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In computational physics and chemistry, the Hartree–Fock (HF) method is a method of approximation for the determination of the wave function and the energy of a quantum many-body system in a stationary state. The Hartree–Fock method often assumes that the exact, ''N''-body wave function of the system can be approximated by a single Slater determinant (in the case where the particles are fermions) or by a single permanent (in the case of bosons) of ''N'' spin-orbitals. By invoking the variational method, one can derive a set of ''N''-coupled equations for the ''N'' spin orbitals. A solution of these equations yields the Hartree–Fock wave function and energy of the system. Especially in the older literature, the Hartree–Fock method is also called the self-consistent field method (SCF). In deriving what is now called the Hartree equation as an approximate solution of the Schrödinger equation, Hartree required the final field as computed from the charge distribution to be "self-consistent" with the assumed initial field. Thus, self-consistency was a requirement of the solution. The solutions to the non-linear Hartree–Fock equations also behave as if each particle is subjected to the mean field created by all other particles (see the Fock operator below) and hence the terminology continued. The equations are almost universally solved by means of an iterative method, although the fixed-point iteration algorithm does not always converge. This solution scheme is not the only one possible and is not an essential feature of the Hartree–Fock method. The Hartree–Fock method finds its typical application in the solution of the Schrödinger equation for atoms, molecules, nanostructures and solids but it has also found widespread use in nuclear physics. (See Hartree–Fock–Bogoliubov method for a discussion of its application in nuclear structure theory). In atomic structure theory, calculations may be for a spectrum with many excited energy levels and consequently the Hartree–Fock method for atoms assumes the wave function is a single configuration state function with well-defined quantum numbers and that the energy level is not necessarily the ground state. For both atoms and molecules, the Hartree–Fock solution is the central starting point for most methods that describe the many-electron system more accurately. The rest of this article will focus on applications in electronic structure theory suitable for molecules with the atom as a special case. The discussion here is only for the Restricted Hartree–Fock method, where the atom or molecule is a closed-shell system with all orbitals (atomic or molecular) doubly occupied. Open-shell systems, where some of the electrons are not paired, can be dealt with by one of two Hartree–Fock methods: * Restricted open-shell Hartree–Fock (ROHF) * Unrestricted Hartree–Fock (UHF) == Brief history == The origin of the Hartree–Fock method dates back to the end of the 1920s, soon after the discovery of the Schrödinger equation in 1926. In 1927 D. R. Hartree introduced a procedure, which he called the self-consistent field method, to calculate approximate wave functions and energies for atoms and ions. Hartree was guided by some earlier, semi-empirical methods of the early 1920s (by E. Fues, R. B. Lindsay, and himself) set in the old quantum theory of Bohr. In the Bohr model of the atom, the energy of a state with principal quantum number n is given in atomic units as . It was observed from atomic spectra that the energy levels of many-electron atoms are well described by applying a modified version of Bohr's formula. By introducing the quantum defect ''d'' as an empirical parameter, the energy levels of a generic atom were well approximated by the formula , in the sense that one could reproduce fairly well the observed transitions levels observed in the X-ray region (for example, see the empirical discussion and derivation in Moseley's law). The existence of a non-zero quantum defect was attributed to electron-electron repulsion, which clearly does not exist in the isolated hydrogen atom. This repulsion resulted in partial screening of the bare nuclear charge. These early researchers later introduced other potentials containing additional empirical parameters with the hope of better reproducing the experimental data. Hartree sought to do away with empirical parameters and solve the many-body time-independent Schrödinger equation from fundamental physical principles, i.e., ab initio. His first proposed method of solution became known as the Hartree method. However, many of Hartree's contemporaries did not understand the physical reasoning behind the Hartree method: it appeared to many people to contain empirical elements, and its connection to the solution of the many-body Schrödinger equation was unclear. However, in 1928 J. C. Slater and J. A. Gaunt independently showed that the Hartree method could be couched on a sounder theoretical basis by applying the variational principle to an ansatz (trial wave function) as a product of single-particle functions. In 1930 Slater and V. A. Fock independently pointed out that the Hartree method did not respect the principle of antisymmetry of the wave function. The Hartree method used the Pauli exclusion principle in its older formulation, forbidding the presence of two electrons in the same quantum state. However, this was shown to be fundamentally incomplete in its neglect of quantum statistics. It was then shown that a Slater determinant, a determinant of one-particle orbitals first used by Heisenberg and Dirac in 1926, trivially satisfies the antisymmetric property of the exact solution and hence is a suitable ansatz for applying the variational principle. The original Hartree method can then be viewed as an approximation to the Hartree–Fock method by neglecting exchange. Fock's original method relied heavily on group theory and was too abstract for contemporary physicists to understand and implement. In 1935 Hartree reformulated the method more suitably for the purposes of calculation. The Hartree–Fock method, despite its physically more accurate picture, was little used until the advent of electronic computers in the 1950s due to the much greater computational demands over the early Hartree method and empirical models. Initially, both the Hartree method and the Hartree–Fock method were applied exclusively to atoms, where the spherical symmetry of the system allowed one to greatly simplify the problem. These approximate methods were (and are) often used together with the central field approximation, to impose that electrons in the same shell have the same radial part, and to restrict the variational solution to be a spin eigenfunction. Even so, solution by hand of the Hartree–Fock equations for a medium-sized atom were laborious; small molecules required computational resources far beyond what was available before 1950. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Hartree–Fock method」の詳細全文を読む スポンサード リンク
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