A seven-facet polyhedron captures the composition-only formation-energy landscape of inorganic solids

TL;DR

This work reveals that the convex hull of formation energies across a 92-element composition space is governed by a remarkably simple, seven-facet geometric structure. By training an interpretable max-affine model with seven facets, plus a minimal set of element-wise coefficients, the authors reproduce hull formation energies to near-DFT accuracy and assign each material to a chemically coherent facet, enabling a compact, composition-only description of stability. The seven facet coefficients also generalize to predict defect energetics, capture experimental spatial correlations in high-entropy phosphides, and guide electrochemical stability via Pourbaix diagrams, all without retraining or structural input. This geometry-free thermodynamic framework unifies stability, defects, mixing, and electrochemistry, enabling rapid, interpretable screening across vast chemical spaces and informing materials design decisions with a small, physically meaningful parameter set.

Abstract

This work demonstrates that the convex hull of formation energies for solid compounds involving elements from hydrogen to uranium admits a remarkably simple description over the 92-dimensional space of chemical compositions, despite the enormous combinatorial complexity of possible atomic structures. By training an interpretable max-affine model directly on near-hull formation energies from the Materials Project density-functional theory (DFT) database, we find that the hull can be reconstructed to DFT accuracy using a polyhedron with only seven facets. These facets define seven chemically coherent materials classes, with just seven coefficients per element sufficing to capture the dominant energetic trends across composition space. Remarkably, this compact, composition-only representation generalizes far beyond bulk formation energies. Without retraining or structural input, the same model reproduces trends in DFT-calculated defect formation energies, captures experimentally observed elemental mixing correlations in high-entropy materials, and enables the construction and optimization of Pourbaix diagrams for electrochemical stability. Together, these results show that many materials properties governed by energy differences can be expressed as simple linear combinations of a small set of interpretable, element-specific parameters. The result is a bonding-geometry-free thermodynamic framework that unifies stability, defects, mixing, and electrochemistry, and enables rapid, interpretable screening across vast chemical spaces.

Figures (5)

• Figure 1: Performance of composition-only models on formation-energy prediction.a Test-set neural network predictions versus DFT-computed formation energies from the Materials Project materials_project. The red line indicates perfect agreement ($y=x$). b Test-set mean absolute error (MAE) of the max-affine model as a function of the number of facets $F$, with the chosen value $F=7$ indicated by the red lines. c Test-set max-affine model predictions versus DFT-computed formation energies from the Materials Project materials_project. The red line indicates perfect agreement ($y=x$).
• Figure 2: Chemical interpretation of max-affine facets for binary compounds.a--g Facet-resolved periodic-table maps for facets 1–7 of the max-affine hull. As indicated by the enlarged element key, the border color of each element denotes its valence category (blue: $s$ block; green: $d$/$f$ block; orange: early $p$ block; red: late $p$ block). Within each element box, the four colored quadrants indicate how frequently that element forms binary compounds with elements from each category on the corresponding facet. h Facet–materials-class correlation matrix, where color intensity denotes the probability that a binary compound from a given materials class activates each facet.
• Figure 3: DFT-calculated versus predicted defect energies. Perfect agreement, up to a constant offset, is indicated by the red dashed reference lines ($y=x+C$). These are true out-of-distribution predictions with no fitting other than a single constant shift. In all panels, max-affine and neural network predictions are shown as blue and red points, respectively. Defect energies predicted by the models are computed using finite-difference approximations with step size $\delta=0.5$. a DFT-computed interstitial, vacancy, and substitution energies (all defect classes) versus max-affine and neural network predictions (offset $C=$+1.24 eV/atom). b,c DFT-computed interstitial energies versus predictions (Eq. \ref{['eq: inf interstitial']}) for nitrogen (b) and oxygen (c) interstitials. Offsets of $C=+1.75$ eV/atom in b and $C=+1.24$ eV/atom in c. d DFT-computed oxygen-vacancy energies versus predictions (Eq. \ref{['eq: inf vacancy']}) for a range of metal oxides (offset $C=+1.28$ eV/atom). e DFT-computed substitution energies versus predictions (Eq. \ref{['eq: inf substitution']}) for a variety of impurities in Nb (offset $C=+0.09$ eV/atom).
• Figure 4: Elemental spatial correlations in high-entropy phosphide nanoparticles.a High-angle annular dark-field (HAADF) STEM image of high-entropy phosphide nanoparticles. b--h Scanning transmission electron microscopy coupled with energy-dispersive X-ray spectroscopy (STEM–EDS) elemental maps of the $\mathrm{(FeCoNiRhPd)_2P}$ high-entropy phosphide, showing the spatial distribution of all metal elements in b and individual elements in c--h, respectively, for the same nanoparticles shown in a. All elemental maps are acquired from the same particles and are spatially registered to the HAADF image. i Intersection-over-union (IOU) values (Eq. \ref{['eq: iou']}, threshold 0.55) extracted from the STEM–EDS images, plotted against the correlation energy $E_{\mathrm{corr}}$ descriptor (Eq. \ref{['eq: correlation']}) computed using the max-affine (blue) and neural network (red) models. Note that the neural network points are not individually labeled to avoid overcrowding the plot. The red dashed line denotes the line of best fit for the max-affine model, which exhibits a p-value of 0.026, indicating statistical significance.
• Figure 5: Predicted versus experimental and DFT-calculated Pourbaix diagrams.a Pourbaix diagram for Ti constructed using max-affine model predictions for solid phases. b Pourbaix diagram for Ti constructed using experimental formation energies for solid phases. c Pourbaix diagram for $\mathrm{Ti}_{0.1}\mathrm{Mn}_{0.9}$ constructed using max-affine model predictions for solid phases. d Pourbaix diagram for $\mathrm{Ti}_{0.1}\mathrm{Mn}_{0.9}$ constructed using the Materials Project Pourbaix diagram tool Persson_pourbaixpatel2019pourbaixsingh2017pourbaix, which employs DFT-calculated formation energies. e Pourbaix diagram for $\mathrm{Ti}_{0.5}\mathrm{Mn}_{0.5}$ constructed using max-affine model predictions for solid phases. This composition is identified by rapid, composition-only optimization as having the largest stable region within the electrochemical window of water at high pH. f Pourbaix diagram for $\mathrm{Ti}_{0.5}\mathrm{Mn}_{0.5}$ constructed using the Materials Project Pourbaix diagram tool. In all panels, blue shaded regions indicate thermodynamically stable solid phases, while white regions correspond to stable aqueous ionic species.