A closer look at the algorithm reveals that the computational effort for the evaluation of local expansions is of the same magnitude as for the forces from the IIZ. It is possible to use the Multiple Time Step Method described in Section 6.2.2 to avoid unnecessarily frequent computations of the local expansions and of forces from the IIZ. In our approach we divided the IIZ in two distance classes. The forces between all atoms which are closer then 5Å are calculated exactly in every integration step. The sum of forces from atoms of the second distance class (5 - 10Å) however is calculated exactly only every other integration step. The values of two previously calculated integration steps are used to extrapolate the forces for the steps between two exact integration steps. As a result of this the computational effort for the IIZ is reduced by a factor of almost two.
As explained above, the influence of atoms separated by more then 10Å is represented through the local expansions. Analogous to the time development of forces the time development of the coefficients of local expansions varies slowly and can be extrapolated for a certain number of integration steps without risking too large errors. Thus the time consuming calculation of local expansions is avoided periodically.
The combination of the Multiple Time Step Method and the Structure Adapted Fast Multipole Method is called FAst MUltiple time step Structure Adapted Multipole Method  (FAMUSAMM). The computational effort of this method scales with O(N).