Bell-Limbach model

The Bell-Limbach model is not designed to give definite interpretations of Arrhenius curves of hydrogen transfer reactions which have to come from more sophisticated methods. However, it provides an opportunity to check whether the number of parameters describing a given set of Arrhenius curves matches or exceeds the number of parameters necessary to describe the same set in terms of sums of single Arrhenius exponentials. This check also tells whether it is useful... 

In this section first different models of single H-transfer will be reviewed, including primary kinetic H/D isotope effects, where the focus is on the Bell-Limbach model. Then formal kinetics will be used to describe multiple hydrogen transfers and their kinetic isotope effects. 

The fit of the experimental data to Eq. (6.48) is very satisfactory, as illustrated in Eig. 6.22(a), where the solid lines were recalculated here using the Bell-Limbach model, with the parameters included in Table 6.4. This result also means that there is no substantial decrease in the zero-point energies of the two protons in the cis-intermediate states as compared to the initial and final trans-states, as this would increase the HD/DD isotope effect beyond the value of 2 as was illustrated in Fig. 6.14(c). 

The scope of this chapter is, therefore, (i) to review the Bell-Limbach tunneling model in comparison with other models and its use for describing single steps of multiple hydrogen transfer networks and (ii) to review applications of this approach in a number of cases which have been studied mainly by NMR. A description of the techniques used for the determination of rate constants of Id-transfer will not be included in this chapter readers interested in this problem are referred to a recent review [4]. 

The simplest tunnel model which allows one to calculate Arrhenius curves of H-transfer reactions is the Bell tunneling model [7] which has been modified in our laboratory [9]. The model has been reviewed recently by Limbach et al. [26]. It is visualized in Fig. 6.8 which will be explained in the following. 

Figure 6.9 Arrhenius curves of H and Dtrans-fer calculated according to the Bell-Limbach tunneling model. Minimum energyfortunnel-ing to occur E/ = 12.55 kj moh, barrier heights = 20.9 kJ moh E° = 27.2 kJ mol", tunnel-...

This means that at low temperature where P is large the HD reaction is ca. twice as fast as the DD reaction. Equation (6.31) has been used in connection with the Bell-Limbach tunneling model to describe the stepwise double proton transfer in porphyrins, azophenine, and oxalamidines, as will be discussed in Section 6.3. Smedarchina et al. [16] used the same equations for their quantum-mechanical treatment of the porphyrin tautomerism. 

Table 6.4 Bell-Limbach tunneling model parameters of various H-transfers )... 

Figure 6.32 Arrhenius curves of the solid state tautomerism of benzoic acid dimers. Data for the HH and HD reactions taken from Ref [74], and for the DD reaction from Ref [72cj. The solid lines were calculated using the Bell—Limbach tunneling model with the parameters listed in Table 6.4.

Figure 6.34 Arrhenius diagram for the double proton and deuteron transfer in the cyclic trimers of solid DMP. Adapted from Ref [25aj. The solid curves were calculated using the Bell-Limbach tunneling model as described in the text.

Using dynamic solid-state CP MAS NMR spectroscopy, the kinetics of the degenerate intermolecular double and quadruple proton and deuteron transfers in the cyclic dimer of N labelled polycrystalline 3,5-diphenyl-4-bromopyrazole (DPBrP) and in the cyclic tetramer of N labelled polycrystalline 3,5-diphenylpyrazole (DPP) have been studied in a wide temperature range at different deuterium fractions in the mobile proton sites. Rate constants were measured on a millisecond time scale by line shape analysis of the doubly N labelled eompounds and by magnetisation transfer experiments on a second timescale of the singly N labelled compounds in order to minimise the effeets of proton-driven N spin diffusion. The Arhenius curves of all processes were found to be nonlinear and indicated tunneling processes at low temperatures. In a preliminary analysis, they were modelled in terms of the Bell-Limbach tuimeling model. 

Solid state proton transfer (SSPT) occurs between tautomers and, even if the initial and the final are the same (degenerate tautomerism, K = 1), it constitutes one ofthe best known kinetic processes. The loss of freedom due to the crystal structure allows for accurate kinetic models to be used, including the Car-Parrinello [51] and Bell-Limbach tunneling model [52]. This field owes much to the works of Limbach et /. [53, 54] and of Claramunt ef /. [55, 56]. 

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