Kramers’ relation transitions

According to Kramers model, for flat barrier tops associated with predominantly small barriers, the transition from the low- to the high-damping regime is expected to occur in low-density fluids. This expectation is home out by an extensively studied model reaction, the photoisomerization of tran.s-stilbene and similar compounds [70, 71] involving a small energy barrier in the first excited singlet state whose decay after photoexcitation is directly related to the rate coefficient of tran.s-c/.s-photoisomerization and can be conveniently measured by ultrafast laser spectroscopic teclmiques. 

The Pt + Pt intervalence transitions of such chain complexes occur in the regions 25,000-18,200 cm 1, 23,600-14,300 cm 1 and 20,600-7,500 cm 1 for chloro-, bromo-, and iodo-bridged complexes, respectively, the trend Cl > Br > I being the reverse of that of the conductivity of the complexes. The transition wavenumbers may be determined either by Kramers-Kronig analysis of specular reflectance measurements or from plots of the excitation profiles of Raman bands enhanced at or near resonance with the Pt I-PtIV intervalence band. The maxima have been found to be related to the Pt —PtIV chain distance, the smaller the latter the less being the intervalence transition energy (3). 

Evidently, because of its exponential dependence on the parameter ct = KVm/T, the quantity x(a) may grow virtually unboundedly. This is the specific property of the solution of any Kramers problem that the probability of transition between the potential minima (wells) depends exponentially on the barrier height related to the thermofluctuational energy. In Section III.B we revisit the problem of calculation of the overbarrier relaxation time and using a mathematically correct asymptotic expansion procedure, show how the preexponential factor in the overbarrier relaxation time may be evaluated with the accuracy far higher than that in Eq. (4.50). 

We must remark that the broadening is in general nonlorentzian and asymmetric, particularly when the optical transition occurs at the boundary of the excitonic band (as in the case of the anthranene crystal). Lastly, the analyticity of the CPA method assures that (4.118) satisfies the Kramers-Kronig relations, and that the total oscillator strength of the transition, redistributed on the two bands, is conserved. 

The electronic absorption characteristics of chromophores within potential gela-tors can provide an important experimental monitor of the microscopic environment in which they reside. This is especially true when the information includes optical rotatory dispersion (ORD) and circular dichroism (CD) data for potential gelalors that arc chiral. Dichroism relates to the absorptivity difference between the two components of circularly polarized light, w-hich constitutes the incident plane of linearly polarized light as described by the Kronig-Kramers transform. The intensity of UV/vis absorption depends on corresponding quantum transition. The wavelengths at which nonzero circular dichroism may be observable in the CD spectrum can be discerned from the shape of the absorption bands. The... 

The traditional approach to evaluating RR intensities involves a summation over all unperturbed eigenstates of the resonant electronic state. This is a direct consequence of the quantum-mechanical derivation of the polarizability tensor components employing second order perturbation theory as given by the Kramers-Heisenberg-Dirac (KHD) relation for the transition polarizability tensor ... 

The theory discussed until now is based on the Kramers-Heisenberg-Dirac dispersion relation for the transition polarizability tensor as given in Eq. (6.1-1). The expression shown in this equation describes a steady state scattering process and contains no explicit reference to time. Therefore, the resonance Raman theory which is based on the KHD dispersion relation is sometimes also termed as time-independent theory (Ganz et al., 1990). 

Dynamic medium effects in solution kinetics were first recognized by Kramers [41], He treated the problem on the basis of the Langevin equation [42] according to which the velocity of the reactants along the reaction coordinate and the friction of the surrounding medium play a role. Details of Kramers theory are not given here but an introduction to this subject can be found elsewhere [G3], The parameters involved in quantitatively assessing the dynamic solvent effect are the frequency associated with the shape of the barrier of the transition state and a friction parameter which is related to solvent viscosity. 

The absorption constant k and the refractive index n are related via the Kramers-Kronig relations. Hence, an increase of the optical density, either due to an increase of the oscillator strength or a spectral red shift of the transition, leads to an increase of the refractive index. Since n is larger than k by two orders of magnitude and usually decreases more slowly with increasing distance from the absorption maximum, it yields the main contribution to the optical susceptibility. Therefore, the effect of... 

The elements of the transition rate matrix A in the DRIS model are estimated from the multidimensional energy surface associated with the interdependent rotation of neighboring bonds u g Kramers rate theory, as described above. Accordingly, the probability of occurrence of a given isomeric state for a bond depends on the state of its first neighbors aloi the chain. Likewise, in the kinetic Ising model the transition rate Wi(Oi) of the i-th spin is assumed to be coupled to the state of its first iKighbors by the relation... 

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