The rule of four: anomalous distributions in the stoichiometries of inorganic compounds

The efficient identification of materials with specific characteristics is crucial for advancements in various technological sectors. Traditional approaches to materials discovery face challenges due to the vast compositional and configurational space. This research addresses a fundamental question in materials science: why are some materials more abundant than others? The study focuses on an observed anomaly—the disproportionate prevalence of inorganic compounds with a number of atoms in their primitive unit cell that is a multiple of four. This phenomenon, herein termed the 'rule of four,' has not been previously reported. The paper's primary objective is to explore the existence and possible origins of this rule, its relationship to established structural descriptors, and potential implications for materials discovery. The research uses large-scale datasets from experimental and computational sources to analyze the abundance and correlations with various properties, aiming to uncover underlying principles that govern the distribution of inorganic compounds.

Computational materials discovery, driven by advancements in density-functional theory (DFT) and high-throughput studies, is a rapidly developing field. Materials informatics and machine learning (ML) techniques are increasingly used to analyze materials' data and predict properties, enabling more efficient screening of potential candidates for applications. However, the success of these data-driven approaches depends heavily on the quality and comprehensiveness of the underlying datasets. Anomalous correlations within these datasets can offer valuable insights into the underlying principles governing material formation and properties. Existing literature extensively explores materials properties, structure prediction, and machine learning applications in materials science. However, the 'rule of four' phenomenon, focusing on the peculiar abundance of compounds based on the number of atoms in their primitive unit cells, is a novel observation warranting detailed investigation.

The research utilizes two distinct databases of inorganic crystal structures: the Materials Project (MP), containing DFT-relaxed structures, and the Materials Cloud 3D source (MC3D-source), incorporating experimental data from COD, ICSD, and MPDS. The analysis begins by verifying that the observed 'rule of four' is not an artifact of data processing or the method of representing primitive unit cells. The influence of the 'symprec' parameter in the primitivization process using spglib is investigated. Next, the correlation of the 'rule of four' with traditional materials science metrics is explored, including formation energy and crystal symmetries (space and point groups). The study analyzes the distribution of formation energies in RoF and non-RoF structures and investigates their correlation with various symmetry descriptors. To analyze the global and local structural features, several geometric properties are considered: the number of atomic species, the relative abundance of small to large atomic radii, the ratio of smallest to largest atomic radii, and the packing fraction. To gain deeper insights into local atomic environments and their relationship to the 'rule of four,' the study employs machine learning techniques. Smooth Overlap of Atomic Positions (SOAP) vectors, a powerful representation for local structural environments, are used. Principal Covariates Regression (PCovR) is used to examine correlations between stability (formation energy) and local symmetries. Random Forest (RF) classification is applied to the species-invariant SOAP vectors to predict the 'rule of four' based solely on local structural descriptors. The performance of the RF classifier is evaluated by varying the interaction cutoff radius to determine the scale of local structural features involved. The accuracy and learning curve of the classifier are then analyzed.

The analysis reveals a statistically significant overabundance of inorganic compounds following the 'rule of four' in both the MP and MC3D-source databases. This overabundance is particularly notable when considering primarily experimentally known compounds. The study finds no significant correlation between the 'rule of four' and formation energy, suggesting that the rule does not stem primarily from thermodynamic stability. Further analysis indicates that structures obeying the 'rule of four' are not characterized by high-symmetry structures but instead tend to exhibit low symmetries and loosely packed arrangements, maximizing free volume. The analysis of point groups and packing fractions supports this conclusion. The machine learning analysis using SOAP vectors and PCovR reveals no strong correlation between the rule of four and formation energy, even when comparing structures with similar local environments. However, a Random Forest classifier trained on species-invariant SOAP vectors achieves an accuracy of ~87% in predicting whether a compound follows the rule of four, demonstrating that local structural symmetries are key determinants of this phenomenon. The classifier's performance plateaus at an interaction cutoff radius of ~4Å, suggesting that the differentiating features are primarily within the first two neighbor shells.

The findings challenge the intuitive expectation that abundant materials would correlate with high symmetry and low energy. Instead, the 'rule of four' suggests that other factors beyond simple energetic stability and global symmetry play a significant role in the distribution of inorganic compounds. The high accuracy of the Random Forest classifier based on local structural descriptors indicates the importance of local atomic environments and symmetries in determining the prevalence of the rule of four. This highlights the need for more detailed investigations into local structural effects in materials science. The absence of a clear energetic explanation suggests that kinetic factors, formation pathways, or other non-thermodynamic factors may play a crucial role in determining the observed abundance. Further research is needed to explore the interplay between local symmetries and packing efficiency in relation to this phenomenon.

This work unveils the 'rule of four,' a previously unrecognized pattern in the abundance of inorganic compounds. The analysis shows that this rule is not primarily driven by energetic stability or global symmetry but rather by local structural features. The high accuracy of the machine learning model in predicting the rule of four solely from local descriptors emphasizes the significance of local atomic environments. Future work should focus on elucidating the underlying mechanisms responsible for this intriguing observation, possibly exploring kinetic effects, synthesis pathways, or other factors beyond thermodynamic stability. A more complete understanding of the 'rule of four' could significantly impact materials discovery and design by guiding the search for new compounds with specific properties.

While the study utilizes large databases, the datasets themselves may have inherent biases reflecting the prevalence of certain types of compounds in experimental and computational studies. The interpretation of local structural descriptors relies on a specific choice of parameters in the SOAP vector calculations, which may affect the results. The study focuses on inorganic compounds, and its findings may not be directly applicable to organic or other types of materials. Further investigation is needed to fully explain the reasons behind the rule of four and to explore potential exceptions or deviations from this trend.