Quantum Theories of Adsorption

V. L. Bonch-Bruevich

Introduction

Certain problems in the theory of adsorption, as one of the forms of interaction of gas molecules with the surface of a solid, unquestionably require quantum-mechanical treatment. At present there are already quite a number of works devoted to quantum-mechanical problems in the theory of adsorption, and it seems timely to review these works from a certain unified point of view.

In order to be able to assess correctly one work or another (or a group of works), it is first of all necessary to formulate the minimal requirements that a quantum theory of adsorption must satisfy.

In view of the enormous mathematical difficulties, it seems to us inappropriate to demand from a quantum theory of adsorption (as, in general, from a quantum theory of the solid state) the obtaining of exact numerical results. The order of magnitude should, of course, be obtained correctly, but the exact values of the constants are easier to measure than to calculate; quantum mechanics, however, must provide a microscopic picture of the phenomena.

In other words, it must:

a) Reveal the mechanism of elementary processes on the surface (in particular, reveal the physical meaning of terms such as “active center”).

b) Establish the dependence of the adsorption properties of the surface on various observable parameters (on temperature, on the composition and concentration of impurities, etc.).

c) Relate the adsorption characteristics of the surface to its other properties (establishing, in so doing, precisely which ones).

Let us note that renouncing the obtaining of exact numerical results by no means signifies renouncing comparison with experiment. Those qualitative dependences that quantum mechanics is to provide can, of course, be subjected to experimental verification.

Existing quantum-mechanical works on the theory of adsorption may be divided into three main groups:

1. The theory of the forces of physical adsorption.
2. The theory of adsorption kinetics (including surface migration and selective adsorption).
3. Electronic phenomena in adsorption.

We shall proceed to examine the works by groups.

I. THEORY OF THE FORCES OF PHYSICAL ADSORPTION

As usual, by physical adsorption we shall mean adsorption due primarily to the forces of intermolecular interaction (van der Waals forces). In the language of quantum mechanics this means that the overlap of the wave functions of the electrons of the adsorbent and of the adsorbed atoms (or molecules) may be neglected. In fact, otherwise we would be obliged to consider the adsorbent and adsorbate as a single system, and the very concept of van der Waals interaction would lose its meaning. Conversely, by chemical adsorption we shall understand that adsorption for which the overlap of wave functions plays a decisive role. Obviously, the indicated distinction is not entirely definite; in fact there is always some overlap of wave functions, and the so-called physical and chemical adsorptions are only two idealized limiting cases. Nevertheless, such a distinction is fully justified experimentally[^53].

The task of the theory of the forces of physical adsorption consists in calculating the energy of the system \(W\) as a function of the distance \(R\) between the surface of the adsorbent and the adsorbed molecule. Let us note that the qualitative course of the curve \(W(R)\) is known to us in advance: it must evidently give attraction at sufficiently large distances, repulsion at very small ones, and have a minimum for \(R\) of the order of several angstroms. Having calculated the curve \(W(R)\) exactly, one can compute the heat of adsorption, which is usually what interests experimentalists. Let us note, however, that by neglecting the overlap of wave functions we shall be able to calculate only the part of the curve corresponding to sufficiently large \(R\); the repulsive potential will have to be taken into account in a purely empirical way, approximating it by some curve verified in other cases. This circumstance may greatly reduce the reliability of the results obtained.

In calculating \(W(R)\) one must distinguish two main limiting cases: adsorption on metals and on ionic crystals. (The word “metal” is used in a somewhat conventional sense: the case meant is that in which one cannot neglect the interaction between the adsorbed molecule and the conduction electrons of the crystal.)

A. Adsorption on Ionic Crystals

This problem in the theory of adsorption is historically the first to which quantum mechanics was applied[^1]. In our literature there is a good (though somewhat outdated) review on this subject[^2], and therefore we shall confine ourselves to only a few remarks.

There is essentially nothing specifically adsorptional in the quantum-mechanical calculations belonging here: quantum mechanics is used only to elucidate the nature of the van der Waals forces. Once the latter have been specified and their additivity has been proved, there remains the purely mathematical problem of summing over all ions of the lattice. This problem was solved in a number of works[^3][^4][^5]; the most detailed calculations were carried out in Orr’s article[^6], where both the van der Waals and the electrostatic potentials are taken into account, as well as the potential of the repulsive forces. However, even there the calculated values of the heats of adsorption are smaller than the experimental ones by 500 cal (25%). This is not surprising in view of a number of approximations that Orr was necessarily forced to admit in the very formulation of the problem (allowance only for dipole–dipole interaction, whereas it is known[^5] that dipole–quadrupole terms are significant at small distances; a purely empirical formula for the repulsive potential, whose validity in the case of adsorption has not been proved; a semiempirical formula for the van der Waals forces, giving an accuracy of about 15%). As has already been indicated, one should not demand exact numerical figures from the quantum theory of adsorption; but an explanation of the nature of the forces responsible for physical adsorption was already given in London’s first work[^1]. Therefore Orr’s calculation (which required no small amount of computational work), in which the problem of obtaining exact numerical values for the heats of adsorption is posed specifically, seems to us to be to a considerable degree useless and even harmful, since scrupulous computations there mask the crudeness of the initial assumptions.

It seems to us that in this question quantum mechanics has already given everything that can be required of it. The situation is somewhat different in the case of adsorption on metals, to which we now turn.

B. Adsorption on Metals

In the works of London and others, the free electrons of the crystal were not taken into account. Therefore these works apply only to the case of a dielectric, where there are very few conduction electrons. In the case of adsorption on a metal, on the contrary, one may expect the conduction electrons to play a decisive role.

The quantum-mechanical problem of the interaction of an atom with a metal was considered by Lennard-Jones[^7], who obtained the following

formula for the binding energy of a spherically symmetric atom with a surface

\[ W=-\frac{e^2 \overline{r^2}}{12 R^3}. \tag{1} \]

Here \(R\) is the distance (along the normal) from the atomic nucleus to the surface, \(\overline{r^2}\) is the mean value of the square of the distance from the valence electron in the adsorbed atom to the nucleus, and \(e\) is the electron charge (we note that in Lennard-Jones the denominator is not 12 but 6, which is connected with an arithmetic error in the calculation). However, Lennard-Jones made an entirely inadmissible approximation by treating the electrons in the metal classically. He calculated the energy of interaction between the virtual dipole moments of the atom and their electric images in the metal. It was thereby assumed that the metal has infinite conductivity, which is entirely inconsistent with reality in the frequency range close to the resonance frequencies of most adsorbed atoms.

Lennard-Jones’s calculation was refined by Margenau and Pollard \(^{8}\), Prosen and Sachs \(^{9}\), and Bardeen \(^{10}\). In all these works the problem is solved as follows: there is a molecule (or atom) without a permanent dipole moment, at such a distance from the surface that the overlap of the electron wave functions in the molecule and in the crystal may be neglected (the condition of physical adsorption). The energy of interaction of the molecule with the crystal is calculated as a function of the distance between them (\(R\)). The calculation is carried out in the second approximation of perturbation theory, with only the virtual dipoles of the molecule taken into account, while multipole moments of higher orders are neglected (which, incidentally, is by no means justified for small \(R\)). The three cited works differ from one another only in details of the calculation and in additional neglects and simplifications.

Margenau and Pollard first calculate the energy of interaction of the molecule with a certain “elementary volume” of the metal, \(d\omega\).

This volume is chosen sufficiently large that it can still be regarded as a metal, and at the same time sufficiently small that the wavelengths corresponding to the most important natural frequencies of the molecule appreciably exceed its linear dimensions. (Then the field in the given volume may be regarded as constant, and the concept of the polarizability of the volume has meaning.) In view of the first assumption, the final result is obtained by summing over all elementary volumes. It is further assumed that the linear dimensions of the elementary volume are much smaller than the distance of the molecule from the surface \(R\). This makes it possible to calculate the van der Waals energy \(W_\omega\)

van der Waals interaction between the molecule and the volume just as between two ordinary molecules\(^5\). We obtain

\[ W_{\omega}=-\frac{e^4}{R^6}\sum_{m_1}\sum_{m_2} \frac{\left|(n_1|X_i^{(1)}|m_1)(n_2|X_i^{(2)}|m_2)\right|^2} {E(m_1)+E(m_2)-E(n_1)-E(n_2)} , \tag{2} \]

where \(m_1\) and \(n_1\) denote sets of quantum numbers characterizing the intermediate and initial states of the molecule; \(m_2\) and \(n_2\) are the same for the crystal; \(E(m_1)\), \(E(m_2)\), \(E(n_1)\), \(E(n_2)\) are the corresponding energies; \(X_i^{(1)}\) and \(X_i^{(2)}\) are the coordinates of the electrons of the molecule and the crystal \((i=1,2,3)\); \((n_1|X_i^{(1)}|m_1)\) and \((n_2|X_i^{(2)}|m_2)\) are the corresponding matrix elements (summation over \(i\) is performed). An accurate evaluation of this sum presents great difficulties. It is therefore necessary to make some additional simplifications. Let us first show, following\(^8\), how formula (1) can be obtained from this. Let us assume for a moment that the energy levels in the molecule are situated much more closely than in the crystal:

\[ E(m_1)-E(n_1)\ll E(m_2)-E(n_2). \]

Then approximately we have

\[ W_{\omega}=-\frac{e^4}{R^6}\frac{1}{3}(n_1|r_1^2|n_1) \sum_{m_2}^{6} \frac{\left|(n_2|X_i^{(2)}|m_2)\right|^2} {E(m_2)-E(n_2)} . \tag{3} \]

(We have used the known relation

\[ \sum_{m_1}(n_1|\mathbf r_1|m_1)^2=(n_1|\mathbf r_1^2|n_1)\equiv \overline{r_1^2}; \]

\(\mathbf r_1\) is the radius vector with components \(X_1^{(1)}, X_2^{(1)}, X_3^{(1)}\).) As is known, the polarizability of any system in a field of frequency \(\nu\) is given by the formula

\[ \alpha_{xx}(\nu)=2e^2\sum_m \frac{|(n|X|m)|^2\{E(m)-E(n)\}} {\{E(n)-E(m)\}^2-h^2\nu^2}. \tag{4} \]

This gives

\[ W_{\omega}=-\frac{e^2}{R^6}\,\overline{r_1^2}\,\alpha^{(2)}(0), \tag{5} \]

where \(\alpha^{(2)}(0)\) is the static polarizability of the volume element under consideration \(d\omega\). It, as is known, is equal to \(d\omega/2\pi\). (In fact, \(\alpha(0)\) is determined by the relation \(P=\alpha E\), where \(P\) is the dipole moment created in the system by the field \(E\). In our case, evidently, \(P=qd\), where \(d\) is the thickness of the piece of metal in the direction of the field, and \(q\) is the charge induced by the field. This latter is equal to \(S\sigma\), where \(S\) is the surface of our piece and \(\sigma\) is the surface charge density. In an ideal conductor \(\sigma=E/2\pi\). Combining these formulas...

we obtain \(P=\dfrac{dS}{2\pi}E\), whence \(a=\dfrac{V}{2\pi}\), where \(V\) is the volume of the metal).

Summing over all elementary volumes (i.e. integrating over the whole volume of the metal, which for simplicity is assumed to be infinite in two directions—perpendicular to \(R\)), we obtain

\[ W=-\frac{e^2\overline{r_1^2}}{12R^3}. \tag{6} \]

This is formula (1), obtained by the image method. As was to be expected, it proves to be valid only if the energy levels in the molecule are much more closely spaced than in the metal (the electron in the molecule is “almost at rest”). In reality, however, one should expect precisely the opposite relation, since the field of the molecule changes little over the distance of the lattice constant, and therefore in its Fourier expansion the amplitudes of the short waves will be small; consequently, the principal role is played by electronic transitions between neighboring levels in the metal:

\[ E(m_1)-E(n_1)\gg E(m_2)-E(n_2). \]

Thus formula (1) gives greatly overestimated values of \(W\) and can be used only for a rough estimate of the upper limit of the interaction energy.

A more accurate result can be obtained either by adopting some model of the metal (which will make it possible to calculate explicitly the sum (2)), or by introducing certain experimentally determined constants. Margenau and Pollard followed the second route. In addition, they restricted themselves to considering only those molecules which can be described by a dispersion formula with one term. Then one obtains

\[ W=-\frac{1}{8R^3}\left\{\frac{e^2\hbar}{m}A^{(2)}(\nu)\frac{f_1}{\nu}+\Delta E^{(2)}\alpha^{(1)}(0)\right\}, \tag{7} \]

where \(\nu\) is the resonance frequency of the molecule; \(f_1\) is the corresponding oscillator strength \(\left(\dfrac{\hbar f_1}{m\nu}=\dfrac{4\pi}{3}\overline{r_1^2}\right)\); \(\alpha^{(1)}(0)\) is the static polarizability of the molecule; \(A^{(2)}(\nu)\) and \(\Delta E^{(2)}\) are constants characterizing the metal (\(\Delta E^{(2)}\) is an energy of the order of several electron-volts, \(A^{(2)}(\nu)\) is a dimensionless quantity). Bearing in mind that \(\overline{r_1^2}\) is related to the diamagnetic susceptibility \(\chi\) by the formula

\[ \chi=-\frac{Ne^2}{6mc^2}\overline{r_1^2} \]

(\(N\) is Avogadro’s number), and \(\alpha^{(1)}(0)\) can be calculated if the dielectric constant \(\varepsilon\) of the gas is known:

\[ \alpha^{(1)}(0)=\frac{\varepsilon-1}{4\pi}, \]

we see that formula (7) does indeed give a connection between adsorption characteristics and the properties of the adsorbed gas. The appearance of the empirical constants \(A^{(2)}(\nu)\) and \(\Delta E^{(2)}\) is inevitable, since no model of the metal has been specified. This does not seem to us a serious shortcoming, since both \(A^{(2)}\) and \(\Delta E^{(2)}\) can probably be related to observed quantities characterizing the metal (in particular, to its optical properties). It is much worse that, in view of the approximations made, the calculation is suitable only for molecules very far from the surface (the linear dimensions of the elementary volume must be much smaller than the distance from the molecule to the surface and much larger than the lattice constant). Thus the most interesting region, \(R\) of the order of several ångströms, falls outside the consideration. Therefore the work of Margenau and Pollard may be of interest, for example, for calculating the Knudsen accommodation coefficient, but not for the problem of adsorption.

Prosen and Sachs\(^9\) (see also \({}^{51}\)) calculate the interaction of a molecule directly with the entire metal. In doing so they specify a model of the electrons in the metal, treating them within the framework of Sommerfeld theory. This enables them to calculate a sum of type (2). For the energy of interaction of the molecule with the electrons of the metal one obtains

\[ W=-\frac{\pi a e^{2} k_{m}^{2}}{(2\pi)^{3}}\frac{\ln(2k_{m}R)}{R^{3}}, \tag{8} \]

where \(k_m\) is the wave vector corresponding to the Fermi energy of the given metal, and \(a\) is the polarizability of the molecule. In the calculation of Prosen and Sachs the value \(R\) is bounded from below only by the requirement that the electron wave functions in the molecule and in the metal not overlap. We see that something quite new is obtained in comparison with earlier works—a different law for the variation of \(W\) with distance. This result, however, cannot be regarded with any special confidence. The fact is that in the derivation the interaction of the metal electrons with one another was not taken into account, which is permissible only if it is small in comparison with their interaction with the molecule. For this the molecule must be quite close to the surface—at a distance of the order of the lattice constant. But at such distances the wave functions of the electrons of the molecule and of the crystal certainly overlap, and the calculation loses its meaning (exchange forces enter into play). According to the authors’ estimate, their calculation is valid under the condition

\[ 2\pi e^{2}R^{2}\ll 1\ \text{ev} \tag{9} \]

($\rho$ is the density of electrons in the metal). Taking here $\rho = 10^{22}\ \mathrm{cm}^{-3}$, we obtain from (9) $R \ll 2\,\text{\AA}$, which confirms our reasoning. Thus, the work of Prosen and Sachs on the calculation of the heat of adsorption on a metal is, strictly speaking, not applicable.

Finally, Berdin’s calculation,^10 like the work of Margenau and Pollard, is suitable only for distances considerably exceeding the lattice constant. Moreover, it is assumed there that the electron density is everywhere constant in the metal, which seems to us inadmissible in considering surface phenomena: it is known that near the surface the electron density may be increased (see III). Thus, the works listed can at best lay claim to determining the course of the curve $W(R)$ for values of $R$ considerably exceeding the lattice constant $a$ (and even then only if one assumes that the interaction of the metal electrons with one another, for some—as yet unclear—reasons, does not affect the course of this curve). The absolute values of the heats of adsorption can in no way be obtained from these calculations; therefore the numerical figures cited there deserve no confidence, and comparison of them with experiment is meaningless. However, in such a qualitative formulation the problem of the interaction of an atom with a metal had already been solved in 1935 by Ya. B. Zel’dovich^55 (whom, incidentally, the American authors did not even consider it necessary to cite). Zel’dovich used the Sommerfeld model of the metal and restricted himself to the region $R \gg a$. He showed that, owing to the allowance for the Pauli principle, for the interaction of an adsorbed atom with a metal only electrons with energies close to the Fermi energy are essential: their momentum $p$ is bounded by the inequality $\hbar k_{\max}-|p|<\dfrac{\hbar}{R}$. In this case the function $W(R)$ has the form

\[ W \sim \frac{a e^2}{a^2 R^2}. \tag{8′} \]

This formula does not differ in essence from (8), since $k_m \sim 1/a$, and $\ln(k_m R)$, within the approximations made, may be regarded as a constant.

Thus, the course of the curve $W(R)$ for $R \gg a$ is more or less clarified. However, the question of calculating the heat of adsorption on a metal and of its connection with other quantities characterizing the system remains entirely open.

II. THEORY OF ADSORPTION KINETICS

Most of the works in this section are devoted to calculating the lifetime of a molecule on a surface and the probability of adsorption when it collides with the surface, as functions of temperature and of the microscopic parameters characterizing the adsorbate and the adsorbent. In addition, three works are devoted to so-called selective adsorption, and one to the surface creep of molecules,

Strictly speaking, the problem of calculating the adsorption coefficients that occur in the Langmuir isotherm should in all cases be solved as follows. There is a system “bounded crystal + a molecule at its surface.” Owing to the quasi-classical nature of the motion of the heavy particles, the distance \(R\) from the center of inertia of the molecule to the crystal surface may be regarded as a parameter on which the mean energy of the system of electrons \(W(R)\) depends; this latter plays the role of potential energy for the motion of the center

Fig. 1. Dependence of the energy of interaction of a molecule with a surface on the distance between them.

of inertia of the molecule. (It is precisely this quantity that is calculated in the works discussed in the preceding section.) The function \(W(R)\) is periodic with the lattice period in the direction parallel to the crystal surface; the section of \(W(R)\) by a plane perpendicular to the surface is represented schematically in Fig. 1. (Qualitatively this is correct both for physical and for chemical adsorption.) A particle in a field of this type may, depending on its energy, either oscillate near the surface or go off to infinity. The former evidently corresponds to an adsorbed molecule (or atom), the latter to a free one.

As a result of the exchange of energy between the molecule and the crystal, transitions of the system from one state to another are possible; in particular, transitions are possible from the “free” state to the “bound” one (adsorption) and back (desorption). The problem is reduced to calculating the probabilities of these transitions.

However, in such a formulation the problem is mathematically very complicated. To simplify it, one neglects either the periodicity of the surface field (thereby reducing the problem to a one-dimensional one), or

thermal vibrations of the crystal. The first simplification is usually applied in calculating Langmuir adsorption coefficients, the second—in the theory of selective adsorption and surface migration of atoms and molecules.

A. Adsorption coefficients:

the lifetime of a molecule on the surface

and the probability of adsorption when a molecule collides with the surface

The calculation of adsorption coefficients is the subject of the work of Blokhintsev and Schechter \(^{11}\), as well as several papers by Lennard-Jones and his collaborators \(^{12-18}\).

In all these works, except one, which will be discussed at the end of this section, the process considered is the exchange of energy between the crystal and the adsorbate molecules, caused by thermal vibrations of the lattice. (For simplicity, a crystal with a simple or face-centered cubic lattice is considered.)

In order to reduce the problem to a one-dimensional one, it is assumed that the adsorbed particle effectively interacts only with one of the surface atoms and can move only perpendicular to the surface. Let us note that this assumption can be justified only in the case of a surface whose various regions differ sharply in their properties: in some places the heat of adsorption is much greater than over the rest of the surface, and the departure of molecules from these places (active centers) by surface migration requires a very large activation energy (compared with the heat of adsorption).

Choosing the coordinate system so that the \(z\)-axis is perpendicular to the surface, we can write the interaction energy \(W(R)\) in the form \(W = U(z - z_0)\), where \(z\) is the coordinate of the adsorbed atom, \(z_0\) is the coordinate of the surface atom (the latter undergoes thermal vibrations, and the point \(z_0 = 0\) corresponds to the equilibrium position); the curve \(U(z - z_0)\) is shown schematically in Fig. 1. Assuming that the amplitude of thermal vibrations is small in comparison with \(z_{\min}\)—the distance from the surface to the point at which the potential curve has a minimum—we may write

\[ U(z-z_0)=U(z)-z_0\left(\frac{dU}{dz}\right)_{z_0=0}. \tag{10} \]

(This assumption is quite legitimate, since otherwise we would obtain adsorption not on a crystal but on a liquid. In fact, one can speak of a crystalline structure only so long as the amplitude of thermal vibrations of the atoms is small in comparison with a certain characteristic dimension, which in our

case is the distance \(z_{\min}\). Since in what follows the Debye picture of the crystal vibrations is used, this assumption is essential.)

The first term in (10) corresponds to the motion of the molecule in the field of an immobile crystal; the second is a small addition characterizing the interaction of thermal vibrations of the lattice with the adsorbed atom. The problem is solved by the perturbation method. In the zeroth approximation the second term in (10) is altogether discarded, and the problem reduces to the investigation of:

a) the free vibrations of the crystal,

b) the possible states of the particle in the field \(U(z)\).

With regard to the vibrations of the crystal it is assumed that they can be described within the framework of the usual Debye picture, thereby neglecting the influence of the surface on the vibrations of the lattice.

The possible states of the particle in the field \(U(z)\) are determined by solving the corresponding one-dimensional Schrödinger equation. It, as was already indicated, has both discrete and continuous portions of the spectrum, corresponding to the adsorbed and the free particle.

In the next approximation the term \(-z_0 \left(\dfrac{dU}{dz}\right)_{z_0=0}\) is taken into account, causing transitions between the states of the unperturbed system both in the discrete spectrum (activation and deactivation of adsorbed molecules) and from the discrete spectrum to the continuous spectrum and back (desorption and adsorption). Transitions between states of the continuous spectrum are also possible, but they have no direct relation to adsorption and therefore do not interest us (they are essential for calculating the Knudsen accommodation coefficient).

As is known\(^{19}\), in the first approximation of perturbation theory the transition probability (per unit time) from one state to another is given by the formula

\[ \Gamma_{pl,rs}=\frac{2\pi}{\hbar}\left|(rs|U_1|pl)\right|^2_{E_r+E_s=E_p+E_l}; \]

\[ U_1=-z_0\left(\frac{dU}{dz}\right)_{z_0=0} \tag{11} \]

(the indices \(p,r\) and \(l,s\) denote sets of quantum numbers characterizing the initial and final states of the crystal and the molecule, respectively).

Noting that \(\left(\dfrac{dU}{dz}\right)_{z_0=0}\) depends only on the coordinates of the molecule, we can represent the matrix element \((rs|U_1|pl)\) in the form

\[ (rs|U_1|pl)=-\int \bar{\chi}_r z_0 \chi_p\,d\tau_{\mathrm{cr}}\int \bar{\varphi}_s\left(\frac{dU}{dz}\right)_{z_0=0}\varphi_l\,d\tau_{\mathrm{mol}}, \tag{12} \]

where \(\chi\) is the wave function describing the vibrations of the atoms of the crystal, and \(\varphi\) is the wave function of the electron in the molecule. Thus

Thus, the matrix element is split into two factors describing the transitions in the molecule and in the crystal. Since the lattice vibrations are described within Debye theory, \(p\) and \(r\) are the numbers of phonons \(n_i^0\) and \(n_i^1\) in the initial and final states of the crystal; the wave functions \(\chi_r\) and \(\chi_p\) have the form

\[ \chi_r=\prod_i \psi_{n_i}(\xi_i), \tag{13} \]

where \(\psi_{n_i}(\xi_i)\) is the wave function of a harmonic oscillator with energy \(\hbar\omega_i\left(n_i+\dfrac12\right)\) and coordinate \(\xi_i\), \(n_i\) is the number of phonons of the \(i\)-th type; finally, the displacement of a surface atom from the equilibrium position, \(z_0\), is represented in the form

\[ z_0=\sum_i (\mathbf e_i)_z\left(e^{i\mathbf k_i\mathbf r}\xi_i+\xi_i e^{-i\mathbf k_i\mathbf r}\right) \tag{14} \]

(\(\mathbf r\) is the radius vector characterizing the equilibrium position of the atom; the index \(i\) denotes a vibration with frequency \(\omega_i=\omega_i(\mathbf k_i,j)\) and wave vector \(\mathbf k_i\), polarized in a plane perpendicular to the axis \(x_j\) \((j=1,2,3)\); \(\xi_i\) is the amplitude of this vibration; \(\mathbf e_i\) is the unit vector in the direction of the vibration). Using the known expression for the matrix elements of Debye oscillators (see, for example, \({}^{20}\)), we obtain

\[ \int \chi_r^* z_0 \chi_p\,d\tau_{\mathrm{cr}}= \begin{cases} \displaystyle \sqrt{\frac{\hbar(n_i+1)}{2MN\omega_i}}\,e^{-i\omega_i t}, & \text{in the case of phonon emission,}\\[1.2em] \displaystyle \sqrt{\frac{\hbar n_i}{2MN\omega_i}}\,e^{i\omega_i t}, & \text{in the case of phonon absorption} \end{cases} \tag{15} \]

(here \(M\) is the mass of a crystal atom, \(N\) is the number of atoms in the crystal). In order to obtain the corresponding factor in the transition probability \(\Gamma_{pl,rs}\), one must take the square of the modulus of the quantity (15), average it over all possible values of the phonon numbers \(n_i\), and sum the result over all frequencies \(\omega_i\), taking into account the resonance condition

\[ E_r+E_s=E_p+E_l. \tag{16} \]

Using the relations

\[ \omega_i=c_j|\mathbf k_i| \]

and

\[ \frac{2\pi a^3}{3}\left(\frac{1}{c_l^3}+\frac{2}{c_{tr}^3}\right)=\frac{3\pi}{\omega_0^3} \quad \text{(for a cubic lattice)} \]

(\(\theta\) is the Debye temperature, \(\omega_0=\dfrac{k\theta}{\hbar}\), \(k\) is Boltzmann’s constant; \(a\) is the lattice constant, \(c_l\) and \(c_{tr}\) are the longitudinal and transverse sound velocities), we ultimately obtain the following factor in \(\Gamma_{pl,rs}\), due to the change in the number of phonons in the crystal:

\[ \frac{12\pi\hbar}{M}(k\theta)^{-2}\frac{\omega_{ls}}{\omega_0}\times \begin{cases} \bar n(\omega_{ls})+1 & \text{for emission of a phonon,}\\ \bar n(\omega_{ls}) & \text{for absorption of a phonon;} \end{cases} \tag{17} \]

\(\bar n(\omega_{ls})\) is the mean number of phonons of frequency

\[ \omega_{ls}=\frac{E_p+E_l-E_0}{\hbar} \]

and is, of course, given by Planck’s function

\[ \bar n(\omega)=\frac{1}{e^{\hbar\omega/kT}-1}. \tag{18} \]

For what follows it is very important that, in the processes considered, only one phonon can be emitted or absorbed, i.e. the maximum energy that the lattice can give up or receive is \(k\theta\). Transitions with the emission or absorption of a larger number of phonons become possible when the next terms of expansion (10) are taken into account, or in higher approximations of perturbation theory. It is not difficult to see, however, that the terms thereby obtained are proportional to \(N\) to some negative power (the amplitudes \(\xi_i\) are proportional to \(N^{-1/2}\); see \(^{20}\)). Therefore the probabilities of multiphonon transitions are very small. In work \(^{17}\), where this question was investigated specially, it was shown that the relative probability of transitions with emission and absorption of a large number of phonons in a simple cubic lattice is determined essentially by the factor \((10^{-8}T^2)^{n_1+n_2}\), where \(n_1\) and \(n_2\) are the numbers of emitted and absorbed phonons (the temperature \(T\) must be expressed on the absolute scale).

Let us now consider the other factor—

\[ \int \varphi_s\left(\frac{dU}{dz}\right)_{z_0=0}\varphi_l\,dz \]

—which characterizes the probability of transition of the adsorbed atom (molecule) from the state \(l\) to the state \(s\). (These states may belong to either the discrete or the continuous spectrum.) The functions \(\varphi_l(z)\) and \(\varphi_s(z)\), and the corresponding energies \(E_l\) and \(E_s\), are the eigenfunctions and eigenvalues of the equation

\[ \left\{-\frac{\hbar^2}{2m}\frac{d^2}{dz^2}+U(z)-E\right\}\varphi=0. \tag{19} \]

The difficulty lies in the fact that the function \(U(z)\) is known only qualitatively; its exact form for all \(z\) (especially in the most important region \(z\sim 1\text{--}3\,\text{\AA}\)) is not known. We know, however,

that it represents strong repulsion at very small distances, attraction for comparatively large \(z\), and has a minimum at \(z\) of the order of several angstroms. The situation is exactly the same in the theory of diatomic molecules, where the potential energy of the nuclei as a function of the distance between them has the same form as our function \(U(z)\), and likewise practically cannot be calculated exactly. It is usually approximated by some empirical formula, chosen so that it correctly conveys the course of the curve both at small and at large distances and, at the same time, is not too complicated. In a number of problems this device has proved very useful; it is therefore reasonable to try to apply it in our case as well. Let us note, however, that the use of such an “adjustable” potential can be justified only if the parameters entering into it can be determined from some other independent measurements. Thereby, incidentally, the connection between the adsorption characteristics of a surface and its other properties will also be established. Moreover, the final formulas must be sufficiently simple and transparent; otherwise the use of an “adjustable” potential loses its meaning.

Blokhintsev and Shekhter, in calculating the lifetime of adsorbed molecules on a surface, take as the functions of the discrete spectrum the wave functions of a harmonic oscillator of some frequency \(\omega\), while for calculating the functions of the continuous spectrum they approximate \(U(z)\) by a rectangular well of depth \(u_0\) and width \(d\). In addition, they restrict themselves to the case of very low temperatures, when it may be assumed that all adsorbed particles are on the lowest vibrational level. Under these assumptions, for the mean lifetime of a molecule on the surface \((\tau)\) one obtains the formula

\[ \tau=\tau_0 e^{-\frac{(U_0-\varepsilon)}{kT}}, \tag{20} \]

where \(\varepsilon\) is the zero-point energy of the adsorbed particle, and \(\tau_0\) is a quantity only weakly dependent on temperature (in the cited work \(\tau_0\) is explicitly expressed in terms of material constants; however, in view of the extreme crudeness of the calculation, which has the character of an estimate, there is no point in writing out the corresponding formula). A formula of this type had long ago been obtained by Ya. I. Frenkel\({}^{21}\) with the aid of classical statistics, and in his exponent there stood the heat of adsorption. From (20) it is seen that the heat of adsorption, as was to be expected, is not simply the depth of the potential well \(u_0\), but the difference \(u_0-\varepsilon\).

A completely analogous problem was considered somewhat later by Strakhan\({}^{13}\). The difference consists only in the fact that Strakhan used a more complicated curve to represent the function \(U(z)\),

\[ U(z)=D\left\{e^{-2x(z-z_0)}-2e^{-x(z-z_0)}\right\} \tag{21} \]

QUANTUM THEORIES OF ADSORPTION

(\(D\) and \(\varkappa\) are empirical constants). This function is well suited for representing curves of the type \(U(z)\), but it leads to rather complicated calculations. As a result, Strachan obtained a very unwieldy answer, practically excluding the possibility of physical applications of the theory.

The most substantial shortcoming of both works considered is the fact that they do not take into account the excited states of adsorbed molecules. These states, of course, are less “populated” than the ground level, but the probability of desorption from the ground level may be considerably smaller than from the excited one\({}^{14}\). Then it will turn out that the mean lifetime \(\tau\) cannot by any means be represented by a simple formula of the type (20), since \(\tau_0\) will depend strongly on temperature.

The role of excited levels was studied in detail by Lennard-Jones and Devonshire\({}^{14}\), who found that \(\tau\) is indeed not represented in the form (20). Unfortunately, the formulae of Lennard-Jones and Devonshire for \(\tau\) and for the probability of adsorption upon collision of a molecule with the surface—\(c\)—are so complicated that it is quite impossible to survey them. It is even difficult to understand how \(c\) and \(\tau\) depend on temperature. The complexity of the calculations, as in Strachan’s work, arises because the authors approximate \(U(z)\) by the function (21). The latter, as already noted, is well suited for representing curves of this type and therefore is rather popular in molecular theory. There, however, one usually does not have to use the solution of Schrödinger’s equation with such a potential. This solution can be obtained in explicit form, but has a rather complicated form, and further calculations with it (calculation of matrix elements) are very laborious.

In other works Lennard-Jones and his collaborators investigated the probability of activation and deactivation of adsorbed molecules (without desorption)\({}^{12}\), and also tried to take into account the possibility of lateral motion of adsorbed molecules\({}^{15}\), assuming that they can also oscillate parallel to the surface (free migration is still excluded in this case). The formulae obtained are just as obscure as in\({}^{14}\), and there is no point in writing them out, all the more so because, as will be shown below, all the works of the Lennard-Jones school contain one internal contradiction.

The only physically interesting results which, apparently, can be extracted from the works considered by Lennard-Jones and his collaborators are that, first, the probability of adsorption upon collision of a molecule with the surface is by no means equal to unity, as is often assumed (for hydrogen on a substance with \(\theta = 300^\circ\mathrm{K}\) at \(T = 300^\circ\mathrm{K}\), \(c\) is of order 0.3); second, for isotopic molecules both \(c\) and \(\tau\) increase with an increase in

mass of the adsorbed particle. It is unclear, however, to what extent these results are independent of the approximations made.

It must be noted that the complexity of the calculations in the present case is completely unjustified and even harmful, since it masks the very crude initial assumptions of the theory: neglect of surface migration, neglect of the influence of the surface on lattice vibrations, and the use of a “fitting” potential. One must also not forget that the calculation is carried out only in the first approximation of perturbation theory, the smallness parameter of which itself depends on temperature.

In addition to these objections of a “technical” nature, two further comments must be made concerning the substance of the work of Lennard-Jones and his school. First, by virtue of the law of conservation of energy, desorption from the ground state (and adsorption into this state) may turn out to be almost impossible, since the phonon energy has an upper limit equal to \(k\theta\), which may be lower than the heat of adsorption. In fact, \(\theta \sim 300^\circ\mathrm{K}\) (for an NaCl crystal \(\theta = 287^\circ\mathrm{K}\)). Consequently, \(k\theta\) is of the order of \(4.2\cdot 10^{-14}\) erg, which corresponds to 630 kcal/mole. This is small even for physical adsorption! Thus the figures given in work \(^{14}\) essentially have no meaning and cannot be compared with experiment. Since multiphonon transitions (in which the indicated difficulty could have been avoided), as was shown above, are extremely improbable, one has to think of a multistage process of desorption: first—excitation, then—desorption from the excited level (if a suitable level exists at all). The process of adsorption, evidently, will proceed in the reverse order. Such multistage processes have not yet been investigated in detail.

Second, the assumptions under which the calculations are carried out are essentially internally contradictory. On the one hand, it is assumed that the adsorbed molecule (or atom) is closely bound to some single point of the surface, which, as already indicated, makes sense only for a surface whose different parts possess sharply different properties. On the other hand, the influence of the surface on the lattice vibrations is neglected. This may still perhaps be justified to some extent in the case of a homogeneous surface (although even there such an assumption is doubtful), but in the case of a heterogeneous surface this neglect is completely illegitimate, since, depending on the type and distribution of defects, the distortions of the ordinary Debye picture will be different at different points of the surface.

In a calculation carried out under such conditions, one may at best claim to estimate very roughly the order of magnitude of the quantity under investigation and to find its qualitative dependence on the microscopic parameters of the system and on temperature (similar to the way this is done by Blokhintsev and Schechter). To obtain же

“an answer in the form given by Lennard-Jones, Devonshire, and Strachan” means, in essence, not calculating anything.

Thus, the very long and laborious calculations of the authors named have only a remote relation to reality.

B. Selective adsorption and surface migration

Up to now we have neglected the lateral motion of adsorbed molecules. Let us now take it into account and investigate, first of all, under what conditions migration of molecules over the surface is possible. Here, obviously, it is necessary to consider precisely the three-dimensional motion of the adsorbed particle.

This problem was considered by Lennard-Jones and Devonshire \(^{26}\) for the case of a homogeneous surface. They neglected the thermal vibrations of the lattice and showed that then, at any temperature, free migration of molecules over the surface is possible (without activation energy). It differs from ordinary free motion only in that the true mass of the particle is replaced by an effective one (in the special case considered, He on LiF, the effective mass exceeds the ordinary one by 8%).

This result, obtained after very lengthy calculations, is in essence quite trivial. Indeed, if one neglects the inhomogeneity of the surface and the thermal vibrations of the lattice, then what we have here is a problem mathematically very similar to the well-known problem of an electron in a periodic field (see, for example, \(^{19}\)). The only difference is that here the field is periodic not in space, but in a plane. As there, it follows from the symmetry of the problem (the periodicity of the potential along the surface) that the mean velocity of the particle is constant (just as an electron in a purely periodic field moves without experiencing resistance). Moreover, it is by no means necessary that the energy of the particle exceed the height of the barrier (the particle “tunnels” through the barrier instead of moving over it). Thus, on an ideally periodic surface migration without activation energy is possible. It should be noted, however, that the mean velocity of such migration of a heavy particle must be small (it is inversely proportional to the mass of the particle). Therefore, at sufficiently high temperatures, when thermal transfers of the particle into states above the barrier become significant, the main role is probably played by “classical” migration associated with activation energy. For the energy, as always, a band spectrum is obtained; for states at the bottom of the band we have

\[ E = E_n + \frac{p_x^2}{2\mu_x} + \frac{p_y^2}{2\mu_y}, \tag{22} \]

where \(p_x\) and \(p_y\) are the components of the quasimomentum; \(E_n\) is a certain constant, depending on the state of motion perpendicular to the surface

(\(E_n < 0\) for an adsorbed particle and \(E_n > 0\) for a particle that can go off to infinity); \(n\) is the zone number; \(\mu_x\) and \(\mu_y\) are the effective masses in the directions of the corresponding axes (for a cubic lattice \(\mu_x = \mu_y\)). Let us recall that the effective mass, generally speaking, has nothing in common with the true mass \(m\). For heavy particles, whose motion is “almost classical,” the numerical value of \(\mu\) may very appreciably exceed \(m\), if the energy of the particle is less than the height of the barrier. \(\bigl(\mu \sim e^{B\sqrt{m}}\), where \(B\) is a positive constant.\(\bigr)\)

If, however, the energy of the particle is considerably greater than the height of the barrier, then \(\mu_x \simeq \mu_y \simeq m\). The figure given by Lennard-Jones and Devonshire does not deserve confidence, since the authors named, expanding the lattice field in a double Fourier series,

\[ V(x,y,z)=V_0(z)+2V_1(z)(\cos bx+\cos by)+\ldots \tag{23} \]

(\(b\) is a vector of the reciprocal lattice),

discard all terms except those explicitly written in formula (23), the second term being treated as a perturbation. This can be done when the field of the crystal changes little over the de Broglie wavelength of the particle considered, \(\lambda\), since in essence we have here an expansion in \(\lambda/d\) (in the case of an ionic crystal \(d\) is the distance between nearest unlike atoms). For helium atoms at normal temperature (\(T=300^\circ K\)) \(\lambda = 1.5\,\text{\AA}\), and \(\lambda/d\) is by no means a small parameter (in the crystal LiF \(d=2\,\text{\AA}!\)).

It is very important that now the energy of the adsorbed particle is by no means obliged to be negative (we count the energy as zero when the particle is at rest at infinity). Speaking in rough classical language, to the energy of oscillations perpendicular to the surface, \(E_n\), there is added the positive kinetic energy of motion along the surface. Therefore one may think that adsorption without transfer of energy to the crystal is possible. (Previously this was impossible, since the energy of a free particle is positive.) This phenomenon, called selective adsorption (the meaning of the name will become clear presently), is closely connected with the diffraction of molecular beams at the surface of a crystal. It will evidently manifest itself in the fact that, under certain conditions (formulated below), the intensity of the molecular beam reflected from the crystal will experience a sharp minimum: molecules incident on the crystal will pass into the adsorbed state. In this case the transition process will be practically instantaneous, since it is not connected with the transfer of energy.

The theory of this effect was given by Lennard-Jones and Devonshire\(^{22—24}\), who, using the same method as in the consideration of surface migration (the expansion (23) with the indicated

above simplifications), by means of rather lengthy calculations the following condition was obtained for the possibility of transition from the free state to the adsorbed one without transfer of energy to the crystal:

\[ E_{n'}=-\frac{2\pi^2\hbar^2 b_x^2}{m}-\frac{2\pi\hbar b_x p_x}{m}+\frac{p_z^2}{2m}, \tag{24} \]

where \(E_{n'}\) is the energy of the vibrational level at which adsorption takes place; \(p_x\) and \(p_z\) are components of the momentum of the incident particle (instead of \(p_x\) one may also have \(p_y\)); \(b_x\) is a component of the reciprocal-lattice vector; \(m\) is the true mass of the particle. \(E_{n'}\) is an eigenvalue of the unperturbed one-dimensional Schrödinger equation with potential \(V_0(z)\); in the next approximation the level \(E_{n'}\) is broadened into a band.

This result can be obtained with the aid of very simple arguments, from which, incidentally, the essence of the matter is evident. It is known\({}^{19}\) that in a periodic field there is in general no conserved momentum; its role is played by the quasimomentum, which is conserved in collisions only up to a reciprocal-lattice vector (to each component of the quasimomentum one may, without changing the properties of the system, add \(2\pi\hbar bk\), where \(b\) is the corresponding component of a reciprocal-lattice vector and \(k\) is an integer; these are the so-called umklapp processes). Let the energy of the incident particle be

\[ E_{\infty}=\frac{p_x^2+p_y^2+p_z^2}{2m} \tag{25} \]

(\(p_x, p_y, p_z\) are the components of the true momentum, \(m\) is the true mass of the particle, which is considered free at an infinite distance from the surface).

In the periodic field of the surface the energy (at the bottom of the band), as was indicated above, is

\[ E_f=E_n+\frac{(p_x+2\pi\hbar b_x k_1)^2}{2\mu}+\frac{(p_y+2\pi\hbar b_y k_2)^2}{2\mu}. \tag{26} \]

(We have taken into account here the possibility of umklapp processes.) For definiteness let \(k_2=0;\ k_1=k\ne0\). Then, requiring that \(E_{\infty}=E_f\), we obtain

\[ E_n=\frac{p_z^2}{2m}+\frac{p_y^2}{2}\frac{\mu_y-m}{\mu_y m}+\frac{p_x^2}{2}\frac{\mu_x-m}{\mu_x m} -\frac{2\pi^2\hbar^2 b_x^2 k^2}{\mu_x} -\frac{2\pi\hbar p_x b_x k}{\mu_x}. \tag{27a} \]

If \(E_f\) considerably exceeds the height of the potential barriers at the surface, then \(\mu\simeq m\), and (27a)

\[ E_n\simeq\frac{p_z^2}{2m}-\frac{2\pi^2\hbar^2 k^2 b_x^2}{m}-\frac{2\pi\hbar b_x p_x}{m}. \tag{27b} \]

\[ \text{*) The components referred to are, of course, those along the axes in which the field is periodic. In the given case, since the field is periodic only in the planes }z=\text{const.},\text{ the quasimomentum is a two-dimensional vector.} \]

If the impulses satisfy this “selection rule,” then adsorption to the levels (26) is possible (hence the name “selective”). For \(k=1\) condition (27.6) becomes (24); in general, however, here one obtains not a single parabola on which the “allowed” values \(p_x\) and \(p_z\) lie, but an entire family of them. As Lennard-Jones and Devonshire themselves note, they obtained only one parabola (24) because they took into account only one periodic term in the expansion (23); the following terms give the remaining parabolas. In our treatment, based on the general properties of motion in a periodic field, all terms of this expansion are taken into account (it is precisely they that make transfer processes possible).

Under the same conditions (27), the reverse process is, naturally, also possible: selective desorption—the transition of a molecule from a bound state to a free one without receiving energy from the crystal.

The probabilities of the processes of selective adsorption and desorption were calculated in the cited work of Lennard-Jones and Devonshire\({}^{24}\) under the assumption that \(V_0\) and \(V_1\) have the form

\[ V_0 = D\left(e^{-2xz} - 2e^{-xz}\right);\qquad V_1 = \beta D e^{-xz} \tag{28} \]

(\(\beta\) is a certain small constant). The result, as always with Lennard-Jones, proved to be quite unwieldy, and there is no point in writing it out, all the more so since, owing to the crude approximations made, it is of a purely qualitative character.

Condition (24) is confirmed by experiment: as Frisch and Stern\({}^{26}\) observed, the intensity of a molecular beam reflected from a crystal does indeed have a sharp minimum at definite angles of incidence. The values of \(p_z\) and \(p_x\) obtained from this, for which this occurs, fit well on the parabola (24). This circumstance can be used for the experimental determination of the quantities \(E_n\). Lennard-Jones and Devonshire\({}^{25}\) constructed in this way a “map” of the surface field, which in their approximation has the form of the function (21), modulated in accordance with the periodicity of the lattice. It was precisely this “map” that they used in calculating the effective mass of helium atoms moving over the LiF surface. The accuracy of the results here, however, need not be reckoned with, since equation (24) itself is approximate in character—in deriving it the effective mass was identified with the true one (Lennard-Jones and Devonshire do not do this explicitly, since they use perturbation theory, regarding \(V_1\) as a small quantity, and in the zeroth approximation the effective and true masses do indeed coincide).

Up to this point all the reasoning has referred to the case of a strictly periodic surface field. However, as was noted in the work\({}^{28}\), on such a surface selective adsorption cannot in principle be observed, since after some time (according to Lennard-Jones and Devonshire’s estimate, of the order of \(10^{-10}\) sec.) the selec-

tively adsorbed molecule is just as selectively desorbed and flies off, by virtue of condition (27), in the same direction in which it flew before adsorption. Thus, what is essential is the presence on the surface of some defects which, while not noticeably affecting (because of their small number) the process of selective adsorption itself, hold the molecules creeping near them, as a result of which the latter have time to transfer part of their energy to the lattice. (This, of course, occurs owing to the interaction with phonons, which we have so far neglected. When thermal vibrations are taken into account, relations (27) are somewhat “smeared out,” but this “smearing” will be of the order of the ratio of the amplitude of the thermal vibrations to the lattice constant and, consequently, will not distort the picture substantially.) Thus, selective adsorption is of interest not so much in itself as as one of the ways of stretching out the process of energy transfer to the crystal over a time sufficient for multiphonon transitions to begin to play a role.

In concluding the section devoted to adsorption kinetics, it is necessary to dwell on one more paper[^18], in which the process of energy exchange between an adsorbed atom and an adsorbent is considered, but on the basis of an entirely special mechanism. It was noted above that, for energetic reasons, a purely phonon mechanism of adsorption and desorption may prove insufficient. Taking this circumstance into account, the authors—Lennard-Jones and Goodwin—investigated the possibility of direct energy exchange between an adsorbed atom and the conduction electrons of the crystal. In essence this work is nothing other than a quantum-mechanical formulation of L. V. Pisarzhevsky’s idea[^27] of the “bombardment” of an adsorbed atom by the electrons of a crystal.

The problem is posed as follows: the adsorbed atom X is chemically bound to one of the atoms of the surface of the crystal M, forming a single complex MX. For simplicity it is assumed that the axis of the complex is directed perpendicular to the surface, and lateral motion is excluded (only vibrations perpendicular to the surface are possible). Since the atoms M and X are different, the complex possesses, generally speaking, a nonzero dipole moment \(\mu(R)\) (\(R\) is the distance between the atoms M and X), owing to which it interacts with the conduction electrons of the crystal. For sufficiently large distances (when exchange forces may be neglected) the interaction energy has the usual form

\[ V(\mathbf r_1,\ldots,\mathbf r_n,R)=\sum_{i=1}^{n}\frac{e\mu(R)\cos\theta_i}{r_i^2}, \tag{29} \]

where \(\mathbf r_1,\ldots,\mathbf r_n\) are the radius vectors of the conduction electrons, \(\theta_1,\ldots,\theta_n\) are their polar angles; \(n\) is the number of conduction electrons ...

most; the origin of coordinates is placed at the center of gravity of the charges of the complex.

As a result of this interaction, an exchange of energy may take place between the complex and the conduction electrons, which leads to activation or deactivation of the adsorbed atom. The probabilities of these processes are subject to calculation (the processes of adsorption and desorption are not considered in the paper, although they apparently can be studied by the same method).

The electrons in the metal are described within the framework of the Sommerfeld model, i.e., it is assumed that they move independently of one another in a potential box formed by the surface of the crystal. It is only necessary to take into account that near the adsorbed atom the jump of the potential has a different value than far from it. In a rough schematic treatment it is assumed that the jump of the potential is equal to \(\chi\) on a hemisphere of some radius \(r_0\), described from the center of gravity of the charges of the complex, and is equal to infinity on the entire remaining surface. Since, in the adopted model, the electrons of the metal move independently of one another, one may consider the interaction of the complex with each electron separately, subsequently summing over all conduction electrons.

Considering the potential (29) as a perturbation and writing

\[ \mu(R)=\mu(R_0)+(R-R_0)\left(\frac{d\mu}{dR}\right)_{R=R_0}, \tag{30} \]

(\(R_0\) is the equilibrium distance between the atoms M and X), the authors obtain the standard quantum-mechanical problem of the probability of transitions in the system “free electrons + complex MX” under the action of the perturbation (29). To solve it one must also know the wave functions of the ground and excited states of the complex; in other words, one must know the potential energy of the complex—\(V(R)\)—as a function of \(R\). The authors consider two cases, approximating \(V(R)\) either by formula (21), or by the curve

\[ V=\text{const.}\left(\frac{R_0}{R}-\frac{1}{2}\frac{R_0^2}{R^3}\right), \tag{31} \]

and, after rather laborious calculations, obtain an answer containing six (!) independent unknown constants: two parameters of the potential curve, \(r_0\), \(\left(\dfrac{d\mu}{dR}\right)_{R=R_0}\), \(\chi\), and the Fermi energy of the given metal—\(\Phi\). Noting that \(\chi-\Phi\) is equal to the work function, we see that two parameters (\(\chi\) and \(\Phi\)) can be determined independently from adsorption measurements; however, the remaining four remain unknown. (True, the result depends little on \(r_0\).) Moreover, the answer is so difficult to survey that it does not seem possible to extract from it any physical consequences of a general nature.

The figures cited by the authors do not deserve confidence, since they were obtained under the assumption that the \(MX\) complex has the same electronic nature as the corresponding diatomic molecule (the values \(R_0\), \(\left(\dfrac{d\mu}{dR}\right)_{R=R_0}\), \(D\), and \(\chi\) were taken from spectral data). Meanwhile this assumption is wholly unjustified, either theoretically or experimentally. Theoretically—because the adsorbed molecule interacts indirectly with the entire crystal, which cannot be ignored; experimentally—because, as is well known \(^{28,29}\), the absorption spectra of adsorbed molecules differ appreciably from the spectra of the same molecules in the gas phase.

It should also be noted that, in the problem of energy exchange, the use of the Sommerfeld model can hardly be justified. Here the structure of the energy spectrum of the system is essential, and therefore one must use at least band theory (taking into account the influence of the surface).

In summary, one may say of this calculation the same thing that was said earlier in connection with the work of the Lennard-Jones school: such crude simplifications as were made here can be justified only if, as a result, they yield a simple and clear answer that qualitatively reflects correctly the most essential features of the phenomenon. But to introduce such crude assumptions in order ultimately to obtain an opaque formula with four adjustable constants makes no sense.

Nevertheless, the mechanism itself, proposed by L. V. Pisarzhevskii and considered quantum-mechanically in the cited work, appears quite reasonable and deserves careful examination. One must only bear in mind that the problem of transferring energy to the lattice is not completely solved here, but only pushed back, since it still remains to clarify how the electronic excitation is “dissipated”*). Let us note, incidentally, that this problem of transferring to the crystal large (in comparison with \(k\theta\)) portions of energy is also highly relevant in the case of luminescence quenching, when an electron thrown into the conduction band likewise makes a nonradiative transition back into the lower band, transferring its energy to the lattice in a very short time.

*) Let us note that if this mechanism is correct, then in the process of adsorption on semiconductors a temporary increase in electrical conductivity should be observed, since lattice electrons will be thrown into the conduction band and remain there for some time. Luminescence of the adsorbent may also occur. In addition, the rate of adsorption will prove to be a structurally sensitive property, since local levels created by various lattice defects may play a very significant role in this mechanism.

Thus, one must state that the mechanism of energy exchange between adsorbed atoms (or molecules) and the adsorbent is still not clear.

We have examined a whole series of quantum-mechanical works on the kinetics of adsorption and have arrived at a very discouraging conclusion: there are almost no physically interesting results, and the little that has been clarified can be obtained from much simpler and more general considerations and in no way justifies the enormous expenditure of labor that we see, for example, in the works of Lennard-Jones and his collaborators.

Such a sad outcome was not difficult to foresee. Indeed, all the works analyzed deal with the behavior of heavy particles. Only the last work concerns electronic phenomena to some extent, but even in it the main emphasis is not placed on them. Meanwhile, it is known that in most cases (at not too low temperatures) heavy particles behave “almost classically,” and their purely wave properties are almost not manifested. It is no accident, therefore, that the more or less interesting results concern precisely selective adsorption and surface migration, for this is the only (diffraction) region in which, even under ordinary conditions, it is precisely the wave properties of heavy particles that are essential. From what has been said it does not follow, of course, that quantum mechanics is altogether unnecessary for the theoretical investigation of the problem of adsorption. It is necessary, but it must be applied mainly to those particles that behave “purely quantum-mechanically,” namely to electrons. It is precisely along this path, by studying the electronic processes accompanying adsorption and responsible for it, that one may hope to obtain fundamentally new results.

It seems to us, however, that it is still altogether premature to pose seriously the question of a quantum theory of adsorption coefficients (as well as of adsorption kinetics in general). The point is that we do not yet understand even simpler things connected with adsorption equilibria. What is an active center in the case of chemical adsorption (when its existence cannot be explained solely by an enhancement of the surface field at edges or vertices of the lattice)? How are the activation energy and the heat of chemisorption related to the microscopic quantities characterizing the adsorbent and the adsorbate (for example, to the number of conduction electrons)? How do they depend on the concentration and properties of impurities? In what state are the adsorbed atoms on the real surface: are they fixed or can they migrate; on what does the corresponding activation energy depend; are atoms on the surface ionized, and if so, to what extent; what determines the degree of ionization? Theory cannot yet give a clear and exhaustive answer to all these questions, and yet an understanding

all these things is necessary in order that the problem of calculating the kinetic coefficients may be posed accurately.

Moreover, the questions raised have, as it seems to us, also great independent interest. The concepts of the heat and energy of activation of adsorption are fundamental in the statistical theory of surface phenomena of S. Z. Roginskii^30. Therefore, establishing their connection with microscopic quantities characterizing the adsorbent and adsorbate, as well as with other properties of the surface, appears to us to be one of the most important tasks of the microscopic theory of adsorption.

It is significant that all the problems listed can apparently be solved only on the basis of an analysis of the electronic processes that take place during adsorption. Thus, they belong to the domain of quantum mechanics, and it is precisely here that one may expect from quantum theory everything it can give—the disclosure of new mechanisms and connections.

Let us also note that the question of the role of electronic phenomena in adsorption has long since been placed on the agenda by a number of experimental facts. We have in mind the change in work function during chemical adsorption on metals, as well as the change in the absorption spectra of adsorbed molecules in comparison with the gas phase ^28, ^29, ^52.

We now turn to the regrettably few works devoted to the study of the role of electronic processes in adsorption and in the phenomena accompanying it.

III. ELECTRONIC PHENOMENA IN ADSORPTION

The few works in this area, belonging mainly to Soviet scientists, are devoted to calculating the activation energy of chemical adsorption, to the theory of surface ionization of atoms*), to studying the nature of the repulsive forces between adsorbed molecules, to elucidating the role of local levels in adsorption, and to the nature of the dipole moments of adsorbed atoms.

In the works of Eyring and his co-workers ^31, ^32 (see also ^33), for calculating the activation energy of chemisorption, the energy-surface method was applied, developed earlier in connection with the theory of absolute reaction rates (see ^41). The adsorption, for example, of hydrogen on carbon (the case considered in ^31) is regarded as a reaction between two carbon atoms and a hydrogen molecule, in the course of which the H—H bond is broken and two new C—H bonds are formed. The problem is thus reduced to the problem of four

*) We do not intend to give a complete review of works on the surface ionization of atoms, but shall touch only upon those which, as it seems to us, are most closely connected with adsorption problems.

electrons, the solution of which, as is known^41, gives for the correction \(\varepsilon\) to the energy of the separated atoms (in the first approximation)

\[ \varepsilon = Q - \sqrt{\frac{1}{2}\{(\alpha-\beta)^2+(\beta-\gamma)^2+(\gamma-\alpha)^2\}}, \tag{32} \]

where \(Q\) is the sum of all Coulomb integrals; \(\alpha=\alpha_1+\alpha_2\); \(\beta=\beta_1+\beta_2\); \(\gamma=\gamma_1+\gamma_2\); \(\alpha_1,\ldots,\gamma_2\) are the exchange integrals for the various pairwise combinations of electrons.

The energies of the individual bonds (between each two atoms), as functions of the distances between the corresponding nuclei, are represented by empirical curves (21), constructed on the basis of spectroscopic data. Using these curves and specifying a certain ratio between the Coulomb and exchange components of the bond energy, one can calculate \(\varepsilon\) as a function of the distances between all the atoms, i.e., construct the potential-energy surface of the system. On it there are always two “valleys,” corresponding to the initial and final states, and a “ridge” separating these states. The height of the ridge determines the activation energy. (Let us take this occasion to point out one rather widespread error. Often the chemisorption process of hydrogen is represented by the graph shown in Fig. 2^42. In this case the distance \(CD\) is identified with the activation energy. However, as F. F. Volkenstein^43 has observed, these quantities have nothing in common. In fact, curves \(I\) and \(II\) are drawn for two different values of the distance between the hydrogen atoms. Therefore point \(C\) in no way corresponds to the ridge on the Eyring diagram.)

Fig. 2. Toward the calculation of the activation energy of chemisorption: \(I\)—the energy of interaction of the surface with two hydrogen atoms as a function of the distance from the surface to the center of inertia of the atoms. \(II\)—the energy of interaction of the surface with a hydrogen molecule as a function of the distance between them.

The results of the calculation proved to be not very satisfactory (for example, the calculated activation energy for the adsorption of hydrogen on carbon is equal to \(8.8\) kcal, which considerably exceeds the experimental value, which is negligibly small). This is not surprising, in view of the extreme crudeness of the calculation. In addition to the usual, very serious objections that are raised against the indicated method in its application to chemical reactions (see the editorial notes

to \(^{41}\)), it must be noted that adsorption by the surface of a crystal can in no case be considered in the way it is done here. A crystal is characterized by the presence of many regularly arranged atoms and, as a consequence of this, by an entirely specific energy spectrum of the electrons, whose peculiarities are responsible for many phenomena connected with chemical adsorption. To reduce the problem to a “reaction” between an \(\mathrm{H}_2\) molecule and two carbon atoms means completely excluding from consideration the influence of the crystal. In essence (as the authors themselves also point out), what is considered here is not adsorption, but the hydrogenation of ethylene (or acetylene). The authors’ attempt to take account of the influence of the lattice indirectly, by estimating the bond energy not from spectral data (which is absolutely incorrect), but from the experimental value of the heat of adsorption, has not led and could not lead to any noticeable success, since the method is defective in its very essence.

Fig. 3.

Thus, the question of calculating the energy (as well as the heat) of activation of chemisorption still remains completely open. It seems to us that this problem is one of the most urgent in the modern quantum theory of adsorption.

The surface ionization of atoms is the subject of papers by N. D. Morgulis \(^{35}\), A. I. Anselm \(^{34}\), Gurney \(^{36}\), and F. F. Vol’kenshtein \(^{37}\).

N. D. Morgulis calculates the probability of ionization of an atom (or neutralization of an ion) adsorbed on the surface of a metal as a function of the temperature, the work function \(\varphi\) of the pure metal, and the ionization potential \(V_i\) of the free atom. The metal is described within the Sommerfeld model. It is assumed that, when the atom approaches the surface, a potential barrier is formed, shown in Fig. 3. Ionization of the adsorbed atom (or neutralization of the ion) takes place by tunneling of the electron through the barrier, which is possible if there are free levels in the metal with energy \(E = \Phi + \varphi - V_i\). In the calculation the complicated potential barrier is approximated simply by a rectangular “hump.” In addition, the calculation is carried out with exponential accuracy (i.e. all pre-exponential factors are regarded as constants). Under these assumptions, for the ionization probability \(a\) one obtains the formula

\[ a = B \frac{ e^{-\frac{2x_0}{\hbar}\sqrt{2mV_i}} }{ 1+e^{\frac{V_i-\varphi}{kT}} }, \tag{33} \]

where \(B\) is a constant, and \(x_0\) is the equilibrium distance between the row of the adsorbed atom and the surface. (By the probability of ionization is meant here the ratio of the total number of ionized atoms to the total number of atoms striking the surface.) As Morgulis showed\({}^{35}\), formula (33) correctly conveys the dependence of \(a\) on \(V_t\) and \(\varphi\). It is essential that ionization is possible not only when \(\varphi > V_i\) (Rb on W), but also in the opposite relation (Na on W). This circumstance (confirmed experimentally by Morgulis\({}^{44}\)) cannot be explained within the framework of classical mechanics and is a purely quantum effect.

Since the work does not set itself the task of obtaining precise numerical values, the crudeness of the calculation can hardly be regarded as a substantial shortcoming, because the qualitative regularities are for the most part conveyed correctly. However, in the calculation the atom and the metal are treated as two almost independent systems (the overlap of the wave functions is small and determines only the ionization, without affecting the other properties of the system). Therefore the work pertains only to the case of physical adsorption. In a number of problems, however (for example, in the theory of catalysis), chemical adsorption is of great interest.

We now turn to works on the surface ionization of chemically adsorbed atoms, where the problem of electronic exchange between the atom and the adsorbent is decisive. So far as we know, the first quantum-mechanical work in this direction belongs to A. I. Anselm\({}^{34}\), who calculated the probabilities of desorption of atoms and ions. In this work it was emphasized for the first time (contrary to the opinion of Becker–de Boer\({}^{29}\) prevailing at the time) that on the surface there do not exist separately completely ionized or completely neutral atoms: all adsorbed atoms are partially ionized (i.e., their electrons are partially drawn into the lattice or, conversely, the lattice electrons are partially drawn onto the adsorbed atoms). Only upon removal of an atom from the surface does the electron become completely localized on it, or else pass completely into the lattice. Unfortunately, in the calculation an inadmissible assumption was made that the valence level of the atom is not shifted and is not deformed during adsorption. In adsorption on a metal, where the energy spectrum is practically continuous, the level of the adsorbed atom is inevitably broadened into a band; in the case of adsorption on a semiconductor, where the energy spectrum consists of a series of bands separated by forbidden regions, this broadening is not obligatory: if the valence level falls in the gap between bands (at a sufficient distance from their boundaries), then it is practically not broadened, but its position is noticeably shifted—instead of the valence level of the free atom there arises a local level in the crystal, caused by the violation of the periodicity of the lattice due to the presence of the adsorbed atom.

This circumstance was taken into account in the works of Gurney \(^{36}\) and A. I. Anselm \(^{34}\), devoted to the theory of the double electric layer in films adsorbed on a metal, and also in the work of F. F. Vol'kenshtein \(^{37}\).

Anselm and Gurney consider the metal–adsorbate within the framework of the Sommerfeld model; the energy level of the atomic electron is assumed to be close to the Fermi energy. At sufficiently large distances between the adsorbed atom and the metal surface, the wave functions of their electrons do not overlap, and the levels of the atomic electrons are sharply defined. As the atom approaches the surface, the atomic level broadens, and the atomic electron (in the case of an electropositive atom) is shared between the atom and the adsorbate lattice. Thus there arises a double electric layer, with its positive side directed outward, as a result of which the work function is lowered. The strength of the double layer is determined by the degree to which the atomic electron is drawn into the metal. This latter depends on the ratio between the Fermi energy of the adsorbate \(\Phi\) and the ionization potential of the adsorbed atom \(V_i\). For \(\Phi > V_i\) the atomic core is almost completely unneutralized, i.e., the drawing-in is large; for \(\Phi < V_i\) the drawing-in is small (Fig. 4).

Fig. 4. Formation of a double layer in adsorbed films. The shaded part characterizes the degree of neutralization of the atom.

It is essential that the neutralization of the atomic cores of monovalent atoms requires half as many electrons as in the case of divalent atoms. Therefore, if case a in Fig. 4 corresponds to a monovalent atom, and case b to a divalent one, then the observed effect will in both cases be approximately the same, independently of the ratio between \(V_i\) and the work function of the pure metal \(\varphi\) (which, according to classical mechanics, should have played the decisive role). This is indeed observed in reality—adsorption of K \((V_i = 4.3\ \mathrm{eV})\) and Ba \((V_i = 6.09\ \mathrm{eV})\) on tungsten \((\varphi = 4.5\ \mathrm{eV})\).

The preceding arguments applied mainly to alkali and alkaline-earth atoms, which are capable of readily giving up their electrons. In the case of atoms of the oxygen type, which possess an affinity for the electron, the picture will be the opposite: there is

level corresponding to the “excess” electron; upon adsorption it “smears out,” and the electron located on it communicates between the atom and the adsorbent (i.e., the lattice electron is drawn onto the atom). We again obtain a double layer, but this time it is turned with its negative side outward, i.e., the work function upon adsorption of such atoms increases. The power of the double layer depends on the degree to which the lattice electrons are drawn into the atom, i.e., ultimately, on the relation between the Fermi energy of the adsorbent and the ionization potential of the “excess” level (not to be confused with the ionization potential of the neutral atom, which plays a role in the adsorption of electropositive elements).

Fig. 5. On the calculation of \(\Delta\varphi\): region \(I\) corresponds to the metal, region \(III\) to the adsorption layer, region \(IV\) to the vacuum; region \(II\) represents the potential barrier between the metal and the adsorption layer.

Gurney confined himself in his work to purely qualitative considerations, set forth above. A. I. Anselm approximated the course of the potential, as shown in Fig. 5, and, by means of quite simple calculations, obtained the following expression for the change in work function upon adsorption:

\[ \Delta\varphi = 4\pi A\sigma_{+} - \frac{V\sqrt{2m^{3/2}Ae}}{2\pi^{3}\hbar^{3}} \int_{0}^{\varphi+\Phi} W(E') \frac{E'-\varphi}{\sqrt{E_{0}+\varphi-E'}} \,dE', \tag{34} \]

where \(\sigma_{+}\) is the charge of positive ions adsorbed on \(1\ \mathrm{cm}^{2}\) of surface, \(W(E')\) is the probability of the electron being in the adsorption layer (it can be calculated explicitly), and the other designations are given in Fig. 5.

As is seen from (34), the change in the work function is the greater, the smaller the ionization potential of the adsorbed atoms; for different adsorbents the change in the work function is the greater, the greater it is itself. This agrees well with experiment, as is seen from Tables I and II, borrowed from the work of A. I. Anselm.

QUANTUM THEORIES OF ADSORPTION

Table I

Change in the work function for platinum upon adsorption of various metals

Adsorbate	Work function $\varphi$: clean surface	Work function $\varphi$: surface covered with an adsorption layer	$\Delta\varphi$	Ionization potential of adsorbed atoms $V_i$
Sodium . . .	6.30	2.08	4.22	5.12
Potassium . . .	6.30	1.60	4.70	4.32
Rubidium . . .	6.30	1.56	4.74	4.16
Cesium . . .	6.30	1.38	4.92	3.88

Table II

Change in the work function for different metals with one and the same adsorbate

Substrate metal	Work function $\varphi$ in eV: clean surface	Work function $\varphi$ in eV: surface covered with an adsorption layer	Change in work function upon adsorption $\Delta\varphi$
Platinum . . .	6.35	4.11	$-2.24$
Gold . . .	4.90	(4.05)	$(-0.85)$
Silver . . .	4.78	3.97	$-0.81$
” . . .	4.78	(3.94)	$(-0.84)$
Thallium . . .	3.68	3.67	$-0.01$
Aluminum . . .	2.81	3.62	$+0.81$

Despite this success, the Anselm–Gurney model is clearly too primitive. It cannot, for example, explain the effect of a change in the sign of the double layer as the degree of filling increases. Moreover, the question remains open of the quantitative—even if approximate—calculation of $\Delta\varphi$ as a function of the microscopic parameters of the system.

The main positive result of the works discussed is that in them the question of the pulling of an electron from the lattice onto the adsorbed atom was, for the first time, clearly posed as a very important factor capable of playing a decisive role in a number of phenomena.

In the work of F. F. Volkenshtein$^{37}$ (devoted mainly to the problem of modification of catalysts and therefore partly going beyond the scope of the present article), the ionization of atoms adsorbed on the surface of a nonmetallic crystal is considered (the ionization coefficient is defined as the charge of the adsorbed

of the atom, expressed in units of the electron charge). This coefficient is calculated as a function of temperature, degree of surface coverage, and impurity concentration. The direct interaction between adsorbed atoms is not taken into account; the dependence of the degree of ionization on surface coverage is obtained owing to the fact that the concentration of the electron gas in the crystal, associated with the degree of ionization of the adsorbed atoms, depends on this latter quantity. In other words, between the adsorbed atoms there arises a quite specific interaction, independent of distance, transmitted through the electron gas of the adsorbent lattice. This interaction, considered by F. F. Vol’kenshtein^45 earlier in connection with purely catalytic problems, is applied in the article under discussion to adsorption problems for the first time.

In the calculation the crystal surface is regarded as a flat semiconductor, in which the adsorbed atoms play the role of defects causing the appearance of local levels in the energy spectrum of the crystal. Indeed, it is known^46 that, owing to the presence of a surface, surface bands appear in the spectrum of a crystal; these correspond to modes of the wave functions that decrease rapidly into the interior of the crystal and are periodic along its surface. When the surface deviates from ideality, additional local levels may appear, with wave functions appreciably different from zero only near the defect. This also justifies the adopted model. Incidentally, it automatically guarantees full allowance for the effect of broadening, since the adsorbent lattice and the adsorbed atoms are considered as a single quantum-mechanical system with a common “electron household.” We note, however, that here the electrons located in the volume of the crystal are not taken into account. Since the adsorbed atoms are partially ionized, their Coulomb interaction with these electrons may prove substantial.

It turned out that the degree of ionization decreases as the surface is filled with adsorbed atoms (in the case of atoms of one kind—as \(Y_0^{-1/2}\), where \(Y_0\) is the concentration of adsorbed atoms). In addition, it depends substantially on the concentration of the impurity added to the crystal. Thus, electropositive impurity atoms lower the degree of ionization of electropositive adsorbed atoms and raise it for electronegative atoms. An electronegative impurity behaves in the opposite manner. These circumstances may prove very important in the theory of the double electric layer in adsorbed films.

The work also investigated the influence of ionization of adsorbed atoms on the adsorption isotherm. Under the assumptions that the probability of ion desorption is very small and that adsorption occurs only on a clean surface, it turned out that filling

the surface grows in proportion to the square root of the pressure in the gas phase (and not to the pressure itself, as follows from the Langmuir theory, which neglects the effect of ionization). It is important to note that this result is obtained for a homogeneous surface and is due exclusively to taking account of the ionization of the adsorbed atoms.

The chief merit of the work considered lies in the investigation of a fundamentally new (and purely quantum) mechanism of indirect interaction between adsorbed atoms, previously proposed by its author. As we have seen, this mechanism leads to new and sometimes quite unexpected consequences. Its further study and development seems to us one of the urgent tasks of the quantum theory of adsorption.

The nature of the forces of interaction between adsorbed atoms is also the subject of a work by A. Kh. Breger and A. A. Zhukhovitskii^38. They assumed that each adsorbed atom creates in the crystal a “nodal surface,” through which conduction electrons cannot pass (their wave functions there become zero). As the authors write, “... according to the general laws of the theory of oscillations,” this means that the energy of the system is increased (apparently, what is meant is Courant’s theorem that the number of nodes of a wave function is equal to the number of the corresponding eigenvalue)^47. When two nodal surfaces approach one another, the energy increases still more, i.e., the situation is as if repulsive forces acted between the adsorbed atoms. These can be calculated as a function of the distance between the atoms. It seems to us, however, that this mechanism is plainly erroneous. First, the initial assumption regarding the formation of nodal surfaces is in no way justified: from the fact that one of the lattice electrons forms a bond with the adsorbed atom, it by no means follows that the other electrons cannot be near the latter. Second, even if the initial assumption were correct, nothing would yet follow from it, since Courant’s theorem was formulated and proved for the case of a discrete spectrum; in the case of a spectrum containing continuous parts, there is no such connection between the number of nodes of a wave function and the magnitude of the corresponding eigenvalue (an example of this is the theory of ferromagnetism^49).

In all the works analyzed (except those of Eyring’s school), what was studied was not so much the adsorption process itself as the secondary phenomena accompanying it. We now turn to the works of F. F. Vol’kenshtein^40 and Pollard^39, devoted specifically to the study of adsorption equilibria themselves, i.e., in quantum-mechanical language, to the investigation of the energy spectrum of the system “crystal + adsorbed atom.”

In Pollard’s work an attempt is made to investigate the role of local levels in chemical adsorption. As was already noted above,

owing to the presence of a surface, surface zones can arise in the crystal with a wave function having a maximum at the surface and decreasing exponentially into the depth of the crystal. In the one-dimensional case these zones are expressed as levels. They were first discovered by I. E. Tamm\(^{48}\), according to whom the condition for their formation is

\[ E \gg V_s - \frac{\hbar^2 p^2}{2ma^3}, \tag{35} \]

where \(E\) is the energy of the local level, \(V_s\) is the jump of the potential at the surface, \(a\) is the lattice constant (a cubic lattice is meant), and \(p\) is a parameter characterizing the magnitude of the change of the potential in the lattice. (Since \(E\) must be smaller than the energy of an electron outside the lattice, local levels are not always formed.) Thus, any external action that lowers \(V_s\) promotes the appearance of a local level. On this basis Pollard considers the following mechanism of chemical adsorption: the adsorbing atom, interacting with the surface, locally lowers \(V_s\), as a result of which a local level is formed in the crystal with a wave function rapidly decreasing in all directions from the point lying directly beneath the adsorbed atom. An atomic electron is drawn into this level (completely or partially, depending on the relation between the energy of the level and the ionization potential of the atom), and a one-electron bond (ionic or covalent) is formed between the surface and the adsorbed atom. On the basis of this mechanism the degree of ionization of the adsorbed atoms and the heat of adsorption are calculated.

In principle such a formulation of the problem is quite possible; however, in calculating the lowering of \(V_s\), Pollard makes an inadmissible error that completely devalues all his further calculations: as the interaction potential he uses the Lennard-Jones formula (1). It is not difficult to see that this is absolutely illegitimate: formula (1) was derived for the case of purely van der Waals adsorption, when the wave functions of the atom and the crystal do not overlap (this was explicitly assumed in its derivation), and the interaction between the adsorbed substance and the adsorbent may be regarded as a small perturbation. In the case under study of chemical adsorption, however, the whole matter lies in the overlap of the wave functions. Here the adsorbent and the adsorbate form a single quantum-mechanical system, and formula (1) is completely inapplicable. Therefore there is no point in considering Pollard’s work in greater detail.

The necessity of considering the adsorbent and the adsorbate as a single quantum-mechanical system (in the case of chemical adsorp-

was first emphasized by F. F. Vol’kenshtein^40. He considered the adsorption of monovalent electropositive atoms \(A\) on an ionic crystal of the type \(M^+R^-\). The electrons were described within the framework of the Bloch model, i.e., the problem was reduced to the study of the behavior of one electron in the field of the lattice ions and of the atomic core \(A^+\). It turned out that the energy spectrum of the system consists of a band and two local levels. The wave function corresponding to the band is nonzero throughout the whole crystal, i.e., in this state the electron propagates freely over the entire lattice (the conduction band). In the local states the electron is located mainly at the surface, near the adsorbed atom. Both in the band and on the local levels the electron is distributed between the lattice and the adsorbed atom, establishing a bond between them (binding).

It is important to note that the local levels referred to here by no means coincide with the usual Tamm levels caused simply by the presence of a surface. They arise from the interaction of the Tamm levels with the proper level of the adsorbed atom.

As the distance between the adsorbed atom and the lattice is increased, one of these levels (\(W_n\)) passes into the proper level of the adsorbed atom, while the other (\(W_i\)) passes into the ordinary surface level of the lattice. The probability of electron binding from the atom to the lattice (\(\eta\)) has been calculated for the local states as a function of the microscopic quantities characterizing our system. It turns out, as was to be expected, to be proportional to the exchange integrals characterizing the overlap of the wave functions of the atom and the lattice. In the limiting transition indicated above, \(\eta_n\) tends to zero (a neutral atom far from a neutral lattice), while \(\eta_i\) tends to unity (an ion far from a charged lattice). Thus, in the limit, one obtains, as should be the case, the two standard problems of the theory of physical adsorption.

Owing to the partial ionization of the adsorbed atom, an electric field is created around it which, at large distances (where the electron wave function practically vanishes), can be represented by the field of a dipole with moment

\[ \mu = e\eta(R)R \tag{36} \]

(\(R\) is the distance between the atomic nucleus and the lattice). Thus, it turns out that adsorbed atoms possess dipole moments of purely quantum origin. As the calculation shows, the dipole moment (36) may considerably (by a factor of 100) exceed the classical dipole moment arising in physical adsorption because of the polarization of the adsorbed atom by the field of the lattice. It is possible that precisely “exchange” dipole moments of this kind play a role in the exchange of energy between

adsorbed atoms and the lattice (cf. the work of Lennard-Jones considered above\(^{18}\)).

In the work discussed, the problem is posed, from the point of view of quantum mechanics, quite carefully. The essential shortcoming, however, is rooted in the very formulation of the problem as a one-electron one. This does not allow one to investigate the role of the lattice electrons in adsorption. Also excluded from consideration are questions connected with the interaction of electrons and with the possibility of the transition of a one-electron bond into a two-electron one (with the corresponding gain in energy). It seems to us that a many-electron formulation of this problem, despite the considerable mathematical difficulties connected with it, is now one of the urgent problems of the quantum theory of adsorption.

We have considered a number of works on the electronic theory of adsorption. As was to be expected, the application of quantum mechanics has proved here to be much more fruitful than, for example, in the majority of works of the Lennard-Jones school. Despite the purely approximate, orientational character of the formulation and solution of most problems, quantum theory has already explained a number of facts that are completely incomprehensible from the point of view of classical mechanics, and has revealed certain new mechanisms—the connection between the ionization potential and the Fermi energy (and not the work function) of the adsorbent as a factor determining the change in work function upon adsorption; the change in the degree of ionization of adsorbed atoms with surface coverage as a consequence of the transmission of interaction through the electron gas of the lattice; the influence of ionization on the adsorption isotherm; the possibility of ionization of physically adsorbed atoms when \(\varphi < V_i\). At the same time it must be noted that the fundamental problems of the microscopic theory of adsorption—the calculation of the dependence of the heat and activation energy of chemisorption on the microscopic parameters of the system, the elucidation of the physical meaning of the concept of an “active center,” and the clarification of the mechanism of energy exchange between adsorbed atoms and the lattice—still remain unsolved. We have seen that quantum mechanics points out certain routes toward the solution of these problems (for example, taking into account the phenomenon of retardation). Further serious investigation of the problems named is highly desirable.

This article was written at the initiative of Prof. F. F. Vol’kenshtein. The author takes the opportunity to express to him his deep gratitude for his exceptional attention to the work and for a number of valuable suggestions.

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