Total Hamiltonian

1. Total Hamiltonian • The Hamiltonian H0of the system incorporates terms relating the eigen-energies of the states and Coulomb interaction energies. Ghosh, Smirnov, Nori,J. Chem. Phys. (2009).

2. フランコ　ノリ　の • “ノリノリ” 　　　　　　　　　　　　　　　　　プレゼンテーション

3. Solar energy conversion mimicking natural photosynthesis: Modeling the light-energy conversion in a molecular triad (inserted between two proton reservoirs or two electrodes). Molecular triad Electrode or proton reservoir Electrode or proton reservoir Photo-sensitive part Donor Acceptor 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.

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) xћω0 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)

10. (ii) For light-to-electricity conversion: Input energy = (number of photons absorbed) xћω0 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)

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 pumpingprotonsuphill. 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 • Moore’s group[Nature 385, 239 (1997)] extensively developed donor-photosensitizer-acceptor 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* C Q 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) D P A S • Simplified ball-and-stick model Lipid layer Inside Outside D P A Aqueous layer Aqueous layer μP μN S μP > 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 > μN The positively charged shuttle is trapped at the interface because it cannot diffuse across the lipid layer. The charged shuttle cannot diffuse across the non-polar lipid layer. Hence, it remains almost static near the lipid-aqueousinterface. 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: The shuttle is deprotonated by donating a proton to the inner aqueous phase. The shuttle gives away an electron to the positively charged donor. The triad and the shuttle return to their initial state, and the process starts again. 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. 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 a photon Represents an electron (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. An electron is thermally transferred from the acceptor to the shuttle. Proton energy 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 Lipid layer Inside Outside D P A Aqueous layer Aqueous layer μP μN S μP > μN μ = proton potential,

23. Stages after the shuttle diffuses • to the inner side of the membrane 23

24. Artificial photosynthesis in a molecular triad Lipid layer Inside Outside D P A Aqueous layer Aqueous layer S μP μN μP > μN

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) Now, this higher energy of the proton in the shuttle permits a spontaneous deprotonation of the shuttle. When the protonated shuttle loses an electron, the proton energy in the shuttle increases. 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 Lipid layer Inside Outside D P A Aqueous layer Aqueous layer μP μN S μP > μN μ = proton potential,

27. The electron-proton system with no leads (the proton reservoirs) can be characterized by the 20 basis states of the Hamiltonian H0 : represents the vacuum state. One electron is located on site D and one on site P. Two electrons on sites A and S and a proton on the site Q. 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.

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 H0of 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 Acceptor • Tunneling elements ∆DS(x) and ∆AS(x) • depend on the shuttle positionx. • Other terms ∆DP,∆DP*, ∆PAand ∆P*A are independent of the shuttle position x. P* 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 = εdP ε = 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 Tkα(x): Aqueous layer Aqueous layer D P A xP+LQ xN - LQ Inside Outside xP xN S N-reservoir P-reservoir LQ defines the proton tunneling length. xP and xN are the coordinates of the proton reservoirs.

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 xjidetermine 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 ρm . • 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 • ς(t) = thermal white noise: Lipid layer Inside Outside D P A 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 (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)]. Ghosh, Smirnov, Nori,J. Chem. Phys. (2009).

40. Robustness of the model • 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: • 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 < 160 meV, and goes to zero when μP> 200 meV (i.e. μP > EQ). • Also, the pumping device does not work when the potential μN is too low μN < EQ − uSQ . • Main parameters: I=0.18 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. Left electrode (L) D P A Right electrode (R) Light-to-electricity energy conversion in a molecular triad Donor Photosensitive part 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 electrodes.

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

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) xћω0 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)