Forster Equation-Theory

The starting point for the Forster-Zuber theory (F4, F5, F6) is the Rayleigh equation (Rl) for a bubble growing in a liquid medium. In this... 

Based on the experimental evidences discussed in sect. 3.6.4 of an effect of the ligand onto the lifetime, numerous publications have appeared that refer to the Forster s theory (De Sa et al., 1993 Beeby et al., 1999 Supkowski and Horrocks, 1999 An et al., 2000). However, this theory is not applied in order to derive the transfer rate constant or the mean interaction distance value but only to justify the search for relationships between the observed decay rate and the number of OH, CH or NH bonds of the ligand, plus a global parameter for the solvent. Thus, although based on a very different theoretical approach, one deals with equations similar to eq. (11), with more terms, as in the following example (Beeby et al., 1999) ... 

To apply the Forster equation, the emission and absorption line shapes must be identical for all donors and acceptors, respectively. However, in many types of condensed-phase media (e.g., glasses, crystals, proteins, surfaces), each of the donors and acceptors lie in a different local environment, which leads to a distribution of static offsets of the excitation energies relative to the average, which persists longer than the time scale for EET. When such inhomogeneous contributions to the line broadening become significant, Forster theory cannot be used in an unmodified form [16, 63]. 

The distance between the donor and the acceptor r is described by Forster s theory. Forster derived an equation for the rate of energy transfer from a specific donor to a spcific acceptro kj ... 

According to Forster s theory, this definition of E, combined with Equation (6.43), gives [39,a] ... 

It is not necessary to assume the liquid film to be completely stagnant. Radial motion can be allowed for, but with some difficulty. It was noted in Sec. IIB2 that Forster and Zuber state that conduction is the chief mode of heat transfer (compared with convection due to radial motion). Eddies or motions of the liquid tangent to the bubble are neglected. The Zwick-Plesset theory likewise excludes eddies. The derivation is lengthy therefore the final typical equations are presented here without proof. 

In the last decade there has been an extraordinary progress towards accurate estimation of each one of the ingredients involved in the Forster rate equation, and comparison of these theories with available single-molecule EET... 

The great success of Forster theory lies on the simplicity of these expressions, which can be applied from purely spectroscopic data. However, the approximations underlying these equations are not evident at first sight. It is better to turn to the Golden Rule expression of the rate ... 

A mixed quantum classical description of EET does not represent a unique approach. On the one hand side, as already indicated, one may solve the time-dependent Schrodinger equation responsible for the electronic states of the system and couple it to the classical nuclear dynamics. Alternatively, one may also start from the full quantum theory and derive rate equations where, in a second step, the transfer rates are transformed in a mixed description (this is the standard procedure when considering linear or nonlinear optical response functions). Such alternative ways have been already studied in discussing the linear absorbance of a CC in [9] and the computation of the Forster-rate in [10]. 

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 value of can also be calculated by using the quantitative theories for D-A multipolar energy transfer given by Forster and Dexter. For instance, the transfer rate for dipole -dipole interaction is given by the Dexter equation [2] ... 

In Section 14.2, we fonnd in Eqnation 14.11 that the interaction matrix element between two chromophores in an excited triplet state decreases exponentially with distance, while the corresponding singlet matrix element (Equation 14.10) decreases more slowly as 1/R with distance. These interactions were derived a long time ago from the old quantnm theory by D. L. Dexter in the triplet case and Theodor Forster in the singlet case. The conclnsions are valid also for the case of unequal chromophores. 

The Marcus equation for nonadiabatic electron-transfer reactions (Eq. B5.3.4), and the Forster theory that we discussed in Chap. 7 apply only to systems with weak intermolecular interactions, which we now can define more precisely as meaning that H21 lh steady-state approximation to the stochastic Liouville equation for a two-state reaction in this limit From Eqs. (BIO.1.15), (10.29a), (10.29b), and (10.30), we have... 

If the donor and acceptor molecules diffuse in solution or in the gas phase, Forster theory predicts that the efficiency of quenching by energy transfer increases as the average distance traveled between collisions of donor and acceptor decreases. That is, the quenching efficiency increases with concentration of quencher, as predicted by the Stern-Volmer equation. 

---