My PhD thesis - Condensed Matter Theory - Imperial College London

Appendix A
The quasi-2D Ewald sum
The generalization of the Ewald summation to a system with periodic repeat in
only two dimensions was first obtained by Parry [65, 66]. However, this original
derivation, although it leads to the correct result, is difficult to follow. An alternative
derivation is presented here in section A.1; in section A.2, the expansion of the result
in the limit of small separation and large cell size is obtained.
A.1 Derivation
The problem is to find the potential due to a charge of unit magnitude at the origin
and all its images in the plane. The charge distribution is therefore
ρ(r) = ∑ R
δ(r − R)
(A.1)
where R is a 2D lattice vector. In fact, as |z| → ∞, the potential tends to −∞
(relative to the potential at z = 0). However, the form of the potential in this limit
can still be deduced, because the array of charges then resembles a uniform sheet;
the potential therefore decreases linearly, and this defines the boundary conditions.
The Ewald method [78] is to rewrite the charge distribution, creating a smooth
term which may be evaluated in reciprocal space and a rapidly-decaying term which
converges quickly in real space. In the quasi-2D system, this is modified slightly,
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