Parabolic barrier dynamics

The parabolic barrier plays a special role in rate theory. The GLE (with space-independent friction) may be solved analytically using Laplace transforms. The two-dimensional Fok-ker-Planck equation derived from the Langevin equation may be solved analytically, as was done by Kramers in his famous paper of 1940. In this section we present some of the analytic results for the parabolic barrier dynamics. These results are important from both a conceptual and a practical point of view. Later we shall see how one returns to the parabolic barrier case as a source of comprehension, approximation, etc. 

The dynamics of the normal mode Hamiltonian is trivial, each stable mode evolves separately as a harmonic oscillator while the imstable mode evolves as a parabolic barrier. To find the time dependence of any function in the system phase space (q,pq) all one needs to do is rewrite the system phase space variables in terms of the normal modes and then average over the relevant thermal distribution. The continuum limit is introduced through use of the spectral density of the normal modes. The relationship between this microscopic view of the evolution... 

A study of the effeets of space and time dependent frietion was presented in Ref 68. One finds a substantial reduction of the rate relative to the parabolic barrier estimate when the friction is stronger in the well than at the barrier. In all eases, the effeets beeome smaller as the redueed barrier height beeomes larger. Comparison with moleeular dynamics simulations shows that the optimal planar dividing surfaee estimate for the rate is usually quite aeeurate. 

The main difference between the two approaches is that PGH consider the dynamics in the normal modes coordinate system. At any value of the damping, if the particle reaches the parabolic barrier with positive momentum i n the unstable mode p, it will immediately cross it. The same is not true when considering the dynamics in the system coordinate for which the motion is not separable even in the barrier region, as done by Mel nikov and Meshkov. In PGH theory the... 

An earlier approacfr was to solve the quantum problem in the high-temperatme limit using Markovian dynamics and assuming a parabolic barrier. The quantum rate has the following formF - ... 

That is, while numerical accuracy of Kramers Eq. (3.41) would validate the slow variable picture of Eq. (3.29), that of Grote and Hynes Eq. (3.45) does not validate any physical model for the reaction dynamics. Rather, it validates only the accuracy of (a) the partial clamping model [21], used to compute the quantities in Eq. (3.45) and (b) the parabolic barrier approximation U x) U x ) — XjlnKsP y to the gas phase reaction coordinate potential. The accuracy of these, however, requires only that y = x — jc remains small prior to the decision to react. 

The separability of the Hamiltonian in the normal mode form implies that the dynamics is in some sense trivial. One must only consider the continuum limit of a collection of independent harmonic oscillators and a single parabolic barrier. As described in Sec. III.D, this simple dynamics leads to some important relations between the Hamiltonian approach and the more standard stochastic theories. Multidimensional generalization of the parabolic barrier case will be discussed briefly in Sec. VIII. 

The parabolic barrier case demonstrates that the effect of the medium is to replace the original reaction coordinate q by a collective mode p along which the dynamics is trivial. It is useful to define a collective bath mode o orthogonal to the unstable mode p as... 

The value of the transmission coefficient kt is shown for each feature in Table 2. (The value of kt for the last feature is greater than 1 because it includes contributions from higher energy transition states that have not been included in the fit.) Many of the values of the transmission coefficients are very close to unity, suggesting that these features correspond to quantized transition states that are nearly ideal dynamical bottlenecks to the reactive flux. Several of the values of kt deviate from unity this could be the result of the assumption of parabolic effective potential barriers or from recrossing or other multidimensional effects. 

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