Abstract
A brief review is given of the current state of the spectroscopy of band spectra, with primary attention devoted to issues of interest from the chemical point of view, such as spectroscopic determination of dissociation energy, isotopy, valence chemistry, chemical constants, etc.; questions of a purely theoretical or experimental-technical nature, such as the conditions for excitation of spectra or their changes under the influence of external effects, are either not considered at all or are considered only very briefly.
Full Text
BAND SPECTRA AND THEIR SIGNIFICANCE FOR CHEMISTRY.¹
R. Mecke, Bonn.
Introduction.
A line spectrum is emitted by an atom; a band spectrum—by a molecule. This fact divides the totality of spectra into two sharply delimited classes which, as their very names indicate, differ characteristically from one another already in outward appearance. On the one hand, in the case of line spectra, we have a certain number of lines, apparently arranged at random, which with greater or lesser success are grouped together into groups (so-called multiplets), and these in turn into series, and in this way the electronic structure of the atom is investigated. On the other hand, in the case of band spectra, we have a dense accumulation of lines in definite regions of the spectrum, and the regular arrangement of these lines into bands and the regular succession of the entire system of bands immediately catches the eye, so that the search for groups and series—apart from a few exceptions—is not a difficult task. But precisely because of this abundance of lines—one usually has to deal with spectra consisting of several thousand lines—it was initially difficult to extract from the enormous numerical material supplied by band spectra
¹ Fortschritte der Chemie, Physik und Physikalischen Chemie. Vol. 20, No. 3. 1929.
information necessary for recognizing the structure of molecules. However, the time when, in the study of atomic structure, line spectra were given preferential interest over band spectra has already passed; now both classes of spectra render equally valuable services to chemistry.
In what follows, a brief survey is given of the present state of the spectroscopy of band spectra, with primary attention devoted to questions of interest from the chemical point of view, such as: the spectroscopic determination of dissociation energies, isotopy, the chemistry of valence, chemical constants, etc.; questions of a purely theoretical or experimental-technical character—such as, for example, the conditions for exciting spectra or their alteration under the influence of external actions—are either not considered at all or are treated only very briefly.
After several remarks concerning the structure and theories of band spectra, we pass to a brief survey of the most important among the known and more or less thoroughly studied band spectra from the point of view of their relation to the periodic system, and, finally, in special chapters individual questions of interest from the chemical point of view are considered in detail.
I. Theory of Band Spectra.
The aim of the investigation of atomic spectra is, on the basis of the study of series of spectral lines, to draw conclusions concerning the mutual arrangement and motion of atomic electrons. The nucleus of the atom is here regarded as at rest; its motions in this case do not interest us. In the case of molecules, which are built of electrons and several nuclei, we take into consideration, in addition, the mutual arrangement of the nuclei and their motions. Corresponding to the greater number of initial elements, we are dealing here with a greater number of degrees of freedom of motion. If we dwell, for example, on the simplest case of a diatomic molecule, with which, in what follows, for quite understandable reasons, we shall chiefly be concerned,
and if we confine ourselves to these, then to the motions of the electrons there are added two further possibilities for the motion of both nuclei: 1) the nuclei may rotate about their common center of gravity; 2) the nuclei may perform oscillations about the position of equilibrium in the direction of the axis joining them, i.e. they are not connected with one another in an absolutely unchanging manner.
At the basis of the whole theory of spectral analysis lies the well-known frequency condition of Sopa \(h\nu = W' - W''\), which represents the frequency of each line as the difference of two energies \(W'\) and \(W''\), or, using the terminology of spectroscopy, as the difference of two terms. Owing to such a close connection between the wavelength of a line and the energy of the atom and molecule, the whole theory of spectra reduces to the calculation of the energy of such motions, carried out from the standpoint of the quantum theory. In the case of molecules, in accordance with the three kinds of motion mentioned, it is natural to divide the whole energy of the molecule into three components: 1) rotational energy, 2) energy of the vibrations of the nuclei, and 3) electronic energy. We shall consider each of these components separately.
- We consider first of all the energy of rotations and take the simplest case of two nuclei, which we shall at first, moreover, suppose to be connected with one another in an unchanging way: the model of a diatomic molecule, resembling a bar for gymnastic exercises. Bohr’s quantum theory requires that the angular momentum,—i.e. the quantity equal to the moment of inertia \(\times\) the angular velocity \(\omega\),—taken on the average over one full revolution of the periodic motion, be equal to an integral multiple \(m\) of the quantity \(\frac{h}{2\pi}\) (Planck’s quantum of action). Since, however, the angular velocity of rotation is constant, we immediately obtain \(J\omega = \frac{mh}{2\pi}\); on the other hand, the energy of such rotational motion is expressed by the formula:
\[ W_r = \frac{1}{2}J\omega^2, \]
whence, excluding \(\omega\), which is of no interest to us in the present
in the case of angular velocity \(\omega^2\), we obtain for the energy the expression:
\[ W_r=\frac{m^2h^2}{8\pi^2J} \tag{1} \]
The calculation of the moment of inertia \(J\) is carried out on the basis of the law of motion of the center of gravity \(m_1 r_1=m_2 r_2\) (\(r_1\) and \(r_2\) are the distances of the masses of the nuclei \(m_1\) and \(m_2\) from the center of gravity, \(r=r_1+r_2\) is the distance between the nuclei). In fact,
\[ J=\mu r^2=m_1r_1^2+m_2r_2^2, \]
where, as a simple calculation shows, one should take for \(\mu\) the so-called “reduced mass”:
\[ \mu=\frac{m_1m_2}{m_1+m_2}, \]
(or \(\frac{1}{\mu}=\frac{1}{m_1}+\frac{1}{m_2}\)). If, furthermore, energy is measured not in ergs but spectroscopically, on the basis of Bohr’s condition, i.e. in units of frequency, then equation (1) must also be divided by \(hc,\)^1 and then for the rotational energy one obtains the quantity:
\[ W_r=\frac{h}{8\pi^2cJ}m^2=Bm^2, \]
in which \(B=\frac{h}{8\pi^2cJ}\) has the value \(27.7\cdot10^{-40}\). The new quantum theory of Heisenberg and Schrödinger gives for the energy a somewhat modified expression
\[ W_r=Bm(m+1), \tag{2} \]
in which, however, the value of the constant \(B\) remains the same. If to the latter expression one adds the quantity \(\frac{B}{4}\), which is not detected spectroscopically, then formula (2) may be written in the form \(W_r=B\left(m+\frac{1}{2}\right)^2\), i.e. the new quantum theory immediately leads to so
^1 In spectroscopy frequencies are always expressed in units reciprocal to wavelength (\(\mathrm{cm}^{-1}\)); therefore energy expressed in ergs must be divided by \(hc\), where \(c=\) the velocity of light \(\left(E=h\nu=hc\cdot\frac{1}{\lambda}\right)\).
often encountered “half-integer” quantum numbers \(m^* = m + \frac{1}{2}\), which were little understood in the old quantum theory.
We shall take a further step and now consider a polyatomic, though initially likewise unchanging, molecule. Such a molecule possesses, generally speaking, three different principal moments of inertia, i.e. it can rotate about three mutually perpendicular principal axes of inertia. In the case when two of these moments of inertia are equal to one another, \(J_1 = J_3\), quantum theory makes it possible to calculate the energy, and one obtains an expression in which two quantum numbers \(m\) and \(p\) already appear \((J_2 < J_1)\):
\[ W_r=\frac{h}{8\pi^2 c}\left[\frac{1}{J_1}m(m+1)+\left(\frac{1}{J_2}-\frac{1}{J_1}\right)p^2\right]; \tag{3} \]
we shall not give the derivation of this formula here.
2. We shall now remove the restrictive condition of the invariability of the molecule. Let the nuclei be able to perform oscillations about the equilibrium position, but initially we shall assume that they do not rotate. We shall consider here again the simplest case, namely, let us assume that between the two nuclei there acts a purely elastic force, i.e. a force proportional to the distance \(x\) of the nuclei from their positions of rest: \(K = 2kx\); this corresponds to the potential energy \(P = -kx^2\). Taking into account the quantum rules, we express the energy of the oscillations by the formula \(W = nhc\nu\), where \(\nu\) is the natural frequency of these oscillations (again measured in units of reciprocal wavelengths). In this case, consequently, the energy increases in proportion to the quantum number, and therefore the amplitude increases in proportion to \(\sqrt{n}\). This expression in the new quantum theory has also undergone a slight change, namely, instead of the “integer” quantum number there is again introduced the “half-integer” \(n+\frac{1}{2}\).
It is clear, however, that the force law written above is excessively simplified. Therefore, as a further approximation to reality, for the potential
energy, an expression of several terms depending on \(x\):
\[ P=-K_1x^2+K_2x^3+K_3x^4+\ldots . \]
Then the expression for the energy may be represented in the form of an analogous series, arranged according to powers of the quantum number,
\[ W_s=a_1n+a_2n^2+a_3n^3+\ldots . \]
Practice shows, however, that in many cases the series can already be cut off at the second term, so that one obtains a formula containing only two constants, \(W_s=an-bn^2\); later we shall encounter this formula quite often. But even this expanded law for the potential assumes that the atoms are bound to one another by elastic forces. In reality this, of course, is not the case, for the bond in a molecule is maintained by electrons, which, by their electric charges, act upon one another with repulsive and attractive forces. For this reason Kratzer1 in his time, starting from the assumption that in the first approximation Coulomb’s law is valid, chose as a further approximation a series arranged in inverse powers of the distance between the nuclei, and put
\[ P=-\left[\frac{e^2}{r}+\frac{c_1}{r^2}+\frac{c_2}{r^3}+\ldots\right] \tag{4} \]
If this series is cut off in good time, then one again obtains the expression already familiar to us for the vibrational energy, \(W_s=an-bn^2\). These premises are, for the case of the so-called polar bond, undoubtedly quite acceptable, since in this case we have a positive and a negative ion which, of course, since their electron shells do not yet deform one another, are attracted according to Coulomb’s law. In the majority of cases, however, there is a “non-polar bond” of neutral atoms, and here the circumstances may be different. In order to obtain, at first, a qualitative picture of the dependence of the potential energy
dependence of the bonding forces on the distance between the atoms, consider Fig. 1, which gives a graphical representation of this dependence. At a large distance between the nuclei, as the atoms approach one another, the potential initially decreases, since attractive forces predominate; after passing through a minimum, however, it increases very rapidly, since now the electron shells of both atoms begin to act upon one another with very rapidly increasing repulsive forces. The abscissa of the potential minimum corresponds to the position of equilibrium in which the atoms establish themselves, for here the repulsive and attractive forces exactly compensate one another; thereby the moment of inertia of the molecule is also determined. The ordinate of the minimum likewise has an intuitive significance: it represents the work that must be expended in order once again to separate the atoms, i.e. the dissociation energy of the molecule \(D_0\), of which we shall speak later in greater detail.
Fig. 1.
This general course of the potential curve can be represented by an extremely simple expression; namely, the potential is decomposed into two parts, of which one represents the potential of the attractive forces (the lower dashed curve in Fig. 1), the other—the potential of the repulsive forces (the upper curve); the difference of these two parts will then give the true potential. By testing individual examples I have shown that even the simple formula
\[ P=-e^2\left[\frac{c_1}{r^p}-\frac{c_2}{r^q}\right] \tag{5} \]
is suitable here.
In using this formula, besides \(r_0\) and \(D_0\), the exponents \(p\) and \(q\), which determine the decrease of the potential, also have a definite meaning from the physical point of view.
of attractive and repulsive forces with distance (the decrease of the forces themselves, in such a case, as is known, is determined by the exponents \(p+1\) and \(q+1\)); meanwhile in the preceding formulas the expansion constants have no special physical significance. Looking somewhat ahead, we shall point out that in the case of homopolar molecules the exponents \(p\) and \(q\) take values respectively from 3 to 4 and from 6 to 9; only the hydride compounds \(XH\) have proved to be quasi-polar, i.e. for them \(p=1,\ q=3\text{—}4\).
Of course, formula (5) also leads to the same value for the energy of nuclear vibrations as before:
\[ W_s=an-bn^2, \tag{6} \]
where, in particular, for the proper vibrations of the nuclei \(\nu=a\), the value obtained is
\[ 2n\nu=\sqrt{\frac{D_0\cdot p\cdot q}{J}} \tag{7} \]
and for \(b\):
\[ 2b=B\left[\frac{11}{12}p\cdot q+\frac{1}{6}(p-q)^2+p+q+1\right]. \tag{8} \]
Up to now we have assumed that the molecule does not rotate; if, however, rotation is also added to the vibrations of the nuclei, it turns out that, although in the first approximation both component parts of the energy (i.e. the vibrational energy and the rotational energy) are added additively, in the second approximation the mutual influence of vibrations and rotations becomes noticeable—an influence which is reflected also in the expression for the energy. Namely, the rotational energy, owing to the vibrations of the nuclei, is somewhat diminished; therefore one has to replace
\[ Bm^2\ [\text{or }Bm(m+1)] \]
by the expression
\[ B_n m^2(1-u^2m^2), \]
where
\[ u=\frac{2B_n}{a} \tag{9} \]
The distance between the nuclei, and consequently also the moment of inertia,
increases together with the increase in the amplitude of the oscillations, i.e., the formula holds
\[ B_n=B_0(1-\alpha n), \tag{10} \]
where \(\alpha\) has the value
\[ \alpha=\frac{2B}{a}(p+q). \tag{10a} \]
In polyatomic molecules, in accordance with the increased number of possibilities of motion, several oscillations of the nuclei exist simultaneously; for example, in a triatomic molecule there will already be three such oscillations. In the first approximation, here too the separate energies of these oscillations are added together; however, in the theoretical study of the interactions of the oscillations with one another, difficulties are encountered.
- With regard to the third kind of energy—electronic energy—we can report few quantitative results. As is known, in the case of atomic spectra the electronic energy can be represented by Rydberg formulas of the form \(W_e=-\frac{R}{n^2}\); among band spectra we know so far only two cases where this has proved possible, namely: in the electronic system of the helium molecule, which has been investigated spectroscopically very accurately, and in some series of the hydrogen molecule. In the case of this energy it is of interest to know not only its absolute magnitude, but also other data that are connected with the physicochemical properties of the molecule; we shall return to these later. Here, however, we shall simply add the electronic energy as an additive quantity \(W_e=\nu_0\) to the two other kinds of energy, so that for the total energy of the molecule we obtain the expression
\[ W=W_e+W_s+W_r=\nu_0+(an-bn^2)+ +B_n(m+1)\cdot[1-u^2m(m+1)]. \tag{11} \]
where the constants \(a\), \(b\), \(B_n\), and \(u\) are connected by equations (5)—(10) with molecular quantities. Regarding the magnitudes of the three energy terms it must be said that always
\[ W_e \gg W_s \gg W_r. \]
In order to pass from this energy of the molecule to the spectral line, it remains only to take into account Bohr’s frequency condition, i.e., the fact that each line can be represented as the difference of two similar energy terms. The emission of a spectral line therefore occurs in such a way that the molecule passes from a more energy-rich initial state, into which it has come by preliminary “excitation,” to a less energy-rich final state. In absorption, which proceeds with the acquisition of energy, the reverse picture naturally takes place: here the initial state is less energy-rich, and the final state is more energy-rich. It is customary to denote quantities that refer to the state richer in energy by letters with one prime, and the state poorer in energy by letters with two primes. Suppose that a graphical representation of the spectrum is known in the form of an energy diagram, in which the distances between energy levels represent the emitted frequencies (e.g., Fig. 5). Then the lowest state, the poorest in energy, which a molecule (or atom) can in general assume and from which in most cases the energies are reckoned, is called the ground or normal state. Since atoms and molecules, when not too strongly excited, i.e., at not too high temperatures, are in most cases in this ground state, it plays a major role in spectroscopic chemistry, to which we shall return.
Alongside Bohr’s frequency condition, the emission of a spectral line is regulated by another equally important law: the selection rule. Whereas the quantum number of the vibrations of the nuclei may change by arbitrary amounts, the quantum number of rotation is subject to a strict limitation, according to which it cannot change by more than one unit. Thus
\[ m'' = m' \]
or
\[ m'' = m' + 1 \]
or
\[ m'' = m' - 1 \tag{12} \]
We shall encounter this selection rule again when discussing the structure of an individual band.
2. The Structure of Banded Spectra.
In order to show the character of a banded spectrum, Fig. 2 gives several typical bands. This figure is an enlargement of photographs made with the highest dispersions presently available.
Fig. 2.
Fig. 3.
We see how from the “gap” in the band—the so-called zero line—regular sequences of lines, series, proceed to the right and to the left; in one of these series the distance between the lines constantly increases, in the other it decreases until the lines merge, forming the “edge” of the band so characteristic of banded spectra. Here the series “turns around” and again goes back (see Fig. 3, which gives a diagram of the CuH spectrum, pre-
placed above in Fig. 2). Depending on which side of the edge the lines lie on, i.e. whether they go toward longer or shorter waves, one says that the band is shaded toward the red or toward the violet side.
Each band spectrum always consists of several such bands (Fig. 4), the number of which, however, may vary very greatly: from a few bands (for example, in OH, CuH, CH, etc.) up to spectra consisting of more than a hundred bands. As an example of spectra of the latter type one may point to the spectra of iodine and of other halides. It is not difficult to establish that all these bands are due to
Fig. 4.
one and the same mechanism of vibrations: when the conditions of excitation are changed they all change in the same way, they all possess the same structure, are shaded to one and the same side, lie in a relatively narrow region of the spectrum, and all vanish together when one may suppose that their carriers have been destroyed. The aggregate of such bands therefore forms a single whole and is called a system of bands. In these bands one may also establish regular distributions—they often gather into groups of bands and form sequences of bands which, as we shall see, obey a law expressed by equation (6). Such, in general outline, is the characteristic picture of a band spectrum. Of course, deviations from this typical picture are encountered. Thus, for example, we know completely diffuse bands which do not allow a fine structure to be detected,
equally there occur bands in which the edges are completely blurred, and here we are dealing only with broad ribbons and very strong crowdings of lines. On the other hand, we again know band spectra in which the lines of the individual bands lie so far apart from one another that, in appearance, these spectra cannot be recognized as band spectra—they give the impression of line spectra. These are the so-called many-line spectra, of which the second spectrum of hydrogen is the characteristic representative. However, in all these cases, after detailed study, we discover a threefold variety. A set of lines gathers into bands; a certain number of the latter again forms a single whole—a system of bands; and finally we find that in one spectrum there may simultaneously be several such systems of bands, all of which belong to one and the same molecule. It therefore seems quite natural to connect this threefold variety with the division of the energy of the molecule into three parts, and experiment has fully confirmed this conclusion. The individual lines of a band represent different states of rotation of the molecule; the individual bands are caused by different vibrations of the nuclei; and the change in electronic energy determines the position of the system of bands in the spectrum.
If we first consider an individual band as a whole, then it is caused, consequently, by a change in the state of vibration of the nuclei and therefore—perhaps, according to equation (6)—can be fixed unambiguously by specifying the pair of numbers \(n', n''\). Therefore we can always regard each system of bands as a double sequence of bands and represent it in a planar scheme of edges (see, for example, “the edge scheme of the cyanogen spectrum” on the next page). This quantum scheme is in such a case arranged so that all bands that stand in one horizontal row possess one and the same initial state \((n')\), and all bands of a vertical row—one and the same final state \((n'')\). All these various longitudinal and transverse series, as they are sometimes called, can be embraced by a single formula:
\[ \nu = \nu_0 + (a'n' - b'n'^2) - (a''n'' - b''n''^2) \tag{18} \]
Quantum scheme of the cyanogen spectrum.
| \(n'\) | \(n'' = 0\) | 1 | 2 | 3 |
|---|---|---|---|---|
| 0 | \(\lambda\) 3884 \(\nu_0\) 25 797,8 (2042,3) (2123,5) |
4216 23 755,4 (2015,9) (2123,6) |
4606 21 739,5 (2123,5) |
|
| 1 | \(\lambda\) 3590 \(\nu_0\) 27 921,3 (2042,3) |
3872 25 879,0 (2016,0) (2063,7) |
4197 23 863,0 (1989,6) (2082,5) |
4578 21 873,4 2083,1 |
| 2 | \(\lambda\) \(\nu_0\) |
3586 27 962,7 (2017,2) |
3862 25 945,5 (1989,0) (2044,2) |
4181 23 956,5 (2043,7) |
| 3 | \(\lambda\) \(\nu_0\) |
3584 27 989,7 (1989,4) |
3855 26 000,3 |
There exist a number of criteria which make it possible to decide with certainty whether the chosen quantum scheme is correct. Thus, for example, the differences between the frequencies of two longitudinal series or of two transverse series must always be respectively equal to one another. As an example I give here the scheme of the cyanogen bands. The indicated criterion follows from the fact that, in forming such differences, in the first case the final term drops out, and in the second case the initial term drops out in equation (13). It is true that this relation of constant differences for the wavelengths of the edges is valid only approximately, since the edges are only the turning points of series of lines, and these turning points for different bands may lie at different places in the series. However, it is valid with complete rigor for the zero lines, i.e. for lines with rotation “0,” or for all other lines in individual bands to which one and the same rotational impulse \(m\) corresponds. Other criteria for the correctness of the distribution we shall report when discussing the structures of bands.
The next question, which often presents difficulties in establishing the scheme of bands, is the question of the absolute values of the vibrational quantum numbers. For ce-
of a whole series of band systems, and precisely in those cases where the bands already, in a purely external way, are grouped into characteristic groups (for example, in CN), one can always indicate the first band, which at the same time will also be the most intense.
To this band one always assigns the vibrational quantum numbers \(n', n'' = 0,0\).
In so-called many-quantum spectra (for example, in iodine), however, the maximum of intensity lies at intermediate
Fig. 5.
quantum numbers, whereas bands with small numbers do not appear at all; here the counting of the quantum numbers is often doubtful. Sometimes in such cases the simultaneous consideration of emission and absorption spectra helps, for in the former the sequence of bands \(n' = 0\) (the horizontal series) has the greatest intensity, while in the latter—the sequence of bands \(n'' = 0\) (the vertical series).
In order to explain this fact, let us consider the diagram of the energy of nuclear vibrations (Fig. 5).
Arrows pointing upward in such a diagram always denote absorption lines; arrows pointing downward—emission lines. Usually the majority of mole-
cules are in the very lowest, unexcited state \(n''=0\). If, however, the absorption of light supplies energy, then the molecules, depending on the absorbed wavelength, rise to various levels of the excited state. In this way one obtains a sequence of bands \(n''=0\),
\[ \nu=\nu_0+(a'n'-b'n'^2). \]
Other sequences of bands are thereby completely suppressed. The situation is different in emission. Here the molecule, as a result of a preliminary supply of energy—for example, by absorption of light—is already raised to a definite state of excitation of the electronic system, with preference given to the state \(n'=0\); in the emission of bands the molecule passes from the state of excitation into separate final states. Thus here one obtains a sequence of bands \(n'=0\) \((1,2\ldots)\),
\[ \nu=\nu_0-(a''n''-b''n''^2). \]
This fact is nothing other than the well-known Stokes rule in a somewhat unusual, more general formulation: absorption bands lie predominantly on the side of shorter wavelengths from the zero position \(n',n''=0\); emission bands (fluorescence bands) lie predominantly on the side of longer wavelengths from the zero position.
The region of wavelengths in which the system of bands lies is determined chiefly by the change in the electronic energy, whose share among the three terms of the molecule’s energy is the largest. If, for example, the electronic energy of the molecule does not change at all, then the entire system of bands naturally shifts into the infrared region, since the energy of vibrations and rotations of the nuclei is relatively small. We are then dealing with the so-called rotational and rotation-vibrational bands, which are observed in a whole series of cases (hydrogen-halogen compounds, CO, CO\(_2\), and most organic compounds). Owing to the difficulty of observation, discussion here can concern only absorption spectra.^1 They
^1 The prerequisite for the appearance of absorption is the presence of an electric moment, which either already exists in the molecule, or ...
are characterized by a series of nearly equally spaced bands, which obey the simple formula
\[ \nu = a(n' - n'') - b(n'^2 - n''^2) \tag{14} \]
Recently Raman1, by an elegant indirect method, succeeded in transferring the study of this very inconveniently located infrared spectrum into the region of short waves, which above all is more easily accessible photographically. Raman caused light from an intense monochromatic source (a mercury lamp) to be scattered in the liquid or gas under investigation. The scattered light, of course, shows the very same mercury lines as the incident light. But, in addition, a quantum absorption of energy also takes place: the scattering molecules, in order to increase their vibrational energy, borrow from the light precisely as much quantum energy as corresponds to these vibrations. As a result, alongside the scattered line, on the side of longer waves, there appears a second line, whose distance on the frequency scale corresponds exactly to the nuclear vibration being studied. Under certain circumstances, though considerably more rarely, the scattering molecule may also impart to the scattered light an additional store of energy at the expense of its own energy. As a result of such scattering there appears a line of shorter wavelength. We thus obtain, under favorable experimental conditions, to the right and to the left of the original line, the entire infrared spectrum of the scattering molecule. This method is still in the stage of development; at present there are still comparatively few results (p. 667); however, it promises to become
arises as a result of the asymmetry of the vibrations. Symmetric molecules such as \(N_2, O_2, I_2\) do not show any absorption in the infrared part of the spectrum.
very fruitful, despite the fact that, owing to the weakness of the scattered light, long exposures are unavoidable here.
We shall now consider the lines of an individual band, to which the various rotational states of molecules must correspond. The simplest case is when neither the electronic energy nor the vibrational energy changes. In this case we are dealing with the so-called rotational spectrum, which, owing to the smallness of the rotational energy, always lies in the far infrared region. To establish the law of distribution of the lines in this spectrum, we shall again choose formula (2) of the new quantum theory, which, if the selection principle is adjoined to it
\[ m''=m'-1 \]
(all the other possibilities of transition, \(m''=m'+1\) and \(m''=m'\), for a readily understandable reason fall away in the present case), gives the serial formula
\[ \nu=2Bm, \tag{15} \]
Thus in this especially simple case the spectrum consists of equidistant lines,\(^1\) which begin at the beginning of the spectrum and follow one another at a distance \(2B\). Such simple spectra have in fact been found in water vapor, ammonia, and have been especially well measured by Czerny\(^2\) in hydrogen chloride. The lines in this spectrum are as follows:
\[ \begin{array}{c|ccccccccc} m= & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ \lambda= & (480\mu & 240\mu & 160\mu & 120\mu) & 96.0\mu & 80.45\mu & 68.95\mu & 60.40\mu & 53.83\mu \\ \end{array} \]
\[ \begin{array}{c|cc} & 10 & 11 \\ & 48.49\mu & 44.15\mu \end{array} \]
(the first four lines were only calculated).
I have mentioned these measurements because they gave direct proof of the correctness of the new quantum theory. For according to this theory the first line must lie at \(2B\), and the following ones at \(2Bm\), whereas the old theory [equation (1a)] fixed the first line at \(B\), and the following ones at \(B(2m+1)\). In the case of water vapor, which gives an example
\(^1\) In reality the distances between the lines are not exactly the same, since according to equation (9) \(\nu=2Bm(1-2u^2m^2)\).
\(^2\) M. Czerny, Z. Physik. 34, 227, 1925.
another molecule well studied in the infrared part of the spectrum, we are dealing with an asymmetric molecule \((I_1 \ne I_2 \ne I_3)\), and therefore we find here in the rotational spectrum three series, namely:
\[ \nu_1 = 24.5m, \qquad \nu_2 = 16.8\left(p+\frac{1}{2}\right), \qquad \nu_3 = 55.5\left(r+\frac{1}{2}\right). \]
Two of them we can directly identify with our energy equation for a polyatomic molecule [equation (3)]:
\[ \nu_1 = 24.5m = \frac{h}{4\pi^2 I_1}\,m \qquad (p=\mathrm{const},\; m \to m-1) \]
\[ \nu_2 = 16.8\left(p+\frac{1}{2}\right) = \frac{h}{4\pi^2}\left(\frac{1}{I_2}-\frac{1}{I_1}\right) \qquad (m=\mathrm{const},\; p \to p-1) \]
for the third series we must take into account the third moment of inertia, for which purpose we set:
\[ \nu_3 = 55.5\left(r+\frac{1}{2}\right) = \frac{h}{4\pi^2 I_3}\left(r+\frac{1}{2}\right) \]
\[ (p=\mathrm{const},\; m=\mathrm{const},\; r \to r-1). \]
From this the three moments of inertia are calculated:
\[ I_3 = 0.98\cdot 10^{-40}, \qquad I_2 = 1.35\cdot 10^{-40}, \qquad I_1 = 2.33\cdot 10^{-40}, \]
which satisfactorily satisfy the condition valid for the water molecule \(I_1 = I_2 + I_3\). Similar measurements are also available in the case of ammonia.^1 It should also be emphasized that the rotational-vibrational spectrum, which from the spacing of the band lines should give, and in fact does give, the same values for the constants, since it contains the very same series.
It has been found, for example:
| Spectrum | HCl | H₂O | NH₃ |
|---|---|---|---|
| Rotational . . . . . | \(2B = 20.8\) | 24.5 16.8 55.5 | 19.96 |
| Rotational-vibrational . . | \(2B = 21.9\) | 24.2 17.9 56.9 | 19.9 |
^1 R. M. Badger, Nature, 121, 942, 1928; R. Robertson, I. Fox, E. S. Hiscocks, Proc. Roy. Soc., 120, 148, 1928.
In other spectra, in which the electronic energy and the vibrational energy also change, in a normally constructed band, according to the selection principle (12), we must expect three series instead of one. They are called branches and are denoted by the letters: \(R\)-, \(P\)-, and \(Q\)-. The \(R\)-branch, which goes toward shorter wavelengths (higher frequencies)—it is also called the positive branch—corresponds to the quantum transition \(m''=m'-1\); the \(P\)-branch (also called the negative branch) corresponds to the transition \(m''=m'+1\); it extends toward longer wavelengths; finally, the \(Q\)-branch (zero branch), for which the quantum number does not change, \(m''=m'\). In order to understand how the “transition” \(m''=m'\) can give rise to a series, it should be borne in mind that when the electronic energy changes simultaneously [according to equation (10), also when the vibrational energy changes], the moment of inertia always also changes, for the electronic system binding the molecule is transformed, as a result of which, of course, a new equilibrium position of the nuclei is obtained. Therefore, in these cases, in contrast to a purely rotational spectrum, \(B''\) will always differ from \(B'\). We must therefore expect, for the three series, the following series formulae (\(\nu_0\) denotes here the result of the change of the two other kinds of energy, which remain unchanged within the band):
\[ \begin{aligned} R\text{-branch}\quad m''&\to m'+1,\quad \nu=\nu_0+(B'+B'')m+(B'-B'')m^2,\\ Q\text{-}\quad\ \ \, m''&\to m',\quad \nu=\nu_0+(B'-B'')m+(B'-B'')m^2,\\ P\text{-}\quad\ \ \, m''&\to m'-1,\quad \nu=\nu_0+(B'+B'')(m+1)+(B'-B'')(m+1)^2 . \end{aligned} \tag{16} \]
We thus see that the series formulae, in the first approximation, are equations of parabolas of the form \(\nu=A+2Bm+Cm^2\), which for the value \(m=-\dfrac{B}{C}\) form turning points, i.e. heads. In Fig. 6 such a normal band is shown as a function of the “running number” \(m\). From this figure it is evident how the individual lines of one series are placed in the intervals between the lines of another series. Depending on whether the quantity \(C=B'-B''\)
positive or negative sign, the moment of inertia in the initial state will be smaller \((B' > B'')\) or larger \((B' < B'')\) than in the final state; in the first case the point of reversal will be on the \(P\)-branch, in the second case—on the \(R\)-branch. For \(B' > B''\) we shall therefore obtain violet shading, for \(B' < B''\)—red shading of the bands.
One should also mention the empirical rule:^1 the shading and the magnitude of the vibrations of the nuclei go parallel to one another: if \(B'\) is greater than \(B''\), then almost without exception also \(u' > u''\), and conversely. This means that the larger vibration of the nuclei corresponds to the smaller moment of inertia,—a rule which, in deciphering bands, often renders essential service.
Fig. 6.
We shall now not adhere to any definite serial formula, but shall simply denote both rotational terms by \(F'(m)\) and \(F''(m)\), in order that from the completely general condition
\[ \left. \begin{aligned} P(m)&=\nu_0+F'(m)-F''(m+1)\\ Q(m)&=\nu_0+F'(m)-F''(m)\\ R(m)&=\nu_0+F'(m)-F''(m-1) \end{aligned} \right\} \tag{17} \]
^1 R. Mecke, Z. Physik. 32, 823, 1925.
obtain two important relations—the so-called combination relations. Namely, the following relations hold with complete exactness:
\[ \begin{aligned} Q(m)-P(m)&=R(m+1)-Q(m+1)\\ &=F''(m+1)-F''(m)\sim 2B''(m+1) \end{aligned} \tag{18a} \]
\[ \begin{aligned} Q(m+1)-P(m)&=R(m+1)-Q(m)\\ &=F'(m+1)-F'(m)\sim 2B'(m+1) \end{aligned} \tag{18b} \]
These relations are important because they make it possible, from branches which after all simultaneously contain both terms, to isolate from one another the initial and final rotational terms. Indeed, the first equation contains only the difference of two consecutive rotational states of the final term, while the second equality contains the same for the initial term. By simple addition of all the differences formed in this way, one can thus compute the rotational terms themselves. In our graphical representation of the band one can, consequently, always connect into a parallelogram four straight lines, of which the two horizontal ones represent the distance between two consecutive rotational terms of a certain state, while the straight lines on the right and on the left represent the same for the initial states. The combination relations also indicate the place where the series begin and whence the counting must start. True, under favorable conditions this zero point \((m=0)\) in the spectrum can already be recognized from the external appearance of the spectrum, for at this place a line is suddenly absent. In Fig. 2, where the \(Q\)-branches are absent, this gap is perfectly clearly visible; one also sees how the intensities of the lines to the right and to the left of the zero lines increase to a maximum and then again decrease. But in many cases the zero positions, owing to an excessively strong crowding of lines, cannot be found, and then the combination relations (18) help in locating them.
Furthermore, the combination relations constitute a very precise and reliable criterion for the correctness of the distribution of bands in the quantum scheme, for all bands which have one and the same final vibrational state.
nuclei, \(n''\), or one and the same initial state \(n'\), i.e. bands of the same longitudinal or transverse series, must give exactly coinciding combination relations (18a and 18b). In many cases the \(Q\)-branch is absent; then, in order to isolate the terms and establish the correct distribution of the bands, one has to confine oneself to somewhat simpler relations:
\[ \begin{aligned} R(m)-P(m)&=F''(m+1)-F''(m-1)\sim 4B\left(m+\frac12\right),\\ R(m+1)-P(m-1)&=F'(m+1)-F'(m-1)\sim 4B\left(m+\frac12\right). \end{aligned} \tag{19} \]
From these relations one can calculate the constant and hence the molecular moment of inertia, which is what is of greatest interest to us.1
Up to this point we have considered the simple case in which in each band there is only one \(P\)-, one \(Q\)-, and one \(R\)-branch. Usually, however, bands do not possess such a simple structure (it occurs, for example, in CuH, CO, etc.), but reveal the existence of several series, often of very complex structure, the so-called fine structure.2 In such a case a simple condition of the type (17) is no longer applicable; in the \(Q\)-branches there appear characteristic “cross combinations,” which obey the equations:
\[ \left. \begin{aligned} P_1(m)&=\nu_0+F_1'(m)-F_1''(m+1)\\ P_2(m)&=\nu_0+F_2'(m)-F_2''(m+1)\\[3pt] Q_1(m)&=\nu_0+F_1'(m)-F_2''(m)\\ Q_2(m)&=\nu_0+F_2'(m)-F_1''(m)\\[3pt] R_1(m)&=\nu_0+F_1'(m)-F_1''(m-1)\\ R_2(m)&=\nu_0+F_2'(m)-F_2''(m-1) \end{aligned} \right\} \tag{20} \]
Thus, in order to obtain combination relations of the type of equation (18), it is necessary, for example, to combi-
to combine \(R_1\) with \(Q_2\) and \(P_2\) with \(Q_1\). I shall give only one of the various possible combinations:
\[ Q_1(m+1)-P_2(m)=R_1(m+1)-Q_2(m)+F_1'(m+1)-F_2'(m). \]
Thus in this case a strict separation of the terms is no longer possible. One must also take into account combinations expressed by equation (19).
As the results obtained so far have shown, such fine structures can be reduced to one of the following three principal types.^1
Fig. 7.
In the first type (as an example I shall mention the well-known cyanogen spectrum), each line has two components, the splitting of which increases proportionally to the running number (Fig. 7). We shall call bands of this type, following Gerlinger, bands with simple series or singlet bands; why we give this name will be said later. In the second group of bands we have, as
Fig. 8.
^1 R. Mecke. Z. Physik. 28, 261, 1924. R. Mulliken. Phys. Rev. 28, 1202, 1926.
once again the opposite picture (Fig. 8). Here the splitting near the zero line has its greatest magnitude and decreases continuously with increasing rotational number. These are the so-called doublet bands (example: the OH and CH bands). But here a combination of type I with type II is also possible by adjoining three further doublet series, so that one obtains a band of highly complex construction, consisting in all of 12 separate series with a subdivision into \(3 \times 4\) series with the arrangement indicated in Fig. 7 (example: the CH bands). Often the splitting near the zero lines may be so large that the convergence of the corresponding \(P\)-, \(Q\)-, and \(R\)-branches at large \(m\) can no longer be observed, and therefore this case may be confused with a double band of type I.
Fig. 9.
However, by exact investigation one can slowly establish the true nature of such bands (example: the HgH, NO bands). In some cases of doublet bands (NO, BO) we find, instead of the expected \(2 \times 6\) branches, only 8 (\(2 \times 4\)); this is caused by the special form of the rotational energy, in which the \(P\)-branch coincides with a \(Q\)-branch, and another \(Q\)-branch with an \(R\)-branch; in this case these branches have doubled intensity. The third type, triplet series (Fig. 9), on which, however, the \(Q\)-branches are not depicted, reveals a structure similar to type II, with the only difference that here three series proceed from different points and gradually approach one another (example: the \(N_2\), \(C_2\), and NH bands). Here, too, three series
in turn may possess fine structure, so that, for example, in the positive bands of nitrogen two \(P\)- or \(Q\)-series, namely the middle one and the one extending toward the violet side, are further split into close doublets with an apparently constant spacing; the same is found in the series of carbon bands. These fine splittings, like the splittings of singlet bands, are caused, as it has turned out, by the influence of rotation on the electronic energy. They have no direct analogues in atomic spectra, in contrast to the large splittings of types II and III, which are ordinary splittings of electronic energy, in complete analogy with the multiplets of atomic spectra, with the only difference that in atomic spectra quartets and quintets are already known, which as yet we do not know in band spectra. The convergence of separate branches is again caused by the influence of rotation on the electronic energy, i.e. by causes of a secondary character.
In molecules that contain two identical atoms, i.e. in molecules of the type \(\mathrm{Me}_2\), a peculiar phenomenon is observed in the bands, characteristic only of such molecules. This phenomenon consists in the fact that the lines in the branches have alternating intensity, i.e. a weak one follows a strong line, then again a strong one, etc. In some molecules, such as, for example, \(\mathrm{He}_2\), \(\mathrm{O}_2\), every second line is even absent. The phenomenon has recently received an explanation on the basis of the new quantum theory; it reduces to the symmetry properties of the molecule, and for such molecules it is very characteristic that they impeccably satisfy the criterion of a diatomic symmetric molecule. This phenomenon has so far been observed in \(\mathrm{H}_2\), \(\mathrm{He}_2\), \(\mathrm{C}_2\), \(\mathrm{N}_2\), \(\mathrm{N}_2^+\) (see Fig. 2), \(\mathrm{O}_2\), and \(\mathrm{O}_2^+\).
3. Band spectra and the periodic system.
In this article, whose purpose is to report the results of investigations of band spectra that are of greatest interest for chemistry, there is, of course, no need to examine in detail all the band spectra studied up to now. In order to give an opportunity to review in the best possible way
the results obtained, we shall confine ourselves to presenting the numerical material (Tables 1–5). The course of the analysis of band spectra has already been briefly outlined in the preceding paragraph. It consists of the following stages: 1) distribution of the bands in the quantum scheme and establishment of the band formula; 2) establishment of the structure of the bands (singlet, doublet, triplet), so that from this one may draw a conclusion concerning the electronic structure of the molecule—we shall dwell on this in more detail in the following paragraph; 3) establishment of combination relations in order to isolate the rotational terms and obtain a reliable criterion for the correctness of the establishment of the series. From the numerical material obtained one can immediately calculate the vibrational frequencies of the nuclei and the moments of inertia in the initial and final states, and from these in turn—if the carrier of the spectrum is known—calculate the distance between the nuclei. But precisely the question of the carrier of a band spectrum was for a long time the subject of lively discussion, since band spectroscopy acquainted us with compounds that until then were unknown to chemists and toward which they were very skeptical until exact criteria for establishing the carriers had been found. Now, when we already have at our disposal a large body of reliably established numerical material, we can easily recognize the carrier of the spectrum. For this purpose we make use of the magnitude of the vibration of the nuclei, the magnitude of the moment of inertia, the character of the fine structure and, finally, in those cases where isotopes exist, also the isotope effect, which we shall consider below in a separate paragraph. In addition, we must of course also take into account the conditions of excitation of the spectra.
In discussing the question of the carrier of bands, two points of view must above all be taken into account. First, the great sensitivity of spectral analysis allows the spectroscopist reliably to detect quantities which are often no longer accessible to physical methods, so that even unstable, rapidly decomposing molecules do not escape spectral analysis. Secondly, spectro-
the spectroscopist in his investigations rarely encounters the normal atoms and molecules which the chemist has at his disposal as the final products of his reactions. On the contrary, the spectroscopist deals almost exclusively with such molecules as, by the prior supply of electrical, thermal, or light energy, have been brought into an excited state, in which they display quite different possibilities of reaction than in the normal ground state. This new chemistry of excited molecules and atoms arose only on the basis of spectral analysis. Thus, for example, helium, such as chemists know it, is not capable of reactions, since both its electrons combine into an especially stable formation—the shell of a noble gas—and therefore cannot act chemically at all. In a Geissler tube, however, these stable electron configurations are destroyed, and we then have a shell consisting of two electrons with a very great capacity for reaction; an atom with such an excited electron shell, upon collision with another similarly excited atom, can enter into combination with it. And in fact we know—though only spectroscopically—the helium molecule $\mathrm{He}_2$.
For convenience in surveying the numerical material compared in the tables given, the spectra are arranged according to the following types of compounds: 1) hydride compounds, forms $\mathrm{MeH}$; 2) oxides and nitrides $\mathrm{MeO}$, $\mathrm{MeN}$; 3) molecules of the elements $\mathrm{Me}_2$; 4) halide compounds $\mathrm{MeX}$; 5) polyatomic molecules. The first column gives the compounds which excite the spectrum, the second—the relation of the bands $R$ and $V$, the third—the interpretation of the electronic transition, which we shall consider in the following paragraph; the remaining five columns—the constants of the band formula:¹ $\nu=\nu_0+(a'n' - b'n'^2)-(a''n'' - b''n''^2)$, followed by the moments of inertia and the distances between nuclei in the initial and final states and, finally, the designation of the spectrum. If one compares within
¹ For practical reasons the old integer values of $n$ are used here. Recalculation to half-integer values is quite simple.
of the aforementioned groups of compounds, above all the moments of inertia of the different molecules, or, still better, the internuclear distances computed on the basis of the moments of inertia, then one can clearly see how the structure of the atoms is reflected in them. Especially characteristic for the hydrides is the dependence of the internuclear distance on the atomic number of the corresponding element (Fig. 10). It is seen that, after passing a noble gas in the periodic system, a jump appears each time on the curve; this is due to the fact that precisely here the atomic electron begins to fill a new shell. Within a period, with increasing atomic number, the internuclear distance continuously decreases, since the attraction, increasing with the growth of the nuclear charge, causes compression of the electron shell.
Fig. 10.
The existence of Bohr closed electron shells is illustrated by these data more vividly than by any other. An analogous picture is also observed for other types of compounds; for oxides and nitrides the magnitudes of the internuclear distances in the period Li—Ne all lie between \(1.0\) and \(1.3 \cdot 10^{-8}\) cm, and in the following period (Na—A) near \(1.6 \cdot 10^{-8}\) cm; the corresponding values are also given by groups 3 and 4, where, however, the observations are very few. By means of such comparisons one can make a good estimate of the moments of inertia of compounds not yet investigated.
A completely analogous course is shown by the data for nuclear vibrations, whose magnitude, according to equation (7), is determined by the mass of the molecules and by the internuclear distance. Light hydrides all have, throughout, very large frequencies of nuclear vibration,
TABLE 1. Hydrides.
| Molecule | Term | \(\nu_0\) | \(a'\) | \(b'\) | \(a''\) | \(b''\) | \(J'\) | \(J''\) | \(r'\) | \(r''\) | Notes | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| BeH | \(V\) | \(^{2}P_i — {}^{2}S\) | 20 032 | 2050 | — | — | — | 2,68 | 2,72 | 1,34 | 1,35 | - |
| CH | \(R\) | \(^{2}S — {}^{2}S\) | 39 059 | 1460,5 | 14,5 | 2182,0 | 41,0 | 3,85 | 2,50 | 1,60 | 1,29 | - |
| CH | \(V\) | \(^{2}D — {}^{2}P\) | 23 163,3 | 2851 | 2797 and 2815 | 1,90 | 1,95 | 1,11 | 1,13 | C + H \(\lambda\) 4300; | ||
| CH | \(R\) | \(^{2}S — {}^{2}P\) | 25 715 | — | — | — | — | 2,21 | 1,95 | 1,20 | 1,13 | C + H \(\lambda\) 3900 |
| NH | — | \(^{3}P_i — {}^{3}S\) | 29 750 | — | — | — | — | — | 1,71 | — | 1,05 | NH\(_3\) bands |
| OH | \(R\) | \(^{2}S — {}^{2}P\) | 32 422,7 | 3085,4 | 97,4 | 3569,8 | 1,633 | 1,498 | 1,02 | 0,97 | Water-vapor bands. | |
| NaH | \(V\) | ? | 21 543 | — | — | — | — | 3,20 | 4,06 | 1,42 | 1,60 | Multiline spectrum. |
| MgH | \(V\) | \(^{2}P_i — {}^{2}S\) | 19 280 | 1603,5 | 34,75 | 1493,5 | 31,2 | — | 4,65 | — | 1,71 | Multiline spectrum. |
| AlH | \(R\) | \(^{1}P — {}^{1}S\) | 23 477,0 | 1082 | 1625 | 4,58 | 4,38 | 1,69 | 1,66 | Multiline spectrum. | ||
| CaH | \(V\) | \(^{2}S — {}^{2}S\) | 28 353,4 | — | — | — | — | 5,733 | 6,546 | 1.886 | 2,009 | |
| CaH | \(R\) | \(^{2}P_i — {}^{2}S\) | 14 392,3 | — | — | — | — | 6,78 | 6,55 | 2,03 | 2,01 | |
| CaH | \(R\) | \(^{2}P_i — {}^{2}S\) | 14 472,2 | — | — | — | — | 6,18 | 6,55 | 1,94 | — | |
| CaH | \(V\) | \(^{2}S — {}^{2}S\) | 15 753,8 | — | — | — | — | 6,30 | — | 1,96 | — | |
| ZnH | \(V\) | \(^{2}P_i — {}^{2}S\) | 23 263,6 | — | — | 1552 | — | 3,80 | 4,23 | 1,53 | 1,61 | |
| ZnH | \(V\) | \(^{2}P_i — {}^{2}S\) | 23 594,0 | — | — | — | — | 3,708 | 4,234 | 1,505 | 1,608 | |
| CdH | \(V\) | \(^{2}P_i — {}^{2}S\) | 22 277,6 | — | — | 1374 | — | 4,647 | 5,201 | 1,679 | 1,776 | |
| CdH | \(V\) | \(^{2}P_i — {}^{2}S\) | 23 279,0 | — | — | — | — | 4,594 | 5,201 | 1,669 | 1,776 | |
| HgH | \(V\) | \(^{2}P^{1}/_{2} — {}^{2}S\) | 24 933,9 | 1938,7 | 1308 | 104 | 4,24 | 5,14 | 1,597 | 1,763 | ||
| HgH | \(V\) | \(^{2}P^{3}/_{2} — {}^{2}S\) | 28 617,1 | 1938,1 | 43,8 | 1308 | 104 | 4,18 | 5,14 | 1,589 | 1,763 | |
| HgH | \(R\) | \(^{2}S — {}^{2}S\) | 33 876,5 | — | — | 1308 | 104 | 6,77 | 5,14 | 2,02 | 1,763 | |
| CuH | \(R\) | \(^{1}S — {}^{1}S\) | 23 311,1 | 1658,8 | 44,71 | 1903,5 | 37,18 | 4,10 | 3,54 | 1,59 | 1,47 | |
| AgH | \(R\) | \(^{1}S — {}^{1}S\) | 29 900 | 1489 | 1693 | 4,57 | 4,37 | 1,67 | 1,63 | |||
| AuH | \(R\) | \(^{1}S — {}^{1}S\) | 27 342,1 | 1631 | 80 | 2249,4 | 34,0 | 4,74 | 4,00 | 1,70 | 1,56 | |
| AuH | \(R\) | \(? — {}^{1}S\) | 38 230 | (1600) | — | 2249,4 | 34,0 | 4,85 | 4,00 | 1,74 | 1,56 | |
| HF | — | \(^{1}S\) | 0 | 3962 | same | — | 1,35 | same | 0,924 | same | Infrared abs. | |
| HCl | — | \(^{1}S\) | 0 | 2940,8 | 53,6 | " | — | 2,645 | " | 1,279 | " | " |
| HBr | — | \(^{1}S\) | 0 | 2559 | " | — | 3,303 | " | 1,418 | " | " |
TABLE 2. Oxides and nitrides.
| Substance | Term | \(\nu_0\) | \(a\) | \(\nu'\) | \(a''\) | \(B''\) | \(J'\) | \(J''\) | \(r'\) | \(r''\) | Notes | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| BeO | \(R\) | \({}^1S-{}^1S\) | 21 230,6 | 1354 | 8,9 | 1465 | 12,7 | 17,65 | 16,85 | 1,37 | 1,33 | |
| BO | \(R\) | \({}^2P-{}^2S\) | 23 960,4 | 1258,1 | 10,6 | 1888,4 | 11,71 | 19,62 | 15,68 | 1,35 | 1,21 | BO \(\alpha\)-group, B II |
| BO | \(R\) | \({}^2S-{}^2S\) | 23 831,2 | 1258,1 | 10,6 | 1888,4 | 11,71 | — | 15,68 | — | 1,21 | BO \(\beta\)-group, B II |
| BO | \(V\) | \({}^2S-{}^2P_i\) | 43 167,4 | 1280,3 | 10,07 | 1259,1 | 10,6 | — | 19,62 | — | 1,35 | Combination bands |
| BO | \(V\) | \({}^2S-{}^2P_i\) | 19 333,2 | 1280,3 | 10,07 | 1259,1 | 10,6 | — | 19,62 | — | 1,35 | |
| BO | \(V\) | \({}^2S-{}^2P_i\) | 19 207,0 | 1280,3 | 10,07 | 1259,1 | 10,6 | — | 19,62 | — | 1,35 | |
| CO | \(R\) | \(2{}^3P-1{}^1S\) | 48 530 | 1724,8 | 14,47 | 2149,7 | 13,70 | — | — | — | — | Camerora bands |
| CO | \(R\) | \(2{}^1P-1{}^1S\) | 64 737 | 1499,8 | 17,24 | 2149,7 | 13,70 | — | — | — | — | 4. Positive group |
| CO | \(V\) | \(1{}^1S-2{}^3P\) | 33 390 | 2214 | [[unclear]] | 1724,8 | 14,47 | — | — | — | — | 3. ″ ″ |
| CO | \(V\) | \(2{}^1S-2{}^1P\) | 22 156 | 2158 | 76 | 1499,8 | 17,24 | 14,9 | 18,2 | 1,13 | 1,27 | Angström bands |
| CO | \(V\) | \({}^1S\) | 0 | 2148 | 9,7 | Same | Same | 14,9 | Same | 1,15 | Same | Infrared bands CO |
| \(C^{+}O\) | \(R\) | \({}^2P-{}^2S\) | 20 471,6 | 1650,5 | 14,07 | 2198,6 | 15,00 | 17,7 | 14,05 | 1,25 | 1,11 | Comet spectrum |
| \(C^{+}O\) | \(R\) | \({}^2P-{}^2S\) | 20 346,1 | 1650,5 | 14,07 | 2198,6 | 15,00 | 17,7 | 14,05 | 1,25 | 1,11 | |
| \(C^{+}O\) | \(V\) | \({}^2S-{}^2P\) | 25 285 | 1697,8 | 24,33 | 1550,5 | 14,07 | 16,2 | 17,7 | 1,20 | 1,25 | Combination bands; Baldet–Johnson |
| \(C^{+}O\) | \(V\) | \({}^2S-{}^2P\) | 25 158 | 1697,8 | 24,33 | 1550,5 | 14,07 | 16,2 | 17,7 | 1,20 | 1,25 | |
| CN | \(R\) | \({}^2S-{}^2S\) | 45 637,3 | 1697,8 | 24,33 | 2198,6 | 15,00 | 16,2 | 13,5 | 1,20 | 1,09 | 1. Negative group C |
| CN | \(V\) | \({}^2S-{}^2S\) | 25 799,8 | 2143,9 | 20,25 | 2055,6 | 13,75 | 14,1 | 14,6 | 1,15 | 1,17 | Violet bands CN |
| CN | \(R\) | \({}^2P_i-{}^2S\) | 14 430 | 1728,5 | 13,5 | 2055,6 | 13,75 | — | 14,6 | — | 1,17 | Red bands CN |
| CN | \(R\) | \({}^2P_i-{}^2S\) | 14 374 | 1728,5 | 13,5 | 2055,6 | 13,75 | — | 14,6 | — | 1,17 | |
| CS | \({}^1P-{}^1S\) | 38 796,8 | 1052,2 | 10,11 | 1276,5 | 6,00 | — | — | — | — | ||
| NO | \(V\) | \({}^2S-{}^2P_i\) | 44 076 | 2345,1 | 14,4 | 1891,97 | 14,45 | 14,06 | 16,55 | 1,07 | 1,16 | 3. Positive group N |
| NO | \(R\) | \({}^2P_i-{}^2P_i\) | 45 394,6 | 1030,88 | 7,45 | 1891,97 | 14,45 | 24,80 | 16,55 | 1,42 | 1,16 | \(\beta\)-bands of active nitrogen |
| NO | \(R\) | \({}^2P_i-{}^2P_i\) | 45 486,1 | 1029,43 | 7,46 | 1891,97 | 14,45 | 24,80 | 16,55 | 1,42 | 1,16 | |
| AlO | \(R\) | \({}^2S-{}^2S\) | 20 646,0 | 866,1 | 4,0 | 971,0 | 7,2 | 46,02 | 43,28 | 1,67 | 1,62 | |
| SiO | \(R\) | Singlet | 42 643 | 844,5 | 5,8 | 1236,0 | 6,04 | — | — | — | — | Si 28 |
| SiN | \(R\) | \({}^2S-{}^2S\) | 24 234 | 1016,8 | 17,7 | 1145,0 | 5,57 | 38,6 | 33,0 | 1,56 | 1,56 | Si 28 |
| PO | \(V\) | \({}^2P-{}^2S\) | 40 384 | 1335,5 | 7,5 | 1228 | 7 | — | — | — | — | |
| PO | \(V\) | \({}^2P-{}^2S\) | 40 509 | 1335,5 | 7,5 | 1228 | 7 | — | — | — | — | |
| SO | \(R\) | — | 36 590 | 623 | 6 | 1094,6 | 6,3 | — | — | — | — | |
| CaO | \(R\) | Singlets | 23 057 | 740 | 3,3 | 848 | 4 | — | — | — | — | |
| ScO | \(R\) | Doublets | 16 561 | 867 | 8,7 | 967 | 3,3 | — | — | — | — | |
| ScO | \(R\) | Doublets | 16 521 | 867 | 8,7 | 967 | 3,3 | — | — | — | — | |
| TiO | \(R\) | Singlets | 19 360 | 833,1 | 4,5 | 1003,5 | 4,5 | 56,75 | 31,87 | 1,69 | 1,62 | |
| VO | \(R\) | — | 17 424 | 852 | 4,5 | 1008 | 6 | — | — | — | — | |
| CrO | \(R\) | Singlets | 16 520 | 732 | 9,5 | 891 | 6 | — | — | — | — | |
| CrO | \(R\) | Singlets | 15 500 | 732 | 9,5 | 891 | 6 | — | — | — | — | |
| MnO | \(R\) | — | 17 906 | 758,9 | 10,2 | 836,7 | 5,06 | — | — | — | — | |
| SrO | \(R\) | Singlets | 24 638 | 516 | 3 | 648 | 3,9 | — | — | — | — | |
| BaO | \(R\) | Singlets | 14 664 | 510 | 2,7 | 664 | 7,5 | — | — | — | — | |
| BiO | \(R\) | — | 22 196 | 410 | 3 | 530 | ? | — | — | — | — | |
| LaO | \(R\) | — | 17 844 | 730 | 1,9 | 813 | 2,4 | — | — | — | — | |
| PbO | \(R\) | \({}^3P-{}^3S\) | 24 770 | 522 | 1,5 | 717 | 3,5 | — | — | — | — | |
| PbO | \(R\) | \({}^3P-{}^3S\) | 23 170 | 496 | 1,4 | 717 | 3,5 | — | — | — | — | |
| PbO | \(R\) | \({}^3P-{}^3S\) | 19 728 | 440 | 1 | 717 | 3,5 | — | — | — | — |
TABLE 3. Molecules of elements.
| Molecule | Term | \(\nu_0\) | \(a'\) | \(b'_{\sim}\) | \(a''\) | \(b''\) | \(y'\) | \(y''\) | \(r'\) | \(r''\) | Notes | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\mathrm{H}_2\) | \(R\) | \(2^1P-1^1S\) \(2^1P-1^1S\) |
99986 91 562 |
2374 1327 |
66.5 11.7 |
\(\} 4264\) | 144.4 | 0.92 | 0.467 | 1.05 | 0.75 | Ultraviolet absorption of \(\mathrm{H}_2\) |
| \(\mathrm{He}_2\) | \(R\) \(V\) |
\(3^1P-2^1S\) \(3^3D-2^1P\) |
21 507.3 17 436.6 |
1688.5 1760 |
36.0 — |
1772.0 1530 |
39.2 — |
3.86 3.66 |
3.64 3.78 |
1.08 1.05 |
1.05 1.07 |
\(\lambda\ 4850\) \(\lambda\ 5730\) |
| \(\mathrm{C}_2\) | \(V\) \(V\) \(R\) |
— \({}^3P_i-{}^3P_i\) Triplet |
25 951 19 374 15 540 15 623 |
\(\} 1773.4\) \(\} 1105\) |
19.3 9.8 |
1590 1629.9 1714 |
— 11.7 — |
\(\} 15.84\) — |
17.03 — |
1.26 — |
1.31 — |
Deslandres–Swan bands Triplet bands |
| \(\mathrm{N}_2\) | \(R\) \(V\) \(V\) |
\(? -{}^1S\) — \({}^3P_i-{}^3P_i\) |
68 956 44 214 29 663 |
1681.4 — 2015.7 |
15.2 — 26.05 |
2345.1 — 1718.45 |
14.4 — 14.5 |
— — \(\}15.7\) |
— — 17.0 |
— — 1.17 |
— — 1.22 |
Absorption in U.V. 4. Positive group 2. “ ” |
| \(\mathrm{N}_2^+\) | \(V\) \(V\) \(R\) |
— \({}^3S-{}^2S\) \(?-{}^2S\) |
9 529 25 546.6 26 142 |
1718.4 2397.7 1459.5 |
14.5 26.2 15.2 |
1474.4 \(\} 2187.1\) |
13.98 16.1 |
13.4 | 14.5 | 1.08 | 1.12 | 1. “ ” 1. Negative group Vegard |
| \(\mathrm{O}_2\) | \(R\) \(R\) |
\(?-{}^3S\) \(?-{}^3S\) |
49 359.3 13 122.9 |
708 1418.7 |
13 18.96 |
\(\} 1565.4\) | 11.37 | 84.2 19.93 |
19.20 19.20 |
1.609 1.22 |
1.20 1.20 |
U.V. absorption Atmospheric O bands |
| \(\mathrm{O}_2^+\) | \(R\) \(V\) |
\({}^2P_i-{}^2P_i\) — |
33 308 38 108 16 592.2 |
\(\} 885.2\) 1180.3 |
13.7 17.8 |
1859.9 1026.1 |
16.58 11.1 |
— — |
— — |
— — |
— — |
1. Negative group O Negative red group O |
| \(\mathrm{S}_2\) | \(R\) | — | 82 140 | 424.4 | 2.7 | 724.5 | 2.94 | — | — | — | (1.81) | Fluorescence and absorption bands |
| \(\mathrm{Se}_2\) | \(R\) | — | 27 307 | 247.2 | 2.3 | 397.5 | 1.32 | — | — | — | (2.20) | Fluorescence and absorption bands |
| \(\mathrm{Te}_2\) | \(R\) | — | 22 671 | 163 | 1 | 250.4 | 0.53 | — | — | — | (2.63) | Fluorescence and absorption bands |
| \(\mathrm{F}_2\) | \(R\) | — | 17 439 | (690) | — | 1071.5 | 9.9 | — | 29.0 | — | 1.35 | Emission bands |
| \(\mathrm{Cl}_2\) | \(R\) | — | 18 976 | 201.3 | 5.4 | 555.0 | 2.6 | — | — | — | 1.93 | Absorption bands |
| \(\mathrm{Br}_2\) | \(R\) | — | 16 463 | 152 | 1.9 | 326.1 | 1.2 | — | — | — | (2.26) | “ ” |
| \(\mathrm{J}_2\) | \(R\) | — | 15 598.1 41 900 |
126.5 70 |
0.85 — |
213.9 90 |
0.57 — |
1200 — |
742 — |
3.4 — |
2.66 — |
“ ” |
| \(\mathrm{JCl}\) | \(R\) | — | 14 825.4 | 182.7 | 3.1 | 381.9 | 0.7 | (995) | (576) | 4.6 | 3.5 | “ ” |
| \(\mathrm{Li}_2\) | \(R\) | — | 20 398.4 | 267.5 | 3.2 | 347.5 | 2.2 | — | — | — | — | |
| \(\mathrm{Na}_2\) | \(R\) | \({}^1P-{}^1S\) | 20 301.7 15 006.7 |
128.8 115.7 |
0.79 0.43 |
157.7 157.7 |
0.55 0.55 |
420.9 — |
179.5 — |
3.41 — |
3.08 — |
Absorption bands |
| \(\mathrm{K}_2\) | \(R\) | — | 15 368.0 11 574.9 |
74.6 71.0 |
0.30 0.40 |
91.5 91.5 |
0.32 0.32 |
212 — |
184 — |
2.56 — |
2.39 — |
“ ” |
| \(\mathrm{NaK}\) | \(R\) | — | 16 967.9 | 69.44 | 1.03 | 122.9 | 0.40 | 78 | 66 | 1.81 | 1.66 | “ ” |
R. MENKE
TABLE 4. Halides.
| Compound | \(\nu\) | \(a'\) | \(b'\) | \(a''\) | \(b''\) | Notes | |
|---|---|---|---|---|---|---|---|
| BeF | \(R\) | 33 217,5 | 1156,1 | 8,4 | 1252,6 | 10,2 | |
| BeF | \(R\) | 33 180 | 1156,1 | 8,4 | 1252,6 | 10,2 | |
| MgF | \(V\) | 27 825,5 | 713,1 | 3,80 | 684,1 | 3,70 | |
| CaF | \(V\) | 16 559 | 588,1 | 2,03 | 583,4 | 2,75 | |
| CaF | \(V\) | 16 485 | 588,1 | 2,03 | 583,4 | 2,75 | |
| SrF | \(V\) | 15 356,3 | 504,6 | 2,22 | 497,7 | 1,85 | |
| SrF | \(V\) | 15 076,8 | 504,6 | 2,22 | 497,7 | 1,85 | |
| BaF | \(R\) | 17 302,6 | 487,1 | 1,91 | 465,0 | 1,85 | |
| BaF | \(R\) | 19 097,3 | 452,9 | 1,75 | 465,0 | 1,85 | |
| CuF | \(R\) | 20 270,0 | 641,0 | 4,15 | 617,8 | 3,78 | Cu 63 |
| CuF | \(R\) | 19 752,9 | 647,1 | 3,70 | 612,4 | 3,58 | Cu 63 |
| CuF | \(R\) | 17 558,7 | 838,7 | 3,82 | 612,4 | 3,58 | Cu 63 |
| CuCl | \(R\) | 25 281,4 | (371) | — | 415,6 | 1,60 | Cu 63 Cl 35 |
| CuCl | \(R\) | 23 071,2 | 401,6 | 1,08 | 415,6 | 1,60 | Cu 63 Cl 35 |
| CuCl | \(R\) | 22 901,7 | 398,4 | 1,87 | 415,6 | 1,60 | Cu 63 Cl 35 |
| CuCl | \(R\) | 20 626,0 | 397,5 | 1,68 | 415,6 | 1,60 | Cu 63 Cl 35 |
| CuCl | \(R\) | 20 479,7 | 398,5 | 1,5 | 415,6 | 1,60 | Cu 63 Cl 35 |
| CuCl | \(R\) | 18 997,7 | 408 | 2 | 415,6 | 1,60 | Cu 63 Cl 35 |
| CuBr | \(R\) | 23 452,4 | 294,3 | 1,25 | 313,6 | 0,90 | Cu 63 Br 79 |
| CuBr | \(R\) | 23 029,3 | 282,7 | 1,18 | 313,6 | 0,90 | Cu 63 Br 79 |
| CuBr | \(R\) | 20 489,3 | 294,4 | 0,94 | 313,6 | 0,90 | Cu 63 Br 79 |
| CuJ | \(R\) | 23 982,7 | 228,9 | 1,00 | 264,3 | 0,76 | Cu 63 |
| CuJ | \(R\) | 22 929,7 | 211,4 | 0,93 | 264,3 | 0,76 | Cu 63 |
| CuJ | \(R\) | 21 851,4 | 230,2 | 0,78 | 264,3 | 0,76 | Cu 63 |
| CuJ | \(R\) | 21 748,3 | 243,4 | 2,18 | 264,3 | 0,76 | Cu 63 |
| CuJ | \(R\) | 19 707,7 | 211,8 | 2,26 | 264,3 | 0,76 | Cu 63 |
| SnCl | \(V\) | 31 262,5 | 431,3 | 1,2 | 353,5 | 1,0 | Cl 35 |
| SnCl | \(V\) | 33 622,6 | 431,3 | 1,2 | 351,4 | 1,3 | Cl 35 |
| SnCl | \(R\) | 26 579,1 | 297,5 | 4,1 | 349,5 | 1,1 | Cl 35 |
| SnCl | \(R\) | 28 665,1 | 246,2 | 4,2 | 350,7 | 1,0 | Cl 35 |
| AgJ | \(R\) | 31 148 | — | — | 206,7 | 0,7 | |
| AuCl | \(R\) | 19 078,5 | 311,3 | 0,70 | 381,5 | 1,30 | Cl 35 |
TABLE 5.
| Mg | Cu | Sr | Ba | |
|---|---|---|---|---|
| F | 684 | 582 | 498 | 465 |
| Cl | 402 | 360 | 301 | 289 |
| Br | 350 | 281 | 215 | 192 |
| J | 316 | 241 | 171 | — |
whereas in oxides and nitrides we encounter average values which in the first period lie between 1300 and 2000, in the second—between 1000 and 1200, in the third—between 800 and 1000, and in the higher periods—always between 500 and 800. In molecules of the elements, the frequency of nuclear vibrations decreases very rapidly with increasing atomic weight; the smallest value is observed for iodine (213) and for the alkalis, which is connected with their large atomic volume. On the basis of these results one can approximately estimate the frequencies of nuclear vibrations, especially in homologous compounds. Namely, the empirical formula
\[ a \sqrt{Z_1 Z_2}=\mathrm{const} \]
is approximately fulfilled, or also
\[ a\sqrt{\mu}\sqrt{Z_1 Z_2}=\mathrm{const} \]
(\(Z_1\) and \(Z_2\) are the atomic numbers of the corresponding elements). The halogen compounds follow this rule especially well. In order to show the dependence of the frequencies of nuclear vibrations on atomic number, I shall give, in a separate table, the frequencies of the alkaline-earth halide compounds, which provide the most complete material for this rule.
A very difficult chapter of the spectroscopy of band spectra is devoted to polyatomic molecules, in which the possibilities of vibration and rotation increase so considerably that the spectrum becomes almost inaccessible to deciphering. In view of this we must, unfortunately, state that up to the present time we cannot yet analyze a single band spectrum of a polyatomic molecule with the same completeness to which we are accustomed in the case of diatomic molecules. We can only point out that in this direction promising first steps have been made, especially thanks to the work of Henri and his collaborators.
Let us consider, as a first example, the simple molecule of carbon dioxide, which, on the basis of various criteria^1 (infrared absorption, specific heat, dispersion, dielectric constant), should be regarded as a rod-shaped molecule with carbon in the middle. Thus three different kinds are already possible here
^1 See the comparison in K. L. Wolf. Z. Phys. Chem. 131, 90, 1927.
oscillations of the nuclei, namely oscillations of carbon in the direction of both oxygen atoms: \(a_1\), oscillations of the carbon atom perpendicular to this direction: \(a_2\), and, finally, the symmetric oscillation of the two oxygen atoms relative to one another: \(a_3\). These three oscillations—the magnitudes of their constants will be \(a_1 = 2350,\ a_2 = 680,\ a_3 = 1330\) (for comparison it should be pointed out that the oscillations \(CO = 2150\) and \(O_2 = 1550\))—are assigned their energy values \(an - bn^2\). As a result, our original two-dimensional quantum scheme (p. 643) is now complicated into a scheme in \(3 \times 2\) dimensions. The spectrum expected in this way must therefore be highly complex, and precisely this actually existing complexity constitutes a reliable criterion that we are dealing with a polyatomic molecule. It should also be noted that in the case of carbon dioxide the carbon oscillation \(a_2 = 680\) is also found in the ultraviolet absorption spectra, whereas the other two are no longer observed here; however, an exact analysis of the spectrum is not yet available.
In the spectrum of polyatomic molecules, not only does the number of bands increase considerably, but also the number of lines within a band—the latter because rotations about several axes are possible here. For the case in which there are only two different moments of inertia, the formula for the rotational energy has already been given [Ur-ye (3)]; on the basis of this formula all the lines of our simple spectrum of a diatomic molecule now become starting points for new series of \(P\)-, \(Q\)- and \(R\)-branches. Thus the band possesses a double structure, which, of course, greatly complicates its analysis. It is to be welcomed that such a case of double structure has been analyzed in detail by Angi and Shue for formaldehyde
\[ O = C \begin{matrix} /H\\ \backslash H \end{matrix}, \]
i.e. for a relatively simple molecule.
Among polyatomic molecules, benzene and its infinite number of derivatives possess the most characteristic spectra and those most often investigated. These spectra
all lie in the region from \(\lambda 3000\) to \(\lambda 2300\); upon substitution they change little and, as a rule, consist of 6–8 sharply protruding groups, which follow one another at intervals on the frequency scale equal, in round numbers, to 930. This is one of the principal vibrations of the molecule in the excited state. Alongside it there is also found a second, which apparently depends to a greater degree on substitutions than the first. In individual derivatives it lies between 300 and 600. Whether other vibrations exist cannot be established with certainty, although it is highly probable. Likewise, we can as yet say little about the vibrations of the ground state, since the latter never appear sufficiently distinctly in absorption spectra, and it is chiefly with these that one has to deal. It is necessary to turn to the spectra of fluorescence, which have very low luminous intensity, since in the case of these spectra the final states most often figure (Stokes’ rule). It is then found that the vibration corresponding to the vibration 920 of the excited state possesses a value differing little from the latter. In addition, in absorption spectra there is found also a vibration at \(63\ \mathrm{cm}^{-1}\), which remains almost unchanged in all derivatives; nothing can yet be said about the nature of this vibration. It is of interest, however, to trace in detail the effect of substitution in the benzene ring on the principal vibration 920. When some element or group of atoms enters the benzene ring, the ring structure, which in itself is symmetrical, acquires asymmetry.
As a consequence of this asymmetry, the vibration 920, which is due to the internal vibrations of the carbon atoms relative to one another, splits into two fundamental vibrations. In simple monoderivatives F, Cl, Br, CH\(_3\), and C\(_2\)H\(_5\) the influence of this asymmetry is still insignificant (930 and 965), but it already becomes greater in phenol, anisole, anilines, nitrile, and aldehyde (OH, OCH\(_3\), NH\(_2\), CN, and COH); here vibrations 950 and 790 are observed. In diderivatives [for example, Cl\(_2\), Br\(_2\), (CH\(_3\))\(_2\)] the asymmetry becomes still greater; here there occurs an increase in the frequencies of the vibrations of the nuclei 1100 and 960
In particular, the para positions are of interest, where the two substituting atoms are located in the benzene ring diametrically opposite one another,
\[ \begin{array}{c} \mathrm{Cl}\\[-2mm] \hexagon\\[-2mm] \mathrm{Cl} \end{array} \]
As a result of such an arrangement in the ring there arises a sharply expressed axis of symmetry, which optically manifests itself in a considerable difference between the vibrations in the direction of this axis and perpendicular to it, 1050 and 780. Detailed data are given in Table 7, from which it is also seen how, as a result of substitution, the spectrum is displaced more and more toward the long-wave side. Thus, the zero point of the spectrum (and at the same time the most intense band) for benzene lies at 38 625 and is shifted to 34 040 for aniline. In Table 6 I use only the serial formulas of toluene, ethylbenzene, and chlorobenzene, since these formulas seem to me the most reliable.
TABLE 6.
Polyatomic molecules.
| Molecule | \(\nu_0\) | \(a_i'\) | \(b'\) | \(a_i''\) | \(J''\) | \(r''\) |
|---|---|---|---|---|---|---|
| Water \(\mathrm{H_2O}\) | 0 | 3750 | 0 | same | \(0{,}98\cdot 10^{-40}\) | H-H \(=1{,}09\cdot 10^{-8}\) |
| Water \(\mathrm{H_2O}\) | 0 | 1850 | — | same | 1,35 | O-H \(=0{,}87\) |
| Carbon dioxide \(\mathrm{CO_2}\) | 0 | — | — | same | 2,32 | |
| Carbon dioxide \(\mathrm{CO_2}\) | 0 | 2350 | — | same | 50 | O-O \(=1{,}95\) |
| Carbon dioxide \(\mathrm{CO_2}\) | 0 | 1380 | — | same | — | C-O \(=0{,}96\) |
| Toluene \(\mathrm{C_6H_5CH_3}\) | 37 428 | 680 | — | same | — | — |
| Toluene \(\mathrm{C_6H_5CH_3}\) | 37 428 | 964 | 0 | same | — | — |
| Toluene \(\mathrm{C_6H_5CH_3}\) | 37 428 | 932 | 1,6 | (890) | — | — |
| Ethylbenzene \(\mathrm{C_6H_5C_2H_5}\) | 37 595 | 570,5 | 0 | 63,9? | — | — |
| Ethylbenzene \(\mathrm{C_6H_5C_2H_5}\) | 37 595 | 960 | — | — | — | — |
| Ethylbenzene \(\mathrm{C_6H_5C_2H_5}\) | 37 595 | 930 | 1,6 | (890) | — | — |
| Chlorobenzene \(\mathrm{C_6H_5Cl}\) | 37 068 | 564,5 | 0 | 64,3 | — | — |
| Chlorobenzene \(\mathrm{C_6H_5Cl}\) | 37 068 | 983,7 | 0 | (900) | — | — |
| Chlorobenzene \(\mathrm{C_6H_5Cl}\) | 37 068 | 932 | 0 | 443 | — | — |
| Chlorobenzene \(\mathrm{C_6H_5Cl}\) | 37 068 | 530 | 0 | 60? | — | — |
| C—H bond | — | 3090 | 58 | — | — | \(\sim 1{,}0\cdot 10^{-8}\) |
In the infrared rotation-vibration spectrum there is also added the vibration \(3090\,n - 58\,n^2\), which corresponds to the CH bond in the molecule. It occurs in all organic compounds that contain a CH group.
and is the vibration most often observed. This vibration has also been established by means of the Raman effect, which shifts the infrared spectrum into the shorter-wavelength, photographically accessible region (p. 646); alongside this, however, the Raman effect has revealed a whole series of other infrared vibrations which have not yet received interpretation. Some of them may be identical with the above-mentioned vibrations of the ultraviolet spectrum.
TABLE 7.
Benzene derivatives.
| Substance | Formula | $\nu_0$ | $a_1$ | $\nu a_2$ | $a_3$ |
|---|---|---|---|---|---|
| Benzene | $C_6H_6$ | 38 624 | 922 | 922 | 465 |
| Toluene | $(C_6H_5)CH_3$ | 37 493 | 932 | 964 | 570 |
| Ethylbenzene | $(C_6H_5)C_2H_5$ | 37 595 | 930 | 960 | 564 |
| Fluorobenzene | $(C_6H_5)F$ | 37 827 | 920 | 970 | 600 |
| Chlorobenzene | $(C_6H_5)Cl$ | 37 065 | 932 | 964 | 530 |
| Bromobenzene | $(C_6H_5)Br$ | 37 008 | 930 | 958 | 512 |
| Phenol | $(C_6H_5)OH$ | 36 365 | 782 | 935 | 475 |
| Anisole | $(C_6H_5)OCH_3$ | 35 210 | 800 | 953 | 408 |
| Aniline | $(C_6H_5)NH_2$ | 34 044 | 797 | 954 | 492 |
| Benzonitrile | $(C_6H_5)CN$ | 36 535 | 706 | 940 | 508 |
| Benzaldehyde | $(C_6H_5)COH$ | 35 210 | 800 | 953 | 408 |
| m-dichlorobenzene | $(C_6H_4)Cl_2$ | 36 199 | 1102 | 958 | 366 |
| m-dibromobenzene | $(C_6H_4)Br_2$ | 36 061 | 1051 | 957 | — |
| m-chlorotoluene | $(C_6H_4)CH_3Br$ | 36 646 | 1220 | 960 | — |
| m-bromotoluene | $(C_6H_4)CH_3Br$ | 36 581 | — | 960 | — |
| o-dichlorobenzene | $(C_6H_4)Cl_2$ | 36 240 | 1089 | 958 | 440 |
| o-dibromobenzene | $(C_6H_4)Br_2$ | 36 123 | 1029 | 955 | 365 |
| o-chlorotoluene | $(C_6H_4)ClCH_3$ | 36 882 | 1190 | 970 | — |
| o-bromotoluene | $(C_6H_4)BrCH_3$ | 36 853 | — | 960 | — |
| p-xylene | $(C_6H_4)(CH_3)_2$ | 35 907 | 1190 | 780 | — |
| p-dichlorobenzene | $(C_6H_4)Cl_2$ | 35 735 | 1063 | 735 | — |
In all these spectra of polyatomic molecules, one favorable circumstance is found which promises to render valuable service to structural chemistry. Namely, it turns out that the vibration between two atoms or groups of atoms is preserved also in complex molecules, so that other groups have little influence on such vibrations; this is confirmed, as we shall see below, also by the dissociation energies.
of these groups. Thus, once such a vibration, for example the vibration of a CH bond, has been established in simple molecules, we can always find it in more complex ones as well; newly arising vibrations can in that case be assigned to definite bonds, which leads to a synthesis of vibrations and, in turn, makes it possible to draw conclusions about the structure of molecules.
The continuous spectra are still difficult to interpret—not those continuous spectra which adjoin, on the short-wave side, band or line spectra (these will be discussed below), but those which often appear, apparently without connection with any other spectrum. Such continuous spectra are observed in haloids, in hydrogen halide compounds, in hydrogen itself, and in alkali-halide compounds. They are interpreted partly as “electron-affinity” spectra, partly as dissociation spectra or, conversely, as recombination spectra. But no unity in their explanation has yet been achieved. A characteristic representative of this type of spectra is the continuous spectrum of hydrogen, which extends from the visible region as far as the far ultraviolet. It has been interpreted in various ways: 1) as the spectrum of recombination of neutral atoms (Schüler and Wolf);¹ 2) as the luminescence of the dissociation of an excited hydrogen molecule (Blackett and Franck);² 3) as luminescence accompanying the reaction
\[ \mathrm{H}_2^+ + \mathrm{H}_2 = \mathrm{H}_3^+ + \mathrm{H} \]
(Dorsch and Kallmann);³ in this case, according to Herzberg’s investigations,⁴ ionized hydrogen molecules should play the principal role, so that the phenomenon, according to Herzberg, should be regarded as the luminescence of the dissociation of \(\mathrm{H}_2^+\). We see how many different interpretations have been proposed in this one case; on the other hand, these spectra play a major role in interpreting the mechanism of reactions.
¹ H. Schüler und K. L. Wolf. Z. Physik. 33, 42, 1925; 35, 477, 1926.
² P. M. S. Blackett und J. Franck. Z. Physik. 34, 889, 1925.
³ K. L. Dorsch und H. Kallmann. Z. Physik. 44, 565, 1927.
⁴ G. Herzberg. Ann. d. Physik. 84, 565, 1927.
4. Chemistry of Valence and Band Spectra.¹
The chemical properties of atoms and molecules, especially their valence, are explained by the electrodynamic actions of their electrons. This is why we shall now dwell in somewhat greater detail on electron shells, since this is of interest for band spectroscopy. From the standpoint of quantum theory, the motion of electrons must be unambiguously characterized by specifying five quantum numbers. There is no need here to proceed from the notion of Bohr’s elliptical orbits, which in the new quantum theory have lost their immediate meaning. We can describe the motion purely formally by means of these five numbers and represent it by means of a term symbol. First of all, we have the principal quantum number \(n\) (not identical with the oscillation number \(n\) used by us up to now). This principal quantum number indicates only in which layer of the electron shell the electron is located. In X-ray spectroscopy these shells are called the \(K\)-, \(L\)-, \(M\)-, \(N\)-shells, and between these designations and the values of the principal quantum number there is the following relation:
| Shell | \(K\) | \(L\) | \(M\) | \(N\) | \(O\) | \(P\) |
|---|---|---|---|---|---|---|
| \(n\) | 1 | 2 | 3 | 4 | 5 | 6 |
| Period | H—He | Li—Ne | Na—A | K—Kr | Rb—X | Cs—Em |
| \(Z\) | 1—2 | 3—10 | 11—18 | 19—36 | 37—54 | 55—86 |
At the same time the principal quantum number also determines the shell in which the chemical activity of the electrons is played out, i.e. the period of the periodic system in which the given element is situated.
¹ A necessary supplement to this chapter is the article by Kondrat’ev: Advances in Physical Sciences 9, 380, 1929. Ed.
If an electron revolves around an atomic nucleus, then according to the laws of physics it always possesses angular momentum (moment of momentum), which, as we have already seen in considering the rotational motion of nuclei, in quantum theory is always an integer if it is measured in units of
\[ \frac{h}{2\pi}. \]
According to Bohr this angular momentum, the so-called azimuthal quantum number \(k\), can take the values \(1, 2, 3, 4\ldots\), with the restriction, however, that \(k \leq n\). However, both empirically and from the new quantum theory, a result was obtained which is very difficult to understand: according to it, contrary to the apparently existing revolution of the electron, the angular momentum averaged over this revolution, under certain circumstances, can take the value \(0\) and in any case turns out to be one unit less than Bohr’s \(k\). This circumstance can in no way be reconciled with Bohr’s conception of elliptical orbits. This new quantum number is at present denoted by \(l\), and moreover \(l = k - 1 \leq n - 1\). In order to be able to introduce this number into the term symbol, the already previously established symbols for terms are adopted: \(s\) (= scharf—sharp series), \(p\) (= prinzipal—principal series), \(d\) (= diffus = diffuse series, etc.), and they are identified with the quantum number \(l\) according to the following scheme:
| \(l\) | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|
| Term | \(s\) | \(p\) | \(d\) | \(f\) | \(g\) |
In this notation the principal quantum number is placed before the corresponding letter, for example, \(2p\) (\(L\)-shell), \(3d\) (\(M\)-shell). To this angular momentum one now adds another; thanks to the discovery of this latter, Goudsmit and Uhlenbeck at one stroke placed the entire theory of spectral series on a broader and, above all, more uniform foundation. This angular momentum is due to the rotation of the electron about its own axis; it is measured by the number \(\frac{1}{2}\) in units of \(\frac{h}{2\pi}\),—a further physically difficult
accessible to understanding the property of the electron. This impulse is usually denoted by \(s\) (not to be confused with the term \(S\)!).
However, the structure of an atom or molecule always includes several electrons, each of which possesses an impulse determined by its revolution in an orbit and by rotation about its own axis. These impulses, as vectors, are added according to the laws of geometrical addition, but in such a way that the resultant impulse is again an integer or, in the case of \(s\), a multiple of one half. We thus obtain the resultant impulse of the orbit \(l=\Sigma l_i\) and the resultant intrinsic impulse \(s=\Sigma\left(\frac{1}{2}\right)\). These resultant impulses must also find expression in the symbol of the term, as a result of which capital letters are used for the total impulse of the orbit instead of small letters. Therefore:
| \(\Sigma l_i\) | . | . | . | . | . | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|---|---|---|---|---|
| Term | . | . | . | . | . | \(S\) | \(P\) | \(D\) | \(F\) | \(G\) |
The next step forward in the characterization of an electronic system: the resultant impulse of the orbit \(l\) and the resultant intrinsic impulse \(s\) of the electrons are combined into the total impulse of the whole electronic system \(j\), but again according to the laws of integers, i.e. \(j\) can then assume only integer values, which lie between the maximum value \(=l+s\) and the minimum value \(=l-s\). If, for example, \(s\) has the value one half, then the total impulse can have only those values which are located between \(l+\frac{1}{2}\) and \(l-\frac{1}{2}\), i.e. for \(s=1\) it can have three values: \(l+1\), \(l\), \(l-1\), and, generally speaking, it is limited to \(2s+1\) different values.
Since the different mutual arrangements of both impulses relative to one another cause differences in energy, the possibilities of arrangement of \(l\) and \(s\) determine the multiplicity of the term (= splitting = number of possibilities of arrangement \(=2s+1\)). Therefore \(\Sigma\left(\frac{1}{2}\right)\) determines the multiplicity of the term, and this \((2s+1)\) is written on the left side above
before the term symbol; thereby the intrinsic angular momentum is also indicated (\(^{2}P=\) doublet, \(^{3}P=\) triplet, etc.). We obtain, consequently.
| \(s=\sum\left(\dfrac{1}{2}\right)\ldots\) | 0 | \(\dfrac{1}{2}\) | 1 | \(\dfrac{3}{2}\) | 2 | \(\dfrac{5}{2}\) |
|---|---|---|---|---|---|---|
| Multiplicity | Singlet | Doublet | Triplet | Quartet | Quintet | Sextet |
| Valence \(^{1}\) | 0 | 1 | 2 | 3 | 4 | 5 |
This relation between multiplicity and the intrinsic angular momentum of electrons immediately leads to an important law, the so-called alternation law, according to which atoms and molecules with an even number of electrons (\(s=0, 1, 2,\ldots\)) always possess odd multiplicity, while atoms and molecules with an odd number of electrons \(\left(s=\dfrac{1}{2}, \dfrac{3}{2}, \dfrac{5}{2}\ldots\right)\) possess even multiplicity. Therefore, in the periodic system, elements with even and odd multiplicity follow one another alternately. Neutral molecules of elements (\(\mathrm{Me}_2\)) always possess odd multiplicity (e.g. \(\mathrm{C}_2=\) triplets, \(\mathrm{N}_2=\) triplets, singlets, \(\mathrm{O}_2=\) triplets). In hydrides (\(\mathrm{MeH}\)) the multiplicity is always the opposite, whereas in oxides it is always the same as in the corresponding elements. I shall give, by way of example, only the multiplicities so far established \(^{2}\) for the elements of the first period and compare them with the corresponding numbers of electrons.
| BeH | BH | CH | NH | OH | HF | |
|---|---|---|---|---|---|---|
| Hydride . . . . . | BeH | BH | CH | NH | OH | HF |
| Number of electrons | 5 | 6 | 7 | 8 | 9 | 10 |
| Multiplicity | Doublet | ? | Doublet | Triplet | Doublet | Singlet |
| Oxide . . . . . | BeO | BO | CO | NO | \(\mathrm{O}_2\) | — |
| Number of electrons | 12 | 13 | 14 | 15 | 16 | — |
| Multiplicity | Singlet | Doublet | Singlet; Triplet | Doublet | Triplet | — |
\(^{1}\) On valence, see below.
\(^{2}\) R. Mecke. Z. Physik. 42, 590, 1927.
But if an atom or molecule is ionized, then at each degree of ionization the multiplicity changes and the element assumes all the properties of that element which has the same number of electrons as it (the so-called displacement law). For example, for molecules it has been established:
\[ \begin{aligned} \mathrm{N}_2&\text{—triplet;} & \mathrm{N}_2^{+}&\text{—doublet, similar to } \mathrm{CN}\\ \mathrm{O}_2&\text{—triplet;} & \mathrm{O}_2^{+}&\text{—}\quad ”\quad ”\quad ”\ \mathrm{NO}\\ \mathrm{CO}&\text{—singlet;} & \mathrm{C}^{+}\mathrm{O}&\text{—}\quad ”\quad ”\quad ”\ \mathrm{BO} \end{aligned} \]
These laws are especially important for molecular spectra, for here they often help to find the correct solution as to the molecule to which a band spectrum belongs.
In order also to include in the designation of a term the characteristic of the individual components of such multiplets, i.e. in order to indicate the individual values of the total angular momentum \(j\), the latter is written as a subscript at the lower right beside the term symbol (in order to avoid fractions in subscripts, for even multiplicities it is increased by one half and, for example, one writes 1 instead of \(1/2\), 2 instead of \(3/2\), etc.).
Thus the symbol \(^{2}{}^{3}P_{2}\) means that in this spectral term electrons of the \(L\)-shell take part, the angular momenta of whose orbits, on adding, give the value \(l=1\), while their intrinsic angular momenta, on adding, give the value \(s=1\), and the total angular momentum from \(l\) and \(s\) is \(j=2\).1 To these four quantum numbers a fifth is also added—and the last one—\(i\), when the electronic system is subjected to the action of external forces, which occurs, for example, in the presence of a magnetic field; this last quantum number is called magnetic. In these cases \(j\) is also established in a definite way with respect to the given chosen direction, namely it is established so that its projection on this direction, in turn, can
change by integral degrees. We have, consequently, here again \(2j+1\) different possible orientations. These possibilities of orientation we must especially take into account in molecules. Fig. 11 presents such a vector diagram of our four quantum numbers of angular momentum, whose specification, together with the principal quantum number, characterizes the electronic system in all its chemical and spectroscopic manifestations. Knowing these four quantum numbers, one can, for example, calculate the intensities of spectral lines or their splitting in a magnetic field; they also make it possible to draw conclusions about the magnetic properties of terms (for example, paramagnetism is always observed in those cases where \(j \ne 0\)); from these same quantum numbers are also determined the statistical weights \(g(=2j+1)\), with which we shall have to deal later, etc. For them, too, the selection rules are valid, with which we have already become acquainted in equation (12), when considering the rotation of nuclei; on these rules, however, we shall not dwell here.
Fig. 11.
The designation of a term gives no indication of the absolute magnitude of the electronic energy. In the case of line spectra it is represented by the Rydberg formulas \(\dfrac{R}{n^{*2}}\), where \(n^*\) denotes the so-called effective quantum number, which differs from the principal quantum number by a small nonintegral amount. For band spectra, as has already been mentioned, this magnitude of the electronic energy can be specified only in exceptional
... cases. This is grounded in the following: in line spectra, each change in electronic energy corresponds to only one spectral line. In the case of band spectra, numerous transitions between separate states of vibration and rotation are added here,
Fig. 12.
so that a definite change in the configuration of the electrons entails not one line, but thousands of lines. As a result of such considerable absorption of energy, which, moreover, is concentrated mainly at small energy steps, i.e. at the energy of rotation and vibration (the excitation occurs proportionally to \(e^{-\frac{kT}{W}}\)), only a few electronic transitions are observed.
Therefore, the calculation of absolute electronic terms is, as a rule, difficult to carry out, unless these terms can be determined in some other way, for example by the method of electron impacts. In cases where there are few rotational and vibrational states and their magnitudes are, moreover, comparable with the electronic energy—namely in the case of hydrogen and helium molecules—we also know most of the electronic terms. Thus, for the helium molecule it was possible to find an entire system of terms constructed quite analogously to the system of atomic terms. Among the remaining molecules,
Fig. 13.
Fig. 14.
the spectrum of the terms of CO is best known; it is very similar to the spectrum of Mg, or, better, to the spectrum of C++, since the two electrons of C drop out owing to the valence bonds (displacement law) (Fig. 12).¹
Although the notation for terms set forth above arose in connection with line spectra,² it can, however, be transferred directly to molecules without any restrictions,³—precisely for this reason we have dwelt on it in such detail. Only a small extension is required in accordance with the increased number of degrees—
¹ For numerical data see Table 2.
² Cf., for example, F. Hund, Linienspektren und Periodisches System der Elemente, or W. Grotrian, Graphische Darstellung der Spektren.
³ Cf. R. Mecke, Z. Physik. 28, 261, 1924; 36, 795, 1926; further, in particular, F. Hund, ibid. 36, 657, 1926; 42, 93, 1927, and R. Mulliken, Phys. Rev. 28, 481, 1202, 1926; 29, 391, 637; 30, 138, 785, 1927.
free molecule. Namely, in molecules both the axis connecting the nuclei and the axis perpendicular to them constitute preferred directions for the electronic momenta, so that here the conditions, at first sight, are apparently not so simple, and first of all the influence of both these directions on the motion of the electrons must be assessed. But here the principal role is played by the magnetic energy of binding, for according to the well-known laws of electrodynamics every electron moving in its orbit about the nucleus, or rotating about its own axis, as a charged body, excites a magnetic field, the magnitude and direction of which are determined by the moment of momentum. The momentum vectors are thus oriented in the magnetic field like magnets. If we take the simplest case, where the momentum of the electron’s orbit is equal to 0 (i.e. an $S$-term), then it turns out that the rotation of the nuclei also carries with it the intrinsic rotation of the electrons: the momentum corresponding to the latter rotation is always set parallel or antiparallel to the axis of rotation, i.e. perpendicular to the axis connecting the nuclei. The total momentum of the molecule, here further increased by the momentum of nuclear rotation $m$, will consequently be $j_m=m\pm s$, and the rotational energy, as before, in the first approximation
\[ W_r=Bm(m+1)=B(j\pm s)(j\pm s+1) \tag{21} \]
The difference in orientation (parallel or antiparallel) in the second approximation entails a small difference in energy, which increases proportionally to $m$ as the rotation of the nuclei increases. Namely:
\[ W_r=Bm(m+1)\pm\delta m; \]
we thus obtain a fine structure, precisely corresponding to type I, which for singlet terms ($s=0$), of course, disappears. If, in addition, there is also orbital momentum, e.g. a ${}^{2}P$-term ($l=1$ and $s=\frac{1}{2}$), then the total momentum of the electrons—in order to avoid confusing it with the total momentum of the molecule $j_m$, into which there also enters
rotation of the nuclei; we shall denote it by \(j_a\)—is established in such a way that the orbital momentum and the intrinsic momentum, according to the law of spatial quantization in the direction of the axis joining the nuclei, can take only the quantized values \(i_l\) and \(i_s\) (\(i=i_l+i_s\)). In the preceding example of a \({}^{2}P\)-term, \(i\) will take two possible values, \(i=\dfrac{3}{2}\) and \(i=\dfrac{1}{2}\), which, just as in atoms, differ very greatly from one another energetically. The doublet splittings are even, in magnitude, almost identical with the splittings of atoms having an equal number of electrons. As an example I shall give some spectra of hydrides, which can be compared with the spectra of the preceding elements, since hydrogen, when a molecule is formed, captures one electron of the element.
TABLE 8.
| CH | NH | OH | MgH | CaH | ZnH | CdH | HgH | |
|---|---|---|---|---|---|---|---|---|
| Molecule \(\Delta \nu\) | 19 | 25 36 | 126 | 22 | 80 | 330 | 1001 | 3688 |
| Atom | B | C | N | Na | K | Cu | Ag | Au |
| \(\Delta \nu\) | 25 | 22 39 | 138 | 27 | 82 | 357 | 1040 | 3610 |
The total momentum of the molecule will here be \(j_m^{2}=m^{2}+i^{2}\) (Fig. 13), and the rotational energy will therefore be expressed by the formula
\[ W_r=B\,[jm(jm+1)-i^2]; \tag{22} \]
in this case we count the lines of the bands according to the values of \(j\), and not according to the values of \(m\)¹ (\(i\) remains constant). As soon, however, as the rotation of the molecule becomes so strong that the magnetic coupling between \(l\) and \(s\) can be weakened,²
¹ The momenta which here and in equation (21) are quantized will now be \(j\) and \(i\), \(m=\sqrt{j^2-i^2}\) and is not quantized. Equation (22) is constructed analogously to equation (3).
² When this occurs depends on the magnitude of the multiplet splitting \(\Delta \nu\) in comparison with the magnitude \(B\). We obtain a pure type II when
\[ \frac{\Delta \nu}{4B}<10; \]
see R. Mecke, Z. Physik. 36, 795, 1925; R. Mulliken, Phys. Rev. 32, 398, 1928.
then the angular momentum of the electrons’ intrinsic rotation again tends to align itself in the direction of its own axis (Fig. 14); and since at the same time the cause of the doublet splitting \(\left(l+\frac{1}{2},\,l-\frac{1}{2}\right)\) disappears, the doublet lines contract as the rotation increases, i.e. we obtain type II of our structure of band spectra. Here also, as a consequence of the presence of different possibilities for the position of \(i\) relative to the axis connecting the nuclei, a fine structure arises, which is indicated in Fig. 8 by the dashed branches. Type III, like the triplet \((s=1)\), corresponds to three values \(i=l+1\), \(i=l\), \(i=l-1\).
I have already pointed out that in the case of type II (and also III), according to equation (22), the counting of lines is carried out according to \(j\), and not, as in case I, according to \(m=j+s\). This has two kinds of consequences. First of all, in the case of doublet terms (quartets, etc.), as a result of the addition of the electronic angular momentum, \(m\) becomes half-integral \(\left(\frac{1}{2},\,1\frac{1}{2},\,2\frac{1}{2}\right)\); as a consequence of this our combination relations, for example those expressed by equation (19), take the form
\[ \Delta^2 F(j)=F(j+1)-F(j-1)=4Bm^*;\quad m^*=j-\frac{1}{2}=1,\,2,\,3\ldots \]
instead of \(4B\left(m^*+\frac{1}{2}\right)\) in case I. The doublet structure here becomes at once noticeable in these differences. Further, the minimal value that \(i\) can in general assume coincides with the magnitude of the electronic angular momentum; consequently, for doublet bands it will in one case be \(i=\frac{1}{2}\), and in another \(i=\frac{3}{2}\); therefore between the \(R\)- and \(P\)-branches in the latter case two lines drop out instead of one. The gap in the band is thus widened, and its width makes it possible in the given case to draw a conclusion concerning the electronic term. To this is added one more criterion for identifying regular terms: if the transitions take place between identical terms (for example, \(S-S\), \(P-P\)), then the \(Q\)-branches are absent, or at least decrease so rapidly in intensity that in the best case
available for observation. In the case of transitions between terms of different names, such as \(P-S\), \(P-D\), the \(Q\)-branches appear very intensely. We see, therefore, that analysis of the structure of bands makes it possible to draw reliable conclusions regarding electronic transitions, and in the following table the various criteria are once again compared: type, number of branches, fine structure, number of lines falling between \(P\) and \(R\), the band formula \(\Delta^2 F=F(m+1)-F(m-1)\), and typical examples.
TABLE 9.
Interpretation of terms.
| Term | Type | Number of branches | Fine structure | Absence of line\(^3\) between \(R\) and \(P\) | \(\Delta^2F(m^1)\) | Examples |
|---|---|---|---|---|---|---|
| \({}^1S-{}^1S\) | I | \(P, R\) | no | 1 | \(4B(m^*+\tfrac12)\) | CuH, HCl |
| \({}^1S-{}^1P\) | I | \(P, Q, R\) | no\(^1\) | 2 | \(4B(m^*+\tfrac12)\) | CO, AlH, He\(_2\) |
| \({}^1P-{}^1D\) | I | \(P, Q, R\) | yes | 4 | \(4B(m^*+\tfrac12)\) | He\(_2\) |
| \({}^2S-{}^2S\) | I | \(P, R\) | ” | 1 | \(4B(m^*+\tfrac12)\) | CH, N\(_2^+\) |
| \({}^2S-{}^2P_i\) | II | \(2P, 2Q, 2R\) | ” | 1 and 2 | \(4B'(m^*+\tfrac12)\); \(4B''m^*\) | NO, HgH |
| \({}^2P_i-{}^2P_i\) | II | \(2P, 2R\)\(^2\) | ” | 2 and 4 | \(4Bm^*\) | NO\(_\beta\), Cu, \(\lambda 3900\) |
| \({}^2P_i-D_i\) | II | \(2P, 2Q, 2R\) | ” | 3 and 5 | \(4Bm^*\) | CH, \(\lambda 4300\) |
| \({}^3S-{}^3S\) | I | \(P, R\) | ” | 1 | \(4B(m^*+\tfrac12)\) | O\(_2\) |
| \({}^4S-{}^3P_i\) | III | \(3P, 3Q, 3R\) | ” | 1 and 2 and 3 | \(4B(m^*+\tfrac12)\) | NH |
| \({}^3P_i-{}^3P_i\) | III | \(3P, 3R\) | ” | 1 and 3 and 5 | \(4B(m^*+\tfrac12)\) | N\(_2\), C\(_2\) |
After this digression into the theory of series, let us return to chemical questions. We have seen that in a spectral term \((S_1P_1D\ldots)\) several electrons usually take part; the motion of these in the atom or molecule we
\(^1\) The term \({}^1P\) possesses fine structure, owing to the possibility of establishing \(\pm i\). As a consequence of the selection rules it does not appear directly, but only in the so-called combination defects when equation (18) is applied; see Mulliken, Phys. Rev. 28, 1202, 1926.
\(^2\) Very weak, rapidly converging to zero \(Q\)-branch.
\(^3\) Their number \(=i'+i''+1\).
must know. However, for the description of terms we have no need to involve all the electrons present in the atom, which would be inconvenient, and in heavy atoms and complex molecules, moreover, difficult to carry out. It turns out, however, that electrons have a strong tendency to arrange themselves in such a way that their momenta annihilate one another, so that \(\Sigma l=0\) and \(\Sigma s=0\); but once such an arrangement has been achieved, these electrons at once fall outside the circle of our considerations, since by such mutual compensation of their momenta they become, electrodynamically, i.e. chemically, inactive and manifest themselves only electrostatically, screening part of the charge of the atomic nucleus. They form a completed shell. Of course, several electrons are always required for this, but their number is limited by the so-called “Pauli prohibition,” according to which two electrons cannot exist that possess completely identical five quantum numbers. Thus, for example, even two \(S\)-electrons form a closed shell, in which the two intrinsic momenta are oriented oppositely to each other. For the same purpose, six \(p\)-electrons \((l=1)\) are required, and already ten \(d\)-electrons \((l=2)\), until both \(\Sigma l=0\) and \(\Sigma s=0\); consequently, in general, \(2(2l+1)\) electrons.^1 If one also takes into account the condition \(l \leq n-1\), then these numbers at the same time determine the maximum number of electrons in the various shells, i.e. the lengths of the separate periods of the periodic system: \(2\), \(2+6=8\), \(2+6+10=18\), and so on. Thus only the remainder of uncompensated electrons has an influence on the chemical and spectroscopic properties of the element. But the vector of the moment of quantity of motion of the proper rotation of the electrons tends to combine with a partner in such a way that the intrinsic momenta compensate one another. If it does not find such a partner (in this
^1 The number of possible settings of \(l\) is \(2l+1\), for each electron: \(+\frac{1}{2}\) and \(-\frac{1}{2}\). Therefore the total number of electrons in the shell is
\[ =2(2l+1). \]
play a role—the energy relations) in its own electron system, then it seeks it in a neighboring atom or molecule, i.e. the atom enters into a valence bond, and the number of uncompensated electrons is thus equivalent to the valence of the element. But since this number is connected with the multiplicity, the valence \(w\) and the multiplicity \(M\) are closely related to one another: \(w=M-1\).
We see from this that singlet terms (\(M=1\)) have valence 0, so that here we have elements or compounds of the noble-gas type, i.e. with saturated valences. Further, we conclude from this that in an atom or molecule the valence can also change by excitation, precisely when the term assumes another multiplicity. But the valences, and likewise the multiplicities, must in this case, according to the transition law, always change by 2 units, i.e. assume the values 1, 3, 5 (doublet, quartet, sextet, term) or 0, 2, 4, 6 (singlet, triplet, quintet, septet term). The multiplicity of the ground term in such a case always determines the principal valence. We see from this that the investigation of the deepest spectral terms of atoms and molecules at the same time permits conclusions to be drawn regarding their structure. Let us consider oxygen as an example. The ground term of its atom is the \({}^{3}P_i\)-term, with the following arrangement of its 8 electrons (by \(+1,0,-1\)
| \(K\) | \(L\) | \(\Sigma l\) | \(\Sigma s\) | |||
|---|---|---|---|---|---|---|
| \(l\ldots\) | \(0\) | \(0\) | \(1\) | |||
| \(i_l\ldots\) | \(0\) | \(0\) | \(+1\quad +1\) | \(0\quad -1\) | \(1\) | |
| \(s\ldots\) | \(+\dfrac12\quad -\dfrac12\) | \(+\dfrac12\quad -\dfrac12\) | \(+\dfrac12\quad -\dfrac12\) | \(+\dfrac12\quad +\dfrac12\) | \(1\) |
are denoted the possibilities of arrangement \(l\)). The two intrinsic momenta of the electrons, therefore, are not compensated. Hence oxygen is divalent. The oxygen molecule has as its deepest term a \({}^{3}S\)-term; consequent-
consequently, with its two uncompensated electrons \((\Sigma s=1,\ \Sigma l=0)\), it must possess two more free valences, and therefore only one valence bond \((-\mathrm{O}-\mathrm{O}-)\), which explains its great chemical activity. London1 established a rule which, in the case of homopolar valence, connects the valence (multiplicity) of molecules with the valences of the atoms and the number of valence bonds by the following relation:
\[ \begin{aligned} M_m &= M_1 + M_2 - 2Z - 1\\ W_m &= W_1 + W_2 - 2Z . \end{aligned} \]
This rule follows directly from what has been said above, for with each realization of a valence bond two electrons drop out. The question arises, however, whether this rule possesses generality, since it says nothing regarding energy relations. It does not say, for example, whether an electron prefers to combine with a partner from the electronic structure of its own atom (internal valence) or with an electron of another atom (a true valence bond).
The fundamental term of the carbon atom is a \({}^{3}P\)-term; consequently, carbon should be divalent. This is also reflected in carbon monoxide, which in its fundamental state possesses a singlet term, i.e. is completely saturated by its double bond \((C=O)\). Nevertheless, the two other electrons of the \(L\)-shell (\(S\)-electrons), which in the atom form a closed shell, prefer to combine with foreign electrons. And indeed carbon, as a rule, is tetravalent. Conversely, nitric oxide NO reveals one more free valence, since it forms a \({}^{2}P\)-term. But spectroscopically it exhibits such a close similarity to other oxides (OH, BO, CO) that this free valence probably belongs to oxygen; consequently, nitrogen is bound by one valence to oxygen, while by means of an internal valence bond the two remaining
of the valence electron have been made unambiguous. (In the atom, the fundamental term is, in all probability, a \(^{4}S\)-term, but there also exist doublet terms, which are deep terms; spectroscopically, the monovalence is entirely comprehensible.) With these examples we shall also conclude the chapter. Its principal task has been to show what significance the study of spectral terms has for structural chemistry.
(To be continued in the next issue).