A Cartesian Encoding Graph Neural Network for Crystal Structures Property Prediction: Application to Thermal Ellipsoid Estimation

TL;DR

CartNet presents a Cartesian-encoded graph neural network for crystal property prediction, focused on anisotropic displacement parameters (ADPs). It uses a cell-less, temperature-aware geometry encoding, a neighbour-equalization scheme, and a Cholesky-based head to guarantee positive-definite ADP outputs, complemented by SO(3) data augmentation. On a curated ADP dataset of over 200k structures, CartNet achieves state-of-the-art MAE, S12, and IoU, while also delivering strong performance on Jarvis and Materials Project benchmarks for formation energy, band gaps, and moduli, with substantial efficiency gains over DFT baselines. The combination of Cartesian geometry, rotation-aware augmentation, and a physics-consistent output layer enables accurate, rotation-sensitive predictions and suggests broad applicability to diverse crystal-property tasks and fast materials screening.

Abstract

In diffraction-based crystal structure analysis, thermal ellipsoids, quantified via Anisotropic Displacement Parameters (ADPs), are critical yet challenging to determine. ADPs capture atomic vibrations, reflecting thermal and structural properties, but traditional computation is often expensive. This paper introduces CartNet, a novel graph neural network (GNN) for efficiently predicting crystal properties by encoding atomic geometry into Cartesian coordinates alongside the crystal temperature. CartNet integrates a neighbour equalization technique to emphasize covalent and contact interactions, and a Cholesky-based head to ensure valid ADP predictions. We also propose a rotational SO(3) data augmentation strategy during training to handle unseen orientations. An ADP dataset with over 200,000 experimental crystal structures from the Cambridge Structural Database (CSD) was curated to validate the approach. CartNet significantly reduces computational costs and outperforms existing methods in ADP prediction by 10.87%, while delivering a 34.77% improvement over theoretical approaches. We further evaluated CartNet on other datasets covering formation energy, band gap, total energy, energy above the convex hull, bulk moduli, and shear moduli, achieving 7.71% better results on the Jarvis Dataset and 13.16% on the Materials Project Dataset. These gains establish CartNet as a state-of-the-art solution for diverse crystal property predictions. Project website and online demo: https://www.ee.ub.edu/cartnet

Figures (23)

• Figure 1: Schematic of the CartNet graph neural network architecture for a 5,5'-dimethyl-2,2'-bipyrazine crystal structure (CSD Refcode: ETIDEQ). The model predicts the ADPs for all non-hydrogen atoms based on the positions of atoms within the unit cell. The architecture separately encodes atomic and edge information using dedicated encoders. This information is then aggregated through N iterations of message-passing via the CartLayer. Finally, the Cholesky Head ensures the output matrix is symmetric and positive-definite, generating a valid ADP matrix. White, light purple, and grey colours represent hydrogen, nitrogen, and carbon atoms, respectively. The parallelepiped represents the unit cell, and the red, green, and blue lines correspond to the a, b, and c unit cell axes.
• Figure 2: Thermal ellipsoids ORTEP representations from experimental ADPs of an 5,5'-dimethyl-2,2'-bipyrazine crystal structure (CSD Refcode: ETIDEQ). Light purple, and grey colours represent nitrogen, and carbon atoms, respectively. Hydrogen atoms have been omitted. The parallelepiped represents the unit cell, and the red, green, and blue lines correspond to the a, b, and c unit cell axis.
• Figure 3: Histogram showing the number of crystal structures within each temperature range in the ADP dataset, displayed on a logarithmic scale on the y axis.
• Figure 4: Histogram illustrating the number of atoms within each ADP volume range in the ADP dataset, presented on a logarithmic scale on both axes.
• Figure 6: Representation of the graph construction process for the atom highlighted in red colour. Covalent bonds are ignored, and a radius around the atom (depicted in green) is defined. Any atom within this radius is considered a neighbour of the red atom and is connected in the graph (depicted using red lines). Periodic boundary conditions are employed to replicate the infinite nature of the crystal.