The rule of four: anomalous stoichiometries of inorganic compounds

Why are materials with specific characteristics more abundant than others?
This is a fundamental question in materials science and one that is traditionally difficult to tackle, given the vastness of compositional and configurational space. We highlight here the anomalous abundance of inorganic compounds whose primitive unit cell contains a number of atoms that is a multiple of four. This occurrence - named here the 'rule of four' - has to our knowledge not previously been reported or studied. Here, we first highlight the rule's existence, especially notable when restricting oneself to experimentally known compounds, and explore its possible relationship with established descriptors of crystal structures, from symmetries to energies. We then investigate this relative abundance by looking at structural descriptors, both of global (packing configurations) and local (the smooth overlap of atomic positions) nature. Contrary to intuition, the overabundance does not correlate with low-energy or high-symmetry structures; in fact, structures which obey the 'rule of four' are characterized by low symmetries and loosely packed arrangements maximizing the free volume. We are able to correlate this abundance with local structural symmetries, and visualize the results using a hybrid supervised-unsupervised machine learning method.

为何具备特定特性的材料相较于其他材料更为丰富？这一问题构成材料科学领域的一项根本性课题，鉴于组成与配置空间的浩瀚，这一问题传统上亦颇具挑战性。在此，我们着重指出一类无机化合物的异常丰富性，其原始晶胞中原子数是四的倍数。这一现象，此处命名为‘四倍规则’，据我们所知，尚未有文献对其进行报道或研究。在此，我们首先强调该规则的存在性，尤其是在仅限于实验已知化合物的情况下，并探讨其与晶体结构的既定描述符之间可能存在的关联，包括从对称性到能量的各个方面。随后，我们通过考察结构描述符来研究这种相对丰富性，这些描述符既包括全局性质（如填充配置）也包括局部性质（如原子位置的平滑重叠）。与直观预期相反，这种过度的丰富性与低能量或高对称结构并不相关；实际上，遵循‘四倍规则’的结构以其低对称性和松散的填充排列为特征，最大化了自由体积。我们能够将这种丰富性与局部结构对称性相联系，并利用混合监督-非监督机器学习方法来可视化这些结果。