Improved Energy Pooling Efficiency Through Inhibited Spontaneous Emission
IImproved Energy Pooling Efficiency ThroughInhibited Spontaneous Emission
Michael D. LaCount and Mark T. Lusk ∗ Department of Physics, Colorado School of Mines, Golden, CO 80401, USA
E-mail: [email protected] a r X i v : . [ phy s i c s . a t m - c l u s ] A p r bstract The radiative lifetime of molecules or atoms can be increased by placing them withina tuned conductive cavity that inhibits spontaneous emission. This was examinedas a possible means of enhancing three-body, singlet-based upconversion, known asenergy pooling. Achieving efficient upconversion of light has potential applicationsin the fields of photovoltaics, biofuels, and medicine. The affect of the photonicallyconstrained environment on pooling efficiency was quantified using a kinetic modelpopulated with data from molecular quantum electrodynamics, perturbation theory,and ab initio calculations. This model was applied to a system with fluorescein donorsand a hexabenzocoronene acceptor. Placing the molecules within a conducting cavitywas found to increase the efficiency of energy pooling by increasing both the donorlifetime and the acceptor emission rate—i.e. a combination of inhibited spontaneousemission and the Purcell effect. A model system with a free-space pooling efficiency of23% was found to have an efficiency of 47% in a rectangular cavity.
Introduction
The utility of a given source of light can be extended by transforming it to a higher frequencythrough processes collectively known as upconversion. This can be exploited to activatemedicine at targeted locations within the body , increase the efficiency of solar energyharvesting , or even to increase the growth rate of plants for biofuels. It may well bethat up/down conversion, within the strong coupling limit, can also be incorporated intoemerging quantum information technologies.Energy pooling is a particular type of upconversion in which two donor molecules areseparately excited by absorption followed by a simultaneous energy transfer to an accep-tor molecule. It has been experimentally observed in a fluorescein-donor/stilbene-acceptorsystem, and a perturbative, quantum electrodynamics framework for the process was sub-sequently established . This, in turn, was used to computationally estimate the rate of2ooling, alongside those of competing relaxation pathways, for the fluorescein/stilbene sys-tem as well as a new hexabenzocoronene(HBC)/oligothiophene assembly. It was demon-strated that energy pooling can be much greater than other competing processes, and a setof design rules were laid out for the requisite properties of ideal donor/acceptor pairs. Veryrecent experimental efforts adopted rhodamine-6G and stilbene-420, which have many ofthese properties, to successfully elicit a relatively high pooling efficiency. A major obstacle to realizing efficient energy pooling is the requirement that two donormolecules be simultaneously excited while in close proximity to each other. This typicallycauses the rate of spontaneous donor emission to surpass the rate at which pairs are excited.The problem can be overcome by using light of sufficiently high intensity but demandingthis level of illumination makes it impractical for many of the applications envisioned.An intriguing and ultimately more practical approach is to extend the singlet lifetime ofthe donor molecules. This amounts to modifying the dominant pathway for de-excitationof the donor molecules. If that pathway is spontaneous emission (SE), then photonicallyconfined environments can be designed to effectively increase exciton lifetimes. In general,such Inhibited Spontaneous Emission (ISE) occurs when the emission energy of a moleculefalls within the photonic bandgap of a waveguide, cavity, or photonic crystal . Thishas been exploited to more easily measure the magnetic moment of electrons, create highquality single-photon sources for photonics, and increase the exciton diffusion length inorganic photovoltaic systems. It has also been used to extend the excited state lifetime ofRydberg atoms by a factor of twenty. The consideration of ISE in association with energypooling is new to the best of our knowledge.SE can be inhibited in photonically constrained settings (PCS) as a result of naturallyoccurring Stokes shifts, as illustrated in Figure 1. The donor molecules absorb photons thatare above the cutoff of the PCS, and the resulting exciton entangles with phonons. Thiscauses a Stokes shift that reduces the exciton energy to below the waveguide cutoff, andSE is inhibited. As will be computationally demonstrated, even a modest Stokes shift is3 inimumWaveguideEnergy t t D D’ D D’AA t D D’A t Figure 1:
Inhibited Spontaneous Emission via Stokes shift.
Energy level diagram for poolingmolecules encapsulated within a geometry that exhibits a photonic bandgap. Top Left:Photons are absorbed by donors (D, D’) at times t and t sufficiently close that excitonsexist on both over a finite time interval. Top Right: Energy pooling results in the excitation ofthe acceptor (A), a simultaneous, three-body process at time t . Lower: Acceptor emissionof a higher energy photon at time t . Solid black lines indicate relevant excitonic energylevels, blue lines reflect the addition of phonons, and the yellow/black line is a virtual state.sufficient to substantially change the SE rate.The result is an exciton lifetime that is functionally similar to triplet-triplet annihilation(TTA), an upconversion method involving two triplet excitons. This is significant becausethe primary advantage of TTA upconversion is the long lifetime of triplets which allows forefficient conversion even at low light intensity. The increased longevity of singlet statesafforded by ISE is not likely to exceed that which can be achieved via the forbidden tran-sitions of triplet states, but ISE-enhanced singlet pooling does offer an advantage overTTA upconversion. Triplets transfer energy through a Dexter process and so require smallseparations to achieve the requisite wave function overlap. ISE-enhanced singlets, though,can hop over longer distances via Förster Resonant Energy Transfer (FRET). The use of ISE4o improve the efficiency of such hops has been previously explored, where ISE was viewedas a means of modifying the local density of states. These FRET rates are unaffected by thepresence of the confining medium so long as the distance to the boundaries is larger thanthe separation between molecules. The present work computationally explores the potential of ISE to increase energy poolingefficiency, the acceptor energy produced per unit of absorbed donor energy. A set of kineticequations is developed that accounts for the primary competitors to energy pooling, mostimportantly spontaneous emission from the donor and FRET from the acceptor back to thedonor (back-FRET). Process rates are obtained from first principles analyses, and the entirekinetic model is embedded within an optimization routine. This allows molecule orientationsand spacings to be tailored for the highest possible efficiency. The analysis procedure isapplied to a well-studied donor/acceptor pair, and it is found that the pooling efficiency issubstantially improved by exploiting ISE.
Theory
The coupling between light and matter is assumed to be sufficiently weak that energy trans-fer processes can be considered within a perturbative Quantum Electrodynamics (QED)setting. The complete Hamiltonian can then be separated into independent light and mat-ter contributions, ˆ H , as well as a small light-matter coupling term, ˆ H . Treating excitonsas indivisible bosonic particles, the base Hamiltonian is thenˆ H = ˆ H ex + ˆ H light , ˆ H ex = X j ε j ˆ c † j ˆ c j , ˆ H light = X λ, k (cid:126) ck ˆ a ( λ, k ) † ˆ a ( λ, k ) . (1)The purely excitonic component, ˆ H ex , is in terms of the exciton annihilation operator, ˆ c j ,of material state, | j i ex , with bosonic commutation relations [ˆ c i , ˆ c † j ] − = δ ij . The excitonic5nergy of molecule, j , is ε j , and c is the speed of light in vacuum. The photon component,ˆ H light , is expressed in terms of the photon annihilation operator, ˆ a ( λ, k ) , for which the modesare parametrized by vector k and polarizations λ = 1 ,
2. It destroys a photon in orientationand mode ( λ, k ) and obeys the following bosonic commutation relations: [ˆ a ( λ, k ) , ˆ a ( γ, p ) ] − = (8 π Ω) − δ ( k − p ) δ λ,γ . (2)Here Ω is a normalization volume.The interaction term is defined within the dipole approximation as ˆ H = − ˆ µ · ˆE , where ˆ µ = e ˆ r is the electric dipole moment operator that acts upon the excitonic states, and ˆE isthe electric field operator that acts on the optical states.ISE relies on the existence of a photonic bandgap, so the environment must be confinedin at least two directions. In planar waveguides, for instance, the TM mode allows forEM-waves of any energy while the lowest modes TM or TE of rectangular waveguideshave a minimum energy related to its dimensions.Three types of confined settings were considered to determine the waveguide geometrythat delivers the greatest improvement in pooling efficiency. The first configuration is asquare cavity tuned to resonate with the absorption of the donor. This will minimize theenergy lost through thermalization while still causing a decrease in the photoluminescence(PL) rate of the donors due to ISE. It will be referred to as a Square-Donor waveguide.The second waveguide also has a square cross-section but is tuned so that the second-lowestenergy level of the cavity is resonant with the lowest acceptor emission, referred to as a
Square-Acceptor waveguide. It will maximize the Purcell effect for acceptor emission and,depending on the energy levels of the encapsulated material, may still allow ISE to beexhibited by the donor. The third waveguide has a rectangular cross-section with one sidetuned to donor absorption energy and the second to acceptor emission energy, a
Rectangular guide that is intended to elicit both a Purcell enhancement of the acceptor emission and ISE6 yz x R a D E x = E Z = E x = E y = b R D’ A
Figure 2: Geometry and relative positions of molecules in a photonically confined setting.of the donor. The free-space case is also considered for the sake of reference.The excitonic operator depends solely on the electronic properties of the molecules inthe assumed weak-coupling setting, while the electric field operator depends only on thewaveguide geometry. For free-space and rectangular cavities, this operator has the followingforms: ˆE free ( r , t ) = ı X λ, k s (cid:126) ck ε ˆ e ( λ, k ) e i k · r e ıckt ˆ a ( λ, k ) + H . c . (3)and ˆE cav ( r , t ) = ı X k s (cid:126) ck Ω ε k η k e ik x x e ıckt (cid:18)(cid:18) sin( k y y )sin( k z z )ˆ x + (4) ik x k y k η cos( k y y )sin( k z z )ˆ y − ik x k z k η sin( k y y )cos( k z z )ˆ z ! ˆ a T M, ( k ) + − ikk z k η cos( k y y )sin( k z z )ˆ y + ikk y k η sin( k y y )cos( k z z )ˆ z ! ˆ a T E, ( k ) ! + H . c . The operator depends on the position vector, r , and time, t . The k x and k y componentsof the wave vector are combined, k η = k y + k z = ( nπ/a ) + ( mπ/b ) , ε is the permittivity offree space, and ˆ e ( λ, k ) are the orthonormal polarization vectors such that ˆ k · ˆ e ( λ, k ) = 0.As shown in the Appendix, this Hamiltonian can be subjected to perturbation theoryto derive rate (Γ) expressions for absorption, photoluminescence (PL), internal conversion(IC), FRET, and energy pooling. Internal conversion was not calculated but was instead7stimated using experimental values of fluorescent quantum yield combined with computedrates of SE. Singlet-Singlet Annihilation (SSA) is an additional process of concern but canbe thought of as a special case of FRET in which one donor transfers energy to the other,creating an excited state above the first excited one that undergoes a subsequent rapid decayinto the first excited state. It is therefore implicitly accounted for in our approach.These rate relations can then be used to populate a kinetic model (Figure 3), a set ofcoupled ordinary differential equations for concentrations, to quantify the efficiency, η EP , ofenergy pooling relative to competing relaxation processes:Γ ( D ) Abs [ D S ] + Γ ( A → D ) F RET [ D S ][ A S ] + Γ ( D → D ) F RET [ D S ][ D S ] = (5)(Γ ( D ) P L + Γ ( D ) IC )[ D S ] + Γ ( D → D ) F RET [ D S ][ D S ] + Γ ( D → D ) SSA [ D S ][ D S ] + Γ ( D ,D → A ) EP [ D S ][ D S ][ A S ]Γ ( D ) Abs [ D S ] + Γ ( A → D ) F RET [ D S ][ A S ] + Γ ( D → D ) F RET [ D S ][ D S ] = (6)(Γ ( D ) P L + Γ ( D ) IC )[ D S ] + Γ ( D → D ) F RET [ D S ][ D S ] + Γ ( D → D ) SSA [ D S ][ D S ] + Γ ( D ,D → A ) EP [ D S ][ D S ][ A S ]Γ ( D ,D → A ) EP [ D S ][ D S ][ A S ] = (Γ ( A ) P L +Γ ( A ) IC )[ A S ]+Γ ( A → D ) F RET [ D S ][ A S ]+Γ ( A → D ) F RET [ D S ][ A S ] (7)1 = [ D S ] + [ D S ] = [ D S ] + [ D S ] = [ A S ] + [ A S ] (8) η EP = Γ ( A ) P L [ A S ] E A Γ ( D ) Abs [ D S ] E D + Γ ( D ) Abs [ D S ] E D . (9)Here [ ξ ζ ] represents the concentration of molecule ξ in state ζ . It was assumed thatthe relative concentration of donors to acceptors was 2:1, as one might expect from a 3-body process. The use of concentrations to form the kinetic model sacrifices informationabout local proximity in favor of statistical averages. While simplifying the problem, sucha coarse-grain model cannot capture effects associated with a non-uniform distribution ofmolecules. All concentrations are normalized such that there is only one of each molecule8 Donor Acceptor Ground States Excited States Abs PL/IC Abs EP SSA PL & Energy Collectionback-FRETPL/IC
Figure 3:
Kinetic Model.
Schematic of energy transfer pathways considered: Absorp-tion (Abs), Photoluminescence (PL), Internal Conversion (IC), Singlet-Singlet Annihilation(SSA), Förster Resonant Energy from acceptor back to a donor (back-FRET), and EnergyPooling (EP).per unit volume (Eq. 8). E D and E D are the excitation energies for donor one and donortwo, and E A is the emission energy of the acceptor.This kinetic model can be used to quantify the steady-state efficiency, η EP , here defined asthe energy emitted by the system per energy unit absorbed (Eq. 9). The system of equationsis too complicated to obtain a general analytic result, so a numerical approach was taken(vide infra) for a specific set of donor and acceptor molecules: fluorescein isothiocyanate(FITC) donors and a hexabenzocoronene (HBC) acceptor. Though not considered together,each molecule has been previously analyzed within a free-space setting with computationalpredictions for ground state properties as well as absorption and emission profiles foundto match experimental measurements. While the free-space pooling efficiency is too lowfor this pair to be technologically interesting, it is well-suited to demonstrate the potentialbenefit of ISE. This is because FITC has a fluorescence quantum yield of 0 . and donorswith a high quantum yield (or a low rate of IC) are promising because it implies that SE maybe a dominant relaxation pathway. The HBC acceptor also has features that serve to enhance9he effect of ISE—a dense excited state manifold and rigid structure. Such molecular stiffnessminimizes the energy loss through geometry reorganization, while the dense excited statemanifold creates more quantum pathways for energy pooling to occur, thus increasing itsrate. As we are concerned with the effect of ISE, we chose to neglect the IC of the acceptor.If its rate is not significantly greater than the SE rate of the acceptor, the efficiency of energypooling will scale linearly with the fluorescence quantum yield of the acceptor.Even prior to a physically accurate numerical implementation, it is possible to artificiallypopulate the kinetic equations and obtain a rudimentary sense of how ISE might influencepooling efficiency. To this end, the following rates were utilized: donor absorption of 10 kHz,donor IC of 10 MHz, acceptor emission of 100 MHz, FRET of 10 GHz, and energy poolingof 100 GHz. It was assumed that the acceptor both absorbs and emits at twice the energyat which the donor emits, E A = 2 (cid:16) E D − ∆ E Stokes (cid:17) . The reduction in radiative lifetimecomes directly from a prescribed Stokes shift, and the larger Stokes shift also means thatless energy is emitted by the acceptor. The increase in Stokes shift drops the donor emissionenergy further below the minimum cavity energy, thus producing ISE.The resulting trends are captured in Figure 4, where units are not shown because they areirrelevant in this qualitative analysis. The radiative lifetime was found by taking the inverseof the inhibited spontaneous emission rate which was varied by altering the Stokes shift ofthe donor. Therefore, increases in radiative lifetime were made indirectly by increasing theStokes shift. As is graphically clear, there is a point at which the energy losses from theStokes shift outweigh any increase in efficiency associated with an increase in donor lifetime.For the particular rates assumed, this happens for a Stokes shift of approximately one-thirdof the donor absorption energy. Several factors were identified that influence both the peakefficiency value and its position. The peak efficiency can be increased by either increasingthe SE rate of the acceptor or by decreasing the rate of back-FRET. These two processescompete, and the impact of the losses are quantified in the Results section that follows. Theposition of the peak can be shifted left (thereby reducing energy loss through the Stokes shift)10 adiative Lifetime E ff i c i e n c y o f E n e r g y P oo li n g τ Free SE τ IC τ optimal η max Higher efficiency as: acceptor PL rate increases back-FRET rate decreasesShorter radiative lifetime as: pooling rate increases donor PL rate increases Figure 4:
Illustrative relation between radiative lifetime and pooling efficiency.
Linear-logplot demonstrating the efficiency gains of energy pooling as the radiative lifetime is extendedusing ISE. The parameters used are identified in the text.by either increasing the rate of energy pooling or by increasing the rate of donor absorption.Increasing the rate of donor absorption causes an increase in excited donor concentration,and this shifts the peak to the left. Increasing the rate of energy pooling makes it moreefficient, also resulting in a leftward shift of the peak. Additionally, a reduction in thedonor emission line-width, γ D , while holding the Stokes shift constant, causes the radiativelifetime to decrease. The line-width was taken to be the full-width half-maximum (FWHM)of the donor emission peak. Narrowing the emission peak will cause a smaller fraction of theemission to fall above the photonic bandgap of the cavity.Using the artificially populated kinetic model, an optimal Stokes shift was determinedfor a prescribed set of rates, excitation energy and donor emission line-width. These rateswere allowed to increase or decrease within an order of magnitude, the excitation energywas allowed to range from 1 eV to 2 eV, and the donor emission line-width was varied overthe range of 60-140 meV. It was observed that the optimal Stokes shift tended to fall within20-30% of the donor excitation energy. These qualitative relationships and rough efficiencyestimates are useful in interpreting the numerical results of the specific calculations to follow.11 omputational Details First principles calculations were used to obtain estimates for the parameters of the kineticsmodel comprised of Equations 6 – 9. Geometry optimization and excited state analyses forHBC and FITC were carried out using the Q-Chem software package. The HBC acceptorwas modeled using a combination of Density Functional Theory (DFT) (ground state geom-etry) and Time-Domain Density Functional Theory (TD-DFT) (spectrum of excited singletstates). It is very rigid, so the Stokes shift was taken to be zero. A series of exchange-correlation functionals were considered, and the B3LYP functional was adopted becauseit predicted absorption properties closest to experimental spectra. The TD-DFT analysis ofHBC predicted a number of low-energy excited states that have no oscillator strength anddo not appear in the absorption or emission spectra. These states were deemed unphysicaland ignored.A similar approach was attempted for FITC, but because it has a combination of local ex-citations and intramolecular charge transfer excitations, no exchange-correlation functionalwas found that accurately captures the properties of the molecule. It was therefore mod-eled using Spin-Opposite Scaling Second Order Møller-Plesset (SOS-MP2) for the groundstate geometry, configuration interaction singles and doubles (CIS(D)) for excited stategeometry, and Spin-Opposite Scaling Configuration Interaction Singles and Doubles (SOS-CIS(D)) for the spectrum of excited singlet states. While more computationally expensivethan DFT/TD-DFT, this methodology was able to generate absorption and emission spectraconsistent with experimental data.The excited state calculations of FITC and HBC included the first 60 excited states ofeach molecule in both the ground state and excited state configurations. This was to ensurethat any excited state within a range of ∼ − Ha, and the self-consistent field (SCF) convergence criteria for the electronic wavefunction12
50 400 450 500 550 6000.00.20.40.60.81.0 N o r m a li z e d C r o ss S e c t i o n Wavelength (nm)
FITC Absorption Spectrum
450 500 550 600 650 700 7500.00.20.40.60.81.0 N o r m a li z e d C r o ss S e c t i o n FITC Emision Spectrum
300 350 400 4500.00.20.40.60.81.0 N o r m a li z e d C r o ss S e c t i o n HBC Absorption Spectrum
Wavelength (nm) Wavelength (nm)Wavelength (nm)
Figure 5:
Absorption Spectra.
Experimental (blue) and computed (red dashed) spectraof HBC and FITC.was set to difference ratios between successive steps of 10 − for HBC and 10 − for FITC.The geometry optimization was considered converged when two of the following are satisfied:total of force magnitudes < . < . < . for ease of comparison. A broadening of 121 meV was used forthe absorption of FITC and 124 meV for the emission of FITC and the absorption/emission ofHBC, empirically determined using the FWHM of the experimental data. These broadeningparameters were also implemented in the kinetic model. Results
The set of kinetics equations, populated with DFT and TD-DFT data, can be numericallysolved after specifying a specific orientation and spacing of the donor/acceptor triad. Thismodel was therefore embedded in an optimization routine which used maximum poolingefficiency for its objective function. The three molecules were treated as point dipoles andtheir relative orientation and separation distances were treated as free parameters. Each was13
BC AcceptorFITCDonorFITCDonor
SideFrontTop Perspectivetransitiondipole
Figure 6:
Optimal Geometry and Orientation of FITC and HBC in Free Space.
Blue arrowsindicate the direction of the lowest-energy transition dipole of the donors. The two-photonabsorption tensor of the acceptor has nonzero elements only in the plane of the HBC.assumed to be illuminated by the electric field found along the centerline of the waveguide(Figure 2). To be specific, the acceptor was positioned at (0,-a/2,b/2), the first donor at at(R,-a/2,b/2), and the second donor at (-R,-a/2,b/2).Molecule orientations and positions were optimized in all three cavity settings, as well asfree space, at an irradiance intensity of 1 MW/m . To give a physical correspondence, thisis approximately 1000 suns, the highest irradiance used in concentrated photovoltaics. Theoptimal free-space geometry is shown in Figure 6. Optimal geometries for the photonicallyconfined settings are visually indistinguishable with donor-acceptor separation distances 0.3nm greater than that obtained for free space. The two-photon absorption (TPA) tensor forHBC has terms only in the plane of the molecule and is nearly isotropic, so that rotation ofHBC within the plane had very little effect. In the optimal orientation, the FITC donors arediametrically opposed. Their emission transition dipoles are completely in the TPA-planeand are parallel to the axis separating the two donors.The rates and efficiencies of the processes in the free-space and cavity settings are sum-marized in Table 1. The optimal configurations within each PCS result in a larger separationbetween donor and acceptors, lowering the rate of energy pooling. However, pooling com-petes with donor relaxation through SE and/or IC, so the negative impact of the decreasein pooling rate is minimal. Instead, the increase in separation lowers the rate of SSA along14ith back-FRET. The overall effect of this is an increase in pooling efficiency.Among the three waveguide designs proposed, those with a square cross-section provideda nearly equal increase in efficiency. The Rectangular geometry gave the greatest increasein efficiency, benefiting from both Purcell enhancement of the acceptor emission and ISE ofthe donor. Table 1: Rates (Hz) and Efficiencies (%) of FITC-HBC System
Environment Γ ( D ) Abs Γ ( D ) P L Γ ( D ) IC Γ ( A ) P L Γ SSA Γ ( A → D ) F RET Γ ( D ,D → A ) EP η EP Free Space 4.9 E4 1.7 E8 1.2 E7 4.1 E8 4.1 E10 1.4 E11 1.0 E12 0.1Square-Donor 1.4 E4 7.6 E6 1.2 E7 1.3 E9 1.0 E10 3.7 E10 6.0 E10 0.8Square-Acceptor 7.3 E3 9.5 E6 1.2 E7 1.8 E9 1.4 E10 4.8 E10 1.1 E10 0.8Rectangular 1.1 E4 8.4 E6 1.2 E7 1.8 E9 1.1 E10 4.1 E10 7.6 E10 0.9With the optimal configuration fixed, the light intensity was subsequently varied as plot-ted in Figure 7. Not surprisingly, the pooling efficiency was found to increase at all lightintensities considered. This is because, as the intensity increases, the excited donor concen-tration rises allowing pooling to better compete against donor decay. However, the exciteddonor concentration eventually saturates, and further increases in intensity do not changethe efficiency.These optimal geometries, and an excitation intensity of 1 MW/m , were then used toanalyze the relative impact of competing relaxation pathways: donor IC, SSA, and back-FRET. The results are summarized in Table 2. Donor IC was decreased such that thefluorescence quantum yield would rise from 0.93 to 0.99. Along the same lines, the SSA andback-FRET rates were decreased by a factor of 100. The resulting pooling efficiencies areonly estimates because the geometry was kept fixed, but it is the trends that are importantin any case. The largest pooling efficiency gains, in decreasing order of importance, areobtained by decreasing back-FRET, using a PCS to create ISE, reducing the rate of IC, anddecreasing the rate of SSA.Damping the SSA pathway offers only a minimal increase in pooling efficiency for allfour environments. This is because the rate of SSA is significantly less than that for pooling15 -1 E ff i c i e n c y ( % ) -2 -3 -4 -1 I (W/cm ) Rectangular cavityDonor-tuned square cavityAcceptor-tuned square cavityFree space
ISE Effect
Figure 7:
Affect of ISE on Pooling Efficiency.
Energy pooling efficiency in free space (black),square-donor cavity (red/dark gray dashed), square-acceptor cavity (blue/light gray dashed),and rectangular cavity (green) for a range of 1-1000 suns.
Table 2: Pooling Efficiencies (%) of FITC-HBC with Other Loss MechanismsDampened
Free Space Square-Donor Square-Acceptor RectangularBaseline 0.1 0.8 0.8 0.9Low SSA 0.1 0.9 0.9 1.1Low IC 0.1 1.2 1.2 1.5Low back-FRET 21 30 29 30Low back-FRET and SSA 22 33 31 33Low back-FRET and IC 22 42 41 42Low back-FRET, IC and SSA 23 48 45 47in the optimized geometries. In other configurations, the losses through SSA may be moresignificant and its suppression therefore more important—e.g. arranging the three moleculesso that they are equidistant in a plane, or if the spacing between donor molecules is less.Increasing the fluorescence quantum yield causes the rate of donor IC to decrease, but theefficiency gains are once again very small in all settings. This is because IC is not relevantwhen radiative decay is fast and explains why the efficiency gains are highest in the PCSs.The analysis shows that the damping of back-FRET has a substantial change on poolingefficiency, increasing it by 21%, 29%, 28%, and 29% for free space, Square-Donor, Square-Acceptor, and Rectangular environments, respectively. This is because, for the FITC/HBC16ystem, back-FRET is much faster than SE from the acceptor. Even so, a PCS increases thepooling efficiency by approximately 9%.Blocking both SSA and back-FRET pathways increases the pooling efficiency to just over30% for both free-space and PCSs. These numbers are impressive, but the highest theoreticalpooling efficiency for the system is 69%. It is the low excited donor concentration that causesthe actual values to be much lower. This can be mitigated by either increasing the donorlifetime or by increasing the light intensity.Damping all three loss mechanisms would result in pooling efficiencies that are substan-tially affected by ISE. While the free-space efficiency is approximately 23%, the PCSs haveefficiencies in the range of 45%—over 20% higher. Stripping out all loss mechanisms but SEshow how useful it can be to create a PCS for pooling.For the FITC/HBC model system, the greatest loss mechanism is from back-FRET be-cause it is significantly faster than acceptor PL as is clear from the following rate ratios:300 (free space), 29 (Square-Donor), 27 (Square-Acceptor) and 23 (Rectangular). This canbe mitigated by choosing a donor that does not have excited states in the energy range ofthe emission of the acceptor (Figure 8). For instance, artificially removing excited donorstates within a band of width equal to the acceptor broadening, γ A =120 meV, decreasesthe back-FRET-to-PL ratios to 13, 1, 1, and 1 while increasing the pooling efficiency to2%, 13%, 13% and 14% for the free space, Square-Donor, Square-Acceptor, and Rectangularenvironments, respectively. Removing excited states within a band of width 3 γ A decreasesthe back-FRET-to-PL ratios to 3, 0.3, 0.3, and 0.2 and increases the pooling efficiency to5%, 22%, 22% and, 24% for the free space, Square-Donor, Square-Acceptor, and Rectangularcavity environments, respectively.While the Stokes shift alone can be used to shift the emission to be within the photonicbandgap, it is also possible to tune the cavity so that the lowest energy mode of the cavityis resonant with a higher excited state on the donor. IC will rapidly lower excited donors tothe first excited state. Such a large shift in energy puts the donor deeper within the photonic17 .53.03.54.0 E n e r g y L e v e l s ( e V ) FITC Donor HBC Acceptor st excited state1 st excited state γ a γ a Figure 8:
Back-FRET Suppression.
Excited state energy levels of FITC donor (left, red) andHBC acceptor (right, blue). The shaded green bands highlight FITC states within γ A and3 γ A of HBC emission from its first excited state. Removal of these donors levels suppressesback-FRET.bandgap and causes ISE to be stronger. Of course, this comes at the expense of a largerloss of energy due to the initial IC, but the approach may still increase pooling efficiency.In the case of FITC, for instance, absorption to the second excited state (2.87 eV) actuallygives a higher pooling efficiency, 2.0% instead of 0.9%. This is because the Stokes shiftfrom the first excited state is only 0.21 eV, an 8.4% shift in energy that is much lower thanthe 20—30% ranged deemed to be optimal. This counterintuitive approach to optimizingpooling efficiency may be easily exploited for donors with the proper spacing between firstand second excited states. Discussion
Molecular assemblies composed of properly matched donor/acceptor pairs has been previ-ously predicted to undergo measurable energy pooling upconversion. That work resultedin a set of
Molecular Design Criteria intended to improve the rate of pooling without con-sidering its overall efficiency within a complete kinetic model: • acceptor with large Two-Photon Absorption (TPA) cross section;18 minimal spectral overlap of donor absorption and acceptor emission; • minimal spectral overlap of donor emission and excited donor absorption; • maximal spectral overlap of dual-donor emission and acceptor absorption, located asclose as possible to the first excited state of the acceptor.A synergistic design strategy, explored in the current work, is to reduce the rate of com-petitive relaxation processes so as to increase the overall efficiency of pooling. In particular,SE was targeted and encapsulation within a photonically confined geometry was consideredas a means of inhibiting this pathway. A donor Stokes shift subsequent to excitation re-duces its energy to below the threshold for waveguide absorption. To quantify the impact ofinhibiting SE, a kinetic model was developed and populated with first principles rate databased on perturbative quantum electrodynamics.When populated with generic but typical rate values, the kinetic model showed that SEwas indeed inhibited by increasing the donor Stokes shift of a hypothetical donor and thatpooling efficiency will rise until the Stokes shift was about one-third the donor absorptionenergy. Beyond this level of relaxation, the energy losses from the Stokes shift overwhelmany advantage gained from reducing the rate of excited donor decay. Increasing the rate ofacceptor emission or decreasing the rate of back-FRET were shown to increase the maximumtheoretical efficiency.First principles data was then used to populate the kinetic model for a previously studieddonor/acceptor pair—FITC donors and an HBC acceptor. Surprisingly, it was found thatmaximum efficiency is obtained by actually increasing the donor/acceptor separation overthe optimal value for free space. This is because the losses associated with back-FRETreduce with increased separation, and this results in an overall efficiency gain even thoughthe pooling rate is lower. The same would not be true for triplet upconversion which requiresa high wavefunction overlap.It was found that any of the three cavity designs considered resulted in a significant19ncrease in energy pooling efficiency. However, a rectangular waveguide design producedthe best results. One side was tuned for high donor absorption and ISE so that its lengthwas L = π (cid:126) c/E D . The second side was tuned for enhanced acceptor emission through thePurcell effect with a length of L = π (cid:126) c/E A . Because the second side is always smaller thanthe first, it will also contribute to ISE.The photonically confined geometries considered in this work were very simple. Settingssuch as nanopits or nanowires are also possible so long as the dimensions are properlytuned and the confinement is in at least two Cartesian directions. Another possibility is toplace the molecules within a photonic crystals. Nanogratings, which are only closed on threesides, do not exhibit the two-dimensional confinement required for ISE.While ISE was found to improve pooling efficiency by roughly an order of magnitude,the efficiency was still less than 1% for the FITC/HBC system studied. Subsequent anal-ysis showed that this is because the rate of back-FRET is high due to substantial donor-absorption/acceptor-emission overlap. An absence of excited donor states within the effectiveemission range of the lowest excited acceptor state puts the rate of back-FRET on the sameorder as that of the desired acceptor emission. Within this setting, a free-space poolingefficiency of 23% is shown to have an efficiency of 47% in a rectangular cavity.Utilizing molecules with minimal donor-emission/excited-donor-absorption overlap canfurther improve pooling efficiency by reducing the rate of SSA. However, this is of minorimportance provided the donors are positioned on opposite sides of the acceptor as in thecurrent study. The matching of donor-emission/acceptor-absorption energy levels is requiredfor energy pooling to occur and is also important for minimizing the efficiency losses associ-ated with acceptor relaxation to its lowest excited state. The FITC/HBC system studiedhere does not meet this requirement as 25% of the energy transferred to the acceptor is lostthrough acceptor internal conversion.In summary, this computational study has shown that two additional Molecular DesignCriteria should be added: 20 high emission quantum yield of donor; • a donor absorption/emission difference, on the order of 20-30% the donor absorptionenergy, accomplished through a combination of Stokes shift and/or relaxation from ahigher energy exciton state.The first criterion reflects the fact that there are both radiative and nonradiative relax-ation pathways, but ISE only targets the former. It is therefore necessary to design a systemwhere the latter is relatively small. The second ensures that donor relaxation reduces theemission energy sufficiently below the photonic cutoff to effectively inhibit SE. The shiftrange was estimated from a series of artificially populated kinetic models.The combined designed criteria suggest that a pooling system composed of Rhodamine101 donors and 1,4-Diphenylbutadiene acceptors is worthy of experimental consideration.Rhodamine 101 has a fluorescence quantum yield that is near unity and a weak absorptionstrength at twice the donor emission energy. The first absorption peak of 1,4-Diphenylbutadieneoccurs at nearly twice Rhodamine 101 emission energy, it has a small Stokes shift and ahigher quantum yield than similar molecules. A recent experimental study of similar butnot ideal pairing, Rhodamine 6G (R6G) and Stilbene-420 (S420), reported efficiencies of3.1-36% in free space for light intensities of between 42000 and 720000 suns. Our designanalysis suggests that encapsulating Rhodamine 101 and 1,4-Diphenylbutadiene in a pho-tonically constrained setting will result in a significant enhancement in pooling efficiency atlower light intensities.On a device level, a relatively high concentration of acceptors to donors can be used toallow excited acceptors to transfer their energy to other acceptors in regions where thereare no donors, so that energy loss through back-FRET is lessened. This is likely to be asignificant factor in the high efficiency found in the R6G/S420 system with a blend ratioof 1:40. However, our concentration-based kinetic model cannot account for local densityon donors relative to acceptors and so cannot accurately predict the effect of modifyingthe donor to acceptor ratio. We can posit a refinement, though, in which a solution of21onded donor-acceptor-donor (D-A-D) moieties are blended a solution of acceptors. Whendeposited as a film, the D-A-D molecules would perform energy pooling with upconvertedenergy efficiently transferred away to unbonded acceptors. This is another means of reducingback-FRET.
Acknowledgements
We are pleased to be able to acknowledge useful discussions with David Andrews on cavityquantum electrodynamics. All computations were carried out using the High PerformanceComputing facilities at the Colorado School of Mines.22
PPENDIX: Derivation of Relaxation Rates
Several relaxation rate expressions are used in our kinetic model. In the free-space setting,these are fairly standard but are listed here for the sake of completeness. Their cavitycounterparts, on the other hand, are subsequently derived.The rate of PL in free space is Γ
P L,F ree = (cid:12)(cid:12)(cid:12) µ ( a,b ) (cid:12)(cid:12)(cid:12) k π (cid:126) ε , (10)where µ ( b,a ) is the transition dipole moment from state a to state b, k is the magnitude ofthe wave-vector of the photon being absorbed, ε is the permittivity of the material.The rate of absorption in free space isΓ Abs,F ree = N X n =1 πI ( k ) (cid:12)(cid:12)(cid:12) µ ( n, (cid:12)(cid:12)(cid:12) (cid:126) cε ρ f ( E n − (cid:126) ck ) , (11)where the sum over n is over all possible excited states, I ( k ) is the irradiance of the light atthe wave-vector k, ρ f is the density of final states, and E n is the energy of the n th excitedstate. The density of the excitonic states was taken to be a Gaussian distribution: ρ f (∆ E ) = 1 √ πγ e − (∆ E )22 γ , (12)where γ is a broadening term dominated by phononic effects. The value is empirically chosenfor each molecular system based on the broadening observed in experimental absorption andemission spectra.The cavity counterparts to these two rates are less standard. The PL rate associated23ith a rectangular cavity is found to be:Γ P L,Cavity = 2 c A ε X k y ,k z (cid:18)(cid:18)(cid:12)(cid:12)(cid:12) µ (0 , x (cid:12)(cid:12)(cid:12) sin( k y y ) sin( k z z ) k η + (13)( (cid:12)(cid:12)(cid:12) µ (0 , y (cid:12)(cid:12)(cid:12) cos( k y y )sin( k z z ) k z − (cid:12)(cid:12)(cid:12) µ (0 , z (cid:12)(cid:12)(cid:12) sin( k y y )cos( k z z ) k y ) (cid:19) Z ∞−∞ ρ f ( (cid:126) ck − E ) q k x + k η dk x + (cid:18)(cid:12)(cid:12)(cid:12) µ (0 , y (cid:12)(cid:12)(cid:12) cos( k y y ) sin( k z z ) + (cid:12)(cid:12)(cid:12) µ (0 , z (cid:12)(cid:12)(cid:12) sin( k y y ) cos( k z z ) ) (cid:19) Z ∞−∞ k x ρ f ( (cid:126) ck − E ) q k x + k η dk x , where A is the cross sectional area of the cavity, and k η = k y + k z .Following along the same lines, the rate of absorption in a rectangular cavity is:Γ Abs,Cavity = N X n =1 πI ( k ) (cid:126) ck ε (cid:18)(cid:12)(cid:12)(cid:12) µ ( n, x (cid:12)(cid:12)(cid:12) sin( k y y ) sin( k z z ) k η + (cid:16) µ ( n, y cos( k y y )sin( k z z ) (14)( k x k y − kk z ) + µ ( n, z sin( k y y )cos( k z z )( k x k z − kk y ) (cid:17) (cid:19) ρ f ( E n − (cid:126) ck )The rates of FRET and Energy Pooling have the same form in both free-space and cavitysettings. The difference lies in the form of the dipole-dipole coupling tensor, V ij , but we haverecently shown that it is reasonable to assume its free-space form holds in both settings: V ij ( k, R ) = − e ıkR πεR (cid:16) (1 − ıkR )( δ ij − R i ˆ R j ) + k R ( δ ij − ˆ R i ˆ R j ) (cid:17) . (15)The FRET expression is then Γ ( A → B ) F RET = N X n =1 π (cid:126) (cid:12)(cid:12)(cid:12) µ B ( n, i V ij ( k ( A ) , R AB ) µ A (0 , j (cid:12)(cid:12)(cid:12) ρ f ( E Bn − E A ) , (16)where µ X ( b,a ) is the transition dipole moment of molecule X from state a to state b, E Xn is theenergy of the n th excited state of molecule X, and R AB is the displacement vector pointingfrom molecule A to molecule B. 24ikewise, the energy pooling rate is Γ ( D ,D → A ) EP = 2 π (cid:126) N X n =1 (cid:12)(cid:12)(cid:12) µ D (0 , i V ij ( k ( D ) , R D A ) α A ( n, jk ( k ( D ) , k ( D ) ) V kl ( k ( D ) , R D A ) µ D (0 , l + (17) µ D (0 , i V ij ( k ( D ) , R D D ) α D (0 , jk ( k ( D ) , − k ( D ) − k ( D ) ) V kl ( k ( D ) + k ( D ) , R D A ) µ A ( n, l + µ D (0 , i V ij ( k ( D ) , R D D ) α D (0 , jk ( k ( D ) , − k ( D ) − k ( D ) ) V kl ( k ( D ) + k ( D ) , R D A ) µ A ( n, l (cid:12)(cid:12)(cid:12) ρ f ( E An − E D − E D )where α X ( b,a ) jk is the two-photon transition tensor of molecule X from state a to state b: α ξ ( fi ) jk ( k , k ) = X ζ µ ξ ( fζ ) k µ ξ ( ζi ) j E ξζ − E ξi − (cid:126) ck + ıγ + µ ξ ( fζ ) j µ ξ ( ζi ) k E ξζ − E ξi − (cid:126) ck + ıγ (18)The rate of absorption in a rectangular waveguide can be derived using Fermi’s GoldenRule: Γ Abs,Cav = 2 π (cid:126) (cid:12)(cid:12)(cid:12) h f | ˆ H | i i (cid:12)(cid:12)(cid:12) ρ f ( E f − E i ) = 2 π (cid:126) (cid:12)(cid:12)(cid:12) h n ; 0 | ˆ µ · ˆE | k i (cid:12)(cid:12)(cid:12) ρ f ( E n − (cid:126) ck ) , (19)where for | A ; B i A represents the electronic state, and B represents the photonic state.Applying the electric dipole operator ˆ µ and the electric field operator ˆE to this expressiongives:Γ Abs,Cav = 2 π (cid:126) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)s (cid:126) ck Ω ε k y + k z ) k N γ ( µ x sin( k y y )sin( k z z )+ (20) µ y ı ( k x k y − kk z ) k y + k z cos( k y y )sin( k z z ) + µ z ı ( k x k z − kk y ) k y + k z sin( k y y )cos( k z z ) !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ρ f ( E n − (cid:126) ck ) , where N γ is the photon mode occupation number. An expansion of the terms results in the25ollowing rate for the rate of absorption within a rectangular cavity:Γ Abs,Cav = (cid:126) c kN γ Ω 4 π ( k y + k z ) (cid:126) c k ε (cid:12)(cid:12)(cid:12)(cid:12)(cid:18) µ x sin( k y y )sin( k z z ) + (21) µ y ı ( k x k y − kk z ) k y + k z cos( k y y )sin( k z z ) + µ z ı ( k x k z − kk y ) k y + k z sin( k y y )cos( k z z ) !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ρ f ( E n − (cid:126) ck )The first term on the right hand side is equivalent to the light intensity for a particularwave-vector.Γ Abs,Cav = I ( k ) 4 π ( k y + k z ) (cid:126) c k ε (cid:12)(cid:12)(cid:12)(cid:12)(cid:18) µ x sin( k y y )sin( k z z ) + (22) µ y ı ( k x k y − kk z ) k y + k z cos( k y y )sin( k z z ) + µ z ı ( k x k z − kk y ) k y + k z sin( k y y )cos( k z z ) !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ρ f ( E n − (cid:126) ck ) . A similar approach is used to derive the rate of spontaneous emission in a rectangularwaveguide. The initial form is similar in form to Eq. 20:Γ
P L,Cav = 2 π (cid:126) X k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)s (cid:126) ck Ω ε k y + k z ) k ( µ x sin( k y y )sin( k z z )+ (23) µ y ı ( k x k y − kk z ) k y + k z cos( k y y )sin( k z z ) + µ z ı ( k x k z − kk y ) k y + k z sin( k y y )cos( k z z ) !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ρ f ( E n − (cid:126) ck )Expanding this expression, and write the k x component of the sum as an integral givesΓ P L,Cav = 4 πc Ω ε X k y ,k z L x π Z ∞ dk x k y + k z k | ( µ x sin( k y y )sin( k z z )+ (24) µ y ı ( k x k y − kk z ) k y + k z cos( k y y )sin( k z z ) + µ z ı ( k x k z − kk y ) k y + k z sin( k y y )cos( k z z ) !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ρ f ( E n − (cid:126) ck ) . The normalization volume Ω and the normalization length in the x-direction L x are simplifiedto the cross-sectional area of the waveguide A . The result is that the rate of PL in the26ectangular cavity isΓ P L,Cav = 2 cAε X k y ,k z Z ∞ dk x k y + k z k | ( µ x sin(k y y)sin( k z z )+ (25) µ y ı ( k x k y − kk z ) k y + k z cos( k y y )sin( k z z ) + µ z ı ( k x k z − kk y ) k y + k z sin( k y y )cos( k z z ) !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ρ f ( E n − (cid:126) ck )From here the density of states Eq. 12 is substituted and the integral is computed numeri-cally. 27 eferences (1) Ang, L. Y.; Lim, M. E.; Ong, L. C.; Zhang, Y. Applications of upconversion nanopar-ticles in imaging, detection and therapy. Nanomed. Nanotech. Biol. Med. , ,1273–88.(2) Chatterjee, D. K.; Gnanasammandhan, M. K.; Zhang, Y. Small upconverting fluores-cent nanoparticles for niomedical applications. Small , , 2781–95.(3) Chen, G.; Qiu, H.; Prasad, P. N.; Chen, X. Upconversion nanoparticles: Design,nanochemistry, and applications in Theranostics. Chem. Rev. , , 5161–5214.(4) Dai, Y.; Xiao, H.; Liu, J.; Yuan, Q.; Ma, P.; Yang, D.; Li, C.; Cheng, Z.; Hou, Z.;Yang, P. et al. In vivo multimodality imaging and cancer therapy by near-infraredlight-triggered trans-Platinum pro-drug-conjugated upconverison nanoparticles. J. Am.Chem. Soc. , , 18920–18929.(5) Xie, X.; Liu, X. Photonics: Upconversion goes broadband. Nat. Materials , ,842–843.(6) Zou, W.; Visser, C.; Maduro, J. A.; Pshenichnikov, M. S.; Hummelen, J. C. Broadbanddye-sensitized upconversion of near-infrared light. Nat. Photonics , , 560–564.(7) Trupke, T.; Shalav, A.; Richards, B.; Würfel, P.; Green, M. Efficiency enhancement ofsolar cells by luminescent up-conversion of Sunlight. Sol. Energ. Mat. Sol. Cells , , 3327–3338.(8) Wondraczek, L.; Batentschuk, M.; Schmidt, M. A.; Borchardt, R.; Scheiner, S.; See-mann, B.; Schweizer, P.; Brabec, C. J. Solar spectral conversion for improving thephotosynthetic activity in algae reactors. Nat. Commun. , .(9) Nickoleit, M.; Uhl, A.; Bendig, J. Non-linear simultaneous two-photon excitation energytransfer in the wrong direction. Laser Chemistry , , 161/174.2810) Jenkins, R. D.; Andrews, D. L. Three-center systems for energy pooling: Quantumelectrodynamical theory. J. Phys. Chem. A , , 10834 – 10842.(11) Jenkins, R. D.; Andrews, D. L. Twin-donor systems for resonance energy transfer. Chem. Phys. Lett. , , 235 – 240.(12) LaCount, M. D.; Weingarten, D.; Hu, N.; Shaheen, S. E.; van de Lagemaat, J.; Rum-bles, G.; Walba, D. M.; Lusk, M. T. Energy pooling upconversion in organic molecularsystems. J. Phys. Chem. A , , 4009–4016.(13) Weingarten, D. H.; LaCount, M. D.; van de lagemaat, J.; Rumbles, G.; Lusk, M. T.;Shaheen, S. E. Experimental demonstration of photon upconversion via cooperativeenergy pooling. Nat. Commun. , .(14) Yablonovitch, E. Inhibited spontaneous emission in solid-state physics and electronics. Phys. Rev. Lett. , , 2059–2062.(15) Flores-Hidalgo, G.; Malbouisson, A. P. C.; Milla, Y. W. Stability of excited atoms insmall cavities. Phys. Rev. A , , 063414.(16) Hanneke, D.; Fogwell Hoogerheide, S.; Gabrielse, G. Cavity control of a single-electronquantum cyclotron: Measuring the electron magnetic moment. Phys. Rev. A , ,052122.(17) Lund-Hansen, T.; Stobbe, S.; Julsgaard, B.; Thyrrestrup, H.; Sünner, T.; Kamp, M.;Forchel, A.; Lodahl, P. Experimental realization of highly efficient broadband couplingof single quantum dots to a photonic crystal waveguide. Phys. Rev. Lett. , ,113903.(18) Kozyreff, G.; Urbanek, D. C.; Vuong, L.; Silleras, O. N.; Martorell, J. Microcavityeffects on the generation, fluorescence, and diffusion of excitons in organic solar cells. Opt. Express , , A336–A354. 2919) Kleppner, D. Inhibited spontaneous emission. Phys. Rev. Lett. , , 233–236.(20) Zang, X.; Lusk, M. T. Designing small silicon quantum dots with low reorganizationenergy. Phys. Rev. B , , 035426.(21) Baluschev, S.; Miteva, T.; Yakutkin, V.; Nelles, G.; Yasuda, A.; Wegner, G. Up-conversion fluorescence: Noncoherent excitation by sunlight. Phys. Rev. Lett. , , 143903.(22) Ogawa, T.; Yanai, N.; Monguzzi, A.; Kimizuka, N. Highly efficient photon upconver-sion in self-assembled light-harvesting Molecular Systems. Scientific Reports , ,10882:1–9.(23) Hulet, R. G.; Hilfer, E. S.; Kleppner, D. Inhibited spontaneous emission by a Rydbergatom. Phys. Rev. Lett. , , 2137–2140.(24) Blum, C.; Zijlstra, N.; Lagendijk, A.; Wubs, M.; Mosk, A. P.; Subramaniam, V.;Vos, W. L. Nanophotonic control of the Förster resonance energy transfer efficiency. Phys. Rev. Lett. , , 203601.(25) LaCount, M. D.; Lusk, M. T. Electric dipole coupling in optical cavities and its impli-cations for energy transfer, up-conversion, and pooling. Phys. Rev. A , , 063811.(26) Andrews, D. L.; Bradshaw, D. S. Optically nonlinear energy transfer in light-harvestingdendrimers. J. Chem. Phys. , , 2445–2454.(27) Andrews, D. L.; Coles, M. M.; Williams, M. D.; Bradshaw, D. S. Expanded horizons forgenerating and exploring optical angular momentum in vortex structures. Proc. SPIE , , 88130Y–88130Y–13.(28) Sj¨oback, R.; Jan, N.; Kubista, M. Absorption and fluorescence properties of fluorescein. Spectrochimica Acta A , , L7–L21.3029) Shao, Y.; Gan, Z.; Epifanovsky, E.; Gilbert, A. T.; Wormit, M.; Kussmann, J.;Lange, A. W.; Behn, A.; Deng, J.; Feng, X. et al. Advances in molecular quantumchemistry contained in the Q-Chem 4 program package. Molecular Physics , ,184–215.(30) Kastler, M.; Pisula, W.; Wasserfallen, D.; Pakula, T.; MÃijllen, K. Influenceof Alkyl Substituents on the Solution- and Surface-Organization of Hexa-peri-hexabenzocoronenes. Journal of the American Chemical Society , , 4286–4296.(31) Becke, A. D. Density-functional thermochemistry. III. The role of exact exchange. TheJournal of Chemical Physics , , 5648–5652.(32) Jung, Y.; Lochan, R. C.; Dutoi, A. D.; Head-Gordon, M. Scaled opposite-spin secondorder Møller-Plesset correlation energy: An economical electronic structure method. The Journal of Chemical Physics , , 9793–9802.(33) Head-Gordon, M.; Rico, R. J.; Oumi, M.; Lee, T. J. A doubles correction to electronicexcited states from configuration interaction in the space of single substitutions. Chem.Phys. Lett. , , 21–29.(34) Head-Gordon, M.; Maurice, D.; Oumi, M. A perturbative correction to restricted openshell configuration interaction with single substitutions for excited states of radicals. Chemical Physics Letters , , 114–121.(35) Rhee, Y. M.; Head-Gordon, M. Scaled second-order perturbation corrections to con-figuration interaction singles:ĂĽ Efficient and reliable excitation energy methods. TheJournal of Physical Chemistry A , , 5314–5326.(36) Tiago, M. L.; Chelikowsky, J. R. Optical excitations in organic molecules, clusters, anddefects studied by first-principles Green’s function methods. Phys. Rev. B , ,205334. 3137) Hu, N.; Shao, R.; Shen, Y.; Chen, D.; Clark, N. A.; Walba, D. M. An electric-field-responsive discotic liquid-crystalline Hexa-peri-Hexabenzocoronene/Oligothiophene hy-brid. Adv. Mater. , , 2066–2071.(38) Du, H.; Fuh, R.-C. A.; Li, J.; Corkan, L. A.; Lindsey, J. S. PhotochemCAD: Acomputer-aided design and research tool in photochemistry. Photochemistry and Pho-tobiology , , 141–142.(39) Dixon, J. M.; Taniguchi, M.; Lindsey, J. S. PhotochemCAD 2: A refined program withaccompanying spectral databases for photochemical calculations. Photochemistry andPhotobiology , , 212–213.(40) Prahl, S. Fluorescein. http://omlc.org/spectra/PhotochemCAD/html/010.html .(41) Bleuse, J.; Claudon, J.; Creasey, M.; Malik, N. S.; Gérard, J.-M.; Maksymov, I.; Hugo-nin, J.-P.; Lalanne, P. Inhibition, enhancement, and control of spontaneous emission inphotonic nanowires. Phys. Rev. Lett. , , 103601.(42) Magde, D.; Rojas, G. E.; Seybold, P. G. Solvent dependence of the fluorescence lifetimesof Xanthene dyes. Photochemistry and Photobiology , , 737–744.(43) Allen, M. T.; Miola, L.; Whitten, D. G. Host-guest interactions: a fluorescence investi-gation of the solubilization of diphenylpolyene solute molecules in lipid bilayers. Journalof the American Chemical Society , , 3198–3206.(44) Jenkins, R. D.; Daniels, G. J.; Andrews, D. L. Quantum pathways for resonance energytransfer. The Journal of Chemical Physics , , 11442–11448.(45) Ford, J. S.; Andrews, D. L. One- and two-photon absorption in solution: The effects ofa passive auxiliary beam. The Journal of Chemical Physics ,141