Off-the-energy-shell effect in the dissociative recombination ofHD+

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The Lippmann-Schwinger (LS) equation for the $K$ matrix is numerically solved for the collision of $\mathrm{H}{\mathrm{D}}^{+}+e$ interacting by the configuration interaction (CI). The realistic CI strength is deduced from ab initio calculation of electron scattering. The LS equation is extended to the negative collision energies in the context of multichannel quantum defect theory (MQDT) for the dissociative recombination (DR). A decoupling property is shown for the LS equation, which is useful for reducing the amount of the calculation. The Chebyshev quadrature is employed for the calculation and a fully converged result has been obtained. Using the result with the MQDT, the DR cross section of $\mathrm{H}{\mathrm{D}}^{+}$ is obtained. It has turned out that the off-the-energy-shell contribution is indispensable for understanding the DR. The contribution from the negative energies largely affects the low-energy DR. The DR at low energy is induced by the indirect process with rotational excitation. The separable approximation on the CI is examined for the realistic CI strength. This approximation has turned out to be inadequate for the DR of $\mathrm{H}{\mathrm{D}}^{+}$. The calculated rate coefficient reproduces the experiment [A. Al-Khalili et al., J. Phys. A 68, 042702 (2003)] both on the absolute magnitude and resonance structure. The off-the-energy shell contribution largely affects on the initial vibrational state $({v}^{+})$ dependence. This contribution increases the rate coefficient for ${v}^{+}=0$ and decreases for ${v}^{+}=2$.