Bidimensional bound states for charged polar nanoparticles

We study the behavior of an electron in the vicinity of a positively charged nanoparticle that has a permanent electric dipole and is of finite size (not point-like) in two dimensions. We place special attention to the fact that a constant of motion, as well as the energy, exists so that the wave functions result from an eigenvalue problem which are classified using two quantum numbers. The (GaAs)3 cluster is used as iconic case. The angular part is solved using Mathieu functions and the radial part by means of a numerical method. The solutions are plotted and the Einstein coefficients are calculated.