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We have developed a novel semiclassical transition state theory ͑SC-TST͒ for truncated parabolic barriers, based on the formulation of Hernandez and Miller ͓Chem. Phys. Lett. 214, 129 ͑1993͔͒. Our SC-TST rate coefficient has the form k SC-TST ϭk TST •⌫, where ⌫ depends on the zero point corrected barrier, ⌬E 0 , and the barrier curvature, ͉ F ‡ ͉. Our rate expression is stable to arbitrarily low temperatures, as opposed to purely harmonic SC-TST, because we identify the maximum possible semiclassical action in the reaction coordinate. For low temperatures, we derive an analytical approximation for ⌫ that is proportional to e ␤ ⌬E 0 . We develop a theory for the tunneling crossover temperature, T x , yielding k B T x Хប͉ F ‡ ͉⌬E 0 /(2 ⌬E 0 Ϫប͉ F ‡ ͉ln 2), which generalizes the harmonic theory for systems with large but finite barriers. We have calculated rate coefficients and crossover temperatures for the O͑1͒→O͑4͒ jump in H-Y and D-Y zeolites, yielding T x ϭ368 K and 264 K, respectively. These results suggest that tunneling dominates proton transfer in H-Y up to and slightly above room temperature, and that true proton transfer barriers are being underestimated by neglecting tunneling in the interpretation of experimental mobility data.

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We have studied the convergence properties of embedded and constrained cluster models of proton transfer in zeolites. We applied density functional theory to describe clusters and ONIOM to perform the embedding. We focused on converging the reaction energy and barrier of the O(1) to O(4) jump in H-Y zeolite as well as vibrational and structural aspects of this jump. We found that using successively larger clusters in vacuo gives convergence of this reaction energy to 14 ± 2 kJ mol(-)(1) and the barrier to 135 ± 5 kJ mol(-)(1) at a cluster size of 5 Å, which contains 11 tetrahedral (Si or Al) atoms. We embedded quantum clusters of various sizes in larger clusters with total radii in the range 7-20 Å, using the universal force field as the lower level of theory in ONIOM. We found convergence to the same values as the constrained clusters, without the use of reactive force fields or periodic boundary conditions in the embedding procedure. For the reaction energy, embedded cluster calculations required smaller clusters than in vacuo calculations, reaching converged reaction energies for quantum systems containing at least 8 tetrahedral atoms. In addition, optimizations on embedded clusters required many fewer cycles, and hence much less CPU time, than did optimizations on comparable constrained clusters.

U nderstanding the polymerization of silicic acid in aqueous solution has been an active area of research for some time. 1,2 This reaction system lies at the heart of solÀgel processing as well as the synthesis of porous materials such as zeolites 3 and ordered mesoporous materials. 4,5 Improved understanding of the fundamental mechanisms involved can be expected to have a significant impact in the control of material structure and proper-