Image states in metal clusters

The existence of image states in small clusters is shown, using a quantum-mechanical many-body approach. We present image state energies and wave functions for spherical jellium clusters up to 186 atoms, calculated in the $GW$ approximation, where $G$ is the Green’s function and $W$ is the dynamically screened Coulomb interaction, which by construction contains the dynamic long-range correlation effects that give rise to image effects. In addition, we find that image states are also subject to quantum confinement. To extrapolate our investigations to clusters in the mesoscopic size range, we propose a semiclassical model potential, which we test against our full $GW$ results.

Figure 1

The classical image potential, $v_{im}^c\left(r\right)$ [Eq. (1)], of a solid sphere with $\epsilon =1000$ becomes more accommodating with increasing radius $R_c$ (in a.u.) and is bound by the planar surface (solid line to the right) and the atomic limit (solid line to the left).

Figure 2

For each of the clusters ${\mathrm{Na}}_{138}$ and ${\mathrm{Na}}_{186}$, a loosely bound image state is found that predominantly resides in the vacuum region outside the cluster and has very little overlap with the cluster region. In the LDA, neither an image state nor a bound state with the same quantum numbers can be obtained. (The energies are referenced with respect to the vacuum and the gray box marks the extent of the clusters.)

Figure 3

The highest image state (solid line) in ${\mathrm{Na}}_{34}$ becomes more tightly bound and more strongly localized with increasing cluster size, ${\mathrm{Na}}_{40}$, and eventually evolves into a surface resonance in ${\mathrm{Na}}_{58}$. For comparison, the LDA wave functions have been included (dashed lines).

Figure 4

The model potential of Eq. (3) of a ${\mathrm{Na}}_{138}$ cluster with $\epsilon =1000$ (solid line) crosses over with the exponentially decaying LDA $v_{xc}$ potential (dashed line) and follows the inverse power-law decay of the classical image potential [Eq. (1)] (dotted line) by construction.