Coherent Energy Transfer: Excitonic Coupling

magnitude of the coupling at zero distance, and S is a constant describing the distance dependence of the considered orbitals.

Since the exchange interaction decreases much faster with distance than the dipole-dipole interaction, the latter is usually the dominant en-ergy transfer mechanism when the intermolecular distance is larger than about 5 Å and the exchange part can be neglected. There are nonetheless some cases where the exchange mechanism becomes important [64]:

1. When the donor and acceptor parts are linked by a bridge that has empty orbitals close in energy to the HOMO of the donor and that of the acceptor. These orbitals will interact with the donor and acceptor orbitals, which may lead to very low S values and thus to efficient en-ergy transfer over long distances [70, 71].

2. When the transition is not dipole-allowed.

3. When during the energy transfer process both molecules change spin, so that the overall spin quantum number is conserved, as in the reac-tion

3D*⁰1A^NNO1D⁰3A*. (1.87)

Since the Förster mechanism is in such case spin-forbidden, the ex-change mechanism will dominate.

1.4.3. Coherent Energy Transfer: Excitonic Coupling

The Förster and Dexter theories examined in the previous section apply only to pairs of molecules for which the intermolecular interactions and thus the coupling are weak. Hopping of the excitation energy from one molecule to the other is slow compared with the vibrational energy relaxa-tion and dephasing and has only little effect on the absorprelaxa-tion spectra of the molecules. If however two molecules are approached more and more so that the time required for the energy transfer gets shorter and shorter, it will at some point become impossible to state which molecule is ex-cited. In this case, the absorption spectrum of the oligomer differs from

that of the individual molecules. The term "7&($%9 refers to an excitation which is delocalised over more than one molecule, or which moves very rapidly from molecule to molecule. It can also be viewed as an excited electron-hole pair. The "7&($%9(& (9$"*?&$(%9) are the intermolecular interactions which cause the excitation to spread over several molecules, and they are actually of the same kind as the interactions causing reso-nance energy transfer, except that they are much stronger because the molecules are closer together.

In this section, the discussion will be limited to A*"95"- excitons, in which the excited electron and the hole migrate together from molecule to molecule. Such excitons are typically found in molecular aggregates and pigment-protein complexes. In J?99("*-M%$$ excitons, on the other hand, the electron and the hole can be on separate molecules. A com-mon feature of all types of excitons is the existence of new accessible energy levels due to the superposition of those of the individual chromo-phores.

The simplest case to consider is that of an excitonically coupled dimer made of a pair of interacting chromophores, A and B, each of them hav-ing only two energy levels. The molecules do not need to be in van der Waals contact or to be bonded. The wavefunctions of such a dimer can be described as a combination of the wavefunctions of the individual molecules. Let NA and NB be the Hamiltonians of the isolated mole-cules, A and B their corresponding eigenstates, and 2A and 2B their corresponding eigenenergies. Superscripts in the following discussion refer to the ground (0) and excited (1) states, respectively. When the two molecules interact to form a dimer, D, a coupling term, D, is included into the Hamiltonian, N:

A B

N"N 0N 0D. (1.88)

Since the two molecules do not exchange electrons, the ground state of the dimer can be described by

0 0 0

D " A B . (1.89)

Its energy, 2_D⁰, is

0 0 0 0 0 0 0

D A B A B A B _{> 4} 0

2 " N 0N 0D "2 02 0D . (1.90)

0 0 0 0

0 A B A B

D " D is the displacement of the ground-state energy due

to the interaction between A and B and is usually negligible.

The excited state is described as a linear combination of two terms in which either molecule is excited:

1 1 0 0 1

A B

D "& A B 0& A B . (1.91)

The coefficients &A and &B determine the relative contribution and phase of each term. The Schrödinger equation for the excited dimer is

#

NA0NB0D

$

D¹ "2¹D D¹ . (1.92)

Since the excited dimer is a two-level system, the Schrödinger equation can be solved exactly. It has two solutions, ²0 and ²!, corresponding to the two eigenstates D0 and D! . With the assumption that the coupling matrix ^D is purely non-diagonal, that is, ^DAA ^"DBB^"0, the eigenenergies are (for a complete derivation, see [13, 64])

2 2 equivalent. Both chromophores that compose the dimer are thus

identi-cal (same chromophores and same environment). In this case, 2¹A "2B¹

These equations illustrate the fact that the excitation is no longer confined to one monomer but equally delocalised over the whole dimer. This de-localisation can alternatively be considered as the ultrafast excitation en-ergy hopping from one monomer to the other, not letting an observer know where the excitation is.

Therefore, the situation is very different in the homodimer relative to that of the monomers: the excited energy levels split apart by an amount of 2D_AB, as shown by equations (1.97) and (1.98). The magnitude of this splitting, also called L?8+=%8 or "7&($%9 )/-($$(9., strongly depends on the magnitude of the coupling DAB; the larger the coupling, the larger the energy difference between the two energy levels.

This energy level splitting has an influence on the optical absorption spectra. Instead of the single transition present in the monomer spectrum, the dimer exhibits in principle two bands, approximately symmetrically displayed around the original monomer band. The transition dipoles and dipole strengths (square of the transition dipole) can readily be related to the spectroscopic properties of the monomers. In the case of a dimer

# $ # $

These relationships show that the transition dipole moments of the dimer bands are proportional to the vector sum and difference of the transition dipoles of the monomers. The dipole strengths of the two transitions

2

+^!D₀ and +^!D_!² can thus range between 0 and 2+^!M², depending on the angle @ between the transition dipoles of the monomers, that is, simply on the relative geometry of the two monomers. The sum of +^!D₀² and +^!D_!²

is however always 2+^!M²: there is no creation or destruction of dipole strength, only redistribution.

The dipole strength being directly related to the oscillator strength through equation (1.18), the dimer absorption and fluorescence spectra will depend on the dipole strength of the monomers and on their relative orientation. If the transition dipole moments are parallel, cos@ "1; then which is more intense than with the monomers.

Figure 1.10. Schematic representation of the effect of excitonic interaction on the energy levels of aggregated and non-aggregated molecules depending on their relative spatial arrangement in two particular cases where the transition dipole moments (represented by small arrows) are parallel. Full vertical arrows indicate allowed transitions, dashed ones forbidden transitions.

Figure 1.10 illustrates this situation for two particular geometries of the two monomeric species with parallel transition dipoles. If the two mole-cules form a sandwich-type complex, the configuration with the lowest energy will be that where +!AA*

and +!BB*

are antiparallel. This corresponds to the D! level and the transition to this state is forbidden. On the other hand, the transition to the higher energy level, D0 in this case, is rein-forced. The excitonic dimer band is thus blue-shifted relative to that of the monomer. This is the situation found in so-called H-dimers, or, if more than two molecules are involved, in H-aggregates [72, 73].

If the two monomer molecules are in a head-to-tail arrangement, a situation corresponding to so-called J-dimers or J-aggregates [73-75], the most favourable energetic configuration corresponds to +^!AA* and +^!BB*

being parallel. Therefore, the lower energy transition to the D0 level is allowed and the higher one to the D! level forbidden. The dimer band is shifted to the red. The effect of the molecular geometry on the absorp-tion spectrum of a forced perylenediimide dimer has been elegantly demonstrated by Wasielewski and co-workers [76].

Exciton formation will also influence the fluorescence characteristics of the dimer, not only in the case where the two molecules it is made of are identical. As the energy levels are shifted and the dipole strengths are redistributed compared with the monomeric species, the position and the shape of the fluorescence spectrum of the dimer is also altered. Further-more, since the oscillator strengths of the transitions change, this influ-ences the radiative rate constant. In turn, as the energy splitting causes a variation of the energy gaps between the different levels, the non-radiative rate constant is in principle also affected by the excitonic interaction.

Since both 5rad and 5nr may vary, the changes on the excited-state lifetime are however difficult to predict (equation (1.39)).

Considering again the excitonically-coupled homodimer, fluorescence will originate in the lower energy level since, even if the higher one is excited, rapid internal conversion to the lower energy level will occur. In the case of the head-to-tail dimer, the transition to the lower level is al-lowed. Consequently, the radiative rate constant of this transition is twice as large as that of the monomer and the excited-state lifetime of the dimer should be shorter than that of the monomer, depending on the amplitude of the non-radiative deactivation paths. The fluorescence quantum yield

should also strongly increase. On the other hand, sandwich-type dimers are essentially non-fluorescent since the transition from or to the lower energy level is forbidden. This leads to a strong (in principle infinite) de-crease of the fluorescence quantum yield and radiative rate constant.

The remarkable fluorescence enhancement of head-to-tail dimers and, more generally, of higher order aggregates with such geometry has lead to numerous applications of J-aggregates in photography. Aggregates play an important role in areas as diverse as silver halide and non-conventional colour photographic processes [77, 78], natural and artificial photosyn-thesis [60, 79], photodynamic therapy [80], laser technologies [81, 82], and are potential non-linear optical materials [83-86].

Although it is not discussed here, excitonic coupling can also be de-tected by circular dichroism spectroscopy [7].

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