Figure 1

Potential energy surfaces, in eV, for a H atom moving in the fixed atomic positions of the Fe atoms; (a) configuration in which the Fe atoms have been relaxed around a H atom in the tetrahedral reactant site; (b) the Fe atoms are relaxed around a H atom held at the saddle point position between two neighboring tetrahedral sites.Reuse & Permissions

Figure 2

Densities of states $\Omega (s_1,s_2)$ (7) for the proton moving in the potential of fixed Fe atoms as shown in Fig. 1: (a) centroid fixed at tetrahedral site; (b) centroid in the vicinity of tetrahedral site; (c) centroid fixed at the saddle point; and (d) centroid on the dividing surface in the region of the barrier top. (a) and (c) are those used in Gillan's theory in which the centroids are fixed at reactant and saddle point positions. (b) and (d) are appropriate to Voth's method; in (b) the centroid is allowed to explore phase space in the region of the tetrahedral lattice site and in (d) the centroid is confined to the Vineyard dividing surface that intersects the saddle point.Reuse & Permissions

Figure 3

Position probability density (PPD) of the hydrogen nucleus when the centroid is fixed at a tetrahedral site at a temperature of (a) $T=20$ K; (b) $T=50$ K; (c) $T=100$ K; (d) $T=200$ K; (e) $T=300$ K; and (f) $T=1000$ K. In each panel the tetrahedral site is located at the center of the image. Note that at high temperature the PPD is spherical and located close to the centroid as expected of a classical particle. At intermediate lower temperatures the PPD spreads out, but at the lowest temperature the PPD contracts due to the freezing of the proton into its lowest oscillator state; however it is no longer spherical as it “feels” the low symmetry of the tetrahedral site.Reuse & Permissions

Figure 4

Position probability density of the hydrogen nucleus when the centroid is fixed at the saddle point at a temperature of (a) $T=20$ K; (b) $T=50$ K; (c) $T=100$ K; (d) $T=200$ K; (e) $T=300$ K; and (f) $T=1000$ K. In each panel the saddle point is at the center of the image and there is a symmetry equivalent tetrahedral site at the upper left and lower right corners. Note how at low temperature the proton splits into two and even though the centroid of the chain of “beads” is held fixed at the saddle point the greatest position probability density is close to the energy minima at the tetrahedral reactant sites.Reuse & Permissions

Figure 5

Position probability density of the hydrogen nucleus when the centroid is in the vicinity of a tetrahedral site at a temperature of (a) $T=20$ K; (b) $T=50$ K; (c) $T=100$ K; (d) $T=200$ K; (e) $T=300$ K; and (f) $T=1000$ K. In each panel the tetrahedral site is located at the center of the image. In comparison to Fig. 3 the particle is more spread out even at the highest temperature. This reflects the greater degree of freedom of the PI-QTST theory compared to the method of Gillan.Reuse & Permissions

Figure 6

PPD of the hydrogen nucleus when the centroid is on the dividing surface in the region of the barrier top at a temperature of (a) $T=20$ K; (b) $T=50$ K; (c) $T=100$ K; (d) $T=200$ K; (e) $T=300$ K; and (f) $T=1000$ K. As remarked in the caption to Fig. 5 the PPD is greatly spread out compared to that in the Gillan formulation of the path integral method. In addition, especially at high temperatures it is seen that the reduced density is allowing the proton to explore the phase space along the Vineyard dividing surface, which is “perpendicular' to the reaction path. As the centroid is no longer constrained to remain at the saddle point the resulting free energy barrier is larger and the diffusivity smaller in Voth's method.Reuse & Permissions

Figure 7

The free energy needed to carry the H atom from an initial stable position to a transition state as a function of temperature, obtained using Gillan's approach. The activation energy is compared with assessments from hydrogen equilibration and permeation tests in Ref. 3.Reuse & Permissions

Figure 8

Quantum transition rate calculated by PI QTST in the temperature interval 20–1000 K.Reuse & Permissions

Figure 9

Diffusion coefficients of H in $\alpha$-Fe in the temperature range 100–1000 K calculated by PI QTST. The bands of data represent an assessment by Kiuchi and McLellan[3] of hydrogen gas equilibration experiments and measurements by Grabke and Rieke,[38] the vertical width of the band reflecting the reported error bars. The remaining lines are data from electrochemical permeation experiments, assessed by Kiuchi and McLellan[3] and measured by Nagano et al.;[6] and measurements using gas and electrochemical permeation by Hayashi et al.[5] The extent of each data set represents the temperature range over which the assessment or measurements are reported. The triangles show theoretical results from centroid molecular dynamics calculations using a classical interatomic potential and are taken from Ref. 19. Note that these are in less good agreement with experiment at high temperature compared to our fully quantum mechanical predictions and that at low temperatures the CMD method predicts diffusivities significantly larger than ours. This may reflect the difficulty in molecular dynamics canonical sampling at low temperature in contrast to the WLMC method which accesses all temperatures as a result of a single sampling.Reuse & Permissions