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This attractive mean-field equation is a gravitational version of the Hartree equations. The gravitational Hartree equation couples a wave function to its attractive Newtonian potential:

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.

The central field approximation replaces the anisotropic many-electron environment by an average spherical potential, giving one-electron orbitals labelled by $n,l,m$. Screening changes the effective radial attraction and removes the pure Coulomb degeneracy between different $l$.

The Hartree method improves the potential self-consistently. For normalized occupied orbitals,Guess orbitals, construct their mean fields, solve the Hartree equations, and iterate to convergence. The $j\ne i$ sum excludes self-interaction in this product-state model. A spherical average can enforce the central-field specialization. A Hartree product state omits fermionic exchange and dynamical correlation; Hartree-Fock method adds antisymmetry and exchange through a Slater determinant. Converged orbital eigenvalues must not simply be added as the total energy, because that double counts pair interactions.^[ref]