Recently, Berman and Greengard [8], introduced a new general method to rapidly evaluate lattice sums of an infinite lattice of certain potential energy functions, e.g. the Ewald sum potential. This method is based on a new renormalization identity and has been developed for both 2D and 3D systems. Periodicity is accomplished by assuming that space is filled with translates of the unit cell and hence each image has the same far field expansion relative to its center. For electrostatic point charge systems, the method utilized multipole and Taylor expansions to arrive at a recursive, infinite sum that required the evaluation of certain finite sums as a function of the coordinates of the lattice points.
The Ewald sum method has been modified in [41,29] to simulate systems that are infinite only in two of the three dimensions with long-range interactions. Examples of such systems are biological membrane and polar fluids. Such quasi-two-dimensional systems cannot utilize 3-D implementations of the Ewald sum as this has been shown to be computationally inefficient and may lead to unrealistic interactions between the sheets of the finite dimension. The authors have presented a reformulation of the Ewald sum that handles such systems and used a system of water trapped between two dielectrics to test their methods. The method is reported to be of reasonable speed and accuracy but requires large memory.
In summary, this section presented an account of Multipole-based Ewald sum methods.
These methods have an attractive computational cost of 
.
Among the methods reviewed above, the Macroscopic multipole method is the better method
since it combines accuracy and efficiency with the flexibility to simulate systems of any
geometrical configuration.