Energetics of Electron Transfer Reactions

Transcription

1 Energetics of Electron Transfer Reactions Dmitry Matyushov Arizona State University MIT, November 17, 24

2 Problems

3 Tunneling between localized states E D A LUMO E HOMO < E> Instantaneous energy gap E becomes the reaction coordinate

4 Reaction Coordinate

5 Linear Relation

6 Marcus-Hush Theory of Electron Transfer Two-parameters model: F act = (λ + F ) 2 4λ 2 X= λ is the reorganization energy F is the driving force. F i (X) F <δx 2 >=2λkT F i (X) = F i + (X X i) 2 4λ X 2λ = X 2 X 1 Energy gap law is the same for charge separation (CS) and charge recombination (CR) - F act -2 CS CR F

7 Charge-Transfer Processes in Condensed Phase Density of energy gaps (X = hω): FC i (ω) spectroscopy = h δ ( hω E(q) q,i = e βf i( hω) charge transfer <X>=ν abs/em optical spectroscopy FC i (X) 1.5 <δx 2 2 >=Γ abs/em ET kinetics X= energy gap/frequency

8 Spectral Band-Shape Intensity solvent-induced gas-phase ν/kk Convolution of gas-phase and solvent-induced bands: FC i (ν) = FC i,solvent (x)fc i,gas (ν x)dx

9 Inhomogeneous Solvent-Induced Broadening I abs/em (ν) = [ 2πσabs/em 2 ] 1/2 exp ( (ν ν abs/em) 2 ) 2σabs/em 2 2 ν gas 1.5 Intensity 1 ν solv R int.5 Γ abs/em frequency/ev σ 2 abs/em = Γ2 abs/em 8 ln(2), λ abs/em = σ2 abs/em 2kT.

10 What do we expect to see? Steady-state spectra. Solvent component of the band: ν st σ 2 /kt.5 solvent polarity.5 1 h ν st Consistency condition: σ 2 abs k B T = σ2 em k B T = h ν st.5 σ em σ abs Total width vs the total Stokes shift: σ 2 /kt 1.5 λ v ν v /kt solvent polarity.5 1 h ν st

11 What do we expect to see? Time-Resolved Spectra Time-resolved excitation: e t = t = g TRF broad-band excitation: t = e t = g Stokes shift correlation function: S Ω,i (t) = Ω(t) i Ω( ) i Ω( i Ω( ) i Equilibrium correlation function: C i (t) = δω(t)δω() i δω() 2 i Linear response (parabolic free energy surfaces): S Ω,i (t) = C i (t) λ i (t) = λ = Const

12 Experimental Evidence: Deviations from the Gaussian Picture. Asymmetry between CS and CR energy gap laws Asymmetry between steadystate absorption and emission lines Change in the time-resolved optical width Coumarin puzzle 14 absorption βσ 2 /kk 12 1 solvent polarity 8 emission ν st /kk

13 The Approximation of Fixed Charges e(1) Charge transfer + j + e(2) + j m + 1 m + 2 Interaction (i) = j e (i) j φ(i) j φ (i) j is the potential of the solvent at charge j.

14 The Approximation of Fixed Charges. Simulations 1: + - βλ i /(q * ) <(δu s ) 2 > 1 <(δu s ) 2 > y :

15 Polarizable Solute H (i) H (i) 1 2 j e(i) j φsolv j W (i) pol dipolar solute H (i) m i R 1 2 α ir 2 α i is the solute dipolar polarizability Two sources of polarizability : D-A coupling through m DA Coupling of D and A states to other electronic states m Dk D k m Am m DA m A

16 Q-Model Hamiltonian: H i = H (i) m i R 1 2 α ir 2 m Dk Free energy surfaces: F (X) = F + D m Am m DA ( α X F + α2 λ α α λ 1 A )2 α = ( 1 λ1 /λ 2 1) 3 When α, the parabolic surfaces are obtained F (X) = F + (X F λ) 2, λ 1 = λ 2 = λ 4λ

17 Q-Model: Derivation

18 Q-Model: Properties F 2 (X)=F 1 (X)+X F i (X) 2 linear asymptote 1 X band boundary X different curvatures, λ 1, λ 2 Connection to spectroscopic observables: λ i = βh 2 δν 2 i /2, α = h ν st + λ 2 λ 2 λ 1 Marcus-Hush term Correction from non-parabolicity F = h ν abs +ν em 2 λ 1 2 α (1+α) 2

19 Reorganization Energy of Polarizable Chromophores nuclear polarization response, µ p = a p m 2 λ i = a p (f i /f ei ) [ m +2a p α m i ] 2 dipole moment change polarizability change α = f e2 α 2 f e1 α 1, m = f e2 m 2 f e1 m 1 f ei = [ 1 2 a e α i ] 1 f i = [ 1 2a p α i ] 1 a e is the response of induced polarization, µ e = a e m 2

20 MC Simulations of Transitions in Polarizable Chromophores CS transition CR transition Energy gap: E = E m R 1 2 α R 2 Reorganization energy: dipole moment m polarizability α diameter σ dipole moment m polarizability α λ i = β [δ( E)] 2 i /2 λ i, ev Q-model.5.1 α /σ 3 JPCA, 18, 24, MC -βµ e -βµ p m=, α/σ 3 =.6, α = slope=a e σ βm /σ α =, α=, βm 2 /σ 3 =5. slope=a p σ βm /σ

21 1.5 TRF band-shape 1. α 2 =5 Å ³ m 2 =15 D I TRF (ω) 1 C 2 (t)= ω/ev m 1 =6 D α 1 =3 Å ³ I TRF (ω, t) e β α(t) ω Ω I 1 (2β ) α(t) 3 λ(t) ω Ω, JCP 21, 115, 8933

22 Time-Resolved Correlation Functions α 2 =1 Å ³ α 2 =4 Å ³ λ(t)/λ( ) 2 1 α 1 =3 Å ³ α 2 =5 Å ³ e (2) g (1) t = t = S Ω,2 (t) C 2 (t) S Ω,2 (t) = Ω(t) 2 Ω( ) 2 Ω( 2 Ω( ) 2 C 2 (t) = δω(t)δω() 2 δω() C 2 (t) S Ω (t) is NOT a good probe of nonlinear dynamics

23 Coumarin absorption.5 m 12 =5.8 D S 1 m 2 =14.9 D α 2 =3.2 Å ³ m=7.53 D βσ 2 /kk 12 1 solvent polarity S m 1 =7.4 D α 1 =25.8 Å ³ βσ i Q-model prediction for α > emission Stokes absorption h ν st, ev m Dk 8 emission k D ν st /kk m Am m DA m A

24 Model and Physical Picture Intensity 1 5 hybrid polarizable two-sphere m 12. R p 1 δ n δn A Nonequilibrium configuration em. abs ν/kk D Equilibrium configuration q n = q n δn γ n 2κ n E change in population q n q n

25 Hybrid Model Coupling between the D and A states is explicitely considered Coupling to all other states is accounted through the dipolar polarizability ᾱ = α 2 m 12 2 E DA Solute-solvent coupling: non-condon coupling m 1 R 1 2ᾱ1R 2 m 12 R m 12 R m 2 R 1 2ᾱ2R 2 Electron-phonon coupling: n γ 1nq n n 1 n γ 2nq n electronic population n 2

26 Calculation Procedure solid - 2MB dashed - gas phase Intensity.2.1 em. abs. 2 3 ν/kk FCWD( ν) = (δn(r) dx 1 δ( ν x E[R])FCWD ref ( ν ref + x/δn(r)) 2 acetonitrile Intensity 1 em. abs

27 Coumarin-153 band-shapes em. theory acn abs. experiment relative width, kk 1-1 em abs ν/kk.2.1 em ν/κκ abs. acet ν st s,v /kk ν st /kk vibrational solvent-induced ν st (calc)/kk

28 Spectral intensity and the Franck-Condon factor Lax, Kubo-Toyozawa, Davydov, 5 s. Spectral intensity: I abs/em (ν) m 12 2 F CW D(diagonal matrix elements), m 12 is the transition dipole arising from the interaction with the external electric field of the radiation m 12 E (t) In a polar medium, m 12 R R is the solvent local field. I abs/em (ν) m 12 2 F CW D(m 12 )

29 Hole Transfer in DNA Hole Acceptor Hole Donor k ET V (R) 2 exp[ E a (R)/kT ] Experiment: k ET exp[ β DA R DA ] β DA = β V + β λ, β λ = 1 4kT β V Å 1, λ R β λ = 1. Å 1 Mg 2+ H 2 O β DA (Exp).9Å 1

30 Energy Gap Law.5.4 G L C T G L A A C F act, ev Marcus-Hush Q-Model λ s, ev α ln(k ET ) CR CS F, ev ( F act = α F λ 1 α 2 /(1 + α) ) 2 α λ 1 β λ (Marcus Hush) = 1. Å 1, β λ (Q Model) =.26 Å 1 JPCB 17, 23,

31 Conclusions β(f i (X)-F 1 ) 8 4 X 2 1 βλ 1 =4 α 1 = 4 3-parameter model: λ 1, λ 2, F -1 1 βx I TRF (ω) C 2 (t)= α 2 =5 Å ³ m 2 =15 D Time-resolved band-shapes: λ(t) Const ω/ev m 1 =6 D α 1 =3 Å ³ βσ 2 /kk emission absorption solvent polarity ν st /kk D-A coupling + polarizability = band-shapes of intense transitions. G L C L T G A A C ln(k ET ) CR CS F, ev Energy gap law with λ 1 λ 2.