The Particle-Mesh Ewald method ( PME) [14] is also inspired by Hockney and Eastwood's particle-particle particle-mesh method (PPPM) [31,20]. Unlike PPPM, PME divides the potential energy into Ewald's standard direct and reciprocal sums and uses the conventional Gaussian charge distributions. The direct sum, Eq.(3), is evaluated explicitly using cutoffs while the reciprocal sum, Eq.(4), is approximated using FFT with convolutions on a grid where charges are interpolated to the grid points. In addition, in contrast to particle-mesh methods, PME does not interpolate but rather evaluates the forces by analytically differentiating the energies, thus reducing memory requirements substantially.
This method is reported to be highly efficient incurring only 
overhead over
conventional truncated list-based (i.e. non-Ewald) methods at a relative force accuracy around
. PME is also capable of achieving higher accuracy (
relative force error)
with relatively little increase in computational cost.
In computing the direct sum, the Ewald parameter 
is chosen large enough so that a fixed
cutoff radius can be applied thus reducing the complexity of the direct sum from 
to 
. To compensate for the truncation
in evaluating the direct sum, the number of reciprocal vectors is increased proportionally to N to
bound errors.
The reciprocal sum is computed using 3D-FFT with an overhead that grows as
. PME is therefore an 
method.
The reciprocal sum, Eq.(4), is given by:
where 
is defined in Eq.(15).
The structure factor can be approximated by:
where 
is the 3D FFT of Q, the charge matrix. The Q matrix is a three dimensional matrix
that is obtained by interpolating the point charges to a uniform grid of dimensions
that fills the simulation cell.
By combining Eq.(25) with Eq.(24), the reciprocal energy can be also approximated by:
The above equation is rewritten, after some manipulation, as a convolution:
where 
is the reciprocal pair potential and ``
'' indicates a convolution.
To evaluate the reciprocal sum, the Q matrix is first computed over a 3D uniform grid
and then transformed using inverse 3D FFT to obtain the structure factors.
The reciprocal energy is then calculated using Eq.(27)
with the aid of FFT to compute the convolution 
.
The charge interpolation function used originally in PME was Lagrange interpolation [14].
However, an enhanced PME [15] utilizes the B-spline interpolation function,
which is smoother and allows
higher accuracy by simply increasing the order of interpolation.
The smoothness of B-spline interpolation allows the force expressions to be evaluated analytically, with high accuracy, by differentiating the real and
reciprocal energy equations rather than using finite differencing techniques.
