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Lecture 3

A. Nitzan, Tel Aviv University SELECTED TOPICS IN CHEMICAL DYNAMICS IN CONDENSED SYSTEMS Boulder, Aug 2007. Lecture 3. Boulder Aug 2007. (1) Relaxation and reactions in condensed molecular systems Kinetic models Transition state theory Kramers theory and its extensions

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Lecture 3

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  1. A. Nitzan, Tel Aviv University SELECTED TOPICS IN CHEMICAL DYNAMICS IN CONDENSED SYSTEMS Boulder, Aug 2007 Lecture 3

  2. Boulder Aug 2007 • (1) Relaxation and reactions in condensed molecular systems • Kinetic models • Transition state theory • Kramers theory and its extensions • Low, high and intermediate friction regimes • Diffusion controlled reactions Chapter 13-15

  3. Boulder Aug 2007 • (2) Electron transfer processes • Simple models • Marcus theory • The reorganization energy • Adiabatic and non-adiabatic limits • Solvent controlled reactions • Bridge assisted electron transfer • Coherent and incoherent transfer • Electrode processes Chapter 16

  4. Theory of Electron Transfer Transition rate

  5. S S S S

  6. Dielectric solvation Born solvation energy

  7. Electron transfer: Activation energy Reorganization energy Activation energy

  8. Electron transfer: Effect of Driving (=energy gap)

  9. Experimental confirmation of the inverted regime Marcus papers 1955-6 Miller et al, JACS(1984) Marcus Nobel Prize: 1992

  10. Donor-to-Bridge/ Acceptor-to-bridge Bridge Green’s Function Franck-Condon-weighted DOS Reorganization energy Marcus expresions for non-adiabatic ET rates

  11. Bridge mediated ET rate Charge recombination lifetimes in the compounds shown in the inset in dioxane solvent. (J. M. Warman et al, Adv. Chem. Phys. Vol 106, 1999). The process starts with a photoinduced electron transfer – a charge separation process. The lifetimes shown are for the back electron transfer (charge recombination) process.

  12. ET rate from steady state hopping

  13. The photosythetic reaction center Michel - Beyerle et al

  14. DNA (Giese et al 2001)

  15. ELECTROCHEMISTRY

  16. Donor gives an electron and goes from state a (reduced) to state b (oxidized). Eb,a=Eb- Ea is the energy of the electron given to the metal Transition rate to a continuum (Golden Rule) D A EF Rate of electron transfer to metal in vacuum M Rate of electron transfer to metal in electrolyte solution

  17. Steady state evaluation of rates • Rate of water flow depends linearly on water height in the cylinder • Two ways to get the rate of water flowing out: • Measure h(t) and get the rate coefficient from k=(1/h)dh/dt • Keep h constant and measure the steady state outwards water flux J. Get the rate from k=J/h • = Steady state rate h

  18. PART C Molecular conduction

  19. Steady state quantum mechanics Starting from state 0 at t=0: P0 = exp(-G0t) G0= 2p|V0l|2rL(Golden Rule) V0l Steady state derivation:

  20. pumping damping V0l

  21. Resonance scattering V1l V1r

  22. Resonance scattering For each r and l j = 0, 1, {l}, {r}

  23. Resonance scattering For each r and l

  24. SELF ENERGY

  25. V1l V1r

  26. V1l V1r V10 Resonant tunneling

  27. Resonant Tunneling Transmission Coefficient

  28. V1r V1l V1l V1r How is current generated? Occupation probabilities (Fermi functions)

  29. Resonant Transmission – 3d 1d 3d: Total flux from L to R at energy E0: If the continua are associated with a metal electrode at thermal equilibrium than (Fermi-Dirac distribution)

  30. CONDUCTION R L m m –|e|f f(E0) (Fermi function) 2 spin states Zero bias conduction

  31. Landauer formula For a single “channel”: (maximum=1) Maximum conductance per channel

  32. Current from classical kinetics I/e = 0 at steady state Find P1 and insert into I Quantum mechanical resalt:

  33. eF fL(E) – fR(E) T(E) eF I T(E) F fL(E) – fR(E)

  34. Molecular level structure between electrodes LUMO HOMO

  35. Cui et al (Lindsay), Science 294, 571 (2001) “The resistance of a single octanedithiol molecule was 900 50 megaohms, based on measurements on more than 1000 single molecules. In contrast, nonbonded contacts to octanethiol monolayers were at least four orders of magnitude more resistive, less reproducible, and had a different voltage dependence, demonstrating that the measurement of intrinsic molecular properties requires chemically bonded contacts”.

  36. General case Unit matrix in the bridge space Bridge Hamiltonian B(R) + B(L) is Self energy Wide band approximation

  37. The N-level bridge (n.n. interactions) G1N(E)

  38. ET vs Conduction

  39. A relation between g and k Electron charge conduction Electron transfer rate Decay into electrodes Marcus

  40. A relation between g and k l0.5eV

  41. Conductance (g(Ω-1)) vs Kinetics ( k0 (s-1) ) for alkane spacers [Marshal Newton] • Conclusions: • • conductance data of Tao et al (g) and rate constant • data (k0) correspond to within ~ 1-2 orders of magnitude • results from the 2 sets of conductance measurements • differ by > 2 orders of magnitude

  42. TOMORROW FACTORS AFFECTING MOLECULAR CONDUCTION

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