Excitation Golden Rule

The first type of interaction, associated with the overlap of wavefunctions localized at different centers in the initial and final states, determines the electron-transfer rate constant. The other two are crucial for vibronic relaxation of excited electronic states. The rate constant in the first order of the perturbation theory in the unaccounted interaction is described by the statistically averaged Fermi golden-rule formula... 

Energy transfer in solution occurs through a dipole-dipole interaction of the emission dipole of an excited molecule (donor) and the absorptive moment of a unexcited molecule (acceptor). Forster<40) treated the interaction quantum mechanically and derived and expression for the rate of transfer between isolated stationary, homogeneously broadened donors and acceptors. Dexter(41) formulated the transfer rate using the Fermi golden rule and extended it to include quadrupole and higher transition moments in either the donor or the acceptor. Following the scheme of Dexter, the transfer rate for a specific transition is... 

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. 

The magnitude of the off-diagonal Hamiltonian (i.e. the energy transfer rate) thus depends on the strengths of the electron-phonon and Coulombic couplings and also the overlap of the two exciton wavefunctions[53]. Energy transfer rates from states to state jx, are calculated via the golden rule [54] and used as inputs to a master equation calculation of the excitation transfer kinetics in PSI, in which the dynamical information is included in the matrix K. 

Fig. 12.2. Schematic illustration of the absorption of an IR photon by the van der Waals complex Ar H2 and the subsequent dissociation into Ar+H2(n = 0). Voo(R) and Vn(i ) are the diagonal elements of the potential coupling matrix defined in Equation (3.6) which serve to define the zero-order radial wavefunc-tions employed in the Golden Rule expression for the dissociation rate. The assignment of the bound levels is (m,n), where m and n denote the number of quanta of excitation in the dissociation mode R and the vibrational mode of H2, respectively.

The rate constant for the radiationless relaxation of the excited states of inorganic compounds has been derived on the basis of the Fermi s golden rule (Equation 6.69). 

The conflicting serial/parallel models for IVR/VP are not readily distinguished until time resolved experiments can be performed on the systems of interest. Both models can relate the relative intensities of the emission features to the various model parameters, but the serial process seems more in line with a simple, conventional [Fermi s Golden Rule for IVR (Avouris et al. 1977 Beswick and Jortner 1981 Jortner et al. 1988 Lin 1980 Mukamel 1985 Mukamel and Jortner 1977) and RRKM theory for VP (Forst 1973 Gilbert and Smith 1990 Kelley and Bernstein 1986 Levine and Bernstein 1987 Pritchard 1984 Robinson and Holbrook 1972 Steinfeld et al. 1989)], few parameter approach. Time resolved measurements do distinguish the models because in a serial model the rises and decays of various vibronic states should be linked, whereas in a parallel one they are, in general, unrelated. Moreover, the time dependent studies allow one to determine how the rates of the IVR and VP processes vary with excitation energy, density of states, mode properties, and isotropic substitution. 

Sa1 excitation, which generates little bare molecule emission, fast IVR for the state pumped is also observed. While concrete proof for a serial mechanism is yet to come in the form of rise and fall times for intermediate states, the inference of this mechanism is quite strong in the results. The discussion below for anilinefNj) clusters, and the simple two parameter serial IVR/VP model based on Fermi s Golden Rule and RRKM theory, will provide the final demonstration for this mechanism. 

Much more is becoming known about the rates of the physical processes in competition with proton exchange reactions in excited states. (For an excellent review see Henry and Siebrand, 1973.) The factors which determine the rate constants (k) for internal conversion and intersystem crossing are neatly summarized in the Golden Rule of time-dependent perturbation theory ... 

An alternative approach widely used in polyatomic molecule studies is based on the Golden Rule and a perturbative treatment of the anharmonic coupling (57,62). This approach is not much used for diatomic molecules. In the liquid O2 example cited above, the Hamiltonian must be expanded to 30th order or so to calculate the multiphonon emission rate. But for vibrations of polyatomic molecules, which can always find relatively low-order VER pathways for each VER step, perturbation theory is very useful. In the perturbation approach, the molecule s entire ladder of vibrational excitations is the system and the phonons are the bath. Only lower-order processes are ordinarily needed (57) because polyatomic molecules have many vibrations ranging from higher to lower frequencies and only a small number of phonons, usually one or two, are excited in each VER step. The usual practice is to expand the interaction Hamiltonian (qn, Q) in Equation (2) in powers of normal coordinates (57,62) ... 

The V-B coupling Hamiltonian to first order in the three HOD dimensionless normal coordinates is Hv b = —2, c], l , where F, is the inter-molecular force due to the solvent exerted on the harmonic normal coordinate, evaluated at the equilibrium position of the latter. This force obviously depends on the relative separations of all molecules, and on their relative orientations. In the most rigorous quantum description of rotations, this term would depend on the excited molecule rotational eigenstates and of the solvent molecules. Instead rotation was treated classically, a reasonable approximation for water at room temperature. With this form for the coupling, the formal conversion of the Golden Rule formula into a rate expression follows along the lines developed by Oxtoby (2,53), with a slight variation to maintain the explicit time dependence of the vibrational coordinates (57),... 

Using a Fermi s Golden Rule approach, if the coupling between the oscillator and the bath modes is weak, then, to first order, the transition rate from the first excited vibrational level to the ground state is given by (3)... 

The basic theoretical framework for understanding the rates of these processes is Fermi s golden rule. The solute-solvent Hamiltonian is partitioned into three terms one for selected vibrational modes of the solute, including the vibrational mode that is initially excited, one for all other degrees of freedom (the bath), and one for the interaction between these two sets of variables. One then calculates rate constants for transitions between eigenstates of the first term, taking the interaction term to lowest order in perturbation theory. The rate constants are related to Fourier transforms of quantum time-correlation functions of bath variables. The most common... 

The contribution to the X-ray absorption coefficient due to the excitation of a deep core level may be expressed as /rc = nco-c, where nc is the density of atoms with the core level of concern and absorption cross section for this level on a single atom. Assuming the X-ray field to be a small perturbation, the latter can be evaluated from the golden rule transition rate per unit photon flux. The general X-ray absorption cross section is given by... 

This is the most direct experimental manifestation of the existence of an electronic interaction. It can occur spontaneously in mixed-valence complexes, but also in bimetallic systems after a photochemical excitation (photoinduced electron transfer). The general theory considers electron transfer as a special case of radiationless transition, with a perturbative treatment based on Fermi s Golden Rule [42]. In the nonadiabatic case, the rate constant can be written as [43] ... 

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