  Taking advantage of the known analytic expression of the eigenfunctions and
eigenenergies of the Morse Hamiltonian, explicit expressions are found for the
scattering length $a$ and the effective range $r_e$ which determine the s-wave
scattering of ultracold atoms. The effects on $a$ and $r_e$ of considering the
radial coordinate in the interval $[0,\infty)$ or in the extended region
$(-\infty, \infty)$ are studied in detail.
