Role of the cut-off function for the ground state variational wavefunction of the hydrogen atom confined by a hard sphere

A variational treatment of the hydrogen atom in its ground state, enclosed by a hard spherical cavity of radius Rc , is developed by considering the ansatz wavefunction as the product of the free-atom 1s orbital times a cut-off function to satisfy the Dirichlet boundary condition imposed by the impenetrable confining sphere. Seven different expressions for the cut-off function are employed to evaluate the energy, the cusp condition, <r^-1>,<r>, <r^2>, and the Shannon entropy, and as a function of Rc in each case. We investigate which of the proposed cut-off functions provides best agreement with available corresponding exact calculations for the above quantities. We find that most of these cut-off functions work better in certain regions of Rc , while others are identified to give bad results in general. The cut-off functions that give, on average, better results are of the form (1- (r/Rc)^n), n=1,2,3