Independent-spin magnetic moment (${\mu }_{\mathrm{eff}}^2=4{\mu }_B^2{\sum }_i{S}_i(S_i+1)$)

For uncoupled centres with approximately equal $g=2$, squared magnetic moments add: ${\mu }_{\mathrm{eff}}/{\mu }_B=2\sqrt{{\sum }_i{S}_i(S_i+1)}$. This does not assume that the individual spins add to a definite total spin; appreciable exchange interaction invalidates the independent-centre approximation.