THE ELECTRIC THEORY OF FORCES BETWEEN ATOMS AND MOLECULES*

F. London, Berlin

The problem of describing the interplay of forces between atoms and molecules, i.e., of reducing them to our conceptions of the structure of matter and of the laws of nature operating at the smallest distances, has a certain similarity to the problems of celestial mechanics. It is instructive to imagine the obstacles that hinder the creation of a theory of atomic and molecular forces by means of such a comparison with astronomy.

In celestial mechanics, in order to describe each of the heavenly bodies the astronomer needs to know only one constant, namely its mass—in the extreme case three further components of the moment of inertia. The fact that the heavenly bodies possess, generally speaking, masses that differ greatly from one another indicates to the astronomer from the very beginning a natural way of solving his problems in stages, namely in the form of successive, rapidly converging approximations. In this procedure one calculates the perturbations in the motions of light bodies caused by heavy ones, neglecting the weaker action of the former on the latter; it is taken into account only at the next stage, and so on. In the end, even if these approximations are not convergent over a long period of time, they entirely satisfy the astronomer, since he is interested in knowing the orbits of the heavenly bodies only over a comparatively short time—several thousand or tens of thousands of periods of revolution.

None of these circumstances can be used by the physicist of the atom: in order to understand fully the motion of atoms, we need far more than the simple knowledge of only one constant for each atom. This is because we are especially interested in atomic forces at very small distances, namely at distances of the order of the size of the atoms themselves. It is clear that in this case the smallest details of atomic structure may become decisive. Exact

* Report at the International Electrotechnical Congress in Paris in 1932; translated by M. V. Vol’kenshtein.

description of the structure of the atom is now a problem of an astronomical type, but with the serious difference that the constituent parts of the atom—the electrons—introduce equal constants of attraction, namely equal charges. Therefore the mutual perturbations caused by the electrons cannot be calculated by successive approximations in the same natural way as perturbations in the motion of the planets. But, on the other hand, a much more exact knowledge of the structure of the atom is necessary, since here we can no longer be satisfied with knowledge over several thousand atomic years, but must know the atom over billions and trillions of its periods. It may be said that, if we had remained on the basis of classical mechanics, the hope of overcoming these difficulties would have been lost forever.

If it has recently proved possible to achieve successes in this field worthy of mention, this has happened above all because it was understood that, in the case of sufficiently small dimensions, one must apply a theory different from the classical one, i.e. quantum mechanics. Quite apart from the fact that this theory makes it possible to deepen to an extraordinary degree our general conception of nature and of the principles by which observations are described, its significance for the field that interests us is characterized above all by the fact that its content is much more limited than the content of classical mechanics, in particular because quantum theory gives, chiefly, relations between statistical data. Quantum mechanics is in many respects more convenient than classical theory, since its descriptions contain fewer details.

Thus, first of all, the principal difficulty indicated above—of obtaining a description of various systems of atoms suitable for sufficiently large intervals of time—was overcome by quantum mechanics in a remarkable way. It showed us that, in order to characterize the normal states of atoms, we must renounce the determination of the trajectories of the various constituent particles. For this purpose the theory gives a statistics of probabilities with which each configuration of the system is compatible. Such a description of the atom can satisfy us, since, we repeat, for the description of interactions of many atoms of this kind only the statistics of configurations is necessary, and not a definite knowledge of trajectories. We thus avoid the difficulties of detailed knowledge for all time and can pass directly to statistics of the system independent of the time that interests us. The fact of the absolute identity of electrons, which makes illusory the application of approximations such as those of celestial mechanics, is, on the other hand, very advantageous for determining the statistical distribution, since it lies at the basis of the characteristic symmetry properties of the distribution functions, which are determined simply, without numerical calculation, by qualitative considerations of symmetry. These symmetrical

the properties of atoms give us, at the same time, a general simple characterization of the properties that are of essential importance for the interplay of forces.

Despite these great simplifications, in quantum mechanics too one has to resort to certain approximate methods, because the state of affairs is too complex and does not admit exact calculation. Along with this there is the practical advantage of eliminating, as far as possible, secondary factors from the final result. Thus approximate methods, at least if they are reasonable, signify a definite understanding of things, since these things have been brought to a definite degree of approximation. Consequently, approximations are not only a means, but also a method for singling out the most important conditions. We could not properly assess the significance of theoretical results if we did not know precisely with what conception they are connected.

A. Statics and dynamics

The principal simplification provided by theoretical study of the questions indicated consisted, above all, in distinguishing a special kind of statics of atomic forces, whose task is to find the energy contained in atoms at definite interatomic distances as a function of the latter. For two atoms we already have a large number of such energy functions (potential curves), depending on the conditions in which the atoms under consideration are found.

Such a static treatment requires, as a supplement, a dynamics of atomic forces, whose task is to find how atoms move under the action of forces. This is the true theory of chemical reactions. Until recently this problem was considered much less closely than the first. It is usually assumed that the energy functions given by the static theory can be used for the motion of atoms as potentials in classical mechanics; in this case quantum mechanics is applied to the constituent parts of the atom—the electrons—and classical mechanics to the atom as a whole. This is justified to some extent by the fact that for large and heavy bodies quantum mechanics asymptotically approaches classical mechanics. However, the sizes and masses of atoms can in no case be regarded as sufficiently large; thus this application of classical mechanics must be regarded only as a first approximation.

The study of such potential curves nevertheless has a certain value for the majority of chemical processes, although these curves do not describe the motion in detail, in the sense of classical mechanics, since they determine the energy relations in a reaction, which are decisive for

its final result and, for the most part, for its rate (the so-called “adiabatic processes”).

These reactions must be contrasted with other, “non-adiabatic” ones, which cannot be defined by a single potential curve.

In the latter case one must take into account the possibility that the internal state of the reacting atoms changes during the process and that they pass, so to speak, from one potential curve to another. It is clear that reactions of this class cannot be even approximately defined with the aid of the simple concept of a potential and classical mechanics. They play an important role in chemistry, above all as intermediate links in reactions to which energy has been supplied in the amount necessary for the course of the process, for its activation.

Finally, let us point to the processes of monomolecular decomposition, which have often been discussed recently. Here there are two possibilities, both proceeding from the concept of a single, single-valued potential. One of them must be recognized as absolutely classical (Polanyi and Wigner); it explains the decomposition of molecules by the interference of their own vibrations, producing strong fluctuations of energy in the various degrees of freedom. Decomposition is the result of the transfer of a sufficient amount of energy to one unit degree of freedom. The other model was first proposed for radioactive decay (Gamow), and then used by Born and Franck to explain heterogeneous catalysis. It is based on the fact that in quantum mechanics potential barriers can always be crossed, even if the energy is insufficient to pass over the barrier in the sense of classical mechanics. The time required for the passage of a potential barrier is comparatively long. Gas-kinetic collisions are too brief for a reaction which, as a consequence of this, must take place between molecules adsorbed by a surface.

We shall not discuss these concepts here in greater detail. At present they are still insufficiently developed for it to be possible to judge their significance in a detailed and definite way.

The statics of atomic and molecular forces, with the exception of a few special cases, indeed cannot give an exact description of the atom, such as is provided by quantum mechanics and, accordingly, by the statistical function of all the constituent elements. And first of all because it is not sufficiently known in all cases. But this is not the only reason, and it will probably cease to exist in the near future.

The chief obstacle hindering the application of exact statistical functions is their great complexity, which makes their application to the majority of atoms almost impossible. Consequently, before us arises the problem of replacing the exact

statistical description of atoms, containing many insignificant details, by another, into which as few constants as possible would enter, but which would fully determine the forces. Thus, in celestial mechanics one may dispense with a detailed study of the heavenly bodies and determine their motion by means of four constants.

B. Forces at Large Distances (Molecular Forces)

Let us consider the forces between two atoms situated at a large distance; it is easy to expand the energy of their interaction \(E\) in a series in powers of the quantity reciprocal to the distance, \(\frac{1}{R}\):

\[ E=\sum_k \frac{t_k}{R^k}; \]

and in this way determine the energy step by step. As is known, we have already proceeded in this way in classical mechanics. In an analogous manner the coefficients \(t_k\) successively contain products of the “mean” moments of the charge distribution: first the total charge, if the atoms are not electrically neutral (ions), then the dipole, quadrupole moments, etc. The moments are constants of atomic systems that are used here for their determination (like mass and moment of inertia in celestial mechanics), and which in the present case can be known only for statistical functions. Let us leave aside ions, whose charge is simply given—their description has long since been carried out successfully, and quantum mechanics has determined them from a different point of view (in particular the repulsive forces; see below). Let us consider rather neutral systems—we shall see that the above expansion in a series gives us very little, since we regard the series as an expression of forces, while the atoms are considered as rigid units.

Most monoatomic gases, the noble gases, vapors of alkali metals, Hg, etc., possess perfect spherical symmetry, as quantum mechanics shows with great precision; that is, the statistics of the state of all component particles possesses perfect spherical symmetry. But this means that the mean dipole and quadrupole moments and all higher moments are equal to zero.

Consequently, a high degree of approximation was useful, and it turned out that, on average, all configurations of electrons in atoms possessing spherical symmetry are formed equally often; nevertheless, these discrete particles may at every instant be arranged without obeying spherical symmetry, and continuously form a certain dipole moment. In reality it does not choose any direction in space—

...space, but we nevertheless can ascribe to it a certain amplitude and period. A monatomic gas, therefore, does not possess an average dipole moment, but it does possess a periodic dipole moment, even a whole series of such moments of various intensities and frequencies. The mean value of these dipoles is zero, but the mean value of their squares is not zero.

Thus, an atom must be characterized by its periodic moments—dipole, quadrupole, etc.—with the corresponding frequencies, not counting constant moments. The periodic dipole moments are well known to us from the optical properties of the atom. They manifest themselves in the intensity of the emitted or absorbed radiation of a given frequency and as the “oscillator strength” in the values of the quantity \(f\) in the dispersion formula. Consequently, we can determine them by optical means and do not need an exact statistical description of the atom in all its details. In many cases, instead of these numerous dipole moments, one can make do with a single constant—the polarizability—or, rather, with two constants, namely also the characteristic frequency corresponding to this polarizability.

If one considers these periodic dipoles of the atom, it becomes evident that in another atom they cause periodic disturbances of the configuration, i.e., disturbances whose mean value must not be considered, but rather the mean value of their squares. It is essential that this disturbance of the configuration is, so to speak, in phase with the disturbing atom, and moreover that it contributes to a decrease in the energy of the whole system when the atoms are in the normal state. This means that when two atoms approach one another from infinity, energy is released, which is usually regarded as a characteristic of the attractive force. And indeed, it can be shown (Wang, London, Eisenchitz) that the potential of these attractive forces is proportional to the sixth power of \(\frac{1}{R}\). This quantity can be determined if the series of dipoles of the atom is known, i.e., its dispersion formula.

Moreover, with the same degree of approximation with which this force is determined, it can be shown that it is additive, i.e., that it acts between two systems independently of whether they are under the influence of a third one or not. It obviously has nothing in common with chemical forces, which have been exhausted by valences since the moment of their appearance. They were rather compared with molecular forces, and it proved possible theoretically to justify the simple notions about these forces that had existed since the time of van der Waals—in particular, the van der Waals pressure correction, the heat of evaporation (Trouton’s rule), and the heat of adsorption. Since these phenomena were

available to calculation, a very satisfactory agreement with measurements was obtained.

Similar considerations apply equally to the forces between polyatomic molecules. The possibility of the existence of a permanent dipole moment naturally plays a large role here. It has, as has been shown, no such definite significance for the action of the forces as it was formerly considered necessary to ascribe to it, when the short-time perturbations of the internal structure of molecules were not taken into account. For molecules consisting of two or more atoms, one must distinguish three parts of the interaction, differing considerably from molecular forces.

1. The interaction of “permanent” moments (the orientational effect of Keesom); it depends on the temperature and at low temperatures disappears owing to the rotation of the molecules.

2. The above-mentioned interaction of rapidly periodically varying dipoles of the internal electronic motion; it is practically independent of temperature.

3. The interaction of permanent and periodic moments. The action of the latter can be determined by the static polarizability (Debye induction effect), since the permanent moments move comparatively slowly. It also does not depend on temperature, but is much smaller than the interactions indicated in 2°, even for strong dipoles (\(\mathrm{H_2O}\) or \(\mathrm{NH_3}\)).

Let us also note that the nature of the forces indicated in 2° is, of course, not specifically quantum-mechanical. Classical mechanics also knew such perturbations with small periods and the forces determined by them. Only two phenomena here belong to the domain of quantum mechanics, and thus it is important for the matter as a whole. First, that even the most stable state of the atom does not exclude the rapid motion of its constituent parts: the so-called zero-point motion; and, secondly, that whereas ordinarily for any periodic perturbations one may expect, under certain possible conditions (phase relations), attractions and repulsions in equal measure, the perturbations of the motion at zero point always have only attraction as their consequence. With the aid of classical mechanics one could have concluded that repelling layers (couches) should predominate, since they are passed through more slowly than the configurations of attraction, which are energetically lower.

The method set forth is not capable of contributing to the understanding of forces at small distances. Even such a crude phenomenon as the impermeability of atoms and molecules (the van der Waals correction to the volume, compressibility, etc.) cannot be understood by means of the above-indicated method of expansion in powers of \(1/R\). And yet it is clear that at sufficiently small distances two atoms must always repel each other, owing to the predominance

ultimately, the influence of the positive charges of the nuclei. We can even account for why it is impossible to understand the coming into play of repulsive forces when using the above-mentioned method of approximation. The probability of the location of the electronic cloud surrounding the positive nucleus of an atom vanishes for large distances \(R\), following an exponential function of this quantity; likewise the unshielded positive charge of the nucleus and the potential of the repulsive forces decrease exponentially for large values of \(R\). But, as is known, it is impossible to expand

\[ e^{-r} \]

in powers of \(\frac{1}{r}\), or

\[ e^{-\frac{1}{s}} \]

in powers of \(s\), since

\[ e^{-\frac{1}{s}} \]

has a singular point at \(s=0\).

C. Forces at intermediate distances (valence forces)

It is clear that this method of subdivision into successive approximations does not apply to forces developing at the periphery of the atom. Therefore another method is used, which, above all, presupposes an entirely different decomposition into successive approximations. In the first approximation the interaction of two atoms is calculated, the charge distribution of which—or, at least, the distribution of “valence electrons”—is taken fully into account and, consequently, is not idealized in the form of dipoles, quadrupoles, and other moments. But at this degree of approximation it is not taken into account that, in reality, atoms situated at such close distances mutually excite one another. The correction for this excitation is made in the next approximation, which naturally entails also a correction to the interaction and thereby a correction to the perturbation of the atomic configuration, which is determined only in the third approximation, and so on.

1. The concept of rigid atoms.

If one is content with the first approximation, then atoms may be regarded as rigid structures, as unchanging units, so to speak—almost as they were depicted in the ancient conception of the atom.

One may expect that the principal properties of the forces at small distances will manifest themselves at this first stage of the method of approximation. Repulsive forces, in any case, do manifest themselves (Unsöld, Pauling, Brück); furthermore, as has been proved, chemical forces manifest themselves (Heitler and London), although, of course, very indistinctly. Nevertheless, for large distances we shall find the action of a force only in the second approximation, only after what was said above, since the first-order effects, based chiefly on the interpenetration of clouds of electric charg—

..., decrease with increasing \(R\) along an exponential curve. They have a smaller value than the forces of molecular attraction, which decrease proportionally to \(\frac{1}{R^6}\), and which cannot yet manifest themselves in first order, since they are based, as we know, on the mutual perturbation of atoms.

Up to the present, an exact calculation of the interaction of atoms has been carried out only for the simplest cases (\(\mathrm{H}+\mathrm{H}\), \(\mathrm{H}_2+\mathrm{H}\), \(\mathrm{He}+\mathrm{He}\), \(\mathrm{Li}+\mathrm{Li}\), \(\mathrm{Li}+\mathrm{H}\)). It is clear that in general it is extremely difficult, since the statistical distribution functions for the various atoms are unusually complicated. For such calculations it is important to find quantities of similar categories, essential for the dynamical conditions, such as dipole moments, etc., for forces acting at large distances, and which make it possible to avoid complicated statistical functions.

Indeed, an atomic constant has been found that is very characteristic of interatomic forces at small distances—namely, the resultant of the impulses of the proper rotation of the electrons. This constant affects the play of interatomic forces in an entirely special way. The Pauli principle states that in each of the cells from which a state is constructed there can be only one electron. This principle fundamentally determines the structure of the atomic shell, requiring that the electrons be grouped around the nucleus in order, beginning with the deeper states. The size of the atom and its compressibility (Frenkel, Kotari and Mahumdar) must, qualitatively reliably, and quantitatively only in general outline, be determined by the capacity of the cells, as the Pauli principle asserts. The cells from which the state is constructed have not only spatial extent, but also, in addition (alongside the extent in momentum space, which interests us less here), an extent in a fourth dimension, representing a fourth degree of freedom—the proper angular momentum, or “spin,” of the electron. This means that each cell is occupied by two electrons with different spins, i.e. the Pauli principle has a limitation.

Obviously, there is a great difference between atoms in which the electrons have already been grouped in this way, in pairs, and atoms still possessing vacant places for other electrons. In the latter case, for two such atoms there is the possibility of combining into a common cell those of their electrons which have not yet been paired; in the former case, on the contrary, the atoms are, to a high degree, isolated.

In the simplest cases, when one wished to determine the interaction of “hard” atoms, it was possible to calculate exactly their behavior and the role played here by the Pauli principle—and to show, for example, how two hydrogen atoms differ from two helium atoms. Thus, in fact, it was establi-

it has been found that two hydrogen atoms in the normal state can come together very closely with a decrease in energy, if their electrons enter a common cell. Such a state undoubtedly also occurs for atoms of helium and other elements, but it must be excluded by the Pauli principle, since it would contain at least three equivalent electrons. For helium there is admissible only a state that includes a considerable repulsive force, which in the second approximation at large distances turns into a very weak van der Waals attraction.

All these assumptions compelled one to regard as valence electrons those electrons of the atom which have not yet saturated their spins pairwise, and to interpret the case of the entry into a common cell of two electrons coming from different atoms as the saturation of valences (London). In this way an explanation was indeed reached of the basic phenomena of the chemistry of exact valences, at least in broad outline; the reproduction of valence values, and an initial understanding of the origin of the energy of chemical affinity, which for hydrogen could be determined numerically with sufficient accuracy.

If, proceeding from this—so far only qualitative—determination of the possibilities of choice furnished by the Pauli principle, one passes to the consideration of energy conditions, then in the characterization of the interaction of “hard” atoms, alongside their own angular momentum, there appears a new constant—an energy which is liberated when atoms completely neutralize the angular momenta of their electrons. This energy (it has not very happily been called the “exchange integral”) naturally depends on the distance between the atoms and, indeed, for large distances decreases along an exponential curve. Moreover, it does not depend on the nature of the atom. It is determined by the integration of the distribution functions of two atoms, and up to now it has been calculated only for the simplest cases. But sometimes it can be determined from band spectra and, in any case, regarded as a function of distance, so to speak, qualitatively known and partly indeterminate.

Now one must take up the complex molecule—to find out whether the energy of its bond is formed from the various bonds of the elementary constituents of the molecule, and in what way. If one neglects certain (insignificant) details, this problem appears quite definite, but very complicated; it has recently been solved in its full generality by Weyl, Heitler, and Rumer. Earlier it was investigated in many cases by elementary methods, in particular in the cases most essential physically.

In its relative complexity this problem is due to the fact that the total energy of the molecule, although it is found with the aid of

of approximations from the energies of various bonds, but is not their additive resultant, as is the case with most of the elementary functions known to us. These different constituent elements rather obey a law of a peculiar algebraic transformation. It is not surprising that something of this kind should follow from quantum mechanics, if it is to explain what corresponds to the “valences” of chemical forces. The mechanism of the transformation just mentioned may in fact explain the relative isolation of an atom with saturated bonds, in comparison with other atoms and molecules. This mechanism indeed shows that the stresses (efforts) between atoms within a molecule are different from the stresses between atoms of different molecules. It shows, in particular, that for the “activation” of chemical-valence forces bound in the simplest way, a quite definite work must be expended, and it gives an estimate of the character and magnitude of the “activation energy” necessary for this phenomenon. Quantum mechanics thus draws a significant distinction between the forces of valence and the additive molecular forces discussed above.

It is noteworthy that the mechanism of saturation and activation of valence forces could already have been determined theoretically by means of potentials, and that, in principle, at least, it is not necessary to picture the isolation of a bond as a quantum jump in the electronic structure of the interacting atoms. Nevertheless, the conditions will in many cases be considerably more complicated.

2. Directed and induced valences. The conception of the atom as an unchanging structure, and its exclusive characterization by means of those components of displacement which did not exist before saturation, prove very useful for an abstract extraction of the laws of chemical valence, which are in fact the subject at issue. But, of course, in this way only the first step is taken toward the theoretical mastery (maîtrise) of the whole variety of chemical phenomena.

The contemporary development of these questions leads to a deeper examination of them, chiefly in two directions: toward a detailed determination of the solid atom by other characteristics, and, on the other hand, toward the consideration of internal changes in the atom that occur as a result of the strong action of the bond.

The angular momentum of the translational motion of the electrons (mouvement du translation) and the total angular momentum that is the resultant of this momentum of translation and the spin momentum enter into consideration, first of all, for a more detailed determination of the atom, as other important characteristics of it. For some time it was assumed that these momenta should be regarded as generalizations of the concept of valence. But, apparently, such an interpretation corresponds little to the present

with the position of things. These valences, the so-called L-valences, can be determined only for isolated atoms; for groups of atoms, radicals, etc., these quantities lose all their meaning. Therefore one cannot speak only of the saturation of these valences. In principle, it would make no sense to extend further the empirical concept of valence now that it is known that it is obvious only under a very crude, though highly characteristic, simplification of real conditions. With such further development nothing would remain of the concept of valence, and it would be useless to apply it. If valence is regarded as an abbreviated notation for the chemist, whose practical utility will not be exhausted for a long time, then it is not the task of quantum mechanics to seek a better magic formula, which it cannot provide. Rather, it must investigate, in a general form, the energetic and, moreover, dynamic conditions for various cases of atoms for which the simple scheme of valences is inapplicable, or for cases lying at the boundary of its applicability.

In particular, the study of the possibilities characterized by rotational angular momentum has shown that the conditions are much more complex, that steric and other restrictions hinder the saturation of valences, and that, for a bond to occur, the symmetry properties of the bond under consideration are essential.

The most important facts that are subject to discussion when considering angular momentum transfer are: the phenomenon of anisotropy in the space of atoms excited by such angular momentum, and the question of the existence of special directions of manifestation of valences. This question, considered by Pauling and Slater, cannot be properly resolved if one adheres to the idea of “rigid” atoms. The concept of rigid atoms is justified only so long as the interaction energy is small in comparison with the difference of the energies of the states under consideration of the atoms and of their nearest neighboring states. Thus, for example, this is the case with hydrogen atoms in the normal state, since here the bond energy reaches approximately 4 V, while the excitation energy is 10 V. If, however, the interaction energy becomes comparable with the excitation energy of the unperturbed atoms, or even exceeds it, then one must reckon with very strong displacements in the internal structure of the atom, and it may be approximately characterized by the fact that the higher states of the atoms, possessing a more or less strong angular momentum and participating in the interaction, are those states whose energy only slightly exceeds the interaction energy. This can occur only in the case when many states of the atoms are very close to the normal one, and such are often the states differing in rotational angular momentum. Thus, these states are, as it were, mixed

thanks to perturbations; this mixing occurs only when the atoms are brought considerably close together, since the interaction at large distances can be arbitrarily small.

The following case is more characteristic and important for the chemist; whereas a charge distribution possessing central symmetry is, in principle, characteristic of solid atoms, these superposed states have absolutely no need of central symmetry, but the charge distributions are subject to elongation in a definite direction. It is clear that these distinguished directions preferentially enter into consideration in connection with other atoms. They should in no case be regarded as immovably fixed in the atom; they are only induced in the presence of other atoms, as was shown above. If, for example, another atom is already bound to the central atom, then this latter, under certain conditions, induces with respect to the central atom a conical zone capable of establishing a bond. Thus Pauling and Slater were able to arrive at an exact explanation of the greater part of the facts relating to directed valencies, and to illustrate it with simple and expressive models (the tetrahedron CH₄, the angle H₂O, etc.; C=C, cis- and trans-isomerism). The superposition of such forced states under certain conditions may manifest itself rather in the appearance of new valencies than in the orientation of already existing valencies. The most essential examples for this line of ideas are trivalent carbon and divalent alkaline-earth substances.

It is obvious that already in application to these questions the conception which confines itself to the picture of solid atoms loses its force. It is not quite correct to regard these deformations as equal to perturbations of the internal structure of the atom which manifest themselves in molecular forces. There the matter concerned periodic perturbations imposed on the internal motion of the atoms; the charge distribution, on the average, changed minimally and, in any case, insignificantly for dynamical effects. Here, on the contrary, the matter concerns a considerable displacement of the average charge distribution, considered as one whole; not phase relations between atomic perturbations, but a continuous change of configuration, which can be defined for the atoms separately.

3. Principles of construction and structure of molecules. Finally, one can take one more step, and completely abandon the direct connection with the atom in the process of its construction, regarding the skeleton (charpente) of the nuclei of the molecule as given, and introducing electrons one after another into its potential field, just as Bohr showed in his principle of the construction of atoms. First of all, one must form an idea of the states of a single electron in the field of stronger nuclei, and then occupy these

states successively by electrons, regarding their interaction as a secondary perturbation.

Obviously, this point of view, which was followed chiefly by Hund, Herzberg, Mulliken, and Lennard-Jones, will be correct only when we are less interested in the manifestation of atomic forces than in the final state of the molecules. In these molecules there often occur such strong perturbations of the atoms that it is very difficult to grasp the reality with the aid of approximations applied to individual atoms. But the possibility of approaching the question from this side as well is, to a lesser extent, an excellent check. The determination required here of the various positions of the electrons in the skeleton of the nuclei is a very difficult problem. Up to now it has been solved for the most part by means of a kind of interpolation, attempting to represent the actual state of the molecules (especially diatomic ones) as intermediate between two limiting cases: the state of united nuclei and that of nuclei infinitely far apart, the state of both being known. Such a treatment was, above all, applied to diatomic molecules comparatively symmetrical with equal or nearly equal nuclear charges. In particular, this applies to those molecules which have the same number of electrons (such as $\mathrm{CO}$, $\mathrm{N}_2$, $\mathrm{NO}^+$ or $\mathrm{CN}$, $\mathrm{BO}$, $\mathrm{CO}^+$, $\mathrm{N}_2^+$); their great similarity could not, of course, have been so directly established by methods whose point of departure is separated atoms.

An important aim of this work is to bring clarity into the question by means of simple characteristics. It would be desirable, above all, to set against the concept of “valence” in a definition proceeding from isolated atoms an interpretation of the “valence stroke” in the finished molecule. But these attempts have not led to so simple a result as could be achieved with the aid of the concept of valence, and mainly because localization of a bond is not in principle possible in all cases, and in fact the situation is considerably more complicated. In simple cases the “bonding” electron is characterized by the fact that its statistical distribution approximately possesses certain symmetry properties which entail a preferred position between the atoms; in this way the atoms are electrostatically bound. On the other hand, “antibonding” electrons avoid such a position between the atoms: the difference between the number of bonding and antibonding electrons is interpreted as the “multiplicity of the bonds.”

Despite numerous successes, this conception suffers from its excessive simplicity and from the fact that it strives to obtain an integral definition where in reality nothing of the kind exists. Already in considering solid atoms, the described construction mechanism shows that in fact there can be no question of a separation of valences

between the different bonds in integral fractions. One cannot expect that, upon closer examination, integer ratios will again prove to appear, and this is all the less probable since precisely the phenomena of activation and saturation of valences are represented by this mechanism as a continual transfer of the bond. The question of the localization of double bonds has been studied more precisely (Lückel), and specifically for benzene—the result was negative. Thus only the possibilities of different atoms or radicals, taken successively, are expressed by whole numbers. When, moreover, the chemist denotes by strokes of bonds the local integer division of valences, this is needed above all so that, by means of a simple symbolism, one may easily calculate the distribution of the different valences among the different atoms; he knows that these symbols must not be taken literally in space. Quantum mechanics has therefore followed still another direction: it has led to a general classification of the states of the electrons of a diatomic molecule by means of invariant characteristics. Quantum mechanics has investigated them one after another, from the point of view of ascertaining how their energy content changes as a function of the distance from the nucleus. Although it has not thereby been possible to arrive at general and simple formulations, this has created something far more valuable: definite views of the actual state of things, the final conception of which will be obtained only after the solution of particular problems.