Application of canonical augmentation to the atomic substitution problem

TL;DR

This paper addresses the combinatorial explosion of atomic-substitution patterns in solid-state supercells by introducing SHRY, a Python implementation of canonical augmentation that yields symmetry-inequivalent structures efficiently. By framing substitutions in terms of group actions and orbit representatives, SHRY generates exactly one pattern per orbit without exhaustive enumeration, achieving linear scaling up to around $N\sim10^9$. The method integrates with CIF input/output and leverages invariants to accelerate the canonical augmentation, delivering performance competitive with established C++ tools on large problems. The approach enables practical high-throughput ab-initio investigations of disordered solids, vacancies, and related phenomena, with open-source availability to the community and potential extensions to multi-site, multi-Wyckoff, and periodic systems.

Abstract

A common approach for studying a solid solution or disordered system within a periodic ab-initio framework is to create a supercell in which a certain amount of target elements is substituted with other ones. The key to generating supercells is determining how to eliminate symmetry-equivalent structures from the large number of substitution patterns. Although the total number of substitutions is on the order of trillions, only symmetry-inequivalent atomic substitution patterns need to be identified, and their number is far smaller than the total. A straightforward solution would be to classify them after determining all possible patterns, but it is redundant and practically unfeasible. Therefore, to alleviate this drawback, we developed a new formalism based on the {\it canonical augmentation}, and successfully applied it to the atomic substitution problem. Our developed \verb|python| software package, which is called \textsc{SHRY} (\underline{S}uite for \underline{H}igh-th\underline{r}oughput generation of models with atomic substitutions implemented by p\underline{y}thon), enables us to pick up only symmetry-inequivalent structures from the vast number of candidates very efficiently. We demonstrate that the computational time required by our algorithm to find $N$ symmetry-inequivalent structures scales {\it linearly} with $N$ up to $\sim 10^9$. This is the best scaling for such problems.

Figures (7)

• Figure 1: Example of the search tree, where three vertices out of eight are colored by red in a cube. $p(X)$ denotes the parent node of $X$, whereas the set of children of $X$ are denoted with $C(X)$.
• Figure 2: Atomic substitutions corresponding to Fig \ref{['fig:tree-example']}. Ce$_{8}$Pd$_{24}$Sb $\rightarrow$ (Ce$_{5}$,La$_{3}$)Pd$_{24}$Sb, where Ce and La atoms are on the 8$g$ Wyckoff position. The crystal structure 1996GOR was obtained from the ICSD database 1983BER*2004HEL (CollCode: 83378). The space group is 221-$Pm\bar{\rm 3}m$; the crystal structures were depicted by VESTA 2011MOM.
• Figure 3: A schematic of the generation via canonical augmentation. The key to canonical augmentation is the relationship between a node $X$ and its parent $p(X)$. SHRY checks if $(X,p(X)) \cong (X,m(X))$; otherwise, $X$ is disregarded. The selection of a subobject $\hat{Z}_{0}$ depends only on the canonical form $\hat{Z}$ and not on $Z$. Thus, the following important relation holds: $\hat{Z} = \hat{W}$ and $Z_{0} = W_{0}$ for any two isomorphic objects.
• Figure 4: An extension of the search tree to replace a single site with three or more elements and/or to replace two or more sites. When the atomic substitution ends for the $j$-th element in the $i$-th Wyckoff position, SHRY proceeds with the substitution for the $(j+1)$-th element in the $i$-th Wyckoff position. Similarly, when the atomic substitution ends for the $i$-th Wyckoff position, SHRY proceeds with the substitution for the $(i+1)$-th Wyckoff position.
• Figure 5: Comparing the normalized CPU times of SHRY and Supercell on a benchmark dataset. On the horizontal axis, we place the number of symmetrically distinct structures whereas the vertical axis indicates the normalized CPU time. The linear fits, shown by the broken lines, were performed using only the $N \ge 10^4$ points.