Theoretical resolution of theH−resonance spectrum up to then=4threshold. I. States of1Po,1Do,and1Fosymmetries

Abstract

We report on a theoretical approach to the calculation of wave functions, energies $E,$ and widths $\ensuremath{\Gamma}$ of high-lying resonances of ${\mathrm{H}}^{\ensuremath{-}},$ with application to the identification of 76 states of ${}^{1}{P}^{o},$ ${}^{1}{D}^{o},$ and ${}^{1}{F}^{o}$ symmetries up to the $n=4$ threshold, with widths down to about $1\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}8}$--$1\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}10}\mathrm{a}.\mathrm{u}.,$ depending on symmetry and threshold. The overwhelming majority of these resonances have not been detected experimentally. Previous calculations by different methods allowed the identification of 35 of these states, with only very few cases having a level of accuracy comparable to the one of the present work. We suggest that the measurement of these resonances might become possible via two-step excitation mechanisms using ultrasensitive techniques capable of dealing with the problems of very small widths and preparation cross-sections. In this work, the ${}^{1}D$ state at $10.872\mathrm{eV}$ above the ${\mathrm{H}}^{\ensuremath{-}}{1s}^{2}{}^{1}S$ ground state, already prepared and measured by electron scattering as well as by two-photon absorption, is considered as the stepping stone for the possible probing of resonances of ${}^{1}{P}^{o},$ ${}^{1}{D}^{o},$ and ${}^{1}{F}^{o}$ symmetries via absorption of tunable radiation of high resolution. By classifying the results according to the Gailitis-Damburg model of dipole resonances (a product of a ${1/r}^{2}$-like potential) we find that there are unperturbed as well as perturbed series, in analogy with the Rydberg spectra of neutrals and positive ions (a product of a $1/r$-like potential). For the former, the agreement with the Gailitis-Damburg predictions as to the relationship of the extent of the outer orbital and of the energies and widths of states is excellent. The perturbed series result from interchannel coupling and the remaining electron correlation. One of the effects is the existence of overlapping resonances. For example, for two ${}^{1}{P}^{o}$ states below the $n=3$ threshold there is degeneracy on the energy axis ${(E}_{1}=\ensuremath{-}0.0555763612\mathrm{a}.\mathrm{u}.$ and ${E}_{2}=\ensuremath{-}0.0555763099\mathrm{a}.\mathrm{u}.)$ but the widths differ $({\ensuremath{\Gamma}}_{1}=1.14\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}4}\mathrm{eV}$ and ${\ensuremath{\Gamma}}_{2}=5.45\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}6}\mathrm{eV}).$ We also comment on whether consideration of the relativistic Lamb shift splitting of the hydrogen thresholds is sufficient for deciding the truncation of the resonance series. Our calculations were carried out by implementing previously published theories, whereby the resonance $E\mathrm{'}\mathrm{s}$ and \ensuremath{\Gamma}'s are determined from properly selected complex eigenvalues of non-Hermitian Hamiltonian matrices constructed in terms of physically relevant square integrable real and complex function spaces representing the localized and asymptotic parts of the resonance eigenfunctions. For the ${\mathrm{H}}^{\ensuremath{-}}$ series of resonances, the physical relevance of the real functions implies the systematic construction of basis sets with average $〈r〉$ extending to thousands of atomic units, in order to account for the extreme diffuseness of the outer orbital as each threshold is approached. The complex one-electron basis sets are Slater-type orbitals of a complex coordinate ${\mathrm{re}}^{\ensuremath{-}i\ensuremath{\theta}}.$ Their inclusion into the overall calculation and their optimization via the variation of nonlinear parameters (including \ensuremath{\theta}) accounts for the contribution of the asymptotic part of the resonance, and for the energy width and shift beyond the real energy ${E}_{o}$ of the localized part.