1. Electronic Tunneling through Dissipative Molecular Bridges Uri Peskin Department of Chemistry, Technion - Israel Institute of Technology Musa Abu-Hilu (Technion) Alon Malka (Technion) Chen Ambor (Technion) Maytal Caspari (Technion) Roi Volkovich (Technion) Darya Brisker (Technion) Vika Koberinski (Technion) Prof. Shammai Speiser (Technion) Thanking:

2. Outline Motivation: Controlled electron transport in molecular devices and in biological systems. Background: ET in Donor-Acceptor complexes: The Golden Rule, the Condon approximaton and the spin-boson Hamiltonian. ET in Donor-Bridge-Acceptor complexes: McConnell’s formula for the tunneling matrix elements. The problem: Electronic-nuclear coupling at the molecular bridge and the breakdown of the Condon approximation. The model system: Generalized spin-boson Hamiltonians for dissipative through-bridge tunneling. Results: The weak coupling limit: Langevin-Schroedinger formulation, simulations and interpretation of ET through a dissipative bridge Beyond the weak coupling limit: An analytic formula for the tunneling matrix element in the deep tunneling regime. Conclusions: Promotion of tunneling through molecular barriers by electronic- nuclear coupling. The effect of molecular rigidity.

3. Motivation: Electron Transport Through Molecules Molecular Electronics Resonant tunneling through molecular junctions Tans, Devoret, Thess, Smally, Geerligs, Dekker, Nature (1997) Reichert, Ochs, Beckmann, Weber, Mayor, Lohneysen, Phys. Rev. Lett. (2002).

4. Long-range Electron Transport In Nature The Photosynthetic Reaction Center Deep (off-resonant) tunneling through molecular barriers Electron transfer is controlled by molecular bridges Tunneling pathway between cytochrome b5 and methaemoglobin

5. Controlled tunneling through molecules? Minor changes to the molecular electronic density High sensitivity (exponential) to the molecular parameters A potential for a rational design based on chemical knowledge Resonant tunneling Deep (off resonant) tunneling Why Off-Resonant (deep) Tunneling ?

6. Electron Transfer in Donor-Acceptor Pairs Donor Acceptor Electronic tunneling matrix element Nuclear factor: Frank-Condon weighted density of states The role of electronic nuclear coupling? The case of through bridge tunneling :

7. Theory: Electron Transfer in Donor-Acceptor Pairs The electronic Hamiltonian: Diabatic electronic basis functions: The Hamiltonian matrix:

8. Theory: Electron Transfer in Donor-Acceptor Pairs A Spin Boson Hamiltonian: The Harmonic approximation:

9. Theory: Electron Transfer in Donor-Acceptor Pairs The Condon approximation Donor Acceptor The golden rule expression for the rate An electronic tunneling matrix element A nuclear factor

10. McConnell (1961): Introducing a set of bridge electronic states; The direct tunneling matrix element vanishes DonorAcceptor Long Range Electronic Tunneling The donor and acceptor sites are connected via an effective tunneling matrix element through the bridge

11. McConnell’s Formula: A tight binding model The deep tunneling regime: First order perturbation theory A simple expression for the effective tunneling matrix element

12. Tunneling oscillations at a frequency : Superexchange dynamics through a symmetric uniform bridge H. M. McConnell, J. Chem. Phys. 35, 508 (1961)

13. Deep tunneling through a molecular bridge The role of bridge nuclear modes? Validity of the Condon approximation?

14. Davis, Ratner and Wasielewski (J.A.C.S. 2001). Molecules 1-5 Charge transfer is gated by bridge vibrations Electronic nuclear coupling at the bridge: Rigid bridges enable highly efficient electron energy transfer Lokan, Paddon-Row, Smith, La Rosa, Ghiggino and Speiser (J.A.C.S. 2001). Breakdown of the Condon approximation!

15. Structural (promoting) bridge modes: Electronically active (accepting) bridge modes:

16. A generalized “spin-boson” model: The nuclear potential energy surface changes at the bridge electronic sites Harmonic nuclear modes Linear e-nuclear coupling in the bridge modes The e-nuclear coupling is restricted to the bridge sites

17. A Dissipative Superexchange Model: A symmetric uniform bridge Introducing nuclear modes with an Ohmic ( ) spectral density The nuclear frequencies: 10-500 (1/cm) are larger than the tunneling frequency!! and a uniform electronic-nuclear coupling : M. A-Hilu and U. Peskin, Chem. Phys. 296, 231 (2004).

18. Coupled Electronic-Nuclear Dynamics A mean-field approximation: The coupled SCF equations: Mean-fields:

19. The Langevin-Schroedinger equation A non-linear, non Markovian dissipation term Fluctuations At zero temperature, R(t) vanishes Initial nuclear position and momentum Electronic bridge population U. Peskin and M. Steinberg, J. Chem. Phys. 109, 704 (1998).

20. Numerical Simulations: Weak e-n coupling The tunneling frequency increases!

21. The tunneling is suppressed ! Simulations: Strong e-n Coupling

22. Interpretation: a time-dependent Hamiltonian The Instantaneous electronic energy: Weak coupling: Energy dissipation into nuclear vibrations lowers the barrier for electronic tunneling A time-dependent McConnell formula

23. Interpretation: a time-dependent Hamiltonian The Instantaneous electronic energy: Weak coupling: Energy dissipation into nuclear vibrations lowers the barrier for electronic tunneling Strong coupling: “Irreversible” electronic energy dissipation Resonant Tunneling

24. Numerically exact simulations for a single bridge mode Tunneling suppression at strong coupling Tunneling acceleration at weak coupling

25. A dissipative-acceptor model: The acceptor population: Dissipation leads to a unidirectional ET The tunneling rate Increases with e-n coupling at the bridge! Introducing a bridge mode A. Malka and U. Peskin, Isr. J. Chem. (2004).

26. A dimensionless measure for the effective electronic-nuclear coupling: Interpretation: Nuclear potential energy surfaces

27. Deep tunneling = weak electronic inter-site coupling Entangled electronic-nuclear dynamics beyond the weak coupling limit A small parameter: The symmetric uniform bridge model: M. A.-Hilu and U. Peskin, submitted for publication (2004).

28. A Rigorous Formulation The Donor/Acceptor Hamiltonian The Bridge Hamiltonian The coupling Hamiltonian (purely electronic!)

29. Introducing vibrational eigenstates: Diagonalizing the tight-binding operator:

30. Regarding the electronic coupling as a (second order) perturbation In the absence of electronic coupling the ground state is degenerate: The energy splitting temperature reads: Frank-Condon overlap factors

31. The energy splitting: Expanding the denominators in powers of and keeping the leading non vanishing terms gives

32. Interpretation: Effective electronic coupling Effective barrier for tunneling McConnell’s expression:

33. Summation over vibronic tunneling pathways: Lower barrier for tunneling Multiple “Dissipative” pathways The effective tunneling barrier decreases

34. An example (N=8) The tunneling frequency increases by orders of magnitude with increasing electronic nuclear coupling 1/cm

35. The “slow electron” adiabatic limit Considering only the ground nuclear vibrational state: A condition for increasing the tunneling frequency by increasing electronic-nuclear coupling:

36. An example (N=8) The slow electron approximation

37. Spectral densities Molecular rigidity = small deviations from equilibrium configuration Flexible vs. Rigid molecular bridges  Increasing rigidity  A consistency constraint:

38. Langevin-Schroedinger simulations: The tunneling frequency increases with bridge rigidity

39. A rigorous treatment: The “slow electron” limit Rigidity = larger Frank Condon factor!

40. Summary and Conclusions A rigorous calculation of electronic tunneling frequencies beyond the weak electronic-nuclear coupling limit, predicts acceleration by orders of magnitudes for some molecular parameters An analytical approach was introduced and a formula was derived for calculations of tunneling matrix elements in a dissipative McConnell model. A comparison with approximate methods for studying open quantum systems is suggested. The way for rationally designed, controlled electron transport in “molecular devices” is still long… The effect of electronic-nuclear coupling in electronically active molecular bridges was studied using generalized McConnell models including bridge vibrations. Mean-field Langevin-Schroedinger simulations of the coupled electronic-nuclear dynamics suggest that weak electronic– nuclear coupling promotes off-resonant (deep) through bridge tunneling

41. Long-range Electron Transport In Nature Deep (off-resonant) tunneling through molecular bridges Electron transfer is controlled by molecular barriers: Fig.6. Calculated path (green) for electron tunneling between an electrostatically docked cytochrome b 5 (left) and methaemoglobin (right)

42. Long-range Electron Transport In Nature This enzyme is used by the bacterium to allow it to inhabit areas of low oxygen concentration when it leads to infections in humans. It contains a calcium ion which appears to be crucial in the control of electron transfer. (Fig 9) Fig. 9. The calculated route of electron transfer between the two haem groups of cytochrome c peroxidase is shown ( in green) together with the close proximity of the bound calcium ion (grey sphere).