# Estimates of Quantum Tunneling Effects for Hydrogen Diffusion in PuO\({}_{2}\)
## 3 Results
The double well results in a splitting of all energy levels, which is on a scale smaller scale than shown in the plot for lower eigenstates. \[k_{\text{QM}}(T)=\frac{1}{2\pi}\frac{\Sigma_{n}\rho_{n}T_{par}^{QM}(E_{n})e^{- \beta E_{n}}}{\Sigma_{n}\rho_{n}e^{-\beta E_{n}}}. \tag{4}\]  
The summation is over the \(n\) computed eigenvalues \(E_{n}\) and eigenstates \(\psi_{n}\) of the system, and \(\rho_{n}\) is the probability of finding the particle on the reactant side, namely:  
\[\rho_{n}=\int_{-\infty}^{x_{r}^{(n)}}|\psi_{n}(x)|^{2}dx. \tag{5}\]  
Similar to Ref [47], we choose to define the reactant region limit for a given eigenstate, \(x_{r}^{(n)}\), as its classical turning point. We note that the computed thermal reaction rate will have some dependence on definition of \(x_{r}^{(n)}\), though the classical turning point is physically intuitive and consistent with well-known semi-classical approaches such as the Wentzel-Kramers-Brillouin (WKB) approximation. Additional accuracy for our model double-well potentials could involve determination of flux-flux autocorrelation functions [49] and/or time-dependent wave-packet propagation to determine the scattering matrix [47; 50]. Full determination of quantum nuclear vibrational effects in the condensed phase would require simulations using path integrals (e.g., Ref. [51]), which would necessitate the development of an accurate interatomic potential due to the extreme computational expense of these simulations [52]. Regardless, results from Ref. [47] indicate that the eigenstate transmission approach we have adopted here likely yields a high degree of accuracy for proton double-well transmission barriers as low as \(\sim\)0.3 eV with respect to scattering matrix calculations. We now make comparison to classical rate constants estimated from our approximate \(E_{\text{A}}\) values from the PBESol AF (3 klong) calculations. Classical diffusion rates were estimated from a modified version of the Eyring equation:  
\[k_{\text{Cl}}=\frac{k_{\text{B}}T}{h}\text{exp}(-E_{\text{A}}/k_{\text{B}}T). \tag{6}\]  
Here, we have assume that the free energy of activation is equal to our estimate for \(E_{\text{A}}\) and that the Eyring transmission coefficient is equal to one. This approach yields a kinetic prefactor of \(6.3\times 10^{12}\) s\({}^{-1}\) at 300 K, which is of the same magnitude as DFT calculations for hydrogen ion diffusion on transition metal oxide surfaces (e.g., Ref. [53]). Our results for the diffusion rates (Figure 4) indicate that int-int diffusion is likely low even at 300 K, where we compute a \(k_{\text{QM}}\) value of \(3.9\times 10^{-5}\) s\({}^{-1}\) and a \(k_{\text{Cl}}\) value of \(2.1\times 10^{-6}\) s\({}^{-1}\). We predict that the rate increases by nine orders of magnitude at 600 K, with values of \(k_{\text{QM}}=2.6\times 10^{4}\) s\({}^{-1}\) and \(k_{\text{Cl}}=7.2\times 10^{3}\) s\({}^{-1}\). These increase by an additional four orders of magnitude at 1000 K, with a values of \(k_{\text{QM}}=1.6\times 10^{8}\) s\({}^{-1}\) and \(k_{\text{Cl}}=6.0\times 10^{7}\) s\({}^{-1}\). In contrast, the oct-int transition is relatively frequent at 300 K due to its relatively low barrier, with values of \(k_{\text{QM}}=9.9\times 10^{8}\) s\({}^{-1}\) and \(k_{\text{Cl}}=5.6\times 10^{6}\) s\({}^{-1}\). These values increase more modestly with increasing temperature, with values of \(k_{\text{QM}}=8.1\times 10^{10}\) s\({}^{-1}\) and \(k_{\text{Cl}}=1.2\times 10^{10}\) s\({}^{-1}\) at 600 K and at \(k_{\text{QM}}=1.0\times 10^{12}\) s\({}^{-1}\) and \(k_{\text{Cl}}=3.2\times 10^{11}\) s\({}^{-1}\) 1000 K. Nonetheless, hydrogen absorption into an octahedral defect site is unlikely at the conditions studied here, given its high formation energy of \(\sim 1.5\) eV.