Diabatic potentials, nonadiabatic transition

An illustration of wave packet propagation on two-coupled electronic states in diabatic representation. Nonadiabatic transition occurs at the intersection region of the two potential energy surfaces, resulting in the observed two portions of the wave packet (initiated on state 1) on the two electronic states, with the larger portion remaining on the initiated state. 

Nonadiabatic transitions play crucial roles in various fields of physics and chemistry [1, 2, 3 4. 5, 7, 8 9 10], and it is quite important to develop basic analytical theories so that we can understand fundamental mechanisms of various dynamics. The most fundamental models among them are the Landau-Zener type curve crossing and the Rosen-Zener-Demkov type non-curve-crossing. Furthermore, there is an interesting intermediate case in which two diabatic exponential potentials are... 

Another kind of radially induced nonadiabatic transition is the Rosen-Zener-Demkov type (1,2). In contrast to the curve-crossing case, the two diabatic potentials have very weak / -dependence (actually, their difference is assumed to be constant in the basic model) and the diabatic coupling has a strong (-exponential) / -dependence (see Fig. 5). The nonadiabatic transition is not as effective as the curve-crossing case, but this case also presents an important transition, especially when the two potentials are near resonance asymptotically. 

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 ... 

The probability of a nonadiabatic transition is essentially determined by the local conditions at the crossing point (x = x ) of the "diabatic" potential curves (x) and Y2M9 rather than those at the transition state (x = x ), i.e., the peak of the adiabatic potential curve V(x)(Pig.21). Therefore, it is necessary to change the energy variable, using (156.111), in the integral expression (78.Ill) in order to obtain... 

When the system is put into the diabatic representation, time evolution of the nuclear wavefunction including nonadiabatic transitions is simple in a split-operator context, which amounts to diagonalizing the potential matrix at each time step [9], In fact, the coding would be the same as for two states coupled though an external field (Sec. 3.3.2), except that the coupling potential would be a time-independent, intrinsic function of the system. 

The formulas derived in the time-independent framework can be easily transferred into the corresponding time-dependent solutions. The formulas in the time-independent linear potential model, for example, provide the formulas in the time-dependent quadratic potential model in which the two time-dependent diabatic quadratic potentials are coupled by a constant diabatic coupling [1, 13, 147]. The classically forbidden transitions in the time-independent framework correspond to the diabatically avoided crossing case in the time-dependent framework. One more thing to note is that the nonadiabatic tunneling (NT) type of transition does not show up and only the LZ type appears in the time-dependent problems, since time is unidirectional. 

Le Roy, et al, (2002) have reviewed all of the different types of experimental observations and theoretical calculations for HI. By an empirical analysis, they have shown that, because in HI the spin-orbit interaction is especially important, the adiabatic relativistic potential curves can explain all of the experimental data without introducing residual nonadiabatic coupling. For the lighter halogen hydrides, the J = 1/2 J = 3/2 branching ratio can be obtained from the solution of inhomogeneous coupled equations with a source term representing the initial vibrational wavefunction multiplied by the electronic transition moment (Band, et al., 1981). These calculations are based on adiabatic electronic (or diabatic relativistic) potential curves (see, for example, for HC1, Alexander, et al., 1993 and for HBr, Peoux, et al., 1997). 

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