# Estimates of Quantum Tunneling Effects for Hydrogen Diffusion in PuO\({}_{2}\)
## 3 Results
\begin{cases}\frac{k(x-x_{1})^{2}}{2}&x<x_{L}(\text{reactant region})\\ \frac{-k_{x}x^{2}}{2}+V_{b}&x_{L}\leq x\leq x_{R}(\text{barrier region})\\ \frac{k(x-x_{2})^{2}}{2}+V_{0}&x>x_{R}(\text{product region})\end{cases}  \tag{2}\]  
Here, \(k\) is the spring constant for each well, \(k_{b}\) is the spring constant for the barrier, \(x_{1}\) is the position of the reactant minimum, \(x_{2}\) is the position of the product minimum, \(V_{b}\) is the parabolic barrier height, \(V_{0}\) is the offset of the product well, and \(x_{L}\) and \(x_{R}\) define the reactant (left) and product (right) regions. For our work, we have chosen \(k=k_{b}\) in \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Pathway** & **Model** & **Magnetic State** & \(\mathbf{E_{A}}\) **(eV)** \\ \hline int-int & DFTB/ChIMES & AF (1 klong) & 1.03 \\  & PBESol & Diamagnetic & 0.36 \\  & & FM (001) & 1.17 \\  & & FM (111) & 1.10 \\  & & AF (3 klong) & 1.10 \\  & PBE & Diamagnetic & 0.32 \\  & & AF (1 klong) & 0.15 \\  & & AF (3 klong) & 1.10 \\ \hline int-int2 & DFTB/ChIMES & AF (1 klong) & 0.36 \\  & PBESol & AF (3 klong) & 0.06 \\ \hline oct-int & DFTB/ChIMES & AF (1 klong) & 0.27 \\  & PBESol & AF (3 klong) & 0.36 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Estimated activation energies for hydrogen diffusion hops from spin polarization calculations, only, where both GGA + U and SOC have been omitted. All calculations were performed on the experimental cubic lattice, excluding the PBE AF (1 klong) results, where the tetragonally distorted cell was used and the hydrogen was moved in the direction of one of the shorter lattice vectors. order to ensure a smooth potential throughout, where \(k=\frac{4V_{b}}{x_{1}^{2}}\). The parameter \(V_{b}\) is set to the approximate \(E_{\text{A}}\) value from the AF (3 klong) calculations. The potential is centered at zero for all three transitions studied here. For the symmetric int-int and int-int2 transitions, \(V_{0}=0\), and \(x_{1}=-\frac{1}{2}\Delta x=-x_{2}\), where \(\Delta x\) is the total reaction path distance as determined from our DFTB/ChIMES NEB calculations. Furthermore, we compute the reaction/product positions as \(x_{L}=\frac{1}{2}x_{1}\) and \(x_{R}=\frac{1}{2}x_{2}\). For the asymmetric oct-int transition, the same equations apply with the exception of \(V_{0}=E_{\text{oct}}-E_{\text{int}}\) and \(x_{2}=\sqrt{4(V_{b}-V_{0})/k}\). We can estimate the quantum tunneling rates by first assuming a quantum harmonic oscillator (Hermite polynomial) basis set and solving for the energy eigenstates. Care was taken to ensure convergence for each transition in terms of basis set size as well as the numerical limits of integration. Results for the int-int transition indicate small splittings between energy levels (i.e., \(<\)1 meV) (Figure 3), typical for a symmetric double well with a relatively large barrier where the particle tends to remain in a single well and tunneling transmission coefficients are small. The probability of hydrogen being found between wells becomes more pronounced as the eigenstates approach the top of the potential barrier, with the state just below the barrier yielding a splitting of slightly less than 0.01 eV. In addition, we compute a zero-point energy correction in each potential well of \(\sim\)0.06 eV, and that the energy levels below the parabolic barrier tend to be spaced by \(\sim\)0.13 eV. The tunneling transmission coefficient for each eigenstate can be computed using the analytical expression for a parabolic barrier [48]:  
\[T_{par}^{\text{OM}}(E)=\Big{(}1+\exp\Bigl{(}-\tfrac{2\pi E}{\omega_{b}}\Bigr{)} \Bigr{)}^{-1},\ \ \ \omega_{b}=\sqrt{\tfrac{k_{b}}{m}}. \tag{3}\]  
The thermal reaction rate for hydrogen diffusion for a given pathway is then computed from the following expression:  
Figure 3: Hydrogen probability densities for the int-int diffusion hop. Probability densities are shifted upward by their eigenvalue and colored for the sake of clarity.