12.1. FÖRSTER RESONANCE ENERGY TRANSFER


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1 ndei Tokmakoff, MIT epatment of Chemisty, 3/5/ FÖRSTER RESONNCE ENERGY TRNSFER Föste esonance enegy tansfe (FR) efes to the nonadiative tansfe of an electonic excitation fom a dono molecule to an accepto molecule: + + (1.1) This electonic excitation tansfe, whose pactical desciption was fist given by Föste, aises fom a dipoledipole inteaction between the electonic states of the dono and the accepto, and does not involve the emission and eabsoption of a light field. Tansfe occus when the oscillations of an optically induced electonic coheence on the dono ae esonant with the electonic enegy gap of the accepto. The stength of the inteaction depends on the magnitude of a tansition dipole inteaction, which depends on the magnitude of the dono and accepto tansition matix elements, and the alignment and sepaation of the dipoles. The shap 1/ dependence on distance is often used in spectoscopic chaacteization of the poximity of dono and accepto. To descibe FR, thee ae fou electonic states that must be consideed: The electonic gound and excited states of the dono and accepto. We conside the case in which we have excited the dono electonic tansition, and the accepto is in the gound state. bsoption of light by the dono at the equilibium enegy gap is followed by apid vibational elaxation which dissipates the eoganization enegy of the dono λ ove the couse of picoseconds. This leaves the dono in a coheence that oscillates at the enegy gap in the dono excited state ( q d ) ω. The timescale fo eg FR is typically nanoseconds, so this pepaation step is typically much faste than the tansfe phase. Fo esonance enegy tansfe we equie a esonance condition, so that the oscillation of the excited dono Th. Föste, Expeimentelle und theoetische Untesuchung des zwischenmoleculaen Uebegangs von Electonenanegungsenegie, Z. Natufosch, 4a, 31 (1949); Zwischenmoleculae Enegiewandeung und Fluoeszenz, nn. Physik, 55 (1948); Tansfe Mechanisms of Electonic Excitation, iscussions Faaday Soc. 7, 7 (1959).
2 ndei Tokmakoff, MIT epatment of Chemisty, 3/5/8 1 coheence is esonant with the gound state electonic enegy gap of the accepto ( q ) ω. Tansfe of enegy to the accepto leads to vibational elaxation and subsequent accepto fluoescence that is spectally shifted fom the dono fluoescence. In pactice, the efficiency of enegy tansfe is obtained by compaing the fluoescence emitted fom dono and accepto. Since the dono and accepto ae weakly coupled, we can wite ou Hamiltonian fo this poblem in a fom that can be solved by petubation theoy H H + V H H + H eg (1.) Hee H is the Hamiltonian of the system with the dono excited, and H is the Hamiltonian with the accepto excited. epesents the electonic and nuclea configuation fo both dono and accepto molecules, which could be moe popely witten dn an. The inteaction between dono and accepto takes the fom of a dipoledipole inteaction: 3( ˆ)( ˆ ) V, (1.3) 3 whee is the distance between dono and accepto dipoles and ˆ is a unit vecto that maks the diection between them. The dipole opeatos hee ae taken to only act on the electonic states and be independent of nuclea configuation, i.e. the Condon appoximation. We wite the tansition dipole matix elements that couple the gound and excited electonic states fo the dono and accepto as + (1.4) + (1.5) Fo the dipole opeato, we can sepaate the scala and oientational contibutions as uˆ (1.) This allows the tansition dipole inteaction in eq. (1.3) to be witten as κ + V B 3 ll of the oientational factos ae now in the tem κ: (1.7)
3 ndei Tokmakoff, MIT epatment of Chemisty, 3/5/8 13 ( )( ) κ ˆ ˆ ˆ ˆ ˆ ˆ. (1.8) 3 u u u u We can now obtain the ates of enegy tansfe using Femi s Golden Rule expessed as a coelation function in the inteaction Hamiltonian: π w p V dtv tv 1 k k δ ωk ω I I + ( ) () ( ) (1.9) Note that this is not a Fouie tansfom! Since we ae using a coelation function thee is an assumption that we have an equilibium system, even though we ae initially in the excited dono state. This is easonable fo the case that thee is a clea time scale sepaation between the ps vibational elaxation and themalization in the dono excited state and the timescale (o invese ate) of the enegy tansfe pocess. Now substituting the initial state and the final state k, we find 1 + κ () () ( ) ( ) w dt t t (1.1) ih t ih t whee () t e e. Hee, we have neglected the otational motion of the dipoles. Most geneally, the oientational aveage is ( t) ( ) κ κ κ. (1.11) Howeve, this facto is easie to evaluate if the dipoles ae static, o if they apidly otate to become isotopically distibuted. Fo the static case κ Fo the case of fast loss of oientation: K() t K( ) κ κ 3. Since the dipole opeatos act only on o, and the and nuclea coodinates ae othogonal, we can sepaate tems in the dono and accepto states. 1 + κ 1 () ( ) () ( ) w dt t t + dt κ ( ) ( ) C t C t (1.1) The tems in this equation epesent the dipole coelation function fo the dono initiating in the excited state and the accepto coelation function initiating in the gound state. That is, these ae coelation functions fo the dono emission (fluoescence) and accepto absoption.
4 ndei Tokmakoff, MIT epatment of Chemisty, 3/5/8 14 Remembeing that epesents the electonic and nuclea configuation dn, we can use the displaced hamonic oscillato Hamiltonian o enegy gap Hamiltonian to evaluate the coelation functions. Fo the case of Gaussian statistics we can wite i ( ) ω λ t g t C t e (1.13) () i ( ) t g t C t e ω. (1.14) () Hee we made use of ω ω λ, (1.15) which expesses the emission fequency as a fequency shift of λ elative to the dono absoption fequency. The dipole coelation functions can be expessed in tems of the invese Fouie tansfoms of a fluoescence o absoption lineshape: 1 + ω i t C () t dω e σfluo ( ω) π (1.1) 1 + ω i t C () t dω e σabs ( ω) π. (1.17) To expess the ate of enegy tansfe in tems of its common pactical fom, we make use of Pasival s Theoem, which states that if a Fouie tansfom pai is defined fo two functions, the integal ove a poduct of those functions is equal whethe evaluated in the time o fequency domain: () () ( ) ( ) f t f t dt f ω f ω dω. (1.18) 1 1 This allows us to expess the enegy tansfe ate as an ovelap integal J between the dono fluoescence and accepto absoption specta: 1 κ + w d abs ( ) fluo ( ) ω σ ω σ ω. (1.19) Hee σ is the lineshape nomalized to the tansition matix element squaed: σσ/. The ovelap integal is a
5 ndei Tokmakoff, MIT epatment of Chemisty, 3/5/8 15 measue of esonance between dono and accepto tansitions. So, the enegy tansfe ate scales as, depends on the stengths of the electonic tansitions fo dono and accepto molecules, and equies esonance between dono fluoescence and accepto absoption. One of the things we have neglected is that the ate of enegy tansfe will also depend on the ate of excited dono state population elaxation. Since this elaxation is typically dominated by the dono fluoescence ate, the ate of enegy tansfe is commonly witten in tems of an effective distance R, and the fluoescence lifetime of the dono τ : w 1 R τ (1.) t the citical tansfe distance R the ate (o pobability) of enegy tansfe is equal to the ate of fluoescence. R is defined in tems of the sixthoot of the tems in eq. (1.19), and is commonly witten as R ( ) ( ) 9 ln(1) φ κ σ fluo ν ε ν dν π nn ν (1.1) This is the pactical definition which accounts fo the fequency dependence of the tansitiondipole inteaction and nonadiative dono elaxation in addition to being expessed in common units. ν epesents units of fequency in cm 1. The fluoescence spectum σ fluo must be nomalized to unit aea, so that σ fluo ( ν ) is expessed in cm (invese wavenumbes). The absoption spectum ε ( ν ) must be expessed in mola decadic extinction coefficient units (lite/mol cm). n is the index of efaction of the solvent, N is vagado s numbe, and φ is the dono fluoescence quantum yield.
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