Equation Landau-Zener

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 ... 

In the framework of the Landau-Zener model, P is related to H by means of equation (75). These equations are also valid when both the stretching and solvent reorganization coordinates are taken into account as in the case of dissociative electron transfer. 

DR. MEYER Dr. Schatz has led me to believe in certain things, and I want to ask him a question. It occurs to me that the Landau-Zener equation cannot work except in the nonadiabatic limit. Do you have any comments on that ... 

This mixed classical/quantum expression is valid for classical nuclear behavior and, strictly speaking, for the case of direct two-site interaction rather than superexchange, as the Landau-Zener expression was derived from the time-dependent Schrodinger equation assuming a two-state (reactant/product) electronic system with direct coupling. Nevertheless, it becomes clear on physical grounds that the form of Eqs. 4-5 can serve to define an effective A in the superexchange case in terms of the Rabi precession frequency characteristic of the two trap sites embedded in the complex system wherein 2A/h would be computed from this net effective Rabi precession frequency. 

First, we shall approach the problem as a transition driven by the time variation of a quasi-static field, in other words, in the Landau-Zener terms we used to describe microwave ionization. We can write the Schroedinger equation for this problem as... 

These monolayers provide a significant opportunity to compare the extent of electronic communication across the p3p bridge when bound to a metal electrode as opposed to being coupled to a molecular species, e.g. within a dimeric metal complex. Electronic interaction of the redox orbitals and the metallic states causes splitting between the product and reactant hypersurfaces, which is quantified by HabL the matrix coupling element. The Landau-Zener treatment [15] of a non-adiabatic reaction yields the following equation ... 

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,... 

The transition coefficient k is related to the probability of the transition in the crossing area (P) and is described by the Landau-Zener equation (Landau, 1932, Zener, 1933)... 

Equation (96) differs from those found earlier by Stueckelberg [15] and Bates [89], Stueckelberg [15] used the perturbation theory to calculate PAB (b) at b > jPc and applied a Landau-Zener type of formula for b < Re. It was then found that the main contribution to distant collisions with b > PC. ... 

In connection with the two-state approximation we mention paper [107] giving detailed analysis of the Landau-Zener-Stueckelberg model in which transition probabilities have been calculated for a wide set of parameters of the model, and also papers devoted to extensions of the model [108-110]. The exponential model with the Hamiltonian given by equation (26) was discussed in [111] and extended in [112],... 

The transition probability was calculated using the Landau-Zener equation for homogeneous adiabatic electron transfer, and for the electrode case with very weak coupling, for which an exact solution exists. In many cases, the rate was expressed with a non-calculable... 

In equation (30), Vei is the frequency of electronic motion across the reaction coordinate near the transition state. Other expressions for /Cei have been proposed, for example, based on Landau-Zener approaches. ... 

Equation (16.51) was derived from the Landau-Zener theory for transitions between electronic states at the crossing of the corresponding potential surfaces, however... 

A semiclassical model to calculate the probability pvl that a molecule follows a diabatic pathway, by retaining the electronic stmcture of the initial electronic state as it passes through an avoided crossing, was developed independently in 1932 by Landau and by Zener.154 In the Landau Zener model, the nuclear movements are described classically, that is, nuclear motion enters parametrically and only a single reaction coordinate, as shown in diagram in Figure 2.27, is considered. In the Landau Zener expression given in Equation 2.53 ... 

Equation 2.53 Landau Zener model for diabatic reactions... 

A more recent paper by the same author (1973) evaluates coupling terms of the Hamiltonian between adiabatic vibrational states in a semiclassical fashion, which leads to equations similar to the Landau-Zener one for electronic excitation. The treatment applies to slow motion along the path and fast vibrational motion in A + BC - AB + C, a situation complementary to that treated by Hofacker and Levine(1971). The author indicates the presence of a spurious asymptotic coupling in this other work. 

From now on we will assume that we are dealing with a set of two crossing diabatic states for all R and an interaction matrix element Hl2(R) = <(/>, Hel 02), at least at R = Rc. In the next section we will discuss the well-known Landau-Zener-StUeckelberg (henceforth abbreviated as LZS-) model in order to solve the coupled equations (4). 

Stuckelberg did the most elaborate analysis (15). He applied the approximate complex WKB analysis to the fourth-order differential equation obtained from the original second-order coupled Schrodinger equations. In the complex / -plane he took into account the Stokes phenomenon associated with the asymptotic solutions in an approximate way, and finally derived not only the Landau-Zener transition probability p but also the total inelastic transition probability Pn as... 

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