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THE NATURE OF CHEMICAL AFFINITY.
Ya. I. Frenkel, Leningrad.
1. The Nature of Interatomic Forces.
Bernoulli already expressed the idea that the forces of chemical affinity reduce to electrical attraction between oppositely charged atoms. At the present time, when we know that atoms consist of positive nuclei and electrons, the electrical nature not only of chemical forces, but also of all kinds of cohesive forces both between atoms and between molecules (“van der Waals forces”), cannot give rise to any doubt. It is easy to imagine the emergence of a force of attraction not only between two oppositely charged atoms (ions), but also between two neutral atoms (or molecules). Each of them acts in the opposite manner upon the oppositely charged particles that form the other. Since the attracted particles thereby come closer together, while the repelled ones move farther apart, the forces of attraction acquire, on the average, a preponderance over the forces of repulsion. This preponderance constitutes the force of chemical affinity between the two atoms, insofar as the latter unite with one another into a molecule, or else the van der Waals force in the case where such union does not occur.
Whatever the force of attraction or affinity between two atoms may be, at a sufficiently small distance between the latter it is exceeded by a force of repulsion, which determines the impenetrability of atoms with respect to one another. This repulsive force, increasing as the distance decreases—
repulsion is far stronger than the force of attraction, is usually characterized purely geometrically by means of certain “dimensions” ascribed to the atom. The force of repulsion is directly connected with the motion of the electrons. This connection between the “hardness” of atoms and internal motion can be illustrated by the hardness acquired by a jet of liquid striking under great pressure, or, more generally, by the mutual impenetrability of vortex rings in a liquid or gas. It was precisely in this way that the hardness of atoms was once explained by Kelvin in his very popular vortex theory of atoms. We need only replace Kelvin’s “ether vortices” by “electronic vortices,” formed by the uniform, rapid rotation of electrons around positive nuclei.
2. Calculation of the Interaction Energy in Classical Mechanics
The above considerations concerning the origin of the forces of interatomic attraction and repulsion are of a purely qualitative character. In order to give them a quantitative form, one must be able to calculate the mutual energy of two atoms, i.e. the additional energy \(W\) arising from their interaction.
In classical mechanics this question is approximately solved as follows. An expression is written down for the mutual potential energy \(U\) of the particles forming atom \(A\), with respect to the particles forming atom \(B\), as a function of their mutual distances. This expression is then averaged over time, under the assumption that the motion of the particles remains the same as in the absence of interaction (“unperturbed”), and that the relative position of the systems \(A\) and \(B\) as a whole—i.e. the distance \(R\) between their centers, the inclinations of the electronic orbits to the line \(R\), and so forth—remains unchanged.
The mean value \(U = W'\) thus obtained represents the required interaction energy in the first approximation.
For a more exact calculation it is necessary to take into account the above-mentioned distortion (“perturbation”) caused by the mutual-
…corresponds to an overall “polarization” of both atoms, i.e., to a certain displacement of the electronic orbits with respect to the nuclei. However, the corresponding additional energy \(W''\) can in most cases be neglected in comparison with the “primary” interaction energy \(W'\), and this interaction can be characterized by the dependence of \(W'\) on the distance \(R\) between the two atoms.
The force of interaction is determined by the formula
\[ F=-\frac{dW'}{dR}, \tag{1} \]
representing repulsion when \(F>0\) and attraction when \(F<0\). If the energy \(W'\) has a (negative) minimum at some value \(R=R_0\), then the atoms can form the molecule \(AB\), remaining at the distance \(R_0\) from one another. This state of the molecule may be regarded as normal; alongside it, however, a number of other states are possible, in which the two atoms oscillate with respect to each other or rotate about a common center of gravity. In what follows we shall restrict ourselves to considering only a single normal state, since the transition from it to the others presents no substantial difficulties.
3. Calculation of the Interaction Energy in Wave Mechanics
The “old” quantum theory, i.e., Bohr’s theory, introduced no corrections whatever into the indicated calculation of the energy \(W'\), fixing only the permissible states of the two atoms that serve as the initial states when they combine into a molecule. It substantially changed the calculation of excitation (“mutual polarization”) and of the “polarization” energy \(W''\); however, because of the smallness of \(W''\) in comparison with \(W'\), this circumstance could not have any substantial significance for the theory of interatomic forces.
The new quantum theory, or wave mechanics, radically altered our ideas about the “behavior” of electrons in atoms and, correspondingly, led to entirely new results in the question of the interaction of the latter.
We shall not concern ourselves here with the exposition of the foundations and methods of wave mechanics, and shall consider them only insofar as they are relevant to the question that interests us.
In contrast to classical mechanics, wave mechanics does not give an exact description of the motion of the particles constituting the atom \(A\). All the positions of the latter are admitted by it as possible, and the problem is reduced to computing the probability of one or another of them—arbitrarily chosen positions—or, in other words, the probability of one or another configuration of the system. This configuration is specified by the elements of volume \(dV_A^{(1)}, dV_A^{(2)} \ldots\), in which the individual electrons are located, or by an element of volume of the coordinate space formed by the totality of the coordinates of all these electrons:
\[ dV_A = dV_A^{(1)} . dV_A^{(2)} \ldots \tag{2} \]
The probability of the configuration characterized by the element of volume \(dV_A\) is determined by the product of \(dV_A\) and the square of the absolute value (modulus) of a certain function \(\psi_A\), satisfying a special “wave” equation discovered by Schrödinger and containing, as the function characterizing the system under consideration, its potential energy \(U_A\).
The time average of any quantity \(f\), depending on the configuration \(A\), corresponds in wave mechanics to the expression
\[ \bar f = \int f |\psi_A|^2 dV_A, \tag{3} \]
which may be defined as the “mathematical expectation” of this quantity (in the sense of probability theory), or as the “mean statistical” value of \(f\). The integral in (3) is taken over the whole coordinate space, and it is assumed that for \(f = 1\) it becomes 1 (i.e. that the probability of any configuration \(A\) is equal to 1).
The interaction of two systems \(A\) and \(B\) is determined in wave mechanics—since the matter concerns a first approximation—in the simplest case in exactly the same way
THE NATURE OF CHEMICAL AFFINITY
as in classical mechanics, i.e. by the mean value of the mutual potential energy \(\overline{U}\) for both systems in the absence of interaction between them. A given configuration of \(A\) is then an “event” entirely independent of the configuration of system \(B\). The probability that system \(A\) is in the configuration \(dV_A\) while system \(B\) is in the configuration \(dV_B\) is therefore equal (by the theorem on the probability of independent events) to the product of the probabilities \(|\psi_A|^2 dV_A\) and \(|\psi_B|^2 dV_B\). The mean value of the potential energy \(U\), equal in the first approximation to the interaction energy, is thus determined by the formula:
\[ \overline{W}=\overline{U}=\iint U|\psi_A\psi_B|^2\,dV_A dV_B, \tag{4} \]
It must be noted, however, that this formula (as, incidentally, the corresponding formula of classical mechanics) is valid only in the very simplest case, when one of the following two conditions is fulfilled:
a) besides the state of the system \(AB\), characterized by the functions \(\psi_A, \psi_B\) (in the absence of interaction), or by the product of these functions \(\psi_A\cdot\psi_B\), there exists no other state \(\psi'_A\psi'_B\) with the same total energy \(W'_A+W'_B=W_A+W_B\);
b) in the presence of such states, the equations hold:
\[ \iint U(\psi_A\psi_B)(\psi'_A\psi'_B)dV_A dV_B=0. \tag{5} \]
Let us note that the integral standing on the left-hand side characterizes the probability of a “spontaneous” transition of the system \(AB\) from the state \(\psi_A\psi_B\) to the state \(\psi'_A\psi'_B\) (or conversely). Transitions of this kind can occur practically only when the energies of the two states are equal; when the sums \(W_A+W_B\) and \(W'_A+W'_B\) are unequal, these transitions, if not in principle, are in practice excluded, so that in this case conditions (5) lose their significance.
Let us further note, in order to avoid misunderstandings, that the state of any system is determined in wave mechanics
* Uspekhi fizicheskikh nauk, vol. IX, issue 4.
not by the configuration itself, but by the corresponding “wave function” or “probability amplitude” \(\psi\). In one and the same state the system may be found in any configurations; and, conversely, for different states \(\psi\) and \(\psi'\) in one and the same configuration \(dV\), the probability of the latter is in one case \(|\psi|^2 dV\), and in the other—\(|\psi'|^2 dV\).
4. The simplest heteropolar molecules.
We shall apply the results set forth in the preceding paragraph first of all to a system consisting of the positive and negative ions of hydrogen, i.e. of a proton \((A \equiv \mathrm{H}^+)\) and of a helium-like ion \(\mathrm{H}^-(\equiv B)\). In other words, we shall try to treat the hydrogen molecule \(\mathrm{H}_2\) as a heteropolar molecule, without being disturbed by the circumstance that this point of view does not correspond to reality. The results which we obtain in this way can easily be applied to typically heteropolar molecules.
We shall regard the ion \(\mathrm{H}^+\) as a point charge perturbing by its electric field the motion of both electrons in the ion \(\mathrm{H}^-\). The mutual potential energy of the two ions is equal to
\[ U=\frac{e^2}{R}-e^2\left(\frac{1}{r_1}+\frac{1}{r_2}\right), \tag{6} \]
where \(R\) is the distance between the two nuclei (“centers” \(\mathrm{H}^+\) and \(\mathrm{H}^-\)), and \(r_1\) and \(r_2\) are the distances of \(\mathrm{H}^+\) from the two electrons. The interaction of \(\mathrm{H}^+\) and \(\mathrm{H}^-\) for a given value of \(R\) is determined, according to the preceding, in first approximation by the statistical mean value of \(U\), i.e.
\[ \overline{U}=\frac{e^2}{R}-2e^2\,\overline{\frac{1}{r_1}}, \tag{7} \]
where
\[ \overline{\frac{1}{r_1}}=\int \frac{1}{r_1}\,|\psi|^2\,dV \qquad (dV=dV_1 dV_2) \tag{7a} \]
is the mean value of the reciprocal distance of one of the electrons of \(\mathrm{H}^-\) from \(\mathrm{H}^+\) \(\left(\overline{\frac{1}{r_1}}=\overline{\frac{1}{r_2}}\right)\), and \(\psi\) is the wave function characteriz-
...characterizing the behavior of these electrons in the normal state of \( \mathrm{H}^{-} \) under consideration. In doing so we assume that both electrons behave on the average in exactly the same way (i.e., from the standpoint of ordinary mechanics, they move in identical orbits), and that the state of \( \mathrm{H}^{-} \), characterized by the function \(\psi\), is the only one possessing the given (minimal) energy \(W\).
The exact dependence of \(\psi\) on the coordinates of both electrons can be found only by the method of successive approximations, the starting point of which is the function \(\psi^{0}=\psi_{1}\cdot\psi_{2}\), characterizing the behavior of electrons bound to the proton in the absence of interaction between them. Here \(\psi_{1}\) and \(\psi_{2}\) are functions characterizing the normal state of the hydrogen atom and having the following form:
\[ \psi_{1}=\frac{1}{\sqrt{\pi a^{3}}}\,e^{-\frac{r_{1}}{a}},\qquad \psi_{2}=\frac{1}{\sqrt{\pi a^{3}}}\,e^{-\frac{r_{2}}{a}}, \tag{8} \]
where \(a\) is a constant coinciding with the radius of the one-quantum orbit in Bohr’s theory, and \(r_{1}\) and \(r_{2}\) are the distances of the corresponding electrons from the center of the nucleus \( \mathrm{H}^{+} \). The proportionality coefficients \(\left(\frac{1}{\sqrt{\pi a^{3}}}\right)\) are chosen so that the integrals
\[ \int |\psi_{1}|^{2}\,dV_{1} = 4\pi\int_{0}^{\infty}|\psi_{1}|^{2}r_{1}^{2}\,dr_{1} \quad\text{and}\quad \int |\psi_{2}|^{2}\,dV_{2} = \]
\[ = 4\pi\int_{0}^{\infty}|\psi_{2}|^{2}r_{2}^{2}\,dr_{2} \]
were equal to 1.
In Bohr’s theory the interaction of the two electrons is taken into account approximately by means of a “screening constant,” which determines the fraction of the positive charge of the nucleus compensated by the other electrons. The very same method is also applicable in wave mechanics; moreover, the change in the effective charge of the nucleus is reduced practically to a change of the parameter \(a\) in formula (8). By understanding \(a\) to mean this changed value, we can characterize the normal state of the \( \mathrm{H}^{-} \) ion by the function
\[ \psi=\psi_{1}\psi_{2} = \frac{1}{\pi a^{3}}\, e^{-\frac{r_{1}+r_{2}}{a}}. \tag{8a} \]
In this case, according to (7a), one obtains:
\[ \frac{1}{r'_1} = \iint \frac{1}{r'_1}\,\psi_1^2\psi_2^2\,dV_1\,dV_2 = \int \frac{1}{r'_1}\,\psi_1^2\,dV_1 \cdot \int \psi_2^2\,dV_2, \]
i.e.
\[ \frac{1}{r'_1} = \int \frac{1}{r'_1}\,\psi_1^2\,dV_1. \]
This expression may be interpreted as the potential created at the point \(\mathrm{H}^+\) by an electric charge distributed around the ion \(\mathrm{H}^-\) with volume density \(\psi_1^2\). Since the latter depends only on the distance \(r_1\) from the center \(\mathrm{H}^-\), and since the potential of a spherical shell of radius \(r_1\) and thickness \(dr_1\) is equal to the charge of this shell, \(\psi_1^2 4\pi r_1^2 dr_1\), divided by the distance \(R\) from its center to the point under consideration \((\mathrm{H}^+)\) when \(r_1<R\), and by its radius \(r_1\) when \(r_1>R\), the preceding expression is reduced to the form:
\[ \frac{1}{r'_1} = \frac{1}{R}\int_0^R \psi_1^2\,4\pi r_1^2\,dr_1 + \int_R^\infty \psi_1^2\,4\pi r_1\,dr_1 = \]
\[ = \frac{4}{a^3} \left[ \frac{1}{R}\int_0^R e^{-2r/a}r^2\,dr + \int_R^\infty e^{-2r/a}r\,dr \right] = \]
\[ = \frac{1}{R} - \frac{e^{-2R/a}}{R} \left(\frac{R}{a}+1\right). \]
Hence, according to (7), it follows that
\[ U = -\frac{e^2}{R} + \frac{2e^2}{R}\, e^{-2R/a} \left(\frac{R}{a}+1\right). \tag{8b} \]
For large values of the distance \(R\) \((R\gg a)\), the value of \(U\) reduces to the first term, \(-e^2/R\), which characterizes the attraction of the two ions as point charges. Conversely, in the case \(R\ll a\) we obtain \(U=e^2/R\), i.e. the energy of mutual repulsion of the two positive nuclei in the absence of electrons. Thus the role of the latter reduces to screening the nucleus of the negative ion \(\mathrm{H}^-\), the more complete the greater the distance \(R\), and vanishing entirely at \(R=0\).
5. Generalization and Discussion of the Preceding Results.
The result expressed by formula (8h) coincides with that which we would have obtained by replacing both electrons by a continuous distribution of negative charge with volume density \(\rho=-2e|\psi_1|^2\), i.e., by regarding the ion \(\mathrm{H}^{-}\) as a central positive charge \(+e\), surrounded by a radially symmetric negative atmosphere (electron “cloud”). In an analogous way one may also treat the general case of a more complicated negative ion. Denoting the charge of the latter by \(z'e\) and considering all the outer electrons as identical, we then obtain, instead of (8h),
\[ \bar U=-\frac{z'e^2}{R}+\frac{Ne^2}{R}\,e^{-\frac{2R}{a}}\,f\!\left(\frac{R}{a}\right), \tag{9} \]
where \(N\) denotes the number of outer electrons, and \(f\!\left(\frac{R}{a}\right)\) is a polynomial whose degree depends on the function \(\psi\) characterizing them. If the latter corresponds, from the point of view of the old conceptions of Bohr’s theory, to electronic orbits with principal quantum number \(n\), then the degree of the polynomial \(f\) must equal \(2n\). For \(R\to 0\) we must then have \(f=1\), i.e.
\[ \bar U=+\frac{N-z'}{R}\,e^2 . \]
Formula (9) remains approximately valid also in the case where the positive ion is not the hydrogen ion, but some more complicated ion possessing an electron shell of sufficiently small dimensions. These dimensions, from the point of view of wave mechanics, are determined by the parameter \(a\) in the expression for the wave function \(\psi\), which is always reducible to the product of a certain integral polynomial and an exponential factor of the form \(e^{-r/a}\), where \(a\) plays the same role as the radius of the electron orbit in Bohr’s theory. The dimensions of positive ions are in most cases so small in comparison with the dimensions of negative ions that they may be neglected.
Ya. I. Frenkel
Equating the derivative of \(U\) with respect to \(R\) to zero, we can determine, according to the preceding formulas, the normal dimensions of the molecule, i.e. the distance between the two nuclei \(R_0\), and its dissociation energy \(U_0 = U(R_0)\). Thus, for example, in the case of formula (8b), the following equation is obtained for \(R_0\):
\[ \left(\frac{R^2}{a^2}+\frac{R}{a}+\frac{1}{2}\right)e^{-\frac{2R}{a}}=\frac{1}{4}; \tag{10} \]
moreover \(\overline{U}\) is expressed in terms of \(R_0\) by the formula
\[ \overline{U}_0 = -\frac{e^2\left(\dfrac{R_0}{a}+\dfrac{1}{2}\right)} {a\left[\left(\dfrac{R_0}{a}\right)^2-\dfrac{R_0}{a}+\dfrac{1}{2}\right]}. \tag{11} \]
In Born’s theory of heteropolar crystals, the interaction energy of two oppositely charged ions is expressed by a two-term formula of the form
\[ U=-\frac{e^2}{R}+\frac{b}{R^n} \]
with two directly empirical parameters, \(b\) and \(n\), which are chosen empirically. In reality, as we see, the energy of repulsion is expressed by a term of the form
\[ \frac{1}{R}e^{-\frac{2R}{a}} f\!\left(\frac{R}{a}\right), \]
which can be approximated by a function \(\dfrac{b}{R^n}\) only within more or less narrow limits.
The theory of the heteropolar bond set forth here was developed by Unsöld;1 its application to the theory of heteropolar crystals2 gives much better agreement with experiment than does Born’s theory. The advantage of the new theoretical formula (for the energy \(U\)) consists, among other things, also in the fact that it contains in all only one
empirical constant \(a\), characterizing the sizes of negative ions. In principle this constant too can be calculated, which, however, presents great practical difficulties.
The theory of the heteropolar bond set forth here in principle coincides with the classical theory based on the conception of definite electronic orbits. Time averaging, giving an approximate value of the energy \(W\), as well as the statistical averaging of wave mechanics, is equivalent to replacing the moving electrons by a certain continuous distribution of electric charge with a time-independent density \(\rho\) around the corresponding nucleus. The essential difference between the new (wave) and the old (corpuscular) mechanics consists here in the following two circumstances.
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In the form of the function \(\rho\). In the case of Bohr’s theory, \(\rho\) has a value different from zero in a sphere-like layer of finite dimensions, the inner radius of which is equal to the smallest (perihelial), and the outer radius to the greatest (aphelial) distance of the electron under consideration from the nucleus. In wave mechanics \(\rho\) proves to be different from zero throughout all space; moreover, the exponential decrease of \(\rho\) with increasing distance \((r)\) is the immediate cause of the repulsive forces determined by the second term of formula (9).
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In the applicability of the method of averaging over unperturbed motion for very small distances \(R\). From the point of view of corpuscular mechanics, the penetration of a positive ion (for example, the nucleus \(H^{+}\)) into the region occupied by the electrons of a negative ion should have substantially disturbed the motion of the latter; whereas in wave mechanics this distortion may be ignored for values of \(R\) of the same order as, and even smaller than, the effective radius of the ion \(a\). Let us note, for comparison, that for a radially symmetric distribution of \(\rho\) the action of the electrons of the negative ion on the positive ion should, within the range of applicability of corpuscular mechanics \((R>a)\), reduce to a diminution of the charge of the nucleus of the negative ion by a constant amount corresponding to the total ...
number of these electrons. The reduction of the shielding, which is connected with the introduction of a positive ion into the electronic shell of a negative one and by which the replacement of attractive forces by repulsive forces is brought about, cannot, according to the method set forth above, be treated in corpuscular mechanics (averaging over the unperturbed motion).
6. Homopolar bond (hydrogen molecule).
If in the case of a heteropolar bond the old and the new theories differ from one another rather only quantitatively than qualitatively, then in the case of a homopolar bond, of which the bond between two hydrogen atoms in the molecule \(H_2\) is the prototype, a profound difference of principle is found between the two theories; here the old theory proves to be completely helpless, whereas the new one makes it possible directly to solve this problem, which has remained unsolved until now.
The distinctive feature of the problem of the homopolar hydrogen molecule \(H_2\), as compared with the problem considered above of the heteropolar molecule \(\left(\overset{+}{H}\ \overset{-}{H}\right)\), consists in the fact that both electrons (1) and (2) are “ascribed”—if one may so express it—to two different nuclei \((A)\) and \((B)\), i.e., consequently, in the zero approximation (in the absence of interaction between the two H atoms) they may be characterized by two different pairs of functions:
\[ \psi_{A1}=\frac{1}{\sqrt{\pi a^3}}\,e^{-\frac{r_{A1}}{a}}, \qquad \psi_{B2}=\frac{1}{\sqrt{\pi a^3}}\,e^{-\frac{r_{B2}}{a}} . \tag{12} \]
\[ \psi_{A2}=\frac{1}{\sqrt{\pi a^3}}\,e^{-\frac{r_{A2}}{a}}, \qquad \psi_{B1}=\frac{1}{\sqrt{\pi a^3}}\,e^{-\frac{r_{B1}}{a}}, \tag{12a} \]
where the quantities \(r_{A1}\), etc., denote the distances between the corresponding particles.
The state of the system formed by the totality of the two atoms (irrespective of whether they are bound to one another or not) is then characterized in the first case by the function \(\psi_{12}=\psi_{A1}\psi_{B2}\), and in the second by the function \(\psi_{21}=\psi_{A2}\psi_{B1}\). These two
states, differing from one another only by the transposition of one electron into the place of the other, are, as it were, two twins, completely similar to each other in physical respect and differing only in name. They have, in particular, one and the same energy \(W^0 = W_{A_1}+W_{B_2}=W_{A_2}+W_{B_1}\) (we are speaking, of course, of the zeroth approximation). Under such conditions the calculation of the additional energy, in other words the energy of interaction of the two atoms according to formula (4), i.e. according to the formula
\[ W'=\iint U_{12}|\psi_{12}|^2\,dV_1dV_2,\quad U_{12}=\frac{e^2}{r_{AB}}+\frac{e^2}{r_{12}}-\frac{e^2}{r_{B1}}-\frac{e^2}{r_{A2}}. \tag{13} \]
in the case (12) or
\[ W'=\iint U_{21}|\psi_{21}|^2\,dV_1dV_2,\quad U_{21}=\frac{e^2}{r_{AB}}+\frac{e^2}{r_{12}}-\frac{e^2}{r_{A1}}-\frac{e^2}{r_{B2}} \tag{13a} \]
in the case (12a), is possible only under the condition of equality to zero of the expression
\[ S'=\iint U_{12}\psi_{12}\psi_{21}\,dV_1dV_2 = \iint U_{21}\psi_{12}\psi_{21}\,dV_1dV_2, \tag{14} \]
which characterizes the probability of spontaneous transitions of the type \(\psi_{12}\rightleftarrows\psi_{21}\), i.e. the probability of “exchange by electrons” between the two atoms (see below).
It is easy, however, to convince oneself that this expression is in fact different from zero, and that, consequently, the preceding formulas for the interaction energy are inapplicable in the case under consideration.
With what are they to be replaced?
To this question wave mechanics gives the following very simple answer. The states \(\psi_{12}\) and \(\psi_{21}\) can be realized simultaneously, so to speak superposed on one another or interfering with one another, similarly to how the superposition (interference) of two different types of waves with the same frequency of oscillation occurs (let us recall that the frequency of oscillations is measured by the corresponding energy according to the formula \(\nu^0=\frac{W^0}{h}\)). We may therefore, for the characterization of the unperturbed system formed by the totality of both
of the hydrogen atoms, replace the initial functions \(\psi_{12}\) and \(\psi_{21}\) by two linear combinations of the latter:
\[ \begin{aligned} \psi_{\mathrm{I}}&=\gamma_{\mathrm{I}2}\psi_{12}+\gamma_{\mathrm{I}1}\psi_{21},\\ \psi_{\mathrm{II}}&=\gamma_{\mathrm{II}1}\psi_{12}+\gamma_{\mathrm{II}2}\psi_{21}, \end{aligned} \tag{15} \]
choosing the coefficients \(\gamma\) in such a way that the functions \(\psi_{\mathrm{I}}\) and \(\psi_{\mathrm{II}}\) satisfy the usual conditions of “normality”
\[ \iint \psi_{\mathrm{I}}^{2}\,dV_{1}dV_{2} = \iint \psi_{\mathrm{II}}^{2}\,dV_{1}dV_{2} =1 \]
and orthogonality
\[ \iint \psi_{\mathrm{I}}\psi_{\mathrm{II}}\,dV_{1}dV_{2}=0 \]
and, moreover, the condition
\[ \iint U\psi_{\mathrm{I}}\psi_{\mathrm{II}}\,dV_{1}dV_{2}=0, \tag{15a} \]
where \(U\) denotes the interaction energy.
The functions \(\psi_{\mathrm{I}}\), \(\psi_{\mathrm{II}}\), determined in accordance with these conditions, have the following form:
\[ \begin{aligned} \psi_{\mathrm{I}}&=\frac{1}{\sqrt{1+X}}\left(\psi_{12}+\psi_{21}\right),\\ \psi_{\mathrm{II}}&=\frac{1}{\sqrt{1-X}}\left(\psi_{12}-\psi_{21}\right), \end{aligned} \tag{16} \]
where
\[ X=\iint \psi_{12}\psi_{21}\,dV_{1}dV_{2}. \tag{16a} \]
The first of these is symmetric with respect to both electrons, or more precisely to their coordinates, while the second is antisymmetric (in the sense that it changes sign when the electrons are interchanged). Accordingly, the states of the \(\mathrm{H}+\mathrm{H}\) system characterized by them are called symmetric and antisymmetric.
In the absence of interaction between the atoms, both these states have the same energy, namely the same energy \(W^{0}\) as the initial states. When this interaction is taken into account, the energy of the first of them changes according to (4) by
\[ W'_{\mathrm{I}}=\iint U\psi_{\mathrm{I}}^{2}\,dV_{1}dV_{2}. \]
and the second by
\[ W'_{\mathrm{II}}=\iint U\psi_{\mathrm{II}}^{2}\,dV_1dV_2. \]
Here, just as in (15a), by \(U\) it is necessary to understand different quantities depending on the factor multiplying \(U\), which is obtained if \(\psi_{\mathrm{I}}^{2}\) and \(\psi_{\mathrm{II}}^{2}\) are expanded into a sum of squares and products of the initial functions \(\psi_{12}\) and \(\psi_{21}\). This is explained by the fact that the mutual potential energy of two hydrogen atoms admits of an unambiguous definition only in the case when both electrons are “assigned” to definite nuclei. Accordingly, we must put \(U=U_{12}\) with the factor \(\psi_{12}^{2}\) and \(U_{21}\) with \(\psi_{21}^{2}\); in the case of the factor \(\psi_{12}\psi_{21}\), one may choose either definition of \(U\), since they prove to be equivalent. I have no possibility of dwelling here on a more detailed justification of this (not claiming great accuracy) method of calculating \(W'\). I shall only note that the error associated with it is the smaller, the smaller the integral (16a).
As a result, for the additional energy of the symmetric and antisymmetric states the following approximate expressions are obtained:
\[ \begin{aligned} W'_{\mathrm{I}}&=\frac{W'+S'}{1+I},\\ W'_{\mathrm{II}}&=\frac{W'-S'}{1-I}. \end{aligned} \tag{17} \]
Substitution here of expressions (12), (13), and (19) gives, after rather laborious calculations, formulas of the following form:
\[ W'_{\mathrm{I,II}}=\frac{e^{2}}{R}+\frac{e^{2}}{R}\, \frac{e^{-\frac{2R}{a}}\,f\!\left(\frac{R}{a}\right)} {1\pm e^{-\frac{2R}{a}}\,\varphi\!\left(\frac{R}{a}\right)}, \tag{17a} \]
where \(f\!\left(\frac{R}{a}\right)\) and \(\varphi\!\left(\frac{R}{a}\right)\) are fourth-degree polynomials having different coefficients in the cases \(W'_{\mathrm{I}}\) and \(W'_{\mathrm{II}}\). We note that the first term in (17a) is the potential energy of the two positive nuclei with respect to one another, while the second—
(averaged) potential energies of the electrons according to their relation to one another and to “alien” nuclei.
The form of the functions \(W'_1(k)\), \(W''_n(k)\), and also \(W'(R)\) is represented graphically in Fig. 1.
It follows from the latter that, in the case of the antisymmetric state of the system \(\mathrm{H}+\mathrm{H}\), repulsion occurs between the two atoms, increasing monotonically as the distance \(R\) decreases; in the case of the symmetric state, repulsion predominates only at small distances \(\left(R<\frac{3}{2}a\right)\), whereas at large distances it is replaced by attraction.
Fig. 1.
At a distance \(R=\frac{3}{2}a\) between the two nuclei, the force of interaction becomes zero.
The corresponding minimal value of the energy \(W'_1\) is equal to \(-2.5\) volts. We thus obtain an image of dissociation of the system \(\mathrm{H}+\mathrm{H}\), i.e., of the hydrogen molecule in the normal state.
These results—in particular the dissociation energy—are in satisfactory agreement with experimental data.
7. Discussion of the preceding theory of the homopolar bond.
The considerations set forth in the preceding paragraph and belonging to Heitler and London have a purely formal character. The question of their physical meaning appears in an entirely different light depending on whether we operate with wave or corpuscular representations.
From the wave point of view, the theory of the interaction of two identical atoms (or of any other systems) is perfectly analogous to the theory of oscillations of two coupled pendulums or of two inductively coupled electrical circuits. Each of these identical pendulums (or circuits) corresponds to one of the two states—differing merely by the exchange of electrons—
When there is no coupling between pendulums, each of them can oscillate independently of the other with a definite frequency $\omega^\circ$. When coupling is present, even if it is minimal, the following picture results. If at some initial moment only one pendulum is oscillating, while the other is at rest, then in the course of time the oscillations of the first must gradually be transmitted to the second until they change places, i.e., until the first pendulum comes to rest and the second receives all its energy. Then the process is repeated in the reverse order. The essential circumstance here is the coincidence of the periods of both pendulums (in the absence of coupling between them), i.e., in other words, the presence of resonance. In the case of two pendulums with different periods, the transfer of energy from one to the other occurs, with weak coupling, only to a very small extent—the smaller, the weaker this coupling. In the presence of resonance, however, the magnitude of the coupling plays no role, determining only the time (and not the degree) of transfer of energy from one pendulum to the other.
Simple theory shows that the oscillations of two coupled resonating pendulums remain stationary only in the case in which, at the initial moment, they were both oscillating with equal amplitudes and, moreover, with either identical or opposite phases. These two types of stationary oscillations, the first of which may be called symmetric and the second antisymmetric, correspond to two definite frequencies, $\omega_I$ and $\omega_{II}$, slightly different from one another. Any other nonstationary oscillation of both pendulums can be represented as the sum of a symmetric and an antisymmetric one with suitably chosen amplitudes and phases. The nonstationarity is caused by the interference of two oscillations of different frequencies $\omega_I$ and $\omega_{II}$. In this, periodic amplifications and weakenings, or beats, are observed in the amplitude of the oscillations of each pendulum, the number of which per unit time is equal to the difference $\omega_I - \omega_{II}$.
Analogous phenomena occur in two electrical circuits capable of oscillating with the same frequency, when there is weak coupling between them, and likewise in two atoms
hydrogen. In the latter case, however, the role of the individual pendulums is played, as was already mentioned above, not by individual atoms, but by one or another distribution of the electrons between them. To each of these distributions there corresponds, from the wave point of view, a special oscillatory process. The interaction between the atoms manifests itself in the fact that these processes cannot proceed independently of one another, but that they periodically reinforce one another at the expense of the other. The only exceptions are the cases of a symmetric and an antisymmetric combination of the two processes (with equal amplitudes and with identical or opposite phases). These combination processes possess definite frequencies
\[ \nu_{\mathrm{I}}=\frac{W_{\mathrm{I}}}{h} \quad \text{and} \quad \nu_{\mathrm{II}}=\frac{W_{\mathrm{II}}}{h}, \]
to which there correspond quite definite energies \(W_{\mathrm{I}}=W^{0}+W'_{\mathrm{I}}\) and \(W_{\mathrm{II}}=W^{0}+W'_{\mathrm{II}}\). All the remaining processes of this kind, not being stationary, do not possess a definite frequency and, consequently, a definite energy. They are connected with “beats,” i.e., with periodic reinforcements of one of the component processes \(\psi_{12}\) or \(\psi_{21}\) at the expense of the other, the beat frequency being equal to
\[ \nu_{\mathrm{I}}-\nu_{\mathrm{II}} = \frac{W'_{\mathrm{I}}-W'_{\mathrm{II}}}{h}. \]
From the corpuscular point of view, these results do not admit of a simple and clear interpretation, above all because it excludes the possibility of combining two different states in one and the same system. Thus, for example, according to corpuscular mechanics, a hydrogen atom cannot be simultaneously in the normal state and in one of the excited states. From the wave point of view, however, such a combination—or interference—appears entirely natural, like, for example, the simultaneous sounding of the fundamental tone of a string and one of its overtones. In this circumstance lies the principal difficulty of the corpuscular theory and its principal difference from the wave theory. In the case we have considered of two hydrogen atoms, both electrons must either periodically, \(\nu_{\mathrm{I}}-\nu_{\mathrm{II}}\) times per second, exchange places (Elektronenaustausch), or else must simultane-
may be in both the one and the other position. The latter, of course, cannot be represented to oneself, just as one cannot represent to oneself the difference between the symmetric and antisymmetric states—so long as the matter concerns the instantaneous position of the two electrons. It should be borne in mind, however, that wave mechanics in principle excludes the possibility of an exact determination of the positions of any particles and of their displacement in space. It admits all conceivable positions, confining itself to the determination of the probability of the various positions, or of the mean (statistical) distribution of density, characterized by the square (modulus) of the function \(\psi\). Therefore, from the point of view of the corpuscular interpretation of wave mechanics, an electron assigned to a definite atom may be at as great a distance from it as desired. Under such conditions it is not difficult to imagine that one and the same electron may be simultaneously assigned to two different atoms.
As regards the mean distribution of the electric charge of the electrons, determined by the functions \(\psi_I\) and \(\psi_{II}\), it differs in that in the second case (the antisymmetric state) the mean charge density vanishes at the points of the plane symmetrically situated between the two nuclei, whereas in the first case (the symmetric state) no such “nodal” plane exists.
8. Generalization of the wave theory of the homopolar bond and the relation of the latter to the heteropolar bond.
The theory of the homopolar bond set forth above can be extended from the hydrogen molecule to more complex molecules of the same kind. This generalization was carried out chiefly by London. The guiding principle here is, as in the case of \(H_2\), the combination of two electrons belonging to different atoms into a symmetric pair, i.e. the introduction of such states of the resulting system as are characterized by symmetry of the Schrödinger function \(\psi\) with respect to the coordinates of these electrons. These states,
since the corresponding antisymmetric ones are, above all, stationary, i.e. possess a quite definite energy (which cannot be said of the others); moreover, as the distance between the atoms is decreased, the energy of the symmetric states passes through a negative minimum, characterizing the affinity of both atoms for one another. Thus the units of affinity, long since introduced by chemists, for the first time receive their theoretical interpretation—as the energy of interaction of separate pairs of electrons belonging to these atoms. Let us note that an analogous conception was long ago advanced in the theory of the chemical bond by Bohr, Debye, Kossel, and others. In that primitive theory, however, both electrons were treated as a “ring” rotating about the straight line joining the two atoms. The modern theory of Heitler and London has nothing in common with the representation of such rings, apart from the circumstance that the two electrons symmetrically determine the state of the resulting system.
The preceding conception of symmetric electron pairs requires, however, one very substantial correction. The necessity of the latter is clear from the following considerations. If every electron of one atom could combine in a pair with one of the electrons of another, then the maximum number of units of affinity of any atom would be equal to the total number of electrons in it. In reality it is, generally speaking, much smaller, being determined by the number of so-called valence electrons. How, then, are “valence” electrons distinguished from ordinary ones?
Further, in the case of three or more atoms, it would be possible to combine electrons into symmetric triples, quadruples, etc.; moreover, such symmetric electron groups would have to correspond to stationary states with still lower energy than separate pairs. As a result, alongside binary molecules \(H_2\), there should exist triple \(H_3\), quadruple \(H_4\), etc., still more stable than \(H_2\). Why, then, do such molecules not exist, and why is symmetry limited to separate pairs of electrons?
To this question wave mechanics, as such, gives no answer. It is resolved by a special principle, called the prohibition of equivalence (or symmetry), or else Pauli’s principle, who first formulated it distinctly in 1925 in connection with the quantum (Bohr) theory of the structure of complex atoms.
In complex atoms the electrons are arranged around the nucleus in the form of a series of shells or groups, the innermost group closest to the nucleus containing, as is known, only two electrons. The energy of the atom would decrease if all the electrons were distributed in the form of a single inner group (or “ring”). Such an arrangement, however, proves impossible. A more detailed investigation of the outer groups shows that there too we have subgroups of two electrons each, with identical orbits (equivalent electrons).
Without attempting to explain this circumstance, Pauli elevated it into a general principle, asserting that in any atomic or molecular system there cannot exist more than two equivalent electrons. The physical meaning of this pair consists in the fact that electrons possess a magnetic moment, usually ascribed to their rotation about their own axis, and that the latter, under given conditions, can assume only two opposite directions. In equivalent electrons the magnetic axes must have different, i.e. opposite, directions. This is the more exact formulation of Pauli’s principle, which in essence means that in one and the same atomic or molecular system there do not exist at all completely equivalent electrons, i.e. electrons not only with identical quantum orbits but also with identical direction of axes.
In wave mechanics the equivalence of two electrons is expressed, in the simplest case, by the symmetry of the function \(\psi\) with respect to their coordinates. In this case Pauli’s principle reduces to the exclusion of such states as would be characterized by wave functions symmetric with respect to three or more electrons.^1 This principle, as was already mentioned above, does not follow from the basic prin-
types of wave mechanics, but fully agrees with them in the sense that states excluded at some moment \(t=0\) cannot, according to the fundamental equation of wave mechanics, arise with the passage of time.
With the aid of the Pauli principle, the questions raised above concerning the number of valence electrons in the atom and the impossibility of the existence of triple, quadruple, etc., symmetric groups of electrons are immediately resolved. The second question falls away of itself. As for the first, it is reduced to the second. Namely, in complex atoms the majority of electrons, especially all the inner electrons, form symmetric pairs (for example, two electrons of an inner electron group). In considering the interaction of two different atoms, these, so to speak, “mated” electrons may (in the first approximation) be left out of account. A bond between atoms can be effected only by the pairwise coupling of those electrons which were in them in the “single state” (the principle of monogamy!). These
¹ The most general and exact formulation of the Pauli principle in wave mechanics was given by Dirac in the form of a restriction of all conceivable functions \(\psi\) to functions antisymmetric in the extended sense of the word, connected not only with taking account of the coordinates of the electrons, but also of their orientations. These functions can be constructed by adjoining to the three coordinates of each electron a fourth variable, characterizing its orientation and capable of assuming only two values \(\left(+\frac{1}{2}\ \text{and}\ -\frac{1}{2}\right)\), corresponding to its two possible orientations. Antisymmetric (in Dirac’s sense) functions must change their sign under the interchange of two quadruples of arguments characterizing two different electrons. In the case of complete equivalence of the latter, this interchange should obviously not affect the magnitude of the function \(\psi\), which in the case of antisymmetric functions is possible only for \(\psi=0\) (\(\psi'=-\psi\)). Thus, by introducing these functions, we automatically exclude the possibility of equivalence of two electrons.
It should be noted that if one takes into account only the coordinates (besides the fourth “axial variable”), then Dirac’s “antisymmetric” functions acquire a greater or lesser degree of symmetry. What is essential is the circumstance that to each such function there corresponds a quite definite degree of orientation of the electrons and, conversely, to each degree of orientation there corresponds one definite function.
“Idle” electrons are also valence electrons, and their number represents the maximum number of units of affinity of the given atom.
Let us note that in electronic chemistry the positive or negative valence of some atom is usually determined by the number of electrons that it can, under appropriate conditions, lose or capture. For the most part, the readily detachable electrons are precisely the “idle” electrons, so that “positive valence” practically coincides with valence in the sense of the word given above.
It should further be noted that the formation of a heteropolar molecule, for example the NaCl molecule, is connected not only with the transfer of the valence electron of Na to Cl, but also with the pairing of this electron with one of the electrons of Cl into a symmetrical pair. In this pairing one may see the very essence of the chemical bond between the two atoms, for it is the common feature of every chemical bond, both homopolar and heteropolar. As for “heteropolarity,” from this point of view it should be regarded not as the antithesis of “homopolarity,” but rather as a supplement to the latter. The center of gravity of the electron cloud formed by two “wedded” electrons may lie closer to one atom than to the other. This circumstance accounts for the “polarity” of the molecule formed, i.e. the presence in it of a greater or lesser electric moment. The essence of the matter, however, lies not in the eccentric position of the bonding pair of electrons, but in the presence of this pair.