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.
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