Limitations of variational methods for hydrogenic systems

Abstract

Theoretical methods that give good approximations of the behaviours of hydrogen-like impurity states in semiconductors in uniform external magnetic fields are by now well established. With variational methods in mind the sizes of the determinants that give sufficiently accurate energies greatly depend on the quality of the wave functions chosen as these affect the rapidity of the series convergence. The recently developed hybrid functions enable hydrogenic energies to be calculated up ton=7 using fairly small determinants. However, it has emerged that there is a restricted region of variational parameters which maintains the numerical symmetry of the hamiltonian matrix elements for each group of states. Although each such parameter regime appears to narrow as the magnetic quantum number increases, the accuracies of the calculated eigenvalues remain satisfactory.