  We perform a comparative analysis of different computational approaches
employed to explore the electronic structure of ultralong-range Rydberg
molecules. Employing the Fermi pseudopotential approach, where the interaction
is approximated by an $s$-wave bare delta function potential, one encounters a
non-convergent behavior in basis set diagonalization. Nevertheless, the energy
shifts within the first order perturbation theory coincide with those obtained
by an alternative approach relying on Green's function calculation with the
quantum defect theory. A pseudopotential that yields exactly the results
obtained with the quantum defect theory, i.e. beyond first order perturbation
theory, is the regularized delta function potential. The origin of the
discrepancies between the different approaches is analytically motivated.
