Natural and unnatural parity resonance states in positron–hydrogen scattering

We present an investigation of resonances with natural and unnatural parities in positron scattering with atomic hydrogen. The complex scaling method has been used. Resonance states for natural parity &pgr; = ( − 1 ) J ?> with total angular momenta J = 0 − 2 ?> and unnatural parity &pgr; = ( − 1 ) J + 1 ?> with J = 1 , 2 ?> are calculated. Resonance energies and widths are reported and compared with other theoretical calculations.


Introduction
Over the past few decades, the study of resonance phenomena in positron-hydrogen systems has gained interest [1][2][3][4][5][6][7][8]. These resonances could play an important role in the production of anti-hydrogen atoms through rearrangement scattering of positronium (Ps) by anti-protons [10]. This system consists of three particles: (1) a positron, (2) an electron, and (3) a proton. There is no truly bound state of this system, but there exist many quasi bound states that lie in the + e H scattering continuum. A number of theoretical methods have been used to find the positions and widths of the resonances in + e H scattering, even though there are no experimental observations [10][11][12]. The majority of these results are limited to S-, P-and D-wave resonances. Ho and Yan [4,8] comprehensively analyzed high-partial-wave resonances in positronhydrogen scattering using the complex-coordinate rotation method. They used Hylleraas type wave functions, in which the coordinates of the positronium atom were included explicitly.
Both resonances with natural parity π = − ( 1) J , where J is the total angular momentum of the system, and resonance states with unnatural parity π = − + ( 1) J 1 are of interest. The latter type of states can exist because the angular momenta of the electron and positron, l 1 and l 2 , couple to the total angular momentum J according − ⩽ ⩽ + l l J l l | | 1 2 1 2 , while the parity is π = − + ( 1) l l 1 2 . Thus, if l 1 and l 2 are both larger than 0, J can take both even and odd values for any l 1 and l 2 . Since l 1 and l 2 are both non-zero these states are, to first order, stable against + − e e annihilation, which is blocked by the centrifugal barrier. A few theoretical works have investigated + e H resonances with unnatural parity [16][17][18]. Again using Hylleraas functions and complex scaling, Yan and Ho [16] reported an examination of resonances with unnatural parities in a positron-hydrogen system. Shimamura et al [17] also investigated unnatural parity resonance states in the positron-hydrogen system using hyper-spherical close coupling calculations. Since ground state H is an S-state, unnatural parity states can only be accessed in positron scattering with an excited state of hydrogen, or excited Ps colliding with a proton [8].
The present study reports results for resonance states with natural parity for S-, P-and D-states and unnatural parity for P e and D o state using explicitly correlated Gaussian wave functions. In addition to the resonances calculated in [16,17] we also identify the unnatural parity resonances below Ps (n = 4) which we have not found in previous literature.

Method
The three-body systems were calculated using the coupled rearrangement channels method developed by Kamimura and co-workers [20,21]. In this method the wave function is expanded using Jacobi coordinates ( r R { , } versatile basis set, capable of adapting to states close to breakup thresholds of any pair of particles. We also note that by using Jacobi coordinates we automatically include all socalled mass-polarization terms which, in atomic calculations using coordinates centred on the nuclei, arise as corrections which are non-trivial to calculate. Within this coordinate system we represent the wave function using a partial wave expansion of the angular variables and Gaussians in the radial variables. That is, for a state with total orbital angular momentum J, M the wave function has the form ( 2 ) Here α denotes the three rearrangement channels, α l and α L are the angular momenta along α r and α R respectively, and i, I numbers the Gaussians along the two radial coordinates (see figure 1). The angular momenta α l and α L are chosen consistent with the total J (i.e. − ⩽ ⩽ + α α α α l L J l L | | , up to some maximum values α l max and α L max , which may be different for different rearrangement channels.) The total number of Gaussian trial functions for each rearrangement channel and angular momentum are given by α i l max and α The widths of the Gaussians α α r l i and α α R L I are, for each channel and set of angular momenta, chosen as geometric progressions are set explicitly and used as non-linear variational parameters. In this way most Gaussians will span the short-to medium range, while a few more diffuse Gaussians capture the long range part of the wave function. In this work, resonances with very small binding energies are calculated, and hence it was essential to set a large enough value for the outer radius R i .
Resonances are calculated using the complex scaling method [22][23][24][25]. The complex dilation operator θ U ( ) acting on a function f r ( ) is defined through where the exponential prefactor ensures that the complex scaled function satisfies the normalization condition . The corresponding transformation of the Hamiltonian is, for the special case ∝ V r 1 , of the system, where E r is the resonance energy and Γ the width.  The three rearrangement channels of the + e H system and their Jacobian coordinates. The second rearrangement channel above corresponds for large R 2 to positronium-like states, while the the third channel corresponds to hydrogen-like states.  -dipole series converging to the atomic thresholds. According to [9] the ratio of binding energies of successive resonances should then be a constant for each threshold. We calculated these ratios but found considerable variation, in agreement with other calculations [6]. As was pointed out in [4], the dipole formula only works well for resonances dominated by the long-range forces, i.e. very close to threshold. Most likely, resonances closer to the threshold should follow the dipole formula, but unfortunately these become increasingly difficult to calculate. For states with total angular momentum J = 1, we have different possible combinations of the individual angular momenta l l l ( , , ) 1 2 3 . The total number of configurations used was 24. For P-wave, our results in table 2 are in good agreement with [8,12]; the uncertainty is about 10 −6 atomic units again larger for large n. The results for states with total angular momentum J = 2 are summarized in table 3. The total number of configuration used was 24. We found no literature values for physical proton mass calculation for D-waves. We found some new resonances for infinite proton mass, below the Ps = n ( 2) threshold, at position E = 0.062 928 au and width Γ = × − 9.25 10 6 au, E = 0.062 57 au and Γ = × − 8.99 10 6 au. We find that while rather different numerical approaches were used, the results agree very well. The uncertainty is about 10 −5 au but larger for large n values. In the present work, also resonance states with unnatural parities have been calculated for positron-excited hydrogen scattering. Table 4 shows the total number of basis functions used for a given angular momentum state, as well as the individual (l l l , , 1 2 3 ) pairs for different angular momentum combinations. N is the number of configurations used.
For the D o state resonances, present results are displayed in table 6, along with the other available theoretical results from [16,17]. The resonances below the = n H( 3) and H = n ( 4) thresholds are in good agreement with literature values. The resonances below Ps(n = 3) threshold were not presented in [17].

Conclusion
In summary, we found resonances with natural as well as unnatural parities in positron-excited hydrogen scattering. Our results are in line with prior calculations. This study reports some resonances for the first time. On the experimental front, there is no report on resonances in + e H scattering. Our predictions will be useful in any future experimental search for atomic resonances involving positrons.