Bound Eigenstates of the Static Screened Coulomb Potential

Accurate numerical solutions have been obtained for Schr\"odinger's equation for a two-particle system interacting through a static screened Coulomb potential (SSCP) $V(\mathcal{r})=\ensuremath{-}\frac{Z{e}^{2}{e}^{\ensuremath{-}\frac{r}{D}}}{\mathcal{r}}$. The numerical integration of the wave equation uses a one-dimensional difference method which is simple, accurate, and efficient. Solutions have been computed for 45 eigenstates, $1s$ through $n=9$, $l=8$, yielding the eigenfunctions and energy eigenvalues for a wide range of $D$, the screening length which characterizes the range of the interaction. Under screening, all energy levels are shifted away from their unscreened values toward the continuum, the energy increasing as $D$ decreases. For each $n$, $l$ eigenstate, there is a finite value of the screening length ${D}_{0}(n, l)$, for which the energy becomes zero. The value of ${D}_{0}$ for the ground state of a two-particle bound system in a potential of this type, such as the Debye or Yukawa potential, is 0.83991 $\frac{{a}_{0}}{Z}$ in agreement with certain previous studies. The total number of different energy levels is finite for any finite $D$, and is approximately linearly dependent on $D$. The number of bound $s$ states ${g}^{*}$ is given by the relation ${({g}^{*})}^{2}=1.2677$ $\frac{\mathrm{DZ}}{{a}_{0}}$. For given $n$, the $l$ degeneracy is destroyed, lowest $l$ levels lying lowest in energy. At sufficiently high $n$, this behavior results in level crossing, high $l$ levels of eigenstate $n$ having higher energies than low $l$ levels of eigenstate $n+1$. This produces increasingly complex deviations of the level order from the unscreened order, commencing with the $5s\ensuremath{-}4f$ cross-over. Because of the displacement of high $n$ states into the continuum, the density of states in the SSCP for any finite $D$ is lower than in the unscreened potential, especially near the continuum.