The central field approximation.

The treatment is based on an approximate method where the central part of the electron-electron interaction is included from the beginning. The Hamilton function of the system is written

The corresponding Hamilton operator is

H₁ is uncomplicated, leading to a system of uncoupled differential equations which may be solved using a wavefunction that is a product of one-electron wavefunctions in the usual manner. At the moment, we also restrict the discussion to atoms which are not too heavy, which allows us to treat H₃ as a perturbation. Remaining is H₂, which we know leads to coupled differential equations. However, H₂ may be separated into two parts, one of which is centrally symmetric. In a first approximation, this part is maintained along with H₁ in the Schrödinger equation. This is the important central field approximation, which may be stated as follows:

In the central field approximation, each electron moves independently of the other electrons in the atom in a spherically -symmetric average field created by the nucleus and the other electrons.

This approximation provides a much better description of the atom than if only H₁ is considered. The Hamiltonian may for each electron be written

where the potential energy operator V(r_i) is spherically symmetric. The Hamiltonian reduces to a sum of one-electron operators, each giving rise to a one-electron eigenfunction , which is also referred to as an orbital. For each one-electron operator H_i an eigenvalue equation can be formulated as

The wavefunction for the Hamilton operator H₀ may then be written in terms of a product of one-electron functions, i.e.

while the energy eigenvalue E₀ of H₀ is obtained as a sum of one-electron energies, i.e.

Since the potential is spherically symmetric, each wavefunction will as in the one-electron case be an eigenfunction to the operators l_i², s_i², l_iz and s_iz. The state for each electron is therefore defined by the four quantum numbers n, l, m_l, m_s or n, l, j, m dependent upon which representation is preferred.

The energy for each electron is determined solely by n and l, which means that the wavefunctions are degenerate with respect to m_l and m_s. According to the Pauli exclusion principle we may therefore have a maximum of 2(2l+1) electrons with the same n and l.

The 2(2l+1) electrons with the same n and l are said to form a closed shell, often called subshell. All closed shells form symmetric charge distributions.