  The LiH$^{+}$ molecule is prototypical of the indirect dissociative
recombination (DR) process, in which a colliding electron destroys the molecule
through Rydberg capture pathways. This Letter develops the first quantitative
test of the Siegert state multichannel quantum defect theory description of
indirect DR for a diatomic molecular ion. The R-matrix approach is adopted to
calculate ab-initio quantum defects, functions of the internuclear distance
that characterize both Rydberg states and the zero-energy collisions of
electrons with LiH$^{+}$ ions. The calculated DR rate coefficient agrees
accurately with recent experimental data (S. Krohn et al, Phys. Rev. Lett. 86,
4005). We identify the doorways to fast indirect DR as complex resonance
manifolds, which couple closed channels having both high and low principal
quantum numbers. This sheds new light on the competition between direct and
indirect DR pathways, and suggests the reason why previous theory
underestimated the DR rate by an order of magnitude.
