BAND SPECTRA *

IV. ROTATIONAL SPECTRA

E. I. Rabinovich, Göttingen

1. Vector model of a rotating molecule. 2. Eigenfunctions and eigenvalues, symmetry of a rotating molecule: a) eigenfunctions, b) symmetry properties, c) nuclear spin and alternation of intensity. 3. Rotational terms: a) term formula, b) dissociation under rotation, c) determination of internuclear distance. 4. Structure of bands.

As the last component of the system of terms of a diatomic molecule, rotation remains to be considered. As was already mentioned at the beginning of the third part of this article, the simplest case of a rotating molecule is characterized by the fact that the system of electrons has no rotational angular momentum, so that the rotational angular momentum of the nuclei of the molecule is at the same time also the total angular momentum of the entire molecule. In more complicated cases the total rotational angular momentum is composed of the angular momentum of the electronic system and the angular momentum of the molecule itself. We shall first consider this more complicated case on the basis of the vector model, i.e. qualitatively, and then consider the eigenfunctions and their symmetry properties. Since the sequence of rotational terms depends only on a single integral parameter—the so-called rotational quantum number—subsequently we shall be able rather easily to pass from the qualitative consideration to the representation of the terms by formulae.

1. Vector model of a rotating molecule

The interaction of rotation with the motion of the electrons is never sufficiently strong to have a noticeable effect on the coupling of the orbital moments of the individual electrons \(l\), which form the total moment \(L\), or on the coupling of the spins \(s\), which form the resultant spin \(S\). Therefore, for a rotating molecule as well, we may characterize the state of the electronic system by two angular momenta \(L\) and \(S\). The further development of the vector model must be carried out by investigating the interaction of the moments \(L\) and \(S\) with the electric field along*

* See Uspekhi Fizicheskikh Nauk, XI, 554; XIII, 253.

axes of the molecule and with its rotation. Here we are dealing with interactions of approximately the same magnitude; this circumstance makes the situation especially complicated. In particular, it turns out that the influence of rotation at small rotational quantum numbers is less than the influence of the electric field in the direction of the nuclear axis, whereas at larger rotational quantum numbers the influence of rotation predominates. Thus, as the rotation increases, one must change the order in which the vector model is constructed in order to obtain an approximation to the true relations.

Hund developed a series of typical cases which approximate real molecules to a greater or lesser extent. The first case (usually called “case \(a\)”) is realized when the influence of rotation is small. For this it is necessary that the rotational energy be small. As will be shown in Section 3, the individual quanta of rotation will be the larger, the smaller the moment of inertia of the molecule, consequently, roughly speaking, the lighter the molecule. In very light molecules (\(\mathrm{H}_2\), \(\mathrm{He}_2\), \(\mathrm{BeH}\), etc.) “case \(a\)” cannot occur even at the smallest rotational quantum numbers. But in molecules of intermediate mass (\(\mathrm{N}_2\), \(\mathrm{CN}\)) the prerequisites for “case \(a\)” are realized already at not too large rotational quantum numbers.

Since “case \(a\)” takes place, not only the composition of the vectors \(l\) into the resultant orbital moment \(\mathbf{L}\), but also the coupling of the vector \(\mathbf{L}\) with the molecular axis, which we shall consider in Section 3, undergoes no essential perturbations. In other words, the component \(L\) in the direction of the molecular axis \(\Lambda\) remains a quantized quantity. In the absence of rotation, we must regard as the next step in the construction of the vector model the coupling of \(S\) with \(\Lambda\) and the formation of the total angular momentum \(\Omega\) relative to the molecular axis. In “case \(a\),” in the first approximation, this coupling is also preserved, and along with it the quantization of \(\Omega\). Only as the very last approximation should one take into account the interaction between the total angular momentum relative to the axis \(\Omega\) and the rotational moment \(\mathbf{R}\). Here we are dealing with the interaction of two rotations, each of which occurs about a definite fixed axis: for \(\Omega\) is the angular momentum relative to the molecular axis, while the rotation \(\mathbf{R}\) occurs about a direction perpendicular to the molecular axis. When such two moments interact and give the resultant \(\mathbf{J}\) (Fig. 1), then, as can easily be seen, not all three quantities \(\Omega\), \(\mathbf{R}\), \(\mathbf{J}\) are simultaneously quantized. In fact (Fig. 1),

\[ J^2 = R^2 + \Omega^2 . \tag{1} \]

But the square of an integer cannot, in general, be represented as the sum of two squares. \(\mathbf{J}\)—the total angular momentum—and as such is in any case strictly quantized (i.e.,

has the value of integers or half-integers depending on the corresponding values of $\Omega$). The smoothing of the strict quantization is explained by the common precession of $L$ and $R$ about the direction $J$, in other words, by the precessional motion of the molecular axis. Since the motion of the electron is a considerably faster motion than the rotation of the molecule, the precession of the nuclear axis with respect to the motion of the electron can always be regarded as a slow motion, even if it is not so with respect to rotation. This means that the quantization of the electronic motion (the quantum number $\Omega$) is preserved more strictly than the quantization of the molecular rotation. Thus, to a first approximation in “case $a$,” along with $J$ the vector $\Omega$ may also still be regarded as quantized, whereas the vector $R$, on the contrary, must assume any non-integral values [which can be determined from the values of $\Omega$ and $J$ by (1)].

Fig. 1. Example of the vector model in case $a$: $L=3$, $\Lambda=+2$, $S=1$, $\Sigma=-\frac{1}{2}$, $\Omega=\Lambda+\Sigma=\frac{3}{2}$ (term $^{3}\Delta_{3/2}$); $J=4$, $R=\sqrt{4^{2}-3/2}=\frac{1}{2}\sqrt{55}$ (in reality the vectors should not all lie in one plane).

We now turn to the case of somewhat higher energy of rotation. The first interaction that can be subjected to perturbation under the influence of rotation is the relatively weak coupling of the spin $S$ with the orbital moment $\Lambda$. This phenomenon, which appears in molecules of medium weight with increasing rotation—and in light ones, conversely, already appears at small rotations—as a “decoupling” (Entkopplung) of the spin from the nuclear axis, should, in constructing the vector model in this so-called “case $b$,” be considered first of all as the interaction of $\Lambda$ with the rotation $R$ and only then the influence of the spin. The interaction of $\Lambda$ and $R$ leads to the formation of a quantized resultant, which we shall denote by $K$ (since we reserve the notation $J$ once and for all for the total angular momentum of rotation of the system). For the corresponding quantum number $K$ we have, completely analogously to equation (1), the following relation:

$$ K^{2}=\Lambda^{2}+R^{2}. \tag{2} $$

Again, to strict (or nearly strict) quantization of $K$ there corresponds approximate quantization of $\Lambda$ and nonquantized $R$.

In the final approximation, as stated above, one should take into account the interaction of $K$ with the spin $S$. Since both $K$ and $S$ have free directions, these two moments, without losing their quantization, give the quantized resultant $J$.

In general the vector model in “case b” has the form shown in Fig. 2.

If the rotational energy is increased still further, it may also destroy the coupling of the orbital angular momentum L with the molecular axis; the quantum number \(\Lambda\) then loses its meaning. This so-called “case d” is observed only in light molecules and at strong rotations. The construction of the vector model is now carried out in the following order: first one takes into account the coupling of L with the rotation R. Since the direction of L is free, L and R add to a quantized resultant K, while R does not lose its quantization. In the last approximation K and S add to the total rotational angular momentum of the molecule J (Fig. 3).

Fig. 2. Example of the model in case \(b\): \(L=2,\ \Lambda=1,\ S=1\) (\({}^{3}\Pi\)-term), \(K=5\) \((R=\sqrt{5^{2}-1^{2}}=\sqrt{24})\), \(J=4\).

All this already seems sufficiently complicated and not very intuitive. However, we are dealing only with phenomena of spin uncoupling (the transition from “case a” to “case b”) and, manifested at still stronger rotation, the uncoupling of the orbital angular momentum L (the transition from “a” to “b”) from the molecular axis. It should be borne in mind that the transition does not occur suddenly, from one rotational quantum to another, but proceeds gradually; moreover, in the transition regions none of the idealizations of the vector model considered gives a good approximation to the actual relations. Subsequently, in heavy molecules new complications arise because of the growing coupling between L and S, which enters into the most serious competition with the coupling of these vectors to the nuclear axis. We cannot, however, here go more closely into these separate phenomena.

Fig. 3. Example of the model in case \(c\): \(L=2,\ R=5,\ K=6,\ S=2,\ J=5\).

2. Eigenfunctions and symmetry properties of a rotating molecule

a) Eigenfunctions

A diatomic molecule without electronic rotational angular momentum is, if stretching due to the action of centrifugal forces is neglected, the case of a rigid rotator, i.e. one of the simplest examples for the application of quantum theory. The simplest rotator always consists of one material point rotating

around a fixed point at the origin of coordinates, with the distance between them invariable. A molecule, by contrast, consists of two material points which can rotate jointly about the common center of gravity. This difference, however, introduces no particular complication; the molecule behaves exactly like a simple rotator with moment of inertia:

\[ I=\frac{m_1 m_2}{m_1+m_2}\,r^2=\mu r^2, \tag{3} \]

where \(m_1\) and \(m_2\) denote the masses of the two atoms, \(r\) is the distance from the nucleus; \(\mu\) is the so-called “reduced” mass.

The simplicity of the rigid-rotator problem is explained by the fact that one of the three spatial coordinates \(r,\vartheta,\varphi\)—\(r\)—remains constant, so that quantization must be carried out only with respect to the angular coordinates \(\vartheta\) and \(\varphi\). One obtains eigenfunctions completely analogous to the eigenfunctions of electron revolution considered in Part III (p. 258), except for the terms relating to \(r\). The nodal surfaces are the nodal planes of the function \(\psi_\varphi\), shown in Fig. 6 of Part III, and the nodal cones of the function \(\psi_\vartheta\). The number of nodal planes determines the “azimuthal” quantum number \(n_\varphi\) (\(\lambda\) in Part III), and the number of cones—the quantum number \(n_\vartheta\). Because of the indeterminacy of the direction of the axis \(\vartheta=0\), the angular momentum and all other measurable properties of the system, just like the energy of the electrons, cannot be separated according to the quantum numbers \(n_\varphi, n_\vartheta\), but depend only on their sum—the rotational quantum number \(R=n_\varphi+n_\vartheta\) (which determines the total number of angular surfaces of the different kinds). The rotational states of a molecule of the kind considered (i.e., without electron angular momentum) are thus arranged in a single series of integers \(R\). Since \(R\) can be decomposed in \(R\) different ways into two integers \(n_\varphi\) and \(n_\vartheta\), each such rotational state must be regarded as a superposition of \(R\) separate states; it is “\(R\)-fold degenerate” and has “statistical weight \(R\).”

Whereas in classical mechanics and in the Bohr–Sommerfeld theory rotation is a planar motion, the corresponding problem of quantum mechanics is solved in space; for, when considering the probability of the various positions of the atomic nucleus relative to the center of gravity of the system, there are no grounds for representing these positions in a definite plane, or, what is the same thing, there is no possibility of assigning to the axis of rotation a definite direction in space; rather, all directions must be regarded as equivalent; the eigenfunction must possess (so long as there is no external field) spherical symmetry. We have become acquainted with this transition from the visually representable planar rotation about an axis with angular momentum \(P_\varphi\) to the no longer visually representable quantum-mechanical

“spatial” rotation about a point with angular momentum \(P_{\vartheta,\varphi}\) and with both proper functions in Part III. It was shown there that, in quantum mechanics, to a spatial eigenfunction \(\psi(\vartheta,\varphi)\) with the sum of quantum numbers \(n_\vartheta+n_\varphi\) there corresponds an angular momentum

\[ P_{\vartheta,\varphi}=\frac{h}{2\pi}\sqrt{n(n+1)}, \]

and not \(P_\varphi=\frac{nh}{2\pi}\), as in the two-dimensional case. This also holds for the rotation of a molecule. But, as has already been said in Part III, the replacement of the vector \(P_\varphi=\frac{Rh}{2\pi}\) by the vector \(P_{\varphi,\vartheta}=\frac{h}{2\pi}\sqrt{R(R+1)}\) in no way affects the qualitative data obtained in constructing the vector model.

b) Symmetry properties. To the eigenfunctions of molecular rotation one may almost without difficulty apply the symmetry considerations that we expressed concerning electronic eigenfunctions. If the atoms are different, then symmetry properties can, of course, exist only for the simultaneous reflection of the entire system as a whole (nuclei and electrons) in the center of gravity of the molecule. In this case the rotational functions are alternately positive (for even \(R\)) and negative (for odd \(R\)), since all nodal surfaces (the total number of which is \(R\)) pass through the origin of coordinates and therefore must intersect in passing into the point of their mirror image. The total eigenfunction of the system (nuclei and electrons) is the product of the eigenfunction of the electron system and the eigenfunction of the molecule’s rotation. The symmetry properties of this total function are determined by the symmetry properties of both constituent parts, according to the “minus times minus gives plus” principle used repeatedly in Part III. Thus positive electronic terms are positive for even rotations and negative for odd ones; for negative terms the reverse relation holds. In this section we speak only of \(\Sigma\)-terms, since we have assumed that the molecule has no angular momentum of the electrons. We know from Part III that such terms can be constructed either from continuous \(s\)-electrons, or also from electrons with \(\lambda\ne 0\) (\(\pi,\delta\ldots\)-electrons), whose angular momenta about the molecular axis cancel pairwise. In the first case the electronic term by itself has no definite “positive or negative” symmetry property at all (because of the absence of a definite direction of revolution). The corresponding property of the total state is therefore determined exclusively by rotation. Among \(\Sigma\)-terms formed from pairs of electrons of the type \(\pi,\delta,\ldots\), on the contrary, positive and negative terms are encountered,

energy is different even with the same distribution of electrons. Such \(\Sigma\)-terms are subject only to the rule just formulated concerning the relation between the symmetry of the electronic and rotational terms. The terms \(\Pi\)-, \(\Delta\)-, ... will be considered by us somewhat below.

If the nuclei are identical, then the property “positive or negative” can be decomposed into the two properties “even or odd” and “symmetric or antisymmetric with respect to the nuclei” (see p. 272, Part III). Since electronic terms are even or odd independently of the rotation of the molecule, the rotational terms must be alternately symmetric and antisymmetric with respect to the nuclei. For even electronic terms the positive rotational terms are symmetric; for odd ones—the negative ones are symmetric.

If the molecule possesses angular momentum of the electron with respect to the molecular axis (i.e. is in a \(\Pi\)-, \(\Delta\)-, ... state), then it is no longer a rigid point rotator, but rather resembles a symmetric top. We shall not dwell in greater detail on the proper functions of such a model and shall say only a few words concerning its symmetry properties.

The rotational terms in this case too are alternately positive and negative (and, for identical nuclei, also symmetric and antisymmetric). As we have already said (see p. 284, Part III), non-rotating \(\Pi\)-, \(\Delta\)-, ... terms cannot be regarded as definitely positive or negative; they should rather be considered as superpositions of positive and negative terms possessing the same energy. It can be shown that rotation affects these two constituent parts of the non-rotational state differently. Therefore from one non-rotational \(\Pi\)- or \(\Delta\)-term two sequences of rotational terms are obtained, which diverge more and more as the rotation is intensified. Their splitting is evidently based on the interaction of the angular momenta of the \(\lambda\) electrons with the rotation \(R\); therefore it can be appreciable only in the case when this interaction is sufficiently strong. In other words: the splitting of rotational terms into two components—negative and positive—always occurs simultaneously with the detachment of \(\lambda\) from the molecular axis.

Thus, in summary, we may say the following. The rotational terms of the state \(\Sigma\) are alternately positive and negative; the rotational terms of the states \(\Pi\)-, \(\Delta\)-, ... consist of two components—positive and negative—which at small rotational energy coincide, and at considerable energy diverge.

c) Nuclear spin and intensity alternation. The distinction between terms possessing different symmetry, which seemed to be of only theoretical interest,

BAND SPECTRA

recently led to very important experimental results. First, on the basis of this difference the existence of a new kind of allotropy was predicted, which was confirmed chiefly in the case of hydrogen; second, it made it possible to establish certain important properties of atomic nuclei.

All these consequences were obtained by applying one very general principle that lies at the foundation of modern statistical physics. This principle—one that has not yet found a deeper justification—states the following:

In nature there are two kinds of elementary particles. Particles of the first kind may occur in any number in one and the same quantum state; as for particles of the second kind, in a closed system two such particles can never be in identical quantum states. Particles of the first kind, at low temperatures, may even all be in the same, lowest-energy quantum state; particles of the second kind, on the contrary, are always forced to distribute themselves among different energy states. The first group includes light quanta: in a closed hollow space any number of light quanta of the same energy (i.e., of the same frequency) may be present simultaneously. The second kind of particles includes electrons: in an atom, in a molecule, or in a piece of metal there cannot exist two electrons possessing exactly equal quantum numbers (i.e., having the same frequencies according to de Broglie).

The entire structure of the atom is governed by this property of electrons; it determines the distribution of electrons among the separate “shells.” If electrons possessed the statistical properties of light quanta, then at ordinary temperature they would all be in one and the same “one-quantum” state. It is clear that the statistical treatment of these two kinds of particles must be different. The statistics that allows for the possibility of an unlimited accumulation of identical particles in one definite state is called Bose–Einstein statistics; the statistics based on the premise that each state can be occupied only once is called Fermi–Dirac statistics.

In application to electrons in the atom, the principle of single occupation of each quantum state is known as Pauli’s rule, or the exclusion principle. In wave mechanics, the distinction between the two kinds of statistics corresponds to the distinction between symmetric and antisymmetric eigenfunctions. A symmetric function remains finite also when the parts of the function depending on two variables become identical; thus, for example, for the symmetric function $z=x+y$, when $x=y$, we have $z=2x=2y$. An antisymmetric function, on the contrary,

in the analogous case vanishes; thus, for example, the antisymmetric function \(z=x-y\) gives zero for \(x=y,\ z=0\). Owing to the physical identity of the particles under consideration (for example, electrons), one can, generally speaking, conceive only symmetric or antisymmetric eigenfunctions. (On exchange of two electrons at most the sign may change, but not the absolute value of \(\psi\), i.e. not the charge density \(\psi\psi\).)

Therefore, for particles of the first kind only symmetric eigenfunctions are possible; for particles of the second kind, on the contrary, only antisymmetric functions. Thus all eigenfunctions of atoms and molecules must be antisymmetric with respect to the exchange of two arbitrary electrons. It should be noted that reflection of the electronic system in the origin of coordinates in a diatomic molecule (in contrast to reflection in the nucleus!) is not identical with exchange and that, therefore, the Pauli principle in no way affects the existence of even and odd eigenfunctions of the electrons. The question of which statistics the nuclei obey—Bose statistics or Fermi statistics—we shall for the time being postpone. But there is no doubt that they must be assigned either to one kind or to the other, and that therefore in a molecule there can be either only symmetric terms in the nuclei or only antisymmetric terms in the nuclei; never in one and the same molecule can eigenfunctions of both kinds arise. If we recall the symmetry properties of rotational terms, which were discussed on p. 859, then we must expect that in each sequence of rotational terms of a molecule consisting of two identical atoms, every second term drops out, and this occurs according to the following rule:

In reality, in many molecules a dropping out of half of all rotational terms is observed. Thus, for example, in the molecule He₂, for positive even electronic terms there are only odd rotational terms; for positive odd electronic terms—only even rotational terms, etc. From this situation it follows, moreover, that He nuclei (α-particles) are particles of the first kind and must obey Bose statistics.

But in many other molecules, for example already in the molecules H₂, no dropping out of rotational terms is observed; instead, an alternation of intensity is often observed, in which all even rotational terms form a sequence with greater probability than the odd terms (or conversely); as a result an alternation of the intensity of lines in the bands is obtained: after a weak line there follows a more intense line, then again a weaker one, etc.

If one adheres to the general principle—that either only symmetric or only antisymmetric terms are permitted—then one must assume that up to now our eigenfunctions have been incomplete. We have seen that a symmetric electronic function, owing to the superposition of a rotation, may become antisymmetric, and conversely. Thus one may suppose that the complete eigenfunctions of the molecule contain still another component part, not considered by us, which in a certain sense improves the symmetry of the “forbidden” terms. According to modern views this component part is the nuclear spin (analogously to the spin of electrons), i.e. the angular momentum of the nucleus relative to its own axis.

It is assumed that this spin is quantized and can take either integral or half-integral values. Thus the vector model of a diatomic molecule must be supplemented by two new vectors—the spins of both nuclei. The coupling of these angular momenta with all the other angular momenta of rotation of the component parts of the molecule is very insignificant; in our consideration we may neglect it. Conversely, the spins are coupled with one another comparatively strongly (as a result of “resonance” between the two nuclei); this coupling leads to the formation of a quantized “total nuclear spin” of the molecule. If each nucleus possesses spin \(i\) (in units \(\frac{h}{2\pi}\)), then for the total spin \(I\) there are possible \((2i+1)\) values
\[ 0, 1, 2, \ldots, 2i. \]

Expressing the same thing in the language of eigenfunctions, we shall say that to the spin of a nucleus \(i\) in a symmetric diatomic molecule there correspond energetically \((2i+1)\) several distinct eigenfunctions. Consideration of the symmetry properties of these functions with respect to exchange of nuclei shows that the functions with total spin
\[ I = 2i,\ 2i - 2,\ 2i - 4,\ \ldots \]
in the nuclei are symmetric; conversely, the functions
\[ I = 2i - 1,\ 2i - 3,\ \ldots \]

antisymmetric. In the simplest cases this assertion can be made pictorial. If two nuclei possess spin \(i=\frac{1}{2}\), then \(I=1\) means that both spins have the same direction; \(I=0\) means that they have different directions (Fig. 4). Obviously, in the first case one cannot expect a change of sign upon interchange of the nuclei, whereas in the second case this does occur.

The existence of nuclear spin at the same time explains the simultaneous appearance of even and odd rotational terms: if the proper functions of a rotational term are themselves antisymmetric, then the term can always be “symmetrized” by superposing likewise antisymmetric proper functions of the nuclear spins. Suppose, for example, that the nature of the nuclei permits only symmetric total proper functions; in that case antisymmetric rotational terms enter into combination with antisymmetric proper functions of the nuclear spins, and symmetric terms—with symmetric proper functions of the nuclear spins, etc.

Fig. 4.

Thus we have explained the appearance of all rotational terms and have connected it with the existence of a nuclear spin different from zero. In order also to explain the alternation of intensities, one must apply the statistical principle, which we have already used more than once in the present article. If a definite energy state can be realized, from the point of view of quantum theory, in \(n\) different ways, then one may assert that it has a “quantum weight” (or “statistical weight,” or a priori probability) equal to \(n\), and that, other conditions being equal, it occurs \(n\) times more often than a “simple” state with the same energy. We have already become acquainted with one example of this phenomenon in considering rotational terms. Each rotational state with quantum number \(R\) can be realized in \(R\) different ways; it is “\(R\)-fold degenerate” and has statistical weight \(R\).

A molecular state with nuclear spin \(I\) is likewise a degenerate state. This becomes clear if we mentally place the molecule in a strong magnetic field; then the spin \(I\) must orient itself along the field (“be quantized in direction”) and, according to the principles for constructing the vector model repeatedly used in the present article, can give \((2I+1)\) different quantized components (from \(-I\) to \(I\)) in the direction of the field. Thus a state characterized by spin \(I\) in a magnetic field makes possible the formation of \((2I+1)\) energetically different states. According to one of the fundamental principles of quantum theory (the so-called “principle

adiabaticity”)—this means that, in a space where there is no magnetic field, the state characterized by spin is a superposition of \((2I+1)\) energetically equal states, i.e. has weight \((2I+1)\). We are interested not in the quantum weight of the individual states, but in the total statistical weight \(P_s\) of all states with symmetric spin functions and the total statistical weight \(P_a\) of all states with antisymmetric spin functions. These weights are:

Symmetric states:

\[ P_s=(2\cdot 2i+1)+[2(2i-2)+1]+\ldots=(i+1)(2i+1). \]

Antisymmetric states:

\[ P_a=[2(2i-1)+1]+[2(2i-3)+1]+\ldots=i(2i+1), \]

whence

\[ \frac{P_s}{P_a}=\frac{i+1}{i}. \]

We see that for \(i=0\) there exists in general only a single state with a symmetric spin function; for \(i=\frac12\) a state with a symmetric spin function is three times more probable than one with an antisymmetric one; for \(i=1\) this ratio is \(2:1\), and as the spin increases the ratio of probabilities approaches the value 1.

As was already shown, the spin of the nucleus is almost not connected with the other rotational angular momenta of the molecule; therefore a change in the electronic state or in the rotation of the molecule cannot cause a change of spin. Hence in all spectral processes (and also in gas-kinetic collisions) the total spin of the molecule remains unchanged. Thus all spectral lines of the molecule are obtained by means of combinations of eigenfunctions with the same symmetry in the nucleus. “Intercombinations” between symmetric (without spin!) and antisymmetric (also without spin!) eigenfunctions do not occur. If the molecule is in a “symmetric state,” then it can pass into an antisymmetric state and back only by an indirect route, through dissociation (or through a very strong deformation).

We are now sufficiently prepared to understand the practical consequences of the above. Every diatomic molecule with nuclear spin different from zero exists in two series of states, which cannot pass from one into the other. In each sequence of rotational terms the even terms belong to one class, the odd terms to the other (with the limitation that the rotational terms are in practice often double and must be regarded as superpositions of two terms—one symmetric and one antisym-

methodical, see above, p. 862). Thus a diatomic gas behaves as a mixture of two gases—one “symmetric” and the other “antisymmetric.” If it were possible to separate both modifications from one another, then they would have to remain unchanged for a long time; in this way it is possible to obtain two special “allotropic modifications” of such a gas. This possibility was first theoretically predicted for \(H_2\) and then confirmed experimentally by Bonhoeffer and Harteck, as well as by Eucken and Hiller. The nucleus of hydrogen—the proton—just like the electron, has spin

\[ i = \frac{1}{2}. \]

The “symmetric” gaseous component, the so-called orthohydrogen, must therefore, at sufficiently high temperature, occur three times as often as the “antisymmetric” component, known as parahydrogen. The fundamental term for \(H_2\) is the term \(1s^2 \cdot {}^1\Sigma_g\); this term is even (as a result of the even number of electrons) and in itself is symmetric in the nuclei, as a consequence of which it requires an antisymmetric spin function, since the proton is a “Fermi particle” and permits only antisymmetric complete eigenfunctions. Thus the rotationless state of \(H_2\) belongs to parahydrogen; parahydrogen also includes all even rotational states; the odd rotational states, on the contrary, form the system of orthohydrogen (they possess antisymmetric rotational functions and give the required antisymmetric complete wave functions by combination with symmetric spin functions).

If we take ordinary hydrogen, in which many rotational quanta are already excited at room temperature, and which therefore consists approximately of 25% parahydrogen and 75% orthohydrogen, and cool it while, by means of catalysis or very high pressure, maintaining thermal equilibrium between the two forms, then in the end, when all the rotational quanta “die out,” we must obtain pure parahydrogen. If it is then subjected to rapid heating, then, owing to the difficulty of the reverse transformation, we shall even at room temperature obtain an almost pure para-form, which, in the absence of catalysts, can be preserved practically for as long as desired. Such a gas gives only half the spectral lines of the ordinary \(H_2\) molecule and also possesses somewhat different thermal properties, such as, for example, melting point, boiling point, thermal conductivity, etc. Between the two allotropic modifications there is a “frozen” equilibrium—only the freezing in this case occurs by heating.

Analogous separations are theoretically possible for many other molecules as well; but in practice they cannot be carried out.

as a result of the smallness of the rotational quanta, which makes the formation of a nonrotating state difficult. In addition, it should be borne in mind that separation by cooling is theoretically possible only in molecules with a ground state \(\Sigma\), since otherwise the rotational terms are degenerate and consist of one symmetric and one antisymmetric term of equal energy.

The second practical consequence of the symmetry laws discussed on p. 859 is the possibility of judging the magnitude of the nuclear spin \(i\) from the alternation of intensities in the rotational bands. If half the lines disappear altogether, then the nuclear spin is zero; if the intensity changes in the ratio \(3:1\), then the spin \(i\) is equal to \(\frac{1}{2}\), and so on.

One may draw one more very important conclusion. Suppose we know the symmetry properties of the electronic term (even or odd, negative or positive); then, having established by observation which rotational terms are preferable— even or odd—we can decide whether the complete functions must be symmetric or antisymmetric with respect to the nuclei. In this way it is possible to determine empirically which nuclei obey Bose–Einstein statistics and which obey Fermi–Dirac statistics. If protons and electrons inside nuclei obeyed the same laws as outside them, then one could in advance directly determine the type of statistics that should be applied to individual nuclei. Instead of interchanging nuclei as wholes, one may interchange separately the protons and electrons contained in them. If, as the result of each interchange, the sign changes, then electrons and protons are Fermi particles and require antisymmetric eigenfunctions. Thus, ultimately, for an odd total number of elementary particles one should expect a change of sign, whereas for an even number, on the contrary, the sign should remain unchanged. In other words, nuclei with odd atomic number should obey Fermi statistics, and nuclei with even atomic number—Bose statistics. This result is not confirmed by experiment. Few phenomena allow us to penetrate so deeply into the nucleus and to see the difference between the laws of nature acting there and those already known to us, as does this nonobservance of the basic laws of the ordinary statistics of electrons and protons.

§ 3. Rotational terms

a) Formula for the terms. We now pass from the qualitative description to a quantitative formulation of the rotational terms. This is easiest to do for the case of a molecule not possessing

by the impulse of reversal of the electrons (i.e., being in the state \({}^{1}\Sigma\)). Since we may neglect the stretching of the molecule under the action of the centrifugal force, such a molecule corresponds to the model of a simple rigid rotator. For such a rotator we have the following relation between the angular momentum \(p_{\vartheta,\varphi}\) and the rotational energy \(E_{\mathrm{rot}}\):

\[ E_{\mathrm{rot}}=\frac{p_{\vartheta,\varphi}^{2}}{2I}. \tag{4} \]

Equation (4) is completely analogous to the equation

\[ E_{\mathrm{kin}}=\frac{mv^{2}}{2}=\frac{(mv)^{2}}{2m}=\frac{p^{2}}{2m} \]

for rectilinear motion, only instead of the usual momentum \(p=mv\) there stands the angular momentum \(p_{\vartheta,\varphi}\), and instead of the mass \(m\)—the moment of inertia \(I\)*.

The molecule consists of two material points which rotate about their common center of gravity. If the masses are \(m_1\) and \(m_2\), and the distance between them is equal to \(r\), then the center of gravity lies at a distance \(\dfrac{m_2}{m_1+m_2}\cdot r\) from the material point \(m_1\) and at a distance \(\dfrac{m_1}{m_1+m_2}\) from the material point \(m_2\). The moment of inertia is equal to:

\[ I=m_1\cdot\left(\frac{m_2 r}{m_1+m_2}\right)^2 +m_2\cdot\left(\frac{m_1 r}{m_1+m_2}\right)^2 =\frac{m_1m_2}{m_1+m_2}\cdot r^2=\mu r^2, \tag{5} \]

where \(\mu\), as has already been indicated on p. 858, denotes the so-called reduced mass.

According to the old quantum theory, for the angular momentum the allowed values were \(\dfrac{Rh}{2\pi}\) \((R=0,1,2,\ldots)\). According to the new theory this condition holds only for the two-dimensional case (i.e., for rotation in a plane).

The transition from plane rotation about a definite axis to “three-dimensional rotation about an indefinite axis” (if one may so express it) leads, as we have already seen on p. 859, to the replacement of the angular momentum \(p_\varphi=\dfrac{Rh}{2\pi}\) by the angular momentum

\[ p_{\vartheta,\varphi}=\frac{h}{2\pi}\sqrt{R(R+1)}, \tag{6} \]

where \(R\) denotes the rotational quantum number (the sum of the quantum numbers \(m_\varphi\) and \(m_\vartheta\)).

* It should be remembered that in the preceding section \(I\) denoted the nuclear spin, whereas here it denotes the moment of inertia.

Substituting expressions (5) and (6) for the angular momentum into equation (4), we obtain:

\[ E_{\mathrm{rot}}=\frac{h^{2}}{8\pi^{2}\mu r^{2}}R(R+1)=B'R(R+1) \tag{7} \]

for the rotational energy, and

\[ T_{\mathrm{rot}}=\frac{E_{\mathrm{rot}}}{ch}=\frac{h}{8\pi^{2}c\mu r^{2}}R(R+1)=BR(R+1)\simeq B\left(R+\frac{1}{2}\right)^{2} \tag{8} \]

for the rotational terms. The old quantum theory gave \(R^{2}\) where, in equations (6), (7), and (8), \(R(R+1)\) stands. Since instead of \(R(R+1)\) one may also write

\[ R(R+1)=\left(R+\frac{1}{2}\right)^{2}-\frac{1}{4}, \tag{9} \]

the rotational terms in the new theory, down to the small additive term \(\frac{1}{4}B\), are the same as in the old theory when half-integral quantum numbers are used \(\left(R=\frac{1}{2},\,1\frac{1}{2},\,2\frac{1}{2},\ldots\right)\). Indeed, it had already been noted earlier that the rotational terms can be expressed much better by means of half-integral quantum numbers. Thus, for example, if we calculate the frequencies of the second and fourth members in a pure rotational spectrum, we obtain:

\[ \nu_{2}=T_{2}-T_{1}=3B,\quad \nu_{4}=T_{4}-T_{3}=7B, \]

\[ \frac{\nu_{4}}{\nu_{2}}=2.33 \]

with integral quantum numbers, and

\[ \nu_{2}=T_{5/2}-T_{3/2}=4B,\quad \nu_{4}=T_{9/2}-T_{7/2}=8B, \]

\[ \frac{\nu_{4}}{\nu_{2}}=2.00 \]

with half-integral quantum numbers. Empirically, we find, for example, for HF:

\[ \nu_{2}=82.56\ \mathrm{cm}^{-1}, \]

\[ \nu_{4}=163.44\ \mathrm{cm}^{-1}, \]

\[ \frac{\nu_{4}}{\nu_{2}}=1.97. \]

Of course, equation (8) gives for the line frequencies exactly the same values as the old theory with half-integral quantum numbers, since the additive term \(\frac{1}{4}B\) drops out when differences are formed. The necessity of half-integral quantum numbers for representing rotational terms is one of the simplest proofs of the advantage of the new quantum mechanics in comparison with the old Bohr–Sommerfeld theory.

According to the old theory, the rotational quanta increase exactly, whereas according to the new theory they increase approximately quadratically with increasing

of the rotational quantum number. The first rotational quanta are comparatively not large. Thus, for example, for two atoms with atomic weight 10 and with a distance between the nuclei of 1 Å we have:

\[ \nu_1=T_1-T_0=2B=6.68\ \mathrm{cm}^{-1} \]

\[ \lambda_1=\frac{1}{\nu}\simeq 0.15\ \mathrm{cm}. \]

The wavelength that corresponds to the first “pure” rotational transition thus already lies in the region of short electric waves. If rotational transitions of such an order of magnitude are superimposed on electronic or vibrational transitions, of the order of 1000 to \(10\,000\ \mathrm{cm}^{-1}\), then they give a “fine structure” with spacings between the lines of several Angstrom units. Only in very light atoms, such as, for example, \(\mathrm{H}_2\), \(\mathrm{He}_2\), \(\mathrm{LiH}\), do the rotational quanta reach magnitudes of the order of \(50\ \mathrm{cm}^{-1}\). The individual rotational lines lie in the spectra of such molecules comparatively far from one another and, even with small dispersion, do not merge into “bands.” Such spectra are called “many-line.” Comparatively “well resolved” are also the bands of molecules in which at least one atom is very light, i.e., first of all, hydrides. Thus, for example, the reduced mass of \(\mathrm{H}_2\) is only 0.5, the mass of HgH is still only 1.0, whereas already for \(\mathrm{N}_2\) it reaches the value 7, and for \(\mathrm{Br}_2\) the value 40. Thus, if in the investigation of spectral bands rotational lines are encountered that are situated at comparatively large distances from one another, then one may be certain that they originate from a hydride molecule. In molecules consisting of two very heavy atoms, such as, for example, \(\mathrm{Hg}_2\) or \(\mathrm{PbJ}\), it is almost impossible to determine the rotational structure of the bands.

In conclusion of this section, a few more words should be said concerning the rotational terms of molecules with angular momentum of the electrons. The rotational energy always remains, according to equation (4), connected with the angular momentum of the molecule. But this angular momentum itself only in the “case \(d\)” is determined by the quantum number \(K\), according to equation (6); in all other cases, on the contrary, it is expressed by the quantum numbers \(J\) and \(\Omega\), or \(K\) and \(\Lambda\). If, for example, we have Hund’s case \(a\), then the angular momentum of the electrons about the molecular axis \(\Omega\) and the total angular momentum of rotation of the molecule \(J\) are the most strictly quantized quantities, whereas the angular momentum of the molecule’s rotation \(R\) is determined, according to equation (1), from \(J\) and \(\Omega\). From equations (1) and (4) we obtain for the rotational terms:

\[ T_{\mathrm{rot}}=B\left(\sqrt{J^2-\Omega^2}\right)\left(\sqrt{J^2-\Omega^2}+1\right)\simeq B\left[(J+1/2)^2-\Omega^2\right], \tag{10} \]

[using \((J+\tfrac{1}{2})^{2}\) instead of \(J(J+1)\)]. Thus, if in case \(a\) we wish to express the rotational terms in the form of a function of an integer, this is possible only by means of equation (10) instead of the simple equation (6); the integer entering thereby into the equation denotes not the rotational angular momentum of the molecule \(R\), but the total rotational angular momentum of the system \(J\). Since the latter is composed of \(R\) and \(\Omega\), it must be equal to \(\Omega\) for \(R=0\); in other words, the number \(J\) of the first rotational term in “case \(a\)” is not zero, but is equal to \(J-\Omega\). This relation makes it possible to determine the quantum number \(\Omega\) of the electronic term from the “dropping out” of rotational lines, which should have corresponded to values \(J<\Omega\).

In Hund’s case \(b\), after the spin \(S\) has been separated from the axis, instead of the angular momentum \(\Omega\) there appears a single orbital moment \(\Lambda\) still coupled to the molecular axis, and instead of the total rotational angular momentum \(J\)—the rotational angular momentum \(K\), which is composed only of the orbital moment \(\Lambda\) and the rotation of the molecule \(R\) (without the spin \(S\)). The rotational terms in this case are expressed by the equation:

\[ T_{rot} \approx B\left[(K+\tfrac{1}{2})^{2}-\Lambda^{2}\right], \tag{11} \]

and the number of lines “falling out” at the beginning of the sequence determines the quantum number \(\Lambda\).

Finally, in “case \(d\)” both the orbital angular momentum and the spin are separated from the molecular axis, and the rotation itself is fairly strictly quantized; when the rotational terms are represented by means of integers we obtain the formula:

\[ T_{rot} \approx B(R+\tfrac{1}{2})^{2}, \tag{12} \]

where \(R\) this time really denotes the pure rotational quantum number and, consequently, always begins with the zero value.

b) Dissociation upon rotation. The estimate of the magnitudes of rotational quanta given above refers to the first members of the series. The tenth rotational quantum must already be larger by two tenths of a percent, and the hundredth member would have to reach a magnitude of the order of the electronic terms. In this connection, formula (8) gives us no upper bound for the possible number of rotational quanta. At first sight it seems that a molecule can absorb unlimited amounts of energy in the form of rotational energy, which is, undoubtedly, incorrect. This false consequence arises because we have neglected the stretching of the molecule under the action of centrifugal forces. Just as, by exciting vibrations, a molecule can gradually be brought to dissociation, so too the accumulation of excessively large amounts of rotational energy leads to the rupture of molecular bonds. But the conditions of dissociation upon rotation are less accessible to observation than in dissociation due to

oscillations. This is explained by the fact that the energy of rotation—in contrast to vibrational energy—is to a significant degree kinetic in nature and can never be used entirely to overcome the potential attraction of the atoms; for this only that part of the energy which corresponds to the centrifugal force is used. Therefore dissociation occurs only in the case when the total rotational energy has already considerably exceeded the actual work of dissociation; thus dissociation due to rotation is a process corresponding to a chemical reaction with “activation energy.” We shall obtain a clearer idea of dissociation in rotation on the basis of the following argument, first carried out by Oldenberg. Dissociation obviously arises in the case when the force acting on the atom, beginning from a certain distance between the nuclei, is directed “outward” and not “inward.” This force consists of two components. The first component is attraction under the action of chemical forces; at each given distance between the nuclei this component is determined by the slope of the potential curve, with which we became acquainted in Part II. The second, oppositely directed component is the centrifugal force. According to the generally known formula, it is equal to:

\[ F=\mu \dot{\varphi}^{2} r=\frac{P_{\varphi}^{2}}{\mu r^{3}}, \tag{13} \]

where \(\dot{\varphi}\) denotes the angular velocity, and \(P_{\varphi}\) the angular momentum. (The whole argument is “classical” and therefore considers only plane rotation.) A comparison of equations (13) and (4) with one another shows that the centrifugal force is the negative derivative of the rotational energy. Therefore the behavior of a rotating molecule can be represented visually in the following way: for each given distance between the nuclei the potential and rotational energies are added, and then the slope of the curves thus obtained is considered (see Fig. 5). The lower curve \(ABC\) is the curve for the state having no rotation; the nearest curve above it corresponds to the first rotational quantum, the second curve to the second quantum, etc. As is seen from (4) and (5), the rotational energy decreases quadratically as the distance between the nuclei increases. Thus, with increasing \(r\), the vertical distance of the curve \(J=1\) from the curve \(J=0\) continually decreases. Since the curve \(ABC\) approaches its limiting value much faster than quadratically, then, as shown in Fig. 5, the addition of \(E_{\mathrm{rot}}\) and \(E_{\mathrm{pot}}\) gives curves with a maximum at a certain distance between the nuclei, followed by a fall of the curve. Curves with higher rotational quantum numbers always have a maximum lying somewhat farther to the left, and a minimum, on the contrary, shifted to the right; such curves are—

are always somewhat flatter. At a certain definite number of rotational quanta the curve has only one turning point and, beginning with this point, the curves have a negative slope throughout their entire extent. The curve with a turning point corresponds to the maximum number of rotational quanta that the molecule can accept without becoming unstable. The course of the curves in Fig. 5 shows that in rotational states close to dissociation, a quite insignificant vibration is sufficient to cause the molecule to break up. Franck and Sponer worked on determining the total amount of energy required for dissociation of a molecule as a result of simultaneous rotation and vibration. It turned out that the total energy required for this purpose is the smaller, the higher the vibrational state. In short, a molecule can be dissociated the more “cheaply,” the larger the part of the energy expended on vibration and the smaller the fraction of rotational energy.

The arguments of Oldenberg, Franck, and Sponer found their experimental confirmation in the breaking off of bands, observed in many molecules, at definite rotational numbers. Whereas normal individual rotational lines weaken only gradually as the rotational quantum number increases, in HgH, for example, a completely sudden breaking off of bands is observed; moreover, this phenomenon is observed at the smaller rotational quantum number, the higher the vibrational state.

Fig. 5. Dissociation under rotation.

c) Determination of the distance between nuclei. The investigation of the rotational structure of bands gives one very important collateral result. Knowing the rotational quanta, we can, using equation (8), determine the distance between the nuclei. This method is the simplest and most direct of all the methods known to us for determining the relative position of the nuclei in a molecule; unfortunately, up to the present time it has been applicable only to diatomic molecules.

Table 1 gives a large part of the internuclear distances determined in this way. As is seen from the table, along with molecules of elements, the molecules of hydrides have been studied best in this respect. Table 1 gives a whole series of regular relations on which we have no opportunity to dwell here in detail. Generally speaking, it may be asserted that the nuclear distances in diatomic molecules, analogously to the nuclear distances in crystals, must be regarded as additive quantities; they can be decomposed into two “atomic radii.” These radii increase regularly with the...

advancement through the periodic system, but at the same time they also depend on the kind of bond. In general they are smaller than the radii of the same atoms in crystals; thus, for example, the C—C distance in diamond is 1.54 Å, in graphite 1.42 Å, and in the molecule C₂—only 1.31 Å; this diminution of the radius may be connected with the diminution of the coordination number, which for diamond is 4, for graphite 3, and for C₂ only 1. The more “one-sided” the bond, the more closely the nuclei approach one another; obviously this is caused by additional polarization forces of bonding, which do not act in an all-sided bond.

TABLE 1

Distances of nuclei in the ground states of diatomic molecules in Å-units \((10^{-8}\,\mathrm{cm})\)

4. Structure of Bands

Until now we have spoken of rotational terms and of the properties of molecules determined by them. In this concluding section of the article we must say a few words about the rotational structure of the bands themselves, since it is by it, above all, that the terms and their properties are determined. As we have already seen in Fig. 2 in Part I of the present article, each band consists of separate “rotational lines” situated very close to one another, which correspond to different “rotational transitions” (with the electronic and vibrational transition unchanged). The ordering of this enormous number of lines formerly seemed an insoluble problem; at the present time, in each individual case, it requires only a greater or lesser degree of patience.

The basic principle in the arrangement of the lines is the division of bands into branches. The theoretical basis for the formation

The selection rules for rotational transitions are different. If the molecule after emission (or absorption) has no angular momentum of the electrons, then the selection rule is in full agreement with the rule for jumps of the electronic orbital quantum number \(l\): only those lines can appear which correspond to combinations of terms with rotational quantum numbers differing by \(\pm 1\).

If the molecule before and after emission (or absorption) has the same angular momentum of rotation \(\Lambda\) (or \(\Omega\)), not equal to zero, then, in addition to transitions with rotational quantum numbers differing by 1, transitions without a change of rotation are also allowed, although the latter are much less probable. Finally, if the radiation occurs with a change in the angular momentum of the electrons \(\Lambda\), then transitions without a change of the rotational quantum number are also probable, or even more probable than transitions with a rotational jump of \(\pm 1\).

Thus we have:

\[ \begin{array}{ll} \Delta J=\pm 1\ \text{strongly},\ \Delta J=0 & \text{forbidden for } \Sigma \to \Sigma \text{ bands},\\ \Delta J=\pm 1\ \text{strongly},\ \Delta J=0 & \text{weakly for } \Pi \to \Pi,\ \Delta \to \Delta \text{ bands},\\ \Delta J=\pm 1\ \text{or } 0 & \text{approximately equally strong for } \Sigma \to \Pi,\ \Pi \to \Sigma \text{ bands}. \end{array} \]

We cannot dwell here on the consequences of this law; they represent special cases of the general wave-mechanical theory of the relative probability of different quantum jumps, a theory based on the change, corresponding to these jumps, of the dipole moment of the electron distribution. The stronger this change, the stronger, in classical terms, the interaction with the electromagnetic field of the surrounding radiation; the greater, in the language of quantum mechanics, is the probability of the corresponding jump upon the emission or absorption of a quantum of light.

Thus in each band there are three (in special cases two) kinds of lines, corresponding to the jumps \(\Delta J=+1\), \(\Delta J=0\), \(\Delta J=-1\). Lines of the first kind are combined into the so-called \(R\)-branch (or positive branch), lines of the second kind into the \(Q\)-branch (or zero branch), and lines of the third kind into the \(P\)-branch (or negative branch).

In the simplest cases the assignment of a line to one branch or another is obvious. Let us consider, for example, a rotational-vibrational band. Since in this case both combining vibrational states belong to the same electronic term, one may, in a first approximation, assume that the distance between the nuclei, and hence the magnitude of the rotational quantum, do not change during the jump. If we are dealing with a \(\Sigma\)-term, the \(Q\)-branch is absent altogether (Fig. 6); for \(\Pi\)-, \(\Delta\)-… terms all the lines of this branch fall in one place (Fig. 7).

If the series of rotational terms of the lower vibrational term is represented in the form:

\[ T''=B''J''(J''+1)\simeq B''(J''+1/2)^2, \tag{14a} \]

and that of the upper one in the form:

\[ T'=\nu_0+B'J'(J'+1)\simeq \nu_0+B'(J'+1/2)^2, \tag{14b} \]

where \(\nu_0\) denotes the “pure vibrational transition,” then for \(B'=B''\) we have for the \(Q\)-branch:

\[ \nu_{J''}=\nu_0, \tag{15a} \]

(\(Q\)-branch, \(J''=J'\)),

Fig. 6. Structure of the rotational-vibrational band in the \(\Sigma\)-state.

Fig. 7. Structure of the rotational-vibrational band in the \(\Pi\)-state.

and for the other two branches:

\[ \nu_{J''}=\nu_0+2B(J''+1) \tag{15b} \]

(\(R\)-branch, \(J''=J'+1\)),

\[ \nu_{J''}=\nu_0-2BJ'' \]

(\(P\)-branch, \(J''=J'-1\)).

The \(R\)-branch thus proceeds from the “zero point” \(\nu_0\) toward higher frequencies, i.e. toward shorter waves; the \(P\)-branch proceeds toward lower frequencies, and consequently toward longer waves. The lines of both branches are arranged equidistantly, at intervals of \(2B\ \mathrm{cm}^{-1}\) from one another. If the \(Q\)-branch is absent, then in the middle of the band, between the first line of the \(R\)-branch and the first line of the \(P\)-branch, there is a gap.

For a visual representation of the structure of the band, a very convenient method is provided by the Fortrat diagram. In such a diagram the frequencies of the lines are plotted as abscissae, and the rotational quantum numbers as ordinates. By convention, for ordering purposes the rotational quantum numbers of the lower state \(J''\) are used throughout (and for the upper state \(J'\)). For the rotational-vibrational band considered above, the diagram has the form shown in Fig. 6. The lower part of the figure represents the band spectrum itself (on the same frequency scale).

If the electronic term is a \(\Pi\)- or \(\Delta\)-term, then for the rotational terms equation (10) or (11) applies instead of (8). However, for the frequencies of the lines, after combining in \(\nu_0\) all terms independent of rotation, one again obtains (15), but for \(J\) all values smaller than \(J=\Omega\) are forbidden (in “case \(a\)”); in “case \(b\)” in (15) \(J\) is everywhere replaced by \(K\), and for \(K\) values are allowed beginning with \(K=\Lambda\). In the middle of the band many lines now drop out. If, for example, we have a \(\Pi\) term, then the rotational-vibrational band has two gaps, one on each side of the \(Q\)-branch (Fig. 7); in the case of a \(\Delta\)-term there are two missing lines on both sides of \(\nu_0\), etc.

Electronic bands in which the distance between the nuclei is the same in the two combining electronic states should have the same form as rotational-vibrational bands. This, however, rarely occurs in practice; ordinarily each electronic term corresponds to its own force constant and, consequently, to its own internuclear distance. A consequence of this change of the moment of inertia upon an electronic transition is the characteristic curvature of the branch in the Fortrat diagram.

Let us assume, for example, that the upper excited state has a weaker bond and, therefore, a larger internuclear distance. Thus \(B'\) is smaller than \(B''\). The formula for the \(Q\)-branch is then:

\[ \nu_{J''}^{Q}=\nu_0+(B''-B')(J''+{}^1/2)^2, \tag{16a} \]

where \(\nu_0\) denotes the sum of the electronic and vibrational transitions. In an analogous way, for the \(R\)-branch we have:

\[ \nu_{J''}^{R}=\nu_0+(B''-B')(J''+{}^1/2)^2+2B'(J+1) \tag{16b} \]

and for the \(P\)-branch:

\[ \nu_{J''}^{R}=\nu_0+(B''-B')(J''+{}^1/2)^2-2B'J'' . \tag{16c} \]

Thus, instead of straight lines, parabolas appear in the diagram. At small rotations the branches go in the directions indicated above: the \(R\)-branch toward shorter waves, the \(P\)-branch toward longer waves, the \(Q\)-branch “vertically upward.” As long as the quadratic term in (16) has a magnitude comparable with the linear terms, all three branches curve. The influence of the quadratic term is especially striking for the \(P\)-branch; if the positive quadratic term reaches the absolute magnitude of the negative linear one, the \(P\)-branch turns and proceeds in the same way as the \(R\)-branch, toward shorter waves (Fig. 8). As is seen in the lower part of the figure, the turning of the \(P\)-branch leads to a crowding of lines at the “edge” \(AB\). In this case the band has a sharp boundary on the long-wavelength (“red”) side, while the lines on the “violet”

gradually become rarer and weaker. Such bands are called “shaded toward the violet side.” If the distance between the nuclei in the upper state is smaller than in the lower one, then, in an analogous manner, an edge is obtained on the violet side in the \(R\)-branch, and the band proves to be “shaded toward the red side.” As a consequence of the formation of edges, the lines of the three branches become intermingled; in order to separate the bands into branches, one must apply to the empirically measured lines certain “combination conditions,” which are obtained from equations (16) by forming differences and thus determine their affiliation and number.

Fig. 8. Structure of the band \({}^{1}\Sigma \to {}^{1}\Pi\) (without consideration of \(\Lambda\)-doubling) for \(B' > B''\). (The lines \(Q\) (0) and \(R\) (0) do not fall out, since the sequence of rotational terms of the lower state \({}^{1}\Pi\) begins with \(J=1\).)

Since the band contains only three branches, its ordering with the aid of such “combination conditions” presents no difficulty. But considerable complications often arise, caused by two circumstances. First, by multiplicity. In molecules that are neither too light nor too heavy, the multiplet splitting of electronic terms is of just the same order of magnitude as the rotational terms. If two such multiplet terms are combined with one another, then a sequence of bands included one within another is obtained. Thus, for example, the combination of two \({}^{2}\Pi\)-terms, each of which contains the components \({}^{2}\Pi^{1/2}\) and \({}^{2}\Pi^{3/2}\), gives 4 bands:

\[ {}^{2}\Pi^{1/2} \to {}^{2}\Pi^{1/2} \]

\[ {}^{2}\Pi^{3/2} \to {}^{2}\Pi^{1/2}, \]

\[ {}^{2}\Pi^{1/2} \to {}^{2}\Pi^{3/2} \quad\text{and}\quad {}^{2}\Pi^{3/2} \to {}^{2}\Pi^{3/2} \]

This already gives 12 branches of bands passing through one another.

To this is further added the doubling of electronic terms. As has already been mentioned more than once, all terms with \(\Lambda \ne 0\) are double, since they consist of one positive and one negative term of equal energy. With increasing rotation the degeneracy is removed. The branches of the bands then split into two “sleeves” (“Arme”), which diverge more and more as the values of \(J\) increase. Thus, taking this splitting into account (which, as was indicated on p. 857, represents the accompanying phenomenon

in the “\(\alpha\)-splitting”* case, we shall say that a \({}^{2}\Pi — {}^{2}\Pi\) band has not 12, but 24 branches! Under such conditions, the systematization and numbering of the lines require a certain amount of patience and labor.

As is clear from what has been said, the number of branches and, to an even greater degree, the number of lines falling at the beginnings of individual branches give clear indications concerning the nature of the corresponding electronic terms; if the rotational structure of a band is being investigated, it is always possible to indicate whether we have \(\Pi \to \Sigma\), \(\Pi \to \Pi\), or some other definite combination of terms.

In conclusion let us say a few words concerning the distribution of intensity within the branches of bands. If one does not take into account intensity anomalies (oscillating intensities, so-called perturbation spots, predissociation phenomena, band cutoffs owing to dissociation), then it may be said that the intensities are determined mainly by the Boltzmann temperature distribution in the molecule among the various rotational states. In temperature equilibrium, the \(J\)-th rotational term with energy \(E_J\) is present in the relative concentration

\[ P_J = A (2J + 1) e^{\frac{E_J}{kT}}, \tag{17} \]

where \(A\) denotes a constant characteristic of the electronic and vibrational terms, and \((2J + 1)\) is the “statistical weight” of the rotational terms. Owing to the presence of the factor \((2J + 1)\), the maximum of the distribution lies not at the term with the lowest energy (as is the case in electronic and vibrational terms), but at a certain nonzero value of \(J\), which increases with rising temperature. Accordingly, in the branches of bands the first lines are not the strongest*; the maximum of intensity lies rather at definite rotational quantum numbers. From the position of the intensity maximum one may determine the temperature of the emitting gas. In any case, however, it has been established that such a method of determining temperatures must be applied with caution, since different molecules in one and the same discharge, depending on circumstances, possess very different “temperatures.”

* It should be borne in mind that the relative intensity of lines depends not only on the distribution (17) over rotational terms, but, in addition, also on the relative probability of these terms, which, in turn, is a function of \(J\). The complete intensity formulas (the so-called Hönl–London formulas) are very complicated and vary depending on the angular momentum of rotation of the electrons of the corresponding terms and on the branch under consideration.