# Estimates of Quantum Tunneling Effects for Hydrogen Diffusion in PuO\({}_{2}\)
###### Abstract  
We detail the estimation of activation energies and quantum nuclear vibrational tunneling effects for hydrogen diffusion in PuO\({}_{2}\) based on Density Functional Theory calculations and a quantum double well approximation. We find that results are relatively insensitive to choice of exchange correlation functional. In addition, the representation of spin in the system and use of an extended Hubbard U correction has only a small effect on hydrogen point defect formation energies when the PuO\({}_{2}\) lattice  
hydrogen-containing PuO\({}_{2}\) systems. In addition, use of an onsite Hubbard-like correction (e.g., GGA + U) [14] combined with calculation of spin effects [15] is required to yield an accurate lattice constant and band gap. Collinear spin polarization calculations tend to be more computationally efficient [16], but neglect spin-orbit coupling and relativistic effects that are known to be significant for plutonium [13]. Both GGA + U and spin-orbit coupling are generally computationally cumbersome due to their creation of numerous local minima in the electron density [17; 18; 19], which make it difficult to find the ground-state of the system. The magnetic ground-state of PuO\({}_{2}\) is itself somewhat unknown, with both experiments and DFT calculations yielding a wide range of possible spin states [12; 13], and the search for the true ground-state could require placing constraints on the orbital occupations [18; 19]. In contrast, the 1s orbitals from the hydrogen have negligible overlap with the plutonium 5f electrons [20], indicating that these effects could be less significant for the H/PuO\({}_{2}\) systems of interest here. These issues make it difficult to determine basic kinetic properties of the initial hydride reactions, such as hydrogen diffusion activation barriers and hopping rates. In particular, quantum nuclear vibrational effects (i.e., zero-point energy effects and/or tunneling) could prove large for these systems. Quantum zero-point energies can cause computed hydrogen surface adsorption energies in \(\alpha\)-uranium to shift by as much as 30%[21]. Quantum tunneling has been shown to enhance hydrogen diffusion on transition metal surfaces [22] and on polycrystalline water ices by several orders of magnitude at low temperatures [23]. Recent path integral simulations have shown that quantum nuclear vibrational effects can have similar results for proton diffusion in high-pressure transition metal clathrates [24]. DFT calculations on plutonium containing materials in general are far too computationally intensive to allow for these types of calculations, which can require millions of force evaluations per diffusion pathway. Thus, plutonium hydriding studies could greatly benefit from determination of the appropriate level of DFT theory for computing hydrogen/substrate interactions, as well as a method for placing a bound on the importance of quantum nuclear vibrational effects. In this work, we attempt to reconcile these issues by first assessing different levels of theory in order to determine whether more computationally efficient DFT calculations could yield satisfactory results for hydrogen absorbed in PuO\({}_{2}\). We report on results for bulk PuO\({}_{2}\) from several different functionals and spin states, using GGA + U with either spin polarization or spin-orbit coupling in order to assess the effects of each. We then compute hydrogen absorption energies in the dilute limit and determine the relative importance of GGA + U and treatment of spin on hydrogen-PuO\({}_{2}\) interactions. Finally, we use a newly developed semi-empirical quantum Density Functional Tight Binding (DFTB) model to seed DFT calculations of the hydrogen diffusion activation energies. We use these values in a simple quantum double well formalism in order to place a bound on the quantum tunneling effects on single hydrogen diffusion in a PuO\({}_{2}\) lattice.