Resonance energy transfer between polar charge-transfer dyes: A focus on the limits of the dipolar approximation

Abstract

Resonance energy transfer (RET) is investigated in a pair of polar charge-transfer (CT) chromophores, adopting essential-state models and time dependent density functional theory (TDDFT) calculations. Essential-state models describe in an efficient way linear and nonlinear optical properties of CT dyes, and prove very useful to rationalize the effects of electrostatic interchromophoric interactions on optical properties of multichromophoric systems. In this paper we adopt the same strategy developed for multichromophoric systems to investigate interchromophoric interactions responsible for RET. In the late forties, Th. Förster proposed a powerful method, based on the dipolar approximation, that directly relates the rate of the RET process to experimental accessible quantities. Here we discuss the applicability of the dipolar approximation for RET between CT dyes. The results obtained with essential-state models are confirmed by TDDFT calculations.

Introduction

Resonance energy transfer (RET) is an important and widespread process in nature, involving the transfer of a virtual photon from an energy donor molecule to an energy acceptor molecule. RET plays a key role in light harvesting systems, where the exciton energy is transferred towards the reaction center [1], [2], [3] and lies at the heart of bioluminescence processes occurring in some animals [4]. RET shows enormous potentialities in the fields of energy storage, photovoltaics and light-emitting devices [5], [6], [7], [8]. According to the distance between the energy donor and acceptor molecules, different mechanisms are responsible for energy transfer. Short-range interactions involve intermolecular orbital overlap, and energy transfer can occur via electron exchange (Dexter mechanism) or charge-resonance mechanism [9], [10], [11]. At very large distances, i.e. when the distance between molecules is larger than optical wavelengths, interactions between molecules are negligible and radiative energy transfer occurs, involving the absorption by the acceptor of a photon emitted by the donor. RET is relevant to intermediate distances, is a non-radiative process and involves the exchange of a virtual photon.

Following a perturbative approach, the rate of energy transfer from the excited donor towards the energy acceptor can be expressed by the Fermi Golden Rule:

$$k_{\mathit{ET}}=\frac{1}{{\hslash }^{2}c}|V{|}^2\delta ({\tilde{\nu }}_{\text{don}}-{\tilde{\nu }}_{\text{acc}})$$

where c is the speed of light, the Dirac $\delta ({\tilde{\nu }}_{\text{don}}-{\tilde{\nu }}_{\text{acc}})$ accounts for energy conservation, and V represents the intermolecular electrostatic interaction. In the late forties, Förster expressed the interaction between the energy-transfer partners in terms of dipole–dipole interactions [12], [13], [14], [15], as follows:

$$V_{\text{dip}}=\frac{1}{4\pi {ϵ}_0{n}^2}\left(\frac{{\overrightarrow{\mu }}_{\text{don}}^t·{\overrightarrow{\mu }}_{\text{acc}}^t}{r^3}-3\frac{\left({\overrightarrow{\mu }}_{\text{don}}^t·\overrightarrow{r}\right)\left({\overrightarrow{\mu }}_{\text{acc}}^t·\overrightarrow{r}\right)}{r^5}\right)$$

where ${\mu }_{\text{don}/\text{acc}}^t=〈{\psi }_{\text{don}/\text{acc}}^{\ast }|\hat{\mu }|{\psi }_{\text{don}/\text{acc}}〉$ is the transition dipole moment relevant to the donor/acceptor molecule, r is the distance between the two point dipoles, ϵ₀ is the vacuum permittivity, and n² is the squared refractive index of the solvent (SI units). One of the most important advantages of the Förster theory is to directly relate the transfer rate to experimentally accessible quantities.

In spite of its great success in describing energy transfer processes, the common implementation of the Förster model is based on the dipolar approximation, that limits its applicability. Quantitatively, the dipolar approximation is inadequate for the description of electrostatic interactions when the distance between molecules is similar to their sizes. Furthermore, the dipolar approximation cannot be applied to describe energy transfer that involves optically dark states, i.e. states with strictly zero transition dipole moment. The role of dark states in RET processes has been extensively studied with reference to light-harvesting complexes, where the first excited state of carotenoids is an optically dark state which possibly takes part in RET [3], [6], [16], [17]. The involvement of dark states in RET has also been proposed in aggregates of light-harvesting complexes [10], [11], [5], [6], [18], [19], [20], [21], conjugated polymers or oligomers, and in artificial light-harvesting dyads [22], [23], [5]. Recently, we have discussed the role of dark states in RET, proving the high sensitivity of this process with respect to the relative orientation of the involved partners [24]. Several computational approaches have been devised for the calculation of RET interaction energies, to overcome the limitations of the traditional Förster model. Along these lines the transition density cube [25], [20], [21] and the Renger approach [26] essentially relax the dipolar approximation. Other approaches [22], [27], [28], [29] also relax the perturbative approximation and apply quite naturally to account for environmental effects [30], [31], [32].

In this paper, we mainly focus on the limitations of the dipolar approximation for the description of intermolecular electrostatic interactions relevant to RET between polar charge-transfer (CT) chromophores. Polar CT chromophores are extended π-conjugated systems constituted by electron-donor (D) and acceptor (A) moieties connected by a π-conjugated bridge resulting in polar (D–π–A) dyes. The presence of an intense low-energy charge-transfer transition in CT chromophores makes these systems attracting for several applications in the fields of nonlinear optics [33], [34], [35], [36], [37], [38], [39], [40], [41], [42], [43], [44], [45], [46], [47], [48] and molecular electronics[49], [50], [51], [52], [53], [54]. Moreover CT dyes are very interesting model systems to investigate energy- and electron-transfer processes [55], [56], [57], [58], [59], [60].

Essential-state models are an efficient theoretical tool for the description of optical properties of CT-chromophores [61], [62], [63], [64]. Essential-state models account just for a few electronic states, relevant for the description of the low-energy optical behavior of CT dyes. The same models proved successful in the description of optical spectra of multichromophoric species where electrostatic interchromophore interactions play a major role [65], [66], [67], [68]. Here we apply the same model to investigate the role of electrostatic intermolecular interactions in RET [24]. We focus the attention on polar CT dyes, to investigate the applicability and limits of the dipolar approximation. For this purpose, we consider the two polar dyes P1 and P2 shown in Fig. 1 as energy donor and energy acceptor, respectively. In Section 2, experimental linear optical spectra are reported and discussed, while in Section 3 the two-state model for dipolar dyes is described in detail. Calculations of intermolecular interactions responsible for energy transfer are developed within essential-state approaches, and results are reported in Section 4. In Section 5 essential-state calculations are compared with the results obtained by time-dependent density functional theory (TDDFT) calculations.