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Laser–Induced Control of Condensed Phase Electron Transfer Rob D. Coalson, Dept. of Chemistry, Univ. of Pittsburgh Yuri Dakhnovskii, Dept. of Physics, Univ. of Wyoming Deborah G. Evans, Dept. of Chemistry, Univ. of New Mexico Vassily Lubchenko, Dept. of Chemistry, M.I.T. NH3 H3N +3 Ru H3N CN NH3 NH3 CN N +2 Ru C NC [polar Solvent] CN CN Tunneling in A 2-State System q |L> |R> V(x) tropolone q q Proton Transfer Multi (Double) Quantum Well Structure e- GaAs AlGaAs GaAs |R> |L> q Electron V(x) Transport In Solids 1. . . . . N Ru+2 Atomic Orbitals Ru+3 e|D> |A> |D> Molecular Electron Transfer |A> Acceptor Donor Equivalent 2-state model For any of these systems: |(t)> = cD(t) |D> + cA(t) |A> where HAA HAD cA HDA HDD cD 1 = cA d iħ dt c D For a Symmetric Tunneling System: HAA= HDD= 0; HAD= HAD = (< 0) Given initial preparation in |D>, for a symmetric system: 1 PD(t) = 1 – PA(t) = cos2(t) = ( cos( x) ) 2 0.5 2 x t e- Now, apply an electric field E |D> |A> R 2 0 R 2 Q: How does this modify the Hamiltonian?? A: It modifies the site energies according to “ - • E ” Permanent dipole moment of A Thus: H = HAA HDD -E eoR/2 0 0 -eoR/2 Analysis in the case of time-dependent E(t) = Eocosot Consider first the symmetric case: i d cA dt cD 0 = 0 - E cos t o o 1 0 cA 0 -1 cD Permanent dipole moment difference Letting: cA(t) = e i a sinot cAI(t) ; cD(t) = e i a sinot cDI(t) Eo Where: a = (ħ)o t [ N.B.: E cos t'dt' = 0 o o Eo sin t ] o o Thus, Interaction Picture S.E. reads: I c d A i = dt I cD 0 e2i a sinot e-2i a sinot 0 cAI cDI Now note: So: e i b sint = m=- Jm(b) e i mt e2i a sinot = m=- Jm(2a) e i mot ·Jo(2a) , for /o << 1 RWA Thus, the shuttle frequency is renormalized to |Jo(2a)| NB: Trapping or localization occurs at certain Eo values! 0 [ 0 2.5 <<1 ] 5.5 a [Grossman - Hänggi, Dakhnovskii – Metiu] Add coupling to a condensed phase environment ˆ = Tˆ H 1 0 0 1 + Nuclear coordinate kinetic energy VD(x) VA(x) - Eocosot D(x,t) 0 0 -1 Nuclear coordinate VD(x) 1 VA(x) A(x,t) x Field couples only to 2-level system Now: T+VD(x) T+VA(x) = D(x,t) d i dt A(x,t) - E cos t o o 0 D(x,t) 0 -1 A(x,t) 1 Construction of (Diabatic) Potential Energy functions for Polar ET Systems: A D VD(x) A D VA(x) A few features of classical Nonadiabatic ET Theory [Marcus, Levich-Doganadze…] Ĥ = ˆT+V1(x) non-adiabatic coupling matrix element ˆT+V2(x) Hamiltonian kinetic E of nuclear coordinates |(t)> = 1(x,t) 2(x,t) (diabatic) nuclear coord. potential for electronic state 2 States Given initial preparation in electronic state 1 (and assuming nuclear coordinates are equilibrated on V1(x)) 1 2 1(x,0) x Then, P2(t) = fraction of molecules in electronic state 2 = < 2(x,t)| 2(x,t)> k12t where k12 = (Golden Rule) rate constant In classical Marcus (Levich-Doganadze) theory, k12 is determined by 3 molecular parameters: , Er, k12 = 2 ½ -(Er- )2/4Er kBT e ErkT ( ) w/ Er = “Reorganization Energy” ; = “Reaction Heat” 1 2 Er x ½ -(Er+ )2/4Er kBT e ErkT ( ) For the “backwards” Reaction: k21 = 2 To obtain electronic state populations at arbitrary times, solve kinetic [“Master”] Eqns.: dP1(t)/dt = - k12P1(t) + k21P2(t) dP2(t)/dt = k12P1(t) - k21P2(t) Note that long-time asymptotic [“Equilibrium”] distributions are then given by: Keq P2() k12 = = P1() k21 = e /kBT for Marcus formula rate constants N.B. Marcus theory for nonadiabatic ET reactions works experimentally. See: Closs & Miller, Science 240, 440 (1988) Control of Rate Constants in Polar Electronic Transfer Reactions Via an Applied cw Electric Field The Hamiltonian is: VD VA ˆ = H x ˆhD 0 0 ˆhA + 0 0 + 12Eocosot 1 0 0 -1 The forward rate constant is: [Y. Dakhnovskii, J. Chem. Phys. 100, 6492 (1994) 2 Re 2 i ˆhD t } kDA = Jm(a) · i mot tr {D -iˆhA t dt e ˆ e e 0 m=- a = 2 12 Eo/ ħo 2 kDA = 4 AD ½ ErkT ( ) m=- 2 Jm(212Eo/ħo) • e -(Er –+ + ħmo)2/4Er kBT Rate constants in presence of cw E-field 300 D 200 2 J1 Schematically: Energy 100 A 2 J0 0 -100 2 J-1 ħo -200 0 5 x In polar electron transfer reactions, for Reorganization Energy Er ħo (the quantum of applied laser field) 1 0.8 2 Jo(a) 0.6 2 J1(a) 0.4 2 J2(a) 0.2 0 0 1 2 3 4 5 6 a 7 8 9 10 Dramatic perturbations of the “one-way” rate constants may be obtained by varying the laser field intensity: [Activationless reaction, Er=1eV] Forward rate constant Dakhnovskii and RDC showed how this property can be used to control Equilibrium Constants with an applied cw field: D A “bias” PA(00) eq = PD(00) Results for activationless reaction: Y. Dakhnovskii and RDC, J. Chem. Phys. 103, 2908 (1995) Activationless ET = 100 cm -1 Er = = ħo = 1eV Evans, RDC, Dakhnovskii & Kim, PRL 75, 3649 (95) REALITY CHECK on coherent control of mixed valence ET reactions in polar media (1) “·E” E Orientational averaging will reduce magnitude of desired effects [Lock ET system in place w/ thin polymer films] (2) Dielectric breakdown of medium? To achieve resonance effects for = 34D Er = ħo = 1eV Electric field 107 V/cm Giant dipole ET complex, solvent w/ reduced Er, pulsed laser reduce likelihood of catastrophe] (3) Direct coupling of E(t) to polar solvent [Dipole moment of solvent molecules << Dipole moment of giant ET complex] (4) ħ0 1eV ; intense fields (multiphoton) excitation to higher energy states in the ET molecule, which are not considered in the present 2-state model. ħ 1eV ħ 1eV ħ 1eV ħ 1eV Absolute Negative Conductance in Semiconductor Superlattice - + -Keay et al., Phys. Rev. Lett. 75, 4102 (1995) Absolute Negative Conductance in Semiconductor Superlattice -Keay et al., Phys. Rev. Lett. 75, 4102 (1995) Immobilized long-range Intramolecular Electron Transfer Complex: D Tethered Alkane Chain molecule A E (ħ) k (Inert) Substrate Light Absorption by Mixed Valence ET Complexes in Polar Solvents: Transmitted Photons Ru+3 Incident Photons Ru+2 Solvent Absorption cross section K abs ( ) # Absorbed Photons # Incident Photons Absorption Cross Section Formula: k abs 1 d E (a, ) 2 E dt d E (a, ) ( eq ) U 1 ( eq ) U 2 n1 n2 dt t t U1, 2 t 4 2 E r k BT 1 (1) (2) 2 2 m J m (a) m 1 Er m 2 Er m 2 exp exp 4 E k T 4 E k T r B r B a 2 E (3) # Absorbed Photons Hush Absorption Spectrum ( L ) # Incident Photons L 212 FCF Marcus Gaussian Δ w/ 12 effective " transition dipole moment" res permanent dipole moment difference Barrierless Hush Absorption = 2 E PRL 77, 2917 (1996) J. Chem. Phys. 105, 9441 (1996) absorption emission Stimulated Emission using two incoherent lasers J. Chem. Phys. 109, 691 (1998) J. Chem. Phys. 109, 691 (1998)