Tunneling splittings

Aside from merely calculational difficulties, the existence of a low-temperature rate-constant limit poses a conceptual problem. In fact, one may question the actual meaning of the rate constant at r = 0, when the TST conditions listed above are not fulfilled. If the potential has a double-well shape, then quantum mechanics predicts coherent oscillations of probability between the wells, rather than the exponential decay towards equilibrium. These oscillations are associated with tunneling splitting measured spectroscopically, not with a chemical conversion. Therefore, a simple one-dimensional system has no rate constant at T = 0, unless it is a metastable potential without a bound final state. In practice, however, there are exchange chemical reactions, characterized by symmetric, or nearly symmetric double-well potentials, in which the rate constant is measured. To account for this, one has to admit the existence of some external mechanism whose role is to destroy the phase coherence. It is here that the need to introduce a heat bath arises. 

The long-time behaviour of (,a t) may essentially differ from that of C(r), but this affects mostly the form of the spectral line at (0 0, and seemingly this is immaterial for determining the tunneling splitting [Sasetti and Weiss 1990]. 

Therefore, the tunneling splitting decreases with increasing n, in accord with experiment. The weak-coupling formula holds for C Ql/hmiCol <4 1. 

Coleman s method can be applied to finding the ground state tunneling splitting in a symmetric double well [Vainshtein et al. 1982], for some... 

The formula for the tunneling splitting in two dimensions is a simple generalization of (3.70),... 

In order to better understand the origin of the first term in (5.59) we separate from the Hamiltonian the part proportional to and average it over the equilibrium oscillators. This gives rise to an effective tunneling splitting A n,... 

Tunneling splittings of different vibrational levels in the exeited A Bj elee-tronie state of the tropolon moleeule... 

Band assignment Vibration frequency [cm- ] Tunneling splitting [cm ] ... 

A major role is played by vibrations with frequencies 318cm and 1378cm . The tunneling splitting increases by several times as the quantum numbers of these vibrations increase. The... 

The results for the tunneling splitting calculated with the use of some of the earlier proposed reaction paths for a single PES (4.40) (with the parameters adopted here) are collected by Bosch et al. [1990]. All of them underestimate by at least an order of magnitude the numerically exact value 10.6 cm which is also given in that paper. The parameters C and Q hit the intermediate region between the sudden and adiabatic approximations, described in sections 2.5 and 4.2, and neither of these approximations is quantitatively applicable to the problem. 

A unique example of observation of tunneling splitting is given by Oppenlander et al. [1989]. Upon replacing the host benzoic acid dimer by a thioindigo molecule of nearly the same size, the resulting bias accidentally turns out to be small, of order of A. The 4x4 Hamiltonian of the complex of two dimers and the guest molecule is... 

A calculation of tunneling splitting in formic acid dimer has been undertaken by Makri and Miller [1989] for a model two-dimensional polynomial potential with antisymmetric coupling. The semiclassical approximation exploiting a version of the sudden approximation has given A = 0.9cm" while the numerically exact result is 1.8cm" Since this comparison was the main goal pursued by this model calculation, the asymmetry caused by the crystalline environment has not been taken into account. 

Fig. 53. The temperature dependence of 7", in tiglic acid. Methyl tunnel splitting (ueV)...

When Va varied within the interval 1-8 cm the tunneling splitting was found to depend nearly linearly on Fj, in agreement with the semiclassical model of section 3.5 [see eq. (3.92)], and the prefactor AjA ranged from 0.1 to 0.3, indicating nonadiabatic tunneling. Since this model is one-dimensional, it fails to explain the difference between splittings in the states with the [Pg.127]

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