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Theoretical Chemistry and Computational Modelling

Chemical Reactions

Basic Theory and Computing

TheoreticalChemistryandComputational Modelling

ModernChemistryisunthinkablewithouttheachievementsofTheoreticalandComputational Chemistry.Asamatteroffact,thesedisciplinesarenowamandatorytoolforthemolecular sciencesandtheywillundoubtedlymarkthenewerathatliesaheadofus.Tothisend,in2005, expertsfromseveralEuropeanuniversitiesjoinedforcesunderthecoordinationoftheUniversidad AutónomadeMadrid,tolaunchthe EuropeanMastersCourseonTheoreticalChemistryand ComputationalModeling (TCCM).Theaimofthiscourseistodevelopscientistswhoareableto addressawiderangeofproblemsinmodernchemical,physical,andbiologicalsciencesviaa combinationoftheoreticalandcomputationaltools.Thebookseries, TheoreticalChemistryand ComputationalModeling,hasbeendesignedbytheeditorialboardtofurtherfacilitatethetraining andformationofnewgenerationsofcomputationalandtheoreticalchemists.

Prof.ManuelAlcami DepartamentodeQuímica FacultaddeCiencias,Módulo13 UniversidadAutónomadeMadrid 28049Madrid,Spain

Prof.RiaBroer TheoreticalChemistry ZernikeInstituteforAdvancedMaterials RijksuniversiteitGroningen Nijenborgh4 9747AGGroningen,TheNetherlands

Dr.MonicaCalatayud LaboratoiredeChimieThéorique Université PierreetMarieCurie,Paris06 4placeJussieu 75252ParisCedex05,France

Prof.ArnoutCeulemans DepartementScheikunde KatholiekeUniversiteitLeuven Celestijnenlaan200F 3001Leuven,Belgium

Prof.AntonioLaganà DipartimentodiChimica Università degliStudidiPerugia viaElcediSotto8 06123Perugia,Italy

Prof.ColinMarsden LaboratoiredeChimie etPhysiqueQuantiques Université PaulSabatier,Toulouse3 118routedeNarbonne 31062ToulouseCedex09,France

Prof.OtiliaMo DepartamentodeQuímica FacultaddeCiencias,Módulo13 UniversidadAutónomadeMadrid 28049Madrid,Spain

Prof.IgnacioNebot InstitutdeCiènciaMolecular ParcCientíficdelaUniversitatdeValència CatedráticoJosé BeltránMartínez,no.2 46980Paterna(Valencia),Spain

Prof.MinhThoNguyen DepartementScheikunde KatholiekeUniversiteitLeuven Celestijnenlaan200F 3001Leuven,Belgium

Prof.MaurizioPersico DipartimentodiChimicaeChimica Industriale Università diPisa ViaRisorgimento35 56126Pisa,Italy

Prof.MariaJoaoRamos ChemistryDepartment UniversidadedoPorto RuadoCampoAlegre,687 4169-007Porto,Portugal

Prof.ManuelYáñez DepartamentodeQuímica FacultaddeCiencias,Módulo13 UniversidadAutónomadeMadrid 28049Madrid,Spain

Moreinformationaboutthisseriesathttp://www.springer.com/series/10635

AntonioLaganà • GregoryA.Parker

ChemicalReactions

BasicTheoryandComputing

AntonioLaganà DipartimentodiChimica,Biologiae

Biotecnologie

Università degliStudidiPerugia

Perugia

Italy

GregoryA.Parker HomerL.DodgeDepartmentofPhysics andAstronomy

UniversityofOklahoma Norman,OK USA

ISSN2214-4714ISSN2214-4722(electronic)

TheoreticalChemistryandComputationalModelling

ISBN978-3-319-62355-9ISBN978-3-319-62356-6(eBook) https://doi.org/10.1007/978-3-319-62356-6

LibraryofCongressControlNumber:2017947726

© SpringerInternationalPublishingAG2018

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ToGiovannaandJeanene

Preface

Energyandmasstransfersinchemicalprocessesareanintricatelandofadventure inwhichatomsandmoleculescompeteandcollaborateondifferentpathsinaway thatchallengesintellectualabilitieswhentryingtorationalizeunexpected outcomes.

ThisbookhasbeendesignedtohelpthestudentsoftheEuropeanErasmus MundusMasterin “TheoreticalChemistryandComputationalModeling” (TCCM) tofamiliarizewithboththeoreticalmethodsandcomputetechniquesusefulto handlethemicroscopicnatureofchemicalprocesses.Becauseofthis,thelevelof references,pseudocodes,andtexthasbeenkeptassimpleandasgeneralaspossible byleveragingontheexperiencegainedbyteachingthesubjectforyearsatthe homeUniversityandattheTCCMintensivecourse.Wehavealsotriedtoavoid misprintsandinaccuraciesthroughrepeatedcrosschecks.Despitethat,thebook mightnotbefreeoferrorsandweaskthereadersthefavoroflettingusknow(our emailsaregiveninthefrontpage)aboutpossibleimprovements.

Inthebook,thereaderisdriventodisentangleelementaryeventsoutofthe kineticsofcomplexsystemsinwhichreactiveandnonreactiveprocessescombine andcompeteindifferentwaysdependingontheinteractionsandmomentaofthe speciesinvolved.Outofsuchcomplexity,wegraduallysingleoutanddealwiththe keyfeatures(byleveragingpreferentiallyonelementarygasphaseprocesses)of two-,three-,four-,andmany-bodycollisions.Then,complexityisregainedto extendthetreatmenttolargesystemsbyintroducingsomeapproximations.

Thebookstartsinchapteronebyconsideringthemodelingofratecoefficientsin termsofthetransitionstate(TS)approach.Fromtheanalysisoftheweaknessofthe TSmodel(usefulforaphenomenologicalsystematizationofexperimentaldata althoughuselessforpredictions),theefficiencyofchemicalprocessesisrationalizedintermsofcollisionsoftwostructurelessbodiesusingclassicalmechanics (inwhichatomsareconsideredasmasspoints)andsimplemodelinteractions(like pureCoulombattractionand/orrepulsion,hardsphere,mixedattractionatlong rangeplusrepulsionatshortrange(Sutherland,Morse,andLennard-Jones)). Classicalmechanicscomputationalmachinery,relyingonbothanalyticaland numericalprocedurestailoredtosolverelatedNewton,Hamilton,andLagrange

equations,areanalyzedinthischapterbyassociatingasetoftrajectoriesstarting fromdifferentinitialconditionstothefateofthecollisionprocess(liketheangleof deflection)andworkingoutthevalueofquantitiesofexperimentalrelevance(like crosssectionsandratecoefficients).

Theobservedfailureoftheclassicalmechanicstreatmenttoreproducesomekey featuresofmeasureddata(liketheelasticdifferentialcrosssectionintwobody collisions)istracedbacktothequantumnatureofmolecularprocessesandtothe relateduncertaintyprinciple.Thisdrivesthereaderinchaptertwototheuseof quantumtechniquesforevaluatingthepropertiesofbothboundandelastically scatteredatom–atomsystems.Relatedquantumtreatmentsarethendiscussedand analyticalsolutionsare firstworkedoutforsomeprototypecasestotheendof guidingthereadertouseofspecialfunctions.Then,somebasicnumericaltechniquesandrelatedpseudocodesusefulforintegratingthecorresponding Schrödingerequationforgenericatom–atominteractions,areillustratedandapplied inordertocomparerelatedresultswithcorrespondingclassicalones.

Atthispoint,thereaderisreadytoabandontheconstraintthatatomsare structurelessbodiesanddealinchapterthreewiththeelectronicstructureofatoms andmolecules.Tomoveinthisdirection,wediscusssometechniquesusedfor carryingoutabinitiocalculationsofelectronicenergiesanddiscusstheadoptionof bothoneelectronfunctionsandvariationalprinciple.Alongthisline,theelectronic structureofpolyatomicmolecules,molecularorbitals,Hartree –Fock,and self-consistent field(SCF)molecularorbital(MO)modelsarediscussedinsome detail.Then,weendupbyillustratingpostHartree–Fockconfigurationinteraction, multiconfigurationself-consistent fields,andperturbationmethodsforthecalculationofelectronicenergiesandothermolecularproperties.Tobetterdealwith largersystems,mentionismadealsotosomeempiricalcorrectionssimplifyingthe electronicstructurecalculationsforlargesetsofatomsaswellasforalargenumber ofmoleculargeometriesofthesamemoleculeandalargenumberofmolecules. Finally,thetechniquesusedtoshapepotentialenergyglobalandlocalfunctional formulationsto fitthedistinctivefeaturesofcomputedabinitiovaluesarediscussedwiththespeci ficintentionofattributingtorelatedparametersaphysical correspondence.

Next,inchapterfour,conceptsandtechniquestobeusedforcarryingout dynamicalcalculationsofreactivesystemsstartingfromatom–diatomelementary processesareconsidered.Tothisend,themotionofnucleiisdisentangledfromthat oftheelectronsbyintroducingtheBorn–Oppenheimerapproximation.Then,for atom–diatomsystems,differentsetsofcoordinatesarediscussedforsinglingout thosebettersuitedforrepresentingtheinteractionandforintegratingdynamics equations.Forthelatter,differentchoicesarediscussedforclassicalandquantum treatmentsaswellasfortime-dependentandtime-independenttechniques.The integrationofdynamicsequationsallowsto figureoutthetypicalfeaturesofthe atomisticphenomenologyofatom–diatomsystemssuchastheeffectofadifferent allocationofenergytothevariousdegreesoffreedominpromotingreactivity,the importanceofprovidinganaccuraterepresentationofthepotentialenergysurface, themeritsanddemeritsofadoptingreduceddimensionalityapproaches,ordealing

quantallywithsomedegreesoffreedom(whilehandlingclassicallytheothers). Then,thediscussionisextendedalsototheusefulnessofsinglingouttheperiodic orbitsofdynamicalsystemsforrationalizingtheirreactivebehavior(includingthe categorizationoftransitionstateeffects)anddesigningproperstatisticaltreatments forlonglivingprocesses.

Atthispoint,theroadispavedforconsideringinchapter fivesystemsofhigher complexitystartingwiththefourandmoreatomonesandendingwiththosefor whichtheatomisticgranularityisdifficulttomanagewithsufficientaccuracy.The introductionofadditionaldegreesoffreedom,infact,impactsonthestructureand, accordingly,ontheformulationofthepotential.Forthisreason,thedefinition ofthequantitiestobecomputed,thecomputationaltechniquesadoptedandthe observablestobesimulatedarealsoreconsidered.Theprogressmadeinthis directionisstrictlyrelatedtotheevolutionofcomputeplatformsandthelevelof concurrencyanddistributionachieved.Thishasledtoaradicalchangeofthe organizationofmolecularsciencestowardservice-orientedprocedures,competitive collaboration,datareuse,andopenness.

Accordingly,thebookisarticulatedasfollows:inthe fi rstchapter,wedealwith theclassicalmechanicsconceptsandtheirapplicationtothetwo-bodyproblem;in thesecondchapter,wedealwiththecorresponding(twobody)quantum mechanicalconceptsandtreatments;inthethirdchapter,wemovetowardthe descriptionpolyelectronicandpolyatomicsystems,thecalculationsofrelated eigenenergies,andtheconstructionofpotentialenergysurfacesconnectingthe differentarrangementsofthemolecularsystem;inthefourthchapter,wetacklethe problemofdescribingtheatom–diatomreactivesystemsandpropertiesandillustrateaswellthedifferentmethodsforrationalizingrelatedmechanisms;inthe fifth chapter,wemovetowardmorecomplex(uptomanyatomsandmanymolecules) systemsandfocusonsynergisticmultiscalecompetitivecollaborationinthecontext ofrecentprogressmadeindistributedcomputing.Eventually,particularimportance isalsogiventothepresentevolutiontowardOpenSciencebyreferringtoaHorizon 2020fundingproposalforestablishingaMolecularscienceEuropeanresearch infrastructure.

Perugia,ItalyAntonioLaganà Norman,USAGregoryA.Parker

Acknowledgements

ALthankshisparents(VincenzoandGiuseppina)fortheirsupporttohiseducation evenduringthedifficulttimesofthepost-SecondWorldWarofthemid-twentieth century,hiswifeGiovannaforherlovingunderstandingofhisdedicationtoscience andeducation,hissonLeonardoforhiscommitmenttobothworkandfamily,his relativesandfriendsfortheirlove.

ALalsothanksthelargenumberofcolleaguesheworkedwith(seeJ.Phys. Chem120(27)4595(2016)).Thebookitselfiscoauthoredandistheresultof variouslong-lastingcollaborations.Mostoftheconceptsillustratedinitwerebetter understoodbyhimthankstosuchcollaborations.Particularhelpforthisbookwas givenbyErnestoGarciaandStefanoCrocchiantinotonlyforproducingalarge fractionofthejointlypublishednumericalresultsusedasillustrativeexamplesin thebookbutalsoforadaptingthemas figuresofthebookandbyLeonardo Belpassiforrevisingtheelectronicstructuresection.Importantforthewritingofthe bookwasalsothecontinuousinteractionwiththestudentsoftheTheoretical ChemistryandComputationalModeling(TCCM)ErasmusMundusMasterduring theclasseson “MechanismsandDynamicsofReactiveSystems.”

GAPthankshistwowonderfulparents(ByronandEdna)whotaughthimthe importanceofaneducation.Hethankshisdedicatedwife(Jeanene)andtheireight wonderfulchildren(Steven,Michael,Sheryl,Jennifer,Tamara,Marilyn,William, andChristopher)fortheirlove,patienceandexample.

GAPthankshisPh.D.advisor,lifelongmentor,andfriendProfessorRussellT Packformanyfruitfulcollaborations.GAPalsothankshispostdoctoraladvisors ProfessorsAronKuppermannandJohnC.Lightfortheirmentorshipandguidance.

ThisbookispartoftheseriesthattheteachersoftheTCCMMasterhave plannedtopublishwithSpringer.

2.2.1TheCoulombPotentialsandtheHydrogenAtom

2.2.2TheFormulationofQuantumElasticScattering

2.2.3TheQuantumElasticScatteringCrossSection

2.3.1ContinuumSolutionsforHydrogen-LikeAtomsE>0

2.3.2TheRigidSphere

2.3.3TheMorsePotential

2.4.3ApproximatingtheWaveFunction

2.4.4TheApproximationtothePotential

2.5.1SystemsofLinearAlgebraicEquations

2.5.2TheStructureoftheWaveFunctions

2.5.3TheTime-DependentMethod

3.1.1TheOne-ElectronWavefunctionApproach

3.1.3Many-ElectronWavefunctions

3.1.4TheElectronicStructureofMolecules

3.2HigherLevelAbinitioMethods

3.2.1BeyondtheHartree–FockMethod

3.2.2TheCIandMC-SCFMethods

3.2.3PerturbationMethods

3.3TowardExtendedApplications

3.3.1ComputationofOtherMolecularProperties

3.3.2DensityFunctionalTheoryMethods

3.3.3TheValenceElectronMethod

3.3.4DroppingMulticenterIntegrals

3.4FullRangeProcessPotentials ...........................

3.4.1TheThree-BodyInternuclearCoordinates

3.4.2GlobalFormulationofthePotentialEnergySurface

3.4.3LocalandMobileMethods

3.4.4Process-DrivenLocalandMobileFittingMethods

3.5.1QualitativeProblems

4TheTreatmentofFew-BodyReactions

4.1TheCombinedDynamicsofElectronsandNuclei

4.1.1TheN-BodyDynamicalEquations

4.1.2ADirectIntegrationoftheGeneralEquations

4.1.3TheBorn–OppenheimerApproximation

4.2Three-AtomSystems

4.2.1Three-BodyOrthogonalCoordinates

4.2.2Atom–DiatomReactiveScatteringJacobiMethod

4.2.3Atom–DiatomTime-IndependentAPHMethod

4.2.4TheAtom–DiatomTime-DependentAPHMethod

4.3BeyondFullQuantumCalculations

4.3.1ReducedDimensionalityQuantumTreatments

4.3.2LeveragingonClassicalMechanics

4.4.3ExperimentalObservables

5.1.1FullRangeAbInitioPESsforMany-BodySystems

5.1.2FittingPESsforReactiveandNonreactiveChannels

5.1.3Four-AtomMany-ProcessExpansion

5.1.4Four-AtomQuantumandQuantum-ClassicalDynamics

5.1.5LastMileCalculationsforCrossedBeamExperiments

5.2LargeSystemsStudiesUsingClassicalDynamics

5.2.1TrajectoryStudiesforMany-BodySystems

5.2.2SomePopularMolecularDynamicsCodes

5.2.3ForceFields ..................................

5.2.4TowardMultiscaleTreatments

5.3SupercomputingandDistributedComputingInfrastructures

5.3.1High-PerformanceVersusHigh-ThroughputComputing

5.3.2NetworkedComputingandVirtualCommunities

5.3.3TheCollaborativeGridEmpoweredMolecular

5.4TowardanOpenMolecularScience

5.4.1AResearchInfrastructureforOpenMolecularScience

5.4.2FoundationsandStakeholdersfortheMolecularOpen

5.4.3ComputeResourcesandDataManagementforMolecular

5.4.4MolecularOpenScienceUse-Cases

5.5.1ServiceLayersandDataStorage

5.5.3UserandServiceQualityEvaluation

5.5.4ACreditEconomy

5.6.1QualitativeProblems

5.6.2QuantitativeProblems

Chapter1

FromthePhenomenologyofChemical

ReactionstotheStudyofTwo-Body

Collisions

Thischapterguidesthereaderthroughthephenomenologyofthesimplestkinetics ofchemicalsystemstothemodelingoftheratecoefficientsgoverningtheirtimeevolution.Fromtheanalysisoftheweaknessofthetransitionstate(TS)modelapproach (thatisphenomenologicallyvalidbutuselessforpredicting),therateofchemical processesisrationalizedintermsofcollisionsoftwostructurelessbodiesusingclassicalmechanics.Inthisway,itispossibletofollowthespaceandtimeevolutionof thecollidingpartners.Themachineryoftherelatedclassicalmechanicsequations (Newton,Hamilton,andLagrange)isexplainedandthenumericalproceduresfor associatingclassicaltrajectoriesstartingfromdifferentinitialconditionstothefate ofthechemicalprocessisgivenoncetheinteractionisknown.Applicationstovariouspopularmodelsoftheinteraction(hardsphere,repulsiveCoulomb,attractive–repulsivepotentials,liketheLennard–Jones(LJ)andtheMorse)areconsideredfor ananalyticalandnumericalsolutionoftheproblem.

1.1FromKineticstoBimolecularCollisions

1.1.1ThePhenomenologicalApproach

Inordertobuildarigoroustheoreticalandcomputationalgroundforthedescriptionandunderstandingofchemicalreactions,onehastoscalethetreatmentofthe problemofchemicalprocessesdownfromthemacroscopicphenomenologicallevel (thatreferstothermodynamicsandkineticstreatments)tothemicroscopicone(that referstodynamicstreatments).Thescalingdownstartsfromconfiningtheanalysis togas-phasehomogeneoussystemsinordertomoreeasilyrelatetheparameters characterizingthetimeevolutionofthesystemtothevariationoftheintervening species(sayXofconcentration [X ] orpartialpressure p X )because,asiswellknown, pressure p X isrelatedtotheconcentration [X ] andthetemperature T bytheequation p X =[X ] RT .Thevariationoftheinterveningspeciesisusuallyquantifiedinterms

©SpringerInternationalPublishingAG2018 A.LaganàandG.A.Parker(eds.), ChemicalReactions,TheoreticalChemistry andComputationalModelling,https://doi.org/10.1007/978-3-319-62356-6_1

ofthetime t dependenceofthereactionrates v(t ) asfollows:

where k ( T ) isthetemperature-dependentratecoefficientandthepower m isthe orderofreactionwithrespectto [X ],thereactants’concentration.

Afurthersteptowardascalingdowntoamicroscopic(molecular)levelthe descriptionoftheconsideredreactiveprocessistheformulationof k ( T ) intermsof quantitiesdependingontheenergy E ofthesystem(andwheneverappropriatewe consideralsoitspartitioninginthevariousdegreesoffreedom)likethecrosssection σ (E),theprobability P (E),andthescattering S(E)matrices.Afterestablishingsuch relationshipsandworkingoutthenumericalvalueofthemicroscopicquantitiesusing appropriate abinitio treatments,onecanregainthewaybacktophenomenology byfirstrelatingthecomputedvaluestotheratecoefficientsandthenevaluating theoreticallythemeasuredsignalandconcentrationoftheinvolvedspecies.

Asweshallconsiderindetaillateron,ofparticularimportanceforthatpurpose aretheproductintensitiesmeasuredinbeam-scatteringexperimentsandgenerated bysingle-collisioneventsbecausetheyrefertoquantitiesthatcanbecomputed usingrigorous abinitio techniquesforalargevarietyofsystems.Suchdirecttheory versusexperimentcomparisonpavesthewaytotheunderstandingofthemicroscopic foundationsofchemicalprocessesandtheconsequentaccurateevaluationofthe averagedkineticsandthermodynamicsquantities.

Forthisreason,thestartingpointofthisbookistheanalysisoftheproperties ofrarefied(verylowpressure p )gasesinwhichsingle-collisionprocesseswithno exchangeofenergy(isolatedsystems)andnoexchangeofmass(closedsystems)to theexteriorcanbetreated.Therelatedprocessisusuallywrittenas

withthereactionrate v(t ) beingdefinedatagiventemperature T as

inwhichthevariationofreactants(consumed)hasanegativesignwhilethatofthe products(generated)hasapositivesign.

Amoregeneralcaseistheoneinwhichmorethanonespeciesparticipateto theprocess(thenumberofparticipatingspeciesiscalledMolecularity),like,for example,thebimolecularoneinwhichthereactantspeciesareAandBandthe productspeciesareCandD(thatis αA + β B → γ C + δ D),leadingto

t ) =−

ThevariationoftheinterveningspeciescorrespondingtothatofEq. 1.1 readsnow

where m + n istheorderoftheprocesswith m and n beingnotnecessarilyintegers. Intheparticularcaseof m =0,1,and2,theratecoefficienttakes,respectively,the followinganalyticalforms:

where [A ]o istheconcentrationofAattheinitialtime to and [A ] isitsconcentration attime t ,

asillustratedintheupperrowofFig. 1.1.Amoregeneralformulationcanbeobtained usingthedimensionlessvariables η =[A ]/[A ]o and τ = k ( T )[A ]m 1 o t byplotting

Fig.1.1 Powellplotsform=0,1,and2oftheconcentrationsasafunctionoftime t (upperpanel) andof η asafunctionofln τ (lowerpanel)

η againstln τ (seetheexamplesketchedinthelowerrowofFig. 1.1 againfor m = 0,1,and2).

Undertheassumptionsmentionedabove,onecan,inprinciple,estimatethevalue oftheratecoefficientatadifferentgiventemperature T andinitialconcentration [X ]o values(suchmeasurementsareperformedusingeitherchemicalorphysical propertiesofthereactivesystem)bymeasuringthecurrentconcentration [X ] ofthe involvedspeciesatdifferentelapsedtimes.

1.1.2RealisticKineticModels

However,evenseemingly,simplegas-phasereactionsaredifficulttointerpretinthis wayduetotheuncertaintyoftheexperimentalmeasurementsandtothecomplexity oftheactualreactionmechanisms.Intherealworld,chemicalprocessesgenerally occurthroughacombinationofseveraldifferentsimpler(elementary)stepsthat produceandconnectvariousintermediatesofdifferentstabilitiesandgiveriseto complexreactionmechanismscombininginitiation,propagation,chainpropagation, branching,termination,etc.steps.Asanexample,letusconsiderthecombustionof puremolecularhydrogenwhosemechanism(typicallyconsistingofasetofsome tensofelementarychemicalreactions)hasbeenreduced,forthesakeofsimplicity, tothoselistedinTable 1.1.

InthissimplifiedschemeoftheH2 +O2 combustionprocess,wecanformulate thereactionrateofproducingH2 Oasfollows:

(hereinafterthedependenceof k on T willbedroppedwhennotexplicitlyrequired). Ifweexcludeexplosionregimes,wecanalsomakethestationarystateassumption (equatingtheratesofproductionandconsumption)forthemostimportantintermediatesOH,H,andOand,therefore,wecanwrite

Table1.1 Reducedsetof elementarychemical reactionsinwhichthe combustionofmolecular hydrogencanbedecomposed

Reaction

Role

H2 +O2 → 2OH (1)Initiation

H2 +HO → H+H2 O (2)Propagation

H+O2 → OH+O (3)Chainbranching

H2 +O → OH+H (4)Chainbranching

H+O2 +M → HO2 +M (5)High-pressure termination

H+wall → H-wall (6)Low-pressuretermination

BysummingEqs. 1.11 and 1.12,onecanobtaintheconcentrationofHatthe stationarystate

Totheendofeliminatingthetermscontaining [O],wecanutilizeEq. 1.12 inside (1.10)andreorderthetermscontainingthe [OH ] and [H].Thisleadstotheexpression

thatcanbeusedinEq. 1.9 totheendofformulatingthefinalrateoftheprocess

Asmentionedabove,inordertoobtainEq. 1.14,itwasnotonlynecessaryto reducetheconsideredsetofequationsandadopttheassumptionofthestationary stateregimebutalsotoexcludetheconditionsintheexplosionregime.

Accordingly,itmakesnosensetouseaboveequationstoestimatethevalues ofthe k ’sinvolvedifthereisnostrictcontroloftheoperatingconditions.Amore theoreticallysolidapproachtotheproblemofcomputingthereactionrateisthe oneintegratingintimecomputationallythesystemofkineticequationsdefining theoverallchemicalprocessbypossiblytakingthefullsetofinterveningreactions (includingthelessefficientonesespeciallyiftheyinitiateparallelprocessesgeneratingnewspeciesofpotentialimpactonthefinalfateoftheprocess).Thisismade possiblebytherapidevolutionofcomputerarchitecturesandplatformsprovided thatan,atleasttheoretical,accurateevaluationofthedetailedratecoefficientsofthe interveningprocessescanbeworkedout.

1.1.3TheTransitionStateTheoryApproach

Thesimplestapproachtotheevaluationoftheratecoefficientofachemicalreaction istheuseofthermodynamicsdatawithinatransitionstatetheory(TST)treatment.

Thethermodynamicstreatmentassumestheexistenceofanintermediate( X ‡ )which isinthermalequilibriumwiththereactants.Theintermediatecanthendissociateto formproducts

Theequilibriumconstant K ‡ ofsuchprocessdefinedas

canberelatedto k f viatherelationship

Thisallowsustorelatetheratecoefficient k f tothestandardfreeenergychange

thankstotherelationship

andtheVan’tHoffequation

withthequantities G o‡ , H o‡ ,and S o‡ beingthefreeenergy,theenthalpy (orheat),andtheenthropyofactivationthermodynamicfunctions,respectively.By consideringthat H o‡ isequivalenttotheactivationenergy E o f atconstantpressure, onecanwrite

Forillustrativepurposes,wesketchhere(seeFig. 1.2)thesimplereactivesystem ofanatom,A,andthediatom,BC,givingthediatomABandtheatomC. FromthebasicassumptionsoftheTSTapproach,onehasthefollowing:

1.Thereactiveprocessoccursonasinglepotentialenergysurface(PES)onwhich apathmadeofthelocalminimalocatedonanarrangementcontinuityvariable (namedminimumenergypath(MEP)coordinateandrepresentedasasolidline intheplotofFig. 1.2)connectsreactants(A + BC)toproducts(AB + C);

Fig.1.2 AsketchofthetransitionstatetheorymodelfortheexothermicreactionA + BCgiving AB + C.Thesolidblacklineistheminimumenergypathconnectingreactantsandproducts,which showsinitscentralpartthetransitionstateABC(atthetop)andtheexothermicity Eo (atthe bottom).Ontheleft-handside(LHS),theforwardprocessactivationenergy Eo f isshown,while thebackwardprocessactivationenergy Eo b isshownontheright-handside(RHS)

2.OntheMEP,onecanlocateanintermediateregionsupportinganABCconfiguration(thetransitionstate(TS))separatingreactantsfromproducts1 ;

3.TheTSpopulationisinequilibriumwiththatofreactantsaccordingtorelationship

is,indeed,thedifferenceinenergyinternalbetween theTSandthereactants(oftencalledactivationenergy)and

S isthepartition functionofthe S system;

4.AttheTS,thesystemcrossesover(withoutrecrossingback)totheproductswith afrequency k B T / h ;

5.ThebasicTSTforwardrate k TST ( T ) (hereinafter,weshalldropthelabel f when notstrictlynecessary)issetequaltotheproductofquantitiesassociatedwith steps3and4giving

CorrectionstothebasicTSTratecanbeintroducedbyformulatingrecrossing, tunneling,andstericeffectscorrectionsandbyproviding,aswell,amoreappropriate

1 TheTSisusuallyassociatedwithasaddle.However,thisassociationisanarbitraryassumption because(asitwillbediscussedinsomedetailinChap. 4)theregionsofthePESdividingtrajectories backreflectedfromthosecrossingoverareassociatedwithperiodicorbitsdividingthe(potential energy)surface(PODS).PODScanbemorethanone,donotnecessarilysitonasaddle,andcan berecrossedbythesystem.

definitionofitslocationand/orenergetics(e.g.,variational,includingcentrifugal barriers,etc.)ontheMEPorevenbyincludingstatisticalconsiderationsderivedby modelorreduceddimensionalitydynamicalcalculations.

1.1.4TowardDetailedSingle-CollisionStudies

IthastobestressedherethattheTSTapproachhasnopredictivepower.Accordingly, itsmostpopularuseisasaphenomenological(empirical)equationwhosecoefficients aretreatedasbest-fitparameters.Asamatteroffact,theTSTformulationoftherate coefficientisextensivelyemployedasapracticalwayofimplementingtheintegrationofkineticequationsofthechemicalsubsystemofmultiscale(atmospheric, combustion,etc.)simulations,themanagementofknowledgecontents,andinartificialintelligenceprocedures.

Fortunately,inthelasthalfcentury,bothexperimentalandcomputationaltechnologieshaveprogressedenormously.Forthispurpose,itisworthrecallinghere that,beforeapplyinganymodeloraccuratetreatmenttoreactiveprocesses,itshould betakenintoaccountthat,forathermalizedsystemoccurringonasinglePES,the overall k ( T ) resultsfromstatisticallyweighted(dependingonthetemperatureof interest)sumofthedetailed i (initial)to f (final)statecontributions k if ( T ) (whose internalenergiesare i and f ,respectively)asfollows:

Thestate-to-stateratecoefficients k if ( T ) canbeformulatedintermsofthestateto-statecrosssection σif ( E tr ) asfollows:

where E tr isthetranslationalenergyofthesystemand g ( E tr ) isthetranslational energydistribution.Foragasinthermalequilibriumattemperature T ,thefunction g ( E tr ) hastheform

Accordingly,bysubstituting(1.26)into(1.25),onehas

if ( T ) =

Fig.1.3 LHS:apictureoftheCMBapparatusbuiltattheUniversityofPerugia(IT);RHS:adraft ofitstechnicalscheme

Alongthisline,abigleapforwardinthestudyofchemicalreactionshasbeen representedbytheassemblageofcrossedmolecularbeam(CMB)experimentalapparatusesofwhichasketchofthemachinebuiltattheUniversityofPerugia(IT)is giveninFig. 1.3.AsillustratedbytheRHSofthefigure,theapparatusconsistsofa highvacuumchamberintowhichthebeamsofthetworeactantsspeciesareinjected andcollimatedtointersect.Thankstothehighvacuum,thereactantmoleculescan collideonlyonceandtheproductsaremeasuredinmassanddirectionbyarotating massspectrometer.Theirspeedisinsteadevaluatedfromthetimetheytaketoreach thedetectionpoint.

TheoutcomeofCMBapparatusesisasetofhighlyinformativedataonmainly bimolecularcollisionsprovidingawealthofinformationonthe

• primaryreactionproducts,

• reactionmechanisms,

• structureandlifetimeoftransients,

• internalenergyallocationofproducts,and

• detailednatureoftheinteraction,

thatcanbeobtained(especiallybycouplingtheCMBtechnologywithlaserones whichcannotonlystateselectivelyoperateonreactantsandproductsbutcanalso interactwithtransientspecies)throughcomputations.Computetechnologiesare,in fact,theothervitalingredientsofsuchstudiesbecausethroughheavycomputations (thatarebecomingincreasinglypopular,thankstothehigh-performanceandhigh throughputfeaturesofmodernplatforms),itisbecomingroutinelyfeasibletowork outacomputationalestimate(virtualexperiment)ofthemeasuredpropertiesofthe realexperimentandfullyandaccuratelyprobetheinteractionsofthesystemsunder consideration.

Theinteractionscomingintoplayinmolecularprocessesare,indeed,aspecific featureofchemicalreactionsstudiesbecauseoftherichnessofthevarietyofforces involved.Asamatteroffact,suchavarietyisgeneratedbytheinterplayofionic andcovalent,permanentandinduced,shortandlongrange,andtwoandmanybody

interactions.2 InordertoillustratemostoftheconceptsdevelopedbyMolecular Scienceinitsapproachto“understanding”thenatureofchemicalprocesses,we shallfocusinthefollowingonthetwoparticles(twobodies)beforeundertakingthe studyofmorecomplexsystems.Thegreatersimplicityofthisstudywillallowusto approachinasmootherwaythebasicconceptsofchemicaltransformationsandof therelatedcomputationalapproaches.

1.2ClassicalMechanicsofTwo-ParticleCollisions

1.2.1ReferenceFrameandElementaryInteractions

Atpresent,weassumethatthetwobodiesarerepresentedastwopointsorspherical objectswithmassesinthethree-dimensionalphysicalspace(afterallalargepartof thecollisionprocesstakesplaceatdistancesatwhichthestructureofthecolliding bodieshashardlyanyeffect)andthatthebehaviorofsuchsystemscanbedescribed bythelawsofclassicalmechanics(whichisareasonablestartingpointformostof thedynamicalcomputationalchemistryapplications).

Inanarbitrarylaboratoryfixedreferencesystem(X,Y,Z)lab (thickarrowsof Fig. 1.4),accordingtoclassicalmechanics,asystemformedbytwocollidingparticles AandBisuniquelydefinedinphysicalspacebythetwopositionvectors r A and r B andthetwomomentumvectors p A and p B defined,respectively,asm A v A andm B v B withmi beingthemassand vi thevelocityofeitherparticleAorB).

Thesevectorsareusuallyreferredtoaslaboratory(lab)axisframe(X,Y,Z)lab ,a frameofCartesianorthogonalX,Y,andZaxeshavingafixedoriginandorientation likethoseofthephysicallaboratory(inthefollowing,weshallomitthespecification lab whennotstrictlynecessary).Thepositionvectors r A and r B oftheparticlesA andB,respectively,canberepresentedeitherintermsoftheirprojectionsX A ,Y A , Z A ,andX B ,Y B ,Z B overtheX,Y,andZaxes(notshowninFig. 1.4 forthesake ofsimplicity)orintermsofthecorrespondingsphericalpolarcoordinates(i.e.,the moduli r A and r B ofthetwovectors r A and r B andtherespectiveangles A , A and B , B 3 Similarrepresentationscanbeadoptedforthevectors p A and p B Arepresentationofthetwo-bodysystemisomorphouswiththeabove-described onecanbeobtainedusing rCM ,thepositionvectorofthecenter-of-mass(CM)of thesystem,and r AB (or r forshort),thepositionvectorofparticleBwithrespectto particleA.

2 Ionicistheinteractionbetweenchargedparticles(ions)inwhichthenumberofpositivecomponents (e.g.,protons)differsfromthatofnegativelychargedparticles(e.g.,electrons).Covalentisthe interactionassociatedwithevenlysharedparticles(e.g.,twoatomsequallysharingtheelectrons). Permanentisastablefeatureoftheparticles(e.g.,thedipolemoment).Inducedisatemporary featureassociatedwiththepresenceofanotherparticle.Shortandlongrangesrefertothedistance betweentheparticles.

3 Thesphericalpolarcoordinatesoftheparticle i makeuseof r i (themoduleofthepositionvector ri )andofitsorientationangles i and i

Fig.1.4 LABORATORYFRAME:Thevectors r A and r B definethepositionoftheAandB collidingbodieswithrespecttotheoriginofthe(X,Y,Z)lab frame.Relatedangles A , A , B , and B ofthecorrespondingpolarcoordinatesarealsogiven.Therelatedmomentumvectorsare p A =m A v A and p B =m B v B with vi = ri .CMFRAME:Thevectors rCM and r define,respectively, thepositionoftheCMwithrespecttotheoriginofthecoordinatesystemandthepositionofparticle BwithrespecttoA

Thepositionvector r isrepresentedseparatelyinFig. 1.5 (LHSpanel)usingthe (x,y,z)CM frame.ItisworthpointingoutheretheuseofsmalllettersfortheCM referenceframe(asopposedtothecapitalonesusedforthe lab referenceframe),of thesphericalpolarangles ϑ and ψ ,andoftheorigincoincidingwiththeCM.The CMframemayhaveanarbitraryorientation(usuallydefinedintermsofthevalues ofangles α, β , γ (namedEulerangles)bywhichthe lab frameorientationneedsto be(continuously)rotatedsoastocoincidewiththeplanedefinedbytheposition vector r anditsvelocity ˙ r (thisframeiscalledbodyfixed(BF))andtheangleformed by r andtheBFzaxis(notshownhere)iscalleddeflectionangle θ ).

Inthesamefigure,weshowintheupperRHScorner(usinga(X,Y,Z)lab frame representation)athree-pointstroboscopicpicture(screenshots)ofthe r A (dasheddottedline)and r B (dottedline)positionvectorsofthetwobodiesduringacoplanar repulsivecollision(relatedmomentaaregivenasboldarrowsand r valuesaregiven asdashedlines).InthelowerRHScorner,the r valuesarerepresentedseparatelyin thesequenceofoccurrence.

Asimplifiedillustrationofthesimplestcasesoftwo-bodyinteractionsisgivenin Fig. 1.6:repulsive(centralrow)inwhichthetrajectoriesofthetwoparticlesdiverge andattractive(lowerrow)inwhichthetrajectoriesofthetwoparticlesconverge foracentralpotentialdependingonlyonthedistance r (r =|r A r B |)ofthetwo bodies.Alsoshowninthefigureisthecaseofnointeraction(upperrow),foracentral potentialdependingonlyonthedistance r (r = r A r B )ofthetwobodies.

Fig.1.5 TopRHSpanel:athree-shotstroboscopicviewofthecoplanarA + Brepulsivecollision trajectory(twothin-solidlinesoccurringonaplaneparalleltotheX,Yaxesofthe(X,Y,Z)lab frame)completeoftherelativethree r A (dasheddotted)and r B (dotted)shotsandoftheassociated threevelocityvectors(boldarrows).Thethreeshotsofvector r arealsoshownasdashedlines. LHSpanel:thevector r representedinits(x,y,z)CM referenceframe.LowerRHSpanel:thethree shotsof r shownintheirsequenceofoccurrenceinthe(x,y,z)CM referenceframe •Repulsive interaction •No interaction

Fig.1.6 Hypotheticaltrajectoriesoftwobodiesplacedonthetwoparallelplanesoftheirinitial positionandvelocity.Toppanel:nointeraction,centralpanel:repulsiveinteraction,lowerpanel: attractiveinteraction

Therepulsiveinteraction(liketheonebetweentwochargesofthesamesign)is completelyintuitive.Therepulsionisverylargeatsmalldistance(atthedistance wheretheparticlesnearlycoalesce)and,asthetwobodiesmoveawayfromone another,theinteractionmonotonicallydecreases.Similarly,thesecondtypeofforce attraction(suchasthatbetweentwochargesofoppositesign)isalsonegligible atlargedistances.Theattractiveforcetendstobringthetwobodiestogetherand themagnitudeoftheinteractionmonotonicallyincreasesastheyapproacheach other.Thesamereasoningappliestothemultipoleswithdifferentorientations.The interactionisrepulsiveifthepolaritiesarefacingmainlyofthesamesign,andattract eachotherifthepolaritiesarefacingmainlyofdifferentsigns.

Yet,themostimportantroleinchemistryisplayedbythethirdtypeofinteraction thatcanchangefromattractivetorepulsiveatdifferentinternuclearseparations.Itis lessintuitivebutevenmoregeneralandrealistic.Thistypeofinteractiontendstobe moreattractiveatlargedistancesandthenbecomesmorerepulsiveatsmallerinternucleardistances.Thus,thissystemhasanequilibriumdistancewheretheinteraction ismostattractive.

1.2.2TheEquationsofMotion

Aswehavejustseen,acompletedescriptionofthesystemoftwoparticlesAandB, withmasses m A and m B ,mustbebasednotonlyonthepositionvectors r A and r B butalsoontheirvariationovertime.Forthisreason,itisnecessarytodeterminethe (linear)momenta p A and p B ofthetwoparticlesaswell.Thespacedefinedbythe setofpairsofconjugatedvariables(position ri andmomentum pi forallparticles ofthesystemunderconsiderationiscalled“phasespace”).Differentpointsofthe phasespacearecharacterizedbydifferentstatesofaclassicalsystem(theposition vectors r A and r B andtheirrespectivemomenta pr A and pr B ).InNewton’ssecond law,themotionofeachparticleisdescribedbycoupled6N(whereNisthenumberof particlesofthesystem)mathematicalequations Fi

and

inthecoordinatesofthechosenframe.TheHamiltonianofthetwo-bodysystemis givenby

where E isthetotalenergy, T isthekineticenergy,and V isthepotentialenergy (thatinourcaseis V (r A , r B )).Theenergy E isaconstantwhenweareconsideringa conservativesystem.TheHamiltonian H = E forthissystemalongwiththeinitial conditionsdeterminesthefateofthesystemforalltimes(completelydeterministic). Asweshallseelater,thepositionsandmomentacanbedeterminedbysolving Hamilton’sequationsofmotion,foreachatom i

Inthelaboratoryframe,thesystemmaybedescribedusingeitheranX,Y,Z Cartesiancoordinaterepresentationor,alternatively,othercoordinates.Thetime

dependenceofthevectors pri and ri canbeobtainedusingdifferentclassicalformulations.InsteadofthejustmentionedpopularNewton’sformulation,weshalluse theformulationofHamilton4 whereonehasapairoffirst-orderordinarydifferential equationsoftheconicallyconjugatedvariables priW and r iW (withW=X,Y,Zbeing thesetofchosenorthogonalcoordinatesystem)

foratotaloftwelveequations.Theseequationscanbeintegratednumericallywith standardtechniqueswhichwewillmentionlater.Inonlyafewspecialcaseswillthe equationsofmotion(1.29)haveanalyticalsolutions.Inthevastmajorityofcases,no analyticalsolutionsareknown.Asweshallseelater,analyticalsolutions,whenthey areavailable,havebeengeneratedonlyafterperforminglaboriousanalyticaltransformations(yet,theyhavetheadvantageofallowingusefuldecompositionsofthe problemleadingtobothinterestinginsightsandsignificantreductionofdimensionalityoftheproblem).Veryoften,however,accurateapproximationstothesolution canonlybefoundusingnumericaltechniques.

Inthecaseofconservativesystems,thepotential V dependsonlyontherelative distancebetweenthetwoparticles.Then,theHamiltonianofthesystemassumesa formparticularlyconvenientwhenoneusesCMcoordinates rCM andtheinternal coordinates r AB (ormorefrequentlysimply r ).Theirdefinitionisimmediate:

ExpressingtheHamiltonian(1.28)inthesecoordinates,wehave

ThefirsttermofthisHamiltoniandescribesthemotionoftheCM.Thesecond andthethirdtermsintheHamiltoniandescribeinsteadtherelativemotionofthe twoparticles.Since V (r ) isindependentof rCM ,themotionoftheCMrelative tothelaboratorysystem(X,Y,Z)lab isthatofafreeparticlewhichmovesata constantvelocity(inertialsystem).TheCMcoordinatesystem(x,y,z)CM differs fromthelaboratoryoneinthatitisnotfixedinspacebutmovesataconstantvelocity withrespecttothelaboratorycoordinatesystem(X,Y,Z)lab .Accordingly,theCM coordinatesystem(x,y,z)CM showninFig. 1.5 canalsobeusedastheoriginofan

4 AnotherpopularformulationoftheequationsofmotionistheLagrange’sone

where L = T V istheLagrangianofthesystem.

1.2ClassicalMechanicsofTwo-ParticleCollisions15

inertialcoordinatesystem.TheCMsetofcoordinatesisalsosaidtobebarycentric (center-of-gravity).Theinertialsystemhastheimportantpropertyofbearinganull totallinearmomentum.

Asdescribedintherelatedcaption,theRHSpanels((a)and(b)ofFig. 1.5)compareacollisionoftwoequalmassesoccurringontheX,Yplane(asweshallseelater thisisacollisionundertheeffectofarepulsiveinteraction)bygivingtheevolution ofthe r A and r B pairinaLabframe(panela)andtheevolutionof r inaCMframe havingthesameorientationastheLabonein(panelb).Thelatterillustratesgraphicallythereductionofcomplexityassociatedwiththedecompositionofthe r A and r B problemintoa r anda rCM problem,thankstotheintroductionofconservationlaws (inourcase,theconservationofCMmomentum).Thetwo-centerproblemis,infact, transformedintoaone-centerproblemofaparticleofmass μ,equaltothereduced massofthesystem,subjecttothepotentialorinterparticleinteractionasshownby theuseoftheonlyvector r .Accordingly,thenumberofHamilton’sequationstobe integratedisreducedfromtwelvetosix,thoserelatingtothethreeCartesiancomponentsofthepositionvector r (r x , r y , r z )(ortheirrespectivecomponentsofthe polarrepresentation r , ϑ, ψ (seeFig. 1.5))andthethreecomponentsofitsconjugated momentum pr AB .

1.2.3TheDeflectionAngle θ

Forthecentralfieldproblemunderconsideration,itispossibletofurtherdecompose theproblemusingsymmetrypropertiesofthesystem.Asalreadymentioned,infact, inthecaseofthecentralfield,theinteractionpotentialdependsonlyonthemagnitude ofthecoordinate r andispreferableforconvenienceandclaritytousesystematically, aswealreadydo, r and pr insteadof r AB and pr AB forthediatomicvariables.Forthe samereasons,wealsoassumethat V (r ) → 0when r →∞ (exceptwhenexplicitly said),bysettingthezeroofenergytotheasymptoticvalueofthepotential.Thenby choosing,asisdoneinFig. 1.7,theorientationoftheaxesofthesystemofreference sothattwoofthem(forexample, z and y )lieintheplanedeterminedbytheinitial velocityvector,thesystemwillremainconfinedtothesimpletrajectory(d ψ /d t = 0), sincethepotentialdependsonlyonthemagnitudeofthevector r .Followingthis transformation,theclassicalHamiltoniancanbewrittenexplicitlyinthefollowing way5 :

5 Infact,seeFig. 1.7,wehaveforthecomponents( z , y )of r , z =−r

and y = r sin θ (θ π /2 = ϑ)orbydifferentiatingwithrespecttotime

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Title: Dr. Courtney's guide to happy marriage

Author: A. B. Courtney

Release date: June 24, 2022 [eBook #68396]

Language: English

Original publication: United States: A. B. Courtney, 1894

Credits: Demian Katz, Craig Kirkwood, and the Online Distributed Proofreading Team at https://www.pgdp.net (Images courtesy of the Digital Library@Villanova University.)

*** START OF THE PROJECT GUTENBERG EBOOK DR. COURTNEY'S GUIDE TO HAPPY MARRIAGE ***

Multum in Parvo Library.

Entered at Post Office as Second-Class matter

Vol. I. J, 1894. Published Monthly No. 6.

Dr. Courtney’s Guide TO Happy Marriage.

Smallest Magazine in the World. Subscription price, 50 cts. per year. Single copies, 5 cents each.

PUBLISHED BY A. B. COURTNEY, 671 Tremont Street, Boston.

MARRIAGE GUIDE.

The

Newly Wed.

We will suppose you have read and profited by the excellent suggestions contained in an article published in a previous issue of this library, under the head of “How to Woo, Win and Wed.” You have succeeded in winning the object of your affections, and have set sail upon the sea of matrimony. It is often called a troublous sea. Such indeed it proves in truth to be for those who embark upon it not fully prepared for the voyage. Do you really, married or unmarried, know the full meaning of the word marriage. It is something far more serious than a mere civil contract or religious ceremony. These are only the marks of outward show. There is a good deal behind these —a good deal that you ought to know, and of which, perchance, you are ignorant. The aim of this article is to give some practical advice to those who have already entered into the bonds of matrimony, and it will not, it is hoped, come amiss to those about to marry, but who have not yet taken the step that leads to misery or bliss. “Marriage,” says Selden, “is a desperate thing; the frogs in Æsop were extremely wise; they had a great mind to some water, but they would not leap into the well, because they could not get out again.” Would that most of us possessed the wisdom of the fabled frogs.

Purpose of Marriage.

Marriage ought to be consummated as the result of mutual love and esteem, and not for the purpose of simply gratifying the desires of our lower nature. This last consideration is, it must be acknowledged, a controlling one with a great many people, and marriages contracted in this manner are not generally, and cannot hope to be, happy ones. The sacredness of the marriage relation ought never to be violated. We must not forget that we are rational beings with a will to withstand the weaknesses of our animal natures.

Happy and Unhappy Marriages.

A happy marriage is without doubt the ideal state of living, the end for which mankind has always striven, while an unhappy marriage is a veritable hell on earth. Examples of both of these states need not be given. We see them every day. To one who reads the daily papers regularly with particular note of the records of divorces, assaults of drunken or jealous husbands, the faithlessness of women and the elopements, the thought must present itself that there are more unhappy marriages than happy ones. This, fortunately, is not true. Where we read of one unhappy marriage and its terrible consequences there are ten happy ones of which the world never hears.

“Marriage,” writes Addison, “enlarges the scene of our happiness and miseries.”

“It is a mistake,” says another writer, “to consider marriage merely as a scheme to happiness; it is also a bond of service, it is the most ancient of that social ministration which God has ordained for all human beings, and which is symbolized by all the relations of nature.”

Still another writer says: “Married life appears to me a sort of philosophical discipline, training persons to honorable duties, worthy of the good and wise. Few unmarried people are affected as they ought to be toward the public good, and perceive what are really the most important objects in life.”

How to be Happy.

Those who wish to lead happy married lives cannot do better than to follow a few rules which we present herewith.

Husband and wife ought to maintain entire confidence in each other, have no secrets each from the other; don’t quarrel; have forbearance for each other’s failings; you have neither of you married an angel. Remember, husband, that of every dollar you own, fifty cents belongs to your wife; she is an equal partner with you in the business of life; don’t compel her to become a mere household drudge, working for her board and clothes; she did not marry you for that. Try to see the good points of your husband or wife; don’t magnify the faults; we are all only human; don’t have the first quarrel and the second will never come.

Golden Precepts.

Let the rebuke be preceded by a kiss.

Don’t require a request to be repeated.

Never should both be angry at the same time.

Be lovers all your life. Let the courtship be continued after marriage.

Never neglect the other for all the world beside.

Let each strive always to accommodate the other.

Let the angry word be answered with a kiss.

Bestow your warmest sympathies in each other’s trials.

Criticize as little as possible, but if you find it necessary to criticize, make your criticism in the most loving manner possible.

Make no display of the sacrifices you make for each other.

Never make remarks calculated to bring ridicule upon the other.

Never deceive; confidence once lost can never be wholly regained.

Always use the most loving and gentle words when addressing each other.

Let each study what pleasure he can bestow upon the other during the day.

Always leave home with a tender good-bye and loving words. They may be the last.

Consult and advise together in all things.

When you feel like quarrelling, discuss the whole thing together and you will come to an amicable settlement.

Never reproach the other, especially in the presence of others, for an error which was committed with a good motive and with the best judgment at the time.

Don’t tell your friends the faults of your husband or wife.

The Duty of the Wife.

The wife’s place is in the home. That is her proper sphere of action, and the one in which she ought to be most happy.

Don’t devote time to society matters while your house duties need to be done. Don’t spend your time in making clothes for some naked boys in Dooloboo who do not need them, while your own children go about with patched or torn clothing.

Greet your husband with a smile on his return from work.

Don’t scold your husband.

Don’t think your husband ought to be perfect. Are you yourself perfect?

Be careful and economical in the expenditure of money for household purposes. (By the way, your husband ought to make you the treasurer of the firm.)

Always dress neatly for your husband’s sake. You used to do it before marriage. Imagine your husband is still your lover.

Treat your husband’s friends politely. This is an important part of your wifely duty.

Don’t spend money extravagantly for dress or other personal adornment.

The Husband’s Duty.

Don’t think the woman you have married is yours, body and soul.

Don’t be niggardly in money matters.

Don’t withhold from your wife your tender love and sympathy. Regard her as your dear sweetheart all through life. Always treat her with tender consideration.

Don’t try to argue with her. A woman cannot reason; she is guided rather by her womanly intuition, which is rarely at fault.

Follow the counsel of your wife. Many a man has done so and succeeded. Wisdom and foresight are possessed by women, as well as men.

Let your wife understand fully your business. Don’t let her think you are earning fifty dollars a week when you are earning only thirty dollars, or vice versa.

Give your wife all the pleasure you can. She needs it.

Do not go about enjoying yourself with boon companions while your wife toils at home.

Don’t spend money for rum while your wife has to make over her old dresses.

If your wife is worthy of it (most wives are), praise her. Women like praise.

Women are not as strongly built as men, and are thus likely to be often in delicate health. If so, your wife may be petulant and cross. Make allowances for this and don’t scold her.

Don’t interfere with your wife in the performance of the duties that belong peculiarly to her.

Give her money enough to dress well, even if you have to make sacrifices to do so.

Treat her mother with becoming respect.

Marrying for a Home.

A good many women are foolish enough and dishonorable enough to marry merely for a home—foolish because they cannot expect such a marriage to result happily, and dishonorable because they have deceived the man they married. Love ought to be the basis of all marriages.

Joy and Pleasure.

A married life is not one of unalloyed bliss. We ought not to expect this. It has its pains as well as its pleasures. As Margaret Fuller says: “Deceive not thyself by over-expecting happiness in the marriage state; look not therein for contentment greater than God will give, or a creature in this world can receive, namely, to be free from all inconveniences. Marriage is not, like the hill of Olympus, wholly clear without clouds.” When misfortune comes to us, and all the rest of the world deserts us, we have those at home to whom to look in certain expectancy of sympathy and encouragement—wife and children. As John Taylor says: “A married man falling into misfortune is more apt to retrieve his situation in the world than a single one, chiefly because his spirits are soothed and retrieved by domestic endearments, and his self-respect kept alive by finding that, although all abroad be darkness and humiliation, yet there is a little world of love at home over which he is monarch.”

A married man is more apt to labor for the good of all mankind, while a single man is apt to be more selfish in his aims and endeavors. The interests of a single man centre round himself, while those of a married man embrace his whole family, and in a larger degree the whole community.

Don’t Board.

If you are a newly-married couple, don’t board. Go to keeping house as soon as possible. Don’t get married if you can’t do this. A young woman ought to learn the duties and pleasures of housekeeping as soon after marriage as possible. If she boards out, she has little to occupy her time, and is apt to pass her days in reading silly novels, or to fall into that terrible habit of gossiping. She ought to find pleasure in working for her husband, and she can work for him best only in her own home, preparing his meals and by the performance of other wifely duties. As a mere boarder in somebody else’s home, she cannot do this. There are many drawbacks and perplexities about housekeeping, to be sure, but these ought to be met and contended with with a brave spirit. In the end, having conquered, it will be found that the pleasures far outweigh them. A woman ought to take pleasure in looking after the home nest, and ought not to leave to some unsympathetic boarding-housekeeper the task of preparing the daily meals for her husband.

The Mother-in-Law.

A mother-in-law is often a blessing, but few will admit the truth of this statement. Many men, however, have found her to be a ministering angel in disguise. It is better, nevertheless, for the newly-married couple to live by themselves, if possible, and not be under the sway of a mother-in-law, or other relative, however kind and unselfish she or they may be. They ought to strike out for themselves. A mother-inlaw, however good her intention may be (and no doubt her intentions are always good), is pretty sure to prove meddlesome and dictatorial to a greater or less degree, and end by being an element of discord in what ought to be a happy home. You will find it better to paddle your own canoe, even if you do not make very rapid or easy progress at first.

Marriage a Lottery.

Marriage has been called a lottery Fortunately it is a lottery in which there are a great many prizes. Let us give you this bit of advice in regard to marriage as a lottery: Try to think that you have drawn a prize. Thinking so will go a good way toward making the supposition a pleasant reality. Never lose sight of this fact. If your husband or wife does not come up to the high standard which you have hoped for, try to stifle the disappointment. It might be worse. Start out well on the sea of matrimony. “Many a marriage,” writes Frederika Bremer, “has commenced like the morning, red, and perished like a mushroom. Wherefore? Because the married pair neglected to be as agreeable to each other after their union as they were before it. Seek always to please each other; lavish not your love to-day; remember that marriage has a morrow, and again a morrow.” Another writer on the same subject wisely says: “The happiness of married life depends on a power of making small sacrifices with readiness and cheerfulness. Few persons are ever called upon to make great sacrifices or to confer great favors; but affection is kept alive, and happiness secured, by keeping up a constant warfare against little selfishness.”

Jealousy.

It is said that jealousy is an unmistakable sign of true love. We are supposed to be jealous of those whom we most dearly love. There may be some truth in this assertion, but we incline rather to think that it is open to dispute. Only the weakest are the most jealous. But the “green-eyed” monster has caused too much misery to treat the matter lightly. Where mutual love and respect and perfect confidence exist, and where the married parties are sensible, intellectual people, jealousy rarely exists. Jealousy, such as that which is said to be an infallible sign of true love, is exhibited chiefly by silly young women, married and single, who have no serious grounds for suspicion against the object toward which their jealousy is directed. If you are of an incurably jealous disposition, never marry any one of the opposite sex who is good looking. This will only add fuel to the fires of your jealousy. Marry a homely person—the homelier the better— one who is not likely to receive undesirable attentions from the opposite sex after marriage. But the best advice of all is don’t be jealous. Be sensible. Nine times out of ten you have no real cause for jealousy after all. To be jealous of your husband or wife implies that you have married a partner who cannot be trusted. Have you married such a person? Probably not. Then why be jealous?

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