Electron-screening correction for the proton-proton reaction

We test the Salpeter formalism for the electron screening of the solar proton-proton fusion reaction by solving numerically the relevant Schrödinger equation. We evaluate exactly the square of the overlap integral of the two-proton wave function and the deuteron wave function and compare with the usual analytic approximation. The usual WKB solution agrees with the numerical result to ${O(10\mathrm{}}^{-4})$. The WKB approximation should be even more precise for the other nuclear fusion reactions in the $\mathrm{pp}$ chain and CNO cycles.