When scientists stumble upon unexpected patterns in their data, the first instinct is to explain them. However, when researchers at the Swiss Federal Institute for Technology in Lausanne (EPFL) discovered a recurring pattern in two electronic structures databases, named the “Rule of Four,” they found themselves at a loss for an explanation. The databases, the Materials Project (MP) and the Materials Cloud 3-dimensional crystal structures database (MC3Dsource), contain over 80,000 electronic structures, with a predominant presence of primitive unit cells composed of a multiple of four atoms. This anomaly led the researchers on a quest to uncover the origins of this mysterious rule.

As the scientists delved deeper into the unexpected pattern, various hypotheses were proposed and subsequently tested. Initially, the reduction of atoms when transforming a conventional unit cell into a primitive cell seemed like a plausible explanation for the prevalence of structures following the Rule of Four. However, further analysis revealed that the software responsible for ‘primitivizing’ the unit cell had executed the process correctly. Another theory pointed to the coordination number of silicon, which is four, as a potential factor contributing to the rule. Surprisingly, not all materials adhering to the Rule of Four contained silicon, debunking this hypothesis. Moreover, the formation energies of the compounds failed to correlate with the Rule, indicating that energetically favored materials did not necessarily follow the pattern.

Faced with the complexity of the materials space covered by the databases, the researchers enlisted the expertise of Rose Cernosky, a machine-learning specialist. Cernosky developed an algorithm to categorize structures based on their atomic properties and evaluate formation energies within similar chemical classes. However, this approach yielded no significant insights into distinguishing Rule-of-Four compliant materials from non-compliant ones. Interestingly, the abundance of multiples of four did not align with highly symmetric structures but rather with low symmetries and loosely packed arrangements, further complicating the investigation.

Despite the team’s efforts, the mystery of the Rule of Four remained unsolved. The publication of their findings in npj Computational Materials serves as a rare instance of a scientific paper documenting a negative result. While disappointing, negative results play a crucial role in scientific advancement by highlighting challenging problems that may require novel approaches. The researchers’ inability to pinpoint a definitive cause for the Rule of Four underscores the complexity of the phenomenon and the need for further exploration.

Moving forward, the researchers speculate that undiscovered small chemical groups within the cells may hold the key to unraveling the mystery of the Rule of Four. Through ongoing research and innovative methodologies, such as leveraging local symmetry descriptors, the team hopes to shed light on the underlying mechanisms driving this perplexing pattern in electronic structures. As the scientific community grapples with enigmatic phenomena like the Rule of Four, embracing negative results and embracing the challenges they present ultimately propels the field forward towards new discoveries and breakthroughs.


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