## 4 Interatomic Coulombic Decay  
The fundamental electrodynamic mechanism of EET plays a role in other photonic interactions too numerous to detail here. However, it is interesting to briefly focus on a recently emerging phenomenon, in which EET plays such a role: interatomic Coulombic decay (ICD) [61; 62; 63; 64; 65]. This process bears the hallmark \(R^{-6}\) distance dependence of EET in the near-zone region, but the energy it transfers between atoms is associated with electromagnetic radiation in the X-ray range. Moreover, compared to molecular EET, much more complex prior and posterior processes occur. Figure 4: Representative near-zone graphs for calculating the matrix element for electronic energy transfer between a donor D and an acceptor A, under the influence of a throughput off-resonance laser beam: (**a**) contribution for transfer satisfying two-quantum selection rules at D and A; (**b**) contribution when the input beam is absorbed and emitted at D while it simultaneously interacts with A. Colour coding as in Figure 2; the orange line segment again indicates a virtual intermediate state, which need not be the same as for the yellow segment preceding it. In essence, ICD is the photoionization of an atom leading to photoionization of another nearby atom through exchange of a high energy quantum, mediated in the form of a virtual photon. The steps involved are as follows: (i) on input of incident X-ray radiation, donor D is photoionized via ejection of an inner shell electron; (ii) the energy released as an outer electron relaxes into this vacancy is transferred from D to A, and then; (iii) acceptor A is also photoionized since the transferred energy expels an outer electron from the atom; in consequence (iv) the newly charged atoms repel each other and move apart. A practical example is the decay of Ne\({}_{2}\) via electron emission, in which the interactions of the created ions cause fragmentation of this dimer in a 'Coulomb explosion' [66]. Step (ii) is clearly analogous to molecular EET, but the associated mechanism is not the simple process of Equation (1) since the static field effects of the newly formed donor ion should also be taken into account. Therefore, the relevant mechanism is the one depicted by the Feynman diagram of Figure 3c. Here, following the initial step that expels the electron to form the ion, the donor is in an energetically unfavourable configuration that is analogous to an excited donor in molecular EET. There are then two quasi-simultaneous interactions between D and A, one involving the energy transfer and the other the static field effects on A due to D. In contrast to molecular EET, where the energy migration from D to A simply excites the acceptor from its ground state (i.e., an electron is promoted to a higher quantized state), the acceptor in the ICD mechanism is ionised because the transfer energy is in the high-frequency X-ray range; in consequence, an outer electron is excited to the continuum. The energy transfer step can be represented by the follow matrix element [67]:  
\[M_{fi}^{\rm polar}=\sum_{r,s}\frac{\mu_{i}^{ra}(\rm D)\mu_{j}^{0r}(\rm D)\mu_{ k}^{s0}(\rm A)\mu_{l}^{\beta s}(\rm A)V_{ik}(0,\mathbf{R})V_{jl}(0,\mathbf{R})}{E_{a}(\rm D )-E_{r}(\rm D)+E_{0}(\rm A)-E_{s}(\rm A)} \tag{13}\]  
where \(r\) and \(s\) are the virtual intermediate states that also appear in the explicit form of the \(S\) tensors referred to in previous equations. Moreover, \(\alpha\) and \(\beta\) are the excited states of D and A, respectively; atomic energies are denoted by \(E\) with 0 signifying the ground state. In this general expression, forsking a two-level approximation that is no longer appropriate, the summations over \(r\) and \(s\) naturally excludes any system state that matches the initial or final system state. Equation (13) is in fact the generic precursor to Equation (9), whose derivation explicitly discounts intermediate states that fall foul of this matching rule.