CHEMICAL BONDING AS A DYNAMIC PROBLEM1

Max Born.

Introduction. The question of the physical nature of chemical forces, in the present state of science, must be regarded as a problem of the theory of quanta. In recent years physicists, headed by Niels Bohr, have arrived at definite conceptions of the structure of atoms from nuclei and electrons. The most important auxiliary device in this work has been the interpretation of the line spectra of the optical and X-ray regions in connection with direct measurements of the steps of atomic energy by the electron-impact method of Franck and Hertz. Other physical phenomena, for example magnetic ones, have played only a secondary role. In this way it has been possible to reduce to physical, quantum conceptions the most essential features of the periodic system of the elements, in which chemistry has concentrated its extensive information about atoms. The next step must consist in the interpretation of the formation of molecules and of their structure. The purpose of my report is to communicate the results and views in this field.

1. The Quantum Significance of Chemical Energy

Up to now it has not been possible, by a purely theoretical method, to calculate the stationary states and energy levels for any atoms except hydrogen. The reason is that the quantum laws governing the bond of several electrons are unknown to us. Hence it is clear that an exact, deductive treatment of molecules consisting of several nuclei and electrons is impossible. Here one must resort to the aid of empirical data and be satisfied with approximate solutions.

In molecules the starting point for a rational splitting of the dynamical problem into successive approximations is self-evident: it lies in the smallness of the electronic mass in comparison with the mass of the nuclei. For the hydrogen atom this ratio is approximately \(\frac{1}{1800}\),

for other atoms—even less. On the other hand, the forces of interaction between electrons and nuclei are of one order of magnitude, and the velocities of the nuclei are very small in comparison with electronic velocities. Therefore, in a first approximation, the motions of the electrons may be calculated as though the nuclei were at rest. Hence the following device for determining the energy is obtained:

The total kinetic energy of the molecule is additively decomposed into two parts: the kinetic energy of the nuclei \(T_K\) and the energy of the electrons \(T_E\). The potential energy \(U\) cannot be decomposed in this way. The total energy

\[ W = T_K + T_E + U \]

can be expanded in a series of terms of various orders of magnitude, according to the occurrence in them of the small electronic mass \(\mu\). Mathematical investigation shows that as the expansion parameter one must choose the quantity \(\lambda = \sqrt{\frac{\mu}{m}}\), where \(m\) is approximately the mass of the H atom. The expression for the energy in the first approximation can be obtained by simply assuming the nuclei fixed in some position and neglecting their kinetic energy \(T_K\); then

\[ V = T_E + U. \]

This quantity depends on the constant chosen coordinates of the nuclei and on the variable coordinates of the electrons.

Next, one must apply quantum theory to the electronic motions. In reality these calculations cannot be carried out not only because of their complexity, but above all because the true quantum laws are not known to us. However, the form of the result can be indicated. Namely, \(V\) will be a function of certain quantum numbers \(n_1, n_2,\ldots\). There remains, further, a dependence on the arbitrarily chosen positions of the nuclei. We indicate this dependence by introducing as arguments the distances between the nuclei \(r_{12}, r_{13}, r_{23},\ldots\):

\[ V(n_1, n_2,\ldots; r_{12},\ldots, r_{13},\ldots). \]

The next degree of approximation consists in our again adding the kinetic energy of the nuclei. The total energy of the molecule will be:

\[ W = T_K + V(n_1, n_2,\ldots; r_{12}, r_{13},\ldots). \]

Mathematical analysis [1] shows that this expression is correct if terms of order \(\lambda^2 = \frac{\mu}{m}\) are neglected (i.e., if an error of order of magnitude \(3 \cdot 10^{-7}\) is allowed).

The expression obtained for the energy shows that, for a given electronic configuration, i.e. for given quantum numbers \(n_1, n_2,\ldots\), the molecule behaves as though there were between the nuclei a potential energy depending only on the mutual distances between the nuclei. In order for the molecule to be able to exist at all, the nuclei must be in an equilibrium configuration, which corresponds to a minimum of \(V\). All motions of the nuclei may in this case be regarded as the result of superposed rotations or gyroscopic motions of the nuclear equilibrium configurations (which may be imagined as rigid) and vibrations of the nuclei relative to one another. In doing so one must also take into account the resultant rotational moment of the electronic motions \([2]\).

Each molecule exists in various modifications, differing in the type of electronic orbits and in the values of the corresponding quantum numbers \(n_1, n_2,\ldots\). The most stable is the modification with the smallest quantum numbers, for example \(n_1=1, n_2=1,\ldots\); without external action such a modification can exist indefinitely long and corresponds to the normal state of the molecule. Here we shall have to deal only with this normal state; only it is of significance in ordinary chemistry. The properties of excited states with higher quantum numbers have also recently been thoroughly studied (especially by Franck and his school), but here we leave this “pathological chemistry” without consideration.

Putting in \(V\) \(n_1=1, n_2=1,\ldots\), we obtain the energy of the normal molecule:

\[ W = T_K + V(r_{12}, r_{13}, r_{23}\ldots). \]

The chemical heat of formation at absolute zero corresponds to the value \(V_0\) of the function \(V\) at equilibrium. Further, for chemistry the equilibrium distances \(r_{12}^{0}, r_{13}^{0}\) and the moments of inertia derived from them, which enter into the chemical constants, are important; also essential are the vibration numbers (determined by the second derivative of \(V\)), on which the chemical constant likewise depends and, moreover, the specific heat at high temperatures.

Thus for chemistry it is not important to know the general course of \(V\) as a function of the internuclear distances. It is sufficient to know a limited number of constants obtained from \(V\) (the equilibrium distances \(r_{ik}^{0}\), the values of \(V\) and of its second derivatives at equilibrium).

Some of these constants can be determined directly by purely physical measurements, using the band spectrum. From this spectrum one can obtain the moments of inertia (and hence, in many cases, the quantities \(r_{ik}^{0}\)), the vibration numbers, and often even still higher derivatives of \(V\) at equilibrium. But the value of the energy \(V_0\), the heat of formation, cannot as yet be determined directly in this way.

The aim of the dynamical theory of the molecule consists in the theoretical calculation of the function \(V\) and of the constants derived from it, above all the heat of formation \(V_0\). Before taking this up in greater detail, we shall briefly set forth how, from the point of view of this theory, the doctrine of chemical equilibrium and the rate of reaction is represented.

2. Chemical equilibrium and rate of reaction.

The classical period of the doctrine of chemical equilibrium, the period of the application of thermodynamic methods, received a certain completion in Nernst’s theorem. The calculation of equilibria and affinities could be reduced to quantities determined by purely physical measurements.

Atomic physics poses the broader task of obtaining these constants from dynamically elementary processes. At the same time it demands a statistical justification of thermodynamic formulae with allowance for quantum laws.

This problem may now also be regarded as solved. Quantum statistics, in the form either adjoining Gibbsian distribution functions or based on the new method of Darwin and Fowler [3], operating only with mean quantities, gives reliably and simply any formula needed for the solution of a statistical problem. For this the following data are necessary:

1. The values of the energies \(W_1, W_2\) of the stationary states of all the particles participating in the system. 2. The statistical weights or a priori probabilities \(p_1, p_2, \ldots\) of these states. 3. The proposition, first introduced by Sakur (O. Sakur) and Tetrode (Tetrode), which consists in the following: in the phase space of translational motion (not restricted by any quantum conditions), to every volume of magnitude \(h^3\) (\(h\) is Planck’s constant) there corresponds weight 1. In this case the most general formulation of Nernst’s theorem is contained in the following proposition: quantum states with the smallest quantum numbers in condensed systems possess equal statistical weight.

These data and propositions give not only the theory of equilibrium of ordinary atoms and molecules, but are also suitable for the case of detachment of electrons, ionization. I shall recall only the successful application of these considerations for explaining stellar spectra in the works of Eggert (J. Eggert), Saha (Megh Nad Saha), Fowler (Fowler), and others.

Conversely, chemical kinetics is still far from a satisfactory theory. I shall use this occasion here, however, to point out one matter important for understanding the possibility of the union of two atoms.

In the formation of a compound accompanied by the release of energy, we are dealing with the elementary act of attraction of two ato-

cules, which fly toward one another and remain together; in this case the energy liberated must be removed from the reacting pair of atoms in one way or another. Two ways of accomplishing this have been considered: 1. The release of energy by means of radiation. 2. The release of energy in a “triple collision,” i.e. the transfer of the liberated energy to a third particle (atom or molecule), not participating in the reaction but lying within the sphere of action of the two reacting particles.

The release of energy in the form of radiation can occur only in certain cases, which are not difficult to indicate; this may happen in the combination of two charged particles—a combination accompanied by the appearance of an electric moment (capture of an electron, combination of two ions). We shall not consider such cases, restricting ourselves to the formation of nonpolar molecules from neutral atoms, where radiation is impossible.

Calculations by Polanyi and Herzfeld showed that the frequency of the processes of formation agrees with the assumption of “triple collisions.” In order to justify this theoretically, the usual argument was as follows: a molecule containing energy greater than its work of dissociation immediately decomposes again; since, in the collision of reacting atoms, there must undoubtedly exist energy at least equal to the work of dissociation, these two atoms could not remain together unless the excess energy were removed by a third particle.

Franck and I [5] observed that the premise of such a conclusion is erroneous. A molecule can retain an energy considerably greater than its work of dissociation. Above all, such excess energy may be accumulated by means of an electronic transition: this occurs, for example, upon the absorption of light by a hydrogen molecule, which, as is known, absorbs ultraviolet light (emitting it again in the form of a band spectrum) with an energy many times exceeding the work of dissociation. Further, it can be shown that the energy of motion of the nuclei may be greater than the work of dissociation. Estimating, in a minimal way, the forces in a rotating molecule which is not undergoing vibrations, we find that the rotational energy may exceed the work of dissociation twofold and threefold before the centrifugal forces tear the molecule apart. All this makes the usual argument in favor of the necessity of “triple collisions” unfounded. Franck and I came, however, to the conclusion that collisions of this type are nevertheless necessary, but on another basis. This basis lies in the difference between translational and internal motions with respect to quantum restrictions. Translational motions may have any energy; internal motions are determined by quantum conditions and have discrete energy levels. When two atoms collide, striving to combine, then

their center of gravity, before and after the collision, moves rectilinearly; the energy of the free atoms relative to the center of gravity has, before the collision, some arbitrary value, but after combination the energy can have only one of those discrete values that correspond to the stationary states of the molecule (the vibrations and rotations of the nuclei are also quantized). The probability that these two energy values will coincide is infinitely small; consequently there remains an excess of energy, which must be given up to some third particle in order that a new molecule may arise at all.

We see that here quantum theory essentially determines the mechanism of chemical kinetics.

3. An approximate method for calculating the energy of formation.

The only case for which there exists a serious attempt at an actual calculation of the function \(V\) is the positive ion of the hydrogen molecule, \(H_2^+\). In this problem there are two nuclei and one electron; if, in a first approximation, the nuclei are regarded as at rest, then we have the already solved problem of Jacobi concerning two centers. The motion of the electron is multiply periodic, and it is permissible to apply to it the quantum rules that have proved valid for the hydrogen atom. Pauli (W. Pauli junior) and Niessen (Niessen) [6], independently of one another, carried out these calculations, hoping in this way to find the correct energy levels for the ion \(H_2^+\); these hopes, however, were not justified: the calculated quantities do not agree with the latest measurements of the ionization and resonance potentials of Smith [7]. Such a negative result is not easy to interpret. In any case we arrive at the conclusion that our quantum rules turn out to be unsuitable for bound systems consisting of several nuclei or several electrons. Thus the hope of advancing further by means of an exact calculation is, for the present, very small. The meaning of the difficulties already appears in the second-simplest atom, helium. In their motion the electrons of helium form alternating fields of high frequency; we know, however, that atoms react to the fields of light waves, whose frequency is of the same order of magnitude, in a manner that is not at all mechanical. Therefore one cannot expect the interactions of electrons in one and the same atom to occur according to the laws of classical mechanics. The attempt to change mechanics in the sense of a consistent adaptation to the fundamental ideas of the quantum theory is still at a very primitive stage. Among such attempts I include Heisenberg’s formal method [8], applied to the interpretation of the anomalous Zeeman phenomenon, and the systematic modification of the classical formulas of perturbation theory recently proposed by me [9] (the transformation of differential equations into difference—

…nesses). The direction of all such attempts is clear: the discontinuous physics of the atom, which rests on whole numbers, must be developed seriously and systematically. The solution of such a problem is still far off.

For the time being we must be content with approximate methods, based not on purely theoretical principles but making use of empirical facts. In the establishment of such a point of view I see the merit of Kossel’s views. Kossel [10] noted that there is a limiting case accessible to dynamical treatment without essential application of special quantum ideas. This is the case of extreme polar compounds, arising as the result of the superposition of two ions previously formed by electron exchange. In such a compound of ready-made ions, Coulomb forces of attraction act chiefly. Thus here the energy of combination is determined in its main features by two constants of the ion—its charge and its volume. The progress of such a view, in comparison with the attempt of Berzelius to regard chemical forces as electrostatic, may be noted in two respects. First, a restriction has been made to the case of strictly polar compounds, precisely defined with the aid of the periodic system; second, the concept of ionic volume has been introduced, about which in the time of Berzelius, 100 years ago, as little was known as about the periodic system. This must be noted because Kossel’s theory is often misunderstood, and its significance is therefore underestimated. The representation of ions as charged rigid spheres is of course only a limiting case, which can serve as the starting point for rough approximations; from Kossel’s papers, however, it is evident that he is aware of this circumstance. It is not within my competence as a physicist to judge whether Kossel, with his simplification, has gone too far in applying the theory to complex compounds.

Expressing the approximate method proposed by Kossel by a mathematical formula, one may expand the quasi-potential energy \(V\) of two ions, one relative to the other, in a power series in inverse powers of the distance \(r\) between the nuclei. This series:

\[ V=\frac{e_1 e_2}{r}+\frac{a_2}{r^2}+\frac{a_3}{r^3}+\cdots \]

begins with the term of Coulomb attraction of the full ionic charges \(e_1\) and \(e_2\); the subsequent terms represent forces arising from the interaction of the electrons; these forces may be attractive and repulsive. In the case of rigid spheres one must take only the first term of the series; instead of the remaining terms, the restriction is imposed that \(r\) cannot be smaller than the sum of the radii of the spheres.

Instead, besides the first Coulomb term, one may introduce still another single repulsive term with a high degree of \(r^{-1}\), i.e., write:

\[ V=\frac{e_1 e_2}{r}+\frac{a_n}{r^n}. \]

4. Electrostatic Theory of the Crystal Lattice.

For a quantitative treatment of this assumption it is necessary to make sure that, in limiting ourselves to the two terms just written, we are not making a large error. It is precisely in this that the inclusion of the crystal lattice in the course of our reasoning is rooted. In an ionic lattice in which, as, for example, in rock salt, each ion is symmetrically surrounded by other ions, the deformations of the ions can consist only in all-round contractions or expansions. Conversely, in binary molecules, for example, in NaCl pairs, unilaterally acting forces may produce strong one-sided distortions. For a crystalline ionic lattice it follows already from simple symmetry considerations that, besides the Coulomb term, there must exist a single non-vanishing term with a high exponent \(n\). For the time being we shall be satisfied with this term.

The consideration of ionic lattices has also made it possible to achieve a quantitative calculation of the energy. In doing so, it was necessary to overcome certain mathematical difficulties connected with the poor convergence of the series representing the Coulomb energy of the lattice ions. Madelung and Ewald [11] solved this problem so perfectly that at present the electrical energy for any ionic lattice can be calculated with relatively little labor.

For the exact determination of the repulsive term, i.e. of the constant \(a_n\) and the exponent \(n\), one may use reliable empirical data, namely the röntgenometrically determined lattice constant and the compressibility. It turned out that the value of \(n\) for all alkali halides is close to 9, which is in agreement with the high (cubic) symmetry of the ions and of the lattice.

As a result of such a calculation one obtains a quantity called the “lattice energy.” This energy is equal to the work necessary to destroy the lattice into ions infinitely far removed from one another. The lattice energy is not yet identical with the measured heat of formation. The latter corresponds to the work required to combine neutral atoms. It is therefore necessary also to know the work required for transforming atoms into ions. This work, in turn, consists of the work of ionization of the cations and the energy liberated when electrons are captured by the anions—the so-called electron affinity (Elektronenaffinität) of the anions. Both of these quantities are atomic constants,

in determining them one cannot dispense with quantum theory. However, these constants do not belong to the chapter on the formation of molecules, but to the doctrine of atomic structure. We see, therefore, a substantial advantage in restricting ourselves only to polar compounds; here it is possible to divide the energy of formation into two parts [12]: one part consists of atomic constants (ionization potential, electron affinity), while the second part (lattice energy) can be calculated in an extremely simple way by means of electrostatics, with the quantum properties of the ions being expressed by a simple repulsive force with constants that can be determined empirically. The relation of these partial energies to the heat of formation is best clarified by considering a simple cyclic process.

Is atomic theory in fact capable of giving the atomic constants required here? Unfortunately, this is possible only for the work of ionization of cations. For hydrogen it is calculated on the basis of Bohr’s model; in other cases it is measured by electrical or optical methods. For determining the electron affinity of anions we have no direct methods of determination. The spectroscopic method proposed by Franck [13], according to recent investigations by Oldenberg [14], does not lead to the goal. To test the theory, however, it is necessary somehow, using other measurements, to exclude the electron affinity. This is possible on the basis of the investigation of Franck and Moeller and of the more exact work of Knipping [15], who measured, by the method of electron impacts, the work of decomposition of hydrogen-halide compounds into atomic ions. Of all the empirical data needed for calculating lattice energies, only the heat of dissociation of hydrogen is unreliable. If for this heat one takes 80 large calories, then the agreement between the electrostatic lattice energy and that calculated from measurements proves to be so good that, conversely, it is permissible to infer from it the correctness of the chosen figure (cf. Table I).

Thus the electrostatic calculation is apparently suitable. Unfortunately, for more complicated lattices it is laborious and is limited only to those cases in which the ions do not undergo unilateral deformations (see Table I, p. 347).

The usefulness of the concept of lattice energy for clarifying chemical relationships is not limited only to the electrostatic method of calculation; the lattice energy can be derived with the aid of cyclic processes on the basis of measured quantities and then investigated to see whether any regularities are revealed when comparing different classes of compounds. Grimm [16] followed this path with great success. It indeed turned out that the lattice energy reveals considerably simpler and clearer regularities than the measured heat of formation, which is composed of various parts

TABLE I.

Lattice energy \(U\) in kg-calories.

(ionization potential, electron affinity, heat of sublimation, lattice energy).

Grimm and Herzfeld [17] answered in detail the following question, which they themselves posed: how can the usual theory of valences be interpreted on the basis of the dynamical interpretation, and does not the latter lead further, explaining exceptions to the strict rules of valence? They used the method of considering, in a series with actually existing compounds, “virtual” compounds as well, calculating theoretically (approximately) the lattice energy for them. It turns out that the heat of formation, regarded as a measure of stability, exhibits a sharp maximum for actually existing compounds. Thus, for example, for monovalent chlorine compounds the following figures are obtained (Table II):

TABLE II.

Heats of formation of monovalent chlorides.

From the table it is evident that the chlorine compounds of the noble gases are impossible, the alkali chlorine compounds exhibit high stability, the compounds ClCa and ClSr are possible but possess little stability; according to the theory of invariant valences these compounds cannot exist, but in reality the compound CaCl has been observed.

Unfortunately, for the time being, with such an energetic method one can investigate only those compounds in which larger quantities of energy are transformed; only in this case is the accuracy of our theoretical determinations of energy sufficient. But the idea of a dynamical interpretation of chemical affinity, realized here for the first time, will undoubtedly still play a major role.

5. Deformation of Ions.

The investigations indicated were somewhat encouraging with respect to the correctness of the method; after this it was possible to try to discard Rossel’s initial simplification, taking into account the mutual deformation of the electronic orbits in ions. Such deformation must already exist in sodium-chloride vapor molecules, as we pointed out above.

The great merit of K. Fajans [^18] consists in pointing out the importance of such deformation of ions. From this point of view he has treated a large amount of chemical material, as he has just reported in his communication. I therefore have no need to set forth these interesting questions, which in many respects touch upon our topic. For clarity, however, I should like to note the difference between Fajans’s line of work and ours. Fajans directs attention mainly to the fact that one and the same ion differs somewhat in different compounds; as a criterion for this he uses the deformability (Deformierbarkeit) of the ion, measured by molecular refraction. Thus, for example, the Cl-ion in the compounds HCl and NaCl differs noticeably, as can be concluded from consideration of the refraction constants. On this Fajans bases a systematics of the mutual influences of ions, deriving varied and important chemical consequences.1

The direction of our reasoning leads to a somewhat different point of view. For a quantitative calculation of energies, the mutual deformation of ions at first represented, above all, a disturbing influence: it had to be feared everywhere, and it was necessary to seek cases where the deformation is insignificant and simple. Such cases were found in crystals of salts with high symmetry, where there could be only an all-sided deformation. The residual differences in the refraction of ions in passing from some crystals to others are a sign of this all-sided deformation, which somewhat changes the forces opposing the light field. For energy calculations such deformation has no significance. In those cases where, owing to insufficient symmetry, one has to reckon with one-sided deformations, it must be taken into account in the interaction of ions. Thus the point is not how much

great is the deformation existing in the finished compound and measured by changes in refraction, but what is the “deformability” of the isolated ion. This deformability is measured by one (or several) constants which, along with the charge and volume (or the repulsion exponent \(n\)), enter into the expansion of the energy function \(V\) in powers of \(r\).

The simplest type of deformation is produced by a homogeneous electric field \(E\). In this case an electric moment arises, proportional to the field, \(p=\alpha E\), and the energy is:

\[ W=\frac{\alpha}{2}E^2. \]

The constant \(\alpha\) is a measure of the polarizability (Polarisierbarkeit) of the atom or ion.

From the point of view of the old theory, which operated with quasi-elastically bound electrons (for example, in the theory of dispersion), this polarizability is self-evident. But quantum theory also leads to the same result. Here, however, there is an exception—the neutral hydrogen atom (and hydrogen-like ions). In this case the electric field forms an additional energy proportional to the field, arising because the Kepler ellipse of the outer electron, at rest in the absence of a field (if one neglects the correction of the theory of relativity), begins in the field to execute secular motions. In the spectrum this motion corresponds to the Stark effect in the hydrogen lines. In all other (unexcited) atoms the electrons of the atom move in a non-Coulomb central field, and therefore even in the absence of a field possess a strong motion of the perihelion. The additional energy formed by the external field will therefore not contain a term directly proportional to the field, since it drops out; the energy will have the form \(W=\dfrac{\alpha}{2}E^2\). Here, consequently, there is only a small quadratic Stark effect, similar to that which Ladenburg \([^{19}]\) observed in the sodium atom. The electric moment is obtained from the energy by differentiation; we have:

\[ P=\frac{dW}{dE}=\alpha E, \]

as was written above.

Unfortunately, in not a single case is it yet possible to calculate the constant \(\alpha\) from the atomic model; one has to be satisfied with empirical determinations. In this respect, a method for measuring refraction has long been known, refraction being proportional to \(\alpha\). For neutral atoms and molecules such a procedure leads to good, unambiguous results. In the case of the polarization of ions, optical measurements give only the sum of the values of \(\alpha\) of all the ions present, and the problem arises of deriving from this the \(\alpha\) for the individual ions. Since refraction is in fact additive, it is sufficient—

exactly, obviously, to make an absolute determination for one ion; thence, from measurements of the sum, the values of $\alpha$ for all ions will be found. Such measurements with solutions were carried out by Geideweiler [20] and Wasserstein [21]. Fajans and Joos [18], along with this, make use of refraction in solid salts. The assumption necessary for determining the absolute quantity consists in the fact that the values of the smallest cations, such as, for example, $\mathrm{H}^+$, $\mathrm{Li}^+$, may be neglected in comparison with the values $\alpha$ of the anions. Geisenberg and I [22], in addition, developed a completely independent method which, for simple metallic ions, directly gives the separate values of $\alpha$. In this method one uses the series spectra emitted when ions are neutralized by the capture of electrons. If the ion, which figures here with respect to the external captured electron as the atomic core, were hard, then the spectrum would almost exactly coincide with the hydrogen spectrum, since the electron passes far from the atomic core. The actual deviations of the measured spectrum from the hydrogen spectrum for such outer orbits give a measure of the polarizability, $\alpha$, of the core, i.e. of the ion. Geisenberg and I further found that these quantities $\alpha$ for ionic series of the same type of structure obey a simple law. If the atomic number $Z$ of the ions of such a series is decreased (for example, $\mathrm{O}^{--}$, $\mathrm{F}^{-}$, $\mathrm{Ne}$, $\mathrm{Na}^{+}$, $\mathrm{Mg}^{++}$, $\mathrm{Al}^{+++}$, $\mathrm{Si}^{++++}$), all by one and the same “screening number” $s$¹), and an expression is formed for the “effective” number of nuclear charges:

$$ Z_{\mathrm{eff}} = Z - s, $$

then $\alpha$ turns out to be proportional to $Z_{\mathrm{eff}}^{-3}$. Here $s$ is very close to the “screening number” used for representing X-ray

Fig. 1.

¹) Electrons situated near the atomic nucleus, as it were, “screen” the action of the positive charge on the outer electron. The “screening number” (Abschirmungszahl) shows by how many units one would fictitiously have to reduce the nuclear charge in order, while disregarding the presence of other electrons in the atom, to describe approximately the motion of the outer “hydrogen-like” electron.

Translator’s note.

therms. In essence, this conclusion is nothing other than the proposition of the old Clausius–Mossotti theory that \(\alpha\) has the dimension of a volume, since in Bohr’s theory the linear dimensions of the atom are proportional to \(Z_{\mathrm{eff}}^{-1}\). In Fig. 1 this regularity is represented graphically: \(Z_{\mathrm{eff}}\) and \(\alpha\) are plotted on a logarithmic scale, and therefore the relation between them is represented by straight lines.

The constant \(a\) also plays a role in other areas of atomic physics; I shall mention only the electrical interpretation of van der Waals cohesive forces in the works of Debye and Kazoma \([23]\).

6. Electrical Energy of Polar Molecules

For our theory of chemical bonds we must establish what influence polarizability has on the expansion of the quasi-potential energy \(V\) in a series in \(r^{-1}\). It is easy to see that polarizability leads to the presence of a term of order \(r^{-4}\). If it is added to those terms which are required by the theory of crystals, we obtain:

\[ V=\frac{e_1 e_2}{r}+\frac{a_4}{r^4}+\frac{a_n}{r^n}, \]

where

\[ a_4=-\frac{1}{2}\left(\alpha_1 e_2^2+\alpha_2 e_1^2\right). \]

This formula is directly applicable to the question of the energy of formation of molecules of vapors of binary salts. Their structure, as we have already noted above, is markedly determined by the polarizability of the ions owing to the one-sidedness of the interaction. The same quantity is evidently equal to the difference between the lattice energy and the heat of sublimation. The latter was measured by Wartenberg and his students \([24]\) for alkali-halide compounds. Therefore Heisenberg and I \([22]\) were able to test the theory directly on experimental data. The results are collected in Table III on p. 352.

The agreement is very good, and at the same time it gives a new independent proof of the correctness of the lattice energies for solid salts.

It is also possible to calculate many other properties of these salt molecules, for example the moment of inertia, the number of vibrations, etc.; however, the necessary observations for this are lacking.

Before proceeding to the exposition of further applications, I should like to dwell on one objection which Nernst raised against Kossel’s theory in the new edition of his widely used textbook. Nernst asserts that, according to this theory, for example in hydrogen chloride, in addition to HCl molecules, triatomic ions of the type \(\mathrm{H}^{+}\mathrm{Cl}^{-}\mathrm{H}^{+}\) and \(\mathrm{Cl}^{-}\mathrm{H}^{+}\mathrm{Cl}^{-}\) should also be found. Indeed, if one calculates the heat of the reaction:

\[ 3\mathrm{H}^{+}\mathrm{Cl}^{-}=\mathrm{H}^{+}\mathrm{Cl}^{-}\mathrm{H}^{+}+\mathrm{Cl}^{-}\mathrm{H}^{+}\mathrm{Cl}^{-} \]

TABLE III.

Energy of combination of salt vapors.

on the assumption that these ions are solid charged spheres, it is easy to see that the heat of the reaction is equal to zero; thus at equilibrium these ions should exist in large quantity, which in fact is certainly not the case. Of course, this objection is based chiefly on the assumption of the rigidity of the ions; if rigidity is abandoned and the deformability of the ions is introduced, something quite different is obtained. A calculation with point \(H^+\)-ions, appreciably penetrating into the anions, is too unreliable; therefore Heisenberg \([25]\) carried out a calculation for NaCl vapor; he found the heat of reaction to be approximately 50 kg-calories and, correspondingly, an insignificant degree of dissociation (of the order of magnitude \(10^{-8}\)). In the same work Heisenberg expressed the idea of the existence of electric moments in molecules of the type \(H_2O\) or \(CO_2\); these moments, according to Debye \([26]\), determine the temperature dependence of the dielectric constant; Zahn \([27]\) discovered them experimentally. They are usually explained by saying that the three atoms in the molecule form a triangle, and it cannot in any way be asserted that this supposition is erroneous—many details of the infra-red band spectra speak in its favor. However, it can be shown that under certain circumstances even rectilinear

molecules can possess a moment, since, as a result of deformation of the ions, the asymmetric position may prove more stable than the symmetric one. For water vapor, according to new calculations by Hund (F. Hund), the triangular form is unquestionably the most stable. In any case, these questions of molecular structure can be approached by our methods; with careful use of empirical data, especially absorption bands, very reliable results can be obtained.

Ions of such radicals as \(\mathrm{CO}_3\), \(\mathrm{NO}_3\), \(\mathrm{ClO}_3\), \(\mathrm{SO}_4\), etc., represent another field of application of the theory. The natural vibrations of these radicals were investigated by Schaefer and his students \([28]\). These natural vibrations are recognized by the fact that they appear almost unchanged in all compounds of a given ion, both in solutions and in solid crystals. In the case of crystals, moreover, it is often possible to determine the direction of the vibrations by using polarized light with a plane of vibration definitely oriented with respect to the structure of the crystal. An extensive theoretical study of these vibrations of molecules and crystals on the basis of symmetry relations was carried out, at my suggestion, by Brester \([29]\). On the basis of this work, for any given structure it is possible to predict the number and direction of each natural vibration. The possibility, moreover, of quantitative calculations is indicated further by Kornfeld’s work \([30]\) on the vibrations of the \(\mathrm{CO}_3\) ion; here it is assumed that such an ion has the form of an equilateral triangle with three \(\mathrm{O}^{--}\) ions at the corners and a \(\mathrm{C}^{++++}\) ion at the center. The dimensions of the triangle are known approximately from the X-ray photograph of calcite \((\mathrm{CaCO}_3)\). Further, from Schaefer’s measurements it is known that, of the three optically active natural vibrations of the \(\mathrm{CO}_3\) group, the vibrations with the smallest and the largest wavelength occur parallel to the plane of the triangle, while the vibrations with the intermediate wavelength occur perpendicular to this plane. This follows exactly from Kornfeld’s theory, with a rational choice of the polarizability of the O ions. The numerical agreement of the vibration frequencies is also not bad.

Similar calculations should be made for other radical ions; here, however, the empirical material is not so complete.

As a final area of application of our theory I shall mention the more subtle properties of crystals, in which the polarizability of the ions manifests itself. These include certain elastic constants, piezoelectricity, and so on. The most complete empirical material in this respect is available for zinc blende \((\mathrm{ZnS})\). Heckmann (Heckmann) is engaged in a theoretical treatment of this crystal, and, apparently, his results will provide valuable clarification regarding the applicability of our assumptions.

Conclusion

It is perfectly clear how this theory must develop further. Starting from strictly polar compounds, one will gradually pass to molecules with a less pronounced polar character. By introducing, on the one hand, deformability (polarizability) \(\alpha\), one will have gradually to pass to more complex cases of deformation. There are cases in which the latter can be found optically; for example: in strong electric fields the electric moment depends on the exciting field not only linearly but also quadratically (or, in other words, \(\alpha\) is not constant but is a function of the field). This is discovered, according to Pockels and Voigt \([31]\), in acentric crystals by double refraction proportional to the field. The constants of this phenomenon, for which only scant empirical material is available, must be introduced into the expression for the energy of formation.

One may hope that along this path it will be possible to go very far. But, of course, it will never be possible by this method to reach the extreme nonpolar compounds, the simplest of which is the hydrogen molecule. The general course of the function cannot be elucidated with the aid of a power series (in the present case, the expansion of the function \(V\) in powers of \(r^{-1}\)). Results for such molecules can be hoped for only from the further development of quantum theory.

Must one therefore, as Nernst apparently wishes, abandon the path now laid out, indicated by Kossel? It seems to me that such a restriction cannot be justified by anything. The resistance that our dynamical theory of polar molecules encounters from many chemists is understandable. In this theory there is no convenient image of fixed directed valences; but Werner’s doctrine of coordination and other experimental data have long since shaken the concept of fixed valences. Furthermore, even for simple compounds one must carry out very complicated calculations, making use of physical measurements, for example X-ray structure, infrared vibrations, and band spectra. But the chemical energy of formation is only one of the numerous equivalent constants of a molecule; as for the calculations, we theoretical physicists exist for this purpose. Apparently, in many people the negative attitude toward the course of our ideas is based on an unpleasant feeling of excessive simplicity and crudeness in the fundamental assumption of our theory. Calculations on the basis of electrostatic attractions, deformations, and the like, essentially speaking, could have been made even before the emergence of quantum atomic physics. To this one may reply that only atomic physics provided the necessary concepts and, above all, reliable numerical material. It was unthinkable to test our theory for the concept

ionization potential, without analysis of band spectra, etc. Therefore we are entitled to hope that every new step in atomic physics will also be useful for the doctrine of the structure of molecules.

LITERATURE

1. M. Born und W. Heisenberg, Ann. d. Phys. 74, 1, 1924.

2. H. A. Kramers, ZS. f. Phys. 13, 343, 1923; H. A. Kramers und W. Pauli, ZS. f. Phys. 13, 351, 1923.

3. C. G. Darwin and R. H. Fowler, Phil. Mag. 44, 450 und 823, 1922; Proc. Cambridge Phil. Soc. 21, 391 and 730, 1923; R. H. Fowler, Phil. Mag. 45, 1 and 497, 1923.

4. M. Polanyi, ZS. f. Phys. 1, 337, 1920; K. F. Herzfeld, ZS. f. Phys. 8, 132, 1922.

5. J. Franck und M. Born, Ann. d. Phys. Paschen-Heft, 1925.

6. W. Pauli jun., Ann. d. Phys. 68, 177, 1922; F. F. Niessen, dissertation, Utrecht, 1922.

7. H. D. Smyth, Proc. Roy. Soc. 105 A, 116, 1923.

8. W. Heisenberg, ZS. f. Phys. 26, 291, 1924.

9. M. Born, ZS. f. Phys. 26, 379, 1924.

10. W. Kossel, Ann. d. Phys. 49, 229, 1916.

11. E. Madelung, Phys. Zeitschr. 19, 524, 1918; P. P. Ewald, Ann. d. Phys. 64, 253, 1921.

12. M. Born, Verh. d. Dtsch. phys. Ges. 21, 13 and 679, 1919; K. Fajans, Verh. d. Dtsch. phys. Ges. 21, 539 and 549, 1919.

13. J. Franck, ZS. f. Phys. 5, 428, 1921.

14. O. Oldenberg, ZS. f. Phys. 25, 136, 1924.

15. P. D. Foote and F. L. Mohler, J. Amer. Chem. Soc. 42, 1832, 1920; P. Knipping, ZS. f. Phys. 7, 328, 1921.

16. H. G. Grimm, ZS. f. phys. Chem. 102, 113, 141 and 504, 1922; H. G. Grimm and K. F. Herzfeld, ZS. f. Phys. 16, 77, 1923.

17. H. G. Grimm und K. F. Herzfeld, ZS. f. Phys. 19, 141, 1923.

18. K. Fajans, Die Naturwissenschaften 10, 165, 1923; K. Fajans and G. Joos, ZS. f. Phys. 23, 1, 1924.

19. R. Ladenburg, ZS. f. Phys. 28, 51, 1924.

20. A. Heydweiller, Ann. d. Phys. 41, 499, 1913; 48, 681, 1915; 49, 653, 1916; Verh. d. Dtsch. phys. Ges. 16, 722, 1914.

21. J. A. Wasastjerna, Comm. Fenn. 1, p. 7, No. 37, 1913.

22. M. Born und W. Heisenberg, ZS. f. Phys. 23, 388, 1924. See also D. R. Hartree, Proc. Cambr. Phil. Soc. 22, 409, 461, 1924; Proc. Roy. Soc. A. 106, 552, 1924.

23. P. Debye, Phys. Zeitschr. 21, 178, 1920; W. H. Keesom, Phys. Zeitschr. 22, 129, 1921.

24. H. v. Wartenberg und Th. Albrecht, ZS. f. Elektrochemie. 27, 162, 1921; H. v. Wartenberg und H. Schulz, ZS. f. Elektrochem. 27, 568, 1921.

25. W. Heisenberg, ZS. f. Phys. 26, 196, 1924.

26. P. Debye, Phys. Zeitschr. 13, 97, 1912.

27. M. Jona, Phys. Zeitschr. 20, 14, 1919.

28. Cl. Schaefer und M. Schubert, Ann. d. Phys. 50, 283, 1916; ZS. f. Phys. 7, 297, 309 und 313, 1921; Cl. Schaefer and M. Thomas, ZS. f. Phys. 12, 330, 1923.

29. C. J. Brester, dissertation, Utrecht, 1923; ZS. f. Phys. 24, 324, 1924.

30. H. Kornfeld, ZS. f. Phys. 26, 205, 1924.

31. F. Pockels, Neues Jahrb. f. Min. Beil. 7, 203 and 224, 1890; Gött. Abh. 39, 1893; W. Voigt, Magneto u. Elektrooptik (Leipzig, 1909), Chapter IX.

1. See the preceding article by K. Fajans. Ed. ↩↩