Quantum dynamics in complex environments –towards biological and nanostructured systems

Chris Engelbrecht Summer School on Quantum Biology

Lecture 4

Irene Burghardt

Department of Physical and Theoretical Chemistry, Goethe University Frankfurt

1

Agenda

Lecture 1 Introduction – some general perspectivesLecture 2 Photophysics of extended molecular systemsLecture 3 Electronic, excitonic, and vibrational coherenceLecture 4 Excitation energy transfer (EET) – coherent or not coherent?Lecture 5 Non-Markovian system-environment interactions

2

Excitonic coherence in multi-chromophoric systems

• Two basic (single-excited) configurations:

|φ1〉 = |e(1)〉 ⊗ |g(2)〉 |φ2〉 = |g(1)〉 ⊗ |e(2)〉

• For two identical, non-interacting monomers, these configurations aredegenerate

• once an excitonic coupling is included, the degeneracy is removed

• Frenkel exciton state = superposition of these configurations:

|Ψexciton(t)〉 = c1(t)|φ1〉+ c2(t)|φ2〉3

Exciton model Hamiltonian

• matrix representation in the basis of single (locally) excitedconfigurations (site basis)

Hexciton =

ε(1) JJ ε(2) J

J ε(3) J. . . . . . . . .

J ε(n−1) JJ ε(n)

where J appears as a diabatic coupling

• alternative way of writing the same Hamiltonian employingcreating/annihilation operators:

H =∑n

ε(n)a†nan + J(a†nan+1 + a†n+1an)4

What is the excitonic coupling?

• Coulomb coupling matrix element between donor (D) and acceptor (A)permitting transitions between single-excited configurations:

J = 〈eDgA|VCoulomb|gDeA〉

• this can be re-written in terms of coupled transition densities ρ(ge)A and

ρ(eg)D (noting that D/A exchange interactions have been disregarded!):

J =∫dr3Ddr3

A

ρ(eg)D (rD)ρ(ge)

A (rA)

|rD − rA|

• at large D-A distances, one can approximate transition densities by

transition dipoles: ρ(eg)D µ

(eg)D

5

Alternatively: diabatization approach

0.8 eV

Bright Dark ∆E/2 = 0.86 eV

Bright-Dark exciton gap

Coupling between localized excitons

• start from a “super-molecular” calculation forthe (DA) system

• employ a diabatizationprocedure which trans-forms the adiabatic (Born-Oppenheimer) electronicstates to a diabatic basisof site-local states

6

Exciton model including vibrations

Hexciton =N∑i

ωi

2

(p2i + x2

i

)

+

ε(1) +

∑i κ

(1)i xi J

J ε(2) +∑i κ

(2)i xi J

J ε(3) +∑i κ

(3)i xi J

. . . . . . . . .

J ε(n)+∑i κ

(n)i xi

• here, we assumed that the vibrational modes couple diagonally

• the coupling parameters can be obtained from the monomerHamiltonian; for example, for a mode that couples exclusivelyto monomer 1: h1(x1) ' c(1)

g x1|g(1)〉〈g(1)|+ c(1)e x1|e(1)〉〈e(1)|,

contributions appear for all exciton configurations 7

Vibronic coupling picture of excitation energytransfer (EET)

Donor Acceptor

FRET

De Ag

Donor Acceptor

Dg Ae

Vibrational modes0

Initial state Final state

absorption

D A

Vibrational modes of 

D­A super­molecule

FRET

• FRET = fluorescence resonance energy transfer (equivalent to EET)8

Exciton transfer can be coherent and ultrafast

quantum dynamics simulation:

0

0.2

0.4

0.6

0.8

1

0 500 1000 1500

site

pop

ulat

ion

time [fs]

site 1: initial excitation

23 4 5 6 7 8 9

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

0 200 400 600 800

site

pop

ulat

ion/

cohe

renc

etime [fs]

site 1: initial excitation

23

12

13

23 coherence

• site-site excitation energy transfer |φn〉 |φn+1〉• transfer is mediated by coherences |φn〉〈φn+1|• but the environment could rapidly induce “decoherence”

• coherent transfer can be observed experimentally! 9

Correlations and coherenceǫ1

ǫ2

ǫ3

ǫ4• • •

ǫn• Do correlated fluctuations favor

conservation of coherence?

0

0.2

0.4

0.6

0.8

1

0 500 1000 1500

po

pu

latio

ns

time [fs]

J=0.01 eV

pop 1pop 2pop 3pop 4pop 5

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0 500 1000 1500

co

he

ren

ce

s

time [fs]

Im coh 12Im coh 13

0

0.2

0.4

0.6

0.8

1

0 500 1000 1500

po

pu

latio

ns

time [fs]

J=0.01 eV

pop 1pop 2pop 3pop 4pop 5

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0 500 1000 1500

co

he

ren

ce

s

time [fs]

Im coh 12Im coh 13

(a) site-local couplings by high-frequencyand low-frequency modes

(b) nearest-neighbor correlatedcouplings, with identical overallcoupling strengths as in (a) 10

Excitonic eigenstate picture

electronic densities of stacked

adenine pentamer corresponding to two

ππ∗ states with different degrees of

localisation

Bittner & co-workers, in: Energy TransferDynamics in Biomaterial Systems, Burghardt etal. (eds), Springer (2009).

• the excitonic eigenstates are obtained by diagonalizing the excitonHamiltonian, as a function of the nuclear coordinates:

|Ψexciton(R; t)〉 =∑n

cn(R, t)|φn〉11

Exciton transfer in the FMO complex

Brixner & collaborators, Nature 434, 625 (2005)

7

6

5

5, 6

4

3

3, 7

2

2, 4

1

1

• FMO = Fenna-Matthews-Olson bacteriochlorophyll a (BChl) protein ofgreen sulphur bacteria

• antenna system that collects light and channels excitation to a reactioncenter where charge transfer takes place 12

“Coherence dynamics in photosynthesis: proteinprotection of excitonic coherence”

Lee, Cheng, Fleming, Science 316, 1462 (2007)

k1

k3 k2

ks=-k1+k2+k3

t1t2B

Fig. 1. The 2CECPE experiment. (A) The 77 K

• 2CECPE = two-color electronic coherence photon echo experiment13

2D photon echoes

pulse 1 −−− t1 −−− pulse 2 −−− t2 −−− pulse 3 −−− t3 −−−

ρeg ρee′ ρfe

• e single-exciton manifold ; f double-exciton manifold

• 2D Fourier transform Secho(ω1, t2, ω3), for various t2’s

• cross peaks appear, indicating a transfer process via ρee′ in the t2 interval

cf. Abramavicius & Mukamel, JCP 133, 064510 (2010)

Cheng & Fleming, Annu. Rev. Phys. Chem. 60, 241 (2009)

14

Simulated 2D PE spectrum (RC of photosystem II)

064510-4 D. Abramavicius and S. Mukamel J. Chem. Phys. 133, 064510 ~2010!

• main diagonal peaks: D1 (e1) and D2 (e2)

• main cross peak: C1 (e1 e2)

• left: Lindblad model (QT)

• right: Classical transfer model (CT)

from: Abramavicius & Mukamel, JCP 133, 064510 (2010)

15

Can we use perturbative limits?

consider site-site coupling (J) vs. electron-phonon coupling (κ)

• case 1: J � κ

Forster theory – non-coherent hopping between sites

e.g., Scholes & collaborators, J. Phys. Chem. B, 113, 656 (2009): EET in semiconducting polymers

• case 2: κ� J

excitonic eigenstates in a vibrational bath

e.g., Abramavicius & Mukamel, J. Chem. Phys. 133, 064510 (2010): EET in Photosystem II

• but in most systems of interest: κ ∼ J

in principle, we need the full dynamics on the Born-Oppenheimer surfacesof the oligomer/aggregate species 16

Two examples

• Case 1: Fluorescent markers in biology: Forster theory

• Case 2: EET in conjugated polymers

17

Fluorescent markers in biological applications

18

Derivation of Forster theory, in a nutshell:Fermi’s Golden Rule + dipole-dipole interaction

thermally averaged rate (2nd order perturbation theory):

kDA =2π

h

∑D/A vib. states

f(E∗D)f(E0A) |〈ψ∗Dψ

0A|V

dipDA|ψ

0Dψ∗A〉|

2δ(E∗D+E0A−E

∗A−E

0D)

〈ψ∗Dψ0A|V

dipDA|ψ

0Dψ∗A〉 = 〈ψ∗Dψ

0A|µD · µA|rDA|3

−3(rDA · µD)(rDA · µA)

|rDA|5|ψ0Dψ∗A〉

= κDA〈ψ∗D|µD|ψ0

D〉〈ψ0A|µA|ψ∗A〉

|rDA|3

Now identify the ingredients of this formula with the expressions for thedonor emission / acceptor absorption spectra spectral overlap 19

Forster theory = rate theory for resonant energytransfer between two dipoles

kDA(rDA) = kradD

( rFrDA

)6

r6F =

9c4 κ2DA

8πη4

∫ ∞0

dω1

ω4ID(ω)αA(ω)

“Forster radius” rF ∝ orientational factor × spectral overlap

ID(ω) = donor emission spectrum

αA(ω) = acceptor absorption spectrum

κDA = orientational factor:

κDA = sinθD sinθA cosΦARD − 2 cosθD cosθA

Standard Forster expression: isotropic average κ2DA = 2/3 20

Forster theory, cont’d

kDA(rDA) = kradD

( rFrDA

)6; r6

F =9c4 κ2

DA

8πη4

∫ ∞0

dω1

ω4ID(ω)αA(ω)

21

Exciton migration and long-lived coherences

Collini, Scholes, Science 323, 369 (2009), Bredas & Silbey, Science 323, 348 (2009)

V lin = ∑i

κ(1)

i xi λ (12)i xi

λ (12)i xi κ(2)

i xi λ (23)i xi

λ (23)i xi κ(3)

i xi λ (34)i xi

λ (43)i xi κ(3)

i xi λ (45)i xi

... ... ...

• coherent “surfing” rather than Forster-type hopping?• role of site-local vs. shared (correlated) modes• do shared modes contribute to the conservation of electronic coherence?

22

How do Excitons Migrate across a Torsional Defect (Kink)?

1 2 3 4 5−0.2

0

0.2

0.4

0.6

0.8

1

0 250 500 750 1000

time [fs]

cos(theta)

p 4+5

Im coh(34)

• exciton initially localized on left fragment (1 + 2 + 3)• the transfer to the (4 + 5) fragment sets in as a function of the torsional

displacement at the (34) junction• this transfer is mediated by the excitonic coherence ρ34

23

Electronic Structure of Oligomers (OPV’s)

(Collaboration Lischka group (Vienna, Texas Tech))

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• high-level electronicstructure methods(ADC, CC2, MRCI)

• exciton trapping, due toBLA modes, describedcorrectly

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Panda, Plasser et al., in preparation, see also: Sterpone, Rossky, JPCB 112, 4983 (2008), Nayyar et al., JPCL 2, 566 (2011)

24

Ring Torsions in OPV’s

0 30

monom

er

pote

ntial

torsion [deg]

S1

S0

●FC

• calculate torsion profiles ofoligomer excited states

• effective, correlated torsionalmodes, e.g., (T1-T1a) ≡ θ

nn′

• likewise, correlated BLA modes

0

0 0.5

1 1.5

2 2.5

3 3.5

4 4.5

5

0 30 60 90 120 150 180

Angle(deg)

T3

S0S1S2

Panda,Pla

sser

etal.,to

be

subm

itte

d(2

011)

25

Fit Data to Exciton-Phonon Model

H = H0 +∑n Hmonomern (x1n

, . . . ,xNn)+∑nn′ δn′,n+1H

junction

nn′(θ torsion

nn′,xBLA

nn′)

• express H in the basis of single-excited configurations:|n〉= |G(1)

G(2) . . .E(n) . . .G(N)〉

• initial state = lowest excited (adiabatic) exciton state @ FC geometry; thenevolve in time: |ψFrenkel(t)〉= ∑n cn({xi,θi}, t) |n〉

• diagonal potentials (torsion + BLA):∝ V0(1− cos(σ(θnn′ −θ0)))+ k/2(xnn′ − x

0nn′

)2

• nearest-neighbor couplings modulated by torsions:

〈n|Hjunction

nn′|n′ = n+1〉= Jnm cosθnn′ (1−α(xnn′ − x0))

Sterpone, Martinazzo, Panda, Burghardt, Z. Phys. Chem. 225, 541 (2011)26

Quantum Dynamics based on Model Hamiltonian

0

0.2

0 250 500 750 1000

time [fs]

p 4+5

p 4

Im coh(34)

0 300

600 900

1 2

3 4

5

1

po

pu

latio

n

time [fs]site index

• T = 0 simulations

• fluctuations due to high and low frequency modes reduce excitonic coherencebeyond 200 fs (but, marked dependence on model parameters)

• Ohmic damping (HO bath) acting upon torsional mode @junction

• extend to include (i) non-zero T and (ii) static disorder27

EET Time Scales

tens of fs partial localization/trapping of excitons due to e-ph(BLA) coupling; formation of “spectroscopic units”

∼200 fs initial polarization anisotropy decay due to exciton mi-gration, possibly involving coherent intra-chain EET

100 fs- few ps torsional geometry relaxation interfering with EET

1-10 ps inter-chain EETps-ns thermally assisted hopping, mainly intra-chain

28