Andrews

Bendig

David L. Andrews

Dos Remedios

Förster

Gomberoff

Juzeliūnas

Nickoleit

Power

Robert D. Jenkins

Stryer

English

Crossref

Chemical Physics Letters

Twin-donor systems for resonance energy transfer

Resonance energy transfer involving two identical donors and one acceptor is modelled by quantum electrodynamical calculation. The results account for recent experimental observations of simultaneous excitation energy transfer in the photochemistry of organic dyes. It is shown that a family of three energy transfer mechanisms has to be considered in order to describe fully the energy pooling phenomenon. The chromophore architecture has a pivotal role in determining the dominant mechanism

Jenkins, Robert D.

Andrews, David L.

University of East Anglia digital repository

26 February 1999 .Chemical Physics Letters 301 1999 235–240Twin-donor systems for resonance energy transferRobert D. Jenkins 1, David L. Andrews )School of Chemical Sciences, Uni˝ersity of East Anglia, Norwich NR4 7TJ, UKReceived 11 November 1998AbstractResonance energy transfer involving two identical donors and one acceptor is modelled by quantum electrodynamicalcalculation. The results account for recent experimental observations of simultaneous excitation energy transfer in thephotochemistry of organic dyes. It is shown that a family of three energy transfer mechanisms has to be considered in orderto describe fully the energy pooling phenomenon. The chromophore architecture has a pivotal role in determining thedominant mechanism. q 1999 Elsevier Science B.V. All rights reserved.1. IntroductionWhen the absorption of light leads to electronicexcitation in condensed phase matter, the associatedenergy is, in many cases, rapidly conveyed from thesite of its initial deposition to another. This processis familiar in many areas of photophysics, and itsdistance dependence is increasingly proving of valuefor the detailed analysis of intersite distances inw xmacromolecules or proteins 1,2 . In the case ofenergy transfer between chemically distinct donorand acceptor species separated beyond significantwavefunction overlap, energy migration is generallyconsidered as involving one of two mechanisms: atw xshort distances, radiationless transfer 3 with an) Corresponding author. E-mail: d.l.andrews@uea.ac.uk1 E-mail: robert.jenkins@uea.ac.ukinverse sixth power dependence on the donor–accep-tor separation – and at longer distances, radiativetransfer identifiable with the well-known inversesquare law. The comparatively recent developmentw xof a unified theory 4 for resonance energy transfer .  .RET , based on quantum electrodynamics QED ,has identified these two mechanisms as the long- andshort-range limits of a more general encompassingmechanism. With due regard to the effects of thew xintervening medium 5,6 , the unified theory accom-modates both intermolecular and inter-chromophoreexcitation transfer.Recent studies have revealed the occurrence incertain systems of a fundamentally new three-centreprocess, first termed simultaneous excitation energy . w xtransfer SEET 7,8 . This involves the transfer ofenergy from two electronically excited donors to asingle acceptor – the initial excitation of each donorhaving occurred through normal single-photon ab-sorption. One such unimolecular system, recently0009-2614r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. .PII: S0009-2614 99 00007-X( )R.D. Jenkins, D.L. AndrewsrChemical Physics Letters 301 1999 235–240236characterised, comprises building blocks of fluores-cein or erythrosin as donors, transferring energy tow xan E-stilbene acceptor unit 9 . Here the acceptorand donors are linked by thiourea units which, inthemselves, inhibit through-bond relaxation and con-jugated charge-delocalisation. The two donors arepre-excited using laser light at a frequency off-reso-nant with respect to direct excitation of the acceptor,and the observed intramolecular energy transfer ischaracterised by the ErZ-isomerisation of the stil-w xbene 9,10 .In this Letter we formulate theory for a generalcase of twin-donor resonance energy transfer, follow-ing the initial excitation of two chemically identicaldonor molecules in the vicinity of the acceptor. Tobe of chemical interest, the structure of the energylevels for the donor and acceptor species must dic- .tate that: 1 the energetics preclude acceptor excita-tion through energy transfer from a single donor; and .2 the transfer of energy is a concerted, not a trivialstepwise process, since the latter offers nothing dis-tinctive. As such, the process with which we areconcerned is best represented by the chemical equa-tion:A) qBqA) “ AqB) qA .It is assumed that the vibronic level of each electron-ically excited donor A) , from which energy trans-fers, is a state populated by intramolecular vibra-tional relaxation, within the excited state manifold,prior to the interaction we here consider. The totalenergy transferred to the acceptor B is thus in gen-eral less than the sum of the initial donor excitationenergies.2. QED formalismThe Hamiltonian, H, for a system comprising two X.chemically identical donor molecules A and A andone acceptor B can be written as:HsH A qH B qH AXqH A qH B qH AXqH ,mol mol mol int int int rad1 .with H j the Hamiltonian for molecule j , H themol radsecond-quantised radiation field Hamiltonian and H jintthe molecule-field coupling. In the electric dipoleapproximation the latter operator is given by:H j sy«y1m j Pd H R , 2 .  . .int 0 j .with m j the electric dipole moment operator andR the position vector of molecule j . The transversejH  .electric displacement field operator, d R , can bejexpressed in the usual expansion in terms of radia-tion modes characterised by wave-vector p and po-larisation l:1r2"cp«0  .  .H l l i pPRjd R s i e p a p e .  . . j  /2Vp , ll. †l. i pPRjye p a p e , 3 .  .  .where e represents the polarisation vector and e itscomplex conjugate, a and a† are the annihilation andcreation operators, respectively, and V is the quanti-sation volume.In the two-donor energy pooling processes thedonors A and AX progress from an already excited< : < :state A to the electronic ground state A , anda 0 .the acceptor molecule B is excited from its ground< : < :state B to an excited state B , through transfer0 bof the excitation energies of the donors A and AX.The energetics satisfy the basic relation 2 EA sEB ,a 0 b 0where EA is the energy lost by a donor and EBa 0 b 0that gained by the acceptor. In quantum electrody-namical terms the energy migrates due to the cre-ation and annihilation of virtual photons, entailing asum over all possible radiation modes and polarisa-tions. The initial and final states are given by:< : < X : < : A Bi s A ; A ; B 0 ; E s2 E qE ,a a 0 i a 0X A B< : < : < :f s A ; A ; B 0 ; E s2 E qE ,0 0 b f 0 b< :where E is the energy of system basis state n , ofn< : < X : < :the form n s A ; A ; B rad . For conveniencen n n nwe also define 2 EA sEB s2"ck, the initial excita-a 0 b 0tion energy of each donor represented as "ck. The< :radiation ket 0 , included for completeness, is usedto signify the vacuum state where no photons arepresent.The quantum probability amplitude or matrix ele-< :ment, M , connecting the initial, i , to the final,f i< :f , system state is given by the lowest-order non-zero term in the time-dependent perturbation expan-sion – which is of fourth order since the processes( )R.D. Jenkins, D.L. AndrewsrChemical Physics Letters 301 1999 235–240 237entail two virtual photon creation and two annihila-tion events each:Mf i < < : < < : < < : < < :f H t t H s s H r r H iint int int ints . E yE E yE E yE .  .  .i r i s i tr , s , t4 .Here we can recognise three mechanisms comprisinga family of interaction pathways for energy pooling,the essential energetics of which are illustrated inw xFig. 1. We have one cooperati˝e mechanism 11Fig. 1. Modified Jablonski diagrams showing the essential ener- .getics of the twin-donor energy transfer processes: a cooperative . Xand b accretive via A. Here S represents a ground electronic0 .state and its associated manifold; S or S denotes a higherb aelectronic excited state manifold; S represents a virtual molecularastate. Vertical arrows represent transitions; horizontal arrows de-note channels of excitation.whereby the two donors transfer their excitationenergy directly to the acceptor B, thus initiating itsexcitation through a virtual molecular state. Alongwith this route we can identify two accreti˝e energytransfer pathways where the initial excitation energyof one donor is passed to its partner and the sum ofthe two excitations is transferred to the acceptor.Here it is the intermediary donor which undergoes itsdecay transition through a virtual molecular state. Inthe following the notation ‘ACC1’ and ‘ACC2’ per-tains to the cases where the donors AX and A, respec-tively, play the intermediary role. .Accommodating all possible time-orderings in 4leads to three matrix elements, one for each of thevarious transfer pathways. The result for the coopera- .tive mechanism is shown in 5a , whilst results for .  .the accretive mechanisms are given in 5b and 5c :m0a A .m0a AX .a b 0B. yk , yk .i l jkCOOP .M sf i 2 24« V0=X XX Xi pPR yi p PRe e e e e ei j k lXpp XX X kyp ykyp .  .p , l , p , lX XX Xi pPR i p PRe e e e e ei j k lq Xkyp kyp .  .X XX Xyi pPR yi p PRe e e e e ei j k lq Xkqp kqp .  .X XX Xyi pPR i p PRe e e e e ei j k lq , 5a .X 5ykyp kyp .  .m0a A .m b 0B.a 0a AX . 2k , yk .i l jkACC1.M sf i 2 24« V0=Y X XX Xi pPR i p PRe e e e e ei j k lXpp XX X kyp 2kyp .  .p , l , p , lY X XX Xi pPR yi p PRe e e e e ei j k lq Xkyp y2kyp .  .Y X XX Xyi pPR i p PRe e e e e ei j k lq Xykyp 2kyp .  .Y X XX Xyi pPR yi p PRe e e e e ei j k lq , 5b .X 5ykyp y2kyp .  .( )R.D. Jenkins, D.L. AndrewsrChemical Physics Letters 301 1999 235–240238m0a AX .m b 0B.a 0a A . 2k , yk .i l jkACC2 .M sf i 2 24« V0=Y XX Xyi pPR i p PRe e e e e ei j k lXpp XX X kyp 2kyp .  .p , l , p , lY XX Xyi pPR yi p PRe e e e e ei j k lq Xkyp y2kyp .  .Y XX Xi pPR i p PRe e e e e ei j k lq Xykyp 2kyp .  .Y XX Xi pPR yi p PRe e e e e ei j k lq , 5c .X 5ykyp 2kyp .  .with summation over repeated Cartesian indices im-x y j .  <  j . < : Xplied. Here m s x m y , e and e denoten n .polarisation components of the virtual photons p, l X X.and p , l respectively, and the donor–acceptorseparation vectors are defined as RsR yR , RXB AsR yR X , and RY sR X yR . The use of primeB A A Alabels on virtual photon parameters hereafter onlyserves to distinguish between them, for calculational .convenience. The expressions of 5 are cast in termsfi j . .of polarisability tensors, a k , k , of the general1 2w xform 12 :a f i j . .k , .k .jk 1 2fz j z i j fz j z i j .  .  .  .m m m mj k k js q , 6 .  5˜ ˜E ""ck E ""ckz iz 1 iz 2where z is the virtual state through which moleculej progresses and a tilde is introduced to imply˜complex molecular energies. Explicitly, E sE qiz izi G with G the damping factor associated with statez zz and its sign fully consistent with time-reversalw x w xarguments 13 . Utilising standard methodology 12 ,and following a considerable amount of algebra, wecan write for the completed matrix elements:M COOP .sm0a A .V k , R a b 0B. yk , yk .  .f i i i j jk=V k , RX m0a AX ., 7a .  .k l lM ACC1.sm0a A .V k , RY a 0a AX . 2k , yk .  .f i i i j jk=V 2k , RX m b 0B. , 7b .  .k l lM ACC2 .sm0a AX .V k , RY a 0a A . 2k , yk .  .f i i i j jk=V 2k , R m b 0B. . 7c .  .k l lEach result is cast in terms of a second-rank, index-symmetric, Cartesian tensor representing the retardedresonance electric dipole–electric dipole coupling ofw xexplicit form 14 :i k Reˆ ˆV k , R s ikRy1 d y3R R .  .  /i j i j i j34p« R02 2ˆ ˆyk R d yR R . 8 . /i j i jIn the short-range, this coupling displays an Ry3distance dependence, though the terms linear andquadratic in kR increasingly modify the behaviour asR increases. The term ‘short-range’ in practice indi-˚cates intermolecular separations below Rf100 Ai.e., small compared to the characteristic opticaly1 .distance k , where the radiationless limit of en-ergy transfer dominates. As R increases, retardationw xeffects 15,4 become more prominent, bringing aradiative character of the transfer to the fore. Here,however, we are primarily interested in the short-range, where the coupling exerts its greatest influ-ence.The rate of twin-donor energy pooling, G , can beascertained by substitution of the matrix elements of .7 into the Fermi golden rule:2p 2 .COOP ACC1. ACC2 .Gs M qM qM r , 9 .f i f i f i b"where r is the appropriate density of final molecu-blar states for the acceptor, as determined by itsvibronic structure. The total rate thus encompasses‘diagonal’ terms associated with each of the threeenergy migration pathways, and also ‘cross-terms’resulting from their quantum interference.Use of the Fermi rule enables us to comparetypical rates of energy pooling with the rates of RETmore familiar in other systems. In conventional two-Fig. 2. Three-centre geometry and the mechanism-determiningangle, u .( )R.D. Jenkins, D.L. AndrewsrChemical Physics Letters 301 1999 235–240 239 .body single donor RET, the matrix element ex-hibits a well-known dependence on the square of thetransition dipole moments for donor and acceptor,and an r 3 dependence on intermolecular separationwith r indicative of a typical chromophore separa-. w xtion 4 . Each of the separate matrix elements for .twin-donor energy pooling, given by 7 , yields adependence on the square of two transition dipolemoments, a linear dependence on a two-photon ab-sorption tensor and an ry6 distance dependence.Considering a representative generalised molecularpolarisablility with components of the order of 1=10y39 C 2 m2 Jy1, then for any transfer distance of afew angstrom it is clear that the two-donor process˚ ¨should offer rates comparable to conventionaldonor–acceptor transfer.3. Molecular architectureIn an assembly of donors and acceptors groupedin twin donorrsingle acceptor triads, we assume that .Fig. 3. Variation of the relevance function, Q k, u , with u .( )R.D. Jenkins, D.L. AndrewsrChemical Physics Letters 301 1999 235–240240energy pooling within each triad is the dominantpathway for acceptor excitation, though our theorycan also address inter-triad transfer. In SEET donorexcitation is usually harvested at an acceptor withina single molecular triad. Specifically, we can theninvestigate the relative importance of competing in-tramolecular mechanisms on twin-donor energytransfer by considering the molecular geometryshown in Fig. 2, where u is the angle subtended atB.Assuming that both donors are chemically equiva-lent and equidistant from the acceptor we can de-scribe the inter-donor separation RY in terms of thedonor–acceptor distance R:uYR s2 R sin . 10 . /2If we further assume the short-range form ofcoupling, a function relating to the rele˝ance of the .accretive mechanisms, Q k, u , can be evaluated:y2G uACC 3Q k , u s s 1q4m sin , 11 .  . 5 /G 2TOT .where G is given by 9 , G is the rate of theTOT ACCaccretive mechanisms and m is related to the molec-ular properties of the donors and acceptor. Fig. 3shows how the Q function varies with u . A large mcoefficient is indicative of molecule B having a highpolarisability which, in turn, favours the cooperativemechanism – as illustrated in Fig. 3 by the sharplydecreasing trend in importance of the accretivemechanism. Conversely a low value for m relates toa high polarisability for the donors, in which case theaccretive mechanism becomes dominant.Within the framework of molecular quantum elec-trodynamics we have shown that molecular triadsexhibiting twin-donor energy pooling do so via threeintertwined mechanistic pathways. Nonetheless, theinterplay between the relative polarisabilities of theacceptor and donor, together with the detailed molec-ular architecture of the triad, exert a powerful influ-ence in determining the dominant mechanistic path-way. Within SEET triads the central stilbene’s largepolarisability leads us to believe that cooperativetransfer is mainly responsible for the observed E–Zisomerisation.AcknowledgementsRDJ gratefully acknowledges a research stu-dentship funded by the Engineering and PhysicalSciences Research Council.Referencesw x  .1 L. Stryer, Annu. Rev. Biochem. 47 1978 819.w x2 C.G. Dos Remedios, P.D.J. Moens, J. Struct. Biol. 115 .1995 175.w x  .3 T. Forster, Naturwissenschaften 33 1946 166.¨w x  .4 D.L. Andrews, Chem. Phys. 135 1989 195.w x  .5 G. Juzeliunas, D.L. Andrews, Phys. Rev. B 49 1994 8751.w x  .6 G. Juzeliunas, D.L. Andrews, Phys. Rev. B 50 1994 13371.w x7 J. Bendig, E. Bellmann, S.J. Helm, J. Inf. Rec. Mater. 21 .1994 441.w x8 M. Nickoleit, S. Hecht, A. Uhl, J. Bendig, J. Inf. Rec. Mater. .23 1996 66.w x  .9 M. Nickoleit, A. Uhl, J. Bendig, Laser Chem. 17 1997 161.w x  .10 Y.P. Sun, J. Saltiel, J. Phys. Chem. 93 1989 8310.w x  .11 R.D. Jenkins, D.L. Andrews, J. Phys. Chem. in press .w x12 D.P. Craig, T. Thirunamachandran, Molecular QuantumElectrodynamics, Academic Press, New York, 1984.w x13 D.L. Andrews, S. Naguleswaran, G.E. Stedman, Phys. Rev A .57 1996 4925.w x  .14 E.A. Power, T. Thirunamachandran, Phys. Rev. A 28 19832671.w x15 L. Gomberoff, E.A. Power, Proc. Phys. Soc., London 88 .1966 281.

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Twin-donor systems for resonance energy transfer

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