= Solution
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,
$$
\left[-\frac{\hbar^2\nabla^2}{2m_e}-\frac{Ze^2}{4\pi\varepsilon_0r}+\sum_{j\ne i}\frac{e^2}{4\pi\varepsilon_0}\int\frac{|\phi_j(\mathbf r')|^2}{|\mathbf r-\mathbf r'|}\,d^3r'\right]\phi_i=\epsilon_i\phi_i.
$$
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.<Topics in Quantum Mechanics by David Tong>[Authored Atoms chapter,§7.3.1: Hartree self-consistency and product-state limitations]{ref}
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