Atomic H over crystal surface: a variational approach

We study the behavior of the lowest electron level of a hydrogen atom confined in the semi-infinite space with a plane surface applying variational methods. The properties of the adsorbent are modeled by the parameter of the Robin boundary condition imposed on the electron wave functions at the boundary. We consider the case of crystalline adsorbent, so we assume that the parameter in Robin condition is a periodoc function. It is shown that the periodicity of the parameter should be taken into account when choosing a trial function. Trial wave funtion should provide the correct relationship between the normal derivative and the wave function value at each point of the boundary and have an antinode at the vicinity of the atomic nucleus. We perform a сomparison of the results obtained via the variational approach and via direct numerical calculations.