Analytically derived switching functions for exactH2+eigenstates

Electron translation factors (ETF's) appropriate for slow atomic collisions may be constructed using switching functions. In this paper we derive a set of switching functions for the $\mathrm{H}_2^{}{}_{}{}^{+}$ system by an analytical "two-center decomposition" of the exact molecular eigenstates. These switching functions are closely approximated by the simple form $f=tanhb\eta$, where $\eta$ is the "angle variable" of prolate spheroidal coordinates. For given united atom angular momentum quantum numbers ($l,m$), the characteristic parameter $b_{\mathrm{lm}}$ depends only on the quantity $c^2=\frac{-\epsilon R^2}{2}$, where $\epsilon$ is the electronic binding energy and $R$ the internuclear distance in a.u. The resulting parameters are in excellent agreement with those found in our earlier work by a heuristic "optimization" scheme based on a study of coupling matrix-element behavior for a number of $\mathrm{H}_2^{}{}_{}{}^{+}$ states. An approximate extension to asymmetric cases (He${\mathrm{H}}^{2+}$) has also been made. Nonadiabatic couplings based on these switching functions have been used in recent close-coupling calculations for ${\mathrm{H}}^{+}-\mathrm{H}\mathrm{}\left(1s\right)\mathrm{}$ collisions and ${\mathrm{He}}^{2+}-\mathrm{H}\mathrm{}\left(1s\right)\mathrm{}$ collisions at energies 1.0-20 keV.