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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 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
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
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According to [[Slater–Condon rules]], the expectation value of energy of the [[Molecular Hamiltonian#Clamped nucleus Hamiltonian|molecular electronic Hamiltonian]] <math>\hat{H}^e</math> for a [[Slater determinant]] is
: <math display="inline">\begin{aligned} E[\psi^{HF}] &= \left\langle\psi^{HF}|\hat{H}^e|\psi^{HF}\right\rangle \\
&= \sum_{i=1}^N \int\text{d}\mathbf{x}_i \, \phi_i^*(\mathbf{x}_i) \hat{h}(\mathbf{x}_i) \phi_i(\mathbf{x}_i) \\
&+ \frac{1}{2} \sum_{i=1}^N\sum_{j=1}^N \int \mathrm{d}\mathbf{x}_i \int \text{d}\mathbf{x}_j \phi_i^*(\mathbf{x}_i)
&- \frac{1}{2} \sum_{i=1}^N\sum_{j=1}^N \int \text{d}\mathbf{x}_i \int \text{d}\mathbf{x}_j\phi_i^*(\mathbf{x}_i)\phi_j^*(\mathbf{x}_j) \frac{1}{|\mathbf{r}_i-\mathbf{r}_j|}\phi_i(\mathbf{x}_j)\phi_j(\mathbf{x}_i) \end{aligned}
</math>
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\end{aligned}</math>
The factor 1/2 in the molecular Hamiltonian drops out before the double integrals due to symmetry and the product rule. We may define [[Fock matrix|Fock
: <math>\hat{F}(\mathbf{x}_k)\phi_k(\mathbf{x}_k) \equiv \left[ \hat{h}(\mathbf{x}_k) + \hat{J}(\mathbf{x}_k) - \hat{K}(\mathbf{x}_k) \right]\phi_k(\mathbf{x}_k) = \epsilon_k \phi_k(\mathbf{x}_k),</math>
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The solution <math>\phi_k</math> and <math>\epsilon_k</math> are called molecular orbital and orbital energy respectively.
Although Hartree-Fock equation appears in the form of a eigenvalue problem, the Fock operator itself depends on <math>\phi</math> and must be solved by different technique.
===Total energy===
The optimal total energy <math> E_{HF} </math> can be written in terms of molecular orbitals.
:<math> E_{HF} = \sum_{i=1}^{N} \hat h_{ii} + \sum_{i=1}^{N} \sum_{j=1}^{N/2} [2\hat J_{ij} - \hat K_{ij}] + V_{\text{
<math>\hat J_{ij}</math> and <math>\hat K_{ij}</math> are matrix elements of the Coulomb and exchange operators respectively, and <math>V_{\text{nucl}}</math> is the total electrostatic repulsion between all the nuclei in the molecule.
It should be
If the atom or molecule is [[closed shell]], the total energy according to the Hartree-Fock method is
: <math>E_{HF} = 2 \sum_{i=1}^{N/2} \hat h_{ii} + \sum_{i=1}^{N/2} \sum_{j=1}^{N/2} [2\hat J_{ij} - \hat K_{ij}] + V_{\text{nucl}}.</math><ref name= Levine>Levine, Ira N. (1991). Quantum Chemistry (4th ed.). Englewood Cliffs, New Jersey: Prentice Hall. p. 402-3. {{ISBN|0-205-12770-3}}.</ref>
=== Linear combination of atomic orbitals ===
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