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We study the two-body problem of the ultracold fermionic alkaline-earth(like) atoms in the electronic $^1$S$_0$ state ($g$-state) and $^3$P$_0$ state($e$-state), which are confined in a quasi-one-dimensional (quasi-1D) tube. Inaddition, in the axial direction, the $g$-atom experience a 1D optical latticeand the $e$-atom is localized by a harmonic potential. In this work, we proposetwo appropriate tight-binding models, which are applicable for the cases thatthe odd-wave scattering between the $g$- and $e$-atom is negligible or not,respectively. We further give a microscopic derivation for the inter-atomicinteraction parameters of these tight-binding models, by exactly calculatingthe low-energy inter-atomic scattering amplitude. Our results show that, as onecan predict, these interaction parameters can be efficiently controlled by theconfinement potentials. We further exam the simple "projection approximation"with which one derives the interaction parameters by directly projecting the 3DHuang-Yang pseudopotential on the ground state of the confinement and thelowest band of the optical lattice. We find that one should be very carefulabout determining the interaction parameters in the tight-binding models.Furthermore, we calculate the spin-exchanging rate, which dependents on theincident quasi-momentum $k$ of the $g$-atom, for the recent experiment (L.Riegger, {\it et. al.,} Phys. Rev. Lett. {\bf 120}, 143601 (2018)) of$^{173}$Yb atoms in this quasi-(1+0)D system, and study finite-momentum effectin this experiment. Our results show that in this system the finite-momentumeffect of the $g$-atom is very significant, and the momentum of the $g$-atomsin this experiment may be pretty high (already in the second Brillouin zone ofthe optical lattice)

Tight-Binding Kondo Model and Spin-Exchange Collision Rate of Alkaline-Earth Atoms in a Mixed-Dimensional Optical Lattice