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Computer simulation of metal-organic materials

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Computer simulation of metal-organic materials
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Stern, Abraham
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MOF
Hydrogen storage
Atomic point charges
Monopoles
Dissertations, Academic -- Chemistry -- Masters -- USF   ( lcsh )
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ABSTRACT: Computer simulations of metal-organic frameworks are conducted to both investigate the mechanism of hydrogen sorption and to elucidate a detailed, molecular-level understanding of the physical interactions that can lead to successful material design strategies. To this end, important intermolecular interactions are identified and individually parameterized to yield a highly accurate representation of the potential energy landscape. Polarization, one such interaction found to play a significant role in H2 sorption, is included explicitly for the first time in simulations of metal-organic frameworks. Permanent electrostatics are usually accounted for by means of an approximate fit to model compounds. The application of this method to simulations involving metal-organic frameworks introduces several substantial problems that are characterized in this work. To circumvent this, a method is developed and tested in which atomic point partial charges are computed more directly, fit to the fully periodic electrostatic potential. In this manner, long-range electrostatics are explicitly accounted for via Ewald summation. Grand canonical Monte Carlo simulations are conducted employing the force field parameterization developed here. Several of the major findings of this work are: Polarization is found to play a critical role in determining the overall structure of H2 sorbed in metal-organic frameworks, although not always the determining factor in uptake. The parameterization of atomic point charges by means of a fit to the periodic electrostatic potential is a robust, efficient method and consistently results in a reliable description of Coulombic interactions without introducing ambiguity associated with other procedures. After careful development of both hydrogen and framework potential energy functions, quantitatively accurate results have been obtained. Such predictive accuracy will aid greatly in the rational, iterative design cycle between experimental and theoretical groups that are attempting to design metal-organic frameworks for a variety of purposes, including H2 sorption and CO2 sequestration.
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Dissertation (PHD)--University of South Florida, 2010.
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by Abraham Stern.
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ComputerSimulationofMetal-OrganicMaterialsbyAbrahamC.SternAdissertationsubmittedinpartialfulllmentoftherequirementsforthedegreeofDoctorofPhilosophyDepartmentofChemistryCollegeofArtsandSciencesUniversityofSouthFloridaMajorProfessor:BrianSpace,Ph.D.RandyLarsen,Ph.D.H.LeeWoodcock,Ph.D.PrestonMoore,Ph.D.DateofApproval:July14,2010Keywords:MOF,hydrogenstorage,atomicpointcharges,monopolesCopyrightc2010,AbrahamC.Stern

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ForMariapaz

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TableofContentsListofTables.................................iiiListofFigures.................................vAbstract....................................viChapter1Introduction...........................1Chapter2Polarization:InclusioninMolecularSimulation......72.1Abstract............................... 7 2.2Introduction............................. 8 2.3ModelsandMethods........................ 11 2.3.1MolecularSimulationParameters.............. 11 2.3.2PolarizabilityModel..................... 16 2.3.3NVTMonteCarlo...................... 19 2.4ResultsandDiscussion....................... 21 2.5Conclusions............................. 28 2.6Acknowledgments.......................... 29 2.7SupportingInformation....................... 29 Chapter3ApplicationofaNewHydrogenPotential..........333.1Abstract............................... 33 3.2Introduction............................. 34 3.3Methods............................... 35 3.3.1HydrogenPotential..................... 35 3.3.2MOF-5Potential....................... 36 3.3.3GrandCanonicalMonteCarlo............... 38 3.4ResultsandDiscussion....................... 41 3.4.1HydrogenIsotherms..................... 41 3.4.2IsostericHeatofAdsorptionofHydrogen......... 46 3.4.3IsothermalCompressibilityofHydrogen.......... 46 i

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3.5Conclusions............................. 49 Chapter4DesignMotifstoMaximizevanderWaalsInteractions..504.1Abstract............................... 50 4.2Introduction............................. 51 4.3Methods............................... 52 4.3.1ComputationalMethods................... 52 4.3.2ExperimentalMethods.................... 55 4.4Conclusion.............................. 58 Chapter5SimulationofH2SorptioninaConnedCavity......595.1Abstract............................... 59 5.2Introduction............................. 60 5.3Methods............................... 63 5.3.1AtomicPointCharges.................... 63 5.3.2PolarizabilityModel..................... 68 5.3.3Parameterizationofleadandsulfur............. 69 5.4ResultsandDiscussion....................... 71 5.4.1ChargeModel{PeriodicvsFragmentApproach..... 71 5.4.2HydrogenSorptionviaGCMC............... 80 5.5Conclusions............................. 83 5.6Acknowledgments.......................... 83 Chapter6AtomicPointChargesinCrystallineSolids.........856.1Abstract............................... 85 6.2Introduction............................. 86 6.3Methods............................... 89 6.4ResultsandDiscussion....................... 95 6.4.1ComparisonofEwaldandWolfsummationmethods... 95 6.4.2Eectoftheexclusionradius................ 98 6.4.3Eectofthegridmeshdensity............... 104 6.4.4Restraintapproachforburiedatoms............ 106 6.5Conclusions............................. 107 References...................................108Appendices..................................121AppendixA:PeriodicESP-derivedchargescodelisting........122Abouttheauthor...............................EndPage ii

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ListofTablesTable2.1Partialchargesusedinsimulationofsoc-MOF..........14Table2.2Partialchargesonasecondfragment...............14Table3.1MOF-5potentialparameters....................38Table5.1PartialchargesforME193:periodicmethod...........71Table5.2PartialchargesforME193:fragmentmethod...........74Table6.1ComparisonofatomicchargesforIRMOF-1andCuBTC....96Table6.2AtomicchargesforIRMOF-1andCuBTCasafunctionof..101Table6.3AtomicchargesforIRMOF-1asafunctionofgriddensity...105Table6.4AtomicChargesforFepzNiCN4asafunctionof......107 iii

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ListofFiguresFigure2.1Hydrogendensityinsoc-MOF..................10Figure2.2Fragmentsofsoc-MOF......................13Figure2.3Radialdistributionfunctioncenteredatindium.........22Figure2.4Radialdistributionfunctioncenteredatnitrogen........23Figure2.5Dipoledistribution.........................25Figure2.6Isosurfaceofthedipolepopulationhistrogram.........26Figure3.1FragmentofMOF-5........................37Figure3.2Low-pressureisothermforMOF-5................42Figure3.3High-pressureisothermforMOF-5................43Figure3.4IsotherminexcessweightpercentunitsforMOF-5.......44Figure3.5Comparisontoexperimentaldata................45Figure3.6Isostericheatofadsorption....................47Figure3.7Isothermalcompressibility.....................48Figure4.1Optimizedmolecularsquarebindingenergy...........53Figure4.2Crystalstructureof1.......................55Figure4.3H2sorptionisothermsforstructure1...............56Figure4.4Crystalstructureof2.......................56Figure4.5H2sorptionisothermsforstructure2...............57Figure5.1CrystalstructureofME193....................62Figure5.2ChemicallydistinctatomsinME193...............72Figure5.3SimpliednumberingschemeinME193.............72Figure5.4Fragmentsselectedforgasphasechargetting.........73Figure5.5AtypicalcongurationofH2sorbedintheMOF.........74Figure5.6H2populationhistogramdierenceplot.............75Figure5.7Sorptionisothermscomparingtheoryandexperiment.....76Figure5.8Heatofadsorptioncomparingtheoryandexperiment......77Figure5.9Isothermalcompressibilitycomparingtheoryandexperiment.78Figure5.10Energydecompositionforthefragmentcharges.........79Figure5.11Energydecompositionfortheperiodiccharges.........79 iv

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Figure5.12Energydecompositionforthenonpolarizablesimulation....80Figure6.1CuBTC,IRMOF-1,andFepzNiCN4.............94Figure6.2CuBTCatomicpointcharges...................97Figure6.3ChargeasafunctionofRc....................99Figure6.4Errorasafunctionofdistance..................100Figure6.5Fittedchargesfor[FepzNiCN4................104 v

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ComputerSimulationofMetal-OrganicMaterialsAbrahamC.SternAbstractComputersimulationsofmetal-organicframeworksareconductedtobothinvestigatethemechanismofhydrogensorptionandtoelucidateadetailed,molecular-levelunderstandingofthephysicalinteractionsthatcanleadtosuccessfulmaterialdesignstrategies.Tothisend,importantintermolecularinteractionsareidentiedandindividuallyparameterizedtoyieldahighlyaccuraterepresentationofthepotentialenergylandscape.Polarization,onesuchinteractionfoundtoplayasignicantroleinH2sorption,isincludedexplicitlyforthersttimeinsimulationsofmetal-organicframeworks.Permanentelectrostaticsareusuallyaccountedforbymeansofanapproximatettomodelcompounds.Theapplicationofthismethodtosimulationsinvolvingmetal-organicframeworksintroducesseveralsubstantialproblemsthatarecharacterizedinthiswork.Tocircumventthis,amethodisdevelopedandtestedinwhichatomicpointpartialchargesarecomputedmoredirectly,ttothefullyperiodicelectrostaticpotential.Inthismanner,long-rangeelectrostaticsareexplicitlyaccountedforviaEwaldsummation.GrandcanonicalMonteCarlosimulationsareconductedemployingtheforceeldparameterizationdevelopedhere.Severalofthemajorndingsofthisworkare:Polarizationisfound vi

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toplayacriticalroleindeterminingtheoverallstructureofH2sorbedinmetal-organicframeworks,althoughnotalwaysthedeterminingfactorinuptake.Theparameterizationofatomicpointchargesbymeansofattotheperiodicelectrostaticpotentialisarobust,ecientmethodandconsistentlyresultsinareliabledescriptionofCoulombicinteractionswithoutintroducingambiguityassociatedwithotherprocedures.Aftercarefuldevelopmentofbothhydrogenandframeworkpotentialenergyfunctions,quantitativelyaccurateresultshavebeenobtained.Suchpredictiveaccuracywillaidgreatlyintherational,iterativedesigncyclebetweenexperimentalandtheoreticalgroupsthatareattemptingtodesignmetal-organicframeworksforavarietyofpurposes,includingH2sorptionandCO2sequestration. vii

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Chapter1IntroductionTherelativelyrecentsynthesisandcharacterizationofmetal-organicframeworksMOFsisapromisingsteptowardtrue,functionallydesignedmaterials.[1]MOFshavealreadydemonstratedproof-of-principalapplicationingassorption,catalysis,andsensing.[2,3,4]Complementingthedesignstrategiesofsyntheticchemistshasbeentheparalleldevelopmentofcomputationalmethodstosimulatethesematerials.Computersimulationoersameansofidentifyingthenecessaryattributesanewmaterialsshouldpossesstoachievespecicapplications,characterizingthosematerialswhichshowpromise,andunderstandingthecomplexinterplayoftopologyandintermolecularinteractionspresentintheseexcitingstructures.Ofparticularrecentinterestisthedesignofmaterialsthatsorbhighquantitiesofgasses.[5,1]Severalgasseshavebeentargetedasdesirablecandidatesforstorageand/orsequestration.Carbondioxide,agreenhousegasandbyproductofmanyindustrialactivities,isonesuchgasthatmaybesequesteredfromuegasasameansofreducingenvironmentaldamage.[5] 1

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Hydrogen,consideredapromisingcandidatetofossilfuels,isanothersuchgasthatMOFscouldndapplication.Desirableforitsextremelyhighmolarenergydensity,hydrogenunfortunatelypossessanextremelylowvolumetricenergydensitythusrequiringameansofstoragethatcancircumventtheenergeticrequirementandsafetyconcernsofsimplecompression.Accordingly,theU.S.DepartmentofEnergyhasidentiedseveralmilestonestargetedtowardtheutilizationofhydrogenintheareasoftransportation,distributedstationarypower,andportablepowerapplications.[6]Ifhydrogenistobeaviablereplacementforhydrocarbonfuels,containmentandstorageissuesmustbeaddressed.[6]Asidefromtheimpracticalityofliquefactionofhydrogenbymeansofcoolingandhighpressures,theenergylossassociatedwiththisprocessmakestheuseofhydrogenfarlessattractive.Weakintermolecularinteractionsresultinaboilingpointofaround20Katambientpressureandcontributetothedicultyincondensinghydrogenatmorereasonabletemperaturesandpressures.MOFsshowpromiseinthattheycanprovideascaoldingofadditionalintermolecularinteractiontherebyeectivelyraisingtheboilingpointofhydrogen.Intermolecularinteractionsaregenerallydecomposedintoseveralclasses;vanderWaals{arisingfromelectroncorrelation,repulsive{arisingfromthePauliexclusionprincipleandelectron-electronrepulsion,electrostaticorCoulombicforces,andorbitalinteractions.Allofthese,tosomedegree,arepresentinahydrogensaturatedMOFsystem.Thegoalofthesyntheticchemististomaximizetheinteractionsinacooperativefashiontoenablethehighestcapacitypossible.Computersimulationhasplayedavitalroleinunderstandingthesometimescomplexphenomenathatcanoccurincondensedmedia.Beginninginthelate1950'stherstcomputersimulationswereconductedusingMonteCarloalgorithms.[7]Sobegantherichdevelopmentofthesophisticatedmethodsnowemployedregularly.Assimulationtechniquesandthecorrespondingtheoreticalmethodsevolved,thesystemsamenabletostudyviasimulationalsochanged.AdvancesinhardwarefacilitatedthetheevolutionfromsimplesystemssuchasmonatomicLennard-Jonesuidstomorecomplex,polyatomiccondensedphasesimulations.Likewise,thesizeofsystemsconsideredgrewrapidly,trackingclosely 2

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theextraordinarygainsmadeinprocessorspeed.Asatomisticsimulationofsystemscontainingmillionsofatomsbecameareality,biologicalsystemsbeganreceivingattentionfromthesimulationcommunity.Assuch,methods,algorithms,force-elds,andsoftwareweredevelopedandcontinuetobedevelopedtomeetthedemandsofsimulationsofthesesystems.Specically,biologicalsimulationshaveplayedacentralroleinthedevelopmentofnewintegrationschemes,eectivesolvationmodels,coarse-grainedmethods,mixedquantummechanics/molecularmechanicstechniques,andcompletelyreparameterizedforce-eldsamongmanyothersubstantialcontributions.[8,9,10,11]Likewise,interestinmetal-organicmaterialsandcorrespondingsimulationofthesesystemsisstillinitsinfancybuthasalreadyseenthedevelopmentofnewtechniquesspecicallydesignedtohandletheuniquenatureofthesesystems.[12]Therstsimulationswereconductedcompletelyneglectingelectrostaticcontributions,nowknowntobeextremelyimportantinsomecases.[13]Assophisticationincreased,electrostaticswereparameterizedstillrathercrudelywithfragmentsoftheperiodicstructure.However,agreatdealofimportantworkhasbeenconductedwiththesestillbasicmethods.[12,14,15]Snurrhasdemonstratedkeyparametersthatarecloselyrelatedtouptakeareheatofadsorption,surfacearea,andfreevolume.[16]Atlowpressures,notsurprisingly,theheatofadsorptionisthedominantcharacteristicthatdictatesuptake.ThisessentiallymeansthattheamountofhydrogeninitiallyadsorbedisdirectlycorrelatedtothebindinganityofhydrogentotheMOF.Atmoderatepressures,thesurfaceareadirectlycorrelateswiththeamountofhydrogenadsorbed.Essentially,oncethemostfavorablehydrogeninteractionsitesarelled,thesecondbestsitesarestillonthesurfaceoftheMOF.Finally,atthehighestpressures,thefreevolumeultimatelydeterminestheuptakecapacity.Thiscorrespondstohydrogenoccupyingthevacuumremainingoncethesurfaceissaturated,notinteractingdirectlywiththeframeworkatall.Kubashasdemonstratedthathydrogencanexhibitinterestingorbitalinteractions 3

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withsometransitionmetals.[17]Thediscoveryofintactdihydrogencomplexedtoametalclusterwasadeningeventincoordinationchemistryandanexcitingprospectwhenconsideringdesignprincipleswithhydrogensorptioninmind.Theso-calledKubascomplexesformasaresultofbackdonationofelectrondensityfromalleddorbitaltotheorbitaloftheH2.Thisresultsinalengtheningaround20%oftheH2bondandabindingenergythatdependsstronglyonthechemicalenvironmentandcanbeanywherefrom20uptoashighas120kJ/mol.BhatiaandMyershaveestimatedtheoptimalconditionsforhydrogenstoragematerialsbyassumingsimpleLangmuirsorptionandback-calculatingthenecessaryheatofadsorptionrequired.[18]Theirestimateof15kJ/molrepresentsaminimumrequirementformaterialsiftheestablishedtargetsaretobeachieved.UsingacombinationofestimatesderivedfromtheoreticalmodelsandgrandcanonicalMonteCarlosimulation,theirstudywentfurtheranddeducedoptimaloperatingtemperaturesandpressuresfordesorption,animportantfactorintheultimateimplementation.However,thisresultcementstheimportanceofhighheatofadsorptionvaluestoachievetheDOEmilestones.InthecontextofMOFssynthesizedtodate,15kJ/molrepresentsasignicantchallenge.ThoseMOFsforwhichexperimentaldatahasbeenpublishedtypicallymeasureintherangeof5-10kJ/mol.AlthoughheatofadsorptionisasignicantfactorindeterminingH2uptake,theultimategravimetricuptakedependsnotonlyontheamountofhydrogensorbed,butalsoonthedensityofthescaolding.Thisconstraintmeansthatsuccessfulmaterialsmustalsobelightinadditiontobeingporous,havejusttherightfreevolume,possessahighQst,andhavehighsurfacearea.Synthesizingamaterialthathasaprecisebalanceofthesecharacteristicsrepresentsasignicantchallenge.Synthesisofmetal-organicframeworksistypicallyconductedbymeansof"rationaldesign."Theorganicligands,orlinkers,areselectedbasedoninherentpropertiesthatultimatelydeterminethenaltopologyoftheframework.Linkerlength,theanglesbetweenligatinggroups,andthefunctionalizationofthelinkersallplayaroleindeterminingthenalstructure.Althoughcertainmetalclustersarecertainlycommonandthereforetargeted,thereissignicantlymorevariationintheboththecongurationofthemetalclustersandthemetal-ligand 4

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coordinationbondsultimatelymakingapriorimaterialdesigndicult.Imperfectdesignstrategies,however,donotpreventsyntheticchemistsfromexhibitingafairamountofcontrolovermanyofthecharacteristicsofthenalstructure.Choiceofmetalsandcompositionoftheorganiclinkershaveobviousandinalienableimplicationsshouldareactionbesuccessful.Althoughthetopologymaydierfromthehypotheticaltarget,linkerlengthandcompositionstillinuencetheporesizeandthechemistryatthesurface.Syntheticchemists,however,lackthedesignprinciplesthatcanonlycomefromadetailedunderstandingofthecomplexinterplaybetweenstructure,function,topology,andintermolecularinteractions.SimulationsofgassorptioninMOFsaimtoprovideanunderstandingoftheprinciplesatworkinthosestructureswithprovensorptionfunctionalityandaidinthedesignofbettermaterials.Theworkdetailedinthismanuscriptreectstheprogressmadeoverthelastveyearstowardthisgoal.Chapter2isastudyofsoc-MOFinwhichapolarizationmodelisdevelopedandtested.Priorsimulationsneglectinductioneectswhichthisworkdemonstratestobeofsignicantimpact.Ofnote,anheterogeneousdistributionofdipolesisidentiedandcharacterized.Thehistogramofthisdistributionindicatesthepresenceoftwodistinctpopulations{onewhichhydrogenexhibitsalargedipoleandonewhichexhibitsasmallerdipole.TheregionoftheMOFthatinduceseachspeciesisidentied.Chapter3appliesanewlyparameterizedhydrogenmodelinagrandcanonicalMonteCarlosimulation.Unprecedentedagreementwithexperimentisshown.Interestingly,hydrogensorptioninMOF-5isdominatedbyvanderWaalsinteractionsandpolarizationisfoundtobeaminorcomponentoftheenergydecomposition.Therefore,nonpolarizablesimulationswereconductedusingatransferable,anisotropicH2potential.Atsaturation,thecompressibilityofhydrogenwasfoundtobecharacteristicoftheliquidstate.Thishassignicantimplicationsinthedesignofhydrogenstoragematerials.Theresultsdemonstratethattheisothermalcompressibility,directlyobtainablefromGCMCcalculations,isanimportantparameterworthmonitoringtoassessthenatureoftheconneduidandtheabilitytofurtherimprovesorptioncapacitywithincreasedpressure. 5

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Chapter4exploresthepossibilityofmaximizingheatofadsorptionbymeansofconnementoradditivevanderWaalsinteractions.FollowingtheworkofHead-Gordon,inwhichbindingenergiesofH2andfunctionalizedbenzenederivativeswerecomputed,amodelsystemcomposedoffourbenzenemoleculeswasconstructedandoptimaldimensionswerecalculatedfromelectronicstructure.[19]Oncetherelationshipbetweenboxsizeandbindingenergywasmapped,astructurewassynthesizedmimickingasnearlyaspossibletheoptimalgeometry.Theexperimentalheatofadsorptionshowedexcellentagreementwiththatpredictedusingthemodelsystem.Thisworkhighlightstheexcellentsynergybetweencomputationandmaterialdesign.Chapter5isacontinuationofthestudyofthematerialdevelopedinchapter4,aMOFwithconningonedimensionalchannels.ThegeometryofthisMOF,whilenotidealforgravimetricuptakeofhydrogenduetoitsrelativelyhighdensity,isnonethelessinteresting.Potentially,theconnementofhydrogencouldexhibitinterestingpackingeectsorotherunforeseendeviationfrombulkproperties.TheseareexploredviapolarizableGCMCsimulation.TheaveragepotentialenergyisdecomposedandindicatesanunprecedenteddependenceonthevanderWaalsinteractions.Thisresultisverycounter-intuitiveconsideringthehighchargespresentinthestructure.Polarizableandnonpolarizablepopulationdensityhistogramsarecomputedtoidentifydierentialadsorptionrelativetothehydrogenmodelused.Chapter6developsanewmethodofttingatomicpointchargesformetal-organicmaterialsoranymicroporousperiodicsolid.Priortothiswork,theonlyreasonablemethodavailabletoassignpartialpointchargesforthepurposeofmolecularsimulationwasabinitiocalculationperformedonfragmentsoftheperiodicstructurethataimtomimicthechemicalenvironmentoftheuntruncatedmaterial.Thispracticerequireschemicalterminationofthearticiallyimposedboundariesofthearbitrarilyselectedfragment.Moreover,long-rangeelectrostaticinteractionsarenotaccountedfor;aninherentandunavoidableconsequenceofttingchargesinthismanner.Inthischapter,amethodofttingchargestothefullyperiodicstructure,explicitlyaccountingforlong-rangeelectrostaticinteractionsviaEwaldsummation,isdevelopedandtestedonseveralMOFs.Resultscomparefavorablywithchargestusingafragmentbasedapproach.Thismethodrepresentsasignicantimprovementoverthepreviousapproachtoparameterizingelectrostaticsinmetal-organicframeworks. 6

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Chapter2Polarization:InclusioninMolecularSimulation2.1AbstractMonteCarlosimulationswereperformedmodelinghydrogensorptioninarecentlysynthesizedmetal-organicframeworkmaterialMOFthatexhibitslargemolecularhydrogenuptakecapacity.TheMOFisremarkablebecauseat78Kand1.0atmosphereitsorbshydrogenatadensitynearthatofliquidhydrogenat20Kand1.0atmospherewhenconsideringH2densityinthepores.UnlikemostotherMOF'sthathavebeeninvestigatedforhydrogenstorage,ithasahighlyionicframeworkandmanyrelativelysmallchannels.ThesimulationsdemonstratethatitisbothofthesephysicalcharacteristicsthatleadtorelativelystronghydrogeninteractionsintheMOFandultimatelylargehydrogenuptake.Microscopically,hydrogeninteractswiththeMOFviathreeprincipleattractivepotentialenergycontributions:VanderWaals,charge-quadrupoleandinduction.PrevioussimulationsofhydrogenstorageinMOF'sandothermaterialshavenotfocusedontheroleofpolarizationeects, 7

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buttheyaredemonstratedheretobethedominantcontributiontohydrogenphysisorption.Indeed,polarizationinteractionsintheMOFleadtotwodistinctpopulationsofdipolarhydrogenthatareidentiedfromthesimulationsthatshouldbeexperimentallydiscernibleusing,e.g.Ramanspectroscopy.Becausepolarizationinteractionsaresignicantlyenhancedbythepresenceofachargedframeworkwithnarrowpores,MOF'sareexcellenthydrogenstoragecandidates.2.2IntroductionAmajorobstacleinachievingahydrogen-basedfueleconomyistheabilitytostoreandtransportmolecularhydrogensafely-atreasonabletemperaturesandpressures.Forexample,atoneatmosphere,hydrogendoesnotliquefyuntil20K[20]becauseofitsrelativelyweakintermolecularinteractions,makingthetransportofneathydrogendicult.Thus,ndingmaterialscapableofstoringlargeamountsofdiatomichydrogenisapromisingavenue.Thechallengeisthathydrogentypicallyinteractsweaklywithitsenvironment.However,hydrogenmoleculescaninteractstronglywithsomematerialsundergoingchemisorptionordissociation,butsuchmaterialsaretypicallyinadequateforhydrogenstoragebecauseitisdiculttoreleasethestoredgaswhenitisneeded.[21,22]Ontheotherhand,materialsthatphysisorbmolecularhydrogenoerthepromiseofstoringitundermoderateconditionsandtheabilitytoreleasethehydrogenfacilely.Suchamaterialwouldrequireoptimizingtheattractiveintermolecularinteractionsbetweenthehydrogenandthecondensedphaseenvironmentand,atthesametime,enhancingH2-H2interactions,thusleadingtoafavorablesorptionenthalpy.Thetheoreticalstudyofthisuniqueproblemhasbecomeanareaofintenseresearchinrecentyears.[15,23,24]Metal-organicframeworkmaterialsMOF'sareaclassofmaterialsthathavealreadyshownpromiseforhydrogenstorage.[25,26]MOF'srepresentanovelclassofsolidcrystallinematerialsthatarebuiltwithrigidorganicligandslinkedtometal-containingclustersalsoknowsassecondarybuildingunitsorSBUs[27]Theycanbeconstructedtohavelargesurfaceareas,arerelativelylightweightandcanbeassembledfrommolecularbuildingblockswithdesiredchemicalfunctionality. 8

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Recently,aMOFwasreported[25]referredtohereassoc-MOFthatwassynthesizedusinganovelindiumtrimerbuildingblockthatresultedinananoporousmaterialwithanionicframework,narrowchannelsaround1nmindiameterandnanometerscalecarcerandcapsules.TheMOFhasararesoctopology[28]e.g.notfoundinzeoliteswithanestimated57%extra-frameworkvolume,alargeLangmuirsurfaceareaof1417m2and0.50cm3g)]TJ/F20 7.9701 Tf 6.5865 0 Td[(1porevolume.[25]Figure2.1showssnapshotsofsoc-MOFaloneandwithahydrogendensityisosurfacecalculatedfromsimulationdescribedbelow.Thehydrogenisresidentintheextra-frameworkvolumeandFigure2.1servestovisuallyhighlighttheporetopologywithinsoc-MOF.Hydrogenuptakestudiesonsoc-MOFshowalargestoragecapacitywithreversiblesorption.Forexample,at78Kand1.0atmosphere,thedensityofH2intheporesisapproximately0.05gcm)]TJ/F20 7.9701 Tf 6.5865 0 Td[(3whileliquidhydrogenatitsboilingpointof20Khasadensityof0.07gcm)]TJ/F20 7.9701 Tf 6.5866 0 Td[(3.[25]Theexperimentsareconductedatliquidnitrogentemperatureasasteptowardndingsuperiorhydrogenstoragematerialsthatwillultimatelyoperateatroomtemperature.[6]ThisporedensityofH2representsacompressionfactor,comparedtotheidealgasvolumeunderthesameconditions,ofapproximately100.Hydrogensorptionisothermsweremeasuredat78Konsoc-MOFandshowedthattheporeswerelledatrelativelylowerpressure.[25]TheMOFapproachedsaturationat1.0atmosphere,indicativeofthenearliquiddensityofthesorbedhydrogen.Thus,toinvestigatethephysicalbasisofthelargehydrogenuptake,canonicalMonteCarlosimulationswereperformedonhydrogeninsoc-MOFattheexperimentallyobservedhydrogendensityat78Kand1.0atmosphere.Becausesoc-MOFhasahighlychargedlatticewithnarrowpores,simulationswereperformedwithandwithoutexplicitmany-bodypolarization[32,33,34,35]contributions,incontrasttomostextantmolecularsimulations.Molecularhydrogen'sattractiveinteractionsinaheterogeneouscondensedphasematrixe.g.MOF's,carbonnanostructuresandzeolitesaredominatedbythreeintermolecularcontributions:VanderWaals,charge-quadrupole,andinduction.Asanestimateoftherelativeimportanceofquadrupolethataretypicallyaccountedforandpolarizationusuallyneglectedterms,considerapolarizablesitewithapointquadrupole,bothofthemagnitudeappropriateformolecularhydrogenplaced0.30nmdistantfromadouble-chargedcationrepresenting,e.g.indiuminachargestatesimilartothatinsoc-MOF.Giventhemostfavorablegeometryforthequadrupolarinteractions,thepolarizationenergyisafactorof4largerandisalways 9

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asoc-MOFempty bsoc-MOFpopulatedFigure2.1:Thisillustrationshowsthesoc-MOFbothemptyandpopulatedviasimulationof113H2withapolarizablemodel.Thehydrogendensityisrepresentedbya90%isosurfacegeneratedbyacustommodulewrittenforDataExplorer[29,30,31]andhasbeenrenderedwithaclippingplaneforclarity. 10

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attractive.Clearly,polarizationisnotnegligibleandneedstobeincludedinsomefashion.Below,wedemonstratethatincludingmany-bodypolarizationexplicitlyhasadramaticeectonthephysisorptionofhydrogentosoc-MOF,evencomparedtoexplicitlyincludinginductionasaone-bodyinteraction.Further,theseobservationssuggestthatpolarizationneedstobeincludedinmodelinghydrogensorptioninavarietyofmaterials,essentiallybecausethequadrupoleinteractionsarerelativelyweak.Note,abinitiomoleculardynamicsMDsimulationsofmolecularhydrogeninanotherMOFMOF-5havebeenpreviouslyperformed[36]andimplicitlyincludeareasonablyaccuraterepresentationofpolarizationinteractions.Unfortunately,thehighcostofperformingabinitioMDsimulationslimitssuchnitetemperatureinvestigationstoveryshorttimes,althoughtheyarequiteeectiveatndingminimumenergycongurations.Therestofthepaperisorganizedasfollows.Section2.3.1presentstheparametrizationandpresentationofthehydrogenandsoc-MOFenergyfunction.Section2.3.2presentsthemany-bodypolarizationmethods.Sections2.7and2.3.3presenttheMonteCarlomethods.Section2.4presentstheresultsofoursimulationsandSection2.5reportstheconclusionsdrawnfromourstudies.2.3ModelsandMethods2.3.1MolecularSimulationParametersAvarietyoftheoreticalmethodologieshavebeenusedtostudyhydrogensorptioninnanostructuredmaterials.[37,38,39,40]RecentstudiesincludeMD,grand-canonicalMonteCarlosimulationstopermitcalculationofsorptionisotherms,electronicstructurestudiestoinvestigatebindingmechanisms/anitiesandsemi-classicalsimulationsthatdierentiatetheinteractionoforthoandparahydrogenwitha 11

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materialrelevantatverylowtemperatureswhereneutrondiractionstudiesareperformedtocharacterizeunderlyinghydrogeninteractionsites.[41,42,43,44,36,45,46,47,48,49,15,37,50,51,52,53,54,55,56,57,58,38,59,40,60,61]Here,classicalMonteCarlosimulationmethodswerechosentostudysoc-MOFinordertoperformequilibriumnitetemperaturesimulationsandtobeabletoexplicitlystudytheroleofinductioninhydrogensorption.Criticaltoanyclassicalsimulationbasedonempiricalpotentialsisthecarefulselectionofforceeldparameters.Aminimal,yeteective,forceeldneedstoincludeelectrostatic,repulsiveandvanderWaals-typeinteractions;[62,63]accuratelydescribingthetotalpotentialenergysurfaceisessentialandtherelevantparametersarenotespeciallywell-characterizedforMOF's.Thus,strivingforsimplicityinthisinitialstudy,theneedforframeworkintramolecularinteractionswasavoidedbyholdingthescaoldrigidduringsimulation.Phononsarenotthoughttoplayanimportantroleinhydrogensorption,especiallynotatthetemperaturesconsideredhere.[39]Lennard-Jonesparameters,representingrepulsiveandvanderWaalsinteractionsbetweenhydrogenatomsandframeworkweretakenfromtheUniversalForceField;[64]thissetofparameterswasusedinearlierMOFstudies.[40,47,15,61]TheUFFinteractionsareparameterizedforenergeticsbetweenlikeatomsandallotherinteractionsareaccountedforinastandardwayusingtheapproximateLorentz-Berthelotmixingrules.[64,62]Whenneglectingpolarizability,electrostaticinteractionsinatomisticsimulationsstemfrompointpartialchargesassignedtothecoordinatecorrespondingtothenuclearcenterofeachatom.Becausethetrueelectrostaticpotentialenergysurfaceofsoc-MOFisunknownandabinitiocalculationsonthesoc-MOFunitcellarecomputationallyprohibitive,pointchargesweredeterminedfromelectronicstructurecalculationsonseveralmodelcompoundsthatmimicthechemicalenvironmentoftheMOFatoms.[40]TheGAMESSabinitiosimulationpackagewasusedtoperformtheHartree-Fockquantummechanicalcalculations.[65]Thestructureofsoc-MOFischaracterizedbycorner-sharingoctahedralindiumtrimersjoinedbybent1,3-benzenedicarboxylateorganiclinkers.ThreerepresentativefragmentsareshowninFigure2.2.Theyallproducedsimilarpartialchargestowithin10%onaverage,andtheelectrostaticparametersusedwere 12

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afragmentA bfragmentB cfragmentCFigure2.2:Illustrationsdepictingthefragmentsusedintheabinitiocalculationofthepartialcharges.StructureAisanSBUwithsixboundlinkers,essentiallyonecompletecornerofapore.StructureB,twoSBUscoupledbyonelinker,canberegardedastheedgeofapore.StructureCisessentiallythebareSBUusedinthedesignofsoc-MOF.[25] derivedfromthelargestofthecandidates-theresultsarepresentedinTable2.1.StudyofthelatticeshowninFigure2.1revealstherepetitionofcertainstructuresinavarietyofgeometrieswithintheunitcellforadetaileddiscussionofthestructureseetheliterature.[25]The448atomunitcellmaybeproducedfrom 13

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Atom Label Chargee)]TJ/F15 11.9552 Tf 7.0847 -4.3384 Td[( In 1 2.0697 O 2 -0.7588 C 3 0.9108 C 4 -0.1086 C 5 -0.2252 H 6 0.1366 C 7 0.3785 N 8 -0.2243 C 9 -0.1327 H 10 0.2167 O 11 -1.3978 Nnitrate 12 0.6934 Onitrate 13 -0.4652 Table2.1:Partialchargestakenfromthefragment2.2ausedinsimulationofsoc-MOF Atom Label Chargee)]TJ/F15 11.9552 Tf 7.0847 -4.3384 Td[( In 1 2.228 O 2 -0.8216 C 3 0.9652 C 4 -0.1286 C 5 -0.2622 H 6 0.1443 C 7 0.4631 N 8 -0.2111 C 9 -0.2150 H 10 0.1854 O 11 -1.5424 Nnitrate 12 1.029 Onitrate 13 -0.6424 Table2.2:PartialchargesforthefragmentinFigure2.2bforcomparison 14

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crystallographicsymmetryoperationsfromonlytwentyatoms.[25,28]Althoughthistypeofsymmetrycannotbetakenadvantageofbyquantumsimulationpackages,thisrepetitionwasusedasabasisfordecidingonrepresentativechemicalfragments.Thelargestfragmentswerechosentoincludeatleastonecompletemetalcenterandanazobenzenelinker;thesmallestistheloneSBU.Addinghydrogenatomswhereappropriatewasrequiredforchemicalterminationofthefragmentboundaries.Measurementsfromthecrystalstructureindicatethattheenvironmentsoftheazobenzenelinkersareessentiallychemicallyequivalentinthattheirinterfacewiththemetalcentersdiersveryslightly.Deningtheazobenzenelinkersasallchemicallyequivalentallowstheentireunitcelltobedenedintermsofonlythirteenchemicallydierentatoms.Usingtheelectrostaticpotentialsurfacefromtheabinitiocalculations,atomicpointchargesweretusingastandardalgorithm.[66,65]Becausesoc-MOFandourmodelfragmentscontainmany-electronmetalatomsindium,theinnerelectronsrequiretreatmentviarelativisticmethods.Hereweusesemi-relativisticpseudopotentials,andtwowerecompared,namelySBKJCandLANL2[67,68,69]thatincludeadierentnumberofexplicitelectronsforindium6and12,respectivelybutgavesimilarresults.Thelightatomsweretreatedatthe6-31Glevelthatproducesover-polarizedchargesappropriateforcondensedphasesimulationstoaccount,inaneectiveway,fortheeectofself-inductionoftheunpopulatedlattice.[70]Asafurthertestoftheabinitiocalculations,relativisticelectronicstructurecalculationswereperformedonthesmallestfragmentwithoutneedforpseudopotentialsusingathird-orderDouglas-KrolltransformationandthecorrespondingDK3basisset.[71]Theresultingchargesagreedwithin7.0%ofthoseusedinthisstudy;thechargesusedhereinaretabulatedinTable2.1.Note,thecondensedphasepolarizationoftheneatMOFisincludedimplicitly,whilethepolarizationinteractionsinoursimulationsbetweenthehydrogenandtheMOFwillbetreatedexplicitly.Partialchargesforhydrogenwerechosentoreproducethequadrupolemomentofthemolecule.[43]Thehydrogenisalsotreatedasrigidanditshighfrequencyvibrationisnotexpectedtocontributetosorption.[43]Theelectrostaticmodelisathree-pointmodelwithachargelocatedatthecenterofmass.Thehydrogenatomsareseparatedby0.741Aandinteractbetweendistinctmoleculesviaapolarizablemany-bodypotential,alongwiththeCoulombicandUFF-denedLennard-Jonesintermolecularpairpotentials. 15

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2.3.2PolarizabilityModelMolecularpolarizationwasexplicitlyincludedintheMonteCarlosimulationsbyuseoftheThole-Applequistmodel.[33,32,34]Thismodeltreatsthesystemintermsofsiteatomicpointdipolesthatinteractviamany-bodypolarizationequations.Onceatomicpointpolarizabilitiesarettoatrainingsetofmoleculesthemodelhasbeenshowntoaccuratelyreproducemolecular/systemdipolesinatransferablei.e.system-independentmanner.[34,32]Thismodelofexplicitpolarizationhasbeensuccessfullyappliedinnumerousareaswhereinclusionofpolarizableeectsisparamount,suchasvibrationalspectroscopy,[72,73,35]liquiddynamics,[74,75,76,77]andbiomolecules.[78,79]Thedipoleforsiteiisgivenby:~i=i~Ei .1 whereiisthe33sitepolarizabilitytensorand~Eiistheelectrostaticeldatthesite.IntheThole-ApplequistmodelthesystemistreatedasacollectionofNpointdipoleswhichinitiallyhaveanassociatedscalarpointpolarizabilityiandadipoleeldtensorinEinsteinnotationTijthatcontainsthecompletesetofinduceddipole-dipoleinteractions.Thesitepolarizabilitytensorsareproducedvia:A~=~E .2 ~=B~E .3 whereA=)]TJ/F20 7.9701 Tf 6.5865 0 Td[(1+TijandB=A)]TJ/F20 7.9701 Tf 6.5865 0 Td[(1TheblockelementsofBarethesitepolarizabilitytensors,andsummingovertheijblocksforthevectorcomponentsforanappropriatesetofsitesyieldsthemolecularpolarizabilitytensor:[34] 16

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mol=Xi;jBij .4 SincedirectinversionoftheAmatrixiscomputationallyfeasibleforonlythesmallestsystems,thedipolesmustbesolvedforbyaniterativemethod.Theithdipoleisfoundbythedipoleeldequation:i=iEi=i)]TJ/F22 11.9552 Tf 5.4795 -9.6838 Td[(Estati+Eindi=iEstati)]TJ/F22 11.9552 Tf 11.9552 0 Td[(Tijj .5 whereEstatistheelectrostaticeldvectordeterminedbytheatomicpartialchargesoftheforceeldMOFandH2andEindiseldvectorfromthesurroundingdipoles.TheApplequistdipoleeldtensor[32]canbederivedfromrstprinciplesas:Tij=rr1 rij= r3ij)]TJ/F15 11.9552 Tf 13.1507 8.0877 Td[(3xx r5ij .6 TheTholemodelintroducestheadditionalconsiderationoftreatingeachdipoleasinteractingwithawell-behavedchargedistributionu,whichresultsinamodiedformofthedipoleeldtensor.Onesuchexponentialdistribution[33]foundtoaccuratelyandtransferablyreproducemoleculardipolesforanassociatedseriesofdependentpolarizabilitiesis:uij=3 8e)]TJ/F23 7.9701 Tf 6.5865 0 Td[(uij;uij=xij)]TJ/F22 11.9552 Tf 5.4795 -9.6838 Td[(ij)]TJ/F21 5.9776 Tf 7.782 3.2585 Td[(1 6 .7 wherethefreeparameterhastheeectofdampingthedipoleinteractionsneartheregionsofdiscontinuity.[33]Takingtheexponentialchargedistributionintoaccount,themodieddipoleeldtensorbecomes: r3ij1)]TJ/F27 11.9552 Tf 11.9551 16.8569 Td[(2r2ij 2+rij+1e)]TJ/F23 7.9701 Tf 6.5865 0 Td[(rij)]TJ/F15 11.9552 Tf 13.1506 8.0878 Td[(3xx r5ij1)]TJ/F27 11.9552 Tf 11.9552 16.8569 Td[(3r3ij 6+2r2ij 2+rij+1e)]TJ/F23 7.9701 Tf 6.5865 0 Td[(rij .8 17

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Themany-bodypotentialenergyduetotheinteractionoftheinduceddipolesreferredtoasthepolarizationenergyisdescribedby:Upol=)]TJ/F15 11.9552 Tf 10.494 8.0877 Td[(1 2Xi~i~Estati .9 Calculatingthepolarizationenergyforthesystemamountstoself-consistentlysolvingthedipoleeldequationforeachatomicdipolevector~ithroughaniterativeprocessuntilasucientdegreeofprecisionisachieved.WhileclassicalMonteCarlosimulationsincludingmany-bodypolarizationarecomputationallycheaperthanabinitioMD,solvingtheiterativeequationsisstillveryexpensiveandneedstoberedoneaftereachMonteCarlomove.Thus,tomakethecalculationspractical,ecientmethodsofsolvingtheequationswererequiredseeSupportingInformation,Section2.7AppendixA,forourapproach.Dipoleswerecalculatedtoaprecisionof10-4Debye,andweresubjecttoa11.2Ahalftheunitcelllengthsphericalcuto.SincetheatomicpointchargesoftheMOFwerecalculatedbyabinitiotoimplicitlyincludepolarization,MOF-MOFself-polarizationwasdisallowedbyexcludingMOF-MOFelectriceldinteractionsandonlytheinducedeldinteractionsbetweentheH2andMOFatomswerecalculated.Allsystemdipoleswereallowedtointeractthroughthedipoleeldtensor,subjecttotheconstraintofthesphericalcuto.Avalueof=0:1fortheexponentialover-relaxationalongwiththeGauss-Seideliterativemethodwasfoundtoproduceoptimalconvergencewhilemeetingtheprecisioncriteria.Thepolarizabilitytensorofdiatomichydrogenwithanequilibriumbonddistanceof0.741AwascalculatedbytherestrictedHartree-Fockmethodwithacorrelation-consistentdouble-zetabasisset[80]augcc-pVDZusingGAMESS.[65]TheatomicTholepolarizabilitiesformolecularhydrogenwerethendeterminedbyttingmoltotheHFpolarizabilitytensorformwhileatthesametimeyieldingone-thirdofthetraceequaltotheexperimentallymeasured[81]H2polarizabilityof0.787A3;thevaluesthatbestsatisedbothcriteriawerefoundtobe0.2658A3forHand0.5865A3forthecenterofmasssite. 18

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TheSBUofsoc-MOFcontainsindiumwhich,incomplex,hasapartialchargeofaboutIn2+asttothecalculatedelectrostaticpotentialsurfaceofthegas-phasefragmentseeTable2.1.Whilethepolarizabilityofclosedshellindiumisknown,[82]thepolarizabilityofindiuminthe2+statehasnotbeenparameterizedpreviouslyfortheTholemodel,norhasitbeenexperimentallyelucidated.Inordertoascertainthepolarizability,abinitiosimulationswereperformedusingnite-eldcalculationsonIn0,In1+,In2+andIn3+.Toassurethattheresultsobtainedfromthenite-eldcalculationswerereasonablethedatawascomparedtopolarizabilitiescalculatedwithananalyticHessianfortheIn1+andIn3+states.Thusanestimateof2.0A3forIn2+wasdeterminedfortheseresults;futureresearchwillbedirectedatapplyingfully-relativisticeldequationstotheSBUandttingthisparameterwithintheTholemodel.TheremainderoftheMOFatomsweregiventheexponentialpolarizabilitiesandassociateddampingparameterascalculatedbyDuijenetal.[34]2.3.3NVTMonteCarloMonteCarlo[83]simulationswereperformedontheH2-MOFsystemat78Kandwiththeexperimentallydeterminedhydrogendensityof113moleculesperunitcell,[25]withperiodicboundaryconditionsapplied.Thetotalpotentialforthesystemisdescribedby:U=Uelect+Upol+ULJ .10 whereUelectistheelectrostaticpotentialenergycalculatedfromtheEwaldeld,Upolisthepolarizationenergycalculatedfromequation2.9andULJistheLennard-Jonespotential.MonteCarlomovesweremadebyselectinganH2moleculeatrandom,andperformingarandomrigid-bodytranslationandrotationofthemolecule.TheMC 19

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movewasthenacceptedorrejectedaccordingtotheMetropolisfunction:min[1;exp)]TJ/F22 11.9552 Tf 9.2985 0 Td[(U] .11 UsingsimpleMonteCarlomovesisnotespeciallydesirablegiventheneedtoentirelyreevalulatethemany-bodypolarizationenergyaftereachsmallmove.Inanattempttomakeglobalmovesthatwouldmoreecientlyexplorephasespace,severalhybridMonteCarloschemes[84,85]wereimplementedwithforcesthatwerecomputedfromeitherthenon-polarizablepotentialorfromarstorderdipole-induced-dipoleevaluationoftheforcesevaluatingequation2.5foroneiteration.Unfortunately,theforceeldsaresucientlydissimilarandtheapproachfailed.Thisresultwasanindicationoftheessentialroleofpolarizationinteractionsinthissystem.ThedetailsofthehybridMonteCarloimplementationaregiveninSupportingInformation,Section2.7AppendixB.TheMonteCarloalgorithmandmany-bodypolarizationcodewereimplementedwithinapackageoriginallydevelopedbytheKleingroupattheCenterforMolecularModelingattheUniversityofPennsylvania.[86,87,88,89]Acellularautomata-basedrule30pseudo-randomnumbergeneratorwasimplementedforit'ssuperiorrandomnumberquality.[90,91]ThemagnitudeoftheMCmoveswereadjustedtoyielda25%acceptancerateinordertominimizethesweep-sweepcorrelations.Autocorrelationofthepolarizationenergygaveacorrelationtimeof=25;000MonteCarlosteps.Afterasystemequilibrationtimeof500,000steps,atomiccongurationsandsystemdipoleswerethensampledatintervalsof2forthecollectionofuncorrelatedstatesofH2intheMOF.Atotalof17.5millionMCstepswerecalculatedon48processorsoftheTeragridmachineLoneStarUniversityofTexasatAustintoyield350statisticallyindependentcongurationsavailableforanalysis.Thecalculationtook300wallclockhours15,000CPUhours.Non-polarizableMCruns,calculatingonlytheelectrostaticandLJtermsinequation2.10,werealsoenactedforcomparison.Themagnitudeoftrialmovementwasgreatlyincreased,andthepotentialenergycorrelationtimewas=2;500MCsteps-anorderofmagnitudelessthanthatofthepolarizedsystem.Clearly,inclusionofthemany-bodypolarizablepotentialgreatlydecreasestheeciencyoftheMCtechniqueinadditiontothecomputationaloverheadintroducedbytheTholemodel;thecalculationofthesystemdipolesconsumesapproximately95%ofthetotalCPUtime.However,thecomputationalcostisstillfarbelowthatofperformingabinitiodynamics. 20

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TheradialdistributionfunctiongrwascalculatedbetweenthehydrogenandvarioussitesontheMOFanderroranalysiswasperformedtoensuretheirconvergence.Isosurfaceanalysiswasperformedoverboththehydrogenpopulationdensityandthehydrogendipolemagnitude.2.4ResultsandDiscussionTodemonstratetheeectofpolarizationonsorptioninsoc-MOF,Figure2.3showstheradialdistributionfunctionbetweenthesorbedhydrogenandindiumionsbothwheninductionisincludedandneglected.Thepolarizationinteractionsstronglyinuencethestructureofthesorbedhydrogenintheregionofthemetalions.Thegurealsoshowsthedistributionfunctionwhenbothcharge-quadrupoleandinductioneectsareneglected-inthiscasehydrogenisinteractingasaLennard-JonesspecieswiththeMOFframework.Thus,thecharge-quadrupoleinteractionsarealsomakinganimportantcontributiontothesorptionstructure.Whiletheeectislessdramatic,theradialdistributionfunctionshowninFigure2.4betweenthehydrogenandazobenzenenitrogenshowsthatthepolarizationeectoftheazobenzeneisalsosignicant.FurtherinsightisgainedbyexaminingthedistributionofH2induceddipolesthatareproducedbytheeldfromthechargesontheMOFprevioussimulationsinthisstudyshowthatthecontributiontowardMOFpolarizationfromthequadrupolarhydrogen-hydrogeninteractionsarenegligible-yet,forcompleteness,theywerealsoincludedhere.Figure2.5plotsthedistributionofinduceddipolesthatisapproximatelyabi-modalGaussiandistributionwithadominantlowdipolar%ofthepopulationwithameandipole0.18Debyeandhighdipolarspecies22%ofthepopulationwithameandipole0.33Debye.AnalysisofmoleculardipolemagnitudeisosurfacesrevealsthatthelowdipolarpopulationcorrespondstospatialregionslocalizedinthevicinityoftheopencoordinationsitesoftheSBU'sshowninFigure2.6,whilethehighdipolarpopulationislocalizedinthemiddleofthewindowformedbytheazobenzenelinkers.Thereasonforthehighdipoledistributionbeingassociatedwiththeazobenzenelinkersseemstobeduetothefactthatthelocationofthewindowisgeometricallyproximaltoallofthe 21

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Figure2.3:Radialdistributionfunctionsbetweenthecenterofmassforthehydrogentotheindiumunderexperimentalconditionsforthreedierentpotentials:Thole-Applequistmany-bodypolarizablepotentialblue,non-polarizablepotentialorangeandLennard-Jonesonlyred. 22

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Figure2.4:Radialdistributionfunctionsbetweenthecenterofmassforthehydrogentotheazonitrogenunderexperimentalconditionsforthreedierentpotentials:Thole-Applequistmany-bodypolarizablepotentialblue,non-polarizablepotentialorangeandLennard-Jonesonlyred. 23

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Figure2.5:Bi-modalH2moleculardipolemagnitudedistributionforsoc-MOFat78Kundertheexperimentaldensity.ThegurepresentsatofthedipoledistributionstotwodecomposedGaussiandistributions. dominantlychargedstructures,namelytheindiumcomplex,nitrateanionandazolinkage.Enhancementoftherstneighborpeakoftheradialdistributionfunctiontothenitrogenoftheazobenzene,showninFigure2.4,wouldseemtosupportthis.ThelowdipolarpopulationisassociatedwiththeelectriceldofthepositivelychargedandhighlypolarizableindiumionsoftheSBU.TheopencoordinationsiteoftheindiumalsopermitshydrogentostronglyassociatewiththemetalionsasshowninFigure2.1.Toputthemagnitudeofthedipolesincontext,ifweconsiderthehydrogenintheMOFapolardiatomicliquidwithadipolecharacteristicofthedominantspeciesof0.18Debye,thismagnitudeiscomparabletothepermanentdipolesofNO.16DebyeorCO0.11Debyethathaveambientboilingpointsof121Kand82K 24

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a0.18Disosurface b0.33DisosurfaceFigure2.6:Isosurfacesshowingthelow-dipolarandhigh-dipolarhydrogenspeciescorre-spondingtotherespectivepeaksofthebi-modalhydrogendipoledistribution. 25

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respectively.Thus,whileneathydrogenat78KhasanegligibleinduceddipoleandisessentiallyanidealgaswithPV/NkT=1.00,[20]thehydrogenintheMOFexperiencesmutual,many-body,dipolarattractions.Thus,itisreasonablethatitisnearlycondensedintheMOFporesattheconditionsconsideredhereandintheexperimentat78K.Simulationsincludingmany-bodypolarizationforthewell-studiedMOF-5[26,92]revealnosignicantchangeinradialdistributionfunctionsforthehydrogentothemetal-centeredSBUwhencomparedtosimulationsneglectinginduction.MOF-5,whichsorbssubstantiallylesshydrogenthanthenewmaterialstudiedhereunderlikeconditions,alsodoesnotpossessanopencoordinationsiteonit'szincSBUandthusproducesamoreweaklypolarizingeldincomparisonwithsoc-MOF.Mostimportantly,MOF-5doesnotpossessnarrowchannelsorahighlypolarframework,butratherhasanopentopologywithwithlargervoidspaces.ThiscomparisonstronglysuggeststhatMOF's,likeMOF-5,withlargeporesarenotthebesttargetmaterialsforsuperhydrogenstorage.ThekeyresultofthisstudyisthathydrogenneedstointeractsucientlystronglywithaMOFtoproduceadipolaruidwithacharacteristicallyhighercondensationtemperature.Recently,hydrogenstorageinaMn-containingMOFwasstudied[93]inwhichasimilarsorptionisothermasthatofsoc-MOFwasmeasured;thisMOFalsopossessesrelativelynarrowchannelsandapolarframework.ThiscubictopologyMOFcontainsanMn2+opencoordinationsiteontheSBU;itisnotunreasonablethentoassumethatthehighsorptioncapacityofhydrogeninboththatmaterialandsoc-MOFarecorrelatedwiththestructuralmotifoftheSBU.Thedipoleisosurfacesgeneratedbythisworkwouldsuggestthattheopencoordinationsitesservetopolarizethehydrogenundertheexperimentalconditions.Usingthepolarizablepotential,theenthalpyofadsorptionforsoc-MOFwascalculatedby:H=EMOF+H2)]TJ/F15 11.9552 Tf 11.9552 0 Td[([EMOF+EH2] 26

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andwasfoundtobe10.6kJmol)]TJ/F20 7.9701 Tf 6.5865 0 Td[(1.Whilethisvaluediersfromthevalueof6.5kJmol)]TJ/F20 7.9701 Tf 6.5865 0 Td[(1experimentallymeasured[25],itshouldbepointedoutthattheexperimentalenthalpiesquotedaretakentobethelimitofzerohydrogenloading.SincetheClausius-Clapeyronmethodislimitedbythetemperaturedierentialforwhichtheisothermsarecalculated,experimentalmeasurementsdeterminingtheheatofadsorptionforthesaturatedstatestudiedheredensityat1.0atmhavenotbeendone.Sinceitisknownthattheenthalpyofadsorptionofsoc-MOFincreaseswithloading,[25]thecalculatedvalueoftheenthalpyseemsconsistentwitha6.5kJmol)]TJ/F20 7.9701 Tf 6.5865 0 Td[(1enthalpyforthezerodensitylimit;hydrogen,undertheconditionsstudiedhere,isastronglyinteractingdipolaruid.ModelingoftheMOFunderlowerdensityconditionsisthesubjectofongoinginvestigation.2.5ConclusionsTheresultsofthisstudysuggestdesirableMOFdesigncharacteristicsforhydrogen,andgasstorageingeneral-MOF'sarepromisingcandidatesforgasstorage/sequestration.Forexample,onewouldexpectCO2sorptioninsoc-MOFtobequitestronggiventhesignicantlyhighermolecularquadrupoleandpolarizabilty.ThisstudysuggeststhatMOF'sshouldhaverelativelysmallporesandinterconnectedporeswithhighsurfaceareatocreatestrongMOF-H2interactionsand,thus,indirectlyH2-H2attractions.Topromotetheseinteractions,theMOFalsoneedstobelocallypolarwithlargechargeseparationsonitssurfacesucientlyfaraparttoallowhydrogenmoleculestobesensitivetothedipolarinterface.Further,whilethesurfaceareaneedstobelarge,theopenspatialnetworkshouldnotbesoexpansivethathydrogenmoleculesfarthestfromtheMOFsurfacedonotpossesssignicantinduceddipolesandcharge-quadrupoleforces.Forexample,ifaMOFweretopossessalargesurfaceareaduetosizablepores,hydrogentowardthecenterofthevoidwillbesimilartoneathydrogenwithcharacteristicallyweakintermolecularinteractionsandacorrespondinglylowercondensationtemperature.Thereis,however,atrade-obetweenhavinglargervoidvolumesthatproducealowermolecularweightmaterialbutdonotpromotestrongsorptiveforcesversushavingahighlynanostructedporesystemwithcorrespondinglymorematerialperunitvolume,butstrongscaold-H2interactions. 27

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Theseinitialstudieshaveprovidedsignicantinsightsintothenatureofhydrogeninteractionsinnanporous,polarMOFmaterials.Theresultspresentedalsosuggestseveralfutureavenuesofinquiry.Foremost,wewillproceedtocalculatesorptionisothermsforourmodelusingaWidominsertionmethod[94,95,96]thatcanbecomparedwithexperiment.Thiswillalsoservetofurthercalibratethepotentialenergysurfaceofourmolecularmechanicsmodelby,forexample,givingusthesystempressureattheexperimentallyobservedhydrogenloading.WecanthenproceedtomutatetheMOFinanexperimentallyplausiblefashiontoattempttoincreaseitshydrogenstoragecapacity.Lastly,thisstudysuggeststhatincludingpolarizationinmodelingotherextantandfutureMOF'scangivereliablephysicalinsightintothemechanismofgasstorage.2.6AcknowledgmentsTheresearchatUSFwassupportedbyanNSFGrantNo.CHE-0312834andagrantfromthePetroleumResearchFundtoBrianSpace.TheauthorswouldliketoacknowledgetheuseoftheservicesprovidedbytheTeragridandtheResearchComputingCenteratUSF.TheauthorsalsothanktheSpaceBasicandAppliedResearchFoundationforpartialsupport.MohamedEddaoudiacknowledgesfundingfromanNSFGrantNo.DMR-0548117.MohamedEddaoudiandBrianSpaceacknowledgefundingfromNASANo.NGA3-2751andtheDepartmentofEnergy,BasicEnergySciences.ThismaterialisbaseduponworksupportedbytheNationalScienceFoundationunderthefollowingNSFprograms:PartnershipsforAdvancedComputationalInfrastructure,DistributedTerascaleFacilityDTFandTerascaleExtensions:EnhancementstotheExtensibleTerascaleFacility. 28

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2.7SupportingInformationAppendixA-IterativeSolutiontotheDipoleFieldEquationsRecently,anumberofmultigridtechniquese.g.Gauss-Seidel[97]havebeenusedwiththeTholemodel.[98]Typicallyhowever,asimultaneousover-relaxationschemei.e.linearsolutionmixingisusedtoimprovetheconvergenceratefortheiterativeeldcalculation.[75]InthisworkwehavefoundthatincorporationofGauss-Seidelsmoothing,whilemakinguseofanewrecursiveexponentialsuccessiveover-relaxationalgorithm,enhancestheconvergenceratebytwofoldoverlinearmixinganorderofmagnitudeovernaiveiterationofequation2.5withinthecontextoftheMonteCarlosimulationsperformed-particularlyonsymmetricmulti-processorarchitectureswherethenewlyavailabledipolesetissharedinlocalmemory.Forparallelarchitectures,constanttransferofthenewlyavailabledipolesetviaGauss-Seidelwithinasingleiterationtothenetworkedprocessorswasfoundtoaectthecomputationalperformancenegatively,i.e.theGauss-Seidelupdatesarebestexecutedinlocalmemory.Inthelimitofhighparallelism,therecursiveexponentialsuccessiveover-relaxationschemeplaysmoreofaroleindeterminingthescalabilityasmoreprocessorsareutilized.Therefore,theperformancetradeobetweenthesetwoextremasinglenodehigh-SMPvs.networkedparallelismisoptimallymanaged.Thisnewmethodallowsforgreatercomputationaleciencyinparallelenvironmentsoversimultaneousover-relaxation,makingMonteCarlosimulationsofMOFsystemsontheorderof1000atomswiththeTholemodelfeasibleonexistingcomputationalplatforms.Note,asatest,simulationsofthisMOFsystemwereperformedusingsinglebodypolarizationandtheresultswereinadequatei.e.annealingresultedinverydierentequilibriumstructuresandsosolvingthemany-bodyequationswasanecessity.TheappliedGauss-Seidelnumericaliterationmethodforaslowlyconvergingprocessconsistsofupdatingthecurrentdipolevectorsetforthekthiterationstep 29

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asthenewdipolevectorsbecomeavailable:ki=iEstati)]TJ/F22 11.9552 Tf 11.9552 0 Td[(Tijk)]TJ/F20 7.9701 Tf 6.5865 0 Td[(1+j;=0;ifij .12 Therecursiveexponentialsuccessiveover-relaxationthatwehaveformulatedreectsthecurrentiterationstepk=1;2;:::suchthattheweightingshiftstowardthenewsolutionasafunctionoftheiterativeprogress:k+1i=)]TJ/F15 11.9552 Tf 5.4794 -9.6838 Td[(1)]TJ/F22 11.9552 Tf 11.9551 0 Td[(e)]TJ/F23 7.9701 Tf 6.5865 0 Td[(kkj+e)]TJ/F23 7.9701 Tf 6.5865 0 Td[(kk)]TJ/F20 7.9701 Tf 6.5865 0 Td[(1j .13 whereisafreeparameterthatcanbetunedtoaltertheconvergencerate.AppendixB-HybridMonteCarloWhileMDsimulationswithmany-bodypolarizableforcescanyieldinsightintothebehaviorofcomplexsystems,theyarealsoreputedfortheirinstabilityandcomplexityofimplementation;developmentofstableandecientpolarizabledynamicsiscurrentlyanactiveareaofresearchincomputationalchemistryandphysics.[74,75,76,77]Thepolarizableforcesaretroublesomeandexpensivetocalculate;thepolarizationpotentialenergyisrelativelylessproblematicandiseasiertocomputeviaequations2.5-2.9.OurinitialapproachwastoperformahybridMonteCarlo[84]HMCcalculationwherebythepotentialenergy,especiallyincludingthepolarizationenergy,wasusedintheMetropolisacceptancefunctionbutthesystemcongurationwasgloballypropagatedbyMDusinganon-polarizableforceeld.Inthisway,wewouldcircumventtheneedtocomputepolarizableforcesandyetproperlysamplethecongurationspacethatwouldreectpolarizationeects-allthewhile,gettingthebenetofshortcorrelationtimesandecientsamplingofphasespaceduetotheglobalmovesbeingmadeasopposedtothelongcorrelationtimesthataretypicalofmany-bodypotentialsinthestandardMonteCarloscheme. 30

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Unfortunately,theaboveHMCapproachprovedunworkableforanumberofreasons.WhiletheHMCmethodhasbeendemonstratedtoworkwellforLennard-Jonesuids[85]andfermionicparticles,[84]inasystemwithmanydegreesoffreedomwefoundadicultyinthestandardmomentumresamplingschemethatisnot,toourknowledge,specicallyaddressedintheliterature.Itwasourexperiencethatcompletelyresamplingthemomentahinderedthemolecularrotationsofthesystem,eveninthenon-polarizablescenario,andthestructureforbulkmolecularuidsthatwetestedsuchashydrogenandwaterdidnotaccuratelycorrespondtotheknownstructure,aswasevidentbythecomparisonofradialdistributionfunctionsfromMDandMonteCarlo.However,foraLennard-Jonessystemi.e.possessingonlytranslationaldegreesoffreedomthecompletemomentaresamplingschemeyieldsthecorrectstructure.IthasbeenclariedthatacompleteresamplingofmomentafromaBoltzmanndistributionisnotrequired,[99]andinfactweimplementedanalternativemomentumupdaterecentlygivenintheliterature[100]wherebythemomentumvectorscanbelessdrasticallyperturbedandthenaMetropolisevaluationofthekineticenergyisperformed:~p0=f~pg+f~g .14 min1;exp)]TJ/F22 11.9552 Tf 9.2985 0 Td[(p02 2m)]TJ/F22 11.9552 Tf 15.893 8.0877 Td[(p2 2m .15 wheref~pgisthesetofmomentumvectorsinthesystem,f~gisasetofvectorsrandomlysampledfromaBoltzmanndistributionandisafreeparameterusedtotunetheacceptanceratio.ThisupdateschemerestoredthecorrectrotationaldynamicsandstructureforthebulksystemsstudiedwithHMC.Whiletheprimemotivationfortheinitialdevelopmentofalternativemomentumresamplingschemeshasbeentodispensablyimprovetheeciency,itisourviewthatlessdrasticmomentumresamplingisrequisiteinsimulatingsystemswithrotationaldegreesoffreedom.However,asecondunfortunatelyinsurmountabledicultywithutilizingHMCtostudysoc-MOFwasinusingapolarizableMCpotentialwithnon-polarizabledynamics:insituationswherethepolarizationeectsaresignicante.g.soc-MOFatlowtemperature,thepolarizableandnon-polarizablepotentialenergysurfacesaresimplydiscrepantenoughtopreventtheadequatesamplingofphasespacethisagainstressestheimportanceofpolarizationinteractionsforH2insoc-MOF.Thus,thecomputationallydemandingstandardMonteCarloschemewasimplemented. 31

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Chapter3ApplicationofaNewHydrogenPotential3.1AbstractAbstract:NewlydevelopedhydrogenandMOMMetal-OrganicMaterialspotentialenergyfunctionsformolecularsimulationarepresented.Theyaredesignedtobehighlytransferablewhilestilldescribingsorbate-MOMinteractionswithpredictiveaccuracy.Specically,theyareshowntoquantitativelydescribehydrogensorption,includingisostericheats,inMOF-5overthebroadtemperatureandpressurerangesthathavebeenexaminedexperimentally.TheapproachthatisadoptedisgeneralanddemonstratesthathighlyaccurateandpredictivemodelsofmolecularinteractionwithMOMsarequitefeasible.Molecularinteractionsgivingrisetotheisostericheathavebeencharacterizedandvalidatedagainsttheexperimentallyrelevantdata.Finally,inspectionoftheisothermalcompressibilityofhydrogeninMOF-5revealsthatundersaturatinghigh-pressureconditionsevenattemperatureswellabovetheneatboilingpointthestateofhydrogenischaracteristicofaliquid, 32

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i.e.withacompressibilitysimilartobulkhydrogen.ThisresultisofparticularrelevanceindevelopingMOMsforhydrogen-storageapplications.3.2IntroductionMetal-OrganicMaterialsMOMsareamongthemostwidelystudiedstructuresforgasstorage,andyetveryfewtheoreticalstudieshaveinvestigatedtheaccuracyoftheintermolecularpotentialenergyfunctionsforMOMsinteractingwithhydrogen.Whilerstgenerationinvestigationshaverevealedreasonablygoodagreementwithexperimentusingadhocparameterizations,othershaveraisedquestionsastotheaccuracyofthepotentialparameters.[101,15]Ineithercase,thepotentialsshouldbevalidatedunderhigh-pressureconditionsratherthanthemoretypicalstandardpressurestatepointandoverawidetemperaturerangewherediscrepancieswouldbeevidentifthepotentialswereinaccurate.Thatis,inordertoproduceapredictive/transferablemodelofMOM-guestinteractionsformolecularsimulations,thepotentialmustincludeallsalientintermolecularinteractions.Hereinweexplorethevalidityofanewlydevelopedhydrogenpotential[102]interactingwiththeprototypicalMOF-5a.k.a.IRMOF-1underallextantexperimentallyexaminedconditions,includinghigh-pressuresthataremorerelevanttotheDOEsrequiredstorageconditions.[6]TheexcellentagreementwithexperimentaldataoverthewiderangeoftemperaturesandpressuresillustratesthepointthatquantitativelypredictivemodelsforgassorptioninMOMsarequitefeasiblewhencarefulattentionispayedtotheintermolecularpotentials.Inadditiontothethermodynamichydrogenuptake,wehaveinvestigatedtheisostericheatofadsorption,[103]Qst.ThetypicalexperimentalapproximationemployedistoderiveQstfromisothermsperformedatdierenttemperatures.QstisdeterminedfromtheClausius-Clapeyronequationviaanumericalderivativeusingtwoisotherms.Incomputersimulation,themoleculardetailsofadsorptionareaccessibleforfurtheranalysisandadirectstatisticalmechanicalexpressionforthe 33

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isostericheatisavailablefromthethermodynamicuctuationsinsorbatenumber.Here,goodagreementisfoundforthefrequentlyexamined77Klow-pressurerange.Athigherpressures,ourdataareconsistentwiththelimitedexperimentaldata;unfortunatelytheexperimentalQstdataathigherpressureshaveinherentlylargeerrorsduetotherangeoftemperaturesoverwhichthenumericalderivativesareperformed.Finally,wehavedeterminedtheisothermalcompressibilityofhydrogeninMOF-5asafunctionofpressure.Interestingly,thehydrogenapproachesastateresemblingthatofthebulkliquidastheexcesssorptionweightplateaus.Thisisevidencedbytheisothermalcompressibilityfallingrapidlyoveranarrowpressurerangetoavaluecharacteristicofbulkhydrogen.ThispreliminaryresultisofparticularimportancebecausetheDOErequirementsforhydrogenstorageareinexcessofliquiddensity,andhenceanyinteractionsthatmayservetostabilizetheliquidstateareofpracticalimportanceinelucidatingMOMdesignprinciples.Thispaperisorganizedasfollows.Section3.3presentsthehydrogenandMOFpotentialsaswellasthestatisticalmethodsemployedinthecomputersimulations.Section3.4detailsanddiscussestheresultsfoundintheinvestigationofhydrogenuptakeandotherthermodynamicquantities.Section3.5concludesthemanuscript.3.3Methods3.3.1HydrogenPotentialRecently,anaccurateandtransferablehydrogenpotentialenergyfunctionhasbeendeveloped[102]whichincludesquadrupolarelectrostatictermsaswellasmany-bodypolarization,bothofwhichhavebeenshowntobeimportantinmodelingdensehydrogeninteractingwithachargedsurface.[12]Thisnewhydrogenpotentialhas 34

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beenshowntoyieldanaccurateequationofstateforhydrogenunderhigh-pressuresandlow-temperatures.Moreimportantly,thefunctionalformofthepotentialiseasilytransferableinthesensethatitcandescribehydrogeninteractingwithmaterialsthataredescribablebyLennard-Jones6-12parameters,partialchargesandpointpolarizabilitiesaformfrequentlyusedformolecularpotentials.Incontrast,theextanthydrogenpotentialsthatcandescribebulkhydrogenaccuratelySilvera-Goldman,Buch,etc.arenotreadilysuitabletocomplexandheterogeneouscondensedphasesimulationand/ortoawiderangeofstatepoints.Wehaveutilizedtheaforementionedpotentialinthisworktovalidateit'saccuracyonmodelingvariousthermodynamicobservablesofhydrogeninMOF-5.Theeectsofmany-bodypolarizationhavebeenshowntobeimportantfortheaccuratesimulationofhydrogenuptakeincertainpolarMOFs,[12]andforbulkhydrogenathighpressures.[102]However,MOF-5doesnotpossesstherelativelylargechargeseparationontheframeworkcharacteristicofsomeotherMOFs.[104,25,93,105,106]Therefore,inductioneectsinmodelinghydrogeninMOF-5arenegligibleand,infact,numericalanalysisofthepolarizationinMOF-5conrmsthis.Whenincluded,thepolarizationenergyislessthan5%ofthetotalenergyat77Kanddoesnotsignicantlyaltertheisothermsandassociatedisostericheats.Therefore,thenon-polarparametersforthehydrogenpotentialreferredtoabovehavebeenusedandtheThole-Applequistmany-bodypolarizationcalculationwasonlyperformedinthisworkatselectedstatepointsasacontrol,verifyingthenegligibilityofinduction.Whileinclusionofinductionisdesirable,itsmany-bodiednaturemakesthesimulationsordersofmagnitudemorecomputationallyexpensive.3.3.2MOF-5PotentialPermanentelectrostaticinteractionsinatomisticsimulationsstemfrompointpartialchargesassignedtothecoordinatecorrespondingtothenuclearcenterofeachatom.PointchargesweredeterminedfromelectronicstructurecalculationsonseveralmodelcompoundsthatmimicthechemicalenvironmentofMOF-5.[40]TheGAMESSabinitiosimulationpackagewasusedtoperformtheHartree-Fockquantummechanicalcalculations.[65] 35

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ThestructureofMOF-5ischaracterizedbybenzene-dicarboxylateBDClinkedzinctetramersinacubicoctahedralnet.The424atomunitcellmaybeproducedfromcrystallographicsymmetryoperationsappliedtoonly7atoms.[26]Thissymmetry-uniqueformwasusedasthebasisfordecidingonrepresentativechemicalfragments.Forthepurposesofchargetting,fragmentsoftheinnitenetwereselectedinavarietyofwaystoassesstheeectsofstructuraltruncationonthetcharges.Theadditionofhydrogenatoms,whereappropriate,wasrequiredforchemicalterminationofthefragmentboundaries.Severaldierentbasissetswerechosenandresultscomparedfavorably,withtheresultingchargesbeingwithin0.1e)]TJ/F15 11.9552 Tf 10.9867 -4.3384 Td[(ofeachotheronaveragethecompletecomparisonbetweenchemicalfragmentsandbasissetsisgiveninsupplementaryinformation. Figure3.1:MolecularfragmentoftheMOF-5frameworkforwhichpotentialparametershavebeendeterminedaslistedinTable3.1. Theneedforintramolecularframeworkinteractions[107]wasavoidedbyholdingthescaoldrigidduringsimulation.Phononsarenotthoughttoplayanimportantroleinhydrogensorption,especiallynotatthetemperaturesconsideredhere.[39]Lennard-Jonesparameters,representingrepulsiveandvanderWaalsinteractionsbetweenhydrogenatomsandframeworkweretakenfromtheUniversalForceField;[64]thissetofLennard-JonesparameterswasusedinearlierMOFstudiesas 36

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Atom Label /A /K q/e)]TJET1 0 0 1 435.8713 704.8518 cmq[]0 d0 J0.3985 w0.1992 0 m0.1992 14.7497 lSQ1 0 0 1 -223.9419 -0.3985 cmq[]0 d0 J0.3985 w0 0.1992 m224.1411 0.1992 lSQ1 0 0 1 0 -1.9925 cmq[]0 d0 J0.3985 w0 0.1992 m224.1411 0.1992 lSQ1 0 0 1 -0.1992 -14.4458 cmq[]0 d0 J0.3985 w0.1992 0 m0.1992 14.4458 lSQ1 0 0 1 -211.7302 -688.015 cmBT/F15 11.9552 Tf 225.3838 692.3488 Td[(Zn 1 2.4616 62.3993 1.8529 O 2 3.118 30.19 -2.2568 O 3 3.118 30.19 -1.0069 C 4 3.431 52.84 1.0982 C 5 3.431 52.84 -0.1378 C 6 3.431 52.84 -0.0518 H 7 2.571 22.14 0.1489 Table3.1:TheMOF-5potentialparametersthatwereusedinthisstudy.TheatomiclabelsrefertotheindicesdepictedinFigure3.1. well.[40,47,15,61]ThecompletesetofpotentialparametersusedinthisstudyiscontainedinTable3.1andthemolecularfragmenttowhichtheyreferisdepictedinFigure3.1.3.3.3GrandCanonicalMonteCarloWithrespecttohydrogenuptake,theprimeobservableofinterestistheaveragenumberofhydrogenmoleculessorbed,hNi,viasamplingofthegrandcanonicalensembleoverarangeofchemicalpotentialscorrespondingtotheequilibriumpressureofthereservoir.ThefollowingstatisticalmechanicalexpressionwasnumericallyestimatedbyGrandCanonicalMonteCarlo[7,108]usingacodedevelopedbyourgroup:[109]hNi=1 1XN=0eN3NYi=1Z1dxiNe)]TJ/F23 7.9701 Tf 6.5865 0 Td[(UFHx1;:::;x3Nwherethechemicalpotentialofthegasreservoir,,wasdeterminedforabroadrangeoftemperatures60-300KthroughtheBACKequationofstate.[110,111]QuantummechanicaldispersioneectshavebeenincludedsemiclassicallythroughuseoftheFeynman-Hibbseectivepotential[112]toorder~4viatheexpression: 37

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UFH=U+~2 24U00+2 rU0+2~4 1152215 r3U0+4 rU000+U0000 .1 andthepotentialenergyfunctionusedamountsto:U=Ues+Urd .2 whereUesistheEwald-summedelectrostaticpotentialandUrdaccountsfortheelectronicrepulsion/dispersionenergythroughuseoftheLennard-Jones6-12function.TheMOF-H2LJinteractionparametersweredeterminedusingtheLorentz-Berthelotmixingrules;pairwise'saredeterminedthroughanarithmeticmeanand'sareformedfromthegeometricmean.AfterobtaininghNiweproceededtocalculateboththeabsoluteandexcessweightpercentofhydrogensorbed;theexcessweightcalculationutilizedthefreevolumeof11595.4A3determinedpreviouslyforMOF-5.[26]Experimentally,theisostericheatofadsorptionisdeterminedbynumericalanalysisoftwohydrogenisothermsperformedatdierenttemperaturestypically77and87K.Theisothermdataisthenprocessedeitherviacurve-ttingorinterpolationandtheisostericheatofadsorption,Qst,isdeterminedoverarangeofdensitiesthroughanite-dierenceapproximationtotheClausius-Clapeyronequation:Qst=kT2@lnP @T .3 WhilethemacroscopicClausius-ClapeyronequationcanbeusedwithGCMCisothermdatatoarriveatvaluesforQst,amoredirectstatisticalmechanical 38

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method[113]istorelatetheisostericheattouctuationsofaquantityinvolvingthenumberofsorbedmolecules,N,andthepotentialenergyU:Qst=)]TJ 10.494 8.0878 Td[(hNUi)-222(hNihUi hN2i)-222(hNi2+kT .4 Anotheraccessible,uctuation-derivedquantityistheisothermalcompressibility:T=)]TJ/F15 11.9552 Tf 12.2803 8.0878 Td[(1 V@V @P .5 whichmaybecalculatedviauctuationsofthenumberofmoleculessorbed,hNi,inthegrandcanonicalensemblethroughuseofthestatisticalmechanicalrelation:T=V kThN2i)-222(hNi2 hNi2 .6 BothEquation3.4and3.6havebeenimplementedintotheMonteCarlocodeusedforthisstudyandthesequantitieshavebeenassessedforH2inMOF-5. 39

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3.4ResultsandDiscussion3.4.1HydrogenIsothermsExperimentalvalidationofagivenMOM'shydrogenstoragecapabilitytypicallytakesplaceattheliquidnitrogenboilingpoint77K.WhiletheDOEmilestonesrequireroom-temperatureoperation,evaluationofthesurfaceinteractionsismoreeasilydeterminedatlowertemperaturewheremeaningfulmeasurementscanbemadewithrespecttohydrogenuptakeforevenmarginallysorbingmaterials.Inaddition,theisostericheatofadsorptionismeasuredacrosstherelativelysmalltemperaturevariationof77-87K.Thus,hydrogenuptakeinMOF-5,assimulatedwiththenewlydevelopedpotentials,isreportedinFigure3.2.Furthermore,theabsoluteandexcesssorptionisothermsforhydrogeninMOF-5havebeenmeasuredoverawiderangeoftemperatureandpressure,makingthiswellcharacterizedsystemidealforcomparisonwiththepresenttheoreticalmodel.TheresultsarecomparedwithexperimentinFigure3.3andFigure3.4.Forcomparison,wenotethatinFigure3.5oursimulationresultsat77Kliewithintherangeofavailableexperimentaldata.Thiscomparison,whilenotavailableatawiderangeofstatepoints,suggeststhattheresultsinFigures3.3and3.4areaccuratetowithintheextantexperimentaluncertaintiesincluding,e.g.,standardmeasurementerrorsandvariationinmaterialspreparationprotocol.Note,thetwosetsofexperimentaldatapresentedarerepresentativeofsomeofthemostcarefulmeasurementsofhydrogensorptioninMOF-5todate. 40

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Figure3.2:GCMCcalculatedlow-pressurehydrogenisothermabsoluteweightpercentofMOF-5at77Kvs.experiment.[114,115] 41

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Figure3.3:GCMCcalculatedstarreddatahigh-pressureisothermabsoluteweightpercentofMOF-5overawidetemperatureandpressurerangevs.experimentaldata[116]solidlines.Maximumcalculatederroris0.07wt% 42

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Figure3.4:GCMCcalculatedstarreddatahigh-pressureisothermexcessweightper-centofMOF-5overawidetemperatureandpressurerangevs.experimentaldata[116]solidlines.Maximumcalculatederroris0.07excesswt% 43

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Figure3.5:GCMCcalculatedhigh-pressureisothermexcessweightpercentofMOF-5at77Kvs.experimentaldata[116,115]solidlines.Maximumcalculatederroris0.068excesswt% 44

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3.4.2IsostericHeatofAdsorptionofHydrogenTheisostericheatofadsorptionofhydrogeninMOF-5wascalculatedfromtheuctuationexpressionEquation3.4overthestatisticalmechanicalstatesandisshowninFigure3.6.ThelowpressureresultsinthegurearefoundtobeingoodagreementwithexperimentespeciallyconsideringthattheexperimentaldataforMOF-5iscorrespondinglysimilarbasedupondieringpreparationtechniques,etc.TheisostericheatofMOF-5ischaracteristicofphysisorption5kJ/molandremainsfairlyconstantoveranextendedrangeofdensities.ThecalculatedQstvaluesfortheothertemperaturesoverbroaderpressurerangesarealsofairlyconstantinarangefromabout4-5kJ/molatlowpressure,and3.5-5kJ/molatthehigherpressuresconsidered.Theexperimentaldataathigherpressuresanddiversetemperatures[116]aresimilarbutdiculttocomparequantitativelybecausethenitedierenceapproximationtoQstviatheClausius-Clapeyronequationgivesarangeofvaluesdependinguponthetwosetsoftemperaturedatathatarechosen;itisalsofoundthatthetheoreticalisotherm-derivedQstvaluestakeonasimilarrangeofvalues.Itwouldbeusefulifcalorimetryexperimentshadmappedoursorptionenthalpiesoverarangeofthermodynamicconditions.[103,117]Nonetheless,innocasearethetheoreticalvaluesinconsistentwiththeapproximateexperimentalvalues.Further,thedataalsosupportamodelwhereMOF-5withit'slargesurfaceareaandporevolumeisknowntoretainit'sinteractionswithhydrogenat77Kacrossthepressurerangeupuntilsurfacesaturation-oncethesurfaceiscoveredthenthebulkcharacteristicallyweakH2-H2interactionscontributeagreaterportiontowardthestatisticalaverageofQst.3.4.3IsothermalCompressibilityofHydrogenTheisothermalcompressibilityofhydrogeninMOF-5at77Kwasfoundtoreachaminimumwhentheexcesssorptionisothermsaturated.Furthermore,asshownin 45

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Figure3.6:Comparisonofthecalculatedviathemicroscopicuctuationequationisos-tericheatofadsorption,Qst,forhydrogeninMOF-5at77Kvs.experimental[26,118]data. 46

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Figure3.7,thevalueofthecompressibilitywasfoundtoresemblethatoftheliquidorsupercriticalstatewithadensitysimilartothenormalliquidstate.Thisbulkcompressibilityvalueisitselfrelativelyinsensitivetotheparticularcondensedphasestatepoint.Itisnotsurprisingthattheporedensityofhydrogenunderhigh-pressureinMOF-5ischaracteristicofthebulkphasesgiventheclosepackingofH2inthatcase.However,thecompressibilityhasseveraluniquepropertieswhichareofinterest.Forexample,thecompressibilitywillbeafunctionofbothsurfaceinteractionandbulk-likei.e.porelocalizedhydrogencontributions,therelativeeectsofwhichwillvaryasacomplexfunctionoftheequilibriumstatepoint.Therefore,giventhattheMOF-5connedH2compressibilityiscomparabletobulkphasessuggestsarelativelyminorperturbationtothestructureintheweaklyinteractingMOM.Theresultsdemonstratethattheisothermalcompressibility,directlyobtainablefromGCMCcalculations,isanimportantparameterworthmonitoringtoassessthenatureoftheconneduidandtheabilitytofurtherimprovesorptioncapacitywithincreasedpressure. Figure3.7:CalculatedisothermalcompressibilityofhydrogeninMOF-5at77Kvs.experimentalvaluesofhighdensityT=0.0015atm)]TJ/F20 7.9701 Tf 6.5865 0 Td[(1K/200atm[119,120,121]andT=0.0020K/1atm.[122]Theexcessweight%ofhydrogeninMOF-5at77Kisalsodepictedontheoppositey-axisforcomparison. 47

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3.5ConclusionsAftercarefuldevelopmentofbothhydrogenandMOMpotentialenergyfunctions,quantitativelyaccuratehigh-pressureresultshavebeenobtainedmorespecically,thesorptionuptakeandisostericheats.TheanalogousdevelopmentofpotentialsurfacesforotherMOMsandpossiblesorbentsislikelytoleadtosimilarlyaccuratemodelsoftheseimportantsystems.Suchpredictiveaccuracywillaidgreatlyintherational,iterativedesigncyclebetweenexperimentalandtheoreticalgroupsthatareattemptingtodesignMOMsforavarietyofpurposes,includingH2sorptionandCO2sequestration.Here,anaccurateandtransferable,anisotropichydrogenpotentialhasbeenemployedinstudyingMOMs,anditisexpectedthatthemany-bodypolarizableformoftheH2potentialwillshedfurtherlightoninteractionsinMOMspossessingopen-coordinationsitesandcharged[123,93,106]and/orpolar[25]frameworksthatpolarizehydrogenmoleculesthecontinuingsubjectoffuturework.Suchcharged/polarMOMshavegreatpotentialformultipleapplicationsduetothenecessarilystrongerinteractionsthattheyhavewithguestmoleculesviathepolarizationinducedbytheMOMframework.Finally,thebenetofliquidstateanalysisappliedtohydrogenintheuniquetopologicalandchemicalenvironmentpresentedbyaMOMmayyieldnewinsightsintothedesignpropertiesnecessaryforenhancedgasstorage,ofwhichanalysisoftheisothermalcompressibilitythisworkandradialdistributionfunctions[12]haveplayedarolethusfar.Lastly,thesuccessofthepresentMOM-guestinteractionmodelsuggeststhatmolecularsimulationmethods,withcarefullyconstructedpotentialenergysurfaces,arethemethodofchoiceinmodelingandpredictingthepropertiesofsuchsystems-theyarereasonableincomputationalcostwhileretaininghighaccuracy. 48

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Chapter4DesignMotifstoMaximizevanderWaalsInteractions4.1AbstractNanoporousmaterialsareparticularlyinterestingcandidatesforapplicationsinvolvinggasstorageorsequestration.Gasstoragenotonlyrequiressucientlystrongattractiveintermolecularinteractionstopreventgasmoleculesfromdesorbing,butalsothattheseinteractionsbeweakenoughthatdesorptionisstillfeasibleatpracticaltemperaturesandpressures.Thisseeminglyparadoxicalsetofconstraintsmayrequireprecisecontrolofmolecularinteractionsifmaterialsofpracticalapplicationaretobedesigned.Understandingtheinteractionspresentinsidetheporesofmetal-organicmaterialsand,morespecically,howtheymaybealteredortunedtomeetspeciccriteriaisessentialiffunctionalmaterialsaretobedesigned.Inthepresentwork,wefocuson 49

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exploringtheroleofdispersiveinteractionsbetweenhydrogenandametal-organicframework.Withamodelsystem,weattempttomaximizedispersiveeectswhileminimizingtheeectsofelectrostatics,polarization,andorbitalinteractions.4.2IntroductionMetal-organicframeworks,asfunctionalsolidstatematerials,continuetoreceivewidescienticinterestduetotheirpotentialapplicationsinhydrogenstorage,gasseparation,carbondioxidesequestration,enhancedcatalysis,anddrugdelivery.[124,125,126,127,128]SuchapplicationsarepertinenttothefundamentalattributesofMOFsincludingdualcomposition,highcrystallinity,andopenstructures.Inparticular,porousMOFshavebeenwidelyinvestigatedforhydrogenstorage,demonstratingreversiblephysisorptioninteractions,withinavailablevoidspace,amenabletoahighdegreeoftuneabilityassociatedwiththehighlymodularnatureofMOFs.[129,130,131,132,133,134,135,136,137,138]Hydrogeninteractionswithmetalcomplexes,clusters,orions,ascanbefoundwithintheinorganicpartoftheframework,arelargelycomposedofelectrostaticforcesbetweenthequadrupolemomentofthehydrogenmoleculeandtheinorganiccomplex.Indeed,suchforcesplayamajorroleindeterminingtheH2uptakecharacteristicsofaparticularMOFduetotheirhighinteractionstrengthandhencearethesubjectofconsiderabletheoreticalandexperimentalinvestigations.[139,140,141,142,143,144]Althoughweaker,thedispersiveinteractionsbetweenH2moleculesandtheorganiclinkersinMOFs,bestrepresentedbybenzeneringderivatives,havebeentheoreticallyinvestigated[19,145,146,147,18]andexperimentallydocumented[26]throughinelasticneutronscatteringexperiments.Recentstudiesdemonstratethatsuchinteractionscould,inprinciple,beenhancedthroughchemicalmodicationstotheorganiclinkers,providingapotentialstrategyforamaterialdesignertoenhanceH2sorptioncharacteristicsofMOFs.[19] 50

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Nevertheless,tothebestofourknowledgenostudiesexistwhichaddressthepossibilityofimprovingH2bindinganitytothewallsofMOFsthroughsimultaneousdispersiveinteractions,actingadditively,betweenH2moleculesandmultiplearomaticringsplacedatoptimalinteractiondistancesandwithinaspecicgeometry.Therefore,weoptedtoexplorethisapproachwhichcouldpotentiallyproveusefulasaviabletargettoconsider,amongothers,inrationaldesignstrategiesforfuturehydrogenstoragematerials.Computationalstudies[19]fortheH2bindinganitiestobenzeneandvariousaromaticringsrevealedmoderatebindinganities,mostlyduetodispersiveinteractionsandintherangeof3.4-4.0kJmol)]TJ/F20 7.9701 Tf 6.5865 0 Td[(1foranH2moleculeinteractingwiththebenzeneringofterephthalicacid.[19]AlthoughtheaforementionedbindingenthalpyisbelowtheestimatedtargetforecientH2storagematerialsatambientconditions,intherangeof21-32kJ/mol-20Candpressurerange1-100bar[19],or15-20kJ/molroomtemperatureatpressureupto30bar[147],itispossiblethatsuchinteractionscouldbeadditiveandhencecanleadtoenhancedfavorableinteractionsbetweenanH2moleculeandmultiplearomaticringsintailoredframeworks.4.3Methods4.3.1ComputationalMethodsAsatestmodel,weenvisionamolecularsquareconstructedoffourbenzeneringsinteractingsimultaneouslywithasingleH2molecule,residinginthecenterofthesquare,asapotentialmodelforamaterialwithenhancedH2bindinganity,asdepictedinFigure4.1.Inthismodel,theH2moleculeislocatedatuniformdistance,R,betweenitscenterofmassandthecentroidsofthesurroundingbenzenerings.Itisobviousthat,duetothedependenceofdispersiveinteractionsonR,whichdecreaseas1/R6,anyexpectedenhancementinH2bindinganityduetosimultaneousdispersiveinteractionswithintheoptimizedmolecularsquaregeometrywillbeextremelysensitivetogeometricdeformations.Althoughthisplacesachallengeonexperimentallyattainablestructures,withsuchstrictcongurations,itprovidesmotivationforfurthertheoreticalandexperimental 51

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investigationswhichcouldeventuallyresultinporouscrystallinematerialswithdesirableH2uptakecharacteristics.Inparticular,aporousMOFthatcombineshighsurfaceareaandtailoredwindowsthatmaycontrolaccessofH2moleculestoitsporesystemsappearsasacompellingtarget. Figure4.1:MP2/6-31G*//RI-MP2/cc-pVNZCBSN=D,ToptimizedmodelmolecularsquareshowingthemostfavorableorientationofanH2moleculeinteractingsimultane-ouslywiththefourbenzeneringsandthebindingenergydependenceonH2-benzeneringseparationdistance,R. Inthisstudy,weattempttoaddressthispointbothviacomputationalmodelingofahypotheticalmolecularsquareconstruct,andtheexperimentalinvestigationsoftworelevantMOFscontainingmolecularsquareswithaccessiblevoids,1and2.Toinvestigatethewindowdimensions,reminiscentofthoseinthesynthesizedstructurespresentedherein,withoptimizeddispersiveinteractions,amodelsystemoffourbenzeneringsinasquarearrangementamenabletocomputationalinvestigationswaschosen.PerturbativeabinitioelectronicstructuremethodswereemployedaslongrangedispersiveinteractionsarenotcapturedwellbyconventionaldensityfunctionalmethodsorbyHartree-Fockcalculations.[148]Toaccountexplicitlyforelectroncorrelation,secondorderresolution-of-the-identityMller-Plesset[149,150]RI-MP2perturbationtheorywasusedinthecalculationsconductedherein.Dunningbasissetswereselectedbecausebyconstructionthey 52

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allowtheestimationoftheenergyofasysteminthelimitofaninnitelylargebasisset{theso-calledcompletebasissetCBSlimit.[151]AlthoughithasbeenobservedthatCBSextrapolationsincludingthedouble-zetabasissetisnotoptimal[151],quadruple-zetacalculationsweretootimeconsumingandunnecessaryforaphysicallymeaningfulestimationusingamodelsystem.Atwo-pointextrapolationwasperformedusingthedouble-zetacc-pVDZandtriple-zetacc-pVTZbasissets.[152,153]Double-zeta,triple-zetaandCBSenergiesareillustratedinFigure4.1forthemodelsystempresented.Initially,anindividualbenzeneringwasgeometrically-optimizedattheMP2/6-31G*leveloftheorytoattainareferencegeometryfromwhichthemodelsystemwasconstructed.Fourbenzeneringswerearrangedinasquaregeometry,suchthatthecongurationroughlymimicsachanneloraboxinamicroporousmaterialoccupiedbyaH2moleculewithitscenter-of-masscoincidentwiththatofthesquare.Forsimplicity,thesquaregeometrywasmaintainedwhileitssizewasvaried,from3.0Ato4.5Ainincrementsof0.05Aasmeasuredfromcenter-of-massofthesquaretocenter-of-massofabenzenering.Foreachstep,onehydrogenmoleculewaspositionedinthecenter-of-massofthesquareandoptimizedattheMP2/6-31G*levelwhilethesquarewasheldxed.Usingthisoptimizedgeometry,bindingenergieswerecomputedusingRI-MP2withDunning'scc-pVDZandcc-pVTZbasissetswiththeircorrespondingRI-ttingbasissets.[154]Allbindingenergieswerecounter-poisecorrectedandextrapolatedtothecompletebasissetlimit.AllcomputationswereperformedontheTeraGridusingNWChemseeSIforreferences.ThecalculationsrevealeddispersiveinteractionsbetweenthearomaticwallsofthemodelandtheH2moleculereachingamaximumbindingenergyof13.8kJ/molatR=3.05A,reinforcingtheroledispersioncanplayinhighlyconstrainingsorptionenvironments.Whiledistinctquantities,itisreasonabletocompareexperimentallymeasuredaverageisostericheatsofadsorptionandcalculatedbindingenergies.[12]Consideringthecongurationofthehydrogenmoleculerelativetothefourbenzenerings,thecalculatedenergyrevealsanadditivebehaviorofaromaticring-H2interactionsascomparedtoH2moleculeinteractingwithonearomaticring.ThesendingsencourageexperimentalimplementationofthisnovelapproachtoenhanceH2bindinganityinMOFs,throughadditivedispersiveinteractionsattainableinsimilargeometricconnement.Further,theexperimentalstructuresdescribedbelowindicatethepotentialtoconstructsuchmade-to-ordermaterials,possessingchannelswithsuitabledimensions,althoughonlyslightlylargerthantheoptimalcalculatedones. 53

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4.3.2ExperimentalMethods Figure4.2:Crystalstructureof1,leaddeepgray,carbongray,sulfuryellow,oxygenred,hydrogenwhite.Solventmoleculesomittedforclarity. SolvothermalreactionofPbNO32and4,4'-sulfonyldibenzoicacidresultedincolorlessrectangularcrystalsof1formulatedasf[PbC14SO6H8H2O]2DMFgnusingsinglecrystalXraydiractionstudy,Figure4.2.Compound1crystallizesinthetriclinicP-1spacegroupwithsquare-likechannelsrunningthrougha-axisoccupiedbydisorderedDMFsolventmolecules.Thedistinctiveshapeoftheditopicligandmolecule-SO2-angleof104facilitatesconstructionofsquare-likechains,runningalongthec-axis,uponcoordinationtoPbIIions.Inthecrystalstructureof1,additionalcoordinationofPbIIionsbycarboxylateionsinbridgingbidentatemoderesultsininniterod-shapedmetalcarboxylatesecondarybuildingunitSBUofPbCO22runningalongthea-axis,holdingadjacentsquare-likechainsinappropriatecongurationtoallowformationofsolvent-occupiedsquare-likechannelsinthecrystallinesolid.Theresultedgeometryofsquare-likechannelslinedbyaromaticringswithmoderateinterplanardistances8.78-9.16A,centroid-to-centroidimmediatelycaughtourattentiontopossibleprojectedeectsontheH2interactionswiththearomaticwallsoftheframework.Althoughtheinterplanardistancesanddihedralanglesobservedinthisstructurearenotoptimal,asrequiredbyourcomputationalmodel,toinducemultipleinteractionsofasingleH2moleculewithsurroundingaromaticrings,theobservedH2uptakecharacteristicsofthisframeworkareremarkable,especiallyintermsoftheobservedshapeforH2sorptionisothermsandtheisostericheatofadsorption.TheH2 54

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Figure4.3:H2sorptionisothermsfor1. Figure4.4:Crystalstructureof2,Cadmiumbu,carbongray,sulfuryellow,oxygenred,hydrogenwhite,uorinegreen.Solventmoleculesomittedforclarity. 55

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Figure4.5:H2sorptionisothermsfor2. sorptionisothermsforthesolvent-exchangedframework1a,Figure4.3,demonstratearapidsaturationatearlydosingstageswhichcouldbeattributedtouniformdistributionofH2bindingsites,mostprobablyonthesurfacesofaromaticringspresentin1a.SolvothermalreactionofCdNO32and4,4'-hexauoroisopropylidenebisbenzoicacidresultedincolorlesstopaleyellowcrystalsof2formulatedasf[Cd2C17H8F6O42]DMF2gnusingsinglecrystalX-raydiraction.Compound2crystallizesintheorthorhombicPccnspacegroupwithsquare-likechannelsrunningthroughb-axisoccupiedbyhighlydisorderedDMFsolventmolecules.Inthecrystalstructureof2,inniterod-shapedmetalcarboxylateSBUsarepresent.TwodistincttypesofCdIIionsarepresentandexhibittwodistinctcoordinationspheres.FourbidentateligandmoleculescoordinateaCdIIioninadistortedcubicconguration.ThesecondCdIIionisoctahedrallycoordinatedtoadjacentfourbridgingcarboxylateligandmoleculesandtwoDMFmoleculesasaxialligands.TheH2sorptionisothermsforthesolventexchangedsolid,2a,arepresentedinFigure4.5.Thepresentedisothermsexhibitadistinctshapemarkedbyrelativelysteeperriseintheearlydosingstagesfollowedbyaknee,intherangeof0.30.4wt%ofadsorbedH2,correspondingtotwoH2moleculespercavityenclosedbyfourligand 56

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molecules.Thisbehaviorisobservedforbothisotherms,conductedat77Kand87K,andcouldbeexplainedintermsofanactivationbarrierforadsorbedH2moleculesthatisovercomeafteroccupationofthesquare-likechannelsbytwoH2molecules.4.4ConclusionInconclusion,wepresentanovelapproachwiththepotentialtoenhanceH2bindinganityinmicroporousMOFs.ThisapproachprovidesmeritforfurtherexperimentalandtheoreticalinvestigationstoassesstheextentofadditivedispersiveinteractionsinenhancingH2bindinganityinhydrogenstoragematerials,ingeneral,andMOFs,inparticular. 57

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Chapter5SimulationofH2SorptioninaConnedCavity5.1AbstractInthis,therstofaseriesofrelatedpapers,anattemptismadetocharacterizetheimportanceparticularintermoleculare.g.charge,polarization,dispersionandrepulsivegassorptioninteractionsforthreerecentlysynthesizedmetal-organicframeworkmaterialsMOFswithverydierenttopologies.Here,ahighdelitymolecularmodelisdevelopedforaMOFwithnarrowapproximately7.3Anearlysquarechannels.MOFpotentialmodels,bothwithandneglectingexplicitpolarization,areconstructed.Atomicpartialpointchargesforsimulationarederivedfrombothfragmentbasedandfullyperiodicelectronicstructurecalculations.Themolecularmodelsaredesignedtoaccuratelypredictandretrodictmaterialgassorptionpropertieswhileassessingtheroleofinductionformolecularpackinginhighlyrestrictedspaces.Thus,theMOFisassayedviaGrandCanonicalMonteCarloGCMCforitspotentialinhydrogenstorage.Theconningchannels 58

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arefoundtotypicallyaccommodatebetweentwotothreehydrogenmoleculesincloseproximitytotheMOFframeworkatornearsaturationpressures.Further,thenetattractivepotentialenergyinteractionsaredominatedbyvanderWaalsinteractionsinthehighlypolarMOF{inductionchangesthestructureofthesorbedhydrogenbutnottheMOFstoragecapacity.Thus,narrowchannels,whileprovidingreasonablypromisingisostericheatvalues,arenotthebestchoiceoftopologyforgassorptionapplicationsfrombothamolecularandgravimetricperspective.5.2IntroductionMetal-organicmaterialsMOFshavebecomethefocusofagrowingnumberofcomputationalstudiesparallelingthedramaticincreaseinexperimentallysynthesisedporousMOFstructures.[12,40,39,47,15,101]InterestinthesematerialshasgrownasthepromiseofdiversepracticalapplicationofMOFsbeginstoberealizedinthelaboratory.[61,26]Here,thefocusisontheabilitytopredictandexplainmolecularsorptioninMOFsbydevelopingrobust,transferableforceelds{thespecicapplication,inthiscase,isdesigningandexplaininghydrogenstorageinarecentlysynthesisedMOF.TheMOFofinteresthasnarrow,approximatelysquarechannelsandahighlypolarframework.ThepresentworkassestheassignmentofatomicpointchargesinMOFmolecularmodelsandtheroleofexplicitpolarizationinbothMOFforceeldsandobservedhydrogensorption.Previoustheoreticalinvestigations,[16,101,12]haveexplicitlystudiedtheimportanceofsurfacearea,porevolume,andheatofsorptionforhydrogeninvariousMOFs.FromthosestudiesapictureemergeswiththreemainfactorscontrollingH2sorptioncapacityasafunctionoftheloading.Atlowloadings,notsurprisingly,QstcharacterizestheinitialH2sorptionsitesprovidingthestrongestattractiveinteractions.Atintermediateloadingthesurfaceareaiscontrollingandthesometimesdenseinterfacialpackingofhydrogendominates.Athigherpressures/loadings,thefreevolumecomesintoplaywiththeMOFactingessentiallyasacontainerwiththelargecavitiesllinglastbecausetheyprovidepoorinteractionbetweenanH2andthematerial.Itisthegoalofthisstudyto 59

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explorethedualroleofelectrostaticinteractionsincludingpolarizationandtopologyinthiscasethepresenceofconningpolarchannelsindeterminingthesuitabilityofaMOFasasuperiorgas/hydrogenstoragevehicle.Note,inaMOFwithsuchconnedspaces,thestandardparadigm,outlinedabove,doesnothaveobviousapplicability.Inordertoachievetheseobjectivesaneectivemolecularforceeldmustbeestablished.ComputersimulationsofMOFsarebecomingincreasinglycommonbutstilllackthelongtrackrecordofliquidmoleculardynamicsMDandMonteCarloMCmethods.[155,108]MostMD/MCMOFstudieshaveutilizedparameterizationtechniquesoriginallydesignedforuseinwhollydisorderedsystems.Topoint,permanentelectrostaticinteractionsaretreatedasarisingfrompointmonopolescenteredoneachnucleusandneedtobeparameterizedinsomefashion.Inthecaseofmolecularsimulationofliquids,forexample,thisisaccomplishedbyttingtheatomicpointchargestotheelectrostaticpotentialenergysurfaceofagasphasemoleculecalculatedfromelectronicstructuremethods,[70,156]{anobviousapproachsincethesystemofinterestiscomposedofdiscreteunits.InthecaseofMOFs,however,studiesthatincludeexplicitelectrostaticshavebeenforcedtobreaktheinniteperiodicstructureintofragments,eachofwhichisthentwithatomicpointchargesusinganalogousgasphaseelectronicstructurecalculations.[40,12,157]Thisreasonablepracticehasbeenwidelyappliedbutremainslargelyuntestedandanessentiallyuncontrolledapproximationwithadegreeofarbitrarinessassociatedwithchoiceoffragments,identicationofchemicallyprotected"environments,andchemicalterminationofthearticialfragments.Here,boththefragmentapproachandoneinwhichtheelectronicstructurecalculationsontheentireMOFunitcellareusedtodeterminetheatomicchargesareadoptedandcompared.Note,untiltherecentadventoffastercalculationsthroughbothimprovedhardwareandsoftware,thelaterapproachwasusuallynotpossible.Currently,however,mostMOFunitcellsareamenabletosucientlyaccurateelectronicstructurecalculationsmakingtheperiodicwholecell"approachattractivebutalsolargelyuntested.Also,theinclusionofexplicitpolarization,recentlybecomingmorecommonplaceinliquidsimulations,[158]isessentialindescribinginteractionsbetweenpolar,inhomogeneousMOFsandsorbentsandalsoplaysanimportantroleinthechoiceofpartialatomicpointcharges.Forexample,thewholecell"approachincludespolarizationinteractionsimplicitlyessentially 60

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providingtheequivalentofliquidstateoverpolarized"chargesinthederivedpointchargesgivenanappropriateelectronicstructuremethod.Further,theinclusionofexplicitpolarizabilityviaatransferablemanybodypolarizationmethodisveriedinthecontextofthefragmentbasedapproach.Theremainderofthemanuscriptisconstructedasfollows.First,themethodsfordeterminingthepotentialenergysurfaceoftheMOFofinterestaredescribedemphasizingdeterminingandevaluatingatomicpointchargesandpolarizabilities.NexttheresultingmodelisappliedinthecontextofGCMCtoassesstheecacyofhydrogensorptioninthisprototypicalnarrowporouspolarMOF.Thepaperisnallyconcluded. Figure5.1:CrystalstructureofME193,carboncyan,sulfuryellow,oxygenred,hydrogenwhite. 61

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5.3Methods5.3.1AtomicPointChargesHistorically,atomicpointchargeshavebeenderivedbymeansofattotheelectrostaticpotentialenergysurfaceascomputedviagasphaseabinitiocalculations.Thisapproachhasbeensuccessfullyemployedformanyyearsinthecondensedphasesimulationcommunity.Translationofthistechniquetoinniteperiodicsolidsrequirestruncationofthematerialtobestudiedintofragmentswhicharethenindividuallytwithatomicpointcharges.Theadventandpopularizationofplanewavedensityfunctionalmethodshasmadepossiblethederivationofatomicpointchargespossiblebyemployingattothecomplete,periodicelectrostaticpotentialsurface.OneoptioninhandlingthelongrangenatureofchargeinteractionsintheperiodicsystemistoincludeEwaldsummationinachargettingscheme.[159,160]Startingwiththeformalisminvolvedinttinggasphaseatomicpointcharges,aleastsquaresdierencebetweentheabinitioESP,denotedasSCF,andtheESPresultingfromtheclassicalpointcharges,denotedasclass,isdenedinequation5.1.2=mXi=1SCFi)]TJ/F22 11.9552 Tf 11.9551 0 Td[(classi2 .1 Theleastsquaressumrunsoverasetofttingpoints"thattobemeaningfularetakenoutsidethevanderWaalsradiiofthemolecule.Inthecaseofagasphasemolecule,classiseasilycalculatedviaCoulomb'slawandSCFistakendirectly 62

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fromtheabinitiocalculation.classi=nXj=1qj rij .2 Ifnorestrainingpotentialsortotalchargeconstraintsareadded,thedesignmatrix[161]canbehandleddirectlybyanynumberofmethodsatthispointe.g.singularvaluedecomposition.However,itisdesirabletoconstrainthetotalchargeofthemoleculetothecorrectformalchargeand/oraddarestrainingpotentialtobetterhandleill-behavedorburied"atoms;[70]itisconvenienttouseLagrangemultipliersforconstrainingthetotalcharge.Theresultingexpressiontobeminimizedisgivenbyequation5.4.G=qtotal)]TJ/F23 7.9701 Tf 18.0203 14.9441 Td[(nXj=1qj=0 .3 InclusionoftheLagrangemultiplieryieldsanewafunctiontobeminimizedgivenby5.4.z=mXi=1SCFi)]TJ/F23 7.9701 Tf 18.0204 14.944 Td[(nXj=1qj rij2+nXj=1qj)]TJ/F22 11.9552 Tf 11.9552 0 Td[(qtotal! .4 Substitutingforclassiandsettingthederivativetozeroyieldsthefollowingsystemofequations. 63

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@z @=0=nXj=1qj)]TJ/F22 11.9552 Tf 11.9552 0 Td[(qtotal .5 @z @qk=0=mXi=1classi rik)]TJ/F23 7.9701 Tf 16.8442 14.9441 Td[(mXi=1nXj=1qj rijrik .6 Writteninmatrixform:2666664A11A12:::A1n1A21A22:::A2n1.........:::1An1An2Ann11111037777752666664q1q2...qn3777775=2666664B1B2...Bnqtotal3777775 .7 whoseelementsaregivenby:Ajk=mXi=11 rijrikandBk=mXi=1SCFi rik .8 2=2ESP)]TJ/F22 11.9552 Tf 11.9551 0 Td[(2restr .9 64

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Inclusionofarestrainingpotentialisdonebymeansofanadditionaltermthatservestopenalizethe2meritfunctionasachargedriftsawayfromatargetvalue,qoj.2restr=anXj=1qoj)]TJ/F22 11.9552 Tf 11.9552 0 Td[(qj2 .10 ThisresultsinthemodicationoftheBvectorandonlythediagonalelementsofA.Ajj=nXi=11 r2ij+@2restr @qj .11 Bj=nXi=1SCFi rij+qoj@2restr @qj .12 Toaccountfortheperiodicitypresentinsolids,wemusthandletheconditionallyconvergentlatticesum.Conventionally,Ewaldsummationisemployedforthispurpose.Here,wemaydirectlysubstituteforclass.classi=Xk6=0nXj=14qj k2eikrije)]TJ/F23 7.9701 Tf 6.5865 0 Td[(k2=4+NXj=1qj rijerfcp r .13 65

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MaintainingtheLagrangemultiplierandtherestrainingpotentialterms,equation5.4becomes:@z @qk=2nXi=1SCFi)]TJ/F22 11.9552 Tf 11.9551 0 Td[(classi@classi @qk+@2restr @qk+@G @qk .14 Where:@classi @qk=Xk4qk k2eikrike)]TJ/F23 7.9701 Tf 6.5865 0 Td[(k2=4+qj rikerfcp rik .15 Fittingatomicpointchargesaccordingtothisschemerequiresahandfulofadditionalparameters.ThevanderWaalsradiiormoreprecisely,theexclusionradiusforeachatommustbeset.Inthepresentstudy,thevanderWaalsradiiweretakenasthosetabulatedbyTruhlar[162].Afactorof1.3*RvdWwasexcludedfromthettingprocedure.ThepointsatwhichtheabinitioESPwascalculatedandsubsequentlyttotakenasthosepointscorrespondingtotheFFTgridintheDFTcalculation.Althoughithasbeenreportedthatbetterpointselectionroutinesexist,asamatterofconveniencethedefaultFFTgridwasused.[163]Thegridofttingpointscanbethinnedbyexcludingpointswhilemaintaininganevenlyspacedgrid.Inthiswork,nothinningwasnecessarybecauseoftherelativelysmallunitcell. 66

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5.3.2PolarizabilityModelMolecularpolarizationwasexplicitlyincludedintheMonteCarlosimulationsbyuseoftheThole-Applequistmodel.[33,32,34,12]Thismodeltreatsthesystemintermsofatomicpointdipolesthatinteractinthefullmany-bodyregime.Onceasmallnumberofparametershavebeent,themodelhasbeenshowntoaccuratelyreproducemoleculardipolesinatransferablei.e.system-independentmanner.[34,32]Thismodelofexplicitpolarizationhasbeensuccessfullyappliedinnumerousareaswhereinclusionofpolarizableeectsisparamount,suchasvibrationalspectroscopy[72,73,35],liquiddynamics[74,75,76,77]andbiomolecules[78,79].Themoleculardipoleisgivenby:~mol=mol~E .16 wheremolisthe33molecularpolarizabilitytensorand~Eistheelectrostaticeldappliedtothemolecule.IntheThole-ApplequistmodelthesystemistreatedasacollectionofNatomicpointdipoleswhichhaveanassociatedscalarpointpolarizabilityiandadipoleeldtensorTijthatcontainsthecompletesetofinduceddipole-dipoleinteractions.InEinsteintensornotation,theithdipoleisfoundbythedipoleeldequation:i=iEi=i)]TJ/F22 11.9552 Tf 5.4795 -9.6838 Td[(Estati+Eindi=iEstati)]TJ/F22 11.9552 Tf 11.9552 0 Td[(Tijj .17 whereEstatistheelectrostaticeldvectordeterminedbytheatomicpartialcharges 67

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oftheforceeldMOFandH2andEindiseldvectorfromthesurroundingdipoles.TheApplequistdipoleeldtensor[32]canbederivedfromrstprinciplesas:Tij=rr1 rij= r3ij)]TJ/F15 11.9552 Tf 13.1507 8.0877 Td[(3xx r5ij .18 Themany-bodypotentialenergyduetotheinteractionoftheinduceddipolesreferredtoasthepolarizationenergyisdescribedby:Upol=)]TJ/F15 11.9552 Tf 10.494 8.0878 Td[(1 2NXi~i~Estati .19 Calculatingthepolarizationenergyforthesystemamountstoself-consistentlysolvingthedipoleeldequationforeachatomicdipolevector~ithroughaniterativeprocessuntilasucientdegreeofprecisionisachieved;becausethisiterationrepresentsthevastmajorityofcomputationaleort,extremelyecientschemesweredevelopedtomakethecalculationsfeasible.[12]5.3.3ParameterizationofleadandsulfurThestructurestudiedcontainstwoelementswhichhavenotpreviouslybeenparameterizedforuseinthemodel,[34]leadandsulfur.Reasonablepolarizabilityparametersfortheseelementsshouldmeettherequirementthattheoverall 68

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molecularpolarizabilitytensorbereproduced.Toaccuratelycomputethepolarizabilitytensorrequirestheuseoflargebasissets,makingthecalculationveryexpensive.Dunning'scc-pVTZbasissetwasemployedusingtheGamessabinitiosoftwarepackagefortheniteeldcalculations.[164,165]Inthismethod,severaleldsofvaryingmagnitudesanddirectionsareappliedtothesystemandthepolarizabilitytensorisderivedfromthesystem'sresponse.Fieldstrengthsof0.001and0.002a.u.wereappliedinasequenceofdirectionsandtheenergeticresponsetotheeldwasusedtodeterminethepolarizabilitytensor.Becauseofthecomputationalexpenseinvolved,fragmentsofthestructureweretwithpolarizabilityvaluesforleadandsulfurbycomparingabinitioniteeldcalculatedmolecularpolarizabilitytensortothatoftheTholemodel.Oncethepolarizabilitytensorofeachfragmentwascalculatedfromelectronicstructure,usingthepreviouslydeterminedpointpolarizabilities,[34]thevaluesforsulfurorleadwereadjustedsuchthattheelectronicstructurederivedpolarizabilitytensorwaswellmatchedbythemodelcalculationtowithinthedelityoftheabinitiocalculation.[166]ResultsarepresentedinTables5.20and5.21.Note,thetraceofthepolarizabilitytensor,correspondingtotheisotropicpolarizabilityisreproducedevenbetterthantheindividualelements.249:870:830:510:839:22)]TJ/F15 11.9552 Tf 9.2985 0 Td[(0:260:51)]TJ/F15 11.9552 Tf 9.2985 0 Td[(0:269:8935249:391:740:601:749:57)]TJ/F15 11.9552 Tf 9.2985 0 Td[(0:330:60)]TJ/F15 11.9552 Tf 9.2985 0 Td[(0:3310:1035 .20 2422:21)]TJ/F15 11.9552 Tf 9.2985 0 Td[(0:07)]TJ/F15 11.9552 Tf 9.2985 0 Td[(0:92)]TJ/F15 11.9552 Tf 9.2985 0 Td[(0:0720:01)]TJ/F15 11.9552 Tf 9.2985 0 Td[(2:78)]TJ/F15 11.9552 Tf 9.2985 0 Td[(0:92)]TJ/F15 11.9552 Tf 9.2985 0 Td[(2:7820:32352420:880:52)]TJ/F15 11.9552 Tf 9.2985 0 Td[(0:850:5220:27)]TJ/F15 11.9552 Tf 9.2985 0 Td[(3:46)]TJ/F15 11.9552 Tf 9.2985 0 Td[(0:85)]TJ/F15 11.9552 Tf 9.2985 0 Td[(3:4621:3735 .21 69

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Atom Label Chargee)]TJ/F15 11.9552 Tf 7.0846 -4.3384 Td[( Atom Label Chargee)]TJ/F15 11.9552 Tf 7.0846 -4.3384 Td[( Pb 1 0.970550 C 16 0.632799 S 2 1.028731 C 17 -0.188539 O 3 -0.476968 C 18 -0.042111 O 4 -0.479863 C 19 -0.122877 O 5 -0.529817 C 20 -0.000782 O 6 -0.635979 C 21 -0.000958 O 7 -0.552769 C 22 -0.184685 O 8 -0.548677 H 23 0.112658 C 9 -0.205082 H 24 0.103635 C 10 0.060345 H 25 0.208300 C 11 -0.275558 H 26 0.161275 C 12 -0.025261 H 27 0.112083 C 13 -0.204497 H 28 0.136080 C 14 0.670162 H 29 0.079465 C 15 0.021172 H 30 0.177167 Table5.1:PartialchargesusedinsimulationofME193.AtomnumberingcorrespondstothatshowninFigure5.2. 5.4ResultsandDiscussion5.4.1ChargeModel{PeriodicvsFragmentApproachME193possessesaunitcellwithonlyinversionsymmetrywhichleadsto30chemicallyuniqueatomsintherelativelysmallunitcell.TheelectrostaticpotentialwasrstcomputedusingVASPwithaconvergent450eVkineticenergycutoandaCeparly-Alderfunctional.[167]TheresultantESPwasthenusedasabasistotatomicpointchargesasdiscussedinSection5.3.1.ThecomputedchargesareshowninTable5.1.Ataglance,thechargesresemblewhatonewouldexpectbasedonelectronegativity,theoxygenatomsareallroughly-0.5e,thehydrogenatomsareslightlypositive,andtheleadandsulfuratomsarebothstronglypositive.Meanwhile,thereismorediversityinthechargesforcarbonduetodistinctchemicalenvironments.Note,thepartialchargesrepresentahighlypolarstructureincludingtheatomsmostaccessibletogassorbents,e.g.theoxygen,sulfurandcarbonatoms 70

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areclearlyeasilyreachedasisclearinFigure5.2. Figure5.2:ChemicallydistinctatomsintheMOFdeningthenumberingsystemcorre-spondingtothechargespresentedinTable5.1. Figure5.3:Structurallyrelatedatomsareaveragedaccordingtothisnumberingschemei.e.thosewiththesamenumberaretreatedasequivalent.ThefragmentderivedchargesontheseatomsareshowntobesimilartoperiodicchargesinTable5.2. Chargeswerealsocomputedusinggasphasefragmentsforcomparison.Usefulfragmentsofthisstructureareparticularlydiculttoselectbecauseofthedensityofthestructure{truncationofthestructureintoniteclustersinevitablyleadstoremovingadjacentpiecesoftheframeworkthatareinverycloseproximitytofragmentatoms.Becausetheatomsaresoclosetoeachother,itispossiblethatthechemicalenvironmentisnotadequatelypreservedwhenisolatingportionsoftheMOFinthegasphase.Forcomparisontotheperiodiccharget,someatomsareaveragedovertogiveasimpler,morereducedpictureofthecharges.Atomschosentoaverageoverarethosethatclearlyarecloselyrelatedstructurallyandthosethatexhibitsimilarchargeintheperiodict.TherelatedatomsareshowninFigure5.3.Averaginginthiswaymakesmoremanageablethetaskoffragmentttingandsubsequentcomparisonwithperiodicderivedcharges.SixfragmentswerechosenforcomparisonofchargestothosebasedonattotheperiodicstructureandareshowninFigure5.4. 71

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Figure5.4:Fragmentsselectedforgasphasechargetting. 72

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fragment 1 2 3 4 5 6 periodic atom1 1.0248 1.0673 0.9863 0.8483 1.0673 0.9706 atom2 1.2100 1.1677 1.1623 1.1283 1.1677 1.0287 atom3 -0.5824 -0.6275 -0.6275 -0.8112 -0.6275 -0.5307 atom4 -0.6141 -0.6027 -0.6446 -0.5987 -0.6027 -0.5507 atom5 0.7542 0.8256 0.8256 0.8483 0.8256 0.6515 atom6 0.1421 0.1453 0.1285 0.1484 0.1598 0.1419 0.1363 atom7 -0.1634 -0.1423 -0.2151 0.0182 -0.0639 -0.1256 -0.0917 atom8 -0.0797 -0.1134 -0.0183 -0.2229 -0.2413 -0.0962 -0.1164 atom9 -0.0695 -0.0636 -0.1719 -0.0553 -0.0168 -0.0733 -0.0681 atom10 -0.1992 -0.1703 -0.1486 -0.2179 -0.1807 -0.1838 -0.1236 atom11 0.1520 0.1472 0.1490 0.1377 0.1377 0.1490 0.1224 Table5.2:Partialchargescomputedviagasphasefragments.ThefragmentscorrespondtothoseshowninFigure5.4andtheatomtypecorrespondstothenumberingschemeinFigure5.3.Notethatnotallfragmentscontainallatoms.Theatomsnotpresentinthefragmentsareleftblank. Figure5.5:AtypicalcongurationofH2sorbedintheMOF. 73

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Figure5.6:H2populationhistogramdierenceplot.Shownisthedierencebetweenpolarizableandnonpolarizablesimulation.Theblueisosurfaceencapsulatesregionsthathaveahigherpopulationdensitywhenincludingpolarizationwhiletheredindicatesregionsthatarelesspopulatedcomparedtothenonpolarizableresult. EachfragmentselectedistwithchargesbyrstconductinganabinitiosinglepointpointenergycalculationattheHF/6-31G*leveloftheory.TheleadatomsweretreatedwiththeLANL2DZbasissetandthecorrespondingeectivecorepotential.Comparisonofthechargesderivedbyafragmentapproachversusthosederivedbyaperiodicmethoddemonstratesthatitispossibletocomputereasonablechargeswithoutemployingaperiodicapproachifmultiplefragmentsarechosencarefully,andthechargesareaveragedoveratomtypesinthefragmentsthatareinaprotected"chemicalenvironment.However,thereareclearadvantagesinsimplyemployingaperiodicapproach.Themostevidentadvantagebeingtheavoidanceofthesomewhatarbitraryprocessofselectingandcuttingtheframeworkintopieces.Thispracticeunnecessarilyintroducesarbitraryboundariesinthecalculationandforcesthepractitionertochemicallyterminatetheindividualfragments.Additionally,usingaperiodicmethodsuchastheoneoutlinedhereexplicitlyaccountsforthelong-rangeelectrostaticeectspresent.Incontrast,thefragmentapproachusesanincompletebasisset-31G*resultingineectivelyoverpolarizedchargesthataresystematicallysomewhatlargerthantheperiodiccharges. 74

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Figure5.7:Isothermsattop77Kandbottom87Karepresentedcomparingexperi-mentalandtheoreticalresults. 75

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Figure5.8:Heatofadsorptionattop77Kandbottom87Karepresentedcomparingexperimentalandtheoreticalresults. 76

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Figure5.9:IsothermalCompressibilityattop77Kandbottom87Karepresentedcomparingexperimentalandtheoreticalresults. 77

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Figure5.10:EnergydecompositionofthefragmentbasedchargesGCMCsimulationasafunctionofpressure.Showninthesolidlinesaretheabsolutemagnitudeoftheenergycomponentswhilethedashedlinesrepresentthepercentageoftotalenergyofeachcomponent. Figure5.11:EnergydecompositionoftheperiodicbasedchargesGCMCsimulationasafunctionofpressure.Showninthesolidlinesaretheabsolutemagnitudeoftheen-ergycomponentswhilethedashedlinesrepresentthepercentageoftotalenergyofeachcomponent. 78

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Figure5.12:EnergydecompositionoftheperiodicbasedchargesGCMCsimulationex-cludingpolarizationasafunctionofpressure.Showninthesolidlinesaretheabsolutemagnitudeoftheenergycomponentswhilethedashedlinesrepresentthepercentageoftotalenergyofeachcomponent. 5.4.2HydrogenSorptionviaGCMCGCMCsimulationswereperformedwithbothfragmentbasedandperiodiccharges.Bothpolarizableandnonpolarizablecalculationswerealsoperformedinadditiontocontrolusingthepolarizablemodelwithpolarizationturnedo.Simulationswereperformedat77Kand87K,includingFeynman-Hibbscorrectionstofourthorder,[14]withoutwhichthesimulationsunphysicallyoversorb,atpressuresrangingfrom0to1atmosphereincorrespondencewiththeavailableexperimentaldata.TheH2uptakeasafunctionofpressureisshowninFigure5.7.Higherpressureswerenotconsideredbecausethesystemsaturatesrapidlyduethestronginteractionsbetweenthehydrogenandthenarrowchannels.Atbothtemperaturesforallmodelsconsideredthetheoreticalsorptionisslightlyhigherthanexperiment.Apparentoversorptioncanbeduetoexperimentaluncertaintiese.g.residualmaterialinthenarrowchannelsortheoreticaluncertaintiesinthemodelparametersandthislevelofagreementcanbeconsideredquitereasonableandthemodelresultsmeaningful.Firstconsideringthetheoreticalsorptiondatawherethehydrogenistreated 79

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withoutexplicitpolarizationinductioneectsaretreatedeectivelybymodifyingthepotentialparametersinttingtoabinitiopotentialenergysurfaces[168],at77KthesystematicallysmallerperiodicchargesimulationactuallyshowslightlyhighersorptionovertheentirepressurerangetheerrorsinthesorptioncurvesestimatedasGaussianrandomnotshownforpresentationclarityaresmallcomparedtothedierencebetweenthecurves.At87Ktheperiodicchargesimulationinitiallysorbssomewhatmorebutisovertakenbythatwiththefragmentchargesatapproximately0.2Atm.Whilethisisinitiallysurprising,theelectrostaticenergyinthissystem,aswillbeexplicitlydemonstratedbelow,isarelativelysmallcontributiontothetotalenergy.Thus,repulsiveandvanderWaalsinteractions,thatenterthesimulationviaLennard-Jonespotentials,aredominantincontrollingsorbantstructure.Thisresultsinlessfavorablechargeinteractionsthatareexacerbatedbythelargerfragmentcharges.Thisresulthighlightstheproblemwithnarrowchannelsforhydrogen/gasstorage{althoughhighlyconstrainedenvironmentsleadtorelativelyhighQstvaluesviadispersion,thereisnotenoughorientationalfreedomtosimultaneouslymaximizethetotallyofthefavorablepotentialenergyinteractions.Figure5.8showthatthenonpolarsimulationscapturetheisostericheatmagnitudeandtrendwithloadingquitewellbutagainareslightlylargerthanthatobservedexperimentally.Note,withinGCMCsimulations,isostericheats,Qstcanbedirectlycalculatedbycrosscorrelationsinthepotentialenergyandsorbentnumber.[108,14]Experimentally,isostericheatsaretypicallycalculatedfromtwoormultitemperatureisothermdatabyestimatingthepartialderivativeintheexpressionQst=)]TJ/F22 11.9552 Tf 9.2985 0 Td[(k@lnP @T)]TJ/F21 5.9776 Tf 5.7561 0 Td[(1vianitedierence.[108]GCMCresultsforQstat77Kand87Kwerefoundtobenearlyidenticaltowithinuncertainties,aswouldbeexpectediftheexperimentalprocedureisvalid.Note,however,thatbecausetheriseintheisothermsissosteepandthenrapidlyplateaustogiveroughlyparallelcurvesatthetwotemperatures,nitedierencecalculationofthederivativecanleadtodicultyinaccuratelyextractingthevalues.Indeed,ifthetheoreticalisothermsareusedtocalculateQstinananalogousfashion,thevaluesareroughly10-20%lowerthanthatderivedbythemorepreciseuctuationexpressionforwhichtheerrorbarsarefarsmaller. 80

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Thepolarizablesimulationsleadtoisothermsat77Karesimilarandbothinitiallyrisefasterbutplateauatalowervaluethanthenonpolarizablemodel.Thisisconsistentwiththeinterpretationabovewiththeaddedobservationthatpolarizationmakesthebest"sorptionsites,thatwillbelledrst,alittlemorefavorable.Interestingly,at87Kthepolarizablemodelsarealsosimilarbutsorbhigherthanthenonpolarizableatallpressuresconsidered.Itispossiblethattheslightlyhighertemperatureallowsmoreconformationalfreedomandabetteroptimization"oftheintermolecularinteractions.Inaccord,theQstvaluesforthepolarmodelsarealsoslightlylowerthanthenonpolarmodelsandshowsimilartrends.Thisissurprisingat87Kbecausethepolarmodelsshowlargersorptionbutsmallerisostericheats.ThetotalenergyisdecomposedforeachsimulationtoassesstherelativecontributionsofvanderWaals,electrostatics,andpolarizationenergyandtheresultsareshowninFigures5.10,5.11,and5.12.Giventhesmallporesizeofthestructure,itwashypothesizedthatthehighsurfacecoveragerelativetothebulkwouldleadtosubstantialelectrostaticinteractionsandthuswouldleadtolargecontributionsofboththeelectrostaticpotentialandpolarizationenergy.Surprisingly,theoppositeeectwasobserved,resultinginextremelylowcontributionsfromeach.Theenergydecompositionindicatesroughly1%oftotalenergyarisingfromelectrostatics,6%frompolarization,andtheremainderbeingvanderWaals.Itcouldbearguedthatthisresultshouldalmostnullifydierencesobservedfromvaryingthechargesand/orincludingpolarizationexplicitly.Indeed,studyoftheisothermsandcompressibilityarevirtuallyindistinguishable,dependinglittleonthemethodofchargettingoroninclusionofpolarization.However,insubsequentpapers,wewilldiscusstwootherMOFswhichhavedrasticallydierentenergyproles,increasinglydependantonaccurateelectrostatics.Counterintuitively,weobserveapositivecorrelationbetweenporesizeandtheimportanceofelectrostatics.Thatis,thelargerthepore,thelargertheelectrostaticcontributionasafractionoftotalenergy. 81

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5.5ConclusionsHere,wehavestudiedH2sorptioninanovel,smallporedmetal-organicframeworkandcharacterizedthenatureoftheintermolecularinteractionspresent.Polarizationeectsareexplicitlycontrolledforbycomparingtwomodels{oneincludingpolarizationviatheTholemodel,andonemodelparameterizedseparatelywithoutpolarization.Inaddition,thefragmentbasedapproachofttingchargesistestedagainstthenewer,moreappropriateperiodicmethod.Wedemonstratethatalthoughcarefulparameterizationusingafragmentbasedapproachcanyieldreasonablecharges,theperiodicmethodresultsincomparablechargeswiththebenetofsignicantlylessambiguityandmorestraightforwardapplication.Surprisingly,eventhoughtheframeworkatomsarehighlychargedandthereforeshouldinducesignicantdipolesinH2,wendthattheelectrostaticcontributiontothetotalenergyisrelativelysmall,resultinginproportionallysmallinductionenergy.ThisdierssubstantiallyfromourndingsintwootherMOFswhichpossesslargerporesandexhibitmuchhigherpolarizationenergiesaswillbereportedinsubsequentpapers.Wedemonstratethatthepropensityforperiodicbasedchargestobeslightlysmallerthanthosetbyafragmentbasedapproachissignicantwheninductionenergyplaysaroleinsorption,althoughinthepresentstudythisdoesnotaectuptake.5.6AcknowledgmentsComputationswereperformedontheRangersupercomputerlocatedattheTexasAdvancedComputingCenterunderaTeragridGrantGrantNo.TG-CHE070098N,andalsoattheUSFResearchComputingCenter.TheauthorsacknowledgefundingfromtheDepartmentofEnergyGrantNo.DE0GG02-07ER46470andtheNationalScienceFoundationGrantNo.CHE-0312834.TheauthorsalsothanktheSpaceFoundationBasicandApplied 82

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Researchforpartialsupport.SupportingInformationAvailable:DetailsoftheME193potentialenergyfunctionparameters. 83

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Chapter6AtomicPointChargesinCrystallineSolids6.1AbstractWepresentamethodforttingatomicchargestotheelectrostaticpotentialESPofperiodicandnon-periodicsystems.ThismethodissimilartothemethodofCampa~naetal.[169]Wepresentanewdirectmethodforcomputingtheosetpotentialforperiodicsystems.WecomparetheWolfandEwaldlong-rangeelectrostaticsummationmethodsincalculatingtheESPforperiodicsystems.WendthattheWolfsummationiscomputationallymoreecientthantheEwaldsummationbyaboutafactorofvewithcomparableaccuracy.Ouranalysisshowsthatthechoiceofgridmeshsizeinuencesthettedatomiccharges,especiallyforsystemswithburiedhighlycoordinatedatoms.Wendthatamaximumgridspacingof0.2-0.3isrequiredtoobtainreliableatomiccharges.Theeectoftheexclusionradiusforpointselectionisassessed;wendthatthecommonchoiceofusingthevanderWaalsvdWradiusastheexclusionradiusforeachatommay 84

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resultinlargedeviationsbetweentheESPgeneratedfromtheabinitiocalculationsandthatcomputedfromthettedcharges,especiallyforpointsclosesttotheexclusionradii.Wendthatalargervalueofexclusionradiusthancommonlyused,1.3timesthevdWradius,providesmorereliableresults.Wendthatapenaltyfunctionapproachforttingchargesforburiedatoms,withthetargetchargetakenfromBaderchargeanalysis,givesphysicallyreasonableresults.6.2IntroductionAssigningatomicpartialpointchargesisausefultoolinmodelingthephysicalinteractionsofmolecular,disordered,andcrystallinesystems.ForceeldbasedclassicalmoleculardynamicsandMonteCarlocalculationstypicallyrelyonatomicchargesforreproducingelectrostaticinteractions.Despiteitsusefulness,thechoiceofatomicchargesissomewhatarbitraryinthatasetofpointchargescannotgenerallyreproducethecontinuouselectronicchargedistribution.Asaconsequence,therearemanypossiblemethodsthatcouldbeusedtodeterminereasonableatomiccharges.[170,171,172]Moreover,thereisasignicantdebateaboutthebestwaytodetermineeectiveatomiccharges.[173]Clearly,itiscriticaltomodeltheelectrostaticinteractionstoanacceptablelevelofaccuracyinordertoproperlydescribethesepotentialenergyinteractionsinmolecularsimulations.MethodssuchasMullikenpopulationanalysis1wereneverintendedforproducingaccurateforceelds.Incontrast,derivingthechargesbyttingthemolecularESPisphysicallymotivatedandyieldschargesthatresultinaccurateenergiesandforces.[173]Typically,ESPchargesarettedtoreproducetheelectricpotentialcalculatedfromabinitiomethodsatasucientlylargenumberofgridpointsaroundamolecule,suchthatanacceptableapproximationtothetruechargedistributioniscaptured.ThisapproachwasrstusedbyMomany3andfurtherrenedbyseveralothers.[174,175,176,177]Thewell-knownmethodCHELPwasinitiallydevelopedbyChirlianandFrancl,[178]andthenmodiedbyBrenemanandWiberg[179]astheCHELPGmethod.Also,Kollmanetal.[173]developedawell-behavedRESPmodelthatwaswidelyusedtodescribetheintermolecularinteractionsincondensedphases.Thisapproachhastheadvantage 85

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thatthettedchargesalsomimicthequantummechanicallydeterminedmultipolemoments.Huetal.[180]proposedanewESPttingmethodbasedonaglobaltintegratingthedierenceoftandtrueelectrostaticpotentialsintheentire3-dimensionalphysicalspacethusimprovingthenumericalstabilitywithrespecttothemolecularpositionsandgeometries.Thechargettingmethodsdescribedabovearedesignedformolecularorclusternon-periodicsystems.Incontrasttothemanymethodsdevelopedfornon-periodicsystems,therearefewmethodsavailableforcomputingeectiveatomicchargesofcrystallinesolids.[181]However,obtainingchargesforperiodicsystemsisoftenextremelyimportantandusefulforsimulationssuchasadsorptionanddiusionofuidsinporousmaterials.[182,183,184]Fittingchargesforperiodicsystemsleadstoadditionaldicultiescomparedtottingchargesforsmallmoleculesinthegasphase.Specically,long-rangeelectrostaticsmustbeaccountedforintheperiodicsystem,buriedatomsareamorecommonproblem,andtheESPisonlydeterminedtowithinaconstantvaluethatdependsonthedetailsoftheelectronicstructurecodeusedtogeneratetheESP.Mostsimulationstudiesthatincludechargesforcrystallinesystemshaveobtainedthosechargesfromabinitiocalculationsonrepresentativegasphaseclustersorfragmentstakenfromthecrystallattice.[181,182,183,184,185,186,187,188,189,190,191]Therearetwoobviousproblemswiththisapproach.Convergenceoftheatomicchargeswiththesizeoftheclustermaybeveryslow,sothatthecalculationsbecomecomputationallyinfeasible.Moreover,evenwhenchargesonatomsconverge,therearealwaysterminatingatomsonclustersandtheseterminatingatomscarryatleastsomecharge,makingitnecessarytorenormalizethechargesontheperiodicatomstoachievechargeneutrality.Thisrenormalizationstepisnotunique,whichintroducesadditionaluncertaintyintotheprocess.Thus,thereisaneedforrobustandaccuratemethodsforcomputingatomicchargesforperiodicsystems.Onepossiblesolutionistouseanapproachbasedonpartitioningtheelectrondensityaroundeachatom,ratherthantheelectrostaticpotential.Thesetypesofmethodsaretypicallybasedontheatoms-in-moleculestheorydevelopedbyBader.[171,192]Theatomicchargeisdeterminedbytakingthedierencebetweentheformalvalencechargeandthesumoftheelectronpartitioneddensityinanall-electroncalculation.Henkelmanetal.[181,193,194]developedanalgorithmforperformingBaderchargeanalysisinperiodicsystems.TheircodehasbeenwidelyusedtoanalyzechargedensitiescomputedfromtheViennaAbinitio 86

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SimulationPackageVASP,whereitcanbeimplementedasauseroption.[195,196,197,198]Unfortunately,theBaderchargedecompositionhasseriousshort-comingsfordeningatomiccharges.[182,183,184,188]Talpolskyetal.[183]pointedoutthattheelectrondensitypartitioningwithinatomicbasinsisoftenhighlyanisotropicandhencesimplepointchargescannotsucientlyrepresenttheelectrostaticinteractionsinsuchascheme;higherordermultipolesmustbeusedwhenttingtochargedensitydistributions.Kosovetal.[199]accuratelyreproducedthemolecularelectrostaticpotentialusingatomicmultipolesofhighrankdenedintheBaderquantumtheoryofatoms-in-molecules.Hence,truncationofthemutipoleexpansionatmonopolescanleadtounacceptablylargeerrorsintheelectrostaticpotentialderivedfromBaderchargedecomposition.[184,188]ManzandSholl[200]developedamethodforcomputingatomicchargesforbothmolecularandperiodicsystemsthatisbasedontheatoms-in-moleculesformalism.Theirmethod,calledDensityDerivedElectrostaticandChemicalDDECcharges,overcomesmanyoftheshortcomingsoftheBaderapproachbyrequiringtheatomicchargestobebothchemicallymeaningfulandreproducetheESPatpointsthataresucientlyfarfromtheatomsinthesystem.TheDDECmethodisabletoproducephysicallymeaningfulchargesfornon-poroussolids,porousmaterials,suchasmetalorganicframeworksMOFs,andforcomplexmolecularsystems.IncontrasttoESPttingmethods,theDDECmethoddoesnothaveproblemsassigningchargesforburiedatoms.WhiletheDDECmethodprovidesabalancebetweenchemicallysignicantcharges,thereissomeevidencethatdirectttingtotheESPprovidesamoreaccuraterepresentationoftheESPandhencebetteratomicchargesforuseinmolecularsimulations.[200]Campa~naetal.[169]developedanalgorithmtogenerateESPderivedatomicchargesincrystallinesolidsandmoleculesfromperiodicquantummechanicalQMcalculations.TheycalledtheirmethodREPEAT,anacronymforRepeatingElectrostaticPotentialExtractedATomiccharges.Campa~naetal.pointedoutthattheelectrostaticpotentialcomputedfromaperiodiccalculationisonlydenedtowithinanarbitraryconstantdependentonthedetailsoftheQMcalculation.[169]Therefore,caremustbetakentodealwiththisosetvalue,Voset,whenttingatomicchargestotheESPcomputedfromperiodicQMcalculations.TheREPEATalgorithmdealswithVosetbydeningitasthedierencebetweentheperiodicQMESPandthatcomputedfromtheatomicpointcharges.Intheirwork,theelectrostaticpotentialresultingfromtheatomicpointchargeswascomputedusingtheEwaldsummation.TheREPEATmethodwasshowntogiveexcellentpredictionsofatomicchargesforperiodicporousmaterialssuchassodaliteand 87

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IRMOF-1.Moreover,theydemonstratedthattheREPEATmethodgiveschargesformolecularsystemsthatareingoodagreementwithmolecular-basedESPttingmethods.TheREPEATalgorithmrepresentsabreakthroughincomputingreasonableatom-centeredchargesforuseinmolecularsimulations.However,therearestillsomeissuesthatremaintobeexploredinttingchargestotheESPforperiodicsystems.Inthispaperweuseasimilaralgorithmthatwehavedevelopedtoinvestigatethefollowing:HowdotheEwaldandWolflong-rangeelectrostaticmethodscompareintermsofaccuracyandeciency?WhatistheeectoftheESPgridspacingontheaccuracyofthettedcharges?HowdoesthevalueoftheexclusionradiustheradiuswithinwhichpointsareexcludedfromtheESPforttingthechargeforeachatomaectthettedvaluesofthechargesforvarioustypesofatoms?Canthevalueofthepotentialosetbedirectlycomputedinthettingprocess?5Finally,arethererobustmethodsforestimatingthechargeonhighlycoordinatedorburiedatoms?6.3MethodsTheQMESPsforallsystemsconsideredherewerecomputedusingperiodicplanewavedensityfunctionaltheoryDFTasimplementedwithinVASP.AllDFTcalculationswereperformedwiththeprojectoraugmentedwavePAWmethod[201,202]togetherwithPW91functional[203,204]usingthegeneralizedgradientapproximationGGAinVASP.Anenergycutoofatleast450eVwasusedforallthecalculations.WehaveperformedcalculationsonthreenanoporousMOFs,namelyFepzNiCN,[205]CuBTC,[206]andIRMOF-1alsoknownasMOF-5.[207]TheBrillouinzoneintegrationswereperformedusinga111111Monkhorst-PackgridforthecalculationsofFepzNiCN,whichhasanorthorhombicstructureof6:956:966:75A3.ForthecalculationsofCuBTCandIRMOF-1,[182,183,184,185]theBrillouinzoneintegrationswereperformedusing333Monkhorst-Packgrids.Thelatticeparametersarea=b=c=18.67Aanda=b=c=18.41AforCuBTCandIRMOF-1,respectively,andbothmaterialshave===60.TheESPwascomputedonaFFTfastFourier 88

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transformgridwithinVASPandpointsusedinthettingroutinewereselectedfromthegridonlyiftheyfelloutsidetheexclusionradius,Rex;jofanyatomjj=1,2,...nintheperiodicsystem.PointsthatfellwithinadistanceofRex;jofanyatomwerediscardedduetothelargedistortionscausedbythecloseproximitytothenucleusofthatatom.SpecicationofRex;jvaluesforeachatomtypejresultsinasetofpointsi=1,2,...mfromtheQMESPgridthatwereusedinthettingprocedure.ThevaluesoftheESPonthesamegridscomputedfromthesetofatomicchargesaredenotedasVqi.WeusealeastsquaresprocedureverysimilartotheotherESPttingapproaches[173,169,208]forttingtheatomicchargeqjtoeachatomiccenterjinthesystem.Thefunctiontobeminimizedintheleast-squaresprocedureisdenedasfqi;Voset;=XihVqi+Voset)]TJ/F22 11.9552 Tf 11.9552 0 Td[(VQMii2+Xjqj)]TJ/F22 11.9552 Tf 11.9552 0 Td[(q0j2+Xjqj)]TJ/F22 11.9552 Tf 11.9551 0 Td[(qtot2.1whereVQMiistheQMESPvaluefromVASPcalculation.ThesumofallttedchargesisconstrainedtototalmolecularchargebymeansofaLagrangemultiplierasthethirdtermineq6.1,whichisequivalenttothetotalchargeconstraintinotherESPttingcodes.Theproblemofdeeplyburiedatomsisaddressedbytheadditionofapenaltyfunction,givenasthesecondtermineq6.1,tothechargettingprocedure.ThisisthesameapproachasisusedinRESPmethod.[173]Thequantityisascalefactordeterminingthestrengthofrestraintandq0jisthetargetcharge.Withthispenaltyfunctiontogetherwiththescalefactor,wecansettargetchargestoselectedatomsburiedatomsinourstudyinthesystem.OneofthemajorproblemsoftheQMESPcalculatedfromperiodiccodesistheexistenceofthearbitraryosetvalueVoset,whichhasbeenclearlydemonstratedbyCampa~naetal.[169]Inourstudy,wedirectlycalculatetheosetvalueasoneoftheunknowns,Vosetineq6.1.Theadvantageourapproachisthatthettedosetvaluehelpustounderstandthequalityofthettedatomiccharges,whichwillbediscussedinSection3.2.Theatomicchargesqj,osetvalueofESPVoset,and,canbefoundbysolvingtheequations: 89

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Xj=1@f @qj=0.2@f @Voset=0.3@f @=0.4ThevaluesVqiatgridpointsarecalculatedforperiodicsystemsusingboththeEwald[209,210]andWolf[210,211]summationmethods.TheEwaldsummationisgivenbyVqi=1 40Xjqj"nmaxXnerfcp jrij+nj jrij+nj+1 VkmaxXk6=042 jkj2exp)]TJ/F22 11.9552 Tf 9.2985 0 Td[(2jkj2 coskrij#.5whereisthedampingparameterwithunitsofA)]TJ/F20 7.9701 Tf 6.5865 0 Td[(1,nistherealspacelatticevector,nmaxisthemaximumnumberofunitcellsintherealspacesum,kisthereciprocalspacelatticevector,kmaxisthemaximumnumberofreciprocalspacevectors,Visthevolumeofthebox,erfcisthecomplementaryerrorfunction,andisthepermittivityoffreespace.Ewaldsummationhasbeenwidelyusedtocalculatetheelectrostaticpotentialenergiesofcondensedphasesystems.TheconvergenceoftheEwaldmethoddependsonthesizeofthesystemandthechoiceoftheparameters,nmax,andkmax.Inpracticeandnmaxarenotindependentbutcanberelatedthroughchoosingarealspacecuto,Rc,bycomputingfromRc,andthenchoosingnmaxjustlargeenoughtoaccommodateRc.Inthisworkweuse=:2=Rc2fortheEwaldmethod.TypicalvaluesofRcreportedinthe 90

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literatureare8{12Aandkmaxisusuallysetto5orlarger.[212]Campa~naetal.[169]usedRc=9Aandkmax=7.Inthiswork,wehaveusedRc=10Aanddeterminedtheoptimalvalueofkmaxbyobservingtheconvergenceofthettedchargeswithincreasingvalueskmax.Wefoundthatkmax=5issucienttoconvergethechargestowithinacceptabletolerance.IncontrasttotheEwaldmethod,theWolfsummationinvolvesonlyasumoverthecharges,withouteitherrealorreciprocalspacelatticesums.TheWolfsummationisgivenby[211]Vqi=1 40Xjqjerfcp rij rij)]TJ/F15 11.9552 Tf 13.1506 8.0877 Td[(erfcp Rc Rc+erfcp Rc R2c+r 4 exp)]TJ/F22 11.9552 Tf 9.2985 0 Td[(R2c Rc!rij)]TJ/F22 11.9552 Tf 11.9551 0 Td[(Rc#.6where=:2=Rc2ineq6.6.TheESPttingmethodgivenbyeq6.1canalsobeappliedtonon-periodicsystems,asdemonstratedbyCampa~naetal.[169]fortheREPEATmethod.WehavealsovalidatedthatourapproachgeneratescorrectchargesformolecularsystemsbycomputingchargesforH2Oandcomparingwithchargescomputedbytraditionalmethods,suchasCHELPG.Weusetherelativeroot-mean-squareRRMSerrortocharacterizethequalityofthetofthecomputedchargestothequantummechanicalESP.TheRRMSisdenedasRRMS=vuuuut PihVqi+Voset)]TJ/F22 11.9552 Tf 11.9552 0 Td[(VQMii2 PiVQMi.7 91

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ThequantummechanicalESPvaluesaretakenasthosecomputedontheDFTFFTgrid.TheDFTFFTgridneedstobesucientlynetoensureproperconvergence.WehavetestedtheconvergenceoftheFFTgridandhavefoundthatareal-spacegridof0.2-0.3Aisrequiredtoachieveconvergenceofthettedchargestowithin0.01eforIRMOF-1.Incontrasttothenegridneededforconvergenceoftheelectrondensity,Campa~naetal.haveshownthatonecanuseamuchcoarsergridinthechargettingroutine.Toexplorethesensitivityoftheresultingchargestochoiceofgridmesh,wedeneaparameter,G,tothintheDFTFFTgrid.GdenesthenumberofpointstobeskippedineachdirectionfromtheoriginalFFTgrid.Forexample,considera200200200gridfromacubicsystem.AvalueofG=0meansthatallpointsarekept,whileavalueofG=1reducesthegridto100100100andkeepingthequalityoftheFFTgridexactlythesame.WedeneRex;jintermsofthevanderWaalsradiusofeachatomaccordingtoRex;j=RvdW;j.8whereRvdW;jisthevanderWaalsradiusforatomj,andisascalingfactor.ThevaluesofRvdW;jformaingroupelementsarespeciedinalookuptableconsistingofTruhlarsveryrecenttabulations,45awelcomeupdateofthefrequentlyused1964Bondiradii.[213]ThevaluesofRvdW;jfortransitionmetalsinourstudyareobtainedfromtheCambridgecrystallographicdatabase.[214]Symmetryofthestructuresisexploitedandenforcedinthettingprocedurebymeansofreadinginanadditionaluser-preparedinputleandcondensingtheappropriatematrixelementscorrespondingtosymmetricatoms. 92

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Figure6.1:RepresentationofthestructuresofaCuBTC,bIRMOF-1,andcFepzNiCN4. 93

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6.4ResultsandDiscussion6.4.1ComparisonofEwaldandWolfsummationmethodsMetalorganicframeworkshavebeenextensivelystudiedbecauseoftheirprominentroleinmanylarge-scaleapplicationsofchemicalseparations.[182,206,207]Inadditiontotheexperimentalresearchinthiseld,MOFshavealsobeenthesubjectofmanytheoreticalinvestigationsusingatomisticsimulation.[182,183]Here,thegoalistoprovidereasonableatomicchargeswithESPttingofthedenseperiodicsystems,withwhichonecanreproducetheESPwithaccuracysucientforatomisticsimulations.Astestcases,theMOFsCuBTCandIRMOF-1,whichhavebeenwidelystudiedtheoretically,[182,183,184,185]areinvestigatedinourstudy.TheatomicpositionsandlatticeparameterswereoptimizedusingDFTandESPFFTgridsweregeneratedusingtheserelaxedstructures.TheprimitivecellstructuresofCuBTCandIRMOF-1areshowninFigure6.3.WehavegeneratedchargesforallthesystemsstudiedhereusingboththeEwaldandtheWolflong-rangeelectrostaticsummationmethods.TheWolfsummationistypicallymoreecientthantheEwaldsummationforagivensystemandhasthepotentialtoscalelinearlywithincreasingsystemsizebecauseitonlyrequiresevaluatingapair-wisesum,truncatedinanappropriatefashion.[211]WeheretesttheconvergenceandtheeciencyofobtainingttedchargesbasedonbothEwaldandWolfsummation.WehaveevaluatedtheconvergenceoftheEwaldandWolfsummationasafunctionofkmaxandRc,respectively.TheparameterGissetto1andthevalueisspeciedas1.0foralloftheresultsshowninTable6.4.1.WehavetestedtheconvergenceoftheEwaldsummationwithkmax,holdingRc=10A_TheconvergenceofthechargesforCuBTCareplottedinFigure6.4.1.WeseethattheEwaldsummationiswell-convergedforavalueofkmax=5.TheatomicchargesforCuBTCalongwiththeCPUtimerequiredtogeneratethemaregiveninTable6.4.1.WehavetestedtheconvergenceoftheWolfsummationwithRc.Thettedchargeson 94

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IRMOF-1 Ewalda Wolfa REPEATb ESPMKc CHELPGd Baderc Zn 1.258 1.258 1.28 1.37 1.501 1.35 O1 -1.425 -1.424 -1.57 -1.59 -1.846 -1.27 O2 -0.601 0.601 -0.61 -0.71 -0.724 -1.24 C1 0.496 0.495 0.52 0.70 0.667 1.65 C2 0.185 0.185 0.14 0.06 0.072 0.00 C3 -0.196 -0.197 -0.18 -0.12 -0.132 0.01 H 0.157 0.157 0.17 0.12 0.140 0.02 CPUTime 76 15 CuBTC Ewalda Wolfa REPEATe CHELPGd Baderf Cu 0.904 0.904 0.915 1.098 1.01 O -0.542 -0.543 -0.548 -0.665 -1.07 C1 0.046 0.043 0.017 -0.092 0.06 C2 0.606 0.607 0.619 0.778 1.50 C3 -0.175 -0.173 -0.147 -0.014 -0.05 H 0.157 0.156 0.149 0.109 0.11 CPUTime 180 41 Table6.1:ComparisonofatomicchargesforIRMOF-1andCuBTCobtainedfromourcodebothEwaldandWolfsummation,G=1,=1.0,aswellasotherapproaches.aThiswork;bReference[169];cReference[183];dReference[185];edataareobtainedusingtheRE-PEATcodeinourstudy;fReference[184]. 95

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Figure6.2:AtomicchargesforCuBTCcalculatedfromthisworkusingtheEwaldsum-mationwithRc=10Aasafunctionofkmax. theirreducibleatomsoftheRcforCuBTCareplottedinFigure6.4.1.Wehavecomputedthelargestandaverageabsoluteerrorswithrespecttothewell-convergedEwaldvaluesasafunctionofRc.Thelargestabsoluteaverageerrorsare0.089.030e,0.024.015e,and0.003.001eforRc=5,10,and20A,respectively.WehavealsotestedIRMOF-1andfoundverysimilarbehaviortoCuBTCseeFigure6.4.1,withlargestabsoluteerrorof0.001eforRc=20A.WeuseRc=20AforallWolfcalculationsandkmax=5withRc=10AforallEwaldsummationcalculationsinthisstudy.Forcomparison,wealsolistthepublishedatomicchargevaluesfromclustercalculations,[184,185]Baderchargeanalysis,[184]andtheREPEATmethod[169]inTable6.4.1.AscanbeseenfromTable6.4.1,theatomicchargescalculatedfromdierentapproachestakeonawiderangeofvalues,especiallyforburiedatomssuchasO1inIRMOF-1.TheBaderchargesareinaccurateforsomeatoms,especiallythoseinanespeciallyanisotropicenvironment,ashasbeennotedbefore,[183]andthereforegiveapoordescriptionoftheESPinmolecularsimulations.[184]Thus,thesetofatomicchargesfromBaderchargeanalysisshouldnotbeusedinmolecularsimulations.AsshowninTable6.4.1,thettedchargesarealmostthesamefortheEwaldandWolfsummationmethods.Also,comparedtothosefromtheREPEATcode,ourresultsforIRMOF-1andCuBTCareingoodagreementwithdeviationslessthan0.03e,withthesoleexceptionoftheburiedatom,O1,inIRMOF-1.ThereasonfortherelativelylargedeviationforatomO1isthatchargesonburiedatomsareverysensitivetothedetailsofthettingprocedure,aswillbediscussedlater.Theatomicchargesfrom 96

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clustercalculations[183,185]areinreasonableagreementwiththosefrombothourworkandtheREPEATmethod.TheCPUtimerequiredforthettingprocedureusingEwaldandWolfsummationinourcodeisalsogiveninTable6.4.1.TheCPUtimeforthettingprocessusingtheWolfsummationisreducedbyafactorof5comparedtothatbasedontheEwaldsummation,whiletheaccuracyisessentiallythesame.TheeciencyoftheWolfmethodcomparedwiththeEwaldsummationreportedinourstudyissimilartothatreportedforelectrostaticinteractionsinmolecularsimulation.[211]6.4.2EectoftheexclusionradiusTheeectoftheexclusionradiusonthettedvaluesoftheatomicchargesisconsideredbyvaryingthescalingfactorineq6.8.Ideally,theatomicchargeswillconvergeforarangeofvaluesof.ValuesofthataretoosmallwithinthevdWradiuswillsampleregionsofhighelectrondensitythatcannotbeaccuratelymodeledwithpointcharges.Ontheotherhand,valuesofthataretoolargewillexcluderegionsthataresampledinmolecularsimulationswherereproducingthecorrectESPiscritical.Campa~naetal.[169]investigateddierentvaluesoffor0:651:15indenselypackedsolidslikeZnO,SnO2,andCdTe.Theyfoundthatthettedchargesareverysenstitivetotheexclusionradius.However,theydidnotstudytheeectoftheexclusionradiusformicroporousmaterials,likeMOFs,anddidnotconsiderlargervaluesof.WehavestudiedtheeectoftheexclusionradiusontheatomicchargesofCuBTC,IRMOF-1,andFepzNiCNfor1:02:0.WehaveusedtheWolfsummationinthettingproceedure.ForCuBTC,aswecanseefromTable2,thettedchargesdonotchangedramaticallyasthevalueofincreasesfrom1.0to2.0,whereastheRRMSvalueisgreatlyreducedwithincreasing.WenotethattheRRMSvaluereducesfrom0.039at=1.0to0.008for=1.3,indicatinglargeerrorsinthettedESPforpointsclosetothevdWradiioftheatomsinCuBTC.WehavecharacterizedtheerrorasafunctionofthedistancefromtheatomsbygroupingpointsontheESPintosphericalshellsaroundeachatomwherethediameterofeachshellisdenedasamultipleofthevdWradiusofthatatom.Hencetheshells 97

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Figure6.3:VariationofthettedatomicchargesforCuBTCandIRMOF-1asafunctionofthecutoradius,Rc,intheWolfsummation. 98

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Figure6.4:TheRRMSerrorsandaverageESPerrorsVavgforaCuBTCandbIRMOF-1asafunctionofthedistancefromeachatomreducedbythevdWradiusofthatatom.TheerrorsarecalculatedinshellsaroundeachatomthataremultiplesofthevdWradiusofthatatom.Twodierentvaluesofthescalingfactordenedineqareused,=1.0and1.3.TheleftyaxisisforRRMSerrorsandtherightyaxisisVavg. 99

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Table6.2:FittedatomicchargesforIRMOF-1andCuBTCasafunctionoffor1:02:0.ThepotentialosetandRRMSerrorsarealsoreported.TheWolfmethodwasusedinthettingprocess. 100

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arounddierentatomtypeswillnotbethesamesizebutwillbethesamerelativedistancefromtheatomcenter.Therstshellaroundatomjextendsfrom1.0RvdW;jto1.1RvdW;j,thesecondshellfrom1.1RvdW;jto1.2RvdW;j,andsoforth.TheRRMSandaverageESPerrorsarecalculatedseparatelyineachshell,wheretheaverageESPerror,Vavg,isdenedasVavg=1 nshellnshellXi=1Vqi+Voset)]TJ/F22 11.9552 Tf 11.9552 0 Td[(VQMi.9wherenshelldenotesthenumberofselectedpointsintheshell.WecomputetheRRMSerrorsandaverageESPerrorfordierentshellsusing=1.0and1.3;theresultsareshowninFigure6.4.1.TheRRMSerrorsandaverageESPerrorsareplottedontheleftandrightyaxes,respectively,andthexaxisisthereducedshellradius.For=1.0,wecanseethattheRRMSerrorforthei1.0{1.1shellislarge.138andtheVavgis0.249eV.Foravalueof1.3theRRMSforeachshellissmallerthanthatfromavalueof1.0andlikewiseVavgisgenerallysmallerforthelargervalueof.Therefore,wecanconcludethatthequalityofthetworsensifthepointsarechosenclosertotheatomsandimprovesifthepointsarechosenfartheraway.FromTable6.4.1,wecanseethatthettedosetvalueofthepotentialforCuBTCslowlyconvergeswithincreasing.Forexample,thettedosetvalueis2.137eVfor=1.0,whichis0.049eVsmallerthantheosetvalueof2.186eVfor=2.0.Incontrast,theosetvalueis2.180for=1.5,whichisonly0.006eVsmallerthanthe=2.0value.Thus,Vosetismoreaccurate,i.e.,closertotheactualvalue,forlargervaluesof,atleastfor2.0.Moreover,theRRMSvaluesdecreasewithincreasing.TheadvantageofdeningVosetasattedvalueratherthanthedenitionusedintheREPEATapproachisthatexplicitlyknowingthevalueofVosetallowsonetochecktheconvergenceasafunctionof.IngeneralthesametrendscanbeseenforIRMOF-1asforCuBTCinTable6.4.1andFigure6.4.1.For=1.0and1.3,thetotalRRMSis0.050and0.012,respectively.For=1.0,asshowninFigure6.4.1,theRRMSis0.217inthe1.01.1shellandVavg=0.253eV.For=1.3,theRRMSerrorsineachshellarereducedtosmallvalues,withthelargestbeing0.04.TheVavgvaluesarelikewisegreatlyreducedrelativetothe=1.0case.ThereisastrikingdierencebetweenthebehavioroftheatomicchargesforIRMOF-1andCuBTCasafunctionof.ThechargesonIRMOF-1aremoresensitivetothevalueof,ascanbeseeninTable 101

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6.4.1.ThisisespeciallytrueforthecentralOxygenO1inIRMOF-1,wherethevalueofthechargechangesfrom-1.424at=1.0to-2.304for=2.0.Thismagnitudeofchangeisunphysical.ThereasonforthesensitivityofO1toisthatO1isaburiedatom;ithasbeendemonstratedthatthechargesoftheburiedatomscanchangedramaticallyinthettingprocess.4Inpractice,shouldbesmallenoughtoincludeallregionsofspacethatcanbesampledineitheramoleculardynamicsorMonteCarlosimulation,butlargeenoughtoreducetheerrorassociatedwithttingtheESPinregionsofspaceclosetothevdWradiioftheatoms.Hence,theoptimumvalueofisthelargestpossiblevaluethatensuresthatallregionsofspaceaccessibletoasimulationareincludedinthetting.Thisvalueissystemandevensimulationdependent.Forpracticalpurposeswecanestimateareasonablevalueofbyestimatingthedistanceofclosestapproachforapairofatomsthatarelikelytobeanextremecase.AlkalimetalshaverelativelylargevdWradiicomparedwithotheratoms.[215,214]Forexample,thepotassiumhasavdWradiusof2.75Alargerthanmostoftheelementsfrommaingroupandtransitionmetals.Incontrast,theHatomhasthesmallestvdWradius.10A,whichisabout40%ofthevdWradiusofKatomandsotheK-Hcontactdistanceis1.4RvdW;K.Hence,aconservativeestimateofwouldbetoselectavaluesmallerthan1.4inordertoincludealltheeectivepointsinthettingprocess.BasedonthediscussionsaboveandtheresultsforCuBTCandIRMOF-1seeninTable2,weselectavalueof=1.3asareasonablecompromise.ThisvalueagreeswellwiththatidentiedbySinghandKollman.TheyclaimedthatitissucienttousevaluesoftheESPinashellaroundeachatomthatrangesfrom1.4i.e.,=1.4to2.0timesthevdWradius.Theystatethatthisrangecoverstheimportantdistancesforintermolecularinteractions.Campa~naetal.havepointedoutthatsystemscontaininghighlycoordinatedorburiedatomsareproblematicwhenitcomestoassigningcharges.ESPttingmethodshavebeenknowntoproduceresultsthathaveagreatdealofvariabilityandthatcanbeindisagreementwithchemicalintuition.[183]ThevalueofthechargeonaburiedatomhasbeenalsoshowntobeverysensitivetothevalueofthevanderWaalsradiususedinthetting.Also,asdiscussedabove,thettedchargesforIRMOF-1varysignicantlywithdierentvalues.ThegoodnewsisthettedchargesforIRMOF-1arephysicallyreasonableandtheresultingESPisaccuratewith=1.3. 102

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WenowconsideranothernanoporousMOF,[FepzNiCN4]wherepzispyrazine.ThisMOFcontainsburiedatomsandhenceprovidesanotherexampleoftheeectthatchanginghasontheatomiccharges.ThestructureofthismaterialisshowninFigure6.3;theFeatomisdeeplyburied.AsshowninFigure6.4.2,thettedchargesfortheburiedFeatomchangewidely,from0.006to-1.265,whenchangesfrom1.0to1.6.InSection3.4wediscusstheuseofapenaltyfunctionwithtargetchargestoobtainphysicallyreasonableatomicchargesforburiedatoms. Figure6.5:asafunctionof]Fittedchargesfor[FepzNiCN4]asafunctionof. 6.4.3EectofthegridmeshdensityAsdiscussedpreviously,itisnecessarytohaveafairlynegridmeshinordertoachieveconvergenceoftheESPcomputedfromDFT.However,usingthefullgridinthettingprocedurewillrequireasignicantamountofCPUtimeforlargesystemsatnegligibleincreaseinaccuracy.Wehaveusedagridthinningparameter,G,togaugetheeectofdierentsizesofgridmeshonthettedatomiccharges.WeuseIRMOF-1inourstudy.ThequantummechanicalESPvalueswerecalculatedwithaverynegriddensity00200200.NotethatCampa~naetal.discussedtheeectofdierentgridmeshinadierentway,i.e.,theESPdatawithdierentgridmeshwereobtaineddirectlyfromtheDFTcalculations.InCampa~na'sstudy, 103

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employingasparsegridmeshfortheDFTcalculationpotentiallyintroducederrorsarisingfromthequalityofESPcomputedfromDFT.ThettedchargesforatomsO1andZnseeFigure1havebeencalculatedfor0G7andtheresultsarepresentedinTable6.4.3.Notethatvaluewassetto1.3.BothEwaldandWolfsummationmethodsareemployedforcomparison.WeseethattheresultsforboththeEwaldandWolfmethodsarenearlyidentical.TheresultsforG=1and2arelessthan1%dierentfromtheG=0charges.Interestingly,thechargesonZnaremuchmoreinsensitivetothevalueofGthanthechargesonO1.ForG=7,theerroronO1increasestoabout12%whileitisabout8%forZnnotethechangeinthesignoftheerror.TheRRMSvaluesforallofthesecasesarereasonableandonlychangeslightly,accordingtothedatafromTable6.4.3.Basedontheseresultsweestimatethatagridspacingof0.2-0.3Ai.e.,G=1or2issucientlyaccurateforcomputingatomiccharges.Note,however,thatitissuggestedtoemployanemeshintheDFTcalculationinordertoobtainaccurateESPvalues.TheaveragenumbersofselectedpointsperatomfordierentvaluesofGarealsolistedinTable6.4.3.Theaveragenumberofpointsperatomis2002forIRMOF-1withG=2,whichisingoodagreementwithSigfridssonandRydesinvestigations.[216]TheycarefullyinvestigateddierentESPttingmethodsandsuggestedahighpointdensityisnecessary,i.e.,atleast2000pointsperatom.48FromTable6.4.3,wecanseethedeviationofatomicchargesbasedonEwaldandWolfsummationmethodsislessthan0.003,whilethettingprocessusingWolfsummationrequireslessCPUtimebyaboutafactorofve. ParameterG 0 1 2 3 4 5 6 7 Ewald O1 -1.711 -1.724 -1.709 -1.712 -1.743 -1.739 -1.730 -1.494 Zn 1.358 1.365 1.357 1.362 1.385 1.369 1.356 1.258 RRMS 0.012 0.012 0.012 0.012 0.012 0.012 0.012 0.013 Wolf O1 -1.709 -1.722 -1.707 -1.710 -1.741 -1.736 -1.728 -1.493 Zn 1.357 1.365 1.356 1.361 1.384 1.367 1.356 1.258 RRMS 0.012 0.012 0.012 0.012 0.012 0.012 0.012 0.013 points/atom 53375 6671 2002 836 426 259 162 105 Table6.3:FittedchargesofO1andZninIRMOF-1withdierentvaluesofthethinningparameterG,usingtheEwaldandWolfsummationtechniques.Thenumberofselectedpointsperatomislistedforeachcase. 104

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6.4.4RestraintapproachforburiedatomsWehaveseenthatburiedatoms,suchastheFeatominFepzNiCN4andtheO1atominIRMOF-1,producechargesthatareverysensitivetothedetailsofthettingprocedureandareoftenatoddswithchemicalintuition.Camp~naetal.usedarestraintmethodtodealwithburiedatoms.Wehavealsoimplementedarestraintmethodinourcode,butourmethoddiersfromthatofCamp~naetal.inthattheyuseaphysicallymotivatedpenaltymultiplierthatisanexpansionoftheenergyofanatomasafunctionoftheatomicchargesuptothesecondorder.WiththeREPEATcode,however,thettedatomicchargesforthismaterialarenotimprovedaftertheactivationofitsrestraintmethod.Thechallengeintherestraintformalismforourcodeistoidentifyareasonablerestraintchargefortheburiedatom.Asdiscussedpreviously,theBaderchargepartitioningschemedoesnotperformwellforttingonlymonopolestohighlyanisotropicatoms.[183]However,buriedatomsaretypicallyhighlycoordinatedandhenceareinamoreisotropicenvironment;thereforetheBaderchargesforburiedatomsshouldprovidereasonableguessesfortheatomicchargesandhenceacceptablerestraintcharges.Forexample,theFeatominFepzNiCNissurroundedbysixNatomsandisnearlyisotropic.ThepredictedBaderchargefortheFeatomshouldthereforebechemicallyreasonableandcanbeusedastherestraintvalueinthettingroutine.TheargefortheFeatomwasthereforeselectedtobe1.189e,ascomputedfromBaderanalysis.Inthecalculationsdescribedbelow,=1.3anddierentvaluesofscalefactorareemployed.ThettedatomicchargesfortheFepzNiCNwithdierentvalueofscalefactorfromeq6.1areshowninTable6.4.4.Ascanbeseen,thechargesforburiedNiatomsincreasegraduallyfrom-1.012to1.189asisincreasedfrom0to50000.TheRRMSvaluechangedslightlyaftertherestraintpenaltywasincludedinthettingprocess,indicatingthatthegoodnessofttotheESPsisfairlyinsensitivetotherestraintprocedure.NotethatthevalueofshouldbetestedfordierentsystemstoassurethattheresultingRRMSvalueisreasonable. 105

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Table6.4:FittedatomicchargesforFepzNiCN4asafunctionofthescalefactor. 6.5ConclusionsWehaveinvestigatedamethodforttingatomicchargestotheESPofperiodicandnon-periodicsystemsthatisverysimilartothemethodofCamp~naetal.WehavedemonstratedthattheWolflong-rangeelectrostaticcorrectionmethodisroughlyafactorofvefasterthantheEwaldsummationandhasessentiallyequivalentaccuracywhenusedforttingatomicchargesforperiodicsystems.WehaveshowntheutilityoftreatingVosetasattedparameterforperiodicsystems.WehavedeterminedthatanESPgridwithamaximumspacingof0.2-0.3Aisrequiredtoobtainreliableatomicchargesinthettingprocedure,butthattheelectronicstructurecalculationsshouldemployanermeshinordertoconvergethetargetESP.Wehaveshownthatanexclusionradiusof1.3timesthevdWradiusofeachatominthesystemresultsinagoodcompromisebetweengoodnessoftandcoverageoftheimportantregionsoftheESPsurface.ThettingofchargesforburiedatomsisingeneralproblematicforESPttingmethods.WehaveshownthatBaderchargescanbeusedastargetchargesforburiedatomsinapenaltyfunctionapproachforttingchargesofburiedatoms. 106

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Appendices 120

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AppendixAPeriodicESP-derivedchargescodelistingThisistheCcodeusedtoproduceelectrostaticpotentialderivedatomicpointchargesforaperiodicsystem.Thecodeislistedstartingwiththeheaderlesthatdenethedatastructuresandfunctionprototypesusedthroughoutthecode.Followingisthemainroutine.Finally,theindividualfunctionsandroutinesarelistedtogether. #include #include #include #include #include /*maximumsizeforalineandkeyval*/ #dene MAXLINE 512 #dene MAXKEY 64 /*fromNIST1Bohr=0.529..Angstoms*/ #dene BOHR RADIUS 0.52917720859 /*fromCRChandbook1Eh=27.2114Ev;also,switchestoconventiontestPOSITIVEcharge*/ 121

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#dene Ev2Eh FAC 27.2114 #ifdef MPI #include #dene TAG R 1 /*MPIglobals*/ MPI Status status ; int myrank ; int ntasks ; #endif typedef struct atoms f int natoms ; /*numberofatoms*/ int ntype ; /*numberoftypes*/ int neachion ; /*numofeachiontype*/ int nzion1 ; /*nuclearchargetype*/ int z ; /*nuclearchargeallatoms*/ int ngxf ; /*NGXFvalue*/ int ngyf ; /*NGYFvalue*/ int ngzf ; /*NGZFvalue*/ double ** frac ; /*fractionalcoordsatoms*/ double ** cart ; /*cartesiancoordsatoms*/ char ** elem1 ; /*elementnamentype*/ char ** elem ; /*elementnameallatoms*/ double vdw1 ; /*vdwradiintype*/ double vdw ; /*vdwradiiallatoms*/ double minvdw ; /*smallvdwradii*/ double maxvdw ; /*largevdwradii*/ double tv [ 3 ][ 3 ]; /*basisvectors*/ double rec tv [ 3 ][ 3 ]; /*reciprocalbasis*/ double cell volume ; /*cellvolume*/ double alpha ; /*anglebetweentv1&2*/ double beta ; /*betweentv2&3rad*/ double gamma ; /*angbettv1&3rad*/ double magA magB magC ; /*magsoftvsj1j,j2j,.*/ double scale1 ; /*scalingofsmallVDWshell*/ double scale2 ; /*scalingoflargeVDWshell*/ int qconstr ; /*agtoturnonconstraint*/ int qtotal ; /*totalchargeifconstron*/ double wolfrcuto ; /*cutoforWolfmethod*/ double wolfrcutosq ; /*cutosquareforWolf*/ char ewaldorwolf ; /*agtochooseewaldorwolf*/ int ewald ag ; /*1=ewaldon*/ int wolf ag ; /*1=wolfon*/ char symmornot ; /*agofsymmetryisusedornot*/ int symm ag ; /*1=symmetryon*/ int nelem symm ; /*numberofatomswithdierentsymmetry*/ int nelem symm natoms ; /*numberofequivalentatomsofeachtype*/ char ** elem symm ; /*nameofsymmetricatoms*/ int ** symm lable ; /*lableNO.ofeachatomwithsymm*/ double aa ; /*valueof"a"inpenaltyfunction*/ double charge ; /*ttedcharges*/ struct timeval t0 ; /*timestampatstarttime*/ int esp output ; /*agtoturnon/oesp output*/ char esp lename [ 64 ]; /*lenameofespoutputle*/ int pdb output ; /*agtoturnon/opdboutput*/ char pdb lename [ 64 ]; /*lenameofpdble*/ int xyz output ; /*agtoturnon/oxyzoutput*/ char xyz lename [ 64 ]; /*lenameofxyzle*/ 122

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int normag ; /*agofnormalization*/ g ATOMS ; typedef struct points f int npoints ; /*numberofpoints*/ double esp ; /*ESPforselectedpoints*/ double esp average ; /*averageESPvalue*/ double *** gridesp ; /*ESPforgridpoints*/ double **** gridfrac ; /*fractionalcoordsgrid*/ double **** gridcart ; /*cartcoordsgrid*/ double V ; /*numberofvertices*/ double ** cart ; /*cartesiancoords*/ double ** frac ; /*fractionalcoords*/ int nshells ; /*numberofshells*/ double rst shell ; /*factorofrstshel*/ double shell step ; /*shellstep*/ int prune ; /*agtoincludeorno*/ int counter ; /*index*/ int scalegrid ; /*scalingofgrid*/ double esp tting ; /*calculatedESPbasedonttedatomiccharges*/ double rrms ; /*calculatedrrmsvalue*/ g POINTS ; typedef struct lsq f double ** des ; /*designmatrix,mxn*/ double des average ; /*averageofl->des*/ double des average temp ; /*tempvalues*/ double ** A ; /*reduceddesignmatrixAnxnorsmallr*/ int m n ; /*AisnxnnatomsXnatoms*/ double b ; /*Bvector---nlong*/ double a ; /*answervector-nlong*/ double ** V ; /*workingspaceforSVD*/ double w ; /*workingspacefortheSVD*/ double sum ; /*sumofttedcharges*/ double des oset ; /*averageof**desovernpoints*/ g LSQ ; typedef struct lrc f int kmax ; /*kmaxforewald*/ int nmax [ 3 ]; /*likekmax,butreal-space*/ int nmax in ; /*input,onlyusedtocalcnmax[]*/ double nmaxf ; /*maximumreal-spacedistance*/ double alpha ; /*alphaforewald*/ double alpha wolf ; /*alphaforwolf*/ double realcut ; /*real-spacecut-ostandard*/ double wolfrcuto ; /*cutoforwolf*/ double wolfrcutosq ; /*cutosquareforwolf*/ double wv ; /*pre-computedewaldwavevectors*/ double ** kvec ; /*pre-computedewaldwavevectors*/ g LRC ; #include "./proto.h" void output const char outstring ; void problem with inputle char le ; void calc recip struct atoms a ; 123

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void calc rcut struct atoms a struct lrc lrc ; void svdcmp double ** a int m int n double w [], double ** v ; void svbksb double ** u double w [], double ** v int m int n double b [], double x []; void results struct atoms a struct points p struct lsq l ; void read inputle struct atoms a struct points p struct lrc lrc FILE inputle ; void assign vdw struct atoms a ; void initialization struct atoms a struct points p struct lsq l struct lrc lrc FILE inputle ; void print2pdb struct atoms a struct points p ; void print2xyz struct atoms a struct points p struct lsq l ; void print charges struct atoms a struct points p struct lsq l struct lrc lrc ; void outesp struct atoms a struct points p struct lsq l ; void constrain struct atoms a struct lsq l ; void rstfew struct atoms a struct points p struct lsq l struct lrc lrc ; void solvebySVD struct atoms a struct points p struct lsq l struct lrc lrc ; void sum charges struct atoms a struct points p struct lsq l struct lrc lrc ; void handle nmax or alpha struct atoms a struct lrc lrc ; void symm constr struct atoms a struct lsq l ; void read symm struct atoms a struct lsq l ; void targetcharge struct atoms a struct lsq l ; void build all matrices struct atoms a struct points p struct lsq l struct lrc lrc ; void compute design matrix struct atoms a struct points p struct lsq l struct lrc lrc ; double one design element ewald int point i int atom j struct atoms a struct points p struct lrc lrc struct lsq l ; double one design element wolf int point i int atom j struct atoms a struct points p struct lrc lrc struct lsq l ; double real distance int n [ 3 ], int atom j int point i struct atoms a struct points p ; void performance struct atoms a int count int npoints ; void timestamp struct atoms a ; void compute wave vectors struct atoms a struct points p struct lsq l struct lrc lrc ; double minimum image double f1 double f2 struct atoms a ; /*externallibrarycalls*/ extern char ctime r const time t restrict timer char restrict buf THROW ; extern char strcasestr const char haystack const char needle ; #include "chargefit.h" void welcome message void f /*moretocome*/ output "BUILDXX.XX.XX nn ; g void usage char exe f printf nn ThisprogramgenerateschargesfittotheESPgeneratedbyVASP.etc. nn ; printf "ExecutionassumesthatthereisavalidLOCPOTandPOTCARfilepresentin nn ; printf "theworkingdirectory. nn ; printf nn USAGE: nnnn ; printf "%s nn exe ; printf nn Whereinputfileisformattedas:TBD nnnn ; exit 1 ; g void problem with inputle char le f printf "Problemopeningfile n" %s n" .Non-existent? nn le ; exit 1 ; g 124

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int main int argc char ** argv f struct atoms atoms a ; /*structurewithalltheatomicdataandapointertoit*/ struct points points p ; /*structurewiththettingpointsandapointertoit*/ struct lsq lsq l ; /*structurewiththeSVDmatricesandapointertoit*/ struct lrc longrc lrc ; /*structurecontainingthelongrangecorrectionrelatedinfo*/ FILE inputle ; char tstamp [ 26 ]; char bu [ MAXLINE ]; if argc != 2 usage argv [ 0 ]; #ifdef MPI /*initializeMPIhere*/ MPI Init & argc & argv ; MPI Comm rank MPI COMM WORLD & myrank ; MPI Comm size MPI COMM WORLD & ntasks ; if ntasks == 1 f printf "MPI:error:pleasempirunwithatleast2processors! nn ; exit 1 ; g #endif if inputle = fopen argv [ 1 ], "r" == NULL problem with inputle argv [ 1 ]; a =& atoms ; p =& points ; l =& lsq ; lrc =& longrc ; /*timestamp*/ gettimeofday & a -> t0 NULL ; ctime r & a -> t0 tv sec tstamp ; sprintf bu "Initializingon%s nn tstamp ; output bu ; welcome message ; /*seriouscodebegins*/ initialization a p l lrc inputle ; build all matrices a p l lrc ; constrain a l ; rstfew a p l lrc ; solvebySVD a p l lrc ; print charges a p l lrc ; sum charges a p l lrc ; results a p l ; print2xyz a p l ; output nn ; timestamp a ; output "exitingcleanly nn ; #ifdef MPI MPI Finalize ; #endif g #include 125

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/*Readtheinputle*/ void read inputle struct atoms a struct points p struct lrc lrc FILE inputle f int i ; oat nmaxoralpha ; char bu [ MAXLINE ], line [ MAXLINE ]; char key [ MAXKEY ], val1 [ MAXKEY ], val2 [ MAXKEY ], val3 [ MAXKEY ]; /*setdefaults*/ a -> ntype = 0 ; a -> elem1 = NULL ; a -> nzion1 = NULL ; a -> ewaldorwolf = malloc MAXKEY sizeof char ; a -> symm ag = 0 ; a -> aa = 0 ; a -> qconstr = 0 ; lrc -> alpha = 0 ; lrc -> nmax in = 0 ; lrc -> kmax = 0 ; a -> esp output = 0 ; a -> pdb output = 0 ; a -> xyz output = 0 ; a -> normag = 0 ; memset line 0 MAXLINE ; memset key 0 MAXKEY ; memset val1 0 MAXKEY ; memset val2 0 MAXKEY ; memset val3 0 MAXKEY ; output nn input: nn ============================================= nn ; while fgets line MAXLINE inputle != NULL f sscanf line "%s%s%s%s" char *& key char *& val1 char *& val2 char *& val3 ; sprintf bu nt %-20s%10s%10s%10s nn key val1 val2 val3 ; output bu ; /*KEYWORDSEARCH*/ if strcmp key "n atom types" f a -> ntype = atoi val1 ; g else if strcmp key "inner vdw fac" f a -> scale1 = atof val1 ; g else if strcmp key "outer vdw fac" f a -> scale2 = atof val1 ; g else if strcmp key "lrc method" f strcpy a -> ewaldorwolf val1 ; g else if strcmp key "alpha" f lrc -> alpha = atof val1 ; g else if strcmp key "kmax" f lrc -> kmax = atoi val1 ; 126

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g else if strcmp key "nmax" f lrc -> nmax in = atoi val1 ; g else if strcmp key "grid thin" f p -> scalegrid = atoi val1 ; g else if strcmp key "element" f a -> elem1 = realloc a -> elem1 a -> ntype + 1 sizeof char *; a -> elem1 [ a -> ntype ]= malloc 10 sizeof char ; a -> nzion1 = realloc a -> nzion1 a -> ntype + 1 sizeof int ; strcpy a -> elem1 [ a -> ntype ], val1 ; a -> nzion1 [ a -> ntype ++]= atoi val2 ; g else if strcmp key "constrain" f if strcmp val1 "yes" f a -> qconstr = 1 ; a -> qtotal = 0 ; g else a -> qconstr = 0 ; g else if strcmp key "restrain" f a -> aa = atof val1 ; g else if strcmp key "symmetry" f if strcmp val1 "yes" a -> symm ag = 1 ; else a -> symm ag = 0 ; g else if strcmp key "esp output" f if strcmp val1 "no" a -> esp output = 0 ; else f a -> esp output = 1 ; strcpy a -> esp lename val1 ; g g else if strcmp key "xyz output" f if strcmp val1 "no" a -> xyz output = 0 ; else f a -> xyz output = 1 ; strcpy a -> xyz lename val1 ; g g else if strcmp key "pdb output" f if strcmp val1 "no" a -> pdb output = 0 ; else f a -> pdb output = 1 ; strcpy a -> pdb lename val1 ; g g else if strcmp key "normalization" f if strcmp val1 "yes" a -> normag = 1 ; else f a -> normag = 0 ; 127

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g g memset line 0 MAXLINE ; memset key 0 MAXKEY ; memset val1 0 MAXKEY ; memset val2 0 MAXKEY ; memset val3 0 MAXKEY ; g output "============================================= nnnn ; /*Readnumberoftypes,twoscalingfactorforvdwradii*/ sprintf bu "READ:int%datomtypes nn a -> ntype ; output bu ; sprintf bu "READ:fittingpointswillfallbetweenfloat%.2fandfloat%.2f*VDWradii nn a -> scale1 a -> scale2 ; output bu ; /*Allocatememoryandreadvdwradii,elementname,andZvalueforeachatom*/ for i = 0 ; i < a -> ntype ; i ++ f sprintf bu "READ:elementsymbol%s nn a -> elem1 [ i ]; output bu ; g for i = 0 ; i < a -> ntype ; i ++ f sprintf bu "READ:elementZint%d nn a -> nzion1 [ i ]; output bu ; g /*Readkmaxforkspaceandnmax inforrealspaceewaldsummation*/ sprintf bu "READ:kmaxsetto:%d nn lrc -> kmax ; output bu ; sprintf bu "READ:gridthinning%s nn p -> scalegrid > 0 ? "active" : "off" ; output bu ; if p -> scalegrid f sprintf bu "READ:forevery1read,%dwillbeskipped nn p -> scalegrid ; output bu ; g /*Readthea->qconstranda->qtotal*/ sprintf bu "READ:constrainedminimization%s nn a -> qconstr ? "active" : "off" ; output bu ; if a -> qconstr f sprintf bu "READ:chargewillbeconstrainedto%d nn a -> qtotal ; output bu ; g /*Readthestringofewaldorwolfsummation*/ if strcasestr a -> ewaldorwolf "wolf" f a -> wolf ag = 1 ; a -> ewald ag = 0 ; output "READ:Wolflongrangesummationmethodselected nn ; g else if strcasestr a -> ewaldorwolf "ewald" f a -> wolf ag = 0 ; a -> ewald ag = 1 ; output "READ:Ewaldlongrangesummationmethodselected nn ; g else f output "READ:Error!LRCmethodnotrecognized nn ; exit 1 ; g 128

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/*Readthevalueofainconstraintofpenaltyfunction*/ sprintf bu "READ:penaltyfunctionfactor:%lf nn a -> aa ; output bu ; /*Readthenumberofatomswithdierentsymmetry*/ if a -> symm ag output "READ:symmetryactivated nn ; else output "READ:symmetrynotactivated nn ; output "READ:inputfileprocessed nnnn ; fclose inputle ; g #include #include #include #include #dene FALSE 0 #dene TRUE 1 #dene boolean int #include "chargefit.h" /* *--{-{-{-{-{-{-{-{-{-{-{-{-{-{-{-{-{-{-{-{-{*ReaddatafromLOCPOTle; *--{-{-{-{-{-{-{-{-{-{-{-{-{-{-{-{-{-{-{-{-{*/ void read locpot struct atoms a struct points p struct lsq l struct lrc lrc f char bu [ MAXLINE ]; char title [ MAXLINE ], dorc [ MAXLINE ]; /*somestringsinLOCPOT*/ int i j k ii ; /*somecounters*/ double scalefactor ; /*scalingfactor*/ /*openLOCPOTcheckforexistence*/ FILE fplocpot ; if fplocpot = fopen "LOCPOT" "r" == NULL problem with inputle "LOCPOT" ; fgets title 80 fplocpot ; /*Readscalefactor*/ fscanf fplocpot "%lf nn ,& scalefactor ; /*ReadlatticevectorsfromLOCPOT*/ for i = 0 ; i < 3 ; i ++ fscanf fplocpot "%lf%lf%lf nn & a -> tv [ i ][ 0 ],& a -> tv [ i ][ 1 ],& a -> tv [ i ][ 2 ]; /*Calculatelatticevectormultipliedwithscalfactor*/ for i = 0 ; i < 3 ; i ++ for j = 0 ; j < 3 ; j ++ a -> tv [ i ][ j ] *= scalefactor ; /*OUTPUTprintsthelatticevectors*/ output "LOCPOT:latticevectors nn ; 129

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output "======================================== nn ; for i = 0 ; i < 3 ; i ++ f sprintf bu "%lf nt %lf nt %lf nn a -> tv [ i ][ 0 ], a -> tv [ i ][ 1 ], a -> tv [ i ][ 2 ]; output bu ; g output "======================================== nnnn ; /*Readthenumberofatomsforeachtype*/ /*andcalculatethenumberofallatoms*/ a -> neachion = int calloc a -> ntype sizeof int ; a -> natoms = 0 ; for i = 0 ; i < a -> ntype ; i ++ f fscanf fplocpot "%d" ,& a -> neachion [ i ]; a -> natoms += a -> neachion [ i ]; g sprintf bu "LOCPOT:totalnumberofatoms:%d nn a -> natoms ; output bu ; /*Allocatememoryforarraybothoffracandcartcoordinates*/ a -> frac = double ** calloc a -> natoms sizeof double *; a -> cart = double ** calloc a -> natoms sizeof double *; for i = 0 ; i < a -> natoms ; i ++ f a -> frac [ i ] = double calloc 3 sizeof double ; a -> cart [ i ] = double calloc 3 sizeof double ; g /*Readthestringof"Direct"*/ fscanf fplocpot "%s nn dorc ; /*Readthefraccoordinatesofatoms*/ for i = 0 ; i < a -> natoms ; i ++ fscanf fplocpot "%lf%lf%lf" ,& a -> frac [ i ][ 0 ],& a -> frac [ i ][ 1 ],& a -> frac [ i ][ 2 ]; /*Calculatethecartcoordinatesofallatoms*/ for i = 0 ; i < a -> natoms ; i ++ for j = 0 ; j < 3 ; j ++ a -> cart [ i ][ j ] = a -> frac [ i ][ 0 ] a -> tv [ 0 ][ j ] + a -> frac [ i ][ 1 ] a -> tv [ 1 ][ j ] + a -> frac [ i ][ 2 ] a -> tv [ 2 ][ j ]; /*ReadNGXF,NGYF,NGZFvalues*/ fscanf fplocpot "%d%d%d nn ,& a -> ngxf ,& a -> ngyf ,& a -> ngzf ; sprintf bu "LOCPOT:GridNGXF,NGYF,NGZFare:%d%d%d nn a -> ngxf a -> ngyf a -> ngzf ; output bu ; /*AllocatememoryforESP,fragcoordinates,cartcoordinatesofgridpoints*/ p -> gridesp = double *** calloc a -> ngxf sizeof double **; for i = 0 ; i < a -> ngxf ; i ++ p -> gridesp [ i ] = double ** calloc a -> ngyf sizeof double *; for i = 0 ; i < a -> ngxf ; i ++ for j = 0 ; j < a -> ngyf ; j ++ p -> gridesp [ i ][ j ] = double calloc a -> ngzf sizeof double ; p -> gridfrac = double **** calloc a -> ngxf sizeof double ***; p -> gridcart = double **** calloc a -> ngxf sizeof double ***; for i = 0 ; i < a -> ngxf ; i ++ f p -> gridfrac [ i ] = double *** calloc a -> ngyf sizeof double **; p -> gridcart [ i ] = double *** calloc a -> ngyf sizeof double **; 130

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g for i = 0 ; i < a -> ngxf ; i ++ f for j = 0 ; j < a -> ngyf ; j ++ f p -> gridfrac [ i ][ j ] = double ** calloc a -> ngzf sizeof double *; p -> gridcart [ i ][ j ] = double ** calloc a -> ngzf sizeof double *; g g for i = 0 ; i < a -> ngxf ; i ++ f for j = 0 ; j < a -> ngyf ; j ++ f for k = 0 ; k < a -> ngzf ; k ++ f p -> gridfrac [ i ][ j ][ k ] = double calloc 3 sizeof double ; p -> gridcart [ i ][ j ][ k ] = double calloc 3 sizeof double ; g g g /*ReadESPvalueforeachgridpoint*/ for k = 0 ; k < a -> ngzf ; k ++ for j = 0 ; j < a -> ngyf ; j ++ for i = 0 ; i < a -> ngxf ; i ++ fscanf fplocpot "%lf" ,& p -> gridesp [ i ][ j ][ k ]; fclose fplocpot ; output "LOCPOT:readin nn ; g /* *--{-{-{-{-{-{-{-{-{-{-{-{-{-{-{-{-{-{-{-{-{-*allocatememoryandsetthevaluesofsomeparameters *--{-{-{-{-{-{-{-{-{-{-{-{-{-{-{-{-{-{-{-{-{-*/ void allocate param struct atoms a struct points p f int i j k ii ; int r c ; char bu [ MAXLINE ]; /*Calculatethefraccoordinatesofeachgridpoint*/ for k = 0 ; k < a -> ngzf ; k ++ f for j = 0 ; j < a -> ngyf ; j ++ f for i = 0 ; i < a -> ngxf ; i ++ f p -> gridfrac [ i ][ j ][ k ][ 0 ] = double i / double a -> ngxf ; p -> gridfrac [ i ][ j ][ k ][ 1 ] = double j / double a -> ngyf ; p -> gridfrac [ i ][ j ][ k ][ 2 ] = double k / double a -> ngzf ; g g g /*Calculatecartcoordinatsofgridpoints*/ /*Gridpointsstartfrom,0,0,,0,0......*/ for k = 0 ; k < a -> ngzf ; k ++ f for j = 0 ; j < a -> ngyf ; j ++ f for i = 0 ; i < a -> ngxf ; i ++ f for c = 0 ; c < 3 ; c ++ f for r = 0 p -> gridcart [ i ][ j ][ k ][ c ]= 0 ; r < 3 ; r ++ f p -> gridcart [ i ][ j ][ k ][ c ] += p -> gridfrac [ i ][ j ][ k ][ r ] a -> tv [ r ][ c ]; 131

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g g g g g /*Allocatememory*/ /*AssigntheVDWradii,elementname,andZforeachatom*/ a -> vdw = double calloc a -> natoms sizeof double ; a -> elem = char ** calloc a -> natoms sizeof char *; for i = 0 ; i < a -> natoms ; i ++ a -> elem [ i ]= char calloc 3 sizeof char ; a -> z = int calloc a -> natoms sizeof int ; a -> minvdw = double calloc a -> natoms sizeof double ; a -> maxvdw = double calloc a -> natoms sizeof double ; /*assigna->z*/ for k = 0 ii = 0 ; k < a -> ntype ; k ++ f for j = 0 ; j < a -> neachion [ k ]; j ++, ii ++ f a -> z [ ii ] = a -> nzion1 [ k ]; strcpy a -> elem [ ii ], a -> elem1 [ k ]; g g output nn ============================================ nn ; assign vdw a ; output "VDW:VDWradiiAng,atom,Zforeachatom: nn ; for i = 0 ; i < a -> natoms ; i ++ f sprintf bu "%lf nt %s nt %d nn a -> vdw [ i ], a -> elem [ i ], a -> z [ i ]; output bu ; g output "============================================ nnnn ; /*geta->minvdw,a->maxvdw,a->elem,anda->vdw;*/ for i = 0 ; i < a -> natoms ; i ++ f a -> minvdw [ i ] = a -> scale1 a -> vdw [ i ]; a -> maxvdw [ i ] = a -> scale2 a -> vdw [ i ]; g g /* *--{-{-{-{-{-{-{-{-{-{-{-{-{-{-{-{-{-{-{-{-{-*ThisroutineselectthepointsfallbetweentwoVDWshells. *Periodicboundaryconditionsareenforced. *--{-{-{-{-{-{-{-{-{-{-{-{-{-{-{-{-{-{-{-{-{-*/ void select points shells struct atoms a struct points p struct lsq l f int i j k ii ; int nn ; char bu [ MAXLINE ]; double detdis ; /*distancebetweenpointandatom*/ double accumulator esp = 0.0 ; /*accumulatorofespofselectedpoints*/ boolean test = TRUE ; /*ag*/ /*initializememoryforcoordinatesofselectedpoints.*/ p -> frac = malloc sizeof double *; p -> frac [ 0 ] = malloc 3 sizeof double ; 132

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p -> cart = malloc sizeof double *; p -> cart [ 0 ] = malloc 3 sizeof double ; /*initializeallocationofmemoryforp->esp*/ p -> esp = malloc sizeof double ; /*Allocatememoryfordistances*/ detdis = calloc a -> natoms sizeof double ; /*Starttoloopgridpoints*/ nn = 0 ; /*npointscounter*/ for k = 0 ; k < a -> ngzf ; k ++ f for j = 0 ; j < a -> ngyf ; j ++ f for i = 0 ; i < a -> ngxf ; i ++ f /*minimumimageconventionisusedhere!*/ /*Eachatomwillbedeterminedtohaveanimageornot.*/ for ii = 0 ; ii < a -> natoms ; ii ++ f /*fragcoordinatesdistancein3directions*/ detdis [ ii ] = minimum image p -> gridfrac [ i ][ j ][ k ], a -> frac [ ii ], a ; if detdis [ ii ] > a -> minvdw [ ii ] test = TRUE ; else f test = FALSE ; break ; g g if test == TRUE f for ii = 0 ; ii < a -> natoms ; ii ++ f #ifdef PLUS28 if detdis [ ii ] < a -> vdw [ ii ] + 2.8 f #else if detdis [ ii ] < a -> maxvdw [ ii ] f #endif /*thisisapointwewant,storingtheinfo*/ p -> cart [ nn ][ 0 ] = p -> gridcart [ i ][ j ][ k ][ 0 ]; p -> cart [ nn ][ 1 ] = p -> gridcart [ i ][ j ][ k ][ 1 ]; p -> cart [ nn ][ 2 ] = p -> gridcart [ i ][ j ][ k ][ 2 ]; p -> frac [ nn ][ 0 ] = p -> gridfrac [ i ][ j ][ k ][ 0 ]; p -> frac [ nn ][ 1 ] = p -> gridfrac [ i ][ j ][ k ][ 1 ]; p -> frac [ nn ][ 2 ] = p -> gridfrac [ i ][ j ][ k ][ 2 ]; p -> esp [ nn ] = p -> gridesp [ i ][ j ][ k ]; accumulator esp += p -> esp [ nn ]; /*thisishowmanywehavesofar*/ nn += 1 ; /*makeroomforthenextpoint*/ p -> cart = realloc p -> cart nn + 1 sizeof double *; p -> cart [ nn ] = malloc 3 sizeof double ; p -> frac = realloc p -> frac nn + 1 sizeof double *; p -> frac [ nn ] = malloc 3 sizeof double ; p -> esp = realloc p -> esp nn + 1 sizeof double ; /*pointconrmedgood*/ break ; g g g i += p -> scalegrid ; /*thinningifactive*/ g j += p -> scalegrid ; /*thinningifactive*/ g 133

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k += p -> scalegrid ; /*thinningifactive*/ g p -> npoints = nn ; sprintf bu "LOCPOT:numberofselectedpoints:%d nn p -> npoints ; output bu ; /*averagevalueofselectedpoints*/ p -> esp average = accumulator esp / p -> npoints ; /*specifythedimensionofAmatrix*/ l -> m = p -> npoints ; l -> n = a -> natoms ; if a -> qconstr l -> n ++; l -> b = calloc l -> n + 1 sizeof double ; free p -> gridesp ; free detdis ; g /* *--{-{-{-{-{-{-{-{-{-{-{-{-{-{-{-{-{-{-{-{-{-*allocatememoryforsvdrelatedparameters *--{-{-{-{-{-{-{-{-{-{-{-{-{-{-{-{-{-{-{-{-{-*/ void allocate svd related struct lsq l f int i ; /*Allocateaandw*/ l -> a = double calloc l -> n + 1 sizeof double ; l -> w = double calloc l -> n + 1 sizeof double ; /*AllocateA-n+1Xn+1*/ l -> A = double ** calloc l -> n + 1 sizeof double *; for i = 0 ; i <= l -> n ; i ++ l -> A [ i ]= double calloc l -> n + 1 sizeof double ; l -> V = double ** calloc l -> n + 1 sizeof double *; for i = 0 ; i <= l -> n ; i ++ l -> V [ i ]= double calloc l -> n + 1 sizeof double ; g /* *================================================================== *ThisroutinewillreadthedatafromLOCPOTlefromVASP,such *asscalefactor,latticevectors,fractionalcoordinatesforeach *atoms,NGXF,NGYF,NGZF,aswellastheelectrostaticpotential *ESPatthegridpointsfromVASPcalculations. *Also,thegridpointswithinarangeofVDWradiiwillbe *selectedinthisroutinefromminvdwtomaxvdw. *PBCconditionisused. *Otherparameterssuchasrcut,alpha...arecalculated. *=================================================================== 134

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*/ void initialization struct atoms a struct points p struct lsq l struct lrc lrc FILE inputle f read inputle a p lrc inputle ; /*readtheparametersfrominputle*/ read locpot a p l lrc ; /*readlocpot*/ allocate param a p ; /*setthevaluesforsomeparameters*/ select points shells a p l ; /*selectpointsfallbetweentwoshells*/ free p -> gridfrac ; free p -> gridcart ; outesp a p l ; /*outputespofselectedpoints*/ calc recip a ; /*calculatereciprocalbasisandvolume*/ calc rcut a lrc ; /*calculatetherealcut*/ handle nmax or alpha a lrc ; /*setalpha,ornmaxdepending*/ allocate svd related l ; /*allocatememoryforsvdrelatedparameters*/ print2pdb a p ; /*printthecoordinatesofpointsandatomintoapdble*/ g #include double compute one element A int atom m int atom j struct atoms a struct points p struct lsq l struct lrc lrc f int i j ; /*dummy*/ int point i ; double total = 0 ; /*accumulatorofl->A*/ double total b = 0 total b1 = 0 ; /*accumulatorofl->B*/ double rij ; /*distancebetweenpointandatom*/ for point i = 0 ; point i < p -> npoints ; point i ++ f /*realpartofl->b*/ total b += l -> des [ point i ][ atom m ]* p -> esp [ point i ] p -> esp average ; /*realpartofl->A*/ total += l -> des [ point i ][ atom m ] l -> des [ point i ][ atom j ]; g /*updatethebvector*/ l -> b [ atom m ] = total b ; /*Conversionfactor*/ return 14.3996 total ; g /*Thisroutinerecalculatethel->desbecauseoftheoset*/ void new des struct atoms a struct points p struct lsq l f int i j ; l -> des average temp = double calloc a -> natoms sizeof double ; /*accumulatorforl->des average*/ l -> des average = double calloc a -> natoms sizeof double ; for j = 0 ; j < a -> natoms ; j ++ f l -> des average temp [ j ] = 0.0 ; for i = 0 ; i < p -> npoints ; i ++ f 135

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l -> des average temp [ j ] += l -> des [ i ][ j ]; g g /*averagevalueofl->des*/ for j = 0 ; j < a -> natoms ; j ++ f l -> des average [ j ] = l -> des average temp [ j ]/ p -> npoints ; g /*respecifynewvalueofl->des*/ for j = 0 ; j < a -> natoms ; j ++ f for i = 0 ; i < p -> npoints ; i ++ f l -> des [ i ][ j ] -= l -> des average [ j ]; g g g void buildAmatrix struct atoms a struct points p struct lsq l struct lrc lrc f int atom j ; /*thenumberofrowsofAmatrix*/ int atom m ; /*thenumberofcolumnsofAmatrix*/ void new des struct atoms a struct points p struct lsq l ; new des a p l ; output "Condensingdesignmatrixtofinalformbeforediagonalization nn ; timestamp a ; /*formatrixelementsatom j>=atom m*/ for atom m = 0 ; atom m < a -> natoms ; atom m ++ f for atom j = atom m ; atom j < a -> natoms ; atom j ++ f l -> A [ atom m ][ atom j ]= compute one element A atom m atom j a p l lrc ; g g /*formatrixelementsatom j natoms ; atom m ++ f for atom j = 0 ; atom j < atom m ; atom j ++ f l -> A [ atom m ][ atom j ] = l -> A [ atom j ][ atom m ]; g g g void build all matrices struct atoms a struct points p struct lsq l struct lrc lrc f int i ; int total ; char bu [ MAXLINE ]; output nn ---begincomputationofdesignmatrix--nnnn ; compute design matrix a p l lrc ; buildAmatrix a p l lrc ; /*addthepenaltyfunctionoftargetcharge*/ if a -> aa != 0.0 targetcharge a l ; /*implementtheconstraintofsymmetry*/ if a -> symm ag f 136

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output "Symmetryenabled,compressingmatrix nn ; read symm a l ; symm constr a l ; g else output "skippingsymmetryoptimization nn ; /*totalnumberofelementsinmatrix*/ total = l -> n l -> n ; sprintf bu "Finalmatrixhas%delements nn total ; output bu ; output "Readytodiagonalize nn ; g #include "chargefit.h" void compute r double r [ 3 ], int point i int atom j struct atoms a struct points p f int s ; /*nominimumimaging-rawdistancevector*/ for s = 0 ; s < 3 ; s ++ r [ s ] = a -> cart [ atom j ][ s ] p -> cart [ point i ][ s ]; g double minimum image double f1 double f2 struct atoms a f double d [ 3 ], di [ 3 ], img [ 3 ]; double c [ 3 ]; /*3vectorincartesians*/ int i j ; double distance ; for i = 0 ; i < 3 ; i ++ c [ i ] = f1 [ i ] f2 [ i ]; for i = 0 ; i < 3 ; i ++ for j = 0 d [ i ]= 0 ; j < 3 ; j ++ d [ i ] += c [ j ] a -> tv [ j ][ i ]; /*matrixmultiplywiththeinversebasisandround*/ /*ThismultiplicationassumesBASISVECTORSAREROWS*/ for i = 0 ; i < 3 ; i ++ img [ i ] = rint c [ i ]; /*matrixmultiplytoprojectbackintoourbasis*/ /*thisassumesROWFORMbasisvectors*/ for i = 0 ; i < 3 ; i ++ for j = 0 di [ i ]= 0 ; j < 3 ; j ++ di [ i ] += img [ j ] a -> tv [ j ][ i ]; /*nowcorrectthedisplacement*/ for i = 0 ; i < 3 ; i ++ di [ i ] = d [ i ] di [ i ]; distance = sqrt di [ 0 ]* di [ 0 ] + di [ 1 ]* di [ 1 ] + di [ 2 ]* di [ 2 ]; return distance ; g 137

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/*Thiscomputesintkcdotreciprocal basis.Theresultisstored *inkdotrecip[3],whichispassedin. */ void compute kdotrecip double kdotrecip int k [ 3 ], struct atoms a f int p q ; /*dummies*/ double result = 0 ; /*dotintkwithrecipbasis,theresultisstoredinkdotrforusecomputingksq*/ for p = 0 ; p < 3 ; p ++ for q = 0 kdotrecip [ p ]= 0 ; q < 3 ; q ++ kdotrecip [ p ] += k [ q ] a -> rec tv [ q ][ p ]; g /*thistakesthecompletekvector2*PI*ncdotrbas *anddotswithr */ double compute kdotr double kdotrecip [ 3 ], double r [ 3 ] f int p ; double result = 0 ; /*dotthevectorwithr*/ for p = 0 ; p < 3 ; p ++ result += kdotrecip [ p ] r [ p ]; return result ; g double compute ksq double kdotrecip [ 3 ] f return kdotrecip [ 0 ]* kdotrecip [ 0 ] + kdotrecip [ 1 ]* kdotrecip [ 1 ] + kdotrecip [ 2 ]* kdotrecip [ 2 ] ; g double real distance int n [ 3 ], int atom j int point i struct atoms a struct points p f double r [ 3 ]; int i j ; /*projectn[]intobasis*/ for i = 0 ; i < 3 ; i ++ for j = 0 r [ i ]= 0 ; j < 3 ; j ++ r [ i ] += n [ j ] a -> tv [ j ][ i ]; /*addthesame-cellvectortotheunitcellvectors*/ for i = 0 ; i < 3 ; i ++ r [ i ] += a -> cart [ atom j ][ i ] p -> cart [ point i ][ i ]; return sqrt r [ 0 ]* r [ 0 ] + r [ 1 ]* r [ 1 ] + r [ 2 ]* r [ 2 ] ; g void compute wave vectors struct atoms a struct points p struct lsq l struct lrc lrc f int i nkvecs ; int wv index = 0 ; int k [ 3 ]; double kdotrecip [ 3 ], ksq ; nkvecs = pow 2 lrc -> kmax + 1 3 ; 138

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/*setupwave-vectorsinanarray.ByDoingthis, *weMUSTiterateoverthekvectorsinthesamewayhere *aswedointheewaldloopwhenwelookupthesevalues!!!! */ lrc -> wv = double malloc pow 2 lrc -> kmax + 1 3 sizeof double ; lrc -> kvec = double ** malloc nkvecs sizeof double *; for i = 0 ; i < nkvecs ; i ++ lrc -> kvec [ i ]= malloc 3 sizeof double ; for k [ 0 ]=lrc -> kmax ; k [ 0 ]<= lrc -> kmax ; k [ 0 ]++ f for k [ 1 ]=lrc -> kmax ; k [ 1 ]<= lrc -> kmax ; k [ 1 ]++ f for k [ 2 ]=lrc -> kmax ; k [ 2 ]<= lrc -> kmax ; k [ 2 ]++ f if k [ 0 ] jj k [ 1 ] jj k [ 2 ] f /*computekdotrecip*/ compute kdotrecip lrc -> kvec [ wv index ], k a ; /*computeksq=jkj^2fromkdotrecip[]*/ ksq = compute ksq lrc -> kvec [ wv index ]; /*thisisthepre-computedwavevector*/ lrc -> wv [ wv index ++] = 1 / ksq exp ksq / 4 / lrc -> alpha ; g g g g g double one design element ewald int point i int atom j struct atoms a struct points p struct lrc lrc struct lsq l f double rij ; /*distancebetweenpointandatom*/ int n [ 3 ]; /*loopovertherealspacecells*/ int k [ 3 ]; /*integervector,k*/ int wv index = 0 ; double ksq ; /*jkj^2,thisisintkdotreciprocal basis*/ double r [ 3 ]; /*vectorcontainingnon-minimumimagecomponentsrij*/ double kdotrecip [ 3 ]; /*vectorthatwillcontainkdotrecipbasisdotrij*/ double kdotr ; /*scalarresultofkdotrectvdotrij*/ double real = 0 recip = 0 total ; /*real-spaceewaldpart*/ for n [ 0 ]=lrc -> nmax [ 0 ]; n [ 0 ]<= lrc -> nmax [ 0 ]; n [ 0 ]++ f for n [ 1 ]=lrc -> nmax [ 1 ]; n [ 1 ]<= lrc -> nmax [ 1 ]; n [ 1 ]++ f for n [ 2 ]=lrc -> nmax [ 2 ]; n [ 2 ]<= lrc -> nmax [ 2 ]; n [ 2 ]++ f rij = real distance n atom j point i a p ; real += 1 / rij erfc sqrt lrc -> alpha rij ; g g g /*reciprocal-spaceewald*/ /*computevectorroutsidetheKloop*/ compute r r point i atom j a p ; for k [ 0 ]=lrc -> kmax ; k [ 0 ]<= lrc -> kmax ; k [ 0 ]++ f for k [ 1 ]=lrc -> kmax ; k [ 1 ]<= lrc -> kmax ; k [ 1 ]++ f for k [ 2 ]=lrc -> kmax ; k [ 2 ]<= lrc -> kmax ; k [ 2 ]++ f if k [ 0 ] jj k [ 1 ] jj k [ 2 ] f 139

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/*computekdotrfrompre-computedkvector*/ kdotr = compute kdotr lrc -> kvec [ wv index ], r ; /*thisisusingthepre-computedwavevector*/ recip += lrc -> wv [ wv index ++] cos kdotr ; g g g g recip *= 4 M PI / a -> cell volume ; total = real + recip ; return total ; g #include "chargefit.h" double one design element wolf int point i int atom j struct atoms a struct points p struct lrc lrc struct lsq l f int i j k ; int n [ 3 ]; /*loopoverrealspacecells*/ double rij ; /*distancebetweenpointandatom*/ double wolfcon1 wolfcon2 ; /*twoconstantofWolfsum*/ double result = 0 ; double temp = 0 ; lrc -> wolfrcuto = lrc -> nmaxf ; /*changealphatosqrtalpha*/ /*squareofcutoinWolf*/ lrc -> wolfrcutosq = lrc -> wolfrcuto lrc -> wolfrcuto ; /*calculatetwoconstantsinWolf*/ wolfcon1 = erfc sqrt lrc -> alpha lrc -> wolfrcuto / lrc -> wolfrcuto ; wolfcon2 = erfc sqrt lrc -> alpha lrc -> wolfrcuto / lrc -> wolfrcutosq + 2.0 sqrt lrc -> alpha exp sqrt lrc -> alpha lrc -> wolfrcuto sqrt lrc -> alpha lrc -> wolfrcuto / sqrt M PI lrc -> wolfrcuto ; /*calcualtethepotentailbetweenpointandatomincenterunitcell*/ /*calculatethepotentialarraybetweenpointandatomsinallofunitcells*/ for n [ 0 ]=lrc -> nmax [ 0 ]; n [ 0 ]<= lrc -> nmax [ 0 ]; n [ 0 ]++ f for n [ 1 ]=lrc -> nmax [ 1 ]; n [ 1 ]<= lrc -> nmax [ 1 ]; n [ 1 ]++ f for n [ 2 ]=lrc -> nmax [ 2 ]; n [ 2 ]<= lrc -> nmax [ 2 ]; n [ 2 ]++ f rij = real distance n atom j point i a p ; /*getthedistance*/ if rij <= lrc -> wolfrcuto f /*decidewhetherinsideofrcuto*/ /*thenaccumulatethevalues*/ result += erfc sqrt lrc -> alpha rij / rij + wolfcon1 + wolfcon2 rij lrc -> wolfrcuto ; g g g g 140

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return result ; g #include void constrain struct atoms a struct lsq l f if a -> qconstr f int m n ; output "Settingupconstraints nn ; if a -> symm ag f n = l -> n 1 ; for m = 0 ; m < a -> nelem symm ; m ++ f l -> A [ m ][ n ] = a -> nelem symm natoms [ m ]; g m = l -> n 1 ; for n = 0 ; n < a -> nelem symm ; n ++ f l -> A [ m ][ n ] = a -> nelem symm natoms [ n ]; g l -> A [ l -> n 1 ][ l -> n 1 ] = 0 ; l -> b [ l -> n 1 ] = a -> qtotal ; g else f m = l -> n 1 ; for n = 0 ; n < l -> n 1 ; n ++ l -> A [ m ][ n ]= 1 ; n = l -> n 1 ; for m = 0 ; m < l -> n 1 ; m ++ l -> A [ m ][ n ]= 1 ; l -> A [ l -> n 1 ][ l -> n 1 ] = 0 ; l -> b [ l -> n 1 ] = a -> qtotal ; g g g void sum charges struct atoms a struct points p struct lsq l struct lrc lrc f int i ; char bu [ MAXLINE ]; l -> sum = 0.0 ; if a -> symm ag f for i = 0 ; i < a -> nelem symm ; i ++ l -> sum += l -> a [ i + 1 ]* a -> nelem symm natoms [ i ]; g else f for i = 0 ; i < a -> natoms ; i ++ l -> sum += l -> a [ i + 1 ]; g sprintf bu "sumofcharges:%lf nn l -> sum ; output bu ; g void rstfew struct atoms a struct points p struct lsq l struct lrc lrc f int atom i atom j ; int i ; 141

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char bu [ MAXLINE ]; output "firstfewelementsofthedesignmatrix: nn ; output "12345678.... nn ; for atom i = 0 ; atom i < 10 && atom i < l -> n ; atom i ++ f sprintf bu "%d" atom i + 1 ; output bu ; for atom j = 0 ; atom j < 10 && atom j < l -> n ; atom j ++ f sprintf bu "%8.2f" l -> A [ atom i ][ atom j ]; output bu ; g output nn ; g output "... nnnn ; /*outputtheb-vector fori=0;in;i++f sprintfbu,"l->b[%d]%18.6fnn",i+1,l->b[i]; outputbu; g*/ g #include void compute design matrix struct atoms a struct points p struct lsq l struct lrc lrc f int i j n ; int index = 0 ; int total = p -> npoints a -> natoms ; #ifdef MPI /*MPIrelatedvariables*/ int who computes ; int nworkers ; int mesg size mesg index = 0 ; double result ; nworkers = ntasks 1 ; mesg size = int ceil a -> natoms p -> npoints / double nworkers ; result = double calloc mesg size sizeof double ; #endif /*ifEwald,pre-computewavevectorsonce*/ if a -> ewald ag compute wave vectors a p l lrc ; /*allocatenpointsXnatomsstoragematrix-wereallyshoulddothiselsewhere*/ l -> des = double ** calloc p -> npoints sizeof double *; for i = 0 ; i < p -> npoints ; i ++ l -> des [ i ] = double calloc a -> natoms sizeof double ; /*loopsoverallelementsoftheMxNdesignmatrix*/ for i = 0 ; i < p -> npoints ; i ++ f for j = 0 ; j < a -> natoms ; j ++ f #ifdef MPI /*BEGINMPI*/ if myrank f /*workers*/ if index % nworkers myrank + 1 f /*isitmyturn?*/ /*Ewald*/ if a -> ewald ag result [ mesg index ++]= one design element ewald i j a p lrc l ; /*Wolf*/ 142

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else if a -> wolf ag result [ mesg index ++]= one design element wolf i j a p lrc l ; /*Error*/ else f output "ERRORINLRCFLAG! nn ; exit 1 ; g g g #else /*ENDMPI,BEGINSERIALCODE*/ if a -> ewald ag l -> des [ i ][ j ]= one design element ewald i j a p lrc l ; else if a -> wolf ag l -> des [ i ][ j ]= one design element wolf i j a p lrc l ; else f output "ERRORINLRCFLAG! nn ; exit 1 ; g #endif index ++; performance a index total ; g g #ifdef MPI /*BEGINMPI*/ output nn Collectingmatrixelementsfromallprocesses nn ; timestamp a ; if myrank == 0 f /*loopsoverallelementsoftheMxNdesignmatrix*/ for n = 1 ; n <= nworkers ; n ++ f MPI Recv result mesg size MPI DOUBLE n TAG R MPI COMM WORLD & status ; index = 0 ; mesg index = 0 ; for i = 0 ; i < p -> npoints ; i ++ f for j = 0 ; j < a -> natoms ; j ++ f if index % nworkers n + 1 l -> des [ i ][ j ]= result [ mesg index ++]; index ++; g g g g else MPI Send result mesg size MPI DOUBLE 0 TAG R MPI COMM WORLD ; output "Allinterprocesscommunicationiscomplete nn ; output "CallingMPIfinalizeonallworkernodes nn ; timestamp a ; #ifdef MPI /*onlytheMasterneedstostayalive*/ if myrank f MPI Finalize ; exit 0 ; g #endif #endif g #include 143

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void shiftdata struct lsq l f int m n ; /*shiftallthedatainAfrom[0,0]to[1,1]*/ for m = l -> n 1 ; m >= 0 ; m -for n = 0 ; n < l -> n ; n ++ l -> A [ m + 1 ][ n + 1 ]= l -> A [ m ][ n ]; /*shiftbvectorfrom0->1*/ for n = l -> n 1 ; n >= 0 ; n -l -> b [ n + 1 ]= l -> b [ n ]; g void solvebySVD struct atoms a struct points p struct lsq l struct lrc lrc f int n ; char bu [ MAXLINE ]; #ifdef MPI /*ifweareusingMPIandarenottherootnode,thenreturn*/ if myrank != 0 return ; #endif /*ifweareusingnumericalrecipes,thenwehavetoshifttheAmatrix *sothatit'sindicesrunfrom1-mand1-nratherthan0-m-1likeinC */ shiftdata l ; /*callSVD*/ output "Beginningmatrixsingularvaluedecomposition nn ; timestamp a ; svdcmp l -> A l -> n l -> n l -> w l -> V ; output "Singularvaluedecompositioncomplete nn ; timestamp a ; /*callSVDbacksub*/ svbksb l -> A l -> w l -> V l -> n l -> n l -> b l -> a ; g #if dened STDC jj dened ANSI jj dened NRANSI /*ANSI*/ #include #include #include #dene NR END 1 #dene FREE ARG char void nrerror char error text [] /*NumericalRecipesstandarderrorhandler*/ f fprintf stderr "NumericalRecipesrun-timeerror... nn ; fprintf stderr "%s nn error text ; fprintf stderr "...nowexitingtosystem... nn ; exit 1 ; 144

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g double vector long nl long nh /*allocateadoublevectorwithsubscriptrangev[nl..nh]*/ f double v ; v = double malloc size t nh nl + 1 + NR END sizeof double ; if v nrerror "allocationfailureinvector" ; return v nl + NR END ; g int ivector long nl long nh /*allocateanintvectorwithsubscriptrangev[nl..nh]*/ f int v ; v = int malloc size t nh nl + 1 + NR END sizeof int ; if v nrerror "allocationfailureinivector" ; return v nl + NR END ; g unsigned char cvector long nl long nh /*allocateanunsignedcharvectorwithsubscriptrangev[nl..nh]*/ f unsigned char v ; v = unsigned char malloc size t nh nl + 1 + NR END sizeof unsigned char ; if v nrerror "allocationfailureincvector" ; return v nl + NR END ; g unsigned long lvector long nl long nh /*allocateanunsignedlongvectorwithsubscriptrangev[nl..nh]*/ f unsigned long v ; v = unsigned long malloc size t nh nl + 1 + NR END sizeof long ; if v nrerror "allocationfailureinlvector" ; return v nl + NR END ; g double dvector long nl long nh /*allocateadoublevectorwithsubscriptrangev[nl..nh]*/ f double v ; v = double malloc size t nh nl + 1 + NR END sizeof double ; if v nrerror "allocationfailureindvector" ; return v nl + NR END ; g double ** matrix long nrl long nrh long ncl long nch /*allocateadoublematrixwithsubscriptrangem[nrl..nrh][ncl..nch]*/ f long i nrow = nrh nrl + 1 ncol = nch ncl + 1 ; double ** m ; /*allocatepointerstorows*/ m = double ** malloc size t nrow + NR END sizeof double *; 145

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if m nrerror "allocationfailure1inmatrix" ; m += NR END ; m -= nrl ; /*allocaterowsandsetpointerstothem*/ m [ nrl ]= double malloc size t nrow ncol + NR END sizeof double ; if m [ nrl ] nrerror "allocationfailure2inmatrix" ; m [ nrl ] += NR END ; m [ nrl ] -= ncl ; for i = nrl + 1 ; i <= nrh ; i ++ m [ i ]= m [ i 1 ]+ ncol ; /*returnpointertoarrayofpointerstorows*/ return m ; g double ** dmatrix long nrl long nrh long ncl long nch /*allocateadoublematrixwithsubscriptrangem[nrl..nrh][ncl..nch]*/ f long i nrow = nrh nrl + 1 ncol = nch ncl + 1 ; double ** m ; /*allocatepointerstorows*/ m = double ** malloc size t nrow + NR END sizeof double *; if m nrerror "allocationfailure1inmatrix" ; m += NR END ; m -= nrl ; /*allocaterowsandsetpointerstothem*/ m [ nrl ]= double malloc size t nrow ncol + NR END sizeof double ; if m [ nrl ] nrerror "allocationfailure2inmatrix" ; m [ nrl ] += NR END ; m [ nrl ] -= ncl ; for i = nrl + 1 ; i <= nrh ; i ++ m [ i ]= m [ i 1 ]+ ncol ; /*returnpointertoarrayofpointerstorows*/ return m ; g int ** imatrix long nrl long nrh long ncl long nch /*allocateaintmatrixwithsubscriptrangem[nrl..nrh][ncl..nch]*/ f long i nrow = nrh nrl + 1 ncol = nch ncl + 1 ; int ** m ; /*allocatepointerstorows*/ m = int ** malloc size t nrow + NR END sizeof int *; if m nrerror "allocationfailure1inmatrix" ; m += NR END ; m -= nrl ; /*allocaterowsandsetpointerstothem*/ m [ nrl ]= int malloc size t nrow ncol + NR END sizeof int ; if m [ nrl ] nrerror "allocationfailure2inmatrix" ; m [ nrl ] += NR END ; m [ nrl ] -= ncl ; for i = nrl + 1 ; i <= nrh ; i ++ m [ i ]= m [ i 1 ]+ ncol ; 146

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/*returnpointertoarrayofpointerstorows*/ return m ; g double ** submatrix double ** a long oldrl long oldrh long oldcl long oldch long newrl long newcl /*pointasubmatrix[newrl..][newcl..]toa[oldrl..oldrh][oldcl..oldch]*/ f long i j nrow = oldrh oldrl + 1 ncol = oldcl newcl ; double ** m ; /*allocatearrayofpointerstorows*/ m = double ** malloc size t nrow + NR END sizeof double *; if m nrerror "allocationfailureinsubmatrix" ; m += NR END ; m -= newrl ; /*setpointerstorows*/ for i = oldrl j = newrl ; i <= oldrh ; i ++, j ++ m [ j ]= a [ i ]+ ncol ; /*returnpointertoarrayofpointerstorows*/ return m ; g double ** convert matrix double a long nrl long nrh long ncl long nch /*allocateadoublematrixm[nrl..nrh][ncl..nch]thatpointstothematrix declaredinthestandardCmannerasa[nrow][ncol],wherenrow=nrh-nrl+1 andncol=nch-ncl+1.Theroutineshouldbecalledwiththeaddress &a[0][0]astherstargument.*/ f long i j nrow = nrh nrl + 1 ncol = nch ncl + 1 ; double ** m ; /*allocatepointerstorows*/ m = double ** malloc size t nrow + NR END sizeof double *; if m nrerror "allocationfailureinconvert matrix" ; m += NR END ; m -= nrl ; /*setpointerstorows*/ m [ nrl ]= a ncl ; for i = 1 j = nrl + 1 ; i < nrow ; i ++, j ++ m [ j ]= m [ j 1 ]+ ncol ; /*returnpointertoarrayofpointerstorows*/ return m ; g double *** f3tensor long nrl long nrh long ncl long nch long ndl long ndh /*allocateadouble3tensorwithranget[nrl..nrh][ncl..nch][ndl..ndh]*/ f long i j nrow = nrh nrl + 1 ncol = nch ncl + 1 ndep = ndh ndl + 1 ; double *** t ; /*allocatepointerstopointerstorows*/ t = double *** malloc size t nrow + NR END sizeof double **; if t nrerror "allocationfailure1inf3tensor" ; t += NR END ; t -= nrl ; /*allocatepointerstorowsandsetpointerstothem*/ 147

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t [ nrl ]= double ** malloc size t nrow ncol + NR END sizeof double *; if t [ nrl ] nrerror "allocationfailure2inf3tensor" ; t [ nrl ] += NR END ; t [ nrl ] -= ncl ; /*allocaterowsandsetpointerstothem*/ t [ nrl ][ ncl ]= double malloc size t nrow ncol ndep + NR END sizeof double ; if t [ nrl ][ ncl ] nrerror "allocationfailure3inf3tensor" ; t [ nrl ][ ncl ] += NR END ; t [ nrl ][ ncl ] -= ndl ; for j = ncl + 1 ; j <= nch ; j ++ t [ nrl ][ j ]= t [ nrl ][ j 1 ]+ ndep ; for i = nrl + 1 ; i <= nrh ; i ++ f t [ i ]= t [ i 1 ]+ ncol ; t [ i ][ ncl ]= t [ i 1 ][ ncl ]+ ncol ndep ; for j = ncl + 1 ; j <= nch ; j ++ t [ i ][ j ]= t [ i ][ j 1 ]+ ndep ; g /*returnpointertoarrayofpointerstorows*/ return t ; g void free vector double v long nl long nh /*freeadoublevectorallocatedwithvector*/ f free FREE ARG v + nl NR END ; g void free ivector int v long nl long nh /*freeanintvectorallocatedwithivector*/ f free FREE ARG v + nl NR END ; g void free cvector unsigned char v long nl long nh /*freeanunsignedcharvectorallocatedwithcvector*/ f free FREE ARG v + nl NR END ; g void free lvector unsigned long v long nl long nh /*freeanunsignedlongvectorallocatedwithlvector*/ f free FREE ARG v + nl NR END ; g void free dvector double v long nl long nh /*freeadoublevectorallocatedwithdvector*/ f free FREE ARG v + nl NR END ; g void free matrix double ** m long nrl long nrh long ncl long nch /*freeadoublematrixallocatedbymatrix*/ f free FREE ARG m [ nrl ]+ ncl NR END ; free FREE ARG m + nrl NR END ; g void free dmatrix double ** m long nrl long nrh long ncl long nch 148

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/*freeadoublematrixallocatedbydmatrix*/ f free FREE ARG m [ nrl ]+ ncl NR END ; free FREE ARG m + nrl NR END ; g void free imatrix int ** m long nrl long nrh long ncl long nch /*freeanintmatrixallocatedbyimatrix*/ f free FREE ARG m [ nrl ]+ ncl NR END ; free FREE ARG m + nrl NR END ; g void free submatrix double ** b long nrl long nrh long ncl long nch /*freeasubmatrixallocatedbysubmatrix*/ f free FREE ARG b + nrl NR END ; g void free convert matrix double ** b long nrl long nrh long ncl long nch /*freeamatrixallocatedbyconvert matrix*/ f free FREE ARG b + nrl NR END ; g void free f3tensor double *** t long nrl long nrh long ncl long nch long ndl long ndh /*freeadoublef3tensorallocatedbyf3tensor*/ f free FREE ARG t [ nrl ][ ncl ]+ ndl NR END ; free FREE ARG t [ nrl ]+ ncl NR END ; free FREE ARG t + nrl NR END ; g #else /*ANSI*/ /*traditional-K&R*/ #include #dene NR END 1 #dene FREE ARG char void nrerror error text char error text []; /*NumericalRecipesstandarderrorhandler*/ f void exit ; fprintf stderr "NumericalRecipesrun-timeerror... nn ; fprintf stderr "%s nn error text ; fprintf stderr "...nowexitingtosystem... nn ; exit 1 ; g double vector nl nh long nh nl ; /*allocateadoublevectorwithsubscriptrangev[nl..nh]*/ f double v ; v = double malloc unsigned int nh nl + 1 + NR END sizeof double ; 149

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if v nrerror "allocationfailureinvector" ; return v nl + NR END ; g int ivector nl nh long nh nl ; /*allocateanintvectorwithsubscriptrangev[nl..nh]*/ f int v ; v = int malloc unsigned int nh nl + 1 + NR END sizeof int ; if v nrerror "allocationfailureinivector" ; return v nl + NR END ; g unsigned char cvector nl nh long nh nl ; /*allocateanunsignedcharvectorwithsubscriptrangev[nl..nh]*/ f unsigned char v ; v = unsigned char malloc unsigned int nh nl + 1 + NR END sizeof unsigned char ; if v nrerror "allocationfailureincvector" ; return v nl + NR END ; g unsigned long lvector nl nh long nh nl ; /*allocateanunsignedlongvectorwithsubscriptrangev[nl..nh]*/ f unsigned long v ; v = unsigned long malloc unsigned int nh nl + 1 + NR END sizeof long ; if v nrerror "allocationfailureinlvector" ; return v nl + NR END ; g double dvector nl nh long nh nl ; /*allocateadoublevectorwithsubscriptrangev[nl..nh]*/ f double v ; v = double malloc unsigned int nh nl + 1 + NR END sizeof double ; if v nrerror "allocationfailureindvector" ; return v nl + NR END ; g double ** matrix nrl nrh ncl nch long nch ncl nrh nrl ; /*allocateadoublematrixwithsubscriptrangem[nrl..nrh][ncl..nch]*/ f long i nrow = nrh nrl + 1 ncol = nch ncl + 1 ; double ** m ; /*allocatepointerstorows*/ m = double ** malloc unsigned int nrow + NR END sizeof double *; if m nrerror "allocationfailure1inmatrix" ; m += NR END ; m -= nrl ; 150

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/*allocaterowsandsetpointerstothem*/ m [ nrl ]= double malloc unsigned int nrow ncol + NR END sizeof double ; if m [ nrl ] nrerror "allocationfailure2inmatrix" ; m [ nrl ] += NR END ; m [ nrl ] -= ncl ; for i = nrl + 1 ; i <= nrh ; i ++ m [ i ]= m [ i 1 ]+ ncol ; /*returnpointertoarrayofpointerstorows*/ return m ; g double ** dmatrix nrl nrh ncl nch long nch ncl nrh nrl ; /*allocateadoublematrixwithsubscriptrangem[nrl..nrh][ncl..nch]*/ f long i nrow = nrh nrl + 1 ncol = nch ncl + 1 ; double ** m ; /*allocatepointerstorows*/ m = double ** malloc unsigned int nrow + NR END sizeof double *; if m nrerror "allocationfailure1inmatrix" ; m += NR END ; m -= nrl ; /*allocaterowsandsetpointerstothem*/ m [ nrl ]= double malloc unsigned int nrow ncol + NR END sizeof double ; if m [ nrl ] nrerror "allocationfailure2inmatrix" ; m [ nrl ] += NR END ; m [ nrl ] -= ncl ; for i = nrl + 1 ; i <= nrh ; i ++ m [ i ]= m [ i 1 ]+ ncol ; /*returnpointertoarrayofpointerstorows*/ return m ; g int ** imatrix nrl nrh ncl nch long nch ncl nrh nrl ; /*allocateaintmatrixwithsubscriptrangem[nrl..nrh][ncl..nch]*/ f long i nrow = nrh nrl + 1 ncol = nch ncl + 1 ; int ** m ; /*allocatepointerstorows*/ m = int ** malloc unsigned int nrow + NR END sizeof int *; if m nrerror "allocationfailure1inmatrix" ; m += NR END ; m -= nrl ; /*allocaterowsandsetpointerstothem*/ m [ nrl ]= int malloc unsigned int nrow ncol + NR END sizeof int ; if m [ nrl ] nrerror "allocationfailure2inmatrix" ; m [ nrl ] += NR END ; m [ nrl ] -= ncl ; for i = nrl + 1 ; i <= nrh ; i ++ m [ i ]= m [ i 1 ]+ ncol ; 151

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/*returnpointertoarrayofpointerstorows*/ return m ; g double ** submatrix a oldrl oldrh oldcl oldch newrl newcl double ** a ; long newcl newrl oldch oldcl oldrh oldrl ; /*pointasubmatrix[newrl..][newcl..]toa[oldrl..oldrh][oldcl..oldch]*/ f long i j nrow = oldrh oldrl + 1 ncol = oldcl newcl ; double ** m ; /*allocatearrayofpointerstorows*/ m = double ** malloc unsigned int nrow + NR END sizeof double *; if m nrerror "allocationfailureinsubmatrix" ; m += NR END ; m -= newrl ; /*setpointerstorows*/ for i = oldrl j = newrl ; i <= oldrh ; i ++, j ++ m [ j ]= a [ i ]+ ncol ; /*returnpointertoarrayofpointerstorows*/ return m ; g double ** convert matrix a nrl nrh ncl nch double a ; long nch ncl nrh nrl ; /*allocateadoublematrixm[nrl..nrh][ncl..nch]thatpointstothematrix declaredinthestandardCmannerasa[nrow][ncol],wherenrow=nrh-nrl+1 andncol=nch-ncl+1.Theroutineshouldbecalledwiththeaddress &a[0][0]astherstargument.*/ f long i j nrow = nrh nrl + 1 ncol = nch ncl + 1 ; double ** m ; /*allocatepointerstorows*/ m = double ** malloc unsigned int nrow + NR END sizeof double *; if m nrerror "allocationfailureinconvert matrix" ; m += NR END ; m -= nrl ; /*setpointerstorows*/ m [ nrl ]= a ncl ; for i = 1 j = nrl + 1 ; i < nrow ; i ++, j ++ m [ j ]= m [ j 1 ]+ ncol ; /*returnpointertoarrayofpointerstorows*/ return m ; g double *** f3tensor nrl nrh ncl nch ndl ndh long nch ncl ndh ndl nrh nrl ; /*allocateadouble3tensorwithranget[nrl..nrh][ncl..nch][ndl..ndh]*/ f long i j nrow = nrh nrl + 1 ncol = nch ncl + 1 ndep = ndh ndl + 1 ; double *** t ; /*allocatepointerstopointerstorows*/ t = double *** malloc unsigned int nrow + NR END sizeof double **; if t nrerror "allocationfailure1inf3tensor" ; t += NR END ; 152

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t -= nrl ; /*allocatepointerstorowsandsetpointerstothem*/ t [ nrl ]= double ** malloc unsigned int nrow ncol + NR END sizeof double *; if t [ nrl ] nrerror "allocationfailure2inf3tensor" ; t [ nrl ] += NR END ; t [ nrl ] -= ncl ; /*allocaterowsandsetpointerstothem*/ t [ nrl ][ ncl ]= double malloc unsigned int nrow ncol ndep + NR END sizeof double ; if t [ nrl ][ ncl ] nrerror "allocationfailure3inf3tensor" ; t [ nrl ][ ncl ] += NR END ; t [ nrl ][ ncl ] -= ndl ; for j = ncl + 1 ; j <= nch ; j ++ t [ nrl ][ j ]= t [ nrl ][ j 1 ]+ ndep ; for i = nrl + 1 ; i <= nrh ; i ++ f t [ i ]= t [ i 1 ]+ ncol ; t [ i ][ ncl ]= t [ i 1 ][ ncl ]+ ncol ndep ; for j = ncl + 1 ; j <= nch ; j ++ t [ i ][ j ]= t [ i ][ j 1 ]+ ndep ; g /*returnpointertoarrayofpointerstorows*/ return t ; g void free vector v nl nh double v ; long nh nl ; /*freeadoublevectorallocatedwithvector*/ f free FREE ARG v + nl NR END ; g void free ivector v nl nh int v ; long nh nl ; /*freeanintvectorallocatedwithivector*/ f free FREE ARG v + nl NR END ; g void free cvector v nl nh long nh nl ; unsigned char v ; /*freeanunsignedcharvectorallocatedwithcvector*/ f free FREE ARG v + nl NR END ; g void free lvector v nl nh long nh nl ; unsigned long v ; /*freeanunsignedlongvectorallocatedwithlvector*/ f free FREE ARG v + nl NR END ; g void free dvector v nl nh double v ; long nh nl ; 153

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/*freeadoublevectorallocatedwithdvector*/ f free FREE ARG v + nl NR END ; g void free matrix m nrl nrh ncl nch double ** m ; long nch ncl nrh nrl ; /*freeadoublematrixallocatedbymatrix*/ f free FREE ARG m [ nrl ]+ ncl NR END ; free FREE ARG m + nrl NR END ; g void free dmatrix m nrl nrh ncl nch double ** m ; long nch ncl nrh nrl ; /*freeadoublematrixallocatedbydmatrix*/ f free FREE ARG m [ nrl ]+ ncl NR END ; free FREE ARG m + nrl NR END ; g void free imatrix m nrl nrh ncl nch int ** m ; long nch ncl nrh nrl ; /*freeanintmatrixallocatedbyimatrix*/ f free FREE ARG m [ nrl ]+ ncl NR END ; free FREE ARG m + nrl NR END ; g void free submatrix b nrl nrh ncl nch double ** b ; long nch ncl nrh nrl ; /*freeasubmatrixallocatedbysubmatrix*/ f free FREE ARG b + nrl NR END ; g void free convert matrix b nrl nrh ncl nch double ** b ; long nch ncl nrh nrl ; /*freeamatrixallocatedbyconvert matrix*/ f free FREE ARG b + nrl NR END ; g void free f3tensor t nrl nrh ncl nch ndl ndh double *** t ; long nch ncl ndh ndl nrh nrl ; /*freeadoublef3tensorallocatedbyf3tensor*/ f free FREE ARG t [ nrl ][ ncl ]+ ndl NR END ; free FREE ARG t [ nrl ]+ ncl NR END ; free FREE ARG t + nrl NR END ; g #endif /*ANSI*/ #include "chargefit.h" 154

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/* *--{-{-{-{-{-{-{-{-{-{-{-{-{-{-{-{-{-{-{-{-{-*reciprocalbasisandvolumeofunitcell *--{-{-{-{-{-{-{-{-{-{-{-{-{-{-{-{-{-{-{-{-{-*/ void calc recip struct atoms a f int i j ; char bu [ MAXLINE ]; /*thiscomputestheunitcellvolume*/ a -> cell volume = a -> tv [ 0 ][ 0 ] a -> tv [ 1 ][ 1 ]* a -> tv [ 2 ][ 2 ] a -> tv [ 1 ][ 2 ]* a -> tv [ 2 ][ 1 ]+ a -> tv [ 0 ][ 1 ] a -> tv [ 1 ][ 2 ]* a -> tv [ 2 ][ 0 ] a -> tv [ 1 ][ 0 ]* a -> tv [ 2 ][ 2 ]+ a -> tv [ 0 ][ 2 ] a -> tv [ 1 ][ 0 ]* a -> tv [ 2 ][ 1 ] a -> tv [ 1 ][ 1 ]* a -> tv [ 2 ][ 0 ]; /*theindividualcomponentsofthereciprocalbasismatrix*/ a -> rec tv [ 0 ][ 0 ] = 2 M PI / a -> cell volume a -> tv [ 1 ][ 1 ]* a -> tv [ 2 ][ 2 ]a -> tv [ 1 ][ 2 ]* a -> tv [ 2 ][ 1 ]; a -> rec tv [ 0 ][ 1 ] = 2 M PI / a -> cell volume a -> tv [ 1 ][ 2 ]* a -> tv [ 2 ][ 0 ]a -> tv [ 1 ][ 0 ]* a -> tv [ 2 ][ 2 ]; a -> rec tv [ 0 ][ 2 ] = 2 M PI / a -> cell volume a -> tv [ 1 ][ 0 ]* a -> tv [ 2 ][ 1 ]a -> tv [ 1 ][ 1 ]* a -> tv [ 2 ][ 0 ]; a -> rec tv [ 1 ][ 0 ] = 2 M PI / a -> cell volume a -> tv [ 2 ][ 1 ]* a -> tv [ 0 ][ 2 ]a -> tv [ 2 ][ 2 ]* a -> tv [ 0 ][ 1 ]; a -> rec tv [ 1 ][ 1 ] = 2 M PI / a -> cell volume a -> tv [ 2 ][ 2 ]* a -> tv [ 0 ][ 0 ]a -> tv [ 2 ][ 0 ]* a -> tv [ 0 ][ 2 ]; a -> rec tv [ 1 ][ 2 ] = 2 M PI / a -> cell volume a -> tv [ 2 ][ 0 ]* a -> tv [ 0 ][ 1 ]a -> tv [ 2 ][ 1 ]* a -> tv [ 0 ][ 0 ]; a -> rec tv [ 2 ][ 0 ] = 2 M PI / a -> cell volume a -> tv [ 0 ][ 1 ]* a -> tv [ 1 ][ 2 ]a -> tv [ 0 ][ 2 ]* a -> tv [ 1 ][ 1 ]; a -> rec tv [ 2 ][ 1 ] = 2 M PI / a -> cell volume a -> tv [ 0 ][ 2 ]* a -> tv [ 1 ][ 0 ]a -> tv [ 0 ][ 0 ]* a -> tv [ 1 ][ 2 ]; a -> rec tv [ 2 ][ 2 ] = 2 M PI / a -> cell volume a -> tv [ 0 ][ 0 ]* a -> tv [ 1 ][ 1 ]a -> tv [ 0 ][ 1 ]* a -> tv [ 1 ][ 0 ]; sprintf bu nn PBC:cell volume:%lfA^3 nn a -> cell volume ; output bu ; output "PBC:reciprocalbasiscalculated nn ; output "========================================= nn ; for i = 0 ; i < 3 ; i ++ f for j = 0 ; j < 3 ; j ++ f sprintf bu "%f nt a -> rec tv [ i ][ j ]; output bu ; g output nn ; g output "========================================= nnnn ; g /* *--{-{-{-{-{-{-{-{-{-{-{-{-{-{-{-{-{-{-{-{-{-*Calculatercutcuto *Therearesixpossiblevaluesthatneededtobeconsideredtond *theshortestdistancebetweentheplanesofthelattice.Oncethe *shortestisidentied,halfofthevalueistakentobercut. *--{-{-{-{-{-{-{-{-{-{-{-{-{-{-{-{-{-{-{-{-{*/ void calc rcut struct atoms a struct lrc lrc f int i ; double temp [ 6 ]; double min = 9999.0 ; char bu [ MAXLINE ]; /*CalculatethelengthjAj,jBj,jCjofthreelatticevectorA,B,C*/ a -> magA = sqrt a -> tv [ 0 ][ 0 ]* a -> tv [ 0 ][ 0 ] + a -> tv [ 0 ][ 1 ]* a -> tv [ 0 ][ 1 ] 155

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+ a -> tv [ 0 ][ 2 ]* a -> tv [ 0 ][ 2 ] ; a -> magB = sqrt a -> tv [ 1 ][ 0 ]* a -> tv [ 1 ][ 0 ] + a -> tv [ 1 ][ 1 ]* a -> tv [ 1 ][ 1 ] + a -> tv [ 1 ][ 2 ]* a -> tv [ 1 ][ 2 ] ; a -> magC = sqrt a -> tv [ 2 ][ 0 ]* a -> tv [ 2 ][ 0 ] + a -> tv [ 2 ][ 1 ]* a -> tv [ 2 ][ 1 ] + a -> tv [ 2 ][ 2 ]* a -> tv [ 2 ][ 2 ] ; /*Calculateanglesalpha,beta,gammabetweenlatticevectors*/ a -> alpha = acos a -> tv [ 0 ][ 0 ]* a -> tv [ 1 ][ 0 ] + a -> tv [ 0 ][ 1 ]* a -> tv [ 1 ][ 1 ] + a -> tv [ 0 ][ 2 ]* a -> tv [ 1 ][ 2 ] / a -> magA / a -> magB ; a -> beta = acos a -> tv [ 1 ][ 0 ]* a -> tv [ 2 ][ 0 ] + a -> tv [ 1 ][ 1 ]* a -> tv [ 2 ][ 1 ] + a -> tv [ 1 ][ 2 ]* a -> tv [ 2 ][ 2 ] / a -> magB / a -> magC ; a -> gamma = acos a -> tv [ 0 ][ 0 ]* a -> tv [ 2 ][ 0 ] + a -> tv [ 0 ][ 1 ]* a -> tv [ 2 ][ 1 ] + a -> tv [ 0 ][ 2 ]* a -> tv [ 2 ][ 2 ] / a -> magA / a -> magC ; /*computethedistancebetweentheplanesoftheunitcell*/ /*thereare3dimesionseachwitha'heightandwidth'*/ temp [ 0 ] = a -> magA sin a -> alpha ; /*distance1*/ temp [ 1 ] = a -> magA sin a -> gamma ; /*distance2*/ temp [ 2 ] = a -> magB sin a -> alpha ; /*distance3*/ temp [ 3 ] = a -> magB sin a -> beta ; /*distance4*/ temp [ 4 ] = a -> magC sin a -> gamma ; /*distance5*/ temp [ 5 ] = a -> magC sin a -> beta ; /*distance6*/ for i = 0 ; i < 6 ; i ++ if temp [ i ]< min min = temp [ i ]; sprintf bu "PBC:magnitudeoflatticevector1:%.4f nn a -> magA ; output bu ; sprintf bu "PBC:magnitudeoflatticevector2:%.4f nn a -> magB ; output bu ; sprintf bu "PBC:magnitudeoflatticevector3:%.4f nnnn a -> magC ; output bu ; sprintf bu "PBC:anglealpharadbetweenTV1andTV2:%.4f nn a -> alpha ; output bu ; sprintf bu "PBC:anglebetaradbetweenTV2andTV3:%.4f nn a -> beta ; output bu ; sprintf bu "PBC:anglegammaradbetweenTV1andTV3:%.4f nnnn a -> gamma ; output bu ; sprintf bu "PBC:anglealphadegbetweenTV1andTV2:%.2f nn a -> alpha / M PI 180.0 ; output bu ; sprintf bu "PBC:anglebetadegbetweenTV2andTV3:%.2f nn a -> beta / M PI 180.0 ; output bu ; sprintf bu "PBC:anglegammadegbetweenTV1andTV3:%.2f nnnn a -> gamma / M PI 180.0 ; output bu ; g void calc nmax struct atoms a struct lrc lrc f double decay by = 3.2 / sqrt lrc -> alpha ; char bu [ MAXLINE ]; double dist now = 0 ; int i ; sprintf bu "PBC:alphasetto%f nn lrc -> alpha ; output bu ; sprintf bu "PBC:real-spacefunctionwilldecayby:%f nn decay by ; output bu ; output "PBC:settingreal-spacecutoffbasedonalpha nn ; lrc -> nmaxf = 3.2 / sqrt lrc -> alpha ; sprintf bu "PBC:real-spacecutoffsetto%f nn lrc -> nmaxf ; for lrc -> nmax [ 0 ]= 0 dist now = 0 ; dist now < decay by ; lrc -> nmax [ 0 ]++ dist now = lrc -> nmax [ 0 ] a -> magA ; 156

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for lrc -> nmax [ 1 ]= 0 dist now = 0 ; dist now < decay by ; lrc -> nmax [ 1 ]++ dist now = lrc -> nmax [ 1 ] a -> magB ; for lrc -> nmax [ 2 ]= 0 dist now = 0 ; dist now < decay by ; lrc -> nmax [ 2 ]++ dist now = lrc -> nmax [ 2 ] a -> magC ; /*forloopsincrementthevariableonetoomanytimes*/ for i = 0 ; i < 3 ; i ++ lrc -> nmax [ i ]--; for i = 0 ; i < 3 ; i ++ f sprintf bu "PBC:anisotropicnmaxalonglatticevector%dsetto%d nn i lrc -> nmax [ i ]; output bu ; g g void isotropic nmax struct atoms a struct lrc lrc f char bu [ MAXLINE ]; output "PBC:nmaxsetbyuser,usingisotropicnmax nn ; lrc -> nmax [ 0 ] = lrc -> nmax in ; lrc -> nmax [ 1 ] = lrc -> nmax in ; lrc -> nmax [ 2 ] = lrc -> nmax in ; output "PBC:settingreal-spacecutoffbasedonalpha nn ; lrc -> nmaxf = 3.2 / sqrt lrc -> alpha ; sprintf bu "PBC:real-spacecutoffsetto%f nn lrc -> nmaxf ; g /* *--{-{-{-{-{-{-{-{-{-{-{-{-{-{-{-{-{-{-{-{-{-*calculatealphavalueinewaldsummation *--{-{-{-{-{-{-{-{-{-{-{-{-{-{-{-{-{-{-{-{-{-*/ void calc alpha struct atoms a struct lrc lrc f int i ; double average ; double min = 999999 ; char bu [ MAXLINE ]; output "PBC:nmaxwassetininputfile nn ; output "PBC:computingrequiredalpha nn ; /*searchfortheminimumdistancemainlyincomputingalpha*/ if a -> magA lrc -> nmax [ 0 ] < min min = a -> magA lrc -> nmax [ 0 ]; if a -> magB lrc -> nmax [ 1 ] < min min = a -> magB lrc -> nmax [ 1 ]; if a -> magC lrc -> nmax [ 2 ] < min min = a -> magC lrc -> nmax [ 2 ]; if strcasestr a -> ewaldorwolf "wolf" f lrc -> alpha = 3.2 / min 3.2 / min ; g else if strcasestr a -> ewaldorwolf "ewald" f lrc -> alpha = 3.2 / min 3.2 / min ; g 157

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sprintf bu "PBC:real-spacesummationmustdecayby%.2f nn min ; output bu ; sprintf bu "PBC:alphasetto%.4f nn lrc -> alpha ; output bu ; sprintf bu "PBC:wolfnmaxfsetto%.2f nn min ; output bu ; lrc -> nmaxf = min ; g void handle nmax or alpha struct atoms a struct lrc lrc f int i ; char bu [ MAXLINE ]; /*zeroallrelatedvariables*/ lrc -> nmax [ 0 ]= 0 ; lrc -> nmax [ 1 ]= 0 ; lrc -> nmax [ 2 ]= 0 ; /*ifallthreeareset,donothing*/ if lrc -> alpha && lrc -> nmax in && lrc -> kmax f isotropic nmax a lrc ; output "READ:alpha,kmax,andnmaxallsetbyuser! nn ; output "READ:Ihopeyouknowwhatyouaredoing! nn ; g /*ifalphaisnotset,computeit*/ else if lrc -> alpha && lrc -> nmax in f isotropic nmax a lrc ; calc alpha a lrc ; output "nmaxset,alphacomputed nn ; g /*ifnmaxisnotset,computeit*/ else if lrc -> nmax in && lrc -> alpha f calc nmax a lrc ; output "alphaset,nmaxcomputed nn ; g else f output "ERROR:youmustsetalphaand/ornmax! nn ; exit 1 ; g output "=========================================== nn ; output "LongRangeParameters nn ; output nn ; sprintf bu ntnt LRCmethod:%s nnnn a -> ewaldorwolf ; output bu ; sprintf bu ntnt alpha ntnt %10f nn lrc -> alpha ; output bu ; if a -> ewald ag f sprintf bu ntnt kmax ntntnt %d nn lrc -> kmax ; output bu ; g for i = 0 ; i < 3 ; i ++ f sprintf bu ntnt nmax%d ntntnt %d nn i lrc -> nmax [ i ]; output bu ; g sprintf bu ntnt cutoff ntnt %10f nn lrc -> nmaxf ; output bu ; output "=========================================== nn ; 158

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g #include "chargefit.h" /*Inthisroutine,targetchargeofzeroisconsidered*/ /*Onlythediagonalelementsneedtobechanged*/ /*matrixjAjjQj=jBj ifl.ne.j,then Al,j=[sum i.0/rij*ril] Bl=sum iVi/ril+a*q0l;q0listhetargetcharge elseifl=j,then Al,j=[sum i.0/rij*ril+1.0]*qj Bl=sum iVi/ril+a*q0l; */ void targetcharge struct atoms a struct lsq l f char bu [ MAXLINE ]; int i j k ; int ntargetcharge ; /*numberoftargetchargesequalstothenumberofburriedatomsyouspecify*/ double target charge ; int number temp ; double targetcharge temp ; printf "reallyweired nn ; target charge = double calloc a -> natoms sizeof double ; for j = 0 ; j < a -> natoms ; j ++ f target charge [ j ] = 0.0 ; /*initializethetargetcharge*/ g FILE target q le ; target q le = fopen "TARGET Q" "r" ; fscanf target q le "%d nn ,& ntargetcharge ; number temp = int calloc ntargetcharge sizeof int ; targetcharge temp = double calloc ntargetcharge sizeof double ; for i = 0 ; i < ntargetcharge ; i ++ f fscanf target q le "%d%lf nn ,& number temp [ i ],& targetcharge temp [ i ]; g fclose target q le ; /*revisedAandBaccordingtothetargetchargeofburriedatoms*/ for i = 0 ; i < ntargetcharge ; i ++ f k = number temp [ i ] 1 ; target charge [ k ] += targetcharge temp [ i ]; printf "number,target charge:%d%lf nn number temp [ i ], target charge [ k ]; l -> A [ k ][ k ] -= a -> aa ; l -> b [ k ] -= a -> aa target charge [ k ]; g g #include #dene NRANSI #include "nrutil.h" double pythag double a double b 159

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f double absa absb ; absa = fabs a ; absb = fabs b ; if absa > absb return absa sqrt 1.0 + SQR absb / absa ; else return absb == 0.0 ? 0.0 : absb sqrt 1.0 + SQR absa / absb ; g #undef NRANSI #include "chargefit.h" void results struct atoms a struct points p struct lsq l f void outcharge struct atoms a struct lsq l ; void normalization struct atoms a struct lsq l ; void rrms struct atoms a struct points p struct lsq l ; void dipole struct atoms a ; outcharge a l ; dipole a ; rrms a p l ; if a -> normag f normalization a l ; g g /*Thisroutinegivethettedchargetoeachatom*/ void outcharge struct atoms a struct lsq l f int i j ; char bu [ MAXLINE ]; a -> charge = double calloc a -> natoms sizeof double ; if a -> symm ag f for i = 0 ; i < a -> nelem symm ; i ++ f for j = 0 ; j < a -> nelem symm natoms [ i ]; j ++ f a -> charge [ a -> symm lable [ i ][ j ]1 ] = l -> a [ i + 1 ]; sprintf bu "a->charge[%d]:%lf nn a -> symm lable [ i ][ j ]1 a -> charge [ a -> symm lable [ i ][ j ]1 ]; output bu ; g g g else f for i = 0 ; i < a -> natoms ; i ++ f a -> charge [ i ] = l -> a [ i + 1 ]; g g g void dipole struct atoms a f int i ; double dipole x = 0.0 ; double dipole y = 0.0 ; double dipole z = 0.0 ; double dipole = 0.0 ; double center x = 0.0 ; double center y = 0.0 ; 160

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double center z = 0.0 ; for i = 0 ; i < a -> natoms ; i ++ f center x += a -> cart [ i ][ 0 ]; center y += a -> cart [ i ][ 1 ]; center z += a -> cart [ i ][ 2 ]; g center x = center x / a -> natoms ; center y = center y / a -> natoms ; center z = center z / a -> natoms ; for i = 0 ; i < a -> natoms ; i ++ f dipole x += a -> charge [ i ] a -> cart [ i ][ 0 ]center x ; dipole y += a -> charge [ i ] a -> cart [ i ][ 1 ]center y ; dipole z += a -> charge [ i ] a -> cart [ i ][ 2 ]center z ; /*printf"dipoleinthreedirections:%lf%lf%lfnn",dipole x,dipole y,dipole z;*/ g dipole = sqrt dipole x dipole x + dipole y dipole y + dipole z dipole z ; printf "dipoleofunitcell:%lf nn dipole ; g void normalization struct atoms a struct lsq l f int i ; printf "chargesafternormalization: nn ; for i = 0 ; i < a -> natoms ; i ++ f a -> charge [ i ] -= l -> sum / a -> natoms ; printf "a->charge[%d]:%lf nn i a -> charge [ i ]; g g /*ThisroutinecalculatetherelativerootmeansquareerrorRRMS*/ void rrms struct atoms a struct points p struct lsq l f int i j ; double temp1 temp2 temp3 temp4 temp5 ; double delta esp ; double oset = 0.0 ; char bu [ MAXLINE ]; p -> esp tting = calloc p -> npoints sizeof double ; delta esp = calloc p -> npoints sizeof double ; for i = 0 ; i < p -> npoints ; i ++ f p -> esp tting [ i ] = 0.0 ; for j = 0 ; j < a -> natoms ; j ++ f p -> esp tting [ i ] += l -> des [ i ][ j ] a -> charge [ j ] 14.3996 ; /*Theoriginall->desischangedbecauseoftheoset*/ /*p->esp tting[i]+=l->des[i][j]+l->des average[j]*a->charge[j]*-14.3996;*/ g delta esp [ i ] = p -> esp [ i ] p -> esp tting [ i ]; oset += delta esp [ i ]; g 161

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oset /= p -> npoints ; printf "offsetvalue%lf nn oset ; temp1 = 0.0 ; temp2 = 0.0 ; for i = 0 ; i < p -> npoints ; i ++ f temp1 += p -> esp [ i ]* p -> esp [ i ]; temp3 = p -> esp [ i ] p -> esp tting [ i ] oset ; temp2 += temp3 temp3 ; /*TheESPdierenceforeachselectedpointcanbeobtainedbycommentingoutthefollowingtwolines*/ /*printf"%12.8lf%12.8lf%12.8lfnn",p->esp[i],p->esp tting[i],delta esp[i];*/ g p -> rrms = sqrt temp2 / temp1 ; sprintf bu "TheRRMSis:%lf nn p -> rrms ; output bu ; g #dene NRANSI #include #include #include "nrutil.h" void svbksb double ** u double w [], double ** v int m int n double b [], double x [] f int jj j i ; double s ,* tmp ; tmp = vector 1 n ; for j = 1 ; j <= n ; j ++ f s = 0.0 ; if w [ j ] f for i = 1 ; i <= m ; i ++ s += u [ i ][ j ]* b [ i ]; s /= w [ j ]; g tmp [ j ]= s ; g for j = 1 ; j <= n ; j ++ f s = 0.0 ; for jj = 1 ; jj <= n ; jj ++ s += v [ j ][ jj ]* tmp [ jj ]; x [ j ]= s ; g free vector tmp 1 n ; g #undef NRANSI #include #dene NRANSI #include "nrutil.h" void svdcmp double ** a int m int n double w [], double ** v f double pythag double a double b ; int ag i its j jj k l nm ; double anorm c f g h s scale x y z ,* rv1 ; rv1 = vector 1 n ; g = scale = anorm = 0.0 ; 162

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for i = 1 ; i <= n ; i ++ f l = i + 1 ; rv1 [ i ]= scale g ; g = s = scale = 0.0 ; if i <= m f for k = i ; k <= m ; k ++ scale += fabs a [ k ][ i ]; if scale f for k = i ; k <= m ; k ++ f a [ k ][ i ] /= scale ; s += a [ k ][ i ]* a [ k ][ i ]; g f = a [ i ][ i ]; g = SIGN sqrt s f ; h = f g s ; a [ i ][ i ]= f g ; for j = l ; j <= n ; j ++ f for s = 0.0 k = i ; k <= m ; k ++ s += a [ k ][ i ]* a [ k ][ j ]; f = s / h ; for k = i ; k <= m ; k ++ a [ k ][ j ] += f a [ k ][ i ]; g for k = i ; k <= m ; k ++ a [ k ][ i ] *= scale ; g g w [ i ]= scale g ; g = s = scale = 0.0 ; if i <= m && i != n f for k = l ; k <= n ; k ++ scale += fabs a [ i ][ k ]; if scale f for k = l ; k <= n ; k ++ f a [ i ][ k ] /= scale ; s += a [ i ][ k ]* a [ i ][ k ]; g f = a [ i ][ l ]; g = SIGN sqrt s f ; h = f g s ; a [ i ][ l ]= f g ; for k = l ; k <= n ; k ++ rv1 [ k ]= a [ i ][ k ]/ h ; for j = l ; j <= m ; j ++ f for s = 0.0 k = l ; k <= n ; k ++ s += a [ j ][ k ]* a [ i ][ k ]; for k = l ; k <= n ; k ++ a [ j ][ k ] += s rv1 [ k ]; g for k = l ; k <= n ; k ++ a [ i ][ k ] *= scale ; g g anorm = FMAX anorm fabs w [ i ]+ fabs rv1 [ i ]; g for i = n ; i >= 1 ; i -f if i < n f if g f for j = l ; j <= n ; j ++ v [ j ][ i ]= a [ i ][ j ]/ a [ i ][ l ]/ g ; for j = l ; j <= n ; j ++ f for s = 0.0 k = l ; k <= n ; k ++ s += a [ i ][ k ]* v [ k ][ j ]; for k = l ; k <= n ; k ++ v [ k ][ j ] += s v [ k ][ i ]; g g for j = l ; j <= n ; j ++ v [ i ][ j ]= v [ j ][ i ]= 0.0 ; g v [ i ][ i ]= 1.0 ; g = rv1 [ i ]; 163

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l = i ; g for i = IMIN m n ; i >= 1 ; i -f l = i + 1 ; g = w [ i ]; for j = l ; j <= n ; j ++ a [ i ][ j ]= 0.0 ; if g f g = 1.0 / g ; for j = l ; j <= n ; j ++ f for s = 0.0 k = l ; k <= m ; k ++ s += a [ k ][ i ]* a [ k ][ j ]; f = s / a [ i ][ i ]* g ; for k = i ; k <= m ; k ++ a [ k ][ j ] += f a [ k ][ i ]; g for j = i ; j <= m ; j ++ a [ j ][ i ] *= g ; g else for j = i ; j <= m ; j ++ a [ j ][ i ]= 0.0 ; ++ a [ i ][ i ]; g for k = n ; k >= 1 ; k -f for its = 1 ; its <= 30 ; its ++ f ag = 1 ; for l = k ; l >= 1 ; l -f nm = l 1 ; if double fabs rv1 [ l ]+ anorm == anorm f ag = 0 ; break ; g if double fabs w [ nm ]+ anorm == anorm break ; g if ag f c = 0.0 ; s = 1.0 ; for i = l ; i <= k ; i ++ f f = s rv1 [ i ]; rv1 [ i ]= c rv1 [ i ]; if double fabs f + anorm == anorm break ; g = w [ i ]; h = pythag f g ; w [ i ]= h ; h = 1.0 / h ; c = g h ; s = f h ; for j = 1 ; j <= m ; j ++ f y = a [ j ][ nm ]; z = a [ j ][ i ]; a [ j ][ nm ]= y c + z s ; a [ j ][ i ]= z c y s ; g g g z = w [ k ]; if l == k f if z < 0.0 f w [ k ] = z ; for j = 1 ; j <= n ; j ++ v [ j ][ k ] = v [ j ][ k ]; g break ; g if its == 30 nrerror "noconvergencein30svdcmpiterations" ; x = w [ l ]; nm = k 1 ; 164

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y = w [ nm ]; g = rv1 [ nm ]; h = rv1 [ k ]; f = y z y + z + g h g + h / 2.0 h y ; g = pythag f 1.0 ; f = x z x + z + h y / f + SIGN g f h / x ; c = s = 1.0 ; for j = l ; j <= nm ; j ++ f i = j + 1 ; g = rv1 [ i ]; y = w [ i ]; h = s g ; g = c g ; z = pythag f h ; rv1 [ j ]= z ; c = f / z ; s = h / z ; f = x c + g s ; g = g c x s ; h = y s ; y *= c ; for jj = 1 ; jj <= n ; jj ++ f x = v [ jj ][ j ]; z = v [ jj ][ i ]; v [ jj ][ j ]= x c + z s ; v [ jj ][ i ]= z c x s ; g z = pythag f h ; w [ j ]= z ; if z f z = 1.0 / z ; c = f z ; s = h z ; g f = c g + s y ; x = c y s g ; for jj = 1 ; jj <= m ; jj ++ f y = a [ jj ][ j ]; z = a [ jj ][ i ]; a [ jj ][ j ]= y c + z s ; a [ jj ][ i ]= z c y s ; g g rv1 [ l ]= 0.0 ; rv1 [ k ]= f ; w [ k ]= x ; g g free vector rv1 1 n ; g #undef NRANSI #include "chargefit.h" void symm constr struct atoms a struct lsq l f int i j m ; int ii jj ll ; /* doubletemp combine column[a->natoms][a->nelem symm]; doubletemp combine row[a->nelem symm][a->nelem symm]; 165

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doubletemp b combine row[a->nelem symm]; */ double ** temp combine column ; double ** temp combine row ; double temp b combine row ; temp combine column = malloc a -> natoms sizeof double *; for i = 0 ; i < a -> natoms ; i ++ temp combine column [ i ] = malloc a -> nelem symm sizeof double ; temp b combine row = malloc a -> nelem symm sizeof double ; temp combine row = malloc a -> nelem symm sizeof double *; for i = 0 ; i < a -> nelem symm ; i ++ temp combine row [ i ] = malloc a -> nelem symm sizeof double ; /*changetheelementsofl->Amatrix*/ /*combineelementincolumnandstoredintemp combine columnarrays*/ for ii = 0 ; ii < a -> natoms ; ii ++ f for jj = 0 ; jj < a -> nelem symm ; jj ++ f temp combine column [ ii ][ jj ] = 0.0 ; for m = 0 ; m < a -> nelem symm natoms [ jj ]; m ++ f ll = a -> symm lable [ jj ][ m ]1 ; temp combine column [ ii ][ jj ] += l -> A [ ii ][ ll ]; g g g /*ThencombineelementsinrowandgetnewelmentsinnewAmatrix*/ for jj = 0 ; jj < a -> nelem symm ; jj ++ f for ii = 0 ; ii < a -> nelem symm ; ii ++ f for m = 0 ; m < a -> nelem symm natoms [ ii ]; m ++ f ll = a -> symm lable [ ii ][ m ]1 ; temp combine row [ ii ][ jj ] += temp combine column [ ll ][ jj ]; g l -> A [ ii ][ jj ] = temp combine row [ ii ][ jj ]; g g /*changetheelementsofl->b*/ for i = 0 ; i < a -> nelem symm ; i ++ f for m = 0 ; m < a -> nelem symm natoms [ i ]; m ++ f ll = a -> symm lable [ i ][ m ]1 ; temp b combine row [ i ] += l -> b [ ll ]; g l -> b [ i ] = temp b combine row [ i ]; g free temp combine column ; free temp combine row ; free temp b combine row ; g void read symm struct atoms a struct lsq l f int i j ; char bu [ MAXLINE ]; FILE fpsymm ; fpsymm = fopen "SYMMINP" "r" ; /*Readthenumberofatomsaftersymmetry*/ fscanf fpsymm "%d" & a -> nelem symm ; 166

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sprintf bu "numberofchemicallyuniqueatoms:%d nn a -> nelem symm ; output bu ; /*Allocatememoryandreadthenumberofeachequivalentatom*/ a -> nelem symm natoms = calloc a -> nelem symm sizeof int ; for i = 0 ; i < a -> nelem symm ; i ++ f fscanf fpsymm "%d" & a -> nelem symm natoms [ i ]; sprintf bu "numberofeachchemicallyuniqueatom:%d nn a -> nelem symm natoms [ i ]; output bu ; g /*Allocatememory*/ a -> symm lable = calloc a -> nelem symm sizeof int *; for i = 0 ; i < a -> nelem symm ; i ++ f a -> symm lable [ i ] = calloc a -> nelem symm natoms [ i ], sizeof int ; g /*Readthelableforeachatom*/ for i = 0 ; i < a -> nelem symm ; i ++ f for j = 0 ; j < a -> nelem symm natoms [ i ]; j ++ f fscanf fpsymm "%d" & a -> symm lable [ i ][ j ]; /*sprintfbu,"lableofatom:%dnn",a->symm lable[i][j]; outputbu;*/ g g /*re-specifythenumberofl->nbecauseofthesymmetry*/ l -> n = a -> nelem symm ; printf "l->n:%d nn l -> n ; if a -> qconstr l -> n ++; g #include #dene FREQ 20 #dene TIMING NODE 1 void performance struct atoms a int count int npoints f char bu [ MAXLINE ]; int64 t elapsed ; struct timeval t ; int heartbeat = 1 ; int block len = int npoints / FREQ ; if count % block len f #ifdef MPI if myrank == TIMING NODE MPI Send & heartbeat 1 MPI INT 0 TAG R MPI COMM WORLD ; if myrank == 0 MPI Recv & heartbeat 1 MPI INT TIMING NODE TAG R MPI COMM WORLD & status ; #endif gettimeofday & t NULL ; elapsed = t tv sec a -> t0 tv sec ; sprintf bu "---%2d%%matrixcompleted.totalwalltime%10lds--nn int count / oat npoints 100 elapsed ; output bu ; sprintf bu "---computingelements%10d~%10d--nnnn count count + block len 1 ; 167

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output bu ; g g void timestamp struct atoms a f char bu [ MAXLINE ]; int64 t elapsed ; struct timeval t ; gettimeofday & t NULL ; elapsed = t tv sec a -> t0 tv sec ; sprintf bu "---walltime%10lds--nnnn elapsed ; output bu ; ush NULL ; g #include "chargefit.h" /*--{-{-{-{-{-{-{-{-{-{-{-{-{-{-{-{-{-{-{-{-{-*AssigntheVDWradiiforatoms *ThedataarefromJPCA,113,5806. *--{-{-{-{-{-{-{-{-{-{-{-{-{-{-{-{-{-{-{-{-{-*/ void assign vdw struct atoms a f int i ; output "VDW:vanderWaalsRadiiassignedfromreference: nn ; output "VDW:Mantina,M.,Chamberlin,A.C.,Valero,R.,Cramer,J.,and nn ; output "VDW:Truhlar,D.G.;J.Phys.Chem.A,2009,113,5806-5812. nn ; for i = 0 ; i < a -> natoms ; i ++ f switch a -> z [ i ] f case 1 : a -> vdw [ i ]= 1.10 ; break ; case 2 : a -> vdw [ i ]= 1.40 ; break ; case 3 : a -> vdw [ i ]= 1.81 ; break ; case 4 : a -> vdw [ i ]= 1.53 ; break ; case 5 : a -> vdw [ i ]= 1.92 ; break ; case 6 : a -> vdw [ i ]= 1.70 ; break ; case 7 : a -> vdw [ i ]= 1.55 ; break ; case 8 : a -> vdw [ i ]= 1.52 ; 168

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break ; case 9 : a -> vdw [ i ]= 1.47 ; break ; case 10 : a -> vdw [ i ]= 1.54 ; break ; case 11 : a -> vdw [ i ]= 2.27 ; break ; case 12 : a -> vdw [ i ]= 1.73 ; break ; case 13 : a -> vdw [ i ]= 1.84 ; break ; case 14 : a -> vdw [ i ]= 2.10 ; break ; case 15 : a -> vdw [ i ]= 1.80 ; break ; case 16 : a -> vdw [ i ]= 1.80 ; break ; case 17 : a -> vdw [ i ]= 1.75 ; break ; case 18 : a -> vdw [ i ]= 1.88 ; break ; case 19 : a -> vdw [ i ]= 2.75 ; break ; case 20 : a -> vdw [ i ]= 2.31 ; break ; case 21 : a -> vdw [ i ]= 2.00 ; break ; case 22 : a -> vdw [ i ]= 2.00 ; break ; case 23 : a -> vdw [ i ]= 2.00 ; break ; case 24 : a -> vdw [ i ]= 2.00 ; break ; case 25 : a -> vdw [ i ]= 2.00 ; break ; case 26 : a -> vdw [ i ]= 2.00 ; break ; case 27 : a -> vdw [ i ]= 2.00 ; break ; case 28 : a -> vdw [ i ]= 1.63 ; 169

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break ; case 29 : a -> vdw [ i ]= 1.40 ; break ; case 30 : a -> vdw [ i ]= 1.39 ; break ; case 31 : a -> vdw [ i ]= 1.87 ; break ; case 32 : a -> vdw [ i ]= 2.11 ; break ; case 33 : a -> vdw [ i ]= 1.85 ; break ; case 34 : a -> vdw [ i ]= 1.90 ; break ; case 35 : a -> vdw [ i ]= 1.83 ; break ; case 36 : a -> vdw [ i ]= 2.02 ; break ; case 37 : a -> vdw [ i ]= 3.03 ; break ; case 38 : a -> vdw [ i ]= 2.49 ; break ; case 39 : a -> vdw [ i ]= 2.00 ; break ; case 40 : a -> vdw [ i ]= 2.00 ; break ; case 41 : a -> vdw [ i ]= 2.00 ; break ; case 42 : a -> vdw [ i ]= 2.00 ; break ; case 43 : a -> vdw [ i ]= 2.00 ; break ; case 44 : a -> vdw [ i ]= 2.00 ; break ; case 45 : a -> vdw [ i ]= 2.00 ; break ; case 46 : a -> vdw [ i ]= 1.63 ; break ; case 47 : a -> vdw [ i ]= 1.72 ; break ; case 48 : a -> vdw [ i ]= 1.58 ; 170

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break ; case 49 : a -> vdw [ i ]= 1.93 ; break ; case 50 : a -> vdw [ i ]= 2.17 ; break ; case 51 : a -> vdw [ i ]= 2.06 ; break ; case 52 : a -> vdw [ i ]= 2.06 ; break ; case 53 : a -> vdw [ i ]= 1.98 ; break ; case 54 : a -> vdw [ i ]= 2.16 ; break ; case 55 : a -> vdw [ i ]= 3.43 ; break ; case 56 : a -> vdw [ i ]= 2.68 ; break ; case 81 : a -> vdw [ i ]= 1.96 ; break ; case 82 : a -> vdw [ i ]= 2.02 ; break ; case 83 : a -> vdw [ i ]= 2.07 ; break ; case 84 : a -> vdw [ i ]= 1.97 ; break ; case 85 : a -> vdw [ i ]= 2.02 ; break ; case 86 : a -> vdw [ i ]= 2.20 ; break ; default: a -> vdw [ i ]= 1.80 ; g g #if CONVERT VDW A2BOHR /*convertstoBOHR*/ for i = 0 ; i < a -> natoms ; i ++ a -> vdw [ i ] /= BOHR RADIUS ; #endif g #include /*Inordertokeeptheoutputreasonable,allprint *statementsshouldbedirectedhere.Thewaytodothis *istosprintftoastringbuerthenpassittothisroutine */ void output const char outstring f 171

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#if MPI /*donotwriteoutifI'mnotthemaster*/ if myrank return ; #endif /*ifI'mnotusingMPI,orI'mstillhere,writeout*/ printf "%s" outstring ; g /*--{-{-{-{-{-{-{-{-{-{-{-{-{-{-{-{-{-{-{-{-{-{ *ThissubroutinewritethecartesiancoordinatesandESPof *theselectedpointsto"output"le. *--{-{-{-{-{-{-{-{-{-{-{-{-{-{-{-{-{-{-{-{-{-*/ void outesp struct atoms a struct points p struct lsq l f #ifdef MPI /*ifusingMPIandI'mnotmaster,return*/ if myrank return ; #endif if a -> esp output return ; int i ; FILE fpout ; if fpout = fopen a -> esp lename "w" == NULL f output "ERROR:openingespfileforwriting nn ; exit 1 ; g fprintf fpout "************************************************** nn ; fprintf fpout "****WELCOMETOCHARGE-FITTINGCODE**** nn ; fprintf fpout "************************************************** nn ; fprintf fpout nn ; fprintf fpout "Latticevectormultipliedbyscalefactor: nn ; for i = 0 ; i < 3 ; i ++ f fprintf fpout "%12.6lf%12.6lf%12.6lf nn a -> tv [ i ][ 0 ], a -> tv [ i ][ 1 ], a -> tv [ i ][ 2 ]; g fprintf fpout "Cartesiancoordinatesofatoms: nn ; for i = 0 ; i < a -> natoms ; i ++ f fprintf fpout "%10.6lf%10.6lf%10.6lf nn a -> cart [ i ][ 0 ], a -> cart [ i ][ 1 ], a -> cart [ i ][ 2 ]; g fprintf fpout "Thenumberofselectedpointsis:%d nn p -> npoints ; fprintf fpout "ThecartcoordinatesandESPofselectedpointsare: nn ; fprintf fpout "numXYZESP nn ; for i = 0 ; i < p -> npoints ; i ++ f fprintf fpout "%10d%10.6lf%10.6lf%10.6lf%18.12lf nn i p -> cart [ i ][ 0 ], p -> cart [ i ][ 1 ], p -> cart [ i ][ 2 ], p -> esp [ i ]; g fclose fpout ; g /* *--{-{-{-{-{-{-{-{-{-{-{-{-{-{-{-{-{-{-{-{-{-{ *printcoordinatestoPDBlewhichcanbevisualizedbytoolsuch *asVMD. 172

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*resnamePTS=selectedpoints *resnameMOL=atoms *resnameBOX=unitcell *--{-{-{-{-{-{-{-{-{-{-{-{-{-{-{-{-{-{-{-{-{-*/ void print2pdb struct atoms a struct points p f #ifdef MPI /*ifusingMPIandI'mnotmaster,return*/ if myrank return ; #endif if a -> pdb output return ; int i j k ; double b [ 3 ]; double cart [ 3 ]; int ncount = 0 ; FILE fppdb ; if fppdb = fopen a -> pdb lename "w" == NULL f output "ERROR:openingpdbfileforwriting nn ; exit 1 ; g /*Writecoordinatesofatoms*/ for i = 0 ; i < a -> natoms ; i ++ f fprintf fppdb "ATOM%5d%4sMOL%4d%8.3f%8.3f%8.3f nn 1 a -> elem [ i ], i a -> cart [ i ][ 0 ], a -> cart [ i ][ 1 ], a -> cart [ i ][ 2 ]; g /*Writecartcoordinatesofselectedpoints*/ for i = 0 ; i < p -> npoints ; i ++ f fprintf fppdb "ATOM%4dXPTS0%8.3f%8.3f%8.3f nn 2 p -> cart [ i ][ 0 ], p -> cart [ i ][ 1 ], p -> cart [ i ][ 2 ]; g /*Writecoordinatesofcornersoftheunitcellandconnectthem*/ for i = 0 ; i < 2 ; i ++ f for j = 0 ; j < 2 ; j ++ f for k = 0 ; k < 2 ; k ++ f b [ 0 ]= double i ; b [ 1 ]= double j ; b [ 2 ]= double k ; cart [ 0 ] = b [ 0 ] a -> tv [ 0 ][ 0 ] + b [ 1 ] a -> tv [ 1 ][ 0 ] + b [ 2 ] a -> tv [ 2 ][ 0 ]; cart [ 1 ] = b [ 0 ] a -> tv [ 0 ][ 1 ] + b [ 1 ] a -> tv [ 1 ][ 1 ] + b [ 2 ] a -> tv [ 2 ][ 1 ]; cart [ 2 ] = b [ 0 ] a -> tv [ 0 ][ 2 ] + b [ 1 ] a -> tv [ 1 ][ 2 ] + b [ 2 ] a -> tv [ 2 ][ 2 ]; ncount += 1 ; fprintf fppdb "ATOM%4dXBOX0%8.3f%8.3f%8.3f nn ncount + 2 cart [ 0 ], cart [ 1 ], cart [ 2 ]; g g g fprintf fppdb "CONECT34 nn ; fprintf fppdb "CONECT35 nn ; fprintf fppdb "CONECT37 nn ; fprintf fppdb "CONECT46 nn ; fprintf fppdb "CONECT48 nn ; fprintf fppdb "CONECT56 nn ; fprintf fppdb "CONECT59 nn ; fprintf fppdb "CONECT610 nn ; fprintf fppdb "CONECT78 nn ; fprintf fppdb "CONECT79 nn ; fprintf fppdb "CONECT810 nn ; 173

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fprintf fppdb "CONECT910 nn ; fclose fppdb ; g /*Thisroutinedoesnotworkwithsymmetryyet! *itmustbemodiedsothelastcolumnprintsthe *chargecorrectlywhensymmetryisturnedon. */ void print2xyz struct atoms a struct points p struct lsq l f #ifdef MPI /*ifusingMPIandI'mnotmaster,return*/ if myrank return ; #endif if a -> xyz output return ; int i ; FILE fp ; fp = fopen a -> xyz lename "w" ; fprintf fp "%d nnnn a -> natoms ; /*xyzcharge;chargeisOFFSET+1becauseofN.R.*/ if a -> symm ag f for i = 0 ; i < a -> natoms ; i ++ fprintf fp "%-4s%8f%8f%8f%6f nn a -> elem [ i ], a -> cart [ i ][ 0 ] a -> cart [ i ][ 1 ], a -> cart [ i ][ 2 ], l -> a [ i + 1 ]; g else f for i = 0 ; i < a -> natoms ; i ++ fprintf fp "%-4s%8f%8f%8f%6f nn a -> elem [ i ], a -> cart [ i ][ 0 ] a -> cart [ i ][ 1 ], a -> cart [ i ][ 2 ], a -> charge [ i ]; g /*blankline*/ fprintf fp nn ; g void print charges struct atoms a struct points p struct lsq l struct lrc lrc f char bu [ MAXLINE ]; int n i j ; if a -> symm ag f /*printsolution*/ output "==============charges============= nn ; for n = 1 ; n <= l -> n ; n ++ f sprintf bu "%10s%12.6lf nn a -> elem [ n 1 ], l -> a [ n ]; output bu ; g output "==================================== nn ; g 174

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else f /*printsolution*/ output "================================================= nnnn ; output "elementchargestoicheometry nnnn ; for i = 0 ; i < a -> nelem symm ; i ++ f sprintf bu "%5s%10.6f%d nn a -> elem [ a -> symm lable [ i ][ 0 ]1 ] l -> a [ i + 1 ], a -> nelem symm natoms [ i ]; output bu ; g output nn ================================================= nnnn ; g g 175

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AbouttheAuthorAbrahamCharlesSternwasbornonDecember29th,1979toRonaldandMinervaSterninHouston,Texas.AfterlivingbrieyinsouthernCalifornia,AbeattendedmostofhisearlyschoolinginBrevardCounty,Florida.AttheUniversityofSouthFlorida,AbeearnedaB.S.inChemistryandsubsequentlybeganstudiesforhisdoctoraldegree.Hisresearchhasfocusedmainlyoncomputersimulationofmetal-organicmaterials.Abelovesgolng,shing,tennis,skiing,surng,snowboarding,biking,hiking,cooking,music,andphilosophy.HeandhisbeautifulwifeMariapazlookforwardtotravelingthewholeworldtogether.


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Computer simulation of metal-organic materials
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ABSTRACT: Computer simulations of metal-organic frameworks are conducted to both investigate the mechanism of hydrogen sorption and to elucidate a detailed, molecular-level understanding of the physical interactions that can lead to successful material design strategies. To this end, important intermolecular interactions are identified and individually parameterized to yield a highly accurate representation of the potential energy landscape. Polarization, one such interaction found to play a significant role in H2 sorption, is included explicitly for the first time in simulations of metal-organic frameworks. Permanent electrostatics are usually accounted for by means of an approximate fit to model compounds. The application of this method to simulations involving metal-organic frameworks introduces several substantial problems that are characterized in this work. To circumvent this, a method is developed and tested in which atomic point partial charges are computed more directly, fit to the fully periodic electrostatic potential. In this manner, long-range electrostatics are explicitly accounted for via Ewald summation. Grand canonical Monte Carlo simulations are conducted employing the force field parameterization developed here. Several of the major findings of this work are: Polarization is found to play a critical role in determining the overall structure of H2 sorbed in metal-organic frameworks, although not always the determining factor in uptake. The parameterization of atomic point charges by means of a fit to the periodic electrostatic potential is a robust, efficient method and consistently results in a reliable description of Coulombic interactions without introducing ambiguity associated with other procedures. After careful development of both hydrogen and framework potential energy functions, quantitatively accurate results have been obtained. Such predictive accuracy will aid greatly in the rational, iterative design cycle between experimental and theoretical groups that are attempting to design metal-organic frameworks for a variety of purposes, including H2 sorption and CO2 sequestration.
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