several chemical processes such as manufacturing of certain silicon rubbers [7] and
the reduction of benzene to cyclohexane [8]. Rhodium also finds use in the jewelry
industry and as an agent for hardening and improving corrosion resistance [9].
The main problem to obtain a deep atomic-level understanding of the physical
and chemical properties of clusters relies on an accurate determination of their
equilibrium atomic structure, which is not as simple as it might appear. A direct
identification of the equilibrium atomic structure by experimental techniques is very
difficult and only indirect measurements can provide few clues about the atomic
structure. Thus, the combinations of experimental techniques with first-principles
calculations have been used. For example, vibrational spectroscopies combined
with theoretical calculations have lead to important insights into the atomic structure of small Rh clusters [10, 11]. Isolated metal clusters have also been investigated by Stern-Gerlach molecular-beam deflection experiments [2, 12–17].
However, there are difficulties in the direct identification of the atomic structure
of clusters by experimental techniques. Thus, most of the structural studies have
been based on theoretical calculations, which can directly determine the atomic
structure of clusters using several well-defined algorithms. Several calculations
based on density functional theory (DFT) have focused on these clusters, for
example, on metal particles containing 13 atoms, Rh 13 [18–29]. Furthermore, it is
important to mention that few studies have focused on the search for the
lowest-energy structures with most studies assuming predefined structures.
Sophisticated algorithms have been employed in the search for the lowest-energy
structure, namely, generic algorithms (GA) [30], basin-hopping Monte Carlo
(BHMC) [31–34], Monte Carlo (MC) [35], conformational space annealing [29],
taboo search in descriptor space (TSDE) [23, 36], high-temperature molecular
Dynamics (high-T-MD) [22]. Almost all the studies with these algorithms have
been used in combination with empirical pair potentials. These potentials have
difficulties in providing a correct description of the atomic structure [37–39], and
hence, the ground state structures might not be correct.
In this paper, we present HF and DFT calculations on clusters with 6 and 8
rhodium atoms for comparison with theoretical and experimental results. We also
present results for the 13-atom cluster since it is one of the most studied clusters and
to the best of our knowledge there are no experimental results.
2 Method and Computational Details
It is well known that the method of calculation and the chosen basis set are the two
most important factors in determining the accuracy of results. Ab initio methods
must represent all the electrons in some manner. However, for heavy atoms it is
desirable to reduce the amount of computation burden. One way to do this is by
replacing the core electrons and their basis functions in the wave function by a
potential term in the Hamiltonian. These are called core potentials, effective core
potentials (ECP) or relativistic effective core potentials (RECP). In this work, we
214
M. A. Mora and M. A. Mora-Ramírez
the reduction of benzene to cyclohexane [8]. Rhodium also finds use in the jewelry
industry and as an agent for hardening and improving corrosion resistance [9].
The main problem to obtain a deep atomic-level understanding of the physical
and chemical properties of clusters relies on an accurate determination of their
equilibrium atomic structure, which is not as simple as it might appear. A direct
identification of the equilibrium atomic structure by experimental techniques is very
difficult and only indirect measurements can provide few clues about the atomic
structure. Thus, the combinations of experimental techniques with first-principles
calculations have been used. For example, vibrational spectroscopies combined
with theoretical calculations have lead to important insights into the atomic structure of small Rh clusters [10, 11]. Isolated metal clusters have also been investigated by Stern-Gerlach molecular-beam deflection experiments [2, 12–17].
However, there are difficulties in the direct identification of the atomic structure
of clusters by experimental techniques. Thus, most of the structural studies have
been based on theoretical calculations, which can directly determine the atomic
structure of clusters using several well-defined algorithms. Several calculations
based on density functional theory (DFT) have focused on these clusters, for
example, on metal particles containing 13 atoms, Rh 13 [18–29]. Furthermore, it is
important to mention that few studies have focused on the search for the
lowest-energy structures with most studies assuming predefined structures.
Sophisticated algorithms have been employed in the search for the lowest-energy
structure, namely, generic algorithms (GA) [30], basin-hopping Monte Carlo
(BHMC) [31–34], Monte Carlo (MC) [35], conformational space annealing [29],
taboo search in descriptor space (TSDE) [23, 36], high-temperature molecular
Dynamics (high-T-MD) [22]. Almost all the studies with these algorithms have
been used in combination with empirical pair potentials. These potentials have
difficulties in providing a correct description of the atomic structure [37–39], and
hence, the ground state structures might not be correct.
In this paper, we present HF and DFT calculations on clusters with 6 and 8
rhodium atoms for comparison with theoretical and experimental results. We also
present results for the 13-atom cluster since it is one of the most studied clusters and
to the best of our knowledge there are no experimental results.
2 Method and Computational Details
It is well known that the method of calculation and the chosen basis set are the two
most important factors in determining the accuracy of results. Ab initio methods
must represent all the electrons in some manner. However, for heavy atoms it is
desirable to reduce the amount of computation burden. One way to do this is by
replacing the core electrons and their basis functions in the wave function by a
potential term in the Hamiltonian. These are called core potentials, effective core
potentials (ECP) or relativistic effective core potentials (RECP). In this work, we
214
M. A. Mora and M. A. Mora-Ramírez