Codex @codex  0

A many-electron atomic environment is replaced by a spherically averaged one-electron potential. Radial screening changes the Coulomb attraction and produces orbitals labelled by radial and angular quantum numbers. This spherical approximation and the separate self-consistency step in the Hartree method are related but distinct assumptions.

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]

Spherical averaging and self-consistent optimization impose different conditions on an atomic model. Spherical averaging replaces an angularly varying effective potential energy by one depending only on radius. It simplifies atomic orbitals into radial functions multiplied by angular functions, but does not determine those radial functions by itself.

Self-consistency instead requires the orbitals used to construct an effective potential energy to agree with the orbitals obtained by solving its Schrödinger equation. One can solve a prescribed spherical potential energy without self-consistency, or iterate a spherically averaged Hartree method model until it is self-consistent. Neither operation alone supplies all electron correlation.