## 5 FRET in Ensembles of Nanocrystals  
All the processes considered above may occur in isolated nanocrystals. Meanwhile, usually in experiments, as was already pointed out in the Introduction, one deals with ensembles of NCs, where a non-radiative energy exchange between them takes place and strongly influences the ensemble photoluminescence [53, 54, 55, 56, 179, 180, 181]. Such an energy transfer occurs by means of the electron tunnelling if the NCs are connected [51] or through the Forster-type migration of excitons [49] if the NCs are separated in space. Below we consider the latter mechanism in some detail. Figure 5: Calculated rates of the exciton generation initiated by highly excited (**a**) electron and (**b**) hole (as shown in Figure 4) in Si\({}_{317}\)X\({}_{172}\) crystallite for X = H (small blue dots), X = Cl (medium green dots) and X = Br (big red dots). The Forster resonant exciton transfer (FRET) takes place mainly through the dipole-dipole interaction (It has been shown [182] that, in contrast to the case of FRET between donor and acceptor molecules, where the dipole approximation breaks down at lengthscales comparable to molecular dimensions, it works fairly well when donor and/or acceptor is a spherical QD, even at contact donor-acceptor separations.) of two QDs (a donor and an acceptor). The Quantum Electrodynamics theory of FRET developed in [183] reproduces the results obtained in a simpler way by [PERSON] [49] and [PERSON] [50] who considered the electrostatic interaction of two dipoles in two adjacent crystallites [184],  
\[V(\mathbf{r}_{1},\mathbf{r}_{2})=\frac{\kappa e^{2}}{e_{1}b^{3}}\left[\mathbf{ r}_{1}\cdot\mathbf{r}_{2}-3\frac{(\mathbf{r}_{1}\cdot\mathbf{b})(\mathbf{r}_{2} \cdot\mathbf{b})}{b^{2}}\right], \tag{10}\]  
where \(\mathbf{b}\) is the inter-crystalline centre-to-centre vector. In one crystallite, the electron-hole pair annihilates and transfers its energy into a neighbouring crystallite where a new electron-hole pair is excited. Thus, virtual transfer of excitons between two crystallites can be realized without charge transfer. In order to calculate the rate of the Forster exciton transition from a QD with a radius \(R_{1}\) into a neighbouring one with a radius \(R_{2}\), we use, as before, the Fermi's Golden Rule:  
\[k_{F}=\frac{2\Gamma}{\hbar}\frac{|\langle\Psi_{i}|\hat{V}|\Psi_{f}\rangle|^{2 }}{\Gamma^{2}+(E_{g}(R_{1})-E_{g}(R_{2}))^{2}}\,. \tag{11}\]  
Here, \(\hat{V}\) is the dipole interaction operator identical to (10), \(\Psi_{i}=\psi_{c}(\mathbf{r}_{1})\psi_{v}(\mathbf{r}_{2})\) is the wave function of the initial two-particle state with the energy coinciding with the energy gap of the first crystallite, \(E_{g}(R_{1})\), and \(\Gamma\) is a phenomenological damping parameter. Initially, there is one electron in the conduction band of the first crystallite, with the wave function \(\psi_{c}(\mathbf{r}_{1})\), while a hole exists in the valence band. In the second QD, the electron occupies a valence band state described by the wave function \(\psi_{v}(\mathbf{r}_{2})\). In the final state, the system has the wave function \(\Psi_{f}=\psi_{v}(\mathbf{r}_{1})\psi_{c}(\mathbf{r}_{2})\), corresponding to the electron-hole pair transferred to the second QD. The result is essentially the same as if one considered the interaction between two transient point dipoles, \(\mathbf{d}_{\mathbf{D}}\) and \(\mathbf{d}_{\mathbf{A}}\) (located at points \(\mathbf{r_{1}}\) and \(\mathbf{r_{2}}\)) assuming that they are stationary. The square of the matrix element in (11) (the donor-acceptor coupling parameter) can be written as [185]  
\[j^{2}=v\frac{\kappa^{2}d_{D}^{2}d_{A}^{2}}{\epsilon_{1}^{2}b^{6}}\,, \tag{12}\]  
where \(\
u\) is a factor of the order of unity that takes into account the relative orientation of the dipoles [186], \(\
u=\frac{2}{3}\) if these orientations are completely random.