THE FRANCK-CONDON PRINCIPLE AND RELATED ISSUES*)

E. U. Condon**)

A review of the historical development and present state of certain questions of molecular physics may serve as a reminder of the progress made in this field during the last two decades.

Let us recall that in 1925 nuclear physics did not yet exist, and high-voltage equipment was almost absent, unless one counts X-ray laboratories. Almost everything that was known about cosmic rays was limited to the facts that ionization increases with altitude and that these rays are capable of penetrating far into the depths of lakes. No one had yet heard anything about electron zones in solids, and the word semiconductor was essentially unknown.

The modern young physicist may think: “What, then, was there of interest in those days for physicists?” The answer, of course, is that physicists were then struggling with questions that students now learn with such ease that it is even difficult to imagine that there was a time when these things were not known at all.

Quantum mechanics did not exist in 1925. Although de Broglie had published his dissertation in 1924, at that time I had not yet met anyone who took it seriously. We quantized by means of the Bohr-Sommerfeld \(\int p\,dq = nh\) method and tried to approach the question of the probability of radiative transitions with the aid of an approximate and vaguely formulated device based on Bohr’s correspondence principle, according to which radiative quantum transitions are connected with, or placed in correspondence with—

*) E. U. Condon, American Journal of Physics, 15, 365 (1947). Translated by Belya.

**) Address by the retiring president of the American Physical Society, delivered on January 31, 1947, in New York.

“I apologize for including my name in the title of this article, but this apparently agrees with already well-established terminology.

In the article I shall simply speak of the ‘Principle.’ Prof. Rabi once told me that the ‘Principle’ often proves a boon for lecturers in atomic physics, since it is so easy to understand that one can avoid preparing one or two corresponding lectures.” — (E. Condon).

...accordance with certain terms of the Fourier expansion for quasiperiodic motions of a mechanical system.^1

Bohr’s theory explained very beautifully the spectrum of the hydrogen atom and the arc spectra of the alkali metals. However, the doublet character of the sodium \(D\)-line remained a profound mystery, the solution of which was ultimately provided by the hypothesis of Goudsmit and Uhlenbeck concerning the rotating electron. Further, the old quantum theory shed considerable light on the question of the infrared and electronic spectra of diatomic molecules. Purely rotational, rotational-vibrational, and electronic band systems were identified, analyzed, and used to obtain interesting quantitative data for the most important molecules. Nevertheless, here too there were puzzling matters; thus, for example, the isotopic shift between \(HC^{35}\) and \(HC^{37}\), as well as in certain other molecules, required that half-integer quantum numbers be assigned to the vibrational levels.

FIRST THOUGHTS ABOUT THE “PRINCIPLE”

At that time I was a senior student at the University of California in Berkeley. Ernest Lawrence was a senior student at Yale University, and there was not a single cyclotron anywhere in the world, not even in Berkeley.

There, in the physics department, the scientific supervisors were Professors R. Birge and L. Loeb—the former in the field of molecular spectra, the latter on all questions concerning the passage of electricity through gases.

During 1925–1926, under Birge’s direction, a seminar on molecular spectra was conducted, and it is impossible to overestimate it. Then, as always, his work was distinguished by the most precise and careful study of the available data and of their connection with theory. I still remember the excitement we felt when Birge showed with great accuracy that the swelling of rotating molecules, which can be discerned from their purely rotational spectrum, agrees with the values obtained from vibrational-rotational and electronic spectra. He was the first to gather together all the existing data on the electronic band spectra that had been investigated—there were then about a dozen of them—and he clearly recognized the basic empirical facts concerning the distribution of intensities, which were later explained by the Principle. I dwell on this point in such detail because, without his stimulating guidance, I would never have suspected the existence of the problem of intensity distribution in band spectra.

In Göttingen, Prof. James Franck was very interested in photochemical reactions and, in particular, in the dissociation of iodine vapor upon absorption of light. He delivered a report before the Faraday Society in London,^2 in which the basic idea of the Principle was first expressed in connection with the problem mentioned above.

An offprint of this article was sent by Franck to his pupil, Dr. Hertha Sponer, who at that time was in Berkeley. She gave me the article to read, and since I had just become acquainted, at Birge’s seminar, with the empirical picture of the distribution of intensities in band spectra, it at once became clear to me how Franck’s idea should be generalized a little further in order to obtain the full picture. Moreover, Birge’s seminar provided me with a good critical compilation of the existing data, which made it possible to check the basic ideas quantitatively very quickly.

This work was completed in its entirety within a few days. A week after Dr. Sponer had shown me Franck’s article, I had finished all the computational work for my 1926 paper.^3 But let us see what the state of affairs was then.

Fig. 1. Potential-energy curves taken from Franck’s paper.^2

Fig. 1 is borrowed from Franck’s paper.^2 Franck pointed out that, owing to the large mass of the nuclei in the molecule, an electronic transition cannot directly affect their relative momenta; consequently, the most probable transitions will be those which most nearly correspond to the Principle. Thus, if for the iodine molecule the curves have the form of group I in Fig. 1, then a molecule which initially is not vibrating will most readily absorb light that takes it into states of intense vibration or into states with energies exceeding the dissociation energy, which leads to photochemical dissociation of the molecule.

Of course, from this it was not difficult to take the next step and understand that, if the molecule is initially vibrating, then the Principle asserts that transitions occur to those states which require the least instantaneous rearrangement of the relative position and momentum. Moreover, it was intuitively felt that an electronic transition is sufficiently independent of the vibrations of the nuclei and therefore must occur with equal probability at any phase of the vibrational motion of the nuclei. This is by no means obvious and probably not quite exact. It is conceivable, for example, that the electron transition is excited by the vibrational motion in the molecule, and in such a way that it occurs most often in the phase of maximum relative velocity. However, this is not so. If all instants of the electron jump are regarded as equally probable, the most probable vibrational

transitions will be those associated with the turning points of the nuclear vibration; for here the nuclei move more slowly and a greater share of the time is spent in such regions.

This idea gave rise to Fig. 2, borrowed from my 1926 article; it shows that there are two most favorable changes of the vibrational quantum numbers, associated with unchanged values of the initial vibrational quantum number.

The whole point is that a number of band spectra had been well studied and therefore it proved possible to depict the form of the potential-energy curves fairly accurately, and also to indicate their relative positions. Near a minimum the curve has the form of a parabola

\[ V(r)=\frac{1}{2}k(r-r_0)^2+\ldots \]

The value of \(k\) can be obtained from the vibration frequency \(\nu\), since if \(\mu\) is the reduced mass, then \(2\pi\nu=(k/\mu)^{1/2}\); \(r_0\) is obtained from the value of the moment of inertia, which in turn can be determined quantitatively from the observed rotational energy levels. If \(\nu\) or \(r_0\) changes little, then the curves in Fig. 2 are evidently similar and lie close to one another, so that the principle requires a small or zero change of the vibrational quantum number. But if \(r_0\) changes strongly, then the curves diverge greatly and the principle requires large changes of the vibrational quantum number. Both cases were found in the data available in 1926. The principle was triumphant insofar as cases of large changes of the vibrational quantum number were, as a rule, associated with cases of large changes in the equilibrium internuclear distance, and conversely.

Fig. 2. Graphical representation of favored transitions; taken from the author’s 1926 article.[^3]

It should be recalled that the available data on intensities were only rough estimates, based on blackening of photographic plates without corrections for the dependence of the sensitivity of the plates on wavelength, the latter in some systems po-

Thus, the calculus occupied a rather large interval. Nevertheless, there was no doubt that the Principle was essentially correct.

It is curious that my calculations were in poor agreement with the experimental data for iodine, i.e., precisely for the molecule that had led Franck to the qualitative discovery of the basic idea. This troubled me greatly, and I searched for a long time for an error in the calculations, until at last I sent my 1926 paper for publication, leaving open the question of the large discrepancy for iodine. About a year later the error was uncovered by Prof. U. Loomis. It turned out that I had used the value of the moment of inertia for the 26th vibrational level of the excited electronic state that participates in the fluorescence spectrum of iodine, assuming that it coincided with the value of the moment of inertia of nonvibrating molecules. When this error was corrected, iodine agreed with the Principle no worse than all the other molecules^4.

The Academic Council of the University of California was sufficiently broad-minded to accept the paper as a doctoral dissertation. It is interesting to note, however, that at that time orthodox theoretical physics was so closely tied to Bohr’s correspondence principle that the referee of Physical Review reluctantly recommended the paper for publication. He regarded it as incorrect, since the approach to the problem was not based on the Fourier amplitudes of classical motion.

The simple classical picture of 1926 was, nevertheless, not exhaustive. The principal remaining puzzle could be formulated as follows: how exact is the Principle? Or: what determines the degree of its inaccuracy?

The potential-energy curves are arranged in a definite way, so that the Principle leads to a rather definite conclusion concerning the most favored transition. In fact, although the predictions of the most favored transition were in good agreement with the experimental facts, there were no indications as to why other transitions may also occur, and how their intensities should be calculated. This was a serious shortcoming of the theory, which Prof. Birge did not hesitate to point out to me.

A year later, in 1927, we all became acquainted with Heisenberg’s uncertainty principle as one of the fundamental propositions of quantum mechanics. But in 1926 it occurred to no one that it was fundamentally wrong to speak of certain definite values of the coordinates and momenta of nuclei that do not change during electronic transitions. Looking back, it becomes clear that this could indeed have been anticipated, because statistical mechanics dealt with elementary cells of finite width in phase space, and the uncertainty principle is closely connected with this representation. However, as often happens in other cases as well, in theoretical physics lack of foresight often proves better than foresight.

E. U. CONDON

QUANTUM-MECHANICAL FORMULATION

Everything I have just described took place in the spring of 1926. Matrix mechanics had already been created, but matrix computation was so complicated that it was difficult to put any physical meaning into it*). Schrödinger’s famous papers on wave mechanics had only just begun to appear, and it was only in the autumn of 1926 that Born first gave the probabilistic interpretation of \(|\psi|^{2}\), which follows of necessity from the consideration of collision problems and is opposed to the more hydrodynamic interpretations possible, in general, for closed systems.

In the autumn of 1926 I went to Göttingen.

The output of great ideas was so large in that period (1926–1927) that a quite erroneous conception arose of the normal rate of progress in theoretical physics. A large part of the year one had to suffer from intellectual indigestion, which had a most depressing effect.

In addition to studying the current papers, I tried to solve the fundamental problem of quantum mechanics that underlies all molecular dynamics, namely, to find a justification for the method by which the electronic states are first determined for fixed positions of the nuclei, so that the electron energy levels depend parametrically on the nuclear coordinates and serve as the potential-energy functions that determine the motion of the nuclei.

The justification, of course, is connected with the smallness of the electron mass in comparison with the nuclear mass, but I never succeeded in carrying out the proof. Later this problem was considered in the fundamental paper by Born and Oppenheimer\(^{5}\), which, incidentally, I never was able to understand to the end. But Born had such a great vision of seeing everything in physics as an “expansion in a small parameter kappa” that, obviously, one should suppose that everything is in order. The paper by Born and Oppenheimer belongs to those difficult papers which are cited more often than they are read.

These investigations, however, did not help to clarify the relation of the Franck principle to the general foundations of quantum mechanics. Although this question was discussed in a short paper written in Göttingen, it still did not receive proper attention until the autumn of 1928, when a paper\(^{4}\) was written in Princeton. The essence of the argument is that the wave function of a diatomic molecule is approximately expressed as the product of the electron wave function (with the nuclear coordinates entering as parameters) and the wave function describing the motion of the nuclei, for example,

\[ \psi = u(x_e, x_n)\, v(x_n), \]

*) I remember well how in the autumn of 1926 in Göttingen Prof. David Hilbert said in this connection to his students: “Physics has become too difficult for physicists.”

where \(x_e\) symbolically denotes the coordinates of the electrons and \(x_n\)—the coordinates of the nuclei. Consequently, the matrix element of such a quantity as the dipole moment, which determines the probability of radiative transitions, will have the form

\[ M_{12}=\int_e \int_n \psi_1 M(x_e,x_n)\psi_2\,dx_e\,dx_n =\int_n \overline{v}_1(x_n)M_{12}(x_n)v_2(x_n)\,dx_n, \]

where

\[ M_{12}(x_n)=\int \overline{u}_1(x_e,x_n)\,M(x_e,x_n)\,u_2(x_e,x_n)\,dx_e . \]

The quantity \(M_{12}(x_n)\) is the matrix element of the dipole moment, regarding the nuclear coordinates as parameters of the electronic problem rather than as dynamical coordinates. It characterizes the electronic jump under consideration and, consequently, is the same for all vibrational transitions of a band spectrum.

The nuclear wave function \(v(x_n)\) may prove to be rather complicated for polyatomic molecules, but for a diatomic molecule it has the form of the product of a function of the radial coordinate \(R(r)\) by a spherical harmonic of simple motion under the condition of conservation of angular momentum.

It is difficult to say anything about \(M_{12}(x_n)\). Moreover, up to now no explicit calculation has been carried out for any concrete case. It is natural, however, to suppose that it will be a slowly varying or smooth function of \(r\) in that small interval of variation of \(r\) in which the radial wave functions have appreciable magnitude.

The radial functions \(R(r)\) have a rather close relation to the corresponding classical oscillatory motion in accordance with the general relation established by the Wentzel–Brillouin–Kramers approximation. According to this approximation the wave function has the form

\[ R(r)=\frac{1}{4p^{1/2}}\cos\left[\frac{2\pi}{h}\int^r p\,dr+\alpha\right] \]

in the region of classical motion and rapidly tends to zero outside this region, where a factor of the form predominates

\[ \exp\pm\left[\frac{2\pi}{h}\int^r |p|\,dr\right]; \]

\(p\) is given by the expression

\[ \frac{1}{2\mu}p^2+V(r)=W, \]

where \(V(r)\) is the effective potential energy of the radial motion, including also the effects of rotational energy:

\[ V(r)=V_0(r)+\frac{h^2 I(I+1)}{2\mu r}, \]

where \(I\) is the rotational quantum number and \(V_0(r)\) is the potential-energy function applicable to the case of a non-rotating molecule.

Qualitatively it is easy to see that the value of an integral of the form

\[ \int R_1(r)R_2(r)\,dr \]

with the wave functions given by the Wentzel–Brillouin–Kramers approximation is determined mainly by the conditions expressed in the classical formulation of the principle, as it was given in my paper of 1928, namely:

1) the value of the integral will be small if the wave functions do not overlap, i.e. if there is not a small sudden change of the internuclear distance as a result of the transition;

2) the value of the integral will be small if the wave functions are in such a relation that the non-oscillatory part of one wave function overlaps the rapidly oscillating part of the other, since this means that the transition of the electron must be accompanied by a noticeable change in the radial momentum.

This formulation is much broader than the formulation of 1926 in three main respects:

a) it follows in a definite deductive way from a well-established theory, the other successes of which give reason to regard it as a reliable foundation of atomic mechanics;

b) it gives, in principle, definite results for the relative strength of every possible quantum transition; it therefore goes beyond the simple classical picture based on curves of potential energy and provides the possibility of calculating the probability of the occurrence of less favored transitions, which in fact do take place;

c) it leads to the prediction of certain special effects that are a consequence of the wave nature of matter, for which I should like to propose the term “internal diffraction.” These effects will be considered in more detail in the following paragraph.

When one has to write explicitly the formulas for the integrals

\[ \int R_1(r)R_2(r)\,dr, \]

it proves impossible to obtain results of broad applicability. It is most natural to assume that the initial and final potential-energy functions are Hooke’s-law parabolas, differing from one another only in the force constant and in the distance between the equilibrium positions. Formulas based on this assumption were first obtained by Hutchisson\(^6\). The corresponding results, however, are rather complicated,

A more severe limitation is that, in the physically most interesting cases, owing to the electron transition there occurs a rather strong change in the equilibrium distance, as a result of which large changes in the vibrational quantum number may take place. In these cases it is necessary to know, with sufficient accuracy, the wave functions at some distance from the equilibrium positions. The true force law does not have the form of a parabola, so that the wave functions of the harmonic oscillator are no longer a good approximation for the substantial region of coordinate variation.

Since the actual deviation from the harmonic law is different for each molecule, general formulas are of no special value. Consequently, the only proper test of the theory consists in a direct calculation in accordance with the particular features of each given case.

INTERNAL DIFFRACTION

I especially want to emphasize the nonclassical, or wave-mechanical, features of the theory, since it seems to me that they deserve wider recognition. These features, it seems to me, are something more than simple quantum-mechanical refinements, as is often asserted. On the contrary, internal diffraction is for me just as real and convincing a proof of the wave nature of nuclear motion as any of the experiments on external diffraction, such as, for example, those in which a beam of electrons or neutral hydrogen atoms is scattered by a crystal.

Fig. 3, taken from my 1928 paper, illustrates one case in which internal diffraction can arise. It is immaterial whether curve \(I\) is above curve \(II\) or below it: if it is above curve \(II\), the phenomenon will be observed in emission, and if it is below curve \(II\), in absorption. The wave function of the lower vibrational level in state \(I\) will approximately be, as shown, a Gaussian error function. The radial wave function for a typical value in the continuum above the dissociation limit of \(II\) will have a form similar to that shown in the figure. As the energy of the final state increases, the radial wave function will change in the following way: its first antinode will correspond rather closely to the turning point of the classical motion; however, with the slow change of the de Broglie wavelength, the nodes are situated ever closer to one another and, consequently, the phase of the quasi-sinusoidal wave situated under

Fig. 3. Arrangement of wave functions in internal diffraction in a continuous spectrum; taken from my 1928 paper.\(^4\)

by the center of a wave function in the form of a Gaussian error function, gradually changes. It is clear that when the antinode of a sinusoidal wave function lies under the center of the Gaussian wave function, we shall obtain a large value of the integral determining the probability of a radiative transition. But when a node lies under the Gaussian function, the value of the integral will be very small.

In this way the wave function of the initial state can “see” the wave nature of the wave function of the final state. Transitions may occur to those continuum energies for which an antinode lies under the initial wave function, and transitions to energies having a node at this point are far less probable. As a result, a pulsating variation of the continuous spectrum is obtained. This manifestation of the wave properties of the \(\psi\)-function I shall call internal diffraction, by analogy with external diffraction, which is likewise determined by the phase relation of the initial and final wave functions.

The term “internal diffraction” is conveniently used in a broader sense, to denote characteristic deviations of transition probabilities from those values which follow from the Principle when the restrictions of the latter are softened by certain conditions arising from the uncertainty principle. In the more general case one may suppose that the wave functions for the initial and final states are approximately related in the manner shown in Fig. 4. Obviously, the exact value of the integral of the product of two such functions is very sensitive to the relative position of the nodes and antinodes. The integrand itself, in rough outline, has the form of an oscillating function, taking positive values when both factors have the same sign, and negative values when the signs differ. Thus the values of the integral are very sensitive to the mean relative phase of the oscillations of the wave functions of the initial and final states. This is a specifically quantum-mechanical effect, and certain consequences arising from it may serve as a good illustration of the reality of de Broglie waves associated with the motion of nuclei in molecules.

Fig. 4. Rough representation of wave functions, showing the sensitivity of the integral to the relative position of nodes.

Although at that time it did not seem possible to carry out sufficiently accurate calculations to test this proposition, nevertheless this view was put forward in 1928 to explain the fluctuations of intensity discovered by Wood in the fluorescence spectrum of iodine vapor. Upon excitation by means of the green mercury line, diatomic iodine molecules are raised to the 26th vibrational level

of the excited electronic state, from which they emit a long series of fluorescence doublets in transitions to various vibrational levels of the ground electronic state. These doublets form a series whose intensity varies in a rather disorderly way, namely: 10, 9, 1, 9, 3, 8, 8, 2, 9, 0, 8, 3, 2, 7, 0, 7, 0, 2, 1, 0... All these transitions are allowed by the approximate formulation of the Principle. Undoubtedly, the fluctuations are

![Figure 5 and Figure 6 diagrams]

Fig. 5. Brown’s calculations for the sodium fluorescence bands.^7

Fig. 6. Potential curves and wave functions used by Gaydon and Pearse in their calculations for rubidium hydride.^8

a consequence of special relations among the phases of the nodes in the initial and final states, but because of the large values of the quantum numbers, a direct calculation to check this assertion would be very difficult and therefore was not carried out.

This gap, however, is fully compensated by two direct calculations, which will now be discussed. A similar disorderly change of intensity occurs in the fluorescence bands of Na₂. The calculation of the approximate integrals was carried out by W. Brown in 1933.^7 The comparison he made between the observed and calculated intensities is shown in Fig. 5.

Analogous calculations of the band system of rubidium hydride were carried out by Gaydon and Pearse^8 in 1939. The potential-energy functions and wave functions they used are shown in Fig. 6,

which is borrowed from their work. The dotted parabolas depict the potential energy in the approximation of Hooke’s law, and the solid curves depict more accurate potential-energy functions obtained on the basis of data on the levels. The wave functions used were obtained by a reasonable approximate transformation of the wave functions of the harmonic oscillator, and not by numerical integration of the wave equation.

Fig. 7. Intensities for rubidium hydride, according to Gaydon and Pearse:
a — observed visual estimates on a tenfold scale; b — computed results, reduced to a twenty-fivefold scale.

The result of their calculations is shown in Fig. 7. The table on the left gives the observed values (estimates), and the table on the right gives the results of the calculations. The principal parabolic geometrical locus of the intense bands is a familiar result of the classical Principle. Secondary, weaker geometrical loci are the result of special relations between the phases of the initial and final wave functions, i.e., of internal diffraction.

THE HYDROGEN MOLECULE

Diatomic hydrogen is an especially interesting molecule, since it has the simplest structure, allowing a number of theoretical calculations to be carried out with fairly good accuracy. These calculations, first performed by Heitler and London,^9 led to a whole series of new ideas applicable to chemistry in general. In particular, they clarified the basic nature of the paired-electron, or homopolar, valence bond. Since it is necessary to keep the present survey within some limits, I shall restrict myself here to certain interesting consequences that arose as a result of applying the Principle to the theoretical potential-energy curves given by Heitler and London and by Barrow^10 for \(H_2^+\).

The first application of the Principle to the hydrogen molecule was carried out by Winans and Stueckelberg[^11] in 1928. They showed that radiative transitions from excited triplet levels to the triplet sigma state \(({}^3\Sigma)\), corresponding to the Heitler–London repulsive forces, can explain the very broad ultraviolet spectrum of the hydrogen molecule. At about the same time, in Smith’s article some interpretations were given of the observed critical potentials on the basis of curves of the potential energy of repulsion. It may be said that these two papers gave the first proofs of the physical reality of the repulsive forces that follow from quantum mechanics.

Much more convincing proof of the physical reality of such electronic states in the hydrogen molecule was given by Bleakney[^13] in 1930, when we were together with him at the University of Minnesota. In the older data on the critical potentials of hydrogen, a value close to \(31.5\ \mathrm{eV}\) was usually obtained. Since this agrees rather well with the minimum value of the energy required to remove both electrons from the hydrogen molecule, it was thought that it corresponded to the process of complete dissociation of the hydrogen molecule into two protons and two electrons.

Fig. 8. Curves of potential energy for the hydrogen molecule; borrowed from Bleakney’s 1930 paper.[^13]

How could this be? For according to the Principle, the incident electron must knock out both electrons in such a way that the protons remain practically at the same distance from one another as in the normal molecule. At such a distance the energy of the Coulomb interaction of the two protons is of the order of \(20\ \mathrm{eV}\); consequently, the process of complete dissociation would require \(51.5\ \mathrm{eV}\) instead of \(31.5\ \mathrm{eV}\). Later, of course, both protons could fly apart in different directions under the action of mutual repulsion, each of them receiving an energy of \(10\ \mathrm{eV}\).

This situation is shown in Fig. 8, which is taken from Bleakney’s 1930 paper.[^13] Curves \(a\) and \(b\) were obtained from the fundamental solutions of Heitler and London for the interaction of two normal hydrogen atoms. Curve \(c\) is taken from Barrow’s calculations for the ground estado.

states of \( \mathrm{H}_2^+ \). Curve \(d\) represents the potential-energy curve of the repulsive forces in \( \mathrm{H}_2^+ \), calculated by Morse and Stueckelberg[^14]. Finally, curve \(e\) is simply \(e^2/r\), i.e. the potential-energy curve of the molecule \( \mathrm{H}_2^{++} \).

From these curves several interesting consequences follow at once. First, it is clear that Princ’s principle does not allow the molecule to be dissociated by a simple collision with an electron possessing the minimum necessary energy, in the present case an energy of about \(44\ \mathrm{eV}\), since in this case a large change in the position or momentum of the nuclei would be required, accompanying the electron transition.

However, at an energy of approximately \(11\ \mathrm{eV}\) one must admit the possibility of dissociation of the molecule by collision with an electron through the induction of transitions from state \(a\) to state \(b\). This transition is not allowed for the absorption of light, but is allowed for electron collisions. The excited \( \mathrm{H}_2 \) molecules would at once fly apart, forming two normal hydrogen atoms, each with a kinetic energy of the order of \(3.5\ \mathrm{eV}\). Since both products are neutral, it would be impossible to observe them in a mass spectrograph. Observations were made which indicate the rapid disappearance of hydrogen, i.e. adsorption by the walls (which is possibly the result of the formation of atomic hydrogen) at this, approximately, energy, but not at lower potentials.

As regards state \(c\), the change in the equilibrium distance indicates that the transition from \( \mathrm{H}_2 \) to normal, non-vibrating \( \mathrm{H}_2^+ \) is unlikely, and it is more probable that a good yield of \( \mathrm{H}+\mathrm{H}^+ \) is obtained by a direct transition from the normal state to that part of \(c\) which lies above the dissociation boundary. This was in fact observed.

Considering now curve \(d\), we find an explanation of the critical potential at \(31.5\ \mathrm{eV}\). It has nothing in common with complete dissociation—this would require a transition to curve \(e\). In reality, the potential at \(31.5\ \mathrm{eV}\) is the transition, indicated by Princ’s principle, to curve \(d\). If this explanation is correct, then the \( \mathrm{H}^+ \) ions formed in this process must possess kinetic energy of about \(6.5\ \mathrm{eV}\). By suitable use of retarding fields in the mass spectrograph, Bleakney was able to show that this actually takes place. This was the first case of observation of molecular ions—fragments formed with a definite kinetic energy, and also the most direct and unambiguous of the existing proofs of the physical reality of the molecular states determining repulsive forces, as predicted by quantum mechanics.

Later Tate and Lozier obtained a number of new reliable data which made it possible to establish that molecular fragment ions possessing kinetic energy can also be formed from

other molecules, but quantum-mechanical calculations in this case are too complicated to allow detailed predictions.

In this connection one may mention still another circumstance, which was briefly studied by Gippel in 1936 and which gives yet another interesting example of the quantum-mechanical aspect of the Principle. Let us again consider transitions from curve \(a\) to curve \(c\), caused by collisions with an electron. The minimum of curve \(a\) is situated with respect to curve \(c\) in such a way that the most favored transitions are those which lead to \(\mathrm{H}_2^+\) molecules in a rather high vibrational state. If \(\mathrm{H}_2\) is bombarded, say, with electrons of energy 18 eV, then a certain amount of \(\mathrm{H}^+\) will be obtained, but transitions to \(\mathrm{H}_2^+\) will nevertheless be more favored.

What will happen if deuterium is used instead of hydrogen? Theory says that the potential-energy curves coincide rather accurately for the molecules of both isotopes; it also says that the wave function of the ground zero-point vibrational state of \(\mathrm{D}_2\) will be narrower than that of \(\mathrm{H}_2\), and, owing to the greater mass, the behavior will be more classical. Thus, at 18 eV one may predict a smaller yield of \(\mathrm{D}^+\) relative to \(\mathrm{D}_2^+\) in \(\mathrm{D}_2\) than the yield of \(\mathrm{H}^+\) relative to \(\mathrm{H}_2^+\) in \(\mathrm{H}_2\). Gippel verified this and found a considerable effect. It would be interesting to obtain accurate data on this question and to try to carry out an exact calculation.

In 1931 Finkelnburg and Wenzel tried to obtain definite information on the form of the fundamental potential curve of the Heitler–London repulsive forces by analyzing data on the potentials required for exciting various portions of the continuous spectrum of molecular hydrogen. As Coolidge, James, and Present\(^{15}\) pointed out, such a procedure presupposes a more exact application of the Principle than had generally been allowed up to that time. In order to consider the problem more rigorously than can be done by means of the usual rough graphical construction on the basis of the potential-energy curves, very careful work was carried out, first to improve the calculations of the theory of the potential curve of the repulsive forces and then to numerically integrate the products of the radial wave functions occurring in the theory.

The work of Coolidge, James, and Present showed rather convincingly that in this case the matrix element of the electron dipole moment cannot be regarded as constant. This meant that the rough assumption which I had made in explaining the general idea of the Principle had to be improved. This gave the above-mentioned authors occasion to write in their summary that “the conclusion is reached that the Franck–Condon principle leads to results which are undoubtedly incompatible with experiment.” I cannot let this pass.

statement without protest, although ten years have passed. The authors may say that I have applied the Principle incorrectly, but they cannot say that the Principle itself is incompatible, for the Principle must be judged according to its correct quantum-mechanical formulation, so elegantly developed by them, and not according to the crude criteria which I gave as approximate rules.

In my view, the whole question of these specific properties of the hydrogen molecule is in a very satisfactory state.

CONCLUSION

Despite the competing attractiveness of nuclear physics and the unproductive interruption caused by the war, over the last decade many new band spectra have been studied, so that at the present time we are acquainted with a large number of electronic transitions of a large number of molecules, and all of them are in good agreement with the semi-quantitative intensity relations that follow from the approximate formulation of the Principle. Nevertheless, it remains a fact that the Principle has never yet been subjected to a truly rigorous test by means of careful measurements of intensities using the method of good photographic photometry and comparison with values calculated with great accuracy from wave functions.

I shall conclude this survey by mentioning one more application. The application of the Principle has contributed greatly to the clarification of many questions concerning the predissociation of molecules16. With the exception of a few cases, where weakly bound molecules may dissociate as a result of rotational instability with the assistance of leakage through a potential barrier, here we are dealing with radiationless transitions between two electronic states of the same energy. And here too the Principle has a limiting effect, since such transitions cannot take place if they require too large a change in the motion of the nuclei.

“In principle” the Principle is also applicable to polyatomic molecules17, although everything here is incomparably more complicated than in the case of diatomic ones. Here there are not only much more complex band systems, for the study of which very little has so far been done, but flat potential curves must also be replaced by potential-energy surfaces, and these latter are poorly known both empirically and theoretically.

CITED LITERATURE

1. W. Lenz, Zeits. f. Physik 25, 299 (1924). This article is an attempt to consider the problem of nuclear transitions associated with electronic transitions on the basis of the correspondence principle.
2. J. Franck, Trans. Faraday Soc. 21, 536 (1925). An article in which the Principle was first formulated in application to the question of the photochemical dissociation of iodine vapors.

1. E. U. Condon, Phys. Rev. 28, 1182 (1926). First application of the Principle to the question of the intensity distribution in absorption bands.

2. E. U. Condon, Phys. Rev. 32, 858 (1928). Development of the quantum-mechanical formulation of the Principle, including also the basic idea of internal diffraction.

3. M. Born and J. R. Oppenheimer, Ann. d. Physik, 85, 457 (1927). Quantum-mechanical justification of the applicability of electron potential-energy curves for determining nuclear motion in molecules.

4. E. Hutchisson, Phys. Rev. 36, 410 (1930); 37, 45 (1931). Explicit calculation of transition integrals with the aid of harmonic-oscillator wave functions.

5. W. G. Brown, Zeits. f. Physik, 82, 768 (1933). Direct calculation of changes in the intensity of fluorescence bands of the sodium diatomic molecule caused by internal diffraction.

6. A. G. Gaydon and R. W. B. Pearse, Proc. Roy. Soc. A 173, 37 (1939). A careful calculation of transition probabilities for the band spectrum of rubidium hydride; the effects of internal diffraction are considered quantitatively.

7. W. Heitler and F. London, Zeits. f. Physik 44, 455 (1927). The first quantum-mechanical calculation of the potential-energy curves of the hydrogen molecule.

8. O. Burrau, Danske Videnskab Selskab Mathfysiske Medd., 7, 14 (1927). The first calculation of potential-energy curves for the ground state of the ionized hydrogen molecule.

9. J. G. Winans and E. C. G. Stueckelberg, Proc. Nat. Acad. Sci. 14, 867 (1928). Interpretation of the continuous ultraviolet emission spectrum of molecular hydrogen.

10. E. U. Condon and H. D. Smyth, Proc. Nat. Acad. Sci. 14, 871 (1928). Interpretation of critical potentials of the hydrogen molecule.

11. W. Bleakney, Phys. Rev. 35, 1180 (1930). Experimental proof that, in collisions of electrons with hydrogen molecules, some of the ions formed possess kinetic energy.

12. P. M. Morse and E. C. G. Stueckelberg, Phys. Rev. 33, 932 (1929). The first calculation of the potential-energy curve of repulsive forces for the ionized hydrogen molecule.

13. A. S. Coolidge, H. M. James and R. D. Present, J. Chem. Physics 4, 193 (1936). A careful consideration of the possibility of applying the Principle to the continuous spectrum of molecular hydrogen.

14. L. A. Turner, Zeits. f. Physic 68, 178, (1931). Application of the Principle to the problem of predissociation.

15. J. Franck, H. Sponer and E. Teller, Zeits. f. Physik. Chemie 18, 88 (1932). Application of the Principle to the predissociation of polyatomic molecules.