QUANTUM MECHANICS AND CHEMICAL REACTIONS *

Henry Eyring, Princeton, U.S.A.

Types of Chemical Reactions

The mechanisms of chemical reactions are very diverse. In many reactions light is emitted, undoubtedly connected with transitions of electrons. The overwhelming majority of reactions are accelerated with increasing temperature. The number of endothermic and the number of exothermic reactions must naturally be one and the same, although the latter can more readily occur spontaneously. Reactions in the solid phase are rare, whereas very many examples of reactions in solutions, in the gas phase, and on surfaces can be cited. Reactions in solutions are often instantaneous; such reactions, as is known, are accompanied by the combination of ions. Reactions in solution accompanied by the rupture of a paired electron bond generally proceed slowly, and their rate increases with increasing temperature. Such bonds belong to the class known as homeopolar, and they will constitute the main subject of the present article. Many bonds which in the gas phase have all the properties of homeopolar bonds are polar in solution. Two atoms are in fact connected by a chain of electrons. If the weakest link in it is a paired electron bond, then we speak of a homeopolar bond. If, however, this link is a bond between a positive nucleus and an electron, then we call it polar. Of course, a bond of the latter kind, being electrostatic,

* Report at the Conference on Chemical Kinetics organized by the American Chemical Society. Chem. Reviews, vol. X, Febr. 1932, p. 103. Translated by T. P. Ehrenfest.

is very much weakened by dissolution in a substance with a high dielectric constant. In the gas phase the weakest link is probably always the homeopolar bond.

A broad class of reactions is not accompanied by radiation at any stage. The energy necessary for the breaking and subsequent restoration of homeopolar bonds is supplied by strong collisions. Since in many cases it is known that the amount of energy required in reactions is insufficient to break any one of the bonds directly, the bonds must evidently weaken one another upon mutual approach. Thus, in a collision some bonds pass into new ones by a gradual or adiabatic process. Quantum mechanics has been applied to this very general type of reaction, with very great success, and it is this type that we shall consider here.

If the bond between two atoms depended only on the distance between them, and not on the position of other atoms, then valence and activation energy would not play the outstanding role that they do. In reality an atom attracts other unsaturated atoms until its valence is saturated, after which it repels all other atoms. If we want the reaction

\[ Y + XZ = YX + Z \]

to occur among three atoms of one and the same valence, then we must somehow make \(Y\) approach \(X\) so closely that \(X\) becomes uncertain as to which atom it properly belongs. This state of uncertainty is the state of activation. The energy expended in bringing \(Y\) close to \(X\) is the activation energy. In the liquid or gas phase the molecules (or atoms) move with various random velocities. A certain fraction of the molecules, increasing with temperature, moves with enormous velocities. If a head-on collision occurs between such rapid groups, the atoms of different molecules will approach one another to the same small distance at which they are from their former partners, and as a result of this

partner exchange and a bimolecular reaction may occur. A monomolecular reaction is a strong internal collision with subsequent exchange of partners bound to one another. In some cases this exchange is nothing other than dissociation, occurring as soon as the energy is concentrated on the appropriate bond. Rice^1 considered monomolecular reactions in connection with predissociation.

The Nature of Chemical Bonds

The forces that bind atoms to one another are all due to the fact that atoms consist of positive and negative charges; and indeed, roughly speaking, one tenth of the energy of a homeopolar bond can be calculated simply on the basis that the force between charges is inversely proportional to the square of the distance. In doing this, of course, it is necessary to assume that for the electrons there exists the probability distribution given by quantum mechanics. This bond, due to Coulomb forces, or the “Coulomb integral,” is obtained by integrating the Coulomb potential over the electron clouds. The assumption that a group of atoms, i.e. its positive nuclei and its electrons, must satisfy Schrödinger’s equation and the Pauli principle introduces, in addition, an entirely new kind of potential energy between atoms, called the exchange bond or “exchange integral.” In order to understand more clearly the nature of this exchange integral, we shall make a short digression.

For the hydrogen atom we have the following Schrödinger equation:

\[ \frac{\partial^2 \psi}{\partial x^2}+\frac{\partial^2 \psi}{\partial y^2}+\frac{\partial^2 \psi}{\partial z^2}+\frac{8\pi\mu}{h^2}(E - V)\psi=0. \tag{1} \]

The potential energy \(V\) has the form \(V=-\frac{e^2}{r}\); \(x, y, z\) are ordinary rectangular coordinates, \(\mu\) is the mass of the electron, and \(h\) is Planck’s constant. Equation (1) has solutions \(\psi\) admitting a physical interpretation only for special values of the energy \(E\); \(\psi\) is called the fundamental *

functions, and \(E\)—the characteristic numbers. These values of \(E\) are evidently the energy levels of hydrogen. \(\psi\) is a function of the electron coordinates \(x,y,z\) and of certain integers \(n,l,m\), whereas \(E\) does not depend on the electron coordinates, but depends only on the quantum numbers \(n,l,m\). Thus, for each set of quantum numbers we have our own particular function \(\psi\) and a particular value \(E\). The principal quantum number \(n\) takes all positive integral values, starting with unity. The quantum number \(l\) is always less than \(n\). \(l=0\) for \(s\)-electrons, \(1\)—for \(p\)-electrons, \(2\)—for \(d\)-electrons, and so on. For a definite value of \(n\) and \(l\), the number \(m\) can take \(2l+1\) values, beginning with \(-l\) and ending with \(+l\). These \(2l+1\) values of \(m\) constitute all the permitted projections of the angular-momentum vector on the axis of the applied magnetic field.

The probability that the electron is located at some point of space is proportional to the value of \(\psi^2\) at that point, so that for two electrons for which not all quantum numbers coincide, the probability of being at some point cannot be the same for all points of space. The same system of quantum numbers that serves for the description of hydrogen enters automatically, through the Schrödinger equation, into the description of any atom. For atoms with more than one electron, the potential energy is simply the sum of all terms corresponding to the attraction between the nucleus and an electron, and of all terms corresponding to the repulsion between electrons. In addition, the equation is further changed in the sense that, for the coordinates of each new electron, a Laplace expression is added:

\[ \frac{\partial^2 u}{\partial x_i^2}+\frac{\partial^2 u}{\partial y_i^2}+\frac{\partial^2 u}{\partial z_i^2}. \]

The fundamental function of the new equation will be approximately represented by a product of fundamental functions of the hydrogen type,

\[ U=\psi_1\psi_2\ldots\psi_n, \]

where to each \(\psi\) we may assign a set of integral quan-

of quantum numbers \(n, l, m\), exactly as in the case of hydrogen. It might have been supposed that all electrons would choose the very lowest set of quantum numbers. However, the Pauli principle states that in one atom no more than two electrons may have one and the same set of quantum numbers \(n, l, m\). In order to explain the Zeeman effect for atoms with a large number of electrons, it was necessary to assign to all electrons yet a fourth quantum number \(s\). The spin quantum number \(s\) can take only the values \(\frac{1}{2}\) or \(-\frac{1}{2}\). If two electrons in an atom have the same quantum numbers \(n, l, m\), then for them the spin quantum numbers must have opposite signs. Thus, for two electrons in an atom all four quantum numbers cannot be identical. However, for two electrons in different atoms all four quantum numbers may coincide. The general statement of the Pauli principle in this case is that no two electrons can have one and the same fundamental function \(\psi_s\), where \(s\) is that part of the fundamental function which corresponds to the rotation of the electron.

Zener and Slater\(^2\) showed how one may approximately determine the fundamental function of an electron of any atom if all four quantum numbers are specified. For our purpose there will be no need to determine them. But it is obvious that, if the quantum numbers are specified and it is said to which atom each electron belongs, then one can easily write the fundamental function of the system in the case when the atoms are at a large distance from one another. It will simply be the product of the fundamental functions of each electron separately. But even if one takes one and the same set of quantum numbers, nevertheless, by interchanging the electrons, we obtain many different fundamental functions, which all correspond to one and the same energy and all are equally good approximate solutions of the Schrödinger equation. In the case where the fundamental functions of two electrons partly overlap either within one atom or between two

atoms, the electrons arrange themselves in every possible way so as not to violate the Pauli principle. We have completely neglected this phenomenon in considering the Coulomb bond, so that we may expect a changed bond energy. There are various possibilities. If the electrons simultaneously enter one and the same region more often than they would if they moved independently of one another, then the attraction increases. If, however, they occupy one and the same region less often than if they moved independently, then a decrease of the bond is obtained. This additional potential energy is called the exchange integral. In order for two equivalent electrons to occupy one and the same region, by the Pauli principle they must have oppositely directed angular momenta (spins), since the remaining parts of their fundamental functions are identical. Consequently, for oppositely directed spins the exchange integral increases the bond, since the electrons responsible for the bond can spend a larger fraction of the time between the atoms.

Heitler and London³ first discovered the exchange bond for molecules and calculated it for H₂.

ACTIVATION ENERGY FOR REACTIONS IN WHICH THREE ELECTRONS CHANGE PARTNERS

If, in two univalent atoms, the spins of the electrons are directed oppositely, i.e., if these spins are antiparallel, producing an attractive exchange bond, then, since the Coulomb attraction does not depend on the direction of the spins, it is obvious that in a third atom the electron spin can in no way be antiparallel to each of the other two. The third atom will be repelled by at least one of the other two atoms. Using perturbation theory, London⁴ gave the following approximate expression for the energy of three univalent atoms:

\[ E_3 = A + B + C + \left[\frac{1}{2}(\alpha-\beta)^2 + (\beta-\gamma)^2 + (\gamma-\alpha)^2\right]^{1/2}. \tag{2} \]

Fig. 1 represents three univalent atoms \(Y\), \(X\), and \(Z\),

located at the vertices of a triangle. The quantities written on the straight lines joining pairs of atoms denote the amount of energy that would be required to separate the pair if the third atom were absent. Here an important circumstance is that, although the energy \(E_3\) is not the sum of the three energies corresponding to the pair bonds, it is nevertheless expressed in terms of quantities that determine these bonds between pairs of atoms. For many diatomic molecules the potential-energy curve can be constructed on the basis of known spectroscopic data, with the aid of the Morse curve\(^5\), and they can be used to calculate the bond between pairs of atoms. However, in order to calculate \(E_3\), it is not enough to know \(A+\alpha\), \(B+\beta\), and \(C+\gamma\), obtained from the Morse curve. In addition one must also know \(A\), \(B\), and \(C\), i.e., the part of the bond energy due to Coulomb forces (\(\alpha\), \(\beta\), and \(\gamma\) are the exchange parts of the bonds). In the case of hydrogen, the Coulomb part of the bond energy amounts to approximately 10% of the total bond energy, as shown by Sugiura’s\(^6\) calculations of the Heitler and London\(^3\) integrals.

Fig. 1. Three atoms and the potential energies determining the total potentials.

Farkas\(^7\) found for the reaction

\[ \mathrm{H}_{2\ \mathrm{para}} + \mathrm{H} = \mathrm{H}_{2\ \mathrm{equil}} + \mathrm{H} \tag{3} \]

an activation energy from 4 to 11 kg-cal. Eyring and Polanyi\(^8\) found, starting from equation (3) and using the Morse curve for \(\mathrm{H}_2\), 13 kg-cal. If one uses another Morse curve (namely one that agrees with the theoretical curve at large distances, where it is known that the theoretical curve is correct), then the calculated value of the activation energy lies within the limits of the experimental data.

Let us now consider how such a calculation of the activation energy is carried out. It is necessary to establish in what way a hydrogen atom \(\mathrm{H}\) can be brought so close to the mole-

cule H₂, so that the middle one of the three atoms, being equally close both to one neighbor and to the other, would be equally capable both of accepting a new partner and of keeping the old one. From equation (2) and the potential-energy curve of the hydrogen molecule H₂, the potential energy can be calculated for any configuration, so that our problem is undoubtedly solvable. A study of the energy \(E_3\) shows that less energy is required in the case when

Fig. 2. Position of the three atoms to which the potential surface shown in Fig. 3 refers.

the three atoms remain on a straight line; consequently, we need not consider other configurations. If the distances \(r_1\) and \(r_2\), shown in Fig. 2,

Fig. 3. Curves of equal values of the energy for the case of three atoms situated on one straight line.

are laid off at an angle of \(120^\circ\) with respect to one another and the curves of equal values of the energy \(E_3\) are drawn, Fig. 3 is obtained. In the work of Eyring and Polanyi\(^{8}\) it was shown that if one lets a ball roll over the surface corresponding to Fig. 3, then it will show exactly,

how the distances \(r_1\) and \(r_2\) will change in the \(H_2\) complex. This surface, therefore, solves our problem. The surface resembles two long valleys extending to infinity parallel to the axes. If one goes along each of the valleys toward the origin of coordinates, one slowly rises to 13 kg-cal. Here one descends into a shallow basin 1.6 kg-cal deep. After passing through this basin, one again reaches a pass over the edge of the basin, situated symmetrically to the first, 13 kg-cal high, and then slowly descends into the second valley. The slope of the valley facing the edge of our geographical map rises steeply, in accordance with the fact that the hydrogen atoms repel one another if they are brought closer than \(0.76\ \text{Å}\). The inner slope of the valley rises less steeply, passing into a high central plateau 101.5 kg-cal in height, which corresponds to the dissociation of the \(H_3\) complex into three separate hydrogen atoms. Sea level, i.e. 0 kg-cal, corresponds to the lowest energy available for the hydrogen molecule \(H_2\), when the third hydrogen atom is infinitely far away. The maximum depth of the sea itself reaches 6.1 kg-cal. This is the potential energy corresponding to half a quantum of vibrational energy of the hydrogen molecule \(H_2\), which it cannot give up while remaining a hydrogen molecule. This energy, however, shows up in a decrease of the activation energy of reaction 3. Reaction 3 is indicated on our map by a broken line with arrows, running along the bottom of one valley through the basin into the other adjoining valley, i.e. from large \(r_1\) to large \(r_2\). The activation energy is, of course, the minimum energy permitting passage from one valley into the other. The motion of a ball rolling along the valley in such a way that \(r_1\) increases while \(r_2\) remains constant depicts the change in energy during the separation of a hydrogen molecule \(H_2\) and a hydrogen atom \(H\), if the one moves translationally relative to the other. The periodic motion back and forth across the valley corresponds to the vibration of the molecule \(H_2\). The significance of these motions will become immediately clear if

look at Fig. 2, what the corresponding dimensions \(r_1\) and \(r_2\) denote. To find out whether the activation energy required for reaction 3 corresponds to vibrational or to translational motion of the atoms, we place the ball on the pass between the basin and the valley and let it descend into the valley. Then we determine what fraction of the kinetic energy corresponds to motion along the axis of the valley and what fraction to motion perpendicular to it. This determines how the ball must be launched in the reverse direction so that it reaches the highest point of the pass without excess energy. For the parahydrogen reaction only translational energy can be used. This is clear if one recalls that vibrational energy can be absorbed only in portions of approximately 12 kg-cal, whereas the surface shows that of the 13 kg-cal required for activation, no more than one or two can be effectively used in the form of vibrational energy.

The ball, launched in the most effective manner from one of the valleys, will very slowly cross the first pass into the middle, shallow basin, where it will move back and forth in zigzags before it gets out over the second pass. During this process the three atoms form a quasi-molecule. If, while the ball is in the basin, a collision with a foreign particle takes away its energy, it must remain in the basin until it recovers this energy. If the depth of the basin were 25 kg-cal instead of 1.6 kg-cal, then \(\mathrm{H}_3\) would be stable at room temperature, and the recovery of energy allowing the ball to rush away would constitute a monomolecular decomposition.

It is instructive to consider such a simple example as \(\mathrm{H}_3\), since Fölmer\(^9\) and his collaborators have shown that the decomposition of nitrous oxide is monomolecular and has an especially small rate coefficient for the reaction

\[ k = 10^{10} e^{-\frac{E}{RT}}. \]

Usually the proportionality coefficient in this expression is of the order of \(10^{13}\), which, as Polanyi and Wigner\(^ {10}\) were able to show, should be the case for molecules

several special kinds, consisting of many atoms. The coefficient \(10^{13}\) obtained by them is equal to the highest natural frequency of vibration of the molecules. Their method, based on the presence of a large number of atoms and a large number of natural frequencies of vibration, does not give exact results for such a simple molecule as \(\mathrm{N_2O}\), but precisely for these simple molecules one can construct the surfaces of potential energy and let the ball roll over them. For a monomolecular reaction, the activation energy \(E\) of equation (4) is the depth of the basin (the surface of potential energy), measured from the bottom to the lowest point of the rim of the basin, through which the ball can escape outward. In this connection it is necessary to take into account the half-quantum of vibration. Indeed, in order that the ball may escape from the basin, it must not only have the energy \(E\), but must also vibrate in such a direction as to pass through the lowest point or the lowest points of the rim of the basin. When it has acquired the energy \(E\), then, generally speaking, it will not at once vibrate in such a direction. Depending on the form of the surface, one or several vibrations may be required before it can escape. Thus it is easy to see that each activated molecule has its own individual fate. For the reaction-rate coefficient we shall have to write an expression analogous to the following:

\[ k=\frac{\overline{S}e^{-\frac{E}{RT}}\left(\frac{E}{RT}\right)^{\frac{1}{2}n-1}}{\left(\frac{1}{2}n-1\right)!}, \]

since we assume that the energy is distributed among all the internal degrees of freedom. \(\overline{S}\) denotes the mean value of the reciprocal of the time required for the ball to escape from the basin after it has at least the energy \(E\) (equal to the depth of the basin), distributed among its quadratic terms. One interesting proposition follows directly from the picture of monomolecular decomposition. If the ball has the smallest energy \(E\) that allows it to escape, then it can escape only through the very lowest points of the rim of the basin. In the event of—

the energy distance at its disposal will already be a large part of the region. Therefore we may state the following assumption: \(S\) is the greater, the more \(E\) exceeds the minimal activation energy necessary for decomposition. For the case when \(E\) exceeds the required minimum by a sufficiently small amount, this dependence will, of course, be linear. Rice and Ramsperger\(^{11a}\) and Kassel\(^{11b}\) arrived at this assumption in explaining the observed rates of monomolecular reactions.

The rate of decomposition of activated \(\mathrm{N_2O}\) molecules, amounting to approximately \(1/1000\) of the ordinary rate, differs from the theoretical results of Polanyi and Wigner more than would be expected from mechanical considerations, so that it is natural to look for non-mechanical causes. Wigner\(^{12}\) pointed out that if the electron spins of the molecules that are the reaction products cannot, by algebraic addition, yield a sum which is also obtained by adding the spins of the molecules entering into the reaction, then the probability of the reaction is small (roughly \(1/1000\) of the normal one), even for molecules possessing the necessary activation energy. Normal nitrous oxide is in a singlet state, i.e. it has no resultant spin. It is diamagnetic. The state of the \(\mathrm{N_2}\) formed is undoubtedly singlet, whereas the atom will undoubtedly be in a triplet state, which gives the products a resultant spin equal to unity. Consequently, the initial and final resultant spins are different, and we expect an unusually slow reaction. Such slowness of reactions owing to a change in the total multiplicity also occurs in the following cases, the reactants and products of which are in their normal state: \(\mathrm{CO}+\mathrm{O}=\mathrm{CO_2};\ \mathrm{H_2}+\mathrm{O}=\mathrm{H_2O};\ \mathrm{H_2}+\mathrm{O_2}=\mathrm{H_2O_2}\). This is also the reason why \(\mathrm{O_2}\), with its unpaired electrons, most often does not behave as a free radical.

Perturbation theory, in the form in which it is usually formulated, does not take into account the possibility of a change in multiplicity. If magnetic forces are taken into consideration,

QUANTUM MECHANICS AND CHEMICAL REACTIONS

then it turns out that the probability of this is small, but different from zero. Analogous forbidden transitions in spectra give us the possibility of estimating this probability of a change in multiplicity[^13].

In considering potential-energy surfaces it is useful to employ the terminology already adopted earlier[^8]. We shall call the activation energy β the energy obtained from equation (2), if it is assumed that all the bonding energy is exchange energy \((A = B = C = 0)\), and also that the bond \(A + \alpha\) between two outer atoms of the three atoms arranged in a straight line is equal to zero. The activation energy β for reaction 3 is 14.4 kg-cal. The good agreement of the activation energy β with the activation energy calculated from the unchanged equation (2) is accidental. The increase in the activation energy β in the case where \(\alpha\) is not neglected is called the activation energy α. The activation energy α is sometimes small, but always positive. The Coulomb activation energy is the difference between the sum of the activation energies α and β and the activation energy calculated from equation (2). It is a consequence of Coulomb forces and is always negative. In calculating the activation energy we count it not from the lowest part of the surface, but starting a half-quantum higher. Thus we assume that the activation energy is diminished by this half-quantum. Generally speaking, this is not quite correct, since the half-quanta of the activated state, generally speaking, cannot be wholly neglected.

If one calculates the activation energy β for reaction 3 and constructs the corresponding surface, the result is that the energy of the \(\mathrm{H}_3\) complex is 10 kg-cal less than the energy of \(\mathrm{H}_2 + \mathrm{H}\), and that in order to decompose \(\mathrm{H}_3\), an activation energy of 24 kg-cal would be required, which would make this compound stable at room temperature. This result, of course, is incorrect and shows that, in order to obtain reasonable results, one must use equation (2). London took into account, for the example considered by him, only the activation energy.

For the case of three identical halogen atoms the energy

the activation of reactions of the type \(X_2+X=X+X_2\) is in fact small. If the bond due to Coulomb forces already amounts to 10% of the total bond, as, for example, in the case of \(H_2\), then complexes of the type \(F_3\), \(Cl_3\), \(Br_3\), and \(I_3\) are more stable at room temperature than the diatomic molecule and the atom. Rollefson and Eyring\(^{14}\) pointed out the significance of such complexes for photochemical reactions. The supposition that the reaction will be extremely slow because three molecules participate in it is undoubtedly still open to question, in view of the inelasticity of many types of collisions.

It is interesting to continue the argument applied by Eyring and Polanyi to the reaction

\[ \mathrm{H}_{2\,\mathrm{para}}+\mathrm{H}=\mathrm{H}_{2\,\mathrm{equil}}+\mathrm{H}. \tag{3} \]

They found that if the collision of three atoms takes place along one straight line, then the activation energy is equal to 13 kg-cal; for collisions not in a straight line the activation energy is, of course, higher. But the fraction of collisions taking place along a straight line may be neglected, so that in fact one must take into account, with the corresponding factor, the probability of a collision in which the direction of the approaching atom forms some angle with the axis of the molecule. In this way one can obtain not only the activation energy, but also the absolute rate of reaction 3.

Pelzer and Wigner\(^{15}\) found, by this purely theoretical method, a reaction rate in excellent agreement with the measurements of Farkas. They also consider the probability that electronic transitions might take part in this reaction, and come to the conclusion that the reaction must proceed adiabatically, as had been assumed earlier. If, therefore, the method applied to this reaction is to be changed, it is necessary to change the technique of calculating the activation energy, and not the original idea, namely that the process is adiabatic.

ACTIVATION ENERGY OF REACTIONS IN WHICH FOUR ELECTRONS CHANGE PARTNERS

The equation for the potential energy of an arbitrary configuration of four univalent atoms is the following:

\[ E_4=A_1+A_2+B_1+B_2+C_1+C_2+\left[{}^{1}/_2\{(a_1+\alpha_2-\beta_1-\beta_2)^2+(\alpha_1+\alpha_2-\gamma_1-\gamma_2)^2+(\beta_1+\beta_2-\gamma_1-\gamma_2)^2\}\right]^{1/2}. \tag{4} \]

The meaning of these quantities is clear from Fig. 4, where the bonds that would exist between pairs of atoms in the absence of the other atoms are written along the connecting straight lines; such a bond is a function only of the distance between the pair of atoms in the case when the electrons are in \(S\) states. In the case of directed valences the proper functions will be directed so that the numerical value of \(E_4\) is as large as possible. In one of the earlier papers it was established that equations (2) and (4) can be obtained by Slater’s method for complex atoms. Zener, Gibson, and the author did not publish these results, but the reader may be referred to a recently published paper by Slater\(^{16}\), in which a whole series of molecular problems is developed and which contains our two cases as well, along with many other results of interest and importance to chemists.

Fig. 4. Four atoms and the potential energies determining the total potential.

For six reactions of the type

\[ WX+YZ=XY+WZ \]

the activation energy\(^{17}\) was calculated and compared with the activation energy for the same overall reaction in which atoms took part in the intermediate reactions. In agreement with experiment it was found that reactions between halogens and hydrogen will proceed with the participation of atoms, except in the case of \(I_2\), in which only molecules will participate. It has also been shown that the homogeneous reaction of the conversion of hydrogen bromide into an equilibrium mixture will definitely proceed with the participation of atoms, as Farkas indicated.

Steric Hindrance and Kinetic Diameters

In certain reactions accompanied by the breaking of two bonds and the formation of two new ones (in which, essentially, only four electrons participate), other atoms are forced to approach one another to distances at which they repel each other. This is called steric hindrance, and the corresponding increase in the activation energy is readily calculated by using the following consequence of the Slater method for molecules. The additional repulsive potential is half the sum of the exchange energies between all possible pairs of electrons, one electron of each pair belonging to different molecules (except for those four which participate directly in the reaction). The exchange energy is estimated, as before, on the basis of the corresponding Morse curves. The presence of Coulomb potentials connected with this always diminishes the repulsion. Including these quantities as well in equation (4), one can calculate the activation energy for a reaction in which steric hindrance also plays a role. In the presence of permanent dipoles it is necessary to include also the potential due to them. Mutual polarization of the molecules, or van der Waals forces, gives rise to a still further small potential, proportional to the sixth power of the distance. London[^18], and also Slater and Kirkwood[^19], gave relations for calculating this quantity, so that it is now possible to calculate, in a first approximation, the activation energy for reactions in which 3 or 4 electrons participate, taking into account the steric hindrance due to atoms which themselves do not take part in the reaction. For such a calculation detailed information is necessary concerning the arrangement of the atoms in the molecules, obtained in many cases from analysis by means of X-rays. In addition, one must take into account the spatial effect due to permanent dipoles.

Ordinary collisions, of course, are simply a special case of steric hindrance, and the distance,

onto which the molecules approach one another should be calculated precisely by the method described above. Indeed, Slater and Kirkwood\(^ {19}\) calculated the constants of the van der Waals equation for helium and obtained excellent agreement with experiment. Kirkwood and Keyes\(^ {20}\) investigated other physical properties and also obtained satisfactory agreement with experiment. The process of collision of two H\(_2\) molecules was also calculated theoretically\(^ {21}\). The agreement with experiment is good. Fig. 5 schematically depicts the potential energy between two molecules at a distance approximately equal to their kinetic diameters.

Fig. 5. Potential curves between two colliding molecules.

The potential energy, of course, depends not only on the distance between the centers of gravity of the two molecules, but also on the relative orientation of their axes. Two colliding molecules behave at such large distances almost like two spheres. Curve \(I\) for H\(_2\) was calculated on the basis of equation (4). If equation (4) is expanded in a binomial series for the case of configurations for which \(\alpha_1\), \(\alpha_2\), \(A_1\), and \(A_2\) are large, while the remaining quantities are small, we obtain:

\[ E_4 = Q + \alpha_1 + \alpha_2 - \frac{1}{2}(\beta_1 + \beta_2 + \gamma_1 + \gamma_2). \tag{6} \]

We have already established how equation (6) must be changed for the case of collision of more complex molecules. During an ordinary collision \(\alpha_1 + \alpha_2\) remains almost constant, as do the potentials \(A_1\) and \(A_2\) entering into \(Q\), while the other terms corresponding to attraction increase, as exponential functions of the distance, when both molecules approach one another, and give an exclusively repulsive force, represented by curve \(I\) in Fig. 5. Each term separately has been calculated on the basis of the potential-energy curves for atomic pairs. Thus,

However complex the colliding molecules may be, in order to construct curve I it is necessary only to know the mutual arrangement of the atoms in the individual molecules and the potential-energy curve for pairs of atoms, each atom of the pair belonging to a different molecule. The potential energy will ordinarily depend both on the relative orientation of the molecules and on the distance between their centers of gravity. The repulsive potential existing between molecules (without permanent dipoles) probably arises from exchange forces, which would give attraction if the electron spins were antiparallel. The van der Waals forces, represented by curve III, can be calculated on the basis of relations proposed by London[^18] or by Slater and Kirkwood[^19]. The sum of these two curves gives us curve II, which represents the actual potential energy of two molecules as a function of the distance between them (for a definite relative orientation). It is interesting to see how much information such a curve provides.

For the present we shall not take into account the difference in the energies with which pairs of molecules collide at one and the same temperature. Then it may be assumed that two molecules approach one another with energy \(RT\) in such a way that the abscissa at point \(A\) is equal to their kinetic diameter. The decrease of the diameter with temperature depends on the slope of the curve at point \(A\) and may be compared with Sutherland’s constant \(C\). The abscissa at point \(B\) is equal to the diameter of the molecules in the liquid state at zero degrees. The ordinate at point \(B\) is the heat of sublimation at zero degrees. When the temperature is raised, the molecules of the liquid oscillate about the minimum. This reduces the heat of sublimation and, in addition, causes expansion of the liquid. The expansion occurs because the curve is not symmetrical, since the oscillating molecules spend more time on the less steep portions of the curve. Thus the coefficient of expansion measures the degree of asymmetry of curve II. The slope of curve II to the left of point \(B\) determines the compressibility. The curvature at the point of minimum determines the frequency of oscillations entering into the expression for the specific heat of the liquid-

validity. Thus, we also have these experimental checks on the correctness of our calculations of valence forces, which determine the activation energy.

The obvious way of calculating the repulsive potential for the distances encountered in kinetic theory is a method based on the fact that a certain definite fraction of the potential of a Morse curve gives the exchange bond. It would be too much to expect that such a method of calculation would lead to exact results, although the errors in the result, generally speaking, are not large. The potential energy of H₂ according to Morse does not decrease with the distance between the H atoms as rapidly as the theoretical curve. The character of the decrease of the potential energy for other diatomic molecules, calculated with the aid of eigenfunctions obtained by Slater, indicates that this may be a quite general difficulty connected with the use of Morse curves. This deficiency leads to somewhat increased activation energies and kinetic diameters. The whole question would need to be considered more thoroughly.

The ratio of the exchange bond to the Coulomb bond

In the preceding calculations of the activation energy it was assumed that the Coulomb bond amounts to 10% of the total bond, as calculated for the case of H₂. Rosen²² has recently calculated the potential energy of the bond for Na₂ with the aid of fundamental functions obtained by Slater. For the heat of dissociation, for the distance between the atoms, and for the vibrational frequency of the lowest state he obtains very good agreement with experiment. For the ratio of the magnitude of the Coulomb bond to the total bond in the neighborhood of the minimum he obtains 28.3%. Bartlett and Furry²³ performed a similar calculation for Li₂, which is also in excellent agreement with experiment. For the ratio of the magnitude of the Coulomb and total bonds they obtain 22%. This percentage ratio is constant within 1%, beginning from the minimum up to distances at which the bond energy has fallen to one fifth of its maximum value. This again confirms

assumption that the ratio of Coulomb to exchange bonding is almost independent of the distance between the atoms in the region essential for calculating the activation energies. However, this also shows that this ratio depends very much on the kind of molecule. Therefore we must have some criterion in order to know in which cases one may expect that the fraction of the bond due to Coulomb forces will be large.

Molecules (irrespective of the valence of the constituent atoms) which form molecular crystals in the solid state possess this property because the exchange potentials greatly exceed the Coulomb potentials. On the other hand, the fact that atomic crystals are formed from divalent or monovalent atoms indicates that the exchange bond due to antiparallel spins is of secondary importance compared with the Coulomb bond. This becomes clear if one takes into account that two electrons with antiparallel angular momenta repel all the others, so that only in the case where the pair is more or less isolated will the total exchange bond help to make the crystal stable. Since the alkalis form atomic crystals, we expect that for them the Coulomb part of the bond is comparatively large. The consideration of activation energies leads to the same conclusion. Of course, the manner in which molecules crystallize is a good indicator of how they will behave in collisions in the gas phase. In particular, from this one can judge how much energy is required in order to make a molecule forget who its former partner was. In a recently published article, Kramer and Polanyi[^24] consider certain properties of crystals, including their vibrational frequency, in connection with the magnitude of the exchange bond.

For the case of directed valences, the relations from which the activation energy is calculated are the same as for \(s\)-electrons, with the difference that the value of each individual bond depends on the angle \(\theta\) between the line joining the atoms and the axes of the eigenfunctions.

The bond \(D\) between a \(p\)-electron and an \(s\)-electron depends on the angle \(\theta\) in the following way:

\[ D=A\cos^{2}\theta+kA\sin^{2}\theta. \]

Slater\(^{16}\) estimated \(k\) for the OH bond and found, in a rough approximation, the number 2. Experimentally, \(k\) can be determined from the frequency of transverse vibrations caused by directed valences.

Summary

Let us briefly summarize the general point of view. Most problems concerning the formation of compounds (valence) and concerning activation energy can be solved by constructing the potential-energy surface, the low points of which correspond to compounds stable at ordinary temperature, provided that the barrier separating one minimum from an even lower one is higher than approximately 20 kcal. The height of the barrier is, evidently, the activation energy. To find how the ball will roll over this surface is simply a problem of mechanics. The essence of the theory of adiabatic reactions is that specifying the relative positions of the atoms is sufficient to determine the energy, since it is assumed that the electrons, owing to their more rapid motion, adjust themselves to the configuration of minimum energy. The fact that light is sometimes emitted shows that this is not always true. It must be considered that a system containing excited atoms lies on a surface of higher potential energy. In passing from a higher to a lower surface, kinetic energy appears, associated with motion in some direction, which may be sufficient to carry the ball (representing the system) over the potential-energy barrier. This constitutes the chemical reaction.

Such a graphic representation of compounds and reactions would not be especially useful if we were unable to construct potential-energy surfaces at least—

to a lesser extent, with some approximation. The data for determining the relative heights of the potential minima are supplied to us by thermodynamics in the form of heats of reaction. The positions of these minima, expressed in interatomic distances, we obtain from X-ray analysis and from the arguments of Pauling^25 and Slater^26 that elucidate this question. From spectroscopic data for diatomic molecules we learn the binding energy between atomic pairs as a function of distance, and, finally, from perturbation theory we can calculate the energy of complex configurations, expressed in terms of the energies between atomic pairs, so that we thus obtain the possibility of investigating the barriers between the minima of our potential surface.

Such a method, which appears general in the sense of its applications, is subject to certain limitations. First, we use perturbation theory, restricting ourselves to the first approximation, which in the case of the H$_2$ molecule gives a binding energy amounting to only three quarters of the experimental value. Recently, however, this method has been applied to Na$_2$ and Li$_2$, and results were obtained differing by only a few percent from the experimental values. Likewise, the results obtained for H$_3$ and for other cases are, apparently, strikingly close to the experimental values. Secondly, we need to know not only the potential curve of a diatomic molecule, but, in addition, we must know what fraction of this potential is due to Coulomb forces. Only in three cases is this known from direct calculation. There are many ways of estimating this quantity, so that this cannot be regarded as a major difficulty. Thirdly, the application of the perturbation method often requires a great deal of work. If only three or four electrons change partners during a reaction, then the binding energy for any configuration of atoms is easily calculated by the methods described above. The steric hindrance due to the fact that electrons not participating in the reaction are forced to approach one another can also be calculated without serious difficulty. If, during the reaction, five or six electrons change partners, then the energy of each configu-

...rations can be found by solving an equation of the fifth degree. Kemble and the author have done this for some interesting cases. The calculation is not difficult, but lengthy. If, in a reaction, seven or eight electrons simultaneously change partners, then for each configuration it is necessary to solve an equation of the fourteenth degree. Although it is entirely possible to find the roots of such an equation, finding them is certainly impractical. Fourth, the activation energies caused by sources other than the valence associated with spins—for example, by \(l\)-valence, or by the fact that charged ions are unable to approach one another—must be considered separately. Fifth, Morse curves must be regarded only as a temporary expedient, and they will have to be replaced by better curves of the potential energy of atomic pairs when these can be calculated. In spite of these reservations, we may predict in advance that the application of perturbation theory will lead to a far more exact understanding of chemical reactions and will indicate the path toward a more perfect theory. The idea that valence bonds, conditioned by the presence of spins, gradually (adiabatically) pass into new bonds of the same kind, despite the fact that the system in no case has sufficient energy to break the bonds directly, is apparently confirmed for some of the cases already considered.

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