Resonance-channel quantum numbers in electron-hydrogen and proton-hydrogen scattering from group theory of the long-range dipole interaction

We derive quantum numbers $K$ and $T$ which label asymptotic potential-energy curves in $e^{-}$-H scattering. Zero-order channel states are constructed group theoretically so that the dipole interaction ${\mathcal{r}}_{1}cos{\theta }_{12}$ is diagonal in a product space of hydrogenic orbitals for 1 and spherical harmonics for 2. $K$ and $T$ are analogous to the "electric quantum number" and the $z$ component of angular momentum in the linear Stark effect for H, and are closely related to quantum numbers described recently for doubly excited He. Channels supporting resonances are predicted from a perturbation expansion of eigenvalues of the effective centrifugal barrier operator $2r_{1}cos{\theta }_{12}+l_2^2$ in terms of $K$ and $T$. Comparison with exact ${\mathrm{H}}^{-}$ results and experiment is excellent, and the method accounts for previously unexplained degeneracies with respect to total parity. A rigorous result is that only channels with $K>\mathrm{}0\mathrm{}$ can support an infinity of states below threshold. An approximate selection rule for coupling between neighboring channels suggests that decay of ${\mathrm{H}}^{-}$ resonances goes preferentially to channels such that $\Delta K=\pm 1$, $\Delta T=0\mathrm{}$. Application of the method is also made to $p$-H scattering channels for which the zero-order basis nearly diagonalizes the long-range centrifugal barrier.