Planar and cagelike structures of gold clusters: Density-functional pseudopotential calculations

Abstract

We study why gold forms planar and cagelike clusters while copper and silver do not. We use density functional theory and norm-conserving pseudopotentials with and without a scalar relativistic component. For the exchange-correlation (xc) functional we use both the generalized gradient (GGA) and the local density (LDA) approximations. We find that planar ${\mathrm{Au}}_{n}$ structures, with up to $n=11$, have lower energy than the three-dimensional isomers only with scalar-relativistic pseudopotentials and the GGA. In all other calculations, with more than six or seven noble metal atoms, we obtain three-dimensional (3D) structures. However, as a general trend we find that planar structures are more favorable with the GGA than with the LDA. In the total energy balance, kinetic energy favors planar and cage structures, while xc energy favors 3D structures. As a second step, we construct cluster structures having only surface atoms with ${O}_{h}$, ${T}_{d}$, and ${I}_{h}$ symmetry. Then, assuming one valence electron per atom, we select those with $2(l+1{)}^{2}$ electrons (with $l$ integer), which correspond to the filling of a spherical electronic shell formed by nodeless one-electron wave functions. Using scalar relativistic GGA molecular dynamics at $T=600\phantom{\rule{0.3em}{0ex}}\mathrm{K}$, we show that the cagelike structures of neutral ${\mathrm{Au}}_{32}$, ${\mathrm{Au}}_{50}$, and ${\mathrm{Au}}_{162}$ are metastable. Finally, we calculate the static polarizability of the two lowest-energy isomers of ${\mathrm{Au}}_{n}$ clusters as a means to discriminate isomers with planar (or cagelike) geometry from those with compact structures. We also fit our data to a semiempirical relation for the size-dependent polarizability which involves the effective valence and kinetic energy components for the homogeneous and inhomogeneous electron densities. Analyzing that fit, we find that the dipole polarizability of gold clusters with planar and cagelike structures corresponds to the linear response of 1.56 delocalized valence electrons, suggesting a strong screening of the valence interactions due to the $d$ electrons.