We show how the parameters of the standard fermionic and boson-fermion Hamiltonians, that are widely used in studies of the BCS-BEC cross-over in dilute atomic gases, can be unambiguously determined from the microscopic two-body interactions. These parameters depend in general on just five physical parameters of resonanceenhanced two-body scattering. Furthermore, we show that for the 202 G resonance in a two-component mixture of fermionic 40 K atoms used in the experiment of C.A. Regal et al. [Phys. Rev. Lett. 92, 040403 (2004)], the closed-channel admixture to the highest-excited vibrational diatomic bound state reaches a maximum of only 8 % at 7.2 G below the zero-energy resonance, in contrast to previous predictions of large admixtures based on model potentials. Our mean-field analysis of the many-body equilibrium quantities characteristic for the BCS-BEC cross-over problem shows virtually no differences between the predictions of the standard fermionic and boson-fermion Hamiltonians, provided that the approaches account not just for the magnetic-field dependence of the scattering length but also properly include the highest excited vibrational bound state of the background scattering potential. PACS numbers: 03.75.Ss, 03.75.Nt, 34.50.-s I
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