Trust-Limited Machine-Learning-Accelerated Nudged Elastic Band Calculations for Small-Polaron Hopping in Rutile TiO2

Abstract

Nudged elastic band (NEB) calculations of small-polaron hopping are demanding because nuclear motion is coupled to a self-consistent localized electronic state. Near the charge-transfer region, slow self-consistent-field convergence, localization switching, and competing electronic solutions complicate surrogate-assisted optimization. We develop a Gaussian-process-assisted NEB workflow for this regime and demonstrate electron small-polaron hopping in rutile TiO2 using spin-polarized density functional theory with a Hubbard U correction (DFT+U ) with SIESTA. A joint energy-force Gaussian process (GP) with a Matérn covariance model advances the elastic band, while active-atom root-mean-square and maximum-atom displacement limits constrain surrogate motion relative to evaluated configurations. Reaching these limits triggers first-principles sampling without requiring full surrogate convergence. Three independently DFT-verified 96-atom [001] calculations using 11, 13, and 15 images and different surrogate settings and GP parameterizations yield barriers of 0.1645, 0.1637, and 0.1640 eV, respectively, spanning 0.793 meV. In the 15-image calculation, repeated central-image sampling activates a separately calibrated local GP during surrogate optimization. Final full-band DFT verification gives a maximum projected NEB force of 0.0346 eV/Å. A 324-atom calculation yields a lower barrier of 0.1302 eV, indicating supercell sensitivity and demonstrating feasibility at larger GP observation dimension. The workflow combines controlled surrogate extrapolation, targeted charge-transfer-region interpolation, and independent DFT verification to obtain polaron migration pathways, emphasizing reproducible barriers and full-band force convergence.