To visually demonstrate this α offsetting scheme, the operators involved in our simple 1-D example are plotted in Fig. , with the long-ranged kernels plotted on a linear axis and the short-ranged kernels shown logarithmically. The α value is chosen such that the short-ranged kernel has decayed to 10 -8 by R = 10 (solid orange line), so interactions with R ≤ 10 will comprise both short-and long-ranged terms. Interactions with 10 < R ≤ 20 re- quire only the long-ranged operator (dashed orange line), which matches the Coulomb operator in this window, due to decay of the short-ranged term before this window and the choice to parameterize it to treat up to twice the short-ranged cutoff. The α 2 scale long-ranged operator (dashed blue line) is clearly different to the Coulomb operator at short range, but is negligibly different in the window 20 < R, due to the decay behavior of its shortranged counterpart. The parameterization of this function is sufficient to describe interactions up to R = 40, at which point we switch to α 4 scale. The fact that the α is chosen to ensure that the longranged kernel matches the Coulomb operator in each window leads to a subtle but important difference between Ewald summation and SGT in a periodic setting. Like FMM, SGT implicitly contains a crystal shape dependent dipole surface term that should be added to Ewald summations in order to achieve agreement between the methods. In two-and three-dimensional systems the scale in which a pair of nodes should interact is less clear-cut than for the 1D example, as depicted for two dimensions in Fig. . The box of interest, marked with an "x" is surrounded by purple loci at R = {1, 2, 4} in leaf width units, while the colors depict the scale. The yellow boxes require both short-and long-ranged terms in α scale, so are processed in a direct, pairwise manner with the conventional Coulomb operator, which is the sum of those two terms. The green region in the next ring is processed entirely using the long-ranged terms, also in α scale, but no short-ranged contributions exist through our choice of α, which truncates these terms after one leaf width. Similarly, the surrounding orange region has only longranged components in α 2 , as does the next blue zone, using α 4 . In cases where the locus bisects a remote node, the larger of the two scales is chosen, to avoid incorrectly omitting short-ranged terms; therefore some padding of x FIG. . Example of a two dimensional system that is subdivided to a tree depth D = 4. The box of interest lies in the center of the yellow zone with an "x", while the coloring depicts the nature of the interaction: yellow → direct, green → long-ranged (α scale), orange → long-ranged ( α 2 scale), blue → long-ranged ( α 4 scale).