Adiabatic region

The mixed-state character of a trajectory outside a non-adiabatic region is a serious weakness of the method. As the time-dependent wave function does not... 

The motivation comes from the early work of Landau [208], Zener [209], and Stueckelberg [210]. The Landau-Zener model is for a classical particle moving on two coupled ID PES. If the diabatic states cross so that the energy gap is linear with time, and the velocity of the particle is constant through the non-adiabatic region, then the probability of changing adiabatic states is... 

Stueckelberg derived a similar fomiula, but assumed that the energy gap is quadratic. As a result, electronic coherence effects enter the picture, and the transition probability oscillates (known as Stueckelberg oscillations) as the particle passes through the non-adiabatic region (see [204] for details). 

For diabatic calculations, the equivalent expression uses the diabatic potential matrix elements [218]. When the value of this coupling becomes greater than a pre-defined cutoff, the tiajectory has entered a non-adiabatic region. The propagation is continued from this time, ti, until the trajectoiy moves out of the region at time f2-... 

The functional B[(2(r)] actually depends only on the velocity dQ/dr at the moment when the non-adiabaticity region is crossed. If we take the path integral by the method of steepest descents, considering that the prefactor B[(2(r)] is much more weakly dependent on the realization of the path than Sad[Q(A]> we shall obtain the instanton trajectory for the adiabatic potential V a, then B[(2(t)] will have to be calculated for that trajectory. Since the instanton trajectory crosses the dividing surface twice, we finally have... 

In typical outer sphere electron transfer on metal electrodes, A is in the weakly adiabatic region and thus sufficiently large to ensure adiabaticity, but too small to lead to a noticeable reduction of the activation energy. In this case, the rate is determined by solvent reorganization, and is independent of the nature of the metal [Iwasita et al., 1985 Santos et al., 1986]. 

These results verified that heat transfer in the melt was conduction dominated, except at intense convection levels because of the low-Prandtl-number characteristic of semiconductor melts. The shape of the melt-crystal interface changes with convection only at these higher convection levels. The flows are cellular, with the direction and magnitude of each cell determined by the radial temperature gradients induced by the thermal boundary conditions. In the idealized system studied, the mismatch in boundary conditions at the junction of the hot zone and the adiabatic region (Figure 16) causes the temperature to increase radially and drive a flow up along the... 

Note that the result m jm obtained through (25) does not depend on the value t in the strong-coupling and/or anti-adiabatic region. 

By comparing the result of w /w for the infinite-site system obtained by VED [96] (see. Fig. 2), we are confident that the two-site calculation provides a reasonably good result for m /m in the whole range of g at least in the anti-adiabatic region of t/a>o. The relevance of the two-site calculation has also been seen in the Holstein model [78]. Thus we can expect that the same is true for the r (g) t JT polaron. In Fig. 3, we show the result of m/m for the T (g) r system solid curve) which is obtained in the anti-adiabatic region by implementing an... 

Fig. 3 Inverse of the mass enhancement factor, mim, as a function of g

In addition to the Coulomb interaction, the phonon-mediated interactions C/ph work on the electrons. In the weak-coupling and anti-adiabatic region, the lowest-order perturbation calculation provides C/ph = J2j with obtained as... 

Fig. 20. The border between radiative and adiabatic regions of the shocks in supernovae as a function of shock velocity and ejecta density profile index r] (p oc v v j. The boundaries depend upon the composition of the ejecta through which the shock may be propagating as also the mass loss parameter of the progenitor stellar wind and the ejecta velocity scale (Fig. courtesy T. Nymark [160])...

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