Metastable potential well V(s) with a barrier of height V 0 and unstable frequency ıω b at the barrier top. Dashed line indicates the barrier potential V b (s).
Euclidean action S as a function of the first order Fourier coefficient at temperatures lower than the crossover temperature, β > β c . Classical instanton solution corresponds to the minimum of S at .
WKB action W(E) (a), and exp ( − W(E) − βE) (b) for an Eckart barrier of height V 0 and temperature β > β c .
Instantons (solid lines) on classical PES. Longer instantons correspond to lower temperatures. Solid circle denotes the saddle point of PES. Coordinates s and y correspond to the unstable and stable normal coordinates at the saddle point, respectively.
Reaction probability P ij as a function of total energy E for the 2D Eckart barrier in Eq. (6.1) . The height of the barrier is 0.425 eV.
Thermal rate constant as a function of inverse temperature for 2D Eckart barrier, Eq. (6.1) . Solid line represents exact quantum results of Eq. (6.2) ; dashed line: transition state theory results; open circles: semiclassical instanton expressions in Eqs. (2.18) and (2.19) ; and solid circles: the improved semiclassical instanton expressions (3.65) and (4.42) . Atomic units are used.
Logarithm of thermal rate constant log10 k as a function of inverse temperature. HT and LT stand for high and low temperature expressions. Atomic units are used.
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