Description of a migrating proton embedded in an electron gas

The direct force on a migrating proton embedded in an electron gas is calculated by implementing recently derived theoretical expressions for this quantity. A self-consistent Kohn-Sham calculation of the scattering potential leads to a doubly populated bound state and a sign change of the direct charge at lower electron densities. A self-consistent calculation under the constraint of single occupancy leads to positive values for all densities. This result is supported by employing constrained model potentials as well. A special constraint on the screening of a proton, related to the strength of the dipolar backflow pattern around it, results in a fairly constant value around unity for the direct charge in the whole metallic density range. Results based on square-well model potentials with prefixed width and depth show a lowering of the direct charge as soon as a bound state is formed. An open question remains regarding the completeness of the formalism in view of the fact that no cancellation of the direct charge is found for a system with a bound state