Hartree method

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.