LZ formula

First, we have applied the ZN formulas to the DHj system to confirm that the method works well in comparison with the exact quantum mechanical numerical solutions [50]. Importance of the classically forbidden transitions has been clearly demonstrated. The LZ formula gives a bit too small results... 

Figure 2. Total cumulative charge-transfer probabilities for H2 + D" " — Hj + D. Dashed line exact quantum mechanical numerical solution. Solid line TSH results with use of the Zhu-Nakamura formulas. Dash-dot line TSH results with use of the LZ formula. Taken from Ref. [50].

Reddington et al. (1973) performed a numerical solution of the time dependent coupled equations for K + I in the energy range 20-100 eV. He reproduced all features which we discussed in the previous section. Risking some monotony, we cite here their conclusion that the simple LZ-formula accounts well for the numerically computed transition probabilities. 

The formula nevertheless reflect the essential physics in nonadiabatic transition. The exponential dependence on parameters reflects the non-perturbative nature of this problem. The non-dimensional exponent hv Fi-F2 " hich represents adiabaticity, ensures the correct limiting behavior (the adiabatic limit v —> 0 and its opposite v — 00 correspond to P = 0 and P = 1, respectively). One also sees the correct scale in this problem for example, whether the passage velocity v is fast or not should be measured in unit V / h Fi — F2 ). Qualitative accuracy of this formula is known to be rather robust see for example, Ref. [445], in which they compare exact result with LZ formula as well as their proposed (surface hopping) method. These virtues are favorable for making rough estimate as well as constructing new guiding principle of one s study see, for example, Ref. [404] to see that the LZ estimate is still conceptually important even in the state-of-art studies. 

ET rate via electronic coupling for a multi-dimensional system in the Marcus inverted regime, (a) pEa—6.7, (b) pSa = 10.0, and (c) pEa = 20.0. Ea represents the minimum energy on the seam surface. Solid line present result dashed line the results predicted from the LZ formula dotted line results from perturbation theory. 

The method is composed of the following algorithms (1) transition position is detected along each classical trajectory, (2) direction of transition is determined there and the ID cut of the potential energy surfaces is made along that direction, (3) judgment is made whether the transition is LZ type or nonadiabatic tunneling type, and (4) the transition probability is calculated by the appropriate ZN formula. The transition position can be simply found by... 

The time-dependent problems are generally simpler than the time-independent ones and the corresponding version of the ZN formulas is presented in Section C. This corresponds to the LZ type, since the NT type does not occur in the time-dependent case. 

The formulas derived in the time-independent framework can be easily transferred into the corresponding time-dependent solutions. The formulas in the time-independent linear potential model, for example, provide the formulas in the time-dependent quadratic potential model in which the two time-dependent diabatic quadratic potentials are coupled by a constant diabatic coupling [1, 13, 147]. The classically forbidden transitions in the time-independent framework correspond to the diabatically avoided crossing case in the time-dependent framework. One more thing to note is that the nonadiabatic tunneling (NT) type of transition does not show up and only the LZ type appears in the time-dependent problems, since time is unidirectional. 

T102. Comparing the T02 formula of the starting NH4Y with that of the LZ-225 product, certain properties of the LZ-225 can be calculated. The LZ-225 was 34% dealuminated (N/a X 100), with 95% of the dealuminated sites filled with Ti (c/N 100). The change in defect structure was +0.032. There is a slight decrease in silicon content, a feature that was observed with all acid-sensitive zeolites. 

The LZ-247 product contained 11.4 wt.% Ti02. Comparison of the calculated T02 formulas shows that the LZ-247 was 31% dealuminated with all of the dealuminated sites filled with Ti. In this case some of the silicon was also removed during the treatment and some of the Si sites were apparently filled by Ti as well. The retained 02 capacity suggests that the Ti in excess of the removed A1 is also inserted into the framework since 02 capacity has barely changed as a result of the treatment. The change in defect structure was +0.041. 

The mordenite sample is still 51% Al-depleted, but only 70% of the iron is present as framework Fe, about 5 wt.%. The remaining Fe is presumed to be the Fe(0H)J cation. Ammonium oxalate has been shown to react with and exchange with cationic iron (17). The sample was refluxed three times with 0.1 N ammonium oxalate solution with the analytical results shown in Table 3 used to calculate the following T02 formula for the resulting white LZ-226 product ... 

A second sample of brown LZ-226 containing about 6 wt.% Fe203 was washed with ammonium oxalate and characterized. Following the ammonium oxalate wash, the product, sample L, contained 3.1 wt.% Fe203. The calculated T02 formula is as follows ... 

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