Self-Consistent Field

the interaction, averaged in a certain way, between a given particle and the other particles of a quantum-mechanical system consisting of many particles. Because the problem of many interacting particles is very complex and has no exact solution, calculations are done by approximate methods. One of the most often used approximate methods of quantum mechanics is based on the introduction of a self-consistent field, which permits the many-particle problem to be reduced to the problem of a single particle moving in the average self-consistent field produced by the other particles. The various schemes for introducing self-consistent fields differ in the way in which the interaction is averaged. The self-consistent field method is widely used for the approximate description and the calculation of the states of, for example, many-electron atoms, molecules, heavy nuclei, electrons in a metal, and the spin system in a ferromagnetic.

In a quantum-mechanical system of many interacting particles, the motion of any particle is interconnected, or correlated, in a complicated way with the motion of the other particles in the system. Consequently, each particle is not located in a definite state and cannot be described by means of its own one-particle wave function. The state of the system as a whole is described by a wave function dependent on the coordinate and spin variables of all the particles in the system. The initial assumption of the self-consistent field method is that to describe the system approximately, wave functions can be introduced for each particle in the system. Then, in order to make approximate allowance for the interaction with the other particles, a field is introduced that is averaged over the motion of the other particles in the system through the use of the particles’ one-particle wave functions. The one-particle wave functions must be self-consistent, since, on the one hand, they are a solution of the Schrödinger equation for a single particle moving in the average field produced by the other particles and, on the other hand, they determine the average potential of the field in which the particles are moving, the term “self-consistent field” reflects this consistency.

In the simplest method of introducing a self-consistent field, not the wave functions but the density distribution of the particles in space is defined. This method, called the Thomas-Fermi method, was proposed independently by the British physicist L. Thomas in 1927 and the Italian physicist E. Fermi in 1928. In many-electron atoms the average potential acting on a given electron varies quite slowly. Many electrons therefore are located within a volume where the relative change in potential is small, and the electrons, which obey Fermi-Dirac statistics, can be dealt with as a degenerate Fermi gas by the methods of statistical physics (see DEGENERATE GAS). The action of the other electrons on a given electron here can be replaced by the action of some centrally symmetric self-consistent field that is added to the field of the nucleus. The field is selected so that it is consistent with the distribution of the average charge density (which is proportional to the distribution of the average density of the electrons in the atom), since the electric field potential is related to the charge distribution by the Poisson equation. The average electron density is in turn regarded as the density of a degenerate ideal Fermi gas located in this average field. The average electron density is related to the field through the Fermi level, that is, through the maximum energy of the Fermi distribution at the absolute temperature T = 0. This means that the selection of the average field potential must be self-consistent. The self-consistent Thomas-Fermi field explains the sequence in which electron shells are filled in atoms and consequently accounts for the periodic table of the elements. The method is also applicable in the theory of heavy nuclei. It makes possible an explanation of the order in which nuclear shells are filled by nucleons (protons and neutrons). Here, in addition to a centrally symmetric self-consistent field, a self-consistent field produced by the interaction of the orbital motion of the nucleons with the nucleons’ spin (the spin-orbit interaction) must be taken into account.

Another, more exact, way of introducing the self-consistent field is the Hartree method, which was proposed in 1927 by the British physicist D. Hartree. In this method, the wave function of a many-electron atom is represented approximately as the product of the wave functions of the individual electrons corresponding to the different quantum states of the electrons in the atom. To such an electron distribution there corresponds an average self-consistent field that depends on the selection of the one-electron functions. These functions in turn depend on the average field. The one-electron wave functions are selected on the condition of minimum average energy. This ensures the best approximation for the selected type of wave functions. The self-consistent field in this case is obtained by averaging over the orbital motions of the other electrons. Self-consistent fields are different for different states of the electrons in the atom. The wave functions of the electrons are determined by the same average field potential. This means that the potential and the wave functions must be selected in a self-consistent manner.

The Hartree method does not take into account the Pauli exclusion principle, which requires that the total wave function of the electrons in an atom be antisymmetric in the exchange of equivalent electrons. An improved way of introducing the self-consistent field is given by the Hartree-Fock method, which was proposed by V. A. Fock (Fok) in 1930. This method starts with a wave function (of the electrons in the atom) of the correct symmetry in the form of a determinant of the one-electron orbital wave functions. This ensures fulfillment of the Pauli principle. One-electron functions are found, as in the Hartree method, from a minimization of the average energy. The self-consistent field is obtained with averaging that takes into account a correlation of the orbital electrons that is associated with the electron exchange (see EXCHANGE INTERACTION).

In addition to the simple exchange correlation, the correlation of pairs of particles with opposite spins is possible. In the case of attraction, such a correlation results in the formation of correlated pairs of particles called coupled pairs. A generalization of the Hartree-Fock method that allows for this correlation was made by N. N. Bogoliubov in 1958 and is used in the theory of superconductivity and the theory of heavy nuclei.

The self-consistent field is also used in the theory of metals. It is assumed that the electrons of a metal move independently in the self-consistent field generated by the ions of the crystal lattice and by the other electrons. In the simplest versions of the theory, this field is assumed to be known. The best way of introducing self-consistent fields in the theory of metals is given by the pseudopotential method, which is applicable to the alkali and polyvalent metals. In this case, the self-consistent field is not a potential field.

Another example of self-consistency in solid-state physics is the distinctive behavior of an electron in an ionic nonconducting crystal. The electron polarizes its surroundings with its field, and the polarization associated with the displacement of ions creates a potential well into which the electron falls. Such a self-consistent state of the electron and the dielectric medium is called a polaron. A polaron can move through the crystal and is a charge carrier in ionic crystals. Electrical, photoelectric, and many optical phenomena in these crystals can be interpreted on the basis of polaron theory.

Historically, the first version of a self-consistent field was the molecular field introduced in 1907 by the French physicist P. Weiss to explain ferromagnetism. Weiss assumed that the magnetic moment of each atom in a ferromagnetic is located in an integral molecular field, which itself is proportional to the magnetic moment and thus is self-consistent. In the language of self-consistent approximations, this field actually expresses a quantum exchange interaction. This can be understood by applying to the system of interacting spins of a ferromagnetic the self-consistent field method, which in this case is called an approximation of the molecular field method. Here, the exchange interaction of each spin with the others is replaced by the action of some effective molecular field that is introduced in a self-consistent manner.