# Estimates of Quantum Tunneling Effects for Hydrogen Diffusion in PuO\({}_{2}\)
## 3 Results
Overall, the AF results from PBE are generally closest to the experimental value of 2.82 eV, though all three functionals yield reasonable results. Results for AF (1 klong) GGA + U calculations from DFTB/ChIMES yielded an accurate lattice constant for a value of \(U_{\text{eff}}=4.08\) eV, albeit for a metallic state. We found that increasing the value of \(U_{\text{eff}}\) to as high as 12 eV did not create an insulating state. This is possibly due to either the minimal basis set used in DFTB calculations or the omission of greater than two-center terms in the DFTB Hamiltonian. Use of different confining radii and/or basis set functions could improve upon this result, though those effects are beyond our current scope of work. Finally, in order to test the validity of our SP results, we have performed noncollinear calculations with SOC and the same value of \(U_{\text{eff}}=6.00\) eV. We find that this set of calculations yields the AF (3 klong) SOC state as the energetic minimum, with all three magnetic states yielding a cubic structure and lattice constant very close to that of experiment. However, once again, the AF (3 kltrans) SOC and FM (111) SOC states are higher in energy by <0.01 eV, indicating that these spin geometries are essentially isoenergetic. In addition, we compute a FM (111) SOC band gap of 2.50 eV, which is very similar in value to the SP result. However, we compute AF (3 klong) SOC and AF (3 kltrans) SOC band gaps that are higher than the SP result by 0.17 eV (3.09 vs. 2.92 eV). The SOC relative energy and band gap results reported here differ somewhat from previous PBESol SOC results using the same value for \(U_{\text{eff}}\)[13]. These are possibly due to a combination of the larger k-point mesh used in our study as well as the more stringent convergence criteria used in Ref. [13]. Based on these results, we choose to proceed with the PBESol functional for the rest of our calculations. PBESol provides accurate equilibrium lattice constants, particularly for the nearly cubic AF (3 klong) magnetic state, as well as a band gap only slightly higher than that from experiment. Hence, it appears to provide the best combination of accuracy and computational efficiency, though we note than any of three functionals studied here would appear to provide reasonable results. Next, we have computed hydrogen point defect formation energies for the int and oct sites using different magnetic states and levels of theory. These were calculated according to the equation:  
\[E_{\text{def}}=E_{\text{PuO}_{2}/\text{H}_{\text{def}}}-nE_{\text{PuO}_{2}}- \frac{1}{2}E_{\text{H}_{2}}, \tag{1}\]  
where the subscript 'def' corresponds to either the int or oct site, \(E_{\text{PuO}_{2}/\text{H}_{\text{def}}}\) is the total int or oct defect system energy, \(n\) is the number of formula units in the PuO\({}_{2}\)/H system, \(E_{\text{PuO}_{2}}\) is the bulk energy per formula unit, and \(E_{\text{H}_{2}}\) is the molecular hydrogen formation energy. The goal here is to assess the effects of these parameters on interactions between the lattice and the hydrogen ion itself. For most spin states, we have fixed our simulation cell at the experimental lattice parameters, which are close the optimized values reported in Table 2.  
The only exceptions is the AF (1 klong) calculation where the supercell lattice was held fixed at the dimensions reported in the table in order to provide direct comparison to previous H/PuO\({}_{2}\) collinear SP GGA + U results [16; 20]. One additional AF (1 klong) calculation was performed with the experimental lattice vectors with SP and without GGA + U, which is labeled as AF (1 klong-iso). Our results indicate a relative consistency in the hydrogen interstitial site formation energy, \(E_{\rm int}\) (Table 3). For the SP calculations, results without GGA + U yield a fairly narrow spread of \(E_{\rm int}\), with a value of 0.63 eV for FM (001) at the low end and 0.71 eV for AF (3 klong) at the high end.