Nonlinear optical responses of multiply ionized noble gases: Dispersion and spin multiplicity effects

Dynamic second-order hyperpolarizabilities of atomic noble gases and their multiply ionized ions are computed using ab initio multiconfigurational self-consistent field cubic response theory. For each species, the calculations are performed at wavelengths ranging from the static regime to those about 100 nm above the first multiphoton resonance. The second-order hyperpolarizability coefficients progressively decrease as the electrons are removed from the system, in qualitative agreement with phenomenological calculations. In higher ionization states, the resulting nonlinear refractive index becomes less dispersive as a function of wavelength. At each ionization stage, the sign of the optical response depends on the number of electrons in the system and, if multiple state symmetries are possible, on the spin of the particular quantum state. Thus, for $\mathrm{N}{\mathrm{e}}^{3+}$ and $\mathrm{N}{\mathrm{e}}^{4+}$, the hyperpolarizability coefficients in the low-spin states (${}^{2}P_u$, and ${}^{1}S_g$, respectively) are positive, while in the high-spin states (${}^{4}S_u$, and ${}^{3}P_g$) they are negative. However, for doubly, triply, and quadruply charged Ar and Kr these coefficients do not undergo a sign change.

Figure 1

The DFWM coefficient for neutral He and $\mathrm{H}{\mathrm{e}}^{1+}$ at 800 nm. For a neutral atom, the calculations were performed using both CCSD and MCSCF methods, and for the ion, using MCSCF. The t-aug-cc-pV5Z basis set was used in all calculations.

Figure 2

The DFWM coefficient for Ne atom and ions at 800 nm as a function of active-space size. For neutral Ne and for $\mathrm{N}{\mathrm{e}}^{6+}$, the calculations were performed using both CCSD and MCSCF methods; for the rest of the ions, the MCSCF approach was used. The symmetries used for calculations of Ne, $\mathrm{N}{\mathrm{e}}^{1+},\mathrm{N}{\mathrm{e}}^{2+},\mathrm{N}{\mathrm{e}}^{3+},\mathrm{N}{\mathrm{e}}^{4+},\mathrm{N}{\mathrm{e}}^{5+}$, and $\mathrm{N}{\mathrm{e}}^{6+}$ are ${}^{1}S_g,{}^{2}P_u,{}^{3}P_g,{}^{4}S_u,{}^{3}P_g,{}^{2}P_u$, and ${}^{1}S_g$, respectively. The t-aug-cc-pV5Z basis set was used in all calculations.

Figure 3

The DFWM coefficient for Ar atom and ions at 800 nm as a function of active-space size. For neutral Ar and for $\mathrm{A}{\mathrm{r}}^{6+}$, the calculations were performed using both CCSD and MCSCF methods; for the rest of the ions, the MCSCF approach was used. The symmetries used for calculations of Ar, $\mathrm{A}{\mathrm{r}}^{1+},\mathrm{A}{\mathrm{r}}^{2+},\mathrm{A}{\mathrm{r}}^{3+},\mathrm{A}{\mathrm{r}}^{4+},\mathrm{A}{\mathrm{r}}^{5+}$, and $\mathrm{A}{\mathrm{r}}^{6+}$ are ${}^{1}S_g,{}^{2}P_u,{}^{3}P_g,{}^{4}S_u,{}^{3}P_g,{}^{2}P_u$, and ${}^{1}S_g$, respectively. The t-aug-cc-pV5Z basis set was used in all calculations.

Figure 4

The DFWM coefficient for a Kr atom and ions at 800 nm as a function of active-space size. For neutral Kr and for $\mathrm{K}{\mathrm{r}}^{6+}$, the calculations were performed using both CCSD and MCSCF methods; for the rest of the ions, the MCSCF approach was used. The symmetries used for calculations of Kr, $\mathrm{K}{\mathrm{r}}^{1+},\mathrm{K}{\mathrm{r}}^{2+},\mathrm{K}{\mathrm{r}}^{3+},\mathrm{K}{\mathrm{r}}^{4+},\mathrm{K}{\mathrm{r}}^{+}$, and $\mathrm{K}{\mathrm{r}}^{6+}$ are ${}^{1}S_g,{}^{2}P_u,{}^{3}P_g,{}^{4}S_u,{}^{3}P_g,{}^{2}P_u$, and ${}^{1}S_g$, respectively. The t-aug-cc-pV5Z basis set was used in all calculations.

Figure 5

The DFWM coefficient for a neutral Ne atom and ions at 800 nm, calculated for the various allowed symmetries. The calculations were performed using CAS (8,12), CAS (5,7), CAS (4,12), CAS (3,12), CAS (4,13), CAS (3,13), and CAS (2,4) for Ne, $\mathrm{N}{\mathrm{e}}^{1+},\mathrm{N}{\mathrm{e}}^{2+},\mathrm{N}{\mathrm{e}}^{3+},\mathrm{N}{\mathrm{e}}^{4+},\mathrm{N}{\mathrm{e}}^{5+}$, and $\mathrm{N}{\mathrm{e}}^{6+}$, respectively.

Figure 6

The DFWM coefficients for neutral Ar atom and ions at 800 nm, calculated for the various allowed symmetries. The calculations were performed using CAS (8,14), CAS (7,14), CAS (6,10), CAS (5,13), CAS (4,13), CAS (3,13), and CAS (2,13) for Ar, $\mathrm{A}{\mathrm{r}}^{1+},\mathrm{A}{\mathrm{r}}^{2+},\mathrm{A}{\mathrm{r}}^{3+}{,}^{}\mathrm{A}{\mathrm{r}}^{4+}{,}^{}\mathrm{A}{\mathrm{r}}^{5+}$, and $\mathrm{A}{\mathrm{r}}^{6+}$, respectively.

Figure 7

The DFWM coefficients for a neutral Kr atom and ions at 800 nm, calculated for the various allowed symmetries. The calculations were performed using CAS (8,13), CAS (7,13), CAS (6,10), CAS (5,10), CAS (4,9), CAS (3,13), and CAS (2,9) for Kr, $\mathrm{K}{\mathrm{r}}^{1+},\mathrm{K}{\mathrm{r}}^{2+},\mathrm{K}{\mathrm{r}}^{3+}{,}^{}\mathrm{K}{\mathrm{r}}^{4+}{,}^{}\mathrm{K}{\mathrm{r}}^{5+}$, and $\mathrm{K}{\mathrm{r}}^{6+}$, respectively.

Figure 8

The DFWM coefficient for He and $\mathrm{H}{\mathrm{e}}^{1+}$ versus wavelength ranging from $\sim 100\mathrm{nm}$ above the first multiphoton resonance to the static regime. The calculations were performed using CAS (2,5) for He and CAS (1,5) for $\mathrm{H}{\mathrm{e}}^{1+}$.

Figure 9

The DFWM coefficient for Ne and multiply charged ions versus wavelength ranging from $\sim 100\mathrm{nm}$ above the first multiphoton resonance to the static regime. The calculations were performed using CAS (8,12), CAS (5,7), CAS (4,12), CAS (3,12), CAS (4,13), CAS (3,13), and CAS (2,4) for Ne, $\mathrm{N}{\mathrm{e}}^{1+},\mathrm{N}{\mathrm{e}}^{2+},\mathrm{N}{\mathrm{e}}^{3+}{,}^{}\mathrm{N}{\mathrm{e}}^{4+}{,}^{}\mathrm{N}{\mathrm{e}}^{5+}$, and $\mathrm{N}{\mathrm{e}}^{6+}$, respectively. The symmetries used for calculations of Ne, $\mathrm{N}{\mathrm{e}}^{1+},\mathrm{N}{\mathrm{e}}^{2+},\mathrm{N}{\mathrm{e}}^{3+},\mathrm{N}{\mathrm{e}}^{4+},\mathrm{N}{\mathrm{e}}^{5+}$, and $\mathrm{N}{\mathrm{e}}^{6+}$ are ${}^{1}S_g,{}^{2}P_u,{}^{3}P_g,{}^{4}S_u,{}^{3}P_g,{}^{2}P_u$, and ${}^{1}S_g$, respectively. In all calculations, the t-aug-cc-pV5Z basis set was used.

Figure 10

The DFWM coefficient for Ar and multiply charged ions versus wavelength ranging from ∼100 nm above the first multiphoton resonance to the static regime. The calculations were performed using CAS (8,14), CAS (7,14), CAS (6,10), CAS (5,13), CAS (4,13), CAS (3,13), and CAS (2,13) for Ar, $\mathrm{A}{\mathrm{r}}^{1+},\mathrm{A}{\mathrm{r}}^{2+},\mathrm{A}{\mathrm{r}}^{3+}{,}^{}\mathrm{A}{\mathrm{r}}^{4+}{,}^{}\mathrm{A}{\mathrm{r}}^{5+}$, and $\mathrm{A}{\mathrm{r}}^{6+}$, respectively. The symmetries used for calculations of Ar, $\mathrm{A}{\mathrm{r}}^{1+},\mathrm{A}{\mathrm{r}}^{2+},\mathrm{A}{\mathrm{r}}^{3+},\mathrm{A}{\mathrm{r}}^{4+},\mathrm{A}{\mathrm{r}}^{5+}$, and $\mathrm{A}{\mathrm{r}}^{6+}$ are ${}^{1}S_g,{}^{2}P_u,{}^{3}P_g,{}^{4}S_u,{}^{3}P_g,{}^{2}P_u$, and ${}^{1}S_g$, respectively. In all calculations, the t-aug-cc-pV5Z basis set was used.

Figure 11

The DFWM coefficient for Kr and multiply charged ions versus wavelength ranging from $\sim 100\mathrm{nm}$ above the first multiphoton resonance to the static regime. The calculations were performed using CAS (8,13), CAS (7,13), CAS (6,10), CAS (5,10), CAS (4,9), CAS (3,13), and CAS (2,9) for Kr, $\mathrm{K}{\mathrm{r}}^{1+},\mathrm{K}{\mathrm{r}}^{2+},\mathrm{K}{\mathrm{r}}^{3+},\mathrm{K}{\mathrm{r}}^{4+},\mathrm{K}{\mathrm{r}}^{5+}$, and $\mathrm{K}{\mathrm{r}}^{6+}$, respectively. The symmetries used for calculations of Kr, $\mathrm{K}{\mathrm{r}}^{1+},\mathrm{K}{\mathrm{r}}^{2+},\mathrm{K}{\mathrm{r}}^{3+},\mathrm{K}{\mathrm{r}}^{4+},\mathrm{K}{\mathrm{r}}^{5+}$, and $\mathrm{K}{\mathrm{r}}^{6+}$ are ${}^{1}S_g,{}^{2}P_u,{}^{3}P_g,{}^{4}S_u,{}^{3}P_g,{}^{2}P_u$, and ${}^{1}S_g$, respectively. In all calculations, the t-aug-cc-pV5Z basis set was used.