Quantum Mechanical Tunneling in Chemical Physics

225 Pages·2013·2.964 MB·English

by  Hiroki Nakamura

Quantum
Mechanical Tunneling
in Chemical Physics

Quantum
Mechanical Tunneling
in Chemical Physics
Hiroki Nakamura
Gennady Mil’nikov
Boca Raton London New York
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Version Date: 20121218
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Contents
Preface...................................................................ix
Chapter1 Introduction...................................................1
Chapter2 One-DimensionalTheory.......................................5
2.1 ExactlySolvableCases....................................5
2.1.1 CaseofDelta-FunctionBarrier.....................5
2.1.2 CaseofParabolicPotentialBarrier..................6
2.1.3 CaseofEckartPotentialBarrier....................8
2.2 WKBApproximationandConnectionFormula.............10
2.3 ComparisonEquationMethod.............................11
2.4 DiagrammaticTechnique.................................13
2.5 InstantonTheoryandModifiedWKBMethod..............16
2.5.1 InstantonTheory................................16
2.5.2 ModifiedWKBMethod..........................24
2.6 EnergyLevelsinaDoubleWellPotential..................26
2.6.1 AsymmetricDoubleWellPotential................26
2.6.2 SymmetricDoubleWellPotential.................28
2.7 DecayofMetastableState................................29
Chapter3 Two-DimensionalTheory......................................33
3.1 WKBTheory............................................33
3.2 InstantonTheory........................................40
Chapter4 MultidimensionalEffects:PeculiarPhenomena...................43
4.1 EffectsofVibrationalExcitationonTunnelingSplitting......43
4.1.1 AdiabaticandSuddenApproximations.............43
4.1.2 CaseofSymmetricModeCouplingPotential.......44
4.1.3 CaseofAntisymmetricModeCouplingPotential...49
4.1.4 CaseofSqueezed(Sqz)DoubleWellPotential......50
4.2 InsufficiencyofTwo-DimensionalModel..................54
4.3 ProtonTunnelinginTropolone............................54
4.3.1 AvailableExperimentalData......................54
4.3.2 TunnelingDynamicsintheGround X˜ State........56
4.3.3 AnalysisofTunnelingDynamicsofthe
Excited A˜ State. ................................. 59
Chapter5 NonadiabaticTunneling.......................................61
5.1 DefinitionandQualitativeExplanation.....................61
5.2 One-DimensionalTheory.................................64
5.2.1 Caseof E ≤ E .................................67
t
5.2.2 Caseof E ≤ E ≤ E ............................68
t b
5.2.3 Caseof E ≤ E .................................68
b
v
vi Contents
Chapter6 MultidimensionalTheoryofTunnelingSplitting..................75
6.1 GeneralFormulation.....................................75
6.1.1 MultidimensionalExtensionofthe
InstantonTheory ................................75
6.1.2 WKBApproachinCartesianCoordinates..........82
6.1.3 WKBApproachintheCaseofGeneral
HamiltonianinCurvedSpace.....................85
6.2 HowtoFindInstantonTrajectory..........................89
6.3 HowtoUsetheTheory...................................92
6.3.1 EvaluationofthePre-ExponentialFactor...........92
6.3.2 IncorporationofHighLevelofabinitio
QuantumChemicalCalculations...................95
6.4 CaseofLowVibrationallyExcitedStates...................96
6.4.1 One-andTwo-DimensionalCases.................96
6.4.2 MultidimensionalCaseinTermsof
CartesianCoordinates............................99
6.4.3 CaseofGeneralMultidimensional
CurvedSpace..................................103
Chapter7 NumericalApplicationstoPolyatomicMolecules...............109
7.1 N-DimensionalSeparablePotentialModel................109
7.2 HydroperoxyRadicalHO ..............................111
2
7.3 VinylRadicalC H .....................................120
2 3
7.4 MalonaldehydeC O H .................................128
3 2 4
7.5 FormicAcidDimer(DCOOH) ..........................139
2
Chapter8 DecayofMetastableStates...................................149
8.1 GeneralFormulation....................................149
8.1.1 DeterminationofInstantonTrajectory............149
8.1.2 FormulationinTermsofCartesianCoordinates....151
8.1.3 GeneralCanonicallyInvariantFormulation........154
8.2 NumericalApplication..................................158
Chapter9 TunnelinginChemicalReactions..............................163
9.1 DeterminationofCausticsandPropagationin
TunnelingRegion ......................................163
9.1.1 CausticsinChaoticHenon-HeilesSystem.........166
9.1.2 CausticsinChemicalReactionDynamics.........167
9.2 DirectEvaluationofReactionRateConstant..............174
9.2.1 AdiabaticChemicalReaction....................174
9.2.2 NonadiabaticChemicalReaction.................178
Chapter10 ConcludingRemarksandFuturePerspectives..................183
Contents vii
AppendixA ProofsofEquation(2.95)andEquation(2.110)...............185
AppendixB DerivationofEquation(6.80)...............................187
AppendixC HerringFormulainCurvedSpace...........................189
AppendixD DerivationofEquation(6.97)...............................191
AppendixE ComputerCodetoCalculateInstantonTrajectory..............193
AppendixF DerivationofSomeEquationsinSection6.4.2................201
Bibliography............................................................205
Index...................................................................213

Preface
In this book we mainly discuss the following three tunneling phenomena based on
our recent theoretical developments: (1) energy splitting, or tunneling splitting, in
symmetricdoublewellpotential,(2)decayofmetastablestatethroughtunneling,and
(3)tunnelingeffectsinchemicalreactions.Thecorrespondingphysicalprocessescan
benaivelyunderstoodintermsofareactionpathinwhich(1)twoendpointsarefixed,
(2) one endpoint is fixed and one is free, and (3) both ends are free. Quantum me-
chanicaltunnelingmanifestsitselfinaverywiderangeofnaturalphenomenaandthe
subjectswedealwithhereconstituteonlyasmallportionofthewholearea.Inspite
ofthelonghistoryofstudiesofquantummechanicaltunnelingphenomena,however,
thetheoryofmultidimensionaltunnelinghasnotyetbeensatisfactorilydeveloped.
Recently, we have successfully developed practical and useful methods applicable
torealmultidimensionalsystemsfortheabove-mentionedproblems.Basictheories,
practicalmethodologies,andactualnumericalapplicationstorealmolecularsystems
are presented so that the reader can hopefully comprehend the basic concepts and
dynamics of multidimensional tunneling phenomena and use the methods directly
in various problems of molecular spectroscopy and chemical dynamics, if neces-
sary. Insufficiency of the low-dimensional treatments that are often employed for
practical problems and intriguing effects of multidimensionality are demonstrated
and clarified conceptually as well as numerically. Furthermore, attention is called
to the nonadiabatic tunneling phenomenon, which is quite ubiquitous in molecular
systemsandyetmanifestsuniqueandintriguingphenomenonincomparisonwithor-
dinarytunneling.Inthisbook,quiteabitofmathematicsisusedforthedevelopment
and explanation of basic theories; thus a background knowledge of mathematics
is required at the level of graduate students. This book is intended as a standard
reference for comprehending the phenomena and solving practical problems in the
fields of chemical physics. H. N. mainly wrote the whole contents and thus is re-
sponsibleforallchapters.G.M.sharesresponsibilityforChapters6,7,and8and
Section9.1.
We would like to thank all the collaborators for their contributions in the de-
velopment of the methods and various applications to real molecular systems that
clearly demonstrate the usefulness of our methods. H. N. would like to express his
sincere thanks to National Chiao Tung University in Hsinchu, Taiwan, and people
attheInstituteofMolecularScience,DepartmentofAppliedChemistry,Facultyof
Science of the university for providing this nice opportunity to work on this book.
Especially,heisdeeplyindebtedtoProfessorsS.H.LinandY.P.Leefortheirhos-
pitality. G. M. thanks his colleagues and friends in Okazaki who have turned this
placeintohissecondhometownafterspendingsevenwonderfulyearsintheInstitute
forMolecularScience,NationalInstitutesofNaturalSciences.Acknowledgmentis
alsoduetothefollowingsocietiesandpublishersforpermissionofreproducingvari-
ouscopyrightmaterials:AmericanInstituteofPhysics,AmericanChemicalSociety,
ElsevierScienceB.V.,andWorldScientificPublishing.
ix