## 2 Electronic Energy Transfer  
We begin with a straightforward representation of the electrodynamic coupling between the transition electric dipoles \(\mathbf{\mu}^{\dagger}\) of a donor D, whose transition involves electronic relaxation, and \(\mathbf{\mu}^{\dagger}\) of an acceptor A which undergoes an excitation. The two dipoles are separated by a displacement vector \(\mathbf{R}=\mathbf{R}_{\text{D}}-\mathbf{R}_{\text{A}}\) (i.e., the distance between the positions of D and A). Omitting a long and intricate derivation based on special functions [28], the matrix element for EET in the electric dipole (E1) approximation is as follows;  
\[M_{fi}=\mu_{i}^{\dagger}(\text{D})V_{ij}(k,\mathbf{R})\mu_{j}^{\dagger}(\text{A}) \tag{1}\]  
\[V_{ij}(k,\mathbf{R})=\frac{\text{e}^{ikR}}{4\pi\epsilon_{0}R^{3}}\Big{\{}\big{(}1 -\text{i}kR\big{)}\big{(}\delta_{ij}-3\hat{R}_{i}\hat{R}_{j}\big{)}-\big{(}kR \big{)}^{2}\big{(}\delta_{ij}-\hat{R}_{i}\hat{R}_{j}\big{)}\Big{\}} \tag{2}\]  
In the above expression, subscripts \(i\) and \(j\) signify Cartesian components: here and throughout this paper, any such repeated indices are subject to implied summation (the [PERSON] index convention). Moreover, the subscript \(fi\) on \(M\) is standard notation meaning the matrix element for a transition from the initial to the final system state. The parameter \(k\) is the reciprocal of the reduced wavelength \(\lambda\) notionally associated with the transfer energy, i.e., \(k=1/=2\pi/\lambda\) where the transfer energy is \(hc/\lambda\). The form of coupling represented by Equation (2) is known as the retarded resonance dipole-dipole interaction tensor; neglect of the imaginary part of this tensor leads to the monotonic distance-dependence being modified by an oscillatory function. Such behaviour arises in dimensionally constrained systems, such as cavities of finite length [20; 29; 30], though not in any open system. Equation (2) is routinely derived by quantum electrodynamical calculations adapting Feynman diagrammatic methods. Here, as shown in Figure 1a, colouring assists visualisation of the energy flow. With calculations of this kind, based on virtual photon coupling, the effects of quantum time-energy uncertainty require that two distinct time-orderings be taken into account. This reflects the fact that each dipole transition--the electronic decay of the donor and the excitation of the acceptor--can engage photon creation and annihilation operators only linearly. However, the sharply diminishing strength of coupling with distance, manifest in Equation (2), is such that EET generally occurs over distances where the term proportional to \(R^{-3}\) dominates. If consideration is limited to this regime, then it is possible (and expedient for the analysis that follows) to adopt a simplified, asymptotic representation formally consistent with treating Equation (2) as a first-order perturbation operator. Such a method was previously used to determine the dispersion interactions in the short-range region [31]. The corresponding single diagrammatic representation, exhibited in Figure 1b, essentially represents systems in which the energy transfer time (the interval between the release of electronic energy by the donor and its arrival at the acceptor) is itself immeasurably small, and whose duration is in practice experimentally indiscernible. A more in-depth analysis of the derivation of the transition dipole-transition dipole (E1-E1) coupling tensor, \(V_{ij}\), and the transfer rate of EET (an outline of which follows)--without providing all of the intricate specifics--is delivered by [PERSON] in his recent review [32]. The physical observable derived from the \(V_{ij}\) tensor, via the matrix element, is the transfer rate of EET, symbolised by \(\Gamma\).