STRIPED SPECTRA

E. Rabinovich, Göttingen

I. Introduction.
II. Vibrational spectrum.
A. Diatomic molecules:
1. Theoretical introduction.
2. How the system of vibrational terms is found?
3. What the system of vibrational terms teaches:
    1) isotopy,
    2) zero-point energy,
    3) merging of vibrational quanta; dissociation energy.
4. Potential curves:
    1) potential curves,
    2) the Franck and Condon principle,
    3) predissociation.
B. Polyatomic molecules.

I. INTRODUCTION

The term “striped spectra” dates from the time when spectroscopic phenomena could be classified only by their external features. As in many other cases, it subsequently turned out that the original empirical classification agrees with the theoretical one. In such cases the old empirical name is often retained, while its original meaning is gradually forgotten—this happened, for example, with the words “acid” and “aromatic compound” in chemistry. In exactly the same way, the term “arc spectrum” is now understood to mean not the spectrum that appears in a voltaic arc, but, in general, the spectrum emitted by a free neutral atom. The term “spark spectrum” denotes “the spectrum

positive atomic ion,” irrespective of whether this spectrum appears in a spark, an arc, or a Geissler tube. And finally, any spectrum belonging to a molecule is called a “band spectrum,” even in those cases when this spectrum does not reveal an immediately noticeable arrangement into bands. Thus those spectra which most deserve the name “band spectra,” namely spectra consisting of continuous bands that are not resolved into separate lines even at high dispersions, are excluded from the class of “band spectra proper” and are placed in the class of “true continuous spectra.” However, no sharp distinction is possible in this case either from a theoretical or from an empirical point of view; all possible transitions exist between spectra with a sharply banded structure and completely continuous spectra. In this review we shall deal almost exclusively with band spectra in the narrow sense of the word. Our present knowledge of the—still very little investigated—true spectral continua has been collected in Finkelnburg’s review.*

Band spectra proper, despite the enormous number of works devoted to them during the last 3–4 years, have also not yet been sufficiently investigated. In the theory of band spectra, the chapter “Electronic states of molecules and their correlations with the states of free atoms”—a chapter that has been developed very intensively in recent times—is still unfinished. Experimentally, almost only the field of diatomic molecules has been worked out in detail. But even here, for many molecules only one system of bands has been studied, which corresponds approximately to the knowledge of only one line in an atomic spectrum. It may be said that in general there exist only three molecules—H₂, He₂, and CO—which are known spectroscopically as fully as most free atoms. In the field of band spectroscopy, the experimenter is still faced with an infinite number of unresolved problems.

* Finkelnburg, Phys. ZS, 31, 1, 1930.

Band spectra, like line spectra, consist of individual lines which, although they do have a definite width, may in a first approximation be regarded as monochromatic. In quantum theory, each such line of a definite wavelength \(\lambda\), frequency \(\nu\), and wave number \(\nu\) is reduced to a combination of two states with different energy (Bohr’s frequency condition):

\[ \nu = F' - F''. \tag{1} \]

If a spectral line is characterized by its wave number \(\nu\), then the quantities \(F\) and \(F'\), the so-called spectral terms, are proportional to the energies of the radiating particle before and after emission. The “term system” is equivalent (up to the proportionality factor \(hc\)) to the system of energy states of the atom or molecule.

Fig. 1.

Already the general appearance of typical band spectra permits one to draw certain conclusions about the structure of the term system of a polyatomic molecule. Fig. 1 gives 3 photographs in which (alongside various atomic lines) several so-called band systems can be detected. Spectrum I contains only one band system of the CN molecule; spectrum II, in addition, also contains a band system of the AlO molecule; spectrum III gives the band systems of CO and CH. But the spectrum of each molecule, generally speaking, contains several band systems; thus for CN a second band system is known in the red

part of the spectrum (6000–7000 Å); but this system is not visible in reproduction, which covers the region from 4000 to 5000 Å.

Each system, as is evident from Fig. 1, consists of a certain number of bands (in CN and AlO these bands are further combined into groups of bands—a regularity that has no general significance, but we shall return to it again on p. 573). Each band consists, finally, of separate lines situated very close to one another. Almost always at one end of the band there is an accumulation of lines (the head of the band, or edge), while at the other end the band gradually spreads out, the lines becoming ever sparser and weaker (the tail of the band).

The threefold subdivision of band spectra (system of bands—band—line) suggests that the system of terms of a molecule must also possess a threefold subdivision. Theoretical consideration of the molecule by the methods of quantum theory confirms this idea and leads to the following threefold variety of permissible states of the molecule.

First, every molecule—just like a free atom—may be in various states that differ in the structure of the electron shell. Among them one distinguishes the state of lowest energy—the “ground state”—and a series of “excited states.” The excitation energies, and consequently the corresponding values of the terms, are of the same order of magnitude as the excitation energies of atoms, for the individual electrons in molecules are bound approximately as strongly as in free atoms.

The second series of theoretically expected molecular terms corresponds to the series of allowed states of vibration of the atoms in the molecule. In diatomic molecules only one such series can be expected, since only one type of vibration is possible—vibrations along the line connecting the nuclei. A polyatomic molecule has several possibilities of vibration; it must therefore possess several series of vibrational terms, which may be superposed upon one another in a complex manner. Since the frequency of vibrations depends on the strength of the bond that holds the atoms together, and this force ...

if the bond has a different magnitude in different electronic states, then each electronic term has its own system of vibrational quanta.

The third quantized kind of motion of a diatomic or polyatomic molecule is rotation about an axis passing through the center of gravity. The frequency of this rotation in the “classical” picture and the magnitude of the “rotational quanta” in the quantum-mechanical picture are determined by the moment of inertia of the molecule. The moment of inertia of any body not possessing spherical symmetry is different for each direction of the axis; but mechanics shows that this moment can always be reduced to three “principal moments of inertia,” which are referred to three correspondingly chosen mutually perpendicular principal axes. For a system consisting of two material points (a diatomic molecule), one of these three principal moments of inertia is equal to zero, while the other two are equal to each other; therefore a diatomic molecule has only one series of rotational quanta. The same applies to polyatomic molecules with a linear arrangement of the atoms (for example \(O — C — O\)). Other polyatomic molecules possess two or three different principal moments of inertia, and therefore their spectra must contain two or three sequences of rotational terms.

If the force of the bond between the atoms is known, then it is possible to calculate theoretically the sequence of its vibrational terms. The sequence of rotational quanta can also be calculated in advance if the moment of inertia is known, i.e. if the masses of the nuclei and their distances are known. We shall deal with the exact relations later; here we shall content ourselves with establishing the orders of magnitude. The spacing of successive vibrational quanta for more or less firmly bound molecules has values of approximately \(1000\ \mathrm{cm}^{-1}\); the spacings decrease continuously with increasing vibrational quantum number. The rotational quanta (the spacings of successive rotational terms) have an order of magnitude of only \(10\) to \(100\ \mathrm{cm}^{-1}\); but they increase in proportion to the square of the rotational quantum number. Taken together, there results a schematic picture of the system of terms

molecules, depicted in Fig. 2. For simplicity the drawing shows only two electronic terms \(A\) and \(B\), at a distance of \(10\,000\ \mathrm{cm}^{-1}\). Above each of these states there is

[In the figure: “system of bands \(A \to B\), series \(A_0 \to B_\nu\), with finely structured bands \(A_2 \to B_0, B_\infty\)”; “segments of the rotational spectrum”; “rotational-vibrational spectrum, band \(0 \to 1\)”.]

Fig. 2.

one sequence of vibrational terms (the scheme thus refers to a diatomic molecule). In constructing the scheme it was assumed that the bond in the upper state is weaker than in the ground state, and that therefore the vibrational quanta above \(B\) are spaced more closely than above \(A\). The vibrational terms are denoted \(A_0, A_1, A_2, \ldots, A_i\); each

to each term there corresponds a definite “vibrational quantum number” \(v\) (equal to \(0, 1,\ldots\)). Naturally, only a limited number of vibrational terms can be plotted in the figure; in reality the terms crowd together at a certain convergence limit, indicated in Fig. 2 by a dotted line. We shall soon discuss the significance of this limit. On each vibrational term there is also a sequence of rotational terms. In order not to make the drawing too complicated, these latter are plotted only above the states \(A_0\), \(A_1\), and \(B_{0k}\). They are denoted \(A_0^0, A_0^1,\ldots\), in general \(A_v^k\); to each there corresponds a special value of the rotational quantum number \(K\)* (equal to \(0, 1, 2,\ldots\)). Since the spacing of terms with increasing rotational quantum number becomes ever larger, there is no convergence for rotational terms. Experimentally, however, only a limited number of these terms can be established.

From this distribution of the system of terms there follows the existence of three types of band spectra. The first and simplest kind is the so-called “pure rotational spectra.” In practice they are observed almost exclusively in absorption and in this case correspond to transitions between rotational terms of the ground state without change of the electronic or vibrational state. The change of the rotational quantum number obeys a general, theoretically simply justified selection rule, according to which this quantum number, in emission or absorption, may change** only by \(+1\), \(0\), or \(-1\). In accordance with this, a purely rotational spectrum is represented by the vertical arrows shown in the left-hand part of Fig. 2. These lines may be denoted \(A_0^0 \to A_0^1\), \(A_0^1 \to A_0^2\), and so on.

The middle part of Fig. 2 represents the occurrence of the so-called rotational-vibrational spectrum, or simply

* The designation of the rotational quantum number by the letter \(K\) corresponds to the convention now adopted; previously this number was denoted by \(m\) or \(j\).

** This rule is a simple transfer of the selection rule for the quantum number of the total angular momentum (“internal quantum number”) \(j\) of the atom to the molecule.

vibrational spectrum. And these spectra are also usually observed only in absorption; in this case they correspond to transitions between different vibrational terms of the ground state. To each such separate transition there corresponds a special band; the splitting of the band into separate lines is due to the fact that, simultaneously with the vibrational state, the rotational state of the molecule also changes. Each band contains a “zero place,” which corresponds to a purely vibrational transition, i.e. to a jump from a non-rotating state into another, likewise non-rotating, state. This zero place is shown in the middle part of Fig. 2 by the heavy vertical arrow. The other rotational lines are denoted by thin arrows. Since the selection rule for the rotational quantum number, as a rule, remains applicable also when the vibrational state changes simultaneously, the lines of each band may be divided into 3 groups: to one belong all lines with \(\Delta K=-1\), to another with \(\Delta K=0\), and to a third with \(\Delta K=+1\). The zero line is, of course, the first line of the second group. These groups are called the negative branch, the zero branch, and the positive branch, or also the \(P\)-, \(Q\)- and \(R\)-branches.*

In the right-hand part of Fig. 2 the origin of a system of electronic bands is shown. This system arises because on a definite “electronic jump”—in our case a jump from the ground state \(A\) to the excited state \(B\)—there are superposed various jumps of the vibrational quantum number \(v\). The system of electronic bands has the same structure as the rotational-vibrational system, but is displaced on the wavelength scale by the magnitude of the pure electronic jump \(v_{el}=A_0^0\to B_0^0\). We have indicated the rotational structure only for the first band (the \(0\to0\) band); in the others only the zero places are marked. Likewise, only bands originating from the non-oscillating lower state \((A_0)\) are taken into account; the latter, of course, forms only

* In rotational-vibrational spectra, in the majority of cases the zero branch is absent; this is a particular consequence of the general selection rule, with which we shall become acquainted subsequently.

part of the whole system. This part may be denoted by \(A_0 \to B_{v'}\), whereas the general formula for the whole system will be \(A_{v''} \to B_{v'}\) (\(v''\) is the vibrational quantum number of the lower state, \(v'\) the same for the upper state).

Owing to the already mentioned relation of the order of magnitudes between electronic, vibrational, and rotational quanta, pure rotational spectra lie in the far infrared region of the spectrum, rotational-vibrational spectra in the near infrared region (partly also in the visible); finally, the most important electronic bands lie in the visible and ultraviolet. For the spectroscopist, who is accustomed to working in the photographically accessible region of the spectrum, the latter bands are the most convenient.

The system of vibrational terms causes the system of electronic bands to split into separate bands, the so-called coarse structure, while the rotational terms cause the appearance of individual lines of bands—the fine structure of the band. In very light molecules \(H_2\), \(He_2\), \(LiH\), the rotational quanta, owing to the small magnitude of the moment of inertia, are very large (of the order of magnitude of vibrational quanta). The individual lines move apart, as a result of which the various bands are superposed on one another and the whole spectrum loses the typical appearance of a band spectrum. This explains the peculiar spectrum of hydrogen molecules and other light molecules, which resembles a line spectrum and is therefore called “many-lined.”

We now turn to the consideration of theoretical and empirical results relating to the system of molecular terms. From what has been said above it is clear that electronic terms can be found from the “zero positions” of electronic bands, vibrational terms from the zero lines of electronic or rotational-vibrational bands, whereas rotational terms are determined by investigating the fine structure.

The difficulties of infrared spectroscopy are the reason that rotational and rotational-vibrational spectra are well known only in rare cases. Likewise, these spectra are observed almost exclusively

mainly in absorption, but not in emission, so that on their basis one can draw conclusions only about the vibrational and rotational structures of the molecule in its normal, unexcited state. A further shortcoming of infrared spectroscopy consists in the small dispersion of the instruments, owing to which a complete separation of individual rotational lines from one another is impossible, and at the same time also a complete analysis of the fine structure of the band; this analysis can be carried out only in the region accessible to the photographic plate, i.e., with the newest sensitizers—approximately down to \(1 \mu\).

The study of the Raman effect makes it possible to avoid the difficulties of infrared spectroscopy. The so-called Raman lines, which arise when monochromatic light is scattered by molecules and are situated symmetrically about the exciting line, may be interpreted as the result of the addition or subtraction of vibrational or rotational quanta of the molecule with the quantum of the incident monochromatic light (combination scattering). These quanta can thus be calculated from the distances of the Raman lines from the exciting line. The Raman spectrum is, in a certain sense, the infrared spectrum of the molecule, shifted from its true position on the frequency scale by the magnitude of the quantum of the exciting light and thus shifted into a region easily accessible photographically.

We shall now consider the vibrational, rotational, and electronic terms of molecules separately.

II. VIBRATIONAL SPECTRUM

A. Diatomic molecules

1. Theoretical introduction. In a first approximation one may regard the vibrations of atoms in a molecule as elastic (harmonic vibrations). The force that draws the atoms toward their equilibrium position in this first approximation is proportional to the displacement of the atoms from the equilibrium position (“elongation”)

\[ K = -az, \tag{2} \]

where \(K\) is the force and \(z\) the elongation. The frequency \(\tilde{\omega}\) of the oscillations and their wave number \(\omega\) are determined, according to classical mechanics, by the formula

\[ \tilde{\omega}=\omega\cdot c=\frac{1}{2\pi}\sqrt{\frac{a}{\mu}}, \tag{3a} \]

where \(\mu\) is the so-called “reduced mass of the molecule.” For a molecule \(AB\) with mass \(m_A+m_B\), the reduced mass will be

\[ \mu=\frac{1}{m_A}+\frac{1}{m_B}=\frac{m_A\cdot m_B}{m_A+m_B}. \]

The frequency does not depend on the displacement, analogously to the number of oscillations of a pendulum.

In place of the classical oscillation frequencies, in quantum theory there appear oscillation terms. Their series is determined, according to the new quantum theory, by the equation

\[ F_n=n\omega;\qquad n=\frac{1}{2},\ \frac{3}{2},\ \frac{5}{2}, \tag{3b} \]

which may also be represented in the form

\[ F_v=\left(v+\frac{1}{2}\right)\omega;\qquad v=0,\ 1,\ 2. \tag{3c} \]

The original Planck theory of harmonic vibrators used half-integer coefficients in formula (3b) instead of integer ones. However, the new theory requires, in agreement with experiment, the introduction of an additive constant \(\frac{1}{2}\omega\). What precisely is now to be regarded as the “quantum number of oscillation”—the half-integer number \(n\), or the integer number \(v\), smaller by \(\frac{1}{2}\)—is a matter of definition. We shall use (as we have already done earlier) the integer numbers.

“Vibrational quanta” are the distances between successive vibrational terms; for them the relation holds

\[ \Delta F_v=(v+1)\omega-v\omega=\omega. \tag{3d} \]

Thus the quanta of vibration of a harmonic oscillator are constant and equal to the classical wave number. The radiation emitted by a harmonic oscillator has, in the classical theory, the same wave number \(\nu\) as the oscillator itself \((\nu=\omega)\); in the quantum theory, however, the wave numbers are determined by the differences of pairs of terms of the vibration

\[ \nu=(v'-v'')\omega=\Delta v\cdot\omega, \tag{4} \]

where \(v'\) is the vibrational quantum number of the initial state, and \(v''\) that of the final state.

In the second approximation one must take into account that the bonds in a molecule are not in fact elastic. If the atoms are moved farther and farther apart, then the force drawing them back will not increase without bound, but at a certain distance will reach a maximum value and then fall to zero; the atoms separate, and the molecule dissociates. Thus, instead of the simple condition (2), one should introduce a power series, for example of the form

\[ K=-az+bz^2+cz^3+\ldots \tag{5} \]

In the same way, in equation (3c) for the sequence of terms, higher powers of \(v\) will appear in this case. A good approximation to the true conditions is already obtained with the quadratic expression

\[ F_v=\left(v+\frac{1}{2}\right)\omega-\left(v+\frac{1}{2}\right)^2 x\omega, \tag{6} \]

where \(x\) is a small numerical factor which characterizes the “anharmonicity” of the bond and is therefore called the anharmonicity factor. This factor naturally depends on the coefficients \(b,c\ldots\) in the force law (5), just as \(\omega\), according to equation (3a), is determined by the coefficient \(a\). Formula (6) gives for the magnitude of successive vibrational quanta of the molecule the expression

\[ \Delta F_v=\omega-2vx\omega, \tag{7} \]

where \(v\) refers to the upper of the two terms. Thus the vibrational quanta decrease linearly with the quantum number \(v\);

But even this law (7) is, of course, only an approximation, as is evident from the fact that \(\Delta F\), according to equation (7), beginning with a certain value of \(v\), must become negative, and this has no physical meaning. Exact images of the system of vibrational terms also contain terms with \(v^3, v^4,\ldots\); they must give an asymptotic approach of \(\Delta F\) to zero, corresponding to the disappearance of the binding forces as the distance increases.

2. How the system of vibrational terms is determined. If one studies the absorption spectrum of a molecule in the near infrared region, then in the simplest cases an absorption spectrum is found which is schematically represented in Fig. 8.

Fig. 8.

This drawing represents a sequence of rotational-vibrational bands which lead from the non-vibrating “ground state” \((v = 0)\) of the molecule to the first, second, third, etc., vibrational states (cf. the term schemes in Fig. 2). Owing to the smallness of \(x\) in (6), the wave numbers of these bands, in the first approximation, form an arithmetic progression (in the case of the harmonic oscillator this is strictly true). Thus the bands stand to one another in the same relations as the fundamental tone to the overtones. Therefore one often speaks of the fundamental band and of “harmonic bands.” The width and structure of the bands are explained by the superposition of rotational terms; here we are interested only in the position of the “zero lines,” which correspond to transitions between non-rotating states. For bands with the structure shown in Fig. 3, the zero line lies at the minimum of intensity in the middle of the band; the wave number of this zero point thus gives the magnitude of the vibrational quantum.

At high temperatures absorption is caused not only by a molecule that is not vibrating, but also in states—

upon equilibrium, more and more vibrating molecules appear—first in the first, then in the second and higher vibrational states. The dependence of the relative intensity on temperature makes it possible to distinguish bands that originate from different vibrational states from one another and from bands that originate in a non-vibrating molecule.

The study of infrared absorption spectra thus leads directly to the recognition of the vibrational terms of the ground state of the molecule. However, in this way one can determine only a small number (not more than 4 or 5) of vibrational terms. Indeed, we have not yet mentioned that for a harmonic oscillator there is a selection rule which permits only transitions between neighboring vibrational terms, \(\Delta v = 1\). Therefore the absorption spectrum of a harmonic oscillator consists only of the fundamental band. Owing to the anharmonicity of the atomic bond in the molecule this rule loses its strict applicability; nevertheless, the intensity of the bands decreases very rapidly with increasing \(\Delta v\), even in real molecules. Thus only a few harmonic bands appear with an intensity accessible to measurement (Fig. 3). Only in the case when the change of vibrational state occurs simultaneously with an electronic jump—as happens in the case of electronic bands—do long series of separate bands appear, corresponding to jumps of the vibrational quantum number by 10, 20, and a larger number of units (see, for example, Fig. 5).

It should further be borne in mind that not all vibrations in a molecule can give rise to infrared bands. In the classical picture, the variable electromagnetic field of the incident wave can set an atomic system into vibration only when this system is not electrically quite neutral; for the field acts only on charges, and not on neutral particles of matter. The action is strongest when the system carries a free charge; it remains appreciable when the system at least has a dipole moment. Hence it follows that, among diatomic molecules, only those can possess rotational-vibra-

tion spectrum, which have a constant dipole moment. Symmetric molecules, such as \(\mathrm{H}_2\), \(\mathrm{N}_2\), etc., do not show any absorption in the infrared region.

Vibrational terms are also determined from the Raman spectra of molecules, analogously to the way they are determined from rotational-vibrational spectra. Here, however, in practice almost exclusively only the fundamental bands appear. In addition, in the case of Raman spectra the restriction concerning the necessity of the presence of a dipole moment is removed.

Measurements of the coarse structure of electronic bands also lead to the determination of vibrational terms, and moreover in the general case simultaneously for two electronic states of the molecule. To the initial and final state of the bands there corresponds to each its own system of vibrational terms; both of them may be represented by an equation of the type (6). The combination of these two systems leads to the distribution of the separate bands, which is represented by the equation

\[ \nu=\nu_{\mathrm{el}}+\left[\left(v'+\frac{1}{2}\right)\omega'-\left(v'+\frac{1}{2}\right)^2 x'\omega'\right]- \left[\left(v''+\frac{1}{2}\right)\omega''-\left(v''+\frac{1}{2}\right)^2 x''\omega''\right]. \tag{8} \]

The constant term \(\nu_{\mathrm{el}}\) represents the change in energy which corresponds to a purely electronic jump, i.e. to a transition between two non-oscillating and non-rotating states. The first bracket represents the vibrational energy of the upper state, the second—the vibrational energy of the lower state. Usually, in general, quantities referring to the upper, more energy-rich state are denoted by one prime, and quantities referring to the lower state by two primes. Formula (8) is quite strictly applicable to the zero lines of bands; but since many systems of bands still cannot be exactly analyzed, and their zero lines have not been established, wave numbers of the band heads are often inserted into formula (8) instead of the zero lines (the quantum formula). Since the heads of all bands of the system are approximately equally removed from the zero lines, in this way, according to gra—

…to a lesser extent the differences of the terms, and consequently the vibration quanta are not greatly distorted.

As regards the general distribution of bands in the system, two types of distribution are possible here. The first appears when the strength of the bond, i.e. the coefficients \(a\) in the equations for the force (2) and (5), and at the same time the magnitudes of the vibration terms \(\omega\) according to (3a), in both combining states do not differ too greatly. In these cases the bands of the system form separate groups (“partial systems”), which correspond to identical values of \(\Delta v\). Thus the bands \(0 \to 0,\ 1 \to 1,\ 2 \to 2\), etc., all lie approximately next to one another; likewise the bands \(1 \to 0,\ 2 \to 1,\ 3 \to 2\) \((\Delta v=-1)\), etc. An example may be seen in the cyanogen spectrum reproduced in Fig. 1. Fig. 4a gives the scheme of terms corresponding to this case.

Fig. 4.

Another distribution of bands is obtained when the bond in one state is very weak in comparison with the other. The position of the bands in such a case (see Fig. 4b) is essentially determined by the vibration quanta of the more strongly bound state. In these cases separate sequences of bands are obtained which possess a common vibrational term in the more strongly bound electronic state. In Fig. 4b this is shown by vertical lines. Similar sequences are observed, for example, in the spectra of halide molecules.

When a system of bands has been measured, one first attempts to distribute the wave numbers of the zero lines or band origins into a square scheme (“quantum scheme”) in such a way that the vertical differences in the horizontal rows and the horizontal…

... the inequalities in the vertical series were constant. The vertical inequalities \((\Delta_{pv})\) within the vertical series, as well as the horizontal inequalities in the horizontal series \((\Delta_{pv})\), must, however, decrease slowly and systematically, so that they can be represented by an equation of type (7) (Table 1, p. 597). These inequalities directly give the quanta of vibration of both states. Their course immediately gives the coefficient \(x\) in the quadratic term of formula (6) or (8). In order also to determine \(\omega\) (and together with it the electronic jump), it is necessary to establish the absolute numbering of the vibrational states. For the series of observed vibration terms does not always begin with the non-vibrating state \(v'=0\). This establishment is not always simple;

Fig. 5.

an unfailing criterion in some doubtful cases is provided by the study of the isotope effect (see below).

3. What the vibrational spectrum teaches. a) Isotopy. Besides the analysis of canal rays (mass spectroscopy), the most important method for the investigation of isotopy is the analysis of the vibrational spectrum. Two isotopic molecules, for example \(\mathrm{Cl}_{35}\ \mathrm{Cl}_{35}\) and \(\mathrm{Cl}_{35}\ \mathrm{Cl}_{37}\), possess somewhat different quanta of vibration. Generally speaking, the frequency of vibration (for example, in the case of a pendulum) is determined by the restoring force and by the mass of the vibrating particle. For two isotopic molecules the force is exactly or almost exactly the same, but the masses are substantially different. This leads to a difference of the vibrational frequencies in classical mechanics, or of the vibration terms in quantum theory. All bands thus undergo a splitting, which is excellently visible in Fig. 5 in the case of the \(BO\) bands.

Theory gives, for the distance between two vibration terms corresponding to identical values of the vibrational quantum number \(v\) for two isotopic molecules \(AB\) and \(AB'\) with masses,

\(m_A, m_B, m_{B'}\), and under the condition that \(m_{B'} > m_B\), the following expression

\[ \Delta \nu_v = (\rho-1)\left(v+\frac{1}{2}\right)\omega' - (\rho^2-1)\left(v+\frac{1}{2}\right)x'\omega', \tag{9} \]

where

\[ \rho = \sqrt{ \frac{m_A(m_A+m_B)} {m_{B'}(m_A+m_{B'})} }, \]

and \(x\) and \(\omega'\) are constants of the heavier of the two molecules.

By this formula one can calculate the displacement of the zero lines of individual bands. It turns out that the isotope splitting in a sequence \((v''=\mathrm{const})\), for a definite value of \(v'\), has a maximum. The position of this maximum can be used to determine the absolute values of the vibrational quantum numbers \(v'\) (this possibility has already been indicated above).

Observation of the isotope effect does not make it possible to determine the absolute masses of the individual isotopes, as is done by means of the mass spectroscope. However, it gives with exceptional accuracy the ratio of the masses of the isotopes of the given element. Likewise it makes it possible to find very rare isotopes and in this respect has an advantage over mass spectroscopy. Thus the analysis of bands has led not only to the confirmation of known cases of isotopy, but also to the discovery of new important isotopes—above all the following: C(13), N(15), O(17), Cl(36).

b) Zero energy. Application of formulas (6) and (9) to empirical results shows that the introduction of “half-integral” quantum numbers is confirmed by experiment. At the same time the long-supposed existence of the “zero energy” of vibrations is definitively established. In the so-called “non-oscillating” state \((v=0)\) the molecule still retains, according to equation (6), a reserve of vibrational energy of magnitude

\[ F_0=\frac{1}{2}\omega-\frac{1}{4}\omega x. \tag{10} \]

For the harmonic oscillator this zero-point energy is simply equal to half the quantum of vibration

\[ E_0=\frac{1}{2}\omega;\qquad W_0=\frac{1}{2}\omega hc. \tag{10a} \]

The most striking confirmation of the existence of this zero-point energy is the fact that in electronic bands the zero points of the bands \(0 \to 0\) (which are due to the transition from one non-oscillating state to another, likewise non-oscillating state) also reveal the isotopic splitting required by equation (9). Such splitting definitely indicates that “non-oscillating” molecules in reality are always still oscillating, and that therefore two isotopic molecules, even in the “non-oscillating” state, are energetically different.

c) Coalescence of vibrational quanta. Dissociation energy.* According to equation (6), the vibrational quanta continuously decrease as the vibrational quantum number increases; this is also shown in Figs. 2 and 4. In the end the vibrational quanta must turn to zero. At this moment the bonds prove to be broken, and the molecule dissociates. Determination of the vibrational energy that has been fully taken up by the molecule up to this moment thus makes it possible to find the dissociation energy.

This determination can be carried out in two different ways. The first method is direct; but it can be applied only to a few molecules. These are precisely those molecules in which the band spectrum extends up to very high vibrational numbers, the bands become increasingly crowded, and finally in a certain place the phenomenon of continuous absorption is found. This continuum may be interpreted analogously to the continuous spectra adjoining the limits of the line series of free atoms. In the case of atomic spectra, the appearance of continuous absorption is explained by the fact that the electron is completely separated from the atomic residue.

* H. Sponer, Ergebnisse der exakten Naturwissenschaften 6, 75, 1927; R. T. Birge, Trans. Faraday Soc. 25, 707, 1929.

and may carry with it any already unquantized amount of kinetic energy; here continuous absorption should be attributed to the complete separation of the atoms. The separated particles may in this case also fly apart with any store of kinetic energy, not subject to any quantization.

Observation of the boundary between the discontinuous band spectrum and the region of continuous absorption thus makes it possible to carry out a direct determination of the maximum vibrational energy that a molecule can receive before its dissociation.

Such a “point of convergence of bands” with the subsequent continuous absorption appears, for example, with great distinctness in the spectrum of $J_2$. This case served as the example on which Franck* and Dymond** first showed the possibility of the spectroscopic determination of the work of dissociation.

In most molecules, however, bands even far from such points of convergence become increasingly weaker; from their measurement one can directly determine only a limited number of vibrational quanta. In these cases it is possible to calculate the point of convergence, and consequently also the dissociation energy, by means of the appropriate extrapolation formula, as was first shown by Birge and Sponer.***

Such an extrapolation formula is already formula (6). The constants $\omega$ and $x$ can formally be calculated already from the first 2 vibrational quanta; but as soon as these constants are determined, then by formula (6) one can calculate all higher vibrational quanta and determine the place where these quanta turn into zero. This occurs, as is easy to verify, at the quantum number

\[ v=\frac{1}{2x}, \]

more precisely, for the next integer after $\frac{1}{2x}$. The complete

* J. Franck, Trans. Faraday Soc. 21, 1925.
** E. G. Dymond, Z. Physik 34, 553, 1925.
*** R. T. Birge und M. Sponer, Physical Rev. 28, 259, 1926.

the vibrational energy which the molecule can absorb in this way according to formula (7) will be

\[ \sum_{s=1}^{\frac{1}{2x}} \Delta H'_s = \frac{\omega}{4x}-\frac{\omega}{2}. \tag{11} \]

To this there is further added the zero-point energy in the amount of approximately \(\frac{\omega}{2}\), and we thus obtain for the work of dissociation the expression

\[ D=\frac{\omega}{4x}. \tag{12} \]

As was mentioned on p. 566, formula (6), and together with it formula (12), are inaccurate; extrapolation of the heat of dissociation by formula (12) is thus associated with a known inaccuracy, which will be the greater the smaller the number of vibrational quanta known experimentally. Apparently extrapolation by formula (12) always gives values that are too high; the vibrational quanta near dissociation decrease more rapidly than linearly. Questions relating to this have recently been discussed in detail in Birge’s work.*

Apart from the unreliability of extrapolation of the place of convergence, in the spectroscopic determination of the dissociation energy another difficulty also appears. For each term one obtains, generally speaking, its own convergence limit (sometimes several terms have a common limit). This shows that dissociation by means of vibrations may lead to different products, depending on the initial term of the molecule; atoms may be liberated either in their ground states or in states with an excited electron shell. Equally, under known circumstances dissociation into free ions is possible. Consequently, before deriving the thermochemical energy of dissociation of a molecule from the convergence limits, it is necessary to know the nature of the products of disso-

* K. T. Birge, Trans. Faraday Soc. 25, 707, 1929.

BAND SPECTRA

to express their energy reserve. In many cases it may be assumed that upon dissociation of a molecule in the ground state, atoms in the ground state are also obtained. However, this is not always the case; for example, dissociation of the CN molecule in the ground state leads to a normal N atom and an excited C atom. We shall deal with these relations in more detail elsewhere. We also give a summary table of spectroscopically determined dissociation energies (Table II, p. 593), all the data being recalculated for dissociation into normal atoms and therefore suitable for direct comparison with the corresponding thermochemical data.

4. Potential curves. a) Potential curves. In direct connection with the system of vibrational terms are the potential curves, i.e. curves representing the potential energy of a molecule as a function of the internuclear distance. The coefficients \(\omega, \omega x,\) etc., in the power series representing the vibrational terms according to p. 565, are determined by the coefficients \(a, b \ldots\) in the power representation of the force law. But the force is the negative first derivative of the potential; therefore the expression for the potential energy can be found by integrating the force law.

Fig. 6.

Let us begin again with the simplest example of purely harmonic vibration. The force law in this case will be \(K=-az\); consequently the potential law will be

\[ P=\frac{az^{2}}{2} \tag{13} \]

This is the equation of a parabola with its vertex at \(z=0\), i.e. at the equilibrium position of the molecule (the internuclear distance \(r_{0}\)). For simplicity we may measure both \(P\) and \(z\) in wave numbers, i.e. divide the ordinates of Fig. 6 directly in units of terms. The curve in Fig. 6 gives a visual representation of the course of the vibration: if one imagines one atom fixed—

linear, then the other will oscillate according to the same law as a ball placed in a parabolic bowl and attracted downward by the force of gravity.

One may combine the potential curve with the scheme of oscillatory terms in a single drawing (Fig. 6). The two points of intersection of the level of a given term with the potential curve determine the turning points of the oscillation, at which all the energy of the molecule will be potential. In every other phase of the oscillation (for example, at point \(A\)) the energy is composed of a kinetic part \(T\) and a potential part \(P\), the relative magnitudes of which are determined by the height at which the potential curve intersects the corresponding ordinate of the term.

Fig. 7.

We have already mentioned more than once that the real oscillations of molecules are not strictly elastic. In a first approximation, however, oscillations in the immediate vicinity of the equilibrium position may still be regarded as elastic. Thus, near the equilibrium position, the potential curve of any molecule may be approximated by a parabola.

If it is necessary to trace the potential curve in its further course, then for this it is necessary to know the subsequent terms of the force law (5), i.e., according to p. 565, the subsequent coefficients in the power series representing the quanta of oscillation. Already from the coefficient \(x\) of the quadratic term in formula (6) one can, with a known approximation, determine the deviation of the forces from the elastic law, and at the same time the deviation of the potential curve from the parabolic form. The presence of a negative quadratic term in formula (6) accounts for a change in the form of the potential curve corresponding to the schematic Fig. 7. For small distances between the nuclei the repulsive force rises more steeply than according to the linear law; therefore the left branch of the potential curve will be steeper than the parabola drawn with a dotted line,

corresponding to harmonic vibration. But when the distance is increased, the attractive force does not increase all the time, as it should according to a parabolic law; at a certain distance the force reaches a maximum and then begins to decrease. Therefore the potential curve has, on its right-hand branch, a point of inflection and finally passes into a horizontal straight line (Fig. 17). The height of this horizontal above the top of the curve determines, according to the considerations of the preceding paragraph, the dissociation energy of the corresponding electronic state. The vibration terms, which in harmonic vibration have constant spacings, here crowd together at the convergence limit \(AB\).

Fig. 8.

How is the potential curve of a molecule found in practice? Kratzer gave formulas by means of which one can pass from a stepwise representation of the vibration terms as a function of the vibrational quantum number—a dependence that can be obtained from empirical values of the terms—to a stepwise representation of the force and energy as functions of the displacement. By applying these formulas it is possible to establish the course of the potential, already at a considerable distance from the equilibrium position, with an accuracy corresponding to the empirical knowledge of the system of vibration terms. Often, however, it is of interest to know the entire course of the potential curve up to the very largest separations of the nuclei, when only the usual representation of the system of terms is known, containing, according to formula (6), only 2 terms. Morse* gave an important interpolation formula, which

* A. Kratzer, Z. Physik 3, 289, 1920; Sommerfeld-Festschrift, Verl. S. Hirzel, Leipzig.

* Ph. H. Morse, Physical Rev. 34*, 57, 1929.

allows one to construct approximately the potential curve, if the following parameters are known: the coefficients \(\omega\) and \(x\) in the formula for the vibrational terms and the dissociation energy of the term \(D\). This formula reads

\[ F = De^{-2az} - 2De^{-az}, \tag{14} \]

where

\[ a=\sqrt{\frac{8\pi^2 c\mu\omega x}{h}}=0.2454\sqrt{M\omega x}, \]

with \(\mu\) the reduced mass of the molecules in C.G.S. units, and \(M\) the same in units of atomic weight. The elongation \(z\) is equal to the difference between the normal distance of the nuclei \(r_0\) and the distance \(r\) at which they are at the given moment,

\[ z=r-r_0. \]

If \(r_0\) is determined from an analysis of the fine structure of the bands (by methods that will be set forth below), then from equation (14) one can calculate the potential energy of the molecule for any distance between the nuclei.

If a series of electronic states of the molecule is known, and for each of these states the potential curve is determined, then systems of curves are obtained; Fig. 8 may serve as an example of these. These curves are of great importance for the study of the spectral, and perhaps also the chemical, properties of the molecule.

In Fig. 8 there is also shown a potential curve which rises continuously and exhibits no minimum. The existence of such potential curves, which correspond to a constant repulsion of the atoms, was first predicted by Heitler and London* on theoretical grounds. But from spectra it is possible to obtain evidence that such curves really exist for the most varied pairs of atoms. If, by absorption or emission of radiation, a molecule passes into such a “repulsion state,” it immediately dissociates. These

* W. Heitler and F. London, Z. Physik 44, 455, 1927; J. Sigiura, Z. Physik 45, 484, 1927; R. Eisenschitz and F. London, Z. Physik 60, 491, 1930.

repulsion states are not quantized, so that transitions to them must correspond to continuous absorption or emission bands. Apparently many of the above-mentioned true continuous spectra can be interpreted in this way.

The picture given by Fig. 9 is suitable for a visual representation of the results of the dissociation of a molecule in various electronic states. The horizontals into which the potential curves pass give the system of terms of the free atoms into which the molecule dissociates. If, for example, the ground state of the molecule \(AB\) in Fig. 8 dissociates into two normal atoms, then the lowest horizontal straight line in the right-hand part of the figure may be denoted by \(A+B\). If, however, the first excited state of the molecule leads, for example, to a normal atom \(A\) and an excited atom \(B\), then the corresponding potential curve must pass into a straight line which must lie above the line \(A+B\) by the magnitude of the excitation energy of the atom \(B^*\) (the line \(A+B^*\) in Fig. 8). Fig. 8 also shows terms which lead to dissociation into \(A^*+B\) and into the ions \(A^+ + B^-\). Conversely, by constructing potential curves one can draw conclusions about the nature of the dissociation products, if one compares the separation of the horizontal branches of the curves with the separations in the term systems of the corresponding atoms. This assignment of molecular states to states of free atoms gives rise to a multitude of theoretical problems. Can one predict how many and what molecular states arise from the combination of two known atomic states? What energy must these states possess? Which of them correspond to stable molecules, and which to pure repulsion states in the sense of Heitler and London?

Fig. 9.

We shall still be concerned with all these questions in the second part of this survey. Here, however, we shall dwell above all

on a problem which theoretically has not yet been completely clarified. This problem consists in the question of the uniqueness of assigning molecular states to atomic states. Indeed, it sometimes happens that two potential curves come very close to one another, and first of all it is difficult to decide whether they intersect (Fig. 9a) or diverge (Fig. 9b). One may even doubt whether it has physical meaning to speak of the “intersection” of two potential curves. At the point of intersection the molecule, regardless of the path by which it has arrived there, has the same energy and the same internuclear distance. Can the states corresponding to the two curves differ in anything else? Can the molecule, having reached the point of intersection, “freely” choose between the two possibilities of further behavior that present themselves to it? In such a case it makes no sense to assign to any branch on the right-hand side of the point of intersection a definite branch on the left-hand side of it. Thus, when we come to a crossroads, we must decide along which of the subsequent paths to proceed, independently of the path by which we arrived there.

Fig. 10.

This question, for example, is important for deciding whether a given molecule is “ionic” or “atomic,” for Frank’s criterion for distinguishing them consists in the fact that atomic molecules in the ground state dissociate into neutral atoms, while ionic molecules dissociate into ions. In cases where the potential curve of the ground state intersects other potential curves, as always occurs, for example, in dissociation into ions* (see Fig. 10), one may therefore doubt the possibility of an unambiguous assignment.

The theoretical discussion concerning this problem is not yet finished. It seems, however, that the situation is as follows

* For the magnitudes of the electron affinities are always smaller than the ionization potentials, so that the energy of the system \(A^{+} + B^{-}\) is always greater than the energy of the system \(A + B\) (Fig. 11).

BAND SPECTRA

thus: when the terms intersect, one can carry out, although not with absolute certainty but with a certain probability (in the sense of statistical theory), an assignment of the sections of the curves to one another. In many cases the probabilities of transition from the left branch to one of the right ones are so different from one another that the assignment is practically unambiguous. It may be asserted that, generally speaking, two different states of the molecule correspond to the point of intersection. It is true that there is a possibility that, in passing through the point of intersection, the molecule will “jump” from one state to the other—such jumps, at equal energy and equal distance between the nuclei, are always possible according to wave mechanics. However, the probability of such a jump will be the smaller, the greater the difference between the two electronic states. In many cases it is already possible, from the form of the branches of the curves, to establish their pairwise correspondence.

Fig. 11.

\(v \backslash v'\)	0	1	2	3	4	5	6	7	8
0	9	8	7	5	3	1	x
1	8	3	6	6	5	3	2	x	x
2	6	6	4	4	4	4	3	2	x
3	2	5	4	4	1	3	2	2	x
4	x	3	5	4	4	1	2	2	2

Fig. 12.

b) The Franck—Condon principle. From the consideration of potential curves, Franck derived a very important principle determining the distribution of intensity in band systems. Condon* gave this principle, stat-

* E. U. Condon, Phys. Rev. 27, 640, 1926; 27, 1182, 1926; 32, 856, 1928.

* J. Franck, Trans. Faraday Soc. 21*, 1925.

given for the first time in qualitative form, a quantitative formulation following from the new quantum mechanics.

Let us consider a molecule with two potential curves, shown in Fig. 12. Suppose that the molecule is initially in the ground state. It then passes, by absorption of light (or as a result of an electron impact), into the upper excited state. The question is: are vibrational quanta excited in the molecule in this process? Or, what is the same thing, do there appear in the absorption spectrum only the bands \(0 \to 0, 1 \to 1, 2 \to 2\), etc., which correspond to transitions between “equally vibrating” states, or also the bands \(0 \to 1, 0 \to 2\), etc.? And if so, with what relative intensity?

Franck answered this question by means of the following reasoning. An electronic jump is a process that takes place very rapidly. While the light electron is already rising to its new orbit, the heavy nuclei practically cannot undergo a change in their relative positions or velocities. Thus, at the first instant after the absorption of light (or electron impact), there occurs only a transition of the molecule from some state on the lower potential curve to the state lying immediately above it on the upper potential curve, along the vertical straight line. Such transitions are indicated in Fig. 12 by vertical lines. Let, for example, the molecule initially execute a “half-oscillation” \(AB\) with zero energy. Then for this molecule only the transitions between \(AA'\) and \(BB'\) are allowed. If the transition takes place precisely at the moment when the molecule is at one of the turning points \(A\) or \(B\), then it passes into the states \(A'\) and \(B'\) and, after the transition, likewise has no kinetic energy. But since the states \(A'\) and \(B'\) do not correspond to equilibrium of the nuclei, the excited molecule must begin to vibrate. If the transition occurs from some intermediate phase of the vibration in the lower state, then the distances of the nuclei remain unchanged, and the nuclei retain their kinetic energy. Their position on the upper curve, in conjunction with the magnitude of the kine-

kinetic energy again determines the strength of the vibration which the molecule can execute after absorption. It should, however, be borne in mind that the velocity of motion of the nuclei at the turning points is the smallest and that therefore the molecule spends most of its time near the states \(A\) and \(B\). Consequently it is most probable that absorption will capture molecules precisely in one of these two states. Thus one may say that in absorption the strongest bands should be those which correspond to transitions from \(A\) and \(B\) to \(A'\) and \(B'\), respectively; hence, in the case shown in Fig. 11, the bands \(X_0 \to Y_0\) and \(X_0 \to Y_4\).

From each lower vibrational state there are two analogous most probable transitions. If below there is not just one vibrational state, but a whole series of them, then all these transitions appear simultaneously. As a result, in the system of bands obtained there arises a distribution of intensities which is characterized by the so-called “Condon parabola” and can most simply be represented visually by a “square scheme” of the type given in the table in Fig. 2.

The same, of course, is also true for emission spectra. Everywhere the most intense bands are precisely those which correspond to transitions without a change in the internuclear distance, and among them, in turn, the bands corresponding to the turning points of the vibration.

From the standpoint of this principle, the reason for the different structure of molecular spectra becomes clear. Fig. 13 illustrates several typical cases.

Fig. 13.

Fig. 13a represents a molecule which in the ground and excited states has almost the same internuclear distance and the same bond strength. In the spectrum of this molecule there appear above all bands for which—

the state of vibration remains unchanged (\(0 \to 0\), \(1 \to 1\), etc.; in general, the “diagonal terms” of the square table).

Fig. 13b represents a molecule with a strong bond in the ground state and a loose bond in the excited state (halides). Here, bands with a large change in the vibrational quantum number should predominate in the spectrum. Since the curve of the upper state already rises steeply where the lower curve is still relatively gentle, long sequences of bands may arise in the spectra, leading from the generally non-vibrating or only weakly vibrating lower state to a series of strongly vibrating states in the upper term, or even into the continuum (in reality, these quite different vibrational states correspond to almost identical internuclear distances at the turning points). Molecules must have this character if one is to be able to count on applying extrapolation of the place of band convergence in order to determine the dissociation energy.

An even more extreme case is that of Fig. 13c. Here the vertical transition from the non-vibrating ground state already leads to such a point of the upper curve which lies above the dissociation limit. Therefore, upon absorption, the molecule decomposes (direct photochemical decomposition). The nuclei acquire such an amount of potential energy that they must immediately fly apart. The molecule still carries out half an oscillation (from \(A\) to \(B\)), but thereafter the nuclei no longer change their direction; instead they separate with a kinetic energy corresponding to the height of point \(A\) above the straight line \(BX\).

In the spectra of molecules whose potential curves have the form shown in Fig. 13c, generally speaking, true “bands” do not appear; these spectra show only (or almost only) continuous absorption bands. Thus, the case favorable for determining the system of vibrational terms—a long sequence of bands and the continuum adjacent to it—can be realized only for quite definite potential relationships. Most molecules exhibit-

appear with sufficient intensity either only as the first bands of the sequence, or only as the continuum. In a series of halides, for example, only in iodine is the entire sequence of bands observed and the continuum strongly developed; in bromine, and still more in chlorine, the center of gravity of the absorption lies already far in the continuum, and the bands appear only very weakly. Above we proceeded as though each molecule possessed only two electronic states. In reality it always has several such states, and it is necessary to consider separately the conditions for each pair of such states, i.e., for each system of bands.

c) Predissociation.* Closely connected with the crossing of potential curves is the phenomenon known under the rather inappropriate name “predissociation.” It consists in the following: sometimes a sequence of absorption bands is observed which initially, for small quantum numbers, shows a normal structure. But at a certain point the bands suddenly become diffuse, and the band structure disappears completely or almost completely. On passing to still higher vibrational numbers, one can sometimes observe the opposite change; the bands again reveal their normal rotational structure. In the spectrum of a molecule, under certain circumstances, several such places of “predissociation” appear. The interpretation of this phenomenon, discovered by Henri, is as follows.** Let us imagine two potential curves that intersect at some point, this point on one curve corresponding to binding, while on the other—to dissociation. In Fig. 14 three possibilities for such an intersection are indicated. When, by absorption of light, the molecule is transferred from the state not shown in Fig. 14 to a state corresponding to the points of intersection \(X\), \(Y\), or \(Z\), then, as has been said, it is offered a choice between the two potential curves intersecting at that point. If

* Cf. V. Henri, Trans. Faraday Soc. 25, 765, 1929.

** The interpretation was given by K. F. Bonhoeffer and A. Farkas, Z. physikal. Ch. 134, 337, 1927, and R. de L. Kronig, Z. Physik 50, 347, 1928.

it chooses, for example, in \(X\) a “stable” curve \(AXB\) and repeats this choice at all subsequent passages through the point of intersection, then such a molecule must again return, with emission of light, to the ground state. If, however, the choice immediately or subsequently falls on the curve \(CXD\), then the molecule dissociates.

It can be shown that such a choice between dissociation and preservation of the molecule must lead to a broadening of the bands. This consequence is obtained rigorously by the methods of quantum mechanics; when using visual models of the molecule one has to resort to “heterogeneous” arguments characteristic of the older atomic theory, mixing classical concepts with quantum ones (an interpretation by means of the “correspondence principle”). According to quantum theory, the frequency of the emitted (or absorbed) light has nothing in common with how long the molecule remains in both combining states. Apparently, in particular, it is immaterial whether this “residence time” is shortened by the possibility of decay or not. But in the classical theory the frequency is determined by the harmonic analysis of the motion of the electrons; if the excited atom or molecule dissociates during emission, then the harmonic analysis of such a “prematurely broken off” train of waves gives less monochromatic radiation than in continuous emission. Therefore the emitted or absorbed line will be diffuse. From the point of view of the correspondence principle in quantum theory it follows that, when the “mean lifetime” of an excited atom is shortened for any reason, the lines originating from this state become diffuse; the same applies—though it is evidently still more difficult to understand this intuitively—

BAND SPECTRA

also to the lines leading to this state. Such a shortening of the mean lifetime is obtained as a consequence of the crossing of potential curves. Without crossing, the lifetime is determined by the probability with which the molecule returns to the ground state with the emission of light. During this period of the “mean lifetime” of the excited electronic state (approximately \(10^{-8}\) sec.) the molecule performs many vibrations (period approximately \(10^{-13}\) sec.). At each such vibration the molecule passes through the intersection of the curves, and at each such passage it has a finite probability of jumping to the other curve and undergoing decomposition. It is clear that this possibility greatly shortens the mean lifetime, and this in turn, as was said above, leads to a broadening of the lines. As a result of this, the individual rotational lines merge, and the rotational structure of the bands becomes indistinguishable.

The probability of jumping to the dissociation curve exists not only at the point of intersection itself, but also in its neighborhood. According to quantum mechanics, between any two states of equal energy there is a finite probability of transition. This probability, however, becomes small as soon as the transition must be associated with a considerable change in the distance between the nuclei or with a considerable conversion of potential energy into kinetic energy. This is precisely the reason why the phenomenon of predissociation is limited to the immediate neighborhood of the point of intersection of the potential curves, and why, at higher vibrational quantum numbers, the appearance of the normal structure of the bands is often observed again.

This criterion for the possibility of predissociation was indicated by Franck and Sponer.* If the condition of invariability of the internuclear distance and constancy of the kinetic energy did not have to be fulfilled, then almost all excited states of the molecule would be unstable, for they lie

* J. Franck und H. Sponer, Nachr. Ges. Wiss., Göttingen, 241, 1928.

almost always above the dissociation limit of the molecule in the ground state (Fig. 2); therefore a transition into a dissociated state with equal energy for all these excited states would be possible in itself.

Apart from the condition of incommensurability of the distance between nuclei, the probability of predissociation also depends on the general “similarity” of the two energetically equal states. By this similarity one should understand the equality of certain quantum numbers that characterize the electronic states and of which we shall speak in more detail later. The more “unlike” the states, the less probable the transition without radiation from one to the other, and the more unambiguous the assignment of separate branches and points of intersection of the potential curves to one another.

Thus predissociation may be found not in all cases when the electronic state lies above the dissociation limit of the molecule; but it can never be observed below this limit. At the same time, predissociation makes it possible to determine the upper limit of the dissociation energy of a molecule. The exact value of the dissociation energy can be obtained from predissociation only when the “dissociation curve” intersects already on a segment that runs horizontally (Fig. 14a). Apparently, however, in many cases this is in fact what occurs. At least for many compounds, dissociation energies are calculated from predissociation which come surprisingly close to the values determined by other methods.

B. Polyatomic Molecules

The investigation of the vibrational states of polyatomic molecules has so far made little progress. The discovery of the Raman effect brought great vitality into this field. Analysis of electronic bands has until now proved possible only for a few polyatomic molecules. The spectra of these molecules are too complicated, and many of them exhibit not discrete spectra at all, but continuous bands—evidently owing to the unfavorable forms of their potential ...

curves in excited states (see, for example, the study by Herzberg and Scheibe* on the spectra of halogen derivatives of methane). Therefore, in establishing vibrational terms it was necessary to rely on infrared absorption bands, the study of which, as already mentioned on p. 562, is associated with many experimental difficulties. The discovery of the Raman spectrum for the first time made it possible to carry out a systematic treatment of the numerous organic and inorganic compounds and to establish their fundamental vibrations. This work is at present being carried out intensively in a whole series of laboratories, for example by Raman with his collaborators, Kohlrausch and Dadieu, Wood, Doro, Petrikaln, and Toxberg, and by a number of other investigators.

In diatomic molecules, which possess only one fundamental vibration, we are interested above all in the series of vibrational terms and their convergence. In polyatomic molecules the foremost problem is that of establishing and interpreting the various fundamental vibrations. For a polyatomic molecule is a mechanical system with various possibilities of vibration, which may be reduced to a certain number of fundamental vibrations. To each such fundamental vibration there corresponds in the infrared spectrum its own fundamental band; besides the higher harmonic bands of all these vibrations, each of which can be characterized by a sequence of vibrational quantum numbers, there are also combination bands, corresponding to a simultaneous change of the various vibrational quantum numbers of the molecule.

Thus, the first stage in the analysis of the spectra of polyatomic molecules consists in reducing the numerous vibrational frequencies to combinations of a small number of fundamental vibrational quanta. As an example of such an analysis of the vibrational spectrum of a complex molecule, we give in Table 3 (p. 594) a summary of all the infrared absorption

* G. Herzberg, G. Scheibe, Z. physikal. Ch. (B) 7, 390, 1930.

bands of ammonia to the three fundamental frequencies \(\nu_1, \nu_2\) and \(\nu_3\) according to Ellis.*

For diatomic molecules we have already mentioned that only those among them can possess infrared absorption bands which have a dipole moment. For polyatomic molecules the situation is more complicated. Here the molecule as a whole may not possess a dipole moment, and nevertheless the excitation of individual fundamental vibrations corresponds to the appearance of such a dipole moment. On the other hand, it is also possible that the molecule has a dipole, but that excitation of the fundamental state leaves this dipole unchanged. In the first case the appearance of the corresponding bands in the infrared spectrum is possible; in the second it is impossible. Thus the question is not of the moment of the molecule as a whole, but of the way in which this moment depends on the excitation of the corresponding vibration. Different fundamental vibrations of a molecule may behave differently in this respect. Some fundamental vibrations may be active, others inactive. Inactive fundamental vibrations appear in infrared absorption bands only with the simultaneous excitation of active vibrations—consequently only as terms in the frequency of combination bands. On the contrary, in the Raman effect theory and experiment show in agreement that inactive vibrations are preferentially excited. In this respect too, the study of the Raman effect represents a valuable supplement to the study of the vibrational spectra proper. The Raman effect is especially useful for assigning fundamental frequencies also because in Raman spectra fundamental bands appear almost exclusively; harmonic and combination bands are either entirely absent or appear only very weakly.

The next stage in interpreting the vibrational spectra of polyatomic molecules is the establishment of a model of the molecule which could explain the system of fundamental vibrations found. The already successful establishment—

* J. W. Ellis, Journ. Franklin Inst. 208, 507, 1929.

BAND SPECTRA

such a model endows an empirically chosen system of fundamental vibrations with a certain plausibility. In doing so, one naturally proceeds from the models of molecules of structural chemistry, with their distribution of atoms and valence lines. For relatively simple molecules—for example \(CO_2\), \(H_2O\), or \(C_2H_2\)—one may try to specify completely the distribution of atoms in the molecule and from this to derive the systems of fundamental vibrations. Such considerations at first led to successful results only for a small number of molecules—see, for example, the works of Schaeffer and Matossi, concerning the molecules \(CO_2\) and \(H_2O\), of Badger and Mecke on \(NH_3\), and of Mecke on \(C_2H_2\). In such investigations it turns out that certain fundamental vibrations can be reduced to vibrations associated with a definite valence line (“internal vibrations” according to Kohlrausch, valence vibrations according to Mecke). Other fundamental vibrations do not admit such a simple interpretation (external vibrations according to Kohlrausch, deformation vibrations according to Mecke). The theoretical determination of the fundamental vibrations is in principle possible for any model of a molecule, but for complex molecules it proves mathematically difficult (see, for example, Brester). Some American investigators** recently attempted, for this purpose, to construct actual models of molecules out of heavy spheres and elastic spiral springs and to analyze the vibrational spectrum of such models with the aid of a stroboscope. For complex, especially organic, molecules, a whole series of investigators has successfully tried to reduce, in a purely empirical way, the repetition of identical or almost identical frequencies

* Schaeffer und Matossi, Das infrarote Spektrum.
R. M. Badger und R. Mecke, Z. physikal. Ch. (B) 5, 333, 1929.
R. Mecke, Z. Physik 64, 173, 1930.
** A. Dadieu und K. W. F. Kohlrausch, Ber. Dtsch. chem. Ges. 63, 251, 1930; K. W. F. Kohlrausch, Naturwiss. 18, 527, 1930.
R. Mecke, Z. Physik 64, 173, 1930.
*** Brester, Dissertation, Göttingen-Utrecht 1923.
**** C. F. Kettering, L. W. Shutts und D. H. Andrew, Phys. Rev. 36, 531, 1930.

oscillations in various homologous molecules with the repetition of identical valence bonds. The discovery of such typical vibration frequencies even promises to provide a new route for structural analysis, which should make it possible to draw conclusions about the presence of a definite bond in a molecule from the appearance of known frequencies in the infrared or Raman spectrum. Thus this should make it possible to distinguish between the keto and enol forms, or between a “true” acid (in the sense of Ganch) and a “pseudo-acid.”

In this abstract it is impossible to enter in detail into the very promising results of “spectral structural analysis.” The method is applicable to all three states of aggregation. A comparison of results, especially for solids, which have long since been investigated with respect to their infrared absorption, may be found in the book by Sheffer and Matossi.

The second problem of the analysis of the vibrational spectrum—the establishment of the sequence of vibrational terms and their points of coalescence in polyatomic molecules—has only just begun (see, for example, Ellis’s indications* concerning the extrapolation of the heats of dissociation of NH and CH bonds).

In drawing the conclusions that were made—especially by Andri and his coworkers**—from the positions of predissociation bands in the spectra of polyatomic molecules, an uncertainty is discovered which is due to the fact that a polyatomic molecule may break up into various fragments. For example, one cannot at once say whether the position of predissociation in the spectrum of CH₃Cl corresponds to decomposition into CH₃ and Cl or to the splitting off of one H atom. At the same time, the difficulties concerning the state of the cleavage products and the height of the point of intersection of the curves above the actual place of dissociation—difficulties which we have already emphasized in considering diatomic molecules—remain here as well. Nevertheless, from the positions of predissociation of certain—

* J. W. Ellis, Journ. Franklin Inst. 204, 507, 1929; Trans. Faraday Soc. 25, 888, 1929.

** See, for example, S. A. Schou, Journ. Chim. Physique 26, 1, 1930.

... molecules, for example \(NO_2\),* important thermochemical data are obtained, for example the heat of dissociation of oxygen, with sufficient certainty.

Table 1

Quantum scheme of the “second positive group” in the band spectrum of \(N_2\) (bold numbers denote wave numbers of band heads; the ordinary printed numbers are differences of wave numbers, which are equal to vibrational quanta)

\(v''\)	0	\(\Delta\)	1	\(\Delta\)	2	\(\Delta\)	3
0	29 655,6	(1704,8)	27 950,8	(1615,0)	20 275,8	(1645,2)	24 630,6
\(\Delta\)	(1989,0)		(1994,4)		(1993,8)		(1993,0)
1	31 646,6	(1703,4)	29 945,2	(1645,6)	28 269,6	(1646,0)	26 623,6
\(\Delta\)	(1937,1)		(1937,5)		(1939,8)		(1939,7)
2	33 585,7	(1703,0)	31 882,7	(1673,3)	30 209,4	(1646,1)	28 563,3
\(\Delta\)	(1870,0)		(1870,8)		(1870,7)		(1872,3)
3	35 455,7	(1702,2)	33 753,5	(1673,4)	32 080,1	(1644,5)	30 435,6

Table 2

Spectroscopic determination of heats of dissociation

Reaction	Heat of dissociation, volts	Heat of dissociation, calories
\(H + H = H_2\)	\(4,34 \pm 0,1\)	\(98,1 \pm 2,3\)
\(Li + Li = Li_2\)	\(1,7\)	\(39\)
\(Na + Na = Na_2\)	\(0,8 \pm 0,05\)	\(18 \pm 1\)
\(K + K = K_2\)	\(0,65 \pm 0,15\)	\(15 \pm 34\)
\(C + C = C_2\)	ca. \(7,0\)	ca. \(160\)
\(N + N = N_2\)	\(9,0 \pm 0,3\)	\(207,5 \pm 7\)
\(N^+ + N = N_2^+\)	\(6,7 \pm 0,3\)	\(155 \pm 7\)
\(O + O = O_2\)	\(5 \pm 6\)	\(155 \pm 138\)
\(O^+ + O = O_2^+\)	\(6,9 \pm 0,3\)	\(159 \pm 7\)
\(S + S = S_2\)	\(5 \pm 0,4\)	\(155 \pm 9\)
\(Se + Se = Se_2\)	\(3,6 \pm 0,6\)	\(83 \pm 14\)
\(Te + Te = Te_2\)	\(3 \pm 0,6\)	\(69 \pm 14\)
\(Cl + Cl = Cl_2\)	\(2,466 \pm 0,0008\)	\(56,89 \pm 0,18\)
\(Br + Br = Br_2\)	\(1,961 \pm 0,008\)	\(45,22 \pm 0,18\)
\(J + J = J_2\)	\(1,544 \pm 0,003\)	\(35,605 \pm 0,07\)
\(Na + K = NaK\)	\(0,62 \pm 0,05\)	\(14,3 \pm 1\)
\(J + Cl = ClJ\)	\(2,04 \pm 0,01\)	\(46,97 \pm 0,2\)
\(Li + H = LiH\)	\(2,6 \pm 0,3\)	\(59 \pm 7\)
\(Be^+ + H = (BeH)^+\)	ca. \(3,7\)	ca. \(85\)
\(Zn^+ + H = (ZnH)^+\)	ca. \(2,5\)	ca. \(58\)
\(Cd + H = CdH\)	\(0,67 \pm 0,01\)	\(15,5 \pm 0,2\)

* See V. Henri, Nature 125, 202, 1930; R. Mecke, Nature 125, 526, 1930; V. Kondratjew, Z. physikal. Ch. (B) 7, 70, 1930.

Reaction	Heat of dissociation, volts	Heat of dissociation, calories
Cd⁺ + H = (CdH)+	1.9 ± 0.3	44 ± 7
Hg + H = HgH	0.37 ± 0.01	8.5 ± 0.2
C + N = CN	8.1 ± 0.5	187 ± 10
Si + N = SiN	ca. 5.0	ca. 115
B + O = BO	ca. 7.5	ca. 173
Ti + O = TiO	ca. 6.7	ca. 155
C + O = CO	ca. 11.2	ca. 258
C⁺ + O = (CO)+	8.1 ± 0.5	187 ± 10
N + O = NO	6.8 ± 0.5	157 ± 10
S + O = SO	ca. 6.4	ca. 148
Ag + Br = AgBr	ca. 2.6	ca. 60
Ag + I = AgI	ca. 2.3	ca. 55
Ca + F = CaF	ca. 3.2	ca. 74
Tl + Cl = TlCl	3.77 ± 0.01	87.0 ± 0.2
Tl + Br = TlBr	3.14 ± 0.01	73.5 ± 0.2
Tl + I = TlI	2.64 ± 0.05	61 ± 1

Table 3

Data on the vibrational bands of ammonia and combinations of the three fundamental vibrations: $\nu_1 = 946\ \mathrm{cm}^{-1}$, $\nu_2 = 1630\ \mathrm{cm}^{-1}$, and $\nu_3 = 3420\ \mathrm{cm}^{-1}$

Wavelength in $\mu$	Wave number, $\mathrm{cm}^{-1}$	Interpretation*
10.55	946	$\nu_1$
6.13	1630	$\nu_2$
4.05	2470	$\nu_1 + \nu_2$ or $\nu_3 - \nu_1$
3.00	3330	$2\nu_2$
2.92	3420	$\nu_3$
2.35	4255	$\nu_1 + 2\nu_2$
2.28	4380	$\nu_2 + \nu_3$ or $3\nu_2$
2.01	4940	$\nu_1 + \nu_2 + \nu_3$
1.66	6030	$2\nu_3$
1.51	6625	$4\nu_2$
1.46	6815	$\nu_1 + 2\nu_3$
1.29	7750	$\nu_2 + 2\nu_3$
1.21	8200	$3\nu_3$
1.035	9650	$\nu_1 + 3\nu_3$
0.976	10250	$4\nu_3$
0.795	12580	$4\nu_3 + \nu_1$
0.733	13630	$5\nu_3$
0.652	15380	$6\nu_3$
0.556	18000

* It should be borne in mind that the vibrations are not strictly harmonic and that therefore column 3 contains an interpretation of the bands, but not formulas for their exact calculation; $5\nu_3$ denotes “the 4th harmonic band with respect to the fundamental,” but the wave number of this harmonic band is smaller than $5 \times \nu_3$.