Spectral Functionals of the Elastic Stiffness Tensor as a Physically Grounded Basis for Hardness Prediction and Directed Crystal Structure Search

We propose and systematically compare ten physically motivated hardness descriptors based exclusively on the six eigenvalues (λ₁ ≤ λ₂ ≤ ··· ≤ λ₆) of the elastic stiffness tensor in Voigt notation. The models are trained on a curated 45-material dataset using elastic tensors from the GRACE-2L machine-learning interatomic potential and experimental Vickers hardness traced to primary sources. The best model, CV-λ (HV = 0.123·λ⁰·⁸⁷⁴harm·CV⁻⁰·⁹²⁶), achieves R² = 0.911 and MAE = 3.91 GPa, outperforming the classical Teter (R² = 0.818), Tian (R² = 0.878), and Mazhnik–Oganov (R² = 0.856) models. Cross-validation confirms robust generalisation (CV R² = 0.826). All anisotropy descriptors are near-equivalent (r(CV, 1−PR) = 1.000). A blind DFT transferability test shows GRACE-based predictions systematically outperform DFT-based ones. Beyond hardness prediction, we demonstrate that the eigenvectors of the stiffness tensor define physically motivated search directions for crystal structure exploration: preliminary application to the Nb–B system via eigenvalue directed exploration (EDE) recovers 3 of 6 known experimental phases with exact space group symmetry, identifies hard candidates overlooked by energy-driven evolutionary search (USPEX), and recovers 4 of 5 OQMD reference phases.