Upside-down barrier

Fig. 3. One-dimensional barrier along the coordinate of an exoergic reaction. Qi(E), Q i(E), QiiE), Q liE) are the turning points, coo and CO initial well and upside-down barrier frequencies, Vo the barrier height, — AE the reaction heat. Classically accessible regions are 1, 3, tunneling region 2.

Again, as in the previous section, we look for the stationary points of the path integral, i.e., the trajectories that extremize the Eucledian action (3.11) and thus satisfy the classical equation of motion in the upside-down barrier. 

In addition to the trivial solutions, there is a /S-periodic upside-down barrier trajectory called instanton, or bounce [Langer 1969 Callan and Coleman 1977 Polyakov 1977]. At jS oo the instanton dwells mostly in the vicinity of the point x = 0, attending the barrier region (near x ) only during some finite time (fig. 20). When jS is raised, the instanton amplitude... 

This formula, however, tacitly supposes that the instanton period depends monotonically on its amplitude so that the zero-amplitude vibrations in the upside-down barrier possess the smallest possible period 2nla>. This is obvious for sufficiently nonpathological one-dimensional potentials, but in two dimensions this is not necessarily the case. Benderskii et al. [1993] have found that there are certain cases of strongly bent two-dimensional PES when the instanton period has a minimum at a finite amplitude. Therefore, the cross-over temperature, formally defined as the lowest temperature at which the instanton still exists, turns out to be higher than that predicted by (4.7). At 7 > Tc the trivial solution Q= Q Q is the saddle-point coordinate) emerges instead of instanton, the action equals S = pV (where F " is the barrier height at the saddle point) and the Arrhenius dependence k oc exp( — F ") holds. 

It is noteworthy that eq. (4.15a) is nothing but the linearized classical upside-down barrier equation of motion (8S/8x = 0) for the new coordinate x. Therefore, while x = 0 corresponds to the instanton, the nonzero solution to (4.15a) describes how the trajectory escapes from the instanton solution, when it deviates from it. The parameter X, referred to as the stability angle [Gutzwil-ler 1967 Rajaraman 1975], generalizes the harmonic-oscillator phase co, which would appear in (4.15), if CO, were a constant. The fact that X is real indicates the aforementioned instability of the instanton in two dimensions. Guessing that the determinant det( — -I- co, ) is a function of X only,... 

The dimensionless upside-down barrier frequency equals = 2(1 — and the transverse frequency Qf = Q. The instanton action at = oo in the one-dimensional potential (4.41) equals [cf. eq. (3.68)]... 

In the nonadiabatic limit ( < 1) B = nVa/Vi sF, and at 1 the adiabatic result k = k a holds. As shown in section 5.2, the instanton velocity decreases as t] increases, and the transition tends to be more adiabatic, as in the classical case. This conclusion is far from obvious, because one might expect that, when the particle loses energy, it should increase its upside-down barrier velocity. Instead, the energy losses are saturated to a finite //-independent value, and friction slows the tunneling motion down. 

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