Franck-Condon weighted density of states

The Franck-Condon-weighted density of states (FCWD) reflects the influence of all of the nuclear modes of the system, often represented in terms of effective normal... 

Here, Tda is the tunneling matrix element between the donor (D) and acceptor (A) wave functions and (FC) is the Franck-Condon-weighted density of states. This is a description appropriate for an effective two-level system [3, 4, 5] with weak coupling... 

In Fermi s Golden Rule, FC represents the Franck-Condon weighted density of states. A Franck-Condon formulation is used because the more massive atomic nuclei remain in the same position as the relatively light electron tunnels from the... 

The time-dependent perturbation theory of the rates of radiative ET is based on the Born-Oppenheimer approximation [59] and the Franck Condon principle (i.e. on the separation of electronic and nuclear motions). The theory predicts that the ET rate constant, k i, is given by a golden rule -type equation, i.e., it is proportional to the product of the square of the donor-acceptor electronic coupling (V) and a Franck Condon weighted density of states FC) ... 

Cf. Example 1.8.) This result shows that in the 3x3 model, the spin-orbit coupling vector depends on three factors the coefficient Q + of the in-phase (/d + /d) character of the singlet state, the spin-orbit coupling parameter (heavy atom effect), and the spatial disposition of the orbitals Xa and Xb-actual intersystem crossing rate will also depend on the Franck-Condon-weighted density of states. (Cf. the Fermi golden rule. Section S.2.3.)... 

The summed term in eq. 4.40 is often referred to as the Franck-Condon weighted density of states (FCWD), and the equation written in the compact form of eq. 4.5. This is the master equation for the calculation of rates of nonadiabatic ET where the reactants remain in thermal equilibration with their surroundings. 

Here, = < /, V /y> is the electronic coupling matrix element and (FCWD) denotes the Franck-Condon weighted density of states. In the high-temperature regime — that is, when assuming that all vibrational modes are classical (fico, kgT ), the FCWD obeys a standard Arrhenius type of equation ... 

In Eq. (1), FCWD is the Franck-Condon weighted density of states that describes the nuclear contribution to k, and //daW is the electronic coupling matrix element that defines the interaction strength occurring between the reactant and product wavefunctions at the transition state. In the classical limit, the Franck-... 

FCWDS is the Franck-Condon weighted density of states and V the electronic coupling matrix element. In a semiclas-sical treatment, this equation can be separated into a preexponential factor A and an exponential term that relates et to the driving force AG and nuclear reorganization energies of reactant A. and solvent... 

Considering the charge injection as a non-adiabatic radiationless process [5], expressed by (5), depends on two factors (1) the squared electronic coupling matrix element between the donor and the acceptor (IHP in (5) and (2) the Franck-Condon weighted density of states (FCWD), which is a function of the reorganization energy k and of the driving force AG, see Fig. 2, and equation (6) [5] ... 

Equation (1.5) is the simplest of all Golden Rule expressions because both the electronic matrix element and the Franck-Condon factor are taken as averages over all interacting vibronic states. A better model employs a Franck-Condon weighted density of states, Pf(F), in order to account for the fact that not all states in the dense manifold couple with i> with the same probability. In any case, the variation in the Franck-Condon factors with electronic energy gap, AE = Ej - Ef, determines the relative rates of radiationless transitions in compounds that contain the same chromophores and hence exhibit similar values of Ujf. The relative magnitudes of the Franck-Condon factors for different vibrational modes also determines the nature of the accepting modes populated preferentially by the radiationless transition. 

Introducing this Franck-Condon weighted density of states in the Golden Rule, gives... 

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