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.
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.
