excitation transfer induced spectral diffusion and the

THE JOURNAL OF CHEMICAL PHYSICS 137, 064109 (2012)

Excitation transfer induced spectral diffusion and the influenceof structural spectral diffusion

Daniel E. Rosenfeld and Michael D. Fayera)

Department of Chemistry, Stanford University, Stanford, California 94305, USA

(Received 27 May 2012; accepted 23 July 2012; published online 10 August 2012)

The theory of vibrational excitation transfer, which causes spectral diffusion and is also influencedby structural spectral diffusion, is developed and applied to systems consisting of vibrational chro-mophores. Excitation transfer induced spectral diffusion is the time-dependent change in vibrationalfrequency induced by an excitation on an initially excited molecule jumping to other molecules thathave different vibrational frequencies within the inhomogeneously broadened vibrational absorp-tion line. The excitation transfer process is modeled as Förster resonant transfer, which depends onthe overlap of the homogeneous spectra of the donating and accepting vibrational chromophores.Because the absorption line is inhomogeneously broadened, two molecules in close proximity canhave overlaps of their homogeneous lines that range from substantial to very little. In the absenceof structural dynamics, the overlap of the homogeneous lines of the donating and accepting vibra-tional chromophores would be fixed. However, dynamics of the medium that contains the vibrationalchromophores, e.g., a liquid solvent or a surrounding protein, produce spectral diffusion. Spectraldiffusion causes the position of a molecule’s homogeneous line within the inhomogeneous spectrumto change with time. Therefore, the overlap of donating and accepting molecules’ homogeneous linesis time dependent, which must be taken into account in the excitation transfer theory. The excitationtransfer problem is solved for inhomogeneous lines with fluctuating homogeneous line frequencies.The method allows the simultaneous treatment of both excitation transfer induced spectral diffusionand structural fluctuation induced spectral diffusion. It is found that the excitation transfer processis enhanced by the stochastic fluctuations in frequencies. It is shown how a measurement of spec-tral diffusion can be separated into the two types of spectral diffusion, which permits the structuralspectral diffusion to be determined in the presence of excitation transfer spectral diffusion. Vari-ous approximations and computational methodologies are explored. © 2012 American Institute ofPhysics. [http://dx.doi.org/10.1063/1.4742762]

I. INTRODUCTION

The transfer of an electronic excitation among moleculesin dense media has been studied for over two thirds ofa century.1–4 Using the theoretical framework devised byFörster, electronic excitation transfer has been analyzedin polymer systems,5–7 micelles,8 solutions,9–13 mixedcrystals,14 two-dimensional systems,15, 16 and biologicalsystems.17–21 Recently, there has been considerable interestin developing a firm theoretical understanding of ultrafasttime-dependent vibrational spectroscopy of dense media.The systems studied include proteins and other biolog-ical molecules,1, 17, 20, 21 water (and isotopically labeledwater),22, 23 clusters of ions in solution,24, 25 phospholipidbilayers,26 and most recently self-assembled molecular mono-layers of catalytically active molecules.27 In particular, 2D IRvibrational echo spectroscopy is employed to measure spec-tral diffusion. Spectral diffusion is the time evolution of thevibrational frequencies of molecular vibrational oscillatorswithin an inhomogeneously broadened absorption line.28–30

The interest in spectral diffusion is that it is caused by struc-tural evolution of the system. Therefore, the time dependence

a)[email protected].

of spectral diffusion reports on the time dependence of struc-tural changes of a system such as a liquid,30 or a protein.31

In addition to structural changes, there can be anothersource of spectral diffusion. If the vibrational chromophoresare close to each other, vibrational excitation transfer (ET)can occur.32–34 In general, two molecules that are physicallynear each other will not have the same vibrational frequencybecause they will have different local environments. Whenan excitation jumps from an initially excited molecule to anearby molecule, the frequency of the excitation will change.Therefore, excitation transfer is a source of spectral diffusion.To avoid this source of spectral diffusion in ultrafast IR stud-ies of water dynamics, the OD stretch of dilute HOD in H2Ois studied.35 Because the HODs are dilute, excitation transferis eliminated. In pure H2O, excitation transfer in itself is ofinterest, and it dominates spectral diffusion.23, 36 However, inother systems excitation transfer cannot be avoided,27 but thesource of spectral diffusion that is of interest is the structuralspectral diffusion (SSD). Therefore, it is necessary to be ableto accurately calculate the excitation transfer induced spectraldiffusion so that it can be separated from the structural spec-tral diffusion. As shown in detail here, the problem is complexbecause the structural spectral diffusion has a strong influenceon the excitation transfer induced spectral diffusion.

0021-9606/2012/137(6)/064109/18/$30.00 © 2012 American Institute of Physics137, 064109-1

http://dx.doi.org/10.1063/1.4742762

http://dx.doi.org/10.1063/1.4742762

http://dx.doi.org/10.1063/1.4742762

mailto: [email protected]

064109-2 D. E. Rosenfeld and M. D. Fayer J. Chem. Phys. 137, 064109 (2012)

In addition, the Förster theory of excitation transfer re-quires knowledge of the overlap of the homogeneous spec-tra of the donating and accepting molecules.4, 37, 38 For elec-tronic excitations, it is assumed that the spectroscopic linesare homogeneously broadened (no inhomogeneous broaden-ing). Electronic excitations generally have a large Stokes shiftbetween the absorption and fluorescence spectra. The neces-sary overlap is the overlap of the absorption and fluorescencespectra.4, 37 Vibrational absorption lines usually have substan-tial inhomogeneous broadening. The homogeneous spectra ofthe donor and acceptor are buried under the inhomogeneouslines. (Here we use donor to be the initially excited moleculeand acceptor to be the molecule that receives the excitationalthough the two molecules are chemically identical.) There-fore, it is necessary to perform a 2D IR vibrational echo ex-periment to determine the homogeneous linewidth,39, 40 whichis a necessary input for the Förster theory and the theoreticalmethod developed below.

A preliminary version of the theory, which is developedfully here, was used to interpret 2D IR vibrational echo spec-troscopic data for samples that consisted of approximately amonolayer of molecules bound to a surface.27 In these sys-tems, the proximity of chromophores indicated that excitationtransfer plays a significant role in determining the observedvibrational spectral diffusion, and therefore excitation trans-fer needed to be treated properly to obtain the underlyingmolecular structural dynamics. As mentioned above, in vi-brational systems, spectral lines are inhomogeneously broad-ened, which precludes the use of standard Förster type results,which depend on the underlying transitions being homoge-neously broadened. Not only are the spectral lines in vibra-tional systems inhomogeneous, they also undergo spectral dif-fusion that can be on the same timescale as excitation transfer.A simultaneous solution to the coupled excitation transfer andstructural spectral diffusion processes is necessary. This paperpresents several useful calculations for these systems: (1) ex-citation transfer in the case of static inhomogeneous lines (nostructural spectral diffusion), (2) spectral diffusion induced byexcitation transfer, and (3) spectral diffusion in the case of ex-citation transfer and structural spectral diffusion processes.

Despite the keen interest in developing an understandingof the vibrational spectral dynamics in the condensed phases,there has not been an acceptable framework developed for un-derstanding how excitation transfer occurs in these systems.In fact, researchers have applied Förster theory in situationsin which the theory is not applicable. Although Förster theoryhas been exceptionally powerful in understanding excitationtransfer among homogeneously broadened electronically ex-cited molecules, many of the simplifying assumptions that ap-ply well to electronic spectroscopy simply do not hold whenvibrations are considered. The primary assumption that failsfor most vibrational systems is the assumption that all spec-tral lines are purely homogeneously broadened. This com-plication was observed for the case of electronic transitionsnearly 40 yr ago in the Russian literature,41–43 but these ob-servations have been considered seriously only in the caseof the electronic excited state dynamics of photosyntheticcomplexes44–47 and dispersive excitation transport in liquidsand polymers.7, 9, 48 The work of Stein and Fayer incorporates

spectral diffusion into a theory of dispersive excitation trans-port, yet the theory only applies to the case of narrowband ex-citation experiments (such as hole burning, fluorescence) andis difficult to generalize. Stein and Fayer do not examine thegeneral aspects of excitation transfer induced spectral diffu-sion (ETISD), its coupling to SSD, or computational method-ologies that allow for the calculation of an observable otherthan the self part of the excitation transfer Green’s function.9

Förster theory describes the transfer of excitation be-tween two chromophores through dipole-dipole coupling.4

The transfer rate can be calculated either by Fermi’s goldenrule or second-order time-dependent perturbation theory inthe density matrix formalism.23, 49, 50 The Förster limit cor-responds to weak incoherent transfer. When the coupling issufficiently strong or the transfer is coherent, Förster theorydoes not apply, and a fully quantum mechanical treatment isnecessary to solve the excitation transfer problem.51 If the ex-citation transfer problem is in the Förster limit, then a masterequation formalism can be used to calculate the effects of ex-citation transfer on the excitation probabilities in a system.47

This paper focuses on results derived from the master equa-tion approach, although despite this limitation, that the gen-eral ideas should shed some light on strongly coupled sys-tems.

Structural spectral diffusion is the stochastic fluctuationamong inhomogeneous spectral states.28, 30 The process canbe characterized by its correlation functions. If the fluctua-tions are Gaussian (or reasonably approximated as Gaussian),then the process is fully characterized by its autocorrelationfunction that is called the frequency-frequency correlationfunction (FFCF). If the FFCF is known, then the absorptionand 2D IR spectroscopic line shapes can be calculated,linking the observed spectroscopy to the microscopic dy-namics. Furthermore, in the higher order nonlinear opticalexperiments, that is, 2D IR vibrational echoes, the FFCFcan be directly measured.39, 52 The FFCF includes both thespectral diffusion among the inhomogeneous states and thehomogeneous linewidth.

Within an inhomogeneous line, transfer of an excitationfrom an excited chromophore to a different chromophore witha different frequency will also contribute to the decay of theFFCF. Calculating the full FFCF requires taking into accountboth structural processes and the excitation transfer process it-self. These processes together form a coupled stochastic non-linear dynamic system, the practical solution of which is themajor goal of this paper.

Recently, other researchers have studied spectral diffu-sion in systems undergoing excitation transfer. Yang, Li, andSkinner recently published an analysis of energy transportin liquid H2O, D2O, and isotopically diluted mixtures.22, 23

Their analysis is based upon a solution of the time-dependentSchrodinger equation for the manifold of local vibrationalstates using molecular dynamics (MD) trajectories to definethe atomic coordinates. Their theory satisfactorily agrees withexperimental measurements of the orientational anisotropydecay.23 Due to the nearly first principles approach, Skin-ner and co-workers have validated that perturbation theoryadequately describes excitation transfer even in extremelystrongly coupled situations (like liquid water) and that the


majority of the transfer is incoherent.23 Despite the fact thatthe system is in the Förster limit, a naïve application fails toagree with experiments that probe the underlying excitationtransfer process. Skinner and co-workers say that by includ-ing orientational effects, a proper radial distribution function,and intramolecular excitation transfer, Förster theory can bemade to agree with experimental anisotropy data. However,Skinner and co-workers have illuminated the pitfalls of ap-plying Förster theory to vibrational energy transport. Theirextension of Förster theory utilizes only a single Förster ra-dius, R0, which in principle cannot account for vibrationalexcitation transport between different types of pairs of watermolecules, i.e., for randomly chosen pairs of water molecules,not only do their relative orientations and distances differ butalso the center frequencies of their homogeneous lines willdiffer. Because R0 depends on the overlap of the homoge-neous lines, different pairs of molecules will have differentR0’s. While Skinner’s ab initio theory correctly incorporatesfrequency fluctuations and frequency differences among pairsof exchanging water molecules, no attempt was made to cor-rectly incorporate this fundamental aspect of the excitationtransfer process into a modified Förster theory. However, thenature of water may mask this inadequacy. The homogeneouslinewidth in water is very broad; it is a substantial fraction ofthe total linewidth. In addition, much of the structural spectraldiffusion is very fast, occurring on the order of a few hundredfemtoseconds. These properties of the water hydroxyl stretchspectral dynamics are in many respects similar to those foundin electronic excitation transfer systems consisting of chro-mophores in room temperature liquids in which the simpleFörster theory generally works well. Water is unique in thisrespect. Many vibrations have homogeneous lines that aremuch narrower than the inhomogeneous linewidth and haverelatively slow structural spectral diffusion. In these systems,simple Förster theory with a single R0 will not provide a use-ful description of the vibrational excitation transfer and itscontribution to spectral diffusion.

II. EXCITATION TRANSFER INDUCED SPECTRALDIFFUSION IN THE STATIC FREQUENCY LIMIT

The failure of nearly every attempt to use Förster theoryto explain vibrational excitation transfer can be reduced to thefact that the expectation value of a function of a random vari-able is not equal to the function evaluated at the expectationvalue of the random variable, i.e., E[f(x)] �= f(E[x]). In certainspecial cases (f(x) is linear, x is deterministic), the equalitycan hold, but none of these cases applies to vibrational ET.In the language of ET theory, x is the transfer rate betweentwo randomly selected molecules and f(x) is an experimentalobservable. f(x) could be the frequency-frequency correlationfunction, the anisotropy decay, or simply the amount of pop-ulation left on the initially excited molecule. For the case ofthe frequency-frequency correlation function, the desired ex-pectation value is

〈δω(t)δω(0)〉 = E{r,ω}[f (r, ω; t)]

= E{r,ω} [E [δω(t)δω(0)|{r, ω}]] , (1)

where 〈δω(t)δω(0)〉 is the frequency-frequency correlationfunction which depends on the position related variables ofthe system r, and the frequency related variables of the sys-tem ω.

In this paper we explicate Förster theory in the inhomoge-neous case and demonstrate how the excitation transfer pro-cess can drive not only spectral diffusion but also be drivenby structural spectral diffusion. The initial inhomogeneousFörster theory is modified to account for structural spectraldiffusion, and tractable approximations are made that greatlysimplify the calculations. Due to recent work on surface sys-tems undergoing spectral diffusion and ET simultaneously,extra attention is given to the two-dimensional case.27

A. Excitation transfer for an ensemble of particle pairs

In the case when vibrations do not undergo structuralspectral diffusion, there is only homogeneous and inhomoge-neous spectral line broadening. When two spatially proximatemolecules are chosen at random from the inhomogeneous dis-tribution of vibrational frequencies, they may undergo ET. Be-cause these molecules belong to different sub-ensembles andare not coupled by any mechanism other than dipole-dipolecoupling (which is assumed to be relatively weak and incoher-ent), the classical Förster theory applies.4 Within this theory,the rate of ET depends on the difference in center frequenciesand the homogeneous linewidths. We can calculate the proba-bility of the excitation moving between these two molecules.For two molecules undergoing reversible ET, the coupleddifferential equations which describe the ET process arewritten as

p0 = −kp0 + kp1,

p1 = +kp0 − kp1,(2)

where k is the Förster transfer rate, p0 denotes the probabilityof the excitation residing on the initially excited molecule, andp1 is the probability of excitation residing on the non-initiallyexcited molecule. Vibrational population relaxation has beenomitted as it is assumed to be the same for all molecules andtherefore can be added in later as decay in the overall proba-bility of finding the excitation anywhere. The solution to thesecoupled differential equations for the initial condition p0 = 1;p1 = 0 is p0(t) = 1

2 (1 + e−2kt ). This probability is also de-noted as the self part of the Green’s function, Gs(t), whichcomes into play when there are more than two chromophores.The rate k is determined by the standard Förster method,4, 38

i.e.,

k = 3

2TV R

κ2(�)

(R0

r

)6

, (3)

R60 = 9(ln 10)κ2QD

128π5NAn4

∫ +∞0 λ4εA(λ)εD(λ)dλ∫ +∞

0 εD(λ)dλ, (4)

where R0 is the Förster radius, TVR is the vibrational life-time, QD is the quantum yield, NA is Avogadro’s constant,n is the index of refraction of the medium, λ is the wave-length in nm, ε is the molar decadic extinction coefficient(in L mol−1 cm−1) and κ2(�) is the dipole-dipole coupling


orientation factor. The quantum yield is the fraction of chro-mophore excitations that decay by spontaneous emission, andcan be calculated from the radiative lifetime and the non-radiative lifetime. R0 depends on the frequency distributionof both oscillators and their transition strengths. In Förstertheory, the overlap integral that is used to determine R0 in-cludes the emission line of the donor and the absorption lineof the acceptor. In the case of vibrations there is no apprecia-ble Stokes shift, so for a given vibrational oscillator the ab-sorption and emission line position and shape are consideredto be the same throughout this paper.

The solutions to the ET problem for two molecules intwo different cases (using the parameters from the Appendix)are illustrated in Figure 1. In the first case, the two moleculeschosen have vibrations with their homogeneous lines closein frequency (Figure 1(a), blue curves). The overlap betweenthe two absorption lines is large. Therefore, R0 is large, whichmakes the transfer rate, k, large. The probability p0 thereforedecays quickly (Figure 1(b), blue curve). In the second case,the molecules are chosen such that their vibrations do not lie

1 0

0 5

0.6

0.7

0.8

0.9

1.0

rban

ce

(a)

0.0

0.1

0.2

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0.4

0.5 ab

sor

-20 -15 -10 -5 0 5 10 15 20

Δω (cm-1)

0.9

1.0

(b)

0.7

0.8

p 0(t

)

0 20 40 60 80 1000.5

0.6

t (ps)

FIG. 1. (a) The inhomogeneous distribution (black curve) of homogeneouslybroadened vibrational lines (green, red, and blue curves) plotted about theline center. (b) The probability that an initially excited molecule remains ex-cited after a time t in two cases: (red) the exchanging molecules have cen-ter frequencies denoted by the red lines in panel (a), (blue) the exchangingmolecules have center frequencies denoted by the blue lines in panel (a).

close in frequency (Figure 1(a), red curves). In this case theoverlap between the two absorption lines is small. So R0 issmall, which makes the transfer rate small. The probabilitythat the excitation resides on the initially excited moleculedecays slowly as shown in Figure 1(b) (red curve).

In addition to calculating the probability of ET, ourgoal here is to also determine the frequency-frequency cor-relation function of the system undergoing ET. As a sim-ple example consider a system composed of an ensembleof pairs of molecules. In each pair, the two molecules in-teract with each other but not with other molecules. We cancalculate the frequency-frequency correlation function as weknow the time-dependent probability distribution of the vi-brational frequency. In terms of expectation values, we cancalculate the inner expectation of the dynamical observableδω(t)δω(0)|ω,r. r was given by the setup of the problem(Appendix), i.e., 8 Å with κ2 = 2/3, and the frequencies,which are static at this point, were drawn from the inhomoge-neous distribution. In this simple case there are only two prob-abilities which must be accounted for, the probability that theexcitation lies on the initially excited molecule and the prob-ability that it does not. Therefore, the FFCF expectation valueis given by the formula

〈δω(t)δω(0)〉 = E{δω1,δω2}

[1

2(1 + e−2k(ω1,ω2)t )δω2

1

+ 1

2(1 − e−2k(ω1,ω2)t )δω1δω2

], (5)

where δω1 is the frequency of the initially excited vibra-tion and δω2 is the frequency of the vibration on the othermolecule. The expectation value is the average over the in-homogeneous distribution of frequencies for each vibration.Note that k is a function of δω1 and δω2 and therefore mustbe calculated inside the expectation value. The above solutionfor the frequency-frequency correlation function is only forspectral diffusion caused by ET. There is no structural spec-tral diffusion. The homogeneous contribution to the total vi-brational line shape can be included by an additional deltafunction component in the FFCF.39

Even in this simple system several key properties of ETare clearly visible. To illustrate them we show the results ofETISD calculation for the ensemble of pairs of molecules.For this calculation we use the numerical inputs chosen for allcalculations (see the Appendix). Figure 2 shows the normal-ized FFCF with the homogeneous component omitted (blackcurve), the self part of the Green’s function (blue curve), andthe self part of the Green’s function with the Förster radiuscalculated taking the total spectroscopic line to be homo-geneously broadened (red curve), and therefore having unitoverlap, instead of using the underlying inhomogeneous dis-tribution of homogeneous line shapes. None of the functionsare identical. This figure clearly demonstrates that it is im-perative that one performs a detailed ET calculation using thetrue homogeneous and inhomogeneous line shapes even whencalculating the effect of ET on observables uncorrelated withfrequency. Using the Förster radius calculated using the entireabsorption line yields an incorrect Gs(t) curve that is much toofast. The ETISD itself evolves differently than the ET process


1 00

0.75

0.80

0.85

0.90

0.95

1.00(t

); G

s(t)

0 20 40 60 80 1000.50

0.55

0.60

0.65

0.70

FF

CF

FFCF(t) G

s(t)

Gs

(avg-rate)(t)

0 20 40 60 80 100

t (ps)

FIG. 2. Normalized FFCF with the homogeneous component omitted forexcitation transfer in an ensemble of pairs of particle system (black line). Selfpart of the Green’s function is shown for system studied (blue line) and for ahypothetical purely homogeneously broadened system with the pre-spectrallyaveraged rate constant (red line).

that drives it, namely, slower. The FFCF decays more slowlythan the underlying ET, given as Gs(t) (Figure 2, blue curve)because there is a tradeoff between fast ET and a large fre-quency change. When molecules are far apart in frequency,there is a large change in frequency when ET occurs. How-ever, the molecules undergo slow ET because the overlap ofthe homogeneous lines is small. When molecules are close infrequency, they exchange excitations quickly, but the associ-ated change in frequency is small.

Improperly pre-averaging the Förster radius before calcu-lation of ensemble averaged properties is a serious flaw in themanner that Förster theory is frequently used. Förster theorydepends on the application of the steady-state approximationto the second-order perturbation theory rate of dipole-dipolecoupling induced excitation transfer.49, 50 This approximaterate is itself an ensemble averaged quantity. By ensembleaveraging over the entire inhomogeneous distribution inthe steady-state approximation step, it becomes impossible tothen use the pre-averaged rate to calculate other dynamicalobservables that must themselves be integrated over the en-semble degrees of freedom. In other words, one cannot use anaverage rate constant to calculate a kinetic process betweendistinct sub-ensembles that do not react at the average rate.

The condition that the full underlying homoge-neous/inhomogeneous distribution of frequencies needs to beaccounted for in an ET problem can be relaxed when thehomogeneous linewidth dominates the total absorption lineshape. In Figure 3, the same ensemble of pairs of moleculesproblem is studied but instead of having the homogeneouslinewidth, = 2 cm−1 and the inhomogeneous linewidthequal to 16 cm−1 (for a total linewidth of 17.1 cm−1) as inFigure 2, is varied from 2 cm−1 to 17 cm−1, and the inho-mogeneous linewidth is set between 17.1 cm−1 and 1.2 cm−1

to achieve a total linewidth of 17.1 cm−1 for all values of thehomogeneous linewidth. The black curves are the normalizedFFCF with the homogeneous component omitted. Thus, onlythe time-dependent part of the FFCF is shown. It is important

to keep in mind that as is increased, the range of spectraldiffusion is decreased because the inhomogeneous width isdecreased. The blue curves are the properly averaged Gs(t),and the red curves are G

(avg-rate)s (t) in which the donor and

acceptor are taken to have perfect overlap as if the lines arehomogeneously broadened. Several key trends are clear fromthe figure. First, as the homogeneous linewidth begins to dom-inate as a fraction of the total linewidth, the time dependenceof the frequency-frequency correlation function, and the selfpart of the Green’s function converge. Furthermore, the im-properly pre-averaged Förster radius G

(avg-rate)s (t) curve also

converges to the true one, indicating that in predominantlyhomogeneously broadened lines the traditional calculation ofthe Förster radius (ignoring inhomogeneous line shapes) isaccurate. This is the reason that the simplified Förster theoryworks for water in which the homogeneous linewidth is a sub-stantial fraction of the total linewidth.23 Also note that as longas there is non-negligible inhomogeneous broadening, ETISDdoes not track the ET process itself. In Figure 3, only for

= 17 cm−1 are the time dependences of ET and ETISD thesame, and in this case the range of spectral diffusion is verysmall. The spectral diffusion induced by ET in an inhomoge-neously broadened system can never be described by the timedependence of the ET process, no matter how well it is calcu-lated. The homogeneous linewidth must be measured and the

0 91.0

0.60.70.80.91.0

2 cm-1

0.60.70.80.9

4 cm-1

0 80.91.0

) ;G(avg)

s(t)

0.70.80.91.0

0.60.70.8

8 cm-1

F FCF(t);Gs(t)

0.60.70.80.91.0

17 cm-1

0.60

10 cm-1

F

0 20 40 60 80 1000.5

17 cm

t (ps)

FIG. 3. Normalized FFCF with the homogeneous component omitted (blackcurve) and self part of the Green’s function (actual, blue curve; pre-averaged,red curve), for an ensemble of pairs of particle excitation transfer systems asa function of homogeneous linewidth. The total linewidth is held constant.


inhomogeneous distribution accounted for in calculating thetime dependence of the ETISD.

These simple cases encompass the entirety of the physicsof vibrational ET theory within the Förster limit. The rest ofthe work is to build tools to solve this problem efficiently forphysically realistic systems rather than an ensemble of iso-lated pairs. For the remainder of this section we will workwithin the static frequency limit (no structural spectral dif-fusion). Once the methodology has been developed we willmove on to include stochastically fluctuating structural spec-tral diffusion.

B. N-particle excitation transfer dynamical system

The two molecule problem can be extended to includeN molecules by creating an N-dimensional linear dynamicalsystem (or a master equation; the matrix method is used herefor ease of computation) which represents the N coupled lin-ear ordinary differential equations:

p = A(ω, r)p, (6)

where the vector p is the vector of probabilities of an exci-tation being found on a given molecule. The matrix A is thedynamics matrix and is filled with the rates kij that are all cal-culated by the Förster method. The above equation holds fora given set of positions and frequencies. The solution to thisproblem (with static frequencies and positions) for any ini-tial condition may be given by matrix exponentiation of thedynamics matrix, A,

p(t) = exp [tA] · p(0), (7)

where p(0) is the vector of initial probabilities and p(t) isthe time-dependent probability of finding an excitation onany given molecule. For systems with large N, it becomesintractable to solve this problem analytically; however, forperiodic or quasi-periodic systems (such as crystals, liquids,solids, surfaces) periodic boundary conditions can be en-forced, for example, by the minimum image criterion, on thedynamics matrix. As long as N is large enough, the results forthe periodic systems will be asymptotically equivalent to theresults for the bulk system of infinite extent.

Once the time-dependent vector of probabilities isknown, then it is possible to calculate the frequency-frequency correlation function for this system. For the casewhere only a single molecule begins in the excited state, thep(0) vector is a vector containing a single non-zero elementthat is equal to 1, that is, p(0)(j�=i) = 0; p(0)(i) = 1. Given thatthe initial excitation begins on the ith molecule, the FFCF fora molecular system with static frequencies given by the inho-mogeneous distribution when the frequencies are not corre-lated with spatial position is

E [ω(t)ω(0)|ω,r] = E [ω(t)ω(0)|A(ω,r)] = (p(t) · ωt ) ωt=0(i).

(8)

The superscript (i) denotes the ith element of a vector,and the vector ω contains the center frequency values for eachmolecule. Note that in the case of time-dependent frequencies(structural spectral diffusion) discussed below, the frequency

vectors need not be the same. Also note the clear differencebetween how the FFCF is calculated versus how Gs(t) is cal-culated for this system, Gs(t) = p(i)(t).

C. Monte-Carlo based computation of the FFCF

The above solution to the ET induced spectral diffusionproblem is based upon a single realization of the static ran-dom frequency distribution and a single static realization ofthe geometric distribution of molecules. To calculate the ex-pectation value of the frequency-frequency correlation func-tion over each of these distributions it is suitable to performMonte-Carlo averaging. Subsequently, a procedure that sim-plifies the calculations will be introduced. Thus, returning tothe conditional expectation notation, the expected frequency-frequency correlation function is,

〈δω(t)δω(0)〉 = E{r,ω} [E [δω(t)δω(0)|A(r, ω)]]

MC−−−→ 1

N

∑{ω,r,i}

(p(t) · ω) ω(i) = 1

N

∑{ω,r}

∑i

(p(t) · ω) ω(i).

(9)

The outer expectation is over the joint space-frequencydistribution function, and the Monte-Carlo summation is asum over the randomly drawn space-frequency realizationsand a randomly chosen initial excitation vector.

For the examples simulated in this paper, the Monte-Carlo averaging is not computational intensive, as spatialdistribution functions without higher order structure arechosen. For simulations of real molecular systems, moreaveraging will be required and Metropolis Monte-Carlo oranother Markov Chain Monte-Carlo method will be neces-sary. For example, we use a uniform random distribution.By drawing independent samples from the uniform randomdistribution, under 1% error at 95% confidence in the MC es-timates can be achieved in under 105 samples. When randomfrequency trajectories are sampled later in the paper, againonly simple (i.e., random sampling) Monte-Carlo averagingis necessary, as the higher order structure in the frequencytrajectory is generated through the integrated stochastic lawwe derive for the random process. The sampling is even moreefficient when one takes advantage of the fact that multipleinitial excitation sites are possible.

III. ETISD AS A PROBE OF STRUCTURE

Due to the r−6 dependence of the Förster transfer rate,ET and ETISD are extremely sensitive to the joint space-frequency distribution function of molecules. For the casewhen frequencies and spatial distribution are independent,and for a given frequency distribution, the excitation transferand associated ETISD problem are sensitive to the geometricdistribution of molecules. This sensitivity can exist in boththe distribution of intermolecular distances (through ther−6 dependence) and the distribution of relative molecu-lar orientations (through the κ2 factor). Below we showsome general properties of this sensitivity for 1D, 2D, and3D dimensionalities and implement the Monte-Carlo solution


1.0 (a)

(b)

0 4

0.6

0.8

F(t

); G

s(t)

0 20 40 60 80 1000.0

0.2

0.4

FF

CF

t ( )

2 5

3.0

3.5

4.0

×10

7 cm-1

)

t (ps)

0.5

1.0

1.5

2.0

2.5

nsit

y (×

1.4

3×

0 5 10 15 200.0de

n

r (Å)

FIG. 4. One-dimensional systems. (a) Normalized FFCFs with the homoge-neous component omitted (solid curves) and Gs(t) curves (dashed curves) forrandom distribution with excluded volume (black), one-dimensional lattice(red line), and hard sphere model (blue line). (b) Radial distribution func-tions for the corresponding geometric models (same colors) in panel (a). Thedashed lines correspond to the lattice sites for the one-dimensional latticemodel.

to the ETISD problem discussed above. Except for thecalculations in Figure 6, the angular factors are static andisotropic, that is, there is a random distribution of fixedorientations.

ETISD is sensitive to the dimensionality of the system.Within a given dimensionality different structures are easilydiscernible due to their different ETISD profiles. For each di-mensionality, three different structures are simulated, a lattice,a loosely packed hard sphere model, and a random distribu-tion with an excluded volume effect. The average density isthe same for the three structures. In the 1D systems, the lat-tice is a linear lattice; in 2D systems it a hexagonal lattice; andin 3D systems it is a simple cubic lattice. The structures dif-fer in their associated radial distribution functions which arevisualized in Figures 4(b), 5(b), and 7(b). The different radialdensities cause ETISD to proceed at different rates. All calcu-lations use the Monte-Carlo formalism developed above. TheMonte-Carlo averaging is performed over both the spatial andfrequency dimensions.

1.0 (a)

(b)

0 4

0.6

0.8

(t);

Gs(t

)

0 20 40 60 80 1000.0

0.2

0.4

FF

CF

(

3

4

4 cm-2

)

t (ps)

1

2

ensi

ty (×

2×10

14

0 2 4 6 8 10 12 14 16 18 200

de

r (Å)

FIG. 5. Two-dimensional systems. (a) Normalized FFCFs with the homoge-neous component omitted (solid curves) and Gs(t) curves (dashed curves) forrandom distribution with excluded volume (black), two-dimensional hexag-onal close packed lattice (red line), and hard sphere model (blue line). (b)Radial distribution functions for the corresponding geometric models (samecolors) in panel (a). The dashed lines correspond to the lattice sites for thetwo-dimensional hexagonal lattice model.

Figure 4(a) illustrates the normalized FFCFs with thehomogeneous component omitted (solid curves) and Gs(t)(dashed curves) for the three model one-dimensional systems.The black curves are for the random distribution with ex-cluded volume. The red curves are for the 1D lattice, and theblue curves are for the hard sphere model. The radial distribu-tion functions of the three model systems are shown in panel(b) of Figure 4. The dashed lines show the position on thelattice as the distance is increased from the donor molecule.The FFCFs are normalized and the homogeneous contributionis not shown. Parameters for the simulations are given in theAppendix. The random distribution with excluded volume ef-fect is the slowest ETISD system, followed by the linear lat-tice, followed by the hard sphere model. The differences inETISD rate are clearly correlated with the radial distributionfunctions. Specifically, higher local density near an excitedmolecule drives ETISD due to the strong r−6 dependence ofthe transfer rate.


1.0 (a)

(b)

0.6

0.8

45°60°F

CF

(t)

0 20 40 60 80 1000.0

0.2

0.4 60 90° dynamic - 45° dynamic iso. static iso.

FF

t (ps)

0.8

1.0

45° 60° 90° dynamic - 45°

0.2

0.4

0.6 dynamic iso. static iso.

Gs(t

)

0 20 40 60 80 1000.0

t (ps)

FIG. 6. Two-dimensional systems. (a) Normalized FFCFs with the homoge-neous component omitted for six different angular distributions. Blue curvescorrespond to fixed polar angle models of 45◦ (dotted), 60◦ (dashed), and90◦ (solid) with respect to the surface normal. Red lines correspond to staticisotropic model (solid), dynamic isotropic (dashed), and dynamic with a fixedpolar angle of 45◦ (dotted). (b) Gs(t) curves for the same systems, line desig-nations are the same as in the panel (a).

Figure 5(a) illustrates the normalized FFCFs with thehomogeneous component omitted (solid curves), and Gs(t)(dashed curves) for the three two-dimensional model systems(monolayers). The hard sphere model has the fastest Gs(t) andETISD (dashed blue and solid blue curves) due to large peakin the radial distribution function at 5.8 Å (panel b). This in-creased local density is nearly four times the average den-sity of molecules in the system and serves to enhance ET andETISD for the hard sphere system. Because the densities arethe same, the hexagonal lattice system (red curves) has slowertransfer, despite its more efficient packing due to the increaseddistance between nearest neighbors relative to the hard spheremodel. The random distribution with excluded area has theslowest transfer of the three models due to its flat radial dis-tribution function.

The angular distribution for the orientation of the tran-sition dipoles can substantially change the properties of theETISD and ET process. This effect is illustrated for the two-dimensional system for which several different angular dis-tribution models were analyzed for the random with excluded

1.0

(b)

(a)

0 4

0.6

0.8

F(t

); G

s(t)

0 10 20 30 40 50 600.0

0.2

0.4

FF

CF

( )t (ps)

2 0

2.5

3.0

21 c

m-3)

0.5

1.0

1.5

2.0

nsit

y (×

2.92

×10

0 2 4 6 8 10 12 140.0de

n

r (Å)

FIG. 7. Three-dimensional systems. (a) Normalized FFCFs with the homo-geneous component omitted (solid curves) and Gs(t) curves (dashed curves)for random distribution with excluded volume (black), three-dimensionalsimple cubic lattice (red), and hard sphere model (blue). (b) Radial distri-bution functions for the corresponding geometric models (same colors) inpanel (a). The dashed lines correspond to the lattice sites for the simple cubiclattice model.

volume effect model (see Figure 6). The static isotropicmodel means the transition dipoles are distributed randomlyand uniformly in both polar and azimuthal angle. The dy-namic isotropic model corresponds to each transition dipolesampling all of angular space on a much faster timescale thanexcitation transfer. The fixed polar angle model has each chro-mophore with its transition dipole at an angle θ with respectto the surface normal and is free to lie at any azimuthal angle.The dynamic fixed polar angle model is when the azimuthalangle is sampled much faster than the excitation transferrate. Three different static fixed polar angle distributions areshown in Figure 6, θ = 45◦, 60◦, and 90◦ where θ is the anglebetween the transition dipoles of the chromophores and thesurface normal. The ETISD process is fastest for θ = 90◦ andslowest for θ = 45◦, however, the change is not monotonic, asθ = 30◦ is about as fast as θ = 60◦ and θ = 0◦ is about as fastas θ = 90◦. These latter angles are not shown in the figure.For the static isotropic, dynamic isotropic (corresponding toκ2 = 2/3 for all chromophores) and dynamic fixed polar


angle (corresponding to κ2 = 9/16 for all chromophores)models, the ETISD occurs fastest in the dynamic isotropicand at a nearly equal but slower rate in the static isotropic anddynamic fixed polar angle models. The dynamic averages areeither slower or faster than the static averages depending onthe angular distribution used. Changing the angular distri-bution effectively changes the R0’s in the system by scalingthe rate factors. This scaling changes the sensitivity of bothETISD and Gs(t) to the underlying frequency distributions indifferent and nontrivial ways.

ETISD and Gs(t) curves have also been calculated forthree-dimensional systems. The results are illustrated inFigure 7. The trends are similar to the one- and two-dimensional cases. However, in three dimensions, the hardsphere and lattice models are more similar than in the 1D and2D systems.

IV. APPROXIMATIONS TO THE ETISD PROBLEM

Baumann and Fayer have previously applied the Hu-ber approximation53 for ET in two- and three-dimensionalsystems.16 The Huber approximation is useful when a sim-plified method is required to solve the ET problem. The keyresult from their work is that the natural logarithm of the selfpart of the Green’s function (probability of initial excitationremaining on initially excited molecule) can be related to aconfiguration space integral over the ET probability kernel be-tween the initially excited molecule and any other molecule inthe system:

ln Gs(t) = −ρ

λ

∫ +∞

r0

dr

×∫

�

d�

[1 − exp

(−3λtR6

0

2τr6κ2(�)

)]u(r)v(�).

(10)

Here, ρ is the density, λ is a parameter which is 1 for donor-acceptor transfer and 2 for donor-donor transfer (the typicalvibrational case), τ is the lifetime, u(r) is the radial distribu-tion function, and v(�) is the orientational probability den-sity function. The above equation differs from Eq. (2.12) ofBaumann and Fayer by the fact that the lower limit, whichwas 0 in the reference, is replaced by r0, the exclusion radius.The exclusion radius is the distance below which the proba-bility of finding a nearby molecule is 0.

Baumann and Fayer derive relations only in the case forr0 = 0, but here we will generalize their results to work forany exclusion radius in a nearly analytical fashion. Follow-ing Baumann and Fayer, two substitutions are made: y = μ

r6 ,

μ(�; t) = 3λtR60

2τκ2(�), resulting in the expression

ln Gs(t) = −ρ

λ

∫ y(r=r0)

0dy

(y

μ

)− 76 1

6μ

×∫

�

d�[1 − exp (−y(r,�; t)]u(r)v(�). (11)

This is the most general result and can be further sim-plified in the cases of one, two and three dimensions if the

radial distribution function is completely uniform except forthe excluded volume effect. For dimension, ∈ {1, 2, 3} anduniform radial distribution, u(r) ∈ {2, 2πr, 4πr2}, the log-arithm of the self part of the Green’s function in the casewhere there is a non-negligible molecular volume may beexpressed as

ln Gs(t) = −c λ /6−1

(3

2

) /6 (t

τ

) /6 ∫�

d�κ /3(�)

×[y0(�)− /6(e−y0(�) − 1) + γ

(1 −

6, y0(�)

)],

(12)

where γ ( · , · ) is the lower incomplete Gamma function, and

y0(�) = 1r6

0

3λtR60

2τκ2(�) is the value of the dimensionless pa-

rameter y when r is at the exclusion radius (r0) lower boundand the chromophores have a relative orientation of � in an-gular space. c is the reduced concentration which is equalto the density multiplied by the Förster volume, i.e., c

= ρV , where V is 2R0 for the one-dimensional case, πR02

for the two-dimensional case, and (4/3)πR03 for the three-

dimensional case.16

In the isotropic dynamic limit, the configuration spaceintegral disappears, κ2 = 2/3, and the expression is replacedby

ln Gs(t) = −c λ /6−1

(t

τ

) /6

×[y

− /60,dyn (e−y0,dyn − 1) + γ

(1 −

6, y0,dyn

)],

(13)

where y0,dyn = 1r6

0

λtR60

τ.

Performing the angular average over the relative orien-tation of the chromophores is nontrivial and in the isotropicstatic case depends on four polar angles regardless of thedimension.16 For the case of the fixed polar angle model (seethe Appendix), the orientational average depends on only thetwo azimuthal angles with respect to the separation vector.

The Huber approximation is a two-particle model (inthat the initial excitation is coupled outward to all nearbyacceptors, but the acceptors are not inter-coupled) plus afirst-order cumulant expansion truncation.16, 53 However, fora three-dimensional random distribution of molecules, Gs(t)calculated with the Huber approximation and with a diagram-matic method involving an infinite number of particles andpathways11 showed essentially perfect agreement.6, 54 Thismodel is easily modified to encompass inhomogeneous lines.Instead of the initial excitation being coupled to an accep-tor density, the initial excitation is coupled to many uncon-nected acceptor densities each with a different acceptor fre-quency. For each acceptor density the ET problem is solvedindependently, and then averages over all possible acceptorfrequencies and initial excitation frequencies are performed.Mathematically, Eq. (12) is first exponentiated so that for eachpossible R0 a Gs(t) curve is generated. These Gs(t) are thenused to calculate the FFCF according to Eq. (5) (where the


1 0

0.2

0.4

0.6

0.8

1.0

1.0

(t)

1D

FFCF

Gs

0.2

0.4

0.6

0.8

FF

CF

(t);

Gs(

2D

0 8

1.0

3D

0 10 20 30 40 50 60 70 80 90 1000.0

0.2

0.4

0.6

0.8

t ( )t (ps)

FIG. 8. (1D) 1D Huber approximation (dashed curves) and full Monte Carloapproach (solid curves). The top set of curves in the figure are normalizedFFCFs with the homogeneous component omitted, and the lower set of curvesthe Gs(t) curves. (2D) Same as 1D but for the two-dimensional system. Redcurves are simulations at half the density of the black curves (1 × 1014 cm−2

vs 2 × 1014 cm−2). (3D) Same as 1D but for the three-dimensional system.

two-particle Green’s functions are replaced with the HuberGreen’s functions).

Figure 8 compares the Huber approximate solutions tothe ETISD (FFCF) and ET (Gs(t)) problems to the MonteCarlo results for the random distribution with excludedvolume model. The static isotropic angular distribution isused. In these calculations, the vibrational lifetime is 10 ps.The calculations go out to 100 ps. In a typical experiment,measurements can be made out to 2 to 4 vibrational lifetimes.Gs(t) is only the decay of the probability of finding theexcitation on the initially excited molecule. The decay of theprobability of being on any molecule caused by the lifetimeis a multiplicative exponential factor. The reduced Försterconcentrations cannot be specified since they are based on theassumption of homogeneous lines and a single value of R0.Here because the lines are inhomogeneously broadened andR0 depends on the overlap of the homogenous lines, there isnot a single R0 or reduced concentration. The concentrationsare very high. If the lines were homogeneously broadened,the 1D, 2D, and 3D reduced concentrations would be 1.5,1.8, and 1.9. For the 2D case, a concentration of half this 2Dvalue is also considered. The slower decaying pairs of curvesare the FFCFs and the faster decaying pairs of curves are theGs(t)s. The solid lines are from the full Monte Carlo solutionto the ET problem and the dashed lines were calculated withthe Huber approximation.

In Figure 8, the 1D Huber approximation to the normal-ized FFCF with the homogeneous contribution omitted andGs(t) are virtually identical to the Monte Carlo simulations forthe entire simulation time. For the 2D case, the lower concen-tration curves (red, 1 × 1014 cm−2) are very close. The FFCF

Monte Carlo and Huber approximation calculations are vir-tually identical to 4 or 5 lifetimes, which is generally longerthan experiments can be conducted. At the higher concentra-tion in 2D (black curves, 2 × 1014 cm−2), Gs(t) has some errorafter about one lifetime, but the calculation of the FFCF is stillvery good out to about 3 lifetimes.

The agreement between the Huber approximation and theMonte Carlo simulations is much worse at the high concen-tration used in the calculations, although it improves rapidlyas the concentration is reduced. The FFCF is quite good outto about two lifetimes. To get a feel for the error, the curveswere fit to multi-exponentials. The FFCFs fit essentially per-fectly with a bi-exponential, while the Gs(t) curves requiretri-exponentials to get an excellent fit. The fits to the FFCFsyield time constants of 25.3 ps and 335 ps (Huber) comparedto 27.1 ps and 237 ps (Monte Carlo). The fits to Gs(t) yieldtime constants of 3.6 ps, 20.7 ps, and 208 ps (Huber) com-pared to 3.9 ps, 20.8 ps, and 139 ps (Monte Carlo). For boththe FFCF and Gs(t), the agreement is very good for the shorttime components, but there is an error of ∼30% on the longtime components.

V. ETISD COUPLED TO STOCHASTICALLYFLUCTUATING FREQUENCIES (STRUCTURALSPECTRAL DIFFUSION)

Excitation transfer is only one of the two mechanismsthat can produce spectral diffusion. The other is SSD causedby microscopic changes in the inhomogeneous environmentswhich molecules experience. For systems in which the vibra-tional chromophores are in high concentration, structural fluc-tuations occur simultaneously with the excitation transfer pro-cess. The goal of most nonlinear spectroscopic experiments inwhich the frequency-frequency correlation function is mea-sured is to extract the SSD correlation function and not theETISD correlation function. However, in order to extract theSSD correlation function it is necessary to have a firm under-standing of the ETISD process. In the following, calculationsand methods are presented that include both ETISD and SSD,and it is possible to extract the structural spectral diffusion.First a phenomenological model for SSD is introduced andthen it is combined with ETISD.

A. Treating structural spectral diffusion

Structural spectral diffusion is associated with stochasticfluctuations affecting the underlying micro-states responsiblefor inhomogeneous broadening of absorption lines. Spectraldiffusion processes are noise processes that are characterizedby statistical autocorrelation functions. These statistical au-tocorrelation functions (time correlation functions), are typi-cally measured using time-resolved spectroscopic techniquessuch as hole-burning55 and 2D IR spectroscopy.56 To simu-late total spectral diffusion measured by one of these exper-iments, a phenomenological model of structural spectral dif-fusion must be developed that can be coupled to the ETISDproblem.

Structural spectral diffusion can be modeled as aLangevin process.57 The damping term of the Langevin


process ensures that the absorption frequency returns to linecenter and the random force results in fluctuations. Thisis essentially a filtered noise process with an exponentialautocorrelation. The process is described by the followingassumptions:

d (δω(t))

dt= −γ δω(t) + F (t), (14)

〈F (t)〉 = 0; 〈F (t)F (t ′)〉 = F0δ(t − t ′); 〈δω(t)F (t ′)〉 = 0,

(15)

where F(t) is a white Gaussian noise process with powerspectral density F0 and γ is the relaxation rate. The aboveassumptions state that the center frequency of a chosen oscil-lator evolves according to a Hookean differential equation thatis noise driven, where the noise is uncorrelated with the fre-quency state variable and is a Gaussian random variable. Thefrequency process given by the above equation is a Gaussianrandom process and all moments (time correlation functionsin the continuous time case) are related to the mean and co-variance via the standard Gaussian relationships for moments.The Green’s function solution to the differential equation is

δω(t) = e−γ t δω(0) +∫ t

0F (τ )e−γ (t−τ )dτ . (16)

To ensure strict-sense stationarity of the stochastic pro-cess it may be shown that F0 = 2γ 〈δω(0)2〉, where 〈δω(0)2〉is the variance of the frequency and defines the inhomoge-neous linewidth in the absorption line being modeled. Bysubstitution of

δω(s) = e−γ sδω(0) +∫ s

0F (τ )e−γ (s−τ )dτ

into

δω(t) = e−γ t δω(0) +∫ t

0F (τ )e−γ (t−τ )dτ

for times s < t, it is possible to show that δω(t)= e−γ (t−s)[δω(s)] + ∫ t

sF (τ )e−γ (t−τ )dτ . This relation

implies that it is possible to increment the Langevin processby sampling from the following stochastic process:

ω(t) = e−γ (t−s)[δω(s)] + ε(t − s), (17)

ε(t − s) ∼ N (0, 〈δω(0)2〉(1 − e−2γ (t−s))). (18)

This procedure permits discrete time propagation of thestochastic process describing SSD on a continuous frequencygrid with any time step. The continuous frequency distribu-tion generated by the error term above can also be discretizedonto a frequency grid for ease of simulation. In experimentsthat measure SSD, a mutli-exponential form of the FFCFis usually used in the data analysis.58, 59 The treatmentgiven above yields a single exponential FFCF for the SSD.For multi-exponential FFCFs, multiple independent SSDprocesses can be simulated and their outputs simply added.

Figure 9 illustrates a sample simulation by the abovemethod for an inhomogeneous width of 16 cm−1 and a SSDcorrelation time of 75 ps. Panel (a) shows a random trajectory

25

5

10

15

20

cm-1)

(a)

0 100 200 300 400 500 600 700

-15

-10

-5

0

δω (

c

t (ps)

0.8

1.0

Sample Autocorrelation Theoretical Autocorrelation - T = 75 ps

ω(0

)2 > (b)

0.2

0.4

0.6

ω(t)

δω(0

)>/<

δω

0 50 100 150 200 250 300 3500.0

<δω

t (ps)

FIG. 9. (a) Example of a small piece of a simulated discrete time spectraldiffusion noise process for a time constant of 75 ps and standard deviation of16 cm−1. (b) Theoretical normalized autocorrelation (red dashed curve) andnormalized sample autocorrelation (solid black curve) from the noise processshown in panel (a).

of frequencies. Note that panel (a) is just a short snapshot ofthe total simulation. Panel (b) shows the numerically calcu-lated normalized autocorrelation function (black curve) fromthe full trajectory. The red dashed line is the theoretical nor-malized autocorrelation (exponential decay of 75 ps). Clearlythe two curves are essentially identical. The agreementdemonstrates the accuracy of the simulation method. Onecaveat is that the simulated variance (45.75 cm−2) is slightlylower than the theoretical variance (46.16 cm−2) due to the ef-fect of discretizing the frequency distribution. However, thisdiscretization effect does not impact the simulated correlationtime which is the most important feature of the model.

B. Multiplicative approximation to the coupledstochastic-dynamic system

Using the idea of propagating the SSD noise process byusing a deterministic component plus a noise term, it is possi-ble to build a multiplicative approximation to the FFCF of thecoupled stochastic-dynamic system for coupled ET/SSD. Theassumptions in this approximation are that there is a single


excitation and that there will be a sequential propagation ofET and SSD. First, ET will occur for a short time t, and thenSSD will occur for a short time τ . Clearly, in the limit of smalltime steps this procedure will generate the correct dynamics.

To begin working with this approximation first we writedown the FFCF in terms of iterated conditional expectationvalues:

〈δω(t)δω(0)〉ET(t) = Eδω(0)Eδω(t)|δω(0) [δω(t)ω(0) | δω(0)] ,

(19)

〈δω(t + τ )δω(0)〉ET(t)+SSD(τ )

= Eδω(0)Eδω(t+τ )|δω(0) [δω(t + τ )ω(0)|δω(0)] . (20)

δω(t)|δω(0) is determined by the ET process and is distributedas P(δω(t)|ω(0)), which is p(t) from Eq. (8). P(δω(t)|ω(0)) isthe probability distribution given by the ET dynamic after atime t. Equation (19) is the short time propagation of the exci-tation transfer dynamic process. Equation (20) is the sequen-tial propagation of the short time ET process followed by theshort time SSD process. Expanding the probability law of thesequential ET/SSD process within Eq. (20) we achieve


= Eδω(0)Eδω(t)|δω(0)Eε(τ )[(e−γ τ δω(t) + ε(τ ))ω(0) | δω(0)],

(21)

where the SSD process has been propagated using the deter-ministic exponential multiplication and the noise term ε(τ )which is given by Eq. (18). We take apart the internal sum bylinearity of expectation,


= e−γ τ Eδω(0)Eδω(t)|δω(0)Eε(τ ) [δω(t)ω(0) | δω(0)]

+ Eδω(0)Eδω(t)|δω(0)Eε(τ ) [ε(τ )ω(0) | δω(0)] . (22)

Clearly, the second term is zero since the error term inthe SSD propagator has zero mean. The first term, referringto Eq. (19) is equal to

〈δω(t + τ )δω(0)〉ET(t)+SSD(τ ) = e−γ τ 〈δω(t)δω(0)〉ET(t) ,

(23)

which shows that within a short time approximation the cou-pled ET/SSD system behaves as the ET only FFCF multipliedby the normalized SSD FFCF. This motivates the approxima-tion that the combined ET/SSD FFCF will be equal to that ofthe SSD FFCF multiplied by the ETISD only FFCF. Here weare considering a SSD FFCF that is a single exponential. Asmentioned above, the SSD FFCF can have more terms than asingle exponential. The time constant for the full FFCF willend up being different from the underlying SSD time con-stant as will be shown empirically for a number of systemsin Sec. V C.

This approximation is not true at all times due to thefact that the SSD process intrinsically changes the statistical-dynamical law of ET, i.e., the Pδω(t)|δω(0) is changed at latertimes in the propagation process. The SSD process, in fact,speeds up the ET dynamics because it pulls initially distant

frequencies towards the center of the line due to the expo-nential damping term of the SSD process. This means thatmolecules that initially undergo slow excitation transfer willon average undergo faster excitation transfer at later times.This phenomenon is observed in calculations below.

C. Solving the coupled stochastic-dynamic system

Solving for the dynamics in the case of ETISD and SSDrequires coupling the ET dynamic system with the stochasticfluctuations of SSD. Perhaps the most straightforward way tosolve this problem on a discrete time grid of size N is to recur-sively build the N-dimensional probability distribution of fre-quencies that obey the SSD law. Unfortunately, this methodresults in a need to keep track of mN probabilities and fre-quency vectors where m is the number of elements in the fre-quency grid upon which the SSD and ETISD problems aresolved. This problem is clearly computationally impossibleas N increases, and therefore is more efficiently solved byMonte-Carlo averaging over frequency trajectories sampledfrom the stochastic SSD system.

The method of coupling the two dynamic systems is toco-propagate their state variables. For example, in a singletime step, first new frequencies are drawn from the stochasticSSD process and then, using populations on the vibrationalchromophores from the previous time step, new populationsare calculated according to the dynamic law governing ET.This method will first be developed in an ensemble of particlepairs case, extended to the N-particle case, and finally approx-imations to allow faster calculation of ETISD in the presenceof SSD will be presented.

1. The ensemble of particle pairs case

For the ensemble of pairs of vibrational chromophoreswith no interactions between pairs, the discrete time-frequency SSD coupled ETISD problem can be solved ex-actly given a random discrete time trajectory along the givenfrequency grid. If the probability of the excitation lying on theinitially excited molecule at a time t is denoted by E0(t), thenE0(t+τ ) is given by

E0(t + τ ) = 1

2

(1 + e−2k(ω0(t),ω1(t))τ (2E0(t) − 1)

), (24)

where k(ω0(t),ω1(t)) is the Förster transfer rate at the time tdue to the current values of the stochastic time-dependent fre-quencies ω0 and ω1. In the ensemble of particle pairs case,only a single variable (E0) must be carried between timesteps due to the law of total probability, in the N-dimensionalcase at least N−1 probabilities must be accounted for at ev-ery time step. As it will be useful in developing further ap-proximations to this problem, we also develop a graphicalmethod for propagating the ET system when SSD is occur-ring simultaneously. The process, which is equivalent to thealgebraic method of Eq. (24), is illustrated in Figure 10. Atthe start of a new time step from t to t + τ , the remain-ing initially excited population E0 is found on the new dy-namic system curve for the new frequencies drawn from thestochastic process. The remaining initially excited population


0.95

1.00

0(t)

E0(t; k(ω0(t-τ), ω1(t-τ)) E0(t; k(ω0(t), ω1(t))

E0(t)

0.85

0.90

E0

E0(t+τ)

0 10 20 30 40 50 60 70t (ps)

FIG. 10. Graphical procedure for coupled dynamic/stochastic system propa-gation. At the start of a new time step from t to t + τ the remaining initiallyexcited population, E0, is found on the new dynamic system curve for thenew frequencies drawn from the stochastic process. The remaining initiallyexcited population is then propagated down the new dynamic system curvefor τ . This method is equivalent to the algebraic propagation in Eq. (19). Thismethod may be used for propagation of the Huber approximation coupled toa stochastic spectral diffusion process, or any other method for estimatingGs(t) as a function of frequency pairs.

is then propagated down the new dynamic system curve forτ . For each trajectory and time step the contribution of aparticular trajectory to the Monte Carlo averaged FFCF is cal-culated by Eq. (8).

For the standard two-particle parameters outlined in theAppendix, we have solved the SSD coupled ETISD problemfor various SSD rates. The results (solid black curves) arethe total normalized FFCFs with the homogeneous compo-nent omitted shown in Figure 11(a). The dashed black curve,which is the top curve in each panel, is the FFCF for ETISDin the absence of SSD. The coupled SSD and ETISD prob-lem was also solved for the case of a separation distanceof 5 Å, for which ET and, therefore, the ETISD is faster(Figure 11(b)). The SSD ranges from slow, 800 ps, which iscomparable to the ETISD, to fast, 25 ps, which is much fasterthan the ETISD.

Figure 11 also illustrates the multiplicative approxima-tion to the coupled ETISD and SSD problem. The red dashedcurves are the result of fitting the properly calculated totalspectral diffusion, which has contributions from ETISD influ-enced by SSD and SSD, to the form of an exponential decaymultiplied by the ETISD with no SSD (dashed black curves)as suggested by the approximation discussed above. The SSDexponential time constant obtained from the fit using the mul-tiplicative approximation is not equal to the true time constantof the SSD stochastic process. However, the important resultis that the SSD time constants from the multiplicative approx-imation are close to the true SSD time constants. Figure 12illustrates the true SSD process time constants plotted againstthe multiplicative approximation SSD fit time constants forboth d = 8.0 Å and d = 5.0 Å. The dashed line is at 45◦,which would be the result if the multiplicative approximationwas perfect. It is a guide to the eye to make it easier tosee the error in the multiplicative approximation. For the

1 0(a) N SSD

0.6

0.8

1.0

F(t

)

SSD- 800 ps

No SSD

0.2

0.4FF

CF

SSD - 25 psd = 8.0 Å

0.8

1.0

)

0.0(b)

No SSD

0.2

0.4

0.6

FF

CF

(t)

SSD- 800 ps

0 20 40 60 80 1000.0

t (ps)

SSD - 25 psd = 5.0 Å

FIG. 11. Normalized FFCFs with the homogeneous component omitted forthe ensemble of particle pairs ET models with SSD processes co-propagated.The FFCFs from the coupled ET/SSD process (solid black) are calculated forSSD time constants of 25 ps, 50 ps, 75 ps, 100 ps, 200 ps, 300 ps, 400 ps,and 800 ps. The dashed black curve is the FFCF due to ETISD without anyaccompanying SSD process. The red dashed curves are fits to the coupledET/SSD FFCFs by assuming the profile is created by an exponential decaymultiplied by the non-SSD coupled ETISD curve. Labels lie above their re-spective curves. (a) Pair separation distance, d = 8 Å. (b) Pair separationdistance, d = 5 Å.

d = 5.0 Å the actual time constant is 1.12 times the time con-stant obtained from the multiplicative approximation, and forthe d = 8.0 Å the actual time constant is 1.058 times the ap-proximate time constant. With d = 5.0 Å, excitation transfer isvery fast. Even in this case, the multiplicative approximationdoes not produce severe error in the determination of SSD.

The fact that the multiplicative approximation (red andblue lines in Figure 12(a)) yields SSD that is faster than theinput process is caused by the fact that SSD speeds up ET.Therefore, the ETISD contribution to the total FFCF is fasterthan the FFCF with no SSD that is used in the approxima-tion. The enhancement in the FFCF decay is caused by theenhancement of ETISD by coupling to SSD, as the SSD pro-cess is not modified by excitation transfer. The SSD processcreates a stochastic grid of frequencies upon which the ETISDproblem is solved. The stochastic nature of the frequency gridchanges the nature of the ET process. Figure 12(b) shows thatthe ensemble averaged Gs(t) curves decay faster when SSD isincluded in the model, demonstrating the predicted result ofcoupling between SSD and ETISD.


800(a)

400

500

600

700ct

ual (

ps)

0 100 200 300 400 500 600 700 8000

100

200

300

TS

SD a

c

5 Å8 Å

TSSD

fit (ps)

0.9

1.0

SSD - 800 ps No SSD

(b)

0.6

0.7

0.8

Gs(t

)

SSD - 25 ps

No SS

Å

0 20 40 60 80 1000.5

t (ps)

d = 8.0 Å

FIG. 12. (a) Actual time constant of SSD process versus the one determinedby assuming SSD can be approximated by an exponential decay multipliedby the ETISD curve calculated without the influence of SSD. The system isthe ensemble of particle pairs with the separations d = 8 Å (blue squares)and d = 5 Å (red diamonds). Solid lines are linear fits (R2 > 0.999). Thedashed line is an aid to the eye that would occur if the approximation gavethe correct result. The true time constant is always longer than the fit timeconstant, demonstrating that the ET/SSD coupling results in faster ETISDthan would occur without the two processes being coupled. The faster the EToccurs, the faster the fit time is relative to the actual time. (b) Enhancementof the underlying ET process by the inclusion of SSD is observed through theself part of the Green’s function for the d = 8 Å case. Gs(t) curves are shownwhen SSD is excluded (dashed curve) and for the ET coupled to SSD withtime constants between 25 ps and 800 ps (solid curves; same time constantsas in Figure 11). Gs(t) decays faster as the SSD process becomes faster.

2. N-particle case

The general N-particle case corresponds to any system– 1D, 2D, 3D, periodic, or non-periodic. Before describingresults for a specific system, the general methodology will berevisited.

To solve the N-particle system, at each time step the in-stantaneous frequencies drawn from the SSD process are usedto calculate the matrix propagator given by Eq. (7) for a sim-ulation time step of τ . This propagator is used to update thefrequency probabilities for the initial excitation as it is trackedacross the manifold of acceptors. The new probability vectoris passed to the next iteration and a new propagator is cal-

0 200 400 6000

200

400

600

TS

SD a

ctua

l (ps

)

TSSD

fit (ps)0.6

0.8

1.0

F (

t)

(a)

No SSD

2D Lattice0 0

0.2

0.4FF

CF SSD- 800 ps

SSD - 25 ps

0.0

0.6

0.8

F(t

)

(b)

0 200 400 6000

200

400

600

TS

SD a

ctua

l (ps

)

TSSD

fit (ps)No SSD

SSD- 800 ps

0 0

0.2

0.4

FF

CF

2D Random SSD - 25 ps

0 20 40 60 80 1000.0

t (ps)

FIG. 13. Results of full coupled ET/SSD calculation for two 2D geometries.(a) Normalized FFCFs with the homogeneous component omitted for the 2DHCP lattice system (with fixed polar angle model with θ = 45◦) for 25-800ps SSD time constants (same as in Figure 11). Dashed curve is ETISD pro-file without coupling to SSD. Black solid curves are the FFCFs from the fullMonte Carlo solutions to the coupled ET/SSD problem. Red dashed curvesare the result of assuming that FFCF can be approximated as an exponen-tial decay multiplied by the ETISD curve calculated with no SSD. Labelslie above their respective curves. (a, inset) Comparison of the multiplicativeapproximation time constants to the simulated SSD time constants (squares).Red line is a linear fit with a slope of 1.17. The dashed line is an aid to the eyethat would occur if the approximation gave the correct result. (b) NormalizedFFCFs with the homogeneous component omitted for the 2D random withexcluded volume system (with fixed polar angle model with θ = 45◦) for 25-800 ps SSD time constants. Curves are as in Panel (a). Labels lie above theirrespective curves. (b, inset) Comparison of the multiplicative approximationtime constant to the simulated SSD time constant (squares). Red line is a lin-ear fit with a slope of 1.14. The dashed line is an aid to the eye that wouldoccur if the approximation gave the correct result.

culated using the new frequencies given by the SSD processprogression.

As all N-particle systems are essentially the same prob-lem, we show only a calculation of the SSD coupled ETISDproblem for a two-dimensional hexagonal lattice model andthe random distribution with excluded volume model usingthe parameters from the Appendix. The angular model isa fixed polar angle model with a 45◦ angle with respect tosurface normal. Figure 13(a) shows the results for the 2Dhexagonal lattice case. The dashed black curve is the normal-ized FFCF with the homogeneous component omitted withno SSD. The solid black curves are the FFCFs for the systemswith SSD time constants ranging from 25 to 800 ps. Clearly,as the SSD rate increases, the FFCF decays more rapidly. The


dashed red curves are the result of assuming that the FFCFcan be approximated by an exponential decay multiplied bythe ETISD FFCF with SSD omitted. A single exponential isused due to the underlying single exponential SSD process.This approximation agrees reasonably well with the fullsimulations. The inset of Figure 13(a) shows the relationshipbetween the time SSD constants used in the multiplicativeapproximation to the FFCF and the underlying SSD timeconstants. A smaller time constant in the fit is indicative of en-hancement of the underlying ET process as discussed above inthe ensemble of particle pairs case. In this case the true under-lying time constant is 1.17 times the fit constant. Figure 13(b)shows similar results for the 2D random distribution withexcluded volume model. Again, the normalized FFCF withhomogeneous component omitted is shown for the case of noSSD (dashed curve) and SSD time constants between 25 psand 800 ps (solid black curves). The red dashed curves showthe approximation that the FFCF is equal to an exponentialdecay multiplied by the ETISD FFCF with SSD omitted.Again this approximation agrees well with the FFCFs fromthe full simulation. The inset in Figure 13(b) shows theagreement between the fit time constant and the true timeconstant input into the simulation. For this system the trueunderlying time constant is 1.14 times the fit time constant.There is a common trend in the enhancement factors acrossall systems simulated, both pairs of particles and N particles.The faster ETISD occurs without SSD, the faster the apparentexponential decay time constant is relative to the underlyingSSD process time constant due to the enhancement effect ofSSD on ET.

The multiplicative approximation (that the measuredFFCF is equal to the SSD FFCF multiplied by the ETISDonly FFCF) allows for the estimation of SSD process ratesfor systems undergoing simultaneous ETISD and SSD evenwhen SSD has not been included in the calculation of ETISD.Therefore, it is possible for the experimentalist to assess therate of SSD in the system once a model for ET has been cho-sen. The error in using this approximation has been shown tobe on the order of 5%–20% for systems undergoing ET at therates illustrated in this paper.

3. Coupled ETISD and SSD withinthe Huber approximation

The Huber approximation can be used to solve thecoupled ET/SSD problem using the graphical propagationmethod illustrated in Figure 10. First, two frequency vectorsare independently drawn from the SSD noise process. Us-ing the Gs(t) curves from the Huber calculation for each fre-quency, the ET process is solved. Specifically, the intersectionbetween the remaining probability of excitation on the ini-tially excited molecule is denoted as E0(t) (E0(0) = 1), withthe G

{ω1,ω2}s (t) curve for the two SSD drawn frequencies ω1,

ω2 at the time t. The initial excitation is then updated to thenew value E0(t + τ ) = G

{ω1,ω2}s (t + τ ) where τ is the simula-

tion time step and the process is repeated. The FFCF is calcu-lated in the same manner as for the ensemble of particle pairscase.

Γ = 1 × 1014 cm-2

0.6

0.8

1.0

(t)

(a)

SSD- 800 ps

No SSD

0 20 40 60 80 1000.0

0.2

0.4FF

CF

SSD - 25 ps

0.8

1.0

Γ = 2 × 1014 cm-2

t (ps)

(b)

No SSD

0.2

0.4

0.6

FF

CF

(t) SSD- 400 ps

exp(-t/25)

0 20 40 60 80 1000.0

t (ps)

SSD - 25 ps

FIG. 14. (a) Comparison of normalized FFCFs with the homogeneous com-ponent omitted for combined ET/SSD models from a 2D random spatial dis-tribution (with fixed polar angle model, θ = 45◦) using the full Monte Carlomethod (black curves) and the Huber method (dashed red curves) for SSDtime constants from 25 ps to 800 ps (same as in Figure 11). The slowest de-caying curves (both dashed and solid) have no SSD process. Surface density() is set to 1 × 1014 cm−2. Labels lie above their respective curves. (b) Samequantities except for a surface density () of 2 × 1014 cm−2 and SSD timeconstants of 25 ps, 100 ps, and 400 ps. The blue curve shows an exponentialdecay of 25 ps, illustrating that even when SSD is fast compared to ETISDcalculated with no SSD the use of an ET model significantly alters the curveshape. The slowest decaying curves (both dashed and solid) have no SSDprocess. Except for the bottom most curve, labels lie above their respectivecurves.

Figure 14(a) illustrates the agreement between the nor-malized FFCFs with the homogeneous component omittedfor the coupled ETISD/SSD problem solved within the Huberapproximation and the full Monte-Carlo solution for the half-density (1 × 1014 cm−2) 2D system. The lines are plottedin pairs of solid (full Monte Carlo calculation) and dashed(Huber approximation) curves. The top pair (dashed blackcurve; solid black curve) are for the case of no SSD (thesecurves are the same as the red curves shown in Figure 8 - 2D).The next eight pairs of curves (red dashed curves; black solidcurves) correspond to simulations with SSD time constantsof 800–25 ps. The Huber approximation in this low densitycase agrees excellently with the full Monte Carlo calculation.Interestingly, as the SSD rate increases the Huber approx-imation improves in agreement which is best seen in thedeviation of the dashed and solid curves at 100 ps. The SSD


dependent agreement is due to the fact that as the ETISDbecomes less of a factor in determining the overall FFCF,the accuracy of the ETISD solution is less important.Figure 14(b) shows the comparison between the higherdensity (2 × 1014 cm−2) SSD/Huber approximation andthe full Monte-Carlo solution. As in panel (a), the top pairof curve (dashed – Huber; solid – full Monte Carlo) isfor the case of no SSD (these are the black curves fromFigure 8 - 2D). The next three pairs of curves (red dashed– Huber; black solid – full Monte Carlo) are for the SSDtime constants 400ps, 100ps, and 25 ps. The agreement ofthe Huber approximation in the case of no SSD is seen to bequite good up to about 30 ps, after which it rapidly degrades.This is discussed above and is due to the high concentrationchosen for the simulations. The agreement however improvesas the SSD rate is increased. At a time constant of 25 ps, theagreement is essentially quantitative. To illustrate that theSSD coupled Huber approximation with a fast time constantis still different from just assuming that ETISD is negligible(i.e., the FFCF is just the SSD component), an exponentialdecay of 25 ps is shown as well (blue curve). Even when SSDis fast relative to ETISD, including the ET process is impor-tant, as fitting the FFCF with a single exponential decay SSDonly model results in a decay time of 21 ps, a 16% additionalerror. These results indicate that the SSD coupled Huber ap-proximation is a valid approximation to the full Monte-Carlosimulation approach, and is essentially exact when SSD ismuch faster than ETISD without SSD. It is conducive to usethis method when the Monte-Carlo method requires excessivecomputation and the radial distribution function of the systemallows for easy calculation of the integral in Eq. (11).

VI. CONCLUDING REMARKS

The theory developed above to describe ETISD in thecase of both static and stochastic fluctuating inhomogeneitiesis broadly useful throughout spectroscopy. In any densesystem in which excitation transfer can occur, our theorycan help elucidate the underlying spectral dynamics. Further-more, other experimental observables often depend implicitlyupon the excitation transfer process such as the orientationalanisotropy. Observables can only be fully understood after ac-counting for the impact of ET, which depends on SSD. Sincethe ET is itself modified by any underlying SSD processes,a full understanding of the spectral dynamics is necessaryto interpret any experiment in systems where the distancebetween chromophores approaches the Förster radius thatwould apply for homogeneously broadened lines. Formeasurements of SSD, even when the SSD is much fasterthan ETISD determined in the static limit, substantial errorsare possible if ET is not accounted for. Fast SSD does notoverwhelm ETISD calculated in static limit because SSDcauses ETISD to become faster than the static limit. As therate of SSD increases, so does the rate of ETISD. To properlyaccount for ET, it is necessary to know the homogeneouslinewidth and the SSD. Only in systems in which the homo-geneous linewidth is close to the total linewidth the standardFörster method can be used because there is essentially noSSD.

Specifically, we have solved the “forward” problem;given a set of parameters that describe excitation transfer andstructural spectral diffusion, the FFCF (or any other observ-able) may be calculated. To solve the “inverse” problem, thatis, given the data, it is desired to determine the dynamicalparameters, which is the problem faced by the experimental-ist, one would simulate a combined ET/SSD system muchlike the above cases and vary the parameters of interest un-til agreement with the experiment is obtained. Specifically,we recommend the following protocol given that an experi-mentalist wishes to measure an observable which depends onexcitation transfer. First, measure the parameters required tosimulate the excitation transfer, which includes the transitiondipole moment, the non-radiative lifetime (which combinedwith the transition dipole gives the quantum yield60), the ho-mogeneous linewidth, and the inhomogeneous linewidth. Ifthe system undergoes spectral diffusion, then the experimen-tal spectral diffusion must be measured using 2D IR or an-other method. By simulating the coupled ET/SSD system andcomparing the FFCF to the experimental FFCF, it is pos-sible to extract the underlying SSD time constant(s). Onlynow is it correct to calculate another observable that de-pends on the excitation transfer. For example, suppose theobservable of interest is the orientational anisotropy. Onlyafter accounting for coupled ETISD and SSD is it possibleto determine the amount of anisotropy decay due to exci-tation transfer. Excess anisotropy decay beyond the amountdue to excitation transfer can be attributed to orientationalrelaxation.

The theory presented here also illustrates the broad utilityof ultrafast nonlinear optical spectroscopy and specificallytechniques such as two-dimensional infrared spectroscopy.Only such techniques can extract the underlying homo-geneous and inhomogeneous linewidths of chromophores.These linewidths are necessary inputs into a microscopicmodel of the ET or ETISD process. Furthermore, techniquessuch as 2D IR can probe the spectral diffusion, anothernecessary input for calculating the ETISD process. Withoutsuch techniques it is impossible to take apart the underlyingdynamics.

The Monte-Carlo based techniques developed here alsohave broad applications in the molecular simulation commu-nity. Instead of full quantum mechanical models, it is pos-sible to build our dynamic system approach on top of clas-sical MD. The structures upon which the ET dynamic sys-tem is solved can be sequential snapshots from a MD simu-lation. Excitation transfer can therefore be incorporated intoMD models of spectroscopic line shapes quite easily. Thisextension of the methods discussed above into MD simula-tions will permit the study of systems where translational androtational motions happen on the time scales of excitationtransfer.

ACKNOWLEDGMENTS

We thank the Air Force Office of Scientific Research un-der Grant No. FA9550-12-1-0050 for support of this research.D.E.R. also thanks the Fannie and John Hertz Foundation,


the National Science Foundation, and the Stanford GraduateFellowship programs for graduate fellowships.

APPENDIX: PARAMETERS USED FOR EXCITATIONTRANSFER CALCULATIONS

To illustrate trends and the underlying physics of excita-tion transfer induced spectral diffusion, consistent model pa-rameters were used across all simulations. The values usedare representative of many molecular systems but do not cor-respond to a specific one.

The “chromophore” modeled has a vibrational reso-nance at 5 μm (2000 cm−1) with an integrated absorptionstrength of 1.8 × 104 M−1 cm−2 (integration performedin wave numbers), in a medium of index of refraction1.4 corresponding to a transition dipole moment of0.34 D in a medium of index of refraction 1.4. The“measured” lifetime is 10 ps which is taken to be thenon-radiative lifetime. Combining the non-radiative lifetimewith the transition dipole moment it is calculated60 that thequantum yield is 4.1 × 10−9. The homogeneous (Lorentzian)linewidth, except where noted otherwise, is taken to be2 cm−1. The inhomogeneous (Gaussian) linewidth, exceptwhere noted otherwise, is taken to be 16 cm−1. The fulllinewidth using these two linewidths is 17.093 cm−1. Thefrequencies were gridded on a 0.5 cm−1 mesh and the gridwas allowed to extend to 2.5 inhomogeneous half widths ineither direction from the line center leaving a 0.32% chanceof finding a center frequency outside of the simulated region.

For ensemble of pairs of particle systems, the two par-ticles are taken to be 8 Å apart and κ2 is taken to be in thedynamic limit and equal to 2/3.

For one-dimensional systems, a density of 1.43 × 107

molecules/cm is used, which corresponds to a lattice constantof 7 Å for the one-dimensional lattice. The hard sphere modelsimulation was run with a hard sphere radius of 2.62 Å and theexclusion radius in the random simulation was set to 5.0 Å.All simulations were performed in boxes of ∼120 Å length.Angular models for the 1D system were statically averagedand isotropically distributed.

For the two-dimensional systems simulated, the densityis assumed to be 2 × 1014 molecules/cm2 except where oth-erwise noted. The molecular diameter (exclusion radius) istaken to be 6 Å for the random distribution with excluded vol-ume model, except where otherwise noted. The hard sphereradius for the loosely packed hard sphere model was 2.92 Å.Prior to Sec. V, except where otherwise noted, the distributionof angles is static and isotropic. For the coupled SSD/ETISDcalculations a fixed polar angle model is assumed. The polarangle, θ , of the transition dipole away from the surface nor-mal is assumed to be 45◦. The azimuthal angle that describesthe orientation of the transition dipole in the surface planehas a uniform random distribution. For this fixed polar an-gle model, given the angles ϕd and ϕa denoting the azimuthalangles of the donor and acceptor chromophores away fromthe donor-acceptor separation vector, κ(ϕ1, ϕ2, θ ) = cos 2(θ )+ sin 2(θ )(sin (ϕ1)sin (ϕ2) − 2cos (ϕ1)cos (ϕ2)). Simulationswere run in 60 Å by 60 Å boxes.

For three-dimensional systems, the transition momentsare isotropically distributed in either the dynamic or staticlimits and a single density is simulated 2.92 × 1021

molecules/cm3. This corresponds to a 7 Å lattice constant ina simple cubic lattice. The hard sphere model for 3D systemswas run with a hard sphere radius of 3.0 Å and the exclusionradius in random with excluded volume models was set to6.0 Å. Random and hard sphere models were run in a cubewith 30 Å sides.

For all simulations other than the ensemble of particlepairs simulations, periodic boundary conditions were en-forced by the minimum image criterion. Hard sphere distribu-tions were generated by filling a volume with fixed positionhard spheres randomly until no more spheres could beadded.

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