FRET - Förster Resonance Energy Transfer || Förster Theory

3F€orster TheoryB. Wieb van der Meer

3.1Introduction

Theory without experiment is like the sound of one hand clapping. F€orster theory isnot like that at all. It is a thundering ovation linking theory and experiment byexplaining the relationship between spectral overlap, energy transfer, and proximity.This chapter explains F€orster’s contributions to the theory of resonance energytransfer. The readers of this chapter form, no doubt, a highly diverse group ofpeople. Most readers are probably only interested in the bottom line. Others maywant to know details. But which details? There are so many. To help students andspecialists find what they need, the chapter is presented as a sequence of a largenumber of sections that are short and focused.

3.2Pre-F€orster

This section is based on some of the information in the most popular papers byF€orster [1–5], Chapter 5 of Ref. [6], and Clegg’s history of FRET [7]. The emphasishere is on the contributions of F€orster’s predecessors and contemporaries.If you want to know who the scientists were who inspired F€orster and what the

science was thatmotivated him, you should read hismost important papers.Hismostimportant, that is, his most cited papers are his papers published in 1946 [1] 1948 [2],and 1949 [4] and reviews published in 1959 [3] and 1965 [5]. F€orster’s papers are noteasy to understand. The language is not a problem because four of the five are inEnglish or translated into English. They are difficult because they use a lot ofmath andcomplicated spectroscopic concepts. Nevertheless, if you are serious about FRET, youshould study them. Start with his 1946 paper [1] and the 1959 review [3]. These papersare much more readable than F€orster’s most cited paper [2], because his 1946 paperpresents a very clear verbal description of the essential ideas on which the theory isbased and a thorough review of the experimental evidence of the importance ofresonance andhis 1959paper is designed toprovide a conceptual understandingof the

FRET – Förster Resonance Energy Transfer: From Theory to Applications, First Edition.Edited by Igor Medintz and Niko Hildebrandt.� 2014 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2014 by Wiley-VCH Verlag GmbH & Co. KGaA.

j23

FRET phenomenon with a minimum of equations, whereas his 1948 paper is arigorous treatment of the theory. In Ref. [3], he gives an overview of the then availableliterature. For example, he mentions experiments proving that the observed transfermechanism could not be due to trivial reabsorption. His discussion of the varioustransfer mechanism is very interesting. He compares, in Table 1 in Ref. [3], FRET,reabsorption, donor–acceptor complex formation, and collisional quenching. Thistable has been adopted in a slightly modified form in Section 3.7. In Refs [1,2], hefocuses on the theory of homotransfer, FRET between likemolecules, whereas in Ref.[4] (only available in German), he emphasizes heterotransfer, FRET between unlikemolecules. If one is interested in the fundamental aspects of the theory, refer to Refs[1,2,5]. Reference [5] is a review. In this paper, he doesmore than just rehash his FRETtheory. Of the 10 sections, only the last one is about FRET. In his 1965 paper, he alsodiscusses exciton theory, strong coupling,weak coupling, andveryweak coupling (veryweak coupling is the basis for FRET). His 1965 paper introduces an extension of histheory put forward in his 1948 paper. This extension allows a description of the timedependence of the donor fluorescence and the relation between the quantum yield ofthe donor and the acceptor concentration (see Sections 3.16 and 3.17).Newton said, “If I have seen further it is only by standing on the shoulders of

giants” [8]. Who were the giants for F€orster? J. Perrin and F. Perrin, Cario, Franck,Kallmann, and London. In the late 1940s when F€orster started his work on energytransfer, the phenomenon of sensitized fluorescence was well established [2–5,7].Cario had shown in 1922 that transfer of energy had taken place from excitedmercury atoms to thallium atoms in a mercury–thallium vapor mixture [9]. Carioand Franck had presented similar results in a mercury–hydrogen system [10]. Manyother experimentalists had presented evidence for sensitized fluorescence from thevapors of silver, cadmium, lead, zinc, and indium in the presence of mercury vapor[7,9,10]. The starting point for a theoretical framework showing the role of resonancein energy transfer was Franck’s principle: If effective energy transfer is to take placefrom initially excited molecules to quenching molecules, the excited states of thequenchers must be in energy resonance with the primarily excited states [11].Kallmann and London [7,12] proposed a theory of energy transfer that can beconsidered to be a precursor of F€orster’s theory in that it is based on the idea ofresonance and has the correct distance dependence of the transfer efficiency.However, unrealistically sharp spectra were assumed and the link between thedistance dependence and spectra had limited significance. J. Perrin [13,14] intro-duced a classical theory modeling the fluorophores as electrical dipoles oscillating ata single frequency with a rate of transfer proportional to the inverse of the distance tothe third power, not the sixth power as F€orster later found. As a result, the predicteddistance over which energy transfer would take place is much too large [7]. F. Perrin[15,16] designed a quantummechanical version of this theory extending the work byKallmann and London to transfer in solutions. The molecules are assumed to havetwo states, a ground state and an excited state, so that the spectra would show sharppeaks. F. Perrin found the same distance dependence as J. Perrin did. However, F.Perrin did invoke collision broadening of the spectra by the solvent, decreasing thepredicted range of transfer, but it was still too large [7]. An interesting overview of the

24j 3 F€orster Theory

contributions of J. Perrin and F. Perrin, father and son, is available [17]. Oppen-heimer and Arnold [7,18,19] pointed out that the phenomenon of resonance energytransfer is very similar to internal conversion in radioactivity where an excitednucleus transfers energy without radiation to one of the orbital electrons resulting inejection of this electron. Using this similarity, they derived an expression of the rateof energy transfer for the case where the acceptors are randomly distributed arounda donor. Clegg showed that modifying their expression for the rate of transfer fromone dominant frequency to a spectrum of frequencies leads to F€orster’s famousequation for the donor–acceptor distance at which the rate of transfer equals that ofdonor emission [7] (see the equation forR0 in Section 3.3). However, the fact is thatOppenheimer and Arnold did not make these modifications. They did not come upwith the idea to incorporate experimentally obtained spectra into their theory.F€orster did [1–5]. This is what sets F€orster apart from his predecessors andcontemporaries. They all assumed one dominant frequency and ignored experi-mental data on spectra. F€orster’s most important innovation was to incorporateexperimentally obtained parameters such as spectra, quantum yield, and lifetimesinto his theory, making it refutable, accessible, and extremely useful.

3.3Bottom Line

To observe FRET, the following conditions must be met:

1) Donor and acceptormust have strong electronic transitions in the UV, visible, or IR.2) Spectral overlap must exist between donor emission and acceptor absorbance

(see Section 3.5).3) Donor and acceptor must be close, but not too close (see Sections 3.6 and 3.7).4) The orientation factor should not be too small (see Section 3.8).5) The donor emission should have a reasonably high quantum yield (see Equa-

tion 3.3 and also data in Chapter 14).

The following are the key quantities in F€orster’s theory:

kT ¼ rate of energy transfer (see Equation 3.2).tD ¼ lifetime of the donor excited state in the absence of acceptor (see Equa-tions 3.1 and 3.2).rDA ¼ distance between donor and acceptor.R0 ¼ F€orster distance, that is, the donor–acceptor distance at which kT ¼ 1=tD, so thatat that particular distance, theprobability of the exciteddonor tofluoresce is equal to theprobability of transfer of energy to the acceptor (see Equations 3.3 and 3.3a–3.3c).E¼ efficiency of transfer (see Equation 3.1).J¼ overlap integral (see Section 3.5 and Equation 3.4).k2 ¼ orientation factor (see Section 3.8 and Equation 3.3).WD ¼ quantum yield of the donor fluorescence in the absence of acceptor (seeEquation 3.3).n¼ refractive index of the medium (see Equation 3.3).

3.3 Bottom Line j25

Constants in F€orster’s theory are as follows:

p ¼ 3:141592654;

ln 10 ¼ 2:302585093;

NA ¼ 6:0221415� 1023 per mole ðactually; F€orster usedN 0

¼ 6:0221415� 1020 per millimoleÞ:The conclusion of F€orster’s theory can be conveniently written as the following set

of three equations:

E ¼ kTkT þ 1=tD

¼ R60

R60 þ r6DA

: ð3:1Þ

kT ¼ 1tD

R60

r6DA

� �: ð3:2Þ

R60 ¼

9 ln 10ð Þk2WDJ128p5n4NA

: ð3:3Þ

For a derivation of these equations from classical theory, see Sections 3.5–3.14(Sections 3.5–3.7 introduce basic ideas, the derivation starts in Section 3.8), andfrom quantum mechanical theory, see Section 3.15. Note that the second equationfollows from the first (and the first from the second). An alternative expression forthe rate of transfer kT is

kT ¼ k2

r6DA

� �1n4

� �CDA; ð3:2aÞ

with CDA ¼ 9 ln 10ð ÞWDJ128p5tDNA

: ð3:2bÞ

This formulation, which is often used in photosynthesis, establishes a clearseparation between spectral properties (CDA), geometric properties (k2=r6DA), andenvironmental factors (n) [20]. When rDA is in nanometers and kT is in inversepicoseconds, CDA is expressed in nm6/ps.Figure 3.1 illustrates the relations between efficiency and donor–acceptor distance

and F€orster distance and between F€orster distance and kappa-squared, overlapintegral, refractive index, and quantum yield.

3.49000-Form, 9-Form, and Practical Expressions of the R60 Equation

F€orster used N 0 instead of NA in Ref. [2], but used NA with 9000 instead of 9 in Ref.[3]. However, N 0 ¼ NA as both have a unit and represent the same amount of

particles per mole: N 0 ¼ 6:02� 1020 mmol�1 ¼ 6:02� 1023 mol�1 ¼ NA. Braslav-sky et al. pointed out that the frequently quoted 9000-form of Equation 3.3 (with afactor of 9000 instead of 9) is incorrect [21]. Simplifying Equation 3.3 by substituting


Figure 3.1 (a) The transfer efficiency E versusthe donor–acceptor distance rDA between 0 and2 times the F€orster distance. (b) The transferefficiency E versus the F€orster distance R0between 0 and 2 times the donor–acceptordistance. (c) The relative F€orster distanceversus the orientation factor, k2, which variesbetween 0 and 4. R0 [2/3] is the F€orster distanceat k2 ¼ 2=3. (d) The relative F€orster distanceR0/R0( J/J1) versus the relative overlap integralJ/J1, where J is the overlap integral and J1 is atypical value of this integral, say 1 OLI.R0( J¼ J1) is the F€orster distance that at J¼ J1.

J-values vary over a wide range (see Chapter14). (e) The relative F€orster distance versus n,the refractive index of the medium in which thedonor and acceptor are embedded. R0(n¼ 1.4)is the F€orster distance for the refractive indexequal to 1.4. All refractive index values in theliterature are in the 1.33–1.6 range. The values1.34 and 1.6 are the ones used most frequently.(f) The relative F€orster distance versus FD, thequantum yield of the donor in the absence ofthe acceptor R0 (FD ¼ 0:5) is the F€orsterdistance at FD ¼ 0:5. FD varies between 0 and1 (see Chapter 14 for data).

3.4 9000-Form, 9-Form, and Practical Expressions of the R60 Equation j27

in the constants with units shown explicitly yields [21]

R0

nm¼ 0:02108

k2WD

n4Jl

mol�1 dm3 cm�1 nm4

� �� 1=6: ð3:3aÞ

Equation 15 in Ref. [22] and the expressions given on page 16 of Ref. [6] are equallyvalid, but the expression (3.3a) is preferable as it has the advantage that the choice ofunits is absolutely clear, so that mistakes can be easily avoided. In practice, the OLI(overlap–integral unit, introduced in Chapter 7 of Ref. [6]) is convenient. This unit is

defined as OLI ¼ 1014 mol�1 dm3 cm�1 nm4 ¼ 10�14 mol�1 dm3 cm3, and is usedin Chapter 14. Alternative forms of (3.3a) using the OLI are as follows:

R0

nm¼ 2:108

k2WD

n4100Jl

OLI

� �� 1=6: ð3:3bÞ

R0

nm¼ 4:542

k2WD

n4Jl

OLI

� �� 1=6: ð3:3cÞ

The following is a step-by-step derivation of Equation 3.3a from Equation 3.3:

1) Divide both sides of Equation 3.3 by nm6 and substitute in all the constants,including the unit of Avogadro’s number, and Equation 3.3 thus becomes

R60

nm6¼ 9ð2:302585Þ

128ð306:0197Þð6:022� 1023Þmol�1

( )k2WD

n4

� �J

nm6

� �:

2) Move mol�1 over to J and convert nm6 to dm3 cm�1 nm4 using1017 nm2

� = dm3 cm�1� ¼ 1.

R60

nm6¼ 9ð2:302585Þ

128ð306:0197Þð6:022� 1023Þ� �

k2WD

n4

� �J

mol�1 nm6

� �1017 nm2

dm3 cm�1

� �:

3) Simplify yielding:R60

nm6¼ 87:8533� 10�12

� k2WD

n4

� �J

mol�1 dm3 cm�1 nm4

� �:

4) Take the sixth root and arrive at Equation 3.3a.

3.5Overlap Integral

The overlap integral can be calculated using wavelength (3.4), wave number (3.6), orfrequency (3.8). Most frequently, the wavelength form is used. This is often referred

to as Jl and defined as:

Jl ¼ J ¼ðf D lð ÞeA lð Þl4dl: ð3:4Þ

The integral in Equation 3.4 extends over the region that encompasses the lineshapes of the relevant donor emission and acceptor absorption bands. Extending the


integrals from zero to infinity may add irrelevant addenda to the integrals whenother areas of overlap occur at very high and/or low wavelengths not relevant to the

specific energy transfer process of interest.1) The overlap integral J ¼ Jl is conve-niently expressed in OLIs, l is the wavelength of the light, most often expressed innanometers, eA lð Þ is the molar extinction coefficient of the acceptor, usually in M�1

cm�1, and f D lð Þ is the fluorescence spectrum of the donor normalized on thewavelength scale:

f D lð Þ ¼ FDllð ÞÐ

FDllð Þdl ; ð3:5Þ

where FDllð Þ is the donor fluorescence per unit of wavelength interval and the

integral extends over the relevant donor emission band(s). A schematic illustrationof overlap is Figure 3.2.The green area in Figure 3.2a is not the overlap integral. The green area only

shows that there is overlap and is certainly not a reliable measure for the magnitudeof the overlap integral. The donor emission f D lð Þ (blue curve in Figure 3.2) isusually expressed in 1/nm. The acceptor extinction eA lð Þ (red curve) is inM�1 cm�1,and the overlap curve (green) is in OLI/nm. The area under the green curve inFigure 3.2b is the overlap integral. There are three different vertical scales inFigure 3.2b. Therefore, by adjusting one scale with respect to the others, theappearance of this figure can be adjusted. However, whatever scale adjustmentis made, Figure 3.2a can never be made to resemble Figure 3.2b, because the

Figure 3.2 (a) Donor emission spectrum (bluecurve) and acceptor extinction spectrum (redcurve) versus wavelength. The green areaindicates that there is overlap between the twospectra. The green area is not the overlapintegral. (b) This schematic graph has threedifferent vertical scales, one in nm�1 for the

blue curve (normalized donor fluorescence, thearea under this curve is one), another in M�1

cm�1 for the red curve (acceptor extinction),and the third scale is in OLI/nanometer for thegreen curve (overlap curve). The area under thegreen curve is the overlap integral.

1) Andrews, D.L. (2012) Private communication.

3.5 Overlap Integral j29

wavelength to the fourth factor will cause the peak of the overlap curve to be always ata wavelength larger than the one where donor emission and acceptor extinctionintersect. The overlap integral in wave number form is

J ¼ðf D ~nð ÞeA ~nð Þ

~n4d~n: ð3:6Þ

Here, the integral extends over the region that encompasses the relevant donoremission and acceptor absorption bands in terms of wave number, ~n ¼ 1=l,conveniently expressed in 1/nm, eA ~nð Þ is the molar extinction coefficient of theacceptor, usually in M�1 cm�1, and f D ~nð Þ is the fluorescence spectrum of the donornormalized on the wave number scale:

f D ~nð Þ ¼ FD~n ~nð ÞÐFD~n

~nð Þd~n; ð3:7Þ

where FD~n ~nð Þ is the donor fluorescence per unit of wave number interval and theintegral extends over the relevant donor emission band(s).The overlap integral in frequency form is the one appearing in F€orster theory (see

Section 3.12):

J ¼ c4ðf D nð ÞeA nð Þ

n4dn: ð3:8Þ

Here, the integral extends over the region that encompasses the relevant donoremission and acceptor absorption bands in terms of frequency, n ¼ c=l, c is the

speed of light in vacuo, which is equal to about 3� 108 m=s (more precisely

c ¼ 2:99792458� 108 m=s). The frequency is conveniently expressed in hertz(¼1/s), eA nð Þ is the molar extinction coefficient of the acceptor, usually in M�1

cm�1, and f D nð Þ is the fluorescence spectrum of the donor normalized on thefrequency scale:

f D nð Þ ¼ FDnnð ÞÐ

FDnnð Þdn ; ð3:9Þ

where FDnnð Þ is the donor fluorescence per unit of frequency interval, and the

integral extends over the relevant donor emission band(s).At first sight, the conversions from Equations 3.4–3.6, and to 3.8 look inconsistent

with the rules of calculus, but they are actually correct and follow from the “first lawof photophysics” (Chapter 2 of Ref. [6]):

FDldl ¼ FD~nd~n ¼ FDn

dn: ð3:10ÞThe intensities are measured using monochromators or filters with a certain

resolution or bandwidth. The reading on the instrument is proportional to thisbandwidth if it is sufficiently small. The intensity is taken to be proportional to thereading per wavelength or wave number or frequency. This proportionality is theidea behind the first law of photophysics. Note that the extinction coefficienttransforms as eA lð Þ ¼ eA ~nð Þ ¼ eA nð Þ, because it is proportional to the logarithmof a ratio of intensities.


The overlap integral depends on both the absorption spectrum of the acceptor andthe emission spectrum of the donor. Absorption spectra vary as a rule relatively littlewith change of solvent or temperature, but emission spectra may be very sensitive toenvironment, a well-studied example being the fluorescence of indole derivatives[23]. Analytical results for the overlap integral have been derived for bands ofGaussian and log normal line shapes [24].

3.6Zones

F€orster visualized a donor as a group of electrical oscillators close together. Theseelectrical oscillators produce an electrical field in the space around the donor. Thisspace consists of four zones: the contact zone or Dexter zone [25], the near zone orthe near field, the intermediate zone, and the far zone (also called the far-field or theradiation zone). The concept of zones, illustrated in Figure 3.3, dates back to Hertz[7] who actually considered three zones: the near, intermediate, and far, because heset out to confirm Maxwell’s prediction of electromagnetic waves [7] and was notinterested in distances very close to the electrical oscillators.The zones can be defined in terms of a distance b:

b ¼ l

2pn; ð3:11Þ

Figure 3.3 The space around a donorfluorophore can be divided into four zones.These zones are shown here on a logarithmicscale with the outer radius of each ring being afactor of 10 larger than the inner radius. Thedonor occupies the center of the contact zone,which extends up to about a nanometer ormore depending on the donor size (seeTable 3.1). Around this zone is the near field,about 1–10 nm from the donor. The near field isthe only zone where F€orster theory applies.

Around the near field is the intermediate zonefrom 10–1000nm. Outside the intermediatezone is the far field where electromagneticradiation takes place. If the acceptorconcentration is sufficiently small – so that theprobability of finding an acceptor in the contactzone is very small – and the sample is not toolarge – so that the probability of reabsorption issmall, FRET is the dominant mode of energytransfer.

3.6 Zones j31

where l is the wavelength of the donor fluorescence that is usually in the 300–800 nm range and n is the index of refraction of the medium in which the donor andacceptor are embedded [26]. This index has values typically between 1.3 and 1.6. So, bis about 100 nm or a little smaller. Properties of these zones are listed in Table 3.1.FRET happens in the near field, that is, roughly in the 1–10 nm range. If it is less

than about 1 nm, F€orster theory does not apply for at least two reasons: First, thecomplex formation may occur between donor and acceptor at such a proximity (seealso Section 3.7 and Refs [3,20,25]). Second, the F€orster’s theory is based on the idealdipole approximation (IDA) and the IDA breaks down if the donor–acceptor distanceis on the order of 1 nm [28]. If the distance is larger than about 10 nm, contributionsto the electric field that are ignored in F€orster’s theory become relevant. In theradiation zone, the acceptor is capable of reabsorbing light emitted by the donor (seeSection 3.7).

Table 3.1 Zones around the donor.

Name Alternativename

Innerradius

Outerradius

Characteristics

Contact zone Dexterzone

0 0.01b(�1 nm)

The ideal dipole approximationbreaks down [25]. An acceptor inthis zone may form a complex withthe donor [3]. F€orster theory doesnot apply. For larger molecules(chlorophyll and porphorin), theouter radius is about 2–3 nm [27].The distance dependence of transferis reviewed in Ref. [20]

Near-fieldzone

Near field 0.01b(�1 nm)

0.1b(�10 nm)

F€orster theory is valid only in thiszone. The electric field due tooscillating donor charges can beconsidered as a sum of dipole termswith a 1/distance3 dependence. Theinner radius may be bigger forlarger molecules (see above)

Intermediatezone

0.1b(�10 nm)

10b(�1000 nm)

The electric field due to oscillatingdonor charges has three terms withdifferent distance dependence,none of which is dominant. F€orstertheory does not apply

Radiationzone

Far field 10b Infinite Electromagnetic donor emissiontakes place in this zone. The electricfield due to oscillating donor chargeshas a 1/distance dependence. Theelectric field lines are pinched offand transverse waves are formed [7].F€orster theory does not apply.Reabsorption will occur


3.7Transfer Mechanisms

F€orster was well aware of the fact that there are at least four different mechanisms bywhich excitation energy can be transferred from a donor to an acceptor. These areresonance energy transfer, reabsorption, complex formation, and collision quench-ing. Resonance energy transfer (also called FRET2)) [29], the main topic of this book,is the radiationless transmission of an energy quantum from its site of absorption tothe site of its utilization in a molecule, or system of molecules, by resonanceinteraction between chromophores, over distances considerably greater than inter-atomic without conversion to thermal energy and without the donor and acceptorcoming into kinetic collision. Reabsorption or “trivial reabsorption” is the emissionof a photon by the donor with the subsequent absorption of that photon by theacceptor. Complex formation is the creation of an excited-state complex of a donorand an acceptor that are in proximity, essentially in molecular contact with eachother. Collision quenching can occur when an excited molecule loses its excitationenergy to another molecule by colliding with it. As F€orster pointed out, these fourtransfer mechanisms have different characteristics and can, therefore, be distin-guished experimentally [3]. Table 3.2, adopted with minor modifications, from oneof F€orster’s papers [3], summarizes these different characteristics. Quantum

Table 3.2 Characteristics of transfer mechanisms.

Resonanceenergytransfer

Reabsorption Complexformation

Collisionquenching

Sample volume: with increasingvolume, transfer exhibits

No change Increase No change No change

Viscosity: with increasingviscosity, transfer exhibits

No change No change No change Decrease

Donor lifetime: because oftransfer, the donor lifetime shows

Decrease No change No change Decrease

Donor fluorescence spectrum:comparing transfer and notransfer. This spectrum is

Unchanged Changeda) Unchanged Unchanged

Donor absorption spectrum:comparing transfer and notransfer. This spectrum is

Unchanged Unchanged Changeda) Unchanged

a) Changes only apply to the wavelength, not to the intensity.

2) There is general agreement about theFRETacronym. However, there is noconsensus yet about the meaning of theletters in FRET. Many authors read it as“fluorescence resonance energy transfer,”while many others as “F€orster resonance

energy transfer.” The author of this chapterprefers “fluorescence with resonanceenergy transfer.” “Fluorescence-detectedresonance energy transfer” was proposedby Vanbeek et al. [29].

3.7 Transfer Mechanisms j33

electrodynamics teaches that all electric and magnetic interactions are mediated byphotons even in the near field. The photons in the near field are actually virtualphotons. As a result, FRET and trivial reabsorption can be interpreted as twodifferent aspects of the same phenomenon (see Section 3.19) [30].

3.8Kappa-Squared Basics

The previous sections can be considered to provide an introduction to F€orster theory.This is the first of a group of sections (Section 3.8–3.14) forming F€orster’s classicalderivation of his equations (Equations 3.1–3.3).The orientation factor, kappa-squared, is the square of k, which is defined as

k ¼ cos qT � 3cos qDcos qA: ð3:12ÞHere, qD is the angle between the donor emission transition moment and thedonor–acceptor connection line, qA is the angle between the acceptor absorptiontransition moment and the donor–acceptor connection line, and qT is the anglebetween the donor emission transition moment and the acceptor absorptiontransition moment. Kappa-squared varies between 0 and 4 and is discussed, inmore detail, in Chapter 4. The relation between the F€orster distance and the kappa-squared is shown in Figure 3.1c. In F€orster theory, k appears in terms of dot products

between d, a, and r, which are unit vectors: d along the donor dipole, a along the

acceptor dipole, and r along the line from the donor to the acceptor. d � r ¼ cos qD,

a � r ¼ cos qA, and d � a ¼ cos qT. Kappa in terms of dot products is

k ¼ d � a� 3 d � r �

r � að Þ: ð3:13Þ

The angles and unit vectors are illustrated in Figure 3.4.The amplitude of the donor dipole moment is qeDD (qe is the charge of an electron

andDD is the displacement of the charge, both shown below). It oscillates along the ddirection at frequency nDONOR (in general, there is a distribution of frequencies) and

Figure 3.4 Illustration of the angles uD, uA, and uT and the unit vectors d, a, and r.


generates an electric field at the location of the acceptor. Donor and acceptor areembedded in a medium with refractive index n, with a distance rDA between them.At time t, the electric field generated by the donor dipole at the location of theacceptor, in the near field, is given by

~E ¼qeDD 3 d � r

�r � d

h icos 2pnDONORtð Þ

4pe0n2r3DA: ð3:14Þ

The component of this field along the acceptor dipole is

~E �a ¼ �qeDDkcos 2pnDONORtð Þ4pe0n2r3DA

: ð3:15Þ

Themks unit system is used here. In the cgs system, the 4pe0 factor is replaced by 1.

3.9Ideal Dipole Approximation

The donor is a group of oscillating charges. We can imagine a sphere drawn aroundthe center of the donor with radius RD containing all these charges. Similarly, theacceptor is a group of oscillating charges contained in a sphere around its centerwith radius RA. In the near field, the distance rDA between the center of the donorand that of the acceptor is much larger than RD and also much larger than RA. As aresult, the ideal dipole approximation holds: The electromagnetic interactionbetween donor and acceptor is a dipole–dipole interaction, and all interactionsdue to higher multipoles can be ignored [31]. The ideal dipole approximation isillustrated in Figure 3.5. It must be emphasized that the relevant donor and acceptordipoles are not permanent dipoles but oscillating dipoles; in quantum mechanicalterms, they are transition dipoles.Consistent with this dominance of the dipole moment above all other multipoles,

F€orster visualized a donor or an acceptor molecule as a group of coupled electricaloscillators [32]. Each electrical oscillator consists of an electron elastically bound to anucleus. The nucleus is stationary,3) but the electron can oscillate along a certaindirection (not along other directions). The charge of the electron is qe(qe ¼ 1:60217646� 10�19 C) and its mass is me (me ¼ 9:10938188� 10�31 kg).The values qe and me do not appear in the final results because they are takenup by the spectral properties of the donor and the acceptor. The donor dipole is

situated at the center of the donor and has a direction d, a unit vector, and dipole

moment~p D ¼ qeDDd, whereDDd is a vector sum of fluctuating vectors oscillating at

a range of frequencies. The vector DDd is from the center of all positive charges tothe center of all negative charges in the donor. Similarly, the acceptor dipole issituated at the center of the acceptor and is along the unit vector a and has dipolemoment~p A ¼ qeDAa, whereDAa is a vector sum of fluctuating vectors oscillating at

3) In reality the nuclei do oscillate, but at frequencies that are irrelevant for FRET.

3.9 Ideal Dipole Approximation j35

a range of frequencies. It points from the center of all positive charges to the centerof all negative charges in the acceptor. We must realize that when a donor is excitedby an electromagnetic wave at a certain frequency, the donor will start oscillating at arange of frequencies and not only at the frequency of the wave, because theoscillators that make up the donor are coupled.F€orster used the cgs unit system, which was the system of choice in the 1940s and

1950s. Today (2012) this system is hardly ever used, and in this presentation of F€orstertheory, the mks system is used, which is also known as the SI system. In the cgssystem, theunit of charge is the statcoulomb (which is equal to the esu); but in themkssystem, the unit of charge is the coulomb (C ¼ A� s). One statcoulomb is equal to3.3356� 10�10C [33].Aconsequence of this difference inunits is thatmany equationsin electromagnetism using the cgs systemdiffer from the corresponding equations inthemks system [33].However, thefinal equations derived by F€orster do not depend onthe system of units, but intermediate equations in the theory, for example, those forenergy, power, and intensity, have different forms in the two systems.

3.10Resonance as an All-or-Nothing Effect

Resonance occurs when an oscillator capable of vibrating at a natural frequencyinteracts with an external system that forces this oscillator to vibrate at an external

Figure 3.5 A cartoon of a donor molecule, onthe left, and an acceptor molecule, on the right.In reality, both donor and acceptor containmany charges, which vibrate and oscillate inseveral directions and at a range of frequencies

on the order of 1 million GHz. When the size ofthe donor is much smaller than the donor–acceptor distance and the size of the acceptor isalso much smaller than this distance, the idealdipole approximation is valid.


frequency. The oscillator will pick up significant energy only if the externalfrequency and the natural frequency are equal or nearly equal. Resonance iseverywhere: accomplished singers can break a wine glass by hitting the right note[34], a sturdy bridge can collapse when jolted at the right frequency [35], and mostadults walk at a frequency of about 1Hz because adult legs swing at a naturalfrequency of about 1Hz. The resonance phenomenon that concerns us hereoccurs when acceptor charges oscillate in phase with donor charges at a distance.In his classical theory of energy transfer [32], F€orster visualized a donor or anacceptor molecule as a group of coupled electrical oscillators. When the donoroscillators are in an external electromagnetic field caused by light that can excitethe donor, the oscillators in the donor will oscillate and cause their own electro-magnetic field. Acceptor oscillators often will not respond to the external electro-magnetic field, but may be sensitive to the electric field from the donor oscillators.Let us first focus on one electrical oscillator inside an acceptor responding to anelectric field generated by a donor at some distance away. We will call the directionin which the acceptor oscillator can swing the acceptor direction. The naturalfrequency of this oscillator is nACCEPTOR. The electric field is also along a certaindirection, the donor field direction, which is not necessarily the same as theacceptor direction. However, this electric field must have a component alongthe acceptor direction, otherwise there will be no response. The amplitude ofthis component is EDF (which is equal to the amplitude of the donor field timesthe cosine of the angle between the two directions). The frequency of the donorfield is nDONOR (which is also equal to the frequency of the donor oscillator). Theelectric field starts at time 0 and lasts for a certain amount of time t, which canvary. The energy WA that the acceptor dipole has at time t as a result of itsinteraction with the donor field depends on the frequency differencenACCEPTOR � nDONOR. As a function of frequency, this energy has a strong maxi-mum when this frequency difference is zero, but also has weak secondary maximaat other values. F€orster made the approximation to replace this intricate resonancebehavior by a sharp rectangular peak. In other words, he assumed that either thereis resonance or there is nothing:

WA ¼

0; if nDONOR � nACCEPTOR <�12t

;

q2eE2D

8me� t2; if

�12t

� nDONOR � nACCEPTOR � 12t;

0; if nDONOR � nACCEPTOR >12t:

8>>>>><>>>>>:

ð3:16Þ

This approximation is schematically illustrated in Figure 3.6.From Equation 3.16 we see that the value of this peak increases drastically with

time, but the width of the peak decreases with time. For example, doubling the timeyields a fourfold increase in the value at the peak, but leads to a reduction by a factorof 2 in the width, as illustrated in Figure 3.7.

3.10 Resonance as an All-or-Nothing Effect j37

Figure 3.6 F€orster replaced the resonancecurve by a rectangular peak. Vertically theenergy of the acceptor is plotted andhorizontally the frequency. The center of the

peak (for the curve as well as the rectangularpeak) corresponds to the donor frequencybeing equal to the acceptor frequency.

Figure 3.7 The height of the resonance peak is proportional to time-squared, but the width isproportional to the time. As a result, doubling the time yields a fourfold increase in the height ofthe peak, but leads to a reduction by a factor of 2 in the width.


3.11Details About the All-or-Nothing Approximation of Resonance

The reader who has no problem accepting F€orster’s all-or-nothing approximation ofresonance and is not interested in the reasons why this approximation is goodshould skip this section.It turns out that F€orster’s all-or-nothing approximation of resonance is excellent. It

is not immediately obvious why. To prove that it is indeed very good, we must set upan equation of motion for the acceptor oscillator, solve it, calculate the energy of theacceptor, and compare this rigorous expression of the energy with the approximateequation displayed in Equation 3.16.Consider the forces acting on each acceptor oscillator. An electrical oscillator in

the acceptor can oscillate along a certain line, the acceptor direction, which we willidentify as the x-axis. The natural frequency of this oscillator is nACCEPTOR. Thisfrequency is related to the spring constant k and the mass of the oscillator me by

nACCEPTOR ¼ 12p

ffiffiffiffiffiffikme

s: ð3:17Þ

The x-component of the electric field generated by a donor oscillator hasamplitude ED and frequency nDONOR and at time t is

x-component of the electric field generated by donor ¼ EDcos 2pnDONORtð Þ:ð3:18Þ

From Equation 3.15, we know ED is equal to

ED ¼ �qeDDk

4pe0n2r3DA: ð3:19Þ

According to Newton’s second law, the net force on the oscillating charge, the sumof the elastic force�kx and the electric forceQeEDcos 2pnDONORtð Þ, equals themasstimes the acceleration:

med2

dt2x ¼ �kx þ qeDDcos 2pnDONORtð Þ; ð3:20Þ

where x is the displacement and qe is the charge of the oscillator. Substituting k ¼4p2men

2ACCEPTOR (from Equation 3.17) into (3.20), dividing by me, and utilizing the

abbreviations u ¼ 2pnACCEPTOR and w ¼ 2pnDONOR transform (3.20) into

d2

dt2x ¼ �u2x þ qeDD

mecos wtð Þ: ð3:21Þ

The solution of Equation (3.21) for the case where the initial displacement and theinitial velocity are zero is

x ¼ qeED

me u2 � w2ð Þ cos wtð Þ � cos utð Þð Þ; ð3:22Þ

3.11 Details About the All-or-Nothing Approximation of Resonance j39

so that the speed of the oscillator motion is

dxdt

¼ uqeED

me u2 � w2ð Þ sin utð Þ � wusin wtð Þ

�: ð3:23Þ

The energy of the acceptor, WA, the sum of the potential energy,

ð1=2Þkx2 ¼ ð1=2Þmeu2x2, and the kinetic energy, ð1=2Þme dx=dtð Þ2, is

WA ¼ q2eE2D

8me� t2

4u2

w2 � u2ð Þ2t2

" #cos wtð Þ � cos utð Þð Þ2 þ w

usin wtð Þ � sin utð Þ

�2� �

:

ð3:24ÞFor w ¼ u� ewith e � p=t, the term in square brackets equals 1= etð Þ2 and the termin curly brackets equals etð Þ2. Therefore, at very small differences between thedonor and acceptor frequencies, this energy is equal to its maximum value ¼WA;PEAK ¼ ½ðq2eE2

DÞ=8me � t2. Relevant values of t are on the order of the lifetime ofthe excited state, so that 2p=t is on the order of 1–10GHz, whereas u and wcorrespond to frequencies in the UV or visible and are, therefore, on the order of amillion gigahertz. As a result,WA has a series of minima that are essentially equal tozero for w ¼ u� Np=t, where N is an even integer larger than 0. The case w ¼u� p=t corresponds to the border of the rectangular F€orster peak. At that frequency,the actual WAvalue is about 0:4�WA;PEAK. With w ¼ u� Np=t where N is an odd

integer larger than 1,WA has secondary maxima equal to about 0:4�WA;PEAK=N2.The width of the all-or-nothing peak equals 1=t. The width of the actual resonancecurve can be defined in terms of the area under the curve. The total area under theall-or-nothing peak and that under the actual resonance curve turn out to beessentially the same. Relevant intervals for the resonance curve are as follows:

Width of frequencyinterval

Frequency interval in termsof vACCEPTOR � vDONOR

Approximate area underthe curve (% of total)

1/t �1/(2t) � vACCEPTOR � vDONOR � 1/(2t) 77%

20/t �10/t � vACCEPTOR � vDONOR � 10 99%

Most of the energy is transferred near the end of the lifetime of the excited state ofthe donor, where t is on the order of nanoseconds, so that the width of the 99%interval is about 20GHz. However, the relevant frequency values in the spectra areon the order of millions of gigahertz. This means that over 20GHz, the spectra donot vary at all. Since the all-or-nothing peak and the actual resonance peak have thesame area under the curve, the total amount of energy transferred in the all-or-nothing approximation is equal to that transferred according to the actual resonancecurve when t is on the order of a nanosecond. The all-or-nothing approximation canonly fail if there is significant spectral variation over the 99% interval. Such variationis expected when t is very small, less than a femtosecond. However, early in the


process, the energy is extremely small because of the time-squared factor in themagnitude of the peak (energy after a femtosecond¼ 10�12� energy after a nano-second). We conclude, therefore, that F€orster’s all-or-nothing approximation, replac-ing Equation 3.24 by Equation 3.16, is excellent.

3.12Classical Theory Completed

FromEquation (3.16) we know that if one donor oscillator and one acceptor oscillatorhave the exact same frequency n ¼ nDONOR, the acceptor has, due to a resonanceinteraction with the donor, t s after the donor has started oscillating, an amount ofenergy WA equal to

WA ¼ q2eE2Dt

2

8me: ð3:25Þ

Expressing WA in terms of the donor–acceptor distance using Equation 3.19, wefind

WA ¼ q4eD2Dk

2t2

8me 4pe0ð Þ2n4r6DA: ð3:26Þ

D2D is proportional to WD, the energy of the donor oscillator, because

WD ¼ 12kD2

D ¼ 2p2men2D2

D; ð3:27Þ

meaning that

D2D ¼ WD

2p2men2: ð3:28Þ

Substituting this into Equation 3.26 yields

WA ¼ q4ek2WDt2

16p2men2 4pe0ð Þ2n4r6DA: ð3:29Þ

Now we must generalize this to the case where there is not just one frequency butdistributions of frequencies for donor and acceptor. Such distributions can bedescribed using oscillator strengths f eD ¼ f eD nð Þ for the donor and f aA ¼ f aA nð Þ forthe acceptor. Specifically,

f eD ¼ f eD nð Þ ¼ probability to find a frequency between n and nþ dn: ð3:30ÞRemember that the width of the resonance peak, using F€orster’s all-or-nothing

approximation, is 1=t (see Equation 3.16). Therefore, relating Equation 3.30 to thecorresponding acceptor frequency interval, we find

f aA nð Þ 1=tð Þ ¼ acceptor frequencies resonating with donor: ð3:31Þ

3.12 Classical Theory Completed j41

It follows, therefore, that we should multiply both sides of Equation 3.29 withf eD nð Þf aA nð Þ 1=tð Þdn and integrate over all frequencies:

WA ¼ q4ek2WDt

16p2me 4pe0ð Þ2n4r6DA

ð10

f eD nð Þf aA nð Þdnn2

: ð3:32Þ

Differentiating this with respect to time gives us F€orster’s rate equation:

dWA

dt¼ kTWD; ð3:33Þ

with

kT ¼ q4ek2

16p2m2e 4pe0ð Þ2n2r6DA

ð10

f eD nð Þf aA nð Þdnn2

: ð3:34Þ

Substituting Equations 3.42 and 3.46 into Equation 3.34 yields

kT ¼ 9 ln 10ð Þk2WD

128p5N 0tDr6DA

ð10

c4f D nð ÞeA nð Þdnv4

: ð3:35Þ

Using the definitions of the F€orster distance and the overlap integral, we find

R60 ¼

9 ln 10ð Þk2WDJ128p5n4N 0 ; ð3:36Þ

and this equation is identical to Equation 3.3, becauseN 0 ¼ NA. In (3.36) J stands for

J ¼ c4ð10

f D nð ÞeA nð Þn4

dn; ð3:37Þ

which is Equation 3.8. And, we arrive at F€orster’s equation:

kT ¼ 1tD

R60

r6DA

� �; ð3:38Þ

which is Equation 3.2. Note that in this section the integrals extend from zero toinfinite frequency as they are based on the theoretical model of coupled chargedoscillators. However, integrals based on experimentally obtained spectra should onlyrefer to the relevant part of the spectra as discussed in Section 3.5.

3.13Oscillator Strength–Emission Spectrum Relation for the Donor

Consider a donor molecule capable of emitting light. The electromagnetic energythis donor has is WD. If the quantum yield is WD, then the energy availablefor fluorescence is WDWD. This can be emitted over a range of frequencies.The normalized fluorescence spectrum is f D ¼ f D nð Þ, so that

Ð10 f D nð Þdn ¼ 1.


The energy available for emission between n and nþ dn is equal toWDWDf D nð Þdn.The average lifetime of the excited state is tD. Therefore,

the rate of emission between n and nþ dn ¼ WDWDf D nð ÞdntD

: ð3:39Þ

On the other hand, one electric oscillator having energy WD, mass me, charge qe,and frequency n, embedded in a medium with refractive index n, must radiateenergy in accordance withMaxwell’s theory of electromagnetism, at a following rate:

the rate of emission of one oscillator ¼ 8p2nn2q2eWD

3 4pe0ð Þmec3: ð3:40Þ

We have a distribution of oscillators with a fraction of f eD nð Þdn between n andnþ dn. As a result, we have

the rate of emission between v and vþ dv ¼ 8p2nn2q2eWDf eD nð Þ3 4pe0ð Þmec3

: ð3:41Þ

CombiningEquations3.39and3.41yields ½WDWDf DðnÞ=tD ¼ ½8p2nn2q2eWDf eD nð Þ=½3 4pe0ð Þmec3Therefore, the relation between the donor oscillator strength and the donor

emission spectrum is

f eD nð Þ ¼ 3 4pe0ð Þmec3WDf D nð Þ8p2nn2q2etD

: ð3:42Þ

3.14Oscillator Strength–Absorption Spectrum Relation for the Acceptor

Imagine electromagnetic radiation falling upon 1 cm2 of a layer containing acceptormolecules at a concentration cA moles=l. This layer has a very small thickness of ‘cm. Consider the spectral energy density s nð Þ, defined such that s nð Þdn representsthe electromagnetic energy per unit of volume in the frequency range between n andnþ dn. From the Lambert–Beer law, we find that the spectral energy densityabsorbed in this layer is equal to the transmitted minus the incident spectral

density, that is, absorbed spectral energy density¼s nð Þ 1� e�eAcA‘ �

, where eA ¼eA nð Þ is themolar extinction coefficient of the acceptor in units 1/(cm�M). Because‘ is very small, and therefore eAcA‘ is very small, the following simplification is valid:

s nð Þ 1� 10�eAcA‘ � ¼ s nð Þ 1� e� ln 10ð ÞeAcA‘

h i¼ s nð Þ 1� 1� ln 10ð ÞeAcA‘f g½

¼ s nð Þ ln 10ð ÞeAcA‘:Here, ln 10 is the natural logarithm of 10 (see Section 3.3). Therefore,ln 10ð Þs nð ÞeAcA‘ is the spectral energy density per unit of volume absorbed.This energy is absorbed in an extremely short time. The speed at which thisradiation propagates in the medium is c=n (c is the speed of light in vacuo and n

3.14 Oscillator Strength–Absorption Spectrum Relation for the Acceptor j43

is the refractive index of the medium). Therefore, if we are interested in the energyabsorbed in 1 s, we must visualize a cylinder with a length of c=n and a cross-sectional area of 1 cm2 just in front of the layer. The energy distributed over thiscylinder is the energy per second entering the layer. So, the energy per secondentering the layer is s nð Þ � c=n, and the energy absorbed in it isc=nð Þ ln 10ð Þs nð ÞeAcA‘. This layer has a volume of ‘� 1 cm3 ¼ ‘� 1ml. Sincethe concentration of acceptor is cA moles=l, this volume contains‘� cA millimoles, which is N 0‘cA acceptor molecules (N 0 is the number of mole-cules per millimole) (see Section 3.3). Thus,

energy absorbed per second and per molecule ¼ ln 10ð Þcs nð ÞeAnN 0 : ð3:43Þ

On the other hand, from classical electromagnetic theory, we know that one electricoscillator with mass me, charge qe, and frequency n, embedded in a medium withrefractive index n, where the spectral energy density is s nð Þ will absorb energy at apredictable rate,

the energy absorbed per second by one oscillator ¼ pq2es nð Þ3n2 4pe0ð Þme

: ð3:44Þ

We have a distribution of oscillators with a fraction of f aA nð Þdn between n andnþ dn. As a result, we have

the energy absorbed per second per molecule ¼ pq2es nð Þf aA nð Þ3n2 4pe0ð Þme

: ð3:45Þ

Combining Equations 3.43 and 3.45 yields

ln 10ð ÞceA nð Þs nð ÞnN 0 ¼ pq2es nð Þf aA nð Þ

3n2 4pe0ð Þme:

Therefore, the relation between acceptor oscillator strength and acceptor extinctionspectrum is

f aA nð Þ ¼ 3 ln 10ð Þ 4pe0ð ÞnmeceA nð ÞpN 0q2e

: ð3:46Þ

3.15Quantum Mechanical Theory

When charges are bound to each other inside a molecule, the energies available tothem do not form a continuous spectrum, but the energy values are quantized.Resonance energy transfer can, therefore, be understood as coupled transitions, asshown in Figure 3.8. This diagram is essentially the same as the energy leveldiagram introduced by F€orster in his 1959 paper [3].According to quantum mechanics, a system can adopt a number of different

states. Considering a donor and an acceptor, it is clear that transfer may take place


from the state in which the donor is excited and the acceptor is not, DAij , to a statein which the acceptor is excited and the donor is not, DAij . The interactionresponsible for transfer at donor–acceptor distances larger than the sizes of thecharge distributions is the dipole–dipole interaction U, which is given by

U ¼ 14pe0n2r3DA

~p D �~p A � 3 ~p D � rð Þ r �~p Að Þ½ ; ð3:47Þ

where n is the refractive index of the medium, rDA denotes the distance between thedonor and the acceptor, r is a unit vector pointing from the donor to the acceptor,~p D

and~p A are the dipole moment vectors of the donor and acceptor charge distribution,respectively,~p D �~p A is the dot product of these two vectors, that is, the projection ofone on the other. According to the time-resolved perturbation theory, the rate oftransfer in the “very weak coupling” [5] limit is

kT WD ;WAð Þ ¼ 1h

ðDA Uj jDAh i2dW; ð3:48Þ

where kT WD ;WAð Þ is the rate of transfer from an excited donor molecule withinitial WD to an acceptor with initial energy WA, h is Planck’s constant, and theintegral is over all possible values of the transferred energy W. (The meaning of“very weak”, “weak”, and “strong coupling” in this context is discussed by F€orster [5],Kasha [36], and Knox [20]. It is safe to assume that the Born–Oppenheimerapproximation is valid. This approximation states that the electronic motion and

Figure 3.8 Simplified energy-level diagram of resonance energy transfer. D refers to the donorand A to the acceptor; asterisks denote excited states. (Adapted from Ref. [2].)

3.15 Quantum Mechanical Theory j45

the nuclear motion in molecules can be separated, so that the wave function for themolecule can be considered to be a product of an electronic wave function times anuclear wave function. As a result, the expectation value for the energy in Equa-tion 3.48 can be expressed in terms of electronic transition dipole moments ~m D and~m A, of the donor and acceptor, respectively. These vectors can be written as

~m D ¼ mDj jd and ~m A ¼ mAj ja; ð3:49Þ

where d and a are unit vectors in the direction of the donor and acceptor transitionmoments, respectively, and, mDj j and mAj j represent the magnitudes of thesemoments. In this approximation, the integrand in Equation 3.48 can be expressedin terms of the electronic transition moments as follows:

DA Uj jDAh i2 ¼ k2m2Dm

2A

4pe0ð Þ2n4r6DAS2DS

2A; ð3:50Þ

where k2 is the orientation factor defined in Section 3.7 and m2D m2

A

� �is the square of

mDj j mAj jð Þ. The factors SD and SA represent vibrational overlap integrals: SD ¼SD W

D;WD �W

� �is the overlap integral between the initial vibrational donor state

with energyWD and thefinal state with energyW

D �W, and SA ¼ SA WA;W

A þW

� �is the overlap integral between the initial vibrational acceptor state with energyWA andthe final state with energy WA þW. From the transfer rate kT WD ;WAð Þ of Equa-tion 14.2, we can obtain the total transfer rate of thermal equilibrium by introducingsuitable Boltzmann factors and integrating over all energies W

D and WA. These

Boltzmann factors g WD

� �for the excited donor and g WAð Þ for the acceptor in the

ground state are continuous functions and arenormalized on anenergy scale. Therefore,bymultiplyingboth sidesofEquation3.48by g W

D

� �dW

D and g WAð ÞdWA, integratingover allW

D andWA, andchanging integrationvariable fromenergy to frequency:W ton,we obtain the following expression for the total transfer rate kT:

kT ¼ k2

4pe0ð Þ2n4h2r6DA

ð10

MD nð ÞLA nð Þdn; ð3:51Þ

with

MD nð Þ ¼ m2D

ðg W

D

� �S2D W

D;WD � hn

� �dW

D ð3:52aÞ

and

LA nð Þ ¼ m2A

ðg WAð ÞS2A WA;WA þ hnð ÞdWA: ð3:52bÞ

Analyses similar to that done in Sections 3.12 and 3.13 [32] show that MD nð Þ isrelated to the normalized fluorescence spectrum of the donor and LA nð Þ isproportional to the extinction spectrum of the acceptor:

MD nð Þ ¼ 4pe0ð Þ3hWDc3f D nð Þ32p3ntDn3

: ð3:53aÞ


LA nð Þ ¼ 4pe0ð Þ3 ln 10ð ÞnhceA nð Þ4p2N 0n

: ð3:53bÞ

Substituting Equations 3.53a and 3.53b into Equation 3.51 yields F€orster’sequation for the rate of transfer:

kT ¼ 9 ln 10ð Þk2QD

128p5N 0tDr6DA

ð10

c4f D nð ÞeA nð Þdnv4

: ð3:54Þ

This equation is the same as Equation 3.35. As noted in Section 3.12, thetheoretical integrals extend from zero to infinite frequency, but integrals basedon experimentally obtained spectra should only refer to the relevant part of thespectra as discussed in Section 3.5.

3.16Transfer in a Random System

Consider an ensemble of donor and acceptor molecules, belonging to differentspecies, for which the following assumptions hold:

1) The molecules are randomly distributed through three-dimensional space.2) Resonance energy transfer is possible at an appreciable rate from donor to

acceptor, but transfer in the opposite direction is negligible.3) Translational diffusion is slow compared to the rate of transfer, so that the

distances between donor–acceptor pairs do not change significantly during thetime transfer takes place.

F€orster has pointed out [3] that these conditions are approximately met insolutions of moderated viscosity, in which case the Brownian rotational motionfor both donor and acceptor molecules is also much faster than the transfer and isunrestricted, so that the orientation factor can be set equal to 2/3. However, in manybiological systems, these assumptions may not be correct (see Chapter 4). In asystem in which these assumptions apply, consider a donor molecule that is alreadyexcited at time t ¼ 0. If no acceptor molecules had been present, it would lose itsexcitation energy after an average lifetime tD through radiation or nonradiativedeactivation processes. Its natural rate of deactivation is, therefore, 1=tD. Thepresence of an acceptor molecule at a distance rk provides another deactivationpathway for the excited donor molecule. The rate of transfer from the donor to the

acceptor is, according to F€orster theory, 1=tDð Þ R0=rkð Þ6. Because of these twocompeting processes, the probability r ¼ r tð Þ that the donor molecule is still excitedat time t is given by the following equation:

ddtr ¼ � 1

tDþXNk¼1

R0

rk

� �6" #

r; ð3:55Þ

where the summation is over allN acceptor molecules in a spherical volume aroundthe excited donor molecule with a radius much larger than the F€orster distance R0.

3.16 Transfer in a Random System j47

The solution of this differential equation with the condition that r 0ð Þ ¼ 1 is

r tð Þ ¼ exp � 1tD

þXNk¼1

R0

rk

� �6" #

ttD

( ): ð3:56Þ

The donor fluorescence is proportional to the average of this quantity, r tð Þh i.F€orster [4] showed that this average can be written as in Equation 3.57. Thederivation of Equation 3.57 is explained in Section 3.17, where other details abouttransfer in a random system are also given:

r tð Þ ¼ exp �s� 2 c=c0Uð Þ ffiffis

p� ; ð3:57Þ

where s ¼ t=tD and c0U is the “critical concentration for heterotransfer,” which isgiven by

c0U ¼ 3

2ffiffiffiffiffip3

pN 0R3

0

¼ 3

2ffiffiffiffiffip3

pNAR3

0

; ð3:58Þ

Note that there are two critical concentrations: c0U and c0L . c0U is the criticalconcentration for heterotransfer, that is, transfer between unlike molecules (Ustands for unlike), and c0L is the critical concentration for homotransfer, that is,transfer between like molecules (L stands for like), discussed in Section 3.17. Theefficiency E can also be calculated (see Section 3.17) and is given by

E ¼ ffiffiffip

px exð Þ2 1� erf xð Þf g; with x ¼ c=c0U ; ð3:59Þ

where erf is the error function, which is defined below, in Equation 3.76. Theefficiency is plotted in Figure 3.9 versus x ¼ c=c0U . It turns out that when c ¼ c0U ,the efficiency is equal to 76%.

3.17Details for Transfer in a Random System

The average of r tð Þ, defined in Equation 3.56, plays a key role in F€orster’s theory ofheterotransfer in a random system of donors and acceptors [4]. This average can bewritten as

r tð Þh i ¼ e�t=tD H tð Þ½ N ; ð3:60Þwith

H tð Þ ¼ðRV

0

e� R=R0ð Þ6t=tDw Rð ÞdR; ð3:61Þ

whereRV is the radius of the sphere around the excited donormolecule that containsthe N acceptor molecules to which transfer can occur, w Rð ÞdR represents theprobability for finding an acceptor molecule at a distance between R and Rþ dRfrom the excited donor molecule. The assumption of randomness (the first


assumption in Section 3.15) implies that w Rð Þ is such that every point in the spherehas equal probability for being occupied. Therefore, w Rð Þ must be equal to

w Rð Þ ¼ 3R2=R3V: ð3:62Þ

Substituting this into Equation 3.61 allows us to transform H tð Þ into

H tð Þ ¼ 12

ffiffiffiffiffizV

p ð1zV

e�zffiffiffiffiffiz3

p dz; ð3:63Þ

where z and zV are defined as

z ¼ RR0

� �6 ttD

; ð3:64Þ

zV ¼ R0

RV

� �6 ttD

: ð3:65Þ

Through integration by parts, we obtain

ð1zV


p dz ¼ 2e�zVffiffiffiffiffizV

p � 2ð10


p dzþ 2ðzV0


p dz: ð3:66Þ

Note that for all relevant values of the time t, zV is much smaller than 1, becauseRV is assumed to be much larger than R0. Therefore, it is a very good approximationto expand the right-hand side of Equation 3.66 in powers of

ffiffiffiffiffizV

pand to keep only the

Figure 3.9 The efficiency versus concentrationfor a system of random donors and acceptorswhere the rotational diffusion is fast, but thetranslational diffusion is slow compared to therate of transfer. Here, C is the acceptor

concentration and C0 is the criticalconcentration for heterotransfer, C0¼C0U,defined in Equation 3.58. The graph followsfrom Equation 3.59.

3.17 Details for Transfer in a Random System j49

first two terms. Applying this approximation and substituting Equation 3.66 intoEquation 3.63 yields

H tð Þ ¼ 1� ffiffiffiffiffiffiffiffipzV

p: ð3:67Þ

As a result, the average of r tð Þ in Equation 3.60 can be rewritten as

r tð Þh i ¼ e�t=tD 1� ffiffiffiffiffiffiffiffipzV

pð ÞN : ð3:68Þ

Because the number N can be assumed to be extremely large, Equation 3.68 can besimplified to

r tð Þh i ¼ e�t=tD limN!1

1� 1N

NffiffiffiffiffiffipzV

p� �N

¼ e�t=tD�NffiffiffiffiffiffipzV

p: ð3:69Þ

Employing the definition of zV and introducing x defined by

x ¼ffiffiffip

pNR3

0

2R3V

; ð3:70Þ

the average of r tð Þ becomes

r tð Þh i ¼ e�s�2xffiffis

p; ð3:71Þ

where s ¼ t=tD. The quantum yield in the presence of acceptor, WDA, is

WDA ¼ ~Cð10

r tð Þh idt; ð3:72Þ

where ~C is a constant. The quantum yield in the absence of acceptor, WD, can becalculated from Equations 3.72 and 3.69 for N ¼ 0:

WD ¼ ~Cð10

e�t=tDdt ¼ ~CtD: ð3:73Þ

Combining Equations 3.72 and 3.73 yields

WDA

WD¼

ð10

e�s�2xffiffis

pds ¼

ð10

ex2�y2ds; ð3:74Þ

where y ¼ ffiffis

p þ x, so that s ¼ y � xð Þ2 and ds ¼ 2 y � xð Þd y � xð Þ. Therefore,

WDA

WD¼ 2

ð10

ex2�y2 y � xð Þd y � xð Þ ¼ 2ex

2ð10

e�y2ydy � xex2 ffiffiffi

pp 2ffiffiffi

pp

ð1x

e�y2dy:

ð3:75ÞThe first integral on the right-hand side equals 1 and the second can be expressed interms of the error function, which is defined as


erf xð Þ ¼ 2ffiffiffip

pðx0

e�y2dy ¼ 1� 2ffiffiffip

pð1x

e�y2dy: ð3:76Þ

Combining Equations 3.75 and 3.76 allows us to express the efficiency in terms of x:

E ¼ 1�WDA

WD¼ ffiffiffi

pp

xex21� erf xð Þf g: ð3:77Þ

Note that if we choose N ¼ NA in Equations 3.60 and 3.70, the concentrationbecomes c ¼ 3= 4pR3

V

� �. As a result, x in Equation 3.70 is equal to

x ¼ 2=3ffiffiffiffiffip3

pNAR

30c; ð3:78Þ

and because of the definition of c0 ¼ c0U in Equation 3.58, we see that x ¼ c=c0,confirming Equation 3.59.

3.18Concentration Depolarization

Concentration depolarization is a homotransfer phenomenon. In a system where thefluorophores belong to a single species, FRET results in a strong depolarization of thefluorescence. The excitation energy of a molecule that absorbs a photon at a certainmoment may jump from molecule to molecule until emission occurs at a later time.Thus, a fluorescence photon, which in dilute solution is emitted by the absorbingmolecule, may in concentrated solutions be emitted by another molecule. This processdoes not affect the time dependence of the fluorescence intensity, but it broadens theangular distribution of the emission transitionmoments and consequently gives rise todepolarization of the emission. A graph of the fluorescence polarization versus thelogarithm of the concentration shows a constant level of depolarization at low concen-trations and a sharp drop at higher concentrations [2]. In his theory of concentrationdepolarization, F€orster assumed that only the photons emitted by the primarymoleculeare maximally polarized and that the other photons are completely unpolarized [2]. Hederives the polarization as a function of the concentration c for c � c0 and for c � c0,where c0 ¼ c0L , that is, the critical concentration for homotransfer:

c0L ¼3

4pN 0R30

¼ 3

4pNAR30

: ð3:79Þ

The depolarization depends on p1, the probability that the fluorescence is emittedby the initially excited molecule. If depolarization is due to concentration quenchingonly and rotational motion can be ignored, p1 is equal to r=r0, the ratio of theanisotropy and the fundamental anisotropy (¼ anisotropy in the absence of motionor transfer). In the low concentration limit, only the interaction between the primarymolecule and one other is considered, and p1 is given by

p1 ¼ð10

1þ t kT1þ 2t kT

e�jdj; c � c0; ð3:80Þ

3.18 Concentration Depolarization j51

where t is the fluorescence lifetime in the absence of transfer, kT is the rate oftransfer, and j ¼ 0:001� NAcV , whereNA ¼Avogadro’s number, c¼ concentrationinmoles/l, andV ¼ 4pR3=3– hereR denotes the distance betweenmolecules. In thehigh concentration limit, the excitation energy is thought to “diffuse” away from theprimary molecule and p1 is given by

p1 ¼ 1� 1þ 1:55=cð Þe�1:55=c; c � c0: ð3:81Þ

Here c ¼ c=c0ð Þ ffiffiffiffiffiffiffiffiffiffiffiffiffiW=W0

p, where W is the quantum yield and W0 is the quantum

yield in the absence of transfer. This theory has been further developed by Knox andothers (see Section 3.19).

3.19FRET Theory 1965–2012

F€orster’s work inspired an enormous volume of both experimental and theoreticalwork, not to mention applications and patents. The concluding section of thischapter is an overview of the theoretical work inspired by his results. However, thisoverview does not include work on kappa-squared. This has been dealt with inChapter 4.The theory of concentration depolarization and quenching has been further

developed by Knox, Craver, and others [36–44] (see Refs [45,46] for reviews). Craverand Knox compared different theories for concentration quenching in threedimensions and showed that experimental data were in good agreement with theirextension of F€orster’s theory [39]. Craver [40] has proposed a “universal” curve forconcentration depolarization in three dimensions. This curve, which is shown inFigure 3.10, fits experimental data quite well [40,47].The GAF theory deals with the time dependence of transport of electronic

excitation between like molecules that are randomly distributed [44]. The theory

predicts the time dependence of GS tð Þ, the probability of finding the excitation onthe initial site as a function of the time t after excitation. This function can beobserved in a picosecond transient (holographic) grating experiment. In this experi-ment, a delayed picosecond probe pulse is Bragg diffracted by a grating that isoptically produced in the sample by the interference of two coherent picosecondexcitation pulses. Absorption by the sample in the overlap region of the twoexcitation pulses results in a spatially varying sinusoidal distribution of excitedstates resulting in Bragg diffraction of the probe pulse. The intensity of the diffractedprobe pulse is proportional to the square of the difference in the absorption betweenthe grating peaks and nulls. Time-dependent processes that reduce this peak–nulldifference result in the decay of the diffracted signal [44,48]. In the GAF theory, the

Laplace–Fourier transform of GS tð Þ is expanded as a diagrammatic series. Topologi-cal reduction of the series establishes an analogy of diffusion. This diagrammatictechnique also suggests an interesting class of self-consistent approximations. Oneof these self-consistent approximations is applied to the specific case of the F€orster


transfer rate. The solutions obtained are well behaved for all times and all sitedensities and indicate that transport is nondiffusive at short times, but diffusive atlong times. The mean-squared displacement of the excitation and its time derivativeare calculated. These calculations illustrate that the time regime in which diffusivetransport occurs is dependent on density. In systems with low density, transport ofelectronic excitation becomes diffusive only at times longer than a few minutes;whereas for high densities, transport becomes diffusive within one lifetime ofthe excited state [44]. Baumann and Fayer have discussed excitation transfer in thedisordered two-dimensional and anisotropic three-dimensional systems for thecases of heterotransfer (direct trapping) in two-component systems and homo-transfer (donor–donor transfer) in one-component systems [43]. Using the two-particle model proposed by Huber et al. [49], Baumann and Fayer calculate the

configurational average of GS tð Þ. For the isotropic three-dimensional case treated byHuber et al., excellent correspondence is found with the GAF theory. The anisotropyof the dipole–dipole interaction is included in the averaging procedure. Two regimesof orientational mobility are considered: the dynamic and static limit, rotationsbeing much faster or slower, respectively, than the energy transfer. Several geomet-rical distributions are investigated. The fluorescence anisotropy decay, which can bestudied in a transient grating experiment or in a florescence depolarization experi-

ment, is a useful observable for GS tð Þ in homotransfer [43]. Baumann and Fayerfocus on nonradiative transport [43]. A unified treatment of radiative and non-radiative transport was introduced by Berberan-Santos et al. [50] and the role ofradiative transport has been reviewed by the same authors [51]. Huber et al. [49]report on the time dependence of fluorescence line narrowing. In the system

Figure 3.10 The “universal curve” proposed byCraver for concentration depolarization. Theratio r=r0, the steady-state anisotropy over thatin the absence of transfer, is plotted versusx 0 ¼ 2

ffiffiffiffiffiffiffiffiffiffiffiffiFD=p

pc=c0ð Þ, where FD is the quantum

yield in the absence of transfer, c is theconcentration, and c0 ¼ c0L is the criticalconcentration for homotransfer, defined inEquation 3.79. This curve is in good agreementwith experimental data [40,47].

3.19 FRET Theory 1965–2012 j53

studied, a background is observed around a narrow band of frequencies. Theappearance of this background in the fluorescence spectra is indicative of thetransfer of excitation from fluorophores inside the band to molecules whose opticalfrequencies lie outside the band. The authors treat back-transfer effects in a varietyof approximations and compare their theory with experimental data [49]. Hart et al.

[52] used time-correlated single-photon counting to measure GS tð Þ by monitoringthe fluorescence concentration depolarization for a dye in glycerol. Hart et al. foundthat the three-body GAF theory accurately describes the fluorescence depolarization

at the lower dye concentrations. At higher concentrations, the measured GS tð Þ wasfound to be perceptibly slower than that predicted by GAF theory. The authorssuggest that this deviation may arise from nonrandom dye distributions in solution,rather than from errors in the three-body GAF theory. They note that the exper-imental decay can also be described at all concentrations by the Huber–Hamilton–Barnett model [49]. The review by Knox [37] is primarily concerned with the theory ofexcitation migration. However, he also discusses experimental data on the rate ofpairwise excitation transfer between like molecules, with particular attention tochlorophyll a. In a more recent review [20], Knox notes that the CDA (seeEquation 3.2b) for chlorophyll a is about 68� 4 nm6/ps. Kawski’s review [45] on“excitation energy transfer and its manifestation in isotropic media” is thorough andlists a large number of relevant references. It discusses essentially all aspects ofenergy transfer. However, the paper devotes particular attention to the effect ofexcitation energy migration on fluorescence anisotropy [45]. Fluorescencedepolarization due to homo- and hetero-FRET was analyzed by Berberan-Santosand Valeur [53] and reviewed in Ref. [54], which elegantly describes many otherenergy transfer phenomena [54].The theory of FRET on surfaces and membranes has been an active field [55–64].

Of these references, the work by Wolber and Hudson [56] probably had the mostimpact. These authors have found an analytical solution of the FRETproblem in twodimensions for the case where the orientation factor is independent of the donor–acceptor distance and both donors and acceptors are randomly distributed in a plane[56]. In Refs [55–64], the emphasis is on heterotransfer. Homotransfer allowsstudying the accumulation of proteins in membranes. The theoretical frameworkthat relates fluorescence anisotropy to cluster size has been provided by Runnels andScarlata [65], who employ a theoretical analysis of homotransfer in clusters of likemolecules all containing the same fluorophore. In its simplest form, the Runnels–Scarlata theory predicts that the anisotropy of a cluster of N molecules equals theanisotropy of the monomer divided byN [65]. Towles et al. have applied Monte Carlosimulations to study microheterogeneity and domain size in membranes. Theyconclude that Monte Carlo calculations clearly indicate that FRET is indeed sensitiveto domain sizes in the range of 5–50 nm, but that a specific model is required toobtain a value for the domain size [66]. The idea of tryptophan imaging ofmembraneproteins has been proposed and analyzed by Kleinfeld [67]: tryptophans in mem-brane proteins serve as donors and anthroyloxy fluorophores serve as acceptors withthe anthroyloxy group attached to lipids at various distances from the midplane of


the membrane. Kleinfeld and Lukacovic successfully applied this idea to locate thetryptophan-109 in cytochrome b5 [68] confirming an earlier conclusion by Fleminget al. [69]. Other interesting membrane FRETstudies are mentioned in Refs [70–75].Fluorescence studies of membrane heterogeneity has been reviewed by Davenport[76]. A theory of FRET in micelles has been developed [22].Photosynthesis motivated substantial theoretical work [77–79,104,105]. Yang et al.

[80] point out that in the theory of energy transfer in photosynthesis, F€orster’s ideascan be successfully applied, but that Redfield theory [81] is more appropriate whenthe Coulombic coupling is greater than the electron–phonon coupling strength.Actually, there are three levels of discourse to consider when reviewing the theory: (i)F€orster process, which is the transfer or delocalization of an initially localized excitedstate. (ii) F€orster theory, which is his selection of a definition of rate of transfer and amethod to calculate it. (iii) F€orster’s equation itself, which is a result of his applyinghis theory to the dipole–dipole case [20].Table 3.3, adapted from Scholes [28] with minor changes, presents a history of

coupling models in FRET.Hauser et al. generalized F€orster’s equation for energy transfer in three dimen-

sions to the case of one, two, or three dimensions [106]. Often it is necessary toconsider excluded-volume effects due to the geometry of the system, which preventsthe acceptor from penetrating a certain volume surrounding the donor. Suchexcluded-volume effects have been discussed by Blumen et al. [107], Wolber andHudson [56], Duportail et al. [108], and Tcherkasskaya et al. [109]. The authors of Refs[108,109] made use of the stretched exponential model introduced by Drake et al.[110]. Dewey [111] has reviewed the relations between FRET and fractals.If the donor–acceptor distance can change because of the lateral diffusion during

the excited-state lifetime of the donor, FRET can be enhanced [112–116]. Theparameter determining the degree of this enhancement is

Z ¼ DtD=s2; ð3:82Þ

where D is the sum of the lateral diffusion coefficients of donor and acceptor, tD isthe donor lifetime in the absence of transfer, and s is the mean donor–acceptordistance. Three regimes can be distinguished [113]:

1) Z � 1, the static limit, where the transfer is low and constant, that is, there isessentially no variation with diffusion.

2) Z � 1, the intermediate regime, where the efficiency is sensitive to diffusion.3) Z � 1, the rapid diffusion limit, where the efficiency approaches a maximum

value and again becomes independent of diffusion.

Since distances of interest are in the 1–10 nm range and the diffusion coefficientof a typical fluorophore in aqueous solvents is on the order of 10�6 cm2/s, the donorlifetime in the case of rapid diffusion should be several orders of magnitude abovethe conventional nanosecond range. This technique of rapid diffusion FRET isreviewed by Stryer et al. [117]. Kouyama et al. [118] and Mersol et al. [119] havediscussed the effects of restricted rotation in diffusion-enhanced FRET.

3.19 FRET Theory 1965–2012 j55

If the parameter Z in Equation 3.82 is on the order of 1, the rate of translationalmotion is of the same order as the rate of transfer, and FRET can be employed tomeasure the lateral diffusion of the donor and/or acceptor [112,115,120–123] orfrom fluctuations in the FRET efficiency [124].

Table 3.3 A history of coupling models in resonance energy transfer.

Authors Application Comments

Craig and Walmsley [82],Davydov [83], Kasha et al.[84], McClure [85]

Exciton states Dipole–dipole coupling is used to definethe electronic states of molecularaggregates. This dipole–dipole interactionarises from Coulomb forces. Highermultipoles are ignored

F€orster [2,5] Electronic energytransfer

Development of a theory for the rate ofenergy transfer, through the dipole–dipolecoupling

Buckingham andDalgarno [86]

The interaction ofground-and excited-state helium atoms

Interaction is between a ground-statehelium atom and a helium atom in the firsttriplet or singlet metastable state. Heitler–London method is used

Dexter [25] andMerrifield [87]

Triplet–tripletenergy transfer

Orbital overlap effects considered to arisevia an exchange integral obtained from theCoulombic integral by permutation of twoorbitals

Koutecky and Paldus[88,89]

Transannularinteractions

Calculation of the interactions betweenclose molecules and perturbations of theirabsorption spectra

Andrews [90], Avery [91],Craig andThirunamachandran[92], McLone and Power[93], Scholes andAndrews [94]

Very long-rangecoupling

Quantum electrodynamical theories for theform of dipole–dipole coupling over verylarge distances, including the near field, theintermediate zone, and the far field

Azumi and McGlynn[95], Murrell and Tanaka[96]

Excimers Calculation of spectra based on the LMO(localized molecular orbital) prescriptioninvolving locally excited and charge transferconfigurations

Naqvi [97], Naqvi andSteel [98]

Exchange-inducedresonance energytransfer

Theory based on the exchange interactionsin singlet/triplet–singlet/doublet energytransfer

Harcourt et al. [99],Scholes and Ghiggino[100], Scholes andHarcourt [101], Scholeset al. [102]

LMO couplingmodel

Orbital overlap-dependent coupling (LMOmodel), revealing that the significantoverlap-dependent coupling is mediated viacharge transfer configurations

Scholes et al. [103] Special cases inphotosynthesis

A study of couplings involving thecarotenoid S1 state


Tanaka and Mataga [125] studied theoretically the effects of internal rotation onthe decay and the anisotropy of a donor and an acceptor bound to a sphericalmacromolecule. The model system they considered is one in which the donor isinternally rotating around an axis fixed at the macromolecule and the acceptor has afixed position and orientation [126]. The relevant parameter is kT=DR, the averagerate of transfer over the rotational diffusion constant. If this parameter is small, boththe donor fluorescence and the anisotropy are single exponentials. However, if thisparameter increases, the deviation from single-exponential behavior becomes moreand more pronounced [125]. In general, rotational motion when present in combi-nation with FRET will affect the time dependence of both the fluorescence and theanisotropy. This does not mean, however, that time dependence in the fluorescenceanisotropy must result from rotational motion. Energy transfer or other excited-statereactions can give rise to a strong time dependence in the anisotropy in the absenceof rotational motion [126–128]. This relation with time is due to coupling betweentwo states with different transition moments [126–128]. Van der Meer et al. [129]proposed a general method to take into account the effects of motion on FRET. Thismethod is applicable to both rotational and translational motion and is based on theidea that a system exhibiting both motion and FRETcan be modeled by specifying anumber of “states” and the rates of transitions between them. A state in this contextis a set of conditions and specific coordinates that describe the system at a certainmoment in time. There are excited-donor states (in which the donor is excited, butnot the acceptor), excited-acceptor states (in which the acceptor is excited, but not thedonor), and the states without excitation (neither donor nor acceptor is excited). Atransition from an excited-donor state to an excited-acceptor state represents energytransfer, whereas a transition from an excited-donor state to another excited-donorstate with a different position or orientation portrays translational or rotationalmotion. Fluorescence corresponds to a transition from an excited state to a statewithout excitation. Photoselection determines the initial occupation values of thestates. This method results in matrix equations that are linear differential equationsin time. The time dependence of the intensities and anisotropies of donor andacceptor can be expressed in terms of eigenvectors and eigenvalues of matrices[129]. The simplest example of this approach is to have a donor–acceptor distancethat can only be equal to R1 (short) or R2 (long) with the distance changing betweenthese two values at a rate J, and both donor and acceptor having isotropicallydegenerate transition dipoles (or undergoing very fast rotations). For this model, thedonor fluorescence at time t after excitation with a short pulse is

I ¼ I0exp � JtD þ 1ð Þt=tDð Þ�a� exp � t=tDð Þ R0=R2ð Þ6 � Jt=p

�þ 1� að Þ

� exp � t=tDð Þ R0=R1ð Þ6 þ Jt=p ��

;

with p ¼2JtD= R0=R1ð Þ6 � R0=R1ð Þ6

h i1þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ 2JtD= R0=R1ð Þ6 � R0=R1ð Þ6

h i �2r and a ¼ p� 1ð Þ2

2 p2 þ 1ð Þ ;

ð3:83Þ

3.19 FRET Theory 1965–2012 j57

where I0 is the initial intensity, tD is the donor lifetime in the absence of acceptor,

and R0 is the F€orster distance. If 2JtD � R0=R1ð Þ6 � R0=R2ð Þ6, the fluorescenceintensity becomes

I ¼ 12I0exp � JtD þ 1ð Þt=tDð Þ exp � t=tDð Þ R0=R2ð Þ6

�þ exp � t=tDð Þ R0=R1ð Þ6

�h i;

ð3:84Þbut for 2JtD � R0=R1ð Þ6 � R0=R2ð Þ6, one finds

I ¼ I0exp � JtD þ 1þ R0=R1ð Þ6 �

t=tD �

; ð3:85Þ

which depends only on the distance of closest approach, as expected, and containsthe FRET enhancement factor JtD.Canley et al. discussed FRET efficiency at high excitation intensity. They derived

the following equation for the FRET efficiency:

E ¼ 1

Lþ R=R0ð Þ6 ; ð3:86Þ

whereL ¼ 1 represents the well-known weak field case. They showed thatL can besignificantly larger than 1. The caseL > 1 reflects the inability of doubly excited dyepairs to undergo energy transfer [130].Raicu has developed a theoretical model for FRET from a single donor to multiple

acceptors and frommultiple donors to a single acceptor [131]. Bojarski et al. studiedthe possibility of FRET from a single donor to multiple acceptors using Monte Carlotechniques [132].Rolinski and Birch have introduced new ideas about donor–acceptor distributions

[133] and lifetime distributions [134]. Swathi and Sebastian pointed out that forenergy transfer from a dye to a nanotube, one can use the dipole approximation forthe dye, but not for the nanotube [135]. Consistent with this finding is the conclusionby Wong et al. that the point dipole approximation is inappropriate for use withelongated systems such as carbon nanotubes and that methods that can account forthe shape of the particle are more suitable [136].In the metal-enhanced fluorescence, a new field with a vast potential for

applications [137], resonance energy transfer plays a significant role [138]. Lakowiczintroduced the radiative plasmon model and showed that this model is consistentwith a wide range of experimental results, including FRET from fluorophores tonearby metal surfaces [138].

Acknowledgments

I wish to thank Dr. Bob Knox for stimulating discussions, useful advice, and makingme aware of papers I had overlooked. I am grateful to Dr. JoggiWirz, who discoveredthat the 9000-form of the F€orster equation is incorrect while working on his book[139]. We had highly interesting correspondence about this topic. Thanks toDr. David Andrews who shared with me his insights into QED&FRET and the


relevance of the spectral overlap. I am indebted to Dr. Manuel Prieto for sending merelevant papers and giving me useful suggestions. I am also grateful to Dr. HerbertDreeskamp for helpful suggestions and for giving me insights into the brilliance andwork ethics of Dr. Theodor F€orster.

References

1 (a) F€orster, T. (1946) Naturwissenschaften,33 , 166–175; (b) English translation bySuhling, K. (2012) Journal of BiomedicalOptics, 17, 01102-1–01102-10.

2 (a) F€orster, T. (1948) Annalen der Physik, 2,55 –75; (b) English translation byMielczarek, E.V., Greenbaum, E., andKnox, R.S. (eds) (1993) Biological Physics ,American Institute of Physics, New York,pp. 148– 160.

3 F€orster, T. (1959) Discussions of the FaradaySociety, 27, 7–17.

4 F€orster, T. (1949) Zeitschrift f €urNaturforschung, 4a, 321–327.

5 F€orster, T. (1965) in Modern QuantumChemistry: Istanbul Lectures Part II. Actionof Light and Organic Crystals (ed. O.Sinano�glu), Academic Press, New York,pp. 93–137.

6 Van der Meer, B.W., Coker, G., III., andChen, S.-Y.S. (1994) Resonance EnergyTransfer: Theory and Data, Wiley-VCHVerlag GmbH, Wienheim.

7 Clegg, R.M. (2006) Reviews in Fluorescence,vol. 3 (eds C.D Geddes and J.R. Lakowicz),Springer Science þ Business Media Inc.,pp. 1–45.

8 http://en.wikiquote.org/wiki/Isaac_Newton,Wikipedia, 2002, accessed 2012.

9 Cario, G. (1922) Zeitschrift f€ur Physik, 10,185–199 (in German).

10 Cario, G. and Franck, J. (1922) Zeitschriftf€ur Physik, 11, 161–166 (in German).

11 Franck, J. (1922) Zeitschrift f€ur Physik, 9,259–266 (in German).

12 Kallmann, H. and London, F. (1928)Zeitschrift f€ur physikalische Chemie, B2,207–243 (in German).

13 Perrin, J. (1925) Conseil de Chimie, Solvay,2i�em, Paris, 1924 Paris, Gauthier & Villar,French, pp. 322–398.

14 Perrin, J. (1927) Comptes RendusHebdomadaires des S�eances de l’Acad�emiedes Sciences, 184, 1097–1100 (in French).

15 Perrin, F. (1932) Annales de Chimie et dePhysique (Paris), 17, 283–314(in French).

16 Perrin, F. (1933) Annales de l’Institut HenriPoincar�e, 3, 279–318 (in French).

17 Berberan-Santos, M.N. (2001) New Trendsin Fluorescence Spectroscopy. Applications toChemical and Life Sciences (eds B. Valeurand J.-C. Brochon), Springer, Berlin,7–33.

18 Oppenheimer, J.R. (1941) Physical Review,60, 158.

19 (a) Arnold, W. and Oppenheimer, J.R.(1950) The Journal of General Physiology,33, 423–435; (b) See also Knox, R.S.(1995) Photosynthesis Research, 48, 35–39for a review of the contributions byArnold.

20 Knox, R.S. (2012) Journal of BiomedicalOptics, 17, 01103-1–01103-6.

21 Braslavsky, S.E., Fron, E., Rodríguez, H.B., San Rom�an, E., Scholes, G.D.,Schweitzer, G., Valeur, B., and Wirz, J.(2008) Photochemical & PhotobiologicalSciences, 7, 1444–1448.

22 Berberan-Santos, M.N. and Prieto, M.J.E.(1987) Journal of the Chemical Society:Faraday Transactions II, 83, 1391–1409.

23 Andrews, D.L. and Rodriguez, J. (2007)Journal of Chemical Physics, 127, 084509-1–084509-7.

24 Steinberg, I.Z. (1971) Annual Review ofBiochemistry, 40, 83–114.

25 Dexter, D.L. (1953) Journal of ChemicalPhysics, 21, 836–850.

26 Knox, R.S. and van Amerongen, H. (2002)The Journal of Physical Chemistry B, 106,5289–5293.

27 Chang, J.C. (1977) Journal of ChemicalPhysics, 67, 3901–3909.

28 Scholes, G.D. (1999) Resonance EnergyTransfer (eds D.L. Andrews and A.A.Demidov), John Wiley & Sons, Ltd,Chichester, pp. 212–243.

References j59

http://en.wikiquote.org/wiki/Isaac_Newton

29 Vanbeek, D.B., Zwier, M.C., Shorb, J.M.,and Krueger, B.P. (2007) BiophysicalJournal, 92 , 4168–4178.

30 Juzeli �unas, G. and Andrews, D.L. (1999)Resonance Energy Transfer (eds D.L.Andrews and A.A. Demidov), John Wiley& Sons, Ltd, Chichester, pp. 65– 107.

31 http://scienceworld.wolfram.com/physics/ElectricMultipoleExpansion.html.Weisstein, E., 2002, accessed 2012.

32 F€orster, T. (1951) Fluoreszenz OrganischerVerbindunger, Vandenhoeck & Ruprecht,G€ottingen.

33 http://www.tf.uni-kiel.de/matwis/amat/mw1_ge/kap_2/basics/b2_1_14.html.Foell, H., 1998, accessed 2012.

34 http://www.youtube.com/watch?v ¼PlpXrB4NP3w, Burke, G., 2007,accessed 2012.

35 http://www.youtube.com/watch?v ¼uWoiMMLIvco, Barrett, B., 2011,accessed 2012.

36 Kasha, M. (1991) Physical and ChemicalMechanisms in Molecular Radiation Biology(eds W.A. Glass and M.N. Varma),Plenum Press, New York, pp. 231–255.

37 Knox, R.S. (1975) Bioenergetics ofPhotosynthesis (ed. Govindjee), AcademicPress, New York, pp. 183–221.

38 Knox, R.S. (1968) Physica, 39, 361–386.39 Craver, F.W. and Knox, R.S. (1971)

Molecular Physics, 22, 385–402.40 Craver, F.W. (1971)Molecular Physics, 22,

404–420.41 (a) Bojarski, C. (1972) Journal of

Luminescence, 5, 413–429; (b) Bojarski, C.(1974) Journal of Luminescence, 9, 40–45;(c) Bojarski, C. (1979) Journal ofLuminescence, 20, 373–378.

42 Bodunov, E.N. (1981) Optics andSpectroscopy, 50, 553–554.

43 Baumann, J. and Fayer, M.D. (1986)Journal of Chemical Physics, 85, 4087–4107.

44 Gochanour, C.R., Anderson, H.C., andFayer, M.D. (1979) Journal of ChemicalPhysics, 70, 4254–4271.

45 Kawski, A. (1983) Photochemistry andPhotobiology, 38, 487–508.

46 Bojarski, C. and Sienicki, K. (1989)Photochemistry and Photophysics, vol. 1(ed. J.F. Rabek), CRC Press, Boca Raton,FL, pp. 1–57.

47 Beddard, G.S. (1983) Time-ResolvedFluorescence Spectroscopy in Biochemistry and

Biology (eds R.B. Cundall and R.E. Dale),Plenum Press, New York, pp. 451–461.

48 Eyring, G. and Fayer, M.D. (1984) Journalof Chemical Physics, 81, 4314–4321.

49 Huber, D.L., Hamilton, D.S., andBarnett, B. (1977) Physical Review B, 16,4624–4650.

50 Berberan-Santos, M.N., Nunes Pereira,E.J., and Martinho, J.M.G. (1997) Journalof Chemical Physics, 107, 10480–10484.

51 Berberan-Santos, M.N., Nunes Pereira,E.J., and Martinho, J.M.G. (1999)Resonance Energy Transfer (eds D.L.Andrews and A.A. Demidov), John Wiley& Sons, Ltd, Chichester, pp. 108–149.

52 Hart, D.E., Anfinrud, P.A., and Struve,W.S. (1987) Journal of Chemical Physics,86, 2689–2696.

53 Berberan-Santos, M.N. and Valeur,B. (1991) Journal of Chemical Physics, 95,8048–8055.

54 Valeur, B. and Berberan-Santos, M.N.(2012)Molecular Fluorescence: Principlesand Applications, 2nd edn, Wiley-VCHVerlag GmbH, Weinheim.

55 Fung, B.K.-K. and Stryer, L. (1978)Biochemistry, 17, 5241–5248.

56 Wolber, P.K. and Hudson, B.S. (1979)Biophysical Journal, 28, 197–210.

57 Estep, T.N. and Thompson, T.E. (1979)Biophysical Journal, 26, 195–208.

58 Dewey, T.G. and Hammes, G.G. (1980)Biophysical Journal, 32, 1023–1036.

59 Snyder, B. and Freire, E. (1982) BiophysicalJournal, 40, 137–148.

60 Doody, M.C., Sklar, L.A., Pownall, H.J.,Sparrow, J.T., Gotto, A.M., Jr., and Smith,L.C. (1983) Biophysical Chemistry, 17, 139–152.

61 Kellerer, H. and Blumen, A. (1984)Biophysical Journal, 46, 1–8.

62 Shaklai, N., Yguerabide, J., and Ranney,H.M. (1977) Biochemistry, 16, 5585–5592.

63 Koppel, D.E., Fleming, P.J., andStrittmatter, P. (1979) Biochemistry, 18,5450–5457.

64 Yguerabide, M. (1994) Biophysical Journal,66, 683–693.

65 Runnels, L.W. and Scarlata, S.F. (1995)Biophysical Journal, 69, 1569–1583.

66 Towles, K.B., Brown, A.C., Wrenn, S.P.,and Dan, N. (2007) Biophysical Journal, 93,655–667.

67 Kleinfeld, A.M. (1985) Biochemistry, 24,1874–1882.


http://scienceworld.wolfram.com/physics/ElectricMultipoleExpansion.html

http://scienceworld.wolfram.com/physics/ElectricMultipoleExpansion.html

http://www.tf.uni-kiel.de/matwis/amat/mw1_ge/kap_2/basics/b2_1_14.html

http://www.tf.uni-kiel.de/matwis/amat/mw1_ge/kap_2/basics/b2_1_14.html

http://www.youtube.com/watch?v=PlpXrB4NP3w

http://www.youtube.com/watch?v=PlpXrB4NP3w

http://www.youtube.com/watch?v=uWoiMMLIvco

http://www.youtube.com/watch?v=uWoiMMLIvco

68 Kleinfeld, A.M. and Lukacovic,M.F. (1985) Biochemistry, 24,1883–1890.

69 Fleming, P.J., Koppel, D.E., Lau, A.L.,and Strittmatter, P. (1978) Biochemistry,18, 5458–5464.

70 Wolf, D.A., Winiski, A.P., Ting, A.E.,Bocian, K.M., and Pagano, R.E. (1992)Biochemistry, 31, 2865–2873.

71 Davenport, L., Dale, R.E., Bisby, R.H.,and Cundall, R.B. (1985) Biochemistry, 24,4097–4108.

72 Dobretsov, G.E., Kurek, N.K., Machov,V.N., Syrejshchikova, T.I., and Yakimenko,M.N. (1989) Journal of Biochemical andBiophysical Methods, 19, 259–274.

73 John, E. and J€ahnig, F. (1991) BiophysicalJournal, 60, 319–328.

74 Fernandes, F., Neves, P., Gameiro, P.,Loura, L.M.S., and Prieto, M. (2007)Biochimica et Biophysica Acta, 1768,2822–2830.

75 Coutinho, A., Loura, L.M.S., Fedorov, A.,and Prieto, M. (2008) Biophysical Journal,95, 4726–4736.

76 Davenport, L. (1997)Methods inEnzymology, 278, 487–512.

77 Gnanakaran, S., Haran, G., Kumble, R.,and Hochstrasser, R.M. (1999) ResonanceEnergy Transfer (eds D.L. Andrews and A.A. Demidov), John Wiley & Sons, Ltd,Chichester, pp. 308–365.

78 Van Grondelle, R. and Somsen, O.J.G.(1999) Resonance Energy Transfer (eds D.L. Andrews and A.A. Demidov), JohnWiley & Sons, Ltd, Chichester,pp. 366–398.

79 Demidov, A.A. (1999) Resonance EnergyTransfer (eds D.L. Andrews and A.A.Demidov), John Wiley & Sons, Ltd,Chichester, pp. 435–465.

80 Yang, M., Damjanovi�c, A., Vaswani, H.M.,and Fleming, G.R. (2003) BiophysicalJournal, 85, 140–158.

81 Redfield, A.G. (1965) Advances in Magneticand Optical Resonance, 1, 1–32.

82 Craig, D.P. and Walmsley, S.H. (1968)Excitons in Molecular Crystals, Benjamin,New York.

83 Davydov, A.S. (1948) Journal ofExperimental and Theoretical Physics(USSR), 18, 210–218.

84 Kasha, M., Rawls, H.R., and El-Bayoumi,M.A. (1966) Pure and Applied Chemistry,11, 371–392.

85 McClure, D.S. (1959) Electronic Spectra ofMolecules and Ions in Crystals, AcademicPress, New York.

86 Buckingham, A.D. and Dalgarno, A.(1952) Proceedings of the Royal Societyof London A, 213, 327–349.

87 Merrifield, R.E. (1955) Journal of ChemicalPhysics, 23, 402–412.

88 Koutecky, J. and Paldus, J. (1962)Collection of Czechoslovak ChemicalCommunication, 27, 599–617.

89 Koutecky, J. and Paldus, J. (1963)Theoretica Chimica Acta (Berlin), 1,268–281.

90 Andrews, D.L. (1989) Chemical Physics,135, 195–201.

91 Avery, J. (1984) International Journal ofQuantum Chemistry, 135, 195–201.

92 Craig, D.P. and Thirunamachandran, T.(1984)Molecular QuantumElectrodynamics, Academic Press,New York.

93 McLone, R.R. and Power, E.A. (1964)Mathematika, 11, 91–94.

94 Scholes, G.D. and Andrews, D.L. (1997)Journal of Chemical Physics, 107, 5374–5384.

95 Azumi, T. and McGlynn, S.P. (1965)Journal of Chemical Physics, 42, 1675–1680.

96 Murrell, J.N. and Tanaka, J. (1964)Molecular Physics, 7, 363–380.

97 Naqvi, K.R. (1981) The Journal of PhysicalChemistry, 85, 2303–2304.

98 Naqvi, K.R. and Steel, C. (1970) ChemicalPhysics Letters, 6, 29–32.

99 Harcourt, R.D., Scholes, G.D., andGhiggino, K.P. (1994) Journal of ChemicalPhysics, 101, 10521–10525.

100 Scholes, G.D. and Ghiggino, K.P. (1994)The Journal of Physical Chemistry, 98,4580–4590.

101 Scholes, G.D. and Harcourt, R.D. (1996)Journal of Chemical Physics, 104, 5054–5061.

102 Scholes, G.D., Harcourt, R.D., andGhiggino, K.P. (1995) Journal of ChemicalPhysics, 102, 9574–9581.

103 Scholes, G.D., Harcourt, R.D., andFleming, G.R. (1997) The Journal ofPhysical Chemistry, 101, 7302–7312.

104 Andrews, D.L., Bradshaw, D.S., Jenkins,R.D., and Rodriguez, J. (2009) DaltonTransactions, 45, 10006–10014.

105 Huo, P. and Coker, D.F. (2010) Journalof Chemical Physics, 133, 184108.

References j61

106 Hauser, M., Klein, U.K.A., and G€osele,U. (1976) Zeitschrift f €ur physikalischeChemie, 101, 255–266.

107 Blumen, A., Klafter, J., and Zumofen,G. (1986) The Journal of Physical Chemistry,84 , 1397–1401.

108 Duportail, G., Merola, F., and Cianos,P. (1995) Journal of Photochemistryand Photobiology A: Chemistry, 89 ,135–140.

109 Tcherkasskaya, O., Gronenborn, A.M.,and Klushin, L. (2002) Biophysical Journal,83 , 2826–2834.

110 Drake, J.M., Klafter, J., and Levitz, P.(1991) Science, 251, 1574– 1579.

111 Dewey, T.G. (1992) Accounts of ChemicalResearch, 85, 4087–4107.

112 Haas, E., Katchalski-Katzir, E., andSteinberg, I.Z. (1978) Biopolymers , 17,11 –31.

113 Thomas, D.D., Carlsen, W.F., and Stryer,L. (1978) Proceedings of the NationalAcademy of Sciences of the United Statesof America, 75, 5746– 5750.

114 Katchalski-Katzir, E. and Steiberg, I.Z.(1981) Advances in Optical Biopsy andOptical Mammography, 366, 41–61.

115 Steinberg, I.Z., Haas, E., and K atchalski-Katzi r, E. (19 83) i n Ti m e - R e s o l v e dFluorescence Spectroscop y in Bi ochem istry a ndBiology (e ds R .B. C u nd al l a nd R .E. Dale ),P l enum Pres s, Ne w York, pp. 4 11 –450.

116 Fairclough, R.H. (2006) BiophysicalJournal, 91 , 1143–1144.

117 Stryer, L., Thomas, D.D., and Meares, C.F.(1982) Annual Review of Biophysics andBioengineering, 11, 203–222.

118 Kouyama, T.K., Kinosita, K., Jr., andIkegami, A. (1983) Journal of MolecularBiology, 165, 91–107.

119 Mersol, J.V., Wang, H., Gafni, A., andSteel, D.G. (1992) Biophysical Journal, 61,1647–1655.

120 G€osele, U., Hauser, M., Klein, U.K.A., andFrey, R. (1975) Chemical Physics Letters, 24 ,519–523.

121 Beechem, J.M. and Haas, E. (1989)Biophysical Journal, 55, 1225–1236.

122 Lakowicz, J.R., Szmacinski, H.,Gryczynski, I., Wick, W., and Johnson, M.

L. (1990) The Journal of Physical Chemistry,94 , 8413–8416.

123 Ku�sba, J., Li, L., Gryczynski, I., Piszczek,G., Johnson, M., and Lakowicz, J.R. (2002)Biophysical Journal, 82, 1358–1372.

124 Haas, E. and Steinberg, I.Z. (1984)Biophysical Journal, 46, 429–437.

125 Tanaka, F. and Mataga, N. (1975)Biophysical Journal, 39, 129–140.

126 Tanaka, F. and Mataga, N. (1975)Photochemistry and Photobiology, 29 ,1091–1097.

127 Szabo, A. (1984) Journal of ChemicalPhysics , 81 , 150–167.

128 Cross, A.J., Waldeck, D.H., and Fleming,G.R. (1983) Journal of Chemical Physics,78 , 6455–6467.

129 Van der Meer, B.W., Raymer, M.A.,Wagoner, S.L., Hackney, R.L., Beechem,J.M., and Gratton, E. (1993) BiophysicalJournal, 64 , 1243–1263.

130 Canley, B.A., Brown, F.L.H., and Lipman,E.A. (2009) Journal of Chemical Physics,131, 104509.

131 Raicu, V. (2007) Journal of BiologicalPhysics , 33 , 109–127.

132 Bojarski, P., Kulak, L., Walczewska-Szewc,K., Synak, A., Marzullo, V.M., Luini, A.,and D’Auria, S. (2011) The Journal ofPhysical Chemistry B, 115, 10120–10125.

133 Rolinski, O.J. and Birch, D.J.S. (2000)Journal of Chemical Physics, 112,8923–8933.

134 Rolinski, O.J. and Birch, D.J.S. (2008)Journal of Chemical Physics, 129, 144507.

135 Swathi, R.S. and Sebastian, K.L. (2010)Journal of Chemical Physics, 132, 104502.

136 Wong, C.Y., Curutchet, C., Tretiak, S., andScholes, G.D. (2009) Journal of ChemicalPhysics, 130, 081104.

137 Geddes, C.D. (ed.) (2010)Metal-EnhancedFluorescence, John Wiley & Sons, Inc.,New York.

138 Lakowicz, J.R. (2005) AnalyticalBiochemistry, 337, 171–194.

139 Klan, P. and Wirz, J. (2009) Photochemistryof Organic Compounds: From Concepts toPractice, John Wiley & Sons, New York.http://www.wiley.com/WileyCDA/WileyTitle/productCd-1405161736.html


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