Fermi Golden Rule expression

Let us now consider how similar the expression for rates of radiationless transitions induced by non Bom-Oppenheimer couplings can be made to the expressions given above for photon absorption rates. We begin with the corresponding (6,4g) Wentzel-Fermi golden rule expression given in Eq. (10) for the transition rate between electronic states Ti,f and corresponding vibration-rotation states Xi,f appropriate to the non BO case ... 

Thus, having prepared the system at the initial time t=0 in the state i f >, the probability of finding the system in the state f f > at time t is given, as usual, by Cf(t) 2. The Fermi Golden-rule expression (to first order in TDPT) has the form [47]... 

The manifestation of the dipole-dipole approximation can be seen explicitly in Equation (3.134) as the R 6 dependence of the energy transfer rate. In Equation (3.134) the electronic and nuclear factors are entangled because the dipole-dipole electronic coupling is partitioned between k24>d/(td R6) and the Forster spectral overlap integral, which contains the acceptor dipole strength. Therefore, for the purposes of examining the theory it is useful to write the Fermi Golden Rule expression explicitly,... 

The second theoretical approach is quantum mechanical in nature and is based on the Fermi Golden Rule expression for nonradiative decay processes [45,46]... 

Equation (34) is the Fermi Golden Rule expression, where Hc, is the electronic interaction, and FC is the F ranck-Condon factor. The analytical version of Eq. (34), applicable to high temperature, is given by Eq. (35) ... 

Equation (2.3) is the key result for the use of spectral information to obtain quantitative information concerning IVR. By determining experimentally the ratio of vibrationally redistributed (Iab>) to vibrationally unredistributed (/ .,) emission, together with knowledge of kf, one can obtain kIVR. kIVR, in turn, can provide other information about the dynamics (within the constraints of the kinetic model). For instance, average vibrational coupling matrix elements can be obtained from a Fermi golden rule expression for the rate. 

The Fermi Golden Rule (Merzbacher, 1970) is often used to interpret rate constants for electronically nonadiabatic transitions in polyatomic molecules. Figure 8.17 depicts vibrational/rotational levels for two electronic states 1 and 2. Unimolecular decomposition occurs on the ground electronic state 1. When the system is initially prepared in the electronically excited state, the complete unimolecular rate constant depends on both the rate constant k 2 for the electronic transition 1 <— 2 and the unimolecular rate constant for the ground electronic state. If a single vibrational/rotational state of electronic state 2 is initially excited, the Fermi Golden Rule expression for, 2 is... 

In the large-molecule (statistical) limit, the 1 manifold is composed of randomly distributed, overlapping levels whose individual properties cannot be determined. It may be thus treated only by statistical methods. As long as the s levels form a discrete set, the system may be described in terms of one s> state coupled to a dissipative continuum the resulting nonradiative width y " of the s> state is given by the Fermi golden rule expression, justified in Section II,E,2... 

Forster transfer The rate for direct exciton transfer between a donor molecule, D, and an acceptor molecule, A, is determined by the Fermi Golden Rule expression,... 

The nonadiabatic electron transfer rate is then given by the Fermi Golden Rule expression,... 

In most practical applications, the starting point for interpretation of molecular Auger spectra has mostly been given by the Fermi golden rule expression... 

With a strong orthogonality condition imposed on the scattered electron orbital, fulfilling the killer condition Po) = 0 the Auger amplitude resulting from the Fermi golden rule expression (Eq. 3.37) takes the form... 

From a quanmm mechanical viewpoint, both the photoinduced and back-electron transfer processes can be viewed as radiationless transitions between different, weakly interacting electronic states of the A-L-B supermolecule (Fig. 2.6). The rate constant of such processes is given by an appropriate Fermi golden rule expression ... 

The expression for the rate R (sec ) of photon absorption due to coupling V beriveen a molecule s electronic and nuclear charges and an electromagnetic field is given through first order in perturbation theory by the well known Wentzel Fermi golden rule formula (7,8) ... 

An accurate calculation of the heat conductivity requires solving a kinetic equation for the phonons coupled with the multilevel systems, which would account for thermal saturation effects and so on. We encountered one example of such saturation in the expression (21) for the scattering strength by a two-level system, where the factor of tanh((3co/2) reflected the difference between thermal populations of the two states. Neglecting these effects should lead to an error on the order of unity for the thermal frequencies. Within this single relaxation time approximation for each phonon frequency, the Fermi golden rule yields, for the scattering rate of a phonon with Ha kgT,... 

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... 

Finally, it may be useful to note that the Fermi golden rule and time correlation function expressions often used (see ref. 12, for example) to express the rates of electron transfer have been shown [13], for other classes of dynamical processes, to be equivalent to LZ estimates of these same rates. So, it should not be surprising that our approach, in which we focus on events with no reorganization energy requirement and we use LZ theory to evaluate the intrinsic rates, is closely related to the more common approach used to treat electron transfer in condensed media where the reorganization energy plays a central role in determining the rates but the z factor plays a second central role. 

Recently, the electron-transfer kinetics in the DSSC, shown as a schematic diagram in Fig. 10, have been under intensive investigation. Time-resolved laser spectroscopy measurements are used to study one of the most important primary processes—electron injection from dye photosensitizers into the conduction band of semiconductors [30-47]. The electron-transfer rate from the dye photosensitizer into the semiconductor depends on the configuration of the adsorbed dye photosensitizers on the semiconductor surface and the energy gap between the LUMO level of the dye photosensitizers and the conduction-band level of the semiconductor. For example, the rate constant for electron injection, kini, is given by Fermi s golden rule expression ... 

In eq. (5-2), vt, refers to the frequency of the ith chromophore vibration (populated by the IVR transition), and the proportionality to l/v( is purely phenomenological. This proportionality reflects the expectation that low frequency chromophore modes will couple most efficiently to the (low frequency) vdW modes. Fermi s Golden Rule expression has two important consequences. First, it predicts that... 

According to the Fermi Golden Rule, the non-adiabatic ET rate constant is strongly dependent on electronic coupling between the donor state D and acceptor state A connected by a bridge (VAb) which is given by an expression derived from the weak perturbation theory... 

Cf. the Fermi golden rule. Section 5.2.3.) The density of states pg (number of states per unit energy interval) is related to the spectral overlap J, and using the relations for (i given above the expressions derived by Fdrster (1951) and Dexter (1953) for the rate constant of energy transfer by the Coulomb and the exchange mechanism, respectively, may be written as... 

The golden rule rate expression is a standard quantum-mechanical result for the relaxation rate of a prepared state z) interacting with a continuous manifold of states 1/ The result, derived in Section 9.1, for this rate is the Fermi golden rule... 

This expression will, depending on the value of the time t, assume positive and negative values. In order to assure that a monotonically increasing excited-state population is prepared, the simplest choice for the parameter co is co = (E — Eq)/H, which is the resonance condition obtained from Fermi s Golden rule expression... 

Fermi s Golden Rule expresses the rate of transitions between b and a as... 

An important achievement of the early theories was the derivation of the exact quantum mechanical expression for the ET rate in the Fermi Golden Rule limit in the linear response regime by Kubo and Toyozawa [4b], Levich and co-workers [20a] and by Ovchinnikov and Ovchinnikova [21], in terms of the dielectric spectral density of the solvent and intramolecular vibrational modes of donor and acceptor complexes. The solvent model was improved to take into account time and space correlation of the polarization fluctuations [20,21]. The importance of high-frequency intramolecular vibrations was fully recognized by Dogonadze and Kuznetsov [22], Efrima and Bixon [23], and by Jortner and co-workers [24,25] and Ulstrup [26]. It was shown that the main role of quantum modes is to effectively reduce the activation energy and thus to increase the reaction rate in the inverted... 

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