Structural properties of transition-metal clusters via force-biased Monte Carlo and ab initio calculations: A comparative study

We present a force-biased Monte Carlo (FMC) method for structural modeling of the transition-metal clusters of Fe, Ni, and Cu with sizes of 13, 30, and 55 atoms. By employing the Finnis-Sinclair potential for Fe and the Sutton-Chen potential for Ni and Cu, the total energy of the clusters is minimized using the local gradient of the potentials in Monte Carlo simulations. The structural configurations of the clusters, obtained from the biased Monte Carlo approach, are analyzed and compared with the same configurations from the Cambridge Cluster Database (CCD) upon relaxation of the clusters using the first-principles density-functional code nwchem. The results show that the total-energy value and the structure of the FMC clusters are essentially identical to the corresponding value and the structure of the CCD clusters. A comparison of the nwchem-relax FMC and CCD structures is presented by computing the pair-correlation function, the bond-angle distribution, the coordination number of the first-coordination shell, and the Steinhardt bond-orientational order parameter, which provide information about the two- and three-body correlation functions, the local bonding environment of the atoms, and the geometry of the clusters. An atom-by-atom comparison of the FMC and CCD clusters is also provided by superposing one set of clusters onto another, and the electronic properties of the clusters are addressed by computing the density of electronic states.

Figure 1

The evolution of the total energy of an ${\mathrm{Fe}}_{55}$ cluster in the classical FMC and MC simulations. (a) Total energy versus CPU time. (b) Total energy versus MCS.

Figure 2

The putative ground-state structures of 13-atom transition-metal clusters from a joint FMC-nwchem simulation: (a) ${\mathrm{Fe}}_{13}$ (icosahedron), (b) ${\mathrm{Cu}}_{13}$ (bilayer 2) with a buckled hexagonal layer (yellow), and (c) ${\mathrm{Ni}}_{13}$ (icosahedron). See Table 4 for the ground-state energy of the structures and the possible low-energy isomers of ${\mathrm{Cu}}_{13}$.

Figure 3

The pair-correlation functions of (a) ${\mathrm{Fe}}_{30}$, (b) ${\mathrm{Ni}}_{30}$, and (c) ${\mathrm{Cu}}_{30}$ clusters obtained from ab initio relaxations of FMC (blue) and CCD (red) structures using the density-functional code nwchem.

Figure 4

The distribution of the nearest-neighbor bond angles for (a) ${\mathrm{Fe}}_{30}$, (b) ${\mathrm{Ni}}_{30}$, and (c) ${\mathrm{Cu}}_{30}$ clusters. The results for the FMC and CCD structures, after relaxation using nwchem, are shown in blue and red, respectively.

Figure 5

Histograms showing the coordination numbers of the first-shell atoms in (a) ${\mathrm{Fe}}_{30}$, (b) ${\mathrm{Ni}}_{30}$, and (c) ${\mathrm{Cu}}_{30}$ clusters. The FMC-nwchem and CCD-nwchem configurations are indicated in the plots.

Figure 6

Bond-orientational order parameters $Q_l$ for (a) ${\mathrm{Fe}}_{30}$, (b) ${\mathrm{Ni}}_{30}$, and (c) ${\mathrm{Cu}}_{30}$ clusters for several values of $l$. The results for the FMC-nwchem and CCD-nwchem structures are shown in blue and red, respectively.

Figure 7

Geometry of ${\mathrm{Fe}}_{30}$, ${\mathrm{Ni}}_{30}$, and ${\mathrm{Cu}}_{30}$ clusters obtained from ab initio nwchem relaxations of the FMC (top panel) and CCD (bottom panel) clusters. For comparison, each configuration was subjected to a translation and appropriate rotations, as described in the text.

Figure 8

The structure of 55-atom Fe, Ni, and Cu clusters obtained from the ab initio nwchem relaxation starting from the FMC and CCD configurations. The FMC (top panel) and CCD (bottom panel) clusters are shown in dark red and light blue, respectively. Each configuration was subjected to a translation and rotations for comparison.

Figure 9

Electronic densities of states of 30-atom Fe, Ni, and Cu clusters. The results for the FMC and CCD clusters are shown in blue and red, respectively. The highest-occupied energy level is shown as a dashed vertical line (black) at 0 eV. For the purpose of comparison, we have broadened the eigenvalue distributions using a Gaussian function with a broadening parameter of 0.3 eV.