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In our recent papers, a non-standard quasi-molecular mechanism was suggested and applied to treat the cosmological recombination. It was assumed that, in the pre-recombination stage of evolution of the Universe, an electron combined with two neighbouring protons and created a hydrogen molecular ion, |$\mathrm{ H}_2^+$| in a highly excited state, which then descended into the lower lying state or dissociated. In this work, we implement a quantitative analysis of this quasi-molecular mechanism of recombination; namely, we elaborate the scheme of calculation for a free–bound radiative transition. We show that the quasi-molecular mechanism played a significant role in the pre-recombination and recombination stages of evolution of the early Universe, and hence must be included into the consideration of the description of a thermal history of the Universe. Together with the earlier developed treatment of bound–bound radiative transitions in |$\mathrm{ H}_2^+$|⁠ , the elaborated scheme of calculation can be used for the design of a rapid and complete cosmological recombination code. [ABSTRACT FROM AUTHOR]