Instanton method

In fig. 26 the Arrhenius plot ln[k(r)/coo] versus TojT = Pl2n is shown for V /(Oo = 3, co = 0.1, C = 0.0357. The disconnected points are the data from Hontscha et al. [1990]. The solid line was obtained with the two-dimensional instanton method. One sees that the agreement between the instanton result and the exact quantal calculations is perfect. The low-temperature limit found with the use of the periodic-orbit theory expression for kio (dashed line) also excellently agrees with the exact result. Figure 27 presents the dependence ln(/Cc/( o) on the coupling strength defined as C fQ. The dashed line corresponds to the exact result from Hontscha et al. [1990], and the disconnected points are obtained with the instanton method. For most practical purposes the instanton results may be considered exact. 

Monte Carlo Samphng of Tunneling Paths The Path Integral Instanton Method. .. 67... 

The instanton method takes into account only the dynamics of the lowest energy doublet. This is a valid description at low temperature or for high barriers. What happens when excitations to higher states in the double well are possible And more importantly, the equivalent of this question in the condensed phase case, what is the effect of a symmetrically coupled vibration on the quantum Kramers problem The new physical feature introduced in the quantum Kramers problem is that in addition to the two frequencies shown in Eq. (28) there is a new time scale the decay time of the flux-flux correlation function, as discussed in the previous Section after Eq. (14). We expect that this new time scale makes the distinction between the comer cutting and the adiabatic limit in Eq. (29) to be of less relevance to the dynamics of reactions in condensed phases compared to the gas phase case. 

To find the values of A we shall use the instanton method turning to operate with the Euclidean time t —> —it. Taking initial and final times as Tj = -oo, Tf = 0 we present the amplitude A as... 

In principle, the instanton method should be applied to the amplitude A directly, but it is more convenient to consider the product A A presenting A as... 

We see that the factor (G5E)N (t2 vq vm E)n represents the probability of tunneling of N electrons or holes from the grain to the metal with 5E being the typical energy of one electron (hole) excitation in the metal. Determination of the factor (3 in (95) requires an application of the instanton method which is capable of careful description of quantum tunneling process between... 

Benderskii managed to solve this problem in the deep tunneling limit using the instanton method. Roughly speaking, an instanton is the most probable among the... 

Computational methods now exist that include contributions from all vibrational modes to the H/D-transfer process, thus eliminating the need to introduce any empirical parameters, e.g., variational transition state theory with semiclassical tunneling corrections (Truhlar, D. G. Garett, B. C. Klippen-stein, S. J.J. Phys. Chem. 1996, 100, 12771) and the approximate instanton method (Siebrand, W Smedarchina, Z. Zgierski, M. Z. Femandez-Ramos, A. Int. Rev. Chem. Phys. 1999, 18, 5). 

The instanton method defines the thermal rate constant for tunneling transfer in terms of the action S (T) (expressed hereafter in units h) along this extremal path ... 

The parameter controlling the concertedness is the proton-proton correlation represented by a term that is bilinear in the local proton coordinates. The model is combined with a previously developed approach to single proton transfer based on an approximate instanton method. This leads to the recognition of three coupling regimes governing the mechanism of nondassical double proton transfer. 

Accordingly, a key requisite for treating molecules is the ability to evaluate tunneling in many degrees of freedom. The instanton method... 

The cusp problem offers a curious lesson. We will find that, although in cylindrical coordinates the Schrodinger equation is nonsep-axable and hence involves tunneling in two coordinates, the potentieil is nicely quadratic near its minima and thereby amenable to the standard instanton methods. 

The instanton method is particularly congenial for the large-D limit, in that the zero-energy trajectory in imaginary time corresponds precisely to tunneling between minima in the effective potential with D oo. Despite the infinite effective mass, timneling still occurs because quantum fiuctuations persist in this limit. As usual with semiclassical methods, the dynamical aspects are evaluated by classical mechanics. However, the unusual and striking aspect of our... 

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