Semiclassical limit

Miller W H 1975 Semiclassical limit of quantum mechanical transition state theory for nonseparable systems J. Chem. Phys. 62 1899... 

The STM postulated tunneling matrix element distribution P(A) oc 1 /A implies a weakly (logarithmically) time-dependent heat capacity. This was pointed out early on by Anderson et al. [8], while the first specific estimate appeared soon afterwards [93]. The heat capacity did indeed turn out time dependent however, its experimental measures are indirect, and so a detailed comparison with theory is difficult. Reviews on the subject can be found in Nittke et al. [99] and Pohl [95]. Here we discuss the A distribution dictated by the present theory, in the semiclassical limit, and evaluate the resulting time dependence of the specific heat. While this limit is adequate at long times, quantum effects are important at short times (this concerns the heat condictivity as well). The latter are discussed in Section VA. 

When studying the border of universality, we always need to consider the limit of large graphs, that is, ue —> oo. This limit is in general not well defined, but may often be obvious from the examples considered. We will thus define the semiclassical limit loosely via a family of unitary-stochastic transition matrices Tn and associate USE s and take ue —> oo. The leading term in (13) then gives a condition for a family to show deviations from RMT statistics in terms of the spectrum of T the diagonal term must obey... 

Hence, for finite times t any initial vector from 77 remains in this subspace up to an error of the order h°°. Moreover, for semiclassically large times, t 0, the error is still approaching zero in the semiclassical limit. Since the construction of the projectors fl was based on the classical projection matrices (6) associated with positive and negative... 

The classical spin-orbit dynamics that emerge from the two semiclassical limits are the following (Bolte and Glaser, 2005) In case (i) one obtains the... 

Thus the primary and secondary isotope eifects are all within the semiclassical limits and their relationship is in full accord with the semiclassical Swain-Schaad relationship. There is no indication from the magnitudes of the secondary isotope elfects in particular of any coupling between motion at the secondary center and the reaction-coordinate for hydride transfer. Thus the sole evidence taken to indicate tunneling is the rigorous temperature-independence of the primary isotope elfects. 

The observation of exalted secondary isotope effects, i.e., those that are substantially beyond the semiclassical limits of unity and the equilibrium isotope effect. These observations require coupling between the motion at the primary center and motion at the secondary center in the transition-state reaction coordinate, and in addition that tunneling is occurring along the reaction coordinate. 

Alternatively, in Bonella and Coker s derivation [118,119] the difference between the two classical Hamiltonians in Eq. (116) arises from a different H-dependence in the semiclassical limit > 0. To illustrate the idea, the quantum-mechanical mapping Hamiltonian (112) is rewritten by introducing... 

It should be noted, however, that the limit 0 is only a formal procedure, which does not necessarily lead to a unique or correct semiclassical limit. In the case of the mapping formulation, this is because of the following reasons (i) For a given molecule, the frequencies f)mi(x) will in general also depend in a nontrivial way on h. (ii) A slowly varying term may as well be included in the stationary phase treatment [147]. (iii) As indicated by the term resulting from the commutator = 8 , the effective action constant ... 

Taking the semiclassical limit, these two cases become very different. For thick, classically forbidden barriers, Eq. (2.8) becomes... 

The semiclassical theory of rates has along history.Here, we will just review briefly the final product, a unified theory for the rate in a dissipative system, at all temperatures and for arbitrary damping. Two major routes have been used to derive the semiclassical theory. One is based on the so called ImF method, whereby, one derives a semiclassical limit for the imaginary part of the free energy. This route has the drawback that the semiclassical limit is treated differently for temperatures above and below the crossover temperature. - ... 

Poliak and Eckhardt have shown that the QTST expression for the rate (Eq. 52) may be analyzed within a semiclassical context. The result is though not very good at very low temperatures, it does not reduce to the low temperature ImF result. The most recent and best resultthus far is the recent theory of Ankerhold and Grabert," who study in detail the semiclassical limit of the time evolution of the density matrix and extract from it the semiclassical rate. Application to the symmetric one dimensional Eckart barrier gives very good results. It remains to be seen how their theory works for asymmetric and dissipative systems. 

We start with a simple model of the two-particle intracollisional correlation function in the semiclassical limit [324],... 

In the semiclassical limit, only the classically allowed paths are taken into account, and we get... 

Makri and Miller [1987b], Doll and Freeman [1988], Doll et al. [1988] (see also the review by Makri [1991b]) exploited the stationary phase approximation (i.e., the semiclassical limit) as an initial approximation to the path integral. For example, for the multidimensional integral of the form J dx exp[iS(x)], one may obtain the following approximation [Makri, 1991b] ... 

C. C. Martens and J. Y. Fang. Semiclassical-limit molecular dynamics on multiple electronic surfaces. J. Chem. Phys., 106(12) 4918-4930, 1997. 

The projection integrals (Yjo y>7) can be interpreted as the (discrete) angular momentum representation of the initial bending wavefunction in the electronic ground state. Employing the semiclassical limit for the spherical harmonics,... 

Then, in this spirit, the semiclassical limit of the ACF (141) becomes... 

Besides, in the situation without damping, it has been shown that the semiclassical limit leads us to write... 

Thus, there is the following expression for the semiclassical limit of quantum indirect damping ... 

Equation (153) is the semiclassical limit of the quantum approach of indirect damping. Now, the question may arise as to how Eq. (153) may be viewed from the classical theory of relaxation in order to make a connection with the semiclassical approach of Robertson and Yarwood, which used the classical theory of Brownian motion. 

In Section IE, a theoretical approach of the quantum indirect damping of the H-bond bridge was exposed within the strong anharmonic coupling theory, with the aid of the adiabatic approximation. In Section III, this theory was shown to reduce to the Marechal and Witkowski and Rosch and Ratner quantum approaches. In Section IV, this quantum theory of indirect damping was shown to admit as an approximate semiclassical limit the approach of Robertson and Yarwood. 

Robertson and Yarwood [46], in the semiclassical limit, without Davydov coupling and direct damping. 

P Other dimensionless one related to a° within the semiclassical limit. 

G//(f)]semi Semiclassical limit of the ACF with direct and indirect dampings. 

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