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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.

The classical electrostatic self-energy functional of an electron number density is the expression in the title. The factor one-half prevents counting each pair twice. Its functional derivative is the Hartree potential; integrating density times that potential gives $2E_H$. The functional includes an unphysical one-electron self-interaction unless other terms cancel it. It differs from the Hartree energy atomic unit.

For the unpolarized gas write $x=n/2$. The positive direct Hartree energy before background cancellation and the kinetic energy areTheir ratio is thereforeThis compares the separate direct and kinetic terms. The net Hartree-plus-background contribution after the stipulated cancellation is zero, so it would be incorrect to interpret $r_s$ as that residual contribution divided by $T$. The exchange-energy magnitude in the unpolarized state is $E_H/2$, not $E_H$.

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.

For a $2p$ atomic orbital, $l=1$. The radial factor obeysSubstitution into the radial Schrödinger equation for a central potential givesThus the energy eigenvalue isThe normalization constant cancels; $E_{\mathrm{h}}$ is the Hartree energy.

At this separation the orbital overlap and interatomic two-electron exchange integrals are negligible. Each localized electron then contributes essentially the isolated hydrogen energy, while the interatomic electrostatic terms cancel to leading order for neutral fragments. Including nuclear repulsion in the molecular energy givesThe closed-shell RHF determinant instead doubly occupies $g\simeq (A+B)/\sqrt{2}$. It retains ionic configurations with both electrons on the same atom, even at dissociation. Using atomic units and the Hartree energy in the stated unrelaxed hydrogenic 1s basis, the same-atom two-electron Coulomb integral is $J_{AA}=J_{BB}=5E_h/8$. Consequently at infinite separationwhere energies and integrals inside the first expression are in atomic units. It is much too high compared with two neutral atoms. Retaining the leading intersite Coulomb term at $R=10\text{A˚}=18.90a_0$ gives $E_{\mathrm{RHF}}\simeq -11/16-1/(2R)\simeq -0.714E_h$, still about $0.286E_h$ above UHF. The numerical RHF estimates specify the supplied fixed 1s basis; a larger optimized orbital space changes them. If nuclear repulsion is excluded by an electronic-energy convention, subtract the same $1/R$ from both quoted total molecular energies. The restricted/unrestricted comparison and their dissociation behaviour are unchanged.^[ref]