Nonlinear optical response of multiply ionized noble-gas atoms

Calculation of dynamic polarizabilities and hyperpolarizabilities of ionized species using \textit{ab initio} methods presents computational and conceptual difficulties, as these ionized species often have open-shell electronic system. We use multi-configurational self-consistent field (MCSCF) method with extended basis sets for calculating dynamic polarizability and second-order hyperpolarizabilities of atomic noble gases and their multiply charged cations in non-resonant regime. The calculations were performed at wavelengths ranging from about 100 nm to the red of the first multi-photon resonance all the way toward the static regime. The results were benchmarked to those of CCSD calculations for ions of even-number charge. The second-order hyperpolarizability coefficients were found to decrease when the electrons are progressively removed from the system. At higher ionization states, these coefficients become less dispersive as a function of wavelength. The values and even the signs of the $\gamma^{(2)}$ coefficients were found to depend on the spin of the ionic quantum state. Thus, for Ne$^{\mathrm{+3}}$ and Ne$^{\mathrm{+4}}$, in low-spin states ($^{\mathrm{2}}$P$_{u}$, and $^{\mathrm{1}}$S$_{g}$, respectively) the sign of $\gamma^{(2)}$is positive, whereas in high-spin states ($^{\mathrm{4}}$S$_{u}$, and $^{\mathrm{3}}$P$_{g})$ the sign is negative. The calculated hyperpolarizabilities of multiply ionized atoms relate to experiments on very bright high-order harmonic generation in multiply ionized plasmas (D.~Popmintchev \textit{et al.}, \textit{Science}, \textbf{350} (6265), 1225 (2015)).