Tunnelling anharmonic effects

Rate constants are calculated with our own Python code, which is interfaced to NWChem. Bimolecular rate constants are computed according to Eqs (7.19) and (7.20) except in our earlier work (Section 7.4.3.1) where we did not include tunneling corrections. (Simple tunneling corrections mostly cancel for the relative rate constants in Eq. (7.42).) Unimolecular rate constants are given by Eqs (7.20) and (7.21) and anharmonic effects are incorporated via Eq. (7.28) for low-frequency vibrations up to 110 cm This implies that on the order of 10 modes per transition state of hydrogen abstraction and about 5 modes per transition state of phenyl migration and reactants are anharmonically corrected. We employ the analytical kinetic model described in Section 13.23 to obtain a//3-selectivities in Section 7.4.3. 

This chapter is devoted to tunneling effects observed in vibration-rotation spectra of isolated molecules and dimers. The relative simplicity of these systems permits one to treat them in terms of multidimensional PES s and even to construct these PES s by using the spectroscopic data. Modern experimental techniques permit the study of these simple systems at superlow temperatures where tunneling prevails over thermal activation. The presence of large-amplitude anharmonic motions in these systems, associated with weak (e.g., van der Waals) forces, requires the full power of quantitative multidimensional tunneling theory. 

There are a plethora of isotope effects accompanying hydrogen bond formation [5]. Isotopic substitution leads to a substantial decrease of anharmonicity of stretching vibrations, while also producing major effects when the barrier height is lowered. In this case the fundamental vibrational level is lowered with respect to the barrier top and a marked reduction of the tunnelling effect takes place. 

An additional factor to consider is the magnitude of the anharmonic contribution to the entropy of proteins. Analyses of molecular dynamics simulations have demonstrated that the major anharmonic contributions can be ascribed to multiple conformations for individual atoms.195 In BPTI and lysozyme, an estimate of the change in entropy due to these effects (e.g., 89 atoms in lysozyme have multiple wells) yields a correction of less than 2% for the classical entropy. Thus multiple conformations appear not to be important for the residual entropy at ordinary temperatures.389 However, near absolute zero (1 to 2 K) there are data that suggest that several minima ( tunneling states ) contributes significantly to the entropy.390... 

Similar variations of the isotopic shift factor have been reported for bulk hydroxyls (161,643—644). Various models were proposed and the anomalous isotopic shift factor was associated with the anharmonicity of the OH bond (161,644). In a recent review (643), three main causes of the anomalous OH/OD isotopic effects were identified (i) higher anharmonicity of the OH vibrations as compared to OD (ii) contribution of the bending modes via coupHng, and (iii) tunneling effects (taking place with strongly hydrogen-bonded systems). It was also pointed out that weak H-bonds complicate the situation because under these circumstances, the anharmonicity decreases. 

We need anharmonic energy level calculations with all six vibrational degrees of fteedom. Although we have a qualitatively correct semiclassical picture of the tunneling splitting and its dependence on monomer stretch and rotational excitations, the details are far fiom settled. We need calculations of the effect of 06 excitation on the tunneling splitting. 

Figure 7.8 Schematic illustration of the effects of anharmonicity on the reaction probability (a) the anharmonicity of the barrier shape along the reaction direction reduces the tunneling probability and (b) excitation of the vibrational mode changes the effective curvature felt by the reaction mode and affects the sharpness of the reaction probability.

---