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In three dimensions, let $a=(3/4\pi n{)}^{1/3}$ be the mean-spacing radius and $a_B^{\ast }=4\pi \epsilon {\hslash }^2/(m^{\ast }{e}^2)$ the effective Bohr radius. The Coulomb scale is $E_C\sim e^2/(4\pi \epsilon a)$ and the Fermi energy scale is $E_F\sim {\hslash }^2/(m^{\ast }{a}^2)$, so $E_C/E_F\sim a/a_B^{\ast }=r_s$. Decreasing density favors repulsion relative to delocalization energy, motivating a Wigner crystal. This scaling does not determine the numerical quantum melting value of $r_s$. A classical nondegenerate gas instead compares Coulomb energy with $k_{B}T$.

The 1979 experiment of C. C. Grimes and colleagues studied a two-dimensional sheet of electrons above liquid helium. It reported crystallization at densities $3\times 10^8$ to $9\times 10^8{\mathrm{cm}}^{-2}$, with melting temperatures $0.35$ to $0.65\mathrm{K}$. The sheet was classical: Coulomb energy competed with thermal motion, unlike the kinetic-energy comparison used for a zero-temperature degenerate Wigner crystal. It supplies evidence for electron ordering while keeping these two physical regimes distinct.^[ref]

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.

A Wigner crystal is an ordered electron arrangement favored when repulsion overwhelms the kinetic cost of localization. In a three-dimensional electron gas with a neutralizing background, define a mean spacing $a=(3/4\pi n{)}^{1/3}$ and effective Bohr radius $a_B^{\ast }=4\pi \epsilon {\hslash }^2/(m^{\ast }{e}^2)$. The Coulomb-to-kinetic energy ratio of an electron gas scales asThus low density favors the crystal in the degenerate quantum model. Translational symmetry is broken, and excitations include collective lattice vibrations. The melting criterion depends on dimension, screening, disorder and quantum fluctuations, so this scaling argument does not supply a universal numerical transition density.

The electron crystallization on liquid helium experiment provides an important related observation: a dilute two-dimensional electron sheet above liquid helium crystallized on cooling. It was a classical electron system, with thermal rather than Fermi motion competing with Coulomb energy; its coupling parameter is $\Gamma =E_C/(k_{B}T)$. It therefore illustrates Coulomb ordering without being a direct measurement of the zero-temperature three-dimensional quantum transition.^[ref]