Landau-Zener curve crossings

On the basis of a Landau-Zener curve crossing formalism, Borgis and Hynes derived the nonadiabatic rate constant k, which is similar to that expressed by the DKL model but where the tunneling term Cnm found in Eq. (4) is significantly modified due to the influence of the low-frequency promoting mode Q, with a frequency 0)q, on the tunneling rate. The dependence of Cnm on Q is given by [13]... 

Published theoretical descriptions of the Ca(5 Pi) - Ca(5 Pj) alignment system have considered the formal Landau Zener curve crossing probability [29] and have used foil quantum mechanical descriptions [30]. Unfortunately, all the theoretical descriptions are limited by the lack of accurate potential surfaces for the van der Waals states of the electronic levels. However, in the future, accurate information may become available from recent experiments to investigate metal atom + rare gas van der Waals potentials using supersonic jet spectroscopy [31-34]. Thus there is an excellent chance that it will also be possible to obtain more accurate theoretical descriptions, which will elucidate important subtleties of alignment effects in energy transfer and reactions. 

For the one-dimensional Landau-Zener curve-crossing problem that we discuss, see Nikitin (1975, 1999), Child (1991), Zhu et al. (2001), Nakamura (2002). The more general case of crossing of potential energy surfaces is discussed by many authors. Koppel et al. (1984), Whetten et al. (1985), Lorquet (1996), Yarkony (1996), Ben-Nun et al. (2000), Baer (2002), Worth and Cederbaum (2004). 

In the study of (electronic) curve crossing problems, one distinguishes between a situation where two electronic curves, Ej R), j — 1,2, approach each other at a point R = Rq so that the difference AE[R = Rq) = E iR = Rq) — Fi is relatively small and a situation where the two electronic curves interact so that AE R) Const is relatively large. The first case is usually treated by the Landau-Zener fonnula [87-92] and the second is based on the Demkov approach [93]. It is well known that whereas the Landau-Zener type interactions are... 

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... 

Thus a wavepacket initiated in well A passes to well B by a curve crossing. Prof. Fleming showed an interesting case of persistent coherence in such a situation, despite the erratic pattern of the eigenvalue separations. An alternative, possibly more revealing approach, is to employ Stuckelberg-Landau-Zener theory, which relates the interference (i.e., coherence) in the two different wells via the area shown in Fig. 2. A variety of applications to time-independent problems may be found in the literature [1]. 

The process may then be described by classical trajectories and by transition probabilities for changing the potential surfaces when these trajectories come close to a crossing. The transition probability for changing the diabatic curves at a crossing line R, is given by a Landau-Zener... 

Another clue to why this branching ratio changes in this counterintuitive way with laser intensity is to note that the three-photon signal is peaked near v = 15, while the two-photon signal is peaked near v = 7. This implies that high vibrational excitation of the ion enhances the curve crossing necessary to produce the two-photon signal. This is exactly the trend observed in the Landau-Zener formula calculations performed by Zavriyev et al. [50], In their calculations on H2 the probability to cross the... 

The well-known Landau-Zener [155-158] formula relating to the probability of an electronic jump near the crossing point of two potential-energy curves or surfaces has been seriously critiqued [4, 154], New treatments of greater validity have been formulated [154, 159, 160]. 

In this study we first examine the systematics that exist in the RKR potentials of the state to establish that the potential curves are strongly ionic for R R. (R. is the distance of the pseudocrossing point). Next we evaluate an essentially experimental value of the parameters relevant to the cross section for the charge transfer process in the Landau-Zener approximation, We also construct a model ionic potential which can be used to describe the charge transfer process in the ionic region. 

Fig. 8. Relative velocity dependence of integral cross sections calculated for Na + O collisions for the indicated exit channels. The solid curve is the charge transfer cross section calculated using a multichannel Landau-Zener formalism (see text). The dashed curve is the two-state Landau-Zener cross section. Charge transfer calculations by van den Bos are indicated by triangles. Full circles and squares are the respective excitation channels as determined using the multichannel Landau-Zener model.

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 diabatie exponential potentials arc... 

The basic assumptions of the Landau-Zener theory need to be satisfied. These involve the applicability of classical mechanics (e.g. the neglect of tunneling) for the nuclear dynamics and the locality of the curve crossing event. 

Note that assumptions (2) and (3) are about timescales. Denoting by x, and tlz the characteristic times (inverse rates) of the electron transfer reaction, the solvent relaxation, and the Landau-Zener transition, respectively, (the latter is the duration of a single curve-crossing event) we are assuming that the inequalities Tr A Ts tlz hold. The validity of this assumption has to be addressed, but for now let us consider its consequences. When assumptions (1)—(3) are satisfied we can invoke the extended transition-state theory of Section 14.3.5 that leads to an expression for the electron transfer rate coefficient of the form (cf. Eq. 14.32)... 

To return now to the semiclassical model of nonadiabatic behavior, one can describe reactions on the spin-state (diabatic) PESs as follows The system will move throughout phase space on the reactant PES until it reaches a point where the product PES has the same energy as the reactant one. At that point, it may either remain on the reactant PES or hop over onto the product one. The Landau-Zener formula for curve crossing in one-dimensional systems has often been used in a multidimensional context (10) as a useful approximation for the probability p with which this hop occurs, leaving (1 - p) oi the trajectories to continue on the initial PES (Fig. 1) ... 

The results obtained in our laboratory as well as by other experimentalists [3, 4] have inspired a considerable amount of theoretical work on this system [2, 5-8], Archirel and Levy [7] have calculated a set of potential energy surfaces for the states N2 (X) + Ar, N2(A) + Ar, and N2 + Ar+(2P) as well as the couplings between these surfaces using a novel computational technique. From their results they developed a set of diabatic vibronic potential energy curves, and they assumed that transitions could occur when two curves crossed. Cross sections were computed using either the Demkov or Landau-Zener formula, as appropriate, and good agreement was obtained with the experimental values in most cases. Nikitin et al. [8] have taken a somewhat similar approach to this system. They estimated the adiabatic vibronic interaction curves for this system, and they assumed that transitions... 

Figure 7. The cross section for D" formation calculated taking the quantum interference into account (solid line). The dotted curve represents the experimental cross section. LZS stands for Landau-Zener-Stuckelberg. (Reproduced with permission from Ref. [55].)...

Figure 2 Two basic cases of curve crossing (a) Landau-Zener (LZ) case, (b) nonadiabatic tunneling (NT) case.

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