There is currently great interest in the very large values of the positron annihilation rate that have been observed in low-energy positron scattering by some molecules. The annihilation rate is proportional to , the effective number of electrons in the target available to the positron for annihilation. These very high rates and associated values of have been observed experimentally to occur at energies just below the energies of excited vibrational states of the molecule concerned. This has been explained by Gribakin [Phys. Rev. A Vol. 61 (2000), p. 022720] and Gribakin and Lee [Phys. Rev. Lett. Vol. 97 (2006), p. 193201] as being due to Feshbach resonances involving excited quasi-bound vibrational states. Their explanation is partly phenomenological. In this paper, I describe the results of an ab initio treatment of this resonant behaviour in the case of the scattering of a heavy ‘positron’ by , using the Kohn variational method.