Transcript Stochastic resonance and resonance activation and their

Total Hamiltonian
Eigen-energiesand
Coulomb interactions

H0
H 
Thermal tunnelingbetween electronsites
 , '{ D , P},{ D , P *},{ A, B},{ A, P *}
Light-induced excitation

 
   ' ( x)a a '  H .c  Fei0t aP aP*  H .c
 '
2
2
2

p
m

1
 
j
j j 

  Tk ( x)d k bQ  H .c   

 x j   x ji ni  
2 
2 i
k
j  2m j
 


 

Tunneling coupling with proton
reservoirs.
Interaction with theenvironment.
•
The Hamiltonian H0 of the system incorporates terms relating the
eigen-energies of the states and Coulomb interaction energies.
H 0   Ei ni  EQ nQ  uDP (1  nD )(1  nPTotal )  uDA (1  nD )nA
i
 uPA (1  nPTotal )nA  uSQ nS nQ ,
Ghosh, Smirnov, Nori, J. Chem. Phys. (2009).
where nPTotal  nP  nP*
nS (nQ )  electron(proton)densityon shuttle
1
フランコ
ノリ
の
“ノリノリ”
プレゼンテーション
2
Solar energy conversion
mimicking natural photosynthesis:
Modeling the light-energy conversion in a molecular triad
(inserted between two proton reservoirs or two electrodes).
Electrode
or proton
reservoir
Molecular triad
Donor
Photo-sensitive
part
Acceptor
Electrode
or proton
reservoir
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.
A. 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.
(before I forget):
I would like to thank the organizers
for the kind invitation.
5
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.
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.
7
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.
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(i) For artificial photosynthesis:
Input energy
= (number of photons absorbed) x ћω0
Output energy = (number of protons pumped) x (μP - μN )
Efficiency
= (output energy) / (input energy)
Efficiency
=
(Quantum yield) x (μP - μN ) / ћω0
Quantum yield Φ = (# of protons pumped) / (# photons absorbed)
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(ii) For light-to-electricity conversion:
Input energy
= (number of photons absorbed) x ћω0
Output energy = (number of electrons pumped) x (μP - μN)
Efficiency
= (output energy) / (input energy)
Efficiency
=
(Quantum yield) x (μP - μN ) / ћω0
Quantum yield Φ = (# of electrons pumped) / (# photons absorbed)
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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.
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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:
Stroma
Primary electron
acceptor
 First step: light (of appropriate wavelength) is absorbed by a light-harvesting
complex.
Stroma
 Second step: the electronic excitation
energy is converted into a redox potential,
in the form of transmembrane charge
separation.
e-
Chlorophyll-a
Lumen
Lumen
 Next steps: the energy stored in the
electron subsystem (in red) is used for
pumping protons uphill.
 The first two initial steps involve three constituents:
(a) light-absorbing pigments, (b) electron acceptors, and (c) electron donors.
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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
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 energywasting 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
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.
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A mimicry of natural photosynthesis
Moore’s g roup [Natu re 385, 239 (1997)]
extensively developed donor-photosensitizeracceptor type systems to study light-driven proton
pumps in an artificial photosynthetic system.
• Molecular triad
QS = diphenylbenzoquinone
Naphthoquion
Carotenoid moiety (C)
moiety (Q)
Porphyrin moiety (P)
Inside of liposome
 The
light-induced excitation of triad
molecules generates charge-separated states.
P* Q
C


C
P+ Q-
C+
P
Q-
This triad molecule is incorporated into the
bilayer of a liposome.
freely diffusing quinone molecule alternates
between oxidized and reduced form to ferry protons
across the membrane.
The
• Liposome: is a small artificially
created sphere surrounded by a
phospholipid bilayer membrane.
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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.
A. 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).
Artificial photosynthesis in a molecular triad
• Molecular triad
Donor (D)
Photo-sensitive
part (P)
D
Acceptor (A)
P
Shuttle (S)
S
A
• Simplified ball-and-stick model
Lipid layer
Inside
D
P
A
Aqueous layer
Aqueous layer
μP
μ = proton potential,
Outside
S
μP > μ N
μN
Artificial photosynthesis in a molecular triad
• Initial state:
Lipid layer
Outside
Inside
Aqueous layer Donor
Photo-sensitive group
μP
μP > μN
The
The
positively
charged
shuttle
charged
cannot
shuttle
diffuse
is the
across
The
photo-sensitive
part
that
just
lost
an
electron
to layer
the
The
shuttle
receives
aan
proton
from
The
neutral
shuttle
slowly
diffuses
across
the lipid
The
A
The
quantum
The
shuttle
higher-energy
triad
shuttle
and
is
gives
of
deprotonated
light
accepts
the
away
electron
(a
shuttle
photon)
an
electron
electron
return
is
by
is
transferred
donating
absorbed
to
from
their
the
to
by
Blinking:
The
photo-sensitive
group
is
trapped
theaqueous
non-polar
atnow
thelayer
interface
lipid
layer.
because
Hence,
itit remains
acceptor
is
positively
charged.
This
attracts
an
electron
near
and
becomes
neutral.
and
carries
the
electron
and
proton
to
the
inner
membrane.
a
the
initial
proton
acceptor
acceptor,
photosensitive
state,
to
and
and
the
making
becomes
the
inner
process
part
it
aqueous
negatively
negatively
of
starts
the
phase.
molecule.
charged.
again.
charged.
excited
to
a
higher
electron-energy
state
.
to
the
positively
charged
donor.
cannot
almost
static across
near
thethe
lipid-aqueous
lipiddonor
layer. positively
interface. charged.
from
thediffuse
donor,
making
the
Inside
Outside
Represents a
anphoton
electron
μP
Aqueous layer
μN
Shuttle
μ = proton potential,
• Process view:
Acceptor Aqueous layer
+
+H
+
+
μN
_
Aqueous layer
_
H+
P-reservoir
N-reservoir
As a net result, one proton is translocated from
the outer aqueous layer to the inner aqueous layer.
19
Energy diagram: energy of the electron and proton sites
(a)
P*
S
μN
H+
D
H+
H+
Shuttle (S)
Acceptor (A)
H+
H+
H+
H+
H+
H+
P
Donor (D)
H+
Ground state of
photo-sensitive
group (P)
H+
Proton energy
S
N-reservoir
Electron energy
A
H+
Excited state of
photo-sensitive
group (P*)
20
Energy diagram: energy of the electron and proton sites
Represents
Represents
a photon
an electron
+
_
Lowering of energy of the proton
site makes the protonation
p r o c eThe
s s charging
of the
s h shuttle
uttle
of the
energetically
a result,
by an possible.
electron As
lowers
the
the shuttle
receives
a
proton
from
energy of the proton site.
outside of the membrane.
+
The donor provides
a thermallyThe unstable
excited photoexited electronsensitive
to the positivelygroup transfers the
An electron is thermally
T h e p htransferred
opart
t o - s of
ensitive
charged photosensitive
electron
toshuttle.
the acceptor,
from the acceptor
to
the
group
absorbs a. photon
t h e m o l e c u producing
le
.
an intermediate
and is excited to a higher
charge-separated state.
electron-energy state.
_
μN
H+
H+
H+
H+
H+
H+
H+
H+
H+
H+
Donor (D)
Shuttle (S)
Acceptor (A)
Protonated
shuttle (S)
Ground state of
photo-sensitive
group (P)
H+
N-reservoir
Electron energy
_
Proton energy
(b)
H+
Excited state of
photo-sensitive
group (P*)
21
Artificial photosynthesis in a molecular triad
Lipid layer
Inside
D
P
A
Aqueous layer
Aqueous layer
μP
μ = proton potential,
Outside
S
μP > μ N
μN
• Stages after the shuttle diffuses
to the inner side of the membrane
23
Artificial photosynthesis in a molecular triad
Lipid layer
Inside
D
P
Aqueous layer
μP
A
Outside
Aqueous layer
S
μN
μP > μ N
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)
Now, this
When
thehigher
protonated
energy ofshuttle
the proton
loses
in
the shuttle
an electron,
permitsthe
a spontaneous
proton
energy
d
e p r o tin
o nthe
a t i oshuttle
n o f increases.
the shuttle.
H+
H+
H
_+
μP
H+
H+
H+
H+
H+
Donor (D)
Shuttle (S)
An electron thermally transfers
The molecular triad and shuttle
from the protonated shuttle to
return to their initial states.
the positively charged donor.
+
H+
Electron energy
H
_+
P-reservoir
Proton energy
H+
H+
H+
Acceptor (A)
Protonated
shuttle (S)
Ground state of
photo-sensitive
group (P)
Excited state of
photo-sensitive
group (P*)
Artificial photosynthesis in a molecular triad
Lipid layer
Inside
D
P
A
Aqueous layer
Aqueous layer
μP
μ = proton potential,
Outside
S
μP > μ N
μN
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 = ai+ai, 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 dkα+,dkα , where α = P, N are the indices of the proton reservoirs,
and k has the same meaning of wave vector in condensed matter physics.
 The electron-proton system with no leads (the proton reservoirs) can be
characterized by the 20 basis states of the Hamiltonian H0 :
1  aD aP 0 ,
2  aD aP* 0 ,............., 20  aA aSbQ 0 .
0
represents the vacuum state.
1  aD aP 0
One electron is located on site D and one on site P.
20  aA aSbQ 0
Two electrons on sites A and S and a proton on the site Q.
27
Energy diagram: energy of the electron and proton sites
(a)
P*
S
μN
H+
D
H+
H+
Shuttle (S)
Acceptor (A)
H+
H+
H+
H+
H+
H+
P
Donor (D)
H+
Ground state of
photo-sensitive
group (P)
H+
Proton energy
S
N-reservoir
Electron energy
A
H+
Excited state of
photo-sensitive
group (P*)
28
Total Hamiltonian
Eigen-energiesand
Coulomb interactions

H0
H 
Thermal tunnelingbetween electronsites
 , '{ D , P},{ D , P *},{ A, B},{ A, P *}
Light-induced excitation

 
   ' ( x)a a '  H .c  Fei0t aP aP*  H .c
 '
2
2
2

p
m

1
 
j
j j 

  Tk ( x)d k bQ  H .c   

 x j   x ji ni  
2 
2 i
k
j  2m j
 


 

Tunneling coupling with proton
reservoirs.
Interaction with theenvironment.
•
The Hamiltonian H0 of the system incorporates terms relating the
eigen-energies of the states and Coulomb interaction energies.
H 0   Ei ni  EQ nQ  uDP (1  nD )(1  nPTotal )  uDA (1  nD )nA
i
 uPA (1  nPTotal )nA  uSQ nS nQ ,
Ghosh, Smirnov, Nori, J. Chem. Phys. (2009).
where nPTotal  nP  nP*
nS (nQ )  electron(proton)densityon shuttle
29
Total Hamiltonian
Eigen-energiesand
Coulomb interactions

H0
H 
Light-induced excitation

 
   ' ( x)a a '  H .c  Fei0t a P a P*  H .c
Thermal tunnelingbetween electronsites
 '
2
2
2

p
m

1
 
j
j j 

  Tk ( x)d k bQ  H .c   

 x j   x ji ni  
2 
2 i
k
j  2m j
 


 

Tunneling coupling with proton
reservoirs.
Electron energy
P*
Interaction with theenvironment.
Tunneling elements ∆DS(x) and ∆AS (x)
depend on the shuttle position x.
Acceptor
A
S
Shuttle
D
Donor
P
Photo-sensitive
group
 Other terms ∆DP, ∆DP*, ∆PA and ∆P*A are
independent of the shuttle position x.
The Hamiltonian
Eigen-energiesand
Coulomb interactions
H 

H0
Light-induced excitation

 
   ' ( x)a a '  H .c  Fei0t a P a P*  H .c
Thermal tunnelingbetween electronsites
 '
2
2
2

pj
m j j 
1
 

  Tk ( x)d k bQ  H .c   

 x j   x ji ni  
2 
2 i
k
j  2m j
 


 

Tunneling coupling with proton
reservoirs.
Electron energy
Excited state of
photo-sensitive
group (P*)
P*
Interaction with theenvironment.
Acceptor
The field amplitude is F = ε dP
A
S
ε = strength of external electric field.
Shuttle
dP = dipole moment of P.
D
Donor
P
Ground state of
photo-sensitive
group
Total Hamiltonian
Eigen-energiesand
Coulomb interactions
H 

H0
Light-induced excitation

 
   ' ( x)a a '  H .c  Fei0t a P a P*  H .c
Thermal tunnelingbetween electronsites
 '
2
2
2

pj
m j j 
1
 

  Tk ( x)d k bQ  H .c   

 x j   x ji ni  
2 
2 i
k
j  2m j
 


 

Tunneling coupling with proton
reservoirs.
• Position-dependent
Aqueous layer
xP+LQ
xP
P-reservoir
coefficients T kα (x):
D
Inside
Interaction with theenvironment
P
TkN ( x)  TkN  [ x  ( xN  LQ )];
TkP ( x)  TkP  [ xP  ( LQ  x )],
Aqueous layer
A
xN - LQ
S
LQ defines the proton tunneling length.
xP and xN are the coordinates of the proton reservoirs.
Outside
xN
N-reservoir
Total Hamiltonian
Eigen-energiesand
Coulomb interactions
H 

H0
Thermal tunnelingbetween electronsites
 , '{ D , P},{ D , P *},{ A, B},{ A, P *}
Light-induced excitation

 
   ' ( x)a a '  H .c  Fei0t aP aP*  H .c
 '
2
2
2

pj
m j j 
1
 

  Tk ( x)d k bQ  H .c   

 x j   x ji ni  
2 
2 i
k
j  2m j
 


 

Tunneling coupling with proton
reservoirs, k indicesof protons
in the reservoirs,   N , P .
Interaction with theenvironment, where x j , p j  positions
and momentum of bath oscillators. i  indicesof electron
sites, xij  coupling strength.
• 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.
33
Total Hamiltonian
 Total Hamiltonian can be represented in terms of the basis of
Heisenberg (i.e., transposed density) matrices
H   H mn m,n
Where:
m, n

m,n  m n ; m  m m
Heisenberg equation for the operator ρm
.
i  m   m , H 
.

General form of the master equation
..
 m    nm ( x)  m 
n


mn
( x)  n
n
The total relaxation matrix
tr
 nm ( x)   mn
( x)  kii' mn  k PP* mn
Relaxation matrix
 Total relaxation matrix
tr
 nm ( x)   mn
( x)  kii' mn  k PP* mn
tr
 mn
(x)
tr
 mn
( x) 
proton tunneling rates between the shuttle and reservoirs
 ( x) | b


 (x)
Q ; mn
|2 1  F (Enm )  | bQ;mn |2 F (Enm ) 
resonant tunneling rate
 Fermi distribution function

 E nm  m
F (E nm )  exp
k BT




  1


1
 The chemical potentials related to the pH of the solution:
mP  mN  V 
2.3RT
pH 
F
R and F are the gas and Faraday
constants, respectively.
V = Transmembrane potential.
35
Master equations
..
m
   nm ( x)  m 
n

mn
( x)  n
n
• Total relaxation matrix
tr
 nm ( x)   mn
( x)  kii' mn  k PP* mn
• 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).
kii' nm 
where,
Δii'(x)
2
  ΔEmn  λii' 2 
π
( Aii' ) mn exp

2
λii' kBT 
4
λ
k
T
ii' B



i i'
( Aii' ) mn  (a a )mn
2

i i'
 (a a )nm
2
36
Master equations
..
 m    nm ( x)  m 
n

mn
( x)  n
n
• Total relaxation matrix
tr
 nm ( x)   mn
( x)  kii' mn  k PP* mn
• Marcus rate describing the light-induced excitations
from the ground state P to the excited state P*
k PP* nm
2





E






mn
0
PP*
 F0
(aP aP* )mn exp

2
PP* k BT 
4

k
T
PP* B


 Emn  0  PP* 2 
2

2

 F0
(aP aP* )nm exp

2
PP* k BT 
4

k
T
PP* B


2

2
37
Equation of motion for the shuttle
dx
dU ( x)
drag
 
  (t )
dt
dx
 ς(t) = thermal white noise:
 (t )  0;
 (t ) (t ' )  2dragT (t  t ' ).
Lipid layer
Inside
Aqueous layer
μN
D
P
Outside
A
Aqueous layer
S
μP
x x(Å)
Results
40
20
0
-20
-40
0
N-reservoir side

300
600
Time (ms)
900
P-reservoir
side
Electron density
Proton density
Ps
1.0
0.5
NP
0.0
Stochastic motion of the shuttle with time.

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.
0
32
24
16
8
0
0
300
600
Time (ms)
900
 NP = Number of protons translocated versus time.

900
1800
Time (ms)
2700
Ghosh, Smirnov, Nori, J. Chem. Phys. (2009).
Quantum yield (Φ) of the pumping process is ~ 55%.
 This result is very close to the experimental result,
Φ ~ 60%, obtained by Moore’s group [Nature (1998)].
39
Robustness of the model
(a)
(meV)
535
245
48.60
Quantum yield (%)
680
12.00
150
430
(meV)
990
680
1270
(b)
(meV)
535
245
0
55.60
Quantum yield (%)
100
12.00
150
430
(meV)
990
680
1270
(c)
(meV)
535
245
51.40
12.00
100
150
430
(meV) 990
1270
Variations of the quantum yield with the:
reorganization energy λ = λDP = λDS = λDP* = λAS = λAP
and the energy gap, δ (= EP* −EA = ES − ED).

Our simulation results show:
1. The maximum pumping efficiency is ~ 6.3%
(corresponding to a quantum yield ~ 55%).
2. This maximum can be achieved at the resonant
tunneling conditions.
0
Quantum yield (%)
100

0
 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).
40
Proton current versus temperature
-2
I = 0.026 mWcm
-2
I = 0.080 mWcm
Quantum yield (%)
-1
Proton current (ms)
12
8
I = 0.132 mWcm
 Both the proton-current
and quantum yield
linearly increase with temperature.
-2
4
200
77
250
300
350
Temperature (K)
I = 0.026 mWcm
70
400
-2
1. All the electron and proton transfer rates
change with temperature.
-2
I = 0.080 mWcm
-2
I = 0.132 mWcm
63
2. The diffusion coefficient of the shuttle
increases with temperature.
56
49
200
 The temperature effects appear through
two factors:
250
300
350
Temperatute (K)
400
Ghosh, Smirnov, Nori, J. Chem. Phys. (2009).
41
Quantum yield (%)
-1
Pumping current (ms)
Proton current versus light intensity
12
 The proton current is roughly linear
8
4
T = 373 K
T = 298 K
T = 273 K
0.1
0.2
0.3
-2
Light intensity I (mWcm )
80
60
T = 373 K
T = 298 K
T = 273 K
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
40
0.1
0.2
0.3
-2
Light intensity I (mWcm )
unsuccessful attempts to pump protons
also increases, decreasing the quantum
yield).
42
-1
Proton current (ms)
Proton current versus proton potentials of the leads
mN = - 110 meV
9
mN = - 140 meV
mN = - 200 meV
 The proton current saturates when the P-side
(left) potential is sufficiently low, μP < 160
6
meV, and goes to zero when μP > 200 meV (i.e.
3
 Also, the pumping device does not work
μP > EQ).
when the potential μN is too low
μN < EQ − uSQ .
0
0
120
240
mP (meV)
360
 Main parameters:
I=0.18 mW cm−2,
temperature T = 298 K.
43
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.
44
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.
45
Light-to-electricity energy conversion
in a molecular triad
Left
electrode (L)
D
Donor
P
Photosensitive part
A
Right
electrode (R)
Acceptor
 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 Imahori’s 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
46
electrodes.
Light-to-electricity energy conversion in a molecular triad
Left
electrode (L)
D
P
Donor
Photosensitive part
P*
Energy diagram
e-
Right
electrode (R)
A
Acceptor
The molecular triad is inserted
between two electrodes.
A
e-
1
eD
L
eP
R
47
• Molecular triad for photosynthesis (studied by Imahori et al.)
Donor (D)
Ferrocene
Photosensitive
part (P)
Porphyrin
Acceptor (A)
Fullerene
• Molecular triad attached to a metal surface
For solar cells:
Input energy
= (number of photons absorbed) x ћω0
Output energy = (number of electrons pumped) x (μP - μN)
Efficiency
= (output energy) / (input energy)
Efficiency
=
(Quantum yield) x (μP - μN ) / ћω0
Quantum yield Φ = (# of electrons pumped) / (# photons absorbed)
50
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.
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 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 .
52
Light-to-electricity energy conversion in a molecular triad
(a)
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
Light-to-electricity energy conversion in a molecular triad
(a) Quantum yield Φ as a function of the
гL between the left lead
and the donor molecule at гR = 20 ns-1
tunnel coupling
(b) Quantum yield Φ as a function of the
tunnel coupling
гR
between the right
lead and the acceptor molecule at
гL =
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
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, μ’s,  ’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.
55
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 (lightto-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
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.
57
Thanks for your attention
58
Following slides are for the Q & A period
(also, those slides can be used for longer talks)
59
Light-induced electron transfer in purple bacteria
Inside of chromatophore vesicle
Lumen surface
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)
 The energy of light-quanta is
stored as a redox potential in the
form of transmembrane charge
separation.
Stromal surface
Outside of chromatophore vesicle
 The initial stage of photosynthesis
involves three constituents:
(a) light-absorbing pigments
(b) electron acceptors
(c) electron donors.
60
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
τ = Lifetime
P*
1400 meV, τ ~ 3 ps
e-
Energy
P+
1200 meV, τ ~ 200 ps
A
e-
e-
e600 meV
-
-
P+ Q A
τ ~ 100 μs
Stromal surface
Inside of chromatophore vesicle
- H-
P
-
-
P+ Q
τ ~1 s
0 meV
61
B
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 – Q 2
P* – Q1 – Q2
+
_
P - Q1 - Q 2
+
_
P - Q1 - Q 2
• The lifetime t of a charge-separated state of triads, tt,
is long compared to the one for a dyad system td.
_
+
+
_
P - Q1 - Q 2
P - Q1
τt
τd
τt > τd
62
Proton pump parameters:

Light intensity: I = 0.18 mW cm−2

Resonant electron tunneling rate: Δ/ћ = 15 ns-1

Resonant proton tunneling rate: Γ/ћ = 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: ε = 3
 Distances between electron sites: rAP = rDP = 4 nm, rDA = 8 nm
 Energy levels: EA – ES = 300 meV, EP* - EA = 400 meV, ED - EP = 400 meV

Reorganization energies: λPP* = 80 meV,
λDP = λDS = λDP* = λAS = λAP = 400 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, 621–636 (2007).
63
Quantum yield (%) Quantum yield (%) Quantum yield (%) Quantum yield (%)
52
38
24
10
52
38
24
10
Quantum yield versus
Resonant tunneling rate
 = 100 meV
 = 200 meV
 = 400 meV
= 500 meV
= 1000 meV
52
38
24
10
50
40
30
20
10
0
 = 130 meV
= 800 meV
 = 1200 meV
meV
meV
meV
meV
meV
meV
meV
meV
20
40
60
-1
 (ns )
80
100
64
Coulomb energy (meV)
Quantum yield (%)
Quantum yield (%)
300
200
2
Coulomb interaction energy = e /(4r)
uDA (r = 8 nm)
uDB and uBA (r = 4 nm)
Quantum yield versus
Dielectric constant
100
0
3
6
Dielectric constant ()
9
48
32
100 meV
400 meV
200 meV
600 meV
16
3
6
Dielectric constant ()
9
45
30
 = 100 meV
 = 400 meV
 = 200 meV
 = 600 meV
15
3
6
Dielectric constant ()
9
65
Potential energy the for shuttle motion
U(x)
Aqueous
layer
Aqueous
layer
Lipid
layer
x
U(x) 
U(x) 
1
1 e
( x  xc ) / x r
1
1  e ( x  xc ) / x r


1
1 e
( x  xc ) / x r
1
1  e ( x  xc ) / x r
,
for, x  x c
 cx 2 , for, x  x c
66
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).
E A  ES

Similarly, the energy criterium for resonant tunneling of an electron
from the protonated shuttle to state D is:
ES  uSQ  ED

Condition for jump of proton from reservoir–N to shuttle:
EQ  uSQ  mout
 Condition for jump of proton from shuttle to reservoir-P
EQ  min
67
The total Hamiltonian of the system
 To remove dependency of xjk we use unitary transformation


1
U  ( x j )U   ( x j  x ji ni )
2

 Total Hamiltonian after unitary transform

i   '  / 2 
H  H 0    ' ( x)e
a a '  Fei0t e i  P  P* / 2 a P aP*
 '
 F *e i0t ei  P  P* / 2 aP*a P   Tk ( x)d k bQ   Tk* ( x)bQ d k
k
 p2 m  2 x2
j
j j j



2
j  2m j




i 
k
1
p j x ji is stochasticphaseoperator


68
For artificial photosynthesis:
Input energy = (number of photons absorbed) x ћω0
Output energy = (number of protons pumped) x (μP - μN )
Efficiency = (output energy) / (input energy)
Efficiency
=
(Quantum yield) x (μP - μN ) / ћω0
Quantum yield Φ =
(number of protons pumped) / (number photons absorbed)
69
Current and efficiency (for solar cells)
• The amount of energy absorbed (per unit time) by the triad
 photo  F0

2
PP* k BT 
2

(a
 P aP* )mn
2
m  n

 Emn  0  PP* 2 
 Emn  0  PP* 2  
  exp
  exp
 

4PP* k BT
4PP* k BT





• Current:
I R  R  | ( a A ) mn | 1  FR (E mn )  n  R  | (a A ) mn | FR (E mn )  m
mn
•
Efficiency:
•
Quantum yield:
mn
 
mP  m N I R
 photo
  0
IR
 photo


 pumping protons
 absorbed from photons
N protons pumped
N photons absorbed
70

Light-to-electricity conversion parameters:

Light intensity: I = 20 mW cm−2.

Resonant electron tunneling rate: Δ/ћ = 15 ns-1.

Coupling to electrodes: ΓL/ћ = 100 ns-1 , ΓR/ћ = 100 ns-1.
 Temperature: T = 298 K.
 Energy of light: ћω0 = 2 eV.
 Proton potentials: μN = - 110, μP = 110.
 Dielectric constant: ε = 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.

Reorganization energies:. λPP* = 100 meV, λDP = 600 meV, λAP = 400 meV.
 μL = - 630 meV and μR = 480 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, 621–636 (2007).
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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.
(CH2)n
Left
electrode (L)
Light harvesting
component I
B
D
P
Donor
A
Light harvesting
component II
B*
• Energy diagram
EN
P*
1
eL
e-
Right
electrode (R)
Acceptor
e-
A
e-
e-
2
eD
B
P
R
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 A molecular triad (Fc-P-C60) and an
additional light harvesting complex (B).
 Both are attached to a metal surface.
Marcus rate:
  ΔE  λ  
π
exp 

2
λ k BT 
 4 λ k BT 
2
k Δ
2
Reorganization energy:
λ
∆E
xA
•
Reorganization energy (λ): 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.
xD
75