A very efficient large-order perturbation theory is formulated for the nuclear motion of a linear triatomic molecule. All coupling between vibration and rotation is included. To demonstrate the method, all of the experimentally observed rotational energies, with values of J almost up to 100, for the ground and first excited vibrational states of CO2 and for the ground vibrational states of N2O and of OCS are calculated. The perturbation expansions reported here are rapidly convergent. The perturbation parameter is D-1/2, where D is the dimensionality of space. Increasing D is qualitatively similar to increasing the angular momentum quantum number J. Therefore, this approach is especially suited for states with high rotational excitation. The computational cost of the method scales only in proportion to JN(nu)(5/3), where N-nu is the size of the vibrational basis set. (C) 1997 American Institute of Physics.