Nonadiabatic transitions

Hammes-Schiffer S and Tully J C 1995 Nonadiabatic transition state theory and multiple potential energy surfaces molecular dynamics of infrequent events J. Chem. Phys. 103 8528... 

Zhu C and Nakamura H 1994 Theory of nonadiabatic transition for general curved potentials I. J. Chem. Phys. 101 10 630... 

The topography of a conical intersection affects the propensity for a nonadiabatic transition. Here, we focus on the essential linear tenns. Higher order effects are described in [10]. The local topography can be detennined from Eq. (13). For T] = 3, Eq. (13) becomes, in orthgonal intersection adapted coordinates... 

SPACEEIL has been used to study polymer dynamics caused by Brownian motion (60). In another computer animation study, a modified ORTREPII program was used to model normal molecular vibrations (70). An energy optimization technique was coupled with graphic molecular representations to produce animations demonstrating the behavior of a system as it approaches configurational equiHbrium (71). In a similar animation study, the dynamic behavior of nonadiabatic transitions in the lithium—hydrogen system was modeled (72). 

The transition described by (2.62) is classical and it is characterized by an activation energy equal to the potential at the crossing point. The prefactor is the attempt frequency co/27c times the Landau-Zener transmission coefficient B for nonadiabatic transition [Landau and Lifshitz 1981]... 

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... 

Now we make the usual assumption in nonadiabatic transition theory that non-adiabaticity is essential only in the vicinity of the crossing point where e(Qc) = 0- Therefore, if the trajectory does not cross the dividing surface Q = Qc, its contribution to the path integral is to a good accuracy described by adiabatic approximation, i.e., e = ad Hence the real part of partition function, Zq is the same as in the adiabatic approximation. Then the rate constant may be written as... 

Once the mechanisms of dynamic processes are understood, it becomes possible to think about controlling them so that we can make desirable processes to occur more efficiently. Especially when we use a laser field, nonadiabatic transitions are induced among the so-called dressed states and we can control the transitions among them by appropriately designing the laser parameters [33 1]. The dressed states mean molecular potential energy curves shifted up or down by the amount of photon energy. Even the ordinary type of photoexcitation can be... 

It is getting more and more important to treat realistic large chemical and even biological systems theoretically by taking into account the quantum mechanical effects, such as nonadiabatic transition, tunneling, and intereference. The simplest method to treat nonadiabatic dynamics is the TSH method introduced... 

Figure 1. Two basic elements of dynamics (l) propagation on a single adiabatic potential and (2) nonadiabatic transition. In the classically allowed case, the transition occurs at Ana- In the classically forbidden case, on the other hand, the transition region spans the interval (xi,Xf), where a , and Xf are the turning points. Taken from Ref. [9].

The closer the trajectory approaches the conical intersection, the smaller Cy becomes. Since the nonadiabatic transitions are expected to take place in the close vicinity of the conical intersection, the nonadiabatic transition direction can be approximated by the eigenvector of the Hessian d AV/dRidRj corresponding to its maximum eigenvalue. Similar arguments hold for nonadiabatic transitions near the crossing seam surface, in which case the nondiagonal elements of the diabatic Hamiltonian of Eq. (1) should be taken as nonzero constant. 

As discussed by Miller and co-workers [52,53], it is worthwhile to develop theories that enable us to evaluate thermal reaction rate constants directly and not to rely on the calculations of the most detailed scattering matrix or the state-to-state reaction probabihty. Here, our formulation of the nonadiabatic transition state theory is briefly described for the simplest case in which the transition state is created by potential surface crossing [27]. 

Figure 10. Arrhenius plot of the thermal rate constants for the 2D model system. Circles-full quantum results. Thick solid (dashed) curve present nonadiabatic transition state theory by using the seam surface [the minimum energy crossing point (MECP)] approximation. Thin solid and dashed curves are the same as the thick ones except that the classical partition functions are used. Taken from Ref. [27].

The nonadiabatic transition state theory given in the Section II.C, namely, Eq. (17), can be applied to the electron-transfer problem [28]. Since the electron transfer theory should be formulated in the free energy space, we introduce the... 

The factor k takes into acount the effects of nonadiabatic transition and tunneling properly. Also note that the electronic coupling //ad is assumed to be constant in the Marcus formula, but this is not necessary in the present formulation. The coupling Had cancels out in k of Eq. (126) and the ZN probability can be calculated from the information of adiabatic potentials. 

The electron transfer discussed above corresponds to the so-called normal case in which the NT type of nonadiabatic transition plays the essential role. There is another important case called inverted case, in which the LZ type of nonadiabatic transition plays a role. Since the ZN theory can describe this type of transition also, the corresponding electron-transfer theory can be formulated [114]. On the other hand, the realistic electron transfer occurs in solution and... 

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... 

In the above numerical examples the held parameter F is taken to be the laser frequency and the nonadiabatic transition used is the Landau-Zener type of curve-crossing. The periodic chirping method, however, can actually be more... 

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]-...

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