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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
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