non-orthogonality of the coordinates, expressions for scalar products are more
involved. In particular the kinetic energy operator becomes
^
T ¼ À
2h
2
12p 2 m
@
2
@x 2 þ
@
2
@y 2 þ
@
2
@x@y
À
h
2
8p 2 m
@
2
@z 2
ð7Þ
where h is the Planck constant and m = 1.007825 Da is the mass of the hydrogen
atom used in this work. We shall evaluate Eq. (4) for a momentum transfer along
the surface vector linking two nearest Pd neighbors (which corresponds to the
h11 20i crystallographic direction in hexagonal closed packed cells). In this direction the scalar product is qx ¼ qx c ¼ qðx À y=2Þ. Details related to the calculations
for the H/Pd(111) system will be reported elsewhere.
3 Results
3.1 CO/Cu(100) System
3.1.1 Eigenvalues
Table 1 reproduces the eigenvalues obtained with the different models. All eigenvalues can be collected in bands, which are levels of nearly degenerate states. The
following is to be noted:
1. The lowest eigenvalues are (almost) 3-fold degenerate, which means that there
are 3 (almost) uncoupled states per energy which can hence be localized in each
of the 3 potential wells of the grid considered here. The bandwidth related to
these states is (almost) zero.
2. Degeneracy starts to be lifted quite readily for Model 1, due to the coupling
between the wells, at the level with about 27 hc cm
−1 . The width of the corresponding band becomes 8 × 10
−5 hc cm
−1
% 0:01 leV. The next largest degeneracy lift is for the level at 44.5 hc cm
−1 : 4 × 10
−4 hc cm
−1
% 0:05 leV. The next
degeneracy lift occurs at the level at 60.4 hc cm
−1 : 8 × 10
−3 hc cm
−1
% 1 leV.
3. For model Model 2, a similar first degeneracy lift occurs at the level at
196.98 hc cm
−1 , and for Model 3 at the level at 74.13 hc cm
−1 . For all models,
this degeneracy lift, which is potentially the first to be related to a broadening of
the dynamical structure factor Sðq; EÞ beyond any intrinsic broadening C i ,
occurs at levels located well below the diffusion barrier.
4. For Model 1, levels at or above the barrier (100 hc cm
À1 ) turn rapidly into the
level structure of a periodic potential, with two degenerate states per level. For
Model 2 and Model 3, this feature is somewhat quenched.
Full Quantum Calculations of the Diffusion Rate of Adsorbates
183
involved. In particular the kinetic energy operator becomes
^
T ¼ À
2h
2
12p 2 m
@
2
@x 2 þ
@
2
@y 2 þ
@
2
@x@y
À
h
2
8p 2 m
@
2
@z 2
ð7Þ
where h is the Planck constant and m = 1.007825 Da is the mass of the hydrogen
atom used in this work. We shall evaluate Eq. (4) for a momentum transfer along
the surface vector linking two nearest Pd neighbors (which corresponds to the
h11 20i crystallographic direction in hexagonal closed packed cells). In this direction the scalar product is qx ¼ qx c ¼ qðx À y=2Þ. Details related to the calculations
for the H/Pd(111) system will be reported elsewhere.
3 Results
3.1 CO/Cu(100) System
3.1.1 Eigenvalues
Table 1 reproduces the eigenvalues obtained with the different models. All eigenvalues can be collected in bands, which are levels of nearly degenerate states. The
following is to be noted:
1. The lowest eigenvalues are (almost) 3-fold degenerate, which means that there
are 3 (almost) uncoupled states per energy which can hence be localized in each
of the 3 potential wells of the grid considered here. The bandwidth related to
these states is (almost) zero.
2. Degeneracy starts to be lifted quite readily for Model 1, due to the coupling
between the wells, at the level with about 27 hc cm
−1 . The width of the corresponding band becomes 8 × 10
−5 hc cm
−1
% 0:01 leV. The next largest degeneracy lift is for the level at 44.5 hc cm
−1 : 4 × 10
−4 hc cm
−1
% 0:05 leV. The next
degeneracy lift occurs at the level at 60.4 hc cm
−1 : 8 × 10
−3 hc cm
−1
% 1 leV.
3. For model Model 2, a similar first degeneracy lift occurs at the level at
196.98 hc cm
−1 , and for Model 3 at the level at 74.13 hc cm
−1 . For all models,
this degeneracy lift, which is potentially the first to be related to a broadening of
the dynamical structure factor Sðq; EÞ beyond any intrinsic broadening C i ,
occurs at levels located well below the diffusion barrier.
4. For Model 1, levels at or above the barrier (100 hc cm
À1 ) turn rapidly into the
level structure of a periodic potential, with two degenerate states per level. For
Model 2 and Model 3, this feature is somewhat quenched.
Full Quantum Calculations of the Diffusion Rate of Adsorbates
183