Static electric polarizability and hyperpolarizability of sodium clusters: The case of the sodium tetramer

We have calculated the static polarizability and second hyperpolarizability of sodium tetramer using conventional ab initio and density functional theory based methods. Our best values for the mean and the anisotropy of the dipole polarizability are α¯= 548.81 and Δα=489.14 e$^{2}$a$_{0}^{2}$E$_{h}^{-1}$. The mean second hyperpolarizability is very large at γ¯=5531 × 10$^{3}$ e$^{4}$a$_{0}^{4}$E$_{h}^{-3}$. The differential-per-atom polarizability is negative but the differential-per-atom hyperpolarizability is positive. Density functional theory methods offer a rather fuzzy picture of the second hyperpolarizability of the molecule. Nevertheless, we have found that the widely used B3LYP and B3PW91 methods perform rather reliably for both the polarizability and the hyperpolarizability of Na$_{4}$.