Semiclassical prefactor

The main numerical difficulty in implementing this surface hopping method for condensed phase problems is that the semiclassical prefactor involves the derivatives... 

This formula resembles (3.32) and, as we shall show in due course, this similarity is not accidental. Note that at n = 0 the short action 1 2 ( q) taken at the ground state energy Eq is not equal to the kink action (3.68). Since in the harmonic approximation for the well Tq = 2n/o)o, this difference should be compensated by the prefactor in (3.74), but, generally speaking, expressions (3.74) and (3.79) are not identical because eq. (3.79) uses the semiclassical approximation for the ground state, while (3.74) does not. 

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]

Hancock et al. [1989] used a version of the small curvature semiclassical adiabatic approach introduced by Truhlar et al. [1982] to calculate the temperature dependence of the rate constant, as shown in Figure 6.29. Variations in k(T) below the crossover point (25-30 K) are due to changes in the prefactor due to zero-point vibrations of the H atom in the crystal. Obviously, the gas-phase model does not take these into account. The absolute values of the rate constant differ by 1-2 orders of magnitude from the experimental ones for the same reason. 

Note that for the semiclassical chemical hardness (3.136) the basic definition (3.3) was employed taking account that for neutral atoms we have N=Z, and where the minus sign was as well reconsidered according with the potential (3.135), while the Vz prefactor was formally abolished since at present semiclassical level an integer quantum leap LUMO-HOMO is considered to be in agreement with the integer fluctuation domain of quantum propagation, see below Eqs. (3.38)-(7.43), see Eq. (4.158) of Section 4.2.3.3 as well as the discussion of Eq. (4.252) in Section 4.5 of the present volume. 

Forward-Backward Semiclassical Dynamics Without Prefactors... 

In this, the semiclassical coherent state prefactor has been eliminated. To compensate for its absence, the position, momentum and action jumps arising from the infinitesimal evolution with H have been rescaled. Thus, the trajectories must incur the following increments at time f ... 

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