Also contains definition of: dipole–dipole excitation transfer
https://doi.org/10.1351/goldbook.FT07381
Non-radiative @E02256@ between two molecular entities separated by distances considerably exceeding the sum of their van der Waals radii. It describes the transfer in terms of the interaction between the @T06460@ of the entities in the very weak dipole-dipole @C01025@ limit. It is a Coulombic interaction frequently called a dipole-dipole @C01025@. The transfer @O04322@ from donor to acceptor, \(k_{\text{T}}\), is given by \[k_{\text{T}} = k_{\text{D}}\left ( \frac{R_{0}}{r} \right )^{6}=\frac{1}{\tau _{D}^{0}}\left ( \frac{R_{0}}{r}\right )^{6}\] where \(k_{\text{D}}\) and \(\tau_{\text{D}}^{0}\) are the @E02056@ @O04322@ and the @L03515@ of the excited donor in the absence of transfer, respectively, \(r\) is the distance between the donor and the acceptor and \(R_{0}\) is the @F02488@ or Förster radius, i.e., the distance at which transfer and spontaneous decay of the excited donor are equally probable (\(k_{\text{T}} = k_{\text{D}}\)) (see Note 3). \(R_{0}\) is given by \[R_{0} = Const.\left ( \frac{\kappa^{2}\mathit{\Phi}_{D}^{0}J }{n^{4}} \right )^{1/6}\] where \(\kappa\) is the orientation factor, \(\mathit{\Phi} _{D}^{0}\) is the @F02453@ @Q04991@ of the donor in the absence of transfer, \(n\) is the average @R05240@ of the medium in the @W06659@ range where @S05818@ is significant, \(J\) is the @S05818@ integral reflecting the degree of overlap of the donor @E02060@ with the acceptor @A00043@ and given by \[J = \int _{\lambda }I_{\lambda}^{D}(\lambda)\epsilon _{A}\left ( \lambda \right )\lambda^{4}\text{d}\lambda\] where \(I_{\lambda}^{D}(\lambda)\) is the normalized @S05827@ of the donor so that \(\int_{\lambda}I_{\lambda}^{D}(\lambda)\text{d}\lambda = 1\). \(\varepsilon_{\text{A}}({\lambda})\) is the @M03972@ of the acceptor. See Note 3 for the value of \(Const.\).
Notes:
- The bandpass \(\Delta \lambda\) is a constant in spectrophotometers and spectrofluorometers using gratings. Thus, the scale is linear in @W06659@ and it is convenient to express and calculate the integrals in wavelengths instead of wavenumbers in order to avoid confusion.
- In practical terms, the integral \(\int_{\lambda}I_{\lambda}^{D}(\lambda)\text{d}\lambda\) is the area under the plot of the donor emission intensity versus the emission @W06659@.
- A practical expression for \(R_{0}\) is: \[\frac{R_{0}}{\pu{nm}} = \pu{2.108E-2} \left \{\kappa^{2}\mathit{\Phi}_{D}^{0}n^{-4} \int_{\lambda} I_{\lambda}^{D}(\lambda)\left[ \frac{\epsilon_{A}(\lambda)}{\pu{dm3 mol-1 cm-1}} \right] \left(\frac{\lambda}{\pu{nm}} \right)^{4}{\rm{d}}\lambda \right\}^{1/6}\] The orientation factor \(\kappa\) is given by \[\kappa = \cos \theta_{\text{DA}} - 3\cos \theta_{\text{D}}\cos \theta_{\text{A}} = \sin \theta_{\text{D}}\sin \theta_{\text{A}}\varphi - 2\cos \theta_{\text{D}}\cos \theta_{\text{A}}\] where \(\theta_{\text{DA}}\) is the @A00346@ between the donor and acceptor moments, and \(\theta_{\text{D}}\) and \(\theta_{\text{A}}\) are the angles between these, respectively, and the separation vector; \(\varphi\) is the @A00346@ between the projections of the transition moments on a plane perpendicular to the line through the centres. \(\kappa^{2}\) can in principle take values from 0 (perpendicular transition moments) to 4 (collinear transition moments). When the transition moments are parallel and perpendicular to the separation vector, \(\kappa^{2} = 1\). When they are in line (i.e., their moments are strictly along the separation vector), \(\kappa^{2} = 4\). For randomly oriented @T06460@, e.g., in fluid solutions, \(\kappa^{2} = 2/3\).
- The transfer @Q04988@ is defined as \[\mathit{\Phi} _{\text{T}} = \frac{k_{\text{T}}}{k_{\text{D}}+k_{\text{T}}}\] and can be related to the ratio\(\frac{r}{R_{0}}\) as follows: \[\mathit{\Phi} _{\text{T}} = \frac{1}{1 + \left ( \frac{r}{R_{0}} \right )^{6}}\] or written in the following form:\[\mathit{\Phi} _{\text{T}} = 1 - \frac{\tau_{\text{D}} }{\tau_{\text{D}}^{0}}\] where \(\tau_{\text{D}}\) is the donor excited-state @L03515@ in the presence of acceptor, and \(\tau_{\text{D}}^{0}\) in the absence of acceptor.
- FRET is sometimes inappropriately called @F02453@-@R05333@ transfer. This is not correct because there is no fluorescence involved in FRET.
- Foerster is an alternative and acceptable spelling for Förster.