  Within the framework of the generalized Landau-de Gennes theory, we identify
a $Q$-tensor-based energy that reduces to the four-constant Oseen-Frank energy
when it is considered over orientable uniaxial nematic states. Although the
commonly considered version of the Landau-de Gennes theory has an elastic
contribution that is at most cubic in components of the $Q$-tensor and their
derivatives, the alternative offered here is quartic in these variables. One
clear advantage of our approach over the cubic theory is that the associated
minimization problem is well-posed for a significantly wider choice of elastic
constants. In particular, quartic energy can be used to model
nematic-to-isotropic phase transitions for highly disparate elastic constants.
  In addition to proving well-posedness of the proposed version of the
Landau-de Gennes theory, we establish a rigorous connection between this theory
and its Oseen-Frank counterpart via a $\Gamma$-convergence argument in the
limit of vanishing nematic correlation length. We also prove strong convergence
of the associated minimizers.
