Variational calculations for the bound-unbound transition of the Yukawa potential

We perform LCAO (linear combination of atomic orbitals) calculations for the ground state of the Yukawa potential V(r)=-(${\mathit{e}}^2$/r)${\mathit{e}}^{\mathrm{-}\mathit{q}\mathit{r}}$ as a function of the screening parameter q. We obtain the best variational result so far for the ground-state energy ${\mathit{E}}_0$ as a function of q. We also obtain the critical exponents of both the probability density at the origin and the ground-state energy as functions of (q-${\mathit{q}}_{\mathit{c}}$), where ${\mathit{q}}_{\mathit{c}}$ is the critical q above which V(r) does not have a bound state. The use of the critical exponents permits the so far most precise determination of ${\mathit{q}}_{\mathit{c}}$, ${\mathit{q}}_{\mathit{c}}$=1.190 612 27±0.000 000 04. We also show that it is possible to use the LCAO calculations as a tool to determine the analytical form of very precise variational wave functions. We obtain, in such a way, the wave function ψ=(${\mathit{e}}^{\mathrm{-}\mathit{a}\mathit{r}}$-${\mathit{e}}^{\mathrm{-}\mathit{b}\mathit{r}}$)/r+${\mathit{e}}^{\mathrm{-}\mathit{b}\mathit{r}}$/(r+α). This variational wave function has a bound-unbound transition at ${\mathit{q}}_{\mathit{c}}$=1.190 610 74.