## 3 Controlled Energy Transfer
### Static-Dipole Induced EET  
Typically, EET is described in terms of a coupling between electronically isolated molecules. In practice, however, most systems comprise numerous nearby atoms, molecules or chromophores whose presence may alter the surrounding electronic environment and, thus, affect any pairwise energy transfer events. Such non-participating entities (i.e., those not directly involved in the pairwise EET) are known as a 'third body' and they are often labelled by M. In recent times, the possibility of M exerting an influence on the rate of donor-acceptor transfer has received much more attention [29; 45; 46; 47; 48; 49; 50; 51]. Such work has verified that these nearby entities, especially any that are strongly polar, may substantially affect the energy transfer rate without themselves having an overall change in state (namely, their initial and final state are identical). In consequence, the physical introduction of ancillary molecules, suitably positioned to act in the capacity of a passive third body, offers a route to control the energy transfer efficiency. It is imperative that third body influences are considered in systems that contain closely spaced chromophores, such as those within light-harvesting materials. Indeed, for all condensed phase applications, it should be borne in mind that the single entity termed the third body, in the following analysis, will almost always be one of many close neighbours--representing many species whose effects on a specific energy transfer process will give additive contributions to the matrix element. By the inclusion of any individual third body, the expression for the correction term of the total rate becomes more complicated. For the mechanisms exemplified by Figure 3a, where the static dipole of M interacts with A, and the corresponding situation coupling M with D, represented by Figure 3b, the net result is written as;  
\[ \begin{array}{ll}M_{fi}^{\rm 3-body}&=\mu_{i}^{\downarrow}({\rm D})V_{ij}(0, \mathbf{R})S_{jk}^{\uparrow}({\rm A})V_{kl}(0,\mathbf{R}_{\rm AM})\mu_{l}({ \rm M})\\ &+\mu_{i}({\rm M})V_{ij}(0,\mathbf{R}_{\rm MD})S_{jk}^{\downarrow}({\rm D})V_{ kl}(0,\mathbf{R})\mu_{l}^{\uparrow}({\rm A})\end{array}  \tag{7}\]  
where \(\mu\) is the static dipole moment of M, up and down arrow superscripts denote excitation and decay transitions, and \(\mathbf{R}_{\rm AM}=\mathbf{R}_{\rm A}-\mathbf{R}_{\rm M}\) is the displacement between A and M (\(\mathbf{R}_{\rm MD}\) is for M and D). Note that when constructing each of the terms shown in Equation (7), an additional contribution is also taken into account, in which the temporal order of the interactions is inverted; these must also be considered to achieve the correct results, which are fully provided in ref. [48]. Whenever energy transfer occurs in the vicinity of a third body M, one other term always arises in the matrix element. This term, engaging the electronic polarisation rather than the static dipole of M, can be considered a background, dynamic interaction that is not directly controllable. For any significantly polar third body, this contribution arising from a higher order of perturbation theory can be anticipated to have a relatively small effect on the efficiency of the EET, but for non-polar species it represents the only local-field correction. Although it is seldom significant, for completeness in this account it is fully expressed as follows;  
\[M_{fi}^{3-\text{body}}\Big{|}_{\text{background}}=\mu_{i}^{\downarrow}(\text{D}) V_{ij}(0,\mathbf{R}_{\text{DM}})S_{jk}(\text{M})V_{kl}(0,\mathbf{R}_{\text{AM}})\mu_{l}^{ \uparrow}(\text{A}) \tag{8}\]  
in which \(S_{jk}(\text{M})\) is not a transition tensor but corresponds to the dynamic polarisability of M at the frequency of the conveyed energy. Irrespective of any neighbouring third body, another intriguing effect arises when D and A are both polar, in which case a coupling of the static dipoles can take place alongside the energy transfer, as illustrated in Figure 3c.