Hartree-Fock electron gas

A variational single-Slater-determinant treatment of interacting electrons and a neutralizing positive background. For uniform plane-wave occupations the direct background and electron terms cancel, leaving kinetic and same-spin exchange energy. Allowing spin polarization or spatially modulated orbitals enlarges the variational class.

In a uniform three-dimensional Hartree-Fock electron gas with repulsive contact coupling $\gamma$, a spin-independent neutralizing background can cancel the Hartree energy. The remaining same-spin exchange term is negative and quadratic in each spin density. Integrating the filled single-spin Fermi sphere gives $A=3{\hslash }^2(6{\pi }^2{)}^{2/3}/(10m)$. At fixed total density the exchange term favors spin polarization, whereas the kinetic term opposes it.

Let $r=(\gamma /A)(n/2{)}^{1/3}$ and $P=(n_{\uparrow }-n_{\downarrow })/n$. In units $2A(n/2{)}^{5/3}$ the spin-dependent energy is $[(1+P{)}^{5/3}+(1-P{)}^{5/3}]/2-r(1+P^2)/2$. Comparing only $P=0$ and $P=1$ gives the endpoint crossing $r=2^{5/3}-2$. Expanding near $P=0$ gives a quadratic coefficient $5/9-r/2$, so the unpolarized state first loses local stability at $r=10/9$. The endpoint crossing is therefore not the actual continuous onset; intermediate polarizations matter.

The uniform paramagnetic three-dimensional electron-gas determinant is unstable within Hartree-Fock theory to a spin-density-wave deformation, even at high density. Comparing only spatially uniform polarized and unpolarized determinants therefore does not find the unrestricted Hartree-Fock ground state.^[ref]

In a three-dimensional unpolarized gas, exchange energy per particle is $-3e^2{k}_F/(16{\pi }^2\epsilon )$, with Coulomb convention $e^2/(4\pi \epsilon r)$. It scales as minus the inverse dimensionless spacing radius, while kinetic energy scales as its inverse square. Exchange acts only between equal-spin occupied orbitals.

For number densities $n_{\uparrow },n_{\downarrow }$, $\zeta =(n_{\uparrow }-n_{\downarrow })/(n_{\uparrow }+n_{\downarrow })$. Uniform spin polarization raises kinetic energy by concentrating occupation in one Fermi sphere, while lowering exchange energy.