Semiclassical model states

Miller W H 1974 Quantum mechanical transition state theory and a new semiclassical model for reaction rate constants J. Chem. Phys. 61 1823-34... 

When Va varied within the interval 1-8 cm the tunneling splitting was found to depend nearly linearly on Fj, in agreement with the semiclassical model of section 3.5 [see eq. (3.92)], and the prefactor AjA ranged from 0.1 to 0.3, indicating nonadiabatic tunneling. Since this model is one-dimensional, it fails to explain the difference between splittings in the states with the [Pg.127]

F2 F4 - 2F and F3 F4 - Fi, correspond to the forbidden orbits. From a quantum-mechanical point of view there is no semiclassical closed orbit to explain these frequencies. However, they can be understood in the frame of the quantum interference (QI) model [10] as two-arms Stark interferometers [11]. Within the QI model [10] the temperature damping of the oscillation amplitude is given by the energy derivative of the phase difference ((pi -cpj ) between two different routes i and j of a two-arms interferometer. This model states that 5(cpi - cpj) / de = ( /eB) <3Sk / de, where Sk is the reciprocal space area bounded between two arms. Since 3(difference between the effective masses of the two arms of the interferometer, the associated effective mass is given by m = mj - mj, where nij and mj are the partial effective masses of the routes i and j. In our case an interferometer connected with the frequency F3 consists of two routes, abcdaf and abef and another interferometer, connected with the frequency F2, includes two cyclotron orbits, abcdaf and abebef (see Fig 5). 

W. H. Miller, Quantum Mechanical Transition State Theory and a New Semiclassical Model for Reaction Rate Constants, J. Chem. Phys., 61 (1974) 1823. 

Expectation Values on Coherent States Relation with the Semiclassical Model... 

We have stated that we can recover the evolution of the semiclassical model from the Floquet evolution in the interaction representation by taking initial states in which the photon held is in a coherent state. This can be formulated more precisely by the following statements. 

The last expression is the expectation value calculated with the semiclassical model with initial phase 0O. We thus conclude that, if one considers only observables of the molecule, the Floquet evolution with a coherent state in the initial condition is equivalent to the semiclassical model. We remark that a somewhat related construction, linking the evolution from cavity dressed states directly to the semiclassical model (i.e., without the intermediate level of Floquet states as we do here), was established in Ref. 16. 

In conclusion to this section, band-shape analysis of vibrational spectra and ground state splitting observed with INS demonstrate that proton transfer dynamics are quantal in nature, even at room temperature. Semiclassical models are not relevant. The dramatic failure of quantum chemistry to account for the observed dynamics should be regarded as one of the major unsolved theoretical problems at the present time. 

To return now to the semiclassical model of nonadiabatic behavior, one can describe reactions on the spin-state (diabatic) PESs as follows The system will move throughout phase space on the reactant PES until it reaches a point where the product PES has the same energy as the reactant one. At that point, it may either remain on the reactant PES or hop over onto the product one. The Landau-Zener formula for curve crossing in one-dimensional systems has often been used in a multidimensional context (10) as a useful approximation for the probability p with which this hop occurs, leaving (1 - p) oi the trajectories to continue on the initial PES (Fig. 1) ... 

A semiclassical model to calculate the probability pvl that a molecule follows a diabatic pathway, by retaining the electronic stmcture of the initial electronic state as it passes through an avoided crossing, was developed independently in 1932 by Landau and by Zener.154 In the Landau Zener model, the nuclear movements are described classically, that is, nuclear motion enters parametrically and only a single reaction coordinate, as shown in diagram in Figure 2.27, is considered. In the Landau Zener expression given in Equation 2.53 ... 

S. Guerin, F. Monti, J.M. Dupont, H. Jauslin, On the relation between cavity-dressed states, Floquet states, RWA and semiclassical models, J. Phys. A Math. Gen. 30 (1997) 7193. 

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