The globally uniform semiclassical approximation for energy eigenstates developed by D. Zor and K. G. Kay [Phys. Rev. Lett. 76, 1990 (1996)] is derived explicitly for the case of multidimensional systems and is applied to two test cases. The adiabatic switching approximation is used to obtain various quantities that enter the semiclassical expression. Two formulations of the method are examined : one requiring several trajectories for each desired state and another requiring only a single trajectory per state. The multitrajectory version yields accurate results for all states investigated (overlap between semiclassical and quantum eigenstates >0.98), including some influenced by classical chaos. The single-trajectory treatment, however, is more efficient, gives accurate results for regular states, and is even applicable for certain chaotic states, although the multiple-trajectory method is preferred in such cases. Despite the substantial resemblance of the present theory to the frozen Gaussian approximation (FGA), it is a true semiclassical approximation and is found to produce wave functions that are significantly more accurate than those obtained from the FGA for all states examined.