Precision measurement of the last bound states in H$_2$ and determination of the H + H scattering length

The binding energies of the five bound rotational levels $J=0-4$ in the highest vibrational level $v=14$ in the X$^1\Sigma_g^+$ ground electronic state of H$_2$ were measured in a three-step ultraviolet-laser experiment. Two-photon UV-photolysis of H$_2$S produced population in these high-lying bound states, that were subsequently interrogated at high precision via Doppler-free spectroscopy of the F$^1\Sigma_g^+$ - X$^1\Sigma_g^+$ system. A third UV-laser was used for detection through auto-ionizing resonances. The experimentally determined binding energies were found to be in excellent agreement with calculations based on non-adiabatic perturbation theory, also including relativistic and quantum electrodynamical contributions. The $s$-wave scattering length of the H + H system is derived from the binding energy of the last bound $J=0$ level via a direct semi-empirical approach, yielding a value of $a_s$ = 0.2724(5) $a_0$, in good agreement with a result from a previously followed theoretical approach. The subtle effect of the $m\alpha^4$ relativity contribution to $a_s$ was found to be significant. In a similar manner a value for the $p$-wave scattering volume is determined via the $J=1$ binding energy yielding $a_p$ = -134.0000(6) $a_0^3$. The binding energy of the last bound state in H$_2$, the ($v=14$, $J=4$) level, is determined at 0.023(4) cm$^{-1}$, in good agreement with calculation. The effect of the hyperfine substructure caused by the two hydrogen atoms at large internuclear separation, giving rise to three distinct dissociation limits, is discussed.