Application of $P$-wave hybrid theory to the scattering of electrons from He${}^{+}$and resonances in He and H${}^{-}$

The $P$-wave hybrid theory of electron-hydrogen elastic scattering [Bhatia, Phys. Rev. A 85, 052708 (2012)] is applied to the $P$-wave scattering from He ion. In this method, both short-range and long-range correlations are included in the Schrödinger equation at the same time, by using a combination of a modified method of polarized orbitals and the optical potential formalism. The short-range-correlation functions are of Hylleraas type. It is found that the phase shifts are not significantly affected by the modification of the target function by a method similar to the method of polarized orbitals and they are close to the phase shifts calculated earlier by Bhatia [Phys. Rev. A 69, 032714 (2004)]. This indicates that the correlation function is general enough to include the target distortion (polarization) in the presence of the incident electron. The important fact is that in the present calculation, to obtain similar results only a 20-term correlation function is needed in the wave function compared to the 220-term wave function required in the above-mentioned calculation. Results for the phase shifts, obtained in the present hybrid formalism, are rigorous lower bounds to the exact phase shifts. The lowest $P$-wave resonances in He atom and hydrogen ion have also been calculated and compared with the results obtained using the Feshbach projection operator formalism [Bhatia and Temkin, Phys. Rev. A 11, 2018 (1975)] and also with the results of other calculations. It is concluded that accurate resonance parameters can be obtained by the present method, which has the advantage of including corrections due to neighboring resonances, bound states, and the continuum in which these resonances are embedded.

Figure 1

The lowest ${}^{3}P$ resonance state in a helium atom at 58.3226 eV with respect to the ground state of the helium atom. The phase shifts (radians) are for the scattering of an electron from a helium ion.

Figure 2

The lowest ${}^{1}P$ resonance state in a helium atom at 60.145 eV with respect to the ground state of the helium atom. The phase shifts (radians) are for the scattering of an electron from a helium ion.

Figure 3

The lowest ${}^{3}P$ resonance in a hydrogen negative ion at 9.7399 eV. The resonance position is with respect to the ground state of the hydrogen atom. The phase shifts (radians) are for the scattering of an electron from a hydrogen atom.