Order relations for lattice sums from order relations for theta functions

Let {τ} and {γ} denote mutually reciprocal unit Bravais lattices in an n-dimensional Euclidean space, and consider the Theta Functions (TF's) $\text{V}{}_{\mathrm{\tau }}\text{(t) = t}{}^{\mathrm{<mtext>n</mtext><mtext>4</mtext>}}\text{∑}{}_{\mathrm{\tau }}\text{exp}\text{(?πtτ}{}^2\text{)}$ for all 0 < t < ∞. By showing how to evaluate a larger class of sums $\text{Z}{}_{\mathrm{\tau }}{}^{\mathrm{(K)}}\text{(t)  π}{}^{\mathrm{k}}\text{t}{}^{\mathrm{<mtext>k\; +\; </mtext><mtext>n</mtext><mtext>4</mtext>}}\text{∑}{}_{\mathrm{\tau }}{}_{\mathrm{r}}{}^{\mathrm{2k}}\text{exp (?πtτ}{}^2\text{)}$, k a nonnegative integer, we are able to evaluate any derivative of the V-functions. With this information we find order relations for the TF's on the cubic lattices in three dimensions. Coupling these relations with Ewald's Theta Function method, we secure order relations for Lennard-Jones, Chaba-Pathria, and other lattice sums on cubic lattices. We also sketch extensions to non-Bravais lattices and give an order relation for TF's on the non-Bravais hexagonal closepacked and the Bravais facecentered cubic.