Simultaneous Deep Tunneling and Classical Hopping for Hydrogen Diffusion on Metals

Hydrogen diffusion on metals exhibits rich quantum behavior, which is not yet fully understood. Using simulations, we show that many hydrogen diffusion barriers can be categorized into those with parabolic tops and those with broad tops. With parabolic-top barriers, hydrogen diffusion evolves gradually from classical hopping, to shallow tunneling, to deep tunneling as the temperature ($T$) decreases, and noticeable quantum effects persist at moderate $T$. In contrast, with broad-top barriers quantum effects become important only at low $T$ and the classical-to-quantum transition is sharp, at which classical hopping and deep tunneling both occur. This coexistence indicates that more than one mechanism contributes to the quantum reaction rate. The conventional definition of the classical-to-quantum crossover $T$ is invalid for the broad tops, and we give a new definition. Extending this, we propose a model to predict the transition $T$ for broad-top diffusion, providing a general guide for theory and experiment.

Figure 1

(a) Energy barrier of the H diffusion paths, obtained from NEB calculations using DFT, for several transition metal surfaces. The filled symbols show data for the conventional barriers that are parabolic near the top, and the open symbols are the data points for broad-top barriers. (b) Top view of the (100) surface. (c) Top view of the (110) surface. (d) Top view of the (111) surface. Green arrows show the diffusion paths.

Figure 2

Temperature dependence of H diffusion on metals for a conventional barrier that is parabolic near the top [Ni(111), left] and one that has a broad top [3H-SB-3H path Pd(110), right]. (a), (d) The exact thermal transmission probability $P(E)e^{-\beta E}$ (dimensionless) plotted against the incident energy $E$ for the conventional barrier and the broad-top barrier, respectively. (b), (c) The transmission action (in units of $\hslash$) defined as $-\ln (P)$, as a function of the incident energy $E$ for the parabolic-top barrier and the broad-top barrier. (e) Illustration of the peaks in the thermal transmission probability using tunneling paths represented by the Feynman PI.

Figure 3

Performance of different rate theories on 1D barriers (for H) with different shapes. (a) Rates on a parabolic-top barrier (inset). (b) Rates on a broad-top barrier (inset); the legend is the same as in (a). (c) The $W$ action [Eq. (1)] plotted against energy $E$ for the broad-top barrier in (b). The closed (open) circles show first- (second-) order saddle instantons obtained through ring-polymer instanton searches. (d) The thermal transmission probability at 40 K plotted against energy $E$ for the barrier shown in (c) obtained using the TMI theory and using the SDI theory.

Figure 4

(a) Transition temperature to deep tunneling [Eq. (2)] predicted by the model over a range of barrier parameters. The four broad-top barriers calculated with DFT in Fig. 1 are marked. (b) Comparison of the transition $T$ predicted by the model with previous experiments (filled symbols) and theoretical studies (open symbols) on Cu and Ni. Previous results are taken from STM [12], FEM [10], LOD [11], PI Monte Carlo (PIMC) calculations with an embedded atom model (EAM) potential [21], and RPMD with an EAM4 potential [24]. The PBE WKB points are from this work (Fig. 1).