Microsolvation of the Cyanyl Radical

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MICROSOLVATIONOFTHECYANYLRADICAL: COMPETITIONBETWEENHYDROGENBONDINGAND ELECTROSTATICNON-COVALENTINTERACTIONS BY JACOBBLOOM AThesis SubmittedtotheDivisionofNaturalSciences NewCollegeofFlorida inpartialfulllmentoftherequirementsforthedegree BachelorofArtsinPhysicsandChemistry UnderthesponsorshipofDr.DonColladay Sarasota,Florida April,2009

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Acknowledgments ThisworkwassupportedbyNSFgrantNSF-CHE0749868.ThankstoH.F.Schaefer andtheCCQCatUGAfortheopportunitytoconductthisresearch.ThankstoS. E.WheelerandD.Colladayformentoringmethroughoutthisprocess.ThankstoH. M.Jaegerforthefruitfuldiscussionsregardingelectrostaticnon-covalentinteractions. FiguresweregeneratedusingHFSmol.[1] ii

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TableofContents Acknowledgmentsii ListofFiguresv ListofTablesvi Abstractvii 1Introduction1 1.1CyanideandItsRadical.........................1 2ElectrostaticsandMultipoleExpansion4 2.1Electrostatics...............................4 2.2ScalarPotentialandWork........................6 2.3ACloserLookAtTheScalarPotential.................10 2.4ScalarPotentialInSphericalCoordinates................12 2.5LegendrePolynomials...........................14 2.6SphericalHarmonics...........................17 2.7MultipoleExpansion...........................20 3QuantumMechanics24 3.1SchrdingerEquation...........................24 3.2LinearOperators.............................26 3.3ParticleInABoxAndTheWaveFunction...............28 3.4SpectroscopicModels...........................34 3.5HydrogenAtom..............................40 3.6Approximations..............................42 4ComputationalMethods45 4.1Overview..................................45 4.2Specics..................................46 iii

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5Results48 5.1OneandNoWaterCyanideandCyanylRadicalComplexes.....48 5.2TwoandThreeWaterCyanideandCyanylRadicalComplexes....52 5.3Conclusions................................56 5.4FutureWork................................56 References58 iv

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ListofFigures 3.1PlotsOf sin 2 nx a ............................32 5.1OneWaterStructuresOptimizedAtThecc-pVTZCCSDTLevelOf Theory...................................49 5.2PartiallyOptimizedStructuresIllustratingTheElectrostaticInteractionsInCN H 2 OStructureIAtThecc-pVTZCCSDTLevelOf Theory...................................50 5.3DZP++B3LYPStructures........................51 5.4CN H 2 OIsomerizationEnergySurfaceComputedAtTheDZP++ B3LYPLevelOfTheorykcal/mol...................51 5.5TwoWaterStructuresOptimizedUsingDZP++B3LYPWithRelative EnergiesInkcal/mol...........................54 5.6ThreeWaterStructuresOptimizedUsingDZP++B3LYPWithRelativeEnergiesInkcal/mol.........................55 v

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ListofTables 5.1CN H 2 O n BindingEnergiesE rel ,HydrationEnergiesE hyd ,and CNStretchingFrequencies CNstrcm )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 AtThecc-pVTZCCSDT andDZP++B3LYPLevelsOfTheory.................52 5.2CN )]TJ/F17 11.9552 Tf 7.085 -4.338 Td [(H 2 O n BindingEnergiesE rel ,HydrationEnergiesE hyd ,and CNStretchingFrequencies CNstrcm )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 AtThecc-pVTZCCSDT andDZP++B3LYPLevelsOfTheory.................53 5.3B3LYPAdiabaticElectronAnitiesandCCSDTDipoleMoments.53 vi

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MICROSOLVATIONOFTHECYANYLRADICAL: COMPETITIONBETWEENHYDROGENBONDINGAND ELECTROSTATICNON-COVALENTINTERACTIONS JacobBloom NewCollegeofFlorida,2009 ABSTRACT Atheoreticalstudyofmicrosolvatedcyanylradicalwithwaterhasbeencarried out.Cyanylradicalandwaterareofinteresttobothastrophysicsandbiochemistry. Atheoreticalstudyofthemicrosolvatedsystemshouldprovefruitfulforfuturestudy intheseelds.Furthermore,thisworkillustratesthecompetitionbetweenhydrogen bondingandelectrostaticnon-covalentinteractions.Thisstudyutilizeddensityfunctionaltheoryforbothcyanylradicalandcyanidecomplexedwithuptothreewaters. Theenergiesofcyanylradicalandcyanidewithzeroandonewaterwerebenchmarked usingcoupledclustertheorywithsingle,double,andperturbativetripleexcitation approximations.Relativeenergies,hydrationenergy,frequencies,dipolemoments, andadiabaticelectronanitieswerecomputedforthecyanylradicalandcyanide systems. Dr.DonColladay DivisionofNaturalSciences vii

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Chapter1 Introduction Thischapterprovidesbackgroundonthecyanylradicalmicrosolvatedsystem. Throughoutthiswork,CN isusedtodenotecyanylradical,CN )]TJ/F17 11.9552 Tf 11.193 -4.339 Td [(isusedtodenote cyanide,and isusedtodenoteanintermolecularinteraction. 1.1CyanideandItsRadical Non-covalentinteractionshydrogenbonding, ,andelectrostatic etc. have emergedasafruitfulareaofresearchinrecentdecades,dueinparttothepivotal roleofsuchinteractionsinbiologicalsystems.CN H 2 Oconstitutesanintriguing systemtostudythecompetitionbetweentwoprototypicalnon-covalentinteractions: hydrogenbondinganddipole-quadrupolecomplexes.ThenitrogeninCN isexpected tofunctionasaprotonacceptor,whileconsiderationofsimplemultipolarinteractions suggeststheformationofacomplexbetweenwaterandthecarbonendofthecyanyl radical. ClustersofcyanylradicalCN andwaterareofinteresttoboththeeldsof astrophysicsandbiochemistry.Cyanylradicalisfoundincometaryicesandina varietyofenzymes,thoughtheprecisechemistryofthissimplediatomicradicalhas notbeenfullyelucidatedineithercase.TheoriginofCN incometsisstillun1

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clear.[2]Inbiochemistry,thefateofcyanylradicalsproducedbythereactionof CN )]TJ/F17 11.9552 Tf 12.046 -4.338 Td [(withhorseradishperoxidaseandmitochondrialcytochromecoxidaseissimilarlyunsettled.[3]Arelatedissueisunderstandingtheabilityofthewhiterotfungus Phanerochaetechrysosporium todecomposeenvironmentalpollutantssuchasDDT, benzo[a]pyrene,andevencyanide.[4][5]Thisabilityhasthepotentialfortheenvironmentalcleanupofareascontaminatedbycyanide.[5] InterestinCNradicalsincometaryicewasspurredbyworkpublishedin1984in whichBockele-Morvan etal. showedthattheupperlimitoftherateofhydrogen cyanideproductionwaslowerthanthatofcyanylradicalincomets.[6]Subsequently, manyotherpossibleparentmoleculesforcyanylradicalhavebeenproposed,includingcyanogens,[7]cyanoacetylene,[7][8]anddiacetylene.[8]Anotherpossiblesourceof cyanylradicalincometsisthroughdirectaccumulationfromdust.[9]Denseinterstellarcloudsarecomposedoficesofwater,carbonmonoxide,carbondioxide,methanol, hydrogen,methane,ammoniaandcyanide-likespecies.[10]CyanylradicalisofparticularinterestinthiscontextsinceR-CNcompoundsconstitutepossiblebuildingblocks ofaminoacids.[11]Detailedstudiesofcyanylradicalinamicrosolvatedenvironment willfacilitateourunderstandingofthechemistryofcometaryiceandinterstellar clouds.FurtherstudyofCN andofthemooncouldhelpdeterminefromwhere anywateronthemooncame,sinceonetheoryisthatitaccumulatedfromthevery cometsthatcontaincyanyl.[12] Cyanylradicalisinvolvedinthestudyofproteinsaswellsincecyanideisused asaninhibitorinthestudyofhemoproteins,includinghorseradishperoxidaseand mitochondriacytochromecoxidase.[13]Horseradishperoxidaseconvertscyanideto cyanylinthepresenceofhydrogenperoxide[14]andcytochromecoxidaseisableto convertwithoutthepresenceofhydrogenperoxide.[15]Inbothcases,whencyanide isconvertedtoCN ,theenzymesareinhibitedevenfurtherthanifcyanideremainsunchanged.[3]CyanylisthoughttoreactwiththeironprotoporphyrinIXof 2

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horseradishperoxidaseandacysteineresidueincytochromecoxidase.[3]Thechemistryofcyanylradicalinthepresenceofasmallnumberofwatermoleculeswillbe vitaltounderstandingthefateofCN inthesesystems.Thesetypesofstudiescan alsobenettheenvironmentthroughthewhiterotfungus.LigninperoxidaseH2in whiterotfungusbehavessimilarlytothatofhorseradishperoxidaseinthatitcan oxidizecyanidetocyanylwheninthepresenceofhydrogenperoxide.[5]Thecyanyl isthenfurthermineralizedintocarbondioxide.[5] Wereportatheoreticalstudyofthemicrosolvationofthecyanylradicalinwater, includingdetailedanalysesofnovelclustersofCN andwater.Asacomparison,and tostudytheevolutionoftheelectronanityofCN inamicrosolvatedenvironment, CN )]TJ/F17 11.9552 Tf 7.085 -4.338 Td [(H 2 O n clusterswerealsoexamined.CN H 2 O n providesfertilegroundfor examiningthecompetitionbetweenhydrogenbondingandelectrostaticnon-covalent complexesinarelativelysimplesystem.Denitiverelativeenergiesofisomersof CN H 2 OandCN )]TJ/F28 11.9552 Tf 9.077 -4.339 Td [( H 2 Owerecomputedusing abinitio energies.Carefully calibrateddensityfunctionaltheorymethodswereusedtostudyCN H 2 O n and CN )]TJ/F17 11.9552 Tf 7.085 -4.339 Td [(H 2 O n n=0-3. 3

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Chapter2 ElectrostaticsandMultipole Expansion ThischapterfollowsJohnD.Jackson's ClassicalElectrodynamics .[16]Itprovidesan overviewoftheclassicaldescriptionofhowelectrostaticinteractionswork.Those unfamiliarwithmultipoleexpansionandelectrostaticswillndthischapteruseful. 2.1Electrostatics Coulombshowedthroughexperimentationhowtwobodiesinteractwhenseparated byadistancemuchlargerthantheirsizes.Hediscoveredthat: ~ F = kq 1 q 2 ~x 1 )]TJ/F25 11.9552 Tf 11.761 0 Td [(~x 2 j ~x 1 )]TJ/F25 11.9552 Tf 11.76 0 Td [(~x 2 j 3 ; .1 where ~ F istheforceexertedupontherstpointcharge, q 1 ,duetoofthesecondpoint charge, q 2 .Thepointchargesarelocatedat ~x 1 and ~x 2 respectively. Anotherconceptthatisusefultointroduceatthisearlystageisthatoftheelectric eld.Thisisdenedastheforceperunitchargeduetochargesatotherlocations. Thiseldisnormallydenotedas ~ E .Theelectriceldwillexertaforceonacharge 4

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q 1 asfollows: ~ F = q 1 ~ E: .2 Thefollowingrelationshipcanthenbemadefortheelectriceldgeneratedbythe secondpointcharge, q 2 ,bycomparingequations2.1and2.2.Thefollowingequation assumesthat q 1 isthesamechargeinequations2.1and2.2. ~ E ~x = kq 2 ~x )]TJ/F25 11.9552 Tf 11.76 0 Td [(~x 2 j ~x )]TJ/F25 11.9552 Tf 11.761 0 Td [(~x 2 j 3 : .3 Asthisequationnolongerdependson q 1 ~x 1 hasbeenreplacedbyageneralcoordinate location ~x .TheconstantkthatisusedintheseequationshasanSIvalueof 0 )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 where 0 hasavalueof 8 : 854 10 )]TJ/F23 7.9701 Tf 6.587 0 Td [(12 F/mandaGaussianunitvalueof1.Equation 2.3andlinearsuperpositionofelectriceldsallowsforanelectriceldequationthat incorporatesmanycharges: ~ E ~x = k n X i =1 q i ~x )]TJ/F25 11.9552 Tf 11.761 0 Td [(~x i j ~x )]TJ/F25 11.9552 Tf 11.76 0 Td [(~x i j 3 : .4 Ifthesechargesaresmallandnumerousenough,theycanbedescribedbyadistributionfunction ~x 0 where q = ~x 0 x 0 y 0 z 0 .Thisallowsareplacementofthe sumwithanintegral: ~ E ~x = k Z ~x 0 ~x )]TJ/F25 11.9552 Tf 11.761 0 Td [(~x 0 j ~x )]TJ/F25 11.9552 Tf 11.761 0 Td [(~x 0 j 3 d 3 x 0 : .5 ThedeningsetofrelationsforelectrodynamicsisMaxwell'sequations.These equationsgovernelectromagneticinteractionswithinavacuum,andcanbeextended tobevalidinmaterials.Thisextensionisunnecessaryforthisworksincethe moleculesinteractinavacuum.ThevacuumequationsexpressedinGaussianunits 5

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areasfollows: ~ r ~ E =4 ; ~ r ~ B )]TJ/F23 7.9701 Tf 13.151 4.707 Td [(1 c @ ~ E @t = 4 c ~ J; ~ r ~ E + 1 c @ ~ B @t =0 ; ~ r ~ B =0 ; .6 where isthechargedensity, ~ J isthecurrentdensity, ~ E istheelectriceld,and ~ B isthemagneticeld.Inanelectrostaticenvironment, ~ J =0 ,andeldsandcharge distributionsaretime-independent. ~ r ~ E =4 ; ~ r ~ B =0 ; ~ r ~ E =0 ; ~ r ~ B =0 : .7 Thesecondandfourthrelationsof2.7implythat ~ B isconstantinanelectrostatic environment. 2.2ScalarPotentialandWork Acommontermusedinelectrostaticsisthescalarpotential.Denitionofthisterm startswithequation2.5.Partoftheintegralcanberewrittenusingagradient ~ r : ~x )]TJ/F25 11.9552 Tf 11.76 0 Td [(~x 0 j ~x )]TJ/F25 11.9552 Tf 11.76 0 Td [(~x 0 j 3 = )]TJ/F25 11.9552 Tf 10.434 3.022 Td [(~ r 1 j ~x )]TJ/F25 11.9552 Tf 11.761 0 Td [(~x 0 j : Since ~ r isthederivativewithrespecttox,notx',itcancomeoutoftheintegral: ~ E = )]TJ/F25 11.9552 Tf 9.299 0 Td [(k ~ r Z ~x 0 j ~x )]TJ/F25 11.9552 Tf 11.76 0 Td [(~x 0 j d 3 x 0 : .8 6

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Thiscanberewrittenintermsofanewsymbol, ~x ,calledthescalarpotential: ~x = Z ~x 0 j ~x )]TJ/F25 11.9552 Tf 11.76 0 Td [(~x 0 j d 3 x 0 : .9 Equation2.8becomes: ~ E = )]TJ/F25 11.9552 Tf 10.434 3.022 Td [(~ r : .10 ThisisconsistentwithMaxwell'sequationsof2.7.Thatis,thecurlofthegradient ofawellbehavedfunctioniszero: ~ r ~ E = )]TJ/F25 11.9552 Tf 10.434 3.022 Td [(~ r ~ r =0 : TherstofMaxwell'sequationswillbeexplicitlyveriedlaterinthissectionand section2.3. Itispossibletodescribeworkintermsofthescalarpotential.Workisdened astheintegraloftheforceappliedoveradistance.Theworkdonebyaforceonan objectmovingfrompointAtopointBis: W = )]TJ/F31 9.9626 Tf 11.291 14.058 Td [(Z B A ~ F d ~ l: .11 Thiscanberewrittenintermsofthescalarpotential.Recalltherelationsbetween force,electriceld,andscalarpotentialinequations2.2and2.10.Theworkperformedonapointcharge q i is: W i = )]TJ/F25 11.9552 Tf 9.299 0 Td [(q i Z B A ~ E d ~ l = q i Z B A ~ r d ~ l = q i Z B A d = q i B )]TJ/F22 11.9552 Tf 11.955 0 Td [( A : .12 Theworkcanbeextendedtothatrequiredtoassembleasystemofpointcharges. Theworkequationsimpliesforapointcharge q i goingfrominnityto ~x i inaregion oflocalizedelectricelddescribedby .Thissimpliedformisthencombinedwith 7

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thescalarpotentialforasystemofn-1thischoicewillbecomeevidentshortlypoint chargesgivenby: ~x i = n )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 X j =1 q j j ~x i )]TJ/F25 11.9552 Tf 11.76 0 Td [(~x j j : Theworkequationbecomes: W i = q i ~x i = q i n )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 X j =1 q j j ~x i )]TJ/F25 11.9552 Tf 11.76 0 Td [(~x j j : .13 The q i pointchargebecomesthenthtermofthesystemofchargesinthefollowing twosums: W = n X i =1 X j > > > > > < > > > > > > : 4 q 4 P i q i 4 R V ~x d 3 x: .16 TogetadierentialformofGauss'slaw,thedivergencetheoremisusedinconjunctionwiththeintegralformofGauss'slaw.Thedivergencetheoremstatesthat 8

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forawell-behavedvectoreld,thevolumeintegralofdivergenceisrelatedtouxby: I S ~ A ~nda = Z V ~ r ~ Ad 3 x: ThiscombinedwiththeintegralformofGauss'slawyields: Z V ~ r ~ Ed 3 x =4 Z V ~x d 3 x; whichyieldstherstofMaxwell'sequationsof2.7: ~ r ~ E =4 : .17 Combiningthisrelationwithequation2.10, ~ E = )]TJ/F25 11.9552 Tf 10.434 3.022 Td [(~ r ; yieldsPoisson'sequation: r 2 = )]TJ/F22 11.9552 Tf 9.298 0 Td [(4 : .18 WithPoisson'sequationandequation2.15,aformulacanbedevelopedthatputs emphasisontheelectriceld.Thispermitstheinterpretationthattheenergyis storedintheeldsurroundingtheparticles.Workbecomes: W = )]TJ/F22 11.9552 Tf 9.299 0 Td [(1 8 Z r 2 d 3 x overallspace.ThroughGreen'stheorem,thisbecomes: W = 1 8 Z ~ r 2 d 3 x = 1 8 Z ~ E 2 d 3 x: 9

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2.3ACloserLookAtTheScalarPotential Multipoleexpansionsofelectrostaticpotentialsareusefulinunderstandingmolecularinteractions.Thoughelectrostaticinteractionisnottheonlywaythatmolecules interact,itisamajorpathwayandcanbeagoodapproximationtogeneralbehavior. Legendrepolynomialsandsphericalharmonicsarerequiredforacompletemathematicalunderstandingofmultipoleexpansion.Beforetheseareintroducedtothe scalarpotential,afurtherexaminationofthescalarpotentialisrequired.First,the Laplacian, r 2 ,isappliedtoequation2.9. r 2 = r 2 Z ~ x 0 ~x )]TJ/F25 11.9552 Tf 12.832 2.735 Td [(~ x 0 d 3 x 0 = Z ~ x 0 r 2 0 @ 1 ~x )]TJ/F25 11.9552 Tf 12.832 2.735 Td [(~ x 0 1 A d 3 x 0 : .19 TheLaplaciangoestozeroatallpointsexceptwhere ~x = ~ x 0 .Thiswillbe demonstratedshortlyafterabriefreviewoftheDiracdeltafunction. TheDiracdeltafunction, x ,iszeroatall x otherthan x =0 ,whereitis innity.Ausefulpropertyofthisfunctionisthat R b a x dx =1 if a > < > > : f if A contains x =0 0if A doesnotcontain x =0 : .20 Thisagreeswiththepreviousintegralforthechoicefx=1.Thisrelationship canbeextendedtothree-dimensionsusingthefollowing: Z V f ~ x 0 ~x )]TJ/F25 11.9552 Tf 12.832 2.734 Td [(~ x 0 dx 0 3 = 8 > > < > > : f ~x if V contains ~x 0if V doesnotcontain ~x: .21 TheLaplaciantermactingon 1 j ~x )]TJ/F26 7.9701 Tf 7.582 2.159 Td [(~ x 0 j behavesliketheDiracdelta;itiszeroatall pointsexceptwhere ~x = ~ x 0 .Thiscanbeeasilyshownbymovingtheorigintothat of ~ x 0 ,replacing ~x )]TJ/F25 11.9552 Tf 12.833 2.734 Td [(~ x 0 with r and r 2 withit'sdenitioninsphericalcoordinates. 10

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r 2 1 r = 1 r d 2 dr 2 r 1 r = 1 r d 2 dr 2 1 =0 ; wherer 6 =0 ; .22 and lim r 0 1 r d 2 dr 2 1 isundened : .23 ThisexpressionhasthesamepropertiesastheDiracdelta.Theydonot,however, necessarilyhavethesamevolumeintegralvalues.Todeterminethedierence,itwill beassumedthat r 2 1 j ~x )]TJ/F26 7.9701 Tf 7.582 2.159 Td [(~ x 0 j = C ~x )]TJ/F25 11.9552 Tf 12.833 2.735 Td [(~ x 0 ,where C issomeconstant. Assumingthat r 2 0 @ 1 ~x )]TJ/F25 11.9552 Tf 12.833 2.734 Td [(~ x 0 1 A = C ~x )]TJ/F25 11.9552 Tf 12.833 2.734 Td [(~ x 0 ; .24 Z r 2 0 @ 1 ~x )]TJ/F25 11.9552 Tf 12.832 2.734 Td [(~ x 0 1 A d ~ x 0 = C Z ~x )]TJ/F25 11.9552 Tf 12.833 2.735 Td [(~ x 0 d ~ x 0 ; .25 Z r 2 0 @ 1 ~x )]TJ/F25 11.9552 Tf 12.832 2.734 Td [(~ x 0 1 A d ~ x 0 = C: .26 Tonishtheintegral,alimittrickisused. Z r 2 0 @ 1 ~x )]TJ/F25 11.9552 Tf 12.832 2.734 Td [(~ x 0 1 A d ~ x 0 lim a 0 Z 1 r d 2 dr 2 r p r 2 + a 2 d ~ x 0 # ; lim a 0 )]TJ/F31 9.9626 Tf 11.291 14.059 Td [(Z 3 a 2 r 2 + a 2 5 = 2 d ~ x 0 # ; lim a 0 [ )]TJ/F22 11.9552 Tf 9.299 0 Td [(4 ] ; )]TJ/F22 11.9552 Tf 31.216 0 Td [(4 : r 2 0 @ 1 ~x )]TJ/F25 11.9552 Tf 12.833 2.734 Td [(~ x 0 1 A = )]TJ/F22 11.9552 Tf 9.298 0 Td [(4 ~x )]TJ/F25 11.9552 Tf 12.832 2.735 Td [(~ x 0 : .27 11

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ThisrealizationallowsustoreplacetheLaplaciantermwithaDiracdelta: r 2 = Z ~ x 0 r 2 0 @ 1 ~x )]TJ/F25 11.9552 Tf 12.832 2.734 Td [(~ x 0 1 A d 3 x 0 ; = Z ~ x 0 h )]TJ/F22 11.9552 Tf 9.298 0 Td [(4 ~x )]TJ/F25 11.9552 Tf 12.833 2.734 Td [(~ x 0 i d ~ x 0 ; = )]TJ/F22 11.9552 Tf 9.299 0 Td [(4 ~x = )]TJ/F22 11.9552 Tf 9.299 0 Td [(4 : .28 Equation2.27hassignicantconsequencesinestablishingmultipoleexpansions ofelectrostatics. 2.4ScalarPotentialInSphericalCoordinates TheLaplaceequationcanbewritteninsphericalcoordinateswiththefollowingform: 1 r @ 2 @r 2 r + 1 r 2 sin @ @ sin @ @ + 1 r 2 sin 2 @ 2 @ 2 =0 : .29 Ifthescalarpotentialisassumedtohavetheform: = U r r P Q ; .30 then r 2 = PQ d 2 U dr 2 + UQ r 2 sin d d sin dP d + UP r 2 sin 2 d 2 Q d 2 =0 : .31 Thevariablescanthenbeseparatedbymultiplyingbyafactorof: r 2 sin 2 =UPQ r 2 sin 2 1 U d 2 U dr 2 + 1 r 2 sin P d d sin dP d !# + 1 Q d 2 Q d 2 =0 : .32 12

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The -dependentpartoftheequationmustbeheldataconstantsetequalto )]TJ/F25 11.9552 Tf 9.298 0 Td [(m 2 : 1 Q d 2 Q d 2 = )]TJ/F25 11.9552 Tf 9.299 0 Td [(m 2 ; .33 where Q m = e im : .34 Forthisequation, m isanintegersothat Q m issinglevaluedasphiincreasesby 2 The -independentpartofequation2.32becomes: r 2 sin 2 1 U d 2 U dr 2 + 1 r 2 sin P d d sin dP d !# = )]TJ/F25 11.9552 Tf 9.298 0 Td [(m 2 ; .35 r 2 U d 2 U dr 2 + 1 P sin d d sin dP d )]TJ/F25 11.9552 Tf 19.133 8.088 Td [(m 2 sin 2 # =0 : .36 The r -dependentpartofequation2.36issetequalto l l +1 ,where l isaninteger valuetomake P well-denedat =0 and = .Thisyieldsthetwofollowing equations: 1 sin d d sin dP d + l l +1 )]TJ/F25 11.9552 Tf 19.133 8.088 Td [(m 2 sin 2 # P =0 ; .37 d 2 U dr 2 )]TJ/F25 11.9552 Tf 13.151 8.088 Td [(l l +1 r 2 U =0 : .38 Equation2.38canbeeasilysolvedfor U intheform: U = Ar l +1 + Br )]TJ/F26 7.9701 Tf 6.587 0 Td [(l ; .39 whereAandBarearbitraryconstantsthathaveyettobedetermined. 13

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The -dependentequation,2.37,issolvedusingAssociatedLegendrePolynomials. 2.5LegendrePolynomials Tosolvefor P ofequation2.37itisuseful,andcustomary,toreplace cos with x : d dx )]TJ/F25 11.9552 Tf 11.955 0 Td [(x 2 dP dx # + l l +1 )]TJ/F25 11.9552 Tf 21.59 8.087 Td [(m 2 1 )]TJ/F25 11.9552 Tf 11.955 0 Td [(x 2 # P =0 : .40 ThepreviousequationiscalledthegeneralizedLegendreequation.ThesolutionsofthisequationarecalledtheAssociatedLegendrefunctions.Tosimplifythis discussion,wewilltreatthespecialcasewhere m =0 : d dx )]TJ/F25 11.9552 Tf 11.955 0 Td [(x 2 dP dx # + l l +1 P =0 : .41 Therangeof x inthesefunctionsisdependentonthefactthat x =cos ,allowing P tobesinglevalued,nite,andcontinuousonadomainof )]TJ/F22 11.9552 Tf 9.299 0 Td [(1 x 1 .The solutionsfor P areassumedtobeapowerseries: P x = x 1 X j =0 a j x j ; .42 where istobedetermined.Valuesforalphadependonthe a j valuesthatarenot equaltozero.Thiscanbeshownbysubstitutingfor P inequation2.41: 1 X j =0 f + j + j )]TJ/F22 11.9552 Tf 11.955 0 Td [(1 a j x + j )]TJ/F23 7.9701 Tf 6.587 0 Td [(2 )]TJ/F22 11.9552 Tf 11.955 0 Td [([ + j + j +1 )]TJ/F25 11.9552 Tf 11.956 0 Td [(l l +1] a j x + j g =0 ; .43 = )]TJ/F22 11.9552 Tf 11.955 0 Td [(1 a 0 x )]TJ/F23 7.9701 Tf 6.586 0 Td [(2 + +1 a 1 x )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 + 1 X j =0 + j +2 + j +1 a j +2 )]TJ/F22 11.9552 Tf 11.955 0 Td [([ + j + j +1 )]TJ/F25 11.9552 Tf 11.955 0 Td [(l l +1] a j x + j : .44 Forbothofthesum-independenttermsof2.44,eitherthe a j coecient,orthe 14

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coecienttermneedstobeequaltozero: if a 0 6 =0 ; then )]TJ/F22 11.9552 Tf 11.956 0 Td [(1=0 if a 1 6 =0 ; then +1=0 9 > > = > > ; ; .45 withthegeneralformfor a j +2 6 =0 of: a j +2 = + j + j +1 )]TJ/F25 11.9552 Tf 11.955 0 Td [(l l +1 + j +1 + j +2 # a j : .46 Therelationsofequation2.45areequivalentandonecanchooseeither a 0 or a 1 to benon-zero.Choosingbothtobenon-zerowouldberedundant. a 0 willbearbitrarily chosenasournon-zeroconstantandalltheoddconstantswillbesettozero.This meansthat =0 or =1 .Regardlessofwhichseriesischosen,theserieswill convergewhenever x 2 < 1 andwilldivergeunlessitterminateswhen x = 1 .It isthereforeimperativetorestricttheseriestoonethatterminatesatsomepointso thatthereisanitesolutionat x = 1 Equation2.46,andthefactthatboth and j arepositiveintegersorzero,shows theneedfor l tobezeroorapositiveinteger.If l isevenorzero,thenonlythe =0 caseterminatesat x = 1 .If l isodd,thenonlythe =1 caseterminatesat x = 1 TherstfewoftheresultingLegendrepolynomialsfollow.Thesepolynomialsare normalizedsothat P l =1 P 0 x =1 ; P 1 x = x; P 2 x = 1 2 x 2 )]TJ/F22 11.9552 Tf 11.956 0 Td [(1 ; P 3 x = 1 2 x 3 )]TJ/F22 11.9552 Tf 11.956 0 Td [(3 x ; P 4 x = 1 8 x 4 )]TJ/F22 11.9552 Tf 11.956 0 Td [(30 x 2 +3 ; P 5 x = 1 8 x 5 )]TJ/F22 11.9552 Tf 11.956 0 Td [(70 x 3 +15 x : .47 ThesepolynomialscanbeeasilyreproducedbytheRodrigues'formula: 15

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P l x = 1 2 l l d l dx l x 2 )]TJ/F22 11.9552 Tf 11.955 0 Td [(1 l : .48 Animportantpropertyofthesepolynomialsisthattheyformacompleteorthogonalsetovertheinterval )]TJ/F22 11.9552 Tf 9.298 0 Td [(1 x 1 .Thiscaneasilybeshownbyintegratingthe productofequation2.41intermsof l andaLegendrepolynomial P l 0 x Z 1 )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 P l 0 x d dx )]TJ/F25 11.9552 Tf 11.955 0 Td [(x 2 dP l dx # + l l +1 P l x dx =0 ; .49 )]TJ/F25 11.9552 Tf 11.955 0 Td [(x 2 P l 0 x dP l dx 1 )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 | {z } a + Z 1 )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 2 6 6 6 4 x 2 )]TJ/F22 11.9552 Tf 11.955 0 Td [(1 dP l dx dP l 0 dx | {z } b + l l +1 | {z } c P l 0 x P l x | {z } d 3 7 7 7 5 dx =0 : .50 Partavanishesatthelimitsofintegration,whilepartsbandd,when l and l 0 are exchanged,donotchange.Theonlypartthatdoeschangeispartc. Z 1 )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 2 6 6 6 4 x 2 )]TJ/F22 11.9552 Tf 11.956 0 Td [(1 dP l dx dP l 0 dx | {z } b + l l +1 | {z } c P l 0 x P l x | {z } d 3 7 7 7 5 dx =0 ; .51 Z 1 )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 2 6 6 6 4 x 2 )]TJ/F22 11.9552 Tf 11.955 0 Td [(1 dP l 0 dx dP l dx | {z } b + l 0 l 0 +1 | {z } c P l x P l 0 x | {z } d 3 7 7 7 5 dx =0 : .52 Next,equation2.52issubtractedfromequation2.51toyield: [ l l +1 )]TJ/F25 11.9552 Tf 11.955 0 Td [(l 0 l 0 +1] Z 1 )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 P l 0 x P l x dx =0 : .53 Asshowninthisequation,if l = l 0 ,then l l +1 )]TJ/F25 11.9552 Tf 10.869 0 Td [(l 0 l 0 +1=0 ,andtheintegralis permittedtobeanon-zerovalue.If l 6 = l 0 ,however,then l l +1 )]TJ/F25 11.9552 Tf 11.415 0 Td [(l 0 l 0 +1 6 =0 ,and theintegralmustequalzeroinordertosatisfythisrelationship.Thismeansthatthe 16

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polynomialsmustbeorthogonal. Itcanbeshownthroughsomefurthermanipulationthat: Z 1 )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 P l 0 x P l x dx = 2 2 l +1 l 0 l ; .54 where l 0 l ,theKroeneckerdelta,isonewhen l = l 0 andzerowhen l 6 = l 0 2.6SphericalHarmonics SphericalHarmonicsareausefulmethodtodescribetheangularcomponentsofthe scalarpotential.Todothis,however,the m componentoftheLegendrePolynomials needtobereincorporated.TheLegendrepolynomialswith m> 0 are: P m l x = )]TJ/F22 11.9552 Tf 9.298 0 Td [(1 m )]TJ/F25 11.9552 Tf 11.955 0 Td [(x 2 m= 2 d m dx m P l x : .55 SubstitutingRodrigues'formula,equation2.48,for P l x gets: P m l x = )]TJ/F22 11.9552 Tf 9.298 0 Td [(1 m 2 l l )]TJ/F25 11.9552 Tf 11.955 0 Td [(x 2 m= 2 d l + m dx l + m x 2 )]TJ/F22 11.9552 Tf 11.955 0 Td [(1 l : .56 Thevaluesfor m arerestrictedby l : )]TJ/F25 11.9552 Tf 9.299 0 Td [(l m l .Theseareintegervalues; m hasvaluesof: )]TJ/F25 11.9552 Tf 9.299 0 Td [(l )]TJ/F22 11.9552 Tf 9.299 0 Td [( l )]TJ/F22 11.9552 Tf 11.715 0 Td [(1 )]TJ/F22 11.9552 Tf 9.299 0 Td [( l )]TJ/F22 11.9552 Tf 11.715 0 Td [(2 ,..., 0 ,..., l )]TJ/F22 11.9552 Tf 11.714 0 Td [(2 l )]TJ/F22 11.9552 Tf 11.714 0 Td [(1 l .Theseequationsare denedusingpositive m ,but m canalsobenegative: P )]TJ/F26 7.9701 Tf 6.586 0 Td [(m l x = )]TJ/F22 11.9552 Tf 9.299 0 Td [(1 m l )]TJ/F25 11.9552 Tf 11.955 0 Td [(m l + m P m l x ;m> 0 : .57 Buildinguponequation2.54,arelationinvolving m canbedetermined: Z 1 )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 P m l 0 x P m l x dx = 2 2 l +1 l + m l )]TJ/F25 11.9552 Tf 11.956 0 Td [(m l 0 l : .58 Bothfunctions, P m l and Q m ,equation2.34,formcompleteorthogonalsetsover 17

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theangularvariables.Thesearecombinedintoorthogonalfunctionscalledspherical harmonics: Y lm ; = v u u t 2 l +1 4 l )]TJ/F25 11.9552 Tf 11.955 0 Td [(m l + m P m l cos e im ; .59 Y l; )]TJ/F26 7.9701 Tf 6.586 0 Td [(m ; = )]TJ/F22 11.9552 Tf 9.298 0 Td [(1 m Y lm ; : .60 Theorthonormalrelationshipforsphericalharmonicsbecomes: Z 2 0 Z 0 Y l 0 m 0 ; Y lm ; sin dd = l 0 l m 0 m : .61 Thefollowingaretherstfewsphericalharmonics: 18

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l =0 Y 00 = 1 p 4 ; .62 l =1 8 > > > > < > > > > : Y 10 = s 3 4 cos ; Y 11 = )]TJ/F31 9.9626 Tf 9.298 19.295 Td [(s 3 8 sin e i ; .63 l =2 8 > > > > > > > > > > < > > > > > > > > > > : Y 20 = s 5 4 3 2 cos 2 )]TJ/F22 11.9552 Tf 13.15 8.088 Td [(1 2 ; Y 21 = )]TJ/F31 9.9626 Tf 9.298 19.296 Td [(s 15 8 sin cos e i ; Y 22 = 1 4 s 15 2 sin 2 e 2 i ; .64 l =3 8 > > > > > > > > > > > > > > > > < > > > > > > > > > > > > > > > > : Y 30 = s 7 4 5 2 cos 3 )]TJ/F22 11.9552 Tf 13.15 8.087 Td [(3 2 cos ; Y 31 = )]TJ/F22 11.9552 Tf 10.495 8.088 Td [(1 4 s 21 4 sin cos 2 )]TJ/F22 11.9552 Tf 11.955 0 Td [(1 e i ; Y 32 = 1 4 s 105 2 sin 2 cos e 2 i ; Y 33 = )]TJ/F22 11.9552 Tf 10.495 8.087 Td [(1 4 s 35 4 sin 3 e 3 i : .65 Intermsofsphericalharmonics,thescalarpotentialequationbecomes: r;; = 1 X l =0 l X m = )]TJ/F26 7.9701 Tf 6.586 0 Td [(l [ A lm r l + B lm r )]TJ/F23 7.9701 Tf 6.587 0 Td [( l +1 ] Y lm ; : .66 Asanote:Besselfunctionsareusedforcylindricalcoordinates,however,asmultipoleeectsofsmallmoleculesarethegoalofthischapter,thesphericalandCartesian coordinatesystemsaretheonlyonesofinteresthere. 19

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2.7MultipoleExpansion Multipoleexpansionisagoodmethodforestimatingtheinteractionpropertiesof twomolecules.Ifthemoleculehasnosinglestrongmultipole,thenitwillbehave likeamixtureofitsstrongmultipoles.Multipoleexpansionisvalidatdistances signicantlylargerthanthesizeofthechargedistributionsthatcausethem. Thescalarpotentialatapointoutsidethechargesthatarebeingdescribedhas theformofthefollowing: ~x = 1 X l =0 l X m = )]TJ/F26 7.9701 Tf 6.587 0 Td [(l 4 2 l + l q lm Y lm ; r l +1 ; .67 wherethechoiceofconstantcoecientswillbemadeclearinthenextcoupleofsteps. Thisformulaiscalledthemultipoleexpansion. l =0 isthemonopoleterm, l =1 are thedipoleterms,and l =2 arethequadrupoleterms. Themultipoleexpansioniscomparedtoaslightlymodiedversionofequation 2.9: ~x = Z ~x 0 j ~x )]TJ/F25 11.9552 Tf 11.76 0 Td [(~x 0 j d 3 x 0 : Tomodifythisequation,anexpansionof 1 j ~x )]TJ/F26 7.9701 Tf 6.476 0 Td [(~x 0 j intermsofsphericalharmonicsis pluggedintoequation2.9. 1 j ~x )]TJ/F25 11.9552 Tf 11.76 0 Td [(~x 0 j =4 1 X l =0 l X m = )]TJ/F26 7.9701 Tf 6.586 0 Td [(l 1 2 l +1 r 0 l r l +1 Y lm 0 ; 0 Y lm ; : .68 Inthisexpansion, r 0
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Themultipolemoments,or q coecients,arenowdenedbycomparingequations 2.67and2.69.Themultipolemomentsaregivenbythefollowingequation: q lm = Z Y lm 0 ; 0 r 0 l ~ x 0 d 3 x 0 : .70 Sincetheonly m -dependentcomponentofthemultipolemomentisthespherical harmonic,the q l; )]TJ/F26 7.9701 Tf 6.586 0 Td [(m isrelatedto q lm inthesamefashionassphericalharmonicsin equation2.60: q l; )]TJ/F26 7.9701 Tf 6.586 0 Td [(m = )]TJ/F22 11.9552 Tf 9.298 0 Td [(1 m q lm : .71 Thefollowingarethemonopole,dipole,andquadrupolemomentsofthepositive m valuesexpressedinCartesiancoordinates: q 00 = 1 p 4 Z ~ x 0 d 3 x 0 = 1 p 4 q; .72 q 10 = s 3 4 Z z 0 ~ x 0 d 3 x 0 = s 3 4 p z q 11 = )]TJ/F31 9.9626 Tf 9.298 19.295 Td [(s 3 8 Z x 0 )]TJ/F25 11.9552 Tf 11.955 0 Td [(iy 0 ~ x 0 d 3 x 0 = )]TJ/F31 9.9626 Tf 9.298 19.295 Td [(s 3 8 p x )]TJ/F25 11.9552 Tf 11.955 0 Td [(ip y 9 > > > > = > > > > ; ; .73 q 20 = 1 2 s 5 4 Z z 0 2 )]TJ/F25 11.9552 Tf 11.955 0 Td [(r 0 2 ~ x 0 d 3 x 0 = 1 2 s 5 4 Q 33 q 21 = )]TJ/F31 9.9626 Tf 9.298 19.296 Td [(s 15 8 Z z 0 x 0 )]TJ/F25 11.9552 Tf 11.955 0 Td [(iy 0 ~ x 0 d 3 x 0 = )]TJ/F22 11.9552 Tf 10.494 8.088 Td [(1 3 s 15 8 Q 13 )]TJ/F25 11.9552 Tf 11.955 0 Td [(iQ 23 q 22 = 1 4 s 15 2 Z x 0 )]TJ/F25 11.9552 Tf 11.955 0 Td [(iy 0 2 ~ x 0 d 3 x 0 = 1 12 s 15 2 Q 11 )]TJ/F22 11.9552 Tf 11.955 0 Td [(2 iQ 12 )]TJ/F25 11.9552 Tf 11.956 0 Td [(Q 22 9 > > > > > > > > > > = > > > > > > > > > > ; : .74 Inthesemultipolemoments, q isthetotalcharge, p i isthedipolemomentalonga particularaxis,and Q ij arethequadrupolemomenttensorelements.Thevaluesof ~p and Q ij are: 21

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~p = Z ~ x 0 ~ x 0 d 3 x 0 ; .75 Q ij = Z x 0 i x 0 j )]TJ/F25 11.9552 Tf 11.955 0 Td [(r 0 2 ij ~ x 0 d 3 x 0 : .76 ThroughTaylorexpansionof 1 = j ~x )]TJ/F25 11.9552 Tf 12.606 2.734 Td [(~ x 0 j ,acompletescalarpotentialequationcan bedeterminedintermsofmultipolemoments.StartingwiththegenericTaylor expansionaroundapoint y = a : f y f a + f 0 a 1! y )]TJ/F25 11.9552 Tf 11.955 0 Td [(a + f 00 a 2! y )]TJ/F25 11.9552 Tf 11.955 0 Td [(a 2 + : .77 AmultidimensionalversionoftheTaylorexpansionaroundapoint ~y = ~a is: f ~y f ~a + X i f 0 i ~a 1! y i )]TJ/F25 11.9552 Tf 11.955 0 Td [(a i + X i;j f 00 i;j ~a 2! y i )]TJ/F25 11.9552 Tf 11.955 0 Td [(a i y j )]TJ/F25 11.9552 Tf 11.955 0 Td [(a j + ; .78 where f 0 i = @f @y i and f 00 i;j = @ @y i @f @y j If f ~x =1 = j ~x )]TJ/F25 11.9552 Tf 12.833 2.734 Td [(~ x 0 j and ~a = ~ 0 ,theTaylorexpansionbecomes: 1 j ~x )]TJ/F25 11.9552 Tf 12.833 2.735 Td [(~ x 0 j 1 j ~x )]TJ/F25 11.9552 Tf 12.833 2.735 Td [(~ x 0 j # ~ x 0 = ~ 0 + X i x i )]TJ/F25 11.9552 Tf 11.955 0 Td [(x 0 i j ~x )]TJ/F25 11.9552 Tf 12.833 2.735 Td [(~ x 0 j 3 # ~ x 0 = ~ 0 x 0 i + 1 2 X i;j 2 4 3 x i )]TJ/F25 11.9552 Tf 11.956 0 Td [(x 0 i x j )]TJ/F25 11.9552 Tf 11.956 0 Td [(x 0 j )]TJ/F25 11.9552 Tf 11.955 0 Td [( ij j ~x )]TJ/F25 11.9552 Tf 12.832 2.735 Td [(~ x 0 j 2 j ~x )]TJ/F25 11.9552 Tf 12.833 2.734 Td [(~ x 0 j 5 3 5 ~ x 0 = ~ 0 x 0 i x 0 j + ; .79 1 j ~x )]TJ/F25 11.9552 Tf 12.833 2.734 Td [(~ x 0 j 1 r + X i x i r 3 x 0 i + 1 2 X i;j 3 x i x j )]TJ/F25 11.9552 Tf 11.955 0 Td [( ij r 2 r 5 # x 0 i x 0 j + ; .80 where r = j ~x j .Thefollowingrelationisusefulforexchangingthe x and x 0 terms: 22

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X i;j y i y j = X i;j 3 y i y j )]TJ/F25 11.9552 Tf 11.955 0 Td [( ij j ~y j 2 : .81 Itisnowpossibletogetthequadrupoletensortermsinthescalarpotential: 1 j ~x )]TJ/F25 11.9552 Tf 12.832 2.734 Td [(~ x 0 j 1 r + ~ x 0 ~x r 3 + 1 2 X i;j 3 x 0 i x 0 j )]TJ/F25 11.9552 Tf 11.955 0 Td [( ij r 0 2 r 5 x i x j + ; .82 ~x = Z ~x 0 2 4 1 r + ~ x 0 ~x r 3 + 1 2 X i;j 3 x 0 i x 0 j )]TJ/F25 11.9552 Tf 11.955 0 Td [( ij r 0 2 r 5 x i x j + 3 5 d 3 x 0 ; .83 ~x = Z ~x 0 d 3 x 0 1 r + Z ~x 0 ~ x 0 d 3 x 0 ~x r 3 + 1 2 X i;j Z ~x 0 h 3 x 0 i x 0 j )]TJ/F25 11.9552 Tf 11.955 0 Td [( ij r 0 2 i d 3 x 0 x i x j r 5 + ; .84 Usingthedenitionof q = R ~x 0 d 3 x 0 andthedenitionsofequations2.75and 2.76,thescalarpotentialbecomes: ~x = q r + ~p ~x r 3 + 1 2 X i;j Q ij x i x j r 5 + : .85 23

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Chapter3 QuantumMechanics ThischapterisstyledafterDonaldA.McQuarrieandJohnD.Simon's Physical Chemistry:AMolecularApproach .[17]Thischapterisintendedtobeareviewof quantummechanicsforthosewithsomefamiliaritywiththeconceptsinherentwithin thematerial.ItstartswiththefoundationoftheSchrdingerequationandendswith approximationmethodsforsystemsthataremorecomplexthantheHydrogenatom. 3.1SchrdingerEquation TheSchrdingerequationisnotsomuchanevidenttruth,butmerelyafundamental postulatethatappearstoworkverywellforndingtheenergiesofatomicscale particles.Toderivethe1-DSchrdingerequationfromtheclassicalwaveequation andthedeBroglieformulaforthewavelikebehaviorofmatterisarathersimpletask. Startingwiththeclassicalonedimensionalwaveequationforwaveswithspeed v : @ 2 u @x 2 = 1 v 2 @ 2 u @t 2 : .1 Thesolutionforthisequationcanbefoundusingthetechniqueofseparationof 24

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variables, u x;t = x cos!t: .2 The partofequation3.2iscalledthespatialamplitude,beingindependentof time.Equation3.1becomestime-independentmerelybysubstitutingequation3.2. Thefunction x thensatises: d 2 dx 2 + 2 v 2 x =0 : .3 Giventhat =2 andthat = v ,thisequationbecomes: d 2 dx 2 + 4 2 2 x =0 : .4 Thetotalenergyoftheparticleasdescribedbythewaveequationwouldbe: E = p 2 2 m + V x ; .5 where V x isthepotentialenergy, p isthemomentum,and m isthemassofthe particle. Whensolvedformomentum,thisequationbecomes: p = q 2 m [ E )]TJ/F25 11.9552 Tf 11.955 0 Td [(V x ] : .6 25

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ThisistheninsertedintothedeBroglieformulafordeterminingthewaveproperty ofmatter: = h p = h q 2 m [ E )]TJ/F25 11.9552 Tf 11.956 0 Td [(V x ] : .7 Thiscanthenbesubstitutedintoequation3.4toobtainthetimeindependent Schrdingerequation,where x isnowawavefunctionfortheparticle. d 2 dx 2 + 2 m ~ 2 [ E )]TJ/F25 11.9552 Tf 11.955 0 Td [(V x ] x =0 : .8 TheSchrdingerequation,asitappearsabove,isadierentialequationfor x whichdescribesaparticleofmass m inapotentialeldof V x andwithatotal energyof E .This x iscalledthewavefunctionoftheparticle. Specically,equation3.8iscalledthetime-independentSchrdingerequation. Thewavefunctionsdeterminedbythetime-independentSchrdingerequationare calledstationary-statewavefunctions.Equation3.8canberewrittenas: )]TJ/F35 11.9552 Tf 12.945 8.088 Td [(~ 2 2 m d 2 dx 2 + V x x = E x : .9 3.2LinearOperators Thebeautyoflinearoperatorsforquantummechanicsinvolvestheconceptofan eigenvalue.Inquantummechanics,everyobservablepropertyofaparticlehasan associatedlinearoperatorandsetofeigenvalues.Thegeneralsetupforaneigenvalue problemis: 26

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^ A x = a x ; .10 where ^ A isthelinearoperator, a istheeigenvalue,and x istheeigenvector. Usingthisconcept,itisnowpossibletosetuptheSchrdingerequationasan eigenvalueproblem. ^ H = E: .11 ThenamefortheoperatorinthisequationiscalledtheHamiltonianoperator: ^ H = )]TJ/F35 11.9552 Tf 12.944 8.088 Td [(~ 2 2 m d 2 dx 2 + V x : .12 ThisisnottheonlyHamiltonianoperatorthatcanbeused,but,israther,one ofmany.ThisisbecausetheHamiltonianoperatorthatsomeonemayusecanbe customizedforanysystemthatheorsheisattemptingtosolve.Thisbeingsaid,it iscustomarytorenametheHamiltonianoperatorwhenthepotentialisnon-existent. Theonedimensionalkineticenergyoperatoris: ^ K x = )]TJ/F35 11.9552 Tf 12.944 8.088 Td [(~ 2 2 m d 2 dx 2 : .13 Comparingthisoperatortoitsclassicalcounterpartsuggeststhatthereshouldbe anoperatorformomentumaswell. 27

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K = p 2 2 m ; ^ P 2 x = )]TJ/F35 11.9552 Tf 9.299 0 Td [(~ 2 d 2 dx 2 ; ^ P x = )]TJ/F25 11.9552 Tf 9.299 0 Td [(i ~ d dx : .14 Extendingthemomentumoperatortothreedimensionsgives: ^ P = )]TJ/F25 11.9552 Tf 9.299 0 Td [(i ~ ~ r : .15 Twootheroperatorsworthmentioningarethatofthepositionoperatorandthe angularmomentumoperator. ^ R = ~r = x ^ x + y ^ y + z ^ z; .16 ^ L = ^ R ^ P = ~r ~p = )]TJ/F25 11.9552 Tf 9.298 0 Td [(i ~ ~r ~ r : .17 Theseoperatorsgivethebasisforbeingabletodetermineclassicalquantities throughquantummechanicsofaparticle.Thatis,theygivetheabilitytoreconcile thetwotheories. 3.3ParticleInABoxAndTheWaveFunction Theparticleinaboxproblemisaclassicproblemofintroductoryquantummechanics. Itprovidesaperfectwaytointroducehowwavefunctionsaretreatedinquantum mechanics.Itcanalsobeasimpleapproximationfor -conjugatedelectronsinlinear unsaturatedhydrocarbons.TheSchrdingerequationfortheparticleinaboxisas 28

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follows: d 2 dx 2 + 2 mE ~ 2 x =00 x a: .18 Thisequationdescribesaparticlethatisinaeldofzeropotentialwhenitis betweenpoints 0 and a .Outsideofthisregion,however,thereisapotentialofinnity forthepurposesofhavingabox. MaxBorn'sinterpretationofthewavefunctionisthat x x dx istheprobabilitythattheparticledescribedby x ismeasuredbetween x and x + dx .Furthermore,thewavefunctionisrestrictedtobeingnormalizedsothattheintegralof x x dx is 1 .Thismeansthatbecausetheparticleisonlypresentinsidethe boxandoutsidethisregionitmustbezero,thatthetotalprobabilitythatthe particleisbetween 0 and a is 1 ,or 100% .Thus, R a 0 x x dx =1 Thewavefunctionhasanotherrestrictionplaceduponit.Thisrestrictionisthat thefunctionshallbecontinuous.Fortheparticleinabox,thismeansthatatthe points 0 and a thewavefunctionmustbezerotobecontinuouswiththefactthatit iszerooutsideofthe 0 to a range. = a =0 : Theproblemnowhasspeciedboundaryconditions.Ageneralsolutionforthe wavefunctionis: x = A cos kx + B sin kx; 29

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with k = mE 1 = 2 ~ = 2 mE 1 = 2 h : .19 Duetotheboundaryconditionsimposedbythisproblem, A =0 because cos= 1 B isnotzerobecausethentheparticlewouldnotexist!Thismeansthatinorder forthelimitat a tobeobeyed, sin ka =0 .Thisbecomes: ka = nn =1 ; 2 ;:::: .20 Theenergiesoftheparticleinthisboxcanbedeterminedusingequations3.19 and3.20.Withalittlemath,theenergyis: E n = h 2 n 2 8 ma 2 n =1 ; 2 ;:::: .21 Thisequationshowsthattheenergyisquantized.Theinteger n inequation3.21 iscalledaquantumnumber.Theparticleinaboxnowhasawavefunctionof: n x = B sin kx = B sin nx a n =1 ; 2 ;:::: .22 Aswasmentioned,thewavefunctionneedstobenormalized.Thisishow B is determined.Startingwiththenormalizationcondition: 30

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Z a 0 n x n x dx =1 ; j B j 2 Z a 0 sin 2 nx a dx =1 ; j B j 2 a 2 =1 ; .23 B = 2 a 1 = 2 : .24 Thisyieldsacompletesolutionfortheparticleinaboxwavefunction: n x = 2 a 1 = 2 sin nx a ; 0 x an =1 ; 2 ;::: .25 Withthisnormalizedwavefunction,theprobabilityfortheparticletobefound inanyrange x 1 x x 2 is: Prob x 1 x x 2 = Z x 2 x 1 x x dx: .26 Theprobabilityfortheparticletobefoundinthersthalfoftheboxisindeed onehalf: Prob x a= 2= Z a= 2 0 x x dx = 2 a Z a= 2 0 sin 2 nx a dx = 1 2 : .27 Theprobabilityfortheparticletobefoundintherstfourthoftheboxismore complicated.Foroddvaluesof n ,theprobabilityissmallerthanonefourth.For 31

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an=1 bn=2 cn=3 dn=4 Figure3.1:PlotsOf sin 2 nx a example,theprobabilityfor n =1 hasonlyonepeakinthemiddlethattapersto zeroontheedgescausingtheprobabilitytobelessthanonefourth.Thismaynot seemallthatintuitiveorsimilartoclassicalmechanics,whichwouldsaythereis anequalprobabilityatanyposition.When n islarge,however,themaximagetso numerousthattheprobabilitygetsspreadoutandisnotjustconcentratedatthe middle.Inotherwords,asngetslarger,itapproachestheclassicalmechanicsresult. Thisiscalledthecorrespondenceprinciple. Itisrathertrivialtoextendtheonedimensionalparticleinaboxtoathree dimensionalparticleinabox.Thisiswheretheproblemnallylookslikeabox! 32

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)]TJ/F35 11.9552 Tf 15.602 8.088 Td [(~ 2 2 m d 2 dx 2 + d 2 dy 2 + d 2 dz 2 = E x;y;z ; 0 x a; 0 y b; 0 z c; .28 )]TJ/F35 11.9552 Tf 15.601 8.088 Td [(~ 2 2 m r 2 = E; .29 where r 2 istheLaplacianoperator. Thisproblemissolvablethroughseparationofvariables.Itcanbeassumedthat: x;y;z = X x Y y Z z ; .30 whichwillthenyieldathreepartproblemof: )]TJ/F35 11.9552 Tf 12.945 8.088 Td [(~ 2 2 m 1 X x d 2 X dx 2 = E x ; )]TJ/F35 11.9552 Tf 12.944 8.088 Td [(~ 2 2 m 1 Y y d 2 Y dy 2 = E y ; )]TJ/F35 11.9552 Tf 12.945 8.087 Td [(~ 2 2 m 1 Z z d 2 Z dz 2 = E z ; .31 where E = E x + E y + E z Eachof3.31isthesameastheSchrdingerequationfortheonedimensionalparticleinabox.Thismeansthatthethreedimensionalparticleinaboxwavefunction isjusttheproductof1-Dsolutionsandtheenergyisthesumofthecorresponding energies: 33

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n x n y n z = 8 abc 1 = 2 sin n x x a sin n y y b sin n z z c ; n x =1 ; 2 ; 3 ;:::; n y =1 ; 2 ; 3 ;:::; n z =1 ; 2 ; 3 ;:::; .32 Note: R j j 2 dV =1 E n x n y n z = h 2 8 m n 2 x a 2 + n 2 y b 2 + n 2 z c 2 ; n x =1 ; 2 ; 3 ;:::; n y =1 ; 2 ; 3 ;:::; n z =1 ; 2 ; 3 ;:::: .33 Manyproblemsinquantummechanicscanbesolvedusingseparationofvariables. 3.4SpectroscopicModels Therigidrotatorandtheharmonicoscillatoraretwosystemsinclassicalmechanics thatcanbesolvedinquantummechanicstoyieldspectroscopicallysignicantpredictions.TheharmonicoscillatorusesHooke'slawforamassconnectedtoawall byamasslessspringandtherigidrotatorisasystemoftwomassesconnectedby masslessrodofxedlengththatrotate. Hooke'slawgivestheforceofaspringexerteduponamasstobe: f = )]TJ/F25 11.9552 Tf 9.298 0 Td [(k l )]TJ/F25 11.9552 Tf 11.956 0 Td [(l 0 = )]TJ/F25 11.9552 Tf 9.298 0 Td [(kx; .34 where k isthespringconstant, l isthecurrentlengthofthespring, l 0 istheequilibrium lengthofthespring,and x isthedisplacementfromequilibrium. 34

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Newton'sSecondLawgives: m d 2 l dt 2 = )]TJ/F25 11.9552 Tf 9.298 0 Td [(k l )]TJ/F25 11.9552 Tf 11.955 0 Td [(l 0 ; .35 m d 2 x dt 2 + kx =0 : .36 Thisissolvedby: x t = c 1 sin !t + c 2 cos !t; .37 where = k m 1 = 2 : .38 IfthespringisinitiallypulledtoadisplacementofAandreleasedfromrest, equation3.37simpliesto: x t = A cos !t and v t = )]TJ/F25 11.9552 Tf 9.298 0 Td [(A! sin !t: .39 Theenergiesofthissystemareeasilycalculated: V x = )]TJ/F31 9.9626 Tf 11.291 14.058 Td [(Z f x dx = k 2 x 2 + constant; .40 K x = 1 2 m dx dt 2 ; .41 theconstantinthepotentialenergyterm,thepotentialatspringequilibrium,isset 35

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tozerowhenthespringsystemisinequilibrium. V t = 1 2 kA 2 cos 2 !t; .42 K t = 1 2 m! 2 A 2 sin 2 !t; .43 E t = 1 2 kA 2 cos 2 !t + 1 2 m! 2 A 2 sin 2 !t = kA 2 2 sin 2 !t +cos 2 !t = kA 2 2 ; .44 rememberingthat = k=m 1 = 2 Thetotalenergyoftheharmonicoscillatorhasbeenshowntobeaconstant. Thevalueofthetotalenergyisequaltothatofthepotentialenergyatmaximum displacement. Extensionoftheharmonicoscillatortoasystemoftwomassesconnectedbya springinvolvestheintroductionoftheconceptofreducedmass, = m 1 m 2 m 1 + m 2 ; .45 d 2 x dt 2 + kx =0 : .46 Thetwobodysystem,where x istheseparationbetweenthetwobodies,mathematicallybehavesinthesamewayastheonebodysystem.TheSchrdingerequation foraquantummechanicalproblemis: )]TJ/F35 11.9552 Tf 11.347 8.087 Td [(~ 2 2 d 2 dx 2 + V x x = E x ; .47 d 2 dx 2 + 2 ~ 2 E )]TJ/F22 11.9552 Tf 13.151 8.088 Td [(1 2 kx 2 x =0 : .48 Theenergygivenbythisequationisquantizedasfollows: 36

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E v = ~ k 1 = 2 v + 1 2 = ~ v + 1 2 = h v + 1 2 v =0 ; 1 ; 2 ;:::; .49 Thesequantizedenergiesyieldtheinfraredspectrumformanydiatomicmolecules; diatomichydrogen'sfundamentalvibrationalfrequencyiswellwithintheinfrared frequencyat4401cm )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 .Thefrequencyoflightwhichthemoleculecanabsorband emitisrelatedtopossibleenergychangesofthemolecule.Theenergychangesthat canoccurfortheharmonicoscillatoraremathematicallyrestrictedtothatofadjacent states.Theremustalsobeachangeinthemolecule'sdipolemomentforthemolecule tobeactiveintheinfraredspectrum. Themoleculeabsorbsat =+1 andemitsat = )]TJ/F22 11.9552 Tf 9.299 0 Td [(1 E = h obs = ~ k 1 = 2 ; .50 obs = 1 2 k 1 = 2 : .51 Thismodelisverysimpliedandthereareanharmonictermsthatcanbeusedto improvetheapproximation.Theseterms,however,normallydonotchangethevalue byalargeamount. Similartotheharmonicoscillator,theenergiesoftherigidrotatormodelare quantized.Themodelisthatoftwomasses, m 1 and m 2 ,separatedbyadistance r .Thesemassesareseparatedfromthecenterofmassandrotationby r 1 and r 2 respectively.Thekineticenergyofthissystemisgivenas: K = 1 2 m 1 v 2 1 + 1 2 m 2 v 2 2 = 1 2 m 1 r 2 1 + m 2 r 2 2 2 = 1 2 I! 2 ; .52 37

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wherethemomentofinertia, I ,is: I = m 1 r 2 1 + m 2 r 2 2 = r 2 ; .53 where r = r 1 + r 2 Asthereisnoexternalforcesexerteduponthesystem,theHamiltonianoperator is: ^ H = ^ K = )]TJ/F35 11.9552 Tf 11.346 8.087 Td [(~ 2 2 r 2 r : .54 Forthismodel,sphericalcoordinatesareusefulwithanoriginaboutthecenter ofmassofthetwoatoms.Withthiscoordinatesystem, r isthedistancebetween theatoms.Thevalueof r isaxeddistancesincetheatomsinarigidrotatoronly rotate.TheLaplacianoperatorinsphericalcoordinatesis: r 2 = 1 r 2 @ @r r 2 @ @r ; + 1 r 2 sin @ @ sin @ @ r; + 1 r 2 sin 2 @ 2 @ 2 r; .55 = 1 r 2 1 sin @ @ sin @ @ + 1 r 2 1 sin 2 @ 2 @ 2 r constant : .56 Usingequations3.54and3.56,theSchrdingerequationbecomes: )]TJ/F35 11.9552 Tf 10.876 8.088 Td [(~ 2 2 I 1 sin @ @ sin @ @ + 1 sin 2 @ 2 @ 2 !# Y ; = EY ; : .57 Similartoequation2.37,thesolutionstoequation3.57arethesphericalharmonics denedinsection2.6.Theenergyofthesystemcanbedeterminedsimplywith 38

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sphericalharmonics. 2 IE ~ 2 = J J +1 ;J =0 ; 1 ; 2 ;:::; E J = ~ 2 2 I J J +1 ;J =0 ; 1 ; 2 ;:::: .58 Theselectionrulefortherigidrotatoristhesameasthatoftheharmonicoscillator.Themoleculemayonlymoveinenergytoanadjacentstateduetomathematical constraints.Themoleculemustalsohaveapermanentdipolemoment.Thefrequency oflightthatisabsorbedbyarigidrotatoris: J = h 4 2 I J +1 ;J =0 ; 1 ; 2 ;:::: .59 Thesefrequenciesareinthemicrowaveregion.Microwavespectroscopycommonly replacesthecoecientsofequation3.59. J =2 B J +1 ;J =0 ; 1 ; 2 ;:::; .60 B = h 8 2 I : .61 where B istherotationalconstantofthemolecule. Therigidrotator,liketheharmonicoscillator,isasimpliedmodelofadiatomic molecule.Inreality,nodiatomicmoleculeisrigidlyspaced.Vibrationsofadiatomic moleculearesmallenoughthattherigidrotatorapproximationworks,butthisapproximationcanbeimprovedupon. 39

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3.5HydrogenAtom Oneofthebiggestissuesinquantummechanicsinvolvestherepulsiontermsbetween particles.TheBorn-Oppenheimerapproximationisabletoeliminatethenuclear repulsiontermsbyapproximatingthenucleiasbeingstationary.Thisispossibleas thenucleiaremuchmoremassivethantheelectrons,and,inessence,donotmove muchonthetimescaleofelectronmotion.Thisleavestheonlyproblemtobethe interelectronicrepulsionterms.Withonlyoneelectron,thisissuedoesnotarisein thehydrogensystem. ThepotentialenergyforthehydrogenatomSchrdingerequationisthecoulomb potential. V r = )]TJ/F25 11.9552 Tf 19.407 8.088 Td [(e 2 4 0 r ; .62 where e isthechargeontheproton, 0 isthepermittivityoffreespace,and r isthe distancebetweentheelectronandtheproton.The 4 0 isfromtheuseofSIunits forthisproblem. TheSchrdingerequationforthehydrogenatomis: )]TJ/F35 11.9552 Tf 15.157 8.088 Td [(~ 2 2 m e 1 r 2 @ @r r 2 @ @r + 1 r 2 sin @ @ sin @ @ + 1 r 2 sin 2 @ 2 @ 2 # )]TJ/F25 11.9552 Tf 19.407 8.088 Td [(e 2 4 0 r r;; = E r;; ; .63 )]TJ/F25 11.9552 Tf 13.295 8.088 Td [(@ @r r 2 @ @r )]TJ/F22 11.9552 Tf 21.325 8.088 Td [(1 sin @ @ sin @ @ )]TJ/F22 11.9552 Tf 23.692 8.088 Td [(1 sin 2 @ 2 @ 2 )]TJ/F22 11.9552 Tf 10.494 8.088 Td [(2 m e ~ 2 e 2 4 0 r + E # r;; =0 : .64 Separationofthevariablesyieldstheassumedwavefunction: 40

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r;; = R r Y ; : .65 TheseparatedcomponentsoftheSchrdingerequationare: )]TJ/F22 11.9552 Tf 19.425 8.088 Td [(1 R r d dr r 2 dR dr + 2 m e r 2 ~ 2 e 2 4 0 r + E R r # = )]TJ/F25 11.9552 Tf 9.298 0 Td [(l l +1 ; .66 )]TJ/F22 11.9552 Tf 25.793 8.088 Td [(1 Y ; 1 sin @ @ sin @Y @ + 1 sin 2 @ 2 Y @ 2 # = l l +1 : .67 l l +1 ischosenforconvenienceasequation3.67issolvedwiththespherical harmonicsofsection2.6.Equation3.66issolvedusinggeneralizedLaguerrepolynomials. SolvingtheradialpartoftheSchrdingerequationyields:[17] R nl r = )]TJ/F31 9.9626 Tf 11.291 17.535 Td [( n )]TJ/F25 11.9552 Tf 11.956 0 Td [(l )]TJ/F22 11.9552 Tf 11.955 0 Td [(1! 2 n [ n + l !] 3 1 = 2 2 na 0 l +3 = 2 r l e )]TJ/F26 7.9701 Tf 6.586 0 Td [(r=na 0 L 2 l +1 n + l 2 r na 0 ; .68 where L x = x )]TJ/F26 7.9701 Tf 6.586 0 Td [( e x d dx e )]TJ/F26 7.9701 Tf 6.587 0 Td [(x x + .69 isthegeneralizedLaguerrepolynomialand a 0 = 4 0 ~ 2 m e e 2 .70 istheBohrradius. Combiningbothpartsofthewavefunctionyields: 41

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nlm r;; = )]TJ/F31 9.9626 Tf 11.291 17.535 Td [( n )]TJ/F25 11.9552 Tf 11.955 0 Td [(l )]TJ/F22 11.9552 Tf 11.956 0 Td [(1! 2 n [ n + l !] 3 1 = 2 2 na 0 l +3 = 2 r l e )]TJ/F26 7.9701 Tf 6.587 0 Td [(r=na 0 L 2 l +1 n + l 2 r na 0 Y m l ; : .71 Thesolutionsdependonthreequantumnumbers n l ,and m .Aquantumnumber relatedtothespinofanelectronexistsaswell.DuetothePauliexclusionprinciple, notwoelectronsofthesameatommayhavethesamefourquantumnumbers. 3.6Approximations WithoutmathematicalapproximationsforsolvingtheSchrdingerequation,quantummechanicswouldfailforanythingmorecomplexthanthatofthehydrogenatom. Aswasalreadymentioned,theBornOppenheimerapproximationxesnucleicpositions.Twoothercommonapproximationmethodsarethevariationalmethodand perturbationtheory. Forthevariationalmethod,thegroundstatewavefunctionandenergyfora givensystemaredenotedby 0 and E 0 respectively.StartingwiththeSchrdinger equation: ^ H 0 = E 0 0 ; .72 Z 0 ^ H 0 d 3 x = Z 0 E 0 0 d 3 x; .73 Z 0 ^ H 0 d 3 x = E 0 Z 0 0 d 3 x; .74 E 0 = R 0 ^ H 0 d 3 x R 0 0 d 3 x : .75 Anarbitrarylinearcombinationoftestfunctionsissubstitutedintotheabove equations.Theenergycalculatedbythiscanbeshowntobegreaterthanorequal 42

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tothegroundstateenergy. E = R ^ Hd 3 x R d 3 x ; .76 where = X n c n n : .77 Substitutingfor inequation3.76gives: E = P n c n c n E n P n c n c n ; .78 E )]TJ/F25 11.9552 Tf 11.955 0 Td [(E 0 = P n c n c n E n )]TJ/F25 11.9552 Tf 11.955 0 Td [(E 0 P n c n c n 0 ; since E 0 istheminimumenergy : .79 Thevariationalmethodshowsthatanypostulatedwavefunctionwillhavean energywhoselowerlimitistheenergyofthegroundstate.Thisprincipleisvery usefulsincethecloseranansatzenergyistothegroundstateenergy,thecloserthe ansatzwavefunctionistothetruewavefunction. Anothermethodforapproximationisperturbationtheory.Perturbationtheory worksforsystemsthatarenotsolvable,butaresimilartoonethatissolvable.Itis assumedthattheunsolvableHamiltoniancanbedescribedbythesumofthesolvable Hamiltonianandsomesmallperturbation. ^ H = ^ H + ^ H : .80 Thewavefunction, ,isthenexpandedas: 43

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= + ; .81 where ^ H = E ; .82 isthesolvableSchrdingerequation. Therstorderperturbationtotheenergyis: H = E ; .83 E = Z ^ H d 3 x: .84 Thisworksbecausetheonlyunknownis E H isanansatzvalueand is solvedusingequation3.82.Thismethodcanthenbeextendedtohigherorderinthe perturbations,witheachnewtermbeingsmallerthanthelast. 44

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Chapter4 ComputationalMethods Thischapterprovidesabriefoverviewofthemethodsusedinthiscomputational studyofcyanideandcyanylradical. 4.1Overview Thisworkusesbothcoupledclusterapproximations,CC,anddensityfunctional theory,DFT.Thissectionisintendedasabriefintroductiontowhatthesemethods are. Incoupledclustertheory,theelectronicwavefunctionisexpandedintermsof productsofexcitedHartree-Fockdeterminants.Thisapproachrapidlyconvergesto thefullcongurationlimitFullCI,whichcorrespondstoanexactsolutiontothe electronicSchrodingerequationforagivenbasisset.[18]CCSDisthecoupledcluster approximationusingsinglesanddoublesexciteddeterminantsforapproximations. Abenettothistheoryisthatitissizeconsistent;coupledclustertheorybecomes proportionaltothenumberofparticles N as N !1 .[18]Onepotentialdownside ofthistheoryisthat,sinceitisnotactuallyavariationalmethod,itispossible toobtainmorethan100%ofthecorrelationenergy.Correlationenergyisdueto opposite-spinelectroninteractionsandisthedierencebetweenthefullCIenergy 45

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andtheHartree-Focklimit.[18] Densityfunctionaltheorytreatstheelectronsasauidwithdensity.Because oftheBorn-Oppenheimerapproximation,thenucleioftheatomsareconsidered xedwithinthisuidofelectrons.[19]Theenergyofthesystemcanbecompartmentalizedintothenoninteractingkineticenergy,theelectrostaticenergy,andthe exchange-correlationenergy.Theelectrostaticenergyisthesimplestandneedsno approximation.[19]Theothertermsdoneedapproximations.ThecurrentapproximationsthatexistcausemanyoftheerrorsfoundinDFT.Localdensityapproximation, forexample,yieldsincorrectanalysiswhenvanderWaalsforcesareinvolvedsince theseareduetononlocalcorrelations.[19]TheThomas-Fermimodelusedinapproximationsforthenoninteractingkineticenergyworksbestformoleculesofhigher density.[19]DFT'sadvantageoverothertheoriesisitslowcomputationalcostand, whenitworks,accurateresults.DFTisbenchmarkedwithcoupledclustertheoryin thisworktoshowthatitisaccurateforthissystem. 4.2Specics Ratheraccuraterelativeenergiesandcomplexationenergies,energiesofformingcomplexeswithwater,oftwoisomersofCN H 2 OandCN )]TJ/F28 11.9552 Tf 7.084 -4.338 Td [( H 2 Owerecomputedby optimizingthegeometryusingcoupledclustertheorywithsingle,double,andperturbativetripleexcitations[CCSDT]pairedwiththecc-pVTZbasisset[20]byanalytic gradientsintheMainz-Austin-BudapestMABversionofACESII.[21]Restricted open-shellHartree-FockROHFwavefunctionswereusedforallopen-shellcoupled clustercalculations.[22][23][24][25] DensityfunctionalmethodswereemployedtostudylargerclustersofCN and CN )]TJ/F17 11.9552 Tf 12.31 -4.338 Td [(withwater.Specically,theB3LYPfunctionalpairedwithaDZP++basis setwasused.TheDZP++basissetcomprisesDunning'scontractionofHuzinaga's 46

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doubleprimitives,[26]augmentedwithaneven-temperedsetofdiusefunctions onallatoms.[27][28]Geometrieswereoptimizedviaanalyticgradientsandharmonic vibrationalfrequenciescomputedtoconrmallreportedstructureswereminimaon theB3LYP/DZP++potentialenergysurface.AllB3LYPcomputationsweredone usingQChem3.1.[29] 47

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Chapter5 Results BelowwereportadetailedstudyofthemicrosolvationofCN andCN )]TJ/F17 11.9552 Tf 11.831 -4.338 Td [(inwater. BenchmarkgeometriesandrelativeenergiesarerstprovidedforCN andCN )]TJ/F17 11.9552 Tf 10.53 -4.339 Td [(with onewater,followedbyresultsforclusterswithtwoandthreewaters.Inter-atomic distancesarereportedinAngstroms. 5.1OneandNoWaterCyanideandCyanylRadicalComplexes UsingCCSDTpairedwiththecc-pVTZbasisset,twodisparateisomersof CN H 2 OstructuresIandIIinFig.5.1werelocated.Thehigher-lyingstructure IIfeaturesatraditionalhydrogenbondwiththecyanylnitrogenservingastheprotonacceptor.TheglobalminimumI,ontheotherhand,doesnotinvolvehydrogen bondingbutisinsteadboundbyelectrostaticinteractions.Basedonour abinitio computations,thisstructurelies1.2kcal/molbelowthehydrogen-bondedstructure II.Forcyanide,twolow-lyingstructureswerealsolocated:theglobalminimumI featuresnitrogenastheprotonacceptorwhilestructureIIisheldtogetherbyan interactionofthewaterhydrogenwiththecarbonatomofCN )]TJ/F17 11.9552 Tf 7.084 -4.339 Td [(.StructureIIlies0.7 48

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kcal/molhigherthantheCN )]TJ/F28 11.9552 Tf 7.084 -4.338 Td [( H 2 OstructureI. Figure5.1:OneWaterStructuresOptimizedAtThecc-pVTZCCSDTLevelOf Theory ThegeometryofCN H 2 OstructureIcanbeunderstoodusingsimplemultipolararguments.Ignoringcontributionsfromthehigher-ordermultipoles,onewould predictadipole-dipolecomplexofwaterandCN toadopttheC 2 v -symmetricstructuredepictedinFig.5.2a.ThisispreciselywhatHartree-Focktheorypredictswhen pairedwithadoublebasisset,presumablybecauseatthisleveloftheorythedipole momentofCN isoverestimatedbyaboutafactoroftwo.However,correlated ab initio methodspredicttheC s symmetricgeometryshowninFig.5.2c.Thisisdue primarilytocontributionsfromtheinteractionofthequadrupoleofwaterwiththe dipolemomentofcyanylradical.Thisissupportedbytheorientationaldependence oftheenergy,demonstratedbytheconstrainedoptimizedstructuresshowninFig. 5.2.MotionabouttheO...CNangleisfacile,withchangesinthisanglefromthe equilibriumvalueof142 o to180 o Fig.5.2baccompaniedbyamere0.2kcal/mol increaseinenergy.Ontheotherhand,forcingthissystemtoadoptaC 2 v -symmetric structureFig.5.2araisestheenergybyanadditional3.4kcal/mol.Thiselectrostaticinteractionisenergeticallymorefavorablethanthehydrogenbondingofwater andthenitrogenofCN 49

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Energiesinkcal/mol Figure5.2:PartiallyOptimizedStructuresIllustratingTheElectrostaticInteractions InCN H 2 OStructureIAtThecc-pVTZCCSDTLevelOfTheory Forboththecc-pVTZCCSDTandDZP++B3LYPlevelsoftheory,thestretchingfrequenciesofCN andCN )]TJ/F17 11.9552 Tf 11.564 -4.339 Td [(bothincreasewhenclusteredwithwater.Inboth cases,thepredictedshiftintheCNstretchingfrequencyissmallerforstructureIthan forIIseeTables5.1and5.2.ThesepredictedshiftsintheCNstretchingfrequency shouldaidintheexperimentalidenticationofthesecomplexesinmolecularbeams. TofacilitatecomputationsoflargerclustersofCN andCN )]TJ/F17 11.9552 Tf 10.545 -4.339 Td [(withwater,wehave benchmarkedB3LYP/DZP++againstourcoupledclusterresultsforCN H 2 Oand CN )]TJ/F28 11.9552 Tf 7.085 -4.339 Td [( H 2 O.ForthecomplexesofCN andCN )]TJ/F17 11.9552 Tf 11.528 -4.339 Td [(withonewater,B3LYP/DZP++ predictedgeometriesareinreasonableagreementwiththeCCSDT/cc-pVTZresults discussedabove.OnenotableexceptionistheB3LYP/DZP++predictedstructurefor structureI.WhereasCCSDT/cc-pVTZpredictstheC s -symmetricstructureshown inFig.5.1tobeaminimum,B3LYP/DZP++predictsasimilarC s -symmetricstructuretobeatransitionstateTS.ThisTScorrespondstotheinterconversionoftwo equivalentstructuresinwhichthecyanylradicaliseclipsedwithoneortheother O-HbondsofwaterseeFig.5.3.However,B3LYPpredictstheTStolieonly0.01 kcal/molhigherthantheassociatedminima,consistentwiththeverysmallmagnitudeoftheimaginaryfrequencycomputedattheTScm )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 .Thisenergydierence iswellbelowthezeropointvibrationalenergyand,therefore,B3LYP/DZP++effectivelypredictsaC s -symmetricgeometryconsistentwiththeCCSDTresults.It 50

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shouldalsobenotedthatwhenpairedwiththe6-31Gdbasisset,B3LYPpredicts aC s -symmetricstructureinaccordwiththecoupledclusterresult. aMinimum bTransitionState Figure5.3:DZP++B3LYPStructures Furthermore,atransitionstateconnectingCN H 2 OstructuresIandIIwas locatedattheB3LYPleveloftheory.ThebarrierforconversionofIItoIisonly0.8 kcal/molatthisleveloftheory,sotrappingstructureIIwillrequirecoldtemperature experiments. Figure5.4:CN H 2 OIsomerizationEnergySurfaceComputedAtTheDZP++ B3LYPLevelOfTheorykcal/mol WhiletheB3LYPpredictedfrequencyshiftsforbothCNH 2 Ostructuresare inagreementwiththecoupledclusterresults,forCN )]TJ/F17 11.9552 Tf 7.084 -4.338 Td [(,B3LYPoverestimatesthe frequencyshiftsbyabout30cm )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 forbothIandII.B3LYPalsooverestimatesthe relativeandhydrationenergiesbyabout3kcal/mol. 51

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n E rel kcal/mol E hyd kcal/mol CNstrcm )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 CCSDT B3LYP CCSDT B3LYP CCSDT B3LYP 0 X X X X 2052.6 2121.7 1 I 0.0 0.0 2.4 5.6 2064.2 2130.0 II 1.2 4.2 X X 2067.3 2143.6 2 I X X X 9.8 X 2107.9 3 I X 0.0 X 11.2 X 2117.1 II X 0.4 X X X 2118.9 III X 0.6 X X X 2119.2 Table5.1:CN H 2 O n BindingEnergiesE rel ,HydrationEnergiesE hyd ,andCN StretchingFrequencies CNstrcm )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 AtThecc-pVTZCCSDTandDZP++ B3LYPLevelsOfTheory 5.2TwoandThreeWaterCyanideandCyanyl RadicalComplexes Low-lyingisomersofCN andCN )]TJ/F17 11.9552 Tf 11.478 -4.338 Td [(withtwoandthreecomplexedwatermolecules werelocatedattheB3LYP/DZP++leveloftheory.OptimizedstructuresandselectedbondlengthsandanglesareincludedinFigs.5.5and5.6.Relativeenergies areprovidedinTables5.1and5.2.Foreachsystem,onlyisomerslyingwithin2.8 kcal/moloftheglobalminimumareincludedintheguresandtables.Acutoof2.8 kcal/molischosensince,at298K,structureslyingabove2.8kcal/molwillconstitute lessthan1%oftheisomerpopulation. WhenCN andCN )]TJ/F17 11.9552 Tf 10.805 -4.339 Td [(formcomplexeswithmorethanonewater,therebecomesa noticeabledierenceinthestructuresoftheirminima.ForCN andmultiplewaters, ring-likestructuresarepreferredwhileCN )]TJ/F17 11.9552 Tf 7.085 -4.339 Td [(H 2 O n favorsmorelinearchains.The ringstructureforCN H 2 O 2 andCN H 2 O 3 arefavoreddespiteastrainingofbond anglesfrom141.9 onthecarbonsideofcyanylradicalto130.2 withtwowatersand 130.6 withthreewaters.Thenitrogensidehasgreaterstrain,goingfrom178.3 to 96.8 withtwowatersand133.6 withthreewaters.Thetwocyanylradicalstructures withthreewatersthatarewithin2.8kcal/moloftheminimumarealsoringshaped; 52

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n E rel kcal/mol E hyd kcal/mol CNstrcm )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 CCSDT B3LYP CCSDT B3LYP CCSDT B3LYP 0 X X X X 2068.6 2095.4 1 I 0.0 0.0 15.6 15.7 2082.4 2142.4 II 0.7 0.5 X X 2098.9 2162.2 2 I X 0.0 X 13.4 X 2166.6 II X 2.0 X X X 2139.1 III X 2.3 X X X 2133.2 IV X 2.5 X X X 2150.7 3 I X 0.0 X 10.5 X 2172.9 II X 0.1 X X X 2179.8 III X 2.1 X X X 2121.5 Table5.2:CN )]TJ/F17 11.9552 Tf 7.085 -4.338 Td [(H 2 O n BindingEnergiesE rel ,HydrationEnergiesE hyd ,andCN StretchingFrequencies CNstrcm )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 AtThecc-pVTZCCSDTandDZP++ B3LYPLevelsOfTheory System AdiabaticEAeV Debye H 2 O X 1.9 CN )]TJETq1 0 0 1 232.989 414.724 cm[]0 d 0 J 0.398 w 0 0 m 0 14.446 l SQBT/F17 11.9552 Tf 282.615 419.058 Td [(X 0.7 CN -4.04 1.3 CN H 2 O 1 -4.48 X CN H 2 O 2 -4.64 X CN H 2 O 3 -4.61 X H 2 O CN )]TJET1 0 0 1 427.16 350.72 cmq.4 0 0 .4 0 0 cmq63 0 0 37 0 0 cm/Im12 DoQQ1 0 0 1 -427.16 -350.72 cmBT/F17 11.9552 Tf 456.261 350.72 Td [(CN Table5.3:B3LYPAdiabaticElectronAnitiesandCCSDTDipoleMoments therearenoCN structureswithtwowatersthatarewithin2.8kcal/molofthe minimumenergystructure.Thesecyclicstructuresaremadepossiblebytherelative atnessofthepotentialenergysurfacealongtheN-C Obendingangleinstructure I. Thecyanidestructuresshowthathydrogenbondingonthenitrogensideispreferabletohydrogenbondingonthecarbonside.Thisisshowninitiallyintheonewater structuresasthenitrogenhydrogen-bondedstructureispredictedtolie0.5kcal/mol .7kcal/molwithcc-pVTZCCSDTbelowstructureII.Thetwowaterstructures indicateapreferenceforcarbonhydrogenbondingoveroxygenhydrogenbondingof 53

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Figure5.5:TwoWaterStructuresOptimizedUsingDZP++B3LYPWithRelative EnergiesInkcal/mol therstandsecondwaterssincetheenergeticallyminimumstructurehasahydrogen bondonthecarbonandthenitrogen,ratherthanthesecondlowestenergystructure whereonewatermoleculeisboundtothenitrogenandtheothertotherstwater. Thethirdtwowaterstructureisringshapedwithtwohydrogenbondstoonenitrogen andismorefavorablethanacarbonsidehydrogenbondwithanoxygenhydrogen bond.Thisindicatesthatevenastrainednitrogensidestructureispreferabletoa carbonsidestructure. AsdeterminedbyB3LYP/DZP++,addingwatertobothcyanylradicaland cyanidechangesthestretchingfrequencyoftheCNlittleafterthersttwowaters areadded.ThemostdramaticresultiswhenCN hasasecondwateraddedand 54

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Figure5.6:ThreeWaterStructuresOptimizedUsingDZP++B3LYPWithRelative EnergiesInkcal/mol formsaring.Thiscausesasignicantdropinfrequency.Thisdatashouldprovidea usefultoolasthefrequencyincreasesonlyalittlebetweenthetwoandthreewater structures. ElectronAnitiesEAsofCN H 2 O n canbereadilycomputedusingDFT.[27] AdiabaticEAsforCNH 2 O n areincludedinTable5.3tostudytheevolutionofthe EAofCNuponhydration.TheEAincreasesbyalmost0.5eVuponcomplexation withtherstwater,butthenquicklylevelsotoavalueof4.6eV. 55

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5.3Conclusions Wehaveshownthatcomplexesofthecyanylradicalwithwaterformcyclicstructures, whileCN )]TJ/F28 11.9552 Tf 7.085 -4.339 Td [( H 2 Oadoptslinearcongurations.Thisisduetoamorerigidnatureof hydrogenbonding,comparedtothedipole-quadrupoleinteractions.Cyanylradical iscapableofformingringstructuresbecausethesedipole-quadrupoleinteractions arerelativelyorientationindependentandcanaccommodatelargerdeviationsfrom optimalangleswithlittleincreaseinenergy.Thestretchingfrequencyofboththe cyanylradicalandcyanideincreases,withoneexceptionduetoringcomplexing,as theybecomehydrated.Thechangeinfrequencyandelectronanitybecomessmaller aswaterisaddedandthelimitofmanywatersshouldbeattainableinfuturework bystudyinglargerCNH 2 O n clusters. 5.4FutureWork EnergiesarebeingcomputedfortwowatersystemsusingCCSDTtoshowthereis improvedcorrelationbetweentheDFTandCCSDTasmorewatersareaddedto thesystem.Inadditiontothiswork,computationsofcyanylradicalandcyanideare underwayforsystemsofupto6waterswiththeDFTapproach.Otherworkthat willbedoneisthatofafocalpointmethodtoobtainveryaccuraterelativeenergies forthezeroandonewatercomplexes.[30][31][32][33] Geometrieshavealreadybeenoptimizedusingcoupledclustertheorywithsingle,double,andperturbativetripleexcitations[CCSDT]pairedwiththecc-pVQZ basisset[20]byanalyticgradientsintheMainz-Austin-BudapestMABversionof ACESII.[21]Withinthefocalpointapproach,oneexecutesdualexpansionsofthe one-andN-particlebasissetsatthisxedoptimizedgeometry.Extrapolationsofa seriesofvalenceelectronicenergiesenablethesystematicapproachtothecomplete basissetBorn-Oppenheimerresult.Detailsoftheprocedurehavebeendescribed 56

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previously.[34]Correctionstotheextrapolatedvalenceelectronicenergiesareappendedtoaccountforcore-electroncorrelation,non-Born-Oppenheimereects,zero pointvibrationalenergy,andscalarrelativisticeects. 57

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