  We present a general method for solving the modified Helmholtz equation
without shape approximation for an arbitrary periodic charge distribution,
whose solution is known as the Yukawa potential or the screened Coulomb
potential. The method is an extension of Weinert's pseudo-charge method [M.
Weinert, J. Math. Phys. 22, 2433 (1981)] for solving the Poisson equation for
the same class of charge density distributions. The inherent differences
between the Poisson and the modified Helmholtz equation are in their respective
radial solutions. These are polynomial functions, for the Poisson equation, and
modified spherical Bessel functions, for the modified Helmholtz equation. This
leads to a definition of a modified pseudo-charge density and modified
multipole moments. We have shown that Weinert's convergence analysis of an
absolutely and uniformly convergent Fourier series of the pseudo-charge density
is transferred to the modified pseudo-charge density. We conclude by
illustrating the algorithmic changes necessary to turn an available
implementation of the Poisson solver into a solver for the modified Helmholtz
equation.
