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)=-(e²/r)e^-qr as a function of the screening parameter q. We obtain the best variational result so far for the ground-state energy E₀ 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-qc), where qc 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 qc, qc=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}=(e^-ar-e^-br)/r+e^-br/(r+{α}). This variational wave function has a bound-unbound transition at qc=1.190 610 74.