We varied r c as in Fig. and used r = r f = √ 3(r c + 2 • 15) in Eq. 12 to determine the GLST cubature for different ε. For the dot product with tm , we set ⃗ r = r f x. As explained further below, r f represents the far cutoff where the same α, ζ , and GLST cubature are used in the range r c < r < r f for a cubic simulation box divided by cells of linear size 15 (Å). The number of non-negative Legendre polynomial roots (N gl ), t-degree, M ′ t , and N GLST are shown in Fig. ,C. In the most stringent case of ε = 10 -12 , N GLST = 8.72 × 10 5 when r c = 15 (Fig. ). However, N GLST rapidly decreases for larger ε, which is 1.66 × 10 5 for ε = 10 -8 and 6,355 for ε = 10 -4 at r c = 15. Also, N GLST decreases with r c , suggesting that t (half the number of t-design vectors) and the total number of the GLST cubature nodes smaller sets of GLST cubature are needed to approximate the interaction for larger separations between atoms. We also observe that the t-degree is not a strictly monotonic function of r c (Fig. ). This is because of the oscillatory nature of the spherical Bessel function (Eq. 11) and the degree of polynomial approximation depends on where the argument of j 0 (x) lands in the phase of the sine function. This results in the decrease of N GLST with r c not strictly monotonic. It may cause an accuracy issue since a GLST cubature selected for a given distance may not be accurate enough for a shorter distance where a larger GLST cubature is needed. However, the occasional increase in N GLST in Fig. is not substantial, and a preliminary implementation indicates that the resulting error is comparable to the target accuracy . If more stringent accuracy is needed, the distance range to be covered in simulation can be pre-scanned to select the GLST cubature in such a way that it monotonically decreases with distance. This will increase the number of cubature nodes to calculate slightly but the target accuracy will be guaranteed.