This work is concerned with the lattice energy of periodic assemblies of mass and charge distributions of the form, exp (−αp2
), where α is an adjustable positive variable and
is the vector from the lattice site or average position. The energy of interaction between two distributions is the density-weighted integral of the interactions between the volume elements of each distribution. Reciprocal space lattice summation formulas derived for particles represented by gaussian smeared-out density distributions are applied to the gaussian potential and a bounded version of the soft-sphere potential for a range of exponents. Two types of spatial broadening are considered, continuous or physical broadening (PB) and broadening resulting from the time average of point particle positions, so-called “time” broadening (TB). For neutral mass distributions a reciprocal space lattice summation formula is derived which is applied to the bounded soft-sphere potential. For the charged systems, the methodology described in Heyes [J. Chem. Phys. 74, 1924 (1981)10.1063/1.441285]
is used, which for the PB case gives the Ewald-like formulas derived by Gingrich and Wilson [Chem. Phys. Lett. 500, 178 (2010)10.1016/j.cplett.2010.10.010]
using a different method. Another expression for the lattice energy of the spread out charge distributions is derived which is cast entirely in terms of a summation over the reciprocal lattice vectors, without the arbitrary charge spreading function used in the Ewald method. The effects of charge spreading on a generalized definition of the Madelung constant (M
) for a selection of crystal lattices are shown to be insignificant for route mean square displacements up to values typical of melting of an ionic crystal. When the length scale of the charge distribution becomes comparable to or greater than the mean inter particle spacing, however, the effects of charge broadening on the lattice energy are shown to be significant. In the PB case, M
→ 0 for the uniform charge density or α → 0 limit, and M
ultimately becomes negative in the TB case for a large enough root mean square displacement (or small enough α).