Electronic Tunneling through Dissipative Molecular Bridges Uri Peskin

Electronic Tunneling through Dissipative Molecular Bridges Uri Peskin Department of Chemistry, Technion - Israel Institute of Technology Thanking: Musa Abu-Hilu (Technion) Alon Malka (Technion) Chen Ambor (Technion) Maytal Caspari (Technion) Roi Volkovich (Technion) Darya Brisker (Technion) Vika Koberinski (Technion) Prof. Shammai Speiser (Technion)

Outline Motivation: • Controlled electron transport in molecular devices and in biological systems. Background: • ET in Donor-Acceptor complexes: The Golden Rule, the Condon approximaton and the spin-boson Hamiltonian. • ET in Donor-Bridge-Acceptor complexes: Mc. Connell’s formula for the tunneling matrix elements. The problem: • Electronic-nuclear coupling at the molecular bridge and the breakdown of the Condon approximation. The model system: • Generalized spin-boson Hamiltonians for dissipative through-bridge tunneling. Results: • The weak coupling limit: Langevin-Schroedinger formulation, simulations and interpretation of ET through a dissipative bridge • Beyond the weak coupling limit: An analytic formula for the tunneling matrix element in the deep tunneling regime. Conclusions: • Promotion of tunneling through molecular barriers by electronicnuclear coupling. • The effect of molecular rigidity.

Motivation: Electron Transport Through Molecules Molecular Electronics Tans, Devoret, Thess, Smally, Geerligs, Dekker, Nature (1997) Reichert, Ochs, Beckmann, Weber, Mayor, Lohneysen, Phys. Rev. Lett. (2002). Resonant tunneling through molecular junctions

Long-range Electron Transport In Nature The Photosynthetic Reaction Center Tunneling pathway between cytochrome b 5 and methaemoglobin Electron transfer is controlled by molecular bridges Deep (off-resonant) tunneling through molecular barriers

Controlled tunneling through molecules? Resonant tunneling Deep (off resonant) tunneling Why Off-Resonant (deep) Tunneling ? • Minor changes to the molecular electronic density • High sensitivity (exponential) to the molecular parameters • A potential for a rational design based on chemical knowledge

Electron Transfer in Donor-Acceptor Pairs Donor Electronic tunneling matrix element Acceptor Nuclear factor: Frank-Condon weighted density of states The case of through bridge tunneling : The role of electronic nuclear coupling?

Theory: Electron Transfer in Donor-Acceptor Pairs The electronic Hamiltonian: Diabatic electronic basis functions: The Hamiltonian matrix:

Theory: Electron Transfer in Donor-Acceptor Pairs The Harmonic approximation: A Spin Boson Hamiltonian:

Theory: Electron Transfer in Donor-Acceptor Pairs Donor The golden rule expression for the rate Acceptor The Condon approximation A nuclear factor An electronic tunneling matrix element

Long Range Electronic Tunneling Donor Acceptor The direct tunneling matrix element vanishes Mc. Connell (1961): Introducing a set of bridge electronic states; The donor and acceptor sites are connected via an effective tunneling matrix element through the bridge

Mc. Connell’s Formula: A tight binding model The deep tunneling regime: First order perturbation theory A simple expression for the effective tunneling matrix element

Superexchange dynamics through a symmetric uniform bridge Tunneling oscillations at a frequency : H. M. Mc. Connell, J. Chem. Phys. 35, 508 (1961)

Deep tunneling through a molecular bridge • The role of bridge nuclear modes? • Validity of the Condon approximation?

Electronic nuclear coupling at the bridge: Breakdown of the Condon approximation! Molecules 1 -5 Charge transfer is gated by bridge vibrations Davis, Ratner and Wasielewski (J. A. C. S. 2001). Rigid bridges enable highly efficient electron energy transfer Lokan, Paddon-Row, Smith, La Rosa, Ghiggino and Speiser (J. A. C. S. 2001).

Structural (promoting) bridge modes: Electronically active (accepting) bridge modes:

A generalized “spin-boson” model: • Harmonic nuclear modes • Linear e-nuclear coupling in the bridge modes • The e-nuclear coupling is restricted to the bridge sites The nuclear potential energy surface changes at the bridge electronic sites

A Dissipative Superexchange Model: A symmetric uniform bridge Introducing nuclear modes with an Ohmic ( ) spectral density and a uniform electronic-nuclear coupling : The nuclear frequencies: 10 -500 (1/cm) are larger than the tunneling frequency!! M. A-Hilu and U. Peskin, Chem. Phys. 296, 231 (2004).

Coupled Electronic-Nuclear Dynamics A mean-field approximation: The coupled SCF equations: Mean-fields:

The Langevin-Schroedinger equation Electronic bridge population A non-linear, non Markovian dissipation term Fluctuations Initial nuclear position and momentum At zero temperature, R(t) vanishes U. Peskin and M. Steinberg, J. Chem. Phys. 109, 704 (1998).

Numerical Simulations: Weak e-n coupling The tunneling frequency increases!

Simulations: Strong e-n Coupling The tunneling is suppressed !

Interpretation: a time-dependent Hamiltonian The Instantaneous electronic energy: Weak coupling: Energy dissipation into nuclear vibrations lowers the barrier for electronic tunneling A time-dependent Mc. Connell formula

Interpretation: a time-dependent Hamiltonian The Instantaneous electronic energy: Resonant Tunneling Weak coupling: Energy dissipation into nuclear vibrations lowers the barrier for electronic tunneling Strong coupling: “Irreversible” electronic energy dissipation

Numerically exact simulations for a single bridge mode • Tunneling acceleration • Tunneling suppression at strong coupling at weak coupling

A dissipative-acceptor model: The acceptor population: Dissipation leads to a unidirectional ET A. Malka and U. Peskin, Isr. J. Chem. (2004). Introducing a bridge mode The tunneling rate Increases with e-n coupling at the bridge!

Interpretation: Nuclear potential energy surfaces A dimensionless measure for the effective electronic-nuclear coupling:

Entangled electronic-nuclear dynamics beyond the weak coupling limit The symmetric uniform bridge model: Deep tunneling = weak electronic inter-site coupling A small parameter: M. A. -Hilu and U. Peskin, submitted for publication (2004).

A Rigorous Formulation The Donor/Acceptor Hamiltonian The Bridge Hamiltonian The coupling Hamiltonian (purely electronic!)

Introducing vibrational eigenstates: Diagonalizing the tight-binding operator:

In the absence of electronic coupling the ground state is degenerate: Regarding the electronic coupling as a (second order) perturbation The energy splitting temperature reads: Frank-Condon overlap factors

The energy splitting: Expanding the denominators in powers of and keeping the leading non vanishing terms gives

Interpretation: Mc. Connell’s expression: Effective barrier for tunneling Effective electronic coupling

Summation over vibronic tunneling pathways: • Lower barrier for tunneling • Multiple “Dissipative” pathways The effective tunneling barrier decreases

An example (N=8) 1/cm The tunneling frequency increases by orders of magnitude with increasing electronic nuclear coupling

The “slow electron” adiabatic limit Considering only the ground nuclear vibrational state: A condition for increasing the tunneling frequency by increasing electronic-nuclear coupling:

An example (N=8) The slow electron approximation

Flexible vs. Rigid molecular bridges Molecular rigidity = small deviations from equilibrium configuration Spectral densities Increasing rigidity A consistency constraint:

Langevin-Schroedinger simulations: The tunneling frequency increases with bridge rigidity

A rigorous treatment: The “slow electron” limit Rigidity = larger Frank Condon factor!

Summary and Conclusions • The effect of electronic-nuclear coupling in electronically active molecular bridges was studied using generalized Mc. Connell models including bridge vibrations. • Mean-field Langevin-Schroedinger simulations of the coupled electronic-nuclear dynamics suggest that weak electronic– nuclear coupling promotes off-resonant (deep) through bridge tunneling • A rigorous calculation of electronic tunneling frequencies beyond the weak electronic-nuclear coupling limit, predicts acceleration by orders of magnitudes for some molecular parameters • An analytical approach was introduced and a formula was derived for calculations of tunneling matrix elements in a dissipative Mc. Connell model. A comparison with approximate methods for studying open quantum systems is suggested. • The way for rationally designed, controlled electron transport in “molecular devices” is still long…