Nonadiabatic transition probability

With the scheme of PSANB, one can calculate the nonadiabatic transition probability as follows. For the sake of simpler explanation, we describe the procedure for the systems of two electronic states. Suppose we have an asymptotic wavefunction irrespective of their history of branching such that 

Chemical Theory Beyond the Born-Oppenheimer Paradigm 

The theoretical scheme proposed so far has not yet given the explicit functional form of Xk - Rfc (t),, and hence we have no rigorous way to 

This form of transition probability will be used throughout the present quantum-classical scheme. The explicit incorporation of the nuclear wavepacket Xfc ( R — Rfc (t), t j will be studied later in Sec. 6.7. 

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The problem of nonadiabatic tunneling in the Landau-Zener approximation has been solved by Ovchinnikova [1965]. For further refinements of the theory beyond this approximation see Laing et al. [1977], Holstein [1978], Coveney et al. [1985], Nakamura [1987]. The nonadiabatic transition probability for a more general case of dissipative tunneling is derived in appendix B. We quote here only the result for the dissipationless case obtained in the Landau-Zener limit. When < F (Xe), the total transition probability is the product of the adiabatic tunneling rate, calculated in the previous sections, and the Landau-Zener-Stueckelberg-like factor... 

Here, pj denotes the nonadiabatic transition probability for one passage of the avoided crossing Xt, and / are the dynamical phases due to the nonadiabatic transition atX, is the kth adiabatic Floquet state, Xq = i and X3 = 2- The transition amplimde Eq. (152) can be explicitly expressed as... 

The laser parameters should be chosen so that a and p can make the nonadiabatic transition probability V as close to unity as possible. Figure 34 depicts the probability P 2 as a function of a and p. There are some areas in which the probabilty is larger than 0.9, such as those around (ot= 1.20, p = 0.85), (ot = 0.53, p = 2.40), (a = 0.38, p = 3.31), and so on. Due to the coordinate dependence of the potential difference A(x) and the transition dipole moment p(x), it is generally impossible to achieve perfect excitation of the wave packet by a single quadratically chirped laser pulse. However, a very high efficiency of the population transfer is possible without significant deformation of the shape of the wave packet, if we locate the wave packet parameters inside one of these islands. The biggest, thus the most useful island, is around ot = 1.20, p = 0.85. The transition probability P 2 is > 0.9, if... 

Figure 34. Contour map of the nonadiabatic transition probability Pn induced by quadratically chirped pulse as a function of the two basic parameters a and p. Taken from Ref. [37]-...

The overall nonadiabatic transition probability between the two adiabatic states is given by... 

The one-passage nonadiabatic transition probability pzn and the phases in (5.25) and (5.26) are explicitly given by... 

Following Tully s fewest switches criterion [94] recipe, the nonadiabatic transition probability from state k to state / is... 

Changes in the energy gap, AE, and the nonadiabatic transition probability, P10, in the aqueous solution simulations are dominated in the initial stages by the coupled proton-electron transfer event and the subsequent relaxation of the system into the excited CT state. Similar to the gas phase, variations in AE and P10 at longer time-scales were found to depend strongly on the out-of-plane motions of the system (for instance the dihedral angles 0 and ). However, the presence of... 

The excited state lifetimes determined from the na-AIMD simulations are generally in good agreement with experimental data. In addition, the na-AIMD simulations provide detailed insights into the dynamical mechanism of radiationless decay. The time evolution of the nonadiabatic transition probability could be correlated with certain vibrational motions. In this way, the simulations yield the driving modes of internal conversion. 

A great deal of research has been done recently on approximate solutions of strongly coupled semiclassical equations. Some solutions have been extensively used for cross-section calculations, often without any estimation of possible errors. Final expressions for the nonadiabatic transition probability P between two electronic states will be written down for the most frequently used approximations. 

It is suggested that the two electronic states fa and l °f noninteracting partners provide a good basis set to be used for constructing orthonormal adiabatic electronic functions fa and fa. At R — oo the functions fa and fa adiabatically correlate with fa and fa, so that the nonadiabatic transition probability calculated for a particular trajectory R(f) refers to the collision-induced transition between the two states of the partners. 

E.E.Nikitin, Methods for the ealeulation of nonadiabatic transition probabilities, Optikai Spekt 18 763 (1964)... 

The simplest model is the following the diabatic potentials are constant with V2 - Vx = A > 0 and the diabatic coupling is V e R where A = 2V0. Recently, Osherov and Voronin obtained the quantum mechanically exact analytical solution for this model in terms of the Meijer function (38). In the adiabatic representation this system presents a three-channel problem at E > V2 > Vu since there is no repulsive wall at R Rx in the lower adiabatic potential. They have obtained the analytical expression of a 3 X 3 transition matrix. Adding a repulsive potential wall at R Rx for the lower adiabatic channel and using the semiclassical idea of independent events of nonadiabatic transition at Rx and adiabatic wave propagation elsewhere, they derived the overall inelastic nonadiabatic transition probability Pl2 as follows ... 

Dykhne used this approach to calculate the semiclassical nonadiabatic transition probability for the ballistic case in the strong interaction limit [320]. His final result. 

Equation (4.2) is to be solved as an initial value problem under the conditions Cl(—00) = 0 and C2(—oo) = 1, so that the nonadiabatic transition probability (or, equivalently, the probability of the system remains as the diabatic state 2) is given by P = C2(- -oo)p. Eliminating C2, the problem is reduced to a second order differential equation... 

Nonadiabatic transition probability compared with the full quantum values... 

In Fig. 6.5, panels (c) and (d) depict the nonadiabatic transition probabilities from 5E+ state to the 3E+ and 4E+ states asymptotically observed... 

To deal with the ET rate in such a case, our strategy is to combine the generalized nonadiabatic transition state theory (NA-TST) and the Zhu-Nakamura (ZN) nonadiabatic transition probability.The generalized NA-TST is formulated based on Miller s reactive flux-flux correlation function approach. The ZN theory, on the other hand, is practically free from the drawbacks of the LZ theory mentioned above. Numerical tests have also confirmed that it is essential for accurate evaluation of the thermal rate constant to take into account the multi-dimensional topography of the seam surface and to treat the nonadiabatic electronic transition and nuclear tunneling effects properly. 

With use of the microcanonical reaction mechanism and the nonadiabatic transition probability P E) from the left to right in Figure 12.1 at a given translation energy E, the thermal rate k can be obtained by an average with the Boltzmann thermal distribution as a weight function ... 

With the above simple background, we consider ET in complex systems. In this case, eqn (12.4) cannot be used straightforwardly because the nonadiabatic transition probability is explicitly dependent on the reaction coordinate and one does not know how to select this one-dimensional reaction coordinate from the multi-dimensional systems. Therefore, our strategy is to start from a generalized quantum rate expression. Miller et al. have shown that the... 

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