Coherence Vibrations and Electronic Excitation Dynamics in Molecular

Aggregates and Photosynthetic Pigment-Proteins

L. Valkunas

Department of Theoretical Physics, Faculty of Physics, Vilnius University,

Institute of Physics, Center for Physical Sciences and Technology, Vilnius

Lithuania

VILNIUS

UNIVERSITY

Contents

• Coherent vs. Incoherent Processes

• Coherences and oscillations in 2D ES

• Model systems

– Coupled two-level systems (electronic dimer)

– Displaced harmonic oscillator

– Coupled displaced harmonic oscillators (molecular dimer)

• Real stuff – PSII RC, FCP

Energy transfer and charge separation

Resonance of energy transfer

Electronic

excited state

Electronic

ground state

Vibronic

states

Resonance

energy

transfer to

nearby

molecule

Molecule 1 Molecule 2

Are excitons the excitons?

• The system evolves into new equilibrium with respect to the excited system configuration.

• The initial exciton becomes not the eigenstate

A. Gelzinis, D. Abramavicius, L. Valkunas, Phys. Rev. B. 84, 245430, 2011.

Are excitons the excitons?

• The systems evolves into new equilibrium with respect to the excited system configuration.

• The initial exciton becomes not the eigenstate

A. Gelzinis, D. Abramavicius, L. Valkunas, Phys. Rev. B. 84, 245430, 2011.

Absorption

Are excitons the excitons?

• The systems evolves into new equilibrium with respect to the excited system configuration.

• The initial exciton becomes not the eigenstate

A. Gelzinis, D. Abramavicius, L. Valkunas, Phys. Rev. B. 84, 245430, 2011.

Emission

Are excitons the excitons?

• The system evolves into new equilibrium with respect to the excited system configuration.

• The initial exciton becomes not the eigenstate

A. Gelzinis, D. Abramavicius, L. Valkunas, Phys. Rev. B. 84, 245430, 2011.

Emission

Are excitons the excitons?

• The system evolves into new equilibrium with respect to the excited system configuration.

• The initial exciton becomes not the eigenstate

A. Gelzinis, D. Abramavicius, L. Valkunas, Phys. Rev. B. 84, 245430, 2011.

Emission

Mixed picture where all “excitons” are more localized

Search for the preferred basis

• The exciton basis is given by:

• The global basis is different:

• The density matrix is diagonal in the global basis at equilibrium

• What happens in the molecular representation?

A. Gelzinis, D. Abramavicius, L. Valkunas, Phys. Rev. B. 84, 245430, 2011.

EFFECTIVEPARAMETERS

The effective polaronic Hamiltonian

• The real effective Hamiltonian is a function of the system-bath coupling

• Initial excitons are no longer relevant

A. Gelzinis, D. Abramavicius, L. Valkunas, Phys. Rev. B. 84, 245430, 2011.

Absorption lineshape calculations

A. Gelzinis, D. Abramavicius, L. Valkunas, J. Chem. Phys. 142, 154107, 2015.

Different levels of theory for off-diagonal fluctuations

A. Gelzinis, D. Abramavicius, L. Valkunas, J. Chem. Phys. 142, 154107, 2015.

Coherence vs. Incoherence

FMO

G. S. Engel et al. Nature 446, 782 (2007)G. Panitchayangkoon et al. PNAS 107, 12766-12770 (2010)

2D photon echo spectroscopy

Scheme

(3) (3)

3 2 1 3 2 1

0 0 0

3 3 2 3 2 1

( , ) ( , , )

( , ) ( , ) ( , ).

P t dt dt dt S t t t

E t t E t t t E t t t t

r

r r r

Classification of third order techniques:

1 1 2 2 3 3s u u u k k k k

1 2 3I k k k k

1 2 3II k k k k

1 2 3III k k k k

Possible phase-matching directions:

Assuming that each laser pulse interacts with the system only once, we get 4 linearly independent signals

rephasing

non-rephasing

double quantum coherence

Two-level system

Homogeneous linewidth

Inhomogeneous linewidth

Noninteracting molecules

• Two diagonal peaks present

• Peaks getting round shape due to losing correlation

Interacting molecules (a dimer)

Excited state absorption

Excitation relaxation

Quantum transport in 2D spectroscopy

• Diagonal peaks representstate populations

• Off-diagonal peaks aremixed: populations and coherences

• Off-diagonal oscillate due to quantum coherencesthis does not show quantum transport

• Exponential decay/rise of the diagonal peaks signify Classical Transport regime

• Oscillatory diagonal peaks signify Quantum Transport regime

Oscillation Fourier maps

• One contribution of the rephasing signal

assuming no bath-induced relaxation and Green’s function , gives

Oscillating cross-peak:

Phase and amplitude of oscillations is obtained by Fourier transform of time-dependent spectra,

V. Butkus, D. Zigmantas, L. Valkunas, D. Abramavicius, CPL 545, 40 (2012)

Purely electronic/vibrational systems

Electronicdimer

Vibrationalmonomer

Electronic dimer

Vibrational monomer

V. Butkus, D. Zigmantas, L. Valkunas, D. Abramavicius, CPL 545, 40 (2012)

Oscillations in model systems (rephasing signal only)

Inte

nsy

vum

as

Inte

nsi

ty

Inte

nsi

ty

Inte

nsi

tyDiagonal elements

Diagonal elements

Off-diagonal elements

Off-diagonalelements

Population time (fs) Population time (fs)

Population time (fs) Population time (fs)

Cross-peak oscillations IN PHASE

V. Butkus, D. Zigmantas, L. Valkunas, D. Abramavicius, CPL 545, 40 (2012)

Purely electronic/vibrational systems

Electronic dimer

Vibrational monomer

V. Tiwari et al. PNAS 110 (2013)

V. Butkus, D. Zigmantas, L. Valkunas, D. Abramavicius, CPL 545, 40 (2012)

Oscillation map in experiment

© Jan Alster

http://www.chemphys.lu.se/2Dgroup/

Water-Soluble Chlorophyll-Binding Protein

from Lepidium virginicumJ. Alster, H. Jokstein, J. Dostàl, A. Uchida,

D. Zigmantas, JPCB 118, 3524 (2014)

Time-resolved phenomena

• Coherent beats of vibrationaldegrees of freedom(internal structure of excitons)

V. Butkus, D. Zigmantas, L. Valkunas, D. Abramavicius, CPL 545, 40 (2012)

Excitonvibronic Hamiltonian of a dimer

• Electronic basis of a dimer consists of a common ground state 0 and molecular excited states

𝑚 = 𝐵𝑚†

0 .

• Hamiltonian of a single molecule:

• Of a dimer:

A. Ground-state basis B. Shifted basis

A.

B.

E. Basinskaite, V. Butkus, D. Abramavicius, L. Valkunas, Photosynth. Res. 121, 95-106 (2014)

Choosing the basis

• GS basis and shifted basis are equivalent from the physical point of view

• In the GS basis approach, a larger number vibrational levels have to be considered

• Multi-particle states– 𝑒𝑖𝑔𝑗 first molecule is

vibronically excited; second molecule is vibrationallyexcited

– 𝑔𝑖𝑒𝑗 first molecule is vibrationally excited; second molecule is vibronicallyexcited

• One-particle approximation– 𝑒𝑖𝑔0 first molecule is

vibronically excited; second molecule in zero-quantum GS.

– 𝑔0𝑒𝑗 first molecule is in zero-quantum GS; second molecule is vibronicallyexcited

A. Ground-state basis B. Shifted basis

𝐻 =

(11)

(22)

0

0

0

0 𝐻 =

(12)

(21)0

0

0

0

A.

B.

E. Basinskaite, V. Butkus, D. Abramavicius, L. Valkunas, Photosynth. Res. 121, 95-106 (2014)

In which cases the one-particle approximation works?

• Let’s consider linear absorption dependence on resonant coupling

• One-particle approximation (OPA) can be used to simulate stationary spectra with low values of Huang–Rhys factor or large inhomogeneous broadening

• OPA fails completely if the system is in the vicinity of the exciton–vibronic resonance

• Validity of OPA in calculations with different laser pulse polarizations is highly questionable

• OPA will not give oscillations associated to mixed coherences†.

Huang-Rhys

s=0.05

Huang-Rhys

s=0.5

Multi-particle states

Two-particle approx.

ElectronicVibronic

E. Basinskaite, V. Butkus, D. Abramavicius, L. Valkunas, Photosynth. Res. 121, 95-106 (2014)

Effects of the inhomogeneous disorder on

coherences

• The amplitude of theelectronic-character beats isdramatically reduced by thedisorder

• Vibrational-character beatsweakly depend on thedisorder

ω0=600 cm-1

ΔE=850 cm-1

s1=s2=0.05

σD=0

σD=200 cm-1

σD=20 cm-1σD=50 cm-1

V. Butkus, D. Zigmantas, D. Abramavicius, L. Valkunas, CPL 587, 93 (2013)

Amplitude dependence on disorder

σD=200 cm-1

σD=0

V. Butkus, D. Zigmantas, D. Abramavicius, L. Valkunas, CPL 587, 93 (2013)

Tight binding hamiltonian

A. Gelzinis et al. New J. Phys. 15, 075013 (2013)

CT pathways

A. Gelzinis et al. New J. Phys. 15, 075013 (2013)

CT states

A. Gelzinis et al. New J. Phys. 15, 075013 (2013)

Spectral density

A. Gelzinis et al. New J. Phys. 15, 075013 (2013)

Absorption of PSII RC

A. Gelzinis et al. New J. Phys. 15, 075013 (2013)

Monomer absorption

A. Gelzinis et al. New J. Phys. 15, 075013 (2013)

PSII RC 2D at 77 K

A. Gelzinis et al. New J. Phys. 15, 075013 (2013

A. Gelzinis et al. New J. Phys. 15, 075013 (2013)

PSII RC 2D at 77 K

A. Gelzinis et al. New J. Phys. 15, 075013 (2013)

PSII RC 2D at 77 K

A. Gelzinis et al. New J. Phys. 15, 075013 (2013)

Excited state energy distributions

A. Gelzinis et al. New J. Phys. 15, 075013 (2013)

2D spectrum of the PSII RC at 77K at a waiting time of 170 fs.

F. D. Fuller , J. Pan, A. Gelzinis, V. Butkus, S. S. Senlik, D. E. Wilcox, C. F. Yocum, L. Valkunas, D. Abramavicius, J. P.

Ogilvie, Vibronic coherence in oxygenic photosynthesis. Nature Chemistry 6, 706-711 (2014)

Physical origin of coherencesEx

cito

nic

stat

es

Experiment TheoryVib

ron

ic

Experiment Theory

F. D. Fuller , J. Pan, A. Gelzinis, V. Butkus, S. S. Senlik, D. E. Wilcox, C. F. Yocum, L. Valkunas, D. Abramavicius, J. P.

Ogilvie, Vibronic coherence in oxygenic photosynthesis. Nature Chemistry 6, 706-711 (2014)

Enhancement of the charge transfer

Excitonic splittings

Pop

ula

tio

n o

fC

T st

ate

Spe

ctra

l de

nsi

ty

D. Abramavicius et al. Acceleration of charge separation by oscillations

of the environment polarization. Chem Phys Lett 368 (2003)

Groundstate

CT state

Separated charges

After the excitation, the electron appears in the initial state A. The transition along overdamped and underdamped coordinates are indicated as states B, C and D. The charge transfer may take place at any time while the most probable event happens in the state D.

F. D. Fuller , J. Pan, A. Gelzinis, V. Butkus, S. S. Senlik, D. E. Wilcox, C. F. Yocum, L. Valkunas, D. Abramavicius, J. P.

Ogilvie, Vibronic coherence in oxygenic photosynthesis. Nature Chemistry 6, 706-711 (2014)

Enhancement of charge transfer

D. Abramavicius, L. Valkunas, Photosynth. Res. (2015)

Fucoxanthin-chlorophyll protein - FCP

V. Butkus et al. JCP 142 (2015)

FCP absorption

V. Butkus et al. JCP 142 (2015)

V. Butkus et al. JCP 142 (2015)

Oscillation frequencies

V. Butkus et al. JCP 142 (2015)

Fourier maps

V. Butkus et al. JCP 142 (2015)

Proposed pigment arrangement

V. Butkus et al. JCP 142 (2015)

Conclusions

• Two types of oscillations of either vibronic orelectronic character can be distinguished by the useof Fourier oscillation maps, constructed from thesequences of time-resolved 2D spectra.

• Inhomogeneous disorder influencesvibronic/electronic coherences differently.

• Vibrational coherences evolving in the ground stateand vibronic coherences in the excited state are equally significant.

Acknowledgements

Vilnius: Michigan: Saclay:

• D. Abramavicius J. P. Ogilvie B. Robert

• V. Butkus F. D. Fuller A. Gall

• A. Gelzinis Lund: Frankfurt:

• R. Augulis D. Zigmantas C. Büchel

• E. Basinskaite

Financial supportLSC grant no:

VP1-3.1-ŠMM-07-K-01-007.