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The rule of four: anomalous stoichiometries of inorganic compounds

Elena Gazzarrini1*, Rose K. Cersonsky2*, Marnik Bercx1*, Carl S. Adorf1*, Nicola Marzari1*

1 Theory and Simulation of Materials (THEOS) and National Center for Computational Design and Discovery of Novel Materials (MARVEL), École Polytechnique Fédérale de Lausanne, CH-1015 Lausanne, Switzerland

2 Department of Chemical and Biological Engineering, University of Wisconsin - Madison, Madison, Wisconsin, USA

* Corresponding authors emails:,,,,
DOI10.24435/materialscloud:8b-90 [version v1]

Publication date: Jul 05, 2023

How to cite this record

Elena Gazzarrini, Rose K. Cersonsky, Marnik Bercx, Carl S. Adorf, Nicola Marzari, The rule of four: anomalous stoichiometries of inorganic compounds, Materials Cloud Archive 2023.104 (2023), doi: 10.24435/materialscloud:8b-90.


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 the 'Rule of Four' - has to our knowledge not previously been reported or studied rigorously. 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, this 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.

Materials Cloud sections using this data

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File name Size Description
155.6 MiB Materials Cloud 3-dimensional crystal structures database (MC3D), containing 79854 inorganic structures.
222.4 MiB Materials Project crystal structures database (MP), containing 83989 inorganic structures.
2.5 MiB The MC3D database is not publicly release yet, but we provide here the version and ID of each structure.
2.2 KiB Instructions


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External references

Journal reference (In preparation)
Elena Gazzarrini, Rose K. Cersonsky, Marnik Bercx, Carl S. Adorf, Nicola Marzari, The rule of four: anomalous stoichiometries of inorganic compounds, in preparation to be submitted to npj Computational Materials


SOAP vectors Inorganic databases Machine Learning Local symmetries Classification MARVEL

Version history:

2023.116 (version v2) Jul 27, 2023 DOI10.24435/materialscloud:fm-za
2023.104 (version v1) [This version] Jul 05, 2023 DOI10.24435/materialscloud:8b-90