The Fast Fourier Poisson method ( FFP) [53], recasts Ewald summation in yet another
form that is also evaluated, as in PME, using FFT in
. The method claims to achieve high accuracy without using interpolation
[14] or multipole expansion [44] schemes.
The main difference is in the implementation of the reciprocal space sum, 
.
Unlike PME, FFP does not interpolate the solution of the reciprocal sum; rather, FFP samples
the Gaussian sources associated with each point charge on a grid and solves for the potential.
In FFP, the reciprocal energy is written as follows:
where 
is the reciprocal potential associated with the Gaussian distributions of the
artificial screening charge distribution 
, and 
is the point charge density, see Sec.(2.2).
Initially, 
and its gradient, are determined over the grid's points using FFT
as a solution to Poisson's equation: 
.
The total energy, E is reformulated so that there is no need to evaluate
the reciprocal potential at particles' locations. This can be accomplished by replacing the interaction
of each point charge with 
by the interaction of the introduced charge density
having the same net charge at the same location.
This is achieved by splitting 
of Eq.(28)
into two integrals:
The first integral in Eq.(29) is canceled out by the real-space sum leaving the second integral for evaluation only.
The FFP method is not restricted to orthogonal unit cells and has the advantage of
the energy and gradients being continuous functions of the point charge position.
However, a careful inspection of the timings presented in this paper [53] shows that for
moderate accuracy, i.e. 
relative force error, and a system size of N=5768 particles,
the runtime is about 3 times more expensive than a conventional 9Å cutoff method.
This suggests that the implementation of this method may need to be optimized further.
In summary, this section has provided an account of various methods that perform the Ewald sum using
efficient FFT with a computational complexity of 
.
In the methods discussed above, both PME and Luty's PPPM [37] methods are highly efficient.
It is not clear, however, whether PPPM can easily achieve the high accuracy levels attainable by PME,
while FFP can benefit from a more efficient implementation. Table[1] cites simulation
results of these methods for comparison.
Table: This table lists the relative performance of Fourier- and Multipole-based Ewald methods
as reported in the literature. Note:
Abs-Pot is the Absolute error in potential measured in kcal/mol;
Abs-Force is the Absolute error in force measured in kcal/mol Å;
Rel-Pot is the Relative error in potential;
Rel-Force is the Relative error in force.
