Van Vleck susceptibility

When an ion has low-lying excited states, they can contribute to the susceptibility in twoways. They may be Zeeman split and thermally populated as usual. Otherwise, the applied field may mix some of the excited-state character into the ground state. This produces a term known as van Vleck paramagnetism or temperature-independent paramagnetism (TIP). It is especially important for Eu³⁺, which has a nonmagnetic J = 0 ground state, but a low-lying J = 1 excited state at 330 K which contributes a paramagnetic susceptibility.
The other rare-earth ion for which it may be necessary to consider the excited J -multiplets is Sm³⁺, which has a J = 5/2 ground state and a low-lying J = 7/2 excited state at 1225 K. Among the actinides, Bk⁵⁺ and Cm⁴⁺ have a localized 5f ⁶ configuration, with a J = 0 ground state.
According to perturbation theory, the correction to the energy of the ion is given by the Hamiltonian 

........(1)

where g and e represent the ground and excited states, respectively, and  ε is their energy separation. From the expression for the Gibbs free energy of a paramagnet in an applied field, −1/2 BM with M = χB/μ₀, it follows that χ = 2nμ₀ε1/B², where n is the number of ions per unit volume. Hence the susceptibility depends on the off-diagonal matrix elements of the magnetic moment:

.......(2)