Delocalized excitations, denoted excitons, play an important role in many systems in chemical physics. The characterization of their extent of delocalization is a crucial element in understanding these quasiparticles. In this paper, I will revisit the most common delocalization measures applied to Frenkel-type excitons. Based on this analysis, I propose to use a so-far ignored measure. The key advantage of this measure, which I will denote as the Manhattan exciton size, is that it directly connects with the oscillator strength of the excitons. It provides a strict upper bound on the oscillator strength of any given exciton for linear aggregates. Finally, I demonstrate that this exciton delocalization measure is more sensible for analyzing super-radiant states compared to, for example, the most commonly applied measure, i.e., the (inverse) participation ratio. However, these two measures together provide insight into the degree of exciton confinement.

Topics
Quantum confinement, Transition moment, Excitons, Oscillator strengths, Fluorescence spectroscopy, Delocalization, Dye, Chemical physics, Dark states, Quantum optics

Collective excitations known as excitons, play a role in many important phenomena such as photosynthesis,1,2 organic photovoltaics,3,4 organic light-emitting diodes,5,6 artificial light-harvesting systems,7–9 and vibrational dynamics.10,11 For example, delocalized excitons are known to facilitate efficient energy transfer in natural12–14 and artificial15–17 light-harvesting systems. In quasi-one-dimensional systems, delocalization can facilitate efficient transport by circumventing traps and defects.8,18 The use of excitons in functional materials19 is due to their role in phenomena such as interference20,21 and superradiance.22–26 To understand this role, it is crucial to be able to quantify and characterize the degree of delocalization in these collective excitations. The goal of this paper is to revisit the measures that can be used to characterize the degree of delocalization, identify a measure that connects well with a physical phenomenon, super-radiance, and provide an intuitive picture of how this measure behaves.

Excitons were first discovered when Scheibe27 and Jelley28 independently of each other discovered that when dye molecules aggregate, their optical properties change drastically. This aggregation leads to the formation of very narrow absorption peaks and efficient fluorescence known as super-radiance.24,29 The origin of these effects is that the electronic excitations of the dye molecules interact with each other because of the close proximity of the molecules. The interaction allows the electronic excitations to delocalize over numerous dye molecules, effectively limiting the effect of fluctuations of the excitation energies. For some of these collective states, the transition-dipole moments of the molecules add up, leading to efficient interaction with external light, while the transition dipoles effectively cancel for other collective states, denoted dark states. For linear aggregate arrangements, the systems can be characterized either as J-aggregates, where the lowest energy states are super-radiant, or H-aggregates, where the highest energy states are super-radiant.30 Fluorescence is highly suppressed in the latter type of aggregates as excitons relax to the bottom of the band of states, which are dark and only emit very weakly. For different aggregate arrangements, this simple picture typically does not hold.4,31 Further complications arise in the presence of charge transfer states and strong coupling with vibrations,4,32–35 which will go beyond the present discussion.

Multiple measures to characterize exciton delocalization have been developed,36–41 where the so-called (inverse) participation ratio (IPR) is probably the most commonly used.38 Experiments such as pump–probe,42 fluorescence spectroscopy,24 and two-dimensional spectroscopy25,43–45 can be used to probe exciton delocalization. Considering the importance of the phenomenon, it is already a very well-established field of research.4 

The remainder of this paper is organized as follows: first, in Sec. II, the basic exciton theory will be outlined and a general class of exciton delocalization measures described. The Manhattan exciton size (MES) will be defined and its connection to the phenomena of super-radiance will be explained. In Sec. III, examples will be given, where the IPR and MES are compared. This will focus on linear J-aggregates and ring-shaped aggregates. Furthermore, these measures will be compared for idealized super-radiant-type wave functions to provide further insight. Finally, the conclusions are drawn in Sec. IV.

Frenkel excitons are frequently used to describe the transport and optical properties of delocalized vibrations and electronically excited states. The Frenkel excitons Hamiltonian is defined as46,47
(1)
where Bm† and Bm are creation and annihilation operators, respectively. ϵm is the site energy for site m, and Jmn is the resonance coupling or hopping amplitude for excitations between site m and site n. The eigenstates of the Hamiltonian are denoted as exciton states. These states are delocalized over multiple sites unless the couplings vanish. The wave function coefficients of eigenstate k are denoted as ckm.
The delocalization of an exciton state can be quantified using a delocalization measure. For eigenstate k, this can be defined using a general definition as36 
(2)
For a localized state, that is, one ckm = 1 for each k,
(3)
For a perfectly delocalized state, all ckm=eiϕmN⁠, where ϕm represents relative phases,
(4)
One can use this to determine the effective number of sites contributing to the exciton delocalization by inverting the following formula:
(5)
It is readily obvious that for l = 1, this expression is ill-defined. In that case, Ikl is always one, as this is the normal normalization criteria for the exciton wave functions. For l = 2, we have the commonly used participation ratio36,38 (it should be noted that in the literature, the terms inverse participation ratio and the participation ratio are occasionally used both for this quantity and its inverse36),
(6)
Here, I will use the name inverse participation ratio (IPR) for exciton k for this delocalization measure with
(7)
Here, we further explore the l=12 variant of this measure, which has a tractable physical connection to the super-radiance property of excitons, as I will discuss in the following. For this measure, one has
(8)
As this delocalization measure is the square of the Manhattan norm of the wave function coefficients, I will denote it with the Manhattan exciton size (MES), which is then given by
(9)
for exciton k.
Now, I want to consider the absorption cross section for a super-radiant state of a linear J-aggregate. The transition dipole of eigenstate k is given by
(10)
In the special case where the transition dipoles are identical in length and direction, the largest possible value of the transition dipole of an exciton is
(11)
where |μ| is the length of the molecular transition dipole moment. For a perfect super-radiant state, all wave function coefficients have the same phase (or sign when real). The absorption cross section Ik⁠, also denoted by the oscillator strength, of the super-radiant state is given by the square of the transition dipole moment,
(12)
Compared to the Manhattan exciton size defined in Eq. (9), we have
(13)
which establishes the physically tractable relevance of the Manhattan exciton size. In other words, the MES establishes the largest possible absorption cross section of an exciton state. In practice, of course, for most excitons, the wave function coefficients have different phases (signs) and the actual cross section will be smaller. For optical spectroscopy, however, the delocalization of the optically active states is often of most interest.
The absolute density matrix (ADM) is defined as48,
(14)
where the brackets ⟨⋯⟩ denote an ensemble average (as an average over different disorder realizations originating from the coupling with a bath). This quantity was defined to characterize the delocalization of exciton wave functions and to devise a way of separating systems into incoherently coupled segments.48 In the following, I will show that this quantity is strongly related to the MES. From Eq. (9), we have
(15)
If we average over all eigenstates and average overall disorder, we get
(16)
In other words, the MES averaged over all eigenstates and disorder realization is the same at the ADM summed over all rows and averaged over all columns (or vice versa). Thus, the ADM provides a nice way to get further understanding of the MES measures on top of being a well-suited way to visualize the wave function ensemble. For the IPR, a similar quantity was recently defined49 as the participation ratio matrix: (PRnm = ∑k|ckn|2|ckm|2).

In the following, I will present a few examples of the application of the measure of exciton size. First, linear aggregates will be considered and later ring-shaped aggregates. Furthermore, idealized wave function forms will be examined. Finally, the effect of confinement will be examined.

First, I will consider the typical linear J-aggregate consisting of a collection of identical chromophores with transition dipoles parallel to the aggregate axis. This arrangement results in the formation of exciton states, where the oscillator strength is carried by the lowest-energy exciton state, resulting in a red shift of the spectrum compared to that of the individual chromophores.27,28,50 I will consider point dipole coupling resulting in the couplings, given by
(17)
where J is the nearest-neighbor coupling. On average, all energies of the chromophores are the same, but random Gaussian disorder with a standard deviation of σ is added. This is done by drawing random numbers from a Gaussian distribution independently for the energy of each site using the reported standard deviation for the width of the distribution. The resulting excitons are found by diagonalizing the Hamiltonian [Eq. (1)] and analyzed using Eqs. (7) and (9). Ten disorder realizations were considered for linear aggregates with 1000 chromophores. This was done for σ = J/2 and for σ = 2J, as shown in Figs. 1 and 2, which includes the data from all ten disorder realizations.
FIG. 1.
(a)–(c) Comparison of the delocalization measures for the excitons in a linear aggregate with σ = J/2. In panels a and b, the two measures are compared to each other and to the oscillator strength of the corresponding exciton normalized with the oscillator strength of the monomers (f=Ik/|μ|2). The vertical line at oscillator strength on one marks the division between dark states with lower oscillator strength than a monomer on the left and stronger states on the right. Panel d shows the ratio between the oscillator strength and MES against the exciton oscillator strength.
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(a)–(c) Comparison of the delocalization measures for the excitons in a linear aggregate with σ = J/2. In panels a and b, the two measures are compared to each other and to the oscillator strength of the corresponding exciton normalized with the oscillator strength of the monomers (f=Ik/|μ|2)⁠. The vertical line at oscillator strength on one marks the division between dark states with lower oscillator strength than a monomer on the left and stronger states on the right. Panel d shows the ratio between the oscillator strength and MES against the exciton oscillator strength.

FIG. 1.
(a)–(c) Comparison of the delocalization measures for the excitons in a linear aggregate with σ = J/2. In panels a and b, the two measures are compared to each other and to the oscillator strength of the corresponding exciton normalized with the oscillator strength of the monomers (f=Ik/|μ|2). The vertical line at oscillator strength on one marks the division between dark states with lower oscillator strength than a monomer on the left and stronger states on the right. Panel d shows the ratio between the oscillator strength and MES against the exciton oscillator strength.
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(a)–(c) Comparison of the delocalization measures for the excitons in a linear aggregate with σ = J/2. In panels a and b, the two measures are compared to each other and to the oscillator strength of the corresponding exciton normalized with the oscillator strength of the monomers (f=Ik/|μ|2)⁠. The vertical line at oscillator strength on one marks the division between dark states with lower oscillator strength than a monomer on the left and stronger states on the right. Panel d shows the ratio between the oscillator strength and MES against the exciton oscillator strength.

Close modal
FIG. 2.
Comparison of the delocalization measures for the excitons in a linear aggregate with σ = 2J. The visualization is as shown in Fig. 1.
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Comparison of the delocalization measures for the excitons in a linear aggregate with σ = 2J. The visualization is as shown in Fig. 1.

FIG. 2.
Comparison of the delocalization measures for the excitons in a linear aggregate with σ = 2J. The visualization is as shown in Fig. 1.
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Comparison of the delocalization measures for the excitons in a linear aggregate with σ = 2J. The visualization is as shown in Fig. 1.

Close modal

It is clear from the plots that the value of the MES is indeed never smaller than the exciton oscillator strength normalized with the oscillator strength of the monomers (f=Ik/|μ|2)⁠. The MES is also always larger than the IPR. For σ = J/2, the mean of the IPR was 23.8, while the mean of the MES was 59.4. For σ = 2J, the mean of the IPR was 3.38, while the mean of the MES was 7.17. Thus, NMES,k* is roughly twice as big as NIPR,k*⁠. However, the relationship is not trivial and for large values, it is seen as non-linear. In both cases, a collection of super-radiant states, where NMES,k*∼|μk|2 are clustered close to the diagonal, resulting from exciton wave functions, where all coefficients are in phase. Thus, for the super-radiant state, there is a strong correlation between the MES and the oscillator strength. On the contrary, such a relationship is not present for the IPR, and for σ = 2J, the IPR is nearly flat compared to the oscillator strength.

The overall behavior of the IPR, MES, and oscillator strengths at the two different disorder strengths are rather similar. In general, the increase in disorder leads to more localized wavefunctions and a reduction in the oscillator strength of the super-radiant states. The disorder dependence of the delocalization length, oscillator strength, and other properties has also previously been studied for linear aggregates.47,51,52

To connect the MES with the actual extent of the wave functions, the ADM [Eq. (14)] is plotted for the two disorder realizations in Figs. 3 and 4. The MES is illustrated by lines above and below the diagonals. This shows that most of the wave functions are contained within the range given by the MES in both the cases. Averages of moving slices along the diagonal of the ADM are shown in the two cases in Figs. 5 and 6. The similarity between these figures suggests that at least in some ranges, the MES is a representative measure of the width of the ADM distribution.

FIG. 3.
Visualization of the ADM for the first 100 sites in the linear aggregates with σ = J/2. The black lines parallel to the diagonal are drawn half of the MES below and above the diagonal to illustrate the delocalization length with this measure on the ADM.
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Visualization of the ADM for the first 100 sites in the linear aggregates with σ = J/2. The black lines parallel to the diagonal are drawn half of the MES below and above the diagonal to illustrate the delocalization length with this measure on the ADM.

FIG. 3.
Visualization of the ADM for the first 100 sites in the linear aggregates with σ = J/2. The black lines parallel to the diagonal are drawn half of the MES below and above the diagonal to illustrate the delocalization length with this measure on the ADM.
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Visualization of the ADM for the first 100 sites in the linear aggregates with σ = J/2. The black lines parallel to the diagonal are drawn half of the MES below and above the diagonal to illustrate the delocalization length with this measure on the ADM.

Close modal
FIG. 4.
Visualization of the ADM for the first 100 sites in the linear aggregates with σ = 2J. The visualization is as shown in Fig. 3.
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Visualization of the ADM for the first 100 sites in the linear aggregates with σ = 2J. The visualization is as shown in Fig. 3.

FIG. 4.
Visualization of the ADM for the first 100 sites in the linear aggregates with σ = 2J. The visualization is as shown in Fig. 3.
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Visualization of the ADM for the first 100 sites in the linear aggregates with σ = 2J. The visualization is as shown in Fig. 3.

Close modal
FIG. 5.
Moving average over vertical slices along the diagonal through the ADM for the linear aggregates with σ = J/2. The distance between the black vertical lines indicates the value of the MES.
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Moving average over vertical slices along the diagonal through the ADM for the linear aggregates with σ = J/2. The distance between the black vertical lines indicates the value of the MES.

FIG. 5.
Moving average over vertical slices along the diagonal through the ADM for the linear aggregates with σ = J/2. The distance between the black vertical lines indicates the value of the MES.
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Moving average over vertical slices along the diagonal through the ADM for the linear aggregates with σ = J/2. The distance between the black vertical lines indicates the value of the MES.

Close modal
Now, I proceed to examine a ring with ten chromophores. The point dipole coupling model is used again. The radius of the ring is set to 20 Å, and the transition dipole moment of each monomer is set to 5.5 Debye reflecting the typical value for a bacteriochlorophyll molecule.53 The transition dipoles are first assumed to be perpendicular to the ring plane, with the couplings given by
(18)
where A=5034.7cm−1Å3Debye2⁠. The largest coupling in this geometry then is 80.6 cm−1. Static disorder with σ = 80.6 cm−1 is added. 100 disorder realizations were analyzed and the results are shown in Fig. 7. The mean value of IPR is 4.65, while the mean value of MES is 6.88. As for the linear aggregates, the IPR is always larger than the MES, and for the super-radiant states, the MES is very similar to the oscillator strength but never larger. Interestingly, essentially two types of behavior are observed for the exciton states. For one group, which can be considered the super-radiant states, the oscillator strength and the MES are directly proportional to each other. For the other group, the ratio between the oscillator strength and the MES is proportional to the oscillator strength. We can understand the groups of states observed in Fig. 7(b) as those without a node in the wave function resulting in states, where the MES and the oscillator strength are the same. The group of states to the left of the vertical line has an odd number of nodes in the wave function, resulting in these being essentially dark. The states between the vertical and the diagonal lines have an even non-zero number of nodes in the wave function, resulting in an intermediate oscillator strength. The closer the states are to the vertical line, the more nodes they can be expected to have.
FIG. 6.
Moving average over vertical slices along the diagonal through the ADM for the linear aggregates with σ = 2J. The distance between the black vertical lines indicates the value of the MES.
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Moving average over vertical slices along the diagonal through the ADM for the linear aggregates with σ = 2J. The distance between the black vertical lines indicates the value of the MES.

FIG. 6.
Moving average over vertical slices along the diagonal through the ADM for the linear aggregates with σ = 2J. The distance between the black vertical lines indicates the value of the MES.
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Moving average over vertical slices along the diagonal through the ADM for the linear aggregates with σ = 2J. The distance between the black vertical lines indicates the value of the MES.

Close modal
FIG. 7.
(a)–(c) Comparison of the delocalization measures for the excitons in a ring aggregate with σ = J. The transition dipoles are perpendicular to the ring plane. The two measures are compared to each other and to the oscillator strength of the corresponding exciton normalized with the oscillator strength of the monomers. The vertical line at oscillator strength on one marks the division between dark states with lower oscillator strength than a monomer on the left and stronger states on the right. Panel (d) shows the ratio between the oscillator strength and MES against the exciton oscillator strength.
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(a)–(c) Comparison of the delocalization measures for the excitons in a ring aggregate with σ = J. The transition dipoles are perpendicular to the ring plane. The two measures are compared to each other and to the oscillator strength of the corresponding exciton normalized with the oscillator strength of the monomers. The vertical line at oscillator strength on one marks the division between dark states with lower oscillator strength than a monomer on the left and stronger states on the right. Panel (d) shows the ratio between the oscillator strength and MES against the exciton oscillator strength.

FIG. 7.
(a)–(c) Comparison of the delocalization measures for the excitons in a ring aggregate with σ = J. The transition dipoles are perpendicular to the ring plane. The two measures are compared to each other and to the oscillator strength of the corresponding exciton normalized with the oscillator strength of the monomers. The vertical line at oscillator strength on one marks the division between dark states with lower oscillator strength than a monomer on the left and stronger states on the right. Panel (d) shows the ratio between the oscillator strength and MES against the exciton oscillator strength.
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(a)–(c) Comparison of the delocalization measures for the excitons in a ring aggregate with σ = J. The transition dipoles are perpendicular to the ring plane. The two measures are compared to each other and to the oscillator strength of the corresponding exciton normalized with the oscillator strength of the monomers. The vertical line at oscillator strength on one marks the division between dark states with lower oscillator strength than a monomer on the left and stronger states on the right. Panel (d) shows the ratio between the oscillator strength and MES against the exciton oscillator strength.

Close modal

Furthermore, I consider a ring identical to that described above, but where the transition dipoles are pointing in the tangential direction, this resembles the geometry found in the LH2 complex of purple bacteria.54 The largest coupling is then 153.6 cm−1 and the disorder was again chosen to match this value. 100 disorder realizations were analyzed and the results are shown in Fig. 8. The mean value of IPR is 4.63, while the mean of the MES is 6.86. For this system, the typical oscillator strength is about half the value of the MES for the super-radiant states. This can be understood as the ring has D10h symmetry and the exciton states with transition dipole in the x,y-plane are doubly degenerate. While the disorder breaks the symmetry, the excitons are still so delocalized that two near-degenerate super-radiant states each carrying about half the oscillator strength dominate the picture. Similar groups of states can be identified here, as for the ring aggregate with parallel transition dipoles. However, the selection rules are different due to the change of transition dipole orientation, leading to states with one node in the wave function to have the largest oscillator strength and those with an even number to be dark.

FIG. 8.
(a)–(c) Comparison of the delocalization measures for the excitons in a ring aggregate with ten chromophores and σ = J. The transition dipoles are tangential to the ring. The two measures are compared to each other and to the oscillator strength of the corresponding exciton normalized with the oscillator strength of the monomers. The vertical line at oscillator strength on one marks the division between dark states with lower oscillator strength than a monomer on the left and stronger states on the right. Panel (d) shows the ratio between the oscillator strength and MES against the exciton oscillator strength.
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(a)–(c) Comparison of the delocalization measures for the excitons in a ring aggregate with ten chromophores and σ = J. The transition dipoles are tangential to the ring. The two measures are compared to each other and to the oscillator strength of the corresponding exciton normalized with the oscillator strength of the monomers. The vertical line at oscillator strength on one marks the division between dark states with lower oscillator strength than a monomer on the left and stronger states on the right. Panel (d) shows the ratio between the oscillator strength and MES against the exciton oscillator strength.

FIG. 8.
(a)–(c) Comparison of the delocalization measures for the excitons in a ring aggregate with ten chromophores and σ = J. The transition dipoles are tangential to the ring. The two measures are compared to each other and to the oscillator strength of the corresponding exciton normalized with the oscillator strength of the monomers. The vertical line at oscillator strength on one marks the division between dark states with lower oscillator strength than a monomer on the left and stronger states on the right. Panel (d) shows the ratio between the oscillator strength and MES against the exciton oscillator strength.
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(a)–(c) Comparison of the delocalization measures for the excitons in a ring aggregate with ten chromophores and σ = J. The transition dipoles are tangential to the ring. The two measures are compared to each other and to the oscillator strength of the corresponding exciton normalized with the oscillator strength of the monomers. The vertical line at oscillator strength on one marks the division between dark states with lower oscillator strength than a monomer on the left and stronger states on the right. Panel (d) shows the ratio between the oscillator strength and MES against the exciton oscillator strength.

Close modal

The real LH2 complex is more complicated than the two ring structures discussed54 The transition dipoles in LH2 are mostly lying in the x,y-plane as in the second example; however, this alignment is not perfect. The transition dipoles in the tightly packed chromophore ring of LH2 are furthermore generally considered to be alternating in their direction. This, however, will have no real physical effect on the analysis as flipping the transition dipole direction is paired with flipping the sign of coupling between chromophores where the transition dipole was flipped and where they were not. Overall, it will flip the number of nodes observed in the wave functions; however, this is exactly paired with the flip of the transition dipole directions and there is a direct one-to-one mapping between the two situations, which are physically identical.

The two considered delocalization measures are examined for different ideal wave function shapes in Fig. 9 and Table I. The functions were defined as discrete on a 101-site grid using the following equations:
(19)
where cn are the wave function coefficients for the different functions and n labels the sites from 0 to 100. Each function was normalized numerically, before plotting and further analysis. The intuitive picture of the size of an exciton can perhaps for a one-dimensional function be connected with the full-width-at-half-maximum (FWHM). I found that the IPR is fairly close to the FWHM for all the four tested shapes and the values even coincide for the exponentially decaying wave function and the sine (homogeneous aggregate) one. The MES always gives larger delocalization lengths; however, for the sine and Gaussian functions that are more confined and have shorter tails, the IPR and MES are closer together. In general, one would expect the two measures to coincide for perfectly confined wave functions, where a few sites have a similar coefficient and all other coefficients are zero [following Eq. (6)]. This confinement effect is also observed for the ring aggregates [Figs. 7(c) and 8(c)], where the MES and IPR are more similar for very localized states as well as for states delocalized over the full ring.
FIG. 9.
Delocalization length for different wave function shapes on a 101-site aggregate as defined in Eq. (19). (a) Gaussian, (b) exponential, (c) sine, and (d) Lorentzian shapes. The black double arrows indicate the full-width-half-maximum (FWHM); the blue double arrows indicate the IPR; and the red double arrows indicate the MES. For the Gaussian and exponential wave functions, the FWHM and IPR overlap.
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Delocalization length for different wave function shapes on a 101-site aggregate as defined in Eq. (19). (a) Gaussian, (b) exponential, (c) sine, and (d) Lorentzian shapes. The black double arrows indicate the full-width-half-maximum (FWHM); the blue double arrows indicate the IPR; and the red double arrows indicate the MES. For the Gaussian and exponential wave functions, the FWHM and IPR overlap.

FIG. 9.
Delocalization length for different wave function shapes on a 101-site aggregate as defined in Eq. (19). (a) Gaussian, (b) exponential, (c) sine, and (d) Lorentzian shapes. The black double arrows indicate the full-width-half-maximum (FWHM); the blue double arrows indicate the IPR; and the red double arrows indicate the MES. For the Gaussian and exponential wave functions, the FWHM and IPR overlap.
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Delocalization length for different wave function shapes on a 101-site aggregate as defined in Eq. (19). (a) Gaussian, (b) exponential, (c) sine, and (d) Lorentzian shapes. The black double arrows indicate the full-width-half-maximum (FWHM); the blue double arrows indicate the IPR; and the red double arrows indicate the MES. For the Gaussian and exponential wave functions, the FWHM and IPR overlap.

Close modal
TABLE I.

Overview of the delocalization measures for the different wave function examples is presented in Fig. 9. The ratio IPR/MES is given for comparison.

GaussianExponentialSineLorentzian
IPR 25.1 19.9 67.3 25.0 
MES 35.4 39.4 81.9 48.3 
FWHM 23.5 20.0 66.7 20.0 
IPR/MES 0.71 0.51 0.82 0.52 
GaussianExponentialSineLorentzian
IPR 25.1 19.9 67.3 25.0 
MES 35.4 39.4 81.9 48.3 
FWHM 23.5 20.0 66.7 20.0 
IPR/MES 0.71 0.51 0.82 0.52 
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Finally, I explore the use of the IPR/MES ratio to characterize confinement. This was done by varying the length of the linear J-aggregate considered at the beginning of this section. The disorder was set to σ = J/2 and the length, N, was varied between 1 and 10 000. To ensure a reasonable averaging for all considered aggregate lengths, the data were averaged for 10 000/N disorder realizations. The resulting IPR and MES values averaged over all exciton states are presented in Fig. 10. As the length of the aggregate is reduced toward the respective delocalization lengths, these reduce in size from the plateau values reached in the long aggregate limit. Examining the IPR/MES ratio, it goes from ∼0.4 in the long aggregate limit to one for the single site aggregate length. A substantial degree of confinement is observed as the aggregate length is reduced below the long aggregate limit delocalization lengths. This happens around IPR/MES ratios between 0.5 and 0.6. Defining a linear degree of confinement scale between the long aggregate limit and the fully confined limits, this is between 20% and 40% confinement. The IPR/MES ratio may thus serve as a heuristic measure of confinement. Here, it is demonstrated for a linear aggregate and for other geometries that the long-range limit may be different for other relevant aggregate geometries.

FIG. 10.
Left panel: the IPR and MES as a function of aggregate length shown along with the used aggregate length. Right panel: the IPR/MES ratios (in black) as a function of aggregate length. The vertical lines indicate the IPR (blue) and MES (red) determined in the long aggregate limit. The degree of confinement is defined as a linear scale of the IRP/MES ratios set to zero in the long-range limit and 100% when the ratio is one.
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Left panel: the IPR and MES as a function of aggregate length shown along with the used aggregate length. Right panel: the IPR/MES ratios (in black) as a function of aggregate length. The vertical lines indicate the IPR (blue) and MES (red) determined in the long aggregate limit. The degree of confinement is defined as a linear scale of the IRP/MES ratios set to zero in the long-range limit and 100% when the ratio is one.

FIG. 10.
Left panel: the IPR and MES as a function of aggregate length shown along with the used aggregate length. Right panel: the IPR/MES ratios (in black) as a function of aggregate length. The vertical lines indicate the IPR (blue) and MES (red) determined in the long aggregate limit. The degree of confinement is defined as a linear scale of the IRP/MES ratios set to zero in the long-range limit and 100% when the ratio is one.
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Left panel: the IPR and MES as a function of aggregate length shown along with the used aggregate length. Right panel: the IPR/MES ratios (in black) as a function of aggregate length. The vertical lines indicate the IPR (blue) and MES (red) determined in the long aggregate limit. The degree of confinement is defined as a linear scale of the IRP/MES ratios set to zero in the long-range limit and 100% when the ratio is one.

Close modal

In this paper, I revisited a general class of measures to quantify the delocalization of Frenkel excitons. The MES was identified as a measure closely connected with the oscillator strength of super-radiant states. As such, it appears to be a logical choice to use, when considering the optical and exciton transfer properties of excitons, which are related to the square of the exciton transition dipole. The MES was compared to the commonly used IPR measure for several realistic aggregate types.

It was demonstrated that the MES always provides an upper boundary for the oscillator strength of a given exciton state. This also means that the radiative rate of an exciton state cannot be higher than the radiative rate of the individual dye molecule times the MES delocalization size of the super-radiant exciton state in an aggregate. It was demonstrated that for linear aggregates, the oscillator strength may be exactly that of the individual molecule times the MES. However, for other geometries, including the illustrated ring structures, the oscillator strength of the collective states will always be smaller than this bound.

While the IPR measure is typically used to characterize the delocalization of excitons, calculating both the MES and IPR may provide insight into the degree of exciton confinement. The IPR value for idealized wave function shapes is always smaller than the MES value. For particle-in-a-box type super-radiant wave functions, the ratio between the two is about 0.82. On a ring with perfect delocalization, the ratio can reach 1. However, for exponentially decaying wave functions, the ratio is closer to 0.5. Tested for all exciton states in a linear aggregate, the ratio between the average IPR and MES was reduced from ∼0.4 to 1 as the confinement was introduced by reducing the aggregate length. Therefore, calculating both measures may provide more information on the nature of the exciton states than just examining one. It is, therefore, recommended to examine not only the IPR but also the MES, particularly when studying optical properties as fluorescence spectroscopy and two-dimensional spectroscopy.

The author has no conflicts to disclose.

T. L. C. Jansen: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Funding acquisition (equal); Investigation (equal); Methodology (equal); Project administration (equal); Resources (equal); Software (equal); Supervision (equal); Validation (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.

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