Landau-Zener formula

For the connection of this result to the well-known Landau-Zener formula [23] see [15]. 

Figure 3. Initial vibrational state specified cummulative reaction probabilities for v = 2. Dashed line exact quantum mechanical numerical solution. Solid hue TSH results with use of the Zhu-Nakamura formulas. Dash-dot hue TSH results with use of the Landau-Zener formula. Taken from Ref. [50].

The Landau-Zener formula in the limit J2/ v - oo gives PLZ = 1 whereas Eq. (183) for the case of fast fluctuations in this limit gives P — At small values of J2/ u, both formulas give the same result, P = 2itJ2/ v. ... 

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 = Ro so that the difference AE(R = Ro) = Ei(R Ro) — E 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 formula [87-92] and the second is based on the Demkov approach [93]. It is well known that whereas the Landau-Zener type interactions are... 

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 TOF spectrum of the Cl or Br atom is measured as a function of wavelength, with the polarization of the laser selected so that only TTq is excited. The peaks that are observed in the spectrum can be identified with atoms in either the 3/2 Dr pl/2 state> so that the experimental data can be used to determine the probability, P, for the diabatic crossing. The Landau-Zener formula for this probability is given by... 

In order to apply quantitatively the Landau-Zener formula Eq. (2), we first saturated the crystal of Fes clusters in a field of Hz = -1.4 T, yielding an initial magnetization Min = -Ms. Then, we swept the applied field at a constant rate over one of the resonance transitions and measured the fraction of molecules which reversed their spin. This procedure yields the tunneling rate P 1010 and thus the tunnel splitting A 1010 , Eq. (2), with n = 0, 1, 2,. 

The probability that the system will remain on this lower potential energy surface can be calculated from the Landau-Zener formula (Equation 5). In this expression, 2EIfII is the splitting at the intersection,... 

For diatomic systems with states of different symmetry and spin, the probability of crossing was derived independently by Landau and Zener. Even for diatomics the Landau-Zener formula has been shown to be inapplicable in certain instances and has been revised. For polyatomic molecules the Landau-Zener formula is not applicable as such, and modifications have been made to permit its use for specified cases. 

Coulson derived a new method for computing P(t) that gives the Landau-Zener formula as a special case. He considered three cases (1) transition between two discrete states [approximate internal conversion when the molecule passes from one discrete (bound) state to another], (2) transition from a bound state to the continuum (predissociation), and (3) transitions between two states of the continuum (corresponding to scattering problems). Coulson omitted in his article the computational details for P(t) for case (1). For case (2) he gave... 

Coulson s analyses yielded different results for cases (2) and (3), in conflict with all previous work on this problem. Prior to this work it had been usual to apply the Landau-Zener formula both to predissociation and to scattering. 

He then went on to show that the conventional Landau-Zener formula is scarcely ever justified by its derivation and that the above formula seems to be more frequently applicable. However, its numerical evaluation is very difficult, because of the difficulty of computing J u( ) d ... 

As seen previously, the chemical reactions studied most often are the exchange ones. Those requiring several potential energy surfaces of excited states (diabatic reactions) are worth special mention, since they most certainly define a domain of application with a future for classical trajectories. An electron jump from one surface to another requires either to be given a statistical probability of occurence by the Landau Zener formula (or one of its improved versions " ) or to be described by means of complex-valued classical trajectories as a direct and gradual passage in the complex-valued extension of the potential surfaces (generalization of the classical S-matrix ). 

The distribution function of n[= Q t)] values has already been determined as G v) of Eq. 55. Since v itself, not v, appears in the Landau-Zener formula (Eq. 59), and is regarded as a positive quantity, it is appropriate to estimate the average value of V by... 

This result is due to Dykhne [48] and Davis and Pechukas [49] and has been extended to A-level systems [50]. The conditions of validity of the so-called Dykhne-Davis-Pechukas (DDP) formula has been established in [51,52], This formula allows us to calculate in the adiabatic asymptotic limit the probability of the nonadiabatic transitions. This formula captures, for example, the result of the Landau-Zener formula, which we study below. 

We can approximately characterize the dynamics by linearizing it in time around the avoided crossing r(t) = rc t — tc) and apply the Landau-Zener formula [64,65] to calculate the probability to jump from the branch v(/ to v(/+ [66] ... 

Formula (270) thus defines the efficiency of the diabatic passage. We remark that the Landau-Zener formula gives the information for the whole range of gap distances, from the limit of exact crossings to widely separated ones. 

We remark that if condition (273) is not satisfied, which is the case if one misses the conical intersection in an intermediate regime (f Ac), the Landau-Zener formula shows that the dynamics splits the population into the two surfaces near the intersection. This gives rise afterwards to two states that will have their own adiabatic evolution. 

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) ... 

Paulsen et al. (1972) developed an optical model for vibrational relaxation in reactive systems. Only collinear atom-diatom collisions were considered, i.e. impact parameter dependencies were omitted. The model was applied to vibrational relaxation of electronically excited I2 in inert gases, in which case dissociation of I2 is responsible for flux loss. Olson (1972) used an absorbing-sphere model for calculating integral cross sections of ion-ion recombination processes A++B ->A + B + AE, with A or B atoms or molecules. He employed the Landau-Zener formula to obtain a critical crossing distance Rc, and assumed the opacity to be unity for distances... 

No direct measurement estimated by using the Landau-Zener formula. 

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