Forster rates

FRET is a nonradiative process that is, the transfer takes place without the emission or absorption of a photon. And yet, the transition dipoles, which are central to the mechanism by which the ground and excited states are coupled, are conspicuously present in the expression for the rate of transfer. For instance, the fluorescence quantum yield and fluorescence spectrum of the donor and the absorption spectrum of the acceptor are part of the overlap integral in the Forster rate expression, Eq. (1.2). These spectroscopic transitions are usually associated with the emission and absorption of a photon. These dipole matrix elements in the quantum mechanical expression for the rate of FRET are the same matrix elements as found for the interaction of a propagating EM field with the chromophores. However, the origin of the EM perturbation driving the energy transfer and the spectroscopic transitions are quite different. The source of this interaction term... 

By substituting Eq. (B4.4.5) into Eq. (B4.4.7), we obtain the Forster rate constant kjl (Eq. 4.78 in the text) for energy transfer in the case of long-range dipole-dipole interaction, and substitution of Eq. (B4.4.6) into Eq. (B4.4.7) leads to the Dexter rate constant k fl (Eq. 4.85 in the text) for the short-range exchange interaction. 

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... 

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]. 

Forster theory [1] expresses the rate of EET from a donor D molecule (or atom) to an acceptor A in terms of the mutual orientation of the molecules, their center-to-center separation in units of cm, R, and the overlap, /, of the donor emission spectrum with the acceptor absorption spectrum, as shown in Figure 3.48. The Forster rate expression is... 

Figure 4.S Schematic illustration of the standard point dipole approach to calculating the Forster rate, and the line dipole method. The line dipole method divides each chromophore into several dipoles whose distance from each other and wavefunction...

More recently Andrews and Juzeliunas [6, 7] developed a unified tlieory that embraces botli radiationless (Forster) and long-range radiative energy transfer. In otlier words tliis tlieory is valid over tire whole span of distances ranging from tliose which characterize molecular stmcture (nanometres) up to cosmic distances. It also addresses tire intennediate range where neitlier tire radiative nor tire Forster mechanism is fully valid. Below is tlieir expression for tire rate of pairwise energy transfer w from donor to acceptor, applicable to transfer in systems where tire donor and acceptor are embedded in a transparent medium of refractive index ... 

FORSTER and ZUBER(85,86J who employed a similar basic approach, although the radial rate of growth dr/dt was used for the bubble velocity in the Reynolds group, showed that ... 

The occurrence of energy transfer requires electronic interactions and therefore its rate decreases with increasing distance. Depending on the interaction mechanism, the distance dependence may follow a 1/r (resonance (Forster) mechanism) or e (exchange (Dexter) mechanisms) [ 1 ]. In both cases, energy transfer is favored by overlap between the emission spectrum of the donor and the absorption spectrum of the acceptor. 

The latent heat transport accounts for only 2% of the total heat flux in this case. However, it was observed by several investigators that the total heat transfer rate is proportional to this value, <7, lenl, because it is proportional to the bubble volume and the number of bubbles that cause intense agitation of the liquid layer close to the surface. This agitation, termed microconvection, together with the liquid-vapor exchange, were considered to be the key to excellent characteristics of boiling heat transfer (Forster and Greif, 1959). 

Inasmuch as the interaction energy can be related to the transition moments, Forster has been able to develop a quantitative expression for the rate of energy transfer due to dipole-dipole interactions in terms of experimental parameters<4 aB-30) ... 

It is possible to estimate the rate of vertical singlet energy transfer (9.31) and (9.33) (when Forster-type energy transfer is negligible, spectral overlap integral is very small) from the relation... 

Forster derived the following expression for the transfer rate ... 

Seminal studies on the dynamics of proton transfer in the triplet manifold have been performed on HBO [109]. It was found that in the triplet states of HBO, the proton transfer between the enol and keto tautomers is reversible because the two (enol and keto) triplet states are accidentally isoenergetic. In addition, the rate constant is as slow as milliseconds at 100 K. The results of much slower proton transfer dynamics in the triplet manifold are consistent with the earlier summarization of ESIPT molecules. Based on the steady-state absorption and emission spectroscopy, the changes of pKa between the ground and excited states, and hence the thermodynamics of ESIPT, can be deduced by a Forster cycle [65]. Accordingly, compared to the pKa in the ground state, the decrease of pKa in the... 

The pioneering work Forster and Hoffmann [28] on the viscosity dependence of the fluorescence quantum yield of triphenylmethane dyes (TPM) has set the foundation for several reports in these dyes (Fig. 12). It was found that both an ability to twist around the carbocationic center and the donor-acceptor properties are important [66], Specifically, a strong intramolecular quenching is observed for 34 that is virtually absent (two orders of magnitude slower quenching rate) in the bridged... 

Post-Forster Subsequent derivations of the rate of energy transfer... 

We now focus our attention on the presence of the unperturbed donor quantum yield, Qd, in the definition of R60 [Eq. (12.1)]. We have pointed out previously [1, 2] that xd appears both in the numerator and denominator of kt and, therefore, cancels out. In fact, xo is absent from the more fundamental expression representing the essence of the Forster relationship, namely the ratio of the rate of energy transfer, kt, to the radiative rate constant, kf [Eq. (12.3)]. Thus, this quantity can be expressed in the form of a simplified Forster constant we denote as rc. We propose that ro is better suited to FRET measurements based on acceptor ( donor) properties in that it avoids the arbitrary introduction into the definition of Ra of a quantity (i />) that can vary from one position to another in an unknown and indeterminate manner (for example due to changes in refractive index, [3]), and thereby bypasses the requirement for an estimation of E [Eq. (12.1)]. 

Developed into a power series in R 1, where R is the intermolecular separation, H exhibits the dipole-dipole, dipole-quadrupole terms in increasing order. When nonvanishing, the dipole-dipole term is the most important, leading to the Forster process. When the dipole transition is forbidden, higher-order transitions come into play (Dexter, 1953). For the Forster process, H is well known, but 0. and 0, are still not known accurately enough to make an a priori calculation with Eq. (4.2). Instead, Forster (1947) makes a simplification based on the relative slowness of the transfer process. Under this condition, energy is transferred between molecules that are thermally equilibriated. The transfer rate then contains the same combination of Franck-Condon factors and vibrational distribution as are involved in the vibrionic transitions for the emission of the donor and the adsorptions of the acceptor. Forster (1947) thus obtains... 

Forster (1968) points out that R0 is independent of donor radiative lifetime it only depends on the quantum efficiency of its emission. Thus, transfer from the donor triplet state is not forbidden. The slow rate of transfer is partially offset by its long lifetime. The importance of Eq. (4.4) is that it allows calculation in terms of experimentally measured quantities. For a large class of donor-acceptor pairs in inert solvents, Forster reports Rg values in the range 50-100 A. On the other hand, for scintillators such as PPO (diphenyl-2,5-oxazole), pT (p-terphenyl), and DPH (diphenyl hexatriene) in the solvents benzene, toluene, and p-xylene, Voltz et al. (1966) have reported Rg values in the range 15-20 A. Whatever the value of R0 is, it is clear that a moderate red shift of the acceptor spectrum with respect to that of the donor is favorable for resonant energy transfer. 

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