Issues in Modern Quantum Theory of Electronic Conductors

S. V. Vonsovskii

Contents

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290
2. Criterion for a metal and a dielectric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292
3. Classical and quantum theory of the electron “gas” . . . . . . . . . . . . . . . . . . . . 296
4. One-electron theory of crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 300
5. General many-electron theory of crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313
6. The method of second quantization as applied to the problem of the crystal . . 319
7. Exchange, polar, and exciton many-electron models of the crystal . . . . . . . . . . 326
8. The case of transition elements—the model of interacting external and internal electrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377
9. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384

This article is a critical review of the modern quantum-mechanical theory of crystalline electronic conductors. In the review, the main attention is devoted to the many-electron treatment of the problem of the crystal, which was developed in the works of Soviet physicists (N. N. Bogolyubov, S. V. Tyablikov, S. I. Pekar, S. V. Vonsovskii, and others). At present, the quantum theory of crystals is experiencing a certain crisis owing to the fact that the most widespread one-electron (so-called “band”) model has already exhausted its possibilities, and its further application to the explanation of the real properties of crystals leads to difficulties of a fundamental character. The reason for these difficulties lies mainly in the simplifying assumption of this model—the neglect of the interaction between electrons in the crystal. Forgetting this circumstance inevitably leads to qualitatively incorrect conclusions of the theory. The absolutization of the conclusions of the one-electron theory, which is especially characteristic of American and English physicists (Slater, Van Vleck, Mott, Stoner, and others), is a typical example of an epistemological source

idealistic distortions in science, of which V. I. Lenin warned.\(^1\) Soviet science must be at the forefront in this field of theoretical physics as well, pointing out the paths for the further development of the quantum theory of crystals, which is at present undergoing a crisis in its development.

1. INTRODUCTION

The creation of a modern physical theory of crystals is one of the problems of the quantum mechanics of many microparticles. It is now established that the principal, leading role in the very formation of the crystalline (ordered, condensed) state of matter is played by the electrical interaction between the nuclei and electrons of which all crystalline bodies consist. Other known types of interactions—magnetic, nuclear, gravitational—play a secondary, subordinate role. The quantum-mechanical equation of motion for the system under consideration (a homoeopolar crystal), in the usual coordinate representation, has the form

\[ i\hbar \frac{\partial \Psi(\mathbf r_1 \ldots \mathbf r_n, \mathbf R_1,\ldots,\mathbf R_N)}{\partial t} = \hat H \Psi(\mathbf r_1 \ldots \mathbf r_n, \mathbf R_1 \ldots \mathbf R_N), \tag{1,1} \]

where the energy operator of the system is equal to:

\[ \hat H = -\frac{\hbar^2}{2M}\sum_{i=1}^{N}\Delta_{\mathbf R_i} - \frac{\hbar^2}{2m}\sum_{j=1}^{n}\Delta_{\mathbf r_j} + V(\mathbf R_1 \ldots \mathbf R_N, \mathbf r_1 \ldots \mathbf r_n). \tag{1,2} \]

Here the first and second terms are, respectively, the operators of the kinetic energy of the nuclei and of the electrons, while the third is the operator of the potential energy of the electrical interaction of all \(N\) nuclei and all \(n\) electrons with one another. It has so far not been possible to solve such a general equation exactly, or even approximately, because of quite understandable mathematical difficulties. The chief cause of the latter is the “entanglement” of the coordinates of the microparticles in the function \(V\) of equation (1,2). Nor is it possible to obtain any information about the general properties of the solutions of equation (1,1) without making some simplifying assumptions about the form of the function \(V\). Moreover, some authors express the supposition that quantum mechanics in its present nonrelativistic form is in general unable “at the present time to explain either the process of formation of a crystal or its periodic structure.”\(^2\) It seems to us that such a conclusion is at least premature. The point is that the “criticism” of quantum mechanics which leads to the conclusion that it is incapable of explaining the existence of a crystal is based on an analysis not of the general equation (1,1), but of approximate equations obtained by the method of the “self-consistent” field. S. V. Tyablikov\(^3\) showed that from the equations of quantum mechanics, though under simplifying assumptions, one can obtain periodic solutions of the type of a crystal lattice.

QUANTUM THEORY OF ELECTRONIC CONDUCTORS

Undoubtedly, there is as yet no sufficiently developed theory of the crystal as a whole. Usually one uses the approximate conception that a crystal is an aggregate of two “autonomous” systems—the crystal lattice, at whose nodes positively charged ions are situated*), and the system of electrons collectivized in the lattice (the former valence electrons and the electrons of the isolated gas atoms following them). Such an approximate separation of the crystal into two parts, by its very nature, can be useful for the knowledge of the real properties of matter only provided that this separation is not made absolute. The program of the indicated approximation is as follows. It is assumed that the potential energy of the system \(V\) in equation (1,2) can be represented as the sum of three terms:

\[ V(\mathbf{R}_1 \ldots \mathbf{R}_N, \mathbf{r}_1 \ldots \mathbf{r}_n) = \]

\[ = V_1(\mathbf{r}_1 \ldots \mathbf{r}_n) + V_2(\mathbf{R}_1 \ldots \mathbf{R}_N) + V_3(\mathbf{R}_1 \ldots \mathbf{R}_N, \mathbf{r}_1 \ldots \mathbf{r}_n), \tag{1,3} \]

where \(V_1\) is the potential energy of interaction between electrons, \(V_2\) the potential energy of interaction between ions, and, finally, \(V_3\) the potential energy of interaction between electrons and ions. Substituting (1,3) into (1,2) and then into the wave equation (1,1), we seek the solution of the latter in the form of a product

\[ \Psi = \Psi_{\mathrm{el}}(\mathbf{r}_1 \ldots \mathbf{r}_n;\ \mathbf{R}_1 \ldots \mathbf{R}_N)\cdot \Psi_{\mathrm{ion}}(\mathbf{R}_1 \ldots \mathbf{R}_N), \tag{1,4} \]

where \(\Psi_{\mathrm{el}}\) is the wave function of the electrons, depending on the coordinates of the ions as on parameters, and \(\Psi_{\mathrm{ion}}\) is the wave function of the ions. Such an approximate solution of (1,1) is customarily called adiabatic, since at each given moment the electronic system is described by such a wave function as corresponds to a stationary distribution of the ions. Naturally, such an approximation is legitimate only when considering those phenomena in a crystal for which, owing to the large difference in mass between ions and electrons, the latter move much faster than the former. It can be shown easily\(^{4}\) that, with accuracy up to terms of order \(\dfrac{m}{M}\), we obtain for the partial functions of the electrons and ions two separate wave equations—for determining the stationary states of the system of electrons:

\[ \hat{H}' \Psi_{\mathrm{el}} = E_{\mathrm{el}}(\mathbf{R}_1 \ldots \mathbf{R}_N)\Psi_{\mathrm{el}}, \tag{1,5} \]

*) The possibility of such a separation of a crystal into a lattice of ions and the outer electrons, and not into a lattice of nuclei and the aggregate of all electrons, is suggested by observations of the characteristic X-ray spectra of solids, which have practically the same line character as the spectra of isolated gas atoms. This experimental fact indicates that the inner layers of the electron shell of isolated atoms, when they are combined in a crystal, are preserved to a considerable degree, as a result of which one may speak of an ionic lattice.

where the energy of the electron system \(E_{\mathrm{el}}\) is a function of the parameters—the coordinates of the nuclei, and

\[ \hat H'=\hat H-\frac{\hbar^2}{2M}\sum_{i=1}^{N}\Delta_{R_i}, \]

and for the ions:

\[ \frac{\hbar^2}{2M}\sum_{i=1}^{N}\Delta_{R_i}\Psi_{\mathrm{ion}}+E_{\mathrm{el}}(\mathbf R_1\ldots \mathbf R_N)\Psi_{\mathrm{ion}}=-E\Psi_{\mathrm{ion}}. \tag{1,6} \]

It is evident from this that the electronic energy \(E_{\mathrm{el}}\) in the wave equation for the ions plays the role of potential energy, while \(E\) is the total energy of the system.

The adiabatic approximation makes it possible to consider separately the electronic and ionic problems in the theory of crystals. However, it is always necessary to remember the approximate character of this division of a real crystal into two subsystems and to justify specially the possibility of applying this approximation in each particular case.

The interaction between the system of electrons and ions can, in first approximation, be regarded as a small perturbation. In particular, in kinetic problems (electrical conductivity, thermal conductivity, etc.) this perturbation is taken into account as the cause of transitions of both interacting systems between their zero stationary states (collisions of electrons and phonons, etc.).

In the present review we shall be chiefly interested in questions connected with the properties of the electronic part of the crystal, and only at the end shall we briefly dwell on the second aspect of the problem of the crystal (see Section 9).

Before turning to the general quantum-mechanical treatment of the electron system in a crystal, we shall dwell on questions of the classification of various types of crystalline electronic conductors (Section 2), and also on a critical examination of the simplest models of these crystals—the model of the electron “gas” (Section 3) and the one-electron, so-called “band” quantum theory (Section 4). Thereafter the main subject of the review is presented—the many-electron theory of crystals (Sections 5–8).

2. CRITERION OF A METAL AND A DIELECTRIC

In every attempt to construct a theory of electronic conductors, the question quite naturally arises whether one can indicate one or several physical characteristics that would make it possible to carry out an unambiguous classification of the type of crystals under consideration. Analysis of the empirical material shows that there exists a physical criterion that makes it possible to divide electronic

conductors into two classes—metals and semiconductors (dielectrics). It should be emphasized from the very beginning that one and the same substance, under different conditions (temperature, pressure, state of the crystal lattice, etc.), may be either a metal or a nonmetal; therefore it is more correct to speak not of a metal (as a substance of definite chemical composition), but of the metallic state of various substances.

To establish the basic physical criterion of a metal, let us turn to the electrical properties of electronic conductors, more precisely, to an analysis of their behavior in an external constant electric field. It is known from experience that, among electronic conductors, metals possess the greatest values of specific electrical conductivity. However, a simple indication of the magnitude of the absolute values of electrical conductivity cannot yet serve as a clear criterion for dividing electronic conductors into qualitatively different groups. For this, one should turn to the consideration of the temperature dependence of the electrical conductivity of these substances in the region of sufficiently low temperatures (near \(0^\circ\mathrm{K}\)). Here two types of different regularities can be indicated quite clearly, namely: in substances classed as nonmetals, the electrical resistance, as the temperature tends to \(0^\circ\mathrm{K}\), always increases, tending in the limit to an infinitely large value. Conversely, in metals at a temperature close to \(0^\circ\mathrm{K}\), a sharp decrease of the electrical resistance \(\rho\) is always observed. Moreover, the purer the metal, the smaller the value of the resistance to which the extrapolation of the curve \(\rho(T)\) leads as \(T \to 0^\circ\mathrm{K}\). The ratio of the absolute values of the electrical resistances of typical metals at \(273^\circ\mathrm{K}\) and \(1^\circ\mathrm{K}\) may reach values of the order of \(10^6\). These facts make it possible to choose, as the physical criterion of a metal and a dielectric, the temperature behavior of the electrical resistance at low temperatures.

Physically, the process of the passage of current through a crystal consists in the fact that some of its electrons (conduction electrons), under the action of an external electric field, acquire an ordered motion with respect to the crystal as a whole. Therefore the criterion given above can be formulated as follows: in metals, already at low temperatures there are conduction electrons; in nonmetals, at low temperatures these electrons are practically absent, and with increasing temperature they do appear, but in insignificant quantities. In other words, the characteristic feature of the metallic state of a substance is that, in it, the minimum energy (\(0^\circ\mathrm{K}\)) of the crystal corresponds to the presence of conduction electrons, whereas in semiconductors and insulators there are no such electrons at the energy minimum.

The totality of theoretical and experimental data on solids leads to the conclusion that this criterion can be

indeed be regarded as the principal feature of electronic conductors. Therefore, for example, for a metal all of its physical properties are, in one form or another, connected with the presence in it of conduction electrons. The deeper meaning which modern theory invests in this statement will become clear from what follows.

The question of the number of conduction electrons cannot be resolved without a detailed consideration of the problem of the motion of electrons in a crystal lattice. Qualitatively, one may assume that in a metal this number is not vanishingly small in comparison with the number of lattice sites, whereas in dielectrics or semiconductors, at least at low temperatures, the number of conduction electrons is small. A very vivid illustration of this assumption may be provided by the fact that the majority of chemical elements possessing metallic properties in the solid and liquid states belong to the first groups of D. I. Mendeleev’s periodic system, i.e. have a certain number of valence electrons; at the same time inert gases and haloids, without exception, are dielectrics in all states of aggregation. It is therefore reasonable to suppose that it is precisely the outer—valence—electrons that play the role of conduction electrons in the solid and liquid states, and that, in order of magnitude, the density of these electrons is equal to the number of lattice sites per unit volume of the crystal. However, it should be emphasized that a large value of the electrical conductivity cannot serve as an unambiguous indication of a larger number of conduction electrons. These electrons may have different effective masses owing to their interaction with the lattice and with one another, and hence also different mobilities. Thus the seemingly simple question of the number of conduction electrons acquires a complex character and, for its solution, requires the development of a much more exact theory.

In order to obtain a clearer idea of the properties of conduction electrons, let us recall what influence an external constant electric field exerts on the motion of electrons in two simpler and, in a certain sense, limiting cases—namely, in the case of a free electron and of an electron in an isolated atom.

In the case of a free electron, the field, regarded as a small perturbation, causes a continuous increase of the component of momentum along the field. The electron continuously increases its energy in the original continuous spectrum*).

In an isolated atom the electron is in the strong electrostatic field of the nucleus \((10^7—10^8\ \mathrm{V/cm})\), and therefore the character of the influence of the external field depends essentially on the ratio

*) In an exact solution of this problem, of course, there can be no talk of any “transitions”1.

QUANTUM THEORY OF ELECTRONIC CONDUCTORS

of this field and of the internal field of the atom. Experience shows that in the case of weak external fields ($\lesssim 10^7 \ \mathrm{V/cm}$), which are the only ones of interest to us in the present case, no changes increasing with time occur in isolated atoms. The electrons of the atom remain in stationary states, differing little from the unperturbed states.

It is easy to understand the physical reason for such an action of a constant field on an electron in an atom. It is known from quantum mechanics that, in the interaction of any two systems, only those transitions can lead to changes increasing with time for which the sum of the energies of both systems does not change. A constant electric field, which may be regarded as the limit of an alternating field with frequency $\omega \to 0$, is capable of transferring energy to an electron only by the “classical” route, i.e. in infinitely small portions ($\hbar \omega \to 0$); therefore under its influence transitions can occur only between infinitely close states. Hence it is clear that a necessary condition for the appearance of the accelerating effect of the field, as is observed for a free electron, is the existence of a continuous energy spectrum. Atomic electrons have a discrete spectrum, and therefore the accelerating effect is absent for them. An intra-atomic electron can take part in a current only by being torn out of the atom, but, owing to the stability of the atom, its ionization always requires the expenditure of a finite portion of energy (the electrostatic energy which the electron receives from the field $F_{\text{ext}}$, while being accelerated over a distance of the order of atomic dimensions $a_0$, equal to $eF_{\text{ext}}a_0$, is, in fields of the magnitude under consideration, always less than the ionization energy).

In order to determine to which of the cases considered the behavior of conduction electrons is closer, let us first turn to experimental facts. Consider a metallic wire to the ends of which a constant potential difference is applied. Let the wire be placed in a thermostat, so that its temperature, despite the liberation of Joule heat, is kept constant at all times. Under these conditions the state of the wire does not change with time (if there are any changes, they are of a rapidly varying character and therefore are not detected macroscopically), i.e. the wire as a whole is in a stationary state. At the same time, however, a current passes through it, which appears in an arbitrarily weak field and, throughout the range of applicability of Ohm’s law, depends linearly on the field. Thus the properties of conduction electrons, in the sense of the accelerating action of the field, are something intermediate between the properties of free electrons, on the one hand, and intra-atomic electrons, on the other. For $F_{\text{ext}}=0$, the stationary state of the conductor corresponds to a current equal to zero. Thus, conduction electrons differ from the electrons of a cathode beam in vacuum in that

they, even after the field is switched on, continue to be in some stationary state, and are not subjected to unlimited acceleration.

Thus, one may arrive at the conclusion that in a metal, in the immediate vicinity of the lowest energy state of the conduction electrons, without current, there exist other possible states with a current different from zero. It is sufficient to apply an infinitely weak external field to the metal for it to pass into one of these “current-carrying” states. It follows at once from this that, in the absence of an external field, each conduction electron is not localized near some definite lattice site of the crystal; otherwise a weak field could not make it take part in the current passing through the metal as a whole.

From a simple analysis of the experimental facts one cannot obtain more concrete statements about the properties of conduction electrons. Before proceeding to the exposition of the rigorous quantum theory of electronic conductors, let us recall the earlier stages in the development of the theory of metals. Such a historical excursion will make it easier to understand the content of the present state of the theory.

3. CLASSICAL AND QUANTUM THEORIES OF THE ELECTRON “GAS”

a) Classical theory

The simplest explanation of phenomena connected with the electrical conductivity of metals can be obtained by introducing the idea of the so-called electron gas. This idea played a major role in the development of the atomic theory of electronic conductors, and in a certain form a number of conclusions obtained with the aid of this idea have retained their significance to this day. We shall dwell precisely on these conclusions, omitting everything that is only of historical interest.

If we return to the example considered at the end of Section 2 and recall that the Joule heat is liberated uniformly throughout the entire volume of the wire, it will become clear that the conduction electrons of a metal give up to the lattice the energy accumulated in the field not at the end of the interval during which the accelerating field acts, as, for example, at the anticathode of a discharge tube with free electrons, but at shorter intervals. Therefore one may say that conduction electrons receive a finite additional energy even when the length of the wire is increased without bound. Classical electron theory explains this by the fact that each conduction electron for the greater part of the time behaves (in the sense of acceleration in the field) exactly like a free electron, i.e. its velocity increases linearly with the field. The processes by which the energy accumulated in the field is transferred to the crystal lattice are ...

separate, comparatively short-lived events. In other words, the finite value of the electrical conductivity of metals is due to the fact that conduction electrons are assigned a certain mean free time \(\bar{\tau}\), analogous to that possessed by molecules of ordinary gases. This assumption makes it possible to apply the entire apparatus of the kinetic theory of gases to the study of the behavior of conduction electrons. It is clear that, in the absence of an external field, because of the chaotic nature of thermal motion, the stationary state of the conduction electrons is devoid of a resulting current. However, in the immediate vicinity of this state there lie states with current, which are realized in the presence of an external field.

Here two essential assumptions have been introduced. First, it is assumed that when a current passes through a metal, the behavior of each conduction electron may be considered separately. In reality, electrons interact strongly with one another, and also with the ions at the sites of the crystal lattice; therefore such an individual consideration of them is, strictly speaking, illegitimate. It is necessary to consider the entire ensemble of interacting electrons as a whole. Second, it is assumed that collisions of conduction electrons with the ions of the lattice are short-lived, i.e., the collision time is much less than the free time

\[ \left(t_{\text{coll}} \ll \bar{\tau}\right). \]

It will be shown below that these simplified, but quite natural, assumptions are in a certain sense preserved also in the rigorous theory. This is what still permits the electron-gas model to be used as a first orientation in the theory of metals.

The electron-gas model makes it possible to obtain also a quantitative estimate of those properties of conduction electrons which distinguish them from the electrons of isolated atoms. Such a quantity is, first of all, the mean free time. To estimate it, let us recall the formula given by the classical theory of the electron gas for the specific electrical conductivity \(\sigma\) of a metal,

\[ \sigma = \frac{ne^2}{2m}\bar{\tau}, \tag{3,1} \]

where \(n\) is the density of conduction electrons. It should be noted that in deriving formula (3,1) one obtains a linear relation between the field and the current density, which was the first theoretical derivation of Ohm’s law. Moreover, if one derives Lenz’s law for the amount of heat released per unit volume of a conductor carrying a current, the same expression for \(\sigma\) is obtained as in (3,1). This coincidence alone already indicates that the given model, despite its evident crudeness, reflects certain properties of real metals.

In order to obtain an estimate of \(\bar{\tau}\) from expression (3.1), it is necessary to make some assumptions about the density of the conduction electrons and their mass. Without going for the moment into a detailed analysis, let us assume that the number \(n\) is of the order of the number of lattice sites in \(\text{cm}^3\), i.e. \(n \sim 10^{22}\ \text{cm}^{-3}\), and that \(m\) is equal to the mass of a free electron, \(10^{-28}\ \text{g}\) (\(e\) is the elementary charge of the electron, equal to \(4 \cdot 10^{-10}\) CGSE). Then, for example, in the case of pure gold we obtain the following values for the time \(\tau\) at different temperatures:

\(T^\circ K\)	273.2	90.1	20.4	11.1	4.2
\(\tau\) sec.	\(5.9 \cdot 10^{-14}\)	\(2.2 \cdot 10^{-13}\)	\(9.8 \cdot 10^{-12}\)	\(1.4 \cdot 10^{-10}\)	\(6 \cdot 10^{-9}\)

Numbers of the same order are also obtained for other metals. To have an idea of how the properties of the conduction electrons considered here change in passing from the gaseous state to the crystalline one, let us compare the value of \(\bar{\tau}\) with the mean duration of atomic periods for isolated atomic systems. These periods are \(\sim 10^{-15}\) sec. From the table given above it is evident that the accelerating action of the field on an electron in a metal is measured by times of an entirely different order of magnitude. Even at room temperatures \(\bar{\tau}\) is greater than \(10^{-15}\) sec by almost a factor of one hundred. At low temperatures \(\bar{\tau}\) exceeds atomic periods by millions of times. Such a large value of the free-path time is a concrete expression of the fact that, in contrast to atomic electrons, the conduction electrons are not localized near definite lattice sites.

It is curious to note that this fundamental conclusion about the “freedom” of conduction electrons within the framework of the classical model was a kind of difficulty. From the point of view of crude classical notions, it would be natural to suppose that the mean free path of an electron in the lattice (it may be obtained by simply multiplying the free-path time by the mean thermal velocity of the electrons) should be of the order of the lattice constant. The estimates given above, however, give for the mean free-path length values hundreds of times greater than the lattice constant at room temperatures, while at low temperatures macroscopic values are obtained (\(\sim 10^{-2}\ \text{cm}\)).

The classical theory of the electron gas also led to other difficulties. While satisfactorily explaining the Wiedemann–Franz law, it was unable to give temperature dependences of the kinetic coefficients (electrical conductivity and thermal conductivity) in agreement with experiment, especially at low temperatures; the sharpest contradiction with experiment was the well-known “catastrophe” with the heat capacity.

These difficulties pointed to fundamental inaccuracies of the model and did not allow this model to be considered suitable for quantitative calculations.

b) Quantum theory. The “catastrophe” with the heat capacity undermined confidence in the concept of the “gas” of conduction electrons and, in general, in the electronic theory of metals. The beginning of the resolution of these difficulties was the well-known theory of Ya. I. Frenkel\(^6\) on “wandering” electrons. When atoms come together in the process of condensation, the average distances between nuclei become of the same order, or even smaller, than the diameter of the extended “orbits” of the valence electrons. Therefore in a crystal such electrons can no longer remain firmly attached to a single nucleus. There arises a high probability that an electron can pass from one site to another. Thus, according to Frenkel, all valence electrons, upon condensation of a metallic vapor, spontaneously turn into “wandering” electrons, which he identified with conduction electrons. A “wandering” conduction electron becomes, as it were, the property of the entire crystal as a whole. Frenkel showed, on the basis of the virial theorem, that the velocity of these wandering electrons must be of the order of the velocity of valence electrons in the orbits of isolated atoms, i.e. \(\sim 10^8\ \text{cm/sec}\). Frenkel further succeeded in obtaining all the positive results of the electron-gas model (the expressions for electrical conductivity, thermal conductivity, the Wiedemann–Franz law, etc.). The most important advantage of the theory of wandering electrons over the classical theory of the electron gas, however, was that in it the difficulty with the heat capacity was automatically eliminated. The wandering electrons do not participate actively in thermal motion, and therefore they do not appreciably affect the heat capacity of the crystal (at least at not very high temperatures, when the effect of “ionization” is absent). Furthermore, in this model a direct connection was established between the conduction electrons and the quantum structure of the atom, though within the framework of the old quantum theory.

A drawback of the theory of wandering electrons was that it could not give a quantitative explanation of the large magnitude of the mean free time at low temperatures, since it did not take into account the quantum nature of electrons. Nor could it explain why some crystals are metals, while others are insulators.

The further development of the theory proceeded along two paths. First of all, it was taken into account that electrons obey not classical statistics, but quantum antisymmetric statistics. On this basis Sommerfeld\(^7\) developed the quantum theory of the electron gas. This theory preserved all the positive results of the classical model, but at the same time eliminated the difficulty with the heat capacity.

A high degeneracy of the dense gas of conduction electrons leads to the fact that the fraction of thermally active electrons is determined by the magnitude of the ratio \(T/\theta\), where \(\theta\) is the degeneracy temperature (of the order of \(10^4\)—\(10^5{}^\circ\mathrm{K}\)). In view of this, at room temperatures about \(1\%\) of the conduction electrons participate in thermal motion, and therefore the share of their participation in the heat capacity, in comparison with lattice vibrations, is also small.

The quantum model of the electron gas also has a number of shortcomings, namely: it does not give the correct temperature dependence of the electrical conductivity and thermal conductivity, does not explain the anomalous sign of the Hall constant in almost half of the metals, the magnitude of the change in electrical resistance in a magnetic field, etc.

However, the weakest point of this model is that it does not explain, but postulates the “freedom” of the conduction electrons, i.e., the enormous magnitude of the mean free time. In analyzing this phenomenon, one may either abandon attempts to give a physical explanation of the large free-path time and unconditionally accept the hypothesis of a free electron gas, or, following the path indicated by Frenkel in his model of wandering electrons, turn to a consistent quantum-mechanical study of the motion of electrons in a crystal lattice. In doing so one can obtain an answer to why, despite the strong interaction between the electron and the crystal lattice, the conduction electron possesses considerable freedom and does not undergo collisions at every lattice site. We now turn to this consideration.

4. ONE-ELECTRON THEORY OF CRYSTALS

Ya. I. Frenkel\(^8\) was the first to point out that the specific properties of electrons in a crystal lattice and, in particular, the anomalously large value of the mean free time should be explained by using ideas about the wave nature of the electron. He succeeded in showing that one can obtain a good qualitative idea of the character of electron motion in a crystal even if the calculations are carried out not with the aid of a consistent quantum-mechanical theory, but by making use of a visual quasiclassical wave picture of electron motion. Frenkel showed that an ideal crystal lattice with a periodic potential is “transparent” for electron waves. Only violations of the correct periodicity of the lattice (inclusions, lattice distortions, thermal vibrations, etc.) can lead to scattering of electron waves (collisions—in the language of corpuscular theory). A quantitative calculation showed that the mean free-path length, which in this case is determined by the effective cross section for scattering of electron ...

waves by thermal vibrations, at room temperatures, is determined by a quantity of the order of hundreds of interatomic distances. Thus, Frenkel’s work for the first time gave a fundamentally correct explanation of the great importance of the mean free time of conduction electrons in crystals. However, many unresolved questions remained in the theory, for example the distinction between a metal and an insulator, etc.; to answer them it was necessary to resort to more exact methods of quantum mechanics.

In Section 1 an approximate method for treating electronic processes in crystals was formulated (the adiabatic approximation). The electronic wave function \(\Psi_{\mathrm{el}}\) must be determined from equation (1,5). The energy operator \(\hat H\), in expanded form, has the form

\[ \hat H=-\frac{\hbar^2}{2m}\sum_{j=1}^{n}\Delta_j+\sum_{i,j}G(\mathbf r_j-\mathbf R_i)+\frac{1}{2}\sum_{j\ne j'}V(\mathbf r_j,\mathbf r_{j'}'); \tag{4,1} \]

here the second term represents the interaction of all \(n\) electrons with all ions, and the third term is the interaction between the electrons. This last term of the operator \(\hat H\) introduces a great complication into the solution of equation (1,5). This is connected with the fact that the interaction operator does not make it possible to separate the variables in the equation and makes the problem of finding the solution extremely difficult. Therefore it is quite natural to strive to transform the expression for the interaction operator so that separation of variables becomes possible. The simplest method is to replace the interaction operator between electrons

\[ \frac{1}{2}\sum_{j\ne j'} V(\mathbf r_j,\mathbf r_{j'}') \]

by the additive operator

\[ \sum_j V'(\mathbf r_j). \]

Physically, such a replacement means that the interaction of each individual electron with all the others, which depends on the instantaneous positions of all the electrons, is replaced by the action of a certain stationary potential field, which represents a kind of averaged effect of the actual motion in the system of interacting electrons. When such a program is carried out exactly, the linear many-electron problem described by equation (1,5) is reduced to the solution of a nonlinear integro-differential equation. To solve this equation is as difficult as the equation (1,5) itself. Moreover, as Pekar\(^9\) showed, in doing so one may lose a number of solutions of the exact many-electron problem. To simplify the problem, approximate methods of solution are used. The most accurate of these methods is the self-consistent field method of V. A. Fock\(^ {10}\), which

is successfully used for studying the quantum states of many-electron atoms. However, in the case of crystals the application of this method can hardly be considered fruitful, since its implementation in this case requires such great simplifications that they do not guarantee even the qualitative correctness of the results. It is therefore advisable to abandon taking into account the operator of interaction between electrons and to consider the motion of electrons in the periodic potential field of the crystalline ionic lattice. As a result of such a simplification, equation (1.5) breaks up into separate one-electron equations, which have the form

\[ \left[-\frac{\hbar^2}{2m}\Delta_r+\sum_i G(\mathbf r-\mathbf R_i)\right]\psi_E(\mathbf r)=E_{\text{el}}\psi_E(\mathbf r). \tag{4.2} \]

The solution of this equation will depend on the particular form of the potential

\[ G(\mathbf r)=\sum_i G(\mathbf r-\mathbf R_i). \]

However, one can make a number of the most general statements about the character of the motion of an electron in a crystal without making any assumptions about the form of the potential \(G\), apart from its periodicity properties. If one considers an infinite ideal lattice with fundamental vectors \(\mathbf a_1,\mathbf a_2,\mathbf a_3\), then the potential \(G(\mathbf r)\) possesses the principal property of periodicity—translational invariance

\[ G(\mathbf r+l_1\mathbf a_1+l_2\mathbf a_2+l_3\mathbf a_3)=G(\mathbf r), \tag{4.3} \]

where

\[ l_1\mathbf a_1+l_2\mathbf a_2+l_3\mathbf a_3=\mathbf a_l \tag{4.4} \]

is an integral lattice vector (\(l_1,l_2,l_3\) are arbitrary integers). The energy operator in the left-hand side of (4.2) is also invariant with respect to these translations. By virtue of condition (4.3), the function \(G(\mathbf r)\) can be expanded in a series

\[ G(\mathbf r)=\sum_{m_1m_2m_3}G_m e^{i\mathbf m\mathbf r}, \tag{4.5} \]

where

\[ \mathbf m=2\pi(m_1\mathbf b_1+m_2\mathbf b_2+m_3\mathbf b_3), \tag{4.6} \]

where \(\mathbf b_1,\mathbf b_2,\mathbf b_3\) are the fundamental vectors of the reciprocal lattice, and \(m_1,m_2,m_3\) are integers. If we immediately restrict ourselves to considering the allowed solutions, it can readily be shown that they have the form of plane waves modulated in the rhythm of the lattice. Indeed, let \(\psi(\mathbf r)\) be one of the allowed solutions of equation (4.2); it can be represented in the form

\[ \phi(\mathbf r)=\int_{-\infty}^{+\infty}a(\mathbf k)e^{i\mathbf k\mathbf r}\,dk_1\,dk_2\,dk_3, \tag{4.7} \]

where the vectors \(\mathbf{k}\) have the form

\[ \mathbf{k}=k_1\mathbf{b}_1+k_2\mathbf{b}_2+k_3\mathbf{b}_3 \quad\text{and}\quad k_i=\mathbf{k}\mathbf{a}_i . \tag{4,8} \]

By virtue of (4,2) and (4,5), the wave equation in the \(\mathbf{k}\)-representation is written as follows:

\[ \left(E-\frac{\hbar^2}{2m}k^2\right)a(\mathbf{k}) = \sum_{m_1m_2m_3}G_m a(\mathbf{k}-\mathbf{m}). \tag{4,9} \]

From (4,9) it is immediately clear that if all \(G_m=0\), i.e., if the electron were free, then (4,9) would be satisfied for \(a(k)=0\) for all \(\mathbf{k}\), except for some one value. In other words, the solution would have the form of a plane monochromatic wave. If \(G_m\ne0\), then the \(a\)’s for different \(\mathbf{k}\) are connected with one another, and each of them cannot be chosen independently of the others. However, from (4,9) it is evident that, owing to the periodicity of the lattice, only those \(a(\mathbf{k})\) are connected with one another which correspond to values of \(\mathbf{k}\) differing from one another by one of the reciprocal-lattice vectors \(\mathbf{m}\). Therefore our solution (4,7) has the form of a series, not an integral. Let \(\xi\) be one of the vectors \(\mathbf{k}\),

\[ \xi=\xi_1\mathbf{b}_1+\xi_2\mathbf{b}_2+\xi_3\mathbf{b}_3, \tag{4,10} \]

whose components are constrained by the inequalities

\[ -\pi\le \xi_i<\pi\quad (i=1,2,3), \tag{4,11} \]

then any vector \(\mathbf{k}\) has the form \(\mathbf{k}=\xi+\mathbf{m}\). When \(\xi\) runs through all values according to (4,11), and \(\mathbf{m}\) through all integers, the components \(\mathbf{k}\) run through all values from \(-\infty\) to \(+\infty\). We seek solutions of (4,9) in the form

\[ a(\mathbf{k})=\sum_{\mathbf{m}}\alpha_{\mathbf{m}}\delta[(\mathbf{k}-(\xi+\mathbf{m}))], \tag{4,12} \]

where \(m_1,m_2,m_3\) are integers, including 0, and \(\xi\) is a definite vector of the type indicated above. Substituting (4,12) into (4,7), we obtain the desired wave function of the electron in the form of a series

\[ \psi(\mathbf{r})=e^{i\xi\mathbf{r}}\sum_{m_1m_2m_3}\alpha_m e^{i\mathbf{m}\mathbf{r}}. \tag{4,13} \]

The coefficients \(\alpha_m\) satisfy the equations (see (4,9))

\[ \left[E-\frac{\hbar^2}{2m}(\xi+\mathbf{m})^2\right]\alpha_m = \sum_{m'_1m'_2m'_3}G_{m'}\alpha_{m-m'} . \tag{4,14} \]

These equations undoubtedly have solutions, since the determinant composed of their coefficients, by virtue of the condition \(G_m=G^*_{-m}\), is Hermitian. Thus, we shall find the energy spectrum of the electron and the wave function (4,13) if we set equal to zero

determinant of the system (4.14). Introducing the notation

\[ \sum_{m_1,m_2,m_3} a_m e^{i m r}=u(r), \]

we obtain, instead of (4.13), the expression

\[ \psi(r)=e^{i\xi r}u(r), \tag{4.15} \]

where the function \(u(r)\) is periodic with the period of the lattice. From the form obtained for the wave function there follows the fundamental conclusion that, owing to translational invariance, an electron in an ionic crystal lattice cannot be localized at any particular node; with equal probability it belongs to all nodes of the lattice. This conception of a peculiar “freedom” of the electron in the periodic field of the lattice, of its “transparency” for electron waves, is precisely the new element that quantum mechanics has introduced into the theory of crystals.

From equations (4.14) it follows that to each given \(\xi\) there corresponds an innumerable set of equations (in accordance with the innumerable set of possible choices of the vectors \(m\)), and consequently, for each \(\xi\) they undoubtedly have an innumerable set of solutions. Thus, specifying the components \(\xi_1,\xi_2,\xi_3\) is insufficient for a unique determination of the state of the electron. Fixing a definite vector \(\xi\), we obtain a discrete spectrum of energies \(E_\zeta\), where the index \(\zeta\) denotes the number of the zone. Running through all possible values of \(\xi\), restricted by the inequality (4.11), we obtain, for each zone, the energy \(E\) in the form of a continuous function of the vector \(\xi\), as well as the wave functions \(\psi(\xi;r)\). In passing from one zone to another, the energy will undergo a jump. Thus we arrive at the conclusion that the structure of the energy spectrum of an electron in a crystal has the form of bands. Each stationary state of an electron in a crystal lattice can be uniquely determined by specifying the quasimomentum vector \(\xi\) and the zone number \(\zeta\). The term quasimomentum is introduced by analogy with the momentum of a free electron, but it should be emphasized that the momentum of an electron in a lattice, of course, cannot be identified with the vector \(\xi\) (see below). To count the number of states in a zone it is necessary to take into account the finiteness of the dimensions of the crystal.

The method of constructing zones given above is not the only one; another well-known method is also used—the Brillouin method[^11]. It differs from the one given above only in the order in which the states in a zone are numbered. In this connection one clarification should be made. Very often in the literature a zone is taken to mean not the totality of states, i.e. wave functions, but the totality of the energy levels corresponding to these states; in this usage one encounters the expressions

“overlap of zones” instead of “overlap of the energy bands of different zones,” and so on. Such use of the word “zone” may lead to the very concept of a zone acquiring a certain vagueness. In fact, however, the classification of states into zones can always be carried out quite precisely and unambiguously. The point is that the division of electron states in a crystal into zones is not a mere mathematical device, but reflects the real properties of the electron in the crystal. It turns out that the transitions of electrons from some states to others, occurring under the influence of external fields, obey substantially different rules depending on whether the initial and final states belong to one and the same zone or to different zones; moreover, this distinction is preserved independently of whether or not the energy bands corresponding to different zones overlap. This is precisely why the totality of electron states in a lattice should always be regarded as subdivided into zones.

Without making any assumptions about the concrete form of the periodic potential of the lattice (4,5), one can compute the mean value of the electron velocity in the stationary state \(\xi,\zeta\). As is known, this quantity is equal to

\[ \overline{\mathbf v}_{\xi,\zeta}=\frac{1}{\hbar}\nabla_{\xi}E(\xi,\zeta). \tag{4,16} \]

Formula (4,16) is a generalization of the well-known de Broglie relation to the case of an electron moving in an ideal crystal lattice. It shows that the mean velocity of an electron in stationary states in the lattice is, as a rule, different from zero; in other words, an electron in an ideal lattice can “move freely” through it, analogously to a free electron in vacuum. The magnitude and direction of the mean velocity are determined by the character of the isoenergetic surfaces \(E(\xi,\zeta)=\mathrm{const}\) in the space of quasimomenta \(\xi\). In stationary states of an electron belonging to certain planes of quasimomentum space, the components of the current in the direction normal to the corresponding plane are equal to zero. These planes, as a rule, coincide with the “edges” of a zone, and therefore the mean velocity somewhere inside the zone has an extremal value.

Let us clarify what the mean acceleration of an electron in the lattice is equal to under the action of weak external fields. Although we are not yet taking into account the interaction between electrons in the crystal, by virtue of quantum laws a system even of free electrons will possess specific quantum properties (antisymmetry of the total wave function—the Pauli principle). In order to solve the problem of the acceleration of an electron in the lattice at once taking electron statistics into account, it is convenient to use the description

electronic states not by means of wave functions, but by means of the density matrix, first introduced into quantum mechanics by L. D. Landau. Knowing the density matrix (the statistical operator of the system) \(\hat u\), one can compute the mean value of any dynamical variable of the system, namely:

\[ \bar A=\int (q'_1\ldots q'_n|\hat u|q''_1\ldots q''_n) (q''_1\ldots q''_n|\hat A|q'_1\ldots q'_n)\times \]
\[ \times dq''_1\ldots dq''_n dq'_1\ldots dq'_n, \tag{4,17} \]

where \(\hat A\) is the operator of the corresponding dynamical quantity, and \(q_i\) is the set of spatial and spin coordinates of the \(i\)-th particle of the system under consideration. In the case of a system of noninteracting electrons all dynamical quantities are represented by additive operators of the type

\[ \hat A=\sum_{i=1}^{n}\hat a[qi]. \tag{4,18} \]

In this case we have

\[ (q'_1\ldots q'_n|\hat A|q''_1\ldots q''_n)= \]
\[ =\sum_{i=1}^{n}\delta(q'_1-q''_1)\ldots \delta(q'_{i-1}-q''_{i-1})\, \delta(q'_{i+1}-q''_{i+1})\ldots \]
\[ \ldots \delta(q'_n-q''_n)(q'_i|\hat a|q''_i), \tag{4,19} \]

and therefore for a system of identical particles instead of (4,17) we obtain

\[ \bar A=n\int dq'dq''(q'|\hat a|q'')\int dq_1\ldots dq_{i-1}dq_{i+1}\ldots \]
\[ \ldots dq_n\,(q_1\ldots q'\ldots q_n|\hat u|q_1\ldots q''\ldots q_n). \tag{4,20} \]

The quantity

\[ (q'|\hat\rho|q'')=n\int (q_1\ldots q'\ldots q_n|\hat u|q_1\ldots q''\ldots q_n)\,dq_1\ldots \]
\[ \ldots dq_{i-1}dq_{i+1}\ldots dq_n \tag{4,21} \]

is an element of the density matrix \(\hat\rho\) of one particle (in \(q\)-space); therefore the mean value of an additive operator is equal to

\[ \bar A=\int dq'dq''(q'|\hat a|q'')(q''|\hat\rho|q')=\operatorname{Sp}(\hat a\hat\rho), \tag{4,22} \]

where \(\operatorname{Sp}\hat b\) is the operation of taking the trace of the operator \(\hat b\). If one chooses

QUANTUM THEORY OF ELECTRONIC CONDUCTORS

not the coordinate representation, but the diagonal representation for the matrix \(\hat{\rho}\), then (4,22) takes the form

\[ \overline{A}=\sum_{\eta',\,\eta}(\eta|\hat{a}|\eta')(\eta'|\hat{\rho}|\eta) =\sum_{\eta}\rho(\eta)(\eta|\hat{a}|\eta), \tag{4,23} \]

where the eigenvalue \(\rho(\eta)\) denotes the number of particles in the state \(\eta\).

To determine the required acceleration of the electron it is necessary to find the change of the operator \(\hat{\rho}\) with time under the influence of an external electric field \(F\). The equation of motion for the operator \(\hat{\rho}\) in the presence of a uniform field \(F\), directed along the \(x\)-axis, has the form

\[ i\hbar\dot{\hat{\rho}}=\hat{H}\hat{\rho}-\hat{\rho}\hat{H} +eF\left(x\hat{\rho}-\hat{\rho}x\right). \tag{4,24} \]

Usually the second term on the right-hand side of (4,24) is regarded as small, the value \(\hat{\rho}_0\) for \(F=0\) is substituted there, and the method of successive approximations is developed. However, this would be a correct procedure if the additions to \(\hat{\rho}\) from the field actually decreased. But this is not always so. Restricting the problem to first order in \(F\) already implicitly assumes collisions with the lattice, which destroy the result of acceleration by the field. We shall solve this problem differently. From the general theory of the motion of an electron in a lattice it is known that there may be two types of transitions: transitions within one band, which obey strict selection rules,

\[ (\xi'\zeta|x|\xi''\zeta)=ia\delta'(\xi'-\xi'') \tag{4,25} \]

and transitions from band to band with the selection rules

\[ (\xi'\zeta'|x|\xi''\zeta'')\ne 0;\quad (\xi'\zeta'|x|\xi'''\zeta'')=0 \quad \text{for }(\xi'\ne \xi''). \tag{4,26} \]

Transitions of the type (4,26) for a constant or slowly varying field are improbable, since they occur with a large energy gap, even, as was indicated above, for overlapping energy bands. We shall solve equation (4,24) exactly, excluding transitions between bands from consideration. This is our approximation. Equation (4,24) for the initial moment of time in such a representation has the form

\[ i\hbar(\xi'\zeta'|\hat{\rho}_1|\xi''\zeta'') =eF\sum_{\zeta'''}\int d\xi''' \left[ (\xi'\zeta'|x|\xi'''\zeta''') (\xi'''\zeta'''|\hat{\rho}_0|\xi''\zeta'') -\right. \]

\[ \left. -(\xi'\zeta'|\hat{\rho}_0|\xi'''\zeta''') (\xi'''\zeta'''|x|\xi''\zeta'') \right]; \tag{4,27} \]

the operator \(\hat{\rho}_0\) here is diagonal; therefore, instead of (4.27) we shall have

\[ i\hbar\,(\xi'\zeta'|\hat{\rho}_0|\xi''\zeta'')= eF\,(\xi'\zeta'|x|\xi''\zeta'')\,[\rho_0(\xi''\zeta'')-\rho_0(\xi'\zeta')]. \tag{4.28} \]

Instead of the matrix element of the coordinate we substitute (4.25) and, moreover, make the replacement

\[ \rho_0(\xi''\zeta')-\rho_0(\xi'\zeta') \simeq (\xi''-\xi')\nabla_{\xi}\rho_0(\xi'\zeta') \]

and use the property of the \(\delta'\)-function

\[ -x\delta'(x)=\delta(x); \]

then from (4.28) we obtain

\[ (\xi'\zeta'|\hat{\rho}_0|\xi''\zeta'')= \frac{ieFa}{\hbar}\,\nabla_{\xi}\rho_0(\xi',\zeta')\, \delta(\xi'-\xi'')\,\delta_{\zeta'\zeta''}. \tag{4.29} \]

It follows from (4.29) that the operator \(\hat{\rho}_0\) is also diagonal and its eigenvalues are equal to

\[ \hat{\rho}(\xi\zeta)=\frac{eFa}{\hbar}\,\nabla_{\xi}\rho(\xi,\zeta). \tag{4.30} \]

The solution of this equation with the initial conditions \(\hat{\rho}=\hat{\rho}_0\) at \(t=0\) has the form

\[ \hat{\rho}=\hat{\rho}_0\left(\xi+\frac{eFa}{\hbar}\,t\right). \tag{4.31} \]

Thus, under the influence of a constant electric field and neglecting transitions to other bands, the entire distribution of electrons as a whole “rotates” over the band with “angular velocity” \(\dfrac{eFa}{\hbar}\). If the band is filled uniformly, i.e. \(\nabla_{\xi}\rho(\xi\zeta)=0\), then such rotation produces nothing; that is, if at the initial moment the current was equal to zero, it remains zero at all subsequent moments of time. It is often incorrectly asserted that the absence of current is due to the Pauli principle, but, as is clear from the calculation given, it is determined only by the uniformity or nonuniformity of the filling of the band. The period of rotation over the band is, in order of magnitude,

\[ T_F=\frac{h}{eFa}=\frac{10^{-27}}{10^{-10}10^{-8}F} =\frac{10^{-9}}{F}\ \text{sec}. \]

For real fields in metals (\(F\simeq 10^{-5}\) CGSE), the quantity \(T_F\) turns out to be appreciably larger (\(\sim 10^{-4}\) sec) than the free-path time determined by collisions with the lattice; therefore an asymmetry is established in the electron distribution, which gives

the resulting current in the metal. In an ideal lattice the current may, on the average, be equal to zero even in a field \(F\ne 0\). Making use of the smallness of the free-path time in comparison with the period of the field \(F\), one may expand (4.31) in a series in powers of \(F\) and confine oneself to terms of zeroth and first order

\[ \rho=\rho_0+\frac{eFa}{\hbar}\nabla_\xi\rho_0 t+\ldots \tag{4.32} \]

The mean velocity is equal to

\[ \overline{\mathbf v}=\operatorname{Sp}(\hat{\mathbf r}\rho), \tag{4.33} \]

and the mean acceleration

\[ \frac{d\overline{\mathbf v}}{dt} =\operatorname{Sp}(\hat{\mathbf r}\hat{\rho}) =-\operatorname{Sp}(\dot{\hat{\mathbf r}}\rho) =-\frac{eFa}{\hbar}\sum_{\zeta}\int d\xi\,(\xi\zeta|\dot{\hat{\mathbf r}}|\xi\zeta)\nabla_\xi\rho(\xi\zeta)= \]

\[ =\frac{eFa}{\hbar}\sum_{\zeta}\int d\xi\,\rho(\xi\zeta)\nabla_\xi(\xi\zeta|\dot{\hat{\mathbf r}}|\xi\zeta); \tag{4.34} \]

here integration by parts has been carried out twice. The result (4.34) may be interpreted in such a way that \(\nabla(\xi\zeta|\dot{\hat{\mathbf r}}|\xi\zeta)\) is the acceleration of an electron in the state \(\xi,\zeta\). Using the generalized de Broglie formula (4.16), we obtain instead of (4.34)

\[ \frac{d\overline{\mathbf v}}{dt} =\frac{eFa^2}{\hbar^2}\sum_{\zeta}\int d\xi\,\rho(\xi\zeta)\nabla_\xi\nabla_\xi E(\xi,\zeta). \tag{4.35} \]

Thus, the mean acceleration of an electron in the lattice in the state \(\xi,\zeta\) is equal to

\[ \frac{eFa^2}{\hbar^2}\nabla_\xi\nabla_\xi E(\xi,\zeta). \tag{4.36} \]

The acceleration of a free electron under the same conditions is equal to \(eF/m\); therefore one may define the effective mass of an electron in the lattice as

\[ m^*=\frac{\hbar^2}{a^2\nabla_\xi\nabla_\xi E(\xi,\zeta)}. \tag{4.37} \]

It is seen from (4.37) that the effective mass \(m^*\) is a function of the state of the electron \(\xi,\zeta\). It was indicated above that the mean velocity of an electron in the lattice at the “edges” of the band is, as a rule, equal to zero, while in the “middle” of the band it reaches an extremal value; therefore the acceleration of an electron in the lattice, and consequently also the effective mass, changes sign. In the lower part of the band the mass \(m^*\) is positive, while in the upper parts it is negative. Thus it is seen that taking into account the interaction between the electron and the lattice leads to the fact that, although one may in a certain sense speak of a “free” electron, nevertheless all its

characteristics also include the properties of the lattice, as is clearly seen in the example of the effective mass.

Formula (4.35) can be compared with the acceleration of an ensemble of free electrons, and the concept of the effective number of conduction electrons can be introduced. In the case of free electrons,

\[ \frac{\overline{dv}}{dt}=n^*\frac{eF}{m}, \]

therefore, from comparison with (4.35) and (4.34), we obtain

\[ n^*=\frac{ma^2}{\hbar^2}\sum_\zeta\int d\xi\,\rho(\xi,\zeta)\nabla_\xi\nabla_\xi E(\xi,\zeta)= \]

\[ =-\frac{ma}{\hbar}\sum_\zeta\int d\xi\,(\xi,\zeta|\dot r|\xi,\zeta)\nabla_\xi\rho(\xi,\zeta). \tag{4.38} \]

It is seen from (4.38) that, for uniform filling of the band \((\nabla_\xi\rho=0)\), the effective number of conduction electrons is equal to zero.

The calculation carried out above is generalized without particular difficulty also to the case of an alternating electric field of not very high frequency \(\omega\), when the inhomogeneity of the field with respect to the coordinates may be neglected. In this case, for the dielectric constant of the crystal, we obtain

\[ \varepsilon=1+4\pi e^2\sum_\zeta\int d\xi\,\rho_0(\xi,\zeta) \left[ -\frac{a^2}{\omega^2\hbar^2}\nabla_\xi\nabla_\xi E(\xi,\zeta)+ \right. \]

\[ \left. +\frac{2}{\hbar}\sum_{\zeta'}\frac{1}{\omega(\xi\zeta,\xi\zeta')}\cdot \frac{|(\xi\zeta|\hat r|\xi\zeta')|^2}{\omega^2(\xi\zeta,\xi\zeta')-\omega^2} \right], \tag{4.39} \]

where

\[ \omega(\xi\zeta',\xi\zeta'')=\frac{1}{\hbar}\left[E(\xi\zeta')-E(\xi\zeta'')\right] \]

is the frequency of transition from the band \(\zeta'\) to the band \(\zeta''\) at an unchanged value of the quasimomentum \(\xi\). This formula is obtained from the expression for the current, which, according to macroscopic electrodynamics, is equal to

\[ j=\frac{\varepsilon-1}{4\pi}\frac{\partial F}{\partial t}+\sigma P \]

\((\sigma\) is the specific electrical conductivity). The second term in the square brackets on the right-hand side of formula (4.39) gives the part of the dielectric constant that depends on transitions from band to band, while the first gives the part due to transitions within one and the same band. If quantum transitions are neglected, then the dielectric constant will have the same form as for free electrons:

\[ \varepsilon=1-\frac{4\pi n^*e^2}{m\omega^2} =1-\frac{4\pi ne^2}{m^*\omega^2}, \tag{4.40} \]

if by \(n^*\) one understands the effective number or by \(m^*\) the effective mass of the conduction electrons. From formula (4.40) it is clear that the most direct way of measuring the effective number of conduction electrons or the effective mass in crystals is the determination of the dielectric constant for comparatively slowly varying fields, when the quantum effect of transitions to other bands may be neglected. As will be shown below (see Section 5), such a determination of the effective values \(n^*\) or \(m^*\) can also be carried out in the case of a system of arbitrarily strongly interacting electrons.

These fundamental results of the one-electron quantum-mechanical theory, and especially the possibility of introducing the concepts of an effective mass or of the number of conduction electrons, in a certain sense justified some ideas of the elementary classical electron theory of metals. It is now clear to us, for example, why, despite the crudeness of the classical model, formula (3.1) and the consequences following from it had such “success” when compared with experiment. This happened because the electrons in the lattice, although they experience a strong interaction with the ions, are nevertheless in a certain sense “free.” Quantum mechanics, of course, not only preserved but also refined all the concepts of the classical electron theory, giving them a new and deeper physical content. While preserving all the achievements of the classical theory, the one-electron quantum model also resolved a number of its difficulties—the catastrophe with the heat capacity, the two signs of the Hall effect, etc.

The results presented exhaust the general properties of an electron in a lattice according to the one-electron model. To obtain more detailed information one must know more precisely the form of the periodic potential, which makes it possible to determine the dependence of the energy and of the density matrix on the quasimomentum in the various bands, i.e. the functions \(E(\xi,\zeta)\) and \(\rho(\xi,\zeta)\). Usually two types of approximation are used—the methods of weakly and strongly bound electrons. In the case of weak binding one starts from a system of completely free electrons as the zeroth approximation, and in the first approximation introduces a weak periodic potential as a perturbation. Using the usual methods of perturbation theory for quantum systems with a continuous energy spectrum, one obtains a splitting of this spectrum into bands, each of which corresponds to its own band of states. At the edges of the energy bands, for weak or, conversely, almost complete filling, one may expand the electron energy in powers of the quasimomentum and, using the fact that at the band edges

\[ \frac{\partial E(\xi,\zeta)}{\partial \xi}=0, \]

represent the energy in the form of a quadratic function of the quasimomenta with the effective mass as coefficient. In the other limiting case—strong binding—in the nu-

in the left-hand approximation one proceeds from a system of isolated atoms with a discrete energy spectrum, which, in the first approximation, is split owing to the interaction of the electron with the other ions of the lattice. In this case one may also use the effective-mass approximation. Obviously, the weak-binding approximation is reasonable for the states of the former valence electrons of the crystal, and the strong-binding approximation for the inner electrons*).

As is well known from the literature, within the framework of the one-electron theory it has been possible to explain an enormous totality of the static and kinetic properties of metals and semiconductors. However, strictly speaking, this explanation should be regarded only as qualitative, since quantitative results cannot be obtained because the strong effect of the interaction between electrons is neglected. A number of phenomena, whose very existence is determined by this interaction, such as, for example, ferromagnetism, superconductivity, the peculiarities of transition metals, etc., cannot in principle be explained within the framework of the one-electron approximation. At the same time, in the literature, especially in the American and English literature (the works of the schools of Slater, Van Vleck, Mott, Stoner, etc.), the conclusions of a crudely approximate one-electron theory are very often metaphysically absolutized; unnecessary “refinements” of almost numerical character begin to be introduced, which create only the appearance of rigor in a theory essentially unsuitable for a quantitative explanation of electronic phenomena in crystals.

What has been said above naturally leads to the conclusion that there is a pressing need to construct a many-electron theory of crystals, one that would be free of the indicated shortcomings of the one-electron model and would make it possible to give a more adequate explanation of the real properties of electrons in crystals. At the same time, a many-electron theory makes it possible to elucidate the physical reason why, in a number of cases, the one-electron theory, despite its crudeness, gives a quite satisfactory explanation of the actual state of crystalline bodies.

*) It should also be pointed out that, when the finiteness of real crystals is taken into account, along with the quasi-continuous energy spectrum of the electron there appear discrete levels of special states of electron motion localized at the surface of the crystal (Tamm levels\({}^{56}\)). In addition, in the presence of local violations of the regularity of the crystal lattice—impurity atoms, holes, etc.—as was first shown by Landau,\({}^{55}\) localized states of an electron arise in a distorted crystal lattice, as if the electron were “sticking” to the places where the correct periodicity of the lattice is disturbed. These states play a very important role in phenomena occurring in electronic semiconductors. A. A. Smirnov\({}^{61}\) developed a one-electron theory of metallic alloys taking into account the phenomenon of ordering in them. A. N. Orlov\({}^{62}\) extended it to the treatment of the elastic properties of alloys.

5. GENERAL MANY-ELECTRON THEORY OF CRYSTALS¹²

According to estimates, the energy of interelectronic interaction in crystals is not small in comparison with the “kinetic” energy of the electrons \((k\theta \sim 10^{-12}\) erg, \(\theta\) being the degeneracy temperature of the electron gas, \(\sim 10^4\,^\circ\mathrm{K})\), or with the energy of their interaction with the ions of the lattice \(\left(\dfrac{e^2}{a} \sim 10^{-20}:10^{-8} \sim 10^{-12}\right.\) erg, where \(a\) is the lattice constant). Therefore there are grounds for believing that, by neglecting this interaction, we deprive ourselves of the possibility of taking into account the most essential properties of real crystals. It is precisely for this reason that, despite the great successes of the one-electron theory, the problem of taking account of electronic interaction is an urgent task of the quantum theory of crystals.

Electronic interaction is sometimes treated as a small correction to the “zero” one-electron approximation and is taken into account by the ordinary method of perturbation theory. Such a treatment of the problem of the interaction of electrons in itself presents no difficulties. However, because this interaction is comparable even with the “kinetic” energy at the Fermi surface of the “zero” problem and, moreover, the latter has a continuous energy spectrum, it remains wholly unclear what is to play the role of the “small parameter.”

Attempts have also been made to justify the neglect of electronic interaction by reference to the fact that, for the properties of crystals, an essential role is played only by electrons in a narrow energy band—near the Fermi surface¹³. The small number of these “active” electrons supposedly allows their interaction to be regarded as small simply because of the large average distance between them. Even if one accepts this point of view literally, it has by no means been proved that electronic interaction in such a “rarefied” gas cannot exert a substantial influence on the physical properties of the crystal. Still less is there sufficient reason to believe that, when electronic interaction is taken into account consistently, it is in general legitimate to speak of the Fermi surface in the sense of the one-electron model.

Sometimes electronic interaction is taken into account as the cause of transitions of the electrons of a crystal from some stationary “zero” states into others¹⁴. In such a formulation the assumption of the weakness of the interaction is introduced in an even more rigid form, namely: it is assumed that the interaction is so small that it affects only the kinetics of the irreversible processes occurring in the crystal. Therefore there is a danger of obtaining conclusions which in no approximation follow from the exact equations of the problem.

It has already been indicated above that there have been attempts to solve the many-electron problem in the one-electron approximation by introducing a “self-consistent

field” (see Section 4). In view of the crude approximations which usually accompany the introduction of a self-consistent field, there is no reason to regard this method even as an approximate account of the electronic interaction.

The ideal would be to construct a theory which would not depend on the particular form of the electronic interaction. However, the construction of such a theory is apparently connected not only with substantial “technical” difficulties, but the very general formulation of the problem itself has no meaning, i.e. a system of interacting electrons in a lattice in fact does not possess such general properties as would be independent of the form of the interaction. Nevertheless, it is possible to establish a number of properties of the system of electrons in a crystal which are insensitive to the particular form of the interaction. Namely, it proves possible: 1) to express in the language of the general theory the fact that the electrons are not localized in an ideal lattice, and 2) to take into account the accelerating action of an external electric field.

The most convenient method for solving these questions is the density-matrix method, which we have already used in Section 4. However, in the case of a system of interacting electrons, the energy operator, along with additive terms \(\hat H_0[\mathbf q_i]\), also has binary terms \(\hat V[\mathbf q_i \mathbf q_j]\) of pair interaction, i.e.

\[ \hat H[\mathbf q_1\ldots \mathbf q_n] = \sum_{i=1}^{n}\hat H_0[\mathbf q_i] + \frac{1}{2} \sum_{i\ne j=1}^{n} \hat V[\mathbf q_i \mathbf q_j]. \tag{5,1} \]

Therefore the mean energy of the system in the present case will be equal to

\[ \bar H = \operatorname{Sp}\left(\hat \rho \hat H_0\right) + \operatorname{Sp}\left(\hat \sigma \hat V\right), \tag{5,2} \]

where

\[ \sigma[\mathbf q_i \mathbf q_j] = \frac{n(n-1)}{2} \operatorname{Sp}_{(\mathbf q_i,\mathbf q_j)} \hat U[\mathbf q_1\ldots \mathbf q_i\ldots \mathbf q_j\ldots \mathbf q_n] \tag{5,3} \]

is the reduced density matrix for a pair of electrons. (The indices in parentheses at the symbol \(\operatorname{Sp}\) mean that no integration is performed over the corresponding variables.) The relation between the density matrix of one electron and that of the whole system, according to (4,21), has the form

\[ \hat \rho[\mathbf q_i] = n\,\operatorname{Sp}_{(\mathbf q_i)} \hat U[\mathbf q_1\ldots \mathbf q_i\ldots \mathbf q_n]. \tag{5,4} \]

Using (5,3), (5,4) and recalling that the equation of motion for the density operator of the whole system \(\hat U\) has the form

\[ i\hbar \dot{\hat U}=\hat H\hat U-\hat U\hat H, \]

for the equation of motion of the operator \(\hat{\rho}\) we obtain

\[ i\hbar \dot{\hat{\rho}}=\hat{H}_0\hat{\rho}-\hat{\rho}\hat{H}_0+\operatorname{Sp}_{q}\bigl(\hat{V}\hat{\sigma}-\hat{\sigma}\hat{V}\bigr); \tag{5,5} \]

in contrast to the one-electron theory, in (5,5), along with \(\hat{\rho}\), the matrix \(\hat{\sigma}\) also enters.

The mathematical expression of the periodicity of the crystal lattice is the equality

\[ \left(q_1' + a_l,\ldots,q_n' + a_l\left|\hat{H}\right|q_1'' + a_l,\ldots,q_n'' + a_l\right)= \]

\[ =\left(q_1'\ldots q_n'\left|\hat{H}\right|q_1''\ldots q_n''\right), \tag{5,6} \]

where \(a_l\) is one of the enumerated lattice vectors. From this condition of translational invariance it follows at once, as in Section 4, that the wave functions of a system of interacting electrons in a crystal possess the property

\[ \psi_m(q_1+a_l,\ldots,q_n+a_l)=e^{i(\omega\cdot a_l)}\psi_m(q_1,\ldots,q_n). \tag{5,7} \]

At first sight it appears that in this way no more detailed information about the properties of the electron system can be obtained.

However, if one passes to the matrix \(\hat{\rho}\), then a number of essential conclusions can immediately be drawn. Substitution of (5,7) into (4,21) immediately gives

\[ (q'+a_l|\rho|q''+a_l)=\left(q'\left|\hat{\rho}\right|q''\right). \tag{5,8} \]

It follows directly from this that the eigenfunctions of the density operator \(\hat{\rho}\) in the most general theory of interacting electrons in a crystal can be represented in the same form as the eigenfunctions of the energy operator in the one-electron theory, i.e. in the form

\[ \chi(\xi,q)=e^{i\frac{\xi q}{a}}\varphi(\xi,q), \tag{5,9} \]

where the function \(\chi\) is periodic in \(q\) with the period of the lattice, and the possible values of the “quasimomentum” \(\xi\) are determined from the usual periodicity conditions. In deriving (5,9) it is assumed that the quantity \(\xi\) can serve as a unique characteristic of each eigenstate of the operator \(\hat{\rho}\). Thus the functions \(\chi\) can be grouped into zones in such a way that within each zone \(\chi\) is uniquely determined by specifying the parameter \(\xi\), and the functions \(\chi\) corresponding to two infinitely close values of \(\xi\) differ infinitesimally from one another. Each zone may be assigned its own

region of variation of $\xi$ and its own definite number $\zeta$. Therefore (5.9) may be written more precisely as follows:

\[ \chi(\xi,\zeta;q)=e^{\,i\frac{\xi q}{a}}\varphi(\xi,\zeta;q). \tag{5.10} \]

Formula (5.10) expresses a very important fact. To each stationary state of a system of interacting electrons in a crystal there corresponds a completely definite set of functions (5.10), just as was the case in Section 4 for noninteracting electrons of the one-electron model. The fundamental difference is that in the one-electron model the functions (5.10), being eigenfunctions of the operator $\hat{\rho}$, are at the same time also eigenfunctions of the energy operator of each individual electron $\hat{H}_0$; therefore their form is characteristic of the given crystal, whereas in the general theory of interacting electrons this is not so, and therefore the set of functions (5.10), generally speaking, is different for different states of the electron collective in the crystal. Nevertheless, despite this difference, the proved property of the matrix $\hat{\rho}$ expresses an essential physical fact, namely that even in the presence of an arbitrarily strong interaction the electrons are not localized in the lattice. This fact is connected precisely with the properties of the operator $\hat{\rho}$, and not of the energy operator $\hat{H}$. Indeed, the mean value of the additive operator of the total momentum of the system is equal to

\[ \overline{P}=\operatorname{Sp}\bigl(\hat{\rho}\hat{P}\bigr) =\sum_{\xi,\zeta}\rho_0(\xi,\zeta)\bigl(\xi,\zeta\bigm|\hat{P}\bigm|\xi,\zeta\bigr). \tag{5.11} \]

The quantity $\bigl(\xi\zeta\bigm|\hat{P}\bigm|\xi\zeta\bigr)$—the diagonal matrix element of the momentum operator of an individual electron in the system of functions (5.10)—is, as a rule, different from zero. Since, by virtue of the Pauli principle, the condition $0\leq\rho_0(\xi,\zeta)\leq 1$ holds, the total mean momentum of the system in the general case is also not equal to zero. It vanishes only in certain special cases—when the distribution of electrons (specified by the spectrum of the quantities $\rho_0(\xi,\zeta)$) is such that in the sum (5.11) the terms with positive $\bigl(\xi\zeta\bigm|\hat{P}\bigm|\xi\zeta\bigr)$ exactly compensate the terms with negative $\bigl(\xi\zeta\bigm|\hat{P}\bigm|\xi\zeta\bigr)$. Consequently, in an ideal lattice a system of interacting electrons possesses, as a rule, a current different from zero. Thus, the interaction between electrons does not change the basic property of the “transparency” of an ideal lattice for electrons.

Let us now consider the question of the accelerating action of a constant electric field \(F\), directed along the \(X\)-axis. Equation (4.24) for \(\hat{\rho}\) in this case, according to (5.5), will have the form

\[ i\hbar \dot{\hat{\rho}}=\hat{H}_0\hat{\rho}-\hat{\rho}\hat{H}_0 +\operatorname{Sp}_{(\mathbf q)}\left(\hat{V}\hat{\sigma}-\hat{\sigma}\hat{V}\right) +eF\left(\hat{X}\hat{\rho}-\hat{\rho}\hat{X}\right). \tag{5.12} \]

It can be shown, by analogy with Section 4, that, if transitions between different bands of the states of the operator \(\hat{\rho}\), and not of the operator \(\hat{H}\), are neglected, the solution of (5.12) has the form (4.30) and (4.31), while for times small in comparison with the period of circulation of the electron distribution over the band, we shall have the approximate solution (4.32). In other words, it can be shown that, up to terms linear in \(t\), the accelerating action of a constant electric field is determined in the general theory by the same formula as in the one-electron theory. It follows from (4.32) that, under the influence of a constant field, the distribution of electrons over bands, as a rule, changes. An exception occurs only when, in all states of the given band, \(\nabla_{\xi}\rho_0=0\), i.e., when the band is uniformly filled.

Simultaneously with the change in the value of \(\hat{\rho}\), the mean momentum of the system of electrons also changes, i.e., the electrons undergo acceleration. The increment of the mean momentum during the time \(t\) after the field is switched on, according to (4.34), is equal to

\[ \Delta \bar{P}=\frac{eFa}{\hbar}\,t\,\operatorname{Sp}\left(\hat{P}\nabla_{\xi}\rho_0\right). \tag{5.13} \]

From (5.13) it is seen that, in those states of the system in which part of the bands is uniformly filled and the remaining bands are empty, the electric field does not cause acceleration of the electrons. In particular, if the energetically lowest states of the crystal belong to such states, then it is, evidently, a dielectric or a semiconductor. In the opposite case, the constant field gives rise to the appearance of an electric current increasing with time and, consequently, the crystal will be metallic.

Apparently, there is no simple criterion that would make it possible to establish unambiguously when one of these cases is realized and when the other is. One can only assert that if the energetically lowest state of the crystal is isolated, i.e., if there is no state infinitely close to it, then the crystal does not conduct, since for an isolated state each band is either filled with maximum density or empty (otherwise, for this state there would be found an infinitely close one). However, the converse assertion, generally speaking, is false. Examples can be given (see below, Section 7) in which the lowest energy state corresponds to a uniformly

...to a uniform filling of the zones and, along with this, the energy spectrum is continuous. The question of how real and typical these examples are can apparently be decided only on the basis of energetic considerations. In the present state of the theory, it is hardly possible to make statements here of a general nature.

Integrating (5,13) by parts and calculating the current density, we obtain

\[ \overline{J} = \frac{e^{2} a\, \operatorname{Sp}' \left(\left\langle \hat{\rho}_{0}\nabla_{\xi}\hat{P}\right\rangle\right)t} {m\hbar} \,F . \tag{5,14} \]

Comparing this formula with the classical expression for the electrical conductivity (3,1), we arrive at the definition of the effective mass in the state \((\xi,\zeta)\):

\[ m^{*} = m\, \frac{n\hbar} {a\,\operatorname{Sp}'\left(\left\langle \hat{\rho}_{0}\nabla_{\xi}\hat{P}\right\rangle\right)} \tag{5,15} \]

or of the effective number of conduction electrons of the crystal:

\[ n^{*} = \frac{a}{\hbar}\, \operatorname{Sp}'\left(\left\langle \hat{\rho}_{0}\nabla_{\xi}\hat{P}\right\rangle\right). \tag{5,15'} \]

(In contrast to the conclusions of Section 4, the state \((\xi,\zeta)\) is now not always an eigenstate of the energy of the crystal.) It can also be shown in the general theory that the dielectric constant is just as simply related to \(n^{*}\) or \(m^{*}\), as was shown in Section 4 (see (4,39) and (4,40)).

From the examples presented it is clear that the general theory gives conclusions just as definite as the one-electron theory. Only in the general theory are these conclusions not connected with any special assumptions about the behavior of the electron system in the crystal (for example, about the smallness of the interaction energy of the electrons or with the fact that this interaction can be described by the self-consistent-field method, etc.). The one-electron model likewise gives only the order of magnitude of \(n^{*}\). The fact that \(n^{*}\) for a given substance is indeed practically constant (and does not depend, for example, on temperature), in the most general form follows from the fact that in experiment there is almost always realized such a distribution \(\hat{\rho}(\xi,\zeta)\) which corresponds to a minimum of the total energy of the crystal (a strong statement). It must be emphasized that the present state of the theory does not make it possible to give an expression for \(n^{*}\) in terms of the atomic number of the element and universal constants, simply because of the extreme complexity of the problem. However, the general qualitative content of the conclusions obtained above is a positive result of the theory.

For obtaining detailed results, more detailed information is needed on the character of the energy spectrum of electrons in crystals. In this case, at least in the present state of the theory, the use of one or another approximation is unavoidable.

The conclusions obtained are at the same time an answer to the question posed at the end of Section 4: why the one-electron model gives a satisfactory explanation of a number of phenomena in crystals. The many-electron theory preserves, in form, the results of band theory, but deepens and broadens its content.

6. THE METHOD OF SECOND QUANTIZATION IN APPLICATION TO THE CRYSTAL PROBLEM2

In constructing a model many-electron theory, we shall again turn to the original quantum-mechanical equation (1.5) for determining stationary states with the energy operator \(\hat H\) in the form (4.1). To solve this equation exactly, as has already been indicated, is practically impossible because of mathematical difficulties. Therefore one has to use approximate methods for its solution. However, we shall choose such methods as do not destroy the many-electron character of the problem under investigation. First of all, here one should pass to another representation, choosing, instead of the usual spatial and spin coordinates, other dynamical variables—the generalized coordinates of the system \(\alpha_1,\ldots,\alpha_n\). For the time being we shall not specify the character of these “coordinates.” We shall seek the solution of (1.5), with the energy operator (4.1), in the form of a series in the complete system of orthonormal functions \(\psi_{\alpha_1\ldots \alpha_n}(\mathbf q_1\ldots \mathbf q_n)\), for which the chosen generalized coordinates are quantum numbers, while \(\mathbf q_i\) is the aggregate of the spatial and spin coordinates of the \(i\)-th electron. Thus, the series will have the form

\[ \Psi_{\mathrm{el}}(\mathbf q_1\ldots \mathbf q_n) = \sum_{\alpha_1,\ldots,\alpha_n} a(\alpha_1,\ldots,\alpha_n) \psi_{\alpha_1\ldots \alpha_n}(\mathbf q_1\ldots \mathbf q_n); \tag{6.1} \]

the coefficients \(a(\alpha_1,\ldots,\alpha_n)\) give the wave function of the system in the \(\alpha\)-representation. By standard methods, instead of (1.5) we obtain a system of equations for the amplitudes \(a\):

\[ E a(\alpha_1\ldots \alpha_n) = \sum_{\alpha'_1\ldots \alpha'_n} (\alpha_1,\ldots,\alpha_n|\hat H|\alpha'_1,\ldots,\alpha'_n) a(\alpha'_1,\ldots,\alpha'_n). \tag{6.2} \]

Usually the function \(\psi_{\alpha_1\ldots \alpha_n}(\mathbf q_1,\ldots,\mathbf q_n)\) is chosen in the form of a product—

of individual wave functions \(\varphi_a(\mathbf q)\) of the separate particles; here the quantum numbers \(a\) denote the set of separate numbers for the individual states. Taking into account that, for a system of electrons, this product must be antisymmetric with respect to the interchange of each pair of coordinates, we shall have, for \(\psi_{a_1\ldots a_n}\):

\[ \psi_{a_1\ldots a_n}(\mathbf q_1\ldots \mathbf q_n) = \sum_{(P)}(-1)^P \varphi_{a_1}(\mathbf q_1)\ldots \varphi_{a_n}(\mathbf q_n'), \tag{6,2'} \]

where \(P\) denotes the symbol for the permutation of the coordinates of a pair of electrons. It follows from \((6,2')\) that if at least two functions \(\varphi_{a_i}\) in the product are identical, then \((6,2')\) is equal to zero; therefore among the indices \(a_i\) there cannot be a single pair of equal ones. Moreover, not all functions \((6,2)\) will be linearly independent, since a permutation of the same indices only adds a factor \(\pm 1\). To take this into account, when summing in \((6,1)\) one must establish a definite order of succession of the indices, namely:

\[ \Psi_{\mathrm{el}}(\mathbf q_1\ldots \mathbf q_n) = \sum_{\substack{a_1\ldots a_n\\(a_1<a_2<\ldots<a_n)}} a(a_1\ldots a_n) \psi_{a_1\ldots a_n}(\mathbf q_1\ldots \mathbf q_n). \tag{6,3} \]

In order to avoid certain inconveniences of the representation \((a_1\ldots a_n)\), let us pass to the representation of second quantization, in which the role of the coordinates of the system is played by the occupation numbers of the individual states \(n_{a_1},\ldots,n_{a_n}\). In the present case \(n_a\) is either equal to zero, if \(a\) is not equal to any of the numbers \(a_1,\ldots,a_n\), or equal to unity, if \(a\) coincides with one of them. \(n_a\) can have no other values (the Pauli principle). Then we have

\[ n_{a_i}=0,1;\qquad \sum_{(a_i)} n_{a_i}=n. \tag{6,4} \]

To each possible system of occupation numbers there corresponds only one function from the system \(\psi_{a_1\ldots a_n}(\mathbf q_1\ldots \mathbf q_n)\). Introducing a new numbering, we shall have, instead of \((6,3)\),

\[ \Psi_{\mathrm{el}}(\mathbf q_1\ldots \mathbf q_n) = \sum_{(\ldots n_\alpha\ldots)} a(\ldots n_\alpha\ldots) \Psi_{\ldots n_\alpha\ldots}(\mathbf q_1\ldots \mathbf q_n), \tag{6,5} \]

where

\[ \Psi_{\ldots n_\alpha\ldots}(\mathbf q_1\ldots \mathbf q_n) = \frac{1}{\sqrt{n!}}\, \psi_{\ldots n_\alpha\ldots}(\mathbf q_1\ldots \mathbf q_n) \tag{6,6} \]

is a system of orthonormal functions. In addition, we have

\[ \sum_{(\ldots n_\alpha\ldots)} \left|a(\ldots n_\alpha\ldots)\right|^2=1, \tag{6,7} \]

therefore the square of the modulus of the wave function of secondary quantization may be interpreted as the probability of occurrence, for the given system, of the occupation numbers of the individual particle states.

Let us now proceed to the definition of the operators of dynamical quantities of the system in the secondary-quantization representation. The wave equation of the system takes the form

\[ \left(\hat H_a - E\right)a(\ldots n_a \ldots)=0, \tag{6,8} \]

and the mean value of any operator can be determined from the formula

\[ \bar A = \sum_{(\ldots n_\alpha \ldots)} a^*(\ldots n_\alpha \ldots)\,\hat A_a\,a(\ldots n_\alpha \ldots); \tag{6,9} \]

here the energy operator \(\hat H_a\) and the operator \(\hat A_a\) are expressed in the secondary-quantization representation. In order to do this, for example for an arbitrary operator \(\hat A\), it is necessary to be able to compute matrix elements of the type

\[ (\ldots n_\alpha \ldots|\hat A|\ldots n'_\alpha \ldots)= \]

\[ = \int \Psi^*_{\ldots n_\alpha \ldots}(q_1\ldots q_n) \hat A \Psi_{\ldots n'_\alpha \ldots}(q_1\ldots q_n)\, dq_1\ldots dq_n. \tag{6,10} \]

In the general case, the deciphering of formula (6,10) is very complicated, but if one confines oneself to the case of additive and binary operators (by virtue of (4,1), these are the only operators of interest to us), the computations present no difficulty. For the purpose at hand one introduces the auxiliary operator matrices of secondary quantization

\[ \hat a_{x_1,\alpha_1},\qquad \hat a_{\alpha_1\alpha_2;\,x_1x_2}^{\prime\prime}, \tag{6,11} \]

which act on the wave functions of secondary quantization. We define the matrix elements of these operators in the chosen representation as follows:

\[ \left( \ldots n_\alpha \ldots \left| \hat a_{x_1,\alpha_1} \right| \ldots n'_\alpha \ldots \right) = \int \varphi^*_{x_1}(q'_1)\, \varphi_{\alpha_1}(q_1)\, \Psi^*_{\ldots n_\alpha \ldots}(q_1q_2\ldots q_n) \times \]

\[ \times \Psi_{\ldots n'_\alpha \ldots}(q'_1q_2\ldots q_n)\, dq'_1\,dq_1\ldots dq_n; \tag{6,12} \]

\[ \left( \ldots n_\alpha \ldots \left| \hat a_{x_1x_2;\,\alpha_1\alpha_2}^{\prime\prime} \right| \ldots n'_\alpha \ldots \right) = \]

\[ = \int \varphi^*_{\alpha_1}(q'_1)\, \varphi^*_{\alpha_2}(q'_2)\, \varphi_{\alpha_1}(q_1)\, \varphi_{\alpha_2}(q_2)\, \Psi^*_{\ldots n_\alpha \ldots}(q_1q_2\ldots q_n) \times \]

\[ \times \Psi_{\ldots n'_\alpha \ldots}(q'_1q'_2q_3\ldots q_n)\, dq'_1\,dq'_2\,dq_1\,dq_2\ldots dq_n. \tag{6,13} \]

Let us first consider the additive operator \(\hat A=\sum_{r=1}^{n}\hat A_r\); for it we shall have (using the identical character of the terms of this operator)

\[ \begin{aligned} \left(\ldots n_\alpha\ldots \left|\hat A\right|\ldots n'_\alpha\ldots\right) &=\sum_{r=1}^{n}\int \Psi^{*}_{\ldots n_\alpha\ldots}\hat A_r \Psi_{\ldots n'_\alpha\ldots}\,dq_1\ldots dq_n \\ &= n\int \Psi^{*}_{\ldots n_\alpha\ldots}\hat A \Psi_{\ldots n'_\alpha\ldots}\,dq_1\ldots dq_n . \end{aligned} \tag{6,14} \]

Recalling the expression for the matrix element in the ordinary representation

\[ \left(\alpha_1\left|\hat A_1\right|\alpha'_1\right) = \int \varphi_{\alpha_1}^{*}(q_1)\hat A_1 \varphi_{\alpha'_1}(q_1)\,dq_1, \]

it is easy to show (see \(^{15}\)) that (6,14) takes the form

\[ \begin{aligned} \left(\ldots n_\alpha\ldots \left|\hat A\right|\ldots n'_\alpha\ldots\right) &= n\sum_{\alpha_1,\alpha'_1} \left(\alpha_1\left|\hat A_1\right|\alpha'_1\right) \int \Psi^{*}_{\ldots n_\alpha\ldots}(q_1q_2\ldots q_n) \\ &\quad \times \varphi_{\alpha_1}(q_1)\varphi^{*}_{\alpha'_1}(q'_1) \Psi_{\ldots n'_\alpha\ldots}(q'_1q_2\ldots q_n)\, dq'_1\,dq_1\ldots dq_n . \end{aligned} \tag{6,15} \]

or, by virtue of (6,12), one may write

\[ \begin{aligned} \left(\ldots n_\alpha\ldots \left|\hat A\right|\ldots n'_\alpha\ldots\right) &= n\sum_{\alpha_1,\alpha'_1} \left(\alpha_1\left|\hat A_1\right|\alpha'_1\right) \left( \ldots n_\alpha\ldots \left|\hat a_{\alpha'_1,\alpha_1}\right| \ldots n'_\alpha\ldots \right). \end{aligned} \]

Thus, in the representation of second quantization the additive dynamical quantity has the form

\[ \hat A_a = n\sum_{(\alpha_1,\alpha'_1)} \left(\alpha_1\left|\hat A_1\right|\alpha'_1\right) \hat a_{\alpha'_1,\alpha_1}. \tag{6,16} \]

In exactly the same way, for the binary operator we have

\[ \sum_{r_1,r_2=1}^{n}\hat A_{r_1,r_2} = \frac{n(n-1)}{2} \sum_{(\alpha_1,\alpha_2;\,\alpha'_1,\alpha'_2)} \left(\alpha_1\alpha_2\left|\hat A_{12}\right|\alpha'_1\alpha'_2\right) \hat a_{\alpha'_1\alpha'_2;\,\alpha_1\alpha_2}, \tag{6,17} \]

where

\[ \left(\alpha_1\alpha_2\left|\hat A_{12}\right|\alpha'_1\alpha'_2\right) = \int \varphi^{*}_{\alpha_1}(q_1) \varphi^{*}_{\alpha_2}(q_2) \hat A_{12} \varphi_{\alpha'_1}(q_1) \varphi_{\alpha'_2}(q_2)\,dq_1\,dq_2 \]

are the matrix elements of the operator \(\hat A_{12}\) in the ordinary representation.

It can be shown (see \(^{15}\)) that the operator matrices \(\hat a_{\alpha_1\alpha'_1}\) are closely connected with the density matrices introduced above; namely, the density matrices \(\hat\sigma\) and \(\hat\rho\) are averages of the matrices \(\hat a_{\alpha_1\alpha_2;\,\alpha'_1\alpha'_2}\) and \(\hat a_{\alpha_1\alpha'_1}\) with respect to the variables of second quantization.

It is also easy to establish (see \(^{15}\)) the connection between the binary and the additive operator matrix. This connection has the form

\[ n(n-1)\hat a'_{\alpha_1\alpha_2;\,\alpha'_1\alpha'_2} = n^2 \hat a'_{\alpha_1\alpha'_1}\hat a'_{\alpha_2\alpha'_2} - n\delta(\alpha'_1-\alpha_2)\hat a'_{\alpha_2\alpha'_1}. \tag{6,18} \]

Finally, using the symmetry property of the operator

\[ \hat a_{\alpha_1\alpha_2;\,\alpha'_1\alpha'_2} = \hat a_{\alpha_2\alpha_1;\,\alpha'_2\alpha'_1}, \tag{6,19} \]

which follows directly from definition (6,13), we find the permutation relations of these operator matrices:

\[ n^2\left[ \hat a'_{\alpha_2\alpha'_2}\hat a'_{\alpha\alpha_1} - \hat a'_{\alpha\alpha_1}\hat a'_{\alpha_2\alpha'_2} \right] = \]

\[ = n\left[ \delta(\alpha'_1-\alpha_2)\hat a'_{\alpha\alpha_1} - \delta(\alpha'_2-\alpha_1)\hat a'_{\alpha\alpha_2} \right]. \tag{6,20} \]

Up to now no restriction has been introduced that our system must necessarily be described by antisymmetric functions. Let us now clarify what the operator matrices will look like in this particular case. For this purpose, in (6,12) we substitute the functions (6,6), which have the form of the well-known determinant. In order to determine the operator matrices, we expand this determinant in the elements of its first column. This gives:

\[ \psi_{\ldots n_\alpha\ldots}(q_1q_2\ldots q_n) = \sum_{r=1}^{n}(-1)^{r-1}\varphi_{\alpha_r}(q_1)\Phi_r(q_2\ldots q_n), \tag{6,21} \]

where \(\Phi_r(q_2\ldots q_n)\) is the corresponding minor obtained by deleting in (6,6) the first column and the row containing the function \(\varphi(q_1)\). The sum (6,21), which gives the expansion of (6,6) in the functions \(\varphi_g(q)\) with \(g=\alpha_1\ldots\alpha_n\) from the given set of these numbers, may formally be written as a sum over all numbers, without restriction to the given set, if only the multiplier \(n_g\) is introduced, for \(n_g=1\) for \(g\) belonging to the given set, and \(n_g=0\) for \(g\ne \alpha_1\ldots\alpha_n\), i.e. not belonging to the given set. Thus, instead of (6,21) we obtain

\[ \sqrt{n!}\,\Psi_{\ldots n_g\ldots}(q_1q_2\ldots q_n) = \]

\[ = \sum_{(g)} (-1)^{\sum_{\alpha<g} n_\alpha} \,n_g\varphi_g(q_1)\, \Psi_{\ldots \beta_g n_\alpha\ldots}(q_2\ldots q_n), \tag{6,22} \]

where it has been used that

\[ \Phi_r(\mathbf q_2 \ldots \mathbf q_n)= \psi_{\ldots n_\alpha \ldots}(\mathbf q_2 \ldots \mathbf q_n) =\sqrt{(n-1)!}\,\Psi_{\ldots n_\alpha \ldots}(\mathbf q_2 \ldots \mathbf q_n), \qquad n'_\alpha=n_\alpha-\delta(\alpha-\alpha_2) \tag{6.23} \]

and the notation has been introduced

\[ \beta_g n_\alpha=n_\alpha-\delta(\alpha-g). \tag{6.24} \]

Substituting (6.22) into (6.12), we shall have

\[ n\left(\ldots n_\alpha \ldots \left| \hat a_{\alpha_1\alpha'_1} \right| \ldots n_{\alpha'} \ldots\right) = \]

\[ = \sum_{g,g'} (-1)^{\sum_{\alpha<g}n_\alpha} (-1)^{\sum_{\alpha<g'}n_{\alpha'}} \,n_g n_{g'}\, \int \varphi^*_{\alpha_1}(\mathbf q'_1)\varphi_{g'}(\mathbf q'_1)\,d\mathbf q'_1 \times \]

\[ \times \int \varphi_{\alpha'_1}(\mathbf q_1)\varphi^*_{g}(\mathbf q_1)\,d\mathbf q_1\, \Psi^*_{\ldots \beta_g n_\alpha \ldots}(\mathbf q_2\ldots \mathbf q_n) \times \]

\[ \times \Psi_{\ldots \beta'_{g'} n'_\alpha \ldots}(\mathbf q_2\ldots \mathbf q_n) \,d\mathbf q_2\ldots d\mathbf q_n . \tag{6.25} \]

As a result of simple calculations (see \(^{15}\)) we find that the action of the operator \(n\hat a_{\alpha_1\alpha'_1}\) on an arbitrary wave function \(a(\ldots n_\alpha\ldots)\) gives

\[ n\left(\hat a_{\alpha_1\alpha'_1}a\right)= \]

\[ = (-1)^{\sum_{\alpha<\alpha_1}n_\alpha} \delta(1-n_{\alpha'_1}) (-1)^{\sum_{\alpha<\alpha'_1}\left(n_\alpha-\delta(\alpha-\alpha'_1)\right)} \delta\!\left(n_{\alpha_1}-\delta(\alpha'-\alpha_1)\right) \times \]

\[ \times a\{\ldots n_\alpha+\delta(\alpha-\alpha_1)-\delta(\alpha-\alpha'_1)\ldots\}. \tag{6.26} \]

From (6.26) it is seen that this action reduces to the following operations: first, we replace the arguments \(n_\alpha\) of the wave function by

\[ n_\alpha+\delta(\alpha-\alpha_1) \]

and multiply it by

\[ (-1)^{\sum_{\alpha<\alpha_1} n_\alpha}\delta(n_{\alpha_1}); \]

second, in the function so obtained we again replace the arguments \(n_\alpha\) by \(n_\alpha-\delta(\alpha-\alpha'_1)\) and multiply by

\[ (-1)^{\sum_{\alpha<\alpha'_1} n_\alpha}\delta(1-n_{\alpha'_1}). \]

To simplify the notation in (6.26), let us introduce displacement operators by the formulas

\[ (\beta_\alpha F)_{n_\alpha}=\delta(n_\alpha)F(n_\alpha+1), \tag{6.27} \]

\[ (\beta^+_\alpha F)_{n_\alpha}=\delta(1-n_\alpha)F(n_\alpha-1). \tag{6.28} \]

It is useful to introduce one more type of operators—the so-called Fermi amplitudes:

\[ \hat a_f=(-1)^{\sum_{g<\alpha} n_g}\hat\beta_\alpha =\hat\beta_\alpha(-1)^{\sum_{g<\alpha} n_g}, \tag{6.29} \]

\[ \hat a_f^{*}=\hat\beta_\alpha^{*}(-1)^{\sum_{g<\alpha} n_g} =(-1)^{\sum_{g<\alpha} n_g} =(-1)^{\sum_{g<\alpha} n_g}\hat\beta_\alpha^{*}. \tag{6.30} \]

Then, instead of the operator \(n\hat a_{\alpha_1,\alpha'_1}\), from (6.26) we shall have

\[ n\hat a_{\alpha_1,\alpha'_1}=\hat a^{+}_{\alpha'_1}\hat a_{\alpha_1}. \tag{6.31} \]

It is easy to establish, on the basis of the equalities (6.27)—(6.31), the commutation relations for the Fermi amplitudes

\[ \hat a_\alpha^{+}\hat a_{\alpha'}+\hat a_{\alpha'}\hat a_\alpha^{+} =\delta(\alpha-\alpha') \tag{6.32} \]

and their relation to the occupation numbers

\[ \hat a_\alpha^{+}\hat a_\alpha=n_\alpha. \tag{6.33} \]

The matrix definition of the Fermi amplitudes, by virtue of (6.29) and (6.30), has the form

\[ (\ldots n_\alpha\ldots|\hat a_{\alpha_1}|\ldots n'_\alpha\ldots) = \]

\[ =(-1)^{\sum_{g<\alpha_1} n_g}\delta(n_{\alpha_1}) \prod_{(\alpha)} \delta\bigl(n_\alpha-n'_\alpha+\delta(\alpha-\alpha_1)\bigr). \tag{6.34} \]

For a binary operator matrix, by virtue of (6.18) and (6.31), we have

\[ n(n-1)\hat a_{\alpha_1\alpha_2,\alpha'_1\alpha'_2} = \hat a^{+}_{\alpha'_1}\hat a^{+}_{\alpha'_2} \hat a_{\alpha_2}\hat a_{\alpha_1}. \tag{6.35} \]

The second-quantization representation for additive and binary operators will, according to (6.16), (6.17), (6.31), and (6.35), have the form

\[ \sum_{r=1}^{n}\hat A_r = \sum_{(\alpha_1,\alpha'_1)} \langle \alpha_1|\hat A_1|\alpha'_1\rangle \hat a^{+}_{\alpha_1}\hat a_{\alpha'_1}, \tag{6.36} \]

\[ \sum_{r_1,r_2=1}^{n}\hat A_{r_1,r_2} = \frac{1}{2!} \sum_{(\alpha_1\alpha_2;\,\alpha'_1\alpha'_2)} \langle \alpha_1\alpha_2|\hat A_{12}|\alpha'_1\alpha'_2\rangle \hat a^{+}_{\alpha_1}\hat a^{+}_{\alpha_2} \hat a_{\alpha'_2}\hat a_{\alpha'_1}. \tag{6.37} \]

In particular, for the energy operator of a system of interacting electrons (4.1), we obtain, in the representation of second quan-

obtaining the following expression:

\[ \hat H_a=\sum_{\alpha_1\alpha'}\int \varphi_\alpha^{*}(\mathbf q_1)\hat H_0\varphi_{\alpha'}(\mathbf q_1)d\mathbf q_1\,\hat a_\alpha^{+}\hat a_{\alpha'} \]

\[ +\frac{1}{2}\sum_{\left(\substack{\alpha_1\alpha_2\\ \alpha'_1\alpha'_2}\right)} \int \varphi_{\alpha_1}^{*}(\mathbf q_1)\varphi_{\alpha_2}^{*}(\mathbf q_2)\hat V_{12} \varphi_{\alpha'_1}(\mathbf q_1)\varphi_{\alpha'_2}(\mathbf q_2)d\mathbf q_1d\mathbf q_2\, \hat a_{\alpha_1}^{+}\hat a_{\alpha_2}^{+}\hat a_{\alpha'_2}\hat a_{\alpha'_1}. \tag{6.38} \]

Formula (6.38) should serve as the starting point for constructing concrete many-electron models of a crystal.

7. EXCHANGE, POLAR, AND EXCITON MANY-ELECTRON MODELS OF A CRYSTAL[^15][^19][^20][^21][^22][^23][^24]

1. To find the exact solution of equations (6.2), or to determine the eigenvalues of the energy operator (6.38) in the representation of second quantization, without solving these equations, is in practice almost as difficult as solving exactly the original equation (1.5). Therefore one has to resort to approximate methods. In choosing an approximation, difficulties arise because in the problem under consideration it is often unclear how to single out a small parameter that would make it possible to develop the standard scheme of calculation by the method of small perturbations. Ultimately this difficulty is due to the fact that the interaction between electrons is not small. By means of a special analysis of the energy operator of the system it is sometimes possible to isolate in it a small additional term that does allow the application of perturbation theory. (This “small addition” does not coincide with the energy of interaction between the electrons.)

The most stringent approximation made in all many-electron models is that, in the expansion (6.1), one uses an incomplete system of orthonormalized functions \(\psi_{\alpha_1\ldots \alpha_n}^{X}(\mathbf q_1\ldots \mathbf q_n)\), artificially cuts it off, and treats such a “cut-off” expansion as complete. Here one may proceed in two ways: either use, as the individual functions in the antisymmetrized products (6.2), atomic functions and correlate the set of indices \(\alpha_1\ldots \alpha_n\) with the indices of individual sites of the crystal lattice; or choose, as these functions, the wave functions of electrons in the lattice (4.15) from one-electron theory. If these two approaches are compared with approximate methods for solving problems in the quantum mechanics of molecules, then the first of them corresponds to the method of atomic orbitals, and the second to the method of molecular orbitals.

Let us dwell on the method of atomic functions. In this case (6.2) has the form

\[ \psi_f=\psi_{\alpha_1\ldots \alpha_n} =\sum_p(-1)^p P\,\varphi_{\alpha_1}(\mathbf r_1s_1)\ldots \varphi_{\alpha_n}(\mathbf r_ns_n), \tag{7.1} \]

where the indices \(\alpha_i\) include the number (vector) of the lattice site \(f\) (see 4.4), the orbital \(l\) and magnetic \(m\) quantum numbers of the electron, as well as the spin quantum number \(\sigma\); \(\mathbf r_i\) are the spatial coordinates and \(s_i\) the spin coordinates of the \(i\)-th electron. Truncating the expansion (6.1) will in the present case amount to taking, as the functions \(\varphi_{\alpha_i}(\mathbf r_i s_i)\), the wave functions of the valence electron of the atom in the normal state, i.e. for the lowest energy level, whereas for a complete system of functions it would be necessary to take into account also the functions of all excited states, including the continuous part of the atomic energy spectrum. In accordance with the criterion for the applicability of ordinary perturbation theory, such neglect of the excited states in the expansion (6.1) can have a certain justification only in the case when the “splitting” of the atomic energy levels in the crystal is small in comparison with the distance between the lowest energy level in the atom and the first excited level. In the opposite case it is necessary, along with the functions of the lowest state, to include also the functions of excited states. Taking excited states into account sharply complicates the calculation. Therefore it is first of all necessary to investigate in full detail the simplest case, in which only the functions of the energetically lowest atomic state are taken into account.

In the expansion (6.1) in the wave functions (7.1), all products of atomic functions allowed by the Pauli principle must be taken into account. Therefore (6.1) includes also such terms in which there are several functions \(\varphi_{\alpha_i}(\mathbf r_i s_i)\) with identical \((f,l,m)\) and different \(\sigma\). These terms correspond to “polar states,” in which several electrons may simultaneously be located at one and the same lattice site. In the first version of the many-electron model of a crystal, proposed by Frenkel\(^{25}\) and then by Heisenberg\(^{26}\), which was a generalization of the Heitler–London method in the problem of the hydrogen molecule\(^{27}\), only “homeopolar states” were taken into account, i.e. in the sum of equation (6.1) only the terms with different \((f,l,m)_i\) remained. Thus, in this so-called “exchange” approximation, at each lattice site of the crystal, as in an isolated atom, there is always one valence electron\(*\).

Thus, the wave function of the system in the chosen “polar” approximation will have the form

\[ \Psi(\mathbf r_1 s_1,\mathbf r_2 s_2,\ldots,\mathbf r_n s_n)= \]

\[ = \sum_{\alpha_1\alpha_2\ldots\alpha_n} a(\alpha_1\ldots\alpha_n)\Psi_{\alpha_1\ldots\alpha_n}(\mathbf r_1 s_1,\ldots,\mathbf r_n s_n), \tag{7,2} \]

\[ \alpha_1\alpha_2\ldots\alpha_n \]

\(*\) The spin quantum number \(\sigma_i\), in this case, of course, may be either the same or different.

the wave functions of the new representation are determined from the secular equations

\[ \sum_{s_1\ldots s_n}\int \Psi_{\alpha_1\ldots \alpha_n}^{*}(r_1s_1\ldots r_ns_n) [\hat H-E]\Psi(r_1s_1\ldots r_ns_n)\,dr_1\ldots dr_n, \tag{7,3} \]

where summation is carried out over the spin coordinates. We note that the wave functions \(\varphi_{a_i}\) with different node numbers are not exactly orthogonal; therefore the products formed from them are not exactly orthogonal either. In the general case we have

\[ \sum_{(s)}\int \varphi_{a_i}(rs)\varphi_{a_j}(rs)\,dr = \delta_{a_i a_j}+\varepsilon S(a_i a_j), \tag{7,4} \]

where \(S(a_i a_j)\) is the nonorthogonality integral, and \(\varepsilon\) is a small dimensionless parameter. The greater the distance between ions, the less the wave functions in the integral (7,4) “overlap,” and, consequently, the smaller \(\varepsilon S(a_i a_j)\) is. The incomplete orthogonality of atomic wave functions makes it impossible to regard integrals of the type

\[ \int \varphi_{\alpha_1}^{*}\ldots \varphi_{\alpha_n}^{*} \varphi_{\alpha_1'}\ldots \varphi_{\alpha_n'}\,dr_1\ldots dr_n, \]

\[ \int \varphi_{\alpha_1}^{*}\ldots \varphi_{\alpha_n}^{*}\hat H \varphi_{\alpha_1'}\ldots \varphi_{\alpha_n'}\,dr_1\ldots dr_n, \]

which have to be evaluated in solving the secular equations (7,3), as equal to zero every time the system of functions \(\varphi_{\alpha_1}\ldots \varphi_{\alpha_n}\) differs from the system \(\varphi_{\alpha_1'}\ldots \varphi_{\alpha_n'}'\) by no fewer than one, two, or three functions; this leads to very great complications in the calculations. In order to circumvent these complications in solving the problem, some integrals with overlapping functions are retained, while others, as small, are discarded without proper justification. As N. N. Bogolyubov has shown \(^{15}\), one can always avoid this difficulty and choose orthogonal functions \(\theta_\alpha\), which are linear combinations of atomic functions,

\[ \theta_\alpha=\sum_\beta U_{\alpha\beta}\varphi_\beta, \tag{7,5} \]

whereupon the sought function (6,1) will have the form

\[ \Psi(r_1s_1\ldots r_ns_n) = \sum_{\alpha_1\ldots \alpha_n} b(\alpha_1\ldots \alpha_n)\, \theta_{\alpha_1\ldots \alpha_n}(r_1s_1\ldots r_ns_n), \tag{7,6} \]

where

\[ \theta_{\alpha_1\ldots \alpha_n}(r_1s_1\ldots r_ns_n) = \sum_P(-1)^P\theta_{\alpha_1}(r_1s_1)\ldots \theta_{\alpha_n}(r_ns_n), \tag{7,7} \]

and the amplitudes \(b\) are determined from the secular equations

\[ \sum_{s_1\ldots s_n}\int \theta^{*}_{z_1\ldots z_n}(\mathbf r_1s_1\ldots \mathbf r_ns_n) (\hat H-E)\Psi(\mathbf r_1s_1\ldots \mathbf r_ns_n)\,d\mathbf r_1\ldots d\mathbf r_n. \tag{7,8} \]

The transformation matrix \(\|C_{\alpha z}\|\) is a series in increasing powers of the small parameter \(\varepsilon\), which makes it possible to develop a rigorous mathematical method of successive approximations. We emphasize, however, that this method concerns only the rigorous allowance for the deviation from exact orthogonality of the atomic functions; it does not concern and does not improve the basic approximation, which consists in truncating the expansion (6,1) or (7,6).

Let us pass to the representation of second quantization. In view of the Pauli principle, for the indices \(\alpha_i\) one may establish in (7,6) a definite order (see Section 6). With the established order, for example,
\(\alpha_1<\alpha_2<\alpha_3<\cdots<\alpha_n\), the system of numbers \(\alpha_1\ldots\alpha_n\) is completely determined by specifying the occupation numbers \(n_{\alpha_1}\ldots n_{\alpha_n}\), equal in our case to one or zero. The functions \(b(\alpha_1\ldots\alpha_n)\) are then transformed into wave functions of second quantization (see (6,5)) in \((\ldots n_\alpha\ldots)\), determined from an equation of type (6,8) with the energy operator in the second-quantization representation (6,38), which here takes the form

\[ \hat H = U_0 + \sum_{(\alpha,\alpha')} L(\alpha,\alpha')\hat a^{+}_{\alpha}\hat a_{\alpha'} + \frac{1}{2} \sum_{\alpha_1\alpha_2;\alpha'_1\alpha'_2} F(\alpha_1\alpha_2,\alpha'_1\alpha'_2) \hat a^{+}_{\alpha_1}\hat a^{+}_{\alpha_2}\hat a_{\alpha'_2}\hat a_{\alpha'_1}; \tag{7,9} \]

the operator amplitudes \(\hat a^{+}_{\alpha}\) and \(\hat a_{\alpha}\) satisfy the permutation relations (6,32). The matrix elements in (7,9) are determined by the formulas:

\[ L(\alpha,\alpha') = \int \theta^{*}_{\alpha} \left[ -\frac{\hbar^2}{2m}\Delta_{\mathbf r} + \sum_{(f)}G(\mathbf r-\mathbf R_f) \right] \theta_{\alpha'}\,d\mathbf r, \tag{7,10} \]

\[ F(\alpha_1\alpha_2,\alpha'_1\alpha'_2) = \iint \theta^{*}_{\alpha_1}\theta^{*}_{\alpha_2} V(\mathbf r_1\mathbf r_2) \theta_{\alpha'_1}\theta_{\alpha'_2}\,d\mathbf r_1\,d\mathbf r_2. \tag{7,11} \]

The principal problem is the determination of the eigenvalues of the energy operator (7,9). In addition, it is essential also to know the momentum operator of the system, which belongs to the class of additive operators and has the form

\[ P_\gamma = - \sum_{(\alpha,\alpha')} \left( i\hbar\int \theta^{*}_{\alpha}\frac{\partial}{\partial r_\gamma}\theta_{\alpha'}\,d\mathbf r \right) \hat a^{+}_{\alpha}\hat a_{\alpha'}. \tag{7,12} \]

N. N. Bogolyubov\(^{15,19}\) showed how one can compute in any

in the approximation the matrix elements (7,10) and (7,11) with the aid of atomic one-electron functions. The transformation matrix \(U_{\alpha\beta}\), as already indicated, is sought in the form of a series in powers of the small parameter \(\varepsilon\). As a result of elementary calculations we obtain:

\[ \theta_\alpha=\varphi_\alpha-\frac{\varepsilon}{2}\sum_{\alpha'} S(\alpha',\alpha)\varphi_{\alpha'}+ \]

\[ +\frac{3}{8}\varepsilon^3\sum_{\alpha'}\left\{\sum_\beta S(\alpha',\beta)S(\beta,\alpha)\right\}\varphi_{\alpha'}+\ldots \tag{7,13} \]

or, by virtue of (7,4),

\[ \theta_\alpha=\varphi_\alpha-\frac{1}{2}\sum_{(\alpha'\ne \alpha)} \left(\int \varphi_{\alpha'}^*\varphi_\alpha\,d\mathbf r\right)\varphi_{\alpha'}+ \]

\[ +\frac{3}{8}\sum_{\alpha'}\left\{ \sum_{(\beta\ne \alpha,\alpha')} \int \varphi_{\alpha'}^*\varphi_\beta\,d\mathbf r \int \varphi_\beta^*\varphi_\alpha\,d\mathbf r' \right\}\varphi_{\alpha'}+\ldots \tag{7,14} \]

In the particular case of \(s\)-functions (7,14) takes the form\(^*\)

\[ \theta_f(\mathbf r)=\varphi_f(\mathbf r) -\frac{1}{2}\sum_{(f'\ne f)} \varphi_{f'}(\mathbf r)\int \varphi_{f'}(\mathbf r')\varphi_f(\mathbf r')\,d\mathbf r' + \]

\[ +\frac{3}{8}\sum_{f'}\varphi_{f'}(\mathbf r) \sum_{(f''\ne f,f')} \int \varphi_{f'}(\mathbf r')\varphi_{f''}(\mathbf r')\,d\mathbf r' \times \]

\[ \times \int \varphi_{f''}(\mathbf r'')\varphi_f(\mathbf r'')\,d\mathbf r'' +\ldots \tag{7,15} \]

Substituting (7,15) into (7,10) and (7,11), after laborious calculations we obtain, to accuracy up to \(\varepsilon^3\), the following expression:

\[ \begin{aligned} L(f,f')={}& \left[E_0+\int\bigl(G(\mathbf r)-G_f(\mathbf r)\bigr)\varphi_f^2(\mathbf r)\,d\mathbf r\right]\delta_{ff'} +(1-\delta_{ff'})\times \\ &\times\Bigg\{ \frac{1}{2}\int \left[ G(\mathbf r)-G_f(\mathbf r) -\int \varphi_f^2(\mathbf r_1)\bigl(G(\mathbf r_1)-G_f(\mathbf r_1)\bigr)\,d\mathbf r_1 \right] \varphi_f(\mathbf r)\varphi_{f'}(\mathbf r)\,d\mathbf r \\ &\quad+ \frac{1}{2}\int \left[ G(\mathbf r)-G_{f'}(\mathbf r) -\int \varphi_{f'}^2(\mathbf r_1)\bigl(G(\mathbf r_1)-G_{f'}(\mathbf r_1)\bigr)\,d\mathbf r_1 \right] \varphi_f(\mathbf r)\varphi_{f'}(\mathbf r)\,d\mathbf r \\ &\quad+ \frac{1}{8}\sum_{(g\ne f,f')} \int \varphi_g(\mathbf r)\varphi_f(\mathbf r)\,d\mathbf r \times \int \varphi_g(\mathbf r)\varphi_{f'}(\mathbf r)\,d\mathbf r \times \\ &\quad\times \left[ 3\int \varphi_f^2(\mathbf r)\bigl(G(\mathbf r)-G_f(\mathbf r)\bigr)\,d\mathbf r + 3\int \varphi_{f'}^2(\mathbf r)\bigl(G(\mathbf r)-G_{f'}(\mathbf r)\bigr)\,d\mathbf r \right. \\ &\qquad\left. + 2\int \varphi_g^2(\mathbf r)\bigl(G(\mathbf r)-G_g(\mathbf r)\bigr)\,d\mathbf r \right] +\ldots \Bigg\} \end{aligned} \tag{7,16} \]

\(^*\) The atomic functions \(\varphi_\alpha(\mathbf r,s)\) are equal to the product of the spatial part \(\varphi_{f,l,m}(\mathbf r)\) by the spin function \(\delta(s-\sigma)\) (in the approximation in which we neglect magnetic interactions). The spin functions are automatically orthogonal (delta-functions), and therefore only the functions \(\varphi_{f,l,m}(\mathbf r)\) need be orthogonalized; in the case of \(s\)-states these reduce to \(\varphi_f(\mathbf r)\), since \(l=m=0\).

and an even more cumbersome expression for \(F(f_1 f_2, f_1' f_2')\), which is given in \({}^{15}\). From (7.16) it is easy to see that the following symmetry relations hold:

\[ L(f,f')=L(f',f); \tag{7.17} \]

it can also be shown that

\[ F(f_1 f_2,\ f_1'f_2')=F(f_1'f_2',\ f_1 f_2)=F(f_1 f_2',\ f_1' f_2)= \]

\[ =F(f_2 f_1,\ f_2'f_1'). \tag{7.18} \]

It is important to note that the expressions \(L(ff')\), \(F(fg,\ f'g)\), and \(F(gf,\ gf')\) for \(f\ne f'\) are quantities of first order of smallness in \(\varepsilon\), whereas \(F(f_1 f_2,\ f_1'f_2')\) for \(f_1\ne f_1'\) and \(f_2\ne f_2'\) are quantities of second order of smallness.

2. Let us first obtain the results of the one-electron theory. Evidently, in the present case the “atomic functions” are obtained in the tight-binding approximation (see Section 4), when the electron wave function in the lattice is sought in the form of a series in atomic functions. If the interaction terms in the operator (7.9) are discarded, then the wave equation in the second-quantization representation takes the form

\[ \left\{ \sum_{(f,f',\sigma)} L(f,f')\,\hat a_{f\sigma}^{+}\hat a_{f'\sigma} -(E-U_0) \right\}C_0=0. \tag{7.19} \]

By virtue of the identity of the lattice sites, the equality

\[ L(ff')=L(f-f') \tag{7.20} \]

holds, where

\[ L(f)=\int \theta(\mathbf r-\mathbf f) \left[ -\frac{\hbar^2}{2m}\Delta_{\mathbf r}+G(\mathbf r) \right]\theta(\mathbf r)\,d\mathbf r. \tag{7.21} \]

Replacing the operators \(\hat a_{f\sigma}\) by their Fourier coefficients

\[ \hat a_{f\sigma}=\frac{1}{\sqrt n}\sum_{(k)} e^{i(\mathbf{k f})}\hat a_{k\sigma}; \qquad \hat a_{k\sigma}=\frac{1}{\sqrt n}\sum_{(f)} e^{-i(\mathbf{k f})}\hat a_{f\sigma} \tag{7.22} \]

and substituting them into (7.19), we diagonalize it:

\[ \sum_{(f,f',\sigma)} L(f,f')\,\hat a_{f\sigma}^{+}\hat a_{f'\sigma} = \sum_{k,\sigma} \left\{\sum_{(f)}L(f)e^{-i(\mathbf{fk})}\right\} \hat a_{k\sigma}^{+}\hat a_{k\sigma}, \]

and consequently, for the energy (up to an additive constant) we obtain

\[ E(\mathbf k)=\sum_{(f)}L(f)e^{-i(\mathbf{fk})}, \tag{7.23} \]

where \(L(f)\), up to a quantity of second order of smallness relative to...

with respect to \(\varepsilon\) is equal to

\[ L(f)=\left\{E_0+\int [G(\mathbf r)-G_0(\mathbf r)]\varphi^2(\mathbf r)\,d\mathbf r\right\}\delta(\mathbf f)+ \]

\[ +\{1-\delta(\mathbf f)\}\int\left[\,G(\mathbf r)-G_0(\mathbf r)-\int \varphi^2(\mathbf r_1)\bigl(G(\mathbf r_1)- G_0(\mathbf r_1)\bigr)d\mathbf r_1\right]\varphi(\mathbf r)\varphi(\mathbf r-\mathbf f)\,d\mathbf r . \tag{7.24} \]

From (7.23) and (7.24) it is seen that the result obtained completely coincides with the results of the well-known tight-binding approximation of the one-electron model (see Sec. 4). Here the inconsistency of the one-electron model, in which electronic interactions are entirely neglected, is once again evident.

3. N. N. Bogoliubov and S. V. Tyablikov \(^{15,19}\) proposed a very elegant method of perturbation theory in operator form. The energy operator of the system is sought in the form of a series in powers of the small parameter \(\varepsilon\) (see (7.4)), namely:

\[ \hat H=\hat H_0+\varepsilon \hat H_1+\varepsilon^2\hat H_2 . \tag{7.25} \]

Transforming expression (7.9), using the anticommutation relations for Fermi amplitudes and the symmetry properties of the coefficients \(L(f,f')\) and \(F(f_1f_2,f'_1f'_2)\) (7.17) and (7.18), we find:

\[ \hat H_0=\sum_{(f)}\left[L(f,f)-\frac12 F(ff,ff)\right]N_f+ \]

\[ +\frac12\sum_{(f_1f_2)}F(f_1f_2,\ f_1f_2)N_{f_1}N_{f_2}+U_0, \tag{7.26} \]

\[ \varepsilon \hat H_1=\sum_{\substack{ff'\sigma\\(f\ne f')}}\hat a^+_{f\sigma}\left[L(ff')+\sum_{\substack{(f'')\\(f\ne f')}}F(ff'',\,f'f'')N_{f''}\right]a_{f'\sigma}. \tag{7.27} \]

Formula (6.38) must serve as the starting point for constructing specific many-electron models of the crystal

\[ \varepsilon^2\hat H_2= \sum_{\substack{f_1f_2 f'_1 f'_2;\,\sigma_1\sigma_2\\ f_1\ne f'_1,\ f_2\ne f'_2}} F(f_1f_2,\ f'_1f'_2)\, \hat a_{f_1\sigma_1}\hat a_{f_2\sigma_2}\hat a^{+}_{f'_2\sigma_2}\hat a^{+}_{f'_1\sigma_1}. \tag{7.28} \]

The equation of the “zero” approximation will then be

\[ (\hat H_0-E)a=0 . \tag{7.29} \]

It follows from (7.29) that the energy levels of the “zero” approximation are determined by the set of values \(\ldots N_f^0 \ldots\) of the occupation numbers ...

\[ E_{\ldots N_f^0 \ldots} = \sum_f \left[ L(ff)-\frac{1}{2}F(ff,ff) \right]N_f^0 + \frac{1}{2}\sum_{f_1,f_2}F(f_1f_2,f_1f_2)N_{f_1}^0N_{f_2}^0 . \tag{7,30} \]

These sets contain, in the general case, numbers \(N_f^0\) equal to 0, 1, 2 and satisfying the condition

\[ \sum_{(f)} N_f^0=n. \]

The quantities \(N_f\) themselves are related to \(n_{fs}\) and to the Fermi amplitudes \(a_{fs}\) by the formulas

\[ N_f=\sum_{(s)} n_{fs}=\sum_s \hat a_{fs}^{+}\hat a_{fs}, \tag{7,31} \]

i.e. \(N_f\) denote the numbers of particles in the states \(\Theta_f\) (if the nonorthogonality of the atomic functions is neglected, when \(\Theta_f=\varphi_f\), then \(N_f\) denotes the total number of electrons of “different” spins at one lattice site). To each such set \(\ldots N_f^0 \ldots\) there correspond wave functions of the zeroth approximation of the form \(c(\ldots n_{fs}\ldots)\times \prod_{(f)} \sigma(N_f,N_f^0)\), where \(c(\ldots n_{fs}\ldots)\) are arbitrary functions of the numbers \(n_{fs}\). Thus the energy level (7,30) of the zeroth approximation is degenerate, since it corresponds to the linear set of functions given above. The energy level for which all numbers \(N_f^0=1\) will naturally be the lowest (in the zeroth approximation). Solutions of this type should naturally be called “quasihomeopolar” (they are not exactly homeopolar, since they contain a small “admixture” of polar states owing to the use of the functions \(\Theta_\alpha\) instead of \(\varphi_\alpha\)). The levels, however, in which, alongside \(N_f^0=1\), there are numbers \(N_f^0=2\) and an equal number of quantities \(N_f^0=0\), will be excited. One should not forget, however, that such a “discrete” spectrum occurs only in the zeroth approximation. Owing to the translational invariance of the crystal lattice, the distribution of spins and electrons (pairs and holes with \(N_f=2\) and \(N_{f'}=0\)) will not be localized. As a result of the perturbation (the electron interaction), the degenerate “zeroth” discrete levels (energy centers of gravity) will split. The magnitude of this splitting will be determined by the integrals \(L(f,f')\) and \(F(f_1f_2,f_1'f_2')\). If the lowest level of the bands arising in this way belongs to the band with \(N_f=1\), then the given crystal will be a semiconductor or a dielectric; the states of its lowest energy levels will not possess current, and the accelerating action of an external electric field will be

equal to zero. If, conversely, the lowest level belongs to the band with twos and holes, then such a crystal will be a metal. We shall dwell on this in more detail below.

To take into account the influence of the terms of the first and second approximations (7.27) and (7.28), it is necessary to resort to perturbation theory. Let us consider the operator method of this theory, proposed by Bogolyubov and Tyablikov\(^{15,19}\).

The functions \(a_0\) of the zeroth approximation form a linear space \(\mathcal L\). Introduce the projection operator \(\hat P\), which projects an arbitrary function \(a\) onto the linear space \(\mathcal L\). Consequently, for an arbitrary function \(a\) we have \(\hat H_0 \hat P a = E_0 \hat P a\). Any function \(a\) can be represented as the sum

\[ a=\hat P a+(1-\hat P)a=a_0+a_1 . \]

Then the wave equation with the energy operator (7.25) can be written as

\[ \left(E-\hat H_0-\varepsilon \hat H_1-\varepsilon^2 \hat H_2\right)\hat P a + \left(E-\hat H_0-\varepsilon \hat H_1-\varepsilon^2 \hat H_0\right)a_1=0. \tag{7.32} \]

Next we multiply (7.32) by the operator \(\hat P\) from the left, taking into account the relations:

\[ \hat P^2=\hat P,\qquad \hat P\hat H_0=\hat H_0\hat P,\qquad \hat P(E-\hat H_0)a_1=(E-\hat H_0)\hat P(1-\hat P)a=0; \]

we have:

\[ \left(E-\hat H_0-\varepsilon \hat P\hat H_1\hat P-\varepsilon^2 \hat P\hat H_2\hat P\right)\hat P a -\varepsilon \hat P\hat H_1 a_1 - \varepsilon^2 \hat P\hat H_2 a_1=0. \tag{7.33} \]

Subtracting (7.33) from (7.32), we obtain

\[ \left(E-\hat H_0-\varepsilon \hat H_1-\varepsilon^2 \hat H_2 +\varepsilon \hat P\hat H_1+\varepsilon^2 \hat P\hat H_2\right)a_1 + \varepsilon\left(\hat P\hat H_1\hat P-\hat H_1\right)\hat P a + \varepsilon^2\left(\hat P\hat H_2\hat P-\hat H_2\right)\hat P a =0. \tag{7.34} \]

We seek the energy and the correction to the wave function in the form of series in powers of the small parameter \(\varepsilon\)

\[ \begin{aligned} E&=E_0+\varepsilon\Delta_0+O(\varepsilon^3),\\ a_1&=\varepsilon K+\varepsilon^2 L+O(\varepsilon^2). \end{aligned} \tag{7.35} \]

Using this expansion, on the basis of (7.34) we find:

\[ \left. \begin{aligned} (E_0-\hat H_0)K &= \left(\hat H_1-\hat P\hat H_1\hat P\right)\hat P a,\\ (E_0-\hat H_0)L &= \left(\hat H_2-\hat P\hat H_2\hat P\right)\hat P a + \left(\hat H_1-\hat P\hat H_1-\Delta_0\right)K . \end{aligned} \right\} \tag{7.36} \]

QUANTUM THEORY OF ELECTRONIC CONDUCTORS

Hence we have

\[ K=(\hat H_0-E_0)^{-1}(\hat P\hat H_1\hat P-\hat H_1)\hat P a, \tag{7.37} \]

\[ \begin{aligned} L={}&(\hat H_0-E_0)^{-1}(\hat P\hat H_2\hat P-\hat H_2)\hat P a+\\ &+(\hat H_0-E_0)^{-1}\times (\Delta_0+\hat P\hat H_1-\hat H_1)(\hat H_0-E_0)^{-1} (\hat P\hat H_1\hat P-\hat H_1)\hat P a . \end{aligned} \tag{7.38} \]

In view of (7.35), the equation of the problem (7.33) assumes the form

\[ \bigl(E-E_0-\varepsilon \hat P\hat H_1\hat P-\varepsilon^2 \hat P\hat H_2\hat P\bigr)\hat P a -\varepsilon^3\hat P\hat H_1K-\varepsilon^3\hat P\hat H_1L -\varepsilon^3\hat P\hat H_2K+O(\varepsilon^4)=0; \tag{7.39} \]

here it has been taken into account that

\[ (\hat H_0-E_0)\hat P a=(\hat H_0-E_0)a_0=0. \tag{7.40} \]

We note further that

\[ \hat P K=\hat P L=0, \]

therefore one may write:

\[ \hat P\hat H_1K=\hat P(\hat H_1-\hat P\hat H_1\hat P)K, \]

\[ \hat P\hat H_1L=\hat P(\hat H_1-\hat P\hat H_1\hat P)L, \]

\[ \hat P\hat H_2K=\hat P(\hat H_2-\hat P\hat H_2\hat P)K. \]

Consequently, equation (7.39) may be represented in the form

\[ \begin{aligned} (E-E_0)\hat P a ={}&\hat P\Bigl\{ \varepsilon \hat H_1+\varepsilon^2\hat H_2 -\varepsilon^2(\hat H_1-\hat P\hat H_1\hat P)(\hat H_0-E_0)^{-1} \times\\ &\times(\hat H_1-\hat P\hat H_1\hat P) -\varepsilon^3(\hat H_2-\hat P\hat H_2\hat P)(\hat H_0-E_0)^{-1} (\hat H_1-\hat P\hat H_1\hat P)-\\ &-\varepsilon^3(\hat H_1-\hat P\hat H_1\hat P)(\hat H_0-E_0)^{-1} (\hat H_2-\hat P\hat H_2\hat P)+\\ &+\varepsilon^3(\hat H_1-\hat P\hat H_1\hat P)(\hat H_0-E_0)^{-1} (\hat H_1-\hat P\hat H_1-\Delta_0)(\hat H_0-E_0)^{-1} \times\\ &\times(\hat H_1-\hat P\hat H_1\hat P)+O(\varepsilon^4) \Bigr\}\hat P a, \end{aligned} \]

if expressions (7.37) and (7.38) are taken into account. Hence, for determining the wave function \(a_0=\hat P a\), we obtain the equations

of the first, second, and third approximations:

\[ (E-E_0)a_0=\varepsilon \hat P \hat H_1 \hat P a_0, \tag{7.41} \]

\[ (E-E_0)a_0=\hat P\left\{\varepsilon \hat H_1+\varepsilon^2\hat H_2-\varepsilon^2\left(\hat H_1-\hat P\hat H_1\hat P\right) (\hat H_0-E_0)^{-1}\right. \]
\[ \left. {}\times\left(\hat H_1-\hat P\hat H_1\hat P\right)\right\}\hat P a_0, \tag{7.42} \]

\[ (E-E_0)a_0=\hat P\left\{\varepsilon\hat H_1+\varepsilon^2\hat H_2-\varepsilon^2\left(\hat H_1-\hat P\hat H_1\hat P\right) (\hat H_0-E_0)^{-1}\right. \]
\[ {}\times\left(\hat H_1-\hat P\hat H_1\hat P\right) -\varepsilon^3\left(\hat H_2-\hat P\hat H_2\hat P\right) (\hat H_0-E_0)^{-1} \left(\hat H_1-\hat P\hat H_1\hat P\right) \]
\[ {}-\varepsilon^3\left(\hat H_1-\hat P\hat H_1\hat P\right) (\hat H_0-E_0)^{-1} \left(\hat H_2-\hat P\hat H_2\hat P\right) \]
\[ {}+\varepsilon^3\left(\hat H_1-\hat P\hat H_1\hat P\right) (H_0-E_0)^{-1} \left(\hat H_1-\hat P\hat H_1-\Delta_0\right) \]
\[ \left. {}\times(\hat H_0-E_0)^{-1} \left(\hat H_1-\hat P\hat H_1\hat P\right) \right\}\hat P a. \tag{7.43} \]

In all approximations the wave equation has been reduced to an equation in which the wave function belongs to the space \(\mathscr L\). Therefore these equations may be regarded as “projections” of equation (7.32) onto the space \(\mathscr L\).

In exactly the same way one can obtain a formula of successive approximations for computing the mean value of any dynamical variable \(\hat A\). By definition we have:

\[ \hat A=\int a^*\hat A a\,d\tau. \]

Substituting here \(a_0+a_1\), taking (7.35) into account, we find:

\[ \bar A=\int a_0^*\hat A a_0\,d\tau +\varepsilon\int K^*\hat A a_0\,d\tau +\varepsilon\int a_0^*\hat A K\,d\tau +\varepsilon^2\int K^*\hat A K\,d\tau+ \]
\[ {}+\varepsilon^2\int a_0^*\hat A L\,d\tau +\varepsilon^2\int L^*\hat A a_0\,d\tau +O(\varepsilon^3). \]

Hence, using (7.37) and (7.38), we obtain*:

\[ \bar A=(a_0^*,\hat A a_0) -\varepsilon\left(a_0^*, \left(\hat H_1-\hat P\hat H_1\hat P\right) (\hat H_0-E_0)^{-1}\hat A a_0\right) \]
\[ {}-\varepsilon\left(a_0^*, \hat A(\hat H_0-E_0)^{-1} \left(\hat H_1-\hat P\hat H_1\hat P\right)a_0\right)+ \]

\[ \underline{\hspace{2.5cm}} \]

\[ \text{*) Here and below the abbreviated notation is used} \]

\[ \int a^*\hat A a\,d\tau=(a^*,\hat A a). \]

\[ \begin{aligned} &+\varepsilon^2\left(a_0^*,\left(\widehat H_1-\widehat P\widehat H_1\widehat P\right)(\widehat H_0-E_0)^{-1}\widehat A(\widehat H_0-E_0)^{-1}\times\right.\\ &\left.\qquad\qquad\times\left(\widehat H_1-\widehat P\widehat H_1\widehat P\right)a_0\right) -\varepsilon^3\left(a_0^*,\left(\widehat H_2-\widehat P\widehat H_2\widehat P\right)(\widehat H_0-E_0)^{-1}\widehat A a_0\right)-\\ &-\varepsilon^2\left(a_0^*,\widehat A(\widehat H_0-E_0)^{-1}\left(\widehat H_2-\widehat P\widehat H_2\widehat P\right)\right)+\\ &+\varepsilon^3\left(a_0^*,\left(\widehat H_1-\widehat P\widehat H_1\widehat P\right)(\widehat H_0-E_0)^{-1} \left(\widehat H_1-\widehat H_1\widehat P-\Delta_0\right)\times\right.\\ &\left.\qquad\qquad\times(\widehat H_0-E_0)^{-1}\widehat A a_0\right) +\varepsilon^2\left(a_0^*,\widehat A(\widehat H_0-E_0)^{-1} \left(\widehat H_1-\widehat P\widehat H_1-\Delta_0\right)\times\right.\\ &\left.\qquad\qquad\times(\widehat H_0-E_0)^{-1}\left(\widehat P\widehat H_1\widehat P-\widehat H_1\right)a_0\right)+O(\varepsilon^3). \tag{7.44} \end{aligned} \]

It is easy to see that if in all terms, beginning with the terms of order \(\varepsilon\), one replaces \(\widehat A\) by \(\widehat P\widehat A\widehat P\), then they vanish identically; therefore we make the replacement \(\widehat A\) by \(\widehat A-\widehat P\widehat A\widehat P\). Then, with accuracy up to terms of order \(\varepsilon^3\), we shall have

\[ A=\left(a_0^*,\,M\left(\widehat A\right)a_0\right), \tag{7.44'} \]

where

\[ M\left(\widehat A\right)=\widehat P\widehat A\widehat P+D_+\left(\widehat A\right)+D_-\left(\widehat A\right) \]

and

\[ \begin{aligned} D_+\left(\widehat A\right)=\widehat P\{& -\varepsilon\left(\widehat H_1-\widehat P\widehat H_1\widehat P\right)(\widehat H_0-E_0)^{-1} \left(\widehat A-\widehat P\widehat A\widehat P\right)-\\ &-\varepsilon^2\left(\widehat H_2-\widehat P\widehat H_2\widehat P\right)(\widehat H_0-E_0)^{-1} \left(\widehat A-\widehat P\widehat A\widehat P\right)+\\ &+\varepsilon^3\left(\widehat H_1-\widehat P\widehat H_1\widehat P\right)(\widehat H_0-E_0)^{-1} \left(\widehat H_1-\widehat H_1\widehat P-\Delta_0\right)(\widehat H_0-E_0)^{-1} \left(\widehat A-\widehat P\widehat A\widehat P\right)+\\ &+\frac{\varepsilon^2}{2}\left(\widehat H_1-\widehat P\widehat H_1\widehat P\right)(\widehat H_0-E)^{-1} \left(\widehat A-\widehat P\widehat A\widehat P\right)\times\\ &\qquad\qquad\times(\widehat H_0-E)^{-1}\left(\widehat H_1-\widehat P\widehat H_1\widehat P\right)\}\widehat P, \tag{7.45} \end{aligned} \]

\[ D_-\left(\widehat A\right)=\{D_+\left(\widehat A\right)\}^{+}. \tag{7.46} \]

From these formulas it is seen that the mean value of a dynamical variable, which is obtained with the full wave function \(a\), is equal to the mean value of the “deformed” operator \(M(\widehat A)\), obtained with the aid of the zero wave function \(a_0\) of the space \(L\). Therefore, for the calculation of these mean values there is no need to determine the full wave function \(a\). Formula (7.45) greatly

simplifies in the second and first approximations, namely:

in the first approximation:
\[ D_{+}\binom{\hat A}{\,}=0, \tag{7,47} \]

in the second approximation:
\[ D_{+}\binom{\hat A}{\,} = -\hat P \varepsilon \left(\hat H_{1}-\hat P \hat H_{1}\hat P\right) \left(\hat H_{0}-E_{0}\right)^{-1} \left(\hat A-\hat P\hat A\hat P\right)\hat P . \tag{7,48} \]

From the normalization condition for the complete function \(a\), i.e. \((a^{*}\cdot a)=1\), on the basis of (7,35) and (7,37) we obtain the following normalization condition for the wave function of the zeroth approximation:
\[ (a_{0}^{*},a_{0})+ \varepsilon^{2} \left( a_{0}^{*}, \hat P \left(\hat H_{1}-\hat P\hat H_{1}\hat P\right) \left(\hat H_{0}-E_{0}\right)^{-2} \left(\hat H_{1}-\hat P\hat H_{1}\hat P\right) \hat P a_{0} \right) +O(\varepsilon^{3})=1 . \tag{7,49} \]

4. The version of perturbation theory set forth above can now be applied to the solution of equation (7,32), the energy operator in which is given by expressions (7,26)—(7,28). In the present case the zeroth approximation will consist of wave functions of the form

\[ a(\ldots n_{f\sigma}\ldots) = \Psi(\ldots n_{f\sigma}\ldots)\prod_{(f)}\delta(N_{f}-1), \tag{7,50} \]

where \(\Psi(\ldots n_{f\sigma}\ldots)\) is an arbitrary function of \(n_{f\sigma}\). In other words, the elements of the linear space of zero functions are characterized by the condition that all occupation numbers \(N_{f}\) are equal to unity. From formulas (7,41)—(7,43) and (7,27), (7,28) it follows that, first of all, it is required to compute the expressions

\[ \hat P\,\hat a^{+}_{f_{1}\sigma_{1}}\hat a_{f'_{1}\sigma'_{1}}\,\hat P; \qquad \hat P\,\hat a^{+}_{f_{1}\sigma_{1}}\hat a^{+}_{f_{2}\sigma_{2}}\hat a_{f'_{2}\sigma'_{2}}\hat a_{f'_{1}\sigma'_{1}}\,\hat P . \tag{7,51} \]

To compute the first relation in (7,51) we take into account that

\[ \hat a^{+}_{f_{1}\sigma_{1}}\hat a_{f'_{1}\sigma'_{1}}\,N_{f} - N_{f}\hat a^{+}_{f_{1}\sigma_{1}}\hat a_{f'_{1}\sigma'_{1}} = \]

\[ = \hat a^{+}_{f_{1}\sigma_{1}}\hat a_{f'_{1}\sigma'_{1}} \sum_{(\sigma)}\hat a^{+}_{f\sigma}\hat a_{f\sigma} - \sum_{(\sigma)}\hat a^{+}_{f\sigma}\hat a_{f\sigma} \hat a^{+}_{f_{1}\sigma_{1}}\hat a_{f'_{1}\sigma'_{1}} = \]

\[ = \sum_{(\sigma)} \left\{ \hat a^{+}_{f_{1}\sigma_{1}}\hat a_{f'_{1}\sigma'_{1}}\, \hat a^{+}_{f\sigma}\hat a_{f\sigma} - \hat a^{+}_{f\sigma}\hat a_{f\sigma} \hat a^{+}_{f_{1}\sigma_{1}}\hat a_{f'_{1}\sigma'_{1}} \right\}. \]

From the commutation relations (6,32) we have

\[ \hat a^{+}_{f_{1}\sigma_{1}}\hat a_{f'_{1}\sigma'_{1}}\, \hat a^{+}_{f\sigma}\hat a_{f\sigma} - \hat a^{+}_{f\sigma}\hat a_{f\sigma}\hat a^{+}_{f_{1}\sigma_{1}}\hat a_{f'_{1}\sigma'_{1}} = \hat a^{+}_{f_{1}\sigma_{1}}\hat a_{f'_{1}\sigma'_{1}}\hat a^{+}_{f\sigma}\hat a^{+}_{f\sigma} + \]

\[ + \hat a^{+}_{f_{1}\sigma_{1}} \left\{\hat a^{+}_{f\sigma}\hat a_{f'_{1}\sigma'_{1}},\,\hat a_{f\sigma}\right\} + \hat a^{+}_{f\sigma}\hat a^{+}_{f_{1}\sigma_{1}}\hat a_{f'_{1}\sigma'_{1}}\hat a_{f\sigma} + \hat a^{+}_{f\sigma}\hat a^{+}_{f_{1}\sigma_{1}}\hat a_{f\sigma}\hat a_{f'_{1}\sigma'_{1}} + \]

\[ +\hat a_{f\sigma}^{+}\left(\hat a_{f\sigma}\hat a_{f_1\sigma_1}^{+} -\delta(f-f_1)\delta(\sigma-\sigma_1)\right) \hat a_{f'\sigma'}-\hat a_{f\sigma}^{+}\hat a_{f\sigma}\hat a_{f_1\sigma_1}^{+}\hat a_{f'\sigma'} = \]

\[ =\delta(f-f_1')\delta(\sigma-\sigma_1')\hat a_{f_1\sigma_1}^{+}\hat a_{f\sigma} -\delta(f-f_1)\delta(\sigma-\sigma_1)\hat a_{f\sigma}^{+}\hat a_{f'\sigma'} . \]

Consequently, one may write:

\[ \hat a_{f_1\sigma_1}^{+}\hat a_{f'\sigma'}N_f - N_f\hat a_{f_1\sigma_1}^{+}\hat a_{f'\sigma'} = \left[\delta(f-f_1')-\delta(f-f_1)\right]\hat a_{f_1\sigma_1}^{+}\hat a_{f'\sigma'} . \tag{7,52} \]

Let us now consider an arbitrary function \(a\). By virtue of (7,52) we have

\[ \left(a_0^{*},\hat a_{f_1\sigma_1}^{+}\hat a_{f'\sigma'}N_f a_0\right) - \left(a_0^{*},N_f\hat a_{f_1\sigma_1}^{+}\hat a_{f'\sigma'}a_0\right) = \]

\[ = \left[\delta(f-f_1')-\delta(f-f_1)\right] \left(a_0^{*},\hat a_{f_1\sigma_1}^{+}\hat a_{f'\sigma'}a_0\right) = \]

\[ = \left[\delta(f-f_1')-\delta(f-f_1)\right] \left(a^{*},\hat P\hat a_{f_1\sigma_1}^{+}\hat a_{f'\sigma'}\hat P a\right), \]

but

\[ N_f a_0=a_0 \quad\text{and}\quad a_0^{*}N_f=a_0^{*}, \]

therefore

\[ \left[\delta(f-f_1')-\delta(f-f_1)\right] \left(a^{*},\hat P\hat a_{f_1\sigma_1}^{+}\hat a_{f'\sigma'}\hat P a\right)=0. \]

Putting here \(f=f_1\), we find that

\[ \left(a^{*},\hat P\hat a_{f_1\sigma_1}^{+}\hat a_{f'\sigma'}\hat P a\right)=0, \qquad f_1\ne f_1', \]

and, by virtue of the arbitrariness of the function \(a\), we obtain

\[ \hat P\hat a_{f_1\sigma_1}^{+}\hat a_{f'\sigma'}\hat P=0, \quad \text{if } f_1\ne f_1'. \tag{7,53} \]

In the case \(f_1=f_1'\) the operator \(\hat a_{f_1\sigma_1}^{+}\hat a_{f'\sigma'}\) does not change the number \(N_{f_1}\), since it corresponds to the simultaneous appearance of spin \(\sigma_1'\) in the state \(f_1\) and annihilation of spin \(\sigma_1\) in the same state. Therefore the operator \(\hat a_{f_1\sigma_1}^{+}\hat a_{f'\sigma'}\) commutes with the projection operator \(\hat P\), since it transforms every quasihomeopolar function also into a quasihomeopolar one. Consequently, we shall have:

\[ \hat P\hat a_{f_1\sigma_1}^{+}\hat a_{f_1\sigma_1'}\hat P = \hat a_{f_1\sigma_1}^{+}\hat a_{f_1\sigma_1'}\hat P . \tag{7,54} \]

The second of expressions (7,51) is calculated in a similar way; as a result we shall have:

\[ \hat P\hat a_{f_1\sigma_1}^{+}\hat a_{f_2\sigma_2}^{+} \hat a_{f_2'\sigma_2'}\hat a_{f_1'\sigma_1'}\hat P =0 \quad (f_1f_2)\ne(f_1'f_2'). \tag{7,55} \]

The symbolic inequality \((f_1 f_2) \ne (f'_1 f'_2)\) means that the pair of points \((f'_1 f'_2)\) differs from the pair \((f_1 f_2)\), with the order of succession not being taken into account. In the case \((f_1 f_2)=(f'_1 f'_2)\) we shall have:

\[ \hat P \hat a^{+}_{f_1\sigma_1}\hat a^{+}_{f_2\sigma_2}\hat a_{f'_2\sigma'_2}\hat a_{f'_1\sigma'_1}\hat P = \hat a^{+}_{f_1\sigma_1}\hat a^{+}_{f_2\sigma_2}\hat a_{f'_2\sigma'_2}\hat a_{f'_1\sigma'_1}\hat P . \tag{7,56} \]

For the application of perturbation theory it remains to expand expressions of the form

\[ \left(\hat H_0-E_0\right)^{-1} \hat a^{+}_{f_1\sigma_1}\hat a^{+}_{f_2\sigma_2}\hat a_{f'_2\sigma'_2}\hat a_{f'_1\sigma'_1}\hat P \tag{7,57} \]

under the condition

\[ (f_1 f_2)\ne(f'_1 f'_2). \]

The operator
\(\hat a^{+}_{f_1\sigma_1}\hat a^{+}_{f_2\sigma_2}\hat a_{f'_2\sigma'_2}\hat a_{f'_1\sigma'_1}\)
corresponds to the disappearance of electrons with spins \(\sigma'_1,\sigma'_2\) in the states \(f'_1,f'_2\) and to the simultaneous appearance of electrons with spins \(\sigma_1,\sigma_2\) in the states \(f_1,f_2\). In expression (7,57) the mentioned operator, owing to the factor \(\hat P\), is always applied to the quasihomeopolar wave function; therefore it creates “holes” in the states \(f'_1,f'_2\) and simultaneously “pairs” in the states \(f_1,f_2\). The increase acquired by the unperturbed energy \(\hat H_0\) upon the appearance, in the quasihomeopolar state, of a complex of holes \(f'_1 f'_2\) and pairs \(f_1 f_2\), according to (7,30), is equal—for one pair—to

\[ \Delta(f_1,f'_1)=\frac{1}{2}\left[C(f_1,f_1)+C(f'_1,f'_1)-C(f_1,f'_1)-C(f'_1,f_1)\right] \tag{7,58} \]

and for two pairs

\[ \Delta(f_1 f_2; f'_1 f'_2) = \Delta(f_1,f'_1)+\Delta(f_2,f'_2) - C(f_1,f'_2)-C(f_2,f'_1), \tag{7,59} \]

where

\[ C(f_1 f_2)=F(f_1 f_2;\,f_1,\,f_2). \]

Thus, we have

\[ (\hat H_0-E_0)^{-1} \hat a^{+}_{f_1\sigma_1}\hat a^{+}_{f_2\sigma_2}\hat a_{f'_2\sigma'_2}\hat a_{f'_1\sigma'_1}\hat P = \]

\[ = \frac{1}{\Delta(f_1,f_2;\,f'_1,f'_2)} \hat a^{+}_{f_1\sigma_1}\hat a^{+}_{f_2\sigma_2}\hat a_{f'_2\sigma'_2}\hat a_{f'_1\sigma'_1}\hat P . \tag{7,60} \]

and analogously

\[ \hat P \hat a^{+}_{f_1\sigma_1}\hat a^{+}_{f_2\sigma_2} \hat a_{f'_2\sigma'_2}\hat a_{f'_1\sigma'_1} \left(\hat H_0-E_0\right)^{-1} = \frac{1}{\Delta(f_1,f_2;\,f'_1 f'_2)} \hat P \hat a^{+}_{f_1\sigma_1}\hat a^{+}_{f_2\sigma_2} \hat a_{f'_2\sigma'_2}\hat a_{f'_1\sigma'_1}. \tag{7,61} \]

The formulas found make it possible to proceed to obtaining the explicit form of the equations of the various approximations.

To obtain the equation of the first approximation, multiply (7,27) on the right and on the left by the operator \(\hat P\); then, taking (7,53) into account, we find

\[ \hat P\hat H_1\hat P=0. \tag{7,62} \]

Hence it is seen that the equations of the first approximation (7,41) do not differ from the equation of the zeroth approximation (7,40) and, consequently, the degeneracy is not removed. It is easy to show, on the basis of (7,44), that the mean value of the total momentum of the system in this approximation is also equal to zero.

The equation of the second approximation (7,42), after using (7,62), (7,54), (7,27), (7,61), will take the form

\[ \begin{aligned} (E-E_0)a_0 &=\frac{1}{2} \sum_{\substack{(f_1,f_2;\,\sigma_1\sigma_2)\\ f_1\ne f_2}} F(f_1,f_2;\,f_2,f_1)\, \hat a^{+}_{f_1\sigma_1}\hat a^{+}_{f_2\sigma_2} \hat a_{f_1\sigma_1}\hat a_{f_2\sigma_2}a_0 \\ &\quad+ \sum_{\substack{(f_1 f_2;\,f'_1 f'_2;\,\sigma_1\sigma_2)\\ f_1\ne f'_1;\, f_2\ne f'_2}} \hat P\hat a^{+}_{f_1\sigma_1} \left[ L(f_1,f'_1)+ \sum_{f''}F(f_1,f'';\,f'_1,f'')N_{f''} \right]\hat a_{f'_1\sigma_1} \\ &\qquad\times \frac{1}{\Delta(f_2,f'_2)} \hat a^{+}_{f_2\sigma_2} \left[ L(f_2,f'_2)+ \sum_{f''}F(f_2,f'';\,f'_2,f'')N_{f''} \right]\hat a_{f'_2\sigma_2}\hat P . \tag{7,63} \end{aligned} \]

We shall next use the following equalities:

\[ N_{f''}\hat a_{f_1\sigma_1}\hat P = \hat a_{f_1\sigma_1}N_{f''}\hat P = \hat a_{f_1\sigma_1}\hat P \quad (f''\ne f_1); \]

\[ \begin{aligned} N_{f_1}\hat a_{f_1\sigma_2}\hat P &= \sum_{(\sigma)} \hat a^{+}_{f_1\sigma}\hat a_{f_1\sigma} \hat a_{f_1\sigma_2}\hat P = -\sum_{(\sigma)} \hat a^{+}_{f_1\sigma}\hat a_{f_1\sigma_2}\hat a_{f_1\sigma}\hat P \\ &= -\sum_{(\sigma)} \left[ \hat a_{f_1\sigma_2}\hat a^{+}_{f_1\sigma} -\delta(\sigma_2-\sigma) \right]\hat a_{f_1\sigma}\hat P = \hat a_{f_1\sigma_2}(N_{f_1}-1)\hat P =0, \end{aligned} \]

\[ \hat P\hat a^{+}_{f_1\sigma_1}N_{f''} = \hat P\hat a^{+}_{f_1\sigma_1} \quad (f''\ne f_1); \qquad \hat P\hat a^{+}_{f_1\sigma_1}N_{f_1}=0, \]

and also

\[ \hat P\hat a^{+}_{f_1\sigma_1}\hat a_{f_2\sigma_2}\hat a^{+}_{f_2\sigma_2}\hat a_{f_1\sigma_1}\hat P = \hat P\hat a^{+}_{f_1\sigma_1}\left[\delta(\sigma_1-\sigma_2)-\hat a^{+}_{f_2\sigma_2}\hat a_{f_2\sigma_1}\right]\hat a_{f_1\sigma_1}\hat P = \]

\[ = \hat P\hat a^{+}_{f_1\sigma_1}\hat a_{f_1\sigma_1}\hat P\delta(\sigma_1-\sigma_2) + \hat P\hat a^{+}_{f_1\sigma_1}\hat a^{+}_{f_2\sigma_2}\hat a_{f_1\sigma_1}\hat a_{f_2\sigma_2}\hat P. \]

Thus, finally, the equation of the second approximation (7.42) is written in the following form:

\[ (E-E^{(0)})a_0 = \sum_{\substack{(f_1f_2),\,\sigma_1\sigma_2\\ f_1\ne f_2}} I(f_1f_2)\, \hat a^{+}_{f_1\sigma_1}\hat a^{+}_{f_2\sigma_2} \hat a_{f_1\sigma_2}\hat a_{f_2\sigma_1}a_0, \tag{7.64} \]

where the abbreviated notation has been introduced

\[ E^{(0)} = E_0 - \sum_{\substack{f_1f_2\\(f_1\ne f_2)}} \left[ L(f_1f_2)+ \sum_{f''(\ne f_2)}F(f_1,f'';\,f_2,f'') \right] \left[ L(f_2f_1)+ \right. \]

\[ \left. + \sum_{f''(\ne f_1)}F(f_2f'';\,f_1f'') \right] \frac{1}{\Delta(f_1f_2)}, \tag{7.65} \]

\[ I(f_1f_2) = \frac12(f_1f_2;\,f_2f_1) - \frac{1}{\Delta(f_1,f_2)} \left[ L(f_1f_2)+ \right. \]

\[ \left. + \sum_{f''(\ne f_1)}F(f_1,f'';\,f_2,f'') \right] \left[ L(f_2,f_1)+ \sum_{f''(\ne f_1)}F(f_2,f'';\,f_1,f'') \right]. \]

Using (7.44) again, we find that the mean value of the total momentum operator of the system is also equal to zero in the second approximation.

It can be shown that the equations of the second approximation of the Bogoliubov–Tyablikov method coincide exactly with the results of the exchange multielectron model of Frenkel–Heisenberg. Naturally, of course, the Bogoliubov–Tyablikov method, which takes into account the nonexact orthogonality of the atomic one-electron functions, gives a more accurate expression for the exchange integrals entering the expression for the energy of the system.

In order to show this, let us introduce, instead of the Fermi operator amplitudes, the Bose operator amplitudes \(\hat b_{fs}\), which are related to \(\hat a_{fs}\) by means of the sign-changing Wigner functions \(\eta_{fs}\), namely,

\[ \hat a_{fs}=\eta_{fs}\hat b_{fs};\qquad \hat a^{+}_{fs}=\hat b^{+}_{fs}\eta_{fs}, \tag{7.66} \]

where

\[ \eta_{fs} = \prod_{(f',s')<(f,s)} (1-2\hat n_{f',s'}). \tag{7.67} \]

It is easy to see that

\[ \hat b^{+}_{f\sigma}\hat b_{f\sigma} = \hat a^{+}_{f\sigma}\hat a_{f\sigma} = n_{f\sigma}. \tag{7,68} \]

Moreover, by virtue of the quasi-molecular approximation, the amplitudes \(\hat b_{f\sigma}\) are subject to the condition

\[ \sum_{(\sigma)} \hat b^{+}_{f\sigma}\hat b_{f\sigma}=1. \tag{7,69} \]

In defining the Wigner function the set of indices \((f,\sigma)\) is numbered in a definite order. We shall assume that \((f',\sigma')<(f,\sigma)\), if \(f'<f\). When \(f'=f\), then \((f,\sigma')<(f,\sigma)\) for \(\sigma'<\sigma\). Then one may write

\[ \hat a^{+}_{f,-\frac12}\hat a_{f,\frac12} = \hat b^{+}_{f,-\frac12}\eta_{f,-\frac12}\eta_{f,\frac12}\hat b_{f,\frac12} \]

and

\[ \eta_{f,-\frac12} \left[1-2n_{f,-\frac12}\right] = \eta_{f,\frac12}; \]

whence we have

\[ \hat a^{+}_{f,-\frac12}\hat a_{f,\frac12} = \hat b^{+}_{f,-\frac12} \left[1-2\eta_{f,-\frac12}\right] \hat b_{f,\frac12} = \]

\[ = \left[1-2\left(n_{f,-\frac12}-1\right)\right] \hat b^{+}_{f,-\frac12}\hat b_{f,\frac12}. \]

From the definition of the operators \(\hat b_{f,\sigma}\) and \(\hat b^{+}_{f,\sigma}\) it follows that

\[ \hat b_{f,-\frac12}F\left(n_{f,-\frac12}\right) = \sqrt{n_{f,-\frac12}}\, F\left[n_{f,-\frac12}-1\right], \]

where \(F(n_{f\sigma})\) is an arbitrary function of the numbers \(n_{f\sigma}\). Since \(n_{f\sigma}=0,1\), we obtain:

\[ \left[1-2\left(n_{f,-\frac12}\right)\right] \hat b^{+}_{f,-\frac12} F\left(n_{f,-\frac12}\right) = \]

\[ = \left[1-2\left(n_{f,-\frac12}-1\right)\right] \sqrt{n_{f,-\frac12}}\, F\left[n_{f,-\frac12}-1\right] = \]

\[ = \sqrt{n_{f,-\frac12}}\, F\left[n_{f,-\frac12}-1\right] = \hat b^{+}_{f,-\frac12} F\left(n_{f,-\frac12}\right), \]

and therefore, owing to the arbitrariness of \(F(n_{f\sigma})\), we find:

\[ \left[1-2\left(n_{f,-\frac12}-1\right)\right]\hat b^{+}_{f,-\frac12} = \hat b^{+}_{f,-\frac12}. \]

As a result we shall have

\[ \hat a^+_{f,-\frac12}\hat a_{f,\frac12} = \hat b^+_{f,-\frac12}\hat b_{f,\frac12} \tag{7,70} \]

and, analogously,

\[ \hat a^+_{f,\frac12}\hat a_{f,-\frac12} = \hat b^+_{f,\frac12}\hat b_{f,-\frac12}. \tag{7,71} \]

From (7,64), the energy operator in the second approximation, up to the additive term \(E^{(0)}\), and after expanding the sums over spin coordinates, will have the form

\[ \begin{aligned} \hat H ={}& -\sum_{\substack{f_1,f_2\\(f_1\ne f_2)}} I(f_1,f_2) \Bigl[ \hat a^+_{f_1,-\frac12}\hat a_{f_1,-\frac12} \hat a^+_{f_2,-\frac12}\hat a_{f_2,-\frac12} \\ &\quad +\hat a^+_{f_1,-\frac12}\hat a_{f_1,\frac12} \hat a^+_{f_2,\frac12}\hat a_{f_2,-\frac12} +\hat a^+_{f_1,\frac12}\hat a_{f_1,-\frac12} \hat a^+_{f_2,-\frac12}\hat a_{f_2,\frac12} \\ &\quad +\hat a^+_{f_1,\frac12}\hat a_{f_1,\frac12} \hat a^+_{f_2,\frac12}\hat a_{f_2,\frac12} \Bigr]. \end{aligned} \tag{7,72} \]

Using (7,68), (7,70), and (7,71) instead of (7,72), we find:

\[ \begin{aligned} \hat H ={}& -\sum_{\substack{f_1,f_2\\(f_1\ne f_2)}} I(f_1,f_2) \Bigl[ \hat b^+_{f_1,-\frac12}\hat b_{f_1,-\frac12} \hat b^+_{f_2,-\frac12}\hat b_{f_2,-\frac12} \\ &\quad +\hat b^+_{f_1,\frac12}\hat b_{f_1,\frac12} \hat b^+_{f_2,\frac12}\hat b_{f_2,\frac12} +\hat b^+_{f_1,-\frac12}\hat b_{f_1,\frac12} \hat b^+_{f_2,\frac12}\hat b_{f_2,-\frac12} \\ &\quad +\hat b^+_{f_1,\frac12}\hat b_{f_1,-\frac12} \hat b^+_{f_2,-\frac12}\hat b_{f_2,\frac12} \Bigr] \end{aligned} \]

or, by virtue of (7,69) and introducing new notation,

\[ \hat b^+_{f,\frac12}=\hat\varphi^+_f,\qquad \hat b_{f,\frac12}=\hat\varphi_f;\qquad \hat b^+_{f,-\frac12}=\hat\psi^+_f,\qquad \hat b_{f,-\frac12}=\hat\psi_f, \tag{7,73} \]

we finally obtain

\[ \hat H = E^{(0)} -\sum_{\substack{f_1,f_2\\(f_1\ne f_2)}} I(f_1,f_2) + \sum_{\substack{f_1,f_2\\(f_1\ne f_2)}} I(f_1,f_2) \left( \hat\psi^+_{f_1}\hat\varphi^+_{f_2} - \hat\varphi^+_{f_1}\hat\psi^+_{f_2} \right) \left( \hat\psi_{f_1}\hat\varphi_{f_2} - \hat\varphi_{f_1}\hat\psi_{f_2} \right), \tag{7,74} \]

QUANTUM THEORY OF ELECTRON CONDUCTORS

which coincides with the energy operator of the ordinary exchange model. The exchange integral according to (7.65), (7.16) has the form

\[ \begin{aligned} I(f_1,f_2)=&\ \frac{1}{2}\int \Phi(\mathbf r_1,\mathbf r_2)\, \varphi_{f_1}(\mathbf r_1)\varphi_{f_2}(\mathbf r_1) \varphi_{f_1}(\mathbf r_2)\varphi_{f_2}(\mathbf r_2)\,d\mathbf r_1\,d\mathbf r_2 \\ &-\frac{1}{2}\int \Psi(\mathbf r_1,\mathbf r_2)\, \varphi_{f_1}(\mathbf r_1)\varphi_{f_2}(\mathbf r_1) \left[\varphi_{f_2}^{2}(\mathbf r_2)+\right. \\ &\left.\qquad\qquad\qquad\qquad +\varphi_{f_1}^{2}(\mathbf r_2)\right]\,d\mathbf r_1\,d\mathbf r_2 \int \varphi_{f_1}(\mathbf r)\varphi_{f_2}(\mathbf r)\,d\mathbf r \\ &+\frac{1}{2}\left[\int \varphi_{f_1}(\mathbf r)\varphi_{f_2}(\mathbf r)\,d\mathbf r\right]^2 \int \Phi(\mathbf r_1,\mathbf r_2) \left[\frac{\varphi_{f_1}^{2}(\mathbf r_1)+\varphi_{f_2}^{2}(\mathbf r_2)}{2}\right] \\ &\qquad\qquad\qquad\qquad\times \left[\frac{\varphi_{f_1}^{2}(\mathbf r_1)+\varphi_{f_2}^{2}(\mathbf r_2)}{2}\right] \,d\mathbf r_1\,d\mathbf r_2 -\frac{\Lambda(f_1f_2)\Lambda(f_2f_1)}{\Delta(f_1f_2)}, \end{aligned} \tag{7.75} \]

where

\[ \Lambda(f_1f_2)= \iint\left\{\widetilde V(\mathbf r)-\widetilde U_{f_1}(\mathbf r) -\int \varphi_{f_1}^{2}(\mathbf r_1)\left[\widetilde V(\mathbf r_1)-\right.\right. \]

\[ \left.\left. -\widetilde U_{f_1}(\mathbf r_1)\right]d\mathbf r_1\right\} \varphi_{f_1}(\mathbf r)\varphi_{f_2}(\mathbf r)\,d\mathbf r; \]

\[ \widetilde V(\mathbf r)=\sum_{(f)} \widetilde U_f(\mathbf r); \qquad \widetilde U_f(\mathbf r)=U_f(\mathbf r)- \int \Phi(\mathbf r,\mathbf r')\varphi_f^{2}(\mathbf r')\,d\mathbf r'. \]

In ordinary exchange theory the integral \(I(f_1,f_2)\) has a somewhat different expression, namely:

\[ \begin{aligned} I(f_1,f_2)=&\ \frac{1}{2}\int \Phi(\mathbf r_1,\mathbf r_2) \varphi_{f_1}(\mathbf r_1)\varphi_{f_2}(\mathbf r_1) \varphi_{f_1}(\mathbf r_2)\varphi_{f_2}(\mathbf r_2)\,d\mathbf r_1\,d\mathbf r_2 \\ &-\frac{1}{2}\int \left[\Phi(\mathbf r_1,\mathbf f_1)+\Phi(\mathbf r_1,\mathbf f_2)\right] \varphi_{f_1}(\mathbf r_1)\varphi_{f_2}(\mathbf r_1)\,d\mathbf r_1 \int \varphi_{f_1}(\mathbf r)\varphi_{f_2}(\mathbf r)\,d\mathbf r \\ &+\frac{1}{2}\Phi(\mathbf f_1,\mathbf f_2) \left[\int \varphi_{f_1}(\mathbf r)\varphi_{f_2}(\mathbf r)\,d\mathbf r\right]^2 . \end{aligned} \tag{7.76} \]

It is appropriate to point out here that in determining the magnitude and sign of the exchange integral one must exercise great caution. According to (7.75), this integral in a more exact theory turns out to be much more complicated than is usually assumed when far-reaching conclusions are drawn concerning the dependence of this integral on interatomic distances, etc. As we have already noted earlier\(^{25}\), these conclusions do not have a proper theoretical foundation.

The perturbation method set forth can be developed further and the third approximation can be calculated both for the energy operator (7.43) and for the total-momentum operator (7.45). In this case it turns out that the mean value of the total momentum is different from zero because of the admixture of polar states, even though the system is in a quasihomeopolar state.

The expression for the energy operator has the same form as in the second approximation; only the exchange integral acquires a correction additive term.

5. In order to investigate the case in which the lowest states are not quasihomeopolar but polar, one may consider a somewhat different variant of the many-electron polar model\(^{20}\). We shall, exactly as before, seek the wave function of the system in the form of the series (7.2). We shall likewise restrict ourselves to taking into account only the \(s\)-states of the individual atoms. The indices \(\alpha_1,\alpha_2,\alpha_3,\ldots,\alpha_N\) in expression (7.1) will be divided into four groups: the numbers \(f_1,f_2,\ldots,f_s\) are the numbers of the \(s\)-sites of the crystal lattice near which there are two valence electrons with oppositely oriented projections of the spin moment (“pairs”); the numbers \(g_1,g_2,\ldots,g_s\) are the numbers of the \(s\)-sites near which there is no valence electron at all (“holes”); \(h_1,h_2,\ldots,h_t\) are the numbers of the \(t\)-sites near which there is one electron with right spin projection (“simple right”); and, finally, \(k_1,k_2,\ldots,k_u\) are the numbers of the \(u\)-sites with one valence electron whose spin projection has left orientation (“simple left”). The number of pairs \(s\) is at the same time also the number of holes. To specify the distribution of the electrons it is sufficient, for example, to specify any three of these sequences of numbers; then the fourth will be determined uniquely. These three sequences of numbers may be chosen as the dynamical variables of our system of interacting electrons. The amplitudes \(a\) in (7.2) will be functions of these numbers. It is easy to show that the secular equations (7.3) in this case take the form

\[ \begin{aligned} &\left\{E' - s(A+D) - \left[\sum_{f<f'}(B_{ff'}-I_{ff'}) +\sum_{g<g'}(B_{gg'}-I_{gg'}) \right.\right.\\ &\left.\left.\qquad\qquad\qquad -\sum_{f,g}(B_{fg}+I_{fg})\right]\right\}a(fgh) +\sum_{h,k} I_{hk}\,[a(T_{hk}\mid fgh)-a(fgh)]\\ &\quad+\sum_{f,g} J_{fg}\,[a(T_{fg}\mid fgh)-a(fgh)] -\sum_{f,p}\beta'_{fp}\,a(T_{fp}\mid fgh)+\\ &\quad+\sum_{g,p}\beta''_{gp}\,a(T_{gp}\mid fgh) +\sum_{h,k}\gamma_{hk}\,[a(S_{h\to f,\;k\to g}\mid fgh)-\\ &\qquad\qquad -a(S_{k\to f,\;h\to g}\mid fgh)] +\sum_{f,g}\delta_{fg}\,[a(S_{f\to h,\;g\to k}\mid fgh)-\\ &\qquad\qquad -a(S_{f\to k,\;g\to h}\mid fgh)] = 0; \qquad (7.77) \end{aligned} \]

here the quantity \(E'\), up to an additive constant, coincides

with the energy of the system \(E\)

\[ E' = E - N E_0 - C - \frac{1}{2}\sum_{q\ne q'} B_{qq'} + \frac{1}{2}\sum_{q\ne q'} I_{qq'} . \tag{7.78} \]

\(E_0\) is the unperturbed energy of the \(s\)-state of the isolated atom, \(C\) is the sum of all quasicalssical energies. The energy of electrostatic repulsion of two valence electrons in a pair is

\[ A=\int \varphi^2(\mathbf r)\, v(|\mathbf r-\mathbf r'|)\, \varphi^2(\mathbf r')\, d\mathbf r\, d\mathbf r', \tag{7.79} \]

\(B_{qq'}\) is the same energy for electrons of two different sites

\[ B_{qq'}=\int \varphi_q^2(\mathbf r)\, v(|\mathbf r-\mathbf r'|)\, \varphi_{q'}^2(\mathbf r')\, d\mathbf r\, d\mathbf r' . \tag{7.80} \]

For \(s=1\) and \(s=2\) this group of integrals in (7.77) coincides respectively with expressions (7.58) and (7.59) in our previous notation and gives the increase of energy due to the formation of pairs and holes for the degenerate levels of the zeroth approximation. The following integrals in (7.77) correspond to electronic transitions. \(I_{qq'}\) is the familiar exchange integral (see formula (7.75) or (7.76)) between “simple” sites. \(\beta'_{fp}\) is the integral of transfer of an electron from site \(f\) to site \(p\), equal to

\[ \begin{aligned} \beta'_{fp} &= \beta'^{(0)}_{fp} + \sum_{f_1\ldots f_{s-1}} \beta'_{fp f_i} - \sum_{g_1\ldots g_s} \beta'_{fp g_i};\\ \beta''_{gp} &= \beta''^{(0)}_{gp} + \sum_{f_1\ldots f_s} \beta''_{gp f_i} - \sum_{g_1\ldots g_{s-1}} \beta''_{gp g_i}; \end{aligned} \tag{7.81} \]

where

\[ \beta'^{(0)}_{fp} = \int \varphi_f(\mathbf r)\varphi_p(\mathbf r) \left[ \sum_{q(\ne p)} G(\mathbf r-\mathbf R_q) + \sum_q v(|\mathbf r'-\mathbf r|)\varphi_q^2(\mathbf r')\,d\mathbf r' \right]d\mathbf r; \]

\[ \beta''^{(0)}_{gp} = -\int \varphi_g(\mathbf r)\varphi_p(\mathbf r) \left[ \sum_{q(\ne p)} G(\mathbf r-\mathbf R_q) + \right. \]

\[ \left. +\sum_{q(\ne p)} v(|\mathbf r'-\mathbf r|)\varphi_q^2(\mathbf r')\,d\mathbf r' - v(|\mathbf r'-\mathbf r|)\varphi_p^2(\mathbf r')\,d\mathbf r' \right]d\mathbf r \]

and

\[ \beta'_{fpq} = \int \varphi_f(\mathbf r)\varphi_p(\mathbf r) v(|\mathbf r'-\mathbf r|)\varphi_q^2(\mathbf r')\,d\mathbf r'\,d\mathbf r; \]

\[ \beta''_{gpq} = -\int \varphi_g(\mathbf r)\varphi_p(\mathbf r) v(|\mathbf r'-\mathbf r|)\varphi_q^2(\mathbf r)\,d\mathbf r\,d\mathbf r' . \]

Hence it is clear that, strictly speaking, one cannot consider that the “kinetic energies” \(\beta'_{fp}\) and \(\beta''_{gp}\) of transfer of a pair or a hole depend only on the two coordinates \(f,p\) or \(g\) and \(p\) (\(p\) here is the number of a simple site, independently of the sign of the spin). However, if the number \(s\) is small, then the influence of the last two sums in (7.81) is small, and the quantities \(\beta'\) and \(\beta''\) are practically equal to their asymptotic values \(\beta'^{(0)}\) and \(\beta''^{(0)}\). Below only these asymptotic values of the transfer integrals are considered.

The integral \(\gamma_{hk}\) corresponds to the creation of a pair and a hole from two simple sites, the right \(h\) and the left \(k\); it is equal to

\[ \gamma_{hk}=\gamma^{(0)}_{hk} +\sum_{f_1\ldots f_{s+1}}\gamma_{hkf_i} -\sum_{g_1\ldots g_{s+1}}\gamma_{hkg_i}, \tag{7.82} \]

where

\[ \gamma^{(0)}_{hk} =\int \varphi_h(\mathbf r)\varphi_k(\mathbf r) \left[ \sum_{q(\ne k)}G(\mathbf r-\mathbf R_q) + \sum_{q(\ne k)} v(|\mathbf r'-\mathbf r|)\varphi_q^2(\mathbf r')\,d\mathbf r' \right]d\mathbf r \]

and

\[ \gamma_{hkg} = \int \varphi_h(\mathbf r)\varphi_k(\mathbf r) v(|\mathbf r'-\mathbf r|)\varphi_q^2(\mathbf r')\,d\mathbf r\,d\mathbf r'. \]

The integral \(\delta_{fg}\) corresponds to the decay of a pair \(f\) and a hole \(g\) into two simple sites—the right and the left. This integral is equal to

\[ \delta_{fg}=\delta^{(0)}_{fg} +\sum_{f_1\ldots f_{s-1}}\delta_{fgf_i} -\sum_{g_1\ldots g_{s-1}}\gamma_{fgg_i}, \tag{7.83} \]

where

\[ \delta^{(0)}_{fg} = \int \varphi_f(\mathbf r)\varphi_g(\mathbf r) \left[ \sum_{q(\ne g)}G(\mathbf r-\mathbf R_q) + \sum_{q(\ne g)} v(|\mathbf r'-\mathbf r|)\varphi_q^2(\mathbf r')\,d\mathbf r' \right]d\mathbf r \]

and

\[ \delta_{fgq} = \int \varphi_f(\mathbf r)\varphi_g(\mathbf r) v(|\mathbf r'-\mathbf r|)\varphi_q^2(\mathbf r')\,d\mathbf r\,d\mathbf r'. \]

The symbol \(T_{qq'}\) denotes the operation of interchanging the coordinates \(q\) and \(q'\) in the function \(a(fgh)\); the symbol \(S_{h\to f,\;k\to g}\) denotes the operation under which the simple right site \(h\) is transformed into the pair \(f\) and, simultaneously, the simple left site \(k\) into the hole \(g\), etc. \(D\) is the sum of exchange integrals for one site:

\[ \sum_{q'} I_{qq'}=D. \]

1. In the general case, solving these equations is practically impossible because of great mathematical difficulties. However, analysis of these equations in certain limiting cases yields much new information about the properties of a system of interacting electrons in a crystal. Thus, for example, in the case of a large polarization energy \(A\), when the ratio of the energies \(B, I, \beta, \gamma, \delta\) to the integral \(A\) may be regarded as a small quantity and the solution of equation (7.77) is sought in powers of these small quantities, it can be rigorously shown that the states of the electron system split into two classes. The first class of states turns out to be close to the states described by the exchange model—quasi-homeopolar states, which we encountered above in considering Bogoliubov’s method. In these states there is, of course, an “admixture” of polar states, because of the nonzero probability of pair and hole creation \((\gamma_{hk} \ne 0)\), but the corresponding amplitudes in the expansion (7.2) for this admixture are small quantities of order \(\gamma/A\), etc. As an illustration, let us consider the simplest model—a linear chain of \(N\) atoms, in which one site has an electron with right spin, and the remaining sites have electrons with left spin. Each of the unperturbed nonpolar states of this model is completely determined by specifying the number \(h\) of the single site with right spin. Therefore the corresponding wave function is simply equal to \(a_0(h)\). Of the polar states, in the first approximation only those functions \(a_1(f,g)\) for which the sites \(f\) and \(g\) are nearest neighbors, i.e. \(f, f \pm 1\), will differ from zero. This state can obviously arise only from two nonpolar states \(a_0(f)\) and \(a_0(f+1)\). From the general perturbation theory it follows that the relation between these functions must be linear. Indeed, from (7.77) we find approximately:

\[ a_1(f, f+1)=\frac{\gamma}{A}\,[a_0(f+1)-a_0(f)]. \tag{7.84} \]

\[ a_1(f+1, f)=\frac{\gamma}{A}\,[a_0(f)-a_0(f+1)]. \tag{7.85} \]

Since \(\gamma \ll A\), it is seen from this that the amplitudes of the polar states \(a_1(f, f \pm 1)\) are smaller than for the initial nonpolar states. If we now pass to polar functions \(a_1(f,g)\), for which \(|f-g|>1\), then they turn out to be still smaller than the functions just considered for “neighbors.” To see this, let us write equation (7.77) for the function \(a_1(f,g)\). This gives

\[ |E' - A + B(|f-g|)|a_1(f,g)= \]

\[ =\beta'[a_1(f+1,g)+a_1(f-1,g)]+ \]

\[ +\beta''[a_1(f,g+1)+a_1(f,g-1)] \tag{7.86} \]

\[ |f-g|>1. \]

For the case of “neighbors,” equations (7.86) take the form

\[ [\varepsilon-A+B_1]a_1(f+1,f)= \]

\[ =\beta'_1 a_1(f+2,f)+\beta''_1 a_1(f+1,f-1)+Ja_1(f,f+1)+ \]

\[ +\gamma[a_0(f+1)-a_0(f)], \tag{7.87} \]

\[ [\varepsilon-A+B_1]a_1(f,f+1)= \]

\[ =\beta'_1 a_1(f-1,f+1)+\beta''_1 a_1(f,f+2)+Ja_1(f+1,f)+ \]

\[ +\gamma[a_0(f)-a_0(f+1)], \tag{7.88} \]

where \(\beta'_1\) and \(\beta''_1\) are the values of the transfer integrals for \(|f-g|=1\). Equations (7.87) and (7.88) for large \(A\) give the already known formulas (7.84) and (7.85). In finding the solution of equation (7.86), let us recall (see Section 5) that a system of interacting electrons in a crystal always possesses a total quasimomentum by virtue of translational invariance. Therefore we shall seek the solution of (7.86) in the form

\[ a_1(f,g)=e^{i\omega f}b(f-g). \tag{7.89} \]

On the other hand, from the usual exchange theory\({}^{39}\) we have

\[ a_0(f)=\frac{1}{\sqrt{N}}e^{i\omega f}; \tag{7.90} \]

therefore, from (7.84) and (7.85) we find:

\[ b_1=\frac{1}{\sqrt{N}}\frac{\gamma}{A}(1-e^{i\omega}),\qquad b_{-1}=\frac{1}{\sqrt{N}}\frac{\gamma}{A}(e^{i\omega}-1). \tag{7.91} \]

Next, substituting (7.89) into (7.86), we have

\[ (E'-A+B_k)b_k=(\beta'e^{i\omega}+\beta'')b_{k+1}+(\beta'e^{-i\omega}+\beta'')b_{k-1}. \tag{7.92} \]

Let first \(1<k<n\); then for the case \(k=2\), from (7.92) we obtain

\[ (E'-A+B_2)b_2=(\beta'e^{i\omega}+\beta'')b_3+(\beta'e^{-i\omega}+\beta'')b_1. \tag{7.93} \]

The left-hand side of this equation in the first approximation is equal to \(-Ab_2\). Suppose that \(b_k\) decreases with increasing \(k\). Then the largest term in the formula for \(b_2\) must, according to (7.93), have the form

\[ b_2=-\frac{1}{A}(\beta'e^{-i\omega}+\beta'')b_1. \]

Next, for \(k=3\) we have

\[ [E'-A+B_2]b_3=(\beta'e^{i\omega}+\beta'')b_4+(\beta'e^{-i\omega}+\beta'')b_2, \]

whence, in the same approximations, we obtain

\[ b_3=\frac{1}{A^3}(\beta'e^{-i\omega}+\beta'')^3 b_1. \]

Direct substitution of this expression into (7.93) shows that taking into account the term with \(b_3\) indeed gives in \(b_2\) a correction

of higher order of smallness. Therefore the assumption that \(b_k\) decreases with increasing \(k\) is justified.

Thus, for all \(k\) constrained by the inequality \(1<k<n\), we obtain

\[ b_k=(-1)^{k-1}\left(\frac{\beta' e^{-i\omega}+\beta''}{A}\right)^k b_1, \tag{7,94} \]

and for \(-1>k>-n\) we find

\[ b_k=(-1)^{|k|-1}\left(\frac{\beta' e^{i\omega}+\beta''}{A}\right)^{|k|} b_{-1}. \tag{7,95} \]

In order to take account of the finiteness of the crystal, we introduce periodicity conditions \(b_{k+N}=b_k\), in particular \(b_n=b_{-n}\) \((n=\tfrac{1}{2}N)\), and therefore

\[ b_n=b_{-n}=-\frac{1}{A}\left[(\beta' e^{i\omega}+\beta'')b_{-n+1}+(\beta' e^{-i\omega}+\beta'')b_{n-1}\right]. \tag{7,96} \]

Formulas (7,89), (7,91), (7,94), (7,95), and (7,96), taken together, give the complete solution of the problem.

To find the corrections to the energy of the crystal caused by the admixture of polar states, let us consider the equation for the functions \(a_0(h)\). From (7,77) we find:

\[ \varepsilon a_0(h)+I[a_0(h+1)+a_0(h-1)-2a_0(h)]= \]
\[ =\delta[a_1(h,h+1)+a_1(h,h-1)-a_1(h+1,h)- \]
\[ -a_1(h-1,h)]. \tag{7,97} \]

Substitution of (7,84) and (7,85) into (7,97) shows that, in the first approximation, the influence of the processes of annihilation of a doublet and a hole on the states \(a_0(h)\) can be described by introducing an “effective exchange integral”

\[ I'=I-2\frac{\gamma\delta}{A}; \tag{7,98} \]

the coefficients \(a_0(h)'\) do not change in this approximation, while the energy acquires the increment

\[ \Delta E=-4\frac{\gamma\delta}{A}(1-\cos\omega). \tag{7,99} \]

It is easy to see that the calculations can be carried out to any degree of approximation. The result obtained coincides completely with the second approximation in Bogoliubov’s theory (see the expression for the exchange integral (7,5)).

The physical meaning of the result obtained is the following: among the states corresponding to the presence of one electron with right spin \(h\), there are \(N\) such states that are close to the states of the usual exchange model, which ignores polar states. Although in the refined model under consideration the probability of the existence of doublets and holes is taken into account also in these states, under the adopted assumption of large energy \(A\) this probability is small and rapidly

decreases as the distance between a pair and a hole increases. (It is clear that if all the functions \(a_1(f,g)\) were quantities of order \(A^{-1/2}\), their influence would still predominate, since the number of these functions is equal to \(N(N-1)\), whereas the number of functions \(a_0(h)\) is equal to \(N\).)

If the “degree of polarity” of a given state can be characterized by the mean number of pairs (or holes) \(\bar{s}\) in this state, then for the class of quasihomeopolar states considered we have

\[ \bar{s}\sim \frac{1}{A}\ll 1. \tag{7,100} \]

A known exception is the case \(a_0(h)=\mathrm{const}\), corresponding to the lowest energy level of a ferromagnetic semiconductor crystal; in this case the states are purely nonpolar, for which \(\bar{s}=0\).

It is not difficult to show that this general conclusion is valid not only for the particular case considered, but also for any number of right spins, and likewise for the case of a three-dimensional lattice.

In order to prove that the considered class of states of electrons in a crystal can indeed be called quasihomeopolar, it is necessary to clarify the question of the mean value of the operator of the total momentum of the system, and also to calculate the accelerating effect of an external electric field.

The total mean current \(\bar{P}_\gamma\), carried by the entire system of electrons throughout the crystal, is given by the expression

\[ \bar{P}_\gamma = \frac{eh}{2mi} \sum_{j=1}^{N} \int \left[ \Psi^{*}\frac{\partial \Psi}{\partial (r_\gamma)_j} - \Psi\frac{\partial \Psi^{*}}{\partial (r_\gamma)_j} \right]\,d\tau . \tag{7,101} \]

Substituting in (7,101) the complete wave function

\[ \Psi(\mathbf r_1\ldots \mathbf r_N) = \sum_{fgh} a(fgh)\, \Psi_{fgh}(\mathbf r_1\ldots \mathbf r_N), \tag{7,102} \]

we find, instead of (7,101),

\[ \bar{P}_\gamma = \frac{eh}{2mi} \sum_{j=1}^{N} \sum_{fgh<f'g'h'} \left[ a^{*}(fgh)a(f'g'h') - a^{*}(f'g'h')a(fgh) \right] \int \left[ \psi^{*}_{fgh} \frac{\partial \Psi_{f'g'h'}}{\partial (r_\gamma)_j} - \psi_{f'g'h'} \frac{\partial \Psi^{*}(fgh)}{\partial (r_\gamma)_j} \right]\,d\tau . \tag{7,103} \]

It is easy to show\(^{20a}\) that in the zeroth approximation, when \(a_1(fg)=0\) and the wave function of the system is equal to \(a_0(h)\), the mean current is exactly

is equal to zero, just as was indicated above in the discussion of Bogolyubov’s method. Further, if polar states are taken into account, but in the first approximation (7.84) and (7.85), then the mean current will also be equal to zero. A current different from zero is obtained only in higher approximations (see above, p. 345).

7. Let us consider the accelerating action of an external electric field. The consideration of this question is made difficult in our model by the fact that the wave functions of the adopted representation do not form a complete system and are not strictly orthogonal. (However, the latter circumstance can be taken into account by the method indicated by Bogolyubov—see above, p. 329.) Therefore the application here of the usual quantum-mechanical methods of taking into account the influence of an external electric field requires special justification. The following method of consideration is the most consistent. The operator of the potential energy of an external constant electric field \(F\), directed along the \(x\)-axis, is equal to

\[ \hat W = eF \sum_{j=1}^{N} x_j . \tag{7.104} \]

If one neglects effects of the type of excitation of an atom by a constant field, then the complete wave equation

\[ i\hbar \frac{\partial \psi}{\partial t} = (\hat H + \hat W)\Psi \]

must be written in the new representation, taking as the “coordinates” of the system the numbers of the sites \(f, g, h\). The operator (7.104) has, above all, nondiagonal elements corresponding to the various possible types of electronic transitions. These elements are simply added to the corresponding elements of the energy operator. The largest of these elements corresponds to the simple transition of an electron from one atom to a neighboring one. Neglecting the orthogonality of the functions, this element will be equal to

\[ eF \int \varphi_f(\mathbf r)\, x\varphi_{f+1}(\mathbf r)\, d\mathbf r . \]

Under the assumptions made, this expression does not depend on the site number \(f\) and is, in order of magnitude, smaller than \(eFd\), where \(d\) is the lattice constant. For weak fields \(eFd\) is certainly smaller than all the other energies in equation (7.77). Thus, taking into account the nondiagonal elements of the operator \(\hat W\) reduces merely to the introduction of relatively very small additions to the energies of types \(\beta\) and \(\gamma\) (some of these additions entering with the sign \(+\), and others with the sign \(-\)). It is clear, therefore, that we lose nothing essential if we do not consider these elements at all. The diagonal elements of the operator \(\hat W\), however, differ from zero only in

polar states and, as can easily be shown \(^{20a}\), are equal to

\[ eFd \sum_{j=1}^{s} (f_j-g_j), \tag{7,105} \]

where the summation is over the numbers of all twos and holes in the given state. The appearance of these expressions constitutes, in mathematical form, the most essential result of switching on the electric field.

To illustrate the method, let us consider the simpler problem of the influence of an external field on the motion of a single “strongly bound” electron in the one-electron model (see Section 4). In this case the wave equation takes the form

\[ i\hbar \dot a(f,t)=\beta [a(f+1,t)+a(f-1,t)]+eFd\cdot f a(f,t), \]

where \(\beta\) is the energy of transition of the electron from site to site. The exact solution of this equation is

\[ a(f,t)=\frac{1}{\sqrt{N}} e^{\,i\left(\omega-\frac{eFd}{\hbar}t\right)f} e^{-\frac{i}{\hbar}2\beta \int_{0}^{t}\cos\left(\omega-\frac{eFd}{\hbar}t'\right)\,dt'} . \]

The meaning of this solution may be briefly expressed by saying that under the influence of the field the quasimomentum \(\omega\) changes to \(\omega-\dfrac{eFd}{\hbar}t\). But this is already known from the exact theory (see Sections 4 and 5), and therefore one may state that the approximation adopted has reasonable grounds.

Let us clarify the influence of the perturbation (7,105) on the quasihomeopolar states under consideration. It is not difficult to convince oneself that, under the assumptions made, the system considered will possess stationary states even in the presence of an electric field, and moreover these states will differ practically in no way from those that existed before the field was switched on.

What is essential is that, in the zeroth approximation, the matrix elements of the operator (7,104) are exactly equal to zero \(^{20a}\). In the first approximation the influence of the field will appear only in the fact that in the diagonal terms of equations of the type (7,87) and (7,88) one must add the expression \(-eFd\), negligibly small in comparison with all the other energies. It is clear that the perturbation (7,105) would give something significant only in those equations which correspond to very large differences \((f-g)\). But these terms play no role, because for \((f-g)\sim \dfrac{A}{eFd}\) this will lead only to a more rapid decrease of the functions \(a_1(fg)\) with increasing \((f-g)\), while such functions, as was shown above, are practically zero even without this.

The results obtained can be given the following intuitive explanation. It was found above that the wave functions of higher approximations for states with \(s \ne 0\) have a form analogous to (7.89), namely, \(\sim e^{i\omega f} b(f-g)\). The appearance of the factor \(e^{i\omega f}\), where \(\omega\) is the total quasimomentum of the “pair–hole” system, indicates that this system as a whole can propagate through the crystal lattice (translational invariance). However, this stable system is electrically neutral, and therefore a constant electric field cannot exert any accelerating action on it, although a weak electric field does lead to a slight increase in the probability of transition of the electron to a neighboring site.

Under the assumptions made (large energy \(A\)), these probabilities themselves are so small that the effect of the electric field in essence changes nothing. One may say visually that the “cutting” of the potential barrier, which is caused by a constant field, takes effect only at distances very large compared with the lattice constant \(a\), for which the transition probabilities are practically equal to zero.

Thus we arrive at the conclusion that in states of the type of a disassembled quasihomeopolar state, up to very small quantities, no current arises under the influence of an external electric field. But for sufficiently large \(A\) the energetically lowest states will certainly belong to them. Consequently, under the assumptions considered, our model represents a dielectric or a semiconductor. The most significant consequence of this conclusion is that the existence of semiconductors (or dielectrics) with a continuous energy spectrum is possible in principle (see above, Section 5).

The existence of bodies of this kind is in principle not excluded also in the one-electron theory (Section 4). Indeed, let us imagine that the lowest energy state of the “gas” of noninteracting electrons in this model corresponds to a uniform distribution of electrons over an energy band, one electron on each level with a given quasimomentum. Such a state is not energetically isolated; nevertheless, by virtue of the known selection rules (see Sections 4 and 5), there can be no accelerating action here on the part of the electric field. This analogy can be carried somewhat further. If, for example, one takes only that part of the complete many-electron wave function which corresponds to purely nonpolar states, and constructs from it the density matrix (see Sections 4 and 5) for one electron, reduced over the spin coordinates—\((xyz|\hat{\rho}|x'y'z')\), then this matrix will coincide exactly with the density matrix of the one-electron model for the case of uniform filling of bands.

The difference consists only in the fact that, whereas in the one-electron model the indicated state can turn out to be the energetically lowest only in an entirely exceptional case, in the many-electron model the situation, as is clear from the preceding discussion, is essentially different.

8. Thus, in order to consider phenomena associated with conductivity, it is necessary to abandon the consideration of quasi-homeopolar states and pass to the consideration of polar states. To simplify the calculation, let us first assume that the effects of creation and annihilation of pairs and holes have no very substantial influence on the form of the stationary states of the system, so that, in a known approximation, the total number of these excitations of the system may simply be regarded as a constant of the motion. As can be seen at least from the preceding example, under the assumption that \(A\) is much greater than all the other energies, such an assumption is quite legitimate. But, on the other hand, as follows from the general properties of the energy spectrum of the system and as will be shown below, polar states can prove to be energetically the lowest only in the case when the energies of type \(\beta\) are greater than \(A\), or at least of the order of \(A\). Thus, strictly speaking, the following reasoning applies not to the lowest state of a metal, but to the excited states of a semiconductor or a dielectric.

In addition to disregarding the effects of annihilation of pairs and holes, let us introduce the following radical simplifications: we shall refuse to consider all exchange effects, i.e. we shall take as the coordinates of the system exclusively the numbers of the pairs and holes, \(f\) and \(g\). For a first orientation such a simplification is quite reasonable. However, even under these assumptions the problem cannot be solved exactly. First, here, in contrast to the quasi-homeopolar case, the transition from a one-dimensional model to a three-dimensional one is connected with substantial difficulties. Further, great complications are introduced into the problem by the presence of electrostatic interaction between a pair and a hole, the possibility of their exchanging places, and also by the circumstance that the “transfer energy” \(\beta'\) and \(\beta''\) depends on the mutual arrangement of these individuals.

It is expedient for the time being to abstract from these difficulties and return to them later.

The simplest model is as follows: in a linear chain of ions two excitations move—a pair and a hole. They cannot enter one and the same site and therefore are, as it were, impenetrable to one another. Their interaction is reduced exclusively to this. The wave equation for such a system can be obtained from equations (7.86)—(7.88) after the corresponding alteration of them in accordance with the simplifications introduced. Since only purely polar states with \(s = 1\) are considered, for brevity of notation one may omit, in the diagonal terms, the energy \(A\), which plays the role of an addi-

QUANTUM THEORY OF ELECTRON CONDUCTORS

tive constant. As a result we find:

\[ \begin{gathered} E'a(f,g)=\beta'[a(f+1,g)+a(f-1,g)]+{}\\ +\beta''[a(f,g+1)+a(f,g-1)];\quad |f-g|>1;\\ E'a(f+1,f)=\beta'a(f+2,f)+\beta''a(f+1,f-1). \end{gathered} \tag{7,106} \]

In solving this system it is convenient to proceed from the following. Equations (7,106) establish no relation between those functions \(a(f,g)\) for which \(f>g\), and those for which \(g>f\). It is clear that, under the assumptions we have made, such a relation can be established by the periodicity conditions. Introduce the notation

\[ \begin{aligned} a(f,g)&=a^{(1)}(f,g)\quad (f>g),\\ a(f,g)&=a^{(2)}(f,g)\quad (f<g). \end{aligned} \tag{7,107} \]

Then the periodicity conditions are written as

\[ \begin{aligned} a^{(1)}(f+N,g)&=a^{(2)}(f,g);\\ a^{(2)}(f,g+N)&=a^{(1)}(f,g). \end{aligned} \tag{7,108} \]

The solutions of (7,106) have the form

\[ a^{(1)}(f,g)=P_1\left[e^{i(\xi f+\eta g)}-e^{i(\eta f+\xi g)}\right]e^{i\varphi\frac{f-g}{2}}, \tag{7,109} \]

\[ a^{(2)}(f,g)=P_2\left[e^{i(\xi f+\eta g)}-e^{i(\eta f+\xi g)}\right]e^{i\varphi\frac{f-g+N}{2}}. \tag{7,110} \]

The phase \(\varphi\) is determined from the condition

\[ e^{i\varphi}=\frac{\beta'e^{i(\xi+\eta)}+\beta''}{\beta'+\beta''e^{-i(\xi+\eta)}}, \]

and the amplitudes \(P_1\) and \(P_2\)—from (7,108). It is easy to see that the number of different solutions is \(N(N-1)\), as was required to be determined. The energy of the system is equal to

\[ \begin{aligned} E'-A&=2\left[\beta'\cos\left(\xi+\frac{\varphi}{2}\right)+\beta''\cos\left(\eta-\frac{\varphi}{2}\right)\right]=\\ &=2\left[\beta'\cos\left(\eta+\frac{\varphi}{2}\right)+\beta''\cos\left(\xi-\frac{\varphi}{2}\right)\right]=\\ &=2\cos\frac{\xi-\eta}{2}\left[\beta'^2+\beta''^2+2\beta'\beta''\cos(\xi+\eta)\right]^{\frac12}. \end{aligned} \tag{7,111} \]

The solution obtained can be explained visually by the fact that a stationary state is established in the chain, representing the result of the collision of a doublet with a hole. They are impenetrable to one another; therefore, upon collision, they exchange quasimomenta and scatter. The width of the corresponding energy band (7,111) is determined by the magnitude of the integrals \(\beta'\) and \(\beta''\). In this case of polar states the doublet and the hole are, as it were, independent particles with their own values of quasimomenta, whereas in the case of quasihomopolar states (see (7,89)) there existed a complex of a doublet and a hole, which moved as a single system.

The solutions obtained are readily generalized, in the same approximations, to the case of large values of the number \(s\). In particular, the formula for the energy, if one sets \(\beta'=\beta''\), will have the form

\[ E' = sA + 2\beta \left\{ \sum_{i=1}^{s} \cos \xi_i + \sum_{i=1}^{s} \cos \eta_i \right\}. \tag{7,112} \]

As long as \(s < \dfrac{N}{2}\), the width of the band increases as \(s\) grows. But when \(s\) becomes comparable with \(\dfrac{N}{2}\), the restrictions imposed on the choice of the possible values \(\xi_1,\xi_2,\ldots\) and \(\eta_1,\eta_2,\ldots\) begin to play a role.^205 This leads to the fact that, beginning with some \(s\), the energy band narrows again (in our approximations, at \(s=\dfrac{N}{2}\) the spectrum should turn into an isolated energy level). However, if the phenomenon of exchange between pairs and holes is taken into account, then even at \(s=\dfrac{N}{2}\) the spectrum will have the form of a band whose width is determined by the exchange integral.

The picture obtained for the energy spectrum makes the existence of metals understandable from the point of view of the many-electron polar model. When the transfer energies \(\beta'\) and \(\beta''\) are comparable with \(A\), the effect of broadening of the energy band with increasing \(s\) may “overlap” the effect of growth of the energy center of gravity. Therefore the energy minimum may turn out to belong to a state with finite \(s\).

One can dispense with a number of the simplifying assumptions made above in the consideration of polar states. If the electrostatic interaction between a pair and a hole, and between two pairs or between two holes, is taken into account, then one can find such stationary states in which a pair and a hole constitute a single complex. Obviously, in these states, just as in quasi-homeopolar states, there is no current. There will also be states of “merged” pairs or holes, as one of the varieties of polar states. In a somewhat different connection, analogous states (the so-called doublons) were considered by Volkenstein and Bonch-Bruevich.^30

9. One may also consider excited states, when one of the functions in the product \(\psi_{fgh}(\mathbf{r}_1 \ldots \mathbf{r}_N)\) belongs not to an \(s\)-state but to an excited state. Such “exciton” states were first considered by Frenkel.^31

The most severe approximation, however, is the assumption of the large magnitude of the energy \(A\). So far it has been possible to solve exactly

only a special case for one right spin, without making any assumptions about the magnitude of the energy \(A\). It is essential that in the general case too one can distinguish two classes of states which only in the limit of large \(A\) pass into the quasihomeopolar and excited polar states considered above. At the same time it must by no means be assumed that states close to \((7,89)\) will necessarily be the lowest in energy. This may be regarded as a fundamental proof of the explanation of the metallic state in the polar many-electron model.

10. Turning to the question of the mean current in polar states, it is easy to show that in the case under consideration \((7,103)\) takes the form

\[ \overline{P}_{r}=-\frac{e\hbar R}{m i}\sum_{f,g}\{a^{*}(f,g)[a(f+1,g)-a(f-1,g)-a(f,g+1)+ \]
\[ +a(f,g-1)]-a(f,g)[a^{*}(f+1,g)-a^{*}(f-1,g)- \]
\[ -a^{*}(f,g+1)+a^{*}(f,g-1)]\}, \qquad (7,113) \]

where

\[ R=\int \varphi_f(\mathbf r)\frac{\partial\varphi_{f+1}(\mathbf r)}{\partial(r_\gamma)}\,d\mathbf r. \]

Substituting solutions of the type \((7,109)\) into \((7,113)\), we find that the current is equal to

\[ \overline{P}_{r}=\frac{2e\hbar}{m}R(\sin \xi-\sin \eta). \qquad (7,114) \]

Naturally, in the energetically lowest state the mean current is zero, for \(\xi=0\) and \(\eta=0\), but infinitely close excited states will already have a current. The same expression for the current was recently obtained by Tissa in his theory of superconductivity on the many-electron model[^32]. The polar model set forth here also includes the states postulated by Tissa. These are the states in which pairs and holes are distributed in the crystal in an ordered manner, and such ordered “lattices” of pairs and holes, by virtue of translational invariance, propagate “freely” through the crystal. With increasing temperature the system passes into more excited states, to which a lesser “order” in these electronic “lattices” corresponds.

Returning to the problem of determining the accelerating action of an external electric field on quasihomeopolar states, the following remarks may be made. If the electrical conductivity is expressed by the classical formula \((3,1)\), then for the effective mass we obtain enormous values \(10^{-6}—10^{-1}\,\mathrm{g}\), or, for the effective number of conduction electrons, correspondingly very small quantities. Therefore, for pure semiconductors or dielectrics at not very high temperatures the electrical conductivity lies within the limits-

in the range from \(10^{-15}\) to \(10^{-20}\ \mathrm{ohm}^{-1}\!\cdot\mathrm{cm}^{-1}\). In strong fields \((\gtrsim 10^{-5}\ \mathrm{CGSE})\) Ohm’s law is violated owing to the enhancement of the tunneling effect through the potential barrier “cut off” by the external field. The electrical conductivity in this case will obey the law

\[ \sigma=\sigma_{0}e^{a_{1}\tau F}, \tag{7,115} \]

where \(a_{1}\) is a constant quantity. This formula is in qualitative agreement with experiment, as follows, for example, from the well-known work of A. V. and A. F. Ioffe\(^{38}\).

11. The attempts described above to solve equations (7,77) of the polar model are still very far from their broad use for concrete calculations of various properties of crystalline bodies. Much more powerful for this purpose is the method of second quantization. It turns out that here it is more convenient not to proceed by Bogoliubov’s general method, but to introduce several other second-quantization operators\(^{20a}\), chosen specially to take account of polar states. If, as before, the system of atomic \(S\)-functions is used as the basis of the representation, then it is convenient to introduce the following coordinates:

\[ P_q \begin{cases} =1, & \text{when site } q \text{ is a doublet,}\\ =0 & \text{in all other cases;} \end{cases} \]

\[ Q_q \begin{cases} =1, & \text{when site } q \text{ is a hole,}\\ =0 & \text{in all other cases;} \end{cases} \]

\[ R_q \begin{cases} =1, & \text{when site } q \text{ is simple right,}\\ =0 & \text{in all other cases;} \end{cases} \]

\[ S_q \begin{cases} =1, & \text{when site } q \text{ is simple left,}\\ =0 & \text{in all other cases.} \end{cases} \]

In the model under consideration these quantities satisfy the conditions

\[ P_q+Q_q+R_q+S_q=1 \quad (\text{for all } q), \]

moreover,

\[ \sum_q P_q=\sum_q Q_q=s,\qquad \sum_q (R_q-S_q)=2m. \]

Next, let us introduce the shift operators

\[ \Gamma_q^{+}\Gamma_q^{-},\qquad \gamma_q^{+}\gamma_q^{-},\qquad \Delta_q^{+}\Delta_q^{-},\qquad \delta_q^{+}\delta_q^{-}, \]

which are defined by the conditions

\[ \Gamma_q^{+} f(P_q\ldots)=f(P_q+1,\ldots);\qquad \Gamma_q^{-} f(P_q\ldots)=f(P_q-1,\ldots); \]

and so on.

Finally, we introduce the Bose amplitudes of secondary quantization:

\[ \Phi_q=\sqrt{P_q}\Gamma^{-},\quad \Phi_q^{+}=\Gamma_q^{+}\sqrt{P_q},\quad \Psi_q=\sqrt{Q_q}\gamma_q,\quad \Psi_q^{+}=\gamma_q^{+}\sqrt{Q_q}, \]
\[ \varphi_q=\sqrt{R_q}\Delta_q^{-},\quad \varphi_q^{+}=\Delta_q^{+}\sqrt{R_q},\quad \psi_q=\sqrt{S_q}\delta_q,\quad \psi_q^{+}=\delta_q^{+}\sqrt{S_q}. \]

It is easy to see that

\[ \Phi_q\Phi_q^{+}=P_q,\quad \Psi_q\Psi_q^{+}=Q_q,\quad \varphi_q\varphi_q^{+}=R_q,\quad \psi_q\psi_q^{+}=S_q, \tag{7,116} \]

\[ \left. \begin{gathered} \Phi_q\Phi_q^{+}+\Psi_q\Psi_q^{+}+\varphi_q\varphi_q^{+}+\psi_q\psi_q^{+}=1,\\ \sum_q \Phi_q\Phi_q^{+}=\sum_q \Psi_q\Psi_q^{+}=S,\\ \sum_q\left(\varphi_q\varphi_q^{+}-\psi_q\psi_q^{+}\right)=2m. \end{gathered} \right\} \tag{7,117} \]

These operators satisfy the commutation relations

\[ \Phi_q^{+}\Phi_{q'}-\Phi_{q'}\Phi_q^{+}=\delta_{qq'},\quad \Phi_q\Psi_{q'}-\Psi_{q'}\Phi_q=0\ \text{etc.} \tag{7,117'} \]

The operators of simple \(\varphi_q\) and \(\psi_q\) exactly coincide with the Bose amplitudes introduced above (7,66), when the number of twos and holes is equal to zero.

As a result of standard transformations, from equation (7,77) we obtain the energy operator in the secondary-quantization representation:

\[ \begin{aligned} H={}&s(A+D)+\frac12\sum_{q\ne q'} (\Phi_q\Phi_q^{+}-\Psi_q\Psi_q^{+})(\Phi_{q'}\Phi_{q'}^{+}-\Psi_{q'}\Psi_{q'}^{+})B_{qq'}\\ &-\frac12\sum_{q\ne q'} (\Phi_q\Phi_q^{+}-\Psi_q\Psi_q^{+})(\Phi_{q'}\Phi_{q'}^{+}-\Psi_{q'}\Psi_{q'}^{+})I_{qq'}+\\ &+\frac12\sum_{q\ne q'} (\varphi_q\psi_{q'}-\psi_q\varphi_{q'})(\varphi_q^{+}\psi_{q'}^{+}-\psi_q^{+}\varphi_{q'}^{+})I_{qq'}+\\ &+\frac12\sum_{q\ne q'} (\Phi_q\Psi_{q'}-\Psi_q\Phi_{q'})(\Phi_q^{+}\Psi_{q'}^{+}-\Psi_q^{+}\Phi_{q'}^{+})I_{qq'}+\\ &+\frac12\sum_{q\ne q'} (\varphi_q\varphi_{q'}^{+}+\psi_q\psi_{q'}^{+})\Phi_{q'}\Phi_q^{+}\beta_{qq'}'\\ &-\frac12\sum_{q\ne q'} (\varphi_q\varphi_{q'}^{+}+\psi_q\psi_{q'}^{+})\Psi_{q'}\Psi_q^{+}\beta_{qq'}''+\\ &+\frac12\sum_{q\ne q'} (\varphi_q\psi_{q'}-\psi_q\varphi_{q'})(\Phi_q^{+}\Psi_{q'}^{+}-\Psi_q^{+}\Phi_{q'}^{+})\gamma_{qq'}+\\ &+\frac12\sum_{q\ne q'} (\varphi_q^{+}\psi_{q'}^{+}-\psi_q^{+}\varphi_{q'}^{+})(\Phi_q\Psi_{q'}-\Psi_q\Phi_{q'})\delta_{qq'}. \end{aligned} \tag{7,118} \]

One can obtain a more general expression for this operator also by taking excitons into account[^23]. Separating out small excitations—for example, small deviations from magnetic saturation or weak polarization, or, conversely, small deviations from almost complete polarity—makes it possible to linearize the operator (7,118) and immediately find its eigenvalues, without solving the wave equation.

If one sets \(\Phi_q=\Psi_q=0\), then the operator (7,118) immediately becomes the operator of the ordinary exchange model. It can be shown that the operator (7,118) is invariant with respect to coordinate transformations. Just as in Bogolyubov’s method, any other dynamical quantity of the electronic system under consideration can be expressed in terms of the operators (7,116). In particular, the operator of a component of the total momentum of the system has the form

\[ \dot P_r= \frac{i\hbar}{2}\sum_{q\ne q'} \left[ (\varphi_q\varphi_{q'}^{+}+\psi_q\psi_{q'}^{+}) (\Phi_q\Phi_{q'}^{+}+\Psi_q\Psi_{q'}^{+})- \right. \]

\[ \left. -(\varphi_q\varphi_{q'}^{+}+\psi_q\psi_{q'}^{+}) (\Phi_q\Phi_{q'}^{+}+\Psi_q\Psi_{q'}^{+})+ \right. \]

\[ \left. +(\varphi_q\psi_{q'}-\psi_q\varphi_{q'}) (\Phi_q^{+}\Psi_{q'}^{+}+\Psi_q^{+}\Phi_{q'}^{+})- \right. \]

\[ \left. -(\varphi_q^{+}\psi_{q'}^{+}-\psi_q^{+}\varphi_{q'}^{+}) (\Phi_q\Psi_{q'}+\Psi_q\Phi_{q'}) \right]R_{qq'}, \tag{7,119} \]

where

\[ R_{qq'}=\int \varphi_q(\mathbf r)\, \frac{\partial \varphi_{q'}(\mathbf r)}{\partial(r_r)}\,d\mathbf r. \]

The quantity \(\hat P\) vanishes for \(\varphi_q=\psi_q=0\) and for \(\Phi_q=\Psi_q=0\). However, the first condition is essentially connected with the fact that, in deriving this formula, the functions \(\varphi_q(\mathbf r)\) and \(\varphi_{q'}(\mathbf r)\) (for \(q\ne q'\)) were assumed to be strictly orthogonal, whereas the second condition remains valid also under an exact treatment and expresses the fact that in nonpolar states the total current carried by the system of electrons is equal to zero.

The energy operator (7,118) makes it possible to write equations of motion for any operator that can be expressed as a function of the operators (7,116). Owing to the noncommutativity of these operators, the equations obtained are so complicated that it is practically impossible to solve them. Therefore one usually uses the quasiclassical approximation[^20a][^15][^31]. In formula (7,118) the noncommutativity of the quantities \(\Phi_q\), \(\Psi_q\), \(\varphi_q\), \(\psi_q\) is neglected, and the equations of motion for them are considered as nonlinear equations analogous to the equations of the Fock-type self-consistent-field method. Such a simplification, of course, deprives us of the possibility of obtaining exact information about the number and distribution of the energy levels in the energy spectrum of the system. However, the boundaries of the energy spectrum can then be determined correctly. Quantitatively, the quasiclassical approximation may be justified better,

if the processes of creation and annihilation of pairs and holes are neglected and the number \(s\) is regarded as a constant of the motion. If, however, in expression (7,118) the terms with “interchanged” \(s\)’s are also taken into account, then the application of the quasi-classical method will be still less justified. Nevertheless, in this case as well one can obtain qualitative information on the boundaries and mutual disposition of the energy bands.

12. As an example, let us consider the case \(s=n\), i.e. the case of maximum “polarity.” If the exchange integrals are neglected, then from the basic equation of the problem (7,77) we obtain the usual classical expression for the energy of electrostatic interaction

\[ E' = nA + \sum_{f\cdot f'} B_{ff'} + \sum_{g\cdot g'} B_{gg'} - \sum_{f,g} B_{fg}. \tag{7,120} \]

Depending on the relation between the absolute values of the quantities \(A\) and \(B\), formula (7,120) gives for \(E'\) either positive or negative values. It follows that in some cases, on the basis of simple electrostatic considerations, one may expect the maximum polarity of the states to be energetically most favorable. For simplicity we shall restrict ourselves to taking into account only those integrals \(B_{qq'}\) which refer to neighboring lattice sites (this is not very accurate, since the energy \(B_{qq'}\) decreases comparatively slowly with increasing distance between the sites \(q\) and \(q'\); but for a qualitative discussion this approximation is admissible, especially since the total electric charge of any macroscopically infinitely small volume of the crystal is always equal to zero). Then the energy (7,120) can be expressed as a function of a single parameter—the number of neighboring pairs \(\alpha\). In particular, for a linear chain we have

\[ E' = nA + 2(2\alpha - n)B, \tag{7,121} \]

where \(B = B_{q,q+1}\). It is evident that the minimum and maximum values of (7,121) are respectively equal to:

\[ E'_{\min} = n(A - 2B);\qquad E'_{\max} = n(A + 2B). \tag{7,122} \]

Two cases may be distinguished: \(A < 2B\)—in this case the band of states with \(s=n\) overlaps the nonpolar band with \(s=0\); \(A > 2B\)—in this case the energy minimum of the band with \(s=n\) lies above the minimum of the band with \(s=0\). The inequalities given have a strikingly simple physical meaning. From (7,122) it is seen that the minimum level of the band with \(s=n\) corresponds to the case of the most nonuniform distribution of the electronic charge over the lattice. Comparing this distribution with the case \(s=0\), we see that although in the case \(s=n\) each occupied site contains two electrons, each electron is deprived of neighbors in two

nearest sites. It is clear that for \(A<2B\) the first distribution will be energetically more favorable than the second. The calculation given above shows that this result remains in full force in quantum mechanics as well, irrespective of the values that the integrals \(I\) and \(\beta\) may have. Thus, from the point of view of the approximation adopted, one can quite distinctly divide metals into two classes. The first class comprises those metals for which \(A<zB\), where \(z\) is the number of nearest neighbors in the crystal lattice. To this class, in all probability, belong metals with “elongated” orbitals of the outer electrons. One should expect that for them a theory will prove valid which takes the number of conduction electrons to be equal to the number of sites (of which one half are negative electrons—doublets—and one half are positive charges—holes). This may occur for alkali metals. The second class comprises metals with \(A>zB\). For them one cannot take, in a single linear combination, all states from \(s=0\) to \(s=n\), and therefore the number of conduction electrons will constitute only some fraction of the number of lattice sites. In the case under consideration\(^{20a}\) one may also take into account the influence of terms with exchange integrals; however, this will not alter the qualitative general conclusions given above.

13. Let us now obtain general formulae for the case of any fixed \(s\). It can be shown\(^{20a}\) that the solutions of the quasiclassical equations have the form

\[ \left. \begin{aligned} \Phi_q &= \sqrt{\frac{s}{2n}}\, e^{\,i\left(\nu_1qa+\frac{\lambda_1 t}{\hbar}\right)};\\ \Psi_q &= \sqrt{\frac{s}{2n}}\, e^{\,i\left(\nu_2qa+\frac{\lambda_2 t}{\hbar}\right)};\\ \varphi_q &= \sqrt{\frac{n-s+m}{2n}}\, e^{\,i\left(\omega_1qa+\frac{\lambda_3 t}{\hbar}\right)};\\ \psi_q &= \sqrt{\frac{n-s-m}{2n}}\, e^{\,i\left(\omega_2qa+\frac{\lambda_4 t}{\hbar}\right)}. \end{aligned} \right\} \tag{7,123} \]

Substituting (7,123) into (7,118) and restricting ourselves to neighbors for the integrals \(I\) and \(\beta'=-\beta''=\beta\), we obtain:

\[ \begin{aligned} H={}&sA+4s\beta \sin\frac{\nu a}{2} \left[ \frac{n-s}{n}\sin\frac{\nu_1+\nu_2-(\omega_1+\omega_2)}{2}\,a\cos\frac{\omega a}{2} \right.\\ &\left. +\frac{m}{n}\cos\frac{\nu_1-\nu_2-(\omega_1+\omega_2)}{2}\,a\sin\frac{\omega a}{2} \right] +\frac{n^2-m^2}{n}\,I\\ &-\left[ \frac{s^2}{n}\cos \nu a +\frac{(n-s)^2-m^2}{n}\cos \omega a \right]I, \tag{7,124} \end{aligned} \]

where \(\omega=\omega_1-\omega_2,\ \nu=\nu_1-\nu_2\). The terms containing the integrals of elec-

trostatic interaction \(B_{qq'}\), have exactly vanished. This is connected with the fact that pairs and holes, propagating through the crystal in the form of plane waves (7,123), assume the most diverse mutual positions. The mean value of the electrostatic interaction energy over all these positions is equal to the energy \(sA\). To estimate the influence of less uniform distributions of pairs and holes, one should use the “point” type of solutions, when the operators \(\Phi_q,\ \Psi_q,\ \varphi_q,\ \psi_q\) may take the values zero or one in accordance with condition (7,117). Then the energy of the system (7,118) takes the form

\[ H=s(A+2I)+B(\alpha-\beta)-I(\alpha-\gamma), \tag{7,125} \]

where the quantities \(\alpha,\ \beta\), and \(\gamma\) have the following meaning: \(\alpha\) is equal to the number of those pairs followed by pairs plus the number of those holes followed by holes; \(\beta\) is equal to the number of pairs followed by holes plus the number of those holes followed by pairs; \(\gamma\) is equal to the number of those simple right ones followed by simple left ones plus the number of those simple left ones followed by simple right ones.

Formulas (7,124) and (7,125) give an approximate idea of the form of the energy spectrum of the system. Of course, such a division of the spectrum into two branches is an artificial consequence of the approximations used. But one circumstance is captured correctly by these formulas: whereas for large \(\beta\) the pairs and holes tend to be uniformly “smeared out” over the whole metal, as a result of which the total energy of the electrostatic interaction between them is on the average equal to zero, for large \(B\) the energetically more favorable situation is the presence of successive condensations and rarefactions of charge, which likewise propagate through the whole crystal by virtue of translational invariance.

14. Formulas (7,124) and (7,125) make it possible to reveal certain typical cases of the arrangement of the energy minima of the spectrum of the system.

a) The energy minimum occurs at \(s=0\). Suppose that \(I>0\) (the ferromagnetic case). Then case a) corresponds to the value of the minimum energy

\[ H_{\min}=0. \]

This level lies at \(\omega=0\); it is the same for all values of the resultant moment \(m\) from \(-n\) to \(+n\).

b) The energy minimum occurs at \(0<s<n\). This can be obtained, for example, at

\[ \omega=0,\quad \nu=-\frac{\pi}{a},\quad \nu_1+\nu_2-(\omega_1+\omega_2)=\frac{\pi}{a}, \]

if the condition

\[ A+2I-4B<0. \]

is satisfied.

Indeed, then the value \(s\), equal to

\[ s_0=\frac{n(A+2I-4\beta)}{8\beta}, \]

will correspond to the energy

\[ H_{\min}=\frac{n(A+2I-4\beta)^2}{16\beta}, \tag{7,126} \]

which is less than zero and may in fact turn out to be the lowest. Here it was assumed that \(\beta>0\); for \(\beta<0\) the lowest level would correspond to the value \(\nu=+\dfrac{\pi}{a}\). This energy level is the same for all \(m\) from \(-(n-s_0)\) to \(+(n-s_0)\).

c) The energy minimum is attained for \(s=n\). From formula (7,125) it is seen that this case is possible if the energy

\[ H_{\min}=n[A-2(B-I)] \]

is less than zero and less than the other possible minima.

Besides the three cases indicated, others are not excluded; however, they are less typical and give nothing essentially new. In considering formula (7,124) it may seem that case b) is characteristic only of ferromagnets, since the level (7,126), as was already indicated, is the same for a wide range of values of \(m\). Further, it would seem to follow from this that in the polar model there are more possibilities for the appearance of ferromagnetism than in the nonpolar one. This, however, is incorrect, for in deriving formula (7,126) the last terms of (7,118) with permuted \(s\) were discarded; in the general case they are not small in comparison with terms containing, for example, the integrals \(\beta'\) and \(\beta''\). From (7,115) it is directly seen that precisely for \(\omega=0\), when \(\varphi_q=\psi_q=\mathrm{const}\), such a discarding is entirely legitimate, and therefore the formula for the lowest energy level of a ferromagnet indeed has the form (7,126). However, from the same expression (7,118) it is seen that precisely for nonferromagnets, i.e., at values of \(\omega\) close to \(\pi\), the terms with the integrals \(\gamma\) and \(\delta\) become substantial and broaden the energy band, thereby lowering the energy minimum of the nonferromagnetic states.

15. After these general considerations on the character of the energy spectrum, let us turn to methods of exact solution of the quantum-mechanical problem, without resorting to the quasiclassical approximation. The most effective method here is the method of small perturbations or quasiparticles. For illustration let us consider several examples.

Let us first dwell on a qualitative definition of the quasiparticle method. In the general case the energy operator is not a sum of one-particle terms because of the interaction between particles. However, for weak excitations of the interacting

system, the operator of its energy can be represented in the form of an additive sum of similar operators of these excitations. To each elementary excitation one can associate a definite discrete image, which is also called a quasiparticle. Proceeding from the general relation between effective mass and energy, one can determine the effective mass of the quasiparticle and regard the original system of strongly interacting particles as an ideal gas of quasiparticles, with all the advantages of such a consideration. The possibility of introducing quasiparticles is one of the manifestations of atomism in systems of many microparticles.

Historically, one of the first examples of the application of the quasiparticle method was the problem of vibrations of crystals, and also the quantization of the electromagnetic field. In the first case the quasiparticles are phonons—elastic, or thermal quanta, and in the second—photons, or light quanta.

For the application of the usual gas-kinetic treatment to ideal gases of quasiparticles, it is necessary to determine the type of statistics to which they must obey. This can be done if the spin characteristics of the elementary excitations or the type of commutation relations for the operators of secondary quantization are known. There may be two types of quasiparticle gases—obeying symmetric (Bose) or antisymmetric (Fermi) statistics.

In this possibility of introducing “gases” of quasiparticles lies a peculiar “justification” of the conclusions of the one-electron theory, which in an “incomprehensible” way agreed quite satisfactorily with experimental data. In fact, in these cases the “electron” of the one-electron model, in its properties, is very close to the quasiparticles of the many-electron system.

Let us first consider a ferromagnetic crystal in states close to magnetic saturation (“to the left”), and under the condition that it has few conduction electrons (few “twos” and “holes”);²⁴ under these assumptions, by virtue of (7,117) one may put

\[ \Phi_q \Phi_{q'}^{+},\ \Psi_q \Psi_{q'}^{+},\ \varphi_q \varphi_q^{+} \ll \psi_q \psi_q^{+} \simeq 1. \]

Then, if we use (7,117′) and (7,118), the simplified energy operator takes the form

\[ \hat H = sA + \sum_{q<q'} I_{qq'}(\varphi_q \varphi_{q'}^{+} - \varphi_q^{+}\varphi_{q'}) + \]
\[ + \sum_{q<q'} \beta'_{qq'} \Phi_q \Phi_{q'}^{+} - \sum_{q<q'} \beta''_{qq'} \Psi_q \Psi_{q'}^{+}. \tag{7,127} \]

To this operator one may also add a term taking into account the influence of excited electrons, assuming at the same time that the number of “left” excitons (i.e., sites with excited electrons with left spin) is greater than the number of “right” excitons. This

the additional term has the form

\[ \Delta E\,(s'_r+s'_l)+\sum_{q<q'} I^*_{qq'}\bigl(\varphi^*_q\varphi^{*+}_{q'}-\varphi^*_q\varphi^{*+}_{q'}\bigr), \tag{7.128} \]

where \(\varphi^*_q,\ \varphi^{*+}_q\) are second-quantization operators for right excitons, and \(I^*_{qq'}\) is the exchange integral between electrons of sites with excitations and sites without excitations (“simple” ones); \(\Delta E\) is the excitation energy of a site with an exciton (right or left), \(s'_r\) is the number of right excitons, and \(s'_l\) the number of left excitons. The wave character of the elementary excitations appears still more clearly if, instead of the operators \(\Phi_q,\ \Psi_q,\ \varphi_q,\ \varphi^*_q\), one introduces their Fourier amplitudes \(\Phi_\mu,\ \Psi_\mu,\ \varphi_\mu,\ \varphi^*_\mu\):

\[ \begin{aligned} \Phi_\mu &= N^{-\frac12}\sum_{q=1}^{N} e^{\,i\left(k^{(1)}_\mu R_q\right)}\Phi_q;\\ \Psi_\mu &= N^{-\frac12}\sum_{q=1}^{N} e^{\,i\left(k^{(2)}_\mu R_q\right)}\Psi_q;\\ \varphi_\mu &= N^{-\frac12}\sum_{q=1}^{N} e^{\,i\left(k^{(3)}_\mu R_q\right)}\varphi_q;\\ \varphi^*_\mu &= N^{-\frac12}\sum_{q=1}^{N} e^{\,i\left(k^{(4)}_\mu R_q\right)}\varphi^*_q, \end{aligned} \tag{7.129} \]

where \(N\) is the number of lattice sites, \(R_q\) is the radius vector between an arbitrary origin of coordinates and the site \(q\), in units of the lattice constant, and \(k^{(1)}_\mu,\ k^{(2)}_\mu,\ k^{(3)}_\mu\) and \(k^{(4)}_\mu\) are the quasi-momenta, respectively, of pairs, holes, simple right excitons, and “right” excitons. The periodicity conditions require that the components of these vectors satisfy relations of the type

\[ k^{(1)}_{\mu x}=\frac{2\pi\lambda^{(1)}_x}{G_x};\quad k^{(1)}_{\mu y}=\frac{2\pi\lambda^{(1)}_y}{G_y};\quad k^{(1)}_{\mu z}=\frac{2\pi\lambda^{(1)}_z}{G_z} \quad\text{etc.}, \tag{7.130} \]

where \(\lambda^{(1)}_x,\ \lambda^{(1)}_y,\ \lambda^{(1)}_z,\ldots\) are integers lying in the ranges \(-\tfrac12 G_x\) and \(\tfrac12 G_x-1\), \(-\tfrac12 G_y\) and \(\tfrac12 G_y-1\), \(-\tfrac12 G_z\) and \(\tfrac12 G_z-1\), while \(G_x,\ G_y,\ G_z\) are the dimensions of the crystal along the axes \(x,\ y,\ z\) in units of the lattice constant. The commutation relations in the new operators have the form

\[ \Phi_\mu\Phi_{\mu'}^{+}-\Phi_{\mu'}^{+}\Phi_\mu=\hat{\delta}_{\mu\mu'} \quad\text{etc.} \tag{7.131} \]

Using (7.131) and the expressions for the new operators (7.129), instead of (7.118) we obtain the energy operator reduced to diago-

to the final form,

\[ \hat H=\Delta E\left(s_f' + s_l'\right)+sA+\sum_\mu A_\mu \varphi_\mu \varphi_\mu^+ +\sum_\mu B_\mu \Phi_\mu \Phi_\mu^+ +\sum_\mu C_\mu \Psi_\mu \Psi_\mu^+ +\sum_\mu D_\mu \varphi_\mu^* \varphi_\mu^{*+}, \tag{7,132} \]

where the abbreviated notation for the quasiparticle energies has been introduced:

\[ \left. \begin{aligned} A_\mu&=\sum_h I_h(R_h)\left[1-e^{i\left(k_\mu^{(3)}R_h\right)}\right],\\ B_\mu&=\sum_h \beta_h'(R_h)e^{i\left(k_\mu^{(1)}R_h\right)},\\ C_\mu&=-\sum_h \beta_h''(R_h)e^{i\left(k_\mu^{(2)}R_h\right)},\\ D_\mu&=\sum_h I_h^*\left[1-e^{i\left(k_\mu^{(4)}R_h\right)}\right],\\ \mathbf R_h&=\mathbf R_q-\mathbf R_{q'}. \end{aligned} \right\} \tag{7,133} \]

From the definition of the operators \(\Phi_\mu\Phi_\mu^+\), \(\Psi_\mu\Psi_\mu^+\), \(\varphi_\mu\varphi_\mu^+\), \(\varphi_\mu^*\varphi_\mu^{*+}\) it follows that their eigenvalues are integers: \(n_\mu^{(1)}\), \(n_\mu^{(2)}\), \(n_\mu^{(3)}\), \(n_\mu^{(4)}=0,1,2,\ldots\). Consequently, the eigenvalues of the energy operator (7,132) for a mixture of “gases” of pairs, holes, simple right excitons, and right excitons are equal to the additive sum of the partial energies (7,133) of the four named types of quasiparticles. This case pertains rather not to ferromagnetic metals, but to ferromagnetic semiconductors. Using the expression for the energy and symmetric statistics, one can calculate the temperature dependence of the spontaneous magnetization \(M_s\) at low temperatures. Using the usual method of calculation (see, for example, \(^{29}\)), we find:

\[ M_s=N\mu_B\left\{1-\left(\frac{T}{\theta_1}\right)^{\frac32} +e^{-\frac{\Delta E}{kT}}\times \left[1-\left(\frac{T}{\theta_2}\right)^{\frac32}\right] -e^{-\frac{A+4\beta}{kT}}\left(\frac{T}{\theta_3}\right)^{\frac32}\right\}, \tag{7,134} \]

if

\[ \beta'=\beta''=\beta \quad\text{and}\quad \theta_1\simeq \frac{I}{k},\qquad \theta_2\simeq \frac{I^*}{k},\qquad \theta_3\simeq \frac{|\beta|}{k}, \]

\(\mu_B\) is the Bohr magneton.

If the excitons in the semiconductor and the polar states have a very high excitation energy

\[ \left(\frac{\Delta E}{kT}\to\infty,\quad \frac{A}{kT}\to\infty\right), \]

then

they practically make no contribution to the spontaneous magnetization of the semiconductor, and formula (7,134) in this case coincides with the well-known expression \(M_s\) of the exchange model\({}^{49}\). If, however, the spontaneous magnetization is due only to excited states (the exchange integral for ferromagnons is simply negative), then the temperature dependence \(M_s(T)\) will be quite different, namely:

\[ M_s(T)=N\mu_B e^{-\frac{\Delta E}{kT}} \left[1-\left(\frac{T}{\theta_2}\right)^{\frac{3}{2}}\right], \tag{7,135} \]

i.e., as \(T \to 0^\circ\) the exciton magnetization tends to zero, and, consequently, ferromagnetic semiconductors of this type will have a second low-temperature Curie point at \(0^\circ\) K. There are not yet sufficient experimental data to make it possible to test this prediction of the theory\({}^{23}\).

In addition to the calculation of static properties, the result obtained above may also serve as a starting point for calculating kinetic processes, for example electrical conductivity. In this case one may apply the usual scheme of the kinetic equation, in which the distribution functions of the quasiparticle numbers enter as the unknown functions. Such a method of treating kinetic processes in crystals was successfully applied by Akhiezer\({}^{55}\), and also by Akhiezer and Pomeranchuk\({}^{56}\), for calculating the relaxation time between the lattice and the spin system and within the spin system, both for the case of paramagnets and for ferromagnets, using the exchange model without allowance for polar states.

16. As a second example, let us present the calculation of the temperature dependence of the electrical conductivity of a metal according to the many-electron model in the case when the role of conduction electrons is played by quasiparticles—doubles and holes, of which there are few in comparison with the number of lattice sites (“poor metals”)\({}^{37}\). In this case

\[ \frac{s}{N}\ll 1, \]

but nevertheless the lowest energy level in the crystal belongs to that band of the energy spectrum for which \(s \ne 0\), although \(s \ll N\). In view of the condition \(s \ll N\), (7,117) may approximately be written as

\[ \varphi_q\varphi_q^{+}+\psi_q\psi_q^{+}\simeq 1 \quad\text{and}\quad \sum_{q=1}^{N}\left(\varphi_q\varphi_q^{+}+\psi_q\psi_q^{+}\right)\simeq N. \]

Moreover, we shall assume that the crystal is not ferromagnetic; therefore, in the lowest energy state \(m=0\). This condition, together with ((7,117), permits one, to a sufficient approximation, to assume that

\[ \varphi_q\sim \psi_q\sim \psi_q^{+}\sim \psi_q^{+}\sim \mathrm{const}\sim \frac{1}{\sqrt{2}}. \]

Using these approximations, and also omitting terms with the product of four operators \(\Phi_q\) and \(\Psi_q\), and replacing the number \(s\) in the first term of (7.118) by (7.117), we find an expression for the energy of a weakly polarized metal:

\[ \hat H=\frac{1}{2}A\sum_{q=1}^{N}\left(\Phi_q\Phi_q^{+}+\Psi_q\Psi_q^{+}\right) +\frac{1}{2}\sum_{\substack{q=1\\(q\ne \alpha)}}^{N} I_{\alpha q} \sum_{q'=1}^{N}\left(\Phi_{q'}\Phi_{q'}^{+}+\Psi_{q'}\Psi_{q'}^{+}\right) + \]

\[ +\sum_{(q\ne q')}\left[(\Phi_{q'}\Phi_q^{+}+\Phi_q\Phi_{q'}^{+})\beta'_{qq'} -(\Psi_{q'}\Psi_q^{+}+\Psi_q\Psi_{q'}^{+})\beta''_{qq'}\right]. \tag{7.136} \]

The exchange and transfer integrals entering (7.136) are functions of the distance between lattice sites. Denoting by \(\mathbf R^0_{qq'}\) the distance between the equilibrium positions of the sites \(q\) and \(q'\), one may write

\[ \mathbf R_{qq'}=\mathbf R^0_{qq'}+\Delta \mathbf R_{qq'},\qquad \Delta \mathbf R_{qq'}=\Delta \mathbf R_q-\Delta \mathbf R_{q'}, \tag{7.137} \]

where \(\Delta \mathbf R_q\) is the displacement of the \(q\)-th site caused by thermal motion. Considering the displacement to be small, one may expand the integrals \(I_{qq'}\) and \(\beta'_{qq'}, \beta''_{qq'}\) in powers of this displacement, which makes it possible to separate out in the energy operator (7.136) the perturbation responsible for the interaction of doublons and holes with phonons of the thermal motion of the lattice. The subsequent calculation proceeds in the usual way. Instead of the operators \(\Phi_q\) and \(\Psi_q\) we introduce their Fourier amplitudes (see (7.12)), thereby linearizing the zero-order term of the operator (7.136) and finding the energy of the elementary excitations. Then we determine the perturbation operator. For this purpose the displacement \(\Delta \mathbf R_{qq'}\) is expanded, as usual, in plane waves. We find the matrix elements of the perturbation-energy operator and, according to the standard rules, set up the kinetic equation of the problem. Solving this equation is very difficult because of the mathematical complications, but one can determine the temperature dependence of the perturbation of the distribution function of doublons and holes, which in the end makes it possible to find the temperature dependence of the electrical resistivity. As a result of all the indicated calculations\(^{37}\) we find that at temperatures near \(0^\circ\mathrm K\), or, more precisely, for \(T\ll \dfrac{\theta^2}{\theta'}\) or \(T\ll \dfrac{\theta^2}{\theta''}\),

\[ \rho\sim T, \tag{7.138} \]

whereas for \(T\gg \dfrac{\theta^2}{\theta'}\) or \(T\gg \dfrac{\theta^2}{\theta''}\), but, of course, for \(T\ll \theta,\theta',\theta''\), we have

\[ \rho\sim T^3, \tag{7.139} \]

where \(\theta\) is the ordinary Debye characteristic temperature, while \(\theta'=\dfrac{\beta'}{k}\), \(\theta''=\dfrac{\beta''}{k}\) are, respectively, the critical temperatures of doublons and holes. The indicated temperature regions have a simple physical meaning. Indeed, for the mean energy of doublons or holes

we have (up to an additive constant)

\[ \varepsilon'=-\frac{1}{2}(A_1+D)=\overline{\beta'(ak')^2}\sim kT; \]

\[ \overline{ak'}\sim \left(\frac{T}{\theta'}\right)^{\frac12} \quad \text{and} \quad \overline{ak''}\sim \left(\frac{T}{\theta''}\right)^{\frac12}. \]

For the mean energy of phonons we also have \(\varepsilon_f=\hbar\omega_f=\hbar\omega_0 f\); \(\overline f\sim \dfrac{T}{a\theta}\), or \(a\overline f\sim \dfrac{T}{\theta}\). Consequently, for \(T\gg \dfrac{\theta^2}{\theta'}\) the case \(\overline f\gg \overline{k'}\) is realized, and for \(T\ll \dfrac{\theta^2}{\theta'}\) the case \(\overline f\ll \overline{k'}\). The boundary between these regions, \(\dfrac{\theta^2}{\theta'}\) or \(\dfrac{\theta^2}{\theta''}\), lies in the temperature interval from 1 to \(10^\circ\) K. Indeed, \(\beta', \beta''\sim 10^{-12}\)—\(10^{-13}\) ergs, \(\theta\sim 10^2\), and therefore, since \(\theta'=\dfrac{\beta'}{k}\); \(\theta''=\dfrac{\beta''}{k}\), we have \(\theta'\sim\theta''\sim 10^3\)—\(10^4\) and \(\theta^2/\theta'\sim \theta^2/\theta''\sim 1^\circ\)—\(10^\circ\) K.

Experiment confirms the result obtained. In fact, in a large number of “poor” metals such as graphite, platinum, hafnium, zirconium, cesium, tungsten, etc., the temperature dependence of the electrical resistance by no means follows the result predicted by the one-electron theory (\(\sim T^5\)), but gives very good qualitative agreement with the results obtained above. More careful low-temperature studies are required, however, and first of all a study of the temperature dependence of the electronic part of the heat capacity, in order to draw a final conclusion about the type of statistics obeyed by the electronic elementary excitations in these and other metals.

It can be shown that, in the case of strongly polar crystals, when \(s\sim n\), the elementary excitations (which in this case are small deviations from complete polarity) also obey Bose statistics, and the temperature dependence of the electrical resistance likewise turns out to be linear or quadratic.

1. The variants of the many-electron polar model considered are not the only ones. As a system of one-electron functions for the antisymmetrized products (6.2′), one may choose not atomic functions, as has been done up to now, but one-particle functions of the one-electron theory (4.15). This method of choosing the functions (6.2′) was developed in greatest detail in the work of Heilmann\({}^{21}\) (see also \({}^{38,39,40}\)). If one confines oneself to considering only one band and assumes that the number of electrons is equal to the number of lattice sites, then the levels in the band, each of which is characterized by a definite value of the quasi-momentum, \(\mathbf{k}_i(i=1,2,\ldots,N)\), split into four groups: 1) levels occupied by one electron with “right” spin, 2) levels occupied by one electron with “left”

spin, 3) levels occupied by two electrons with opposite spins, and 4) levels not occupied by electrons.

Let us denote the numbers of the first group by \(h_1 h_2 \ldots h_t\) (\(t\) is the number of such levels), the numbers of the second group by \(k_1 k_2 \ldots k_u\) (\(u\) is their number), the numbers of the third group by \(f_1 f_2 \ldots f_s\) (\(s\) is their number), and the numbers of the fourth group by \(g_1 g_2 \ldots g_S\) (\(S\) is their number). The wave function (6,2) will have the form

\[ \begin{aligned} \psi_{f_1 \ldots f_s;\, g_1 \ldots g_S;\, h_1 \ldots h_t}(\mathbf r_1,\ldots,\mathbf r_N) &= \frac{1}{(N!)^{1/2}}\sum_P \eta_P P \psi_{k f_1}(\mathbf r_{f_1})\psi_{k f_1}(\mathbf r_{g_1})\ldots \\ &\quad \ldots \psi_{k f_s}(\mathbf r_{f_s})\psi_{k f_s}(\mathbf r_{g_s}) \psi_{k h_1}(\mathbf r_{h_1})\ldots \psi_{k h_t}(\mathbf r_{h_t})\psi_{k k_1}(\mathbf r_{k_1})\ldots \\ &\quad \ldots \psi_{k k_u}(\mathbf r_{k_u}) C_R(\sigma_{f_1})C_L(\sigma_{g_1})\ldots C_R(\sigma_{f_s})\ldots C_L(\sigma_{g_S})C_R(\sigma_{h_1})\ldots \\ &\quad \ldots C_R(\sigma_{h_t})C_L(\sigma_{k_1})\ldots C_L(\sigma_{k_u}). \end{aligned} \tag{7,140} \]

Substituting these functions into (6,1) and then into the wave equation (1,5), we find a system of equations of the type (6,2). The matrix elements in the resulting system of equations will in this case be not functions of the numbers of the sites, as was the case in (7,77), but functions of the quasi-momenta. In particular, the diagonal matrix element will be equal to

\[ \begin{aligned} (fgh|\hat H|fgh) &= 2\sum_f E(\mathbf k_f)+\sum_h E(\mathbf k_h)+\sum_k E(\mathbf k_k) \\ &\quad +4\sum_{f<f'} B(\mathbf k_f,\mathbf k_{f'}) +2\sum_{p,f} B(\mathbf k_p,\mathbf k_f) +\sum_{p<p'} B(\mathbf k_p,\mathbf k_{p'}) \\ &\quad +\sum_f B(\mathbf k_f) -2\sum_{f<f'} I(\mathbf k_f,\mathbf k_{f'}) -\sum_{p,f} I(\mathbf k_p,\mathbf k_f) \\ &\quad -\sum_{p<p'} I(\mathbf k_p,\mathbf k_{p'}), \end{aligned} \tag{7,141} \]

where \(E(\mathbf k)\) is the electron energy determined by equation (4,2) of the one-electron theory, \(B(\mathbf k,\mathbf k')\) and \(B(\mathbf k)\) are, respectively, the energies of the electrostatic interaction of two electrons in different levels and in one level,

\[ B(\mathbf k,\mathbf k')=\int |\psi_{\mathbf k}(\mathbf r)|^2 |\psi_{\mathbf k'}(\mathbf r')|^2 v(|\mathbf r-\mathbf r'|)\,d\mathbf r\,d\mathbf r', \]

\[ B(\mathbf k)=\int |\psi_{\mathbf k}(\mathbf r)|^2 |\psi_{\mathbf k}(\mathbf r')|^2 v(|\mathbf r-\mathbf r'|)\,d\mathbf r\,d\mathbf r', \]

and \(I(\mathbf k,\mathbf k')\) is the exchange integral:

\[ I(\mathbf k,\mathbf k')=\int \psi_{\mathbf k}(\mathbf r)\psi_{\mathbf k'}^{*}(\mathbf r) \psi_{\mathbf k'}(\mathbf r')\psi_{\mathbf k}^{*}(\mathbf r') v(|\mathbf r-\mathbf r'|)\,d\mathbf r\,d\mathbf r'. \]

Because of the orthogonality of the spin functions, the off-diagonal matrix elements in (6.2) vanish if the total spin momenta in the different states are different. Therefore the equations split into systems with the same spin number \(m\). The set of indices \(f'g'h'\) may differ from the set \(fgh\) in the off-diagonal matrix elements \((fgh|\hat H|f'g'h')\), for constant \(m\), by simple or double permutations. Triple permutations always give zero because of the orthogonality of the wave functions (4.15). Of the simple permutations, only the permutation between two singly occupied spins of different names gives elements different from zero. Schematically it may be represented as

\[ \psi_{k_h}(\mathbf r)\,C_R(\sigma_h)\,\psi_{k_k}(\mathbf r')\,C_L(\sigma_k) \to \psi_{k_k}(\mathbf r)\,C_R(\sigma_k)\,\psi_{k_h}(\mathbf r')\,C_L(\sigma_h). \]

The matrix element corresponding to this permutation is equal to the exchange integral with a negative sign. For all other simple permutations (a double level with an empty one, a double level with a right or left single one, an empty level with a single one, the decay of a double level into two simple ones, and the formation of a double level from two simple ones) for the functions (4.15) the matrix elements are equal to zero. Indeed, for example, in the permutation of a double level with a left single one, the matrix element has the form

\[ \int \psi_{k_f}(\mathbf r)\psi^{*}_{k_k}(\mathbf r)|\psi_{k_s}(\mathbf r')|^{2} v(|\mathbf r-\mathbf r'|)\,d\mathbf r\,d\mathbf r'; \]

using (4.15), we find:

\[ \int e^{i(k_f-k_k)\mathbf r}\, u(k_f,\mathbf r)u^{*}(k_k,\mathbf r)\, |u(k_f,\mathbf r')|^{2} v(|\mathbf r-\mathbf r'|)\,d\mathbf r\,d\mathbf r'. \]

We introduce relative coordinates \(\mathbf r-\mathbf r'=\rho\) and shall measure \(\mathbf r\) from the lattice sites \(a_i\), i.e. \(\mathbf r=a_i+\mathbf R\). This gives

\[ \sum_{a_i} e^{i(k_f-k_k)a_i} \int e^{i(k_f-k_k)\mathbf R} u(k_f,\mathbf R)u^{*}(k_k,\mathbf R)|u(k_f,\mathbf R-\rho)|^{2} \times \]

\[ \times v(|\rho|)\,d\rho\,d\mathbf R. \]

In view of the fact that \((k_f-k_k)\ne 0\), this integral is equal to zero. However, for double permutations (the simultaneous exchange of the positions of two single levels with a hole and one empty level, the decay and formation of two double levels, the simultaneous permutation of two single levels with two double and two empty ones), the matrix elements will be different from zero if the difference of the quasimomenta of the new and initial states for the two electrons is the same in absolute value (the law of conservation of quasimomentum):

\[ \mathbf k_1-\mathbf k'_1=\pm(\mathbf k_2-\mathbf k'_2). \]

QUANTUM THEORY OF ELECTRON CONDUCTORS

Heitlikman \(^{21}\) considered the case of a ferromagnet at low temperatures, when the total number of left spins \(u+s=l\) is small (i.e., \(l \ll N\)). If for the functions (4.15) one adopts the tight-binding approximation, namely,

\[ \psi_{\mathbf{k}}(\mathbf{r})=\frac{1}{\sqrt{N}}\sum_{i=0}^{N} e^{i(\mathbf{k}\mathbf{a}_i)}\,\varphi_0(\mathbf{r}-\mathbf{a}_i), \tag{7.142} \]

then equation (6.2) (see also (7.141)) takes the form

\[ \begin{aligned} E'a(\mathbf{k}_{f_1}\ldots \mathbf{k}_{f_s},\ \mathbf{k}_{g_1}\ldots \mathbf{k}_{g_s},\ \mathbf{k}_{h_1}\ldots \mathbf{k}_{h_l})={}& \\ ={}&(fgh|\hat H|fgh)\, a(\mathbf{k}_{f_1}\ldots \mathbf{k}_{f_s},\ \mathbf{k}_{g_1}\ldots \mathbf{k}_{g_s},\ \mathbf{k}_{h_1}\ldots \mathbf{k}_{h_l})\\ &-\sum_{\mathbf{k}_h,\mathbf{k}_k} I(\mathbf{k}_h,\mathbf{k}_k)\, a(\mathbf{k}_{f_1}\ldots \mathbf{k}_{f_s},\ \mathbf{k}_{g_1}\ldots \mathbf{k}_{g_s},\ \mathbf{k}_{h_1}\ldots \mathbf{k}_k\ldots \mathbf{k}_{h_l})\\ &+\sum_{\mathbf{k}_f<\mathbf{k}_{f_1};\ \mathbf{k}_h,\mathbf{k}_{h_1}} I(\mathbf{k}_{f_1},\mathbf{k}_h)\, \delta\!\left[\,|\mathbf{k}_f-\mathbf{k}_h\pm(\mathbf{k}_{f_1}-\mathbf{k}_{h_1})|\,\right]\times\\ &\qquad \times a(\mathbf{k}_{f_1}\ldots \mathbf{k}_h\ldots \mathbf{k}_{h_1}\ldots \mathbf{k}_{f_s};\ \mathbf{k}_{g_1}\ldots \mathbf{k}_{g_s};\ \mathbf{k}_{h_1}\ldots \mathbf{k}_f\ldots \mathbf{k}_{f_1}\ldots \mathbf{k}_{h_l})\\ &+\sum_{\mathbf{k}_g<\mathbf{k}_{g_1};\ \mathbf{k}_h,\mathbf{k}_{h_1}} I(\mathbf{k}_g,\mathbf{k}_h)\, \delta\!\left[\,|\mathbf{k}_g-\mathbf{k}_h\pm(\mathbf{k}_{g_1}-\mathbf{k}_{h_1})|\,\right]\times\\ &\qquad \times a(\mathbf{k}_{f_1}\ldots \mathbf{k}_{f_s};\ \mathbf{k}_{g_1}\ldots \mathbf{k}_h\ldots \mathbf{k}_{h_1}\ldots \mathbf{k}_{g_s};\ \mathbf{k}_{h_1}\ldots \mathbf{k}_g\ldots \mathbf{k}_{g_1}\ldots \mathbf{k}_{h_l})\\ &-\sum_{\mathbf{k}_f<\mathbf{k}_g;\ \mathbf{k}_h\ne \mathbf{k}_{h_1}} I(\mathbf{k}_f,\mathbf{k}_g)\, \delta\!\left[\,|\mathbf{k}_f-\mathbf{k}_h\pm(\mathbf{k}_g-\mathbf{k}_{h_1})|\,\right]\times\\ &\qquad \times a(\mathbf{k}_{f_1}\ldots \mathbf{k}_h\ldots \mathbf{k}_{f_s};\ \mathbf{k}_{g_1}\ldots \mathbf{k}_{h_1}\ldots \mathbf{k}_{g_s};\ \mathbf{k}_{h_1}\ldots \mathbf{k}_f\ldots \mathbf{k}_g\ldots \mathbf{k}_{h_l}). \tag{7.143} \end{aligned} \]

As in the polar model, equations (7.143) can be translated into the language of second quantization by means of operators of type (7.116). Carrying out analogous calculations, we obtain

\[ \begin{aligned} \hat H_a={}&(fgh|\hat H|fgh) -\sum_{\mathbf{k}_1\ne\mathbf{k}_2} I(\mathbf{k}_1,\mathbf{k}_2)\, \varphi_{\mathbf{k}_1}\varphi_{\mathbf{k}_2}^{+}\psi_{\mathbf{k}_2}\psi_{\mathbf{k}_1}^{+}\\ &+\sum_{\substack{\mathbf{k}_1<\mathbf{k}_2\\ \mathbf{k}'_1,\mathbf{k}'_2}} I(\mathbf{k}_1,\mathbf{k}'_1)\, \Phi_{\mathbf{k}_1}\Phi_{\mathbf{k}'_1}^{+} \Phi_{\mathbf{k}_2}\Phi_{\mathbf{k}'_2}^{+} \varphi_{\mathbf{k}'_1}\varphi_{\mathbf{k}_1}^{+} \varphi_{\mathbf{k}'_2}\varphi_{\mathbf{k}_2}^{+}\times\\ &\qquad \times \delta\!\left[\mathbf{k}_1-\mathbf{k}'_1{}^{+}-(\mathbf{k}_2-\mathbf{k}'_2)\right]+\\ &+\sum_{\substack{\mathbf{k}_1<\mathbf{k}_2\\ \mathbf{k}'_1,\mathbf{k}'_2}} I(\mathbf{k}'_1,\mathbf{k}_1)\, \Psi_{\mathbf{k}_1}\Psi_{\mathbf{k}'_1}^{+} \Psi_{\mathbf{k}_2}\Psi_{\mathbf{k}'_2}^{+} \varphi_{\mathbf{k}'_1}\varphi_{\mathbf{k}_1}^{+} \varphi_{\mathbf{k}'_2}\varphi_{\mathbf{k}_2}^{+}\times\\ &\qquad \times \delta\!\left[\mathbf{k}_1-\mathbf{k}'_1{}^{+}-(\mathbf{k}_2-\mathbf{k}'_2)\right] -\sum_{\substack{\mathbf{k}_1,\mathbf{k}_2\\ \mathbf{k}_1\ne\mathbf{k}_2}} I(\mathbf{k}_1,\mathbf{k}'_1)\, \Phi_{\mathbf{k}_1}\Phi_{\mathbf{k}'_1}^{+} \Psi_{\mathbf{k}_2}\Psi_{\mathbf{k}'_2}^{+} \varphi_{\mathbf{k}'_1}\varphi_{\mathbf{k}_1}^{+}\times\\ &\qquad \times \varphi_{\mathbf{k}'_2}\varphi_{\mathbf{k}_2}^{+}\, \delta\!\left[\mathbf{k}_1-\mathbf{k}'_1\pm(\mathbf{k}_2-\mathbf{k}'_2)\right]. \tag{7.144} \end{aligned} \]

Since the number of right single levels is large, one may put

\[ \varphi_k \varphi_{k'} \sim 1; \]

moreover, using \(\delta\)-functions, instead of (7,144) we shall have

\[ \begin{aligned} \hat H_a = (fgh|\hat H|fgh) &- \sum_{k_1 \ne k_2} I(k_1,k_2)\psi_{k_1}\psi_{k_2} + \\ &+ \sum_{\substack{k_1<k_2\\ k}} I(k)\Phi_{k_1}\Phi^+_{k_1-k}\Phi_{k_2}\Phi^+_{k_2+k} + \sum_{\substack{k_1<k_2\\ k}} I(k)\Psi_{k_1}\Psi^+_{k_1-k}\Psi_{k_2}\Psi^+_{k_2+k} - \\ &\qquad\qquad - \sum_{\substack{k_1,k_2\\ k}} I(k)\Phi_{k_1}\Phi^+_{k_1-k}\Psi_{k_2}\Psi^+_{k_2-k}. \qquad (7,145) \end{aligned} \]

\[ (k = k_1 - k'_1 = k'_2 - k_2). \]

Replacing the operators \(\Phi_k,\ \Phi^+_k,\ \Psi_k,\ \Psi^+_k\) by their Fourier amplitudes according to (7,129), and also neglecting, following Geilikman, the term with \(\psi_{k_1}\psi^+_{k_2}\), instead of (7,145) we find, if we also use the expression for the exchange integrals in the strong-coupling approximation

\[ I(k)=\frac{1}{N}\sum_n e^{ikn} I(n), \]

where \(n\) are integer lattice vectors:

\[ \begin{aligned} H_a = (fgh|\hat H|fgh) &+ \frac{1}{2}\sum_{n,m} A(n-m)\Phi_n\Phi^+_n\Phi_m\Phi^+_m + \\ &+ \frac{1}{2}\sum_{n,m} A(n-m)\Psi_n\Psi^+_n\Psi_m\Psi^+_m - \\ &\qquad\qquad - \sum_{n,m} A(n-m)\Phi_n\Phi^+_n\Psi_m\Psi^+_m . \qquad (7,146) \end{aligned} \]

Thus, the energy spectrum of ferromagnets consists of two branches: the translational energies of double and empty levels entering into the diagonal element are the conduction energies—the quasiparticles of electrons obeying antisymmetric quantum statistics, while the terms with exchange integrals and the operators \(\Phi_q,\ \Psi_q\) represent the interaction between pairs and holes—quasiparticles obeying symmetric quantum statistics. Indeed, from (7,141) we have:

\[ (fgh|H|fgh)=2\sum E(k_f)+\sum_h E(k_h)+\sum_k E(k_k)+ \]

\[ +\sum_f E(k_f)-\sum_g E(k_g) \]

Tyablikov and Bonch-Bruevich\(^ {32}\) considered an analogous problem with elementary excitations of the Fermi type according to Bogolyubov’s general scheme. In doing so, they regarded the electron density as small compared with the number of lattice sites per unit volume, and all integrals of type (7.18) as small of the first order. The diagonalization of the energy operator (7.9) is carried out by means of a unitary transformation of a special form. The effective mass of the Fermi quasiparticles turns out to depend on the density of the electron gas in the lattice. In Heilkman’s calculation, the effective masses of holes and electrons coincide with the effective masses of these quasiparticles in the one-electron model.

The examples given show in sufficient detail the general content of existing many-electron models of crystals. From comparison with Section 4 it is clear that taking electron interaction into account is not trivial; it makes it possible to encompass much more fully the complex phenomena occurring in the crystal lattices of solids.

The development of general methods on the basis of second quantization and the introduction of the method of elementary excitations—quasiparticles—open broad possibilities for constructing already quantitative calculations of various static and kinetic phenomena in metals and semiconductors.

8. THE CASE OF TRANSITION ELEMENTS—A MODEL OF INTERACTING EXTERNAL AND INTERNAL ELECTRONS\(^ {41}\)

A large number of metals and semiconductors belong to the group of so-called transition elements, i.e., elements whose isolated atoms have an inner unfilled electron shell. When atoms condense into a crystal, the special structure of their electron shell manifests itself in the distinctive character of the electronic energy spectrum of the crystal lattice, which is the basis of all the “anomalies” in the physicochemical properties of metals of the transition groups in comparison with the analogous properties of simple metals.

Usually, works on the quantum theory of transition metals\(^ {42}\) have used a one-electron model. The author undertook an attempt\(^ {41}\) to construct a model in which electron interaction would be taken into account at least partially. In this model it is assumed that in the crystal, just as in an isolated atom, one can distinguish between “external” (valence) electrons and “internal” \(d\)-electrons. Such a separation, of course, has no literal meaning, since all electrons in a metal undoubtedly interact strongly with one another. Therefore this model may be regarded as a first, more or less consistent approximation. In the zeroth approximation, the external and internal electrons will be regarded as noninteracting, and the interaction as a small perturbation.

S. V. Vonsovskii

There is every reason to suppose that ferromagnetism is due to the exchange interaction of the internal electrons. Therefore the exchange interaction between them and the external electrons must have a nontrivial influence on the behavior of the latter. It is clear that the presence of exchange interaction between the external and internal electrons does not introduce any fundamental changes into the usual concepts of the one-electron model. In particular, for example, in the absence of thermal vibrations of the lattice, the electrons will still pass through it without resistance, etc., since these basic properties follow, as we have seen, from the most general assumptions (see Sections 4 and 5). However, exchange forces of the indicated type can affect the energy spectrum of the electrons in the crystal.

In the adopted model, the internal electrons are described by a many-electron exchange model, and the external ones by a one-electron model. The interaction between these two groups of electrons is regarded as a small perturbation. In accordance with formulas (6.1) and (6.2), the complete wave function of our system can be written in the form

\[ \Psi_{s+d}(\mathbf r,\mathbf r_1,\ldots,\mathbf r_N;\sigma,\sigma_1,\ldots,\sigma_N)= \]

\[ =[(N+1)!]^{-\frac12}N^{-\frac12}\sum_{h_1,\ldots,h_r} \left\{a(h_1\ldots h_2)\times\right. \]

\[ \times\left[\sum_Q \eta_Q Q e^{i\mathbf k\mathbf r}u(\mathbf k;\mathbf r)\cdot \varphi_1(\mathbf r_1)\ldots \right. \]

\[ \left.\left. \ldots\varphi_N(\mathbf r_N)C_R(\sigma_{h_1})\ldots C_R(\sigma_{h_r})C_L(\sigma_{k_1})\ldots C_L(\sigma_{k_n}) \right]\right\}, \tag{8.1} \]

where \(h_1,h_2,\ldots,h_r\) are the numbers of the \(r\) lattice sites with right spins of the internal electrons, \(\varphi_j(\mathbf r)\) is the spatial wave function, \(C_R\) or \(C_L\) are the spin wave functions of the \(d\)-electron of site \(j\), \(e^{i\mathbf k\mathbf r}u(\mathbf k\mathbf r)\) is the wave function of the external electron in the one-electron model (see (4.15)), and \(Q\) is any of the \((N+1)!\) permutations of all the electrons of the system.

The mean energy of interaction between the electrons is equal in our approximation to

\[ \overline E_{s+d} = \int \Psi^{*}_{s+d} \left[ \sum_{j=1}^{N} v(\mathbf r_j-\mathbf r) \right] \Psi_{s+d}\,d\tau . \tag{8.2} \]

Substituting (8.1) into (8.2), we find

\[ \overline E_{s+d} = N^{-1} \sum_{h_1,\ldots,h_r} |a(h_1\ldots h_2)|^2 \int e^{-i\mathbf k\mathbf r}u^*(\mathbf k,\mathbf r) \varphi_j^*(\mathbf r_1)\ldots C_L^*(\sigma_{k_n})\times \]

\[ \times \left[ \sum_{j=1}^{N} v(|\mathbf r_j-\mathbf r|) \right] \sum_Q \eta_Q Q e^{i\mathbf k\mathbf r}u(\mathbf k,\mathbf r) \varphi_1(\mathbf r_1)\ldots C_L(\sigma_{k_n})\,d\tau . \tag{8.3} \]

for, by virtue of the assumption of exact orthogonality between the atomic functions \(\varphi_i\), all terms in (8,3) containing factors of the type \(a^*(h_1\ldots h_r)a(h'_1\ldots h'_r)\), where \(h_i \ne h'_i\) \((i=1,2,\ldots,r)\), automatically drop out. By virtue of this same orthogonality, in (8,3) only a comparatively small number of terms is not annulled, namely only those for which the permutation \(Q\) is the identity or a simple transposition of the coordinates of the external electron with the coordinates of one of the internal ones. The identity permutation (see Section 7) gives simply the energy of the quasiclassical interaction between the external and internal electrons. The second type of permutation gives the required exchange energy. For definiteness let us choose the right orientation for the spin of the external electron; then, by virtue of the orthogonality of the spin functions, in the expression for the exchange energy only those terms will be different from zero which correspond to a permutation of the coordinates of the \(s\)-electron and of one of the \(d\)-electrons with right spin. This gives

\[ \overline{E}_{s+d} = -\,N^{-1}\sum_{h_1\ldots h_r} |a(h_1\ldots h_r)|^2 \sum_{j=1}^{r} \int_{0}^{N_a} e^{i\mathbf{k}(\mathbf{r}'-\mathbf{r})} u^*(\mathbf{k},\mathbf{r})\varphi_{h_j}^{*}(\mathbf{r}') \times \]

\[ \times v(|\mathbf{r}-\mathbf{r}'|) u(\mathbf{k},\mathbf{r}') \varphi_{h_j}(\mathbf{r})\,d\mathbf{r}\,d\mathbf{r}' . \tag{8,4} \]

By the basic periodicity property of the function \(u(\mathbf{k},\mathbf{r})\), (8,4) does not depend on the number of the site; therefore one may write that

\[ \int_{0}^{N_a} e^{i\mathbf{k}(\mathbf{r}'-\mathbf{r})} u^*(\mathbf{k},\mathbf{r})\varphi_j^{*}(\mathbf{r}') v(|\mathbf{r}-\mathbf{r}'|) u(\mathbf{k},\mathbf{r}') \varphi_j(\mathbf{r})\,d\mathbf{r}\,d\mathbf{r}' = I(\mathbf{k}), \tag{8,5} \]

where \(I(\mathbf{k})\) is a function only of the quasimomentum \(\mathbf{k}\). Substituting (8,5) into (8,4), we see that the unknown amplitudes \(a(h_1\ldots h_r)\) enter only in the combination \(\sum_{h_1\ldots h_r}|a(h_1\ldots h_r)|^2\), which, by the normalization condition for the function (8,1), is always equal to unity. As a result we find that \(E_{s+d}=-\frac{r}{N}I(\mathbf{k})\). Carrying out analogous calculations for the case when the spin of the \(s\)-electron is directed to the left, we obtain: \(E_{s+d}=-\frac{N-r}{N}I(\mathbf{k})\), or, in the general notation,

\[ \overline{E}_{s+d} = -\frac{1}{2}(1+\mu\sigma)I(\mathbf{k}), \tag{8,6} \]

where \(\mu\) is the average magnetic moment of the internal electrons per site, and \(\sigma\) \((\sigma=\pm 1)\) is the magnetic moment of the \(s\)-electron. From (8,6) it is seen that the exchange energy is determined by the magnitude of the magnetization of the internal electrons. The term with \(\mu\sigma\) in (8,6) can plainly be understood as

effective “quasimagnetic” field, which “magnetizes” the outer electrons on the part of the inner ones possessing spontaneous magnetization. The magnitude of this “field” is determined by the electrical electron interaction; it is comparable with the ordinary exchange interaction between the inner electrons and depends on the state of the outer electron. If, for the one-electron function, one uses the strong-binding approximation (7.142), then the exchange integral (8.5), in the nearest-neighbor approximation and for a simple cubic lattice, will be equal to

\[ I(\mathbf{k})=I_0+2I(\cos k_1+\cos k_2+\cos k_3), \tag{8.7} \]

where

\[ I_0=\int \psi_j^*(\mathbf{r})\chi_j^*(\mathbf{r}')v(|\mathbf{r}-\mathbf{r}'|)\psi_j(\mathbf{r}')\chi_j(\mathbf{r})\,d\mathbf{r}\,d\mathbf{r}' \tag{8.8} \]

is the exchange integral between \(d\)- and \(s\)-electrons at one site \((\chi_j(\mathbf{r})\) is the atomic function of the \(s\)-electron), and

\[ I=\int \varphi_j^*(\mathbf{r})\chi_{j\pm1}^*(\mathbf{r}')v(|\mathbf{r}-\mathbf{r}'|)\varphi_j(\mathbf{r}')\chi_{j\pm1}(\mathbf{r})\,d\mathbf{r}\,d\mathbf{r}', \tag{8.9} \]

is the analogous integral for two neighboring atoms. The energy of a valence electron according to the ordinary theory in this same approximation is equal to

\[ E_0=b_0+2b(\cos k_1+\cos k_2+\cos k_3). \tag{8.10} \]

From a comparison of (8.9) and (8.10) it is seen that the action of the exchange forces reduces to the appearance, in the integrals \(b_0\) and \(b\), of terms depending on \(\mu\sigma\).

In the effective-mass approximation (when one may put

\[ \cos k_i=1-\frac{1}{2}k_i^2+\cdots \]

), the total energy of the \(s\)-electron, taking into account its exchange energy with the \(d\)-electron, will have the form

\[ E=E_0+\overline{E}_{s+d}=\alpha-\alpha'\mu\sigma+(\beta+\beta'\mu\sigma)(k_1^2+k_2^2+k_3^2), \tag{8.11} \]

where

\[ \begin{aligned} \alpha&=b_0+6b-\frac{1}{2}(I_0+6I), \qquad \alpha'=\frac{1}{2}(I_0+6I),\\ \beta&=-b+\frac{1}{2}I, \qquad \beta'=\frac{1}{2}I. \end{aligned} \tag{8.12} \]

The appearance of terms with \(\mu\sigma\) leads to the fact that in the “gas” of outer electrons of a ferromagnet near the Curie point, where the magnitude of the magnetization \(\mu\) changes strongly, there occurs a noticeable redistribution of the outer electrons over velocities, as well as a change in their effective mass, which according to definition (4.37) and by virtue of (8.11) is equal to:

\[ m^*=\frac{\hbar^2}{2(\beta+\beta'\mu\sigma)a^2}. \tag{8.13} \]

For an estimate of the results obtained, let us consider the free energy of the entire system of \(s\)- and \(d\)-electrons. In the present approximation its

can be regarded as a sum

\[ F=F_0(\mu,T)+F_1, \tag{8.14} \]

where \(F_0(\mu,T)\) is the free energy of the \(d\)-electrons at given \(\mu\) and \(T\), calculated without allowance for the exchange interaction (8.6), while \(F_1\) is the free energy of the “gas” of \(s\)-electrons, in which to each of the \(s\)-electrons one must assign the energy (8.11). If the strong degeneracy of this gas is taken into account, then in place of the free energy \(F_1\) one may simply take the energy.

Let us denote the number of \(s\)-electrons per unit volume with right spin by \(n_+\), and the number of such electrons with left spin by \(n_-\). Then we have

\[ n_+ + n_- = n,\qquad n_+ - n_- = n\mu', \tag{8.15} \]

where \(n\) is the total number of \(s\)-electrons per unit volume, and \(\mu'\) is their mean magnetization per atom. The energy of the gas, according to the usual formulae, is equal to

\[ \frac{3}{5}(n_+\zeta_+ + n_-\zeta_-), \tag{8.16} \]

where \(\zeta_+ \ne \zeta_-\) are, respectively, the limiting energies (chemical potentials) of the \(s\)-electrons with right and left spin. It is easy to show that they are equal to

\[ \zeta_+ = \frac{\hbar^2}{2m_+^*}\left(\frac{3n_+}{4\pi}\right)^{2/3} \quad\text{and}\quad \zeta_- = \frac{\hbar^2}{2m_-^*}\left(\frac{3n_-}{4\pi}\right)^{2/3}. \tag{8.17} \]

Here \(m_+^*\) and \(m_-^*\) are the effective masses of the \(s\)-electrons, respectively, with right and left spins. In the first case \(\mu\sigma=+\mu\), and in the second \(\mu\sigma=-\mu\). Using formulae (8.11), (8.13)—(8.17), it is not difficult to find that

\[ F(\mu,\mu',T)=F_0(\mu,T)+n\left[-\alpha'\mu\mu' + \lambda(\beta+\beta'\mu)(1+\mu')^{5/3} +\lambda(\beta-\beta'\mu)(1-\mu')^{5/3}\right], \tag{8.18} \]

where the abbreviated notation has been introduced

\[ \lambda=\frac{(3\pi^2)^{5/3}}{10\pi^2}\sim 3. \tag{8.19} \]

Starting from the requirements that the free energy (8.18) be a minimum,

\[ \frac{\partial F}{\partial \mu} = \frac{\partial F}{\partial \mu'} =0;\qquad \frac{\partial^2 F}{\partial \mu^2}>0,\qquad \frac{\partial^2 F}{\partial \mu'^2}>0, \tag{8.20} \]

one can, in principle, find the magnitudes of the magnetizations \(\mu\) and \(\mu'\), i.e. determine the state of the “gas” of \(s\)-electrons.

Since the general form of the function \(F_0(\mu,T)\) is unknown, it is impossible to write (8.18) in the general case. From symmetry considerations one can only say that \(\mu'\) will have the form of a series in

to odd powers of the magnetization of the \(d\)-electrons \(\mu\), i.e.

\[ \mu' = k_1\mu + k_3\mu^3 + \cdots, \tag{8,21} \]

where the order of magnitude of the coefficients \(k_{2n+1}\) is determined by the ratios \(\dfrac{\alpha'}{\beta}\) and \(\dfrac{\beta'}{\beta}\), i.e. will lie somewhere between 0.1 and 1.0. In the adopted approximations, the calculation of these coefficients does not require knowledge of the free energy \(F_0\) of the \(d\)-electrons.

The theory set forth makes it possible to explain a number of “anomalies” of ferromagnetic metals of the transition groups. In particular, it yields a natural explanation of the fractional character of the atomic magnetic moments of ferromagnetic metals. Indeed, those values of the atomic moments which are solutions of (8,20) and, consequently, correspond to the minimum (8,18), even at \(0^\circ\) K, by no means have to be integers. This fractional character at \(0^\circ\) K and at high temperatures (determined through the Curie constant in the expression for the paramagnetic susceptibility) will, generally speaking, manifest itself in different ways.

One can also find, for temperatures close to the Curie point (where \(0 - T \ll 0\)), the magnetization of the \(s\)-electrons, proportional to the magnetization of the \(d\)-electrons. Indeed, from (8,20), for small magnetizations (\(\mu \ll 1\) and \(\mu' \ll 1\)), if the terms with \(\mu'^2\) and higher powers are discarded, we obtain

\[ \mu' = k_1\mu = \left( \frac{9}{20\lambda}\frac{\alpha'}{\beta} - \frac{3}{2}\frac{\beta'}{\beta} \right)\mu. \tag{8,22} \]

The coefficient on the right-hand side of (8,22) before \(\mu\) does not depend on temperature, and therefore the total magnetization near the Curie point has the temperature dependence \((\mu+\mu') \sim \sqrt{\theta - T}\), which can be determined by a purely thermodynamic method\({}^{29}\).

The dependence of the effective mass and of the chemical potential of the \(s\)-electrons on temperature near the Curie point must show itself in the anomalous course of the curve of electrical resistance versus temperature. Indeed, at high temperatures \((T > \theta_D\), where \(\theta_D\) is the Debye temperature of the metal) one can always introduce a relaxation time \(\tau\) and express the specific electrical conductivity by the generalized classical formula (3,1):

\[ \sigma = \frac{n_+ e^2}{m_+^*}\tau(\zeta_+) + \frac{n_- e^2}{m_-^*}\tau(\zeta_-). \tag{8,23} \]

The expression for the relaxation time, as is known, is, first, inversely proportional to the temperature (because of the usual interaction of electrons with phonons of the thermal vibrations of the lattice) and, second, inversely proportional to the density of states of the \(s\)-electrons near the Fermi “drop.” Namely, up to universal factors and factors not depending on \(m^*\) and the magnetiza-

QUANTUM THEORY OF ELECTRON CONDUCTORS

For systems \(\mu, \mu'\), the following expressions are valid:

\[ \tau(\zeta_+)=-\frac{A}{T}(k_0^+)^2\left(\frac{dE_+}{dk}\right)_0;\qquad \tau(\zeta_-)=-\frac{A}{T}(k_0^-)^2\left(\frac{dE_-}{dk}\right)_0, \tag{8,24} \]

where the subscript “0” means that the given quantity is to be taken for the Fermi surface. Substituting (8,24), (8,13), and (8,15) into (8,23), we obtain

\[ \sigma=\frac{A^*}{T}\left[1+\gamma(\mu+\mu')^3\right], \tag{8,25} \]

accurate up to terms no higher than second degree in \(\mu\) and \(\mu'\); here the abbreviated notation has been introduced

\[ \gamma= \frac{ \left(\dfrac{9a'}{10\lambda}\right)^2+\dfrac{3}{5\lambda}\alpha'\beta'-11\beta'^2 }{ \left|2\beta(1+k_1)\right|^4 }. \tag{8,26} \]

Finally, passing from electrical conductivity to resistance, we obtain

\[ \frac{\Delta\rho}{\rho_0}=-\gamma(\mu+\mu')^3; \tag{8,27} \]

here \(\Delta\rho\) denotes the difference between the resistance actually observed below the Curie point and that value which is obtained by extrapolating the electrical-resistance curve above the Curie point, \(\rho_0\). Experiment \(^{46}\) leads to the same dependence of \(\Delta\rho\) on the spontaneous magnetization of a ferromagnet \((\mu+\mu')\) as follows from (8,27). The quantity \(\gamma\), according to measurements, proves to be of order \(0.8\). From an estimate of formula (8,26) it follows that this quantity is of order 1. The sign of this quantity remains undetermined here. The theoretical value of this calculation, which is not connected with far-reaching model assumptions, consists chiefly in the fact that it makes it possible to separate those properties of ferromagnets whose explanation constitutes a real task of theoretical physics from accidental “properties,” the appearance of which is connected with crude approximations (for example, in the one-electron theory).

The anomaly of electrical conductivity was also investigated at low temperatures \(^{44}\). Its cause is collisions between external conduction electrons and ferromagnons (spin waves) of the system of internal electrons. The temperature dependence of this “anomalous” part of the electrical resistance follows the law \(\sim T^3\). Just as the thermal conductivity of ferromagnets at high \(^{25}\) and low temperatures experiences ferromagnetic anomalies. At high temperatures these anomalies are analogous to (8,27), while at low temperatures, alongside the “phonon” part, which varies as \(\sim T^{-2}\), there appears also a “ferromagnon” additional term, which varies with temperature according to the law \(\sim T^{-1}\). On the basis of this same \(s\)—\(d\)-exchange model one can show that the thermoelectric properties of ferromagnetic metals possess corresponding anomalies \(^{45}\). The absorption of sound in ferromagnetic metals also

has an additional term varying with temperature according to a law \(\sim T^{-3}\), whereas the “phonon” part varies as \(\sim T^{-5}\) 46.

In addition, the \(s—d\)-exchange model has been applied to explain ferromagnetic anomalies in optical 47, magneto-optical 48, photoelectric 49, and galvanomagnetic 50 phenomena. In all these cases it was shown that the principal influence on the conduction electrons of a ferromagnetic metal is exerted not by the external magnetic field, but by a certain effective internal field, whose magnitude is proportional to the resultant magnetization of the specimen. The quantities determining all the effects enumerated above turn out to be functions of the spontaneous magnetization, and therefore at temperatures close to the Curie point they must undergo appreciable changes (maxima of their temperature coefficient), which is indeed observed experimentally.

The problem of a system of \(N+1\) \(s\)- and \(d\)-electrons can be considered more accurately for the case in which the internal electrons are close to magnetic saturation. The electron system can then possess, depending on the relative magnitude of the \(s—d\)-exchange energy, two types of quantum states: for weak coupling, the scattering of waves of the external electrons by the spin waves of the system of internal electrons also occurs weakly; while for strong coupling this scattering is intense. Therefore, in the first case, the external electrons and ferromagnons practically form two mutually independent ideal gases. In the second case such a representation can no longer be introduced. In both these states the system possesses a current different from zero, and in the presence of an external electric field an accelerating effect takes place. Thus Bethe’s statement 51 that the exchange interaction can by itself serve as the cause of electrical resistance is incorrect. At \(0^\circ\) K the system has no resistance; the latter arises only through the scattering of electrons by ferromagnons with increasing temperature, i.e., by the elementary excitations of the internal electrons.

A weak point of the \(s—d\)-exchange model is that it does not take into account the interaction among the external electrons themselves. Taking this interaction into account will make it possible to study still more fully the physical nature of electronic phenomena in transition metals and their alloys.

9. CONCLUSION

The several general questions of the modern quantum theory of crystals set forth in the present review show that this branch of atomic physics is still far from completion even within the framework of the existing theory of microphenomena.

The principal shortcoming of the theory considered is that in it, in fact, only part of the crystal is studied, namely, its

the electronic part, while the ionic crystal lattice itself is taken into account as an external passive source of a periodic potential field. The active participation of the lattice in the life of the crystal is taken into account only in kinetic phenomena as a small perturbation, somewhat distorting the equilibrium distribution functions of the electrons or elementary excitations in the system of electrons.

Such a “forgetting” of the active role of the ionic lattice in the atomic theory of a solid is a potential source of all sorts of metaphysical and, in the final analysis, idealistic distortions. By absolutizing certain particular conclusions of the theory obtained with complete neglect of the active participation of ions in the motion of a solid, one can arrive at such “conclusions” that correspond decisively to nothing in real actuality. Especially indicative in this sense are the complicated constructions of the band theory of metals in the works of the school of Slater, Stoner, and other foreign physicists.

The adiabatic approximation at present deprives us of the possibility of considering, from the standpoint of a consistent microscopic theory, phenomena so important for understanding the nature of a solid as diffusion, plastic deformation, etc. From the field of view of the theory of crystals there is excluded an enormous and practically very important class of phenomena whose theory is still in an unsatisfactory state. Nor should one think that a more profound account of the motion of the lattice is unnecessary also for phenomena in crystals usually regarded as “purely electronic.” As an example one may cite the phenomenon of superconductivity. Until recently there existed the false conviction that this is a purely electronic phenomenon and that the lattice here indeed plays only the passive role of a source of a periodic potential field. However, the discovery of the isotope effect in superconductors, i.e. of the dependence of the temperature of transition from the superconducting state to the nonsuperconducting state on the isotopic composition of the metal, forced us to reconsider our former notions of the “purely electronic” character of this phenomenon and to include, as its active participant, alongside the system of interacting electrons, the ionic lattice of the metal itself[^32].

In recent times Soviet physicists have shown initiative in developing the atomic theory of crystals with allowance for the active role of the crystal lattice. One may point to the extensive investigations of A. A. Vlasov[^3] on the theory of many bodies, in which the author attempts to solve the problem of the formation of crystalline systems. Although the method proposed by A. A. Vlasov is quite debatable and raises many objections, the general formulation of the many-body problem with allowance for all the specificity of interacting systems, in which individual particles lose to a considerable degree their “free” individuality, is correct.

Of great interest for the development of a consistent microscopic theory of the solid state are the works of N. N. Bogolyubov and S. V. Tyablikov3, 52, 53, which, from the standpoint of the rigor of the mathematical treatment and the physical generality of the method, are the most perfect and determine the leading place of Soviet physics in this field.

Very promising are the works on the theory of semiconductors by S. I. Pekar54, who took into account the active influence of the lattice in electronic processes, considering the polarization of a crystal caused by electrons during their motion. Pekar’s theory of polarons outlines a definite path toward a more complete treatment of the problem of the crystal, developing the general theory of the motion of electrons in a deformed crystal lattice, the foundations of which had already been laid by the works of other Soviet authors: Landau55, Tamm56, Blokhintsev57, and others.

E. I. Adirovich58, in considering questions of the luminescence of crystals, made a successful attempt to take into account the dynamical action of the lattice and gave an explanation of the nature of radiationless electronic transitions. This example also clearly shows that what would seem to be a “purely electronic” effect is in reality due to the active action of the lattice. An important step in the development of the quantum theory of molecular crystals is represented by the works of A. S. Davydov59.

Despite these very promising attempts to solve the problem of the crystal as a whole, there is as yet no basis for thinking that we have achieved great successes in this direction. Soviet physicists working in this field face a very large number of difficult, but rewarding, problems.

From the survey given above it is evident that even in the purely electronic theory of crystals, even in the adiabatic approximation, there still remain very many unclear and unsolved questions.

The method of elementary excitations (quasiparticles) opens before us fairly broad prospects for the further development of the many-electron theory of crystals.

The very important problem of investigating various types of energy spectra (of the Bose and Fermi types) of elementary excitations in crystals was first posed by Landau and Lifshitz. It received its fullest development in their fundamental course of Statistical Physics60. The concretization of these general propositions for various phenomena in crystals constitutes one of the principal tasks of the many-electron theory of solids. It is highly desirable to develop the one-electron theory of metallic alloys, worked out in the studies of A. A. Smirnov61 and A. N. Orlov62, with allowance for electronic interaction.

When using the quasiparticle method, it is by no means necessary every time to traverse the whole complicated path of calculation similar to those set forth in Section 7. It is possible, using certain

physical notions about the properties of the system of microparticles under study, immediately proceeding from a definite form of the diagonalized energy operator that corresponds to certain elementary excitations of the system. But here, of course, it is necessary to carry out with extreme care an analysis of the physical justification for the chosen form of the energy operator of the many-electron system. The rigorous derivations presented in this review may serve as guides for such a choice of diagonalized operators. It is in this direction that the paths for the development of the many-electron theory of crystals are being outlined in the works of Soviet physicists.

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1. 5. ↩

2. The exposition of the content of Section 6 and the beginning of Section 7, paragraphs 1–4, is given mainly following the book of N. N. Bogolyubov[^15]. ↩