1. Lecture 1 Theoretical models for transport, transfer and relaxation in molecular systems A. Nitzan, Tel Aviv University SELECTED TOPICS IN CHEMICAL DYNAMICS IN CONDENSED SYSTEMS

2. INTRODUCTION

3. Chemical dynamics in condensed phases Molecular relaxation processes Quantum dynamics Time correlation functions Quantum and classical dissipation Density matrix formalism Vibrational relaxation Electronic relaxation (radiationaless transitions) Solvation Energy transfer Applications in spectroscopy Condensed phases Molecular reactions Quantum dynamics Time correlation functions Stochastic processes Stochastic differential equations Unimolecular reactions: Barrier crossing processes Transition state theory Diffusion controlled reactions Applications in biology Electron transfer and molecular conduction Quantum dynamics Tunneling and curve crossing processes Barrier crossing processes and transition state theory Vibrational relaxation and Dielectric solvation Marcus theory of electron transfer Bridge assisted electron transfer Coherent and incoherent transfer Electrode reactions Molecular conduction Applications in molecular electronics

4. electron transport in molecular systems Reviews: Annu. Rev. Phys. Chem. 52, 681– 750 (2001) Science, 300, 1384-1389 (2003); J. Phys.: Condens. Matter 19, 103201 (2007) – Inelastic effects Phys. Chem. Chem. Phys., 14, 9421 - 9438 (2012 ) – optical interactions Molecular Plasmonics Solar cells, OLEDs

6. Chemical processes Gas phase reactions Follow individual collisions Follow individual collisions States: Initial  Final States: Initial  Final Energy flow between degrees of freedom Energy flow between degrees of freedom Mode selectivity Mode selectivity Yields of different channels Yields of different channels Reactions in solution Reactions in solution Effect of solvent on mechanism Effect of solvent on mechanism Effect of solvent on rates Effect of solvent on rates Dependence on solvation, relaxation, diffusion and heat transport. Dependence on solvation, relaxation, diffusion and heat transport.

7. I 2  I+I A.L. Harris, J.K. Brown and C.B. Harris, Ann. Rev. Phys. Chem. 39, 341(1988) molecular absorption at ~ 500nm is first bleached (evidence of depletion of ground state molecules) but recovers after 100- 200ps. Also some transient state which absorbs at ~ 350nm seems to be formed. Its lifetime strongly depends on the solvent (60ps in alkane solvents, 2700ps (=2.7 ns) in CCl4). Transient IR absorption is also observed and can be assigned to two intermediate species.

9. The hamburger-dog dilemma as a lesson in the importance of timescales

11. TIMESCALES Typical molecular timescales in chemistry and biology (adapted from G.R. Fleming and P. G. Wolynes, Physics today, May 1990, p. 36).

12. Molecular processes in condensed phases and interfaces Diffusion Relaxation Solvation Nuclear rerrangement Charge transfer (electron and xxxxxxxxxxxxxxxxproton) Solvent: an active spectator – energy, friction, solvation Molecular timescales Diffusion D~10 -5 cm 2 /s Electronic 10 -16 -10 -15 s Vibraional 10 -14 s Vibrational xxxxrelaxation 1-10 -12 s Chemical reactions xxxxxxxxx10 12 -10 -12 s Rotational 10 -12 s Collision times 10 -12 s

13. VIBRATIONAL RELAXATION

14. Frequency dependent friction WIDE BAND APPROXIMATION MARKOVIAN LIMIT Golden Rule

15. Molecular vibrational relaxation “ENERGY GAP LAW”

16. Molecular vibrational relaxation Relaxation in the X 2 Σ+ (ground electronic state) and A 2 Π (excite electronic state) vibrational manifolds of the CN radical in Ne host matrix at T=4K, following excitation into the third vibrational level of the Π state. (From V.E. Bondybey and A. Nitzan, Phys. Rev. Lett. 38, 889 (1977))

17. Molecular vibrational relaxation The relaxation of different vibrational levels of the ground electronic state of 16 O 2 in a solid Ar matrix. Analysis of these results indicates that the relaxation of the  < 9 levels is dominated by radiative decay and possible transfer to impurities. The relaxation of the upper levels probably takes place by the multiphonon mechanism. (From A. Salloum, H. Dubust, Chem. Phys.189, 179 (1994)).

18. DIELECTRIC SOLVATION

19. Dielectric solvation Emission spectra of Coumarin 153 in formamide at different times. The times shown here are (in order of increasing peak- wavelength) 0, 0.05, 0.1, 0.2, 0.5, 1, 2, 5, and 50 ps (Horng et al, J.Phys.Chem. 99, 17311 (1995)) Born solvation energy

20. Continuum dielectric theory of solvation How does solvent respond to a sudden change in the molecular charge distribution? Electric displacement Electric field Dielectric function Dielectric susceptibility polarization Debye dielectric relaxation model Electronic response Total (static) response Debye relaxation time (Poisson equation)

21. Continuum dielectric theory of solvation WATER:  D =10 ps  L =250 fs

22. “real” solvation The experimental solvation function for water using sodium salt of coumarin-343 as a probe. The line marked ‘expt’ is the experimental solvation function S(t) obtained from the shift in the fluorescence spectrum. The other lines are obtained from simulations [the line marked ‘Δq’ –simulation in water. The line marked S 0 –in a neutral atomic solute with Lennard Jones parameters of the oxygen atom]. (From R. Jimenez et al, Nature 369, 471 (1994)). “Newton” dielectric

23. Electron solvation The first observation of hydration dynamics of electron. Absorption profiles of the electron during its hydration are shown at 0, 0.08, 0.2, 0.4, 0.7, 1 and 2 ps. The absorption changes its character in a way that suggests that two species are involved, the one that absorbs in the infrared is generated immediately and converted in time to the fully solvated electron. (From: A. Migus, Y. Gauduel, J.L. Martin and A. Antonetti, Phys. Rev Letters 58, 1559 (1987) Quantum solvation (1) Increase in the kinetic energy (localization) – seems NOT to affect dynamics (2) Non-adiabatic solvation (several electronic states involved)

24. Electron tunneling through water 1 2 3 Polaronic state (solvated electron) Transient resonance through “structural defects”

25. Electron tunneling through water Time (ms) STM current in pure water S.Boussaad et. al. JCP (2003)

26. CHEMICAL REACTIONS IN CONDENSED PHASES

27. Chemical reactions in condensed phases  Bimolecular  Unimolecular diffusion Diffusion controlled rates R

28. excitation reaction Thermal interactions Unimolecular reactions (Lindemann)

29. Activated rate processes KRAMERS THEORY: Low friction limit High friction limit Transition State theory (action) 00 BB

30. Effect of solvent friction A compilation of gas and liquid phase data showing the turnover of the photoisomerization rate of trans stilbene as a function of the “friction” expressed as the inverse self diffusion coefficient of the solvent (From G.R. Fleming and P.G. Wolynes, Physics Today, 1990). The solid line is a theoretical fit based on J. Schroeder and J. Troe, Ann. Rev. Phys. Chem. 38, 163 (1987)). TST

31. The physics of transition state rates Assume: (1) Equilibrium in the well (2) Every trajectory on the barrier that goes out makes it

32. The (classical) transition state rate is an upper bound Assumed equilibrium in the well – in reality population will be depleted near the barrier Assumed transmission coefficient unity above barrier top – in reality it may be less

33. Quantum considerations 1 in the classical case

34. What we covered so far Relaxation and reactions in condensed molecular systems Timescales Relaxation Solvation Activated rate processes Low, high and intermediate friction regimes Transition state theory Diffusion controlled reactions

35. Electron transfer

36. Electron transfer in polar media Electron are much faster than nuclei  Electronic transitions take place in fixed nuclear configurations  Electronic energy needs to be conserved during the change in electronic charge density Electronic transition Nuclear relaxation (solvation)

37. Electron transfer ELECTRONIC ENERGY CONSERVED Electron transition takes place in unstable nuclear configurations obtained via thermal fluctuations Nuclear motion

38. Electron transfer Solvent polarization coordinate

39. Transition state theory of electron transfer Adiabatic and non-adiabatic ET processes Landau-Zener problem Alternatively – solvent control

40. Solvent controlled electron transfer Correlation between the fluorescence lifetime and the longitudinal dielectric relaxation time, of 6-N-(4-methylphenylamino-2-naphthalene-sulfon-N,N- dimethylamide) (TNSDMA) and 4-N,N-dimethylaminobenzonitrile (DMAB) in linear alcohol solvents. The fluorescence signal is used to monitor an electron transfer process that precedes it. The line is drawn with a slope of 1. (From E. M. Kosower and D. Huppert, Ann. Rev. Phys. Chem. 37, 127 (1986))

41. Electron transfer – Marcus theory They have the following characteristics: (1) P n fluctuates because of thermal motion of solvent nuclei. (2) P e, as a fast variable, satisfies the equilibrium relationship (3) D = constant (depends on  only) Note that the relations E = D-4  P; P=P n + P e are always satisfied per definition, however D   s E. (the latter equality holds only at equilibrium). We are interested in changes in solvent configuration that take place at constant solute charge distribution 

42. Electron transfer – Marcus theory  Free energy associated with a nonequilibrium fluctuation of P n “reaction coordinate” that characterizes the nuclear polarization

43. The Marcus parabolas Use  as a reaction coordinate. It defines the state of the medium that will be in equilibrium with the charge distribution  . Marcus calculated the free energy (as function of  ) of the solvent when it reaches this state in the systems  =0 and  =1.   

44. Electron transfer: Activation energy Reorganization energy Activation energy

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

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

47. Electron transfer – the coupling From Quantum Chemical Calculations The Mulliken-Hush formula Bridge mediated electron transfer

48. Bridge assisted electron transfer EBEB

49. V DB D A B V AD EE D A V eff

50. V DB D A B1B1 V AD D A EE V eff B2B2 BNBN … V 12 Green’s Function

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

52. Bridge mediated ET rate  ’ ( Å -1 ) = 0.2-0.6 for highly conjugated chains 0.9-1.2 for saturated hydrocarbons ~ 2 for vacuum

53. Bridge mediated ET rate (J. M. Warman et al, Adv. Chem. Phys. Vol 106, 1999).

54. Incoherent hopping

55. ET rate from steady state hopping

56. Dependence on temperature The integrated elastic (dotted line) and activated (dashed line) components of the transmission, and the total transmission probability (full line) displayed as function of inverse temperature. Parameters are as in Fig. 3.

57. The photosythetic reaction center Michel - Beyerle et al

58. Dependence on bridge length

59. DNA (Giese et al 2001)

60. 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 SUMMARY

61. IRREVERSIBILITY

62. What is the source of irreversibility in the processes discussed? Vibrational relaxation Activated barrier crossing Dielectric solvation Electron transfer V V0lV0l Starting from state 0 at t=0: P 0 = exp (-   t)   = 2  |V 0l | 2  L (Golden Rule)

63. 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: (1)Measure h(t) and get the rate coefficient from k=(1/h)dh/dt (1)Keep h constant and measure the steady state outwards water flux J. Get the rate from k=J/h = Steady state rate h

64. Steady state quantum mechanics V0lV0l Starting from state 0 at t=0: P 0 = exp(-   t)   = 2  |V 0l | 2  L (Golden Rule) Steady state derivation:

65. pumping damping V0lV0l

66. Resonance scattering V 1r V 1l

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

68. Resonance scattering For each r and l

69. SELF ENERGY

70. V 1r V 1l

71. Resonant tunneling V 1r V 1l V 10

72. Summary V 1r V 1l V 1r V 01 V0rV0r V 1r V 1l

73. Lecture 2 electron transfer, energy transfer, molecular conduction, inelastic spectroscopies, heat conduction, optical effects… A. Nitzan, Tel Aviv University TOMORROW: