# Estimates of Quantum Tunneling Effects for Hydrogen Diffusion in PuO\({}_{2}\)
## 3 Results
We now approximate the activation energies, \(E_{\text{A}}\), for these pathways using a two-step approach. First, we have computed minimum energy pathways (MEP) from our DFTB/ChIMES model using the climbing image nudged elastic band (NEB) method [43]. All calculations were run within the Atomic Simulation Environment python interface [44; 45], with the image dependent pair potential method [46] used to generate an initial guess for each pathway. Here, our SCC convergence was held to \(2.72\times 10^{-4}\) eV (\(10^{-5}\) au) and the NEB forces were converged to \(0.05\) eV/A. In addition, the simulation cell was held fixed at the experimental PuO\({}_{2}\) density with cubic symmetry. Decreasing the SCC convergence to \(2.72\times 10^{-5}\) eV (\(10^{-6}\) au) and the NEB force convergence to \(0.01\) eV/A resulted in an increase in the activation energy of \(0.01\) eV for the int-int2 rotation. Next, we performed constrained geometry DFT optimizations starting from the DFTB/ChIMES computed transition state geometries using PBESol with different spin states, where the hydrogen ion was held fixed and the surrounding PuO\({}_{2}\) ionic positions were allowed to relax. In doing so, we were able to approximate \(E_{\text{A}}\) for several different spin states relatively efficiently at a DFT level. All PBESol optimizations were kept at the experimental PuO\({}_{2}\) cubic lattice, excluding AF (1 klong) where the ionic coordinates and lattice vectors were scaled to the optimized tetrahedral distorted cell in order to facilitate comparison with previous DFT  
Figure 2: Hydrogen diffusion pathways investigated in this study. The hydrogen in the octahedral site in (**c**) is colored blue for the sake of clarity. NEB results [20]. More exact calculations of MEPs in H/PuO\({}_{2}\) systems is the subject of future work. Our results indicate a high degree of consistency between results for different spin states (Table 4). All magnetic states from PBESol conducted on a cubic simulation cell yield an \(E_{\text{A}}\) of \(\sim\)1.1 eV for the int-int transition, while the diamagnetic calculation yields a value of only 0.36 eV. DFTB/ChIMES yields a value in agreement with the PBESol magnetic results, with a value of 1.03 eV. Additional constrained optimization calculations were performed with PBE in order to examine the effect of choice of functional on our results. We find that results from both the diamagnetic and AF (3 klong) states yield close agreement with results from PBESol, with values of 0.32 eV and 1.10 eV, respectively. Our result for the tetragonally deformed AF (1 klong) geometry computed with PBE yields a value of 0.15 eV along the short lattice length, very close to the previously reported PBE activation energy of 0.13 eV for this transition [20]. Given the relative agreement between results from different magnetic states with a cubic simulation cell, we continue with the AF (3 klong) spin geometry for PBESol, only. We find that for the int-int2 transition DFTB/ChIMES determines a relatively large \(E_{\text{A}}\) relative to PBESol, with values of 0.36 eV vs. 0.06 eV. In contrast, we observe improved agreement for oct-int, where DFTB/ChIMES yields an activation energy of 0.27 eV compared to the value of 0.36 eV from PBESol. It is possible that the agreement of DFTB/ChIMES with PBESol could be improved with additional and more accurately computed training data, which is the subject of future work. We choose to use the PBESol AF (3 klong) estimates for \(E_{\text{A}}\) for the remainder of this work due to their likely higher accuracy than those from DFTB/ChIMES. We estimate quantum effects on hydrogen diffusion by treating the diffusion pathways as a one-dimensional quantum double-well with a parabolic barrier, using an approach adapted from Ref. [47]. We start with the following piece-wise quadratic function:  
\[V(x)=