Exponential variational bound for a capped Coulomb potential ($V_R(r)=-\kappa /\max (R,r)$)

For $\kappa >0$ and the radial exponential trial state $e^{-\alpha r}$, $\alpha >0$, put $a={\hslash }^2/(m\kappa )$. Direct radial integration gives the Rayleigh quotientThe kinetic expectation is ${\hslash }^2{\alpha }^2/(2m)$; the potential expectation follows by splitting its radial integral at $R$. The common prefactor cancels from the stationary condition, giving $a\alpha =(1+2\alpha R)e^{-2\alpha R}$. Its positive solution is unique, since the left side increases and the right side decreases. For $\epsilon =R/a\to 0$, it satisfies $a\alpha =1-2{\epsilon }^2+O({\epsilon }^3)$.

The quantum variational principle makes $E(\alpha )$ an upper bound for the ground energy. The explicit choice $\alpha =1/a$ gives $E_0(R)\le (\kappa /a)[1/2-\{1-(1+\epsilon )e^{-2\epsilon }\}/\epsilon ]$, whose first correction to $-\kappa /(2a)$ is $2\kappa R^2/(3a^3)$. The true energy is no lower than the point-Coulomb value because capping makes the potential less attractive.