# Controlling Electronic Energy Transfer: A Systematic Framework of Theory

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Electronic Energy Transfer

**μ**

^{↓}of a donor D, whose transition involves electronic relaxation, and

**μ**

^{↑}of an acceptor A which undergoes an excitation. The two dipoles are separated by a displacement vector

**R**=

**R**

_{D}−

**R**

_{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;

^{−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.

_{ij}, and the transfer rate of EET (an outline of which follows)—without providing all of the intricate specifics—is delivered by Salam 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 Γ. This rate is determined from the Fermi rule [33], which is given by $\mathrm{\Gamma}~{\left|{M}_{fi}\right|}^{2}$ when omitting a proportionality constant that corresponds to the spectral overlap of the donor emission and acceptor absorption [7,34]; the same constant of proportionality applies to all of the EET rate equations in our subsequent analysis. Assuming a system of two freely tumbling molecules, meaning that a rotational average is required [35], the following is found;

## 3. Controlled Energy Transfer

#### 3.1. Static-Field Induced EET

_{s}, C

_{n}or C

_{nv}. The most familiar manifestation of alignment in a static field, featuring in the Debye equation, is the dipolar contribution to electronic polarisation [39].

^{2}) selection rules. In fact, for polar molecules of every symmetry type, all of their conventional (i.e., E1-allowed) electronic transitions are also E1

^{2}-allowed, so this condition is automatically satisfied. Nonetheless, the contribution of the latter, static field-engaging channel, will generally be small since it arises from a higher order of perturbation theory. The second mechanism for a field-induced change in the energy transfer rates, i.e., that which occurs through an influence on molecular orientation, arises from the conventional average value of the orientation factor—the 2/3 in Equation (3)—being modified as a result of partial alignment, commonly signified by a Boltzmann-weighted distribution function. As Van der Meer showed in a comprehensive analysis of the orientation factor known as ‘kappa squared’ [41], optimal alignment can enhance the rate of transfer by a factor of up to six (the isotropic value 2/3 increasing to an upper limit of 4). Methods for estimating the value of the orientation factor are discussed in a useful review by Loura [42].

_{0}is the vacuum permittivity,

**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 ${S}_{ij}^{\downarrow}\left(\mathrm{D}\right)\equiv {S}_{ij}^{\downarrow}\left(-ck;0\right)$ and, for the acceptor, we have ${S}_{ij}^{\uparrow}\left(\mathrm{A}\right)\equiv {S}_{ij}^{\uparrow}\left(0;ck\right)$. 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.

**μ**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

_{D}-axis within the donor and the z

_{A}-axis within the acceptor, given that each is polar; T is the absolute temperature and k

_{B}is the Boltzmann constant. The mechanism of mutual orientation that generates the above result applies only when both donor and acceptor are polar; the inverse fourth power dependence on temperature indicates a sharp diminution of the mutual orientation effect as temperature rises, since randomising thermal motions become increasingly dominant. In this result, there is also a notable dependence on the relative internal orientation of each transition dipole with respect to the corresponding static dipole, as signified by the two factors defined as $\mathsf{\Delta}{\mu}_{z}^{2}\equiv 3{\mu}_{z}^{2}-{\left|\mathit{\mu}\right|}^{2}=2{\mu}_{z}^{2}-{\mu}_{x}^{2}-{\mu}_{y}^{2}$ in Equation (6). Evidently the effect is optimised if the transition and static dipoles have a common orientation, signifying that the relevant excited state has the symmetry of a totally symmetric representation in the relevant point group.

#### 3.2. Static-Dipole Induced EET

**μ**is the static dipole moment of M, up and down arrow superscripts denote excitation and decay transitions, and

**R**

_{AM}=

**R**

_{A}−

**R**

_{M}is the displacement between A and M (

**R**

_{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].

#### 3.3. Optically Controlled EET

_{ijl}is a transition hyperpolarisability, i.e., ${T}_{ijl}^{\downarrow}\left(\mathrm{D}\right)\equiv {{T}^{\prime}}_{ijl}^{\downarrow}\left(-ck,-c{k}^{\prime};c{k}^{\prime}\right)$ and ${T}_{ijl}^{\uparrow}\left(\mathrm{A}\right)\equiv {T}_{ijl}^{\prime}{}^{\uparrow}\left(-c{k}^{\prime};ck,c{k}^{\prime}\right)$, while I and

**e**are the intensity and polarisation of the throughput beam, respectively. For entirely nonpolar donor and acceptor molecules, the rate expression for the optically controlled EET (otherwise known as laser-assisted RET or LARET) is then written as;

^{3}) process [55]. Hence, mechanism (

**b**) will always be effective in, to some degree, modifying EET efficiency when D and A are polar, whereas mechanism (

**a**) provides a basis to switch on transfer processes between states that would otherwise not be allowed.

^{2}) allowed. This signifies that the initial excitation of the donor cannot occur via direct one-photon absorption but, for example, may involve initial excitation to a higher energy electronic state followed by relaxation. In such a case, only terms involving the transition polarisability arise, since they represent two-quantum allowed transitions at D and A, i.e., the absorption (or emission) of the input beam and the energy transfer interaction. Here, since this system is electric dipole (E1) forbidden, EET could not, in the absence of a beam, occur—except via extremely weak higher multipole interactions (associated with exceptionally sharp R

^{-n}dependence of the rate, where $n\gg 6$). Therefore, energy transfer can only reasonably occur when the input beam is applied; this is the origin of the switching action. For this scheme, since the input-field-independent contribution does not arise, Equation (11) is null and the leading term of the rate, which is dependent on I

^{2}, then becomes;

## 4. Interatomic Coulombic Decay

^{−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.

_{2}via electron emission, in which the interactions of the created ions cause fragmentation of this dimer in a ‘Coulomb explosion’ [66].

**S**tensors referred to in previous equations. Moreover, α and β 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, forsaking 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.

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Graphs for calculating the basic matrix element for electronic energy transfer, in which time is progressing upwards: (

**a**) two separate time-orderings accounting for causality and (

**b**) representation of the conflated result obtained from a near-zone approximation, in which $R\ll \u019b$. In the vertical world-lines for D and A, red segments indicate an electronic excited state and black is the ground state. Between the two world-lines is a wavy green line that denotes virtual photon propagation; the horizontal dashed line indicates short-range dipole–dipole coupling.

**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. Red and black lines denote electronically excited and ground states, respectively; a yellow line denotes a virtual intermediate state. The blue dashed line indicates the applied static field, while the green dashed line denotes short-range dipole–dipole coupling.

**Figure 3.**Representative near-zone graphs for calculating the matrix element for electronic energy transfer between a donor D and an acceptor A, modified by the electronic influence of a local electric dipole: (

**a**) one of two time-orderings for the effect of a neighbouring third-body static dipole M on the acceptor A; (

**b**) the other contribution, where M engages with the donor D; (

**c**) the effect of static dipoles held by D and A. Colour coding as in Figure 2.

**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.

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Andrews, D.L.; Bradshaw, D.S.
Controlling Electronic Energy Transfer: A Systematic Framework of Theory. *Appl. Sci.* **2022**, *12*, 8597.
https://doi.org/10.3390/app12178597

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Andrews DL, Bradshaw DS.
Controlling Electronic Energy Transfer: A Systematic Framework of Theory. *Applied Sciences*. 2022; 12(17):8597.
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Andrews, David L., and David S. Bradshaw.
2022. "Controlling Electronic Energy Transfer: A Systematic Framework of Theory" *Applied Sciences* 12, no. 17: 8597.
https://doi.org/10.3390/app12178597