The Forster Theory

Suppose we know that the excitation is on molecule 1 at zero time. How fast will it move to molecule 2 Let s start by describing the system by Eq. (7.6) with Ci = 1 and C2 = 0. Then, using Eq. (2.61), we can write 

If H21 is independent of time, and we restrict ourselves to intervals that are short enough so that C2 1, Eq. (7.8) can be integrated immediately  

To obtain the overall rate constant, we must integrate the expression in Eq. (7.10) over the distribution of the initial energies, pAE ), and then divide by r  

The interaction matrix element//21 in the integrand must consider the initial and final nuclear states of the energy donor and acceptor in addition to the electronic wavefunctions. However, to the extent that the Bom-Oppenheimer approximation is valid, the nuclei will not move significantly during the instant when the excitation energy jumps between the molecules. //21 thus can be approximated as a product of a purely electronic interaction matrix element (//2i(eo) and two nuclear overlap integrals [cf. Eq. (4.42)]  

In general, the nuclear wavefunctions Xb nd Xa could represent any of many different vibrational states of the system. We have to weight the contribution from each of these nuclear states by the appropriate Boltzmaim factor. Taking the Boltzmann factors into account gives the following expression for the rate constant  

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A detailed theory of energy transfer by the Coulombic mechanism was developed by Forster, so the process is often referred to as Forster resonance energy transfer (FRET). According to the Forster theory, the probability of Coulombic energy transfer falls off inversely with the sixth power of the distance between the donor and the acceptor. For... 

Figure 6.11 The dependence of the efficiency of energy transfer, Er, on the donor-acceptor distance, R, according to the Forster theory...

May V (2009) Beyond the Forster theory of excitation energy transfer importance of higher-order processes in supramolecular antenna systems. Dalton Trans 45 10086-105... 

Wong KF, Bagchi B, Rossky PJ (2004) Distance and orientation dependence of excitation transfer rates in conjugated systems beyond the Forster theory. J Phys Chem A 108 5752-5763... 

Because this chapter will be mainly concerned with the Forster mechanism of transfer, the results of the Forster theory, given in Section 4.6.3, are recalled here for convenience. The rate constant for transfer between a donor and an acceptor at... 

One would still like to examine the effect of ethidium on the torsional rigidity and dynamics at high binding ratios. One would also like to test the Forster theory for excitation transfer between bound ethidium molecules, since it has been questioned/65- This is possible in principle by deconvoluting the effects of depolarization by excitation transfer on the FPA, as will be shown subsequently. DLS also provides crucial information on this same question. 

Scholes GD, Jordanides XJ, Fleming GR (2001) Adapting the Forster theory of energy transfer for modeling dynamics in aggregated molecular assemblies. J Phys Chem B 105 1640... 

The validity of the Forster theory was tested and confirmed in a number of model studies with compounds that contained a donor and an acceptor separated by well-defined rigid spacers. This work has been reviewed 5] In a classical study, a naphthyl group (donor) was attached to the C-terminal and a dansyl group (acceptor) to the N-terminal of poly-L-proline oligomers (1-12 proline residues) 61 These proline oligomers assume a trans helical conformation in ethanol and thus represent spacers of well-defined length (12-46 A). A continuous decrease in the transfer efficiency from 100% at a donor-acceptor separation of... 

The elucidation of the structure, dynamics and self assembly of biopolymers has been the subject of many experimental, theoretical and computational studies over the last several decades. [1, 2] More recently, powerful singlemolecule (SM) techniques have emerged which make it possible to explore those questions with an unprecedented level of detail. [3-55] SM fluorescence resonance energy transfer (FRET), [56-60] in particular, has been established as a unique probe of conformational structure and dynamics. [26-55] In those SM-FRET experiments, one measures the efficiency of energy transfer between a donor dye molecule and an acceptor dye molecule, which label specific sites of a macromolecule. The rate constant for FRET from donor to acceptor is assumed to be given by the Forster theory, namely [59,61-64]... 

In contrast, studies aiming at determining the exact nature of the non-radiative deactivation mechanism and the corresponding mean distance of interaction appear to be more in line with the Forster theory spirit and may be regarded as a fascinating attempt in this field. However, applications appear rather limited as far as the question of the hydration sphere determination is concerned. 

This relation was first obtained by Forster and is usually called the Forster theory. Rather than expressing W,-. y in the spectral-overlap relation, lT, y can be expressed in terms of the Franck-Condon factors which can be calculated from the potential surfaces as was done for the photo-induced ET or radiationless transitions. 

In the first contribution, Huxter Scholes present a review of the recent evolution of theory of EET in condensed phase from their earliest and simple formulation, based on the Forster theory to the most recent advances of theoretical and computational methods based on continuum solvation models. In the second contribution, Curutchet reviews the recent developments of PCM towards accurate theoretical investigations of EET in solution. In particular, the modelization of the various contributions of solvent effects in the chromophore-chromophore electronic coupling is presented using quantum-mechanical approaches. 

Applying the Forster theory (see 4.1) to the excitation transfer between the bases in a DNA chain a mean distance of the exciton transfer of about 55 base pairs was calculated. 

Figure 18.3 is an example of an experimental verification of the distance dependence predicted by the Forster theory. Some more points are notable ... 

The dipole-dipole coupling, an essential ingredient in the Forster theory, is a convenient approximation, valid when the donor and acceptor molecules are far enough (relative to their molecular sizes) from each other. The obvious correction to take, in situations where the donor-acceptor distance becomes comparable to molecular size, is the inclusion of higher-multipole interactions. Without going into details, one can readily infer that such corrections... 

The efficiency of energy transfer from an excited biphenyl group to the naphthyl group was determined from the fluorescence spectrum and its plotted against the possible range of biphenyl-naphthyl distance in Figure 6. In the same figure, theoretical efficiencies, calculated from the Forster theory are also shown ... 

A very low value of 8.1 A (28) for R° has been obtained for the fluorescence quenching of poly(methylvinyl ketone) by benzo-phenone in the solid phase. This poor agreement between experiment and theory is due to the inapplicability of the Forster theory at such short distances, since under 15 exchange interactions become dominant. Short-range behavior can best be described by, first, the Perrin (29) approach and, secondly, and to a much better extent, by that due to Hlrayana and Inokutl (30). 

The coupling, of course, is different in PIET and PIEET. In the latter case, as we have seen in Chapter 14, the coupling follows the Forster theory for singlet states, and may be written as an interaction between transition dipole moments. The probability for excitation transfer depends on the intensity of the absorption. The dependence... 

According to the Forster theory, the probability of energy transfer falls off inversely with the sixth power of the distance between the donor and the acceptor... 

Figure 6.19 Experimental tests of the Forster theory of long-range transfer of electronic energy, (a) Dependence of transfer rate constant on spectral overlap integral. Plot of logger against log 7. From Ref. [40]. (b) Distance-dependence of the energy transfer efficiencies in dansyl-(L-prolyl)n-a-naphthyl, n = I to 12. Plotof ln( - 1) against In R. From Ref. [42].

Fig. 7.6 In the Forster theory, the rate of energy transfer from molecule 1 to molecule 2 is proportional to the overlap integral, Je2 v)Fi v)v dv. Contributions to the integral come only from the spectral region where the weighted emission spectrum of molecule 1 overlaps the...

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