  Three variational approaches, the hyperspherical-harmonics, Gaussian-basis
and Lagrange-mesh methods involving different coordinate systems, are compared
in studies of $0^+$ bound-state energies in 3$\alpha$ models. Calculations are
performed with different versions of the shallow Ali-Bodmer potential (with and
without Coulomb) and with the deep Buck-Friedrich-Wheatley potential. All three
methods yield very accurate energies. Their advantages and drawbacks are
evaluated. The implications of the disagreement between the obtained results
and the experimental $^{12}$C energies in $3\alpha$ models with Coulomb
interaction are discussed.
