Time-dependent Golden rule treatment

Time-dependent Golden-rule treatment for Electronic predissociation... 

The purpose of this work is to study the electronic predissociation from the bound states of the excited A and B adiabatic electronic states, using a time dependent Golden rule (TDGR) method, as previously used to study vibrational pre-dissociation[32, 33] as well as electronic predissociation[34, 35], The only difference with previous treatments[34, 35] is the use of an adiabatic representation, what requires the calculation of non-adiabatic couplings. The method used is described in section II, while the corresponding results are discussed in section III. Finally, some conclusions are extracted in section IV. 

In the rigorous full quantum treatment (19), one solves for the full time-dependent wavefunction in Eq. (48) which can be computationally expensive. In the golden rule treatment, V t = oo) is obtained by a first-order approximation as follows ... 

In one quantum mechanical approach based on the diabatic approximation , the electron is assumed to be confined initially at one of the reactant sites and electron transfer is treated as a transition between the vibrational levels of the reactants to those of the products. The quantum mechanical treatment begins with the time dependent Schrodinger equation, Hip = -ihSiplSt, where the wavefunction tj/ is written as a sum of the initial (reactant) and final (product) states. In the limit that the Bom-Oppenheimer approximation for the separation of electronic and nuclear motion is valid, the time dependent Schrodinger equation eventually leads to the Golden Rule result in equation (25). 

For a quantitative treatment of establishing connections between vibronic coupling and vibrational progressions in electronic spectra, band profiles from vibronic wavefunctions must be calculated using common procedures of time-dependent perturbation theory and Fermi s golden rule [57], For emission, e.g., the transition rate which is the transition probability per unit time summed over... 

Spectra, P E), can be obtained directly from a time-dependent treatment as the Fourier transform of the autocorrelation function C t), assuming a direct transition from the initial to the final states within the framework of Fermi s golden rule [6,119] ... 

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. 

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