Using the addition theorem of tensor operators, the product of a rank n tensor and a rank m tensor can generate a new tensor with rank n+m ~ |n-m|. Therefore, a high rank tensor can be expressed as the product of low rank tensors. This convention is useful to interpret the high rank multipolar exchange terms as a "multi-exchange" process of dipoles (or pseudospins).  For example, for the spherical harmonic tensor operators of  case, we have
If so, a quadrupole-quadrupole interaction (see next section) can be considered as a two steps dipole-dipole interaction. For example, , so the one step quadrupole transition  on site  now becomes a two steps of dipole transition . Hence not only inter-site-exchange but also intra-site-exchange terms appear (so called multi-exchange). If  is even larger, one can expect more complicated intra-site-exchange terms would appear. However, one has to note that it is not a perturbation expansion but just a mathematical technique. The high rank terms are not necessarily smaller than low rank terms. In many systems, high rank terms are more important than low rank terms.