Promotion of Tunneling

via Dissipative Molecular Bridges

Uri Peskin

Department of Chemistry,

Technion - Israel Institute of Technology

and

The Lise Meitner Center for Computational

Quantum Chemistry

Introduction•Dissipation, de-coherence and heat production due to electronic-nuclear coupling are inevitable during electron transfer through molecular bridges and wires.

•We study the effects of electronic-nuclear coupling on electronic deep-tunneling in donor-bridge-acceptor molecular complexes.

• The involved many body dynamics associated with generalized spin-boson models, requires high dimensional quantum mechanical tools and is computationally challenging.

•We formulate the entangled electronic-nuclear dynamics beyond the weak electronic-nuclear (system-bath) coupling limit, in terms of summations over vibronic tunneling pathways. For limiting cases of physical (and chemical) interest, exact analytic expressions are obtained for dynamical observables.

Bridge

Donor Acceptor

BT0T0T

0E

BE

The Electronic Model

||E|)||(|ETH n

N

1nnBAADD0elec

ˆˆ

||δTc.c)|||(|TT mn

N

1mn,1mn,BNA1D0

ˆ

]E[E|T||,T| 0BB0

N 120 B

D-AB 0 B 0

T T2ν

h E E E E

The deep tunneling frequency:

Bridge

Donor Acceptor

η

BT0T0T

0E

BE

Introducing Vibronic Coupling

nenucelec HHHH ˆˆˆˆˆˆ

nucH

ω

2 2Nj j

jj 1

P Qω( )

2 2

ˆe nH

ωN N

j j n nj 1 n 1

λ Q | |

Electronically active (accepting) bridge modes:

Structural (promoting) bridge modes:

Not Considered

E

Donor/Acceptor

0BQ

Bridge

0B EE

0( )j BE E

02( )j B j jE E

Harmonic modes with an Ohmic ( ) spectral density/j c

je

Nuclear frequencies 5-500 1/cm - larger than the tunneling frequency!

The Langevin-Schroedinger equation

A non-linear dissipation term

Electronic Population

at the bridgeM. Steinberg and U. Peskin, J. Chem. Phys. 109, 704-710 (1998)

A mean field approach

T=0

Coupled Electronic-Nuclear Dynamics

Simulations: Effect of vibronic coupling

Weak coupling: the tunneling frequency increases!

3.2 , 8,BE eV N 0, 1BT T eV

Strong coupling:the tunneling is suppressed !

Interpretation: time-dependent Hamiltonian

)(ˆˆ)(ˆ1

tFHtH n

N

nneleceff

)()()()(ˆ )()( tUttUtH ll

leff

The Instantaneous electronic energy:

Weak coupling: Dissipation lowers the

barrier

Strong coupling:“Irreversible” electronic

energy dissipation

Resonant Tunneling

( 1) ( 1)ˆ ˆ ˆ ˆN Nelec nuc e nH H H H

1

, 11

[ | 1| . .]N

n nn

T n n c c

, ,2 2 2ˆˆ ˆ( ) | |{ [ ( ) ( ) ]}2j n j n j

n n j jj j j

H n n E P Q

Q

On-site Hamiltonians

Beyond weak electronic-nuclear coupling

,,

n jn j

j

Q

2,

,02( )

j n jn j

n

Q

E E

, 1

0

| |1

| |n n

n

T

E E

1

0

ˆ ( )N

nn

H

Q

Vibronic Tunneling Pathways

Recursive Perturbation Calculation

, 1( ) ( , 1) ( 1)'(0)

',

| , | 1| , 1|N NN N N Ng g

N g

TN N N N

E

l,l

ll

l l'

1, ( ), 1, , | , |eff N N N

D A N N gH T N N 0 l

l

l

The effective tunneling matrix element

, 1

, 0

| |1

| |N N

N

T

E

l

( ) ( 1)

( ) (0)

( ) ( 1) (0) ( )

( , 1), 1 ,( 1, )

, 1, , ,1 0,,

n n

N

N N n

n nNn neff N N

D A N Nn n

TH T

l l

0 l 0 ll l l 0l

M. A.-Hilu and U. Peskin, J. Chem. Phys. 122 (2005).

Promotion of Tunneling:

3.2BE eV 0.125BT eV

0 0.1T eV12000 cm

22 1,0

[ ]2

[ (1 ) ]

NlB

N M MB B

T T

hE E l

0

l

•Lower barrier for tunneling

•Multiple “Dissipative” pathways

(1 )B BE E

•Frank Condon integrals

The “slow electron” “adiabatic” limit: 2 1

02

(1 )

BENB

ad N MB

T T e

hE

2 102 N

BRigid N

B

T T

hE

(1 ) /BM E 0ad

Condition for tunneling promotion:

“Site-directed” Electronic Tunneling

1 1

, '0 , ' 0

ˆ ˆ ˆN N

n n nn n n

H H H

Bridges are perturbations

1 1,2 1,3

2,1 2 2,3

3,1 3,2 3

H H H

H H H

H H H

A reduced N-level system

A Linear D-A1-A2 Complex

0 0 0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0 0 0

B

B

B

e

B

B

B

T

T E t

t E t

t E T

T T

T E t

t E t

t E T

T

H

Contact

2 2

2

2 2 2 2 2

2 2 3

2 2 2

3

0

2 (1 )

0

B B B

B B B B B

B B

T TTt

E E E

TTt T t T t

E E E E E

T t T

E E

H

The reduced matrix Hamiltonianin the deep tunneling regime:

, , ,B BT T t E E

Site Directing in a D-A1-A2 Complex

0 0 0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0 0 0

B

B

B

e

B

B

B

T

T E t

t E t

t E T

T T

T E t

t E t

t E T

T

H

2 2

2B B

T T

E E

2 2

B B

T T

E E

DA2 DA1

Site Directing by e-n Coupling

3.2B BE E eV 0.97t eV

0.85T T eV 1500cm

0 0.4

0.8

DA2 DA1DD

1.0 0 2.0

A single mode:

An Ohmic bath:

,

,

,

,

,2 3.6BE

,

,4 2.8BE

,

,5 2.4BE

,0 ,41.0691 /B BT T T E E

.

Site directing in a multi-acceptor network

,3 3.2BE

,0 3.2BE

Tunneling to a selected electronic site

Summary and Conclusions

• A rigorous approach was introduced for calculations of electronic tunneling frequencies beyond the weak electronic-nuclear coupling, predicting acceleration by orders of magnitudes in the realistic regime of molecular parameters

• Off-resonant (deep) tunneling (super-exchange) in long-range electron transfer through molecular barriers was studied.

• Simulations of the coupled electronic-nuclear dynamics suggest that a pollaronic effect at weak electronic–nuclear coupling promotes off-resonant tunneling through molecules.

• A generalized McConnell model was introduced for studying the role of electronic-nuclear coupling at bridges in molecular Donor-Bridge-Acceptor complexes.

• Site directed tunneling was demonstrated in models of molecular networks. The rigorous formulation would enable to predict the effect of electronic nuclear coupling on site-directed tunneling in such complex networks.