Property prediction of crystalline solids from composition and crystal structure
Bruno A. Calfa and John R. Kitchin, “Property prediction of crystalline solids from composition and crystal structure”, AIChE Journal 62(8), 2605-2613 (2016). https://doi.org/10.1002/aic.15251
This summary was written by a language model from the paper and has not been checked line by line. Rely on the published paper for details.
Question
Group contribution methods predict properties of molecules from counts of functional groups, and because those counts are explicit, the same models can be run in reverse in computer-aided molecular design to find a molecule with target properties. Machine learning models for crystals published before this work used descriptors such as electronegativity, ionization potential or orbital radii, which predict well but cannot be mapped back to a specific material. The paper asks whether a property model for crystalline solids can be built on descriptors that identify the crystal directly, so that a design (inverse) problem returns a composition and structure (Calfa 2016).
Approach
Each crystal is described by two kinds of integer predictors: its three-dimensional space group number (1 to 230) and the number of atoms of each element in its chemical formula. These carry composition and structural arrangement in the spirit of “groups” in a group contribution method. The work is restricted to atomic crystals, and the space group is treated only as a label: no geometry or structure optimization is done.
Because all predictors are discrete, the authors use nonparametric kernel regression with the Racine-Li kernel for categorical data. The prediction for a new crystal is a weighted average of the training responses, where each predictor contributes a factor equal to its bandwidth (a value between 0 and 1) whenever a training point’s value differs from the query value. One bandwidth per predictor and property is fitted by least-squares leave-one-out cross-validation, using the npreg routine of the R np package (in its parallel npRmpi version) with five multistart runs on a 32-CPU Google Compute Engine virtual machine. A noted consequence of the nonparametric form is that prediction requires the full training set, not just fitted parameters.
Training data came from the Materials Project (DFT calculations, accessed through its REST API). Two cases were studied: band gap and Fermi level of 746 binary metal oxides, and Voigt and Reuss averages of the bulk and shear moduli of 1173 crystals. Total energy, formation energy per atom and density were modeled for both sets. Accuracy was reported as RMSE, MAE and MAPE between the DFT data and model predictions; the text does not describe a separate held-out test set.
Key findings
For the metal oxides, the band gap model gave an RMSE of 0.053 eV and an MAE of 0.013 eV, and formation energy per atom an RMSE of 0.0024 eV. The Fermi level was the least accurate property (RMSE 0.24 eV, MAPE about 10%), with the model tending to overestimate negative Fermi levels; the largest residuals were for TiO2 in space groups 14 and 35 (about 1.27 and -1.75 eV). The largest band gap residuals were for ThO2 and Rb2O in space group 225 (about 1.0 and 0.84 eV). Fitting the bandwidths took roughly 1300 to 2500 s per property.
For the elastic data set, which spans one to four elements per compound and 63 elements as predictors, the moduli were predicted with RMSEs of about 0.7 to 2.2 GPa and MAPEs of about 0.6 to 1%. Total energy was much worse here (RMSE 3.8 eV, MAPE 7.3%), which the authors attribute to the chemical diversity of the set. Fitting took about 2700 to 5000 s per property. Overall MAPE stayed at or below about 10% for the oxides and below 7.5% for the elastic set.
As a first route to the inverse problem, the authors describe an exhaustive enumeration algorithm that predicts properties for every valid combination of space group and element counts present in the data (each oxide candidate must include a space group, a metal and oxygen), using a matrix of unique predictor values to avoid repeats. Applied to the oxide set, it produced 1,153,504 unique candidates in about 19.7 h. Filtering this list for target band gaps and Fermi levels returned candidates such as Bi2O3 (space group 197) for a band gap near 1.23 eV, including compositions not in the original data. The authors caution that model error can prevent recovering a known crystal from its own property values.
Significance
The paper frames crystal property prediction as an extension of group contribution and computer-aided molecular design ideas from chemical engineering, choosing descriptors for invertibility rather than physical insight. The cost of brute-force enumeration (over a million evaluations for one small data set) is presented as motivation for optimization-based inverse design formulations, which the authors describe as ongoing work. Since no geometry optimization is performed, a returned space group is a suggested atomic arrangement rather than a computed structure.