This journal is c the Owner Societies 2012 Phys. Chem. Chem. Phys., 2012, 14, 13835–13840 13835

Cite this: Phys. Chem. Chem. Phys., 2012, 14, 13835–13840

Ballistic charge transport through bio-molecules in a dissipative

environment

Daria Brisker-Klaiman and Uri Peskin*

Received 13th April 2012, Accepted 28th June 2012

DOI: 10.1039/c2cp41197k

The question whether dissipative bio-molecular systems can support efficient coherent

(phase-conserving) charge transport is raised again following recent experiments on electron-

energy transfer in bio-molecules. In this work we formulate conditions under which the current

due to coherent ballistic resonant charge transport through DNA molecular junctions can be

measured in spite of coupling to a dissipative environment.

The rational design of single molecule electronic devices

requires the ability to control charge flow through molecules.

Understanding the charge transport mechanism is paramount

in order to achieve such control. In a molecular junction setup,

a molecule is placed between two macroscopic leads and the

current is measured as a function of the applied voltage.1,2

Classical mechanics is often inadequate to account for the

transport mechanism as the current may manifest quantum

mechanical phenomena such as coherent (phase preserving)

transport through the molecule,3 including, e.g., tunneling4

and interference effects.5 However, coupling of the moving

charges to the vibrational degrees of freedom of the molecule

and the surrounding medium results in inelastic scattering and

de-phasing,6 so that coherent elastic (i.e., ballistic) transport is

often obscured. Recently, it was demonstrated that in contrast

to this general perception electron energy transfer in a bio-

molecular system maintains coherence on the atomic motion

time scale, in spite of coupling to a dissipative molecular

environment.7,8 This intriguing finding suggests that coherent

transport in natural systems may be underemphasized.

The study of quantum transport in bio-molecules is indeed

enlightening in this context. DNA for example is a bio-molecule

with a double stranded arrangement of nucleo-bases and has

versatile charge transport (CT) properties.9–11 Most experiments

which aim to identify the CT mechanism in DNA focus on the

number of base pairs,12–16 or on the dependence on the

temperature.17,18 Past experiments and theoretical considera-

tions shed light on the regimes in which the transport is coherent

(typically in the off-resonant, tunneling regime) or kinetic

(hopping) and on the tunneling-to-hopping transition,19–24 by

considering a homological series of molecules at different temp-

eratures. Nevertheless, it is still an open challenge to reveal the

existence of coherent ballistic resonant transport in DNA,

considering the coupling to the dissipative molecular environment.

Recently,25 a new experimental setup was introduced in

which the connection strategy between a double stranded

DNA structure and the electrodes in a junction was controlled.

In this work we propose that for appropriate DNA sequences

the connection strategy can be chosen to selectively block

charge transport channels associated with dynamical disorder

due to nuclear motion, and thus the measured currents reflect

ballistic, phase conserving, charge transport.

Our model for hole transport through DNA is based on a

tight-binding ladder molecular Hamiltonian, used in our ear-

lier work26 on coherent elastic transport in ordered DNA

sequences. The model takes explicit account of the double

stranded nature of the structure, beyond the 1D sequence of

base pairs. This level of detail is often unnecessary for simulat-

ing transport through DNA22,27 but it is essential for modeling

different connection strategies between the four terminals of

the double stranded structure and the electrodes.25,26 The

model parameterization is based on the work by Voityuk

et al.28–30 for the on-site hole energies and hopping integrals:

HM ¼X2Nn¼1

endn ydn þXN�1n¼1

an;nþ1dn ydnþ1 þ h:c:

( )

þX2N�1

n¼Nþ1fan;nþ1dn ydnþ1 þ h:c:g

þXNn¼1fbndn ydnþN þ h:c:g:

ð1Þ

The operators dnyðdnÞ represent a creation (annihilation) of a

hole at the nth molecular site. It is convenient to rewrite the

molecular Hamiltonian in terms of the creation ðam yÞ and

annihilation (am) operators of a single hole in the mth mole-

cular orbital (MO), HM ¼P2Nm¼1

emam yam. Each (many body)

molecular eigenvector is then associated with a unique set of

hole occupation numbers {nm} (nm A 0,1) in the MOs.Schulich Faculty of Chemistry, Technion – Israel Institute ofTechnology, Haifa 32000, Israel. E-mail: uri@tx.technion.ac.il

PCCP Dynamic Article Links

www.rsc.org/pccp PAPER

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13836 Phys. Chem. Chem. Phys., 2012, 14, 13835–13840 This journal is c the Owner Societies 2012

In the molecular junction scenario the molecule is coupled

to two charge (electronic) reservoirs, which enables flow of

holes into and out of the molecule. The coupling between the

system and the reservoirs is introduced here by the leads

Hamiltonian,33 Hleads = HR + HL; HJ ¼PjJ

ejJ bjJybjJþ

PjJ

xjJ bjJyP

n

ln;Jdn þ h:c:

( ), where bjJ

yðbjJ Þ is the creation

(annihilation) operator of a hole at the jth orbital of the Jth

electrode, and dn is the annihilation operator of a hole at the

nth nucleobase site. The connection strategy between the

double stranded structure and the two leads is captured in a

vector {ln,J} which obtains the value 1 when the nth site is

coupled to the Jth electrode and zero otherwise.

Nuclear vibrations of the molecule and its environment are

modeled in terms of interactions with collections of harmonic

modes. The charge at each single base (site) is coupled to a set

of local vibrations associated with a local nuclear Hamiltonian

(Hn), where the equilibrium position of the jn vibration mode

depends on the hole population at the respective nth base.

We emphasize that within a time-dependent framework as

practiced in quantum-classical schemes,22,31 this form of

coupling is sufficient to introduce uncorrelated fluctuations

in the on-site (local) hole energies along the molecule. The

corresponding nuclear Hamiltonian reads, Hnuc ¼Pn

Hn;

Hn¼PNn

jn

�hojnðcjn ycjn þ 12ÞþPNn

jn

Zjnffiffi2p ðcjn yþcjnÞdn ydn, where cjn

yðcjnÞ

are the creation (annihilation) operators of a vibration

quantum at the jth nuclear mode associated with the nth

nuclear bath, and {Zjn} are microscopic vibronic coupling

parameters. The full Hamiltonian then takes the form,

H = HM + Hleads + Hnuc. (2)

In the limit of weak coupling each one of the electrodes

(hole reservoirs) and the local nuclear environments (phonon

reservoirs) maintains a quasi-equilibrium density. Invoking a

Markovian second order approximation in the system–

reservoirs coupling, and accounting for rapid de-phasing

(decay of coherences) between molecular eigenstates,32,33 the

time evolution of the molecular (reduced) density matrix in the

presence of interaction with the reservoirs is cast into popula-

tion transfer rates between the electronic eigenstates of

the molecular system. Denoting the hole population at the

mth eigenstate as Pm(t), the following set of equations is

obtained:34

@

@tPmðtÞ ¼

XJ2R;L

½kJ ðeleÞ�m;m0 þX2Nn¼1½kn ðnucÞ�m;m0

!Pm0 ðtÞ:

ð3Þ

The rate constants for electrode-induced molecular transitions

are given by33

½kJ ðeleÞ�m;m0 ¼ ð1� dm;m0 ÞðGJ;em;m0 þ GJ;h

m0 ;mÞ

� dm;m0Xm00am

GJ;em00;m þ GJ;h

m;m00

� �;

ð4Þ

where GJ;h=em;m0 ¼

Pn

ln;Jhmjdnjm0i����

����2

JJðEm0 � EmÞfJ ðh=eÞðEm0�EmÞ=�h are single hole hopping rates out of or into the

molecule. These rates depend on the voltage via the Fermi

occupation numbers for holes at the two electrodes,

fJðhÞðEÞ � 1

1þ eðE�mJ Þ=KBT; fJ

ðeÞðEÞ � 1� fJðhÞðEÞ

and on the microscopic coupling parameters, fxjJ 2g, via the

electrode conductance band spectral density.33

The rate constants for nuclear-induced molecular transi-

tions are given by34

½kn ðnucÞ�m;m0 ¼ ð1� dm;m0 ÞðGn;emm;m0 þ Gn;ab

m0;mÞ

� dm;m0Xm00am

ðGn;emm00 ;m þ Gn;ab

m;m00 Þ;ð5Þ

where Gn;em=abm;m0 ¼ jhmjdn ydnjm0ij2JnðEm0 � EmÞgðem=abÞðEm0 �

EmÞ=�h are rates of phonon emission and absorption during

the respective molecular transitions. These rates are related to

the phonon thermal occupation factors at the respective

nuclear reservoir, gðabÞð�hoÞ ¼ 1e�ho=kBT�1; gðemÞð�hoÞ ¼ e�ho=kBT

e�ho=kBT�1,

and they depend on the microscopic vibronic coupling para-

meters fZjn 2g via the nuclear bath spectral density.

Transient currents35 are associated with the net rate of hole

transitions from the left electrode into the molecule. The

steady state current is associated with the infinite time limit,33

and depends explicitly on the steady state populations of the

molecular eigenstates (coherences, if present, do not appear in

the current formula), i.e.

IL!R ¼ limt!1

Xm;m0

2e½kL ðeleÞ�m;m0Pm0 ðtÞNm; ð6Þ

where Nm ¼P2Nn¼1

dnm ;1 is the hole occupation number at the mth

eigenstate. Notice that the infinite time limit assures that the

entire frequency band of the dynamical fluctuations is

accounted for in the current calculation.22

Two different DNA models were considered in detail and

are shown in Fig. 1. Both contain 4 base pairs and share the

same ‘‘edge’’ base pairs. Two connection strategies of the

double strand to the electrodes were considered, a linear and

a diagonal connection, as depicted in Fig. 2.

Fig. 1 Two selected DNA sequences. The DNA is represented by a

tight binding model Hamiltonian, where each DNA base is associated

with its own site energy and both intra- and inter-strand coupling are

accounted for. The parameters (in eV units) were taken from ref. 28–30.

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I–V curves were calculated for hole chemical potentials,

mL = m0 + eF/2, mR = m0 � eF/2, where m0 = 7 eV. The

electrodes were introduced using a semi-elliptical model,33

JJðEÞ ¼ 2pPjJ

xjJ2dðE � ejJ Þ ffi

xJ 2

gJ 2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4gjJ

2 � ðE � mJÞ2q

, with a

conductance band width parameter, gJ = 5 eV, and a coupling

parameter xJ = 0.02 eV. The nuclear reservoirs were modeled

as identical Ohmic baths36 with a cutoff frequency, �hoC =

0.25 eV, i.e., Jnð�hoÞ ¼ 2pPjn

Zjn2dð�ho� �hojnÞ ffi 2pZ2

�hoC2 oe�o=oC

for o > 0, and zero otherwise. We start by setting the

reservoirs temperature to zero (a numerical value of 1 �10�6 K was used). This choice fixes the direction of energy

flow from the system into the bath. Energy flow into the

molecular system, following thermal activation of low

frequency modes (�ho { kBT), is therefore explicitly blocked

(finite temperatures are addressed later below).

The present calculations rely on a second order approxi-

mation in the coupling to the electronic reservoirs. In the

experiment, the coupling strength is controlled by the linking

group between the (carbon nanotube) electrodes and the DNA

strand. The presence of long linkers and the experimentally

measured currents in the 10–100 nA regime justify the weak

coupling assumption. To ensure consistency of our perturba-

tive treatment, the calculated currents in the absence of

electronic–nuclear coupling were reproduced by numerically

exact (non-perturbative) calculations based on Landauer’s

elastic transport equation.3,33

The vibronic coupling was restricted to the weak coupling

limit by ensuring that Z { �hoC. The present system–bath

treatment excludes effects of strong vibronic coupling, such as

transport through vibronic pathways37 or vibrationally

induced coherences. If the electronic transport is strongly

coupled to particular nuclear motions (e.g., low frequency,

anharmonic modes) then the treatment can be extended to

include the relevant nuclear degrees of freedom within the

molecular (system) Hamiltonian such that the system eigen-

states are vibronic states.

In Fig. 3 I–V curves are plotted for sequence A. The current

through the direct connection vanishes in the absence of vibronic

coupling, and increases dramatically (compare plots b, c to

plot a), as this coupling term is turned on. The current through

the diagonal connection seems to be unaffected by the introduc-

tion of vibronic coupling to the dissipative nuclear environment.

The same qualitative trend is observed for sequence B, as

presented in Fig. 4. The current through the diagonal connection

is insensitive to the change in the vibronic coupling parameters,

while the current through the direct connection increases with

increasing coupling to the dissipative baths.

To rationalize these results let us first analyze the case of

purely elastic transport, for Z = 0. For weak molecule–

electrodes coupling as considered here, the current includes

additive contributions of transport through single MOs.26,33

Given the large differences in the on-site energies between

different DNA bases in comparison to the electronic couplings

(see Fig. 1), each MO primarily populates DNA bases of the

same type. The molecular symmetry of sequences A and B

implies that the MOs are delocalized among the bases, i.e.,

there are MOs populating the guanine bases that will be

referred to as ‘‘G type’’ MOs, and similarly, there are ‘‘C

type’’, ‘‘A type’’ and ‘‘T type’’ MOs. Each orbital contributes

to the current according to its ‘‘reduced contact probability’’,26,33

i.e. the properly weighted amplitudes on the two contact sites

to the electrodes. Therefore, the current is pronounced in the

diagonal connection strategy, where the two electrodes

are connected to ‘‘C type’’ orbitals. In contrast, the current

in the linear connection strategy is vanishingly small since

none of the MOs are delocalized over both C’s and G’s, which

are coupled to the left and to the right electrodes, respectively

(see Fig. 3a and 4a).

Fig. 2 Two ways of connecting the DNA double strand to the leads

were considered. (i) Linear connection strategy where the upper strand

is connected to the leads at both ends. (ii) Diagonal connection strategy,

where the two strands are connected to the leads at one of their ends.

Fig. 3 Current as a function of bias voltage for sequence (A), in the diagonal (dashed blue) and direct (solid red) connection strategies. The rise in

the currents reflects the onset of resonant charge transmission through the molecular orbitals. The threshold voltage is controlled by the gap

between the Fermi level of the electrodes and the molecular band. The left, middle and right panels correspond to the vibronic coupling

parameters, Z= 0, 0.1, 0.1 eV, respectively, where the middle and right panels are associated with full Ohmic spectral density and a filtered density

at the C-to-G inter-band transition frequencies (1.2 o �ho o 1.3 eV), respectively. The other parameters are: kBT = 0, oC = 0.25 eV, m0 = 7 eV,

gJ = 5 eV, xJ = 0.02 eV.

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Additional (inelastic) transport channels become available

for Z > 0, corresponding to incoherent hopping between MOs

(‘‘MO-Hopping’’, see Appendix A). For example, the charge is

injected from the left electrode into one orbital, and leaves to

the right electrode from another one at a different MO energy.

These processes do not depend on each orbital being deloca-

lized over the two contact bases. Rather, the first orbital

should have significant amplitude at the contact to the source

electrode, and the other at the contact to the drain. The

requirements for significant inelastic rates (eqn (5)) define

the necessary conditions for transport by MO-hopping.

Theoretical analysis of the rates (see Appendix A) in the low

temperature limit shows that efficient inelastic transport

requires: (i) strong coupling of one orbital to the source lead

and another orbital to the drain. (ii) The orbital coupled to the

drain should be lower in energy than the orbital coupled to the

source. (iii) The two MOs should have a significant ‘‘spatial

overlap’’ between them. (iv) The spectral density of the nuclear

bath should be significant at the MOs transition frequency.

The connections between the double stranded structure and

the leads control the validity of the first necessary condition

for incoherent inelastic transport (see Fig. 1 and 2). The

diagonal connection strategy couples both the source and

the drain to ‘‘C-type’’ orbitals, so that the first requirement

for transport by inelastic MO-hopping does not hold. Indeed,

the onset of inelastic transitions (Z > 0) hardly changes the

calculated currents for this connection strategy (see Fig. 3 and 4).

This is our central result. The coupling to the dissipative

nuclear baths does not seem to affect the current, in spite of

the possibility for inelastic MO-hopping from the ‘‘C-type’’ to

lower energy orbitals. This is understood considering that the

‘‘non-C-type’’ orbitals are disconnected from the electrodes,

and charge accumulation in them effectively blocks any

inelastic transition. Therefore, the observed current in the case

of diagonal connection is due to purely elastic ballistic trans-

port in the molecule.

In contrast, a linear connection strategy implies that the

source and drain electrodes are coupled primarily to ‘‘C-type’’

and ‘‘G-type’’ orbitals, respectively, so that condition (i) is

valid. Considering the respective orbital energies, the spatial

overlap between the orbitals (the proximity between the C and

G bases) and the broad frequency range of the nuclear bath,

inelastic transport should be efficient for the linear connection

strategy. This is indeed reflected in the increase in the currents

for this connection strategy with the onset of the vibronic

coupling, for Z > 0 (see Fig. 3 and 4).

The nuclear bath-induced transport in the direct connection

reveals an apparent difference between the two sequences,

where for sequence A the current is larger, to the extent that

the (inelastic) current through the direct connection exceeds

the (elastic) current through the diagonal one. The difference

between the two sequences is attributed to the difference in

their inelastic MO-hopping frequencies. In sequence A there is

a wealth of inelastic MO-hopping pathways. The ‘‘C-type’’ to

‘‘G-type’’ transition can be either direct or broken into

sequences of inelastic hops, e.g., ‘‘C-type’’ -‘‘A-type’’ -

‘‘T-type’’ - ‘‘G-type’’. In contrast, only the direct transition

exists for sequence B. Considering that the efficiency of inelastic

transitions increases with the bath spectral density at the

respective transition frequency, the direct transition is less

efficient since the relatively large frequency is at the decaying

tail of the bath spectral density, while the indirect transitions are

at lower, and thus more abundant, frequencies. Hence, by

choosing the DNA sequence one controls the relative contribu-

tion of the inelastic transport mechanism. The importance of

matching between the bath spectral density and the electronic

transition frequencies is demonstrated in Fig. 3c and 4c, where

the Ohmic bath spectrum was filtered to include only the direct

‘‘C-type’’ - ‘‘G-type’’ inter-band transition frequencies. The

inelastic current through sequence B is shown to be hardly

affected, while the inelastic current through sequence A drops

significantly in the absence of the indirect, ‘‘C-type’’ -

‘‘A-type’’ - ‘‘T-type’’ - ‘‘G-type’’, transitions. Note that

experimental measurements of the currents and a comparison

between the measured currents at the two different connection

strategies can point to the relative strength of vibronic coupling

in the different sequences. Using the current in the diagonal

connection strategy (elastic transport) as a reference for each

sequence, the current intensity in the linear connection strategy

reflects the relative efficiency of inelastic transport.

Fig. 4 Current as a function of bias voltage for sequence (B), in the diagonal (dashed blue) and direct (solid red) connection strategies. The rise in

the currents reflects the onset of resonant charge transmission through the molecular orbitals. The threshold voltage is controlled by the gap

between the Fermi level of the electrodes and the molecular band. The left, middle and right panels correspond to the vibronic coupling

parameters, Z= 0, 0.1, 0.1 eV, respectively, where the middle and right panels are associated with full Ohmic spectral density and a filtered density

at the C-to-G inter-band transition frequencies (1.2 o �ho o 1.3 eV), respectively. The other parameters are: kBT = 0, oC = 0.25 eV, m0 = 7 eV,

gJ = 5 eV, xJ = 0.02 eV.

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Considering the presence of inelastic transition (Z > 0), it is

interesting to inspect the effect of finite temperatures on the

currents, as depicted in Fig. 5. In the diagonal connection

strategy, the currents (attributed to ballistic transport) hardly

change over the entire range from zero to room temperature.

As analyzed above, the transport is dominated in this case by

delocalized orbitals with significant contacts to the two elec-

trodes, and inelastic transitions to other orbitals which are

disconnected from the electrodes would be effectively blocked

due to charge accumulation in such orbitals. In the direct

connection strategy the dominant transport mechanism is

inelastic, and increasing the temperature indeed changes the

current, but the change is significant only for sequence B.

Notice that inter-band (e.g., C to G) transitions are associated

with Em0 � Em c kBT even at room temperature, and are not

activated. The temperature effect on the transport is therefore

restricted to intra-band transitions. Analysis of the G band of

sequence B reveals that orbitals with larger contact to the right

electrode are higher in energy. This implies that thermal

activation of intra-band transitions opens additional inelastic

transport channels in this case, and the current increases with

increasing temperature.

Finally, let us address possible effects of changing the length

of the molecule on the transport mechanism as well as on the

selectivity of the current to the connection strategy. In Fig. 6

the calculated current is plotted for a homological series of

(CG)n sequences. For even n values the calculated currents are

nearly invariant to the length. In view of the common sym-

metry, the above analysis of sequence B (n = 4) applies also

for n= 2,6. For odd n values different symmetry of the double

stranded structure is reflected in different calculated currents,

and different dependence on the connection strategy. In this

case, the diagonal connection strategy implies that one elec-

trode is coupled to C while the other to G, and the direct

connection strategy implies that both electrodes are coupled to

C. One might thus expect to see enhancement of the current in

the direct (and not in the diagonal) connection strategy for

odd n values. However, the change in the molecular symmetry

from the n = 2,4,6 to the n = 3,5 cases turns out to be

associated with localization of the C band orbitals and there-

fore resonant ballistic transport through the C-type orbitals is

less efficient for odd n values.

Notice that even though the present calculations suggest that

the length dependence is minor for sequences of the same

symmetry, in practice sequences of increasing length are more

prone to have defects, which should induce length dependence of

the resonant transport even if the basic symmetry is preserved.

In summary, our model studies of resonant charge transport

through DNA sequences suggest that the transport mecha-

nism is controlled by the connection of the bio-network to the

electrodes. Considering both coherent elastic and incoherent

inelastic transport pathways on the molecules, we demonstrate

that for appropriate DNA sequences and connection strategy,

inelastic processes should be effectively blocked, and the

coherent ballistic transport is revealed even in the presence

of coupling to the dissipative molecular environment.

Appendix A: inelastic MO-hopping rates

Considering the rate constants for nuclear-induced molecular

transitions, Gn;em=abm;m0 ¼ jhmjdn ydnjm0ij2JnðEm0 � EmÞgðem=abÞ

ðEm0 � EmÞ=�h, in the low temperature limit (�ho { kBT),

phonons emission processes are favored over phonon absorp-

tion. This amounts to setting the respective phonon occupa-

tion factors, g(em)(�ho) E 1 and g(ab)(�ho) E 0 in the rate

expression. The overall rate of an inelastic transition from a

system eigenstate, |m0i, to another eigenstate, |mi, reads in

this case,

Km;m0 �X2Nn¼1½kn ðnucÞ�m;m0 ¼

X2Nn¼1

Gn;emm;m0

¼X2Nn¼1

1

�hjhmjdn ydnjm0ij2JnðEm0 � EmÞ;

ðA1Þ

Fig. 5 Calculated current as a function of temperature. (a) and (b)

correspond to sequences A and B, respectively (see Fig. 1). In each plot

dashed blue and solid red refer to diagonal and linear connection

strategies, respectively. Notice the negligible temperature-dependence

of the currents in the diagonal connection strategy, attributed to

resonant ballistic transport through the C band. The calculations are

for a full Ohmic spectral density, with the parameters Z = 0.1 eV,

eF = 4 eV, �hoC = 0.25 eV, m0 = 7 eV, gJ = 5 eV, xJ = 0.02 eV.

Fig. 6 Calculated current as a function of molecular length (n) for a

poly-CG sequence (CG)n. Dashed blue and solid red refer to diagonal

and linear connection strategies respectively. Notice the negligible

length-dependence of the currents for n = 2,4,6. The calculations

are for a full Ohmic spectral density with the parameters eF = 4 eV,

kBT=0, Z=0.1 eV, �hoC= 0.25 eV, m0= 7 eV, gJ=5 eV, xJ=0.02 eV.

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where Em0 > Em. Using the expansion of single hole molecular

orbitals (MOs) in the local sites basis, amy �

P2Nn¼1

un;ldny, the

electronic coupling term at the nth site can be expressed in

terms of the MOs creation and annihilation operators, i.e.,

hmjdn ydnjm0i ¼Pk;l

u�n;lun;khmjal yakjm0i. It follows that the rate

vanishes unless the two eigenstates, |m0i and |mi, are identicalexcept for the (hole) occupation in precisely two of the orbitals,

one of which is occupied only at the mth state while the other is

only occupied at them0th state. Denoting these orbital indexes as

lm and km0 respectively, it follows that hmjdn ydnjm0i¼ u�n;lmun;km0 .

A non-vanishing transition between the many-body states |m0iand |mi would therefore involve a single ‘‘MO Hopping’’ event,

and the corresponding hopping rate reads,

Km;m0 ¼X2Nn¼1

1

�hjun;lm j

2jun;km0 j2Jnðekm0 � elmÞ; ðA2Þ

where ekm0 and elm are the respective orbital energies, and an

emission process implies ekm0 4 elm . For further simplicity, we

assume that the spectral density of the nuclear mode does not

depend on the particular site index (n) (implying that the nuclear

environment is the same for all the bases sites). In this case the

MO-hopping rates become,

Km;m0 ¼ Skm0 ;lmJðekm0 � elmÞ

Skm0 ;lm �X2Nn¼1jun;lm j

2jun;km0 j2:

ðA3Þ

The rates are shown to depend on two terms. The first of which is

a generalized ‘‘spatial overlap’’ between the probability densities

(not amplitudes!) of the two orbitals. The second term is the

spectral density of the phonon modes, which must be significant

at the required frequency for the particular electronic transition.

Acknowledgements

This research was supported by the Israel Science Foundation,

the US–Israel Binational Science foundation and by the fund

for promotion of research at the Technion – Israel Institute of

Technology. DB acknowledges the Schulich scholarship.

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