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Microsolvation of the Cyanyl Radical

Permanent Link: http://ncf.sobek.ufl.edu/NCFE004053/00001

Material Information

Title: Microsolvation of the Cyanyl Radical Competition Between Hydrogen Bonding and Electrostatic Non-Covalent Interactions
Physical Description: Book
Language: English
Creator: Bloom, Jacob
Publisher: New College of Florida
Place of Publication: Sarasota, Fla.
Creation Date: 2009
Publication Date: 2009

Subjects

Subjects / Keywords: Computational Chemistry
Cyanyl
Cyanide
Water Microsolvation
Genre: bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: A theoretical study of microsolvated cyanyl radical with water has been carried out. Cyanyl radical and water are of interest to both astrophysics and biochemistry. A theoretical study of the microsolvated system should prove fruitful for future study in these fields. Furthermore, this work illustrates the competition between hydrogen bonding and electrostatic non-covalent interactions. This study utilized density functional theory for both cyanyl radical and cyanide with up to three waters. The zero and one water energies were benchmarked using coupled cluster theory with single, double, and perturbative triple excitation approximations. Relative energies, hydration energies, frequencies, dipole moments, and adiabatic electron affinities were computed for the cyanyl radical and cyanide systems.
Statement of Responsibility: by Jacob Bloom
Thesis: Thesis (B.A.) -- New College of Florida, 2009
Electronic Access: RESTRICTED TO NCF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE
Bibliography: Includes bibliographical references.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The New College of Florida, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Local: Faculty Sponsor: Colladay, Donald

Record Information

Source Institution: New College of Florida
Holding Location: New College of Florida
Rights Management: Applicable rights reserved.
Classification: local - S.T. 2009 B65
System ID: NCFE004053:00001

Permanent Link: http://ncf.sobek.ufl.edu/NCFE004053/00001

Material Information

Title: Microsolvation of the Cyanyl Radical Competition Between Hydrogen Bonding and Electrostatic Non-Covalent Interactions
Physical Description: Book
Language: English
Creator: Bloom, Jacob
Publisher: New College of Florida
Place of Publication: Sarasota, Fla.
Creation Date: 2009
Publication Date: 2009

Subjects

Subjects / Keywords: Computational Chemistry
Cyanyl
Cyanide
Water Microsolvation
Genre: bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: A theoretical study of microsolvated cyanyl radical with water has been carried out. Cyanyl radical and water are of interest to both astrophysics and biochemistry. A theoretical study of the microsolvated system should prove fruitful for future study in these fields. Furthermore, this work illustrates the competition between hydrogen bonding and electrostatic non-covalent interactions. This study utilized density functional theory for both cyanyl radical and cyanide with up to three waters. The zero and one water energies were benchmarked using coupled cluster theory with single, double, and perturbative triple excitation approximations. Relative energies, hydration energies, frequencies, dipole moments, and adiabatic electron affinities were computed for the cyanyl radical and cyanide systems.
Statement of Responsibility: by Jacob Bloom
Thesis: Thesis (B.A.) -- New College of Florida, 2009
Electronic Access: RESTRICTED TO NCF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE
Bibliography: Includes bibliographical references.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The New College of Florida, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Local: Faculty Sponsor: Colladay, Donald

Record Information

Source Institution: New College of Florida
Holding Location: New College of Florida
Rights Management: Applicable rights reserved.
Classification: local - S.T. 2009 B65
System ID: NCFE004053:00001


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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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