Variational Bound States of Screened Potentials

A number of years ago, a calculational scheme was introduced by Stubbins (Phys.~Rev.~\textbf{A}48, 220 (1993)) to compute the energies of both the Hulth\'{e}n and Yukawa potentials. The method introduces a particular ansatz for solving the Schr\"{o}dinger equation with screened Coulomb type potentials. In this work we wish to review the method of Stubbins and to show that it is, in fact, equivalent and a subset of a more systematic (and hence more useful) variational scheme (Zhou et al.~Phys.~Rev.~\textbf{A}51, 3337 (1995)). This variational approach involves the construction of a basis by taking derivatives of the variational parameters of the system. The eigenvalues of the Hamiltonian matrix are then minimized with respect to these parameters yielding a ``best guess" upper bound on the energies.