  A new method for Ewald summation in planar/slablike geometry, i.e. systems
where periodicity applies in two dimensions and the last dimension is "free"
(2P), is presented. We employ a spectral representation in terms of both
Fourier series and integrals. This allows us to concisely derive both the 2P
Ewald sum and a fast PME-type method suitable for large-scale computations. The
primary results are: (i) close and illuminating connections between the 2P
problem and the standard Ewald sum and associated fast methods for full
periodicity; (ii) a fast, O(N log N), and spectrally accurate PME-type method
for the 2P k-space Ewald sum that uses vastly less memory than traditional PME
methods; (iii) errors that decouple, such that parameter selection is
simplified. We give analytical and numerical results to support this.
