Full Text
QUANTUM THEORY AND THE CHEMICAL BOND1
F. London, Berlin.
§ 1. Introduction.
One of the most profound and interesting problems of contemporary atomic theory is that clear-cut and at the same time essentially mysterious regularity which underlies an enormous body of factual material and is symbolically expressed in the language of chemical formulae. In the first period of the study of atoms, chief attention was directed to the structure of the individual atoms of the periodic system. Now we are already faced with the task of testing our knowledge of the structure of atoms and of the dynamic laws governing them on the problem of the mutual force interactions of atoms with one another; of determining whether this knowledge is sufficiently complete to decipher the meaning of the rules found by chemists by semi-empirical means; of placing these rules on a deeper theoretical foundation; of establishing their limits and, as far as possible, also of constructing a quantitative theory.
The characteristic interaction of the so-called chemical forces, which differ so sharply from other forces known to us, their excitation after a preceding “activation,” their sudden extinction after “saturation” of the available “valences,” as chemists describe it, without providing explanations—all this at first even raises doubts as to whether, in general, the principles of the description of nature that have hitherto been applied in the investigation of atoms are sufficient, not
are manifested in chemical actions that are entirely new forces of nature, which by no mathematical subtleties can be derived from the assumptions known to us.
We shall see that the formal (valence-chemical) content of the system of rules provided by chemistry is based chiefly on two new principles, which belong to the domain of quantum mechanics and which recently have already played an outstanding role in the qualitative explanation of the structure of individual atoms of the periodic system: the Pauli principle and the phenomenon of the so-called rotation of the electron. But these purely formal categories made it possible, in fact, to disclose the energetic content of the basic modes of interaction only thanks to a new phenomenon, which is characteristic of quantum mechanics and has no analogue in classical theory.
§ 2. The Problem of Chemical Interaction in Bohr’s Theory.
For a part of chemical compounds, the so-called heteropolar bonds, a certain explanation had already been given in connection with the Bohr model of the atom in the theory of Kossel and Lewis,^1 according to which electrostatic forces determine the nature of the compound. But attempts^2 to study the interactions of atoms more precisely and in greater detail on the basis of Bohr’s theory encountered serious difficulties, since this problem proved to be indeterminate in the sense of “classical” quantum theory.
If one specifies the Bohr orbits of both atoms or ions interacting with one another, then it still remains completely unknown what the relations between the phases of the electrons of the different atoms on their orbits are. But, depending on the accidental positions of the electrons on the orbits (phases), attraction or repulsion may continuously pass into one another, especially for neutral atoms. For
^1 W. Kossel, Ann. d. Phys. 49, 229, 1916; G. N. Lewis, Journ. Am. Chem. Soc. 38, 762, 1916.
^2 In particular M. Born und A. Landé, Berl. Ber. 1048, 1918; Verh. d. D. Phys. Ges. 20, 210, 1918.
ions likewise leads to an extreme uncertainty in the interaction, which is altogether incomparable with the clear relations established by chemistry for reactions between atoms. Even at that time it was supposed that the goal could be reached by means of a generalization of Bohr’s theory, and such an attempt was made1. However, this did not lead to satisfactory results.
I should also like to recall that Kossel’s theory, in its fundamental propositions, gave rather a formulation of the problem than an answer, since it postulated a certain “preference” of atoms for forming so-called completed electron shells, and this preference could not be satisfactorily explained on the basis of Bohr’s theory. In any case, assuming this postulate, one could determine those forces which bind heteropolar compounds, and thus a broad generalization was given of those interrelations which occur in the combination of ions. But for a considerable number of so-called homeopolar compounds this mode of explanation proved inapplicable.
Attempts to explain electrostatically the forces between neutral atoms proved unsuccessful precisely because “neutral” atoms, as opposed to ions, are in fact neutral, because they do not constitute rigid force centers, as chemistry requires. If, however, one takes into account the circumstance that atoms are mutually polarized, and that statistical preferences of certain phase relations appear in the sense of a certain “directing” action, then in this case one can speak only of weak forces between neutral atoms. These forces were compared with van der Waals forces. But such an aggregate of forces has no resemblance to those forces which have already been established in chemistry empirically. Apparently the most fundamental phenomenon itself still remained unknown.
§ 3. Remarks on the Content of the Propositions of Quantum Mechanics1
One of the most remarkable features of quantum mechanics, and one rich in consequences, is that its propositions are in principle poorer in content than the propositions of classical mechanics, and that its laws constitute the necessary logical connections between these propositions with their limited content. As a result, those problems which were still insufficiently determined in classical mechanics have become determinate problems only from the point of view of the new mechanics.
In quantum mechanics the concept of the phase in a given state2 is completely abolished. Quantum mechanics refrains from indicating an individual phase; it considers all possible phases simultaneously and merely describes their frequency statistically by means of a certain continuous quantity, the so-called spatial charge of Schrödinger. In order to form a vivid, though rough, idea of the distribution of this charge, one may imagine that all those orbits which can be obtained in classical mechanics by varying the phases while keeping the quantum numbers constant now exist simultaneously. The difference consists only in the fact that the state and dynamics of this “electric cloud” are obtained not by extending the domain of classical motions, but are described directly—as in any dynamics of a continuous medium—by means of a differential equation with partial derivatives, the wave equation of Schrödinger. Correspondingly, the “ball” of classical orbits turns into a uniformly distributed cloud. Its main part is localized in the region of the classical
of motion, but the remaining part extends without sharp boundaries, gradually becoming more rarefied, to infinity. For example, for a hydrogen atom in the normal state the density of this cloud decreases in proportion to \(e^{-\frac{2|r|}{a_0}}\) (\(a_0\) is the radius of the Bohr orbit).
The essence of the matter is that the establishment of quantum numbers gives, from the point of view of quantum mechanics, a quite definite answer to all questions about the stationary state of an atom, while the determination of a further series of constants of integration, such as the phases in the classical theory, proves to be superfluous. Correspondingly, the investigation of the interaction between two quantum systems could become a quite definite problem. But, owing to the absolute identity of electrons in quantum mechanics, a new phenomenon appears, which again introduces ambiguity into the manner of interaction between atoms. This ambiguity, however, is again restricted in a peculiar way if the statements of Pauli’s principle are introduced. We shall examine these circumstances in more detail using the example of the interaction between two hydrogen atoms, which served as the starting point for the present discussion.
§ 4. Interaction between two hydrogen atoms1.
Let us suppose that two hydrogen atoms are given in the ground state. Each is described by the corresponding Schrödinger wave function \(\psi\) or \(\varphi\). Let the nucleus of the first atom be at the origin of coordinates, and let the position of the electron be determined by the vector \(r_1\). For this atom we have:
\[ \psi(r_1)=e^{-\frac{|r_1|}{a_0}}\cdot e^{-\frac{2\pi i}{h}E_0t} \tag{1a} \]
(\(E_0\) is the energy of the ground state of hydrogen). For the second atom, the position of whose nucleus is determined by the vector \(\mathfrak{R}\),
and the position of the electron by the vector \(\mathbf r_2\); the oscillatory function is equal to:
\[ \psi(\mathbf r_2)=e^{-\frac{|\mathbf R-\mathbf r_2|}{a_0}}\cdot e^{-\frac{2\pi i}{h}E_0 t}. \tag{1b} \]
The spatial functions \(|\psi|^2\) and \(|\varphi|^2\) determine at each point of space the density of the aforementioned spatial charge. They describe the spatial distribution of the square of the amplitude of the stationary oscillatory state with frequency \(\frac{E_0}{h}\), which accompanies each atom, according to the ideas of de Broglie and Schrödinger. The frequency of this process thus depends on the energy of the corresponding stationary state.
We shall now consider both atoms as a single system, for the time being disregarding the dynamical interaction.1 From the theory of linear partial differential equations it follows that, as the general oscillation of the whole system, one must consider the product of both oscillations \((1a, 1b)\), with the frequencies (and with them the energies) adding additively.
Since there are two possibilities for distributing electrons 1 and 2 between the two nuclei, there also exist two oscillations which, owing to the complete identity of the electrons, have the same frequency \(\frac{1}{h}(E_0+E_0)\) and the same energy \((2E_0)\), namely:
\[ \psi(\mathbf r_1)\cdot\varphi(\mathbf r_2) \]
\[ \psi(\mathbf r_2)\cdot\varphi(\mathbf r_1). \]
If, however, the coupling of the two systems is taken into account, a “resonance displacement” of the two oscillations, initially of equal frequency, will appear. These oscillations are no longer stationary: if at first one of them is excited, it gradually passes into the other. Resonance beats are formed (similar to those in coupled systems of ordinary mechanics), which may be represented as a superposi-
…of two stationary fundamental vibrations with somewhat different frequencies.
Calculation gives, for these fundamental vibrations, two resultant vibrations, one symmetric and the other antisymmetric1 with respect to the electrons. The frequencies of these vibrations (and, together with them, the energies) are different as a consequence of the resonance shift.
\[ \{\psi(\mathbf r_1)\varphi(\mathbf r_2)+\psi(\mathbf r_2)\varphi(\mathbf r_1)\}\cdot e^{-\frac{2\pi i}{h}E_s t} \tag{2a} \]
\[ \{\psi(\mathbf r_1)\varphi(\mathbf r_2)-\psi(\mathbf r_2)\varphi(\mathbf r_1)\}\cdot e^{-\frac{2\pi i}{h}E_a t}. \tag{2b} \]
\(\frac{1}{h}E_s\) and \(\frac{1}{h}E_a\) give the frequency shift. It depends, of course, on the strength of the bond between the two atoms, i.e. on the distance between the nuclei \(R=|\mathfrak R|\). As \(R\to\infty\), \(E_s\) and \(E_a\) tend to zero. Superposition of these fundamental vibrations (2a) and (2b) gives the original nonstationary vibrations, alternately passing into one another, with frequency
\[ \frac{E_a-E_s}{h}. \]
From the point of view of the so-called statistical explanation of the propositions of quantum mechanics, such a statement of “both—and” about the simultaneous existence of fundamental vibrations, superposed on one another, with different frequencies must be interpreted as a statistical “either—or.” The statement “both—and” refers to a virtual ensemble of identical systems, while the statistical alternative “either—or” refers to the given individual case.2
From this point of view, each of the fundamental vibrations has an independent significance. Each, taken by itself, describes one of the stationary states of the system,
formed from both atoms. The appearance of such a state is established only statistically.
The result of the calculations, which have so far been carried out only in the first approximation and therefore claim only approximate accuracy, is shown in Fig. 1. The energies \(E_s\) and \(E_a\) of both fundamental oscillations are plotted in the drawing as functions of the distance between the nuclei \(R\). As is evident from the drawing, the energy of the symmetric oscillation has a clearly pronounced minimum. Its magnitude (3.2 volts) and its position (\(R = 0.80 \mathring{\mathrm A}\)) are in very satisfactory agreement with the measured dissociation energy of \(\mathrm H_2\) (4.3 volts) and the distance between the nuclei of the molecule (\(R = 0.76 \mathring{\mathrm A}\)). The oscillation which is antisymmetric with respect to the electrons always gives repulsion. Thus we obtain the remarkable result that two identical atoms in the normal state can act upon one another in two ways. Only one of these modes of action can be regarded as the chemical bond \(\mathrm H_2\).
Fig. 1. Interaction of two neutral hydrogen atoms.
(In the figure: vertical axis \(V\); horizontal axis \(R/a_0\); curves labeled “symmetric” and “antisymmetric.”)
§ 5. Pauli’s principle and the rotation of the electron.
This result appears to be in contradiction with our assertion in § 3 that all questions in quantum mechanics receive an unambiguous answer if the states of the atoms are specified by quantum numbers. In fact, the meaning of this assertion must be explained more precisely: in the present formulation of quantum mechanics, the ambiguity we have noted is obtained at first. This ambiguity is based, as we have seen, on the absolute identity of electrons, as a result of which one more state is introduced, which
differs from the original only in the distribution of electrons 1 and 2 over both nuclei. If there were more than two electrons, then the ambiguity, as we shall see below, would become still more complicated.
The so-called Pauli principle1 is important in that it subsequently again eliminates this ambiguity, excluding the existence of all those states in which Schrödinger’s wave function depends in the same way on several entirely identical particles, i.e. is symmetric with respect to their coordinates. On this basis it reduces, as is easy to show, the totality of all possible vibrations to a single one, namely to that which is antisymmetric with respect to all identical particles. This subsequent prohibition of the initially admitted superfluous possibilities is undoubtedly a shortcoming of the modern formulation of quantum mechanics. I cannot fail to mention that there already exists such a formulation of the theory2 in which this fundamental aesthetic shortcoming is absent. But it seemed to me inappropriate to use here this very abstract and, apparently, still preliminary method. In any case, the Pauli principle is a proposition whose correctness is based on the entire body of experience concerning the structure of atoms in the periodic system, and which, in view of its qualitative character, may be regarded as one of the best substantiated in atomic physics.
The assertion of § 3 concerning the unambiguity of the definitions of quantum mechanics becomes true only when the Pauli principle is applied. If this were sufficient for the description by means of quantum mechanics, then in the case considered by us of two hydrogen atoms the Pauli principle would allow only the antisymmetric vibration, i.e. precisely that vibration which gives repulsion, and there would be no bond forming the molecule H$_2$. The possibility of homeopolar chemistry is based, as we shall show, essentially on the circumstance that electrons can differ from one
from one another by one more property, which we have so far completely ignored, namely by the orientation of the axes of rotation of their own rotational motion1. There exist two different sorts of electrons, which differ in the orientation of the rotation vectors (parallel or antiparallel to the chosen magnetic direction). And since the Pauli principle applies only to absolutely identical particles, it imposes no restriction in the case when the rotation vectors are oppositely directed.
Therefore, for our two hydrogen atoms such a symmetric oscillation is permissible, which leads to a homeopolar bond. It is only necessary to assume that the axes of rotation of the two electrons are antiparallel. Thus there exist side by side both possibilities of interaction, which are shown in Fig. 1, depending on the relative disposition of the rotating electrons. If the relative orientation of the rotation vectors is given, then the character of the interaction is determined unambiguously.
When the number of particles is greater than two, the Pauli principle leads to peculiar restrictions. Since, according to the hypothesis of Goudsmit and Uhlenbeck, rotation can have only two different values, only two electrons can differ from one another in their rotation. Therefore the Pauli principle may be formulated as follows. If the rotation of the electrons is taken into account, then the oscillatory function may have arguments symmetric only in pairs; those pairs of electrons on which the oscillatory function depends symmetrically have antiparallel rotation.
This formulation is essentially equivalent to the one originally communicated. It would have been more systematic to take into account the rotation of the electron at the very beginning of the solution of the problem; then it would have been possible to make use of the original simple formulation of Pauli’s principle. But it seems to me more expedient not to burden the problem under consideration with questions of fine structure, which are connected with the rotation of the electron, since they have no essential significance for the energy relations in chemical interactions. Therefore I take the phenomenon of electron rotation into account only in the final formulation of Pauli’s principle.
§ 6. Mechanism of Saturation and Activation of the Chemical Forces of Valence.
Having convinced ourselves that quantum mechanics makes it possible to explain the process of binding of two hydrogen atoms, one may ask why, then, there is no molecule \(H_3\). Let us recall that, according to Pauli’s principle, no more than two electrons, equivalent (symmetric) in their manner, can enter into the description of one state. Hence it follows that, when a third hydrogen atom appears, it is necessary to exclude such a state in which all three electrons are equivalent. Consequently, the third atom can attach itself only in the case where the electrons of the first two atoms do not enter symmetrically into the process of oscillation, i.e. if the first two atoms do not combine, but repel one another.
From this reasoning it is easy to understand that Pauli’s principle proves capable of explaining the fact of saturation of valences, which has received such a distinct expression in the language of chemical symbols. It is natural to assume that in all cases where two electrons of different atoms combine in a symmetric Schrödinger oscillation, a bond arises. This assumption can be formulated more precisely (§§ 7, 8) and receives considerable confirmation from experiment (§ 9).
Before proceeding to the general consideration of the possible reactions that are obtained for arbitrary atoms from quantum-
…of mechanics, the Pauli principle, and the rotation of the electron; and for the chemical interpretation of these possibilities, I should like to show graphically, in several diagrams, how the mechanism of saturation of the chemical forces of valence acts on the basis of the conclusions of quantum mechanics. For this I shall take the already mentioned case of three hydrogen atoms \((A, B, C)\). Let us suppose that two of them \((A\) and \(B)\) are fixed in the equilibrium position of the molecule, and let us plot the energy as a function of the position of the third atom. We shall determine this position by means of the distance \(R\) between the atom \(C\) and the center of gravity of the atoms \(A\) and \(B\). Although the energy relations depend to a considerable degree on the direction, this is not essential for the qualitative character of the curves.
Fig. 2. Interaction of two atoms with a third (the molecule is not illustrated).
In the curves of Fig. 2 the sign \(+\) indicates that both fixed atoms \(A\) and \(B\) are bound to each other (\(E_{\text{sym}}\) in Fig. 1), while the sign \(-\) indicates that they repel each other (\(E_{\text{antisym}}\) in Fig. 1). The asymptotic value of these curves at a large distance from the third atom directly gives the interaction energy of the first two atoms. From the drawing the following is evident: if \(A\) and \(B\) repel each other, then \(C\) may also be repelled (the upper curve; in this case the vibrational function is antisymmetric with respect to all particles) or else may be attracted (the second curve from above; the vibrational process is symmetric with respect to two particles). To which of the atoms, \(A\) or \(B\), the third atom is attracted is not visible in the drawing; it is a matter of chance.
If, however, the atoms \(A\) and \(B\) enter into a molecular bond, then \(C\) is repelled (the third curve from above; the vibrational pro-
cess is symmetric with respect to two particles). The lower curve, drawn with a dotted line in Fig. 2, along which atom \(C\) is attracted to the molecule \(AB\), corresponds to the case when the vibration is symmetric for all three electrons. The Pauli principle excludes this case, which energetically would correspond to allowing the formation of the molecule \(\mathrm{H}_3\), and thus makes evident the saturation of univalent hydrogen.
The physically important case is that in which atoms \(A\), \(B\), and \(C\) are not identical. Then curves are obtained similar to those shown in Fig. 2. But if the resonance phenomenon that determines the bond of the molecule \(AB\) is smaller than in the formation of \(BC\), then the relation between the potential curves is different, and a possible substitution exists according to the scheme:
\[ AB + C \to A + BC. \]
The reactions possible in this case are shown in Fig. 3. The signs \(+\) and \(-\) have the same meaning as in Fig. 2. The dotted curve is likewise excluded on the basis of the Pauli principle. Of chief interest to us is the third curve from above, which represents the energy relations for the only possible reaction between the molecule \(AB\) and the atom \(C\). At large distances, as before, there is repulsion of \(C\), but upon approaching \(A\) or \(B\) a limit is reached beyond which attraction already begins. As soon as the region of attraction is reached, the bond between atoms \(A\) and \(B\) breaks, and the atom that is farther from \(C\) is repelled and completely separates (to simplify the drawing we assume that such an external action does not hold this atom).
The condition under which the case shown in Fig. 3 becomes possible consists, as mentioned, in the fact that
\[ \text{the resonance phenomenon of } AB < \text{ the resonance phenomenon of } BC. \]
This case always pertains to a reaction in one of two directions. In order to cause the reaction in the opposite direction, it is necessary to move the atoms in the molecule apart. In this way one can diminish the resonance interaction of the ato-
… of atoms in the molecule by any amount. However, there is no possibility of directly moving the atoms apart in the molecule; but one can, by exciting the vibration of the nuclei, increase the average distance between the nuclei and thus make the molecule capable of reacting with an atom.
We have before us, evidently, one of those mechanisms that play such a large role in the kinetics of reactions and have received in it the name of activation.^1
Fig. 3. Interaction of two atoms with a third (activated molecule).
In order to explain statistically the course of gas reactions, Arrhenius introduced the hypothesis that only a very small, but rapidly increasing with temperature, fraction of all gas molecules can enter into reaction; moreover, this reacting fraction of “active” molecules differs from the remaining molecules by possessing a large store of energy. As we have seen (and experimental data also lead to this assumption), this process of activation, mysterious until quite recently, consists chiefly in an increase of vibrational energy. A reaction can occur only after these preliminary conditions have been realized and under favorable collisions: the energy of translational motion is sufficient to cross the threshold of attraction.
The investigation can be carried out still more precisely, but this would take us too far. I wished here only to show—
^1 S. Arrhenius, ZS. f. phys. Chem., 4, 226, 1889. For more detail on questions connected with the problem of activation, see the book: C. N. Hinshelwood, Reaktionskinetik gasförmiger Systeme, Leipzig, 1928.
to say, without detailed proof, how deeply and peculiarly the processes of saturation and activation are explained in quantum mechanics.
§ 7. Reactions between any two atoms from the point of view of quantum mechanics.
With the aid of mathematical methods, which I shall not set forth here, one can obtain an exhaustive survey1 of all possible reactions between any two atoms and of the characteristic symmetry properties of Schrödinger vibrations, and also, in principle, obtain in the first approximation (i.e. reduce to quadratures) the energy relations.2 These considerations, based chiefly on group theory, contain the most essential part of the problematics and, methodically, the most interesting aspect of the whole circle of questions, and bring the doctrine of chemical valence into direct connection with one of the deepest and most elegant achievements of mathematics, with the theory of irreducible representations of the permutation group.3 Here I shall confine myself only to presenting the results without proof.
Each of the two atoms consists, according to the Pauli principle, of a certain number of pairs of equivalent electrons and of a certain number of pairs of nonequivalent electrons. When two atoms approach one another, there appear, as we saw in the example of \( \mathrm{H}_2 \), new normal vibrations with a new symmetry, whose energy is unequal as a consequence of “resonance splitting.” But according to the Pauli principle one must again exclude all those vibrations which depend in the same way on more than two electrons.
The various mathematically derived types of reactions between the two atoms can be described exhaustively on the basis of the symmetry properties of the vibrational process, in the following way—
together. Along with a state whose wave function has as many kinds of symmetry as the individual atoms taken together, all those and only those vibrational states are possible which are symmetric with respect to pairs of electrons that originally belonged to different atoms and were not originally bound in an equivalent manner. The enumeration of such states completely gives the totality of all possible reactions.
A pair of electrons that is already equivalently bound can never give a new symmetry with respect to other atoms. For atoms that consist only of such pairs of equivalent electrons (as, for example, the inert gases), there is only one possibility of reaction with any other atom, in which the number of symmetric pairs does not increase. For helium this mode of interaction has been approximately calculated.^1 It was found that those forces which helium atoms exert on one another in the normal state do not lead to a molecular bond, but manifest themselves at most in the form of “van der Waals forces.” Since in this case the regulating mechanisms of saturation do not operate, these forces can lead at very low temperatures only to a chaotic accumulation of all atoms in random quantitative ratios, i.e. to liquefaction.
§ 8. Chemical interpretation of the varieties of reactions based on quantum mechanics.
The circumstances considered above lead to the conclusion that there is a relationship between the extremely characteristic possibilities of symmetry governing interatomic reactions, derived from quantum mechanics, on the one hand, and the empirically obtained possibilities of chemical bonding, which are determined by the formalism of the doctrine of valence, on the other hand. From this the following interpretation is obtained. If, upon the approach of initially
^1 L. Pauling, Chem. Rev., 5, 173, 1928.
THE THEORY OF QUANTA AND THE CHEMICAL BOND
If, for separated atomic systems, the number of symmetrically bound pairs of electrons increases by one, then this signifies the appearance and saturation of a homeopolar valence.
An electron that can enter into such a symmetrical bond is called a “valence electron.” To it corresponds a “free homeopolar valence.” Only those electrons can be valence electrons which have not yet entered into such a symmetrical bond.
Consequently, the “number of valences” of an atom is determined by the number of those electrons which are not yet bound in equivalent pairs in the inner parts of the atom. A free valence that is saturated by the corresponding free valence of another atom cannot enter into other processes of combination, as we shall see in § 6. It can form a new bond only in the event that the original one is destroyed.
It can be shown in detail that the quantum interpretation we have proposed of the concept of valence is equivalent to its chemical prototype in all details, i.e. that it indeed satisfies the same formalism of the rules of combination. Of course, such an interpretation does not claim to give a detailed explanation of the energetic relationships in the various kinds of bonds. Indeed, after it has been clearly shown what quantum formations are hidden behind the chemical concept of the homeopolar bond, and on what the heuristic applicability of this concept in chemistry is based, the task of the quantum theory of the bond will not be to justify the entire formalism of the doctrine of valence. On the contrary, it will now be required of it to establish the energetic or other dynamical reasons why in many cases the molecular bond does not correspond to this scheme, and to determine the limits of applicability of this scheme.
It seems to me necessary also to show that our definition leads essentially to the same numerical values of valence that are obtained in chemistry for the atoms of the periodic system.
F. London
§ 9. Valence and Spectroscopic Multiplets.
Those electrons which do not take part in valence, being pairwise equivalently bound to one another, must, according to Pauli’s principle, have pairwise antiparallel axes of rotation. Therefore they make no contribution to the resultant angular momentum of the rotating electrons.
The remaining electrons, however, i.e. the valence electrons, arrange themselves so that all their rotation vectors are parallel1, the resultant moment for \(n\) electrons being equal to \(n \dfrac{h}{4\pi}\). Since there are several possible spatial orientations for this resultant moment of the electrons, each state splits into a series of states very close to one another2. In this case the number of such states, the so-called “multiplicity” \(M\), is equal to:
\[ M = n + 1, \]
where \(n\) is the number of equally directed rotating electrons. The multiplicity of any state is in most cases known exactly from spectroscopic data. It is directly related to the number of free homeopolar valences of the system in the following way.
The valence is one less than the number of lines in the multiplet of the corresponding state of the atom or molecule, and is equal to the resultant rotational moment of the electron, measured in units \( \dfrac{h}{4\pi} \). Upon saturation of the valence
rotational moment of each of the joining systems is decreased by one. Homopolar valences are saturated if the resultant moment of rotation of the electrons of the whole system is equal to zero, so that there is no fine structure.
From this dependence there is obtained directly the law of evenness and oddness of valences, in full agreement with the spectroscopic law on the alternation of even and odd multiplets in the various groups of the periodic system.
If one makes the quite natural assumption that the inner closed electron shells are not opened in chemical processes, then, for example, for the outer seven electrons of the halogens the following possibilities are obtained with respect to symmetry: 1) 3 symmetric pairs and 1 valence electron; 2) 2 symmetric pairs and 3 valence electrons; 3) 1 symmetric pair and 5 valence electrons; 4) no symmetric pairs and 7 valence electrons.
For elements in the same group as oxygen the valences obtained are: 0, 2, 4, 6. But valence 0 drops out, since the ground state of oxygen, theoretically and on the basis of experiment, is a triplet, which indicates the divalence of this state characterizing chemical processes. On the same basis, for elements of the nitrogen group one may expect valences 1, 3, 5, but valence 1 is excluded, since the ground state proves, theoretically and experimentally, to be in the system of quartets and therefore trivalent. The carbon group proves to be di- and tetravalent.
In particular, for tetravalent carbon there is obtained a configuration which corresponds to the configuration of neon, with the sole difference that the four states of the outer shell are not double, but simple. The proposed configuration of carbon has the same spatial structure as the structure of the completed shells of the noble gases. It possesses spherical symmetry and resists the formation of ions and polar bonds. But all the outer electrons may take part in homopolar bonds. The similarity between the configurations of carbon and of the noble gases appears, apparently, also in the fact that neighboring atoms, like
as if, under the influence of ionization, they tend toward such a “state of a noble gas,” for example \((\mathrm{NH}_n)^+ \mathrm{Cl}^- \mathrm{BN}\) and others.
The law of alternation of the numbers of homeopolar valence (the same apparently applies to any definitely expressed homeopolar bond) extends chiefly to those elements which precede the formation of closed octet shells, and in which the valence bond is formed only by electrons with the same principal quantum number. These are, in essence, chiefly those columns of the periodic system which we mentioned above. It is understandable that for the transition elements, in which the formation of inner shells is still continuing, the magnitude of the valence is expressed less sharply, since unpaired electrons may be found not only in the outer but also in the inner layers, and may participate in valence in a more or less noticeable way.
I shall allow myself to confine myself here to these summary considerations. In order to penetrate more deeply into the correlations of the valences of atoms, it is necessary to examine in greater detail the structure of individual atoms. The connection between the magnitude of the valence and spectroscopic multiplets must play a decisive role in such an investigation.
§ 10. The boundary between the homeopolar and ionic bond.
The type of bond which we have considered as resonance from the point of view of quantum mechanics, and to which we have applied the commonly used name of a “homeopolar” bond, by no means exhausts all cases of chemical bonding. We have no intention at all of driving the former conceptions of bonding out of that region where they were justified. On the contrary, the task arises of establishing the boundary with respect to other types of bonding, above all with respect to the ionic, or heteropolar, bond according to Kossel.
Quantum mechanics not only does not contradict the model conceptions proposed by Kossel, but it may even be asserted that only from the point of view of the new theory
these ideas were in fact derived from the principles of the structure of the atom, whereas Bohr’s theory, on the basis of which they, of course, arose, did not provide a complete justification for them. Only quantum mechanics could prove that ions do indeed possess spherical symmetry1, that at small distances between them repulsive forces arise2, which balance the forces of electrostatic attraction, as is postulated by Born and Landé for the forces in the lattice of a solid. Finally, quantum mechanics proved capable of explaining the hitherto mysterious “tendency” toward the formation of closed shells.
Fig. 4a.
Fig. 4b.
Fig. 4c.
In order to establish which of the two kinds of bond occurs in a given particular case, it is first of all necessary to calculate, for both models, the magnitude of the interaction as a function of distance, and to compare them with one another. For this, a rather rough calculation is sufficient.
On the curves presented (Figs. 4a, 4b, 4c) are plotted the potentials of the homopolar bond \(H\) and of the ionic bond \(J\), calculated for various cases of interaction between neutral atoms or the corresponding ions, beginning with infinitely large distances between them. Characteristic of the course of the curves is that already at small distances between the atoms, in comparison with the equilibrium position, the homopolar potential becomes constant and equal to
zero \(\left(e_{\infty}^{-\frac{2R}{a}}\right)\), whereas the ionic potential varies as \(E_{\infty}-\dfrac{e^{2}}{R}\) (Coulomb interaction), where \(z\) is the degree of ionization, \(E_{\infty}\) is the ionization potential of the cation minus the electron affinity of the anion. The relative position of the two curves therefore depends on the magnitude of the ionization energy, the electron affinity, and the degree of ionization.
The curves in the diagrams have been calculated on the assumption that the two kinds of bond do not influence one another. This is true, of course, in the case where both curves (coming from infinity) do not intersect or do not come too close to one another before the equilibrium position. Then the character of the bond depends simply on which of the curves lies lower. Thus, for example, in the case of Fig. 4a the homopolar bond predominates; in the case of Fig. 4c the polar bond predominates. In view of the different course of the two curves as they approach the asymptote, a case is also possible in which the two curves intersect. Here a more detailed investigation was already required. The latter showed that, in the case where the point of intersection lies in a region in which the homopolar potential is close to zero, the curves do not influence one another. In this case the ionic bond also predominates, since the ionic curve has the smaller potential minimum.
From Fig. 4b it is evident that an ionic bond is possible even in the case when the electron affinity is insufficient for ionization of the cation (i.e., the electron affinity is less than the ionization potential and \(E>0\)). Previously this was unclear precisely because the course of the curve of the homopolar potential, which plays a decisive role, was not known. The usual polar bonds do indeed correspond to case 4.
For the transitional cases from Fig. 4b to Fig. 4c there is always an ionic bond. On the other hand, in the intermediate region between Figs. 4a and 4 there exists a transition from homopolar to ionic bonding, which from the point of view of quantum mechanics (“either—or”) must be understood statistically in the following way. Suppose that we have separated the bond by means of a prearranged adiabatic¹ process (for example, by inducing oscilla-
¹ This means: without causing electronic jumps.
nuclei). Then either a pair of ions or a pair of atoms will result, and the probability of each case is precisely established on the basis of quantum mechanics. As we see, the transition region is comparatively insignificant, and the limiting cases of a purely homopolar or purely ionic bond stand out clearly.
If this criterion is applied, we find that the hydrogen halides (especially HJ, HBr, and also HCl) must be regarded as homopolar compounds. Recently Frank and Kuhn1 expressed the same assumption, proceeding from other experimental data.
On the other hand, the hydrides of the alkali metals turn out to be polar compounds, in which hydrogen plays the role of an anion, like a halide. This is also consistent with other properties of the alkali hydrides.
§ 11. “Valence Strokes.”
In conclusion we shall show, with the aid of visual drawings (Fig. 5 and Fig. 6), how the structure of the volume-charge density reflects the processes of manifestation of homopolar valence.
Fig. 5. Distribution of density under elastic reflection.
Fig. 5 depicts the case of two atoms that do not enter into combination. In the plane passing through both nuclei, curves of equal density of one of the electrons are drawn for the antisymmetric oscillation of two hydrogen atoms (the atoms are unperturbed, polarization does not exist),
The numbers denote the corresponding densities in arbitrary units. The second electron is represented in the same way. The densities are noticeably displaced outward, as if striving to separate from one another as much as possible. If the two nuclei, which here are at the same distance as in the molecule \(H_2\), are brought closer together, then the interception on the plane curves will become still sharper. With further approach of the atoms, the density at that point will fall to zero. The same figure will be obtained for two interacting helium atoms.
Fig. 6 depicts two \(H\) atoms that are in a state of homeopolar bond. Here both densities seem to approach one another and strive to merge with each other. I do not attach especially profound significance to these pictorial representations, but with their aid one may form an idea of how, by means of such a binding bridge—the \(|\psi|^2\)-density—those atoms are joined in complex molecules which mutually saturate their valences, while the remaining atoms are sharply separated from one another.
Fig. 6. Distribution of density in a homeopolar bond.
It should be noted that these figures are correct only approximately. They are drawn on the basis of the unperturbed functions of the atoms (1a) and (1b), superposed on one another according to (2a) and (2b). In reality the atoms are further deformed as a result of polarization and other influences. This is reflected both in the distribution of density and in the energy of interaction. Our drawings and the calculations of energy in § 4 must be regarded as the first step on the path of the method of successive approximations.
It is interesting to note that already in this first approximation the typical categories in valence receive a primary—
more distinct expression and are differentiated as symmetry properties of the unperturbed atoms. It may be foreseen that, with further application of the perturbation method, no new separation of categories will appear, but only a more accurate numerical approximation will be obtained. Therefore all possibilities of this kind may be regarded as definitively exhausted, and already in the first approximation we have a complete qualitative survey of all the existing chemical interrelations of atoms.
The phenomena of the physics of aggregate states are obtained only after the kinds of interaction of the homopolar forces of valence, as perturbing phenomena of higher order (polarization phenomena, van der Waals forces). This circumstance makes it understandable why chemical compounds exist in general independently of the aggregate state, in the form of solid, liquid, and gaseous bodies, and thus appear primary in comparison with the kinds of physical states.