Transition pathways on the energy landscape of atactic polystyrene (aPS) glassy specimens are probed below its glass-transition temperature. Each of these transitions is considered an elementary structural relaxation event, whose corresponding rate constant is calculated by applying multidimensional transition-state theory. Initially, a wide spectrum of first-order saddle points surrounding local minima on the energy landscape is discovered by a stabilized hybrid eigenmode-following method. Then, (minimal-energy) "reaction paths" to the adjacent minima are constructed by a quadratic descent method. The heights of the free energy, the potential energy, and the entropy barriers are estimated for every connected triplet of transition state and minima. The resulting distribution of free energy barriers is asymmetric and extremely broad, extending to very high barrier heights (over 50 kBT); the corresponding distribution of rate constants extends over 30 orders of magnitude, with well-defined peaks at the time scales corresponding to the subglass relaxations of polystyrene. Analysis of the curvature along the reaction paths reveals a multitude of different rearrangement mechanisms; some of them bearing multiple distinct phases. Finally, connections to theoretical models of the glass phenomenology allows for the prediction, based on first-principles, of the "ideal" glass-transition temperature entering the Vogel-Fulcher-Tammann (VFT) equation describing the super-Arrhenius temperature dependence of glassy dynamics. Our predictions of the time scales of the subglass relaxations and the VFT temperature are in favorable agreement with available experimental literature data for systems of similar molecular weight under the same conditions.