## 3 Controlled Energy Transfer
### Static-Field Induced EET
Simultaneous interactions with both are also possible, but their description entails a yet higher order of perturbation theory that contributes negligibly to the rate of energy transfer. The Fermi rate expression is again used to determine an observable, but in principle it now includes an extra term that represents a second-order molecular interaction, signifying one molecule (either the donor or the acceptor) coupling with both its neighbour and the input static electric field. This is written as \(\Gamma^{\prime}\sim\left|M_{fi}+M_{fi}^{\rm stat}\right|^{2}\), in which \(M_{fi}^{\rm stat}\) accordingly representsan overall third-order correction term due to the static field. The structure of the correction term is readily ascertained from the simplified (near-zone) time-ordered diagrams shown in Figure 2. Hence, we find:  
\[M_{fi}^{\text{stat}}=-\varepsilon_{0}D_{i}\Big{\{}S_{ij}^{\downarrow}(\text{D}) \mu_{k}^{\uparrow}(\text{A})+\mu_{k}^{\downarrow}(\text{D})S_{ij}^{\uparrow}( \text{A})\Big{\}}V_{jk}(0,\mathbf{R}) \tag{4}\]  
where \(\varepsilon_{0}\) is the vacuum permittivity, \(\mathbf{D}\) is the local electric displacement field and \(S_{ij}\) is a second rank (E1\({}^{2}\)) response tensor. Specifically including the state connections and frequency arguments of the latter, the donor tensor takes the form \(S_{ij}^{\downarrow}(\text{D})\equiv S_{ij}^{\downarrow}(-ck;0)\) and, for the acceptor, we have \(S_{ij}^{\uparrow}(\text{A})\equiv S_{ij}^{\uparrow}(0;ck)\). Both tensors are fully defined in the original paper [40], which majors on the selection rules and an orientational averaging. In Equation (4), the first term relates summed contributions from both (a) and (b) in Figure 2; the second term is derived from both (c) and (d). Moreover, since the system resides in the near-zone region, the static limit (for which \(k=0\)) of Equation (2) can be used--while the explicit frequency dependence of the \(S\) tensors, representing their dispersion property, needs to be retained. In the relatively unusual case of entirely nonpolar donor and acceptor molecules, the rate of transfer delivered by the matrix element (4) without any input-field-independent contributions will clearly lead, through the Fermi Rule, to a transfer rate with a quadratic dependence on the applied field strength \(D\), so that:  
\[\Gamma_{\text{STAT}}\,\sim\,\frac{D^{2}}{3(4\pi R^{3})^{2}}\Big{|}\mathbf{\mu}^{ \downarrow}(\text{D})\Big{|}^{2}\Big{|}\text{Tr}S^{\uparrow}(\text{A})\Big{|} ^{2}+\Big{|}\text{Tr}S^{\downarrow}(\text{D})\Big{|}^{2}\Big{|}\mathbf{\mu}^{ \uparrow}(\text{A})\Big{|}^{2} \tag{5}\]  
More generally, in the case of orientable molecules, the general outcome from the original study [40] results from a derivation requiring the evaluation of Boltzmann-weighted tensor averages [43], and it is duly cast in a form that entails spherical Bessel functions. However, simplification can now be effected for the short-range effects that dominate energy transfer, where E1 processes are pre-eminent. With a Taylor series expansion of Equation (31) from ref. [40], we thus find for partially oriented systems:  
\[\Gamma^{\prime}_{\text{STAT}}\,\sim\,\frac{\big{|}\mathbf{\mu}^{\downarrow}(\text {D})\big{|}^{2}\big{|}\mathbf{\mu}^{\uparrow}(\text{A})\big{|}^{2}D^{4}}{12\big{(} \pi c_{0}^{3}R^{3}\big{)}^{2}(k_{\text{B}}\text{T})^{4}}\Delta\mu_{z}^{ \downarrow 2}(\text{D})\mu^{2}(\text{D})\Delta\mu_{z}^{\uparrow 2}(\text{A})\mu^{2}( \text{A}) \tag{6}\]  
Here, the absence of superscript arrows on \(\mathbf{\mu}\) designates a _static_ dipole moment, whose direction within each molecule represents an internal axis chosen to define its intrinsic \(z\)-direction--i.e., the \(z_{\text{D}}\)-axis within the donor and the \(z_{\text{A}}\)-axis within the acceptor,  
Figure 2: Near-zone graphs for calculating the matrix element for electronic energy transfer that linearly engages with a static electric field: in (**a**,**b**) the static field influences the donor decay transition; in (**c**,**d**) it affects the acceptor excitation.