use the ECP LANL2DZ [40–42] potential, which is one of the most widely used for
heavy elements.
Another factor to consider in selecting the method of calculation is the spin
contamination. A high spin contamination can affect the geometry and population
analysis and significantly affect the spin density. The error introduced by spin
contamination is unacceptable when systems with transition metals are investigated.
The restricted open-shell Hartree-Fock (ROHF) ab initio method is one of the best
ways to avoid spin contamination and obtain a reliable wave function.
We performed ab initio molecular orbital calculations with full optimization at
the ROHF level with a LANL2DZ basis set. Also in this research we employed the
B3LYP hybrid functional with the small-core quasirelativistic approach of Wood
and Boring [43] MWB ECP, and, the General Gradient Approximation
(GGA) formulated by Perdew, Burcke, and Hernzerhof (PBE) [44, 45] since it has
been shown that these hybrid functional can yield reliable energetics and structural
results for other metal compounds. The geometries were adjusted until a stationary
point on the potential energy surface was found, using the Berny algorithm [46, 47]
for the minimization.
3 Results and Discussion
Figure 1 shows the Rh 6 optimized geometry calculated with the B3LYP/MWB
chemical model obtained from different initial geometries; Fig. 1a shows the final
geometry obtained from an initial octahedron, Fig. 1b is a final triangular prism
conformer, Fig. 1c is a deformed pentagonal pyramid which converges to an distorted trigonal prism or capped tetrahedron as in Fig. 1d.
Table 1 shows the relative energy (eV) of the Rh 6 cluster obtained with the
B3LYP/MWB chemical model; we searched the spin multiplicity up to 23. The
lowest energy isomer is an octahedron with a multiplicity equal to 9, followed by
the distorted pentagonal pyramid with a multiplicity of 11. Finally the trigonal
prism isomer has M = 9. The minimum energy isomer is in complete agreement
with the experimental results [10, 48], and with previous theoretical results [21, 24,
25, 48–56]. The relative energy of ROHF-isomers with different geometry is 1.039,
3.973 eV for pentagonal pyramid and octahedron respectively. For the
B3LYP-isomers with different energy the relative energy is 0.218 for pentagonal
pyramid, and 0.490 eV for trigonal prism. With the ROHF/LANL2DZ method, the
triangular prism has a minimum energy conformation with multiplicity equal to 15.
The more stable pentagonal pyramid is when M = 17, and the square bi-pyramid
with M = 11. A geometry formed by two perpendicular squares (an incomplete
cube) with M = 15 is the fundamental state for this particular geometry. Among
these four geometries, the minimum energy is the triangular prism. All calculations
in this series were performed with the ROHF/LANL2DZ methods and with multiplicities from 1 to 23 and full optimization.
Small Rhodium Clusters: A HF and DFT Study–III
215
heavy elements.
Another factor to consider in selecting the method of calculation is the spin
contamination. A high spin contamination can affect the geometry and population
analysis and significantly affect the spin density. The error introduced by spin
contamination is unacceptable when systems with transition metals are investigated.
The restricted open-shell Hartree-Fock (ROHF) ab initio method is one of the best
ways to avoid spin contamination and obtain a reliable wave function.
We performed ab initio molecular orbital calculations with full optimization at
the ROHF level with a LANL2DZ basis set. Also in this research we employed the
B3LYP hybrid functional with the small-core quasirelativistic approach of Wood
and Boring [43] MWB ECP, and, the General Gradient Approximation
(GGA) formulated by Perdew, Burcke, and Hernzerhof (PBE) [44, 45] since it has
been shown that these hybrid functional can yield reliable energetics and structural
results for other metal compounds. The geometries were adjusted until a stationary
point on the potential energy surface was found, using the Berny algorithm [46, 47]
for the minimization.
3 Results and Discussion
Figure 1 shows the Rh 6 optimized geometry calculated with the B3LYP/MWB
chemical model obtained from different initial geometries; Fig. 1a shows the final
geometry obtained from an initial octahedron, Fig. 1b is a final triangular prism
conformer, Fig. 1c is a deformed pentagonal pyramid which converges to an distorted trigonal prism or capped tetrahedron as in Fig. 1d.
Table 1 shows the relative energy (eV) of the Rh 6 cluster obtained with the
B3LYP/MWB chemical model; we searched the spin multiplicity up to 23. The
lowest energy isomer is an octahedron with a multiplicity equal to 9, followed by
the distorted pentagonal pyramid with a multiplicity of 11. Finally the trigonal
prism isomer has M = 9. The minimum energy isomer is in complete agreement
with the experimental results [10, 48], and with previous theoretical results [21, 24,
25, 48–56]. The relative energy of ROHF-isomers with different geometry is 1.039,
3.973 eV for pentagonal pyramid and octahedron respectively. For the
B3LYP-isomers with different energy the relative energy is 0.218 for pentagonal
pyramid, and 0.490 eV for trigonal prism. With the ROHF/LANL2DZ method, the
triangular prism has a minimum energy conformation with multiplicity equal to 15.
The more stable pentagonal pyramid is when M = 17, and the square bi-pyramid
with M = 11. A geometry formed by two perpendicular squares (an incomplete
cube) with M = 15 is the fundamental state for this particular geometry. Among
these four geometries, the minimum energy is the triangular prism. All calculations
in this series were performed with the ROHF/LANL2DZ methods and with multiplicities from 1 to 23 and full optimization.
Small Rhodium Clusters: A HF and DFT Study–III
215