Landau-Zener probability

In this case, P is equal to the Landau-Zener probability PLZ. 

Herein a is the rate of change of the lower dressed state energy i(t) (black dashed line in Figure 6.10c) evaluated at the inflection points at t = +15 fs, and the Rabi frequency H22 is evaluated at the crossing times. For symmetry reasons, the Landau-Zener probability is the same for both avoided crossings. Now the second requirement concerning the field amplitude is to tailor the Rabi frequency of the main pulse such that = 0.5. Then 50% of the population is transferred... 

When the Landau-Zener probability, - ex -lnJ / h v ti) )] with v ti) = Q ti), is written as Pi, this trajectory Q t) has a reaction yield given by / i(l - Pu) in the first channel and (1 — Pi)P2 in the second channel. How this result can be derived directly from the Hamiltonians in Eqs. 37 and 38 is shown in Ref. [24]. Since velocities v t ), r(tn), and v t2) have approximately the same absolute value, they can be replaced by velocity v tc) at the first crossing time tc = t. Therefore, the reaction takes place with a yield given by... 

Newton determined the transmission coefficient, k, for the homogeneous electron transfer reaction between Fe " (H20)6 and Fe " (H20)6 redox couples in solution. The transmission coefficient was expressed in terms of the Landau-Zener " ° probability factor, P12, i.e.,... 

The probability for passing from the lower PES to the upper is expressed in the Landau-Zener approximation. We will use this approximation to calculate k in Equation 10.51. The Landau-Zener probability P for crossing from the ground state PES to the upper state may be expressed in terms of the slopes of the matrix elements, Hji and H22, just outside the crossing region and the coupling H,2 = 1/2A ... 

Before this expression is inserted into the expression for the Landau-Zener probability, it is convenient to introduce an electronic frequency (without particular interpretation), which has the same dimension as the nuclear frequency ... 

Finally we find for both complexes that the Landau-Zener probabilities are much larger than the ones obtained in the mixed quantum-classical approach (Table 5.2). These values would lead to reactions occurring in the adiabatic... 

The Landau-Zener transition probability is derived from an approximation to the frill two-state impact-parameter treatment of the collision. The single passage probability for a transition between the diabatic surfaces H, (/ ) and R AR) which cross at is the Landau-Zener transition probability... 

The motivation comes from the early work of Landau [208], Zener [209], and Stueckelberg [210]. The Landau-Zener model is for a classical particle moving on two coupled ID PES. If the diabatic states cross so that the energy gap is linear with time, and the velocity of the particle is constant through the non-adiabatic region, then the probability of changing adiabatic states is... 

This dependency is seen in the Landau-Zener expression for the probability of a classical particle changing states while moving through a non-adiabatic... 

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... 

That is, the semi-classical approximation to the photon absorption rate is equivalent to a Landau-Zener treatment of the probability of hopping from Vj -i-hco to Vf induced by the electronic coupling perturbation p, f (s,0,Q). 

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 influence of the fluctuational motion along the reaction coordinates on the probability of the electron transition has been considered recently in the framework of the Landau-Zener method.102 A Hamiltonian of the form... 

The probability P in equations (61) and (62) may be related to the electronic coupling matrix element through equation (63) by application of the Landau-Zener model ... 

The probability may be related to H by application38 39 of the Landau-Zener model40 42 ... 

The probability of a diabatic hop, P, is given to good approximation by the Landau-Zener (LZ) relation, ... 

Section 6.5.1), two avoided crossings arise in Figure 6.10c at t = +10 fs between states /) and 1 3), the first of which is marked by a gray circle. Due to the highly nonadiabatic time evolution, diabatic transitions between these dressed states are likely to occur. The Landau-Zener model [48, 104, 105] estimates the probability for a diabatic transition at the avoided crossings as... 

Figure 15.7 Near approach (or avoided crossing) of two electronic states as a function of nuclear coordinate Q. The inset expands the region of the avoided crossing to facilitate the definition of quantities appearing in the Landau-Zener surface-hopping-probability model...

Various models to compute the probability of hopping exist. One of the simplest is the Landau-Zener model for avoided crossings in a single coordinate. The probability of the hop is determined as... 

The loss of the B state population depends on both the dynamics of the vibrational relaxation and the strength of the coupling of the B state to the repulsive a/a states. In the simulations, the predissociation of the B state molecules is described with the help of the Landau-Zener theory. There, the coupling strength is given by the perturbation energy. Furthermore, the probability for predissociation also depends on the (classical) velocity v t) which relates this process to the vibrational relaxation dynamics of the B state. The theoretical model uses a friction coefficient a to describe the latter process. 

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