Landau-Zener treatment

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). 

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 ... 

Although, AE is not generally equal to AG° unless the frequencies of the reactant and product are the same, this assumption is almost universally made. With this assumption classical Marcus ET theory combined with a quantum mechanical (Landau-Zener) treatment of the barrier crossing also yields Eq. (4) [2,36-39]. This derivation of Eq. (4) is called semiclassical ET theory, and therefore in the rest of this paper Eq. (4) will also be referred to as the semiclassical rate expression or the semiclassical model. 

On the other hand, for a non-adiabatic reaction, k i 1, /CeiVn = Vgi and the rate constant is given by Eq. 23 where Vei is the electron hopping frequency in the activated complex. The Landau-Zener treatment yields Eq. 24 for Vei [16, 17]. 

Electron transfer in proteins generally involves redox centers separated by long distances. The electronic interaction between redox sites is relatively weak and the transition state for the ET reaction must be formed many times before there is a successful conversion from reactants to products the process is electronically nonadiabatic. A Landau-Zener treatment of the reactant-product transition probability produces the familiar semiclassical expression for the rate of nonadiabatic electron transfer between a donor (D) and acceptor (A) held at fixed distance (equation 1). Biological electron flow over long distances with a relatively small release of free energy is possible because the protein fold creates a suitable balance between AG° and A as well as adequate electronic coupling between distant redox centers. 

Bates criticized the Landau-Zener " treatment for finding the probability of transition. He mentioned that the assumption by Landau-Zener that the transition occurs only in the zone of crossing of two potential energy surfaces is not necessarily correct the transition may occur away from the crossing, and he calculated the width of the transition zone as... 

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... 

At this point, we make two comments (a) Conditions (1) and (2) lead to a well-defined sub-Hilbert space that for any further treatments (in spectroscopy or scattering processes) has to be treated as a whole (and not on a state by state level), (b) Since all states in a given sub-Hilbert space are adiabatic states, stiong interactions of the Landau-Zener type can occur between two consecutive states only. However, Demkov-type interactions may exist between any two states. 

This is, beyond all doubt, the most important process and the only one which has been already tackled with theoretically. Nevertheless, the prediction given by the classical overbarrier transition model is not correct for this collision [9] and the modified multichaimel Landau-Zener model developed by Boudjema et al. [34] caimot explain the experimental results for collision velocities higher than 0.2 a.u.. With regard to the collision energy range, we have thus performed a semi-classical [35] collisional treatment... 

For k(r) we shall assume at first, as in (19), that the reaction is adiabatic at the distance of closest approach, r = a, and that it is joined there to the nonadiabatic solution which varies as exp(-ar). The adiabatic and nonadiabatic solutions can be joined smoothly. For example, one could try to generalize to the present multi-dimensional potential energy surfaces, a Landau-Zener type treatment (41). For simplicity, however, we will join the adiabatic and nonadiabatic expressions at r = a. We subsequently consider another approximation in which the reaction is treated as being nonadiabatic even at r = a. 

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]. 

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. 

If we further assume that the diabatic coupling V(t) is constant, then Eq. (12) can be solved exactly in terms of the Weber function. Then the final transition probability is exactly equal to Eq. (10). The linearity in time t is very much different from the linearity in coordinate R and the effects of turning points are completely neglected in the former approximation. In Landau s treatment this corresponds to the assumption of the common straightline classical trajectory with constant velocity. Thus, the Landau-Zener formula Eq. (10) is valid only at collision energies much higher than the crossing point. 

Ao and Rammer [166] obtained the same result (and more) on the basis of a fully quantum mechanical treatment. Frauenfelder and Wolynes [78] derived it from simple physical arguments. Equation (9.98) predicts a quasiadiabatic result, = h k/ v 1 and the Golden Rule result, Pk = k/ v, in the opposite limit, which is qualitatively similar to the Landau-Zener behavior of the transition probability but the implications are different. Equation (9.98) is the result of multiple nonadiabatic crossings of the delta sink although it does not depend on details of the stochastic process Xj- t). This can be understood from the following consideration. For each moment of time, the fast coordinate has a Gaussian distribution, p Xf, t) = (xy — Xj, transition region, the fast coordinate crosses it very frequently and thus forms an effective sink for the slow coordinate. 

Serious error is likely to arise in the treatment of Landau and Zener unless the wave functions i and 2 are spherically symmetrical and unless the velocity of the relative motion is low. For example, in a charge transfer process which involved electron transfer between p and d states, the Landau-Zener result will be less applicable. 

Stueckelberg also introduced two-state model but adopted time-independent formulation and used semiclassical approach for solution. The latter is in contrast to constant velocity assumptions in the treatment of Landau Zener, but is essential for analytical derivation of correct adiabatic phase factors. Semiclassical contom integral method and analysis of accompanying Stokes phenomena is used for deriving transition amplitude in time-independent formulation of this problem [395], which will be briefly mentioned in the next subsection (also see Ref. [99] for more details including corrections). 

Holstein then proceeded to calculate transition probabihties between polaronic states localized at different sites (molecules). This treatment is essentially identical to that of Landau and Zener [15,36,37] and also has a lot of similarities with the more elaborate Marcus theory [16]. For small polarons at high temperatures the result is... 

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