Molecule formation in tenuous media: Quantum effects in spontaneous radiative association

Abstract

The semiclassical theory for the spontaneous radiative association coefficient is reviewed. The example of ${\mathrm{H}}_{2}^{+}$ formation during a proton collision with a hydrogen atom is considered in both the Kramers and Ter Haar and the Bates versions of the semiclassical theory. These two semiclassical predictions are identical for $T>50\ifmmode^\circ\else\textdegree\fi{}\mathrm{K}$. Below this temperature range they differ and show the small-$T$ limiting behavior expected from asymptotic considerations. The quantum theoretical rate coefficient for the same process is also evaluated. For $500\ensuremath{\le}T\ensuremath{\le}32000\ifmmode^\circ\else\textdegree\fi{}\mathrm{K}$, the quantum and semiclassical rates are in quantitative agreement. The quantum rate and the Bates version of the semiclassical rate show identical temperature dependence as $T$ becomes small, $O({T}^{0})$, but their magnitudes differ for $T\ensuremath{\le}500\ifmmode^\circ\else\textdegree\fi{}\mathrm{K}$. The quantum rate always exceeds the Bates semiclassical rate in this temperature range, with the maximum difference being nearly an order of magnitude. This enhancement of the rate of molecule formation due to quantum effects is anticipated to be important in the context of the interstellar media.