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DIFFUSE SPECTRA AND CHEMICAL PROBLEMS
E. V. Shpolsky, Moscow
- Introduction. 2. Potential curves. 3. Construction of potential curves. 4. Quantum-mechanical resonance. 5. Auger effect. 6. Passage through a potential barrier. 7. Franck–Condon principle. 8. Franck–Condon principle and the structure of spectral bands. 9. Determination of the energy of dissociation. 10. Optical dissociation and the type of chemical bond. 11. Ionic molecules. 12. Atomic molecules. 13. Polarization (van der Waals) molecules. 14. Unstable states. Continuous spectrum of hydrogen. 15. “Predissociation” spectra. 16. How to understand predissociation. 17. Explanation of predissociation: a) shortening of the lifetime of the molecule, b) spontaneous dissociation. 18. Conditions for the possibility of predissociation. 19. Franck–Condon principle in predissociation. 20. Consideration of various cases of predissociation with the aid of the Franck–Condon principle. 21. Dissociation by rotation. 22. Magnetic quenching of fluorescence. 23. Induced predissociation. 24. Chemical applications:
1. Introduction
In gas spectra one often observes—both in absorption and in emission—individual sections or entire regions that cannot be resolved at any resolving power of the instruments. The best known among such continuous spectra are the so-called limiting continua in atomic spectra: at the limit of an absorption series in an atomic spectrum there is always adjacent a continuous region of absorption. The interpretation of continua of this type is at present entirely trivial: the series limit corresponds to the work of freeing the electron from its atomic bond, while the continuum corresponds to the same work plus a certain kinetic energy of the ejected electron; since the latter has an unquantized character, the result is the diffuse form of the spectrum. Analogous continua were found later also in the spectra of diatomic molecules, where the continuum adjoins the place at which band quanta merge. As is known, Franck explained these continua as a consequence of the optical (photochemical) dissociation of the molecule into atoms flying apart with kinetic energy.
DIFFUSE SPECTRA AND CHEMICAL PROBLEMS
…value, owing to the unlimited increase of the repulsive forces at unlimited approach of the atoms. Thus, the entire curve of potential energy has the form shown in Fig. 1 at the top.
Of course, we could have taken as the zero of potential energy its value for \(r=\infty\), as is usually done. In that case our curve would simply have been shifted parallel to itself, so that the abscissa axis would serve as its asymptote and the minimum would be situated below the abscissa axis. Since for us in what follows the most essential thing is only the mutual arrangement of the various potential curves, either method of representation may be used with equal success. The method we have adopted is somewhat more convenient, and therefore we shall chiefly use it in what follows.
If, besides the potential energy, for some reason the course of the magnitude of the force as a function of the distance is also of interest, then the corresponding curve can be constructed by using the well-known relation
\[ F=-\frac{dU(r)}{dr} \]
and carrying out a graphical differentiation with the aid of the curve \(U(r)\). In Fig. 1 (lower half) such a force curve is given, corresponding to the upper curve of potential energy. Both curves are given for the real molecule HCl \(^{49}\).
Fig. 1. Force curve and potential-energy curve.
Let us now suppose that we already have a completed molecule. Obviously, the nuclei in it will rest at a distance \(r_0\). If we remove them from the equilibrium position, the nuclei will begin to oscillate, periodically converting their potential energy into kinetic energy. We know, however, that not every amplitude and not every value of the energy of oscillation is possible, but only selected ones. These selected levels of vibrational energy we may, for convenience, plot on the same drawing as the potential curve, as is shown in Fig. 2. It is clear that only at the points of intersection of the levels with the potential curve will all the energy of the molecule be potential; at the remaining points it will be partly kinetic and partly potential. Thus, for example, if the molecule oscillates with two quanta of vibration (Fig. 2), then only at the internuclear distances \(r_2=OQ\) and \(r_2'=OP\) will all its energy be potential—these will be the turning points in the oscillations; in some intermediate position, determined—
But, in addition to limiting continua, diffuse spectra of another kind are also observed. Thus, for example, sometimes the entire spectrum is continuous* (the absorption spectrum of HCl, HBr, HI, the continuous emission spectrum of molecular hydrogen, etc.); in other cases, on the contrary, the continuous bands are so narrow that they differ comparatively little from lines; there are cases when, within an extensive diffuse region, fluctuations of the spectrum intensity are observed, producing a picture resembling the usual sequence of bands, and so on.
The study of continuous spectra in gases is of considerable interest not only from the optical, but also—and perhaps chiefly—from the chemical point of view. The very circumstance that one of the two states which combine to emit a continuous spectrum is non-quantized leads one to expect that the study of these spectra will make it possible to approach more closely the processes of dissociation of molecules, as well as the reverse process of formation of molecules from colliding atoms. It is precisely from this chemical point of view that the mechanism of the origin of continuous spectra is considered in the present article.** For the convenience of the reader, in the first paragraphs a brief exposition is given of a number of questions familiarity with which is necessary for understanding what follows.
2. Potential curves. In what follows we shall make extensive use of the so-called potential curves, i.e., curves representing the dependence of the potential energy of a molecule on the distance between its nuclei. The qualitative character of such curves is easily established from the following most general considerations. The interaction between two atoms forming a molecule, as is known, is characterized by the fact that at large distances the atoms attract one another, while at small distances this attraction turns into repulsion. At some distance \(r = r_e\) the forces of attraction and repulsion mutually balance one another, and the atoms are at stable equilibrium. Obviously, this distance \(r_e\) corresponds to a minimum of the potential energy \(U(r)\). If we agree to take this minimum as the zero of potential energy, then the quantity \(U(r)\) for any other distance gives us the total energy of the molecule whose nuclei are at the distance \(r\). Consequently, for \(r = \infty\) the quantity \(U(r)\) will be equal to the total energy of the separated atoms, i.e., to the dissociation energy of the molecule. On the contrary, for \(r = 0\) the potential energy tends to infinity—
* It should be remembered that throughout this entire article the discussion concerns spectra of gases at reduced pressure, so that broadening of lines at high pressure is left aside.
* For acquaintance with the earlier literature on continuous spectra see R. Mecke, Handb. d. Physik, Bd. XXI, and also the review by Finkelnburg: Phys. Z.*, 1930.
...by a distance between the nuclei \(OR=r\) \((r_2>r>r_1)\), part of the energy will be kinetic and part potential. We can say at once that at this distance the potential energy will be \(RB\), and the kinetic energy \(B'B\), so that \(RB' + B'B = U\). The entire process of oscillation can in this case be vividly represented by the motion of a heavy ball raised to a certain level and performing oscillations inside a hollow whose cross-section has the form of the potential curve.
The potential curves of the type considered up to now correspond to interatomic interactions which until recently seemed the only possible ones. Wave mechanics has shown, however, that there exists still another, entirely peculiar kind of interaction. Namely, it turns out,* for example, that two hydrogen atoms interact according to the ordinary potential curve \(I\) only when the spins of their electrons are antiparallel. If, however, the spins are parallel, then the interaction is characterized by curve \(II\) (Fig. 2), which has no minimum. Since there is no minimum, equilibrium is not established at any distance between the nuclei: curve \(II\) is a curve of pure repulsion; a molecule brought into a state corresponding to this curve slowly dissociates.
Fig. 2. Potential curves of a stable and an unstable molecule.
3. Construction of potential curves. For constructing potential curves one may make use of any expressions for the potential energy of a molecule as a function of the distance between its nuclei. Suppose, as the simplest example, that we have a molecule built of ions with charges \(\pm e\). Then, in a first approximation, such ions will attract each other with a force inversely proportional to the square of the distance, and their mutual potential energy will be:
\[ U=-\frac{e^2}{r}. \]
* This follows from the theory of homopolar valence of Heitler and London, which has received exhaustive treatment in the pages of Uspekhi. Cf. London’s articles: IX, 167, 1929; Frenkel et al.: IX, 515, 1929.
However, the Coulomb law of interaction is too crude an approximation. At small distances, owing to attraction, a repulsive force arises between ions, varying inversely as a higher power of the distance, and, in addition, mutual polarization of the ions is manifested. Therefore, instead of the simple Coulomb formula, one should take the expression for the potential energy in the form of a series, which, in its most general form, has the following form:
\[ U=-\frac{e^{2}}{r}+\frac{c_{1}}{r^{2}}+\frac{c_{2}}{r^{3}}+\frac{c_{3}}{r^{4}}+\cdots \]
Kratzer gave an expansion in a series for the potential energy of an ionic molecule, assuming that the vibrations in the molecule are harmonic. Kratzer’s formula has the form:
\[ U=C-k\left[\frac{r_{0}^{2}}{r}-\frac{r_{0}}{2r^{2}}+c_{3}\left(\frac{r-r_{0}}{r_{0}}\right)^{3}\right], \]
where \(C\) and \(k\) are molecular constants which can be calculated from band spectra, \(r_{0}\) is the distance between the nuclei in the molecule at equilibrium. Although this formula was derived for ionic molecules, in the case of small vibrations, to which it alone is applicable, it can with equal success also be used for constructing the potential-energy curve of atomic (i.e., homonuclear) molecules. The drawback of Kratzer’s formula is that it gives the correct result only for the case of harmonic vibrations, i.e., for distances close to \(r_{0}\).
Free from this drawback is Morse’s formula,\(^1\) which is used almost exclusively at the present time in constructing potential curves. Morse’s formula gives the expression for the potential energy not in the form of a series in inverse powers of \(r\), but in the form of an exponential function
\[ U=D[1-e^{-a(r-r_{0})}]^{2}, \]
where \(D\) is the work of dissociation of the molecule. Substituting this expression for \(U\) into the corresponding Schrödinger equation and, by finding the relative eigenvalues of this equation, one can find a formula connecting \(a\) with the constants of the molecule:
\[ a=\sqrt{\frac{2\pi^{2}\mu c\omega^{2}}{D}}, \]
where \(\mu\) is the reduced mass of the molecule \(\left(\frac{1}{\mu}+\frac{1}{m_{1}}+\frac{1}{m_{2}}\right)\), \(\omega\) is the frequency of the mechanical vibrations (in \(\mathrm{cm}^{-1}\)). The quantities \(r_{0}\) and \(\omega\) can be obtained from analysis of band spectra.\(^*\) If for some reason—
\(^*\) How these calculations are made, see, for example, Mekke, Band Spectra and Their Significance for Chemistry (“Advances in Physical Sciences,” vol. IX, 757, 1929).
either only one of these quantities is directly accessible to determination, then the other can be calculated using the relation:
\[ r^{2}\cdot \omega=\mathrm{const}. \]
The quantity \(D\)—the dissociation energy—in many cases is also determined with great accuracy from band spectra (cf. below, § 9). In those cases where the exact value of \(D\) is unknown, it may be found approximately from the relation known from the theory of band spectra:
\[ D=\frac{\omega^{2}}{4x\omega}, \]
where \(x\) is a fraction characterizing the deviation of the molecule’s vibrations from harmonic ones. The quantity \(x\omega\) is determined from the analysis of band spectra simultaneously with \(\omega\).
Thus band spectra make it possible to find all quantities necessary for constructing potential curves. One should not, of course, forget that Morse’s formula is only approximate. However, for the purposes for which potential curves are constructed, the accuracy it provides is quite sufficient. If for some reason the force curve is of interest along with the potential curve, it can always be constructed by using the relation between force and potential.
4. Quantum-mechanical resonance. In molecular phenomena leading to the occurrence of diffuse spectra, quantum-mechanical resonance plays an essential role. Let us consider the simplest case of a many-electron system—an atom of helium with its two electrons. Let one of these electrons be on the normal orbit, the other on some excited one. Let the state of the first electron be described by the wave function \(\Psi_{1}(1)\), and let the corresponding energy be \(E_{1}\); for the second electron let the same quantities be \(\Psi_{2}(2)\) and \(E_{2}\). As a first approximation, assume that there are no interactions between the electrons; then the state of the atom as a whole will be described by the wave function \(\Psi\), satisfying Schrödinger’s equation for the given system, with
\[ \Psi=\Psi_{1}(1)\cdot\Psi_{2}(2), \]
and the energy of the whole atom will be equal to the simple sum of the energies of the individual electrons:
\[ E=E_{1}+E_{2}. \]
If we now interchange the electrons, then, owing to their complete identity, the energy \(E\) will retain its value, while the wave function will now be:
\[ \Psi'=\Psi_{1}(2)\cdot\Psi_{2}(1). \]
We have here the case in which different functions \(\Psi\) and \(\Psi'\), each of which describes its own distinct oscillatory state, correspond to one and the same energy \(E = E_1 + E_2\), i.e., a case of degeneracy. As is well known, a system of two oscillators—for example, two pendulums or two electrical oscillatory circuits—tuned to resonance, i.e., having one common frequency but not coupled to one another, is also degenerate (the number of natural frequencies is smaller than the number of degrees of freedom). In the case of an atom with two electrons, the degeneracy is due to the complete identity of the electrons themselves and to the identical character of their connection. This latter circumstance gave Heisenberg occasion to call this degeneracy also resonant.
If, however, we pass to the next approximation and take into account the coupling between the oscillators, then the well-known resonance beatings appear: the energy periodically wanders from one oscillator to the other. Such fluctuations of energy are the result of the splitting of one common oscillator frequency into two nearby frequencies, of which one is larger than the original one (symmetric oscillation, the pendulums oscillate in the same direction), while the other is smaller (antisymmetric oscillation, the pendulums oscillate in opposite directions). The energy of a system of such two oscillators will no longer be equal to the sum of the energies of each separately, but will contain one more term—the mutual potential energy of both oscillators, or the energy of their coupling. By formal analogy with this case, for our atom with two electrons we must expect the occurrence of analogous fluctuations in the atom as well. This means that if electron 1 was initially on the normal level and electron 2 on the excited one, then after some time electron 1 will be on the excited level and electron 2 on the normal one, and so on. The stationary states in which these fluctuations are absent correspond to the two possible stationary oscillations of the oscillators—symmetric and antisymmetric. At the same time, just as in the case of coupled oscillators, the energy of the whole atom, in addition to the sum of the energies \(E_1 + E_2\), contains an additional term—the potential energy of the coupling of the electrons, which has different values in the symmetric and antisymmetric states. Under certain circumstances this coupling energy can have a very considerable magnitude. It accounts for a whole series of spectroscopic and chemical facts.
5. Auger effect. The phenomenon, discovered by Auger and having an extraordinarily general significance, consists in the following: if one ionizes any of the inner shells of an atom (for example, the \(K\)-shell), then the return of the resulting excited atom to the normal state may occur in two ways: 1) by the filling of the vacant place formed in the \(K\) group by an electron
from a neighboring shell (for example, \(L\)); this process is accompanied by the release of excess energy in the form of an X-ray quantum and is the usual mode of origin of X-ray fluorescence, and 2) by means of a peculiar process of spontaneous ionization, in which the atom returns to the normal state without radiation, while the excess energy is spent on ejecting from the same atom another electron belonging to a more weakly bound shell. In Wilson photographs this process becomes noticeable because in some places it is seen that the paths of two or four electrons begin simultaneously from one atom.
Fig. 3. Toward the explanation of the Auger effect.
The quantitative theory of this phenomenon on the basis of wave mechanics was first given by Wentzel, whose calculations were based on the following physical picture. Let \(n_1\) and \(n_2\) be the principal quantum numbers of the two electrons of the helium atom. In Fig. 3 are shown the energy levels of the second electron when the first is in various excited states \((n_1 = 1\) and \(2)\).
Let us now consider the case when one of the discrete levels of some system of terms coincides with the continuous region of another system. As is seen from the figure, this occurs, for example, for the level \(n_1 = 2,\ n_2 = 2\). In this case resonance beats arise between the two systems of terms, in which the atom from the initial state \((n_1 = 2,\ n_2 = 2)\) will pass into a state of equal energy from the other system of terms \((n_1 = 1,\ n_2 =\) corresponding values from the continuum), i.e. the first electron will return to the normal state without radiation, and the second will fly out beyond the atom with kinetic energy the greater, the higher above the condensation point of the system \(n_1 = 1\) the level \(n_1 = 2,\ n_2 = 2\) lies.
Fig. 4. Passage through a potential barrier.
6. Passage through a potential barrier. The last specifically quantum-mechanical phenomenon that must be recalled is passage through a potential “barrier.” This phenomenon was revealed by Fowler and Nordheim in constructing the quantum-mechanical theory of the Richardson effect and, especially, by Gamow in connection with the theory of radioactive processes.
According to classical mechanics, a particle can make transitions between regions of equal energy separated by a potential barrier, as in Fig. 4, only on condition that its total energy is greater than the height of the mound. Some particle of energy \(W\), according to classical mechanics, cannot overcome the barrier, since in the shaded region its potential energy would have to be greater than its total energy, i.e. its kinetic energy would become negative and, consequently, its velocity imaginary. In reality, however, such transitions are nevertheless observed. It is known, for example, that there are groups of \(\alpha\)-particles of uranium I which fly out with an energy insufficient for overcoming the potential barrier around the nucleus. Analogous facts are also known for the liberation of electrons from a metal under the influence of an intense electric field.
All these facts, completely incomprehensible from the point of view of classical mechanics, are quite natural from the point of view of wave mechanics. Indeed, according to wave mechanics a particle has a finite probability of passing through a potential barrier even in the case when its energy is less than the height of the barrier. This feature is a complete analogue of the optical phenomenon of total internal reflection, in which, as is known, light nevertheless penetrates into the second medium, although with an intensity decreasing very rapidly (according to an exponential law) with distance.
Fig. 5. Various types of potential barriers.
\(a\)—emission of an electron from the surface of a metal, \(b\)—atom in an electric field, \(c\)—diatomic molecule, \(d\)—predissociation.
According to Franck’s apt remark, particles can not only pass over a potential barrier, rising above its summit, but also travel through a tunnel right through the barrier itself. As we shall see, these wave properties of particles explain a whole series of features of diffuse spectra—features interesting from the chemical point of view. In Fig. 5 various forms of potential barriers are shown.
7. The Franck–Condon principle. J. Franck expressed an important principle governing transitions between the vibrational levels of different electronic states of a molecule. Condon gave this principle a quantitative formulation and a quantum-mechanical justification. The essence of the Franck–Condon principle reduces to the following. Let the lower potential curve in Fig. 6 correspond to the normal state of the molecule, and the upper one to the excited state. Let us first consider the question from the point of view of the old quantum theory. The process of excitation of molecules
conditioned by the transition of the electron to a higher energy state. This process proceeds very rapidly, so that the relatively heavy nuclei, at the moment when the transition has already been completed, practically occupy their former positions. We may therefore say that in the first moment after excitation the molecule passes from the lower potential curve to that point of the upper potential curve which corresponds to the unchanged internuclear distance, i.e. lies directly above its former position in the unexcited state. Now we must also take into account that the vibrating nuclei must spend the greater part of the time near the turning points of the vibrations, where they have the smallest velocity. Therefore the most probable transitions will be those starting from the turning points of the vibrating nuclei and leading to those points of the upper curve which lie with them on one vertical line, i.e. transitions occurring without change in the distances and velocities of nuclear motion.
Fig. 6. The Franck–Condon principle.
The quantum-mechanical interpretation of Franck’s principle not only gives it a rigorous justification, but also introduces certain fundamentally new features. If we have any two states \(m\) and \(n\), then the probability of transition between these states, according to quantum mechanics, is determined by the integral:
\[ q_{mn}=\int q\psi_m\psi_n^{*}\,d\tau, \tag{1} \]
where \(q\) denotes the coordinate, \(\psi_m\) and \(\psi_n\) are the eigenfunctions, i.e. the solutions of Schrödinger’s equation for the corresponding states, \(d\tau\) is the volume element in phase space; the asterisk indicates that the complex conjugate quantity \(\psi\) is to be taken. (For example, for a linear oscillator the transition probabilities are expressed by the integral:
\[ X_{mn}=\int x\psi_m\psi_n^{*}\,d\omega.) \]
Let us consider two electronic states of a molecule \(E'\) and \(E''\), and let each of them correspond to its own system of vibrational levels with quantum numbers \(v'\) and \(v''\). To find the probabilities of transitions between different states characterized by the parameters \(E',v'\) and \(E'',v''\), we must form integrals of type (1).
The totality of the numerical values of all these integrals,
calculated for all possible combinations of energy levels forms an entire table, or “matrix,” of probabilities of transitions between the corresponding levels. If any element of this matrix is equal to zero, this means that the transition between the corresponding states is impossible.
It is precisely in this way that various “selection rules,” found semi-empirically with the aid of Bohr’s correspondence principle, are obtained in quantum mechanics.
Let \(E_e\) and \(E_v\) be, respectively, the electronic energy and the vibrational energy, and let \(\psi_e\) and \(\psi_v\) be the components of the eigenfunction depending, respectively, only on the electronic and only on the vibrational state of the molecule. Then, to a first approximation, one may put:
\[ E = E_e + E_v, \qquad \psi=\psi_e\cdot\psi_v. \]
Substituting these products instead of \(\psi\) and \(\psi'\) in integral (1) and, after making certain simplifications, we obtain for the probability of a transition between the vibrational levels of the upper and lower states an expression whose most essential part is the integral
\[ \int \psi_v'\cdot\psi_v''\,dr, \tag{1a} \]
where \(r\) is the distance between the nuclei. If the potential curves of the upper and lower states have exactly the same form, then the vibrational eigenfunctions \(\psi_v'\) and \(\psi_v''\) in the one and the other state will be identical and, by virtue of the orthogonality condition, integral (1a) will be equal to unity for \(v'=v''\), and zero in all other cases. This is what the Franck—Condon principle requires.
Fig. 7. Energy levels and vibrational eigenfunctions.
If, however, under electronic excitation the form of the potential curve changes substantially, the situation is somewhat more complicated. We shall consider this case only qualitatively. In Fig. 7, alongside the potential curves and energy levels, graphs are given of the values of the eigenfunctions \(\psi_v\) as functions of the distance \(r\) for two vibrational states: \(v''=0\) (dotted curve) and some higher one (solid curve). It is seen that for the “zero vibration” the function \(\psi_v\) has a broad maximum with its peak falling over the minimum of the potential curve. For any higher level, \(\psi_v\) has maxima at the classical turning points of the vibration, while at all other points it passes through a number of
rapidly alternating maxima and minima close to the axis. The eigenfunctions for the excited state have exactly the same character. It is clear that the integral \(\int \psi_v' \psi_{v'}''\,dr\) will have large values only when, for identical \(r\), the maxima of the functions \(\psi_v\) coincide or almost coincide. In all other cases it will be close to zero. But this means that the most probable transitions will be those satisfying the Franck—Condon principle (cf. Fig. 6). What is essentially new in the quantum-mechanical treatment is that the maxima have an appreciable width (sometimes a considerable one), as a result of which there is a certain indefiniteness in the distances at which transitions occur with the greatest probability: the positions of the oscillating nuclei for such transitions are determined by an entire probability curve. This feature, as we shall see, has a quite definite effect on the structure of diffuse spectra.
In what follows we shall make use primarily of Franck’s original formulation, because of its clarity and simplicity, and only in special cases shall we resort to the quantum-mechanical formulation.
Fig. 8. Franck’s principle and the structure of band spectra.
8. Franck—Condon principle and the structure of band spectra.
It is not difficult to see that, with the aid of the Franck—Condon principle, on the basis of the mutual arrangement of the potential curves in the normal and excited states, many features of band spectra can be theoretically justified. In Fig. 8 four characteristic cases are presented. In case I, excitation results in a strengthening of the bond; the equilibrium position is shifted to the left relative to the curve of the normal state. Examples of molecules belonging to this case will be given below. In case II the strength of the bond does not change upon excitation; the minima of the potential curves of the normal and excited states lie on the same vertical straight line. It is obvious that in this case the transitions occurring with greatest probability are those without a change in the state of vibration, for example the transitions \(0—0\) (\(v' = 0,\ v'' = 0\)), \(1—1\), \(2—2\), etc. Examples are the bands of SiN and the violet bands of the CN molecule.
Case III is encountered most often: here the potential—
DIFFUSE SPECTRA AND CHEMICAL PROBLEMS
the curve of the excited state is shifted to the right relative to the curve of the normal state. This means that the internuclear distance at equilibrium in the excited state is greater than in the normal one; upon excitation the molecule is loosened. Examples: the halogen molecules, the oxygen molecule. The Franck–Condon principle shows that in all such cases, upon excitation a considerable part of the energy of electronic excitation is converted into vibrational energy, since the most probable transitions will be those from the “non-oscillating” state \(v' = 0\), or from a state with a small number of vibrational quanta \((v' = 1, 2)\), into states with a large number of vibrational quanta, or even into the continuum. Still more extreme is case \(IV\). Here the upper potential curve is shifted so far to the right that the most probable transitions from the state \(v' = 0\) lead into the region lying above the dissociation limit, i.e. into the continuum.
In case \(III\) long series of quanta are often obtained, which can be followed up to their passage into the continuum. Such is the spectrum of iodine vapor \(J_2\); in bromine the displacement of the potential curve of the excited state to the right is more considerable; therefore here only a few discrete bands appear with noticeable brightness before the continuum. Finally, in chlorine the displacement is so great that no discrete bands are obtained at all, and the maximum of intensity lies far in the continuum.
Fig. 9. Origin of a diffuse spectrum.
Let us finally consider the case in which one of the combining states corresponds to a repulsion curve (Fig. 9). It is obvious that in this case we shall always be dealing with spectra of diffuse character, since transition to a repulsion curve always entails dissociation of the molecule. We can, however, apply the Franck–Condon principle to this case as well, in its extended formulation. Suppose we have the case shown in Fig. 9, when between two potential curves \(Ia\) and \(II\) there lies a curve \(Ib\), corresponding to an unstable state. Then, upon excitation, the molecule may pass from state \(Ia\) into state \(II\), but upon the reverse return to the normal state the molecule may fall onto the repulsion curve \(Ib\). As a result, part of the excitation energy will be given off in the form of a quantum of radiation, while the remaining part will go into dissociation of the molecule plus the kinetic energy of the decomposition products. If in the upper state the mole-
where it was located, for example, at the zero level \((v' = 0)\), then—according to the quantum-mechanical interpretation of the Franck principle—this means that no precisely determined value can be assigned to the distance between its nuclei (owing to the presence of the “zero-point vibration” \(\frac{1}{2}h\omega\)); rather, it is characterized by the probability curve shown in Fig. 9 by a dotted line. It is obvious that, in order to obtain the width of the continuum emitted in the elementary act, it is sufficient to project the “half-width” of the probability curve onto the repulsion curve \(Ib\), as is shown in Fig. 10. It is clear that the continuum will be the broader, the greater 1) the uncertainty in the initial position of the nuclei and 2) the steeper the repulsion curve. The application of these general considerations to the specific case of the hydrogen spectrum is given in § 14.
Fig. 10. Scheme of the absorption spectrum of iodine.
9. Determination of the dissociation energy. Franck first showed in what way the dissociation energy of a molecule can be determined from absorption spectra. As was already indicated, for a certain mutual arrangement of the potential curves of the ground and excited states, long series of quanta are obtained, which can sometimes be followed all the way to the place of coalescence and to the continuous spectrum adjoining this place. Such favorable conditions are found in the molecule \(\mathrm{J}_2\), where Pringsheim and Mecke isolated and analyzed several series of quanta in the absorption spectrum. In Fig. 10 the shortest-wavelength of these series is represented schematically (to simplify the figure, only the positions of the quanta are given, without rotational structure). The quanta of the bands gradually converge until complete coalescence at \(\lambda = 4995\ \text{Å}\); farther toward shorter waves there is a continuous region. The origin of this place of coalescence is clear from consideration of the left half of Fig. 11. It is assumed that the absorbing molecules are at low temperature, so that all transitions occur from the initial state \(v'' = 0\). It is clear that the place of coalescence \(\nu_k\) indicates the dissociation of the molecule, and the magnitude of the quantum \(h\nu_k\) characterizes the dissociation energy. In order, however, to find from this quantity \(h\nu_k\) the dissociation energy itself in the ground state, it is necessary also to take into account that only in relatively rare cases does the molecule break up into two unexcited atoms. In most cases at least one of the products of decomposition proves to be excited. Consequently, as a rule, the quantity \(h\nu_k\) includes the dissociation energy plus the excitation energy of the decomposition products. These relations are clearly visible from Fig. 12. Here \(D''\) and \(D'\) are the dissociation energies in the ground
… and in the excited state, \(h\nu_k\) is the quantum corresponding to the place of convergence, and \(h\nu_e\) and \(h\nu_a\) are the quanta of electronic excitation of the molecule and of the product of its decomposition. It is clear that the quantum \(h\nu_k\) is greater than the quantity that interests us. We can obtain the value \(h\nu_k\) in the following two ways: 1) the molecule is given the energy \(D'\), and immediately after this one of the dissociation products is given the excitation energy \(h\nu_a\), or 2) the electron shell of the molecule is excited by the quantum \(h\nu_e\), after which the excited molecule is given the dissociation energy \(D'\). In either case we arrive at the same result \(h\nu_k\); whence, expressing \(D'\) and \(D''\) directly in spectroscopic units, \(\mathrm{cm}^{-1}\), we obtain the relation:
Fig. 11. Toward the determination of the work of dissociation.
Fig. 12. Toward the determination of the work of dissociation from absorption spectra.
\[ D' + \nu_e = D'' + \nu_a = \nu_k. \]
In the case considered of the \(J_2\) molecules, the place of convergence corresponds to a quantum of 2.4 V (\(2050\ \mathrm{cm}^{-1}\)). What is the energy state of the dissociation products? From physicochemical measurements the dissociation energy of \(J_2\) is known with rather great accuracy—about 1.5 V (34.5 kg-cal). Consequently, about 0.9 V remains for the excitation energy of the atom \(J\). Such a level is known for the atom \(J\); it corresponds to the metastable state \(2^2P_1\), located above the normal state \(2^2P_2\) by 0.94 V. Thus, if one assumes that the optical dissociation of the \(J_2\) molecule leads to one normal atom \(J\) and one atom excited to the state \(2^2P_1\), then for \(D'\) we obtain:
\[ D' = 2.4 - 0.94 = 1.06\ \mathrm{V} = 34.2\ \text{kg-cal}, \]
which agrees well with the value of the energy given above.
of dissociation of \(J_2\) from physico-chemical measurements (34.5 kg-cal.). Determination of the convergence point for other halides, \(Br_2\) and \(Cl_2\), also made it possible to find accurately the corresponding dissociation works. The results obtained are given in the following table.
TABLE 1
| Molecule | Convergence limit, Å | \(h\nu_k\) | State of the products of decomposition | \(D\) spectroscopic | \(D\) chemical |
|---|---|---|---|---|---|
| \(Cl_2\) | 4785 | 3.58 | \(^{2}P_1 = 0.11\) | \(2.47 \pm 0.02\nu = 57\) kg-cal | 57 kg-cal |
| \(Br_2\) | 5107 | 2.41 | \(^{2}P_1 = 0.45\) | \(1.96 \pm 0.02 = 45.2\) ” | 46 ” |
| \(J_2\) | 4995 | 2.47* | \(^{2}P_1 = 0.94\) | \(1.53 \pm 0.1 = 35.2\) ” | 34.5 ” |
As we see, the agreement is satisfactory everywhere, and since the magnitudes of the excitation energies are known with spectroscopic accuracy, the magnitude of the work of dissociation is obtained with the same accuracy, if, of course, \(\nu_k\) can be directly measured. Unfortunately, direct determination of the convergence point is possible only in comparatively rare cases. Much more often either a small group of discrete bands is observed, or a single continuum. In the first case, to find the convergence point one may use linear extrapolation by the method of Birge and Sponer.* In the second, orienting data are obtained from an estimate of the position of the long-wavelength boundary of the continuum.
Finally, in those cases when one of the products of optical dissociation is obtained in an excited state, one may use the fluorescence method proposed and developed by A. N. Terenin. The method consists in the following. The vapors under investigation are illuminated by a series of frequencies, and that frequency is found beginning with which atomic fluorescence is obtained. Knowing the magnitude of the exciting quantum and the excitation energy of the resulting fluorescing atom, one can directly find \(D\); for example, for NaJ, beginning with \(\lambda = 2460\) Å, upon illumination yellow fluorescence of Na vapor, \(\lambda = 3800/86\), is obtained. Hence \(D = 2.9\) V, or 69 kg-cal, is calculated.
In conclusion it should be indicated that one may also imagine the process reverse to that considered in this paragraph, namely the combination of two atoms into a molecule, emitting the excess energy in the form of a limiting continuum in emission. It is precisely in this way that Kondrat’ev and Leipunsky\(^6\) interpret
* See the article by Mecke, Uspekhi fizicheskikh nauk, IX, 620, 1929; Brown has recently given an accurate value for \(h\nu_k\), namely \(h\nu_k = 2.472 \pm 0.001\) V (Phys. Rev., 38, 709, 1931).
continuous spectra, not emitted upon heating, approximately up to \(1000^\circ\), of vapors of halides (\(\mathrm{Cl_2, Br_2, I_2}\)).
10. Optical dissociation and the type of chemical bond. The simple conception of Abegg of two types of chemical bond—homeopolar and heteropolar—has recently undergone a number of substantial complications. In a number of cases it turned out, for example, that molecules which ought to be typically heteropolar, i.e., built of oppositely charged ions, are by no means such in the field itself. The feature to which decisive significance was originally attached—the presence or absence of an electric moment in molecules—proved to be far from final. The study of optical dissociation played a very important role in clarifying the concept of the type of chemical bond.
Since the electric moment loses its significance as the only criterion of the type of chemical bond, it is expedient, together with Franck, to speak not of homo- and heteropolar molecules, but of ionic and atomic molecules. Franck gives the following definitions to both types of chemical bond. Ionic molecules are those molecules which, upon a gradual increase of oscillations in the ground state, adiabatically dissociate into oppositely charged ions. Atomic molecules are molecules which adiabatically dissociate in the ground state into uncharged atoms.
With respect to both types of bond, the study of optical dissociation leads to the results set forth in the following paragraphs.
11. Ionic molecules. In ionic molecules the bond is due to the electrostatic interactions of the ions forming the molecule. According to Franck’s definition, the characteristic feature of such molecules is dissociation in the ground state into oppositely charged ions. Consequently, the state which in the limit leads to uncharged normal atoms will already be an excited* state. We can represent this excitation as the transfer of an electron from the anion to the cation. It is obvious that the result of such a transition must be a strong weakening of the bond in the excited state, so that the corresponding potential curve will display only a weakly expressed minimum (Fig. 13). But since in the ground state there is a very strong bond (a deep minimum), the curves of the normal and excited states, as a rule, must intersect.
* Cf. the summary devoted to the electrostatic nature of the bond: A. van Arkel and J. H. de Boer, Chemische Bindung als elektrostatische Erscheinung, Leipzig 1931.
Experience shows that, in fact, at the very first stage of excitation ionic molecules usually dissociate into neutral unexcited atoms. What we mean by the dissociation energy of a molecule, from the chemical point of view, is the work that must be expended in order to separate the molecule \(MX'\) into normal uncharged atoms \(M\) and \(X\). Therefore
\[ D_{\text{chem.}} = D(0) + A_y . \]
Thus the quantity \(D(0)y + A_y\), which characterizes the place where the quanta converge in the absorption spectrum, or—as the discrete part in the absorption spectra of ionic molecules is usually absent—the position of the beginning of the continuum, gives us a certain \(D_{\text{chem.}}\).
The most characteristic representatives of ionic molecules are molecules of alkali-halide salts. Their absorption spectra have been studied in detail by Born’s coworkers; the conclusions that follow directly
Fig. 13. Potential curves for ionic molecules.
Fig. 14. Potential curve of alkali-halide salts.
were subsequently refined by Kuhn. These spectra, at sufficiently low vapor density or with a large layer thickness, have a peculiar form: a continuum with a number of maxima, sufficiently crowded toward the red side up to complete merging.
In Fig. 14 typical potential curves of alkali-halide salts are given, constructed on the basis of the study of these absorption spectra. Here the appearance of the upper curves is characteristic. In them the minimum is expressed so weakly that it is practically difficult to distinguish. The left branch of the curve rises very steeply upward, whereas the right branch runs almost parallel to the abscissa axis. This form of the potential curves fully explains the described character of the absorption spectra. At the high temperatures at which these experiments are carried out, in the vapors there are always suffi-
in a definite number of molecules with excited vibrational levels. According to the Franck–Condon principle, transitions from the points \(A_0, A_1, A_2\) of the lower potential curve occur with the greatest probability as indicated by the vertical arrows in the drawing. The lengths of these arrows are proportional to the absorbed frequency.
Since the upper potential curve has only a small inclination to the abscissa axis, a slight uncertainty in the value \(r\) near \(A_0, A_1 \ldots\) leads only to small fluctuations in the length of the arrow. Therefore the absorption maxima are expressed so sharply that they have the form of a sequence of narrow spectral bands. If, moreover, the upper potential curve in its right-hand part runs parallel to the abscissa axis, then, as is seen from the drawing, the spacing of the narrow bands in the absorption spectrum must directly give the vibrational quanta of the ground state.
Indeed, the values of the fundamental vibrational quantum found by Sommermeyer from the potential curves constructed by him agree satisfactorily with the corresponding values calculated by Born and Heisenberg according to the old quantum mechanics and by Van Vleck according to the new (see Table 2).
TABLE 2
| Salt | Fundamental vibrational quantum, according to Sommermeyer | Fundamental vibrational quantum, according to Born and Heisenberg |
|---|---|---|
| CsJ | 140 | 153 |
| RbJ | 170 | 188 |
| KJ | 240 | 252 |
| NaJ | 379 | 334 |
| CsBr | 192 | 189 |
| KBr | 283 | 300 |
| RbCl | 255 | 296 |
Different conditions obtain when a molecule absorbs light at the moment of the greatest convergence of the nuclei (points \(a_1, a_2, \ldots\) in Fig. 14). Since the left part of the upper potential curve rises steeply upward, here even small variations in the value \(r\) will lead to a very considerable change in the length of the arrow. Therefore, instead of narrow bands, we obtain whole regions of absorption, which overlap one another and together create a broad continuum. This extensive continuum is one of the most characteristic features of the absorption spectrum of alkali-halide salts. It is interesting that in this continuum there are also broad maxima, the spacings between which correspond to different stages of excitation of the dissociation products of the molecule; whence it follows that the potential curves of the different excited states run approximately parallel to one another.
As an example, in the following table we give, along with the spacings of the maxima in the CsJ continuum, also the frequency differences for the atomic terms of the dissociation products, in the corresponding interpretation of the dissociation processes associated with the different maxima.
TABLE 3
| No. | Position of maximum (exp.), Å | Interpretation | Δν of maxima, cm⁻¹ (exp.) | Δν of maxima, cm⁻¹ (exp.) | Δν of atomic terms, cm⁻¹ | Δν of atomic terms, cm⁻¹ |
|---|---|---|---|---|---|---|
| 1 | 3240 | Cs (1S) + J (²P_{?}) | 1—2 | 7900 | J (²P_{?}) — (²P_{?}) | 7600 |
| 2 | 2580 | Cs (1S) + J (²P_{1}) | 1—3 | 10900 | Cs (1S — 2P_{?}, ²P_{?}) | 11450 |
| 3 | 2195 | Cs (2P_{1}, 2P_{?}) + J (²P_{?}) | 1—4 | 16200 | Cs (1S — 3D_{?}, ³D_{?}) | 14550 |
| 4 | 2125 | Cs (³D_{?}, ³D_{?}) + J (²P_{?}) | 1—5 | 19400 | Cs (1S — 2S) | 18550 |
| 5 | 1980 | Cs (2S) + J (²P_{?}) | 1—6 | 23200 | Cs (1S — ³P_{?}, ³P_{?}) | 21850 |
| 6 | 1850 | Cs (3P_{?}, 3P_{?}) + J (²P_{?}) |
12. Atomic molecules. The binding forces in atomic or, in the old terminology, homopolar molecules have a quantum-mechanical origin and are “exchange” or resonance forces.* According to Franck’s definition, atomic molecules are characterized by dissociation in the unexcited state into uncharged atoms. Upon absorption of light (photochemical dissociation) an atomic molecule decomposes into atoms, of which, in the first stage of excitation, one is usually excited. The normal relations of the potential curves of the ground and excited states are such as shown in Fig. 15. Here, as a rule,
Fig. 15. Potential curves for atomic molecules.
\[ D(X) < D(Y) + Ay, \]
and the curves do not intersect.
The principles set forth were applied to the study of the nature of the chemical bond in a number of molecules, and it then turned out that some molecules, despite the presence of an electric moment, are (in the gaseous state) not ionic but atomic. Such are the halide salts of thallium and silver. The molecules of hydrohalic acids HCl, HBr, HI also belong here.** There are, however, exceptions to the indicated general rule. Thus, in some cases a normal atomic molecule is a combination of a normal and an excited atom. In these cases, dissociation in the first stage of excitation may lead to two unexcited neutral atoms. Ta—
* Cf. the article on the question of the quantum-mechanical nature of the chemical bond.
* It is very interesting that, in studies of the infrared absorption spectrum of solid HCl, Gettner (Z. Physik 78*, 141, 1932) came to the conclusion that the lattice of solid HCl is atomic, not ionic.
molecules, apparently, are \( \mathrm{N_2}, \mathrm{SiN}, \mathrm{CN}.\) Finally, Terenin* found a curious case in which an atomic molecule, upon photochemical dissociation in the first stage of excitation, breaks up into oppositely charged ions. Such a case is observed in thallium chloride vapors, which exhibit anomalous photoionization.
13. Polarization (van der Waals) molecules. Alongside the two principal types of molecules considered, there also exist “loose,” weakly bound molecules, in which the bond is maintained by mutual polarization. A very instructive example of such molecules is provided by the “non-true molecules” discovered by Kuhn^10 in the vapors of alkali metals. Alkali-metal atoms possess one valence electron, and therefore they can interact with one another in the same ways as hydrogen atoms: when the electron spins are antiparallel, the formation of true molecules with an atomic bond is possible—the corresponding potential curve has a sharply pronounced minimum (curve 1, Fig. 2); if, however, the spins are parallel, the atoms experience repulsion at all distances, the potential curve rises continuously upward, and stable molecules cannot arise. All this reasoning, however, would be perfectly strict if the atoms were unchangeable systems. In reality they are characterized by a certain capacity for displacement of charges, for polarization. Therefore, when such atoms with identically directed spins (the triplet state) are brought together, alongside the repulsion due to quantum-mechanical exchange forces, there also arises an attraction, since the molecules deformed by polarization act upon one another as dipoles. And since the exchange forces decrease with distance considerably more rapidly than the polarization forces,*** at large distances the latter prevail, and a weakly expressed minimum appears on the potential curve, making the formation of molecules possible. In the case of the interaction of two hydrogen atoms this minimum is so insignificant that it plays practically no role: according to London and Eisenchitz,^11 the polarization forces here amount to only \( \frac{1}{400} \) of the exchange forces. On the contrary, in the case
* Birge expressed doubt as to the correctness of the extrapolation of the quanta of the excited state of CN, carried out by Herzberg (cf. Sponer, Leipz. Vorles.).
* Sov. Phys., 2*, 290, 1932.
*** According to London and Eisenchitz^11 the polarization forces decrease with increasing distance \(r\) as \( \frac{1}{r^7} \), whereas the exchange forces decrease in proportion to \( e^{-ar} \).
of the alkali metals, the exchange forces that lead to the formation of true molecules are considerably smaller than for hydrogen, whereas the electrical polarizability is many times greater. As a result, the minima on the potential curves of the true bond (antiparallel spins, singlet state—\(D^{(1)}\) in Fig. 16) and of the non-true, polarization bond (parallel spins, triplet state \(D^{(3)}\) in Fig. 16) prove to be of the same order of magnitude. This picture is especially sharply expressed for cesium, where the work of dissociation of the “true” molecules \(D^{(1)}\) is only \(0.25\,V\), while the work of dissociation of the polarization molecules \(D^{(3)}\), apparently, exceeds \(0.1\,V^{8,9}\).
Fig. 16. Potential curves of polarization molecules.
Thus, by increasing the elasticity of the vapor of an alkali metal, we may expect the following sequence of states: at low pressures the atoms exist in the free state—the absorption spectrum consists of the principal series of the atomic spectrum; at higher pressures true (singlet) molecules appear, giving the typical band absorption spectrum. At still higher pressure, a sufficient number of non-true, polarization molecules appear—at the same time diffuse bands are found in the absorption spectrum, characterized by the fact that, owing to the small magnitude of the work of dissociation \(D^{(3)}\), they closely adjoin the atomic lines of the principal series and have a very small (for bands) width. Finally, at still higher pressures, entire diffuse regions appear in the spectrum, the origin of which will become clear from what follows.
Somewhat different conditions for the occurrence of polarization molecules are presented in the case of mercury vapor. As is known, these vapors are monatomic. In the normal state their atoms interact along a repulsion curve, i.e., in collisions they undergo elastic reflection. This repulsion curve in the normal state is deformed only to a slight extent by polarization forces, as shown in Fig. 17 (curve \(I\)). If, however, the unstable system of two atoms undergoes absorption of light at the moment of collision, then the atom that has absorbed the light quantum, owing to the violation of symmetry in the distribution of electric charges, acquires a strong dipole moment, thanks to which the loose molecule is considerably strengthened, and the corresponding potential curve acquires a sharply pronounced minimum (curve \(II\) in Fig. 17).
We have here, therefore, an interesting case in which the bond in the excited state proves to be considerably greater than in the normal state—the molecule, upon excitation, does not loosen, but, on the contrary, becomes strengthened.
Spectroscopically, the formation of such molecules as Hg\(_2\), Zn\(_2\), Cd\(_2\) manifests itself in the appearance of diffuse regions in the absorption spectrum. Thus, for mercury there is a known band immediately adjacent to the resonance line 2537 Å (the sharp edge of the band is located at 2540 Å). This band, however, is not structureless: in it there are observed fluctuations\(^{12}\) of the same character as in the case of vapors of alkali-halide salts (see p. 342). Comparing Fig. 15 with Fig. 18, it is clearly seen that the explanation
![Figure 17 and Figure 18]
Fig. 17. Origin of the continuous spectrum of mercury vapor.
Fig. 18. Origin of the continuous spectrum of molecular hydrogen.
of these fluctuations in both cases must be the same; only for Hg\(_2\), Cd\(_2\), Zn\(_2\), in contrast to the alkali-halide salts, the lower curve runs almost parallel to the abscissa axis, and, consequently, in this case the distance between the fluctuations characterizes the vibrational quanta of the excited state, which here is more stable.
From observation of the dependence of the intensity of the band on temperature, it was possible to calculate the heat of dissociation of the molecule Hg\(_2\). For the normal state, as was to be expected, very small values are obtained, \(D_I = 1.4\)—\(1.6\) kcal,\(^{13,14}\). We have, consequently, in the normal state very weakly bound molecules; but the excited state proves to be considerably more stable: for \(D_{II}\) values are obtained of \(0.84\) V, i.e. about 19 kcal.\(^{13,14}\).
The continuous absorption regions mentioned above in the absorption spectra of alkali-metal vapors at high pressures are explained in the same way as the Hg bands. At high pressures, in addition to singlet and triplet molecules, the absorption centers must be “quasimolecules,” i.e. the aggregate of two atoms at the moment of collision. They produce extensive regions of absorption along with the narrow bands belonging to triplet molecules.
14. Unstable states. The continuous spectrum of hydrogen. Let us now consider the case in which the final state in the emission of a spectrum is unstable, i.e. such that it corresponds to a potential curve having no minimum. In this case, obviously, the whole spectrum will be continuous, since the transition always ends with dissociation of the molecule, the parts of which fly apart with kinetic energy. An excellent example of such transitions is provided by the well-known continuous spectrum of molecular hydrogen, which has an enormous extent (it begins in the visible region at \(\lambda = 5000\,\text{\AA}\), reaches a maximum at \(\lambda = 3000\,\text{\AA}\), and extends far into the ultraviolet, ending approximately at \(\lambda = 1500\,\text{\AA}\)). The Franck–Condon principle can be applied also to such transitions; here, however, it is expedient to use an extended quantum-mechanical interpretation of this principle.
Let us consider, from this point of view, the origin of the continuous spectrum of hydrogen. The upper state corresponding to this spectrum, according to Weizel and Stueckelberg, is the stable state* \(1s\sigma\,2s\sigma\,{}^{3}\Sigma_g\); the lower state—\(1s\sigma\,2p\sigma\,{}^{3}\Sigma_u\)—is characterized by a repulsion curve and is unstable. In Fig. 18 the potential curves of both these states are represented. According to the Franck–Condon principle, the most probable transitions will be \(F'F''\), \(D'D''\), \(C'C''\), ... Since the curve \({}^{3}\Sigma_u\) falls very steeply and since, on the other hand, the position of the electron near \(E'\), \(D'\), \(C'\) is not precisely fixed, but there is a certain probability of finding the electron within an entire region (see Figs. 7 and 9), as a result of the transitions we obtain, instead of lines, continuous intervals of frequencies which, overlapping one another, give an extensive continuum. Considering the potential curves, we can derive a number of consequences from the outlined picture:
- The continuum must have a boundary on the long-wavelength side. Indeed, the transition \(X'Y'\) corresponds to the lowest possible frequency, since upon excitation above the level \(X'Y'\) the molecule dissociates, i.e. ceases to exist as a whole
* For the notation used here, see Rabinovich’s article, Uspekhi fizich. nauk, 13, 1933.
also in the upper state. This limiting wavelength must correspond to the highest excitation energy (12.6 V).
- As the excitation energy decreases, the region of emitted frequencies must shift toward shorter wavelengths.
Considering the course of the potential curves relative to one another, we see that the gradual decrease of the emitted wavelength (the increase of the segments \(E'E''\), etc.) corresponds to the sequence of transitions \(E'E''\ldots, A'A''\), \(\beta'\beta''\ldots\). For these transitions to take place, the excitation energy must at first decrease so as to reach a minimum at \(A'\), and then begin to increase again.
All these conclusions are fully confirmed by experiment,\(^{16}\) so that the interpretation of the continuous spectrum of hydrogen may be considered complete.
- “Predissociation” spectra. We now turn to the consideration of a special type of diffuse spectra—the so-called “predissociation” spectra. In 1923 V. Henri\(^{17}\) showed that in a whole series of cases, when bands are followed toward short wavelengths, it is found that, beginning with a certain wavelength, sometimes suddenly and in other cases gradually, the rotational structure of the bands disappears. The bands, as such, continue to exist, but acquire a completely diffuse character. In some cases, on following farther into the ultraviolet region, the individual bands also disappear, merging into extensive regions of the continuous spectrum. Sometimes, on the contrary, it is observed that the rotational structure which had disappeared reappears in the region of shorter wavelengths.
Henri also showed that when vapors of a substance are illuminated with wavelengths at which blurring of the band structure is observed, the molecules acquire a special chemical activity; at the same time, if fluorescence was observed under the action of longer wavelengths, then on passing to the regions under consideration the fluorescence disappears completely or is very strongly quenched. These phenomena were found both in complex molecules (benzene, pyridine, naphthalene, etc.) and in simple ones (\(S_2\), \(P_2\), \(SO_2\), \(NH_3\), etc.).
All the facts described gave Henri grounds to suggest that the phenomenon he had found indicates that, under the influence of certain wavelengths, a molecule passes into a special loosened state preceding dissociation, in view of which he called this phenomenon “predissociation.” This name has been retained for the corresponding spectra to the present time, although Henri’s explanation proved to be incorrect.
- Several examples. We shall briefly describe several cases of predissociation.\(^{18}\)
Acetaldehyde $\mathrm{CH_3COH}$. The absorption spectrum of acetaldehyde consists of a large number of sharp bands with a distinct rotational structure between $\lambda = 3484\,\text{\AA}$ and $3050\,\text{\AA}$. At $3050\,\text{\AA}$ the bands become blurred rather quickly, and about 60 diffuse bands can still be measured down to $\lambda = 2823\,\text{\AA}$. In addition, approximately at $3080\,\text{\AA}$ there is superposed a broad region of continuous absorption with a maximum at 2850, which gradually decreases toward shorter wavelengths.
The investigation of the photochemical properties of acetaldehyde showed that at $3050\,\text{\AA}$ no decomposition occurs, whereas, beginning at 3050, acetaldehyde decomposes quantitatively into carbon oxide and methane:
\[ \mathrm{CH_3COH \to CO + CH_4}. \]
Thus, in acetaldehyde the onset of the photochemical reaction coincides with the disappearance of the fine structure of the bands, i.e., with the appearance of a predissociation spectrum.
Ammonia. The absorption spectrum of ammonia consists of a series of bands located between 2860 and $1935\,\text{\AA}$. Beginning with the band $2167\,\text{\AA}$, the fine structure disappears, and the bands become increasingly blurred as one proceeds farther into the ultraviolet region.
Photochemical investigation gave the following results: at room temperature the Cd lines (2329, 2321, 2313, 2307, 2288, 2265, 2195) have no effect; the Zn lines (2025, 2062, 2100, and 2139) give a photochemical effect. Thus the photochemical reaction begins at the same wavelengths ($\lambda < 2140$) as does predissociation.
Sulfur. The banded absorption spectrum of sulfur vapor is very complex and has not yet received an exhaustive interpretation. Henri distinguishes four regions in it: 1) From the visible to the near ultraviolet region, $\lambda = 8000—3175\,\text{\AA}$. Henri attributes this absorption region to $\mathrm{S_4}$ molecules. 2) A continuous spectrum in the far ultraviolet region. Henri attributes it to $\mathrm{S_3}$ and $\mathrm{S_8}$ molecules. 3) The region from 4100 to the extreme ultraviolet (measured by Henri down to 2300). In this region, at $2798.2\,\text{\AA}$, a sudden change of the spectrum is observed. Up to this wavelength the bands are sharp and reveal a rotational structure; after it the bands immediately become blurred. In the region from 2798.2 to $2614\,\text{\AA}$ a considerable number of narrow but completely continuous bands without any fine structure is observed. At $2614\,\text{\AA}$ a new change of the spectrum occurs: the individual bands disappear and are replaced by a continuous region that extends to $2501\,\text{\AA}$, after which individual bands again appear. All these changes
...tion can be traced from the microphotograms presented in Fig. 19. At 2580 Å a new absorption region begins, superimposed on the preceding one and consisting of a series of narrow bands. This region has not yet been interpreted.
Thus, despite the mutual overlap of different regions of the spectrum, which makes interpretation difficult, the absorption spectrum of sulfur vapor quite clearly reveals two predissociation points at 2798.2 Å and at 2614 Å.
As for chemical activity, the observations of Taylor and Avery showed that, upon illumination with \(\lambda\lambda > 2800\), a mixture of sulfur vapor with hydrogen does not produce \(H_2S\), whereas upon illumination of the mixture with rays of the first predissociation region, the formation of \(H_2S\) is observed at once. Thus here too predissociation is associated with increased chemical activity.
Fig. 19. Microphotograms of the absorption spectrum of sulfur.
17. Explanation of predissociation. a) Shortening of the lifetime of the molecule. A complete explanation of the phenomena of predissociation has been given by a number of investigators—mainly Born and Franck, Herzberg, Bonhoeffer and Farkas, and de Kronig. The essence of this explanation, briefly stated, comes down to the fact that predissociation is caused by a very strong shortening of the lifetime of the excited molecule.
It is well known from classical optics that an ideally monochromatic line could be produced only by an infinitely long wave train. A real atom or molecule radiates for a limited interval of time and therefore gives a wave train of limited length, which can always be represented, with the aid of the Fourier integral, as a superposition of waves of a certain finite interval of frequencies. This is the reason for the “natural” width of lines (in actual fact, the natural width is increased further, by about a factor of 100, owing to the Doppler effect of the emitting molecules). Further, it can be rigorously shown that the shorter the wave train, the larger the frequency interval over which the Fourier integral must be spread in order to represent this wave train, and, consequently, the shorter the emission time of a spectral line, the less “monochromatic” that line is.
If now the lifetime of an excited molecule is shortened, for example as a result of spontaneous dissociation, this must lead to a broadening of the corresponding lines of the fine...
structure, which may, in the end, be so considerable that neighboring lines merge with one another, the fine structure disappears, and the band becomes continuous.
All the preceding reasoning was constructed on a classical analogy. Kronig[^19] showed by exact calculations based on quantum mechanics that, under certain conditions, which will be discussed below, the merging of fine-structure lines must occur when the lifetime of the molecule becomes shorter than its period of rotation. In this way we obtain an entirely clear picture: the quantization of rotation during predissociation becomes impossible precisely because the molecule breaks up before it has time to complete even a single revolution. The quantization of vibrational motions, however, remains possible, since even during the shortened lifetime of the excited molecule it has time to execute, for example, 100 vibrations.
Fig. 20. On the origin of predissociation.
In conclusion to this paragraph we shall further show that the blurring of energy levels owing to the shortening of the lifetime of the excited molecule follows directly from the basic principles of quantum theory. According to Heisenberg’s “uncertainty principle,” the time interval \(\Delta t\) and the accuracy of the determination of energy \(\Delta E\) are connected by the relation:
\[ \Delta E \cdot \Delta t \geq h. \]
It follows directly from this that a decrease in \(\Delta t\) leads to an increase in \(\Delta E\), i.e. it produces a blurring of the energy level.
6) Origin of dissociation. Up to now we have concentrated our attention on the shortening of the lifetime of the excited molecule and have left aside the physical cause of this shortening. We shall now turn to the consideration of the latter.
Let us construct systems of vibrational levels for different states of excitation of the electron shell. If, in a first approximation, we assume that there is no coupling between the motions of the electrons and the vibrations of the nuclei, then we obtain independent systems of terms \(A, B, C,\ldots\) (Fig. 20). In the second approximation, one must take into account the coupling between electronic and nuclear vibrations; this coupling causes the occurrence of resonance transitions between some discrete level of system \(B\) and the corresponding continuum of system \(A\). But the result of a transition from level \(B\) into the continuum \(A\) is the dissociation of the molecule. Wenzel under-
assumed that the number \(Z\) of such transitions per unit time is equal to
\[ Z=\frac{4\pi^{2}}{h}N\cdot v_{hk}v_{kd}=\frac{4\pi^{2}}{h}N\cdot [v_{dk}]^{2}, \]
where \(N\) denotes the number of molecules in the state corresponding to the discrete system of terms; the indices \(d\) and \(k\) refer respectively to the discrete and continuous systems; \(v_{dk}\) and \(v_{kd}\) are the probabilities of transition from one system to the other. The latter probability can be calculated by wave mechanics if the eigenfunctions \(\psi_d\) and \(\psi_k\) of both states are known and, in addition, the “perturbation function” \(W\), characterizing the interaction of the motions in the molecule, namely:
\[ v_{dk}\cong \int \psi_d W\psi_k\, d\tau, \]
where \(d\tau\) is the volume element in phase space.
For an experimental verification of the correctness of the above explanation of predissociation, the following criteria may be used:
a. Fluorescence quenching. A molecule excited to a certain discrete level has a twofold possibility of returning to the normal state; namely, it may: a) return to it directly, giving up the excess energy in the form of radiation; b) pass into a state where its energy level lies in the continuum, and use its excess energy for dissociation. In the first case we shall observe molecular fluorescence; in the second, the transition takes place without radiation.
Thus, if the transition from the discrete system of terms to the continuous one is allowed by the corresponding selection rules (see § 18), then the onset of predissociation must be reflected not only in the broadening of absorption lines, but also in a decrease in the intensity of emission, i.e. in the quenching of the fluorescence of the predissociating molecule. Such quenching of fluorescence is in fact observed and is one of the most characteristic signs of predissociation. Moreover, it is not difficult to see that this sign is more sensitive than the disappearance of fine structure in absorption. \({}^{20}\) Indeed, in order for the broadening of absorption lines to be detectable, the latter must in any case exceed the width of the spectral lines; but since the natural width of the lines is increased 100-fold owing to Doppler broadening, the effect of predissociation must exceed this broadening in order to be detected. Meanwhile, fluorescence quenching can be detected at once and, consequently, is accordingly a more sensitive sign.
6. The existence and spontaneous character of photochemical dissociation. It is necessary to make sure that the observed dissociation does not depend on possible collisions, which would also be proof of its spontaneous character.
Bonhoeffer and Farkas[^21] were the first to apply these criteria to the predissociation of ammonia. They showed that, when ammonia is illuminated by rays from the predissociation region: 1) it is in no way possible to excite fluorescence of $\mathrm{NH_3}$, and 2) dissociation of $\mathrm{NH_3}$ takes place, which is completely independent of pressure and is observed down to the very lowest pressures. Thus the explanation of predissociation given above, in the example of ammonia, received entirely convincing experimental proof.
Analogous observations were later made on sulfur vapor[^22],[^23],[^24] and $\mathrm{NO_2}$[^25].
18. Conditions for the possibility of predissociation. The only condition for predissociation that we have discussed so far consisted in the coincidence of a discrete level of one system with the continuous region of another system of levels. In reality such a coincidence occurs very often in molecules, so that, if this condition were not only necessary but also sufficient, predissociation would be observed much more frequently than it actually is. If this is not so, it is primarily because—as has already been indicated earlier—for predissociation to occur certain special conditions must be satisfied.
These conditions reduce both to restrictions on the possibility of electronic transitions and to certain special restrictions relating to the state of vibration of the nuclei. The restrictions on the possibility of electronic transitions are formulated in the form of certain selection rules found by Kronig[^19]; the conditions imposed on the vibrational state of the nuclei are given by the Franck–Condon principle, modified for the present case. Below we give Kronig’s rules without special commentary. The Franck–Condon principle, modified for the case of predissociation, is set forth in the following paragraph. Kronig’s rules state:
1) Both levels of equal energy must have the same total angular momentum $J$ (i.e., in the transition the condition $\Delta J = 0$ must be satisfied).
2) They must belong to such electronic states in which the angular momenta of the electrons relative to the line joining the nuclei (the quantities $\Lambda$) differ by no more than unity ($\Delta \Lambda = 0$ or $\pm 1$).
3) They must belong to electronic states of the same multiplicity ($\Delta S = 0$).
4) They must possess the same symmetry with respect to reflection at the origin of coordinates.
5) In molecules consisting of two identical nuclei, they must also possess the same symmetry with respect to the nuclei.
19. The Franck–Condon Principle in Predissociation
The quantum-mechanical interpretation of the processes of spontaneous dissociation is analogous to the interpretation given in § 7 of the probabilities of radiative transitions. The probability of a spontaneous transition without radiation, according to the Wentzel formula given on p. 353, depends on the integral of the product of the eigenfunctions of the states between which the transition takes place and the factor \(W\)—the “perturbation function,” characterizing the mutual coupling of motions in the molecule:
\[ v_{dk} \sim \int \psi_d^{*} W \psi_k\, d\dot{r}. \]
This integral has an appreciable value only in those cases when \(\psi_d\) and \(\psi_k\), for identical values of \(r\), have maxima. This can most simply be verified by considering Fig. 21, where, together with the potential curves of the discrete and dissociated states, the curves of the course of the eigenfunctions of the one and the other state for a certain energy level are plotted; the solid curves refer to the discrete state, the dotted ones to the continuum. We see that the probability of spontaneous dissociation will be greatest in the case shown in Fig. 22, i.e., when both curves intersect at the height of the energy level under consideration or somewhat below it. In other words, transitions without radiation will most often take place when they can occur without a change in the distance and kinetic energy of the nuclei of the molecule. But this is precisely the Franck–Condon principle for the case of interest to us.
Fig. 21. The Franck principle in predissociation.
20. Consideration of Various Cases of Predissociation by Means of the Franck–Condon Principle
The Franck–Condon principle makes it possible to understand various cases of predissociation. The application of this principle to the interpretation of the phenomenon of predissociation was made by Franck and Sponer \(^{26}\) and subsequently, especially by Herzberg \(^{27, 28}\).
In Fig. 22 are shown various cases in which predissociation is possible. In all these cases \(n\) is the potential curve of the normal state, while \(a\) and \(a'\) are the curves of various excited states. According to the Franck principle, transitions from the normal state with \(v''=0\) lead to the region \(A—B\) of the curve \(a\) and to the region \(F—H\) of the curve \(a'\). Since \(F\) already lies above the asymptote of the curve \(a'\), the latter transitions give a continuous absorption region situated somewhere far in the ultraviolet part. As for transitions to the curve \(a\), they correspond to bands of a completely normal appearance,
Fig. 22. Various cases of predissociation.
until the upper vibrational level lies below \(D\). Beginning with this level and above it, the appearance of the spectrum changes sharply. Indeed, let, for example, upon excitation the molecule fall onto the level \(E\). The vibrations of the nuclei of the molecule in this state can be vividly represented by the motion of a heavy ball placed in the saddle of the potential curve \(a\). A ball raised to the level \(E\) and left to itself will move with ever increasing speed and, having passed the lowest point of the potential curve with maximum kinetic energy, will again rise to the point \(E'\), lying at the level \(E\). On the return motion, at the point \(C\), the ball is offered two possibilities: either to roll down along the previous curve, or to pass onto the curve \(a'\) without change of its position and kinetic energy, i.e. in accordance with the Franck–Condon principle. If the latter possibility is realized, then the ball, rolling along \(a'\), will rise above the level of the asymptote of the curve \(a'\) and therefore, on the return motion, will jump beyond the limits of \(a'\)—in other words: the molecule undergoes dissociation, and its constituent parts fly apart with a certain kinetic energy. Since during the mean lifetime of a molecule in the predissociation state (\(10^{-8}\) sec) it nevertheless manages to execute approximately 100 vibrations, it is clear that the molecule has at its disposal a sufficient number of occasions when the described transition from one curve to the other can take place. *
Let us now pay attention to the following essential circumstance. If, upon excitation, the molecule falls onto the level \(D\), then, in predissociation, the products of the decomposition of the molecule possess zero kinetic energy. The higher the upper level lies above \(D\), the greater is the kinetic energy with which the parts of the molecule fly apart. But at the same time, the higher the level is situated above \(D\), the greater the speed possessed by the vibrating nuclei when passing the point of intersection of the potential curves, and therefore the greater the chance that the molecule will “slip through” this dangerous point and remain on the former curve. Hence follows a natural explanation of the already mentioned phenomenon, consisting in the fact that at some distance from the boundary of predissociation the bands again acquire a fine structure. It should also be noted that in the cases under consideration the boundary of predissociation must be perfectly sharp, since below \(D\) predissociation is impossible for energetic reasons.
The case (Fig. 22, \(c\)) differs substantially from those considered. Here the curve \(a'\) is a curve of pure repulsion, not
* More precisely, this very mean lifetime, or, still better, its reciprocal, is a measure of the probability of transition from one curve to another.
possessing minima. Therefore transition onto this curve always entails the breakup of the molecule: a ball placed at any point of the curve rolls down and acquires kinetic energy. The Franck—Condon principle requires, however, that the transition from curve \(a\) to curve \(a'\) take place only at point \(C\) and slightly above it. At these points the breakup is already accompanied by the scattering of the products with considerable kinetic energy, since point \(C\) is always situated above the asymptote of curve \(a'\).
The quantum-mechanical treatment of this case leads to the conclusion that a certain probability of dissociation also exists below point \(C\). In fact, it is clear that a molecule excited to some level \(D\) has a certain probability of “leaking through” under the potential barrier \(C\), i.e. of passing to curve \(a'\) and, consequently, of undergoing dissociation.
Further, from the same quantum-mechanical considerations it follows that the probability of this “leakage” will be the greater, the closer the energy level \(D\) lies to the summit \(C\). It follows from this that the blurring of the fine structure of the bands here must begin earlier than the predissociation limit \(C\), and must gradually increase, so that the fine structure does not disappear at once \(^{29}\). Such a gradual blurring is indeed observed in the case of the first place of predissociation of \(\mathrm{NO}_2\). There is reason to assume an analogous character of the intersection of the potential curves also in the case of \(\mathrm{S}_2\) \(^{30}\).*
21. Dissociation by rotation.
In certain band spectra of emission (hydrides \(\mathrm{AlH}\), \(\mathrm{CaH}\), \(\mathrm{HgH}\)) a peculiar phenomenon is observed: on tracing the fine structure of such a band, it turns out that, beginning with some value of the rotational quantum number, the band is unexpectedly cut off. Sometimes this cutoff is preceded by a blurring of the lines of the rotational structure. This phenomenon was at first attributed exclusively to dissociation due to rotation. An increase of the rotational quantum number means an increase of the speed of rotation, and one might think that at some limiting speed of rotation the molecule must be torn apart under the action of the centrifugal force. A more detailed analysis has shown, however, that in most cases this cutting off of bands
* Such transitions contradict the Franck—Condon principle in its classical formulation. However, an extended quantum-mechanical interpretation of this principle permits a finite probability also for such transitions. In fact, the fundamental functions \(\psi_\nu'\) and \(\psi_\nu''\) exhibit, although very small, maxima and minima also beyond the classical turning points of the oscillating nuclei. Therefore the integral \(\int \psi_\nu'\psi_\nu''\,dr\) can have a finite value also for transitions excluded by the strict Franck—Condon principle.
** Quite recently doubts have been expressed as to the correctness of this interpretation; see Herzberg, Ann. d. phys.
has the same mechanism as the phenomenon of predissociation considered (cf. quenching of fluorescence in the predissociation region). However, there are cases where we are indeed dealing with dissociation as a result of rotation; such is certainly the case for HgH. Thus, since the breaking off of a band as a result of predissociation adds nothing new in comparison with what has been set forth, we shall confine ourselves to merely mentioning its possibility. We shall dwell somewhat on dissociation under the action of rotation.
Let us consider what influence rotation has on the form of the potential curve. The equilibrium condition for a rotating molecule will be:
\[ \text{centrifugal force}=\text{force of attraction} \]
or
\[ \frac{\mu\omega^2}{r^3}=\frac{p^2}{\mu r^3}=U'(r), \]
where \(p\) is the angular momentum and \(\mu\) is the reduced mass of the molecule. If, in addition to rotation, the molecule also undergoes vibrations, then the equilibrium condition is correspondingly changed:
\[ \frac{p^2}{\mu r^3}=U'(r)+\text{restoring force}. \]
It is easy to see, however, that what has been written can be represented in the following form:
\[ \text{restoring force}=\frac{p^2}{\mu r^3}-U'(r). \]
Fig. 23. Potential curves of a rotating molecule.
It follows from this that the expression
\[ \frac{p^2}{2\mu r^2}-U(r) \]
for a rotating molecule plays the role of a potential; the minimum of this function corresponds to stable equilibrium, and the derivative with respect to \(r\) gives the force. If one constructs a graphical representation of this function for various \(p\), one obtains the family of curves shown in Fig. 23. All these curves in the region of large \(r\) tend to one common asymptote. But in passing to small values of \(r\), in contrast to the potential curves of a non-rotating molecule, we first encounter a maximum and then a minimum corresponding to a stable state. As \(p\) increases, the form of the curves changes, as can be traced in Fig. 24. Finally, for a certain value of the rotational quantum number, the maximum and the minimum merge into a single inflection point \(L\). The next curve will have neither a maximum nor a minimum, i.e. it will be a curve of repulsion. A molecule, pere-
falling on this curve must inevitably break up: it dissociates as a result solely of an increase in the speed of rotation.
It is easy to see that, unlike the potential curves of a non-rotating molecule, here the ordinate of each point of the curve is equal to the sum of the potential energy \(U(r)\) and the kinetic energy \(T(r)\). In dissociation this store of energy \(U(r)+T(r)\) is expended on the actual rupture of the molecule (the energy \(D\)), while the remainder is converted into the kinetic energy of the products of decomposition. By the law of conservation of energy:
\[ U(r)+T(r)=D+K, \]
and since \(U(r)+T(r)\), generally speaking, is considerably greater than \(D\), \(K\) in this mode of dissociation always has an appreciable magnitude. Therefore it is difficult to draw conclusions about the dissociation energy from the directly accessible determination of the quantity \(U(r)+T(r)\).
Calculations made by Oldenberg¹ for the case of HgH showed that the breaking off of bands in this case is quite satisfactorily explained by the considerations set forth. Experimentally it was found that the band breaks off at \(J=31\). According to Oldenberg, a curve with a point of inflection is obtained only for somewhat larger values of \(J\). The agreement may be regarded as satisfactory, since the latter curves already possess very sharply expressed maxima and minima, and therefore for high vibrational levels dissociation may occur below the limiting curve.
It is very interesting that, before the breaking off of a band, a broadening of the lines is observed. This broadening is a quantum-mechanical effect analogous to the gradual onset of predissociation. Let us choose some vibrational level and trace its position on curves corresponding to successively increasing values of the rotational velocity. We shall see that, with increasing \(I\), our level \(J\) will also rise and, finally, will rise above the common asymptote of all the curves. Then dissociation becomes energetically possible; however, according to classical theory it would still be impossible, since the level \(v\) lies below the potential barrier. According to quantum mechanics, however, there is a definite probability of passage beneath the potential barrier (“tunnel transition”), i.e., a definite probability of dissociation of the molecule. As a consequence of this, as soon as the energy level rises above the asymptote, processes of molecular dissociation become possible—the lifetime is shortened, and the levels become blurred.
22. Magnetic quenching of fluorescence. In all the cases considered, the transition from a stable state to an un-
stable, occurred spontaneously. Quite recently it was discovered that sometimes such a transition may also be forced. This means that, if a molecule has two states of equal energy, of which one is stable and the other unstable, and if spontaneous transitions between these states are forbidden by some selection rule, then this rule may be violated if the molecule is placed under certain special conditions—for example, placed in a magnetic field. Let us consider these phenomena of forced predissociation, which are extremely important for understanding chemical phenomena.
Steubing \(^{33-34}\) discovered long ago that the fluorescence of iodine vapor, excited by the green mercury line, can be quenched by a sufficiently intense magnetic field. This effect remained completely mysterious until Turner \(^{35}\) pointed out that this magnetic quenching can be interpreted as predissociation, which under ordinary conditions does not occur, since the transition from a stable potential curve to an unstable one is forbidden by one of Kronig’s rules (see p. 354)—for example, by the rule requiring constancy of the angular momentum \(\Delta J = 0\), whereas the appearance of a magnetic field may lead to the removal of this prohibition.
Fig. 24. Potential curves of the iodine molecule.
Turner studied this phenomenon in detail and showed that it depends very strongly on the frequency of the exciting light. Thus, at \(\nu = 17\,300\ \mathrm{cm}^{-1}\) the effect is not observed at all, whereas already at \(18\,500\ \mathrm{cm}^{-1}\) it reaches its maximum value and then decreases rather slowly with further increase of the exciting frequency. If, from these data, one attempts to trace the conditions for the occurrence of magnetic quenching of fluorescence, it turns out that it sets in unexpectedly when the molecule reaches a definite vibrational level of the excited state (\(v' = 17\) or \(18\)), and then decreases upon excitation to higher vibrational levels. But precisely such a character is possessed, in predissociation, by the probability of transition between different states, as was indicated in § 21.
In order to establish the possibility of such forced predissociation, it is necessary, first of all, to examine the energetic states of the molecule. Such an analysis for the molecule $\mathrm{J}_2$ was carried out by van Vleck1, who based himself on a series of earlier works, chiefly those of Mulliken. In Fig. 24 are given the potential curves of the various possible energy states of the molecule $\mathrm{J}_2$. It is seen that the curves denoted by $0^+$ and $0^-$, of which the first corresponds to the stable state ${}^3\Pi_0^+$ and the second to the unstable ${}^3\Pi_0^-$, beginning from a certain value of $r$, approach one another very closely. Under ordinary conditions, however, the transition from the stable curve to the unstable one is forbidden by the selection rules.* The appearance of a magnetic field creates such perturbations under which a dynamical interaction between these states becomes possible and, consequently, a transition from one to the other.
23. Induced predissociation. Still more interesting from the chemical point of view are those cases in which predissociation is induced by interactions between molecules. Turner2 found that, in the presence of argon atoms, $\mathrm{J}_2$ molecules spontaneously dissociate under illumination both by frequencies from the region of continuous absorption and by frequencies from the region of the discrete absorption spectrum ($\lambda > 5100\ \text{\AA}$). Absorbing light in the region close to the place where the quanta merge ($\lambda = 5100\ \text{\AA}$), the $\mathrm{J}_2$ molecule passes into an excited state, possessing an excess of energy far exceeding the work of dissociation of this molecule. However, if $\lambda > 5100\ \text{\AA}$, then under ordinary conditions this energy is preserved in the form of the energy of electronic excitation and, after a time $\tau$, is emitted again in the form of fluorescence light. Turner's experiments show, however, that in excited $\mathrm{J}_2$ molecules in the presence of argon there occurs such a redistribution of energy over the degrees of freedom as ends in dissociation. Since, however, the excitation energy of argon atoms lies far above the excitation level of the $\mathrm{J}_2$ molecule when $\lambda_2 = 5100$, it is difficult to suppose that such a redistribution could be the simple result of collisions of the second kind. It is much more probable that the argon atoms, by their presence, perhaps through the influence of their electric field, in some way assist the occurrence of spontaneous dissociation, i.e. the transition from the stable curve (cf. Fig. 24) to the unstable one. Indeed, in investigating the fluorescence spectrum of $\mathrm{J}_2$ vapor in the presence of Ar, Turner showed that under these conditions, beginning from a known place, a break in the bands occurs, i.e. the molecule excited to
* For the proof and a detailed analysis of the conditions see the cited work of van Vleck. It is also shown there that perturbations produced by the rotation of the molecule likewise cannot lead to a violation of the selection rules.
of a certain level, at this and higher levels undergoes spontaneous dissociation. Such a picture, as we have repeatedly seen, is characteristic precisely of the phenomenon of predissociation. Thus, under the influence of argon atoms, a violation of the selection rules takes place and, consequently, forced or induced predissociation occurs.
An analogous phenomenon was discovered by Loomis and Fuller ^38, and also by Kondrat’ev and Polak ^39. Namely, it turned out that the addition of oxygen to iodine or bromine vapors very strongly increases the absorption of these vapors—an effect so considerable that, under certain conditions, the change in the color of iodine vapor can be noticed simply by eye. It is characteristic here that the intensification of absorption is observed only beginning with a certain definite band. Thus, for iodine it begins at \(v > 12\) (\(v'\) is the vibrational quantum number in the excited state), and the curve of the intensity of the effect is entirely analogous to the corresponding Terper curve for the magnetic quenching of fluorescence. Hence it is natural to suppose that the increase in absorption is the result of a strong broadening of absorption lines owing to predissociation induced by the presence of \(O_2\) molecules. Kondrat’ev and Polak (loc. cit.) also showed that, in the presence of \(O_2\), the already existing predissociation of \(NO_2\) undergoes considerable enhancement.
24. Chemical applications. The phenomena of predissociation considered in the preceding paragraphs undoubtedly play a large role in various chemical processes. We shall now consider those chemical conclusions that can be drawn from the facts and theories set forth, moreover, as in the case of boundary continua, restricting ourselves to applications to photochemistry and to the determination of dissociation work.
Elementary photochemical processes reduce either to the primary electronic excitation of a molecule that undergoes decomposition upon subsequent collision, or to spontaneous photochemical dissociation ^40. In the case of ordinary band spectra with a boundary continuum, upon illumination of molecules from the region of discrete absorption only processes of the first type can be observed, whereas upon illumination with \(\lambda\lambda\) from the continuum—processes of the second type.
A photochemical criterion for one or the other type of process may be the applicability of the photochemical law of equivalence. For processes of the first type the quantum yield is less than unity and depends on pressure; for processes of the second type it is equal to or greater than unity (in the case of chain reactions) and does not depend on pressure.
Predissociation spectra have, as it were, an intermediate character. Here the band structure is preserved, but the bands themselves are devoid of rotational structure. We know that physi-
chemical cause of the predissociation phenomenon is spontaneous dissociation. We have also seen that Bonhoeffer and Farkas in fact discovered spontaneous photochemical dissociation in the case of NH₃. As was said above, here the rays of the Cd spark (λλ-2339, 2321, 2313, 2307, 2288, 2265 and 2198) do not cause any photochemical reaction at all; the rays of the Zn spark (λλ 2025, 2062, 2100, 2139) cause photochemical dissociation. That the products of this dissociation are hydrogen atoms was discovered by Farkas, Haber, and Harteck⁴¹ in a very ingenious way. If one takes a mixture of ammonia with H₂ + O₂, heats it to a temperature slightly below the explosion temperature of H₂ + O₂, and illuminates it with rays from the region of predissociation of NH₃, then an explosion immediately occurs. The hydrogen atoms arising upon dissociation initiate the chain reaction H₂ + O₂, so that the detonating mixture in this case is a sensitive reagent for hydrogen atoms.
Fig. 25. Potential curves of NO₂.
Predissociation of NO₂ has been investigated in great detail from the photochemical point of view.⁴² Here, as we know, two places of predissociation are observed: at λ = 3800 with a preliminary gradual blurring of the lines and at λ = 2450. In the first case the dissociation leads to NO and a normal O atom, and an energy 4–6 kg-cal greater than that necessary for separating the first oxygen atom from NO₂ is expended*. In the second case—NO and an oxygen atom in the ¹D state are obtained. The potential curves of NO₂ with indication of the dissociation products are given in Fig. 25.
The photochemical data relating to NO₂ are collected in the following table:
| Rays | Reaction | Quantum yield |
|---|---|---|
| 5790—4358 | no reaction | γ = — |
| 4050 | weak reaction | γ = 0.74 |
| 3650 | intense reaction | γ = 2.10 |
| 3131—2967 | intense reaction | γ = 2.07 |
Thus, for λ > the onset of predissociation no reaction is observed at all. In the intermediate region, where already a
* According to Mecke (“Z. Physikal. Ch.”), for separation from NO₂ of the first oxygen atom 71 kg-cal is required, while for separation of the second—already 160 kg-cal. In accordance with this, the structural formula of NO₂ must be such: O = N — O.
a noticeable blurring of the lines is observed; the reaction proceeds with incomplete yield. Here, however, the fluorescence has quite appreciable intensity; consequently, only a part of the excited molecules returns to the normal state with emission, while the rest undergo dissociation, probably by subsequent collision with an unexcited molecule of NO₂. In the region of predissociation the fluorescence is completely quenched, and the photochemical reaction proceeds with full quantum yield.^2 In accordance with this magnitude, the γ-reaction has the following mechanism:
\[ \begin{aligned} \mathrm{NO_2}+h\nu&=\mathrm{NO}+\mathrm{O},\\ \mathrm{NO_2}+\mathrm{O}&=\mathrm{NO}+\mathrm{O_2},\\ \hline 2\mathrm{NO_2}+h\nu&=2\mathrm{NO}+\mathrm{O_2}. \end{aligned} \]
Determination of the work of detachment of the O atom from NO₂ from the first place of dissociation can clearly give only the upper limit of the sought quantity. This is quite evident from consideration of the potential curves in Fig. 26 and from what was said in § 21: the disappearance of the rotational structure of the bands occurs at energies lying above the asymptote of the lower potential curve, and the excess energy expended is transformed into the kinetic energy of the decomposition products.
The second place of predissociation of NO₂ is characterized by a sudden disappearance of the structure of the bands. The place of predissociation can therefore be found very accurately, and it is located at \(\lambda=2459\ \text{Å}\). This wavelength corresponds to a quantum of 116.2 kg-cal, approximately 45 kg-cal greater than the energy of detachment of the O atom from NO₂ calculated theoretically from the heat of combustion of NO to NO₂. And since the level \(\mathrm{O}({}^1D_2)\) lies just 45.4 kg-cal above the normal one, the scheme may be considered reliable:
\[ \mathrm{NO_2}\to \mathrm{NO}+\mathrm{O}({}^1D_2)-116.2\ \text{kg-cal}. \]
Hence the spectroscopic value for the energy of detachment of the first O atom from NO₂ is: \(116.2-45.4=70.8\) kg-cal. Using this value and the thermochemical equation
\[ \mathrm{NO_2}\to \mathrm{NO}+\frac{1}{2}\mathrm{O_2}-13.5\ \text{kg-cal}, \]
we obtain for the dissociation energy of the O₂ molecule the following value:
\[ 2(70.8-13.5)=114.6\ \text{kg-cal}. \]
Of the more complex molecules, from the photochemical point of view, predissociation has been studied in benzaldehyde, acetaldehyde, and formaldehyde. We shall not dwell on these results here, however, since they yield nothing fundamentally new.
In conclusion, it should be noted once again that predissociation makes it fully possible to determine dissociation energies with spectroscopic precision, but only on the condition that the predissociation boundary is sharp. In the case of a gradual blurring of the lines, for the reasons set forth above, we can estimate only the upper limit of the dissociation energy sought.
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