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. H3NH3N H3NH3N RuN NH 3 Ru NC CN C [polar Solvent] +3 +2

Tunneling in A 2-State System q tropolone |L> |R> q V(x) Proton Transfer GaAs AlGaAs q |L> |R> V(x) e-e- q Electron Transport In Solids Multi (Double) Quantum Well Structure

N e-e- |D>|A> Atomic Orbitals Ru +2 Ru +3 |D>|A>  Donor Acceptor Molecular Electron Transfer Equivalent 2-state model For any of these systems: |  (t)> = c D (t) |D> + c A (t) |A> where H AA H DA H AD H DD cAcA cDcD = iħiħ d dtdt cAcA cDcD 1 For a Symmetric Tunneling System: H AA = H DD = 0; H AD = H AD =  (< 0)

Given initial preparation in |D>, for a symmetric system: P D (t) = 1 – P A (t) = cos 2 (  t) = t  22

Now, apply an electric field |D>|A>  e-e- E 0 R 2 R 2 Q: How does this modify the Hamiltonian?? A: It modifies the site energies according to “ -  E ” Thus: H = H AA  H DD  - E e o R/2 -e o R/20 0 Permanent dipole moment of A 

Analysis in the case of time-dependent E(t) = E o cos  o t Consider first the symmetric case: i d dtdt cAcA cDcD = 0 0   -  E o cos  o t 10 0 cAcA cDcD Permanent dipole moment difference Letting: c A (t) = e i a sin  o t c A I (t) ; c D (t) = e i a sin  o t c D I (t) Where: a = EoEo (ħ)  o [ N.B.:  0 E o cos  o t'dt' = ] t EoEo oo sin  o t Thus, Interaction Picture S.E. reads: cAIcAI cDIcDI = 0 e 2i a sin  o t  0 e -2i a sin  o t  cAIcAI cDIcDI i d dt

Now note: e i b sin  t =  J m (b) e i m  t  m=-  So:  e 2i a sin  o t =   J m (2a) e i m  o t m=-     ·J o (2a), for  /  o << 1 RWA Thus, the shuttle frequency is renormalized to  | J o (2a) |  0 0 a [   <<1 ] NB: Trapping or localization occurs at certain E o values! [Grossman - Hänggi, Dakhnovskii – Metiu]

Add coupling to a condensed phase environment H = T ˆˆ V D (x)   - V A (x)  E o cos  o t 10 0 Nuclear coordinate kinetic energy Nuclear coordinate Field couples only to 2-level system V D (x) V A (x)  D (x,t)  A (x,t) x

Now: T+V D (x) T+V A (x)   -  E o cos  o t 10 0  D (x,t)  A (x,t) =  D (x,t)  A (x,t) i d dtdt

Construction of (Diabatic) Potential Energy functions for Polar ET Systems: V D (x) V A (x) A AD D

A few features of classical Nonadiabatic ET Theory [Marcus, Levich-Doganadze…] T+V 1 (x) T+V 2 (x)   ˆ H = ˆ ˆ Hamiltonian non-adiabatic coupling matrix element kinetic E of nuclear coordinates (diabatic) nuclear coord. potential for electronic state 2  1 (x,t)  2 (x,t) |  (t)> = States

Given initial preparation in electronic state 1 (and assuming nuclear coordinates are equilibrated on V 1 (x)) x 1 2  1 (x,0) Then, P 2 (t) = fraction of molecules in electronic state 2 =  k 1  2 t where k 1  2 = (Golden Rule) rate constant

x 2 1  ErEr For the “backwards” Reaction: k 2  1 =  2 ( )  E r kT ½ e -(E r +  ) 2 /4E r k B T In classical Marcus (Levich-Doganadze) theory, k 1  2 is determined by 3 molecular parameters: , E r,  k 1  2 =  2 ( )  E r kT ½ e -(E r -  ) 2 /4E r k B T w/ E r = “Reorganization Energy” ;  = “Reaction Heat”

To obtain electronic state populations at arbitrary times, solve kinetic [“Master”] Eqns.: dP 1 (t)/dt = - k 1  2 P 1 (t) + k 2  1 P 2 (t) dP 2 (t)/dt = k 1  2 P 1 (t) - k 2  1 P 2 (t) Note that long-time asymptotic [“Equilibrium”] distributions are then given by: K eq = P2()P2() P1()P1() k12k12 k21k21 =e /kBT/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: x VAVA VDVD H = ˆ ˆ ˆ ˆ ˆ hDhD hAhA   +  12 E o cos  o t The forward rate constant is: [Y. Dakhnovskii, J. Chem. Phys. 100, 6492 (1994) k D  A =  2  J m (a) · m=-   dt e i m  o t tr {   Re   0  D ee -i h A t ˆ i h D t } a = 2  12 E o / ħ  o 2

k D  A =  J m (2  12 E o /ħ  o ) m=-   2 ( )  E r kT ½ e -(E r –  + ħm  o ) 2 /4E r k B T 2 2 4 + A  D Rate constants in presence of cw E-field

D A ħoħo 05 x Energy J -1 J0J0 J1J Schematically: In polar electron transfer reactions, for Reorganization Energy E r  ħ  o (the quantum of applied laser field)

a J o (a) J 1 (a) J 2 (a) 2 2 2

Dramatic perturbations of the “one-way” rate constants may be obtained by varying the laser field intensity: [Activationless reaction, E r =1eV] Forward rate constant

Dakhnovskii and RDC showed how this property can be used to control Equilibrium Constants with an applied cw field: “bias”  D A

Results for activationless reaction:  eq = P D (00)P A (00) Y. Dakhnovskii and RDC, J. Chem. Phys. 103, 2908 (1995)

Evans, RDC, Dakhnovskii & Kim, PRL 75, 3649 (95)  = 100 cm -1 Activationless ET E r =  = ħ  o = 1eV

REALITY CHECK on coherent control of mixed valence ET reactions in polar media “  ·E”   (1) 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 E r = ħ  o = 1eV  Electric field  10 7 V/cm Giant dipole ET complex, solvent w/ reduced E r, 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) ħ  0  1eV ; intense fields  (multiphoton) excitation to higher energy states in the ET molecule, which are not considered in the present 2-state model. ħħ ħħ ħħ ħ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: A D Tethered Alkane Chain molecule (Inert) Substrate E k (ħ  )  

Light Absorption by Mixed Valence ET Complexes in Polar Solvents: Ru +2 Ru +3 Solvent ħħ ħħ ħħ ħħ ħħ ħħ Incident Photons Transmitted Photons

Absorption Cross Section Formula:

Marcus Gaussian permanent dipole moment difference Δ

Barrierless Hush Absorption = PRL 77, 2917 (1996) J. Chem. Phys. 105, 9441 (1996)

Stimulated Emission using two incoherent lasers J. Chem. Phys. 109, 691 (1998) absorption emission

J. Chem. Phys. 109, 691 (1998)