Coupled-States Method for Scattering Calculations

The eigenfunction-expansion method for calculating scattering of electrons and positrons from atoms has the very desirable property of providing phase-shift bounds, but is slowly convergent in some cases. Phase-shift bounds and improved convergence are combined in recently developed hybrid methods, but these are complicated to apply. An alternative method based on expansion in target states, some of which are chosen for best convergence rather than restricted to eigenstates, is proposed. The coupled equations are similar to those of the eigenfunction-expansion method and provide phase-shift bounds. The method was tested by calculation of $S$-wave positron-hydrogen scattering at energies of $k^2=0,0.04$, 0.16, and 0.36. Target states used in the expansions ($1s2p^{\prime }$ and $1s2p^{\prime }3d^{\prime }$) were of the same functional form as eigenstates, except that orbital exponents were not fixed at $\frac{1}{2}$ and $\frac{1}{3}$. Calculated phase shifts as a function of orbital exponent value have a broad maximum for exponent values larger than those of eigenfunctions, and phase shifts are considerably larger than those calculated with eigenfunction expansions.