Energy-gap law

The fonn of the classical (equation C3.2.11) or semiclassical (equation C3.2.11) rate equations are energy gap laws . That is, the equations reflect a free energy dependent rate. In contrast with many physical organic reactivity indices, these rates are predicted to increase as -AG grows, and then to drop when -AG exceeds a critical value. In the classical limit, log(/cg.j.) has a parabolic dependence on -AG. Wlren high-frequency chemical bond vibrations couple to the ET process, the dependence on -AG becomes asymmetrical, as mentioned above. 

Figure C3.5.6 compares the result of this ansatz to the numerical result from the Wiener-Kliintchine theorem. They agree well and the ansatz exliibits the expected exponential energy-gap law (VER rate decreases exponentially with Q). The ansatz was used to detennine the VER rate with no quantum correction Q= 1), with the Bader-Beme hannonic correction [61] and with a correction based [83, M] on Egelstaff s method [62]. The Egelstaff corrected results were within a factor of five of experiment, whereas other corrections were off by orders of magnitude. This calculation represents the present state of the art in computing VER rates in such difficult systems, inasmuch as the authors used only a model potential and no adjustable parameters. However the ansatz procedure is clearly not extendible to polyatomic molecules or to diatomic molecules in polyatomic solvents.

The exponent in this formula is readily obtained by calculating the difference of quasiclassical actions between the turning and crossing points for each term. The most remarkable difference between (2.65) and (2.66) is that the electron-transfer rate constant grows with increasing AE, while the RLT rate constant decreases. This exponential dependence k AE) [Siebrand 1967] known as the energy gap law, is exemplified in fig. 14 for ST conversion. 

This is usually called the energy gap law expression and can be obtained from Eq. (3.50) by using the Stirling formula for n. Equation (3.51) is often used by Mataga and his co-workers. See, for example, Ref. 76 and the citations therein. 

A closely related test of the energy gap law for Ru complexes has come from temperature dependent lifetime and emission measurements for a series of complexes of the type RuO py I " (L py, substituted pyridines, pyrazine...). From the data, the variation in lnknr with Eem predicted by the energy gap law has been observed and it has been possible to observe the effect of changing the ligands L on the transition between the MLCT and dd states (20). 

One striking prediction of the energy gap law and eq. 11 and 14 is that in the inverted region, the electron transfer rate constant (kjjj. = ket) should decrease as the reaction becomes more favorable (lnknr -AE). Some evidence has been obtained for a fall-off in rate constants with increasing -AE (or -AG) for intermolecular reactions (21). Perhaps most notable is the pulse radiolysis data of Beitz and Miller (22). Nonetheless, the applicability of the energy gap law to intermolecular electron transfer in a detailed way has yet to be proven. 

Application of the energy gap law to the energy conversion mechanism in Scheme 1 leads to a notable conclusion with regard to the efficiency for the appearance of separated redox products following electron transfer quenching. From the scheme, the separation efficiency, sep> is given by eq. 18. Diffusion apart of the... 

Tachiya, M. and Hilczer, M. (1994) Solvent effect on the electron transfer rate and the energy gap law,in Gauduel, Y. and Rossky, P. J.(eds.), Ultrafast reaction dynamics and solvent effects, AIP Press, New York, pp.447-459. 

Understand the importance of the overlap of vibrational probability functions and the energy gap law in determining the rate of internal conversion and intersystem crossing. 

At sufficiently large p this may be recast in the form of an energy gap law,... 

The emitting level must not be at too low an energy. The energy gap law states that radiationless processes become more efficient as the emitting state approaches the ground state.01 12)... 

Fi gu re 4.3. Plot of nonradiative decay constants for an homologous series of Re(bpy)(CObX (X = pyridine or substituted pyridine). The observed lifetime is dose to I /knr. The solid line is the best theoretical fit with the energy gap law. (Reprinted from Ref. 13 with permission. Copyright 1990 American Chemical Society.)... 

Chemical modifications can be used to tune the state energies and enhance properties. CO ligands greatly stabilize the t levels, which results in both an increased A and a higher-energy MLCT transition. For example, the primary MLCT bands of Os(phen)32+ and [Os(phen)2Cl(CO)l+are at 430 and 365 nm, respectively. The emissions are similarly shifted from 710 to 646 nm. In keeping with the expectations of the energy gap law, the r of[Os(phen)3l2+ and [Os(phen)2Cl(CO)J+ are 74 and 234 nsec, respectively,(21)... 

J. V. Caspar and T. J. Meyer, Application ofthe energy gap law to nonradiative, excited state decay, /. Phys. Chem. 87, 952-957 (1983). 

NONRADIATIVE TRANSITIONS IN RARE EARTH IONS THE ENERGY-GAP LAW... 

The experimentally obtained energy-gap law for LaCls has been represented in Eigure 6.5. Erom this figure, one can obtain that the best fit of the experimental data(theblackpoints)to expression (6.l)corresponds too = 0.015 cm and Anr(O) = 4.22 x 10 ° s. Thus, we can write the energy-gap law for the LaCls crystal as follows ... 

Using these AE values and the previous expression for the energy-gap law, the following nomadiative rates are obtained ... 

---