Luty et al. [37] and Rajagopal et al. [40] offered an alternative approach to the Ewald sum by extending the Particle-Particle Particle-Mesh method ( PPPM) developed by Hockney and Eastwood [31]. The method relies on expressing the long-range inter-particle force as the sum of two components: the short-range force, which is only nonzero within some cutoff radius, and the ``reference'' force, that is long-ranged and smooth and can be approximated on a grid. The analogy between PPPM and the Ewald sum is clear. In the traditional Ewald sum, the direct sum, due to the point-charge and Gaussian counter-distributions, is also short-ranged; while the reciprocal sum, due to the Gaussian distributions, is a smooth function and its Fourier transform converges rapidly.
The authors [37] used PPPM's standard charge distribution, a sphere with a uniform decreasing density (the S2 function), rather than the Gaussian distribution used for the Ewald sum. The S2 distribution is given by:
where a is a parameter that adjusts the S2 distribution.
The short-range potential between two particles, each with an S2 charge distribution, is evaluated
with a cutoff radius, 
, given in terms of
by:
where 
The long-range potential is evaluated in Fourier space using :
where `` 
'' indicates the Fourier transform of a function.
The influence function is usually given by
but can be optimized
depending on the system size, charge shape function, and the interpolation function.
The long-range potential is computed using the following steps, see Fig.(4):
which is a function of both the charge
distribution 
and assignment functions. Several charge assignments can
be used depending on the accuracy desired, e.g. triangle-shaped charge function (TSC).
However, for a charge assignment function to be
feasible, it has to cover a relatively small number of grid points. Moreover, the assignment should vary
smoothly with the particle's location, which requires adequate grid spacing and in turn
determines the runtime cost.
, and calculate
using Eq.(23).
Apply the inverse Fourier Transform to obtain 
evaluated at the grid points.
algorithm.
It should be noted that in using this approach, the influence function 
is system-specific
and hence for each new system, and depending on system parameters, e.g. size or charge shape,
a new optimal influence function has to be computed resulting in some loss of generalization.
In addition, the PPPM implementation of [37] implies that in order to increase the
accuracy of potential/force computations, one needs to either
refine the mesh, or use a better weighing/interpolation scheme; both choices can be
computationally expensive. In addition, since the electrostatic force experienced at a grid point is obtained
by numerically differentiating the potential, another source of error is introduced to the results.
In order to improve the accuracy and reduce the error in the above steps,
higher-order differencing schemes have to be incorporated.
There is a rich interaction between all of the above choices
( i.e. weighing/interpolation, differentiation, charge shape, influence function, etc.)
and the user has to experiment with the system of interest in order to utilize this algorithm
efficiently.
Figure 4: A 2D schematic of particle-mesh technique used in most Fourier-based methods (a) A system of charged particles. (b) The charges are interpolated on a 2D grid. (c) Using FFT, the potential
and forces are calculated at grid points. (d) Interpolate forces back to particles and update coordinates.
