14. ENERGY AND CHARGE TRANSFER Electronic Interactions

Transcription

1 14. ENERGY ND CHRGE TRNSFER Electronic Interactions In this section we will describe processes that result fro the interaction between two or ore olecular electronic states, such as the transport of electrons or electronic excitation. This proble can be forulated in ters of a failiar Hailtonian H H V in which H describes the electronic states (including any coupling to nuclear otion), and V is the interaction between the electronic states. In forulating such a proble we will need to consider soe basic questions: Is V strong or weak? re the electronic states described in a diabatic or adiabatic basis? How do nuclear degrees of freedo influence the electronic couplings? For weak couplings, we can describe the transport of electrons and electronic excitation with perturbation theory drawing on Feri s Golden Rule: w p Vk Ek E k, 1 w dtvi tvi This approach underlies coon descriptions of electronic energy transport and non-adiabatic electron transfer. We will discuss this regie concentrating on the influence of vibrational otions they are coupled to. However, the electronic couplings can also be strong, in which case the resulting states becoe delocalized. We will discuss this liit in the context of excitons that arise in olecular aggregates. To begin, it is useful to catalog a nuber of electronic interactions of interest. We can use soe scheatic diagras to illustrate the, ephasizing the close relationship between the various transport processes. However, we need to be careful, since these are not eant to iply a echanis or eaningful inforation on dynaics. Here are a few coonly described processes involving transfer fro a donor olecule D to an acceptor olecule : a) Resonance energy transfer pplies to the transfer of energy fro the electronic excited state of a donor to an acceptor olecule. rises fro a Coulob interaction that is operative at long range, i.e., distances large copared to olecular diensions. Requires electronic resonance. Naed for the first practical derivations of expressions describing this effect: Förster Resonance Energy Transfer (FRET) ndrei Tokakoff, 1/1/14

2 14- b) Electron transfer Marcus theory. Nonadiabatic electron transfer. Requires wavefunction overlap. Ground state: Excited state: Hole transfer: c) Electron-exchange energy transfer Dexter transfer. Requires wavefunction overlap. Singlet or triplet d) Singlet fission

3 Förster Resonance Energy Transfer Förster resonance energy transfer (FRET) refers to the nonradiative transfer of an electronic excitation fro a donor olecule to an acceptor olecule: * * D D (14.1) This electronic excitation transfer, whose practical description was first given by Förster, arises fro a dipole dipole interaction between the electronic states of the donor and the acceptor, and does not involve the eission and reabsorption of a light field. Transfer occurs when the oscillations of an optically induced electronic coherence on the donor are resonant with the electronic energy gap of the acceptor. The strength of the interaction depends on the agnitude of a transition dipole interaction, which depends on the agnitude of the donor and acceptor transition atrix eleents, and the alignent and separation of the dipoles. The sharp 1/r 6 dependence on distance is often used in spectroscopic characterization of the proxiity of donor and acceptor. The electronic ground and excited states of the donor and acceptor olecules all play a role in FRET. We consider the case in which we have excited the donor electronic transition, and the acceptor is in the ground state. bsorption of light by the donor at the equilibriu energy gap is followed by rapid vibrational relaxation that dissipates the reorganization energy of the donor D over the course of picoseconds. This leaves the donor in a coherence that oscillates at the D energy gap in the donor excited state eg qd dd. The tie scale for FRET is typically nanoseconds, so this preparation step is typically uch faster than the transfer phase. For resonance energy transfer we require a resonance condition, so that the oscillation of the excited donor coherence is resonant with the ground state electronic energy gap of the acceptor eg q. Transfer of energy to the acceptor leads to vibrational relaxation and subsequent acceptor fluorescence that is spectrally shifted fro the donor fluorescence. In practice, the efficiency of energy transfer is obtained by coparing the fluorescence eitted fro donor and acceptor. This description of the proble lends itself naturally to treating with a DHO Hailtonian, However, an alternate picture is also applicable, which can be described through the EG Hailtonian. FRET arises fro the resonance that occurs when the fluctuating electronic energy

4 14-4 gap of a donor in its excited state atches the energy gap of an acceptor in its ground state. In other words D eg D eg t t D These energy gaps are tie-dependent with occasion crossings that allow transfer of energy. (14.) Our syste includes the ground and excited potentials of the donor and acceptor olecules. The four possible electronic configurations of the syste are GG D, EG D, GE D, EE D Here the notation refers to the ground (G) or excited (E) vibronic states of either donor (D) or acceptor (). More explicitly, the states also include the vibrational excitation: EG en; gn D D D Thus the syste can have no excitation, one excitation on the donor, one excitation on the acceptor, or one excitation on both donor and acceptor. For our purposes, let s only consider the two electronic configurations that are close in energy, and are likely to play a role in the resonance transfer in eq. (14.) EG D and GE D Since the donor and acceptor are weakly coupled, we can write our Hailtonian for this proble in a for that can be solved by perturbation theory ( H H V ). Working with the DHO. approach, our aterial Hailtonian has four electronic anifolds to consider: H E H E G H G E H E G H G E G D D D D D D E G Each of these is defined as we did previously, with an electronic energy and a dependence on a displaced nuclear coordinate. For instance E D D D D e D e (14.3) H e E e H (14.4)

5 D D e D D D 14-5 H p q d (14.5) D Ee is the electronic energy of donor excited state. Then, what is V? Classically it is a Coulob interaction of the for V D qi qj r r (14.6) D i, j i j Here the su is over all electrons and nuclei of the donor (i) and acceptor (j). s is, this is challenging to work with, but at large separation between olecules, we can recast this as a dipole dipole interaction. We define a frae of reference for the donor and acceptor olecule, and assue that the distance between olecules is large. Then the dipole oents for the olecules are D D D D i i i j J j q r r q r r (14.7) The interaction between donor and acceptor takes the for of a dipole dipole interaction: rˆ rˆ 3 D D V (14.8) 3 r where r is the distance between donor and acceptor dipoles and ˆr is a unit vector that arks the direction between the. The dipole operators here are taken to only act on the electronic states and be independent of nuclear configuration, i.e., the Condon approxiation. We write the transition dipole atrix eleents that couple the ground and excited electronic states for the donor and acceptor as (14.9) * * * * D D D D (14.1) * * D * * DD D D For the dipole operator, we can separate the scalar and orientational contributions as uˆ (14.11) This allows the transition dipole interaction in eq. (14.8) to be written as r * * * * V B D D D D 3 (14.1) ll of the orientational factors are now in the ter :

6 3 u r u D r u u D 14-6 ˆ ˆ ˆ ˆ ˆ ˆ (14.13) We can now obtain the rates of energy transfer using Feri s Golden Rule expressed as a correlation function in the interaction Hailtonian: 1 w p V dt V t V k k k I I (14.14) Note that this is not a Fourier transfor! Since we are using a correlation function there is an assuption that we have an equilibriu syste, even though we are initially in the excited donor state. This is reasonable for the case that there is a clear tie scale separation between the ps vibrational relaxation and theralization in the donor excited state and the tie scale (or inverse rate) of the energy transfer process. * * Now substituting the initial state D and the final state k D, we find 1 * * ET 6 D D w dt D t t D (14.15) r ihdt ihdt where D t e De. Here, we have neglected the rotational otion of the dipoles. Most generally, the orientational average is t (14.16) However, this factor is easier to evaluate if the dipoles are static, or if they rapidly rotate to becoe isotropically distributed. For the static case.475. For the case of fast loss of orientation: K t K 3. * Since the dipole operators act only on or D, and the D and nuclear coordinates are orthogonal, we can separate ters in the donor and acceptor states. 1 * * ET 6 D D 1 dt C 6 * * t C D D t w dt D t D t r r (14.17) The ters in this equation represent the dipole correlation function for the donor initiating in the excited state and the acceptor correlation function initiating in the ground state. That is, these are correlation functions for the donor eission (fluorescence) and acceptor absorption. Reebering that dn * * D represents the electronic and nuclear configuration * D, we can use the displaced haronic oscillator Hailtonian or energy gap Hailtonian to evaluate the correlation functions. For the case of Gaussian statistics we can write

7 14-7 i * * D t gdt DD C t e (14.18) * * DD * DD i * t g t C t e (14.19) * Here we ade use of (14.) D * D DD * D which expresses the eission frequency as a frequency shift of D relative to the donor absorption frequency. The dipole correlation functions can be expressed in ters of the inverse Fourier transfors of a fluorescence or absorption lineshape: 1 i t D C * * t d e fluor DD (14.1) 1 i t C t d e abs (14.) To express the rate of energy transfer in ters of its coon practical for, we ake use of Parsival s Theore, which states that if a Fourier transfor pair is defined for two functions, the integral over a product of those functions is equal whether evaluated in the tie or frequency doain: * * f t f t dt f f d (14.3) 1 1 This allows us to express the energy transfer rate as an overlap integral JD between the donor fluorescence and acceptor absorption spectra: w r 1 ET 6 * * d DD abs fluor D (14.4) Here is the lineshape noralized to the transition atrix eleent squared: /. The overlap integral is a easure of resonance between donor and acceptor transitions. 6 So, the energy transfer rate scales as r, depends on the strengths of the electronic transitions for donor and acceptor olecules, and requires resonance between donor fluorescence and acceptor absorption. One of the things we have neglected is that the rate of energy transfer will also depend on the rate of excited donor state population relaxation. Since this relaxation is

8 14-8 typically doinated by the donor fluorescence rate, the rate of energy transfer is coonly written in ters of an effective distance R, and the fluorescence lifetie of the donor D : w ET 1 R D r 6 (14.5) t the critical transfer distance R the rate (or probability) of energy transfer is equal to the rate of fluorescence. R is defined in ters of the sixth-root of the ters in eq. (14.4), and is coonly written as R D 9ln(1) 6 D fluor d nn (14.6) This is the practical definition that accounts for the frequency dependence of the transitiondipole interaction and non-radiative donor relaxation in addition to being expressed in coon units. represents units of frequency in c -1. The fluorescence spectru D fluor ust be D noralized to unit area, so that fluor is expressed in c (inverse wavenubers). The absorption spectru ust be expressed in olar decadic extinction coefficient units (liter/olc). n is the index of refraction of the solvent, N is vagadro s nuber, and D is the donor fluorescence quantu yield.

9 14-9 ppendix: Transition Dipole Interaction FRET is one exaple of a quantu echanical transition dipole interaction. The interaction between two dipoles, and D, in eq. (14.1) is V e 3 g g D e (14.7) r Here, g D e is the transition dipole oent in Debye for the ground-to-excited state transition of olecule. r is the distance between the centers of the point dipoles, and is the unitless orientational factor 3cos 1cos cos 1 The figure below illustrates this function for the case of two parallel dipoles, as a function of the angle between the dipole and the vector defining their separation. In the case of vibrational coupling, the dipole operator is expanded in the vibrational Q Q, and haronic transition dipole atrix eleents are noral coordinate: 1 c Q (14.8) where is the vibrational frequency. If the frequency is given in c -1, and the transition dipole oent Q is given in units of D Å -1 au -1/, then the atrix eleent in units of D 1/ is Q. If the distance between dipoles is specified in Ångstros, then the transition dipole coupling fro (14.7) in c is V( c ) 534 r. Experientally, one can deterine the transition dipole oent fro the absorbance as N (14.9) 3c Q

10 14-1 Readings 1. Chea, T. C.; Kri, S., Transition dipole interaction in polypeptides: b initio calculation of transition dipole paraeters. Cheical Physics Letters 1984, 17, Forster, T., Transfer echaniss of electronic excitation. Discussions of the Faraday Society 1959, 7, Förster, T., Zwischenolekulare Energiewanderung und Fluoreszenz. nnalen der Physik 1948, 437, Förster, T., Experientelle und theoretische Untersuchung des zwischenolecularen Uebergangs von Electronenanregungsenergie. Z. Naturforsch 1949, 4, 31 37

11 Excitons in Molecular ggregates The absorption spectra of periodic arrays of interacting olecular chroophores show unique spectral features that depend on the size of the syste and disorder of the environent. We will investigate soe of these features, focusing on the delocalized eigenstates of these coupled chroophores, known as excitons. These principles apply to the study of olecular crystals, J- aggregates, photosensitizers, and light-harvesting coplexes in photosynthesis. Siilar topics are used in the description of properties of conjugated polyers and organic photovoltaics, and for extended vibrational states in IR and Raan spectroscopy. Energy transfer in the strong coupling liit Strong coupling between olecules leads to the delocalization of electronic or vibrational eigenstates, under which weak coupling odels like FRET do not apply. Fro our studies of the coupled two-state syste, we know that when the coupling between states is uch larger that the energy splitting between the states (1 V) then the resulting eigenstates ± are equally weighted syetric and antisyetric cobinations of the two, whose energy eigenvalues are split by V. Setting 1 = E V 1 1 If we excite one of these olecules, we expect that the excitation will flow back and forth at the Rabi frequency. So, what happens with ultiple coupled chroophores, focusing particular interest on the placeent of coupled chroophores into periodic arrays in space? In the strong coupling regie, the variation in the uncoupled energies is sall, aking this a proble of coupled quasi-degenerate states. With a spatially period structure, the resulting states bear close siilarity to siple descriptions of electronic band structure using the tight-binding odel. Excitons Excitons refer to electronic excited states that are not localized to a particular olecule. But beyond that there are any flavors. We will concentrate on Frenkel excitons, which refer to excited states in which the excited electron and the corresponding hole (or electron vacancy) reside on the sae olecule. ll olecules reain electrically neutral in the ground and excited states. This corresponds to what one would expect when one has resonant dipole dipole interactions between olecules. When there is charge transfer character, the electron and hole can reside on different olecules of the coupled coplex. These are referred to as Mott Wannier excitons.

12 14-1 bsorption spectru of olecular dier To describe the spectroscopy of an array of any coupled chroophores, it is first instructive to work through a pair of coupled olecules. This is in essence the two-level proble fro earlier. We consider a pair of olecules (1 and ), which each have a ground and electronically excited state ( e and g) split by an energy gap, and a transition dipole oent. In the absence of coupling, the state of the syste can be specified by specifying the electronic state of both olecules, leading to four possible states: gg, eg, ge, ee, whose energies are,,, and, respectively. For shorthand we define the ground state as G and the excited states as 1 and to signify the the electronic excitation is on either olecule 1 or. In addition, the olecules are spaced by a separation r1, and there is a transition dipole interaction that couples the olecules. V J 1 1 Following our description of transition dipole coupling in Eqn. (14.8), the coupling strength J is given by r 3 r r J r r 1 1 where the orientational factor is ˆ ˆ 3 ˆ rˆ ˆ rˆ We assue that the coupling is not too strong, so that we can just concentrate on how it influences 1 and but not G. Then we only need to describe the coupling induced shifts to the singly excited states, which are described by the Hailtonian J H J s stated above, we find that the eigenvalues are E and that the eigenstates are: J

13 These syetric and antisyetric states are delocalized across the two olecules, and in the language of Frenkel excitons are referred to as the one-exciton states. Furtherore, the dipole operator for the dier is M 1 and so the transition dipole atrix eleents are: 1 M M G 1 M+ and M- are oriented perpendicular to each other in the olecular frae. If we confine the olecular dipoles to be within a plane, with an angle between the, then the aplitude of M+ and M- is given by M cos M sin We can now predict the absorption spectru for the dier. We have two transitions fro the ground state and the ± states which are resonant at = J and which have an aplitude M. The splitting between the peaks is referred to as the Davydov splitting. Note that the relative aplitude of the peaks allows one to infer the angle between the olecular transition dipoles. lso, note for θ = or 9, all aplitude appears in one transition with agnitude, which is referred to as superradiant.

14 14-14 Frenkel Excitons with Periodic Boundary Conditions Now let s consider linear aggregate of N periodically arranged olecules. We will assue that each olecule is a two-level electronic syste with a ground state and an excited state. We will assue that electronic excitation oves an electron fro the ground state to an unoccupied orbital of the sae olecule. We will label the olecules with integer values (n) between and N-1: If the olecules are separated along the chain by a lattice spacing, then the size of the chain is L=N. Each olecule has a transition dipole oent, which akes an angle with the axis of the chain. In the absence of interactions, we can specify the state of the syste exactly by identifying whether each olecule is in the electronically excited or ground state. If the state of olecule n within the chain is n, which can take on values of g or e, then,, n N 1 1 This representation of the state of the syste is referred to as the site basis, since it is expressed in ters of each olecular site in the chain. For siplicity we write the ground state of the syste as G ggg,,, g If we excite one of the olecules within the aggregate, we have a singly excited state in which the n th olecule is excited, so that ggg,,,, e,, g n For shorthand, we identify this product state as n, which is to be distinguished fro the olecular eigenfunction at site n, n. The singly excited state is assigned an energy corresponding to the electronic energy gap. In the absence of coupling, the singly excited states are N-fold degenerate, corresponding to a single excitation at any of the N sites. If two excitations are placed on the chain we can see that there are N(N1) possible states with energy, recognizing that the Pauli principle does not allow two excitations on

15 14-15 the sae site. When coupling is introduced, the ixing of these degenerate states leads to the one-exciton and two-exciton bands. For this discussion, we will concentrate on the one-exciton states. The coupling between olecule n and olecule n is given by the atrix eleent V nn. We will assue that a olecule interacts only with its neighbors, and that each pairwise interaction has a agnitude J V nn J If V is a dipole dipole interaction, the orientational factor dictates that when the transition dipole angle < 54.7 then the sign of the coupling J <, which is the case known as J- aggregates (after Edwin Jelley), and iplies an offset stack of chroophores or head-to-tail arrangeent. If > 54.7 then J >, and the syste is known as an H-aggregate. To begin, we also apply periodic boundary conditions to this proble, which iplies that we are describing the states of an N-olecule chain within an infinite linear chain. In ters of the Hailtonian, the olecules at the beginning and end of our chain feel the sae syetric interactions to two neighbors as the other olecules. To write this in ters of a finite NN atrix, one couples the first and last eber of the chain: J, N1 JN 1, J. With these observations in ind, we can write the Frenkel Exciton Hailtonian for the linear aggregate in ters of a syste Hailtonian that reflects the individual sites and their couplings H H V S S N H n n n1 N n1 n, n1 V J n n n n nn, 1 (14.3) Here periodic boundary conditions iply that we replace N and 1 N 1 where they appear. The optical properties of the aggregate will be obtained by deterining the eigenstates of the Hailtonian. We look for solutions that describe one-exciton eigenstates as an expansion in the site basis. N 1 ( x) cn(x) n( xxn) (14.31) n which is written in order to point out the dependence of these wavefunctions on the lattice spacing x, and the position of a particular olecule at xn. Such an expansion should work well when the electronic interactions between sites is weak enough to treat perturbatively. For the

16 14-16 electronic structure of solids, this is known as the tight binding odel, which describes band structure as a linear cobinations of atoic orbitals. Rather than diagonalizing the Hailtonian, we can take advantage of its translational syetry to obtain the eigenstates. The syetry of the Hailtonian is such that it is unchanged by any integral nuber of translations along the chain. That is the results are unchanged for any suation in eqs. (14.3) and (14.31) over N consecutive integers. Siilarly, the olecular wavefunction at any site is unchanged by such a translation. Written in ters of a displaceent operator D, ip x / e that shifts the olecular wavefunction by one lattice constant n x n D x (14.3) These observations underlie Bloch s theore, which states that the eigenstates of a periodic syste will vary only by a phase shift when displaced by a lattice constant. ik ( x ) e ( x) (14.33) Here k is the wavevector, or reciprocal lattice vector, a real quantity. Thus the expansion coefficients in eq. (14.31) will have an aplitude that reflects an excitation spread equally aong the N sites, and only vary between sites by a spatially varying phase factor. Equivalently, the eigenstates are expected to have a for that is a product of a spatially varying phase factor and a periodic function: ( x) e ikx u( x) (14.34) These phase factors are closely related to the lattice displaceent operators. If the linear chain has N olecules, the eigenstates ust reain unchanged with a translation by the length of the chain L = N: ( x L) ( x ) n ikl Therefore, we see that our wavefunctions ust satisfy e 1, or Nk (14.35) where is an integer. Furtherore, since there are N sites on the chain, unique solutions to eq. (14.35) require that can only take on N consecutive integer values. Like the site index n, there is no unique choice of. Rewriting eq. (14.35), the wavevector is k n (14.36) N

17 14-17 We see that for an N site lattice, can take on the N consecutive integer values, so that k varies over a range of angles. The wavevector index labels the N one-exciton eigenstates of an N olecule chain. By convention, k is chosen such that < k. Then the corresponding values of are integers fro N1)/ to N1)/ if there are an odd nuber of lattice sites or N)/ to N/ for an even nuber of sites. For exaple, a olecule chain would have = 9, 8, 9,1. These findings lead to the general for for the one-exciton eigenstates N 1 1 ink k e n (14.37) N n The factor of N -1/ assures proper noralization of the wavefunction, =1. Coparing eqs. (14.37) and (14.31) we see that the expansion coefficients for the n th site of the th eigenstate is 1 1 c e e N N ink i n/ N n, (14.38) We see that for state k, with =, the phase factor is the sae for all sites. In other words, the transition dipoles of the chain will oscillate in-phase, constructively adding for all sites. For the case that k = /, we see that each site is out-of-phase with its nearest neighbors. Looking at the case of the dier, N, we see that = or 1, k = or /, and we recover the expected syetric and antisyetric eigenstates: in 1 k k e n n in 1 k / k1 e n n Scheatically for N =, we see how the dipole phase varies with k, plotting the real and iaginary coponents of the expansion coefficients

18 14-18 I( c ) n, ( n) arctan Re( cn. ) lso, we can evaluate the one-exciton transition dipole atrix eleents, M(k), which are expressed as superpositions of the dipole oents at each site, n : M N 1 (14.39) n n M k M G N 1 1 ink N n e n G n (14.4) The phase of the transition dipoles of the chain atches their phase within each k state. Thus for our proble, in which all of the dipoles are parallel, transitions fro the ground state to the k= state will carry all of the oscillator strength. Plotted below is an illustration of the phase relationships between dipoles in a chain with N =.

19 14-19 k k1 k3 Finally, let s solve for the one-exciton energy eigenvalues by calculating the expectation value of the Hailtonian operator, eq. (14.3) Ek ( ) kh k 1 N N 1 in k e H n n, (14.41) k H k S N 1 1 N n N 1 1 ik ik k V k e n1v n e n1v n N n J cos( k ) Ek ( ) Jcos( k ) (14.4) You predict that the one-exciton band of states varies in energy between J and J. If we take J as negative, as expected for the case of J-aggregates (negative couplings), then k is at the botto of the band. Exaples are illustrated below for the N= aggregate.

20 14- Note that the result in eq. (14.4) gives you a splitting of 4J between the two states of the dier, unlike the expected J splitting fro earlier. This is a result of the periodic boundary conditions that we enforce here. We are now in a position to plot the absorption spectru for aggregate, suing over eigenstates and assuing a Lorentzian lineshape for the syste: M ( Ek ( )) For a oscillator chain with negative coupling, the spectru is plotted below. We have one peak corresponding to the k ode that is peaked at J and carries the oscillator strength of all dipoles. bsorption spectru for N = aggregate with periodic boundary conditions and J <. Open Boundary Conditions Siilar types of solutions appear without using periodic boundary conditions. For the case of open boundary conditions, in the olecules at the end of the chain are only coupled to the one nearest neighbor in the chain. In this case, it is helpful to label on the sites fro n 1, N. Furtherore, 1, N. Under those conditions, one can solve for the eigenstates using use the boundary condition that at sites and N+1. The change in boundary condition gives sine solutions: k N n sin N1 n1 N1 n The energy eigenvalues are E J cos N 1

21 14-1 Returning to the case of the dier (N=), we can now confir that we recover the syetric and anti-syetric eigenstates, with an energy splitting of J. If you calculate the oscillator strength for these transitions using the dipole operator in eq. (14.39), one finds: 11 cot M k M G N 1 N 1 This result shows that ost of the oscillator strength lies in the 1 state, for which all oscillators are in phase. For large N, M1 carries 81% of the oscillator strength, with approxiately 9% in the transition to the 3 state. bsorption spectra for N = 3,7,11 for negative coupling The shift in the peak of the absorption relative to the onoer gives the coupling J. Including long-range interactions has the effect of shifting the exciton band asyetrically about. 1.4J (=1, botto of the band with J negative) N 1.8J (Top of band) Exchange Narrowing If the chain is not hoogeneous, i.e., all olecules do not have sae site energy, then we can odel this effect as Gaussian rando disorder. The energy of a given site is n We add as an extra ter to our earlier Hailtonian, eq. (14.3), to account for this variation. H HS Hdis V Hdis n n n The effect is to shift and ix the hoogeneous exciton states. n n

22 14- kn k khdis k sin n N 1 n N 1 We find that these shifts are also Gaussian rando variables, with a standard deviation of 3 N 1, where is the standard deviation for site energies. So, the delocalization of the eigenstate averages the disorder over N sites, which reduces the distribution of energies by a factor scaling as N. The narrowing of the absorption lineshape with delocalization is called exchange narrowing. This depends on the distribution of site energies being relatively sall: 3/ 3 J / N. bsorption spectra for N =,6,3 noralized to the nuber of oscillators. 3 = J and J <. Readings 1. Knoester, J., Optical Properties of Molecular ggregates. In Proceedings of the International School of Physics "Enrico Feri" Course CXLIX, granovich, M.; La Rocca, G. C., Eds. IOS Press: sterda, ; pp

23 Multiple Particles and Second Quantization In the case of a large nuber of nuclear or electronic degrees of freedo (or for photons in a quantu light field), it becoes tedious to write out the explicit product-state for of the state vector, i.e.,,, 1 3 Under these circustances it becoes useful to define creation and annihilation operators. If refers to the state of ultiple haronic oscillators, then the Hailtonian has the for which can also be expressed as H p 1 q (14.43) H (14.44) 1 aa and the eigenstates represented in through the occupation of each oscillator n1, n, n3. This representation is soeties referred to as second quantization, because the classical Hailtonian was initially quantized by replacing the position and oentu variables by operators, and then these quantu operators were again replaced by raising and lowering operators. The operator a raises the occupation in ode n, and a lowers the excitation in ode n. The eigenvalues of these operators, n n 1, are captured by the coutator relationships: a, a (14.45) a, a (14.46) Eqn. (14.45) indicates that the raising and lower operators do not coute if they are operators in the sae degree of freedo (), but they do otherwise. Written another way, these expression indicate that the order of operations for the raising and lowering operators in different degrees of freedo coute. aa aa (14.47) aa aa aa aa (14.48)

24 14-4 These expressions also iply that the eigenfunctions operations of the fors in eq. (14.47) and (14.48) are the sae, so that these eigenfunctions should be syetric to interchange of the coordinates. That is, these particles are bosons. This observations proves an avenue to defining raising and lowering operators for electrons. Electrons are ferions, and therefore antisyetric to exchange of particles. This suggests that electrons will have raising and lowering operators that change the excitation of an electronic state up or down following the relationship or bb bb (14.49) b, b (14.5) where [ ]+ refers to the anti-coutator. Further, we write b, b (14.51) This coes fro considering the action of these operators for the case where. In that case, taking the Heretian conjugate, we see that eq. (14.51) gives bb or bb (14.5) This relationship says that we cannot put two excitations into the sae state, as expected for Ferions. This relationship indicates that there are only two eigenfunctions for the operators and b, naely n and n 1 bb n bb n n or 1. This is also seen with eq. (14.5), which indicates that bb n bb n (14.53) If we now set n, we find that eq. (14.53) iplies b bb bb bb bb (14.54) gain, this reinforces that only two states, and 1, are allowed for electron raising and lowering operators. These are known as Pauli operators, since they iplicitly enforce the Pauli exclusion principle. Note, in eq. (14.54), that refers to the eigenvector with an eigenvalue of zero, whereas refers to the null vector.

25 14-5 Frenkel Excitons For electronic chroophores, we use the notation g and e for the states of an electron in its ground or excited state. The state of the syste for one excitation in an aggregate can then be written as an G, or siply n gggg,,, e g N 1 a n, and the Frenkel exciton Hailtonian is (14.55) H n n J n n, n n, or bb J bb (14.56) n n n n, n n, Readings 1. Schatz, G. C.; Ratner, M.., Quantu Mechanics in Cheistry. Dover Publications: Mineola, NY, ; p. 119.

26 Marcus Theory For Electron Transfer The displaced haronic oscillator (DHO) foralis and the Energy Gap Hailtonian have been used extensively in describing charge transport reactions, such as electron and proton transfer. Here we describe the rates of electron transfer between weakly coupled donor and acceptor states when the potential energy depends on a nuclear coordinate, i.e., nonadiabatic electron transfer. These results reflect the findings of Marcus theory of electron transfer. We can represent the proble as calculating the transfer or reaction rate for the transfer of an electron fro a donor to an acceptor D D (14.57) This reaction is ediated by a nuclear coordinate q. This need not be, and generally isn t, a siple vibrational coordinate. For electron transfer in solution, we ost coonly consider electron transfer to progress along a solvent rearrangeent coordinate in which solvent reorganizes its configuration so that dipoles or charges help to stabilize the extra negative charge at the acceptor site. This type of collective coordinate is illustrated in the figure to the right. The external response of the ediu along the electron transfer coordinate is referred to as outer shell electron transfer, whereas the influence of internal vibrational odes that proote ET is called inner shell. The influence of collective solvent rearrangeents or intraolecular vibrations can be captured with the use of an electronic transition coupled to a haronic bath. Norally we associate the rates of electron transfer with the free-energy along the electron transfer coordinate q. Pictures such as the ones above that illustrate states of the syste with electron localized on the donor or acceptor electrons hopping fro donor to acceptor are conceptually represented through diabatic energy surfaces. The electronic coupling J that results in transfer ixes these diabatic states in the crossing region. Fro this adiabatic surface, the rate of transfer for the forward reaction is related to the flux across the barrier. Fro classical transition state theory we can associate the rate with the free energy barrier using k G k T. If the coupling is weak, we can describe the rates of transfer between f exp( B )

27 14-7 donor and acceptor in the diabatic basis with perturbation theory. This accounts for nonadiabatic effects and tunneling through the barrier. To begin we consider a siple classical derivation for the free-energy barrier and the rate of electron transfer fro donor to acceptor states for the case of weakly coupled diabatic states. First we assue that the free energy or potential of ean force for the initial and final state, G(q) = kbt ln P(q), is well represented by two parabolas. 1 GD( q) q dd 1 G( q) G q d (14.58) To find the barrier height G, we first find the crossing point dc where GD(dC) = G(dC). Substituting eq. and solving for dc gives 1 1 d d G d d C D C d C G 1 d d d d D G d dd d d D The last expression coes fro the definition of the reorganization energy, which is the energy to be dissipated on the acceptor surface if the electron is transferred at dd, G 1 d G d D d d Then, the free energy barrier to the transfer G is G G d G d D D C D D 1 dc d 1 G 4 So the rrhenius rate constant is for electron transfer via activated barrier crossing is D D (14.59)

28 14-8 k ET exp G 4kT (14.6) This curve qualitatively reproduced observations of a axiu electron transfer rate under the conditions G, which occurs in the barrierless case when the acceptor parabola crosses the donor state energy iniu. We expect that we can ore accurately describe nonadiabatic electron transfer using the DHO or Energy Gap Hailtonian, which will include the possibility of tunneling through the barrier when donor and acceptor wavefunctions overlap. We start by writing the transfer rates in ters of the potential energy as before. We recognize that when we calculate therally averaged transfer rates that this is equivalent to describing the diabatic free energy surfaces. The Hailtonian is H H V (14.61) H D HD D H (14.6) Here D and refer to the potential where the electron is either on the donor or acceptor, respectively. lso reeber that D refers to the vibronic states d,nd. These are represented through the sae haronic potential, displaced fro one another vertically in energy by E E E and horizontally along the reaction coordinate q: H d E d H H a E a H D D D d a (14.63) Hd p ( qdd) (14.64) Ha p ( qd) Here we are using reduced variables for the oenta, coordinates, and displaceents of the haronic oscillator. The diabatic surfaces can be expressed as product states in the electronic and nuclear configurations: D dn,. The interaction between the surfaces is assigned a coupling J V J d a a d (14.65)

29 14-9 We have ade the Condon approxiation, iplying that the transfer atrix eleent that describes the electronic interaction has no dependence on nuclear coordinate. Typically this electronic coupling is expected to drop off exponentially with the separation between donor and acceptor orbitals; E J J exp R R (14.66) Here E is the paraeter governing the distance dependence of the overlap integral. For our purposes, even though this is a function of donor-acceptor separation (R), we take this to vary slowly over the displaceents investigated here, and therefore be independent of the nuclear coordinate (Q). Marcus evaluated the perturbation theory expression for the transfer rate by calculating Franck-Condon factors for the overlap of donor and acceptor surfaces, in a anner siilar to our treatent of the DHO electronic absorption spectru. Siilarly, we can proceed to calculate the rates of electron transfer using the Golden Rule expression for the transfer of aplitude between two states 1 wk dt VI t VI (14.67) iht/ iht/ Using V t e Ve, we write the electron transfer rate in the DHO eigenstate for as I J i Et wet dte F() t ihdt ihat where Ft () e e (14.68) (14.69) This for ephasizes that the electron transfer rate is governed by the overlap of vibrational wavepackets on the donor and acceptor potential energy surfaces. lternatively, we can cast this in the for of the Energy Gap Hailtonian. This carries with is a dynaical picture of the electron transfer event. The energy of the two states have tie-dependent (fluctuating) energies as a result of their interaction with the environent. Occasionally the energy of the donor and acceptor states coincide that is the energy gap between the is zero. t this point transfer becoes efficient. By integrating over the correlation function for these energy gap fluctuations, we characterize the statistics for barrier crossing, and therefore forward electron transfer. Siilar to before, we define a donor-acceptor energy gap Hailtonian H D H HD (14.7)

30 14-3 which allows us to write F t exp dth t and D t i D ihdt/ ihdt/ D H t e H e (14.71) (14.7) These expressions and application of the cuulant expansion to eq. allows us to express the transfer rate in ters of the lineshape function and correlation function i F t exp HD t g t (14.73) t (14.74) g t d d C D C t H t H (14.75) D D D H D (14.76) The lineshape function can also be written as a su of any coupled nuclear coordinates, q. This expression is coonly applied to the vibronic (inner shell) contributions to the transfer rate: D it it 1e 1 e 1 g t d d n i t n i t D d d coth / cost 1 isint t (14.77) Substituting the expression for a single haronic ode into the Golden Rule rate expression eq. gives w ET J dte i Et/ g( t) J ie t/ exp dt e D coth / cost 1i sint (14.78) where D d dd (14.79) This expression is very siilar to the one that we evaluated for the absorption lineshape of the Displaced Haronic Oscillator odel. detailed evaluation of this vibronically ediated transfer rate is given in Jortner. To get a feeling for the dependence of k on q, we can look at the classical liit kt. This corresponds to the case where one is describing the case of a low frequency

31 14-31 solvent ode or outer sphere effect on the electron transfer. Now, we neglect the iaginary part of g(t) and take the liit coth : J ie t DkBT wet dte exp 1cost (14.8) Note that the high teperature liit also eans the low frequency liit for. This eans that we can expand cos t 1 t, and find w ET J E exp kt 4kT (14.81) where D. Note that the activation barrier E for displaced haronic oscillators is E E. For a therally averaged rate it is proper to associate the average energy gap with the standard free energy of reaction, H HD G. Therefore, this expression is equivalent to the classical Marcus result for the electron transfer rate k ET o G exp 4kT (14.8) where the pre-exponential is This expression shows the nonlinear behavior expected for the dependence of the electron transfer rate on the driving force for the forward transfer, i.e., the reaction free energy. This is unusual because we generally think in ters of a linear free energy relationship between the rate of a reaction and the equilibriu constant: ln k ln K eq. This leads to the thinking that the rate should increase as we increase the driving free energy for the reaction G. This behavior only hold for a sall region in G. Instead, eq. shows that the ET rate will increase with G, until a axiu rate is observed for G and the rate then decreases. This decrease of k with increased G is known as the inverted regie. The inverted J 4kT (14.83)

32 14-3 behavior eans that extra vibrational excitation is needed to reach the curve crossing as the acceptor well is lowered. The high teperature behavior for coupling to a low frequency ode (1 c -1 at 3 K) is shown at right, in addition to a cartoon that indicates the shift of the curve crossing at G is increased. Particularly in intraolecular ET, it is coon that one wants to separately account for the influence of a high frequency intraolecular vibration (inner sphere ET) that is not in the classical liit that applies to the low frequency classical solvent response. If an additional ode of frequency and a rate in the for eq. is added to the low frequency ode, Jortner has given an expression for the rate as: w ET o G j D J e j D exp kt j j! 4kT (14.84) Here is the solvation reorganization energy. For this case, the sae inverted regie exists; although the siple Gaussian dependence of k on G no longer exists. The asyetry here exists because tunneling sees a narrower barrier in the inverted regie than in the noral regie. Exaples of the rates obtained with eq. are plotted in the figure below (T= 3K). s with electronic spectroscopy, a ore general and effective way of accounting for the nuclear otions that ediate the electron transfer process is to describe the coupling weighted density of states as a spectral density. Then we can use coupling to a haronic bath to describe solvent and/or vibrational contributions of arbitrary for to the transfer event using g t d coth 1costisintt (14.85)

33 14-33 Readings 1. Barbara, P. F.; Meyer, T. J.; Ratner, M.., Conteporary issues in electron transfer research. J. Phys. Che. 1996, 1, , and references within.. Georgievskii, Y.; Hsu, C.-P.; Marcus, R.., Linear response in theory of electron transfer reactions as an alternative to the olecular haronic oscillator odel. The Journal of Cheical Physics 1999, 11, Jortner, J., The teperature dependent activation energy for electron transfer between biological olecules. Journal of Cheical Physics 1976, 64, Marcus, R..; Sutin, N., Electron transfers in cheistry and biology. Biochiica et Biophysica cta (BB) - Reviews on Bioenergetics 1985, 811, Nitzan,., Cheical Dynaics in Condensed Phases. Oxford University Press: New York, 6; Ch. 1.