Stochastic resonance and resonance activation and their interference effects

1
Total Hamiltonian

• The Hamiltonian H0 of the system incorporates
  terms relating the eigen-energies of the states
  and Coulomb interaction energies.

Ghosh, Smirnov, Nori, J. Chem. Phys. (2009).
2

• ???? ?? ?
• ???? ?????????

3
Solar energy conversion mimicking natural
photosynthesis Modeling the light-energy
conversion in a molecular triad (inserted between
two proton reservoirs or two electrodes).
P. K. Ghosh, A. Yu. Smirnov and F. Nori Advanced
Science Institute, RIKEN, Japan, and Univ. of
Michigan, USA

• P. K. Ghosh, A. Yu. Smirnov, and F. Nori,
• Modeling light-driven proton pumps in artificial
  photosynthetic reaction centers,
• J. Chem. Phys. 131, 035102 (2009). Chosen as the
  Research Highlight of this issue.
• Yu. Smirnov, L. G. Mourokh, P. K. Ghosh, and F.
  Nori,
• High-efficiency energy conversion in a molecular
  triad connected to conducting leads.
• J. Phys. Chem. C 113, 21218 (2009).
  Complimentary color copies of these are online.

4

• (before I forget)
• I would like to thank the organizers
• for the kind invitation.

5
(No Transcript)
6

• Problems
• We are just beginning to work on this
• Thus, this talk will show some initial steps into
  a new direction for us.
• We looked into some published experiments, and we
  wrote the first models for these.
• Molecular Dynamics (MD) can model ps (up to
  ms)
• Kinetic equations can cover from ps to seconds.
• More importantly, MD solves classical equations,
  not quantum, and we are studying quantum
  transport of protons and electrons.

7
Summary of light-driven proton pumps

• Our study is the only theoretical model for the
  quantitative study of light-driven protons pumps
  in a molecular triad.
• Our results explain previous experimental
  findings on light-to-proton energy conversion in
  a molecular triad.
• We compute several quantities and how they vary
  with various parameters (e.g., light intensity,
  temperature, chemical potentials).
• We have shown that, under resonant tunneling
  conditions, the power conversion efficiency
  increases drastically. This prediction could be
  useful for further experiments.

8
Conclusions for (i) proton pumps and (ii)
e- pumps

• Our study models the physics in artificial
  photosynthesis.
• (i) The numerical solutions of the coupled master
  equations and Langevin equation allows
  predictions for the quantum yield and its
  dependence on the surrounding medium, intrinsic
  properties of the donor, acceptor,
  photo-sensitive group, etc.
• (ii) We have also shown that, under resonant
  tunneling conditions and strong coupling of
  molecular triads with the electrodes, the
  (light-to-electricity) power conversion
  efficiency increases drastically. Thus, we have
  found optimal-efficiency conditions.
• Our results could be useful for future
  experiments, e.g., for choosing donors, acceptors
  and conducting electrodes or leads (on the basis
  of reorganization energies and reduction
  potentials) to achieve higher energy-conversion
  efficiency.

8
9
(i) For artificial photosynthesis Input energy
(number of photons absorbed) x ??0
Output energy (number of protons pumped) x
(µP - µN ) Efficiency (output
energy) / (input energy) Efficiency
(Quantum yield) x (µP - µN ) / ??0
Quantum yield F ( of protons pumped) / (
photons absorbed)
10
(ii) For light-to-electricity conversion Input
energy (number of photons absorbed) x
??0 Output energy (number of electrons
pumped) x (µP - µN) Efficiency
(output energy) / (input energy) Efficiency
(Quantum yield) x (µP - µN ) / ??0
Quantum yield F ( of electrons pumped) / (
photons absorbed)
11
Content

• A brief summary of natural photosynthesis.
• A brief summary of artificial photosynthesis
• processes based on molecular triads.
• Our studies Quantum mechanical modeling of
• artificial photosynthesis in molecular
  triads.
• (a) model,
• (b) method,
• (c) results.
• Conclusions.

12
What is photosynthesis?

• Photosynthesis is a process that converts light
  energy into chemical energy
• 6 CO2 6 H2O light ?
  6O2 C6H12O6

• A simple scenario of plant photosynthesis with a
  single pigment Chlorophyll-a

• First step light (of appropriate
  wave-length) is absorbed by a light-harvesting
  complex.

Stroma
Stroma

• Second step the electronic excitation energy
  is converted into a redox potential, in the form
  of transmembrane charge separation.

Primary electron acceptor
e-

• Next steps the energy stored in the electron
  subsystem (in red) is used for pumping protons
  uphill.

Chlorophyll-a
Lumen
Lumen
light

• The first two initial steps involve three
  constituents
• (a) light-absorbing pigments, (b)
  electron acceptors, and (c) electron donors.

13
Some important characteristics of natural
photosynthesis

• The formation of a charge-separated state (using
  the energy of light) is a key strategy in natural
  photosynthetic reaction centers.

• The charge-separated states are stable (with
  long lifetime, increasing quantum yield).
  

• The (distant) charge-separated states are
  produced
• via multi-step electron transfer
  processes.

13
13
14
Some important characteristics of natural
photosynthesis

• In natural photosynthesis, a distant
  charge-separated state is produced via a
  multi-step electron transfer.

• Why a distant charge-separated state ?
• A large separation of the ions (in an ion
  pair) suppresses energy-wasting
  charge-recombination processes.

• Why the multi-step electron transfer processes?
• With increasing distance between the
  donor and the acceptor, the electron-transfer
  rate decreases, so multiple steps are needed for
  a distant charge-separation with a long lifetime
  (and a high quantum yield).

14
15
Artificial photosynthesis mimicking natural
photosynthesis

• Artificial photosynthesis a process for
  converting light-energy into another usable form
  of energy via artificial reaction centers (a
  molecular triad here) mimicking natural
  photosynthesis.

• A molecular triad linking the three components
  donor
  --- photo-sensitive part --- acceptor
• provides a standard protocol for light-energy
  conversion in artificial systems.

• These linked systems have some advantages
• (i) eliminate problems arising from the
  diffusion of individual components.
• (ii) usually, intra-molecular electron-transfer
  processes are faster than
• inter-molecular electron transfer
  processes.

16
A mimicry of natural photosynthesis

• Moores group Nature 385, 239 (1997)
  extensively developed donor-photosensitizer-accept
  or type systems to study light-driven proton
  pumps in an artificial photosynthetic system.

• Molecular triad

QS diphenylbenzoquinone
Naphthoquion moiety (Q)
Inside of liposome
Carotenoid moiety (C)
Porphyrin moiety (P)

• The light-induced excitation of triad
  molecules generates charge-separated states.

membrane
Q-
Q-
P
Q
C
C
P
P
C

• This triad molecule is incorporated into the
  bilayer of a liposome.

• Liposome is a small artificially created
  sphere surrounded by a phospholipid bilayer
  membrane.

• The freely diffusing quinone molecule
  alternates between oxidized and reduced form to
  ferry protons across the membrane.

17
Aim

• The aim of this work is to quantum mechanically
  model
• i) protons climbing their chemical potential
  energy
• (using the energy provided by photons)
  and
• ii) light-to-electricity conversion in a
  molecular triad.
• Theoretical model should be
• (a) simple, but not oversimplified
• (b) useful (i.e., to explain
  experimental results
• and to make testable predictions).

• P. K. Ghosh, A. Yu. Smirnov, and F. Nori,
• Modeling light-driven proton pumps in artificial
  photosynthetic reaction centers,
• J. Chem. Phys. 131, 035102 (2009). Chosen as the
  Research Highlight of this issue.
• Yu. Smirnov, L. G. Mourokh, P. K. Ghosh, and F.
  Nori,
• High-efficiency energy conversion in a molecular
  triad connected to conducting leads.
• J. Phys. Chem. C 113, 21218 (2009).

18
Artificial photosynthesis in a molecular triad

• Molecular triad

Donor (D)
Photo-sensitive part (P)
Acceptor (A)
Shuttle (S)
P
A
D
S

• Simplified ball-and-stick model

Lipid layer

Outside
Inside
P
A
D
Aqueous layer
Aqueous layer
µP
µN
S
µP
gt µN
µ proton potential,
19
Artificial photosynthesis in a molecular triad

• Initial state

Lipid layer
Outside
Inside
Photo-sensitive group
Aqueous layer
Aqueous layer
Acceptor
Donor
µP
µN
Shuttle
µ proton potential,
µP
gt µN
The charged shuttle cannot diffuse across the
non-polar lipid layer. Hence, it remains almost
static near the lipid-aqueous interface.
The positively charged shuttle is trapped at the
interface because it cannot diffuse across the
lipid layer.
The photo-sensitive part that just lost an
electron to the acceptor is now positively
charged. This attracts an electron from the
donor, making the donor positively charged.
The shuttle receives a proton from the near
aqueous layer and becomes neutral.
The neutral shuttle slowly diffuses across the
lipid layer and carries the electron and proton
to the inner membrane.

• Process view

A quantum of light (a photon) is absorbed by the
photosensitive part of the molecule.
The higher-energy electron is transferred to the
acceptor, making it negatively charged.
The shuttle accepts an electron from the acceptor
and becomes negatively charged.
The shuttle gives away an electron to the
positively charged donor.
The shuttle is deprotonated by donating a proton
to the inner aqueous phase.
The triad and the shuttle return to their initial
state, and the process starts again.
Blinking The photo-sensitive group is excited to
a higher electron-energy state.

Outside
Inside
Represents an electron
Represents a photon
µP
µN
_


Aqueous layer
Aqueous layer
_

H
H
P-reservoir
N-reservoir
As a net result, one proton is translocated
from the outer aqueous layer to the inner aqueous
layer.
20
Energy diagram energy of the electron and proton
sites
(a)
P
A
S
S
µN
H
Electron energy
H
Proton energy
D
H
H
H
H
H
H
P
N-reservoir
H

H
H
H
Excited state of photo-sensitive group (P)
Ground state of photo-sensitive group (P)
Donor (D)
Acceptor (A)
Shuttle (S)
21
Energy diagram energy of the electron and proton
sites
Represents an electron
Represents a photon
(b)

Lowering of energy of the proton site makes the
protonation process of the shuttle energetically
possible. As a result, the shuttle receives a
proton from outside of the membrane.
_
The charging of the shuttle by an electron lowers
the energy of the proton site.
_
_
µN
The donor provides a thermally-exited electron to
the positively-charged photosensitive part of the
molecule. .
The unstable excited photo-sensitive group
transfers the electron to the acceptor, producing
an intermediate charge-separated state.

Proton energy
An electron is thermally transferred from the
acceptor to the shuttle.
Electron energy
H
The photo-sensitive group absorbs a photon and is
excited to a higher electron-energy state.

H
H
H
H
H
H
H
N-reservoir
H

H
H
H
Excited state of photo-sensitive group (P)
Ground state of photo-sensitive group (P)
Donor (D)
Acceptor (A)
Protonated shuttle (S)
Shuttle (S)
22
Artificial photosynthesis in a molecular triad
23

• Stages after the shuttle diffuses
• to the inner side of the membrane

23
24
Artificial photosynthesis in a molecular triad
25
Energy diagram energy of the electron and proton
sites (The stages after the shuttle diffuses to
the inside of the membrane)
Denotes an electron
(c)
When the protonated shuttle loses an electron,
the proton energy in the shuttle increases.
Now, this higher energy of the proton in the
shuttle permits a spontaneous deprotonation of
the shuttle.
H
An electron thermally transfers from the
protonated shuttle to the positively charged
donor.
The molecular triad and shuttle return to their
initial states.
_
H
_
H
H
H
µP
Proton energy

Electron energy
H
H
H
H
H
P-reservoir
H

H
H
Excited state of photo-sensitive group (P)
Donor (D)
Acceptor (A)
Ground state of photo-sensitive group (P)
Protonated shuttle (S)
Shuttle (S)
26
Artificial photosynthesis in a molecular triad
27
The model

• Electrons on the five electron-sites and
  protons on the proton-site are characterized by
  the corresponding Fermi operators ai,ai and
  bQ,bQ with electron and proton population
  operators ni aiai, nQ bQ bQ, respectively.

• We assume that each electron and proton site can
  be occupied by a single electron or single proton
  (i.e., the spin degrees of freedom are not
  important).

• The protons in the reservoirs (inner and outer
  aqueous layers) are described by the Fermi
  operators dka,dka , where a P, N are the
  indices of the proton reservoirs, and k has the
  same meaning of wave vector in condensed matter
  physics.

28
Energy diagram energy of the electron and proton
sites
(a)
P
A
S
S
µN
H
Electron energy
H
Proton energy
D
H
H
H
H
H
H
P
N-reservoir
H

H
H
H
Excited state of photo-sensitive group (P)
Ground state of photo-sensitive group (P)
Donor (D)
Acceptor (A)
Shuttle (S)
29
Total Hamiltonian

• The Hamiltonian H0 of the system incorporates
  terms relating the eigen-energies of the states
  and Coulomb interaction energies.

Ghosh, Smirnov, Nori, J. Chem. Phys. (2009).
30
Total Hamiltonian

• Tunneling elements ?DS(x) and ?AS (x)
• depend on the shuttle position x.
• Other terms ?DP, ?DP, ?PA and ?PA are
  independent of the shuttle position x.

P
Acceptor
A
S
Shuttle
Electron energy
D
Donor
Photo-sensitive group
P

31
The Hamiltonian
Excited state of photo-sensitive group (P)
Acceptor
P

• The field amplitude is F e dP
• e strength of external electric field.
• dP dipole moment of P.

A
S
Shuttle
Electron energy
D
Ground state of photo-sensitive group
Donor
P

32
Total Hamiltonian

• Position-dependent coefficients Tka(x)

33
Total Hamiltonian

• The medium surrounding the active sites is
  represented by a system of harmonic oscillators.
  These oscillators are coupled to the active
  sites.

• The parameters xji determine the strengths of
  the coupling between the electron subsystem and
  the environment.

34

Total Hamiltonian

• Total Hamiltonian can be represented in terms of
  the basis of Heisenberg (i.e., transposed
  density) matrices

Where

• Heisenberg equation for the operator ?m

• General form of the master equation

• The total relaxation matrix

35
Relaxation matrix

• Total relaxation matrix

proton tunneling rates between the shuttle and
reservoirs
resonant tunneling rate

• Fermi distribution function

• The chemical potentials related to the pH of
  the solution

R and F are the gas and Faraday constants,
respectively. V Transmembrane potential.
36
Master equations

• Total relaxation matrix

• The Marcus rate describing the thermal electron
  transfers between the pairs of sites (D,P),
  (D,P), (P,A), (P,A), (A,S), and (D,S).

37
Master equations

• Total relaxation matrix

• Marcus rate describing the light-induced
  excitations
• from the ground state P to the excited state P

38
Equation of motion for the shuttle

• ?(t) thermal white noise

Lipid layer

Inside
Outside
P
A
D
Aqueous layer
Aqueous layer
µN
µP
S
39
Results
N-reservoir side
x (Å)

• Stochastic motion of the shuttle with time.

P-reservoir side

• Variation in the electron and proton population
  (almost coincide) on the shuttle.
• Note that the shuttle loads (an e- and a H) in
  the N side and unloads them in the P side.

• NP Number of protons translocated versus
  time.

• Quantum yield (F) of the pumping process is
  55.
• This result is very close to the experimental
  result, F 60, obtained by Moores group
  Nature (1998).

Ghosh, Smirnov, Nori, J. Chem. Phys. (2009).
40
Robustness of the model

• Variations of the quantum yield with the
• reorganization energy ? ?DP ?DS ?DP ?AS
  ?AP
• and the energy gap, d ( EP -EA ES - ED).

• Our simulation results show
• The maximum pumping efficiency is 6.3
  (corresponding to a quantum yield 55).
• This maximum can be achieved at the resonant
  tunneling conditions.

• Parameters Light intensity I 0.18 mW cm-2,
  temperature T 298 K,
• and the energy gaps
• (a) EA-ES 100 meV,
• (b) EA-ES 300 meV, and
• (c) EA-ES 500 meV.
  

Ghosh, Smirnov, Nori, J. Chem. Phys. (2009).
41
Proton current versus temperature

• Both the proton-current
• and quantum yield
• linearly increase with temperature.

• The temperature effects appear through two
  factors
• All the electron and proton transfer rates change
  with temperature.
• The diffusion coefficient of the shuttle
  increases with temperature.

Ghosh, Smirnov, Nori, J. Chem. Phys. (2009).
41
42
Proton current versus light intensity

• The proton current is roughly linear for small
  intensities of light, but it saturates with
  higher light-intensity.
• This is consistent with experiments.

• The pumping quantum efficiency decreases with
  light-intensity, for all temperatures (because
  the number of unsuccessful attempts to pump
  protons also increases, decreasing the quantum
  yield).

42
43
Proton current versus proton potentials of the
leads

• The proton current saturates when the P-side
  (left) potential is sufficiently low, µP lt 160
  meV, and goes to zero when µP gt 200 meV (i.e. µP
  gt EQ).

• Also, the pumping device does not work when
  the potential µN is too low
• µN lt EQ - uSQ .

• Main parameters I0.18 mW cm-2, temperature
  T 298 K.
  

43
44
Summary of light-driven proton pumps

• Our study is the only theoretical model for the
  quantitative study of light-driven protons pumps
  in a molecular triad.
• Our results explain previous experimental
  findings on light-to-proton energy conversion in
  a molecular triad.
• We compute several quantities and how they vary
  with various parameters (e.g., light intensity,
  temperature, chemical potentials).
• We have shown that, under resonant tunneling
  conditions, the power conversion efficiency
  increases drastically. This prediction could be
  useful for further experiments.

45

• Second part of the talk starts here
• ( ten slides)
• High-efficiency energy conversion
• in a molecular triad
• connected to conducting electrodes.

Smirnov, Mourokh, Ghosh, and Nori,
High-efficiency energy conversion in a molecular
triad connected to conducting leads. J. Phys.
Chem. C 113, 21218 (2009). Complimentary color
copies of these are available online.
46

• Light-to-electricity energy conversion
• in a molecular triad

Left electrode (L)
P
A
D
Right electrode (R)
Photosensitive part
Acceptor
Donor

• The molecular triad is inserted between two
  electrodes.
• Here, there are no shuttle and proton
  reservoirs.
• Energy of light is now directly converted to
  electricity.
• Example (from Imahoris group, J. Chem. Phys. B,
  2000)
• Molecular triad ferrocene (D)
  ---- porphyrin (P) ---- fullerene (A)
• Left electrode (L) gold electrode
• Right electrode (R) electrolyte solution
  containing molecules of oxygen, O2,
• or methyl viologen, MV2.
• Our proposed model is valid for arbitrary donors,
  photosensitive parts, acceptors, and electrodes.

47
Light-to-electricity energy conversion in a
molecular triad
Left electrode (L)
P
A
D
Right electrode (R)
Photosensitive part
Acceptor
Donor
The molecular triad is inserted between two
electrodes.
48

• Molecular triad for photosynthesis (studied by
  Imahori et al.)

Photosensitive part (P)
Donor (D)
Acceptor (A)
Porphyrin
Fullerene
Ferrocene
49

• Molecular triad attached to a metal surface

50
For solar cells Input energy (number of
photons absorbed) x ??0 Output energy
(number of electrons pumped) x (µP -
µN) Efficiency (output energy) /
(input energy) Efficiency (Quantum
yield) x (µP - µN ) / ??0
Quantum yield F ( of electrons pumped) / (
photons absorbed)
51
Light-to-electricity energy conversion in a
molecular triad

• (a) Electron current and (b) power conversion
  efficiency versus the chemical potential µL of
  the left lead.
• The current saturates as µL increases however,
  the efficiency, which is proportional to the
  voltage V, decreases linearly.
• Our estimates show that the maximum power-
  conversion efficiency 40 ,
• when µL - 630 meV and µR 480 meV.

52
Light-to-electricity energy conversion in a
molecular triad

• (a) Electron current and (b) power conversion
  efficiency versus the chemical potential µL of
  the left electrode.
• The current saturates as µL increases however,
  the efficiency, which is proportional to the
  voltage V, decreases linearly.
• Note that in (b) the efficiency goes to zero when
  µL approaches µR .

53
Light-to-electricity energy conversion in a
molecular triad

• Electron current as a function of the photon
  energy at different temperatures.
• Note the peak when the photon energy
  matches the P energy gap (minus the
  reorganization energy)
• (b) Temperature dependence of the
  power-conversion efficiency at the resonant
  photon energy. The broad peak includes room temp.
• (c) Linear dependence of the current on the
  light intensity at different temperatures.

µR 480 meV, µL -540 meV. Other parameters
are the same as in previous figures.
53
54
Light-to-electricity energy conversion in a
molecular triad

• Quantum yield F as a function of the tunnel
  coupling ?L between the left lead and the donor
  molecule at ?R 20 ns-1
• Quantum yield F as a function of the tunnel
  coupling ?R between the right lead and the
  acceptor molecule at ?L 100 ns-1.
• Both graphs are plotted at µR 480 meV, T
  298. The light intensity, and other parameters
  are the same as in previous figures.

54
55
Summary (light-to-electricity energy conversion)

• We developed a theoretical model for
  quantitative calculations of the
  light-to-electricity energy conversion efficiency
  in molecular triads.
• We compute several quantities and how they vary
  with various parameters (e.g., light intensity,
  T, µs, G s, etc.).
• Our calculations show that in the case of
  relatively strong coupling of the molecular triad
  to the leads, the power-conversion efficiency can
  exceed 40. This prediction could be useful for
  future experiments.

56
Conclusions

• Our study models the physics in artificial
  photosynthesis.
• The numerical solutions of the coupled master
  equations and Langevin equation allows
  predictions for the quantum yield and its
  dependence on the surrounding medium, intrinsic
  properties of the donor, acceptor and
  photo-sensitive group, etc.
• We have also shown that, under resonant tunneling
  conditions and strong coupling of molecular
  triads with the electrodes, the
  (light-to-electricity) power conversion
  efficiency increases drastically. Thus, we have
  found optimal-efficiency conditions.
• Our results could be useful for future
  experiments, e.g., for choosing donors, acceptors
  and conducting electrodes or leads (on the basis
  of reorganization energies and reduction
  potentials) to achieve higher energy-conversion
  efficiency.

56
57
Summary of light-driven proton pumps

• Our study is the only theoretical model for the
  quantitative study of light-driven protons pumps
  in a molecular triad.
• Our results explain previous experimental
  findings on light-to-proton energy conversion in
  a molecular triad.
• We compute several quantities and how they vary
  with various parameters (e.g., light intensity,
  temperature, chemical potentials).
• We have shown that, under resonant tunneling
  conditions, the power conversion efficiency
  increases drastically. This prediction could be
  useful for further experiments.

58

• Thanks for your attention

59

• Following slides are for the Q A period
• (also, those slides can be used for longer talks)

60
Light-induced electron transfer in purple bacteria
P Bacteriochlorophyl dimer, BA, BB Some
bacteriochlorophyl acts as intermediate electron
acceptor. HA , HB Bacteriopheophytin QA
primary ubiquinone, QB secondary ubiquinone,
C2 cytochome (e- carrier)
Inside of chromatophore vesicle
Lumen surface

• The energy of light-quanta is stored as a redox
  potential in the form of transmembrane charge
  separation.

• The initial stage of photosynthesis involves
  three constituents
• (a) light-absorbing pigments
• (b) electron acceptors
• (c) electron donors.

Stromal surface
Outside of chromatophore vesicle
61
Light-induced electron transfer in purple bacteria
P Bacteriochlorophyl dimer, BA, BB Some
bacteriochlorophyl acts as intermidate electron
acceptor. HA , HB Bacteriopheophytin QA
primary ubiquinone, QB secondary ubiquinone,
c2 cytochome
Outside of chromatophore vesicle
Lumen surface
t Lifetime
P
1400 meV, t 3 ps
e-
1200 meV, t 200 ps
P - H-A
e-
e-
Energy
e-
600 meV
P- Q-A
P- Q-B
Stromal surface
t 100 µs
t 1 s
0 meV
P
Inside of chromatophore vesicle
62
Mimicking natural photosynthesis

• Nishitani et al. J. Am. Chem. Soc. 105, 7771
  (1983), first synthesized a
• donor-acceptor system linking porphyrin (P) to
  two quinones (Q1 and Q2)

Light
P Q1 Q2
P Q1 Q2

• The lifetime t of a charge-separated state of
  triads, tt,
• is long compared to the one for a dyad
  system td.

td
tt
tt gt
td
63

• Proton pump parameters
• Light intensity I 0.18 mW cm-2
• Resonant electron tunneling rate ?/? 15
  ns-1
• Resonant proton tunneling rate G/? 15 ns-1
  
• Temperature T 298 K
• Proton potentials µN - 110, µP 110
• Diffusion coefficient of the shuttle at 298 K
  Ds 2 nm2 µs-1
• Electron tunneling length Ltun 0.5 nm
• Proton tunneling length LQ 0.5 nm
• Dielectric constant e 3

• Parameters are taken from Nature, 392, 479
  (1998) J. Am. Chem. Soc., 123, 2607 (2001) J.
  Am. Chem. Soc., 123, 6617 (2001) J. Am. Chem.
  Soc., 123, 100 (2001) Angew. Chem., Int. Ed. 41,
  2344, (2002) Bull. Chem. Soc. Jpn. Vol. 80, No.
  4, 621636 (2007).

64
Quantum yield versus Resonant tunneling rate
64
65
Quantum yield versus Dielectric constant
65
66
Potential energy the for shuttle motion
U(x)
Aqueous layer
Aqueous layer
Lipid layer
x
66
67
Essential ingredients of the model

• The model must satisfy the following conditions
• The energy EA of the state A and shuttle ES must
  be comparable (for resonant tunneling of electron
  from state A to shuttle S).

• Similarly, the energy criterium for resonant
  tunneling of an electron
• from the protonated shuttle to state
  D is

• Condition for jump of proton from
  reservoirN to shuttle

• Condition for jump of proton from shuttle to
  reservoir-P

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The total Hamiltonian of the system

• To remove dependency of xjk we use unitary
  transformation

• Total Hamiltonian after unitary transform

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For artificial photosynthesis Input energy
(number of photons absorbed) x ??0 Output
energy (number of protons pumped) x (µP -
µN ) Efficiency (output energy) / (input
energy) Efficiency (Quantum yield)
x (µP - µN ) / ??0
Quantum yield F (number of protons pumped) /
(number photons absorbed)
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Current and efficiency (for solar cells)

• The amount of energy absorbed (per unit time)
  by the triad

• Current

• Efficiency

• Quantum yield

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• Light-to-electricity conversion parameters
• Light intensity I 20 mW cm-2.
• Resonant electron tunneling rate ?/? 15
  ns-1.
• Coupling to electrodes GL/? 100 ns-1 ,
  GR/? 100 ns-1.
• Temperature T 298 K.
• Energy of light ??0 2 eV.
• Proton potentials µN - 110, µP 110.
• Dielectric constant e 4.4.
• Distances between electron sites rAP 1.8 nm,
  rDP 1.62 nm, rDA 3.42 nm
• Energy levels ED - 510 meV, EP - 1150 meV,
  EP 750 meV, EA - 620 meV.

• Parameters are taken from Nature, 392, 479
  (1998) J. Am. Chem. Soc., 123, 2607 (2001) J.
  Am. Chem. Soc., 123, 6617 (2001) J. Am. Chem.
  Soc., 123, 100 (2001) Angew. Chem., Int. Ed. 41,
  2344, (2002) Bull. Chem. Soc. Jpn. Vol. 80, No.
  4, 621636 (2007).

72
Future plans

• An extension of our model would be to study
  light-to-electricity energy conversion in a
  molecular triad with additional light-harvesting
  components.

Light harvesting component I
(CH2)n
B
Left electrode (L)
Right electrode (R)
P
A
D
Acceptor
Donor
Light harvesting component II
B
EN

• Energy diagram

P
e-
A
e-
e-
e-
e-
D
e-
L
R
P
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B
73

• A molecular triad (Fc-P-C60) and an
• additional light harvesting complex (B).
• Both are attached to a metal surface.

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Marcus rate
Reorganization energy

• Reorganization energy (?) Energy required to
  displace the system an amount Q XA - XD without
  electron transfer.
• This is the energy required to transfer the
  electron from the bottom of the energy profile of
  the acceptor (product) state up to the energy
  profile of the acceptor state in the same nuclear
  configuration as the energy minimum of the donor
  state.

?
?E
xA
xD