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The Hartree-Fock method minimizes the expectation of an interacting-electron Hamiltonian over normalized Slater determinants of orthonormal one-particle spin orbitals. It is a variational, self-consistent mean-field approximation. The determinant enforces fermionic antisymmetry exactly within that class, while each orbital responds to the mean charge density and a nonlocal exchange operator generated by the occupied orbitals.

For an electron gas, this accounts for kinetic pressure, the average electrostatic field, same-spin exchange and the associated exchange hole, and possible spin polarization. It does not account for general dynamical electron correlation, such as the correlated avoidance of opposite-spin electrons, and cannot capture all screening or collective fluctuations merely by optimizing one determinant. Uniform plane waves are an additional restriction, not the definition of Hartree-Fock itself.

In the energy functional, the one-body term is the sum of electron kinetic energy and ionic-potential energy. The direct two-body term is the Hartree energy: the classical interaction of the mean electron density with itself. Its factor $1/2$ avoids counting every pair twice. The negative Fock term comes from exchanging the two indistinguishable electron labels in the antisymmetric state. For a spin-independent interaction, spin orthogonality makes this exchange act only between equal spin components. For an orbital interacting with itself, direct and exchange contributions cancel exactly; the exchange term removes the spurious one-electron self-interaction. Ionic background terms and their prescribed cancellation must be included consistently with the direct term.

Use a three-dimensional electron gas with a uniform positive background, the jellium model. The Hartree-Fock method minimizes the expectation of the interacting Hamiltonian over normalized Slater determinants. Antisymmetry is exact within that trial class, but correlations requiring a superposition of determinants are omitted. For orthonormal occupied spin orbitals ${\varphi }_i$, the energy iswhere $J_{ij}$ is the direct Coulomb integral and $K_{ij}$ the exchange integral; the spin overlap makes exchange vanish between opposite-spin orbitals. The self-interaction terms $J_{ii}$ and $K_{ii}$ cancel. Variation gives a self-consistent Fock operator $\hat{f}=h_0+{\sum }_j({\hat{J}}_j-{\hat{K}}_j)$; exchange is nonlocal. Its eigenfunctions define occupied orbitals of a stationary determinant, and the determinant must be compared with competing self-consistent solutions to select a variational ground state.

The eigenvalues are orbital Lagrange multipliers. Their sum counts each interaction twice:Frozen-orbital removal energies may be estimated using Koopmans' theorem, but measured quasiparticle energies can contain relaxation and correlations absent from this approximation. Unoccupied orbital energies are not automatically an exact excitation spectrum. In particular, unscreened exchange in a uniform three-dimensional gas produces an unphysical logarithmic slope singularity at its Fermi surface; a one-determinant mean field does not reproduce the interacting Fermi liquid's screened response.

For a spatially uniform plane-wave determinant, electron-background and background-background terms cancel the uniform direct Coulomb contribution. Retain the background when making this cancellation: an isolated infinite electron charge distribution would have a divergent direct energy. The surviving contributions per particle are kinetic energy and negative same-spin exchange energy. Let $(3/(4\pi n){)}^{1/3}=r_s{a}_B^{\ast }$ define the Dimensionless Wigner-Seitz radius, with $a_B^{\ast }=4\pi \epsilon {\hslash }^2/(m^{\ast }{e}^2)$, $E_H^{\ast }={\hslash }^2/(m^{\ast }{a}_B^{\ast 2})$ and $\zeta =(n_{\uparrow }-n_{\downarrow })/n$. ThenFor the unpolarized state this is approximately $1.105/r_s^2-0.4582/r_s$. The kinetic contribution penalizes spin polarization because it enlarges an occupied Fermi sphere; exchange favors polarization because equal-spin occupation gains more exchange energy. As density falls, the $r_s^{-1}$ exchange scale overtakes the $r_s^{-2}$ kinetic scale. The uniform fully polarized and unpolarized energies cross atThis is a variational comparison within the homogeneous Hartree-Fock electron gas, not an exact physical ferromagnetic phase boundary. It demonstrates how Pauli kinetic energy and exchange compete and why the allowed trial states matter.

Allowing spatial symmetry breaking changes the result: the Overhauser instability lowers the paramagnetic determinant's Hartree-Fock energy through a spin density wave, even in the high-density limit. A calculation restricted to uniform Fermi spheres therefore does not give the unrestricted Hartree-Fock ground state.^[ref] At still lower density, localization and a Wigner crystal compete with fluid states. electron-gas correlation energy beyond a single determinant is important in assessing their competition. Quantum Monte Carlo provides a concrete route to comparing correlated fluid and crystal energies; the original stochastic calculation demonstrates why exchange-only energy differences are insufficient.^[ref]

Hartree-Fock is a useful variational baseline with exact exchange, not a controlled complete low-density phase diagram. At high density the kinetic scale dominates, but even a small symmetry-breaking energy gain can invalidate a restricted determinant as the exact Hartree-Fock minimizer. At low density neglected correlations, screening and localization can change both transition locations and the ordering of candidate phases. A numerical phase boundary should therefore state dimension, spin constraints, background model and whether correlation and spatially modulated states were included.

The central field approximation replaces the anisotropic many-electron environment by an average spherical potential, giving one-electron orbitals labelled by $n,l,m$. Screening changes the effective radial attraction and removes the pure Coulomb degeneracy between different $l$.

The Hartree method improves the potential self-consistently. For normalized occupied orbitals,Guess orbitals, construct their mean fields, solve the Hartree equations, and iterate to convergence. The $j\ne i$ sum excludes self-interaction in this product-state model. A spherical average can enforce the central-field specialization. A Hartree product state omits fermionic exchange and dynamical correlation; Hartree-Fock method adds antisymmetry and exchange through a Slater determinant. Converged orbital eigenvalues must not simply be added as the total energy, because that double counts pair interactions.^[ref]