The electron detachment from the hydrogen negative ion in strong fields is studied using the two-electron and different single-electron models within the quasistatic approximation. A special attention is payed to over-the-barrier regime where the Stark saddle is suppressed below the lowest energy level. It is demonstrated that the single-electron description of the lowest state of ion, that is a good approximation for weak fields, fails in this and partially in the tunneling regime. The exact lowest state energies and detachment rates for the ion at different strengths of the applied field are determined by solving the eigenvalue problem of the full two-electron Hamiltonian. An accurate formula for the rate, that is valid in both regimes, is determined by fitting the exact data to the expression estimated using single-electron descriptions.
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