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Dissipative Quantum Molecular Dynamics in Gases and Condensed Media

Permanent Link: http://ufdc.ufl.edu/UFE0024969/00001

Material Information

Title: Dissipative Quantum Molecular Dynamics in Gases and Condensed Media A Density Matrix Treatment
Physical Description: 1 online resource (128 p.)
Language: english
Creator: Leathers, Andrew
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2009

Subjects

Subjects / Keywords: dissipation
Chemistry -- Dissertations, Academic -- UF
Genre: Chemistry thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: We present a study of dissipative quantum molecular dynamics, covering several different methods of treating the dissipation. We use a reduced density matrix framework, which leads to coupled integro-differential equations in time. We then develop a numerical algorithm for solving these equations. This algorithm is tested by comparing the results to a solved model. The method is then applied to the vibrational relaxation of adsorbates on metal surfaces. We also use this model to test approximations which transform the integro-differential equations into simpler integral equations. Our results compare well to experiment, and demonstrate the need for a full treatment without approximations. This model is then expanded to allow for electronic relaxation, as well as excitation by a light pulse. The electronic relaxation occurs on a different time scale, and is treated differently than the vibrational relaxation. Our method is shown to be general enough to handle both cases. Our next model is light-induced electron transfer in a metal cluster on a semiconductor surface. We consider both direct electronic excitation causing electron transfer, as well as indirect transfer, where there is excitation to an intermediate state which is coupled to the electron transferred state. Our results indicate vibrational relaxation plays a small role in the direct transfer dynamics, but is still important in the indirect case. Finally, we present a mixed quantum-classical study of the effect of initial conditions, with the goal of moving towards a method capable of treating dissipation in both quantum and mixed quatum-classical systems.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Andrew Leathers.
Thesis: Thesis (Ph.D.)--University of Florida, 2009.
Local: Adviser: Micha, David A.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2009
System ID: UFE0024969:00001

Permanent Link: http://ufdc.ufl.edu/UFE0024969/00001

Material Information

Title: Dissipative Quantum Molecular Dynamics in Gases and Condensed Media A Density Matrix Treatment
Physical Description: 1 online resource (128 p.)
Language: english
Creator: Leathers, Andrew
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2009

Subjects

Subjects / Keywords: dissipation
Chemistry -- Dissertations, Academic -- UF
Genre: Chemistry thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: We present a study of dissipative quantum molecular dynamics, covering several different methods of treating the dissipation. We use a reduced density matrix framework, which leads to coupled integro-differential equations in time. We then develop a numerical algorithm for solving these equations. This algorithm is tested by comparing the results to a solved model. The method is then applied to the vibrational relaxation of adsorbates on metal surfaces. We also use this model to test approximations which transform the integro-differential equations into simpler integral equations. Our results compare well to experiment, and demonstrate the need for a full treatment without approximations. This model is then expanded to allow for electronic relaxation, as well as excitation by a light pulse. The electronic relaxation occurs on a different time scale, and is treated differently than the vibrational relaxation. Our method is shown to be general enough to handle both cases. Our next model is light-induced electron transfer in a metal cluster on a semiconductor surface. We consider both direct electronic excitation causing electron transfer, as well as indirect transfer, where there is excitation to an intermediate state which is coupled to the electron transferred state. Our results indicate vibrational relaxation plays a small role in the direct transfer dynamics, but is still important in the indirect case. Finally, we present a mixed quantum-classical study of the effect of initial conditions, with the goal of moving towards a method capable of treating dissipation in both quantum and mixed quatum-classical systems.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Andrew Leathers.
Thesis: Thesis (Ph.D.)--University of Florida, 2009.
Local: Adviser: Micha, David A.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2009
System ID: UFE0024969:00001


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Iwouldliketothankmyadvisor,Dr.DavidMicha,forhisguidanceandgreatpatience.IwouldalsoliketothankboththeDepartmentofChemistryandtheQuantumTheoryProjectforprovidingawonderfulenvironmentforstudyforthepastyears.Iwouldliketothankmyparents,StevenandMarilynLeathersfortheirsupport. 4

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page ACKNOWLEDGMENTS ................................. 4 LISTOFTABLES ..................................... 8 LISTOFFIGURES .................................... 9 ABSTRACT ........................................ 12 CHAPTER 1INTRODUCTION .................................. 14 1.1TheReducedDensityMatrix ......................... 14 1.2AlternativeMethods .............................. 16 1.3OurMethod ................................... 16 1.4TwoStateVibrationalRelaxation ....................... 16 1.5VibrationalandElectronicRelaxation .................... 17 1.6DirectandIndirectElectronTransfer ..................... 17 1.7OutlineofDissertation ............................. 17 2DENSITYMATRIXEQUATIONWITHDELAYEDDISSIPATION ...... 20 2.1Introduction ................................... 20 2.2LiouvilleEquationforReducedDensityMatrix ............... 20 2.3AMasterEquation ............................... 22 2.3.1TheInteractionPicture ......................... 22 2.3.2ProjectionOperatorFormalism .................... 23 2.3.3AMasterEquation ........................... 25 2.3.4FastDissipationLimits ......................... 26 2.3.5DiadicFormulation ........................... 27 2.4Quantum-ClassicalTreatment ......................... 27 2.4.1TheWignerTransform ......................... 27 2.4.2PartialWignerTransform ....................... 29 2.4.3Quantum-ClassicalTreatmentwithDissipation ............ 31 3QUANTUM-CLASSICALTREATMENT ..................... 32 3.1Introduction ................................... 32 3.2DissociationofNaI ............................... 32 3.3EectofInitialConditions ........................... 34 4NUMERICALMETHOD .............................. 44 4.1ARunge-KuttaMethodforIntegro-DierentialEquations ......... 44 4.1.1VolterraIntegralEquations ....................... 45 4.1.2VolterraIntegro-DierentialEquations ................ 47 5

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............................ 50 4.2ImplicitFormulation .............................. 52 4.3TestSystem ................................... 53 4.4ScalingandLimitations ............................ 56 4.5TheMemoryTime ............................... 57 4.6Conclusion .................................... 59 5VIBRATIONALRELAXATIONOFADSORBATESONMETALSURFACES 64 5.1Introduction ................................... 64 5.2ModelDetails .................................. 64 5.3TheCorrelationFunction ........................... 67 5.4Results ...................................... 68 5.5Conclusions ................................... 69 6ELECTRONICANDVIBRATIONALRELAXATION .............. 80 6.1Introduction ................................... 80 6.2ModelDetails .................................. 80 6.3PrimaryRegion ................................. 81 6.4SecondaryRegion ................................ 84 6.5UnperturbedDynamics ............................. 86 6.6ResultsWithaPulse .............................. 86 6.7Conclusion .................................... 87 7ELECTRONICALLYNON-ADIABATICDYNAMICSOFAG3SI(111):H .... 92 7.1Introduction ................................... 92 7.2ModelDetails .................................. 92 7.3CorrelationFunction .............................. 95 7.4DirectExcitation ................................ 95 7.4.1InitialDynamics ............................. 95 7.4.2PhotoinducedDynamics ........................ 96 7.5IndirectExcitation ............................... 97 7.5.1InitialDynamics ............................. 98 7.5.2PhotoinducedDynamics ........................ 99 7.5.3MemoryTime .............................. 99 7.6Conclusions ................................... 101 8CONCLUSION .................................... 118 8.1VibrationalRelaxation ............................. 118 8.2ElectronicandVibrationalRelaxation .................... 119 8.3ProgramDevelopment ............................. 119 8.4FutureWork ................................... 119 APPENDIX 6

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................................ 121 A.1Overview .................................... 121 A.2ModelSpecicFunctions ............................ 121 A.3InputAndOutputFiles ............................ 122 REFERENCES ....................................... 124 BIOGRAPHICALSKETCH ................................ 128 7

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Table page 3-1ParametersfortheNaImodel. ........................... 35 4-1NumberofequalitiestobesatisedforagivenorderoftheRunge-Kuttamethod 59 4-2HighestattainableorderofanexplicitRunge-Kuttamethodforagivenm 59 4-3Minimummneededtoattainagivenorderp 59 4-4Examplecoecients ................................. 63 5-1Frequenciesandcouplingparameters ........................ 70 6-1Parametersforthegroundandexcitedstatepotentials .............. 89 6-2Parametersforsecondaryregion ........................... 89 6-3Parametersfortheheatdiusionequations ..................... 89 7-1ParametersforthechosentransitionsinAg3Si(111):H ............... 101 8

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Figure page 1-1Delayeddissipationbetweentheprimaryregion(P)andthesecondaryregion(S).Thereisanexcitationfromthegroundstate(g)inthePregion,thenthereareinteractionsbetweentheexcitedstates(e)andthesecondaryregionduringtimestandt0 19 3-1PotentialcurvesfortheNaImodel ......................... 36 3-2Populationsfortheorbitandgridinitialconditions ................ 37 3-3Realpartofthecoherencefortheorbitandgridinitialconditions ........ 38 3-4Imaginarypartofthecoherencefortheorbitandgridinitialconditions ..... 38 3-5Initialformofthephasespaceforthegridinitialconditions ........... 39 3-6Initialformofthephasespacefortheorbitinitialconditions ........... 39 3-7Finalformofthephasespaceforthegridinitialconditions ............ 40 3-8Finalformofthephasespacefortheorbitinitialconditions ........... 41 3-9AveragePandPfororbitandgridinitialconditions,alongwithfullquantumresults. ......................................... 42 3-10AverageRandRfororbitandgridinitialconditions,alongwithfullquantumresults. ......................................... 43 4-1hzit,Model1:ourresults(solidcurve)andselectedpointsfromGrifonietal.(points) ........................................ 60 4-2hxit,Model1:ourresults(solidcurve)andselectedpointsfromGrifonietal.(points) ........................................ 60 4-3hzit,Model2:ofourresults(solidcurve)andselectedpointsfromGrifonietal.(points) ...................................... 61 4-4hxit,Model2:ourresults(solidcurve)andselectedpointsfromGrifonietal.(points) ........................................ 61 4-5hzit,Model1:thehightemperaturelimit.Exactresults(solidcurve)andcalculatedresults(dashedcurve) ................................ 62 4-6hzit,Model2:thehightemperaturelimit.Exactresults(solidcurve)andcalculatedresults(dashedcurve) ................................ 62 4-7hzit,Model1:delayeddissipation(solidcurve),instantaneousdissipation(dashedcurve),andtheMarkolimit(dottedcurve) .................... 63 9

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.................. 71 5-2Populationofthegroundstate(00)forCO/Cu(001)at150Kand300K .... 72 5-3Populationofthegroundstate(00)forNa/Cu(001)at150Kand300K ..... 73 5-4Populationofthegroundstate(00)forCO/Pt(111)at150Kand300K ..... 74 5-5Populationofthegroundstate(00)forCO/Cu(001)at150Kfornormalcouplingstrengthandat0.8timesthecouplingstrength .................. 75 5-6Populationofthegroundstate(00)forCO/Cu(001)at150Kfornormalcouplingstrengthandat1.2timesthecouplingstrength .................. 76 5-7Populationofthegroundstate(00)forCO/Cu(001)at150Kusingdelayeddissipation,theinstantaneousdissipationlimit,andtheMarkolimit ...... 77 5-8Realpartofthequantumcoherence01forCO/Cu(001)at150K(solidline)and300K(dashedline) ................................ 78 5-9Imaginarypartofthequantumcoherence01atshorttimesforCO/Cu(001)at150Kand300K(upper)andlongtimes(lower) .................. 79 6-1EnergydiagramforCO/Cu(001) .......................... 88 6-2CO/Cu(001),reprintedwithpermissionfromA.SantanaandD.A.Micha,Chem.Phys.Lett.369,459(2003) ............................ 88 6-3Populationswithoutapulseofthegroundelectronicgroundvibrational(g0)state,alongwiththecoherencebetweenthegroundelectronicgroundvibrationalstateandthegroundelectronicsecondvibrationalstate(g02)at300K ..... 90 6-4Populationdierencesat300Kwithapulse,forthethreevibrationalstatesinthegroundelectronicstate .............................. 91 7-1Ag3Si(111):H,reprintedwithpermissionfromD.S.KilinandD.A.Micha,J.Phys.Chem.C113,3530(2009) .......................... 102 7-2EnergydiagramforAg3Si(111):H,directexcitation ................ 103 7-3EnergydiagramforAg3Si(111):H,indirectexcitation ............... 103 7-4SpectraldensityofAg3Si(111):H .......................... 104 7-5RealpartofthecorrelationfunctionofAg3Si(111):H ............... 105 7-6ImaginarypartofthecorrelationfunctionofAg3Si(111):H ............ 106 7-7Theg0populationandg02coherencewithoutapulse ............... 107 10

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........................... 108 7-9Totalpopulationofthegroundandexcitedelectronicstateswithandwithoutdelayeddissipation,el=1ps ............................ 109 7-10Populationofthee4state,withandwithoutdelayeddissipation ......... 110 7-11Quantumcoherencebetweenstatese0ande1 ................... 111 7-12Populationofthevibrationalstatesofthegroundelectronicstatefortheindirectcasewithoutapulse ................................. 112 7-13Totalpopulationoftheground,excited,andnalelectronicstateswithandwithoutdelayeddissipation,el=200fs ........................... 113 7-14Populationsoftheg3andg4stateswithoutalightpulse,usingeitherthefullmemorykernelorthememorytime ......................... 114 7-15Vibrationalpopulationsofthegroundelectronicstatewiththefullmemory(dashedcurve)andusingamemorytimeof50000au(solidcurve) ............ 115 7-16Vibrationalpopulationsoftheexcitedelectronicstate,e,withthefullmemory(dashedcurve)andusingamemorytimeof50000au(solidcurve) ........ 116 7-17Vibrationalpopulationsofthenalelectronicstate,f,withthefullmemory(dashedcurve)andusingamemorytimeof50000au(solidcurve) ........ 117 11

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, 2 { 6 { 20 { 24 26 28 14

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1-1 showsdelayeddissipationfromtheprimaryregiontothesecondaryregion.Thereareinteractionsbetweentheprimaryandsecondaryregionsduringtimestandt0.Thephysicalsystemsconsideredherearemoleculesandclustersadsorbedonsurfaces,andwestudythevibrationalrelaxationofthefrustratedtranslation(T-mode). 15

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, 32 { 35 38 16

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, 39 17

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Delayeddissipationbetweentheprimaryregion(P)andthesecondaryregion(S).Thereisanexcitationfromthegroundstate(g)inthePregion,thenthereareinteractionsbetweentheexcitedstates(e)andthesecondaryregionduringtimestandt0

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^=XnPnjnihnj;(2{1) 20

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@tjni=i ^H=^HP+^HR+^HPR:(2{5)Wecantakethetraceoverthereservoirvariablestogivethereduceddensityoperator,^=trR^.Thisreduceddensityoperatordescribesonlytheprimaryregion.Theequationofmotionforthereduceddensityoperatoris @t=i 2{3 ,withaformalsolution ^(t)=^UP(t;t0)^(t0)^UyP(t;t0)(2{7) 21

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^UP(t;t0)=ei 2{6 thatgovernsenergydissipationfromtheprimaryregiontothereservoir.Itcanbetreatedthroughvariousapproximations. 2.3.1TheInteractionPictureItwillbeconvenientinourderivationtoswitchtotheinteractionpicture.Weassumethatthetotalhamiltonianisbrokenupas^H=^H0+^V,where^Visasmallperturbationand^H0isthehamiltonianforasimpler,possiblysolvablesystem.Astatevectorintheinteractionpicturecomesfrom ^U0(t;t0)=ei @tj(I)n(t)i=i ^O(I)(t)=^Uy0(t;t0)^O^U0(t;t0):(2{12)TurningnowtoourhamiltonianinEquation 2{5 ,weset^V=^HPRand^H0=^HP+^HR,sothat ^U0(t;t0)=ei 22

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2{3 ^P^O=^RtrR(^O);(2{16)where^Oisanarbitraryoperatorinbothprimaryandreservoirvariablesand^Risareservoiroperator,withtrR(R)=1.Inwhatfollows,wewillassumethatthereservoirislargeenoughthatitstaysinthermalequilibrium,whichmeanssetting^R=^Req.Wealsodenetheoperator^Qas ^Q=1^P;(2{17)sothat ^O=^P^O+^Q^O:(2{18)Itisalsoworthnotingthat^P2=^P,^Q2=^Q,and^P^Q=^Q^P=0.Returningtothedensityoperatorintheinteractionpicture,wehave ^(I)(t)=^P^(I)(t)+^Q^(I)(t)=^Req^(I)(t)+^Q^(I)(t): 23

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2{15 ,giving 2{14 gives 2{20 ,theseareacoupledsetofequationsfortheprimaryregion,^(I)(t),andtherestofthedensityoperator,^Q^(I)(t).Thesecond-ordersolutionfortheequationforthereduceddensityoperatorisobtainedbyusingtherst-ordersolutionfor^Q^(I)(t).Therst-ordersolutionofEquation 2{22 comesfromneglectingthe^Q^(I)(t)termontherightside.Thisgives ^Q^(I)(t)=^Q^(I)(t0)i 2{20 gives whichisanintegro-dierentialequationforthereduceddensitymatrix,tosecondorderwithrespectto^HPR.Itispossibletoformulatehigher-orderequations;reinsertingEquation 2{23 intoEquation 2{22 wouldgiveasecond-orderresultfor^Q^(I)(t),whichcouldthenbeusedtogetathird-orderequationforthereduceddensitymatrix.Thismethodcanbeiteratedtogiveanyorderdesired.Inthisworkwewillusethesecond-orderexpression. 24

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^HPR=Xj^Aj(P)^Bj(R);(2{25)where^Ajdependsonlyontheprimaryregionand^Bjdependsonlyonthereservoir,wecandenethereservoircorrelationfunctionas 2{24 thenreducesto @t=i Thiscanbeexpressedasanintegro-dierentialequationwithamemorykernel,leadingtothegeneralform 25

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dtab=i!abab+i with~!ab=(EaEb),and 2{28 involvespreviousstatesofthesystem,fromthecurrenttimetbacktothebeginning(t=0),andwecallitthedelayeddissipationcase.Inthissection,weconsidertwoapproximationswhichcanreducetheintegro-dierentialequationtosimplerequations,greatlyreducingcomputationtime.Therstapproximationiswhatwecalltheinstantaneousdissipationlimit.Theassumptionisthatthedissipationoccursmuchfasterthanchangesinthedensitymatrix.Whenthisisthecase,wecanmaketheapproximation(t0)=(t);wecanthenmoveoutsidetheintegral.ApplyingthistoEquation 2{28 gives 26

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=Cmn[A(m)A(n);]+2A(n)A(m)ab; where[;]+indicatesananticommutator.WecallthisformLindbladdissipation. 42 2{28 intoalinearform, 2.4.1TheWignerTransformWeuseapartialWignertransformtotreatmixedquantum-classicalmechanics.Tobeginourderivationofthismethod,werstdicussthefullWignertransform.TheWignertransformisusedtocreateaphasespacerepresentationofthedensityoperator,throughaFouriertransformofanoperatorincoordinatespace, { 46 27

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^OW(P;R)=ZdyhQyj^OjQ+yie2ipy=~:(2{40)AnalagoustoEquation 2{4 ,theexpectationvalueofanoperatorintheWignerrepresentationis (^A^B)W=AWe~!=2iBW;(2{42)where!isabidirectionaloperatordenedas @P!@ @R@ @R!@ @P;(2{43)sothat ^A!^B=@^A @P@^B @R@^A @R@^B @P:(2{44)ThisisoftenabbreviatedusingthePoissonbracket, [^A;^B]W=AWe~!=2iBWBWe~!=2iAW:(2{46)Thetransformeddensityoperatorisnotitselfaprobabilitydistribution,butitdoeshavemanyusefulproperties.Thedistributioninonevariablecomesfromintegratingout 28

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ForanyfunctionofRorP,wecanobtaintheexpectationvalueofthefunctionthesamewayasforoperators ^(t)=ZdQZdQ0jQi^(Q;Q0;t)hQ0j:(2{51)Wenowdenenewcoordinates,R=(Q+Q0)=2andS=QQ0,whichallowsustodenethepartialWignertransform, ^W(P;R;t)=1 2{3 togive { 50 55 2f^HW;^Wgf^W;^HWg:(2{54) 29

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{ 54 ^H=^Hcl(Q)+^Hqu(q)+^Hqucl(Q;q);(2{55)where^Hcl(Q)isthehamiltonianfortheclassicalpart,^Hqu(q)isthehamiltonianforthequantumpart,and^Hqucl(Q;q)isthecouplingbetweenthem.Theclassicalhamiltonianconsistsofkineticandpotentialterms, ^Hcl=P2 ^HW=P2 ^H0qu=^Hqu(q)+^Hqucl(R;q); whichleadstotheequationofmotion M@^W 2@V0 55 M@^W 30

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, 55 2{62 reducestoEquation 2{3 .Ifweagainpartitionthequantumsectionintoaprimaryregionandareservoir,thersttermontherighthandsideofEquation 2{62 willbereplacedwithanexpressionliketherightsideofEquation 2{27 ,with^replacedby^W.Iftherewereacouplingbetweenthequantumandclassicalsectionswhichleadstodissipation,morecomplicatedexpressionswouldensue. 31

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, 55 55 2{62 inthis22basis,anddiscretizingPandRgivesthematrixequation M@W(P;R) 32

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3{1 areacoupledsetofdierentialequationswhichmustbesolvedtogetherforeachpointinphasespace.ThematrixelementsoftheHamiltonianare State1istheneutralstate,andState2istheionic.Weassumethatthesystemisinitallyinthelowestvibrationallevelintheionicstate,andundergoesaninstantexcitationtotheneutralstate.Thereisnopopulationremainingintheionicstate,andnocoherence,sothat22=12=21=0.NotethatwehaveomittedtheWsubscripthereforclarity.Wecalculatetheinitialvalueof11from 11(P;R)=1 3-1 .AplotofthepotentialsisshowninFigure 3-1 .Theionicpopulationcomesfrom22, 33

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3-5 .Here,wechooseanorbitfortheinitialphasespace,showninFigure 3-6 .ThesepointsevolveintimeaccordingtoEquations 3{2 and 3{3 .TheresultsforthepopulationsareshowninFigure 3-2 .Thepopulationsoftheneutralboundandionicstatesarequalitativelysimilar,withtheorbitmethodshowingsharperoscillations.Thepopulationoftheneutralfree(dissociated)stateareverysimilar.Ifwewereonlyconcernedwiththeyieldofthephotoinduceddissociation,itwouldnotmatterwhichinitialconditionswechose.Figures 3-3 and 3-4 showtherealandimaginarypartsofthequantumcoherenceforboththeorbitandgridinitialconditions.Thegridmethodshowsalargercoherenceatearlytimes,butthetwomethodsdonotgiveverydierentresults.Figures 3-7 and 3-8 showthenalconformationofthephasespaceforthegridandorbitinitialconditions.Theswirlingnatureofthephasespaceissimilartothephasespaceofclassicalparticlesinaharmonicwell,liketheionicpotential.Thefreemotion,whichisevidentinthephasespaceofthegridinitialconditions,doesnotappearhere.InFigures 3-9 and 3-10 weshowtheaveragePandR,aswellastheirdispersionsfororbitandgridinitialconditions,alongwithfullquantumresults. 34

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ParametersfortheNaImodel. ParameterValue(au) V.EngelandH.Metiu,J.Chem.Phys.90,6116(1989) 35

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PotentialcurvesfortheNaImodel 36

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Populationsfortheorbitandgridinitialconditions 37

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Realpartofthecoherencefortheorbitandgridinitialconditions Figure3-4. Imaginarypartofthecoherencefortheorbitandgridinitialconditions 38

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Initialformofthephasespaceforthegridinitialconditions Figure3-6. Initialformofthephasespacefortheorbitinitialconditions 39

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Finalformofthephasespaceforthegridinitialconditions 40

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Finalformofthephasespacefortheorbitinitialconditions 41

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AveragePandPfororbitandgridinitialconditions,alongwithfullquantumresults. 42

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AverageRandRfororbitandgridinitialconditions,alongwithfullquantumresults. 43

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(4{3) wherem(aninteger)isthenumberofstagesofthemethodrequiredtoachieveadesiredaccuracy,andtheaij;bi;andciarerealcoecients.Foranexamplesetofcoecients,seeTable 4-4 .Thecisatisfytherelation 44

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=ZttnK(t;t0)[rn(t0)+qn(t0)]dt0: Notethatqn,calledthelagterm,onlydependsonknownvaluesofy(t)(thoseuptoy(tn)),whilern,calledtheincrementfunction,satisesanintegralequation,oranequivalentdierentialequation.IfweapplytheforegoingRunge-Kuttamethodtotheexpressionforr,weobtain Therstterm,rn(tn),iszero.IfwenowchoosetheYniaccordingto 46

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wewillhaveanequationsimilartoEquation 4{30 47

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Ifwemakethedenitions wehave 48

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Notethatthereisnodependenceontintheexpressionsforr()n(t)andq()n(t),sowemaywrite Ifwesetq()n=n,wehave 49

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2+O(h3):(4{57)WenowconsidertheTaylorseriesoftheapproximatesolutionatt1=h, 2+O(h3):(4{58)Thederivativesofthetruesolutionatt0are (4{60) andfortheapproximatesolution Wenowseth=0andnotethat 50

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Notethaty0=y(t0).ForthetwoTaylorseriestoagreetorstorderinh(p=1),therstderivativesmustagree,andwehave 2:(4{69)Thismethodcanberepeatedforhigherderivatives.Uptoorderp=4,wehave 3; 6; 51

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6; 8; 12; 24: InTable 4-1 weseethetotalnumberofequalitiestobesatised,Np,foragivenorderp.Foragivenm,thereare(m2+m)=2coecients.Atm=5itisnotpossibletoattainamethodoforderp=m=5;thereare17orderconditions,butonly15coecients.InTables 4-2 and 4-3 weseetheattainableorderofaRunge-Kuttamethodforagivenm,andtheminimummneedtoobtainorderp.InTable 4-4 weseeanexamplesetofcoecientsforaRunge-Kuttamethodwithm=p=4. 52

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whichleadsto wherethedimensionsofCaren21,andAandBarebothn2n2. 61 2[(t)x+(t)z+(t)z];(4{82)whereisanenergytransfermatrixelement,(t)isthebiasingenergywhichmayincludeadrivingeld,and(t)isastochasticfactorwhichcontainssystem-bathcoupling.Thex;zarePaulimatrices.Theexpectationvaluesofthesematrices,hiit,aredenedashiit=Tr[(t)i].Theseexpectationvaluesgivethepopulationdierence, 53

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1 2hxit=1 2(12+21);(4{84)andtheimaginarypart 1 2hyit=i 20B@1+hzithxitihyithxit+ihyit1hzit1CA:(4{86)Toaccountforthesystembathcoupling,westartwithabathataconstanttemperatureT,sothat sinh! 61 dthzit=Zt0dt0[Ka(t;t0)Ks(t;t0)hzit0]; (t)d dthzit; 54

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with (t;t0)=Ztt0()d; whereTisthetemperatureofthereservoir.Theantisymmetrickernels,KaandYa,aresimilartothesymmetricones,exceptwitheverysinereplacedbyacosine,andcosinesreplacedbysines.Itshouldbenotedthatthesekernelsusethenoninteracting-blipapproximation(NIBA). 4-1 and 4-2 wecompareresultsusingtheRunge-Kuttamethodwiththosefoundinapreviousstudy 4-3 55

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4-4 .Thereisverygoodagreementinallcases.Asafurthertest,wecalculatehzitinthehightemperaturelimit,whereexactresultsareavailable,presentedinFigures 4-5 and 4-5 .Atshorttimestheresultsagreeverywell,buttherearesomesmalldeviationsatlongtimes.Thesedeviationsaresmallenoughtoconsiderourmethodtobeareliableone.Finally,wecalculatehzitforModel1intwoapproximations-theinstantaneousdissipationlimit,andthemarkovianlimit.Intheinstantaneousdissipationlimit,weassumethekernelsarerapidlychangingrelativetohzi.Wethenassumehzit0=hzitsothat dthziIDt=Zt0dt0Ka(t;t0)hziIDtZt0dt0Ks(t;t0):(4{99)IntheMarkolimit,weassumetheintegralin 4{99 canbeextendedtoinnity,giving dthziIDt=Z10dt0Ka(t;t0)hziIDtZ10dt0Ks(t;t0):(4{100)TheresultsareshowninFigure 4-7 .Allthreemethodsgivethesamevalueatlong(equilibrium)times,howeverneithermethodfullydescribestheshorttimedynamics.TheMarkolimityieldsanexponentialdecay,whileintheinstantaneouscasethereareoscillationsthataredampedcomparedtothedelayeddissipationcase. 56

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InthecaseofEquation 2{30 ,thememorytimewilldependonthecorrelationfunction,C(t).Ifthecorrelationfunctionissmallenoughtobeneglectedaftertmem,thentheentirekernelwillbesmallenoughtobeneglected,sinceeachtermismultipliedbyC(t).The 57

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dtabdiss=XcdZttmemdt0[Mcd;db((tt0))ei!da(tt0)ac(t0)+Mac;cd(tt0)ei!bc(tt0)db(t0)Mdb;ac((tt0))ei!bc(tt0)cd(t0)Mdb;ac(tt0)ei!da(tt0)cd(t0)]: Notethattheminimumvalueoft0isttmem,whichmeansthatwecandiscardanypreviousvaluesof(t0)wheret0
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4{104 .Ifwechooseourmemorytimeproperly,thereshouldbenoinitaldynamics-thesystemwillbeinatrueequilibrium. Table4-1. NumberofequalitiestobesatisedforagivenorderoftheRunge-Kuttamethod Table4-2. HighestattainableorderofanexplicitRunge-Kuttamethodforagivenm m12345678910 Table4-3. Minimummneededtoattainagivenorderp p12345678910 59

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Figure4-2. 60

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Figure4-4. 61

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Figure4-6. 62

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Examplecoecientsc A=faijg 00001 2 1 20001 2 2001 0010 61 31 31 6 63

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, 15 18 37 65 { 68 ^H=^HP+^HR+^HPR; ^HP=~!0^ay^a; ^HR=Xj~!j^byj^bj; ^HPR=~Xjj(^ay^byj+^ay^bj+^a^byj+^a^bj); 64

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^q=1 ^B=~p sothat ^HPR=^q^B:(5{7)Notethat^qisdimensionless,while^Bhasthedimensionsofenergy.Thisgivesacouplingwhichisseparatedintotwoparts,onewhichdependsonlyontheprimaryregion(^q),whiletheotherdependsonlyonthereservoir(^B).Withthistypeofcoupling,wecanuseamasterequationsimilartoEquation 2{30 @tab=i!abab+i where!ab=(EaEb)=~,HPa=Eaa,and Notethatwehaveexpandedinthebasisfag,theeigenstatesof^HP,withenergiesEa=~!0(a+1=2).Inthissystem,h^Bi=0.Equation 5{8 canbesimpliedusingtherelations 65

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dt00=1 2Zt0[4cos(!0(tt0))RefC(tt0)g00(t0)C((tt0))ei!0(tt0)+C(tt0)ei!0(tt0)]dt0; dt11=1 2Zt0[4cos(!0(tt0))RefC(tt0)g11(t0)C((tt0))ei!0(tt0)+C(tt0)ei!0(tt0]dt0; dt01=i!001iZt02RefC(tt0)gImf01(t0)gdt0; dt10=i!010iZt02RefC(tt0)gImf10(t0)gdt0: Foratwo-statecase,withthischoiceofqab,thereisnocouplingbetweenthestates.Thisisbecausewearedealingwithonlya22matrix,andwecansimplifytheequationsusingtherelations00+11=1and01=10,wherethesuperscriptindicatesthecomplexconjugate.However,aswewillseeinlaterchapters,forthreeormorestates,couplingsdoappear.Ifweassumetheinstantaneousdissipationlimit,wecantakeoutsideoftheintegral,andforthegroundstatepopulationwehave dt(ID)00=1 2(ID)00Zt04cos(!0(tt0))RefC(tt0)gdt0Zt0C((tt0))ei!0(tt0)+C(tt0)ei!0(tt0)dt0; whichisanintegralequation.Extendingtheintegraltoinnitygivesthemarkolimit, dt(ID)00=1 2(ID)00Z104cos(!0(tt0))RefC(tt0)gdt0Z10C((tt0))ei!0(tt0)+C(tt0)ei!0(tt0)dt0: 66

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wherepandqareparameterswhichdependonthesystem.ThevaluesoftheseandotherparametersusedinthecalculationsaregiveninTable 5-1 .Theimaginarypartofthecorrelationfunction,whichistemperatureindependent,canbeintegratedexactly.Fort=0,therealpartcanbeintegratedexactlyaswell.Fort6=0,however,thisisnotthecase.Forlargeenoughtemperatures,wecanusetheapproximation 5-1 fortemperaturesof150K,300K,and450K. 67

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5-2 5-3 ,and 5-4 .Ineachcase,weseelargeinitialoscillations,whichrelaxtowardsequilibiumwithin30,000to50,000au.Increasingthetemperatureseemstodamptheoscillations,withrelaxationtoequilibriumcomingsooner.Thisincreaseisinqualtitativeagreementwithexperiment. 5-5 weseetheeectofreducingthecouplingto80%ofnormal,whileinFigure 5-6 thecouplinghasincreasedto120%.Thecouplingislinearinpandq,somultiplyingeachby0.8hastheeectofdecreasingitto80%ofnormal.Toincreaseto120%,wemultiplyby1.2.Decreasingthecouplingmeansweakerinteractionswiththereservoir,leadingtomoreoscillationsandanincreaseinrelaxationtime.Increasingthecouplinghasthereverseeect.Ineachcase,thenalequilibriumvaluesarethesame.Wehavealsotestedtheapplicabilityoftheinstantaneousandmarkovianlimits.ResultsforCO/Cu(001)at150KareshowninFigure 5-7 .Theinstantaneousdissipationlimitshowsverystrongoscillations,evensurpassing1(fullpopulation).Thisshowsthatinstantaneousdissipationdoesnotworkinthiscase.Thatistobeexpected,asthecorrelationfunctiondoesnotdecayrapidlyenough.Themarkovianlimitshowsaveryfastexponentialrisetotheequilibriumvalue.Themarkovianlimitthusgivescorrectvaluesat 68

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5-8 and 5-9 .Thecoherencesarenotcoupledtothepopulations,soiftheyarezeroinitially,thentheywillremainzero.Fortheseruns,theinitialconditionsare01=0:1+0:1i.Boththerealandimaginarypartsofthecoherencedecayoverlongerperiodsoftimethanthepopulations,showingoscillationsontheorderof!0,graduallygoingtozero.Atshorttimes,theimaginaryparthasasimilarpatterntothepopulations.Ourinitialconditionsarethatthepopulationbeginsentirelyintheexcitedstate,withnocoherence.Thiswouldbeachievedbyaveryrapidtransitionfromthegroundstatetotheexcitedstate.Infact,experimentalstudieshaveshownthatthereisaveryfastexcitation,followedbyaslowrelaxation. 67 69

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Table5-1. Frequenciesandcouplingparameters Na/CuCO/CuCO/Pt 70

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Upper:RealpartofC(t)forCO/Cu(001)at150K,300K,and450K.Lower:ImaginarypartofC(t)forCO/Cu(001)at150K 71

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Populationofthegroundstate(00)forCO/Cu(001)at150Kand300K 72

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Populationofthegroundstate(00)forNa/Cu(001)at150Kand300K 73

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Populationofthegroundstate(00)forCO/Pt(111)at150Kand300K 74

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Populationofthegroundstate(00)forCO/Cu(001)at150Kfornormalcouplingstrengthandat0.8timesthecouplingstrength 75

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Populationofthegroundstate(00)forCO/Cu(001)at150Kfornormalcouplingstrengthandat1.2timesthecouplingstrength 76

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Populationofthegroundstate(00)forCO/Cu(001)at150Kusingdelayeddissipation,theinstantaneousdissipationlimit,andtheMarkolimit 77

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Realpartofthequantumcoherence01forCO/Cu(001)at150K(solidline)and300K(dashedline) 78

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Imaginarypartofthequantumcoherence01atshorttimesforCO/Cu(001)at150Kand300K(upper)andlongtimes(lower) 79

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, 8 39 ^H(t)=^H0+^HPL(t)+^HRL(t);(6{1)where^H0wasourhamiltonianintheabsenceofapulse,and^HPLand^HRLaretheinteractionoftheregionswithlight,whichinthedipoleapproximationare ^HPL(t)=^DPEP(t); ^HRL(t)=Zd3rEext(t)^PR; where^DPistheelectricdipoleoftheprimaryregionand^PRisthedipoleperunitvolumeinthereservoir.TheexternaleldisdescribedbyEext,andEPisthelocalvalueinsidetheprimaryregion.Todescribetheelectronicstates,wedividethesystemintoaprimaryregionwhichincludestheadsorbateandadjacentmetalatoms,andthereservoir,whichincludesthe 80

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6-2 .Thismodelhasbeenusedpreviouslyforstudiesofphotodesorption.Here,wewillrestrictthedistancefromthesurfacetotheadsorbate,Z,toitsequilibriumvalueinthegroundexcitedstate,Zg.Theanglesandaresettozero. 71 { 76 2MCOg(Z)2x2; 2MCOe(Z)2x2; whereV0g;e()arepotentialwelldepths,g;e()arethewidthsofthewells,=MO=(MC+MO,withMO,MC,andMCOarethemassesofoxygen,carbon,andcarbonmonoxide.TheC-ObonddistanceisdCO,heretakentobeconstant;Zg;e()istheequilibriumdistancefromthecenterofmassofCOtothecoppersurface.Thefrequencyofthefrustratedtranslationis!g;e.Thepotentialswerefoundtobeonlyweaklydependenton,anditscontributionherehasbeenneglected. 81

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2MCOg(Zg)2x2; 2MCOe(Zg)2x2: Thedipoleis withtheo-diagonalelement 82

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, 42 @teldiss=1 2~[W;(t)]+2W0:5(t)W0:5:(6{13)ValuesoftheconstantsusedareshowninTable 6-1 .TheelectronicdissipationisdonewiththeLindbladoperators,whilethevibrationaldissipationisdoneusingthedelayeddissipationtreatmentoutlinedinChapter 5 .Thesepotentialsarefunctionsofxandmustbeexpandedinaharmonicoscillatorbasisseti(x)toincludethevibrationalstates.Thepotentialsforthegroundandexcitedstatearecastinmatrixform,as (Vg)ij=Zdxi(x)Vg(x)j(x);(6{15)withssimilarexpressionforVe.Thecouplingbetweenthestatesdependsonthedipole,andthesecondaryregion,discussedlater.TheLindbladmatricesare (Wg)ij=Zdxi(x)Wg(x)j(x);(6{17)andsimilarexpressionsforWeandWc.Theequationforthedensitymatrixisthen 2~[W;(t)]+2W0:5(t)W0:5Zt0K(t;t0)(t0)dt0(6{18)whereK(t;t0)isasupermatrixcontainingalloftheelementsfromEquation 5{8 83

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2I+3Xiii;(6{19)with 2(ccuu): ThesecomponentsevolveintimeaccordingtotheopticalBlochequations 8 77 wherecohisthecoherencetime,popisthepopulationdierencedecaytime,and~3isthelongtimeequilibriumvalueof3.Thesetermsarerelatedtotheexcitationrate,+,thedeexcitationrate,,andthedecoherencerate,0,accordingto ~3=pop 84

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1=2 2=2 whered(el)cuistheelectronicdipolebetweenthebands.Thef(t)termisthelasereldenvelope,chosenheretobe wheregc;uarethedensityofstatesofbandscandu.ValuesoftheparametersusedontheopticalBlochequationsaregiveninTable 6-2 .Thevalueoftheelectriceldinsidetheprimaryregiondependsonpropertiesofthesecondaryregion,andisgivenby 85

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, 78 whereClatisthespecicheatofthelattice.ParametersfortheheatdiusionequationsaregiveninTable 6-3 6-3 wehavetheresultsshowingthepopulationofthegroundelectronicgroundvibrationalstate,alongwiththecoherencebetweenthegroundvibrationalstateandthesecondexcitedvibrationalstate.After20,000au,thesystemsettlesintoarepeatingbeatpattern.Theoscillationsarelarge,andtheeectsofaddingalightpulsewouldbediculttodetermine.Thisbehaviorisdictatedbythecorrelationfunction,whichdescribesthecouplingofthemediumtothesystem.Inthiscase,thecorrelationfunctionislarge,indicatingastrongcoupling.Thisaccountsforthelargeoscillations.Inaddition,thecorrelationfunctiondoesnotdecayrapidly;itfallsoastime1.Thismaybethecauseofthelong-livedoscillations. 86

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(t)=(t)0(t)(6{40)whichshowstheoveralleectofthepulse.Thepulseissimilartothatusedinthepreviouschapter,agaussianofwidth100fs,uenceof3.0mJ/cm2,andfrequency620nm.Ingure 6-4 weseethepopulationdierences.Thepulsecausesanexcitationtotheexcitedstate,whichdecaystothegroundstate.Theoverallshapeofthedecayisduetotheelectronicdissipation.Thereisabeatpatternofoscillationsinthepopulations;thisisduetothevibrationaldissipation.Thecoherencesshowthisbeatpatternaswell,decayingtozeroasthepopulationsreturntoequilibrium.Thissamebeatpatternisseenintheunperturbeddynamics. 87

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EnergydiagramforCO/Cu(001) Figure6-2. CO/Cu(001),reprintedwithpermissionfromA.SantanaandD.A.Micha,Chem.Phys.Lett.369,459(2003) 88

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Parametersforthegroundandexcitedstatepotentials ParameterValue(atomicunits) Parametersforsecondaryregion ParameterValue(atomicunits) Parametersfortheheatdiusionequations ParameterValue(atomicunits)

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Populationswithoutapulseofthegroundelectronicgroundvibrational(g0)state,alongwiththecoherencebetweenthegroundelectronicgroundvibrationalstateandthegroundelectronicsecondvibrationalstate(g02)at300K 90

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Populationdierencesat300Kwithapulse,forthethreevibrationalstatesinthegroundelectronicstate 91

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@t=i @tel;diss+@ @tvib;diss:(7{1)Eachofthethreetermsisdenedbelow.TheHamiltonianH0containselementsforeachstateaswellascouplingsbetweenthem.Fordirectexcitation,weexpandinabasisoftwoelectronicstates,thegroundstate,g,andthenalexcitedstate,e,sothat 92

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^H(l)ij=DijE(t);(7{5)whereE(t)isthelightpulseandDijisthedipolebetweenelectronicstatesiandj.WecalculatethefullvibroniccouplingsthroughtheFranck-Condonoverlap 80 93

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7-1 .Thetransitiondipoles(Dij)werecalculatedusingKohn-Shamorbitalsgeneratedfromaugmentedplane-wavedensityfunctionalcalculations,usingxedatomicpositions 7-1 .Theprimaryregionisthesilvercluster,alongwiththeadjacentsiliconandhydrogenatoms.Weagainconsiderbothelectronicrelaxationandvibrationalrelaxationoftheprimaryregion.ElectronicrelaxationiscalculatedusingLindbladdissipation 42 @tel;diss=1 2[LyL;]++LLy;(7{9)where[;]+denotestheanticommutator,andLisamatrixderivedfromtherelaxationratesas 84

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, 80 2{30 .Weassumetherearenointeractionsviadelayeddissipationacrosselectronicstates,meaningthevibrationalstatesinthegroundelectronicstatearecoupledtoeachother,butnottothevibrationalstatesintheexcitedelectronicstate,andviceversa. 5{19 .ThespectraldensitycomingfromtheabinitiotreatmentisshowninFigure 7-4 .Thereisasinglelargepeaknear70meVthatdominatesthespectraldensity.Thisisintheareaofsurfacevibrations;bulkmodesdonotcouplestronglytothevibrationsofthesilvercluster.ThecorrelationfunctionisshowninFigures 7-5 and 7-6 .ThemagnitudeofthecorrelationfunctionismuchsmallerthanthatforCO/Cu(001).Italsohasanearlygaussianshape,anddecaysexponentially.Thisfastdecayshouldleadtoshorterlivedoscillations. 7.4.1InitialDynamicsInthecaseofdirectexcitation,showninFigure 7-2 ,thepulseexcitesthesystemdirectlyfromtheinitialgroundstatetothenalstate.Thenalstatechosenherehas 95

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7-7 .Whenthereisnopulse,theonlydrivingforceisthedelayeddissipation.Inthiscase,thedissipationisweak,andhaslittleeectonthepopulation.Thereissomechangeinthepopulations,shownintheupperpanelofFigure 7-7 ,however,thechangesarenotnearlyasstrongasinCO/Cu(001),andadierencingmethodisnotrequired.Wecanassumethesystembeginsinastablethermalequilibrium. 7-8 and 7-9 .Thepulsecausesalargetransferofpopulationfromthegroundstatetotheexcitedstate,whichthendecayssmoothlybacktothegroundstate.ThisdecayisduetotheLindbladdissipation.Asthetransitionrateisdecreased(byincreasingthetimefrom200fsto1ps),thedecaytakeslonger.Theelectronicrelaxationdominatesthedynamics.Thereissomechangeinthepopulationsatearlytimeswhendelayeddissipationisadded,butthiseectdisappearsastimegoeson. 96

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7-10 wefocusononlythehighestenergystatetoseetheeectsofdelayeddissipation.Whilethereissomeeect,thepopulationoscillatesaroundthecasewithoutdelayeddissipation.Theotherpopulationsshowasimilarpattern.Itmaybethattheseeectsareimportantearlyinthedynamics,howeverthelightpulseandtheelectronicrelaxationarethedominantfactors.Inthiscase,thedynamicsarelargelycapturedbyincludingonlytheelectronicdissipation.Thequantumcoherencebetweenstatese0ande1isshowninFigure 7-11 .Thecoherencepeaksearly,around2500au,justastheestateisbeingpopulatedbythepulse.Thecoherencehasdisappearedby40,000au. 7-3 ,thelightpulseexcitesfromtheinitialgroundstate(g)toanexcitedstate(e),whichthenrelaxesintothenalstate(f).Thisnalstateistheelectron-transferredstate.Theintermediateexcitedstatehasalargeoscillatorstrength,butsmallchargetransfer.Thenalstatecouplesstronglytotheintermediatestate,butweaklytothegroundstate,sothereislittledirectexcitation.Thenalstatehasalargechargetransfer.Thestrongcouplingbetweenthegandestatesleadstoelectronicrelaxation,whichisagaintreatedwiththeLindbladform.Herewehavechosentouseonlythe200fsratefortherelaxation.Thecouplingbetweenthegandfstatesissosmallthattherelaxationisveryslow,andhasbeenneglectedinourcalculations.Thevibrationalcouplingistreatedwithdelayeddissipation,withnovibrationalcouplingacrosselectronicstates.Theeandfelectronicstatesarearecoupledthroughmomentum, 86 @qjfi@ @q;(7{13)wheremisthemassofthesilvercluster.Weneedthecouplingnotjustbetweenelectronicstates,butalsothevibrationalstates,soweexpandthisexpressionintermsofharmonic 97

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@qjfv0fihv0fj@ @qjvfi:(7{14)Inournotationje;veiistheeelectronicstateintheve=f0;1;2:::gvibrationalstate.ThersttermcanbeexpressedasaFranck-Condonoverlapandacouplingfactor,cef,whichdependsonlyontheelectronicstates,togive @qjvfi;(7{15)Thelasttermcanbeexpressedintermsofthecreationandannihilationoperatorsayandaas @qjvfi=1 InsertingthisintoEquation 7{15 gives 2he;vejfvf+1ir 98

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7-12 .Theresultsaresimilartothedirectcase,withsmalloscillationsthatrapidlydecay.Thereisnopopulationineithertheeorfstates. 7-13 ,forresultswithandwithoutdelayeddissipation.Ineachcase,theLindbladdissipationisincluded.Whilethedelayeddissipationhaslittleeectonthetotalpopulationoftheintermediatestate,thereisanoticeableincreaseinthepopulationofthenal,chargetransferredstate.Inthiscase,theelectronicdissipationisstillthedominantdrivingforce,butthedelayeddissipationdoeshaveaneectandshouldbeincluded.Vibrationalrelaxationenhancesthetransferintothenalstate.Thenalpopulationofthechargetransferredstateisaout3.5%.Inthismodel,thispopulationislong-lived,andcouldleadtosurfacephotovoltage. 4.4 .Ifourresultswithamemorytimearethesameasthoseobtainedusingthefullmemorykernel,thenwemaybeabletousethememorytimetospeedupoursimulations,andovercomesomeofthecomputationalproblemswhichcanarisewhenlargesystemsorlongsimulationtimesareconsidered.Thememorytimeapproachalsochangesourinitialstate.Intheprevioussimulationswebeginwiththesysteminathermaldistributionatt=0.Inthiscase,thesystemisnotonlyinathermaldistributionattimet=0,butithasalsobeeninthatstateforthedurationoftmem.Thisshouldchangethedynamicsforthedierentstatesindierentways,dependingonwhetherornottheyareinitiallypopulated.Forthegstates,whichareallinitiallypopulated,weexpecttheinitialoscillationstoeitherdisappear,orat 99

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7-12 ,weseethatmostoftheoscillationsintheg0populationhavelargelysubsidedby50,000au.Wethentakethisvalueforthememorytime,tmem,andtruncatethememorytermasshowninEquation 4{103 .Wethenrepeatthesimulationwiththesameparametersasbefore.Intheabsenceofalightpulse,weexpectmorestableinitialdynamics.Thepopulationsoftheg3andg4statesareshowninFigure 7-14 forthefullmemorykernel,andusingthememorytimecuto.Therearesomeveryslightoscillationsinthepopulationswhenusingthememorytime,butoverallitismuchmorestableinitiallythaninthefullmemorykernelcase.Theinitialstateofthesystemmorecloselyresemblesastationarythermalequilibrium.Whenapulseisadded,thedynamicsshouldbesimilarbetweenthetwoapproaches.ResultsforthegelectronicstatepopulationsareshowninFigure 7-15 ,alongwiththepreviousresultswhichincludetheentirememorykernel.Thepopulationsofthegstatesareverysimilar,althoughtherearesomesmalldierencesinthem.Theg0statehasthelargestdierence,withthememorytimeapproachgivingasmallervalueatlongtimes.However,thesedierencesaresmall,andonecanconsiderthememorytimeapproacheective.ThepopulationsoftheeandfelectronicstatesareshowninFigures 7-16 and 7-17 .Thesestatesarenotpopulatedinitially,soweexpecttoseenodierenceinthepopulations.Thisisexactlywhatisseen-thememorytimeapproachgivesthesameresults. 100

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Table7-1. ParametersforthechosentransitionsinAg3Si(111):H SystemTransition!,auE,aufijq,aun 101

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Ag3Si(111):H,reprintedwithpermissionfromD.S.KilinandD.A.Micha,J.Phys.Chem.C113,3530(2009) 102

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EnergydiagramforAg3Si(111):H,directexcitation Figure7-3. EnergydiagramforAg3Si(111):H,indirectexcitation 103

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SpectraldensityofAg3Si(111):H 104

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RealpartofthecorrelationfunctionofAg3Si(111):H 105

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ImaginarypartofthecorrelationfunctionofAg3Si(111):H 106

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Theg0populationandg02coherencewithoutapulse 107

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Totalpopulationofthegroundandexcitedelectronicstateswithandwithoutdelayeddissipation,el=200fs 108

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Totalpopulationofthegroundandexcitedelectronicstateswithandwithoutdelayeddissipation,el=1ps 109

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Populationofthee4state,withandwithoutdelayeddissipation 110

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Quantumcoherencebetweenstatese0ande1 111

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Populationofthevibrationalstatesofthegroundelectronicstatefortheindirectcasewithoutapulse 112

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Totalpopulationoftheground,excited,andnalelectronicstateswithandwithoutdelayeddissipation,el=200fs 113

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Populationsoftheg3andg4stateswithoutalightpulse,usingeitherthefullmemorykernelorthememorytime 114

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Vibrationalpopulationsofthegroundelectronicstatewiththefullmemory(dashedcurve)andusingamemorytimeof50000au(solidcurve) 115

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Vibrationalpopulationsoftheexcitedelectronicstate,e,withthefullmemory(dashedcurve)andusingamemorytimeof50000au(solidcurve) 116

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Vibrationalpopulationsofthenalelectronicstate,f,withthefullmemory(dashedcurve)andusingamemorytimeof50000au(solidcurve) 117

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118

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119

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120

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funcsThissubroutinehandlesanyoperationsthatmusthappeninthebeginningofthesimulation.Thesemayincludereadingparameters,initializingvariables,oropeningrequiredinputoroutputles. 121

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funcsThissubroutineiscalledattheendofthesimulation.Possibleoperationsperformedherearerecordingendvaluesoraverages,andclosingles. fThissubroutinecalculatesfasinequation A{1 ,wheret;(t)andz(t)areassumedtobegiven. VThissubroutinecalculatesVasinequation A{3 .Thissubroutineisonlycalledwhenimplicitintegrationisused. kThissubroutinecalculatesKasinequation A{1 ,wheret;t0;and(t0)areassumedtobegiven. MThissubroutinecalculatesMasinequation A{3 .Thissubroutineisonlycalledwhenimplicitintegrationisused. rhoThissubroutineiscalledattheendofeachpropagationstep.Itcontrolstheoutput.Inthesimplestcase,itcanrecordthecurrenttand(t)toanoutputle.However,itcouldbeusedformoredetailedoutputifdesired.Theoutputlesneededshouldbeopenedduringinit funcs,andclosedinfinal funcs. 122

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123

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AndrewLeatherswasborninLexington,Kentucky.HisfamilywouldlatermovetoVirginia,wherehisbrotherMattthewwasborn,andafewyearslaterreturntoKentucky,settlinginLouisville.In1994,AndrewwouldbeginattendingtheUniversityofKentucky,thealmamaterofhisparentsStevenandMarilyn.Afterhisjunioryear,hebecameateachingassistantforthephysicalchemistrylaboratory,andbecamearesearchassistantintheeldofquantumchemistryunderDr.RogerGrev.Hegraduatedcumlaudein1999withbachelorsdegreesinbothchemistryandmathematics.Shortlyaftergraduation,AndrewmovedsouthtoAtlantawherehespentafewyearsworkinginanenvironmentallaboratory.Duringthistime,hetaughthimselfsomebasicprogrammingtoautomatedatacollectionandreportinginthelab.Ashelearned,hediscoveredheenjoyedprogramming,anddecidedtoreturntotheeldofchemistry,withanemphasisoncomputationalwork.Inthesummerof2003,hemovedsouthagaintoGainesville,FloridatoenrollingraduateschoolandjointheQuantumTheoryProjectattheUniversityofFlorida.Hetookaninterestindensitymatrixtheory,andbegantoworkunderDr.DavidMicha. 128