Schematic representation of the basic model of double-proton transfer along parallel equivalent hydrogen bonds in a system with (a) or (b) symmetry.
Schematic two-dimensional potential-energy surfaces appropriate to double-proton transfer for weak (a), intermediate (b), and strong (c) hydrogen-bond couplings, respectively, showing the equilibrium configurations (MIN), the stable intermediates (INT), the second-order saddle point (MAX), and the first-order saddle points (TS).
Parameters governing the competition between different tunneling paths on the two-dimensional surface of Fig. 2(a). The solid lines depict the ratio , i.e., of the tunneling barrier for stepwise transfer and the zero-point level of the INT well, evaluated numerically, for (top) a very weak and (bottom) a very strong hydrogen bond and a typical value of . The dashed line depicts the ratio for , which depends only on , and shows explicitly that the 1D instanton corresponding to synchronous tunneling is the dominant tunneling trajectory at zero temperature.
One-dimensional potentials and for the concerted and composite instantons, defined by Eqs. (19) and (18), represented by dot-dash and solid lines, respectively. The corresponding potential for stepwise transfer is half that for the composite instanton, the tunneling path at being indicated by a dashed line.
Inverse (dimensionless) temperatures , marking the boundaries between prevailing transfer mechanisms, plotted as a function of the coupling between the hydrogen bonds. The solid line marks the crossover temperature below which the transfer is dominated by tunneling along the coordinate of synchronous (1D) transfer. The dashed line marks the boundary between 1D and 2D concerted tunnelings, the shaded area indicating the region where 2D transfer is faster. The dot-dash line approximately marks the boundary between stepwise and concerted transfers; it exists only where the coupling supports a stable intermediate.
Transfer rate constants calculated for a 2D model potential governed by three parameters with values chosen to represent porphine (and three of its isotopomers), namely, , , and . Thick solid lines depict overall rate constants for HH, HD, DD, and TT in descending order. Thin lines illustrate the components of the rate constant for HH: solid and dashed lines the concerted and stepwise components, respectively, and the dot-dash line represents in Eq. (29), showing the contribution of the tunneling to the observed activation energy.
(Color online) Comparison of the rate constants of tautomerization of porphine, illustrated in the inset, with the available experimental data for HH (circles), HD (solid triangles), DD (squares), and TT (open triangles).
Parameters of several characteristic configurations of the model surfaces in Eqs. (2)–(5). The energies are measured relative to the energy of the global minimum. Coordinates are in units of , energies in , and frequencies in (see text for details). The frequencies are listed for double-proton transfer of the type, where and and , respectively.
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