y}=\frac{1}{Y}+\frac{n\cos^2\alpha_y}{X}. \tag{5} \]

An analogous formula can be written for the corresponding deformational vibration of this “triatomic molecule”; only, instead of \(\cos^2\alpha\), \(\sin^2\alpha\) will now enter.

In this way it is easy to identify the natural frequencies of various halogen derivatives of methane, which are given in Table 12 in agreement with investigations of infrared and Raman spectra. From these data it would not be difficult to estimate the missing quantities for iodine compounds.

TABLE 12

“Triatomic” halogen derivatives.
Valence vibration \(C—H_y\)

Valence vibration \(X_y—C\)

Deformational vibration \(X—C—H\)

In an analogous manner one may also calculate the \(\nu(\sigma)\)-vibration of the atomic group by the “bending” (“Knickung”) of a linear triatomic molecule. For example, for such molecules as chloroform

and methyl chloride \((Z—X—Y_3)\), we thus obtain one valence vibration (a double vibration):

\[ 2\pi\nu_s=\sqrt{\frac{2k_y}{r_y^2}\left[\frac{1}{Y}+\frac{{}^{3}/_{2}\sin^2\alpha}{X}\right]}, \tag{6} \]

and for such molecules as methylene chlorides \((Z_2—X—Y_2)\), both \(\nu(\sigma)\)-frequencies:

\[ \begin{aligned} 2\pi\nu_s&=\sqrt{\frac{2k_z}{r_z^2}\left(\frac{1}{Z}+\frac{2\sin^2\alpha_z}{X}\right)},\\ 2\pi\nu_s&=\sqrt{\frac{2k_y}{r_y^2}\left(\frac{1}{Y}+\frac{2\sin^2\alpha_y}{X}\right)}. \end{aligned} \tag{7} \]

Since in all these \(\nu(\sigma)\)-vibrations the other valence bond is affected only “in bending,” then, in the approximation chosen here, its bond constant and the mass of the outer atom do not enter into the formula; thus the frequency is almost independent of these bonds. Table 13 shows several examples where, despite a considerable difference in the mass and character of the bond (Tables 6 and 7), the \(\nu(\sigma)\)-frequency indeed remains almost constant. Thus for radicals such as, for example, \(-\mathrm{CH}_2\), \(-\mathrm{CH}_3\), \(-\mathrm{NH}_2\), \(-\mathrm{NO}_2\), it is completely immaterial to which residue of the molecule they are attached.

TABLE 13

Constancy of the \(\nu(\sigma)\)-vibration

The same can be said with respect to the deformation frequency. Tables 14 and 15 give several examples of such “radical vibrations”; Table 16 gives the results of a complete analysis of the halogen derivatives of methane, carried out from the point of view indicated above with the aid of measurements in the infrared region* and Raman spectra.

* The designations \((s)\) and \((a)\) refer here only to the symmetry of the direction of the vibration. Symmetry of vibrations in the usual sense, of course, appears only in a symmetric molecule.

TABLE 14

Bent molecule \(X\begin{matrix}H\\[-2pt]H\end{matrix}\)

TABLE 15

Bent molecule \((H_2C)\begin{matrix}X\\[-2pt]X\end{matrix}\)

TABLE 16

Molecular quantities for halogen derivatives of methane

1. These examples should suffice to show how, by combining the corresponding atoms into groups of “branching” and “bending” (“Winkelung”) of valences, one can arrive at quite simple types of molecules embracing a large number of compounds. It is clear that in this way we do not obtain a complete solution of the problem of the characteristic frequencies, but nevertheless this method makes it possible to calculate the most important frequencies. Table 17 gives the principal types of molecules, and Figs. 7 and 8 give diagrams of their vibrations. I have already spoken about types 1—3,

TABLE 17

Models of Molecules

1. Type \(X - Y - C - H\).

\[ \nu;\qquad \delta_1,\ \delta \]

1. Type \(Z - X - Y\) (extended), HCN.

\[ \nu(s),\qquad \nu(a),\qquad \delta \]

1. Type \(Y - X - Y\) (bent), \(H_2O\).

\[ \nu(\pi),\qquad \nu(\sigma),\qquad \delta(\pi) \]

1. Type \(Y - X - X - Y\) (extended), \(C_2H_2\).

\[ \nu_1(s),\qquad \nu_2(s),\qquad \nu(a),\qquad \delta(s),\qquad \delta(a) \]

1. Type \(Z - X\begin{matrix} /Y \\ \backslash Y \end{matrix}\) (planar), \(H_2CO\).

\[ \nu(\pi,s),\qquad \nu(\pi,a),\qquad \nu(\sigma),\qquad \delta(\pi),\qquad \delta_1(\sigma),\qquad \delta(\sigma) \]

Fig. 7. Types of vibrations.

1. Type \(X\begin{matrix} /Y \\ -Y \\ \backslash Y \end{matrix}\) (pyramid), \(NH_3\).

\[ \nu(\pi),\qquad \nu(\sigma),\qquad \delta(\pi),\qquad \delta(\sigma) \]

1. Type \(\begin{matrix}Y\\Y\end{matrix} - X = Y_2\) (tetrahedron), \(CH_4\).

\[ \nu(s),\qquad \nu(a),\qquad \delta(s),\qquad \delta(a) \]

Fig. 8. Types of vibrations of \(XY_3\) and \(XY_4\).

1. A “diatomic” molecule \(X—Y\) has one valence vibration and two deformation vibrations perpendicular to it, which may degenerate into a double vibration.

2. A linear molecule \(Z—X—Y\) (of the HCN type) has two valence vibrations; in one of them the directions of motion of the two outer atoms are mirror images of one another (symmetric vibrations), in the other they are antisymmetric. Since in this case we cannot speak of \(\sigma\)- and \(\pi\)-vibrations, let us introduce for them the designations \((s)\) and \((a)\). Thus \((s)\) and \((a)\) always refer to the character of the vibration of the outer atoms, while \(\pi\) and \(\sigma\) refer to the central atom. In the case of equal outer atoms the symmetric frequency (active in the Raman spectrum and inactive in the infrared spectrum) is the smaller of the two frequencies. The deformation vibration is always a degenerate double vibration and is always active in the infrared spectrum.

3. In bent, symmetric molecules of the type \(Y—X—Y\) (water type) all three frequencies \(\nu(\pi)\), \(\nu(\sigma)\), and \(\delta(\pi)\) are active in the infrared spectrum and, probably, also in the Raman spectrum. The two valence vibrations are almost equal when the valences are mutually perpendicular, or when there is a large difference in the masses \((X \gg Y)\)*.

4. The next very important type—the acetylene type \(Y—X—X—Y\)—includes many compounds. I shall confine myself to considering the molecules \(C_2H_n\) (Table 18)**. First of all, the benzene ring belongs here, as the molecule \(C—(C_2)—(C_2)—C\). Both \(\nu(s)\)-frequencies are inactive in the infrared spectrum and active in the Raman spectrum, and so is \(\delta(s)\). In hydrocarbons \((Y \ll X)\), \(\nu_2(s)\) can be interpreted as the \(C—C\) vibration, and \(\nu_1(s)\) and \(\nu(a)\) as the \(C—H\) vibration. Both deformation vibrations are degenerate double vibrations, so that only five frequencies can be observed instead of the seven expected.

5. Next comes the formaldehyde type \((Z—X—Y_2)\). Here all the proper frequencies are active and nondegenerate (Table 19) and can easily be calculated according to types 2 and 3 \((Y—X—Y)\) by “splitting the valence and its bending.” Degeneracy appears only

* Approximate formulas obtained by “splitting the valence” \(X—(Y_2)\) and by “bending the valence” (Valenzwinkelung) of a linear molecule of the type \(Y—X—Y\) have the following form:

\[ 2\pi\nu_\pi=\sqrt{\frac{2k}{r^2}\left[\frac{1}{Y}+\frac{2\cos^2\alpha}{X}\right]} \]

and

\[ 2\pi\nu_\sigma=\sqrt{\frac{2k}{r^2}\left[\frac{1}{Y}+\frac{2\sin^2\alpha}{X}\right]}. \]

** The formula for the valence vibrations was given in my article \(^{13}\). I draw the reader’s attention to the fact that Table 18, compiled on the basis of new investigations, deviates in some places from the data of my earlier article; therefore, above all, some objections that had been raised on the basis of the symmetry properties of molecules against my previous interpretation no longer apply.

TABLE 18

Natural frequencies of the molecules \(C_2H_n\)*

TABLE 19

Natural frequencies of the molecules \(ZXY_2\)

in a planar molecule \(XY_3\) (for example, in the \(CO_3\) group, according to Clemens-Schäfer). Here \(\nu(\pi,a)=\nu(\sigma)\) and \(\delta(\pi)=\delta(\sigma)\); moreover, \(\nu(\pi,s)\) is not active in the infrared spectrum.

For spatial molecules \(XY_3\) and \(XY_4\) (Fig. 8) the graphical representation is somewhat more complicated; we made use of the projection of the vibration onto a plane passing through the axis of symmetry.

1. In the pyramidal model (ammonia type) we have two simple vibrations \(\nu(\pi)\) and \(\delta(\pi)\), the group \(Y_3\) vibrating as a whole: either all the atoms vibrate in the direction of the valence bond \(\nu(\pi)\), or in a direction perpendicular to the latter. These vibrations can also easily be derived from the type \(X—Y\). Next two double vibrations appear; here the states of motion of the three \(Y\) atoms as oscillators are different: one motion is again performed in the direction of the valence \(\nu(\sigma)\), the other in the perpendicular direction—\(\delta(\pi)\)—and always in such a way that the central atom \(X\) vibrates perpendicular to the axis of symmetry. All four natural frequencies are active (Table 20).

2. The tetrahedral molecule \(XY_4\) (methane type) is best represented in the form of the molecule \(Y_2=X-Y_2\), with both planes determined by each group \(Y>X\) being mutually perpendicular. This model (Table 21) has, first of all, a symmetric

* Frequently, in \(C_2H_4\) there are also encountered the \(-CH_2\)-frequencies \(\nu(\sigma,a)=2988\), \(\nu(\sigma,s)=2880\), \(\delta(\pi,s)=1342\), and \(\delta(\pi,a)=1444\), corresponding to the vibration scheme \(\nu_4\), \(\nu_5\), \(\delta_4\), and \(\delta_5\) in the cited work.

TABLE 20

Natural frequencies of \(XY_3\) molecules

TABLE 21

Natural frequencies of tetrahedral molecules

a simple vibration \(\nu(s)\), in which both outer atoms have the same state of motion, while the inner atom is at complete rest; thus the molecule pulsates, and therefore it is not active in the infrared spectrum and is active in the Raman spectrum. Further, there is an asymmetric, triply degenerate \(\nu(a)\)-vibration, which is active and in which the central atom moves over the surface of a sphere. Both frequencies can easily be calculated from the symmetric type of molecules \(Y—X—Y\). Finally, there is also a doubly degenerate active \(\delta(a)\) deformation vibration and a triply degenerate vibration \(\delta(s)\), not active in the infrared spectrum. Owing to the spherical symmetry of this type, one cannot speak here of \(\pi\)- and \(\sigma\)-frequencies. However, this does not apply to molecules of the type \(Z—X—Y_3\) (Table 16) and \(Z_3—X—Y_2\), where nine different natural frequencies appear instead of four, six, and eight, respectively. These types have already been discussed by us.

It goes without saying that, besides these seven models of molecules, there are also others that are likewise of interest; thus, for example, the alcohol type \(R—O—H\), corresponding to the ne—

in a symmetrically bent molecule and the acid type \(R—O—OH\). The computational treatment here is somewhat more difficult. Chain molecules as well, which ultimately lead, through the higher paraffins, sugars, proteins, and rubber, to crystals, promise interesting particulars. Cyclic bonds also require more thorough study. Thus, for example, a triple ring has three valence vibrations, but has no deformation vibrations. Of special interest, naturally, is a plane ring of six nuclei (the benzene nucleus). If here there is symmetry in all six atoms, then 12 vibrations can easily be derived in a first approximation from the linear scheme \(C—C_2—C_2—C\) and the triatomic \(C_2—C_2—C_2\), in which the bond constant must be chosen in accordance with the valence splitting. I shall confine myself to these simple examples, which should sufficiently demonstrate the “principle of construction.”

It is, however, advisable to indicate the limits of applicability of the methods briefly set forth here and to warn against excessive expectations, which I have sometimes had occasion to encounter. One must always bear in mind that here we are dealing only with approximate methods, which should be sufficiently accurate only in order to construct a scheme of vibration of the observed frequency. For this purpose it was first of all necessary to investigate, on the basis of the available experimental material, the bond strength of a di- or polyatomic molecule, and then, with the aid of simple mechanical notions, to create a method of calculation making it possible to estimate frequencies. But there can be no question here of a detailed calculation of the spectrum of frequencies belonging to a given chemical structural formula—in the sense in which Andrews attempts to carry this out with his beautiful mechanical models of molecules, which may perhaps facilitate calculation in the future. The calculation method for this is not sufficiently accurate; moreover, we cannot, with due completeness—at least at the present time—become acquainted with the deforming forces determining the structure of the molecule. In conclusion I should like to explain this by a particularly graphic example of the tetrahedral model of carbon tetrachloride, in which four heavy, electron-rich external atoms are bound to a light central atom. Calculation of the bond constant \(k\) from the symmetric valence vibration by the simple formula

\[ 2\pi \nu_s = \sqrt{\frac{2k}{r}\frac{1}{\mu}} \]

gives \(45\,V\)—a value which, as was shown in Table 8, is practically the same for all tetrachlorides, since precisely in this symmetric pulsating vibration the tetrahedral structure of the molecule is fully preserved. But if we consider the asymmetric vibration \(\nu(a)\), with which a considerable change of structure is associated, a change that is already clearly expressed in methane

in the photographed band \(\lambda = 8860\) Å, then from the corresponding formula we obtain

\[ 2\pi \nu_0 = \sqrt{\frac{2k}{r^2}\left[\frac{1}{Y}+\frac{4}{3X}\right]} \]

a value of only \(27\ \mathrm{V}\), which is practically equal to the corresponding value for the monochloride (\(31\ \mathrm{V}\)), and undoubtedly corresponds to a single bond (Tables 5 and 6). But on the basis of Debye’s interferometric investigations we know that, when a chlorine atom is replaced by hydrogen, an increase (Spreizen) of the tetrahedral angle takes place. Consequently, the molecule is readily subject to deformation under the appearance of asymmetries caused here by vibration (Table 7). Almost analogous, though not so striking, is the case of carbonic acid \(O=C=O\). The asymmetric vibration \(\nu(a)\) (2350) gives a bond constant of \(57\ \mathrm{V}\), characteristic of a double carbonyl bond \((=C=O)\) (Table 5); the symmetric frequency, which does not change the structure of the molecule, \(\nu(s)\) (1330), again has the large value \(67\ \mathrm{V}\) (which, however, does not reach the value for a triple bond—\(77\ \mathrm{V}\)). We thus have, in both cases, clear indications of the rigidity of the structure of the molecules (Table 2), and the task for the future will be to advance further the approximation of the calculation method and to obtain the possibility of drawing conclusions, on the basis of the observed frequencies, about the most subtle details of the structure of the molecule. For this, however, it is absolutely necessary that the frequency spectrum of molecules be analyzed and correctly interpreted with respect to the nature of the bonds and vibrations. My report has sought to show here the path leading to this goal.

If one takes into account the difficulties indicated at the beginning which we encounter, it is easy to see how much clarity must first be introduced into the various molecular data of the ground electronic state by means of the combined study of rotational-vibrational bands in the infrared region and Raman spectra, with the simultaneous use of interferometric methods of investigation. On this basis we can proceed to the study of various excited electronic states. Investigations promising success lie ahead of us. If, from this point of view, one looks at the course of development of the spectral analysis of diatomic molecules over the last eleven years, during which I have had the opportunity to work on these questions, then I think one can say with confidence that the analysis of polyatomic molecules will in the near future advance as far as the analysis of diatomic molecules has at the present time.

Literature

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1. Debye. ↩

2. Ammonia molecule. ↩

3. Barker and Stinchcomb. ↩