The calculated magnetic moment (B. M.) for the ground state of a f⁵ion is

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The magnetic moment

of many d-metal ions can be calculated by using the spin-only approximation because the strong ligand field quenches the orbital contribution.
In a free atom or ion, both the orbital and the spin angular momenta give rise to a magnetic moment and contribute to the paramagnetism. When the atom or ion is part of a complex, any orbital angular momentum is normally quenched, or suppressed, as a result of the interactions of the electrons with their nonspherical environment. However, if any electrons are unpaired the net electron spin angular momentum survives and gives rise to spin-only paramagnetism, which is characteristic of many d-metal complexes.
For orbital angular momentum to contribute, and hence for the paramagnetism to differ significantly from the spin-only value, there must be one or more unfilled or half-filled orbitals similar in energy to the orbitals occupied by the unpaired spins and of the appropriate symmetry (one that is related to the occupied orbital by rotation round the direction of the applied field). If that is so, the applied magnetic field can force the electrons to circulate around the metal ion by using the low-lying orbitals and hence it generates orbital angular momentum and a corresponding orbital contribution to the total magnetic moment.

For the lanthanoids, where the spin-orbital coupling is strong, the orbital angular momentum contributes to the magnetic moment, and the ions behave like almost free atoms. Therefore, the magnetic moment must be expressed in terms of the total angular momentum quantum number J:

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The notation for a full-term symbol is:

​The energy and the orbital angular momentum of a multielectron species are determined by a quantum number, L. Energy states for which L=0, 1, 2, 3, 4... are known as S, P, D, F, G... terms, respectively.
For any system containing more than one electron, the energy of an electron with principal quantum number n depends on the value of l, and this also determines the orbital angular momentum which is given by the equation:

The spin quantum number, s, determines the magnitude of the spin angular momentum of an electron and has a value of 1/2. For a 1-electron species, m_s is the magnetic spin angular momentum and has a value of +1/2 or -1/2. The spin angular momentum for a multielectron species is given by the following equation, where S is the total spin quantum number.

​The quantum number M_S is obtained by algebraic summation of the m_s​ values for individual electrons:

​For each value of S, there are (2S+1) values of M_S.

​The interaction between the total angular orbital momentum, L, and the total spin angular momentum, S is defined by the total angular momentum quantum number, J.
The following equation gives the relationship for the total angular momentum for a multi-electron species.​

​The value of J for the ground state is given by (L-S) for a sub-shell that is less than half-filled, and by (L+S) for a sub-shell that is more than half-filled.