Electronic Energy Transfer in Molecular Aggregates: Theoretical Approaches from F

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Electronic Energy Transfer in Molecular
Aggregates Theoretical Approaches from Förster,
Redfield and beyond

• Co-workers Jianshu Cao, Xin Chen, Eric Zimanyi,
  Seogjoo Jang,Yuan-Chung Cheng, Alberto Suarez,
  Irwin Oppenheim
• Experimental colleagues G. Scholes (Toronto),
  J. Kohler (Bayreuth), S. Volker (Leiden)

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Förster 1948

• Formula for Resonance Energy Transfer in terms of
  donor emission and acceptor absorption
• (trad/tD) 1 trad/tET 1 (Ro/R)6
• Ro calculated from the overlap of donor emission
  and acceptor absorption which provides a
  measurement of R
• Single donor and large number of random acceptors
  leads to exponential fluorescence.
• Today, used as a biological ruler in many single
  molecule experiments.

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• Recent extensions multi-chromophoric donors and
  acceptors application to light harvesting
  complexes (Sumi et al Scholes Jang, Newton,
  RJS Cheng RJS) Review of the assumptions of
  model Beljonne, Curutchet, Scholes, RJS (JPCB
  2009)

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Förster Resonance Energy Transfer

• H (1/2) S?PD?2 ? D?2Q D?2 (1/2) SµPAµ2
  ? Aµ2Q Aµ2
• EDS? gDQD? De Aggt ltDeAg EASµgAQAµ Dg
  Aegt ltDgAe
• J De Aggt ltDgAe Dg Aegt ltDeAg
  
• J Coulomb interaction between D and A -- in the
  multipole expansion this becomes a (transition)
  dipole dipole interaction 1/R3
• Note the close resemblance to the Marcus model
  for electron transfer.
• Use 2nd order perturbation theory (Fermi Golden
  Rule) starting from the eigenstates of H with J0
  (polaron or displaced oscillator states).

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Davydov The Theory of Molecular Excitons

• In 1948, Davydovs book was published in Russia
  and translated by Kasha in the 1950s.
• Davydov was only interested in coherences His
  treatment of the electronic states of molecular
  crystals was like band theory in solid state
  physics. The states were superpositions of site
  (excited) states with translational invariance.
• Davydov was interested in low temperature
  spectroscopy of organic solids Förster was
  interested in molecular fluorescence in solution.
  

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Moderate to Strong Electronic Interactions
Between Donor and Acceptor

• When both coherent and incoherent processes take
  place, there are many possible theoretical
  descriptions (which are often rearrangements of
  others), usually involving the reduced density
  matrix (that is, the density matrix for the
  entire system averaged over the thermal
  environment). Some names are stochastic
  Liouville equation (SLE), Generalized Master
  Equation (GME), Bloch equation, Redfield theory,
  Lindblad form, Completely Ordered Perturbation
  (COP), Partially Ordered Perturbation (POP), ...
• In all of these methods, it is difficult to go
  beyond second order in the perturbation, so one
  has to be careful in choosing how to form H0 V
  from H.

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Redfield Model Coherence, decoherence, and all
that

• H H0 V Hsys Hbath V
• dr/dt -i H, r ?(t) Trbath r???
• r(0) ?(0) rbathequ (assumption)
• d?(t)/dt -i Hsys , ?(t) - ?0tK(?) ?(t-??
  d??????Exact!
• Redfield bath relaxes much faster than system,
  so K decays quickly, then
• d?(t) /dt -i Hsys , ?(t) - R ?(t?
• R(ij,kl) given in terms of infinite integrals
  over time correlation functions of matrix
  elements of V (in the system eigenstates
  representation). Note well the Redfield model is
  a weak coupling model, and depends on how you
  choose H0 and V. It has problems if you are not
  careful.

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Redfield Model Coherence, decoherence, and all
that

• Consider a two state system, like the Förster
  model. Hsys has eigenstates 1gt and 2gt
  (eigenstates of Hsys not site states!) , then
  ?11 and ?22 are populations and ?12 and ?21 are
  coherences. The general Redfield model will have
  coupling terms between populations and
  coherences. The secular approximation neglects
  these and then (A exp- (E1-E2)/T)
• d(?11-?22)/dt -(1-A)? - (1A)? (?11-?22)
• d?12/dt -i(E1-E2) - ?pd(1A) ?/2
  ?12(1A)?/2 ?21

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Redfield Model Coherence, decoherence, and all
that

• ?????-88ltV12(t)V21(0)gt expi(E1-E2)tdt
• Fermi Golden Rule for transitions from 1 to 2
• ?pd ?? ?-88ltV11(t)-V22(t)V11(0)-V22(0)gt dt
• Pure dephasing rate (fluctuation of eigenenergy
  due to bath)
• The terms coupling coherences and populations are
  related to
• ?-88ltV11(t)-V22(t)V12(0)gt exp(i?t)dt
• for ? 0 and ? E1-E2

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Redfield Model Coherence, decoherence, and all
that

• Lets naively apply this to the Förster problem
  choose Hsys ED De Aggt ltDeAg EA Dg Aegt
  ltDgAe J De Aggt ltDgAe
  Dg Aegt ltDeAg
• VS? gDQD? De Aggt ltDeAg SµgAQAµ Dg Aegt
  ltDgAe
• Now the zeroth order states are exciton states
  (linear combinations of the site states) and the
  perturbation term only allows one quantum jumps
  in the vibrational modes. Leads to incorrect
  FRET formula!
• What went wrong? We chose the wrong zeroth order
  Hamiltonian. This is a common mistake in
  thinking about energy transfer, coherence and
  decoherence.

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Polaron Transformation

• To repair this, we must choose a different zeroth
  order Hamiltonian. One good way to do this is to
  transform the Hamiltonian by a unitary
  transformation that diagonalizes the V in the
  site representation, that is, form polaron site
  functions. This transforms the electronic
  coupling term to
• J expS? gDPD?????? S? gAPAµ???? De Aggt ltDeAg
  ? H.c.
• Now average this over the bath density matrix to
  get
• ltJgt Jexp(-S) and add and subtract this average
  from the Hamiltonian. Finally,

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Polaron Transformation

• Hsys ED De Aggt ltDeAg EA Dg Aegt ltDgAe
  
• ltJgt De Aggt ltDgAe Dg Aegt
  ltDeAg
• V JexpS? gDPD?????? S? gAPAµ??????J?? De Aggt
  ltDeAg
• ?H.c.
• Note that the zeroth order site energies and the
  electronic coupling have been changed by the
  interaction with the bath and are T dependent.
  The zeroth order eigenstates are excitons (I.e
  coherent superpositions of site states) but with
  renormalized site energies and J.
• Note also, the perturbation is small when g is
  large (Förster) and when g is small (weak
  exciton-phonon coupling).
• The initial condition may have to be changed
  (Jang etal JCP 2008).

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Polaron Transformation

• The renormalized energy transfer matrix element,
  ltJgt, is T dependent. It goes to 0 for large T or
  large g.
• For Ohmic coupling to the bath, this procedure
  leads to ltJgt 0 for all g. In order to improve
  the result, you can do a variational method.

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Variational Polaron Transformation

• We can further improve on this by choosing the
  unitary transformation in a variational manner
  (to minimize either the lowest energy state or
  the free energy at T). When this is applied to
  the spin-Boson model with Ohmic coupling,
  (almost) all the results of Caldeira-Leggett et
  al are reproduced Silbey and Harris, JCP 1984)
• For our discussion, the important point is that
  we almost always do 2nd order perturbation
  theory. Therefore we should be careful that we
  choose H0 and V so that our density matrix
  equation has a chance of working in as many
  situations as possible.

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Polaron Transformation Holstein and
othersEnergy or electron transport in molecular
crystals

• The polaron transformation and a second order
  treatment for electron or excitation transfer was
  first discussed by Holstein (1959). He
  calculated the rate of electron transfer from one
  site to another and thus a formula for the
  diffusion constant valid in the hopping regime .
  
• Others (Lang and Firsov, Grover and Silbey
  Kenkre and Knox) used the full density matrix
  equations (or equivalent) and found a formula for
  the diffusion of excitons that had both band
  (coherent) and hopping (incoherent) terms that
  nicely merged the two regimes.
• D/a2 ltJgt2 ??T) Dhop(T) (a nn distance)
• Where ??T) is the scattering time in the band
  model and Dhop is the hopping rate (similar to
  Holstein)

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Haken-Strobl-Reineker Model

• In this model, the exciton- phonon interactions
  are replaced by classical Gaussian white noise
  (fluctuations in the site energies and transfer
  matrix elements)
• H0 ED Dgt ltD EA Agt ltA
• ltJgt Dgt ltA Agt ltD
• V eD(t) Dgt ltD eA(t)Agt ltA
• j(t) Dgt ltA Agt ltD where
• lt eD(t) eD(t)gt lt eA(t) eA(0)gt 2 ?0 ?(t-t)
• ltj(t)j(t)gt 2?1 ?(t-t)
• lt eD(t) eA(t)gt0 ltj(t) ei(t)gt 0

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Haken-Strobl-Reineker Model

• Using these assumptions, we find an equation for
  the density matrix that is exact (within the
  assumptions) and is identical to the Redfield
  equations for a infinitely fast bath
• d?(t) /dt -i H0 , ?(t) - RHSR ?(t?
• The eigenstates of H0 are tan ? -2ltJgt/(ED -EA)
  
• gt cos(?/2)Dgt sin (?/2) Agt
• -gt - sin(?/2) Dgt cos (?/2) Agt

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Haken-Strobl-Reineker Model

• And the R matrix is
• ? ?? ?????? ????
• ?????? ? ???? ???? ????? ?????
• ????????? ?????? ?pd????? ??
• ???????? ????? ? ?? ?pd??
• Because of the HSR assumptions, A 1, and the ?
  dependence of the matrix elements that would
  appear in the Redfield model is gone. This model
  has coupling between populations and coherences.
• ? s2?02c2 ?1 ?pd??? 2c2 ?0 4s2?1 ?(0)
  sc(?0-?1)

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Haken-Strobl-Reineker Model

• The HSR model assumes no correlation between j(t)
  and ei(t) This can easily be fixed, but the
  equations then are a bit more complex.
• The HSR model predicts equal populations in the
  states at equilibrium (A 1) and so is
  appropriate only for Tgtgt ?. However, one is
  tempted to remedy this by inserting the correct A
  in the matrix elements of R in order to assure
  the proper equilibrium,
• There have been attempts to relax the assumption
  of white noise and calculate the correction for
  short (but not zero) relaxation time of the
  correlation functions, but this has not yielded a
  consistent result.
• One can easily put correlation between sites into
  the equations.

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Haken-Strobl-Reineker Model

• The HSR model predicts that for ?0 lt ?, there
  will be oscillations in the population matrix
  elements (evidence of coherence).
• The HSR model does not have the problems of the
  Redfield model (see later) that is, the
  eigenvalues of ? are between 0 and 1 for all
  initial conditions. This is due to the delta
  function correlations or infinitely fast bath
  relaxation and the fact that A 1.

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Generalized Master Equation

• The equation for the reduced density matrix is
  called a Stochastic Liouville Equation. If we
  reduce the variable set further to consider only
  populations, we can derive a GME
• dP(t)/dt ?0t M(?) P(t-?? d?????
• In the limit that M(t) relaxes very quickly
  compared to P, we get the Pauli master equation.
• It is easy to show that the SLE and the GME are
  equivalent that is, given an SLE (e.g. the HSR
  model or a Redfield model), one can find the
  equivalent GME.
• For these cases, one finds an M(t), memory
  function, that has oscillations and relaxation
  even though the bath relaxes infinitely quickly.
  That is, the memory function in the GME may have
  nothing to do with the memory in the bath!

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Redfield and Slippage of Initial Conditions

• It has been known that the Redfield equations can
  lead, for certain initial conditions, to
  violations in the positivity of the reduced
  dynamics -- that is, it can lead to negative
  values of population. Alicki and Lendl,
  Lindblad, van Kampen, Pechukas, and Dumke and
  Spohn have all looked at this problem.
• Suarez, Oppenheim and RJS (JCP 1992) took another
  tack. We assumed that the problem was that the
  early time dynamics of the system was
  inextricably involved with the relaxation of the
  bath from its assumed initial condition. After
  all, you do not expect the Redfield equation to
  be correct at early times. We solved the
  dynamics of a simple exciton phonon problem by
  perturbation theory for short times and compared
  to the Redfield result.

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Redfield and Slippage of Initial Conditions
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Redfield and Slippage of Initial Conditions
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Redfield and Slippage of Initial Conditions
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Redfield and Slippage of Initial Conditions-
Gaspard and Kagaoka

• Gaspard and Kagaoka (JCP 111, 5668 (1999))
  defined a slippage operator S such that
  ?(0)slipped S ?(0) by solving the early time
  dynamics as in Suarez and comparing to Redfield
  for times tb ltlt t ltlt tsys.
• The non-Markovian evolution at early times leaves
  its signature in the slippage conditions. For
  delta function correlations, the equation reduces
  to a Lindblad form.
• For slipped conditions, all of the initial
  conditions found to give non-positivity for
  unslipped Redfield give good results when slipped
  initial conditions are included (weak coupling)
• Slippage of initial conditions takes into account
  the initial dynamics that can lead to problems
  for particular initial conditions up to second
  order in perturbation. If the bath-system
  interaction gets too large, this simple version
  of slippage may not solve the problem.

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Redfield and Slippage of Initial Conditions- Y-C
Cheng and RJS

• Cheng introduced another version of slippage
  integrate the perturbation theory for a time, t,
  larger than the bath relaxation time and shorter
  than the time that 2nd order pt should work.
  Then at that time, use your results as the
  initial conditions for a Redfield description.
  For a simple spin Boson model, he compared these
  results to a calculation using a non-Markovian
  description that should work for weak coupling.
  For all initial conditions, both approximations
  kept the positivity of the density matrix.

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Efficiency and Optimization of Energy Transfer
and Trapping

• Recently, the question of the how decoherence (or
  dephasing) can help to optimize energy trapping
  has been examined in an interesting way (see
  talks at this workshop). A simple approximation
  that seems to get qualitative results (Cao and
  Silbey, unpublished)
• The off diagonal density matrix equations in the
  site representation, sji(t) have terms like
  
  
  dsji/dt -iEj-Ei- Gij sji - iJji sjj - sii
• Take stationary approx. sji Jji sjj - sii/
  Ej-Ei- iGij
• This approximation yields an equation in terms of
  populations alone -- i.e a kinetic equation with
  effective rates, that is much easier to solve.
  Can compare to exact results for simple model
  systems.

J. Cao and R. Silbey Optimization of exciton
trapping in energy transfer processes JPC
(submitted, 2009)
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Efficiency and Optimization of Energy Transfer
and Trappingsimple linear examples

• ...

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Efficiency and Optimization of Energy Transfer
and Trappingsimple linear examples

• 2 site with trap lttgt 2/kt (?2G2)/(2G J2)
  
• exact for Bloch equations with G Gpd
  ?(kt/2)
  
• N.B. the optimal trapping time is dependent on
  the dephasing rate e.g for fixed kt lt 2? , the
  optimal lttgt 2/kt ?/J2 for G ?- kt/2.
• N site funnel with constant detuning ?, J, and G
  
• lttgt N/kt (N-1)/2 J2N G/2 kt/2 ?2/G
  kt/2
• for G (2/N)1/2?- kt/2.
  
  once again the optimal trapping time depends
  on the dephasing rate.
• We have gone on to look at more complicated
  structures (with loops that require quantum
  corrections, and gone beyond the first order
  approximation), but the message is clear the
  optimal trapping rate depends on the dephasing
  rate.

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Efficiency and Optimization of Energy Transfer
and Trapping

• What is the physics behind the importance of ?
  (noise) for optimizing trapping?
• a) given a set of interacting chromophores, the
  zeroth order eigenstates are excitons (coherent
  super positons). Some of those eigenstates may
  have very weak coupling to the trap state. Noise
  or disorder mixes the eigenstates and allows
  trapping to occur.
• b) ? broadens the energy levels and improves
  energetic overlap and therefore energy transfer
  to the trap.

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The effect of coherence in a real biological
system LH2

• Very briefly, I discuss our work on spectroscopy
  and energy transfer in the light harvesting
  complex LH2 of photosynthetic bacteria.

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How does the B850 coherence effect the B800 -gt
B850 energy transfer?

• Compare the energy transfer rate between B800 and
  B850 using standard Forster theory (FRET) with
  the energy transfer rate when the coherent states
  of B850, including disorder, are used (MC-FRET).

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What happens if we change the relative energies
of the B800 and B850 molecular
transitions? Jang, Newton and Silbey, JPC (2007)
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Our calculations suggest that the coherence in
the multi-chromophoric aggregate of the B850 band
plays a role in keeping the FRET rate from B800
to B850 stable for changes in the relative
excitation energies of the B800 Chlorophyls and
the B850 Chlorophyls. --- This will have to
tested in other aggregates to see if it is
general.
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Conclusions

• Since we almost always do second order theory, we
  should try to choose H0 and V to make the
  analysis as wide ranging as possible.
• Coherent and incoherent interactions have been
  looked at in the context of a unified band and
  hopping picture. Perhaps we can learn something
  from that perspective.
• The interaction between dephasing, population
  decay, static energetic disorder and coherent
  interactions, especially at room T have to be
  explored by a number of methods.
• In LH2, it is clear that coherence in the
  electronic states plays a fundamental role in the
  ( picoseond) energy transfer processes.

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