PHYSICAL REVIEW B
VOLUME 62, NUMBER 23
15 DECEMBER 2000-I
Adsorption energies and bond lengths of adatoms at surfaces simulated by clusters
D. Geschke, S. Fritzsche, W.-D. Sepp, and B. Fricke
Fachbereich Physik, Universität Kassel, Heinrich-Plett-Strasse 40, D-34132 Kassel, Germany
S. Varga
Department of Physics, Chalmers University of Technology, S-41296 Göteborg, Sweden
J. Anton
Department of Atomic Physics, Stockholm University, Fescativägen 24, S-10405 Stockholm, Sweden
共Received 30 December 1999兲
Using a self-consistent relativistic molecular density-functional method we have calculated adsorption energies and bond lengths of adatoms on surfaces simulated by clusters within the full relativistic description.
The results indicate that our method is working so well that first big clusters can be handled in a realistic time
and second the numerical accuracy is so good that there is justified hope that adsorption sites and energies can
be calculated with improved density functionals in the future.
I. INTRODUCTION
Calculations on the electronic and geometric structure of
polyatomic systems are not only important in solid-state and
cluster physics, they are also useful in surface physics to get
a deeper insight and a better understanding of processes
which may occur on surfaces, most notably catalytic reactions and interactions of single atoms or small molecules
with surface atoms.1
The most favorite adsorption site determines which processes may occur 共charge transfer, replacements兲 and how
surface properties of a clean surface may be influenced. For
a long time, calculations of this physical situation have been
performed with different approaches.2–5 Band-structure
methods raise difficulties as soon as periodic boundary conditions are no longer applicable. On the other hand, large
cluster calculations become expensive and unrealistically
time consuming with an increasing number of atoms. However, they offer the possibility of treating the local interactions accurately. Difficulties may arise when heavy elements
are involved, because the influence of relativistic effects may
be significant.
Due to several improvements of our four-component relativistic molecular density functional code, we are now able
to calculate total energies as well as bond distances of small
molecules or highly symmetric systems such as fullerenes
with sufficient accuracy. This enables us to predict their
ground-state energies and geometries as well as the energetic
and structural differences arising from relativistic effects in
heavy elements.6,7 Consequently, we have recently started to
extend our method to apply it to larger molecules and/or
describe reactions of adatoms with atomic clusters towards
an ab initio quantum-mechanical description of moleculesurface interactions.
In this paper we present results of atom-cluster calculations. Our aim is not to reproduce experimental data of a
specific system but to present a relativistic method which can
be used in future to treat problems of more physical relevance. The aim is to show first that big clusters which simulate the solid can be handled in a realistic time, and second
that the numerical accuracy is so good that there is hope that
0163-1829/2000/62共23兲/15439共4兲/$15.00
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adsorption sites and energies can be calculated with improved density functionals in future.
Since barium is a system with dominant outer s-electron
valence character which already exhibits strong relativistic
effects, we have chosen this system, Ba at a Ba共110兲 surface,
as the first test to see if this method can actually be applied,
although there are no experimental results for this system.
Nevertheless, barium and barium compounds are widely
used in material science due to its dielectric properties, for
example in titanate thin films8 or in superconductors.9,10
There exist also some theoretical investigations of barium
fluoride in luminescent materials.11 Rosen et al. reported on
calculations of the electronic structure of BaF2共111兲 surfaces
for laser-induced emission of electrons and ions.12 The
choice of Ba at a Ba共110兲 surface is also a good starting
point because it probably will not reconstruct or relax, so we
do not have these complications in this very first test system.
II. COMPUTATIONAL
Our method is based on a fully relativistic implementation
of the density-functional discrete variational method
共DVM兲,13 which is explained at some length in Ref. 14. In
our calculations we have used the Slater X ␣ approximation
setting ␣ ⫽0.7 to account for correlation. We are aware of
the fact that the Slater X ␣ approximation is not the best xc
potential and there exist better approximations like such as
the generalized gradient approximation 共GGA兲. The results
for Au2 which are reported in Ref. 14 are much improved
with the newer and better density functionals,15 but from a
technical point of view they cannot yet be applied to such big
cluster calculations. So we still use the Slater X ␣ approximation here in order to get answers to the following questions:
Is the description of an adsorption process within this
method possible? Do we get reasonable and numerically accurate potential-energy curves and can we distinguish the
various adsorption sites?
An essential feature to perform calculations of larger molecules and clusters is the frozen core approximation 共FCA兲,
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©2000 The American Physical Society
BRIEF REPORTS
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valence
E FCA⫽
兺v
⫺ 14
⫺
⑀ v ⫺ 12
冕␳
兺
ij
⬘
v
冕␳
v
V Cv 共 ␳ v 兲 d 3 r
V xc共 ␳ c ⫹ ␳ v 兲 ⫹ 34
c
Z cj 共 Z j ⬘ ⫺ 12 Z j ⬘ 兲
i⫽ j ⬘
R j j⬘
冕兺␳
j
M
⫹
兺
m,n
m⬎n
c xc c
3
v
j V 共 ␳ ⫹␳ 兲d r
Z nZ m
.
兩 Rn ⫺Rm 兩
共3兲
⑀ v ⫽ 具 ␺ v 兩 关 t⫹V N ⫹V C ( ␳ c ⫹ ␳ v )⫹V xc( ␳ c ⫹ ␳ v ) 兴 兩 ␺ v 典
Here,
represent the one-particle energies with the external nuclear
potential V N , the Hartree potential V C , and the exchange
potential ( ␣ ⫽0.7),
FIG. 1. Potential-energy curve of an adsobed Ba atom at the top
position of a Ba共110兲 surface simulated by a nine-atom cluster.
which we have implemented here in our fully relativistic
program. Due to the fact that the binding mainly arises from
the valence electrons, one can neglect the SCF part of the
core contributions. These may further be justified by the observation that the main part of the core contributions cancels
out when examining the binding energy. Here, one has to
subtract the total energies of the system adatom plus cluster
and its fragments. The FCA is based on the decomposition of
the Hilbert space into a core 共c兲 and a valence ( v ) part.
Accordingly, the molecular wave functions can be written as
兩 ␺ 典 ⫽ 兩 ␺ c典 ⫹ 兩 ␺ v典 .
共1兲
To ensure the constrained Ritz variation of the valence orbitals, a core-valence orthogonalization is explicitly carried out
in each iteration. The transformed wave functions, after orthogonalization, can be written as
兩 ␺ c⬘ 典 ⫽ 兩 ␰ c 典 ,
兩 ␺ ⬘v 典 ⫽ 兩 ␺ c 典 ⫺S v c 兩 ␺ c 典 .
Here, S v c represents the core-valence orthogonalization matrix. For chemical applications, the atomic-core wave functions ␰ c are used to represent the core parts of the molecular
wave functions. However, this should be proven in every
case. As a simple criterion for an appropriate definition of
core and valence separation, one has to ensure that the overlap of the core states of different atoms can be neglected. As
a consequence, the ansatz 共1兲 leads to a block structure of the
Fock and overlap matrices,
H⫽
冉
H cc
H cv
H vc
H vv
冊
,
共2兲
with the core-core (H cc ), core-valence (H c v ), and valencevalence (H vv ) parts, respectively. In the calculations the core
orbitals are not varied among the SCF iterations and cause
no direct changes in the molecular 共valence兲 potential. In the
FCA, the core-valence matrix elements in the Fock matrix
are neglected. This is the real approximation within the FCA.
By treating the core densities of different atoms as point
charges, one ends up with the expression for the total
energy,16
V xc„␳ 共 r兲 …⫽⫺3 ␣
冉
3
␳ 共 r兲
8␲
冊
1/3
.
共4兲
Z cj is the core charge of atom j located at the position R j ,
while ␳ cj represents the corresponding electronic core density.
For the molecular calculations we used the C 2 v symmetry. The neutral atomic wave functions of the barium atom
were used as basis functions. The core was defined by the
1s 1/2-4d 5//2 atomic orbitals while the 5s 1/2 , 5 p 1/2 , 5p 3/2 ,
6s 1/2 , 6p 1/2 , and 6p 3/2 orbitals were considered as the active
valence states. In order to check this definition of the core
and valence space, we have performed calculations of the
Ba2 dimer, first with all electrons and second within the
FCA. We found a difference of less than 0.02 eV in the
binding energy, which indicates that our choice seems to be
appropriate. Furthermore, looking at the Mulliken occupation numbers near the equilibrium distance, we observed no
significant changes even for the 5s 1/2 orbitals, reflecting the
fact that these orbitals as well as the lower-lying atomic
states do not change their atomic character. After this we
carried out a basis optimization with respect to the total energy and found the best results for slightly ionized atomic
wave functions from Ba⫹0.2. Within this core-valence decomposition and the optimized basis set, we have calculated
the cluster with the highly accurate three-dimensional numerical multicenter integration scheme by Boerrigter et al.17
III. RESULTS AND DISCUSSION
The results of these first relativistic cluster calculations
for the adsorption energy and the adsorption site as a function of the cluster size are given in Figs. 1–3. The choice of
the system Ba at Ba共110兲 and the approximations of the density functional have been discussed above. Figures 1 and 2
show the results of Ba at Ba共110兲 for the top position and the
hollow position, respectively, as functions of the distance to
the surface. In both cases the cluster was modeled by nine
barium atoms. Relaxations of the cluster atoms were not
taken into account in this case. The shaded atoms in both
figures have to be considered as a continuation in the next
surface layer but were not included in the calculations. We
do clearly observe an energetic difference of more than 0.15
eV between the top and hollow position.
These very first results indicate that our method seems to
be working, both from a technical as well as from a numerical point of view.
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FIG. 2. Potential-energy curve of an adsobed Ba atom at the
hollow position of a Ba共110兲 surface simulated by a nine-atom cluster.
In a second step, we are now starting from the hollow
adsorption site which was found to be the most stable position for the nine-atom cluster. Our goal is to study the dependence of the atom-cluster interaction as a function of the
cluster size. By successively increasing the number of atoms
in the first layer of the simulated surface, we end up with
calculations including 14 and 18 atoms 共Fig. 3兲 which corresponds to an 8-atom and a 12-atom first layer, respectively.
This is a simple but very obvious extension of the cluster
which neglects possible contributions of additional atoms in
a second or even a third layer, reflecting the fact that the
adsorption is supposed to be local. To prove this assumption,
we have tested the inclusion of four additional atoms for the
14-atom cluster in the second layer but we did not find any
considerable difference within our accuracy of about 0.02
eV. On the other hand, this result shows the sensitivity of our
method. As one can see, the potential-energy curves near the
equilibrium distance for the 8-atom and 12-atom layer are
very close, which indicates that we have already converged
in cluster size.
The rapid convergence with respect to the cluster size for
this system Ba at Ba共110兲 may be exceptional. Probably the
reason is that the bonding is dominantly done by the isotropic outer s electrons only. For any other atomic species
where the bond character is different and anisotropic, the
convergence with respect to the cluster size may differ very
much and depend on the type of the lattice as well.18,19
Although the absolute value of the adsorption energy
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1
2
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FIG. 3. Potential-energy curves of an adsorbed Ba atom in the
hollow position at a Ba共110兲 surface as function of the number of
atoms which simulate the surface.
probably is not correct due to the simple approximation of
the exchange correlation potential, we nevertheless can conclude the following: 共i兲 such a full relativistic calculation is
possible within an acceptable computing time; 共ii兲 the numerical accuracy is so high that realistic potential-energy
curves can be calculated; 共iii兲 different surface positions can
be calculated which allow a choice of the preferred adsorption site; 共iv兲 a frozen-core calculation within the relativistic
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we can state that our method is appropriate for the description of atom-cluster interactions. As we already mentioned
above, these investigations should be considered as a first
step towards a more sophisticated method. Concerning more
state of the art potentials, we have quite recently implemented nonlocal gradient corrected exchange-correlation
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calculations soon. Furthermore, we will try to embed the
共adatom兲-cluster system in an external potential which
should simulate an infinitely extended crystal and ensure a
proper connection with the rest of the solid.
ACKNOWLEDGMENTS
D.G. gratefully acknowledges support by the Deutsche
Forschungsgemeinschaft 共DFG兲, S.V. acknowledges support
by the DFG and DAAD, and J.A. by GSI and DFG.
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