Hartree equations

The Hartree equations are self-consistent one-particle equations obtained by approximating a many-body wave function by a tensor product of orbital wave functions. Each orbital sees the potential averaged over the other particles' probability densities; attractive Newtonian mean-field variants include the Gravitational Hartree equation.

A variational product of one-electron orbitals produces coupled one-electron equations, each containing the electrostatic density of the other occupied orbitals. Iterating orbitals and potentials gives a self-consistent field. The product omits fermionic exchange; an antisymmetric Slater determinant instead leads to Hartree-Fock theory.

For Hartree direct integrals $J_{ij}$ and orbital eigenvalues ${ϵ}_i$, the product-state energy is $E={\sum }_i{ϵ}_i-{\sum }_{i<j}{J}_{ij}$. Each pair enters two orbital mean fields but only once in the total interaction energy. Orbital energies are therefore not independent additive particle energies.

A many-particle trial state built from one-particle factors, without the additional correlations of a general many-body wavefunction. For identical bosons occupying one normalized orbital $\phi$, the symmetric fixed-number form is $(a_{\phi }^{†}{)}^N|0⟩/\sqrt{N!}$. Its field expectation is zero because annihilation changes particle number, while its one-body density matrix has the macroscopic eigenvalue $N$. An antisymmetrized multi-orbital fermionic trial state instead belongs to the Hartree-Fock construction.

The local mean Coulomb potential generated by the occupied electron density. A separate positive background or external charge contribution must be included where appropriate. Hartree-Fock exchange is nonlocal and removes equal-orbital self-interaction from the total direct-plus-exchange result.

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.