Quantum tunnel splitting

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. 

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... 

Hr(p which depend on the radial, nh, and the deformation, cr, quantum numbers. For 0cr AU the levels are grouped so that the gaps between groups are approximately the same and equal to hwi whereas the tunneling splitting which occurs for p > 1 in each group of p levels is exponentially small in the parameter 4AC//fcfi>/ 6U21... 

Semiclassical techniques like the instanton approach [211] can be applied to tunneling splittings. Finally, one can exploit the close correspondence between the classical and the quantum treatment of a harmonic oscillator and treat the nuclear dynamics classically. From the classical trajectories, correlation functions can be extracted and transformed into spectra. The particular charm of this method rests in the option to carry out the dynamics on the fly, using Born Oppenheimer or fictitious Car Parrinello dynamics [212]. Furthermore, multiple minima on the hypersurface can be treated together as they are accessed by thermal excitation. This makes these methods particularly useful for liquid state or other thermally excited system simulations. Nevertheless, molecular dynamics and Monte Carlo simulations can also provide insights into cold gas-phase cluster formation [213], if a reliable force field is available [189]. 

M. D. Coutinho Neto, A. Viel, and U. Manthe, The ground state tunneling splitting of malonaldehyde Accurate full dimensional quantum dynamics calculations. J. Chem. Phys. 121, 9207 9210 (2004). 

Since the particle (electron) in one redox site can respond to the presence of a second deep well out in the distance only to the extent that this quantum tail is non-zero, the tunneling splitting A decays essentially exponentially just as the wavefunction tail in Eqs. 18-19 above. More precisely, we write ... 

In this case the adjustable parameters of the PES (4.41) are V0 = 18.52 kcal/mol, V = 5.62 kcal/mol, C = 1.07, 11 = 0.91, and w0 = 1.50x 1014s 1 [Bosch et al., 1990], As in the case of malonaldehyde, the PES parameters place this system between the sudden and adiabatic regimes. The PES contour map and the instanton trajectory for this case are shown in Figure 6.11. Benderskii et al. [1993] have utilized the instanton analysis to obtain the prefactor Bt = 54 and the tunneling splitting 1.4 cm-1, which is in excellent agreement with the value 1.30 cm-1 obtained by Bosch et al. [1990] from a quantum mechanical calculation. 

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