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The study of the structure and properties of atoms based on quantum mechanics and the Schrödinger equation. These tools make it possible, in principle, to predict most properties of atomic systems. A stationary state of an atom is governed by a time-independent wave function which depends on the position coordinates of all the particles within the atom. To obtain the wave function, the time-independent Schrödinger equation, a second-order differential equation, has to be solved. The potential energy term in this equation contains the Coulomb interaction between all the particles in the atom, and in this way they are all coupled to each other. See Quantum mechanics
A many-particle system where the behavior of each particle at every instant depends on the positions of all the other particles cannot be solved directly. This is not a problem restricted to quantum mechanics. A classical system where the same problem arises is a solar system with several planets. In classical mechanics as well as in quantum mechanics, such a system has to be treated by approximate methods.
Independent particle model
As a first approximation, it is customary to simplify the interaction between the particles. In the independent particle model the electrons are assumed to move independently of each other in the average field generated by the nucleus and the other electrons. In this case the potential energy operator will be a sum over one-particle operators. The simplest wave function which will satisfy the resulting equation is a product of one-particle orbitals. To fulfill the Pauli exclusion principle, the total wave function must, however, be written in a form such that it will vanish if two particles are occupying the same quantum state. This is achieved with an antisymmetrized wave function, that is, a function which, if two electrons are interchanged, changes sign but in all other respects remains unaltered. The antisymmetrized product wave function is usually called a Slater determinant. See Exclusion principle
In the late 1920s, only a few years after the discovery of the Schrödinger equation, D. Hartree showed that the wave function to a good approximation could be written as a product of orbitals, and also developed a method to calculate the orbitals. Important contributions to the method were also made by V. Fock and J. C. Slater (thus, the Hartree-Fock method). The Hartree-Fock model thus gives the lowest-energy ground state within the assumption that the electrons move independently of each other in an average field from the nucleus and the other electrons.
To simplify the problem even further, it is common to add the requirement that the Hartree-Fock potential should be spherically symmetric. This leads to the central-field model and the so-called restricted Hartree-Fock method.
The Hartree-Fock method gives a qualitative understanding of many atomic properties. Generally it is, for example, able to predict the configurations occupied in the ground states of the elements. Electron binding energies are also given with reasonable accuracy.
Correlation is commonly defined as the difference between the full many-body problem and the Hartree-Fock model. More specifically, the correlation energy is the difference between the experimental energy and the Hartree-Fock energy. There are several methods developed to account for electron correlation, including the configuration-interaction method, the multiconfiguration Hartree-Fock method, and perturation theory.
Strongly correlated systems
Although the Hartree-Fock model can qualitatively explain many atomic properties, there are systems and properties for which correlation is more important, such as negative ions, doubly-excited states, and some open-shell systems. If the interest is not in calculating the total energy of a state but in understanding some other properties, such as the hyperfine structure, effects beyond the central field model can be more important. See Hyperfine structure, Negative ion
The Schrödinger equation is a nonrelativistic wave equation. In heavy elements the kinetic energy of the electrons becomes very large, and calculations are based on the relativistic counterpart to the Schrödinger equation, the Dirac equation. It is possible to construct a Hartree-Fock model based on the Dirac equation, where the electron-electron interaction is given by the Coulomb interaction, a magnetic contribution, and a term which corrects for the finite speed (retardation) with which the interaction propagates. See Antimatter, Relativistic quantum theory
Radiative corrections, which arise when the electromagnetic field is quantized within the theory of quantum electrodynamics, For many-body systems, calculations of radiative effects are usually done within some independent-particle model, and the result is added to a correlated relativistic calculation based on the Dirac equation. See Atomic structure and spectra, Quantum electrodynamics