New geometry optimization techniques are introduced for characterizing local minima, transition states, and pathways corresponding to enthalpy surfaces at constant pressure. Results are obtained for comparison with the potential energy surfaces of model glass formers studied in previous work. The constant pressure condition, where the the box lengths of the simulation cell vary, makes the enthalpy surface less rugged than the potential energy surface corresponding to the same mean density. Analysis of barrier heights as a function of pressure provides insight into transport and relaxation processes. Elementary rearrangements can be separated into "diffusive" and "nondiffusive" processes, where the former involve changes in the nearest-neighbor coordination of at least one atom, and the latter do not. With increasing pressure the barrier heights for cage-breaking rearrangements rise, while those for cage-preserving rearrangements appear relatively unchanged. The "strong" or "fragile" character of the system can therefore change with pressure because the barriers encountered vary in a systematic fashion. The geometric mean normal mode frequencies of a binary Lennard-Jones system decrease with increasing potential energy for constant pressure, rather than increase as they do at constant volume, in agreement with a simple model.(C) 2003 American Institute of Physics.