The magnetic clusters are usually described by an effective Hamiltonian acting in a fixed S space. This single-spin description neglects the mixing between states with different total spin S produced by anisotropic interactions, such as the local crystal field and the magnetic dipole-dipole coupling. We have developed a general and simple procedure, based on a perturbational approach, where the S-mixing effects are included in some new terms to be added to the original effective Hamiltonian. These terms contain some new operators, which depend on the anisotropic interactions and on the difference DeltaS between the spin of the states involved in the mixing. Interestingly, these operators are very similar (in some specific cases equal) to the well known Stevens operator equivalents. A list of them for DeltaS=1 and DeltaS=2 and for second-order anisotropic interactions is presented. The method has been applied to study the S-mixing in two iron nanomagnets, Fe-4 and Fe-8.