Triply excited autodissociating resonant states in the positron-helium system

In this paper, we present a complex-coordinate rotation calculation for high-lying S-wave resonances in positron-helium scattering. Highly correlated Hylleraas wave functions containing all six interparticle coordinates are used. A total of seven resonances, including their positions and widths, are reported, where the lowest one, denoted as $e^{+}\mathrm{He}\left(2s^2\right)$, is formed by a positron attaching to the doubly excited $2s^2{}^{1}S^e$ state of helium, and the other six resonances, denoted as $\mathrm{P}{\mathrm{s}}^{-}\mathrm{H}{\mathrm{e}}^{2+}$ (nS) with $n$ from 2 to 7, are located in the Rydberg series converging to the $\mathrm{P}{\mathrm{s}}^{-}+\mathrm{H}{\mathrm{e}}^{2+}$ threshold. Our results are compared with those available in the literature. The calculated energies for $\mathrm{P}{\mathrm{s}}^{-}\mathrm{H}{\mathrm{e}}^{2+}$ (nS) with $n$ from 2 to 7 show a good fit to a quantum defect formula that describes the interaction between the positively charged $\mathrm{H}{\mathrm{e}}^{2+}$ ion and the negatively charged $\mathrm{P}{\mathrm{s}}^{-}$ ion. The 3S state in this Rydberg series provides an alternative designation for the previously identified $e^{+}\mathrm{He}\left(3s^2\right)$ state in the literature.

Figure 1

Stabilized complex eigenvalue $E_{\mathrm{res}}=E_r+i{E}_i$ near the pole of the $e^{+}\mathrm{He}\left(2s^2{}^{1}S^e\right)$ resonance state, showing $E_r=-0.794892\mathrm{a}.\mathrm{u}.$ and $\Gamma /2=0.001296\mathrm{a}.\mathrm{u}.$ The set for the nonlinear parameters (α,β,γ) used in the wave function Eq. (5) is (0.95, 0.95, 0.65) for curve A and (0.95, 0.95, 0.55) for curve B.

Figure 2

Stabilized complex eigenvalue $E_{\mathrm{res}}=E_r+i{E}_i$ near the pole of the $\mathrm{P}{\mathrm{s}}^{-}\mathrm{H}{\mathrm{e}}^{2+}$(2S) resonance state, showing $E_r=-0.75678\mathrm{a}.\mathrm{u}.$ and $\Gamma /2=0.00533\mathrm{a}.\mathrm{u}.$ The set for the nonlinear parameters (α,β,γ) used in the wave function Eq. (5) is (0.91, 0.91, 0.27) for curve A, (0.91, 0.91, 0.25) for curve B, and (0.91, 0.91, 0.29) for curve C.

Figure 3

Stabilized complex eigenvalue $E_{\mathrm{res}}=E_r+i{E}_i$ near the pole of the $\mathrm{P}{\mathrm{s}}^{-}\mathrm{H}{\mathrm{e}}^{2+}$(3S) state, showing $E_r=-0.48307\mathrm{a}.\mathrm{u}.$ and $\Gamma /2=0.00338\mathrm{a}.\mathrm{u}.$ The set for the nonlinear parameters (α, β, γ) used in the wave function Eq. (5) is (0.63, 0.63, 0.40) for curve A and (0.63, 0.63, 0.45) for curve B.

Figure 4

Energy levels for the $e^{+}\mathrm{He}\left(2s^2\right)$ state and Rydberg states in the $\mathrm{P}{\mathrm{s}}^{-}\mathrm{H}{\mathrm{e}}^{2+}$(nS) series with $n=2$ to 7. Under the heading $e^{+}\text{−}\mathrm{He}$, the solid line is from the present calculation. The two dashed lines are from the SVM-CR results in Ref. [2] for the $e^{+}\mathrm{He}\left(2s^2\right)$ state and in Ref. [3] for the $e^{+}\mathrm{He}\left(3s^2\right)$ state, respectively.

Figure 5

Fitting of the Rydberg states of the $\mathrm{P}{\mathrm{s}}^{-}\mathrm{H}{\mathrm{e}}^{2+}$ system, in logarithmic scale. The circles are from the present ab initio calculations, and the squares are from the data fitted to the quantum defect formula. $n^{*}$ is the effective quantum number.