An example of the hierarchical cell for the tree-based method. The root level cell (the simulation cell) is recursively sub-divided into octants. Each P cell is a level 2 cell, which is divided twice from the root level cell. Cell A, shaded cells, and M cells are level 3 cells. In this figure, level 3 cells mean leaf level cells. Shaded cells are nearest neighbor cells, and M cells are the next nearest neighbor cells from cell A.
(a) In the IPS/Tree method, the simulation cell is recursively sub-divided, like the typical tree-based method. Shaded cells are local region cells that partially include the cutoff radius territory. Cells M and P are both middle-range cells. The scheme of the interaction calculation between A cells and middle-range cells is always the same regardless of the difference of the size of each middle-range cells. (b) All positive point charges inside each divided cell was plotted. If the multipole expansion is not applied, interaction calculations between cells A and numerous point charges are needed. (c) All positive point charges after the P2M2 application are plotted. The multipole expansion up to the pth order can theoretically be reproduced by the expansion of K min(p) = ⌈(p + 1)2/4⌉ pseudoparticles. For p = 2, the minimum number of pseudoparticles necessary is K min(2) = 3 for each cell. (d) The IPS method is applied for point charges after the P2M2 application, which includes pseudoparticles. The corner regions (outside of the large cutoff radius) is an isotropic periodic force field. Since the pseudoparticle is a mere point charge, interaction calculations using approximated IPS potentials by polynomial functions can be easily performed.
(a) θ dependence of e Ewald. IPSp/Tree at r c ⩾ 2.7 nm has nearly the same error as IPSp with a long cutoff radii in comparison with the Ewald sum. On the other hand, the IPSn/Tree gives a relatively large error for estimating the Coulombic interaction. This is from the difference of the interaction treatment between IPSn and IPSp. (b) θ dependence of e IPS. The interaction of the typical tree-based method with second-order multipole expansion has 10−2 − 10−4 order relative error, and this error is roughly proportional to θ−3. The error of IPSp/Tree has almost the same dependence against θ. In the case of IPSn/Tree, however, 10−3 order error remains even if the short cutoff radius has a value larger than 2.7 nm, so θ dependence of the error is weaker than that of the IPSp/Tree and other typical tree-based methods.
The results of oxygen-oxygen, oxygen-hydrogen, and hydrogen-hydrogen radial distribution functions for (a) IPSn/Tree and (b) IPSp/Tree method at r c = 1.5 nm. All radial distribution functions sufficiently follow the results of the Ewald sum.
The magnified figures of oxygen-oxygen, oxygen-hydrogen, and hydrogen-hydrogen radial distribution functions for (a) IPSn/Tree and (b) IPSp/Tree method at r c = 1.5 nm. The detail of the shape of RDFs at 0.86 nm ⩽ r ⩽ 5.0 nm was plotted. There were no special deviation of g IPSn/Tree(r) and g IPSp/Tree(r) around r c and R c. The IPSn/Tree can avoid innate IPSn problems where a sharp deviation appears.25,27,28
Parameters of the Coulombic potential for the IPSn method. The average deviation of the polynomial fitting function, Eq. (7), from the analytic solution Eq. (8), is about 2 × 10−8 q i q j /R c.
Potential energy and self-diffusion coefficient for the IPS/Tree method, IPS method with long cutoff, and Ewald sum.
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