The linear electron-phonon coupling model for molecular nonadiabatic ET. Simple derivations of the electron transfer rate

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1 The linear ectron-phonon coupling mod for molecular nonadiabatic T Simple derivations of the ectron transfer rate Spiros S. Sourtis epartment of Physics, University of Cyprus Nicosia Cyprus Ph Course CN NNO S3 Modena May

2 ssume the transferring ectron in the molecular system interacts with one vibrational degree of freedom, e.g., a collective normal mode of the system or a local vibration (e.g., a charged bond, etc). We represent this degree of freedom by a harmonic oscillator (spring constant, equilibrium length 0 ): Molecular system M +e 0 Fig. (1) S. Sourtis CN Nano S3 Ph course

3 When the ectron is in the onor ectronic state: the equilibrium length of the harmonic oscillator is shifted to due to the force exerted by the ectron on the oscillator, F 0 F / Ψ Ψ -e { ( t) } +e M F (α) For a classical oscillator the time-dep. displacement is () t S. Sourtis CN Nano S3 Ph course

4 When the ectron is in the cceptor ectronic state: () r, the equilibrium length of the harmonic oscillator is shifted to due to the new force exerted by the ectron by the oscillator, F 0 F / Ψ Ψ -e +e F (γ) S. Sourtis CN Nano S3 Ph course

5 nergetics of ectron + oscillator lectron in lec. energy depends on the displacement of the osc., i.e., ( ) Linear ec.-phonon coupling mod (constant force). xpand to first order the ectron s energy wrt the osc. equil. pos. in the absence of the ec.: F 0 ( ) ( 0) 0 ( 0) 0 F d ( ) / d Born-Oppenheimer potential (potential seen by perturbed osc.): ( ) F ( ) ( ) ( ) Total force on the osc.: F ( ) d ( ) / d ( ) F tot 0 S. Sourtis CN Nano S3 Ph course

6 nergetics of ectron + oscillator lectron in () r lec. energy : ( ) Linear ec.-phonon coupling mod : F 0 ( ) ( 0) 0 ( 0) 0 F d ( ) / d Born-Openheimer potential (potential seen by perturbed osc.): ( ) F ( ) ( ) ( ) Total force on the osc.: F ( ) d ( ) / d ( ) F tot 0 S. Sourtis CN Nano S3 Ph course

7 In general the ec. energies for the and states are different : ( ) To have ectron transfer from to the energies must cross (Landau-Zener crossing). This happens for a particular osc. displacement, denoted the resonance displacement. ( ) ( ) res res res res 0 0 ( ) / ( F F ) 0 S. Sourtis CN Nano S3 Ph course

8 LNU-ZN COSSING Ψ Ψ lec. ner. Ψ -e Ε ( ) +e Ε ( ) -e Ψ F t () L Ψ -e For = res, ec. energy of Ψ becomes equal to ec. energy of Ψ Α lectron transfer can tae place Ψ lec. ner. ΔΕ: Tunning barrier +e F Ε ( res ) = Ε ( res ) -e lec. Coupling Ψ / Ψ : res e T e ( m ) L/ Ψ Ψ -e lec. ner. Ε ( ) +e Ε ( ) -e F S. Sourtis CN Nano S3 Ph course

9 Born-Oppenheimer Hamiltonian ˆ H ( ) ( ) ( ) T H ( ) T ( ) T ( ) Born-Oppenheimer potentials written in terms of equil. pos. of osc. in absence of ec. : Born-Oppenheimer potentials written in terms of equil. pos. of osc. in presence of ec. : ( ) F ( ) ( ) ( ) (min) ( ) F 0 (min) / 0 F / ( ) F ( ) ( ) S. Sourtis CN Nano S3 Ph course ( ) (min) ( ) F 0 F / 0 (min) /

10 nergy ( ) ( ) res ( ) act - esonance res res ( ) ( ) res res res ( ) ( ) (min) 0 (min) res ( ) ( ) res res res min act S. Sourtis CN Nano S3 Ph course

11 erivation of the T rate Initially the ectron is in the state. The osc sees the Born-Oppenheimer potential: ( ) (min) ( ) and it oscillates around the equil pos : () t ( (0) ) Cos( t) ( (0) / ) Sin( t) M The motion of the osc maes the ec ener time dep : H ( ( t)) T ( ( t)) T ( ( t)) S. Sourtis CN Nano S3 Ph course For = res : H ( ) T ( ) T ( ) res res re res

12 If the total ener of the osc is greater than the Born-Oppenheimer ener at - resonance, M tot res (min) act res there will be two - Landau-Zener crossings at for every period of oscillation / For every Landau-Zener crossing at time t res, the ec transfer probability in the nonadiabatic nergy ( ) limit will be: ( ) P T d ( t ) / dt d ( t ) / dt res res where d ( t ) / dt [ d ( ) / d] res res res tot res ( ) M res (min) - esonance act d ( t ) / dt [ d ( ) / d] res res res and res [ tot ( res )] / M (min) is the speed of the osc at the crossing S. Sourtis CN Nano S3 Ph course res

13 We use res [ tot ( res )] / M [ d( ) / d] res F [ d( ) / d] res F to write the Landau Zener transition probability in terms of the total ener and the forces exerted by the ec on the osc: P T 1 res F F P tot ( ) T M 1 F tot res F ( ) Since there are two Landau-Zener crossings per osc period, for total ener greater than the resonance ener the ec transfer rate as a function of the total ener will be: tot tot res tot P ( ), ( ) ( ) 0, ( ) tot res S. Sourtis CN Nano S3 Ph course ( ) res min act

14 The ensemble-averaged ectron transfer rate is: tot tot tot d ( ) P ( ) Boltz res ( ) where P Boltz tot ( ) (min) e d tot tot / K T B e tot / K T B fter calculating the integral we get: nergy ( ) T FC ( ) Franc-Condon factor: λ - esonance FC 1 4KT B act exp KT B res ( ) act (min) eorg ener: (min) ctivation ener: act (min) (min) 4 S. Sourtis CN Nano S3 Ph course res

15 e-derivation of nonadiabatic T rate using Fermi s Golden rule and energy-gap fluctuation picture S. Sourtis CN Nano S3 Ph course

16 ifferent members of the ensemble see different oscillator displacements (hence different energy gaps) nergy ( ) ( ) Prob. istr. for oscillator displacement U p K T KT B exp KBT exp ( ) B ns. Mem. 1 onor-acceptor energy gap U nergy ( ) ( ) nergy ( ) ( ) U U ns. Mem. S. Sourtis CN Nano S3 Ph course ns. Mem. 3

17 Since displacement is a random variable, so is energy gap Prob. istr. for oscillator displacement p K T KT B exp KBT exp ( ) B U ( ) (min) ( ) ( ) (min) ( ) 1 U U U exp U U U (min) (min) Prob. istr. for donor-acceptor energy gap KT U B U U S. Sourtis CN Nano S3 Ph course U

18 T rate is written as an average over oscillator displacement which is a gaussian distributed random variable Golden rule rate expression used rather than Landau-Zener approach: T d p U p K T KT B exp KBT exp ( ) B U Integral over oscillator displacement pics resonance position = res res res due to dta function S. Sourtis CN Nano S3 Ph course

19 T rate is written as an average over energy gap which is a gaussian distributed random variable T du U U T U 0 esonance condition 1 U U U exp U U 1 T U exp U U FC Franc-Condon factor: FC 1 4KT B act exp KT B ctivation ener: act (min) (min) 4 eorg ener: S. Sourtis CN Nano S3 Ph course

20 ppendix: Getting the distribution for the energy gap from the distribution for the osc. displacement U d p( ) U p K T KT exp ( ) B B ( ) (min) ( ) F 0 (min) / 0 F / ( ) (min) ( ) 0 ( ) F / 0 F / (min) (min) S. Sourtis CN Nano S3 Ph course

21 U d p( ) U (min) (min) p K T KT exp B B Substitute fourier-transform representation of dta function U (min) (min) 1 = dsexp is U (min) (min) U 1 KT B d ds exp KBT is U (min) (min) S. Sourtis CN Nano S3 Ph course

22 o first integral over φ then integral over s 1/ dx exp ax / ijx a exp J / a 1/ dx exp ax / Jx a exp J / a S. Sourtis CN Nano S3 Ph course

23 Time dependent formulation of T rate 1 T U T exp U U FC Franc-Condon factor: FC 1 4KT B act exp 4KT B ctivation ener: act (min) (min) 4 eorg ener: 1 it T FC T dt e C t C FC FC, (min) (min) i t t CFC t e / FC FC KT B S. Sourtis CN Nano S3 Ph course

24 i t t CFC t e / FC is called the time-dependent Franc-Condon factor To prove that the time-dependent Franc-Condon factor is the Fourier transform of the usual Franc- Condon factor use again the gaussian integral formulas S. Sourtis CN Nano S3 Ph course

25 Homewor Fill in all the missing steps between the equations Questions about homewor: nytime while I am here (until the end of the wee) sourtis@ucy.ac.cy when I am not S. Sourtis CN Nano S3 Ph course

26 Some Textboos S. Sourtis CN Nano S3 Ph course