# Estimates of Quantum Tunneling Effects for Hydrogen Diffusion in PuO\({}_{2}\)
## 3 Results
The effect of using GGA + U appears to be fairly small, with a result of 0.64 eV from AF (3 klong) GGA + U. In this case, additional use of SOC has a similar magnitude of effect in the opposite direction, with a result of 0.70 eV for AF (3 klong) GGA + U SOC. The tetragonal cell distortion in the AF (1 klong) calculation does not appear to have a significant effect on \(E_{\rm int}\), though we note that the breaking of the cubic symmetry likely results in the creation of additional interstitial sites and energies. Our DFTB/ChMES result agrees relatively closely with the spread of results from PBESol with a value of 0.75 eV. Our results at all levels of theory for all magnetic states agree with the previously published SP GGA + U result from PBE of 0.54 eV from Ref. [16]. The exception presented here is the diamagnetic state, which yields a relatively high \(E_{\rm int}\) value of 0.97 eV. Our results also show some agreement with the range of PBE results of 0.48-1.19 eV from Ref. [20], though there is some ambiguity in that publication as some calculations might have been performed on the PuO\({}_{2}\) unit cell, where system size effects are likely significant. Given the relative agreement between levels of theory, we choose to continue our study with the more efficient SP calculations, only. Results for the octahedral site formation energy, \(E_{\rm oct}\), with results from collinear FM (001) and noncollinear AF (3 klong) showing agreement within 0.08 eV, with values of 1.25 and 1.33 eV, respectively. Inclusion of GGA + U in the AF (3 klong) GGA + U calculation yields a modest increase in \(E_{\rm oct}\) of 0.16 eV to a value of 1.49 eV. We note that our AF (3 klong) GGA + U result is smaller than the collinear SP GGA + U results from PBE of 2.24 eV [16] and 2.12 eV [20]. These previous PBE results are in relatively good agreement with the DFTB/ChMES result of 2.45 eV. Regardless, our results indicate overall that both SOC and GGA + U likely play a relatively small role in hydrogen/PuO\({}_{2}\) lattice interactions. Hence, for the remainder of this work, we choose to perform calculations with SP, only and omit these two levels of theory for the sake of computational efficiency. This allows us to more efficiently explore the kinetics of hydrogen diffusion pathways as a function of magnetic state. \begin{table}
\begin{tabular}{c c c} \hline
**Defect** & **Magnetic State** & **E (eV)** \\ \hline
**Interstitial** & Diamagnetic & 0.97 \\  & FM (001) & 0.63 \\  & FM (111) & 0.71 \\  & AF (1 klong) & 0.64 \\  & AF (1 klong-iso) & 0.61 \\  & AF (3 klong) & 0.71 \\  & AF (3 klong) GGA + U & 0.64 \\  & AF (3 klong) GGA + U SOC & 0.70 \\  & DFTB/ChMMES & 0.75 \\ \hline
**Octahedral** & FM (001) & 1.25 \\  & AF (3 klong) & 1.33 \\  & AF (3 klong) GGA + U & 1.49 \\  & DFTB/ChMES & 2.45 \\ \hline \end{tabular}
\end{table} 
Table 3: Hydrogen interstitial and octahedral site formation energies from PBESol. Calculations were performed with SP only and exclude GGA + U and SOC unless otherwise noted. The AF (1 klong) calculations was performed on the tetragonally distorted lattice, whereas the AF (1 klong-iso) calculation was performed with the cubic experimental lattice. We now investigate the kinetics of three fundamental pathways in the H/PuO\({}_{2}\) system (Figure 2), namely:  
1. Diffusion from one int site to its nearest neighbor int site (where an O-H bond is broken), which we label _int-int_. 2. Diffusion between int sites containing an O-H bond rotation (where the bond remains intact), which we label _int-int_2. 3. Diffusion hops from the oct site to the closest int site, which we label _oct-int_. We have not computed int-oct diffusion (from an interstitial to an octahedral site) due to the large increase in energy for octahedral absorption, indicating that this pathway is likely only at extreme temperatures. In addition, oct-oct diffusion is highly unlikely to occur, given the closer proximity of low energy int sites.