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We present a simple two-dimensional model of the indirect dissociativerecombination process. The model has one electronic and one nuclear degree offreedom and it can be solved to high precision, without making any physicallymotivated approximations, by employing the exterior complex scaling methodtogether with the finite-elements method and discrete variable representation.The approach is applied to solve a model for dissociative recombination ofH$_2^+$ and the results serve as a benchmark to test validity of severalphysical approximations commonly used in the computational modeling ofdissociative recombination for real molecular targets. The second, approximate,set of calculations employs a combination of multi-channel quantum defecttheory and frame transformation into a basis of Siegert pseudostates. The crosssections computed with the two methods are compared in detail for collisionenergies from 0 to 2 eV.

Dissociative recombination by frame transformation to Siegert pseudostates: A~comparison with a numerically solvable model