Frequency barrier crossing

The barrier crossing frequency described by Eq. (36) is dependent not only on the barrier height, but also on its shape. Equation (36) was derived under the assumption that there is no donor-acceptor overlap and the barrier has a cusp-like shape. 

The problem of a proper calculation of the barrier-crossing frequency, in such cases, was described by Hynes [200] t, may be used in Vp calculations of a strongly adiabatic reactions. For weakly adiabatic reactions, the effective longitudinal relax-... 

The required result is derived by considering the case of a single particle A in which the volume surrounding A is taken to be a sphere of radius Rg. The number density of B particles is The activation energy for dissociation is Qp = P Rt) — t (Rp), and the barrier crossing dynamics is characterized by the dissociation attempt frequency ojp and the barrier crossing frequency toj. These are defined by thermal averages (partition functions) rather than expansions of the potentials. Thus... 

Weaver M. J., McManis G. E., Jarzeba W. and Barbara P. F. (1990), Importance of fast solvent relaxation components to electron-transfer rates—comparisons between barrier-crossing frequencies and subpicosecond time-resolved solvation dynamics , J. Phys. Chem. 94, 1715-1719. 

The energy of the barrier-crossing frequency, cov, is employed in the pre-exponential term.8 We used cdv = 800 cm-1 for bis(hydrazines), which is both close to the value estimated using semiempirical dynamics calculations on 11+ 33 and obtained in as yet unpublished resonance Raman data for 20 +. 87 We used cov = 1100 cm-1 for bis(diazenium) radical cations. This value is smaller than the 1403 cm-1 obtained from dynamics calculations on sB4T+ (7+)33 but close to the 1053 cm-1 obtained from resonance Raman work in collaboration with Williams and Hupp.88... 

More generally, then, it is useful to consider a net barrier-crossing frequency, Aet=KelVn. with the electronic transmission coefficient electron transfer occuring each time the system reaches the barrier top. An illustrative, albeit simplified, expression for Kei is [2] ... 

Finally, it is worth noting that the nature of the solvent-dependent ET dynamics is also predicted to be affected by the presence of inner-shell (reactant vibrational) contributions to the activation barrier [19]. As might be expected, the presence of higher-frequency vibrational contributions to the activation barrier can yield a marked attenuation in the degree to which overdamped solvent dynamics control the adiabatic barrier-crossing frequency [19]. The experimental exploration of such effects is limited in part by the paucity of redox couples suitable for solvent-dependent studies that exhibit known vibrational barriers, and complicated by the qualitatively similar behavior expected for nonadiabatic pathways. Nevertheless, there is some evidence that vibrational activation can indeed attenuate the role of solvent dynamics, although the theoretical predictions appear to overestimate the magnitude of this effect [10b,20]. 

A common approach for the study of activated barrier crossing reactions is the transition state theory (TST), in which the transfer rate over the activation barrier V is given by (0)R/2jt)e where 0)r (the oscillation frequency of the reaction coordinate at the reactant well) is an attempt frequency to overcome the activation barrier. For reactions in solution a multi-dimensional version of TST is used, in which the transfer rate is given by... 

Figure 15. Calculated values of the transmission coefficient k plotted as a function of the solvent viscosity rf for four barrier frequencies a>b at 7 = 0.85. The squares denote the calculated results for to = 3 x 1012 s I, the asterisks denote results for to = 5 x 1012 s-1, the triangles denote results for to = 1013 s I and the circles denote results for a>b = 2 x 1013 s l. The solid lines are the best-fit curves with exponents a 0.72 for wb = 3 x 1012s l, a 0.58 for wb = 5 x 1012 s-1,a 0.22 for wb = 1013 s l, and a 0.045 for cob = 2 x 10I3s-. Note here that the barrier crossing rate becomes completely decoupled from the viscosity of the solvent at wb = 2 x 10l3s-1. The transmission coefficient k is obtained by using Eq. (326). Note here that the viscosity is calculated using the procedure given in Section X and is scaled by a2/ /mkBT, and a>b is scaled by t -1. For discussion, see the text. This figure has been taken from Ref. 170.

As can be seen from the numbers, the exponent a is clearly a function of barrier frequency (cob) and its value is decreasing with increase in a>b- For cob — 2 x 1013 s-1, its value almost goes to zero (a < 0.05), which clearly indicates that beyond this frequency the barrier crossing rate is entirely decoupled from solvent viscosity so that one recovers the well-known TST result that neglects the dynamic solvent effects. 

In order to complete the above analysis, one needs to solve the full non-Markovian Langevin equation (NMLE) with the frequency-dependent friction for highly viscous liquids to obtain the rate. This requires extensive numerical solution because now the barrier crossing dynamics and the diffusion cannot be treated separately. However, one may still write phenomenologically the rate as [172],... 

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. 

The thermal-induced intramolecular electron transfer rates of mixed-valence biferrocene monocation (Fe(II),Fe(III)) were determined in seven solvents and at various temperatures by the proton paramagnetic relaxation measurements. The rate constants of pico-second order were obtained at 298 K and the frequency factors showed a solvent dependence. The effect of solvent friction on the barrier crossing in the reaction trajectory was examined in the strong adiabatic regime. 

The deviation of nitrobenzene from the solid line (the slope = 1) in Figure 2 is probably attributed to the frequency dependent dielectric friction for the reaction dynamics around the barrier top, i.e., the much slower dielectric fluctuation of nitrobenzene (tl - 6 ps at 298K) compared with the ET rate hardly works as friction for the barrier crossing. In such case, the friction is shows tl (a[Pg.400]

The 10 s order rate constants for the thermal-induced (ground state) intramolecular electron transfer rates of the mixed-valence biferrocene monocation were first elucidated in various solvents by the H-NMR relaxation measurements. The obtained solvent dependent frequency factors indicated significant contribution of the solvent dielectric friction on the barrier crossing. An existence of the faster processes compared with the ET rate such as the internal vibration as an escape route of the reaction dynamics along the solvent coordination was also proposed in some extent. 

Solvent reorientation and isomerization of trans-stilbene in alkane solutions has been studied by ps time scale anisotropic absorption and polarization239 Coupling of solute and solvent decreases as the size of the solvent molecules increases. The applicability of currently favoured models for the activated barrier crossing in the photoisomerization of stilbene is discussed, A method for measuring quantum yields in the photoisomerization of trans-stilbene gives high accuracy without use of a chemical actinometer . Evidence has been found for dynamic solvent effects on the photoisomerization of 4,4 -dimethoxystilbene in which the effects of temperature and hydrostatic pressure were made in n-alkane and n-alkyl alcohol. A ps laser time-resolved study fits frequency dependent solvent shifts but gives results inconsistent with the free volume model. Photophysical and theoretical studies of trans and 9-... 

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