Comparative ab initio study of the structural, electronic, and magnetic trends of isoelectronic late $3d$ and $4d$ transition metal clusters

We present a density-functional comparative study of the electronic properties and the structural trends of the late isoelectronic $3d$ (Fe, Co, Ni) and $4d$ (Ru, Rh, Pd) free-standing transition metal clusters of 13 and 23 atoms. Different types of structures have been analyzed: compact arrangements such as icosahedral shape, open arrangements such as the cubic ones, and structures reminiscent of the crystal lattices. We have calculated total-energy differences between the structural and the spin isomers, as well as the electronic charge and spin-moment distribution within the clusters. We have found that some structural trends correlate with electronic trends, which are consistent with the spectrum of the single atoms in the $d^n{s}^1$ configuration ($n=7,8,9$ for Fe and Ru, Co and Rh, and Ni and Pd, respectively). We have also found that magnetism plays a role in the Fe clusters. The exceptions found in some clusters, which depart from the structural trend obtained for their respective $3d$ or $4d$ family, illustrate the complexity of the bond formation in transition metal systems in which the itinerant but localized $d$ electrons coexist with the delocalized $sp$ electrons in the valence states.

Figure 1

Structural shapes of the isomers of clusters with 13 atoms. We have different cubic isomers (cubic-L, cubic-T, C-BPC), icosahedral (ICO), pentagonal prism (PP), buckled biplanar (BBP), different compact biplanar (BPC-1, BPC-2), fragment of body-centered cubic (bcc), and fragment of hexagonal-closed packed (hcp).

Figure 2

Structural shapes of the isomers of clusters with 23 atoms. We have cubic isomers (cubic), equatior pentagonal prism (EQ-PP), fragment of hexagonal-closed packed (hcp), equatior icosahedron (EQ-ICO), poli icosahedron (PO-ICO), and twined icosahedron (TW-ICO) formed from two 13-atom icosahedra joined though a triangular face.

Figure 3

Comparison of the cubic isomers of 23 atoms of Ru, Rh, and Pd. Notice the deformation of the cubic ${\text{Ru}}_{23}$ and the compacity (more bonds) of the cubic ${\text{Pd}}_{23}$.

Figure 4

(Color online) Electronic levels of the $3d$ and $4d$ isolated atoms in the nonrelativistic and relativistic approximations.

Figure 5

$s$- and $d$-projected densities of states for paramagnetic ${\text{Fe}}_{13}$ and ${\text{Ru}}_{13}$ clusters with BBP and ICO structures. The energies are relative to the Fermi level.