1
Supporting Material for
Reactive scattering of H2 from Cu(100): comparison of dynamics calculations
based on the specific reaction parameter approach to density functional theory
with experiment.
L. Sementa,1,2, a) M. Wijzenbroek,1 B.J. van Kolck,1 M.F. Somers,1 A. Al-Halabi,1 H.F. Busnengo,3
R. A. Olsen1,4 and G.J. Kroes1
1
Leiden Institute of Chemistry, Gorlaeus Laboratories, Leiden University, P.O. Box 9502, 2300
RA, Leiden, The Netherlands
2
CNR-IPCF, Istituto per i Processi Chimico-Fisici of the Consiglio Nazionale delle Ricerche,
Molecular Modeling Laboratory, I-56124 Pisa, Italy.
3
Instituto de Física Rosario (CONICET-Universidad Nacional de Rosario), A. Pellegrini 250,
(2000) Rosario, Argentina.
4
SINTEF Materials and Chemistry, P.O. Box 124 Blindern, N-0314 Oslo, Norway
M. Rutkowski,5 C. Thewes,5 N.F. Kleimeier5 and H. Zacharias5
5
Physikalisches Institut, Westfälische Wilhelms-Universität, D-48149 Münster, Germany
S.1 Theoretical Model
We study the effects of the interaction between molecules and surfaces starting from two principal
assumptions: the adiabatic evolution of the electrons (Born-Oppenheimer approximation) and a
static surface where the surface nuclei are frozen to their ideal lattice positions. In the BornOppenheimer approximation the electronic transitions due to the motion of the nuclei are neglected,
so that the molecule-surface interaction can be represented using just one potential energy surface
(PES). This assumption should be valid for H2 interacting with metallic surfaces which, like the
a
Author to whom correspondence should be addressed. Electronic mail : luca.sementa@ipcf.cnr.it
2
Cu(100) surface, do not involve a deep molecular chemisorption well H 2 in front of the barrier,
while additionally molecular hydrogen has a low affinity for electrons. 1 The second assumption
neglects energy exchange of H2 with the lattice vibrations which is reasonable considering the large
mass mismatch between hydrogen and copper atoms.2
These two assumptions reduce the dynamical variables of the molecule-surface system to the six
molecular degrees of freedom of H2. The Hamiltonian in atomic units is thus

1  2
2
2  1 2
Jˆ 2
 2 

H 



 V ( X , Y , Z , r,  ,  ) .
2 M  X
Y 2 Z 2  2 r 2 2 r 2
(S.1)
Here, X,Y and Z are the center-of-mass coordinates of the molecule, X and Y being coordinates for
motion parallel to the surface (see Fig.S1), and M is the mass of the molecule. The first term on the
RHS represents the kinetic energy in the center-of-mass motion. Furthermore (r ,  ,  ) is the vector
of the internal coordinates of the molecule, where r is the H-H distance,  the polar and  the
azimuthal angle of orientation of the molecular axis (see also Fig.S1).
Fig. S1 : Coordinate system for dissociation of H2 on a surface. In the plot, X, Y and Z are the
center-of-mass coordinates of H2, r is the H-H distance,  is the polar angle of the molecular axis
with the Z-axis, and  is the azimuthal angle.
3
Also,  is the reduced mass of the molecule (   mH2 / 2mH , mH is the mass of a hydrogen atom),
Ĵ is the angular momentum operator, and the second and third terms on the RHS represent the
kinetic energy in the internal (vibrational or dissociative and rotational) motion of the molecule.
The last term of the Hamiltonian is the potential energy of the molecule interacting with the surface
in the electronic ground state.
S.2 Potential Energy Surface.
A slab model and the DACAPO code 3 were used to compute the potential energy surface (PES) of
H2 interacting with Cu(100). Electronic densities and the metallic substrate geometry were obtained
at the DFT-level using the PW91 functional.4 The ion cores were described by ultrasoft
pseudopotentials, 5 with core cutoff radii of rcH  0.46 Å for H and of rcCu  1.16 Å for Cu. A plane
wave basis set was used for the electronic orbitals, with a cutoff energy of 400 eV. The Brillouin
zone was sampled by the Monkhorst-Pack6 method, using a set of 8  8 1 k-points. Tests with both
spin-polarized and spin-unpolarized calculations showed that spin unpolarized calculations could be
used to compute the potential energies in the whole region of the configuration space energetically
accessible for low energy molecules impinging on the surface.
Starting from bulk Cu, the fcc lattice constant has been computed and was found to be a = 3.66 Å,
which is in good agreement with the experimental Cu bulk value that is equal to 3.61 Å.7 Next, the
Cu(100) slab was built, using a 2  2 slab unit cell that consists of 4 layers. A vacuum layer of 12 Å
was placed between the slab and its image in the Z direction to avoid any possible artificial
interactions caused by the periodicity. The slab interlayer distances (initially
a / 2,
where a  3.66 Å) were then relaxed by applying the quasi-Newton (QN) method in the slab
optimization. These relaxed distances were subsequently kept frozen in the calculations of the
4
H2/Cu(100) DFT data points. The maximum force allowed in the QN optimization was set to 0.01
eV/Å. The energies of the constructed Cu slabs were converged to within 0.1 eV/(surface unit cell)
and the slab geometries (distances between layers 1 and 2, and layers 2 and 3) were converged to
within 0.01 Å, according to the results of the tests that were performed by the addition of more Cu
layers at the bottom of the 4-layer slab, going from 4 layers to 8 layers.
For dynamics calculations a global PES is required, i.e, the potential energy surface should be
available throughout configuration space. To ensure that the comparison of the computed dynamics
results to experiment reflects the quality of the electronic structure data, we used the corrugation
reducing procedure (CRP)8,9 to fit the DFT data. In this procedure the interaction of the H-atoms
with Cu(100) is subtracted from the interaction of the H2 molecule with the surface. In this way the
strong corrugation due to the repulsive interaction of the H atoms with the surface is reduced and
much smoother 2D cuts are obtained that can be easily and accurately interpolated. The subtracted
H/Cu(100) potentials, which are also interpolated by applying the CRP to the H/Cu(100) potential,
are then added to the interpolated potential to retrieve the potential for the H2/Cu(100) system:
V6 D ( X , Y , Z , r ,  ,  )  I 6 D ( X , Y , Z , r , ,  )  V3D ( X A , YA , Z A )  V3D ( X B , YB , Z B )
.
( S.2)
Here, I 6 D ( X , Y , Z , r ,  ,  ) represents the interpolated smoother correction function for the
H2/Cu(100) potential, and V3D ( X A , YA , Z A ) , V3D ( X B , YB , Z B )
represent the interpolated 3D
potentials for the H atoms A and B, each interacting with the surface in the absence of the other.
The atom-surface potential of the H/Cu(100) system has been calculated for 6 different sites: the
bridge, hollow, and top sites, and 3 additional sites that are: halfway between top and bridge (t2b),
halfway between bridge and hollow (b2h), and halfway between hollow and top (h2t). For each site,
35 points have been calculated, scanning along the Z coordinate of the H-atom for
5
 2.4  Z 10.0 a0 where Z  0.0 defines the position of the Cu(100) surface top layer. In the case of
the top site, additional data points (47 DFT points) have been calculated.
To construct the 6D PES, the H2/Cu(100) interaction energy has been calculated for 16 2D cuts in Z
and r, using 224 DFT-GGA points per 2D cut. The points of these 2D cuts were interpolated using
2D splines. The resulting I 6 D ( X , Y , Z , r ,  ,  )
freedom
as
described
in
Ref.[9].
was interpolated in the X ,Y ,  , 
These
cuts
include
four
[ ( 180 ,   0 ) , (  90 ,   0 ) , (  90 ,   45 ) , (  90 ,   90 ) ]
2D
that
degrees of
cuts
have
for
been
calculated over the bridge site. Three 2D cuts were calculated over the top and hollow site
[ ( 180 ,   0 ) , ( 90 ,  0 ) , (  90 ,   45 ) ] and six 2D cuts above the h2t site
[ ( 180 ,   0 ) , (  90 ,   45 ) , (  90 ,  135 ) , ( 135 ,   45 ) ,

( 135 ,  135 ) , and ( 135 ,   225 ) ]. For each 2D cut, the DFT-GGA calculations were

performed for 0.5  r  4.8 a0 and for 0.0  Z  8.0 a0 .
To check the quality of the interpolation, we have compared the energies of 39 DFT-GGA data
points that were not used in the process of the interpolation with the values obtained by using the
interpolated H2/Cu(100) PES. The tests showed that the fit is very successful, with a maximum
error of 30 meV in the entrance and barrier regions of the PES.
S.3 Dynamical method and calculation of the observables.
One of the set of observables we are interested in are state selected scattering probabilities
Pv j m j v ' j ' m j ' m n from the initial rovibrational state (v j m j ) to the final state (v' j ' m j ' m n) . The
quantum numbers
(v' j ' m' j ) specify the final rovibrational state and the diffraction quantum
6
numbers ( m n) specify the final translational state parallel to the surface. The Pv j m j v ' j ' m j ' m n are
extracted from a solution  (t ) of the time dependent Schrodinger equation by the time-dependent
wave packet (TDWP) method.10 Because the reaction of H2 on low index copper surfaces exhibits
normal energy scaling11,12 (at least within a good approximation13), we consider scattering at
normal incidence only, for which the collision energy equals the normal incidence energy En . This
means that the symmetry of the metal surface (C4v for Cu(100)) can be used to reduce the
computational costs14. In our method, the wave packet at time t is represented as an expansion in
real basis functions that are adapted to the symmetry of the surface unit cell and the inversion
symmetry of the molecule.15 The wave function is propagated in time using the absorbing boundary
condition Chebyshev propagator method16. The S-matrix elements Sv ' j 'm j 'm nv j m j ( E) are obtained
using the scattering amplitude formalism adapted for molecule-surface scattering.17 The squares of
these elements yield the state-to-state scattering probabilities for rovibrationally and diffractively
(in)elastic scattering Pv j m j v ' j ' m j ' m n . The summation of these probabilities over v' , j ' , m j ' , n, m and
subtraction from unity yields the initial state selective reaction probabilities Rv j m j (E) . The
degeneracy-averaged reaction probability Rv j (E ) can be obtained by averaging the Rv j m j (E) over
m j . The tests conducted indicated that the reaction and scattering probabilities presented here are
converged to within 1% with respect to the basis size and the propagation time.
To make a meaningful comparison of our reaction probabilities with the results of the sticking
measurements of Rendulic and coworkers12 who employed molecular beams, it is necessary to take
into account both the thermal populations of different rovibrational energy levels of the incident H 2
molecules and the spread in incident energies that is inherent to molecular beam experiments.18,19
To account for the different initial rovibrational states involved, we calculate the ‘monoenergetic
reaction probabilities’ R( E, Tn ) as a sum over the Rv j (E ) weighted by a Boltzmann factor
7
FvBj (Tn ) representing the population of the corresponding rovibrational level at nozzle temperature
Tn :
R( E, Tn )   FvBj (Tn ) Rv j ( E )
(S.3)
vj
F (Tn )
B
vj
(2 j  1) w( j )exp(  Evvib / kTn )exp(  Evrot
, j /( 0.8kTn ))
N
N  (2 j  1) w( j ) exp(  Evvib / kTn ) exp(  Evrot
, j /( 0.8kTn )) .
(S.4)
(S.5)
vj
Above, the vibrational energy Evvib of H2 in a (v, j ) state is defined as the internal energy of the
molecule in the (v, j  0) state, and the rotational energy Evrot
, j is the internal energy of H2 in the
(v, j ) state relative to Evvib . In the Boltzmann factors FvBj (Tn ) , we assumed the beams to have a
rotational temperature equal to 0.8 Tn and a vibrational temperature equal to the nozzle temperature
Tn .12 In Eqs. S.4 and S.5 w(j) is a weight factor describing the nuclear spin statistics of H2 and D2,
w(j) being 1 (2) for H2 (D2) with even (odd) j values, and 3 (1) for H2 (D2) with odd (even) j values.
As shown by Eq.S.3, calculation of R( E, Tn ) implies a sum over an infinite number of quantum
(v, j ) states. In practice, in our beam simulations we took into account the initial state resolved
reaction probability for (v  0, j  0  10) and (v  1, j  0  7) states only. At the maximum Tn
(1700 K) we considered, the calculated Boltzmann factors for higher exited states are smaller than
1/1000. This choice leads to convergence with respect to the number of (v, j ) states included for
the beam simulations of the system H2 + Cu(111).18
To account for the spread in incident energies we weight our R( E, Tn ) with a flux-weighted velocity
distribution given by :13
8


f ( , Tn ) d  C  3 exp  (   0 ) 2 /  2 d .
( S.6)
In Eq.S.6, C is a constant and  the velocity. Furthermore, the parameter  0 is typically called the
stream velocity, while  is a parameter that characterizes the width of the velocity distribution. The
parameters  0 and  were determined from both time-of-flight fits (method 1) and from a plot of
the hydrodynamic speed ratio as a function of incidence energy (method 2) presented in Berger’
thesis20, which are characteristic of the beams used in the experiments of Rendulic and coworkers
on H2 scattering from copper surfaces21. The procedure used is described in detail in Ref.[18]. The
sticking probabilities averaged over the energy distribution of the beam are computed from:
 
Rb ( E , Tn ) 
 
 f ( , T ) R( E, T ) d /   f ( , T ) d
n
 0
n
n
( S.7)
.
0
Since desorption is the time-reversed process to adsorption, assuming the validity of detailed
balance the calculated rotational quadrupole alignment parameter A02,v j ( E ) and the mean kinetic
energies Evmj of the molecules reacting on a surface can be compared directly with those obtained
from measurements on the desorbing flux from a surface. For a particular incident energy E and
initial rovibrational (v, j ) state, the quadrupole alignment parameter describing the alignment of the
angular momentum j of the reacting molecules can be calculated from the initial state resolved
reaction probabilities Rv j m j (E ) according to:22
 3m 2j


R
(
E
)
 1

v j mj
 j  j  1 
mj

 .
A02,v j ( E ) 
 Rv j m j ( E )
mj
( S.8)
9
Considering that the probability to find a H2 molecule in the (v, j , m j ) state, with translational
energy E along the surface normal, after desorption from a surface at temperature Ts , is directly
related to the sticking probability via:23

Pv, j ,m j ( E)  Rv, j ,m j ( E) E exp E / kTs  exp  Ev, j / kTs

( S.9)
we calculate the rotational quadrupole alignment parameters and mean kinetic energies measured in
desorption experiments, from:
 3m
2 
A0, v j ( E ) 
2
j
 
 
/ j  j  1  1
 0
mj
 
 

mj
Pv j m j ( ) G (  E ) d
( S.10)
Pv j m j ( ) G (  E ) d
0
and
E 
E 
av
vj

E 0
E 
E Rv j ( E ) exp(  E / kTs ) dE /
2
R
vj
( E ) E exp(  E / kTs ) dE .
( S.11)
E 0
In Eq.S.10, G (  E ) is a normalized Gaussian function with mean value E and standard deviation
equal to 20 meV. G (  E ) is a measure of the finite energy resolution of the experimental
apparatus measuring the desorption probabilities. The standard deviation of 20 meV chosen
correspond to the best available energy resolution in the experiments: the 10-point averaging
procedure used to obtain rotational quadrupole alignment parameters from the TOF-spectra (Fig.6a
in section 3.A) corresponds to an energy resolution of 20 meV at the lowest energies and of 100
meV at the highest energies (  1eV). The computed A02,v j ( E ) is a quantity that reveals information
on the |mj| distribution of those molecules initially in the level (v , j ) that go on to react. The
10
maximum value of the alignment parameter for a given j level, 3j/( j+1) – 1 corresponds to the
situation where only helicoptering molecules react (|mj|=j). The minimum value, -1.0, corresponds
to the situation where only cartwheeling molecules react (|mj|=0). If there is no orientational
preference for reaction the value of the alignment parameter is zero.
Equation S.11 implies that the surface acts as a heat bath that determines the population distribution
of the molecular states in desorption. It also assumes that there is no strong dependence of the PES
or the dynamics on the motion of surface atoms and on the atomic coverage. We nevertheless
accounted for some of these effects, taking advantage of the observations of Rettner et al.23 who
obtained the reaction probability curves for H2 on Cu(111) both from dissociative adsorption
measurements on a ‘cold’ (Ts = 120 K) surface and from associative desorption measurements on a
warmer surface (Ts = 925 K). According to Rettner et al.23, the effect of lowering Ts can be modeled
by decreasing the widths of the adsorption functions.
Using the sticking curves of Rettner et al.23 it has been possible to estimate by how much static
surface calculations should overestimate the average desorption energy. One can approximately
account for the static nature of the surface by using a low temperature (Ts=120 K) value of the
width parameter of the experimental reaction probability curves, while retaining the experimental
high surface temperature (925 K) in the exponential function in Eq.16. Using the width parameter
appropriate for Ts=120 K instead of the value for 925 K increases the computed mean kinetic
energies of reacting molecules by 0.07 eV for H2 in the ground vibrational state and by 0.05 eV for
H2 in the v  1 state, for H2 + Cu(111).18 We assume here that for a reasonable comparison to our
theoretical static surface results for Cu(100), the experimental results obtained at Ts = 1030 K for
Cu(100) need the same corrections, i.e., they need to be increased by the same amounts. The effect
discussed is due to the rapidly decreasing exponential weight function in Eq.S.11, which
emphasizes the contribution of molecules with low translational energy. Molecules with low
11
translational energies contribute less if the width of the reaction probability curve is decreased to
account for the static nature of the surface used in the calculations, the net effect being to increase
the average desorption energy for H2 in a specific rovibrational state.
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