Modeling the Optical Response of Phonondressed Excitons in OLED Simulations
Abstract
We demonstrate the modeling of optical response of excitonpolarons based on the well established Holstein Hamiltonian to model coupled excitonphonon systems in organic molecular chains. Our approach uses Green’s functions to compute the density of states and the linear optical susceptibility, and thus eliminates the conventional and computationally expensive step of diagonalizing a large Hamiltonian matrix. We exploit this technique further to focus exclusively on the optically active states when computing the linear optical response, and significantly reduce the computational effort to construct the optical susceptibility. In this article, we demonstrate the computation of absorption and emission spectra of Alq_{3} at 4.2 K and at room temperature using our model. Using the two parameters of the Holstein model, the inhomogeneous broadening energies, and a phenomenological reorganization energy of the solute, we obtain excellent fits to established experimental results. We then use this model inside the larger simulation of a 3layer organic light emitting (OLED) structure composed of Alq_{3}, Alq_{3}:DCJTB, and αNPD, which are the electron transport, emissive, and hole transport layers respectively. In our methodology, we also couple the optical response into the rate equations for exciton dynamics in addition to computing the spectrum of light output by the device.
Keywords: Frenkel Exciton, Phonon, Organic, OLED, trishydroxyquinoline, Holstein model, Optical emission, Optical absorption
I Introduction
Organic light emitting (OLED) and photovoltaic (PV) technologies are growing at a rapid pace. Compared to the inorganic semiconductor based technology, organics provide much simpler and cheaper fabrication methodologies. With continuing research in this field, a vast number of organic materials have become potential candidates for device applications, and they generally exist in diverse forms ranging from crystalline phases to fully disordered solutions. This provides a challenge for developing reliable and widely applicable models for understanding experimental data and predicting device characteristics.
Yet the optical and transport properties of these materials can often be captured via models with one or more excited states (excitons) hopping on the underlying molecular lattice, and linearly coupled to its internal vibrational modes[1–3]. Each type of vibrational mode can in turn be modeled as a harmonic oscillator. Here we describe a methodology that exploits this fact to simulate exciton dynamics and light emission from OLEDs. The primary purpose of this work is to provide a physically based model to compute the optical response with a small set of parameters.
Materials for which a widely accepted measured spectrum over the desired energies does not exist are a clear target application. The model is also relevant for well known materials since the optical response for most organic systems varies widely due to their sensitivity to their environments and their contact with charge injection layers in devices. With a small parameter set in which parameters are related to fundamental physical mechanisms, the present model calibrated against experimental data acquires predictive value for exploring an entire class of devices. For instance emissive layers with similar vibrational modes, excitonphonon coupling, and inhomogeneous broadening can fall within the range of a single model fit once to reliable experimental data.
Our methodology applies to electroluminescent organic materials such as Alq_{3}, DCM, DCJTB etc., which are generally a mixture of short chains of molecules, or independent units with random orientations for the dipolar charge excitations[2, 4, 5]. Electrons andholes injected into these materials form Frenkel excitons, in which both the electron and the hole reside on the same molecular unit. For example in Alq3, the exciton forms by electron transition from phenoxide to the the pyridyl ring of a single molecule[6, 7].
Radiative recombination of excitons gives rise to luminescence. In the absence of spinorbit coupling (SOC), photons are emitted only by singlets due to the optical selection rules. Heavy metal impurities are increasingly being used to enhance radiative annihilation of triplets with SOC[4, 8, 9]. One of the main attractions of using organic materials is that the color of emitted light can be easily controlled by doping with molecules of different band gaps[4]; the relative population of excitons on each species determines the overall shift in the main emission wavelengths. The transfer of excitons between the host and the dopant controls the relative populations, and this transfer is fundamentally driven by Förster energy transfer[2, 10–13] for singlets and Dexter transfer[2, 14, 15] for triplets. In our methodology, the radiative and transfer rates are computed from the quantum mechanical model for the optical response of each species.
One of the most important aspects of Frenkel excitons in organic materials is their strong modification by the internal vibrational modes of the molecular units[1–3, 16–20]. These modes may vary from bending, stretching, or rotational modes, and they can range from being localized at one molecule to being spread out over an entire polymer chain. Providing great simplification in modeling is the unifying aspect of these modes: their energies tend to be insensitive to the presence of an exciton, and their coupling to the excitons is linear[26]. The magnitude of the linear coupling gives a timescale for intramolecular relaxation. The localized vibrational mode follows the exciton if the intramolecular relaxation is faster than intermolecular charge transfer[3]. This effectively dresses the exciton with a phonon cloud, creating a new quasiparticle, called an excitonpolaron, which has different transport and optical properties than the bare electronhole pair comprising the Frenkel exciton. In our methodology, we compute the optical properties for this composite quasiparticle, thus taking into account the strong phonon dressing exactly within the model of linear coupling.
Figure 1 summarizes the radiative transitions between vibrational modes of typical organic molecules. The vibrational energy of the molecule defines a potential energy surface, which can be approximated as a parabola (in nuclear coordinates) near equilibrium. If no coupling to phonons existed in the molecule, the emission and absorption spectra will both consist of a single peak represented by the green line in Figure 1. called the zerophonon transition, which is between 0 and 1 exciton states containing no phonons. In the presence of coupling to phonons, optical excitation creates both the exciton and its phonon cloud. The blue line indicates transitions from the lowest vibrational state (the only state at 0 K) to 1exciton state containing one or more phonons. These transitions give a progression of uniformly spaced peaks above the zerophonon transition, which merge into a single broad spectrum after inhomogeneous broadening is taken into account.
Figure 1. Schematic illustration of the fundamental optically driven transitions in the presence of excitonphonon coupling. The vertical scale is energy, and the horizontal scale is a set of generalized nuclear coordinates. The lower parabola represents the potential energy surface of the electronic ground state (0 exciton), and the upper parabola represents the potential energy surface of the first excited state (containing 1 exciton). The dashed horizontal lines indicate the quantized energy levels of the vibrational potential (phonons). Both energy surfaces are modeled using the same parabolic potential, and thus the modes in each are those of a harmonic oscillator. The wavefunctions in the upper level are shifted by amount g that parameterizes the excitonphonon coupling. The peaks on the right schematically depict the emission (red), absorption (blue) and zerophonon line (green). 
In the emission spectrum, the phonon occupation of the ground state is probed. The red line indicates luminescence transition in which the exciton, having achieved quasiequilibrium to its lowest energy state, recombines and leaves the molecule in a vibrational state with one or more phonons. These transitions give peaks in emission spectra progressing below the zerophonon line. Finally, the two yellow lines indicate excitations that occur due to thermal excitations of phonons in the ground state, and the excited state at quasiequilibrium. These transitions are suppressed exponentially by the corresponding Boltzmann factors.
As explained in the next section, the zerophonon transition is almost invisible in most systems because the overlap of nuclear wavefunctions in the 0 and 1exciton manifolds is much higher at intermediate phonon occupations. This results in a difference in the location of peaks in the absorption and emission spectra, known as the Stokes shift. The Stokes shift in organics is driven both by the above mentioned overlap, and by the reorganization energy of the environment when an exciton is introduced into it[7, 21–23]. The former is calculated in our model from the exciton phonon coupling, and the latter is used as a parameter since it is impossible to calculate reorganization energy of an arbitrary environment.
In Section II, we provide the theoretical background and discuss the salient aspects of computation of spectra in our approach. In Section III, we discuss results of our computations for Alq_{3} and compare them with experimental data. We then discuss simulation of a 3layer OLED device based on Alq_{3} : DCJTB. Finally we conclude in Section IV.
II Methodology
A. Structure and Setup
At the top level of the simulation in Atlas, we solve the coupled rate equations for densities of electrons, holes, intrinsic excitons and dopant excitons (see Section 15.3 of Atlas manual). Ignoring the Stark effect[21], the energy levels of the molecules do not shift with bias and therefore we compute the optical response for unit density of both the intrinsic and dopant molecules separately at the beginning of the simulation. At subsequent bias steps, this spectrum is used to compute radiative loss of excitons and coupling between the intrinsic and dopant singlets fully selfconsistently with the quantum mechanical model of fluorescence. The total fluorescence from each mesh node is computed by combining the spectra the exciton density on each node. When used in conjunction with ray tracing, transfer matrix, or finite difference time domain algorithms, the total fluorescence spectrum gives the angular and spectral characteristics of the light output by the device [27].
The simulation is setup by dividing a device into regions, and associating a material with each region. We have extended the MATERIAL statement in Atlas to facilitate the addition of up to 10 different exciton species per region. An excitonpolaron species is added by specifying the parameter ADDPOLARON and specifying a name for the species as MIX.NAME=name. At present the rate equations are limited to two species per node only. Specifying the HOLSTEIN parameter for a particular region initiates the quantum calculation of optical response for each species defined in the region. Since the model described below does not depend explicitly on the spatial distribution of excitons, a single model per species is solved for each region, rather than each mesh node. This simplification is correct as long as spatial distribution of energy levels can be captured by inhomogeneous broadening. When this cannot be justified, it is best to divide up a region into smaller pieces over which we expect energy levels and their couplings to lie within inhomogeneous broadening.
Below we describe the Hamiltonian for quantum mechanical description of excitonpolaron dynamics in an organic materials. We then describe our computation of optical response and the main physical quantities calculated in the simulation.
B. Excitonpolaron states
The fundamental description we use here for the excitonphonon system is given by the Holstein Hamiltonian,
(1)
where a_{n}, a annihilate and create an exciton at molecular site n respectively, while b_{n} and b perform the same function for phonons. The parameter J is the hopping energy, g is the excitonphonon coupling, E_{0} is the band gap or the difference between the HOMO and the LUMO levels (see Figure 1), and E_{vib} is the energy of a single excitation (phonon) of the vibrational mode. The term proportional to g^{2} (1) aligns the LUMO level and the band gap to the user specified value. The band gap E_{0} plays no essential role in determining the eigenvalues and eigenstates of the system, except for shifting the resulting spectrum by the band gap energy. Note that the above form of the Hamiltonian does not make reference to the detailed spatial structure of the exciton and nuclear wavefunction. That information has been absorbed into the parameters described above.
Following Hoffman et. al. [3] we compute the energy spectrum of this Hamiltonian in the basis represented as , where represents exciton at site n, and represents a phonon cloud with specifying the phonon occupation in the oscillator at site m. The oscillator at the exciton site is shifted by amount g (Figure 1) as dictated by (1), and we represent its occupation number by a different symbol .
Thus by virtue of the dependence of phonon occupation on the location of exciton, the state of the molecular distortion is coupled fully to the exciton. The resulting Hamiltonian can be diagonalized using the LangFirsov transformation[3], in which the system is described by excitonpolaron whose hopping energy is given by , where where are called FrankCondon factors, and they are equal to the overlap of the phonon clouds at the initial and final site in a hopping event.
The main benefit of using harmonic oscillator to model the vibrational modes is that Frank Condon factors can be computed analytically from the inner product of shifted oscillator wavefunctions. Thus the two parameters, J and g, fully determine the effective mass of the polaron. The same FrankCondon factors also determine the amplitudes and selection rules of optical transitions.
A fully cohrent excitonpolaron in a lattice carries a definite momentum, and the energymomentum relationship yields a set of exciton bands as a function of the momentum k. Since the photon momentum is negligible compared to that of an exciton, optically driven transitions occur only at k = 0. We therefore compute only k = 0 states and exploit the fact that large inhomogeneous broadening rather than band dispersion dominates density of states at k = 0. This formulation of the DOS is also consistent with assumptions underlying the hopping model of exciton dynamics simulated in Atlas.
C. Radiative Emission and Energy transfer
Following the standard treatment of dipole coupling between light and matter, the Hamiltonian for the interaction of an excitonpolaron with a plane wave electric field is,
where the sum includes both positive and negative frequencies, and is the position operator.
A standard approach to compute absorption and emission spectra is in terms of the matrix elements of taken between the exact eigenstates of excitonpolarons. This is an extremely expensive calculation since a very large number of phonon cloud states exist for a given modest size and phonon occupancy. In addition, since most of the states are optically forbidden, their inclusion in the spectrum serves only as an additional broadening mechanism.
In our methodology, we use a much faster Green function based method to compute the absorptive and emissive contributions optical susceptibility: χ_{ab}(ω), χ_{em} (ω) respectively. In this method, only the optically accessible states are referenced explicitly by the calculation, while the presence of the remaining states appears as an additional broadening mechanism as is expected physically. The technique is mathematically equivalent to exact diagonalization, and becomes useful in the presence of sufficient broadening as is the case for organic materials[28].
With homogeneous broadening, specified by η, the susceptibility can be written as,
(2)
(3)
where is the density of molecules, and N(α) is the number of phonons in the excitonpolaron state Ψ_{α}, and Z is the partition function normalizing the Boltzmann factors. Computation of χ_{ab}(ω), χ_{em} (ω) by (2) and (3) is done by solving a series of linear systems. By organizing the basis states according to whether they are dipole allowed or not, we minimize the number of systems that must be solved. The spectra are subjected to energydependent inhomogeneous broadening in the end.
The power radiated per unit volume in energy interval [E, E + ΔE] by spontaneous emission is given by the imaginary component of χ_{em},
(4)
where f_{orient} accounts for averaging over the random orientations of the exciton dipoles in amorphous materials. Thus the radiative rate, normalized to 1 exciton per unit cell volume is,
(5)
In the case of doped materials, the Förster transfer rate of a singlet from donor site D to acceptor A is,
(6)
From this formula we also compute the F¨orster radius, which is intermolecular distance at which the nonradiative transfer equals the radiative decay of excitons. Note that conventional formulas for Forster transfer use slightly different definitions for the spectra used in the overlap integral (6). See Appendix A for equivalence of k_{dd} to conventional formula.
We now turn to results of our simulations with this model.
III Results and Discussion
A. Spectrum of Alq_{3 }Alq_{3} is one of the most important host materials used in OLEDs. It is known to emit green light at wavelengths of approximately 530 nm. In addition, it is also one of the simplest applications of the model described above. Frenkel excitons in Alq_{3} couple to the bending modes of the molecule where the exciton resides. However the intermolecular hopping is weak, and thus the vibrational modes of Alq_{3} are expected to create small phonon clouds only. The large inhomogeneous broadening generally render the vibrational modes unobservable in the emission and absorption spectra of Alq_{3}3.
However, in their experiments, Brinkman et. al. were able to obtain this for crystalline Alq_{3} 4.2 K [23]. From their observations, the authors concluded that the HuangRhys factor, (g/E_{vib})^{2} ≈ 2.6 ± 0.4, and E_{vib} = 0.065[22, 23]. Using the value g = 1.6Evib in our model, and setting the hopping parameter J = 0.1Evib, we computed spectra for a single Alq_{3} region using our model inside Atlas. We varied the phonon cloud sizes from 1 (onsite vibron) to 4 and noticed only small changes in the spectra, which is expected due to the fact that g suppresses hopping exponentially. As was discussed in Section I, two additional parameters are needed: a band gap and a Stokes shift. Here we used a band gap of 2.87 eV, and an additional Stokes shift of 0.15 eV. These parameters alone correctly reproduce the absorption spectrum below the peak and emission spectrum above the peak, in particular the vibronic structure (peaks along the spectral backbone), as shown in Figure(2).
Figure 2. Comparison of the calculated Alq_{3} spectra to those measured at for the crystalline phase at 4.2 K. The dots and crosses are digitized experimental data from [23]. Homogeneous broadening was set to 10 meV and inhomogeneous broadening to 22 meV. A single Voigt lineshape is added for both the emission and absorption to take into account the background dielectric due to higher energy states. 
However high energy modes and the background dielectric add additional featureless profile to the measured spectrum. These are often modeled as highly broadened extra energy levels. We model these as Voigt lineshapes (Gaussian convolved with Lorentzian) by using the parameter ADDSTATE on MATERIALS statement of the corresponding region. The lineshape center is specified using EX.ECEN or GS.ECEN depending on whether the state is added to the 1exciton or 0exciton states respectively. The parameters EX.HOMOBROADENING and EX.INHOMOBROADENING must be used to specify the corresponding values of broadening in electron volts. The extra lineshapes in the present calculation are centered at 3.2 and 2.4 eV, with both the homogeneous and inhomogeneous broadening both set to 10 meV.
We now consider room temperature spectrum at 300 K, the emission spectrum of a thin Alq_{3} film exhibits no vibronic structure and is left as a smooth profile, shown in Figure 3. In this calculation, we used larger inhomogeneous broadening of 150 meV, and did not use extra Voigt lineshapes to account for the background dielectric. Thus a combination of 2parameter Holstein model, and a broad extra energy level can be used to successfully reproduce experimental spectra for this material. We remark that the size and the occupancy of the phonon clouds are not free parameters, as both these should be increased until the results converge.
Figure 3. Comparison of the calculated Alq_{3} absorption spectrum to the measured spectrum for a thin film at 300 K. The dots and crosses are digitized experimental data from [23]. A single Voigt lineshape is added to take into account the background dielectric due to higher energy states. 
B. Light emission from Alq_{3}:DCJTB based OLED
We now demonstrate the ability to apply this model in simulating a typical 3layer OLED. The left panel in Figure 4 shows the structure of the device simulated. The device is composed of a 30 nm wide emission layer (EML) with the host Alq_{3} doped to 1% with DCJTB. At the top of EML is a 30 nm thick electron transport layer (ETL) of pure Alq_{3}, and at the bottom is a hole transport layer (HTL) with material properties corresponding the αNPD. The main energy level requirements to make this device emit are as follows. The LUMO levels of the ETL and EML are aligned to facilitate electron injection into the EML, while that of the HTL is about 300 meV higher. Thus HTL essentially acts as an electron blocking layer and maximizes recombination of electrons with holes in the EML. Similarly, the HOMO level of the HTL is only slightly above the HOMO level of the EML, which facilitates hole injection, while the HOMO level of ETL is much lower to block holes at the EML/HTL interface.

Figure 4. Top: Structure of the three layer device with 30 nm electron transport layer with Alq_{3}, 30 nm emission layer with Alq_{3}:DCJTB and a 40 nm hole transport layer of αNPD. Bottom: CurrentVoltage characteristics of 3layer device, intrinsic and dopant exciton densities, Langevin recombination rates, and the Forster exchange per unit volume. 
We used the organic defect model described in Atlas User’s Manual to simulate exciton transport in each layer. The Holstein model was applied to ETL and EML for computing the emission spectra and the radiative as well as Förster rates. We used the PooleFrenkel fielddependent mobility model, with parameters taken from [24]. The right panel in Figure 4 shows the typical currentvoltage relationship of an LED.
In Figure 5 we show the density of host and dopant excitons in the ETL and EML layers. Good injection of carriers into the active region (EML) is apparent from the much larger magnitude of the densities in the EML. For a given Förster radius following from the device spectrum, the relative fraction of dopant excitons to host excitons is highly sensitive to the ratio of this radius to hostdopant distance. We set the intermolecular distance equal to 3.7 nm in these simulations, corresponding to host density of approximately 4×10^{18} cm^{−3}. The simulation yields a Förster radius to be approximately 3.6 nm for exciton transfer to the dopant. The radius is approximately 1 nm for the reverse process. Thus exciton migration to the dopant is very nearly one directional.
Figure 5. Densities of excitons (cm^{−3}) on Alq_{3} and DCJTB in the electron transport and emissive layers (left). Spectrum of light emitted from the EML layer (right) and computed using reverse ray trace with source terms restricted to the respective layers. The larger density of excitons on DCJTB explains the overall shift in the spectrum from the host to dopant emission wavelength. 
Figure 6. shows the average number of radiative transitions, Förster transfers and Langevin recombinations per unit volume per time. The Förster rate exceeds radiative rates for each species, and thus we expect the dopant to make significant contribution to the output spectrum of light. The large Förster rate is due to a good overlap between the absorption spectrum of the dopant and emission spectrum of the host, as displayed in the left panel of Figure 7.
Figure 6. The radiative emission rates, Forster transfer rates and Langevin recombination rates per unit volume as a function of bias voltage. Vertical scale: 1/(cm^{3} &). 
At the final bias point of 7 V, we performed a reverse ray trace analysis to compute the light output by the device. The resulting spectrum is shown in the right panel of Figure 7. The output spectrum is clearly dominated by emission from the dopant, while the emission from the host contributes the smaller peak on the higher energy side. Thus the effect of Forster transfer on the emission spectrum is captured quite well by the simulation.

Figure 7. Emission and absorption spectra for the host (Alq_{3}) and the dopant DCJTB. The good overlap between host emission and dopant absorption yields high F¨orster transfer rates. The vertical scale on the top panel indicates the response from a volume of 1 molecular unit. The vertical scale on the bottom panel is the power spectral density. 
IV Conclusion
We have implemented the calculation of OLED optical response using the Holstein Hamiltonian. The shape of spectra is determined by only 2 parameters while the additional parameters account for the position and scaling. The computation is fully integrated with the LED device simulation in Atlas. This integration is performed both at the level of light output coupling as well as at the deeper level of determining the radiative and excitation transfer rates in exciton dynamics. With the addition of Voigt lineshapes to model the background dielectric contributions, we have demonstrated the model to fully capture the important physical features of measured spectra for Alq_{3}. We demonstrated the full methodology of using this model in Atlas by simulating a typical 3layer OLED device with a doped emissive layer. We extracted the dynamical rates computed using our model within each region, and verified that the relative magnitudes of the dynamical rates are well correlated with the main qualitative aspects of the emission and absorption spectra of the host and dopant molecules.
References
 M Hoffmann, Z G Soos, K Leo. Absorption Spectra [1] M. Hoffmann, Z. G. Soos, and K. Leo, 237 (2006)
 V. Agranovich, Excitations in Organic Solids (Oxford, 2009)
 M. Hoffmann and Z. Soos, Phys. Rev. B 66, 024305 (2002)
 S. Reineke, M. Thomschke, B. L ̈ssem, and K. Leo, Rev. Mod. Phys. 85, 1245 (2013)
 L. Hung and C. Chen, Mater. Sci. Eng. R Reports 39, 143 (2002)
 M. D. Halls and H. B. Schlegel, Chem. Mater. 13, 2632 (2001)
 O. Gardens, ed., Trends in Optical Materials (Nova, 2007)
 H. Kanno, N. C. Giebink, Y. Sun, and S. R. Forrest, Appl. Phys. Lett. 89, 023503 (2006)
 Y. Sun, N. C. Giebink, H. Kanno, B. Ma, M. E. Thompson, and S. R. Forrest, Nature 440, 908 (2006)
 T. Forster, Discuss. Faraday Soc. 27, 7 (1959)
 D. Beljonne, C. Curutchet, G. D. Scholes, and R. J. Silbey, 113 (2009)
 G. D. Scholes, Annu. Rev. Phys. Chem. 54, 57 (2003)
 G. Scholes and D. Andrews, Phys. Rev. B 72, 125331 (2005)
 A. K ̈hler and H. B ̈ssler, Mater. Sci. Eng. R Reports 66, 71 (2009)
 A. OlayaCastro and G. D. Scholes, Int. Rev. Phys. Chem. 30, 49 (2011)
 I. Vragovi ́, R. Scholz, and M. Schreiber, Europhys. Lett. 57, 288 (2002)
 I. Vragovi ́ and R. Scholz, Phys. Rev. B 68, 155202 (2003)
 J. T. Devreese and A. S. Alexandrov, Reports Prog. Phys. 72, 066501 (2009)
 N. Karl, Synth. Met. 133134, 649 (2003)
 Y. Zhao, G. Li, J. Sun, and W. Wang, J. Chem. Phys. 129, 124114 (2008)
 R. Boyd, Nonlinear Optics (Academic Press, 2003)
 M. Muccini, M. Brinkmann, G. Gadret, C. Taliani, N. Masciocchi, and A. Sironi, Synth. Met., 122, 31 (2001)
 M. Brinkmann, G. Gadret, M. Muccini, and C. Taliani, J. Am. Chem. Soc. 122, 5147 (2000)
 C.C. Lee, M.Y. Chang, P.T. Huang, Y. C. Chen, Y. Chang, and S.W. Liu, Journal of Applied Physics 101, 114501 (2007)
 V. G. Kozlov, V. Bulovic, P. E. Burrows, M. Baldo, V. B. Khalfin, G. Parthasarathy, S. R. Forrest, Y. You, and M. E. Thompson, J. Appl. Phys. 84, 4096 (1998)
 From experimental data, linearity is implied by the presence of uniformly spaced peaks in both photoluminescence (PL) and photoluminescence excitation (PLE) spectra. The insensitivity to excitation follows from the spacing being the same in both PL and PLE spectra.
 This creates a feedback mechanism whereby one can create a strongly nonlinear dependence of the emission spectrum on bias. However, this is not generally seen in common materials
and thus within the most common parameter regime of the model explored here.  The technique requires too many Green function evaluations to be useful when broadening is negligible.
Appendix A: Derivation of conventional formula for Förster radius
Absorption cross section σ = αV/N = Ωχ_{I;abs}(ω)ω/c, as there is 1 molecule in volume Ω. Take the ratio to radiative, while substituting absorption cross section and fluorescence spectrum,
If we normalize F^{D} such that . We now get the traditional formula[10, 25]