Energetics of Electron Transfer Reactions Dmitry Matyushov Arizona State University MIT, November 17, 24
Problems
Tunneling between localized states E D A LUMO E HOMO < E> Instantaneous energy gap E becomes the reaction coordinate
Reaction Coordinate
Linear Relation
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 -2 2 4 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 -2 2 4 - F
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 2 1.5 <X>=ν abs/em optical spectroscopy FC i (X) 1.5 <δx 2 2 >=Γ abs/em ET kinetics 1 1.5 2 2.5 X= energy gap/frequency
Spectral Band-Shape Intensity.8.6.4.2 solvent-induced gas-phase 15 2 25 ν/kk Convolution of gas-phase and solvent-induced bands: FC i (ν) = FC i,solvent (x)fc i,gas (ν x)dx
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.5 1 1.5 2 2.5 3 frequency/ev σ 2 abs/em = Γ2 abs/em 8 ln(2), λ abs/em = σ2 abs/em 2kT.
What do we expect to see? Steady-state spectra. Solvent component of the band: 1 1.5 1 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 1 2 3 4 Total width vs the total Stokes shift: σ 2 /kt 1.5 λ v ν v /kt solvent polarity.5 1 h ν st
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
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 4 4.4 4.8 5.2 5.6 6 6.4 ν st /kk
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.
The Approximation of Fixed Charges. Simulations 1: + - βλ i /(q * ) 2.5.4.3 <(δu s ) 2 > 1 <(δu s ) 2 >.2 1 2 3 4 5 6 7 y :
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
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 1 1 + 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
Q-Model: Derivation
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
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
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 2.5 2 1.5 1 Q-model.5.1 α /σ 3 JPCA, 18, 24, 287-296. MC -βµ e -βµ p 2 15 1 5 m=, α/σ 3 =.6, α = slope=a e σ 3 2 4 6 8 2 3 βm /σ 2 15 1 5 α =, α=, βm 2 /σ 3 =5. slope=a p σ 3 1 2 3 4 2 3 βm /σ
1.5 TRF band-shape 1. α 2 =5 Å ³ m 2 =15 D I TRF (ω) 1 C 2 (t)=.2.4.6.8.5-6 -4-2 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
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).2.4.6.8 1 1.8.6.4 C 2 (t) S Ω,2 (t) = Ω(t) 2 Ω( ) 2 Ω( 2 Ω( ) 2 C 2 (t) = δω(t)δω() 2 δω() 2 2.2.2.4.6.8 1 C 2 (t) S Ω (t) is NOT a good probe of nonlinear dynamics
Coumarin-153 1 14 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 ³.5 1 6 βσ i 2 4 2 Q-model prediction for α > emission Stokes absorption.5 1 1.5 2 2.5 h ν st, ev m Dk 8 emission 4 4.4 4.8 5.2 5.6 6 6.4 k D ν st /kk m Am m DA m A
Model and Physical Picture Intensity 1 5 hybrid polarizable two-sphere m 12. R p 1 δ n δn A Nonequilibrium configuration em. abs. 18 2 22 24 ν/kk D Equilibrium configuration q n = q n δn γ n 2κ n E change in population q n q n
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
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. 1 15 2 25 3
Coumarin-153 band-shapes 2.2.1 em. theory acn abs. experiment relative width, kk 1-1 em abs 12 16 2 24 28 32 ν/kk.2.1 em. 12 16 2 24 28 32 ν/κκ abs. acet ν st s,v /kk -2 4.5 5 5.5 6 5 4 3 2 ν st /kk vibrational solvent-induced 1 4.6 4.8 5 5.2 5.4 5.6 ν st (calc)/kk
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 )
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.7 1.7 Å 1, λ R β λ = 1. Å 1 Mg 2+ H 2 O β DA (Exp).9Å 1
Energy Gap Law.5.4 G L C T G L A A C F act, ev.3.2.1 Marcus-Hush Q-Model.5 1 1.5 2 λ s, ev α 1.2 1.3 ln(k ET ) 3 25 2 CR CS 15-4 -3-2 -1 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, 1459-1452.
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) 1.5 1.5 C 2 (t)=.2.4.6.8 1. α 2 =5 Å ³ m 2 =15 D Time-resolved band-shapes: λ(t) Const. -6-4 -2 ω/ev m 1 =6 D α 1 =3 Å ³ βσ 2 /kk 14 12 1 8 emission absorption solvent polarity 4 4.4 4.8 5.2 5.6 6 6.4 ν st /kk D-A coupling + polarizability = band-shapes of intense transitions. G L C L T G A A C ln(k ET ) 3 25 2 CR CS 15-4 -3-2 -1 F, ev Energy gap law with λ 1 λ 2.