Quantum Theory of Chemical Forces

Ya. I. Frenkel, Leningrad

I. General Principles

It has long been known that the forces between atoms in molecules and between molecules in solid and liquid bodies are electrical in origin, i.e., arise owing to the electric charges of the electrons and nuclei of which neutral atoms are built.

Electronic theory has provided a general understanding of the fact that these forces reduce to attraction at large distances and repulsion at small distances comparable with the dimensions of atoms. A neutral atom creates a strong electric field that displaces the electrons and the nucleus of another atom in opposite directions, so that the attracting particles must come closer together, while the repelling ones move farther apart from one another. As a result, the forces of attraction increase, while the forces of repulsion, on the contrary, decrease; the resulting attraction thereby obtained rapidly disappears as the distance increases. This is the essence of the van der Waals forces of attraction.

At very small distances between the nuclei of two atoms, smaller than the radius of their outer electron shells, they must begin to repel one another, since the two positive nuclei are no longer screened by electrons. This qualitatively explains the repulsive van der Waals forces that determine the hardness of matter.

The explanation given for attraction and repulsion between neutral atoms leaves no room for chemical forces, which are an attraction essentially different from the van der Waals attraction in its specific character and in the presence of the phenomenon of saturation.

Generally speaking, these chemical forces can be explained as the result of a partial collectivization of the outer (valence) electrons of two atoms entering into combination. The collectivized electrons no longer move about a separate nucleus, but each of them circulates around both nuclei and partially compensates the mutual repulsion of the latter, passing from time to time between them. Such compensation is possible only at certain intermediate distances between the nuclei. At distances that are too small, the shared electrons move mainly outside the region lying between the nuclei, not screening ...

of the latter, so that the repulsive forces are no longer compensated. At very large distances, collectivization of the outer electrons does not take place or, more precisely, proves ineffective, as will be shown below. In this case the two atoms may be regarded as two separate systems perturbing one another, this perturbation manifesting itself in the appearance of van der Waals attractions.

A chemical combination of two identical atoms can occur only as a result of a partial collectivization of their outer electrons. Chemical saturation is explained by the limitation on the number of collectivized electrons or pairs of electrons in connection with the Pauli principle.

In the absence of such electrons—as, for example, in the case of the inert gases—a chemical bond is impossible. In the case of unlike atoms, the shared electrons move asymmetrically with respect to both nuclei, being displaced in the direction of the nucleus with the larger effective charge. The resulting molecule acquires a dipole moment along its axis. Such conditions are realized in a sharp form in “heteropolar” molecules, such as, for example, NaCl or HCl. Usually such molecules are represented as the result of the union of two oppositely charged ions, collectivization being replaced by the “expropriation” of one or several electrons of the “stronger” atom by the “weaker” one; the latter is thereby transformed into a positive ion, and the former into a negative one, so that the chemical bond is reduced to the mutual attraction of unlike charges.

However, this idea (first expressed by Berzelius as early as 100 years ago) represents the actual state of affairs rather crudely and simplistically. In fact, the chemical combination of two such atoms as Na and Cl occurs thanks to collectivization of the outer electrons—one in Na and seven in Cl. Since the effective charge of the latter is significantly greater than that of the former, all 8 collectivized electrons turn out to be strongly displaced toward Cl, and the collectivization therefore appears as expropriation. This distribution of electrons loses its polar character as the difference between the two nuclei decreases, becoming completely symmetrical in the case of identical atoms.

Thus we see that the difference between homopolar and heteropolar molecules is purely quantitative, depending on the magnitude of the displacement of the collectivized electrons in the direction of one of the partners.

A diatomic molecule is usually imagined as the sum of two atoms held in equilibrium at small distances from one another by repulsive and attractive forces, the atoms remaining completely unchanged by their mutual proximity. Such a conception of unchanged atoms acting upon one another with forces depending on the distance between their centers goes back to the time when atoms were still regarded as indivisible units that could be described simply as force centers;

In modern physics and chemistry, where the role of force centers is played by electrons and nuclei, this traditional conception of immutable atoms must be replaced by the conception of complex systems that change substantially upon mutual approach and transform into a single new system, which is characterized by the collectivization of a certain number of outer electrons. This change of the combining atoms in the case of heteropolar molecules is roughly described by replacing the concept of “atoms” with the concept of “ions” (which, again, is connected with the idea of two charged force centers that remain unchanged if they are separated).

An understanding of the structure and properties of molecules, as well as of those solids (crystals) in which the atoms are held by chemical forces (such as, for example, diamond, metals, ionic crystals), can be achieved only if we abandon the traditional way of describing these systems as sums of separate immutable atoms and regard them as the sum of nuclei (screened by inner electrons) and collectivized valence electrons, whose motion depends on the distances between the nuclei.

Thus the conception of chemical forces as forces acting between individual immutable atoms is entirely false. The dynamics of a system (molecule or crystal) is determined by the motion of collectivized electrons around nuclei, whose positions are established a posteriori from the principle that the energy of motion in the normal state must have a minimal value.

2. Motion of an electron around two fixed nuclei. The ion H₂⁺

Turning to the quantitative development of the ideas expressed in the preceding paragraph, we must first of all consider the simplest case of the collectivization of one electron by two identical nuclei. This case corresponds to the formation of the ion H₂⁺ (which is a stable system) from an H atom and an H⁺ ion. According to traditional conceptions, the H₂ ion is described simply as the sum of an H atom and an H⁺ ion.

However, such a description is erroneous, since the formation of the H₂⁺ ion from H and H⁺ is caused by a radical transformation of the electron’s motion into one that is symmetric with respect to both nuclei (protons).

We shall not attempt to describe the course of this transformation in time. It must be regarded as a transition from a stationary state in which the electron is bound to one of the protons, while absent from the other or located at an infinite distance, into a stationary state in which it is symmetrically bound to two protons situated at a given distance \(R\) from one another.

The description of such a transition is associated with difficulties, because it constitutes a motion of the protons (or, at least, one of them)

...of them, if the second is immobile) outside the domain of wave-mechanical study.

Under such conditions it is impossible to establish a definite value for the probability of the transition process. In connection with this, one circumstance must be emphasized. If the distance between two nuclei (regarded as force centers) is kept unchanged, then it becomes meaningless to speak of a stationary state of an electron bound to one nucleus, because in the presence of two identical nuclei at a finite distance from one another only those states can be stationary (i.e. bound to a definite value of the energy) in which the electron is symmetrically bound to both nuclei, i.e. for which the probability of finding the electron at symmetrically situated points has one and the same value. Introducing a coordinate system with the \(x\)-axis situated along the line joining the nuclei, and with the origin at the midpoint of the distance between them, we must describe the stationary states of the electron by such wave functions

\[ \psi(x,y,z,t)=\psi^\circ(x,y,z)e^{-i\frac{2\pi}{h}Wt}, \]

which are either symmetric or antisymmetric with respect to \(x\), i.e. which satisfy one of the two relations

\[ \psi^\circ(-x,y,z)=\psi^\circ(x,y,z) \]

or

\[ \psi^\circ(-x,y,z)=-\psi^\circ(x,y,z). \]

Since the function \(\psi^\circ\) must in both cases be symmetric with respect to \(y\) and \(z\) (the \(x\)-axis is an axis of rotational symmetry), the preceding relations may be rewritten in the form

\[ \psi^\circ(-x,-y,z)=\psi^\circ(x,y,z), \tag{1} \]

i.e. the function \(\psi^\circ\) is symmetric with respect to the center \(O\), or

\[ \psi^\circ(-x,-y,z)=-\psi^\circ(x,y,z), \tag{1a} \]

i.e. the function \(\psi^\circ\) is antisymmetric with respect to \(O\).

In the first case the function \(\psi^\circ\) is called even, and in the second—odd.

The probability density \(|\psi^\circ|^2\) in both cases is symmetric with respect to \(x\) (or \(O\)). The general method of finding these functions consists in solving the wave equation

\[ \nabla^2\psi^\circ+\frac{8\pi^2m}{h^2}(W-U)\psi^\circ=0 \tag{2} \]

with potential energy

\[ U=-\frac{e^2}{r_a}-\frac{e^2}{r_b}, \tag{2a} \]

corresponding to the motion of an electron in the field of two fixed protons; \(r_a\) and \(r_b\) denote the distances of the electron from the latter.

However, the exact solution of this problem presents great difficulties, and moreover it is too complicated to be useful. An approximate solution satisfying relation (1) or (1a) can easily be found in the limiting case of very large internuclear distances by means of an even or odd combination of two atomic wave functions

\[ \varphi_a(\mathbf r_a,t)=\varphi_a^\circ(\mathbf r_a)e^{-i\frac{2\pi}{h}Wt} \]

and

\[ \varphi_b(\mathbf r_b,t)=\varphi_b^\circ(\mathbf r_b)e^{-i\frac{2\pi}{h}Wt}, \]

representing stationary states of an electron bound to one nucleus (the other either absent or at an infinite distance). These two functions form two independent approximate solutions of equation (2) and, for the limiting case \(R\to\infty\), correspond to one and the same value \(W\). The linear combination

\[ \psi^\circ=C_a\varphi_a(\mathbf r_a)+C_b\varphi_b(\mathbf r_b) \]

with arbitrary coefficients \(C_a\) and \(C_b\) also forms an approximate solution of (2), corresponding to the same value of the energy. Putting \(C_a=C_b\) or \(C_a=-C_b\), we obtain two solutions satisfying conditions (1) or (1a). Since for \(R\to\infty\) the two functions \(\varphi_a(\mathbf r_a)\) and \(\varphi_b(\mathbf r_b)\) become orthogonal to each other (one of them vanishes in the region where the other is finite), from the condition

\[ \int \psi^*\psi\,dV=1 \]

in connection with

\[ \int |\psi(\mathbf r_a)|^2\,dV=\int |\psi(\mathbf r_b)|^2\,dV=1 \]

it follows that the coefficients \(C_a\) and \(C_b\) are numerically equal to \(\frac{1}{\sqrt{2}}\). Thus we obtain

\[ \psi_+^\circ=\frac{1}{\sqrt{2}}\,[\varphi^\circ(\mathbf r_a)+\varphi^\circ(\mathbf r_b)], \tag{3} \]

\[ \psi_-^\circ=\frac{1}{\sqrt{2}}\,[\varphi^\circ(\mathbf r_a)-\varphi^\circ(\mathbf r_b)]. \tag{3a} \]

It is natural to assume that these expressions will remain approximately valid also for finite values of \(R\) that are not too small.

In this case the functions \(\varphi^\circ(\mathbf r_a)\) and \(\varphi^\circ(\mathbf r_b)\) are no longer exactly orthogonal, and in order to satisfy the normality condition for \(\psi_\pm^\circ\) we must put

\[ C_\pm=\frac{1}{\sqrt{2(1\pm I)}}, \]

where

\[ I=\int \varphi^\circ(\mathbf r_a)\varphi^\circ(\mathbf r_b)\,dV; \]

where the plus sign applies to the even case, and the minus sign to the odd case.

Expressions (3) and (3a) must accordingly be replaced by the corrected ones

\[ \psi_+^{\circ}=\frac{\varphi^{\circ}(r_a)+\varphi^{\circ}(r_b)} {\sqrt{2(1+I)}} , \tag{4} \]

\[ \psi_-^{\circ}=\frac{\varphi^{\circ}(r_a)-\varphi^{\circ}(r_b)} {\sqrt{2(1-I)}} . \tag{4a} \]

It is interesting to test these formulas for the limiting case \(R\to 0\), corresponding to the coalescence of the two hydrogen nuclei into one nucleus with charge \(2e\). If expressions (4a) and (4) were correct, then in this limit they would turn into wave functions representing two different stationary states of the ion \(\mathrm{He}^+\). Since the integral \(I\) in this case is equal to 1, function (4) reduces to \(\varphi^{\circ}(r)\), where \(r=r_a=r_b\). This result is obviously incorrect, since \(\varphi^{\circ}(r)\) is the wave function of the H atom, but not of the \(\mathrm{He}^+\) ion. If, for example, the function \(\varphi^{\circ}(r)\) corresponds to the normal state of H, then

\[ \psi(r)=\varphi^{\circ}(r)=\sqrt{\frac{\alpha^3}{\pi}}\,e^{-\alpha r}, \]

whereas the corresponding state of \(\mathrm{He}^+\), which should be represented by (4), is in fact described by the function

\[ \sqrt{\frac{(2\alpha)^3}{\pi}}\,e^{-2\alpha r}. \]

Expression (4a) for the odd state becomes indeterminate at \(R=0\). In order to determine its form in this limiting case, we first take a very small, but finite, \(R\). Denoting the distance of the electron from the origin by \(r\) (see Fig. 1), we obtain:

\[ r_b \simeq r+\frac{1}{2}R\cos\vartheta,\qquad r_a \simeq r-\frac{1}{2}R\cos\vartheta, \]

where \(\vartheta\) is the angle between \(r\) and the \(x\)-axis. Hence

\[ e^{-\alpha r_a}\simeq e^{-\alpha r}\left(1-\frac{1}{2}\alpha R\cos\vartheta\right),\qquad e^{-\alpha r_b}\simeq e^{-\alpha r}\left(1+\frac{1}{2}\alpha R\cos\vartheta\right) \]

and, consequently,

\[ I=\frac{\alpha^3}{\pi}\int e^{-2\alpha r}\left(1-\frac{\alpha^2R^2}{4}\cos^2\vartheta\right)dV =1-\frac{\alpha^2R^2}{12}, \]

Fig. 1.

Since

\[ \frac{\alpha^{3}}{\pi}\int e^{-2\alpha r}\,dV=1, \]

and the mean value of \(\cos^{2}\vartheta\) is equal to \(1/3\), we thus obtain

\[ \psi^{0}=\sqrt{\frac{\alpha^{3}}{\pi}}\,e^{-\alpha r}\cos\vartheta . \]

This function is similar to the wave function

\[ \sqrt{\frac{\alpha^{3}}{\pi}}\,\alpha r e^{-\alpha r}\cos\vartheta, \]

which describes one of the two-quantum states \(H_{2}^{+}\). It differs from the latter only by the absence of the factor \(\alpha r\). We see, therefore, that expressions (4) and (4a) can be applied only for large and for mean values of the internuclear distance.

One of the properties of expressions (4) and (4a), however, remains valid for all values of \(R\): this is their “evenness” or “oddness” in the sense indicated above (i.e. symmetry or antisymmetry with respect to the point \(O\)). This property becomes especially characteristic in the limiting case of large \(R\). The mean distribution of electric charge, whose volume density is determined by the product \(|\psi^{0}|^{2}e\) (where \(e\) is the absolute value of the electron charge), becomes in this case the sum of distributions with volume densities

\[ \frac{1}{2}e\,|\varphi^{0}(r_{a})|^{2} \]

near nucleus \(a\) and

\[ \frac{1}{2}e\,|\varphi^{0}(r_{b})|^{2} \]

near nucleus \(b\).

The electron behaves as if it spent part of the time near \(a\), and part of the time near \(b\), with the same charge distribution as in the case of one nucleus, but with a “half” electron charge. It does not follow from this that the electron oscillates, passing periodically from one nucleus to the other; this would mean that the probability density \(|\psi|^{2}\) is a function of time, whereas in a stationary state of the electron it must not depend on time. Since the energy \(W\) of the electron is regarded as exactly known (as in the case of any stationary state), the time to which one or another position of it refers must, according to Heisenberg’s uncertainty relation, be completely indeterminate.

At mean internuclear distances we obtain two different types of symmetric distribution of electric charge, corresponding to the even and odd functions \(\psi^{0}\). In the first case the two halves of the electron, each of which is connected with one of the nuclei, are drawn toward one another, forming a threadlike strand of negative electricity connecting the two nuclei. If one draws the family of surfaces corresponding to constant charge density \(\rho=e|\psi|^{2}\), then one obtains a picture resembling equipotential—

tential surfaces in the case of two equal electric charges of the same sign (Fig. 2).

An even closer analogy is presented by the picture of a biological cell dividing before its complete separation; here the atomic nuclei correspond to the nuclei of the daughter cells, and the electronic charge to the protoplasm. A protoplasmic strand holds the cells close to one another until the process of division is completed. In our case the same role is played by the strand of “electronic protoplasm,” causing mutual attraction of the nuclei—as though some part of the electronic charge were concentrated between them.

Fig. 2.

For large and intermediate values of the distance \(R\), these attractive forces, caused by the wave-mechanical collectivization of the electron, more than compensate the mutual repulsion of the nuclei. For small \(R\), the part of the charge concentrated between the nuclei becomes too small; the electron is, as it were, pushed out into the outer region, so that the repulsion of the nuclei is no longer compensated. Thus we see that the force \(F\) between two nuclei is represented as a function of the distance \(R\) by the dashed curve in Fig. 3, where negative values of the force correspond to attraction, and positive values to repulsion.

Fig. 3.

The corresponding values of the potential energy \(U\)

\[ \overline{U}_{+}=\frac{e^{2}}{R}-e^{2}\int\left(\frac{1}{r_{a}}+\frac{1}{r_{b}}\right)\left|\psi_{+}^{\circ}\right|^{2}\,dV \tag{5} \]

are shown by the solid curve. At the minimum value of \(\overline{U}\), \(F\) is equal to zero. It should be noted that for small \(R\), \(U\) asymptotically reduces to the expression \(\frac{e^{2}}{R}\), characterizing the mutual repul-

…tion of the nuclei, whereas for large \(R\) it has the asymptotic form

\[ -\frac{e^2}{R}e^{-\alpha R}, \quad \text{or} \quad e^2\alpha e^{-\alpha R}. \]

The latter is easily obtained from the expression

\[ \psi^\circ_+ = \frac{1}{\sqrt{2}}\sqrt{\frac{\alpha^3}{\pi}}\left(e^{-\alpha r_a}+e^{-\alpha r_b}\right). \]

Squaring, we obtain

\[ |\psi^\circ_+|^2=\frac{\alpha^3}{2\pi}\left[e^{-2\alpha r_a}+e^{-2\alpha r_b}+2e^{-\alpha(r_a+r_b)}\right], \tag{6} \]

which, after substitution into (5), gives for large \(R\) an expression of the form written above.

Fig. 4.

In the opposite case of the odd wave function \(\psi^\circ_-\), the average distribution of the electric charge \(\rho=e|\psi^\circ|^2\) is such as if the two halves of the electron bound to the nuclei repelled one another. The surfaces \(\rho=\text{const}\) are now similar to the equipotential surfaces of two like charges; moreover, the plane passing midway between the two nuclei, perpendicular to the line joining them, corresponds to \(\rho=0\) (Fig. 4). In this case, for any \(R\), the mutual repulsion of the nuclei cannot be compensated, since there is always an insufficient negative charge between them. The resulting force as a function of the distance is represented in this case by the dotted curve in Fig. 5, and the corresponding potential energy

\[ \overline{U}=\frac{e^2}{R}-e^2\int\left(\frac{1}{r_a}+\frac{1}{r_b}\right)|\psi^\circ_-|^2\,dV \]

where

\[ |\psi_-^\circ|^2=\frac{\alpha^3}{2\pi}\left[e^{-2\alpha r_a}+e^{-2\alpha r_b}-2e^{-\alpha(r_a+r_b)}\right], \]

is represented by the solid curve. Thus a stable compound of two protons and electrons in the system \(\mathrm{H}_2^+\) is possible only in the case when the motion of the electron is described by an even wave function.

It is clear that for the same \(R\) the energy of this state is less than that of the other, described by an odd function obtained from the same atomic wave functions \(\varphi^\circ\), as is the even one.

3. Determination of the total energy and force

In the preceding exposition we did not take into account the kinetic energy \(\overline{T}\) of the electron. In a stationary state its mean value \(\overline{T}\), like \(\overline{U}\), is a function of \(R\), and the resultant force between the nuclei must be determined not as \(-dU/dR\), but as \(-dW/dR\), where \(W\) is the total energy, equal to \(\overline{T}+\overline{U}\).

Fig. 5.

This total energy can be determined in two ways: directly, based on the Schrödinger equation, and indirectly, first proposed by Slater, based on the virial theorem.

Multiplying the Schrödinger equation (1) by its solution \(\psi^\circ\), which we shall regard as real, and integrating, we obtain

\[ \int \psi^\circ \nabla^2 \psi^\circ dV+\frac{8\pi^2m}{h^2}\int (W-U)|\psi^\circ|^2\,dV=0; \]

whence, since

\[ \int \psi^{\circ 2}dV=1, \]

\[ W=\int U\psi^{\circ 2}dV-\frac{h^2}{8\pi^2m}\int \psi^\circ \nabla^2\psi^\circ dV. \tag{7} \]

This formula gives the exact value of the energy \(W\) if the function \(\psi^\circ\) is an exact solution of equation (1). Applying an approximate expression for \(\psi^\circ\), with the aid of (7) we can calculate the corresponding approximate value of \(W\). It should be noted that the first term in (7) is the mean potential energy \(\overline{U}\), corresponding

to the distribution of the electronic charge with volume density \(e|\psi^\circ|^2\), whereas the second term gives the mean or probable value of the kinetic energy, which is represented by the operator

\[ -\frac{h^2}{8\pi^2 m}\nabla . \]

If the mean potential energy is known, then by means of the virial theorem one can also find the mean kinetic energy. As is known, in the case of purely Coulomb forces the virial theorem reduces to the equality \(2\overline{T}=-\overline{U}\).

For its applicability in this form it is necessary either to take into account the motion of the positive nuclei, or else to assume that the resultant force acting on the nuclei vanishes, so that the latter remain immobile, as, for example, is the case if the ion \(\mathrm{H}_2^+\) is in the equilibrium state at \(R=R_0\).

For other values of \(R\) some external forces must act on the nuclei or, more precisely, two equal and opposite forces \(\mathbf F_a\) and \(\mathbf F_b\), balancing the electric forces \(\pm\mathbf F\) which they experience from one another and from the electron.

The virial theorem in this case takes the more general form

\[ 2\overline{T}=-\overline{U}-\mathbf F_a\mathbf r_a-\mathbf F_b\mathbf r_b, \]

or, since \(\mathbf F_a=-\mathbf F_b=\mathbf F\) (where \(\mathbf F\) is the electric force acting on \(b\)) and \(\mathbf r_b-\mathbf r_a=\mathbf R\) (the radius vector of \(b\) relative to \(a\)), then

\[ 2\overline{T}=-\overline{U}+\mathbf F\mathbf R, \tag{8} \]

where \(\mathbf F\) must be defined as

\[ \mathbf F=-\frac{d}{dR}\left(\overline{U}+\overline{T}\right). \tag{8a} \]

If the function \(\overline{U}(R)\) is known, for \(\overline{T}\) we obtain the following differential equation

\[ 2\overline{T}+R\frac{d\overline{T}}{dR} = -\overline{U}-R\frac{d\overline{U}}{dR}, \]

which is equivalent to the equation for the total energy \(W\)

\[ 2W+R\frac{dW}{dR}=\overline{U}. \]

Multiplying it by \(R\), we obtain

\[ \frac{d}{dR}\left(R^2 W\right)=R\overline{U}, \]

i.e.

\[ W=-\frac{1}{R^2}\int_0^\infty \overline{U}R\,dR, \tag{9} \]

the upper limit being chosen so that \(W=0\) as \(R\to\infty\).

Returning to the direct method of calculating \(W\), let us substitute in place of \(\psi^\circ\) in (7) one of the expressions (3) or (3a). Since \(\varphi^\circ(r_a)\) and \(\varphi^\circ(r_b)\) are exact solutions of the equations

\[ \nabla^2 \varphi^\circ(r_a)+\frac{8\pi^2 m}{h^2}(W^\circ-U_a)\varphi^\circ(r_a)=0 \]

and

\[ \nabla^2 \varphi^\circ(r_b)+\frac{8\pi^2 m}{h^2}(W^\circ-U_b)\varphi^\circ(r_b)=0 \]

for \(U_a=-\frac{e^2}{r_a}\), \(U_b=-\frac{e^2}{r_b}\), and \(W^\circ\) equal to the energy of the isolated atom H in the state \(\varphi^\circ\), then

\[ \overline{T}=-\frac{h^2}{8\pi^2 m}\int \psi^\circ \nabla^2\psi^\circ dV =\frac{1}{2(1\pm I)}\int(\varphi_a\pm\varphi_b)[(W^\circ-U_a)\varphi_a\pm \]

\[ \pm(W^\circ-U_b)\varphi_b]\,dV, \]

where, for brevity, we have put

\[ \varphi_a=\varphi^\circ(r_a)\quad \text{and}\quad \varphi_b=\varphi^\circ(r_b). \]

This expression is easily brought to the form

\[ \overline{T}=W^\circ-\frac{K\pm L}{1\pm I}, \]

where

\[ K=\int U_a\varphi_a^2\,dV=\int U_b\varphi_b^2\,dV, \tag{10} \]

\[ L=\int U_a\varphi_a\varphi_b\,dV=\int U_b\varphi_b\varphi_a\,dV. \tag{10a} \]

Next

\[ \overline{U}=\int U\varphi^2\,dV =\frac{1}{2(1\pm I)}\int\left(\frac{e^2}{R}+U_a+U_b\right)(\varphi_a\pm\varphi_b)^2\,dV, \]

i.e.

\[ \overline{U}=\frac{e^2}{R}+\frac{K+M\pm 2L}{1\pm I}, \tag{11} \]

where

\[ M=\int U_a\varphi_b^2\,dV=\int U_b\varphi_a^2\,dV. \tag{11a} \]

Hence

\[ W=W^\circ+\frac{e^2}{R}+\frac{M\pm L}{1\pm I}. \tag{12} \]

The integral \(M\) represents the Coulomb energy of an electronic charge distributed around one of the nuclei, under the assumption that the second nucleus is absent. \(L\) corresponds, as it were, to a mixture of two distributions of electronic charge (not additive, but multiplicative) belonging to different nuclei.

The integral \(L\) is usually called the “exchange integral.” \(M\) can be calculated in the following way. Imagine a sphere of radius \(R\) with center at \(a\). The part of the electron charge contained in this sphere, distributed radially symmetrically about \(a\), will act on the nucleus \(b\), located on the surface of the sphere, just as if the charge of this entire part were concentrated at the center; the potential of the outer part of the charge will be the same both at \(b\) and at the center of the sphere \(a\). We thus obtain

\[ M=\int U_b\varphi_a^{\,2}dV =-\frac{e^2}{R}\int_0^R \varphi_a^{\,2}4\pi r^2dr -e^2\int_R^\infty \varphi_a^{\,2}4\pi r\,dr = \]

\[ =-\frac{e^2}{R}\frac{\alpha^3}{\pi}\int_0^R e^{-2\alpha r}4\pi r^2dr -e^2\frac{\alpha^3}{\pi}\int_R^\infty e^{-2\alpha r}4\pi r\,dr = \]

\[ =4e^2\alpha^3\left[ -\frac{1}{R}\frac{\partial^2}{\partial\beta^2} \left(\frac{1-e^{-\beta R}}{\beta}\right) +\frac{\partial}{\partial\beta}\frac{e^{-\beta R}}{\beta} \right], \]

where

\[ \beta=2\alpha, \]

i.e.

\[ M=-\frac{e^2}{R}+2\frac{e^2}{R}(1+\alpha R)e^{-2\alpha R}. \]

The calculation of \(L\) and \(I\) is most easily carried out by introducing the elliptic (more precisely, spheroidal) system of coordinates \(\lambda,\mu,\varphi\), defined by the equalities \(r_a+r_b=R\lambda,\ r_a-r_b=R\mu\), or \(r_a=c(\lambda+\mu),\ r_b=c(\lambda-\mu)\), where \(c=\dfrac{R}{2}\), and \(\varphi\) is the azimuth relative to the \(x\)-axis.

Putting \(y^2+z^2=\rho^2\), we have

\[ r_a^2=(x+c)^2+\rho^2,\qquad r_b^2=(x-c)^2+\rho^2, \]

whence

\[ r_a^2-r_b^2=4cx=4c^2\lambda\mu, \]

\[ r_a^2+r_b^2=2(x^2+c^2+\rho^2), \]

therefore

\[ x=c\lambda\mu,\qquad \rho=c\sqrt{(\lambda^2-1)(1-\mu^2)}. \]

The square of the element of length

\[ ds^2=dx^2+dy^2+dz^2=dx^2+d\rho^2+\rho^2d\varphi^2 \]

in the new coordinate system is equal to

\[ ds^2=c^2(\lambda-\mu^2)\left(\frac{d\lambda^2}{\lambda^2-1}+\frac{d\mu^2}{1-\mu^2}\right) +c^2(\lambda^2-1)(1-\mu^2)d\varphi^2. \]

Hence one finds the expression for the volume element

\[ dV=c^3(\lambda^2-\mu^2)\,d\lambda d\mu d\varphi. \]

It should be noted that, in integration over the whole volume, \(\lambda\) must vary from \(1\) to \(\infty\), \(\mu\)—from \(-1\) to \(1\), and \(\varphi\)—from \(0\) to \(2\pi\). Thus

\[ L=e^2\int \frac{1}{r_a}\varphi_a\varphi_b\,dV =\frac{e^2\alpha^3}{\pi}\int \frac{1}{r_a}e^{-\alpha(r_a+r_b)}\,dV \]

or, since

\[ r_a+r_b=2c\lambda \quad \text{and} \quad r_a=c(\lambda+\mu), \]

\[ L=\frac{e^2\alpha^3c^2}{\pi}\int_0^{2\pi}\int_{-1}^{1}\int_1^\infty e^{-2\alpha c\lambda}(\lambda-\mu)\,d\lambda d\mu d\varphi = \]

\[ =2e^2\alpha^3c^2\int_{-1}^{1}\int_1^\infty e^{-2\alpha c\lambda}(\lambda-\mu)\,d\lambda d\mu =4e^2\alpha^3c^2\int_1^\infty e^{-2\alpha c\lambda}\lambda\,d\lambda \]

i.e.

\[ L=e^2\alpha(1+\alpha R)e^{-\alpha R}. \]

Similarly we find \(I\)

\[ I=\int \varphi_a\varphi_b\,dV =\frac{\alpha^3}{\pi}\int e^{-\alpha(r_a+r_b)}\,dV = \]

\[ =2\alpha^3c^3\int_1^\infty\int_{-1}^{1} e^{-2\alpha c\lambda}(\lambda^2-\mu^2)\,d\lambda d\mu = \]

\[ =\left[1+\alpha R+\frac{1}{3}(\alpha R)^2\right]e^{-\alpha R}. \]

Thus, according to (12),

\[ W=W^0+\frac{e^2}{R}+ \frac{\frac{e^2}{R}\left[-1+2(1+\alpha R)e^{-2\alpha R}\right]\pm e^2\alpha(1+\alpha R)e^{-\alpha R}} {1\pm\left[1+\alpha R+\frac{1}{3}(\alpha R)^2\right]e^{-\alpha R}}. \]

The reader may verify for himself that the same result for the total energy is obtained with the aid of equation (9).

We shall not discuss the question of what the degree of approximation is that is obtained by this method for intermediate values of \(R\). We already know that, as \(R\) decreases, the accuracy of the results must become less and less. The qualitative character of our calculations does not, however, alter their principal result, namely, that the curve \(W(R)\) in the even case lies lower than in the odd case, differing especially from the latter at intermediate values of \(R\), when it has a minimum corresponding to the stable state of the ion \(\mathrm{H}_2^+\).

The forces binding this system owe their origin to the collectivization of the electron by the two protons in connection with the even character of the wave function describing its motion. This

collectivization should not be regarded as the result of the fact that the electron moves from one nucleus to the other. Such an oscillatory motion can be represented by superposing the even and odd states, taking into account the difference of their energies. Putting

\[ \psi_+ = \psi_+^\circ e^{-i\frac{2\pi}{h}W_+ t} \]

\[ \psi_- = \psi_-^\circ e^{-i\frac{2\pi}{h}W_- t}, \]

we see that for sufficiently large \(R\) the function

\[ \psi=\frac{1}{\sqrt{2}}(\psi_+ + \psi_-), \]

which can be written in the form

\[ \psi=\frac{1}{2}\varphi_a \left[ \left(e^{-i\frac{2\pi}{h}W_+ t} + e^{-i\frac{2\pi}{h}W_- t}\right) + \varphi_b \left(e^{-i\frac{2\pi}{h}W_+ t} - e^{-i\frac{2\pi}{h}W_- t}\right) \right] \]

is reduced alternately to \(\varphi_a^\circ\) and \(\varphi_b^\circ\) with a “beat” frequency equal to

\[ \frac{W_- - W_+}{h}. \]

The probability density \(\psi\psi^*\) oscillates, and its maximum passes alternately from one nucleus to the other, which indicates a similar oscillatory motion of the electron. We see that such a motion corresponds not to a stationary state of the system under consideration, but to a superposition (or alternation) of two states with opposite symmetry.

The probability distribution, and hence also that of the electric charge, in each of these stationary states must remain constant in time.

4. The \(H_2\) Molecule and the Exchange Effect

Having established the basic principles which determine the formation of the ion \(H_2^+\), we shall readily understand the nature of the forces binding the simplest neutral molecule \(H_2\).

As in the case of the He atom, which also has two electrons, we can describe the motion of each electron as if it were alone; to take account of their interaction it is necessary correspondingly to modify the electric field produced by the nuclei, for example by introducing a certain screening constant for their effective charges. In this way we obtain a description of the normal state of the \(H_2\) molecule which, as regards the number of electrons, is an exact duplication of the normal state of the \(H_2^+\) ion with a somewhat reduced charge of both nuclei. Were it not for this reduction, the density of the electronic charge in \(H_2\) would be exactly twice as large as in \(H_2^+\) at the same distance between

QUANTUM THEORY OF CHEMICAL FORCES

nuclei; this would be expressed in a doubling of the electric force acting on both nuclei, when \(R \ne R_0\).

Since the electrons in \(\mathrm{H}_2\) are in the same state of motion, their magnetic axes or “spins” must be directed oppositely, so that the \(\mathrm{H}_2\) molecule, like the He atom in the normal state, must be diamagnetic.

In the first excited state of \(\mathrm{H}_2\), the motion of one electron may be described by an even function, and that of the second by an odd function, corresponding to the same normal states of the separate atoms. In this case the binding action of one electron is balanced by the repelling action of the other, so that the molecule will break up into separate atoms.

The normal state of \(\mathrm{H}_2\) may, on the basis of the considerations set forth, be described by a wave function \(\Psi_0\), equal to the product of two identical functions \(\varphi_+^\circ(r_1)\) and \(\varphi_+^\circ(r_2)\) for the two electrons. Substituting expression (3) for \(\varphi_+^\circ\), we obtain

\[ \Psi^\circ = \frac{1}{2(1+J)} \left[ \varphi^\circ(r_{a1})\varphi^\circ(r_{b2}) + \varphi^\circ(r_{a2})\varphi^\circ(r_{b1}) + \varphi^\circ(r_{a1})\varphi^\circ(r_{a2}) + \varphi^\circ(r_{b1})\varphi^\circ(r_{b2}) \right], \]

where \(r_{a1}\) is the distance of the first electron from nucleus \(a\), \(r_{a2}\) is the same for the second electron, etc. This function represents a superposition of four different states, corresponding to the four possibilities for attaching two electrons to two nuclei:

1. Electron 1 is bound to nucleus \(a\), electron 2 to \(b\),
2. ” ” ” ” \(b\), ” ” \(a\)
3. Both electrons are bound to nucleus \(a\),
4. ” ” ” ” \(b\).

The first two states may be called nonpolar, the second two—polar. The four terms in (12) have identical numerical factors, i.e., the four states represented by them are regarded as equally probable.

Therefore this assumption must be hidden in the very form of the wave function (13). It is an obvious consequence of our method of describing the motion of two electrons independently of one another, reducing their interaction only to a change in the effective charge of the nuclei. But in reality, the mutual repulsion of the electrons must make their clustering at one nucleus a considerably less probable event than their distribution between different nuclei. Therefore we shall make a relatively small error if we omit the two polar terms in (13). This, of course, will require a correction of the normalizing factor. The new function, corresponding to the superposition of only nonpolar states with the same probability amplitudes, proves to be equal to

\[ \Psi_+^\circ = \frac{1}{\sqrt{2(1+J^2)}} \left[ \varphi^\circ(r_{a1})\varphi^\circ(r_{b2}) + \varphi^\circ(r_{a2})\varphi^\circ(r_{b1}) \right]. \tag{13} \]

This function was first introduced by Heitler and London in their theory of the \(H_2\) molecule for the following reasons. Two hydrogen atoms, placed at a sufficiently large distance from each other, are approximately described by the product of atomic wave functions corresponding to the association of each electron with a certain nucleus. This association can be achieved in two different ways, which are obtained from one another by interchanging the electrons. Each of the resulting wave functions \(\varphi^\circ(r_{a1})\varphi^\circ(r_{b2})\) and \(\varphi^\circ(r_{a2})\varphi^\circ(r_{b1})\) cannot represent a stationary state of the entire system, since in this case the identity principle—or, more precisely, the Pauli principle—is not satisfied. This principle requires that the wave function describing two electrons be either symmetric or antisymmetric with respect to the geometrical coordinates of both electrons; in the first case the resultant spin must be equal to zero, in the second—to unity (i.e. the spins of the two electrons, equal to one half \(\dfrac{h}{2\pi}\), are parallel to each other). These two functions can be obtained, respectively, by adding and subtracting the two multiplicative functions and multiplying the result by a suitably chosen factor. In the first case we obtain the function (13), and in the second the function (13a)

\[ \Psi^\circ_-=\frac{1}{\sqrt{2(1-I)}}\left[\varphi^\circ(r_{a1})\varphi^\circ(r_{b2})-\varphi^\circ(r_{a2})\varphi^\circ(r_{b1})\right]. \tag{13a} \]

These two functions satisfy the identity principle and possess the further advantage that they are mutually orthogonal for all values of \(R\) (whereas the multiplicative functions \(\varphi^\circ(r_{a1})\varphi^\circ(r_{b2})\) and \(\varphi^\circ(r_{a2})\varphi^\circ(r_{b1})\) are orthogonal only for \(R\to\infty\)). Their further advantages consist in the fact that they represent stationary states of the whole system precisely in the sense that transitions from one state to the other due to the interaction of the electrons are impossible.

Indeed, we know that the probability of such a transition is determined by the expression

\[ \iint \Psi_-\,\frac{e^2}{r_{12}}\,\Psi_+\,dV_1\,dV_2, \]

which is identically equal to zero, since the functions \(\Psi_+\) and \(\dfrac{1}{r_{12}}\) are symmetric with respect to both electrons, whereas \(\Psi_-\) is antisymmetric. It should be emphasized that the functions \(\Psi_+\) and \(\Psi_-\) are symmetric or antisymmetric in a twofold sense: 1) with respect to interchange of the two electrons and 2) with respect to the coordinates \(x\) (or also \(y\) and \(z\)) of the two electrons. Namely, if we simultaneously replace \(x_1\) and \(x_2\) by \(-x_1\) and \(-x_2\), i.e. move each electron from its original position to a position symmetrically located with respect to both nuclei (or to the plane passing between them), then function (13) remains unchanged, while (13a)

changes sign. To avoid confusion, we shall call symmetry of the first type “permutation symmetry,” and of the second type “spatial symmetry” or “parity”; the latter is, obviously, equivalent to symmetry under permutations of the nuclei (which are regarded as identical). An insufficiently clear understanding of this ambiguity in the symmetry properties of the wave functions \(\Psi_{+}\) and \(\Psi_{-}\) often leads to incorrect notions about the nature of the forces binding the atoms in the case of \(\Psi_{+}\), or preventing this bond in the case of \(\Psi_{-}\).

If we calculate the energies \(W_{+}\) and \(W_{-}\) of a system of two electrons as functions of the internuclear distance \(R\), using the direct method applied in the preceding paragraph, we obtain two curves of the same kind as in Fig. 2, which correspond to the even and odd states of each of the two generalized electrons.

We already know that the binding action of each individual electron depends on its parity property. The permutation effect in itself is not the cause of binding forces (which are often incorrectly treated as “volume or permutation forces”); in reality the latter owe their existence to the parity property associated with permutation symmetry. The true dynamical meaning of the “exchange effect” consists in the simultaneous collectivization of two electrons by two identical nuclei, taking into account the impossibility (or improbability) of their accumulation at one nucleus. Thus the chemical bond of two H atoms in the molecule \(\mathrm{H}_2\) must be treated not as an exchange effect (as is usually done), but as the result of the collectivization of each of the electrons in connection with the even character of its motion. If the symmetric wave function \(\Psi_{+}\) corresponds to the even motion of both electrons, it should not be thought that the antisymmetric wave function \(\Psi_{-}\) corresponds to the odd motion of each of the two electrons; in fact, it is impossible to construct a function antisymmetric with respect to permutation of two electrons from identical wave functions of each of them. We thus arrive at the conclusion that the function \(\Psi_{-}\) must correspond to two different individual states of motion—an even one for one electron and an odd one for the other. This conclusion is easily verified by taking the product of the functions (3) and (3a) for two electrons, then interchanging the electrons and subtracting the results from one another. Omitting the normalizing factor, we obtain

\[ (a_1+b_1)(a_2-b_2)-(a_2+b_2)(a_1-b_1)=2(b_1a_2-a_1b_2), \]

where, for brevity, \(a_1=\varphi^{\circ}(r_{a1})\), etc. This expression is nothing other than the function \(\Psi_{-}\). It should be noted that both polar states have completely disappeared in ...

written combination. If we had taken the even combination instead of the odd one, we would have obtained the function

\[ 2(a_1a_2+b_1b_2), \]

containing only polar states. The fact that, in the state of motion described by the function \(\Psi_{-}\), the combination of the atoms into a stable molecule is impossible follows directly from the mutual compensation of the binding action exerted by the “even” electron and the “loosening” action exerted by the odd electron. The energies \(W_{\pm}\) of the symmetric and antisymmetric states in the Heitler–London theory can also be approximately calculated by the method that was applied in § 3 for one electron. From the Schrödinger equation for the whole system we obtain

\[ W=\int\!\!\int U\psi^2\,dV_1dV_2-\frac{h^2}{8\pi^2m}\int\!\!\int \psi(\nabla_1^2+\nabla_2^2)\psi\,dV_1dV_2, \]

where \(\nabla_1^2\) and \(\nabla_2^2\) are the Laplace operators for the coordinates of the first and second electrons. The potential energy is equal to

\[ U=\frac{e^2}{R}+\frac{e^2}{r_{12}}-e^2\left(\frac{1}{r_{a1}}+\frac{1}{r_{a2}}+\frac{1}{r_{b1}}+\frac{1}{r_{b2}}\right). \]

With the aid of the two equations for the separate atoms

\[ \nabla_1^2a_1+\frac{8\pi^2m}{h^2}\left(W^\circ+\frac{e^2}{r_{a1}}\right)a_1=0,\qquad \nabla_1^2b_1+\frac{8\pi^2m}{h^2}\left(W^\circ+\frac{e^2}{r_{b1}}\right)b_1=0, \]

\[ \nabla_2^2a_2+\frac{8\pi^2m}{h^2}\left(W^\circ+\frac{e^2}{r_{b2}}\right)a_2=0,\qquad \nabla_2^2b_2+\frac{8\pi^2m}{h^2}\left(W^\circ+\frac{e^2}{r_{b2}}\right)b_2=0, \]

we easily find the mean kinetic energy of the electrons:

\[ \overline{T}=-\frac{h^2}{8\pi^2m}\int\!\!\int \psi(\nabla_1^2+\nabla_2^2)\psi\,dV_1dV_2 =-\frac{h^2}{8\pi^2m}\frac{1}{2(1\pm I^2)} \]

\[ \int\!\!\int (a_1b_2\pm a_2b_1)\,[(b_2\nabla_1^2a_1\pm\nabla_1^2a_2b_1)+(a_1\nabla_2^2b_2\pm b_1\nabla_2^2a_2)]\,dV_1dV_2= \]

\[ =\frac{1}{2(1\pm I^2)}\int\!\!\int (a_1b_2\pm b_1a_2)\left[\left(2W^\circ+\frac{e^2}{r_{a1}}+\frac{e^2}{r_{b2}}\right)a_1b_2\right. \]

\[ \left.\pm\left(2W^\circ+\frac{e^2}{r_{b1}}+\frac{e^2}{r_{a2}}\right)a_2b_1\right]\,dV_1dV_2, \]

i.e.

\[ \overline{T}=2\left(W^\circ+\frac{K\pm LI}{1\pm I^2}\right). \]

The mean value of the potential energy is equal to

\[ \overline{U}=\frac{1}{2(1\pm I^2)}\int\!\!\int U(a_1^2b_2^2+a_2^2b_1^2\pm2a_1a_2b_1b_2)\,dV_1dV_2= \]

\[ =\frac{e^2}{R}-2\frac{K+M\pm LI}{1\pm I^2}+\frac{N\pm P}{1\pm I^2}, \]

where

\[ P=e^2\int\!\!\int \frac{1}{r_{12}}a_1^2b_2^2\,dV_1dV_2,\qquad Q=e^2\int\!\!\int \frac{1}{r_{12}}a_1b_1a_2b_2\,dV_1dV_2. \]

Adding the expressions found, we obtain

\[ W_{\pm}=2W^\circ+\frac{e^2}{R}+\frac{N\pm P-2(M\pm Ll)}{1\pm l^2}. \tag{14} \]

In comparing \(W_-\) with twice expression (12) (for the even state), corresponding to the method of “independent” electrons, it must be borne in mind that their interaction is taken into account in this method by reducing the effective nuclear charge. We shall not dwell on the calculation of the integrals \(N\) and \(P\), since expression (14) is only a crude approximation for such values of \(R\) at which the atoms actually combine into a molecule. It should only be mentioned that the integrals \(M\) and \(N\) represent the ordinary Coulomb energy of an electron distributed around one nucleus with respect to the second nucleus, and the mutual energy of electrons distributed around the separate nuclei. \(L\) and \(P\) have an analogous meaning for a distribution of the “mixed” type, when each electron is partially associated with each of the two nuclei. The latter integrals are usually called “exchange” (or permutation) integrals. The method of Heitler and London, as well as the simpler method described above, should be regarded as a purely qualitative method for characterizing chemical forces on the basis of the concept of collectivization of electrons.

An important advantage of the Heitler–London method is that it extends without any changes to the case of two different nuclei. In this case, an approximate description of the motion of an electron collectivized by two nuclei by means of the method of § 2 becomes impossible, since considerations based on spatial symmetry, i.e. evenness or oddness, are inapplicable when the nuclei are different. The Heitler–London method can, however, be applied in this case without any changes, except for replacing the two identical atomic functions by two different ones \(\varphi_a^\circ(r)\) and \(\varphi_b^\circ(r)\).* The permutation of two electrons here, as in the particular case of identical nuclei, leads to their simultaneous collectivization; the function obtained by permutation will again represent such a state of motion as corresponds to a certain concentration of charge in the region between the two nuclei.

5. Chemical valence and electron spin

The Heitler–London theory was often called the “spin theory of valence.” The reason for this was the fact that the pair of electrons responsible for the chemical bond has a resultant spin equal to zero, whereas in the case of parallel spins

* If one of the two nuclei has a considerably larger charge than the other, then the charge of the electron will be concentrated mainly near it, and its collectivization will appear as expropriation; the whole system will be strongly heteropolar (ionic).

such a bond is impossible. These results have led many chemists, insufficiently familiar with wave mechanics, to a false understanding of the nature of the chemical bond in the new theory. Some of them suppose, for example, that this bond arises owing to the mutual attraction of the oppositely directed magnetic moments of the two electrons. The erroneousness of this idea is clear from the fact that in the preceding discussion we introduced no forces other than Coulomb forces. In a number of works it is indicated that the “permutation” term, representing the additional energy of attraction in the Heitler–London formula (14), corresponds to special “exchange” forces of quantum mechanics, unknown in classical mechanics. Such a point of view is the result of an incorrect identification of the mathematical device consisting in the permutation of the two electrons in order to obtain a symmetric or antisymmetric function, in agreement with the identity principle, with a real physical process of the joining of atoms into a molecule. In an exact theory of the stationary states of the molecule H$_2$ there could be no question at all of a permutation of the two electrons. This permutation is performed by us only because we start from two isolated atoms, in each of which each electron is bound to the corresponding nucleus; the physical meaning of the permutation of electrons in describing the origin of the chemical bond in reality consists in the simultaneous collectivization of these electrons. The circumstance that the resultant spin of the molecule H$_2$ in the normal state is equal to zero has no direct dynamical relation to the interaction between the two atoms. We encounter exactly the same situation in the case of the He atom in the normal state; the antiparallelism of the spins corresponds in both cases to the identity of the “orbital” motions in accordance with the Pauli principle, and also to the circumstance that the state of two electrons with minimum energy is realized when they move in the lowest individual states with opposite spins. The fact that the chemical bond is effected by a pair of electrons thus follows from the Pauli principle and has no deeper dynamical meaning. A chemical bond could also be effected by one electron, as in the case of the ion H$_2^+$; since, however, the Pauli principle, in connection with the two possible orientations of the spin, permits two electrons to be in identical “bonding” states of motion corresponding to the normal (lowest) state of an individual atom, it is natural that the chemical bond is usually effected by a pair of collectivized electrons with opposite spins in all those cases where such a pair is possible. If a third electron is added to such a “pair,” then it will be compelled to move in another (collectivized) state, corresponding to a somewhat larger (nearer) value of the energy. At sufficiently large internuclear distances, such a state in the case of identical nuclei (only this case will be considered in what follows)

will be the one described by the wave function (3a), with \(\varphi^\circ\) denoting the same atomic states as those forming the symmetric function (3). Such an electron does not help to hold together two atoms, or, more precisely, nuclei, but, on the contrary, so to speak “nullifies” the work of one of the two other bonding electrons. This loosening action is increased still further by the addition of a fourth electron in the same odd state as the third.

These considerations explain the impossibility, under normal conditions, of the existence of such systems as \(\mathrm{He}_2^+\) or of the molecule \(\mathrm{He}_2\). At the same time they show that these systems can be formed if, instead of starting from four normal atomic states, we allow one of the four electrons to pass into an excited atomic state. Let us consider, for example, two excited He atoms, in each of which one electron is in a normal one-quantum state and the other in a two-quantum state. If we suppose that, for some reason, the excited electron cannot return to the normal state (i.e. the excited state is metastable), then the stationary state of the whole system after collectivization will be even or symmetric, since both two-quantum electrons are collectivized into an even state just as the two one-quantum electrons are, and we shall obtain a stable \(\mathrm{He}_2^*\) molecule in an electronically excited state (as indicated by the asterisk), in which the atoms are bound to one another by a double bond (each bond, from the usual point of view, corresponds to a symmetric pair of electrons with oppositely directed spins). If the total energy of such a system, for all values of \(R\), is greater than the energy of two normal He atoms, then in the end the system will separate into two atoms. This is exactly the case with helium. However, it may happen, and in fact often does happen, that two atoms with two or several outer electrons can, by the excitation of several electrons, combine into a molecule whose minimum energy, corresponding to the equilibrium distance between the nuclei, is considerably less than the energy of two separate atoms in the normal state. Such a molecule will be not only relatively stable, like \(\mathrm{He}_2^*\), but stable in the absolute sense of the word. In this case the excitation of two atoms or, at least, of one of them is a necessary condition and at the same time a direct consequence of their chemical combination.

From this point of view, the maximum number of simple bonds that can be formed between two identical atoms is equal to the number of outer electrons in each of them. As a rule, the inner electrons may then be disregarded, since their excitation, i.e. their transition to outer vacant levels, requires amounts of energy that are not compensated by the energy gained in the chemical combination of the atoms.

For this reason the maximum valence of a chemical ele-

element is mainly determined by the number of its outer electrons, which on this basis are often called “valence” electrons. When this number is small (less than 4), it is usually called “positive valence”; when it is greater than 4, “negative valence,” because the combination of an atom \(A\) of the first (metallic) type with an atom \(B\) of the second (metalloid) type leads to the formation of a heteropolar or ionic molecule which is usually described as the result of the expropriation by atom \(B\) of the outer electrons of atom \(A\). As was already indicated above, in reality collectivization takes place, not simple expropriation.

It should be mentioned that the formation of a molecule from two different atoms, especially such as \(A\) and \(B\) (one electropositive, the other electronegative), is not accomplished by withdrawing from individual use by both atoms an equal number of electrons, which would be collectivized in the form of symmetric pairs with opposite spins, corresponding to separate chemical bonds. In fact, some or even all of the outer electrons of the atoms come into play and are mobilized to form the outer electron shell of the molecule. This is the case, at least, with such molecules as NaCl or HCl. In these cases it makes no sense to speak of electron pairs with partners that are made up from different atoms. Of course, the outer electron shell of the molecule can be divided into separate pairs formed by electrons that are in one and the same collectivized state with opposite spins—just as in the case of the outer shell of an atom. But these pairs are not necessarily formed by partners originating from different atoms.

The valence of an atom in the Heitler–London theory is determined not by the total number of outer electrons, but by the resultant spin of the atom, which is equal to the number of unpaired electrons. These “bachelor” electrons, being in different quantum states with equally oriented spins, combine with the bachelor electrons of another atom (of the same kind), with spins oriented in the opposite direction (opposite spin directions correspond here, as it were, to the opposite “sex”), forming the same number of symmetric pairs holding the atoms together, each pair corresponding to a single bond.

This Heitler–London picture of a chemical compound as the result of one or several “marriages” between bachelor electrons with opposite spins is a very schematic description of reality, applicable only to the simplest cases of the combination of two identical atoms, for example to the molecules \(\mathrm{H}_2\) and \(\mathrm{H}^{*}\) already considered above.

It is necessary further to note that the number of bachelor electrons in an atom is not a definite, constant quantity. If the atom is surrounded by several others with which it can combi-

QUANTUM THEORY OF CHEMICAL FORCES

to be counted, then this number may become larger than in the case of an isolated atom, since the interaction of atoms may lead to the “separation” of part of the electron pairs with opposite spins (in one and the same atom), just as occurs in the case of the paramagnetism of metals, where the “separation” is caused by the action of an external magnetic field.

For example, a carbon atom in the isolated state has one pair of electrons in a two-quantum state with subsidiary number \(l = 0\) (an \(s\)-state) and two “singlets,” situated in two of the three different two-quantum states with \(l = 1\).* But in all chemical compounds carbon is tetravalent. This means that the interaction of a C atom with other atoms leads to the separation of the first pair of electrons and transforms all four electrons into singlets (of which one is in the state \(l = 0\), and the rest in three different states with \(l = 1\)). Thus the Heitler–London estimate of valence is valid when among the outer electrons of the atom there are no paired electrons, or when these pairs are separated in the formation of a chemical compound.

At first glance it might seem that the number of unpaired electrons in the outer shell of an isolated atom should be very small, namely zero if the number of outer electrons is even, so that all of them can be combined in pairs, or equal to one if their number is odd. However, this point of view is at variance with experimental data showing that many atoms and ions in the normal state have a permanent magnetic moment, manifested in the paramagnetism of the corresponding substance. But this would seem to mean that in the normal state atoms and ions do not possess the minimum possible energy, since electrons with spin directed in the same way must move in different orbits.

This conclusion would be correct only in the case in which it were possible to determine the energy of an atom as the sum of the energies of individual electrons, taking their interaction into account by introducing some external “self-consistent” field (corresponding, for example, to screening of the nuclear charges). In reality this energy depends in an essential way on the properties of the permutation symmetry of the wave function describing the motion of the whole system of electrons. Let us take, for example, the case of two electrons. If their motion is described by an antisymmetric function (of the geometrical coordinates), then the probability of their meeting at one and the same point is equal to zero.** This means that their motion is such as if they avoided one another. As a result the mean value

* Here only the 4 outer two-quantum electrons are meant. The two inner one-quantum electrons play no role in chemical phenomena.

** When the arguments \(x_1, y_1, z_1\) and \(x_2, y_2, z_2\) coincide, the function \(\psi(x_1, y_1, z_1; x_2, y_2, z_2)\) obviously cannot change upon their permutation, while, on the other hand,

their mutual potential energy \(\dfrac{e^2}{r_{12}}\), and consequently also the total energy, must in this case be smaller than in the opposite case of motion described by a symmetric wave function. In the latter case the electrons do not avoid one another; on the contrary, the probability of their meeting at one and the same point is greater than in the case of motion described by the product of two wave functions \(\varphi_1(\mathbf r_1)\varphi_2(\mathbf r_2)\), corresponding to the independent motion of both electrons. The antisymmetric state described by the function

\[ \varphi_1(\mathbf r_1)\varphi_2(\mathbf r_2)-\varphi_2(\mathbf r_1)\varphi_1(\mathbf r_2), \tag{15} \]

is possible, of course, only if the individual states \(\varphi_1\) and \(\varphi_2\) are different. Under this condition, of the two types of motion compatible with the principle of the identity of electrons, i.e. the antisymmetric (15) and the symmetric

\[ \varphi_1(\mathbf r_1)\varphi_2(\mathbf r_2)+\varphi_2(\mathbf r_1)\varphi_1(\mathbf r_2), \]

the first will correspond to a lower energy than the second. But, according to the Pauli principle, the antisymmetric state (15) is associated with the same direction of the spins of both electrons, i.e. with a resultant magnetic moment of the atom. Therefore, if two different states \(\varphi_1\) and \(\varphi_2\) correspond to the same energy of an individual electron, i.e. if they are degenerate, then the normal state of the system will be the antisymmetric state of both electrons with resultant spin 2. If, however, the energy \(W_1\) of the state \(\varphi_1\) is less than the energy \(W_2\) of the state \(\varphi_2\), then two cases are possible, namely: if the difference of energies \(W_2-W_1\) is greater than the energy gained owing to the decrease of the mutual potential energy of the two electrons in the antisymmetric state, then both electrons will be in one and the same individual state, and the system will be described by the symmetric wave function \(\varphi_1(\mathbf r_1)\varphi_1(\mathbf r_2)\), corresponding to the absence of resultant spin. In the opposite case the electron pair will spontaneously separate, and we shall obtain two electrons in the unpaired state (15).

These results can be generalized to a system of any number of electrons, described in the collective state of motion by a function antisymmetric with respect to all of them:

on the other hand, being antisymmetric, it must in this case change its sign. Hence it follows that, for

\[ x_1=x_2,\quad y_1=y_2,\quad z_1=z_2,\quad \psi=0, \]

i.e. that the meeting of two electrons at one and the same point is impossible.

QUANTUM THEORY OF CHEMICAL FORCES

\[ \frac{1}{\sqrt{n!}} \left| \begin{array}{cccc} \varphi_1(r_1) & \varphi_1(r_2) & \cdots & \varphi_1(r_n)\\ \varphi_2(r_1) & \varphi_2(r_2) & \cdots & \varphi_2(r_n)\\ \cdot & \cdot & \cdots & \cdot\\ \cdot & \cdot & \cdots & \cdot\\ \varphi_n(r_1) & \cdots & \cdots & \varphi_n(r_n) \end{array} \right| \tag{15a} \]

The electrons will thereby avoid one another, which will result in a decrease of their mutual potential energy in comparison with any other state formed from the same individual states \(\varphi_1(r)\), \(\varphi_2(r)\), etc. Therefore, if their energies \(W_1, W_2, \ldots W_n\) are equal or sufficiently close to one another, then the normal state of the whole system will be described by the wave function (15a).

This corresponds to the largest possible value of the resultant spin of the entire system, corresponding to the same orientation of the spins of all the individual electrons. However, if the energies of some of the individual states, for example \(\varphi_1\) and \(\varphi_2\), are much smaller than those of the others, then these states will be occupied by pairs of electrons with resultant spin equal to zero, while the motion of the remaining electrons will be described by a function of the form (15a).

The considerations set forth make it possible to explain the origin of the ferromagnetic properties of iron and similar bodies; the essence of ferromagnetism consists in the spontaneous magnetization of a body (or of its individual regions) owing to the parallel orientation of the spins of all (valence) electrons.

The problem in this case is much more complicated than in the case of a single atom, since we must operate with a large number of nuclei around which collectivized electrons move. In the simplest case of two nuclei, for example in the case of the molecule \(\mathrm{H}_2\), as was shown above, the normal state is described by the function \(\Psi_1\), symmetric with respect to both electrons and corresponding to two identical individual states of motion (even with respect to both nuclei). Several diatomic molecules are known which, as, for example, \(\mathrm{O}_2\), possess a magnetic moment. The two “idle” electrons which determine the magnetic moment of the \(\mathrm{O}_2\) molecule are, in all probability, farthest from the nuclei and therefore do not assist their binding to one another. Under such conditions they behave like external electrons in an individual atom when the difference in energy of two successive individual states is very small.

6. Polyatomic molecules and crystals

In conclusion we shall briefly consider the generalization of the preceding results to the case of a system containing more than two fixed nuclei. To begin with, let us take the simplest case of three protons placed at the vertices of a regular triangle, and consider the stationary states of motion of one electron in the field of these

three nuclei. It is clear that, as in the case of two protons, the electron will be collectivized by all three protons, so that the average distribution of its electric charge will be symmetric with respect to all the nuclei (so long as they are at equal distances from one another).

Denoting the atomic wave functions corresponding to the binding of the electron with the nuclei \(a, b, c\), by \(\varphi(r_a), \varphi(r_b), \varphi(r_c)\), we may represent its motion in the collectivized state, for sufficiently large internuclear distances \(R\), by the linear combination

\[ \psi = p_a\varphi(r_a) + p_b\varphi(r_b) + p_c\varphi(r_c), \]

where \(p_a, p_b, p_c\) are complex numbers with equal moduli (since, under the given conditions, the probabilities of the electron being associated with any one of the three protons must be equal). These coefficients can be determined from the condition that a cyclic permutation of the nuclei \(a, b, c\) must be equivalent to multiplying \(\psi\) by an inessential factor of the form \(e^{i\alpha}\).

Since such a permutation replaces \(p_a\) by \(p_b\), \(p_b\) by \(p_c\), and \(p_c\) by \(p_a\), we have:

\[ p_b = p_a e^{i\alpha}, \quad p_c = p_b e^{i\alpha} = p_a e^{2i\alpha}, \]

whence

\[ e^{3i\alpha} = 1,\quad \text{i.e. } \alpha = \frac{2\pi}{3} k \quad \text{for } k = 0, 1, 2. \]

In the first case one obtains the wave function

\[ \psi_1 = \frac{1}{\sqrt{3}}\,[\varphi(r_a) + \varphi(r_b) + \varphi(r_c)], \]

symmetric with respect to all three nuclei. The other two

\[ \psi_2 = \frac{1}{\sqrt{3}}\,[\varphi(r_a) + e^{i\frac{2\pi}{3}}\varphi(r_b) + e^{i\frac{4\pi}{3}}\varphi(r_c)], \]

\[ \psi_3 = \frac{1}{\sqrt{3}}\,[\varphi(r_a) + e^{i\frac{4\pi}{3}}\varphi(r_b) + e^{i\frac{2\pi}{3}}\varphi(r_c)] \]

are neither symmetric nor antisymmetric. The function \(\psi_1\) corresponds to such a distribution of the density of electricity \(\rho = e|\psi_1|^2\), which is completely analogous to the distribution for the even function in the problem of the ion \(\mathrm{H}_2^+\). One third of the electron charge is concentrated near each nucleus, forming on each side something like a threadlike strand tending to hold the nuclei together, despite their mutual repulsion. In the case of the functions \(\psi_2\) and \(\psi_3\), such binding electron strands practically disappear. A stable ion \(\mathrm{H}_3^{++}\) of this type is unknown; even in the most favorable case of the state \(\psi_1\), the mutual repulsion of the protons cannot be balanced by the binding action of the “electronic protoplasm.”

However, if a second electron, likewise in a symmetric state (with respect to all the nuclei), is added to such a system, then the attractive action will almost double and will become capable of balancing the mutual repulsion of the protons. In fact, J. J. Thomson and other physicists discovered a stable ion \(H_3^+\) in positive rays of hydrogen.

If, however, a third electron is added to such a system, it will not turn into a neutral molecule \(H_3\), but, as is well known, will break up into the ordinary molecule \(H_2\) and a neutral atom \(H\). The reason for this is clear: the third electron cannot move in the same symmetric state as the first two; therefore it will be forced to occupy the next state \(\psi_2\) or \(\psi_3\), which do not provide additional binding forces. On the contrary, owing to the mutual repulsion of the electrons, the binding energy will decrease, and the whole system \(H_3\) will have an energy greater than the energy of the system \(H_2 + H\); therefore \(H_3\) will transform into the latter.

The possibility of the existence of a stable triatomic oxygen molecule, in contrast to hydrogen, depends on the presence of at least two different atomic states from which the collectivized states of the electrons in the molecules \(O_2\) and \(O_3\) are formed. If, in the preceding example, one of the three atoms \(H\) had initially been in an excited state, then the third electron could have moved in the symmetric state \(\psi_1\) corresponding to this excited state; this would lead, for not too small values of \(R\), to a relatively stable molecule \(H_3\)* in an electronically excited state.

These considerations illustrate the wave-mechanical method of explaining chemical forces and their principal peculiarity, absent in van der Waals forces—the property of “saturation.”

If we take a large number of atoms \(H\) and give them the possibility of combining with one another, they will always form pairs in the form of molecules \(H_2\), which are bound to one another only by van der Waals forces depending on mutual polarization. We encounter a different situation when passing to carbon atoms, which, as is known, are capable of combining with one another in any number, forming a crystal of graphite or diamond. Such a crystal, in which each atom is surrounded by four others, may be regarded as a single molecule of gigantic dimensions. Obviously, the reason for this difference lies in the different number of outer, or valence, electrons in the atoms \(H\) and \(C\). According to the ideas of Heitler and London, the formation of diamond may be pictured in the following way. Of the four “unpaired” electrons of the atom \(C\), each serves to bind it to one of its four neighbors, together with an unpaired electron of opposite spin originating from another atom. The simple bond between two neighboring atoms \(C\) in diamond is thus an exact analogue of the bond between atoms \(H\), with each atom \(C\) equivalent to four atoms \(H\).

*

This picture gives a somewhat distorted notion of the actual state of affairs. The latter can be described more precisely by a method analogous to that used in the case of three H atoms. Each electron must then be regarded as collectivized by all the atoms of the crystal. To each atomic state of this electron there correspond, in a crystal consisting of \(n\) atoms, \(n\) collective “substates” (symmetric and antisymmetric in the case \(n=2\), and others, more complicated, for \(n>2\)). In a diamond crystal consisting of \(n\) carbon atoms, there are \(4n\) collectivized electrons, which must be distributed in pairs among the \(2n\) lower states out of the \(4n\) states obtained from the four different individual states of the 4 “valence electrons” in a separate atom.

The principal defect of this scheme is the impossibility of taking sufficiently accurately into account the interaction of the electrons, which prevents them from accumulating (in a number greater than 4) at one and the same nucleus. This phenomenon of accumulation and the appearance, owing to it, of polar states can as it were automatically be excluded by distributing the \(4n\) electrons among the \(4n\) collectivized states considered above; their motion is then described by an antisymmetric wave function. However, this would correspond not to a real, diamagnetic diamond, but to a ferromagnetic crystal, unknown in nature.

The phenomenon of accumulation can be excluded from consideration by applying the generalized Heitler–London method, forming linear combinations of products of all the wave functions representing the states of separate atoms with a definite distribution of electrons. These products are obtained from one another by various permutations of the electrons in accordance with the Pauli principle (for a collective state with resultant spin equal to zero).

Up to now this program has not yet been carried out. There exists only one consistent wave-mechanical investigation of crystals—Gilleraas’s work on the LiH crystal, carried out by the Heitler–London method under the assumption that Li is present as a positive ion and H as a negative one. In Bruch’s investigations of crystals of the same ionic type (similar to the halide compounds of the alkali metals), phenomena connected with the collectivization of electrons were not taken into account. Born and Mayer developed a simplified theory of ionic crystals based on the formula

\[ U=\pm \frac{e^{2}}{R}-\frac{a}{R^{6}}+ee^{-\beta r}, \]

which expresses the mutual energy of two ions (of the same or of opposite sign) as a function of the distance between their centers. Thus in this theory, as also in the works mentioned above—

QUANTUM THEORY OF CHEMICAL FORCES

by the authors mentioned, ions are treated as immutable force centers—in accordance with the old naïve conceptions. A study of ionic crystals free from these objections, i.e. one that properly takes into account the collectivization of electrons, was carried out by Lennard and especially by Jensen on the basis of the Thomas–Fermi equation, with a somewhat modified solution obtained by means of the variational principle.

The crude approximate character of this equation, especially for the outer region of the atom, limits the applicability of this method to ionic crystals, in which the attraction of the ions is attributed only to their resultant charge, regarded as concentrated at the center, while the law of repulsion follows from the spatial distribution of negative charge determined by the Thomas–Fermi equation. For this reason the Lennard–Jensen method is inapplicable to metals.

The earliest attempt to describe a metal on the basis of collectivization of electrons was made by the author in 1924. The metal was then regarded as a liquid with practically constant volume density, formed by free (i.e. valence) electrons, in which positive ions float. The mutual repulsion of neighboring ions could be partly compensated by negative charges distributed in the region between them. In 1928 this theory was supplemented by taking into account the kinetic energy of the electrons on the basis of Fermi statistics.

The fundamental difference between the chemical forces among carbon atoms in diamond and the forces holding together the atoms of a solid metal is that the density of the electronic protoplasm in the region between two neighboring atoms in the first case has an appreciable magnitude only near the line connecting them (“directed valence” of Pauling and Slater), whereas in the second case it remains practically constant everywhere.

The latter result was also obtained by Wigner and Seitz⁴ by means of a wave-mechanical method, taking into account the interaction of neighboring atoms of the crystal lattice by introducing a periodicity condition for the wave functions that describe the motion of an electron about a definite atom or ion.

This theory was successfully applied to the calculation of the specific volume and heat of vaporization of sodium. However, the attempt to improve the theory by introducing the exchange effect into it and by taking account of the Pauli principle proved unsuccessful; the results of the simple theory are in better agreement with experiment than the results of the “improved” theory.⁵

In conclusion it should be noted that Jones⁶ has recently developed a theory of metals, especially of metallic alloys, which is based on individual states of the collectivized electron in the crystal lattice (described in the Bloch–Peierls theory of electrical conductivity). This, although incomplete, theory seems to me especially promising; one of its chief achievements is the explanation of the Hume-Rothery rule for

of the average number of free electrons per atom in the so-called $\gamma$-, $\varepsilon$-, and $\eta$-alloys.

References

1. Slater, Phys. Rev., 1934.
2. Frenkel, Wave Mechanics, vol. II, p. 31.
3. Frenkel, Wave Mechanics, § 30.
4. Wigner and Seitz, Phys. Rev., 1933.
5. Wigner, Phys. Rev., Dec., I, 1934.
6. Jones, Proc. Roy. Soc., 1934.