ElectronicTunnelingthroughDissipativeMolecularBridgesUriPeskin

DepartmentofChemistry,Technion-IsraelInstituteofTechnologyMusaAbu-Hilu(Technion)AlonMalka(Technion)ChenAmbor(Technion)MaytalCaspari(Technion)RoiVolkovich(Technion)DaryaBrisker(Technion)VikaKoberinski(Technion)Prof.ShammaiSpeiser(Technion)Thanking:OutlineMotivation:

Controlledelectrontransportinmoleculardevicesandinbiologicalsystems.Background:ETinDonor-Acceptorcomplexes:TheGoldenRule,theCondonapproximatonandthespin-bosonHamiltonian.ETinDonor-Bridge-Acceptorcomplexes:McConnell’sformulaforthetunnelingmatrixelements.Theproblem:Electronic-nuclearcouplingatthemolecularbridgeandthebreakdownoftheCondonapproximation.Themodelsystem:Generalizedspin-bosonHamiltoniansfordissipativethrough-bridgetunneling.Results:Theweakcouplinglimit:Langevin-Schroedingerformulation,simulationsandinterpretationofETthroughadissipativebridgeBeyondtheweakcouplinglimit:Ananalyticformulaforthetunnelingmatrixelementinthedeeptunnelingregime.Conclusions:Promotionoftunnelingthroughmolecularbarriersbyelectronic-nuclearcoupling.Theeffectofmolecularrigidity.Motivation:ElectronTransportThroughMoleculesMolecularElectronicsResonanttunnelingthroughmolecularjunctions

Tans,Devoret,Thess,Smally,Geerligs,Dekker,Nature(2019)Reichert,Ochs,Beckmann,Weber,Mayor,Lohneysen,Phys.Rev.Lett.(2019).

Long-rangeElectronTransportInNatureThePhotosyntheticReactionCenterDeep(off-resonant)tunnelingthroughmolecularbarriers

Electrontransferiscontrolledbymolecularbridges

Tunnelingpathwaybetweencytochromeb5andmethaemoglobinControlledtunneling

throughmolecules?MinorchangestothemolecularelectronicdensityHighsensitivity(exponential)tothemolecularparametersApotentialforarationaldesignbasedonchemicalknowledgeResonanttunnelingDeep(offresonant)tunnelingWhyOff-Resonant(deep)Tunneling?ElectronTransferinDonor-AcceptorPairsDonor

AcceptorElectronictunnelingmatrixelementNuclearfactor:Frank-CondonweighteddensityofstatesTheroleofelectronicnuclearcoupling?Thecaseofthroughbridgetunneling:Theory:ElectronTransferinDonor-AcceptorPairsTheelectronicHamiltonian:Diabaticelectronicbasisfunctions:TheHamiltonianmatrix:Theory:ElectronTransferinDonor-AcceptorPairsASpinBosonHamiltonian:TheHarmonicapproximation:Theory:ElectronTransferinDonor-AcceptorPairsTheCondonapproximationDonor

AcceptorThegoldenruleexpressionfortherateAnelectronictunnelingmatrixelementAnuclearfactorMcConnell(1961):Introducingasetofbridgeelectronicstates;ThedirecttunnelingmatrixelementvanishesDonor

AcceptorLongRangeElectronicTunnelingThedonorandacceptorsitesareconnectedviaaneffectivetunnelingmatrixelementthroughthebridgeMcConnell’s

Formula:

AtightbindingmodelThedeeptunnelingregime:

FirstorderperturbationtheoryAsimpleexpressionfor

theeffectivetunnelingmatrixelementTunneling

oscillationsatafrequency:

Superexchangedynamicsthrough

asymmetricuniformbridgeH.M.McConnell,J.Chem.Phys.35,508(1961)DeeptunnelingthroughamolecularbridgeTheroleofbridgenuclearmodes?ValidityoftheCondonapproximation?

Davis,RatnerandWasielewski(J.A.C.S.2019).

Molecules1-5Chargetransferisgatedbybridgevibrations

Electronicnuclearcouplingatthebridge:

RigidbridgesenablehighlyefficientelectronenergytransferLokan,Paddon-Row,Smith,LaRosa,GhigginoandSpeiser(J.A.C.S.2019).BreakdownoftheCondonapproximation!Structural(promoting)bridgemodes:Electronicallyactive(accepting)bridgemodes:Ageneralized“spin-boson”model:ThenuclearpotentialenergysurfacechangesatthebridgeelectronicsitesHarmonicnuclearmodesLineare-nuclearcouplinginthebridgemodesThee-nuclearcouplingisrestrictedtothebridgesitesADissipativeSuperexchangeModel:

Asymmetricuniformbridge

IntroducingnuclearmodeswithanOhmic()spectraldensity

Thenuclearfrequencies:10-500(1/cm)arelargerthanthetunnelingfrequency!!

andauniformelectronic-nuclearcoupling:

M.A-HiluandU.Peskin,Chem.Phys.296,231(2019).CoupledElectronic-NuclearDynamicsAmean-fieldapproximation:ThecoupledSCFequations:Mean-fields:TheLangevin-SchroedingerequationAnon-linear,nonMarkoviandissipationtermFluctuationsAtzerotemperature,R(t)vanishesInitialnuclearpositionandmomentumElectronicbridgepopulationU.PeskinandM.Steinberg,J.Chem.Phys.109,704(2019).NumericalSimulations:Weake-ncouplingThetunnelingfrequencyincreases!Thetunnelingissuppressed!Simulations:Stronge-nCouplingInterpretation:atime-dependentHamiltonianTheInstantaneouselectronicenergy:

Weakcoupling:EnergydissipationintonuclearvibrationslowersthebarrierforelectronictunnelingAtime-dependentMcConnellformulaInterpretation:atime-dependentHamiltonianTheInstantaneouselectronicenergy:

Weakcoupling:EnergydissipationintonuclearvibrationslowersthebarrierforelectronictunnelingStrongcoupling:“Irreversible”electronicenergydissipation

ResonantTunnelingNumericallyexactsimulationsforasinglebridgemodeTunnelingsuppression

atstrongcouplingTunnelingacceleration

atweakcoupling

Adissipative-acceptormodel:Theacceptorpopulation:DissipationleadstoaunidirectionalETThetunnelingrateIncreaseswithe-ncouplingatthebridge!Introducing

abridgemodeA.MalkaandU.Peskin,Isr.J.Chem.(2019).Adimensionlessmeasurefortheeffectiveelectronic-nuclearcoupling:Interpretation:NuclearpotentialenergysurfacesDeeptunneling=weakelectronicinter-sitecouplingEntangledelectronic-nucleardynamics

beyondtheweakcouplinglimitAsmallparameter:Thesymmetricuniformbridgemodel:M.A.-HiluandU.Peskin,submittedforpublication(2019).ARigorousFormulation

TheDonor/AcceptorHamiltonianTheBridge

HamiltonianThecouplingHamiltonian(purelyelectronic!)Introducingvibrationaleigenstates:Diagonalizingthetight-bindingoperator:Regardingtheelectroniccouplingasa(secondorder)perturbation

Intheabsenceofelectroniccouplingthegroundstateisdegenerate:Theenergysplittingtemperaturereads:Frank-CondonoverlapfactorsTheenergysplitting:Expandingthedenominatorsinpowersof

andkeepingtheleadingnonvanishingtermsgivesInterpretation:EffectiveelectroniccouplingEffectivebarrierfortunnelingMcConnell’sexpression:

Summationover

vibronictunnelingpathways:LowerbarrierfortunnelingMultiple“Dissipative”pathwaysTheeffectivetunnelingbarrierdecreasesAnexample(N=8)Thetunnelingfrequencyincreasesbyordersofmagnitudewithincreasingelectronicnuclearcoupling1/cm

The“slowelectron”adiabaticlimitConsideringonlythegroundnuclearvibrationalstate:Aconditionforincreasingthetunnelingfrequencybyincreasingelectronic-nuclearcoupling:Anexample(N=8)TheslowelectronapproximationSpectraldensitiesMolecularrigidity=smalldeviationsfromequilibrium configuration

Flexiblevs.RigidmolecularbridgesIncreasingrigidity

Aconsistencyconstraint:

Langevin-Schroedingersimulations:ThetunnelingfrequencyincreaseswithbridgerigidityArigoroustreatment:

The“slowelectron”limit

Rigidity=largerFrankCondonfactor!SummaryandConclusionsArigorouscalculationofelectronictunnelingfrequenciesbeyondtheweakelectronic-nuclearcouplinglimit,predictsaccelerationbyordersofmagnitudesforsomemolecularparametersAnanalyticalapproachwasintroducedandaformulawasderivedforcalculationsoftunnelingmatrixelementsinadissipativeMcConnellmodel.Acomparisonwithapproximatemethodsforstudyingopenquantumsystemsissuggested.Thewayforrationallydesigned,controlledelectrontransportin“moleculardevices”isstilllong…Theeffectofelectronic-nuclearcouplinginelectronicallyactivemolecularbridgeswasstudiedusinggeneralizedMcConnellmodelsincludingbridgevibrations.Mean-fieldLangevin-Schroedingersimula