Within the hypothesis that DC ¼ C d is the diffusion related broadening, its
variation on top of the much larger, and constant “friction” related broadening must
be explained by the occurrence of the very first degeneracy lifts of eigenstates far
below the barrier for diffusion. These splittings are due to the tunneling motion of
the adsorbates, and hence, the variation of the diffusion broadening with q cannot
really be simulated by classical mechanics.
Note that the full 6D treatment should be closest to Model 3 from the present
study. We could henceforth conclude that the expected effective barrier for the
diffusion of CO on Cu(100) should be about 30 meV, which is indeed the value
estimated by Toennies and Graham [18].
The hypothesis presented in this section can be rationalized both within a kinetic
model and from a statistical theory of the quantum dynamics, the details of which is
the object of separate works.
3.2 H/Pd(111) System
While the extension of the aforementioned study on the CO/Cu(100) system to 6
dimensions is in progress, we can report here on the full 3 dimensional study of the
H/Pd(111) system and check, whether the aforementioned hypothesis is correct.
The section of the potential energy surface for this system is represented in Fig. 2.
One clearly sees the stable adsorption sites, “fcc” and “hcp”, which are also indicated in the scheme of Fig. 3. On this PES, the hcp site is about 190 hc cm
−1 less
stable than the fcc site and the barrier between the two sites is at about
1,150 hc cm
−1 above the fcc site.
There are 4 fcc and 4 hcp sites per unit cell. The hcp/fcc occupation ratio is about
0.37 at room temperature, and we can therefore assume that the occupation of sites
is approximately homogeneous, which makes the present study mimic a coverage
degree of 12.5 %.
Figure 8 shows the form of Sðq; EÞ for this system, when C i ¼ 1 meV is
assumed. As for the CO/Cu(100) system, the variation of the width is very feeble on
the scale of the intrinsic broadening, but nicely structured when magnified. Figure 9
shows this variation in terms of the corresponding diffusion rate, defined here as
a ¼ pDC=h % 0:7596 ps
À1
 DC=meV, where DC ¼ C À C i . There is currently no
experimental result for this function. We may compare the present theoretical result,
however, with experimental results for systems that should be rather similar, i.e.
H/Ru(0001) [2, Fig. 1 therein], and H/Pt(111) [1]: the general behavior of the rate
function shown in Fig. 9 reproduces qualitatively very well the form of the
experimental functions; however, the variation range for the rate is a factor
10 smaller than that observed for H/Ru(0001), and a factor 100 smaller than what is
observed for H/Pt(111). Also, the dip occurring at q % 1:6 ˚
A
À1 cannot possibly be
related to the “de Gennes” narrowing of the quasi-elastic broadening [19], which
would be expected to be around 1.14 Å
−1 for palladium.
Full Quantum Calculations of the Diffusion Rate of Adsorbates
189
variation on top of the much larger, and constant “friction” related broadening must
be explained by the occurrence of the very first degeneracy lifts of eigenstates far
below the barrier for diffusion. These splittings are due to the tunneling motion of
the adsorbates, and hence, the variation of the diffusion broadening with q cannot
really be simulated by classical mechanics.
Note that the full 6D treatment should be closest to Model 3 from the present
study. We could henceforth conclude that the expected effective barrier for the
diffusion of CO on Cu(100) should be about 30 meV, which is indeed the value
estimated by Toennies and Graham [18].
The hypothesis presented in this section can be rationalized both within a kinetic
model and from a statistical theory of the quantum dynamics, the details of which is
the object of separate works.
3.2 H/Pd(111) System
While the extension of the aforementioned study on the CO/Cu(100) system to 6
dimensions is in progress, we can report here on the full 3 dimensional study of the
H/Pd(111) system and check, whether the aforementioned hypothesis is correct.
The section of the potential energy surface for this system is represented in Fig. 2.
One clearly sees the stable adsorption sites, “fcc” and “hcp”, which are also indicated in the scheme of Fig. 3. On this PES, the hcp site is about 190 hc cm
−1 less
stable than the fcc site and the barrier between the two sites is at about
1,150 hc cm
−1 above the fcc site.
There are 4 fcc and 4 hcp sites per unit cell. The hcp/fcc occupation ratio is about
0.37 at room temperature, and we can therefore assume that the occupation of sites
is approximately homogeneous, which makes the present study mimic a coverage
degree of 12.5 %.
Figure 8 shows the form of Sðq; EÞ for this system, when C i ¼ 1 meV is
assumed. As for the CO/Cu(100) system, the variation of the width is very feeble on
the scale of the intrinsic broadening, but nicely structured when magnified. Figure 9
shows this variation in terms of the corresponding diffusion rate, defined here as
a ¼ pDC=h % 0:7596 ps
À1
 DC=meV, where DC ¼ C À C i . There is currently no
experimental result for this function. We may compare the present theoretical result,
however, with experimental results for systems that should be rather similar, i.e.
H/Ru(0001) [2, Fig. 1 therein], and H/Pt(111) [1]: the general behavior of the rate
function shown in Fig. 9 reproduces qualitatively very well the form of the
experimental functions; however, the variation range for the rate is a factor
10 smaller than that observed for H/Ru(0001), and a factor 100 smaller than what is
observed for H/Pt(111). Also, the dip occurring at q % 1:6 ˚
A
À1 cannot possibly be
related to the “de Gennes” narrowing of the quasi-elastic broadening [19], which
would be expected to be around 1.14 Å
−1 for palladium.
Full Quantum Calculations of the Diffusion Rate of Adsorbates
189