A DFT + <i>U</i> Study on the Stability of Small Cu_N Clusters (<i>N</i> = 3–6 Atoms): Calculation of Phonon Frequencies

Abstract

Despite the interest in copper clusters, a consensus on their atomic structure is still lacking. The experimental observation of isolated clusters is difficult, and theoretical predictions vary widely. The latter is because one must adequately describe the closed shell of d electrons both in its short- and long-range effects. Herein, we investigate the stability of small copper clusters (CuN, N = 3–6 atoms) using spin-polarized DFT calculations under the GGA approximation, the Hubbard U correction, and the van der Waals forces. We found that the spin-polarized and vdW contributions have little effect on the binding energies of the isomers. The inclusion of U represents the most relevant contribution to the ordering of the CuN isomers, and our calculated binding energies for the clusters agreed with the experimental values. We also found that atomic relaxations alone are not enough to determine the stability of small copper clusters. It is also necessary to build the energy landscape or calculate the vibrational frequencies of the isomers. We found that the vibrational frequencies of the isomers were in the THz range and the normal modes of vibration were discrete. This approach is relevant to future studies involving isolated or supported copper clusters.

Keywords

• copper clusters
• DFT + <i>U</i>
• van der Waals
• Hubbard
• phonon frequencies