# Estimates of Quantum Tunneling Effects for Hydrogen Diffusion in PuO\({}_{2}\)
## 3 Results  
We have computed a survey of results from PBE, PBESol, and SCAN by determining optimized ground-state PuO\({}_{2}\) properties for a variety of spin geometries and with GGA + U (Table 2). All of our results for all functionals and levels of theory yield lattice constants that are close to the experimental value and lattice volumes that agree within \(\sim\)3%. We find that all magnetic states (FM and AF) are essentially isoenergetic for all three functionals, while the diamagnetic state exists over 3 eV per formula unit higher in energy for PBE and PBESol, and over 5.5 eV for SCAN. In addition, the FM (111) state is \(1.4\times 10^{-2}\) eV higher than all AF states for PBE and PBESol, but is \(8.8\times 10^{-3}\) eV lower in energy for SCAN. However, these energy differences are negligible at 298 K (\(\sim\)0.025 eV). \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multicolumn{1}{c}{\(\mathbf{E_{Xc}}\)} & \multicolumn{1}{c}{**Magnetic State**} & \multicolumn{1}{c}{**Lattice Const (Å)**} & \multicolumn{1}{c}{**Volume (Å\({}^{3}\))**} & \multicolumn{1}{c}{**Band Gap (eV)**} & \multicolumn{1}{c}{**Relative Energy to**} \\  & & & & & **AF (3 klong)** \\ \hline PBESol with GGA & & & & & **(eV/E.U. )** \\ + U & DM & 5.39, 5.35, 5.37 & 154.85 & 3.07 & 3.06 \\  & FM (111) & 5.41, 5.39, 5.37 & 156.59 & 2.54 & \(1.4\times 10^{-2}\) \\  & AF (1 klong) & 5.37, 5.37, 5.43 & 156.58 & 2.92 & \(-7.5\times 10^{-4}\) \\  & AF (3 klong) & 5.38, 5.40, 5.38 & 156.30 & 2.92 & \(-7.5\times 10^{-4}\) \\  & AF (1 kltrans) & 5.37, 5.43, 5.37 & 156.58 & 2.92 & \(-7.5\times 10^{-4}\) \\ PBESol SOC with & & & & & \\ GGA + U & FM (111) & 5.39 & 156.59 & 2.50 & \(7.5\times 10^{-3}\) \\  & AF (3 klong) & 5.39 & 156.59 & 3.09 & \(-\) \\  & AF (3 kltrans) & 5.39 & 156.59 & 3.09 & \(5.0\times 10^{-4}\) \\ PBE with GGA + & DM & 5.43 & 160.10 & 2.60 & 3.06 \\ U & FM (111) & 5.45 & 161.88 & 2.60 & \(1.4\times 10^{-2}\) \\  & AF (1 klong) & 5.42, 5.42, 5.48 & 160.98 & 2.80 & \(1.8\times 10^{-3}\) \\  & AF (3 klong) & 5.42, 5.47, 5.44 & 161.28 & 2.85 & \(-\) \\  & AF (1 kltrans) & 5.42, 5.48, 5.42 & 160.98 & 2.80 & \(1.8\times 10^{-3}\) \\
**SCAN with GGA** & & & & & \\ + U & DM & 5.39 & 156.59 & 0.0 & 5.52 \\  & FM (111) & 5.40 & 157.46 & 2.16 & \(-8.8\times 10^{-3}\) \\  & AF (1 klong) & 5.41, 5.41, 5.47 & 160.10 & 2.68 & \(3.4\times 10^{-2}\) \\  & AF (3 klong) & 5.43 & 160.10 & 2.75 & \(-\) \\  & AF (1 kltrans) & 5.41, 5.47, 5.41 & 160.10 & 2.67 & \(3.4\times 10^{-2}\) \\
**DFTB/ChIMES** & & & & & \\ \cline{1-1}  & **with GGA  
As expected, all three AF calculations with SP yield virtually isoenergetic structures, though there are differences in the lattice constants and in some cases the final lattice volume. In all cases, the AF (1 klong) and AF (1 kltrans) spin states resulted in a tetragonal lattice distortion of similar magnitude (0.06 A). In contrast, the AF (3 klong) state yields a smaller tetragonal distortion for PBESol (0.02 A), an orthorhombic unit cell for PBE, and a cubic unit cell for SCAN. In general, PBESol appears to yield the best overall agreement with the experimental lattice constant and volume for the magnetic states studied here, though the differences between PBE and PBESol do not appear to be substantial. We note that \(U_{\text{eff}}\) can be adjusted in order to yield results close to the experimental lattice volume. Some additional differences are observed with some of the band gap results. All three AF band gaps from PBESol agree with a value of 2.92 eV. However, both PBE and SCAN yield a slightly higher band gap for AF (3 klong) state relative to AF (1 klong) and AF (1 kltrans), with values of 2.85 vs. 2.80 eV for PBE and 2.75 vs. 2.68 eV for SCAN. PBESol yields a higher gap for the DM state relative to the AF states (3.07 eV), whereas PBE yields a lower gap (2.60 eV), and SCAN yields a metallic system. All three functionals yield a lower gap for FM (111) compared to the AF states, with a value of 2.54 eV from PBESol, 2.60 eV from PBE, and 2.16 eV for SCAN.