Editing Hartree–Fock method (section)

{{Short description|Method in quantum physics}}
{{Electronic structure methods}}

In [[computational physics]] and [[Computational chemistry|chemistry]], the '''Hartree–Fock''' ('''HF''') method is a method of approximation for the determination of the [[wave function]] and the energy of a [[Many-body problem|quantum many-body system]] in a [[stationary state]]. The method is named after [[Douglas Hartree]] and [[Vladimir Fock]].

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 [[fermion]]s) or by a single [[Permanent (mathematics)|permanent]] (in the case of [[boson]]s) of ''N'' [[spin-orbital]]s. 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. Hartree–Fock approximation is an instance of [[mean-field theory]],<ref>{{cite book |last1=Bruus |first1=Henrik |last2=Flensberg |first2=Karsten |title=Many-body quantum theory in condensed matter physics: an introduction |date=2014 |publisher=Oxford University Press |location=Oxford New York |isbn=9780198566335 |edition=Corrected version |url=https://www.phys.lsu.edu/~jarrell/COURSES/ADV_SOLID_HTML/Other_online_texts/Many-body%20quantum%20theory%20in%0Acondensed%20matter%20physics%0AHenrik%20Bruus%20and%20Karsten%20Flensberg.pdf}}</ref> where neglecting higher-order fluctuations in [[Phase_transition#Order_parameters|order parameter]] allows interaction terms to be replaced with quadratic terms, obtaining exactly solvable Hamiltonians.

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]], [[Douglas Hartree|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 [[Hartree–Fock#The Fock operator|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.<ref>{{cite journal|journal = Computer Physics Communications
|title = General Hartree-Fock program
|last = Froese Fischer | first = Charlotte
|volume = 43 |issue = 3
| pages =355–365 | year = 1987 |doi=10.1016/0010-4655(87)90053-1
|bibcode = 1987CoPhC..43..355F }}</ref>
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<ref>{{cite journal |first=Mudar A. |last=Abdulsattar |title=SiGe superlattice nanocrystal infrared and Raman spectra: A density functional theory study |journal=[[Journal of Applied Physics|J. Appl. Phys.]] |volume=111 |issue=4 |pages=044306–044306–4 |year=2012 |doi=10.1063/1.3686610 |bibcode = 2012JAP...111d4306A |doi-access=free }}</ref> and solids but it has also found widespread use in [[nuclear physics]]. (See [[Bogoliubov transformation|Hartree–Fock–Bogoliubov method]] for a discussion of its application in [[nuclear structure#Nuclear pairing phenomenon|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 number]]s 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|Open-shell]] systems, where some of the electrons are not paired, can be dealt with by either the [[Restricted open-shell Hartree–Fock|restricted open-shell]] or the [[Unrestricted Hartree–Fock|unrestricted]] Hartree–Fock methods.