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TheoreticalChemistryandComputational Modelling
EditorialBoard
ManuelAlcami,DepartmentofChemistry,UniversidadAutónomadeMadrid, Madrid,Madrid,Spain
RamónSayós,DepartamentdeCiènciadeMaterialsiQuímicaFísica,Universidad deBarcelona(UB),Barcelona,Spain
IñakiTuñón,DepartamentodeQuímicaFísica,UniversidaddeValencia,Burjassot -València,Spain
StefanoEvangelisti,LaboratoiredeChimieetPhysiqueQuantiques,Université PaulSabatier(ToulouseIII),Toulouse,France
NicolasSuaud,LaboratoiredeChimieetPhysiqueQuantiques,IRSAMC, UniversitéPaulSabatier,Toulouse,France
MonicaCalatayud,LaboratoiredeChimieThéorique,SorbonneUniversité,Paris Cedex05,France
NoeliaFaginasLago,DipartimentodiChimica,BiologiaeBiotecnologie, UniversityofPerugia,Perugia,Italy
MauroStener ,DepartmentofChemicalandPharmaceuticalSciences,University ofTrieste,Trieste,Italy
ShirinFaraji,DepartmentofTheoretical&ComputationalChemistry, RijksuniversiteitGroningen,Groningen,TheNetherlands
DanielEscudero,DepartmentofChemistry,KULeuven,Leuven,Belgium
JeremyHarvey,DepartmentofChemistry,KULeuven,Leuven,Belgium
MinhThoNguyen,DepartmentofChemistry,KULeuven,Leuven,Belgium
ModernChemistryisunthinkablewithouttheachievementsofTheoreticaland ComputationalChemistry.Asamatteroffact,thesedisciplinesarenowamandatory toolforthemolecularsciencesandtheywillundoubtedlymarkthenewerathat liesaheadofus.Tothisend,in2005,expertsfromseveralEuropeanuniversities joinedforcesunderthecoordinationoftheUniversidadAutónomadeMadrid,to launchtheEuropeanMastersCourseonTheoreticalChemistryandComputational Modeling(TCCM).ThiscourseisrecognizedbytheECTNA(EuropeanChemistry ThematicNetworkAssociation)asaEuromaster;ithasbeenpartoftheErasmus MundusMastersProgramoftheEACEA(EuropeanEducation,Audiovisualand CultureAgency)since2010.Theaimofthiscourseistodevelopscientistswhoare abletoaddressawiderangeofproblemsinmodernchemical,physical,andbiologicalsciencesviaacombinationoftheoreticalandcomputationaltools.Thebook series,TheoreticalChemistryandComputationalModeling,hasbeendesignedby theeditorialboardtofurtherfacilitatethetrainingandformationofnewgenerations ofcomputationalandtheoreticalchemists.
ÁngelMartínPendás · JuliaContreras-García TopologicalApproaches totheChemicalBond ÁngelMartínPendás
DepartamentodeQuímicaFísicay
Analítica
UniversidaddeOviedo Oviedo,Spain
JuliaContreras-García LaboratoiredeChimieThéorique SorbonneUniversité Paris,France
ISSN2214-4714ISSN2214-4722(electronic)
TheoreticalChemistryandComputationalModelling
ISBN978-3-031-13665-8ISBN978-3-031-13666-5(eBook) https://doi.org/10.1007/978-3-031-13666-5
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Foreword WithinthisseriesdedicatedtoTheoreticalChemistryandComputationalModelling (TCCM),thesubjectdevelopedinthisbookisofspecialrelevancewithregards preciselytocomputationalmodeling,sincethesimulationoftherealworldthrough theuseofquantum-chemicalmethodsisunavoidablybasedonthemathematical itemthatdescribesthatreality,thatis,onthewavefunction.WhentheSchrödinger equationforanychemicalsystem, H ψ = E ψ ,issolved,theenergyofthesystem andthewavefunctionthatdescribesit,areobtainedsimultaneously.Theproblem, however,isthattheSchrödingerequationcanonlybesolvedexactlyinveryfew situations,whichimpliesthatforalmostthetotallyofthesystemswewishtodescribe, bothenergyandwavefunctionareapproximateentitiesThismeansthat,althoughin mostcaseswhentheHamiltonian, H ,intheSchrödingerequationistheexactone, thereisnowayofobtainingtheexactwavefunctionofthesystemtobedescribed, andthereforeneithertheexactenergycanbeevaluated.Butitispreciselythewave functiontheentitythatallowsusto“see”,understandandinterpretthepropertiesof thesystem,simplybecauseaccordingtooneofthepostulatesofquantummechanics, althoughnophysicalmeaningcanbeassociatedwiththewavefunction,sucha physicalmeaningcanbeascribedtoitssquare,admittingthatthesquareofthe wavefunctionhasaprobabilisticinterpretation.Thus,ifonewishestounderstand thepropertiesoftheparticles,nucleiandelectronsensemble,thatmakeupany chemicalsystem,onemustanalyzethepropertiesofamathematicalfunctionof manyparticles,whichisthesquareofthewavefunction.Theconsequencesofthis assertionareobvious,sothatinquantummodelingthedifferentwaystowithdraw usefulinformationfromthesquareofthewavefunctionbecomecrucialtoobtain apreciseview,oratleastaspreciseaspossible,oftheparticlesdistributionand, consequently,ofthepropertiesofthesystemthatwearetryingtomodel,andthisis preciselytheobjectiveofthebookyouhaveinyourhands,toprovideatimelyand completeoverviewofthedifferentformalismsabletogetusefulinformationfrom thewavefunctionobtainedbysolvingtheSchrödingerequation.
Letmeaddthatthemethodologicalcompilationthatisthecontentsofthisbook itisnoteasilyfound,atpresent,inotherpublicationsinthisarea.Here,youwillbe
abletofollowrigorousdeductionsandexplanationsofthedifferentmathematicaland physicalconceptsbehindasignificantlylargevarietyofthemethodologiesnowadays available,toanalyzeawavefunction,specificallyadaptedforpostgraduatestudents.
Onecanconsiderthatthestorybeganinthefiftiesofthepastcentury,whenprofitingthatintheframeworkoftheBorn–Oppenheimerapproximation,thesquareof thewavefunctionprovidesaprobabilisticdescriptionofthedistributionoftheelectrons,thefirstmethodsabletocalculatetheprobabilityoffindingelectronsassociated withthedifferentnucleiinthephysicalspaceweredeveloped.Theseformalisms, usuallyknownasPopulationAnalyses,werethefirstattempttounderstandtherules thatgovernthepolaritywithinthedifferentchemicalentities,andtoformulaterules abletounderstandthevaluesofmultipoles(dipoles,quadrupoles,octuples…)within therealmofmolecularchemistry.
However,wecannotforgetthat,afterall,theinformationtobeobtainedisassociatedwithamathematicalfunctionthat,asmentionedintheIntroductiondwellsin Hilbert,ratherthaninthephysicalspace.Theconsequenceisthatthemethodology tobeusedtoobtaintheinformationspeaksalanguageverydifferentfromthatused bychemistsintheireverydayactivity,simplybecausemoleculesliveinthephysicalspace,butassaidabovethefunctionthatprovidesthatinformationdwellsina Hilbertspace.Thisisoneofthevirtuesofthisbook,toleadusbythehand,without losingthenecessarymathematicalrigorinthedescriptionofthemethodologytobe used,toadeepunderstandingandclearcomprehensionofthepropertiesofatoms, molecules,bonds,non-covalentinteractionsandotherendless“real”worldfacts. Frommypersonalviewpoint,aparadigmaticexampleofthisvirtuewouldbe,for instance,thedeductioninChap. 2 ofMorse’sequalitythatestablishestherelationshipbetweenthenumberofcagepoints,ringpoints,bondpointsandthenumber ofnucleiforanychemicalsystem;buttherearemanymoreallalongthefollowing chapters.
Theorganizationofthebook,ontheotherhand,isperfecttoachievethisgoal, startingwiththeIntroductioninwhich,itismadecleartothereaderthattheanalysisoftheelectrondensityallowstoobtainaveryrichinformationcontainedin thewavefunction.Anintroductorychapterontopologicalspaces,includingscalar fieldsandsomeveryinterestingadvancedmaterial,suchasthesectiondevotedto topologicalinvariants,isindeedessentialtounderstandcorrectlytheanalysisofthe electrondensity,whichispreciselyascalarfieldwhosetopologicalcharacteristics areofaparamountimportancetoachieveasoundandcompleteviewofaconcept centraltochemistry:thechemicalbond.Inthissensethesectiondedicatedtotopologicalinvariantsconstitutesanoveltythatisdifficulttofindcompletelydeveloped inpreviouspublications,whichthroughtheanalysisofthecriticalpointsofadynamicalsystemsprovidesaperfectbackgroundtounderstandthebasisoftheQTAIM (QuantumTheoryofAtomsinMolecules)developedinthefollowingchapter,and thatculminateswiththedeductionofMorse’sequalityImentionedabove.
Veryinterestingisthepresentationofchemicalrealityasafascinatinggamein whichboth,localizationanddelocalizationoftheelectrondensity,playfundamental roles,andhowtheyareconnectedwithotherpioneeringideasas,forinstance, thetheoryoflogesofR.Daudel,thelinkbetweenlocalizationcriteriaandthe
formationofelectronpairs,theequivalencebetweenhole-localizationandelectronlocalization,theusefulnessofthelocalizationanddelocalizationindexes,orhow theseeffectscharacterizewell-knownchemicalconcepts,amongthemtheinductive andthemesomericeffects,orwhyratheroldmodels,suchastheBerlin’sregions, arerecoveredwhenmanyofthesenewformalismsareapplied.
Manypeople,amongthemmanychemists,arecompelledtothinkthationicand covalentbondingarethefundamentalnecessaryconceptstorationalizethewhole chemicalworld,andtheanalysisofchemicalbondinginthisbookisreallyanice andcompletepieceofwork,butinthisbookemphasisismadeonthefactthat nowadaysismoreandmoreevidentthatitisimpossibletounderstandchemical realityignoringthenon-covalentinteractions.However,non-covalentinteractions alwaysrepresentachallengebecause,bynature,theyareweakandthereforevery difficulttobeadequatelyandaccuratelydescribedbytheory.Theyareforsureessentialtounderstandmanybiochemicalphenomena,amongotherreasonsbecausethey arebehindchemicalrecognitionandmolecularassembly;butmorerecentlythey werealsofoundtoplayimportantrolesinmaterialscienceandcrystals,aswell asinatmosphericchemistryandastrochemistry.Anotherrichnessofthebookyou haveinyourhandsisthatitexploresadvancedmethodologyfortheanalysisof non-covalentinteractionswhichcannotbefoundinotherbooksontheoreticaland computationalchemistry.Indeed,someoftheapproachesdescribedherearevery recentcontributionsinthefield.TothisgroupbelongmethodssuchastheLocalized OrbitalLocator(LOL)formulatedintheyear2000,theElectronLocalizationIndicator(ELI)proposedfouryearslater,theElectronPairLocalizationFunction(EPLF) firstdefinedin2004,butmodifiedin2011tomakeitanalyticallycomputablewith standardwavefunctions,obtainedwithbothDFTorabinitiomethods,theLocalizedElectronDetector(LED)initiallyproposedin2010,ortheDensityOverlap RegionsIndicator(DORI)formulatedin2014;butthebookalsopresentsthefirst successfulattemptstowelldescribeandclassifyintermolecularinteractionsthrough theuseoftheQuantumTheoryofAtomsinMolecules(QTAIM),theElectron LocalizationFunction(ELF)andtheNon-CovalentInteractions(NCI)formalism, inwhosedevelopment,oneoftheauthorsofthisbook(J.C.-G.)hadanimportant role.Tocompletethisperspective,theauthorswereabletodescribeindetailhow theseformalismschangetheperspectiveofmodernchemistryusingrepresentative casesofweakinteractionssuchashydrogenbonds,thecoordinationofbiomolecules aroundmetalcenters,stericrepulsionsorvanderWaalsinteractions.
AlthoughQTAIMhasbeen,Iwouldsay,heavilyusedbytheoreticalandcomputationalchemistsalongthelasttwodecadesofthelastcenturyandalongthefirsttwo decadesofthepresentoneand,thereforeitcouldbeconsidereda“wellknown”tool inthisdomain,thepresentationofsuchaformalisminthisbookisoriginalindeed andveryrigorous.ManyreadersperhapsareawarethatQTAIMtheoryisrelated withthetopologicalanalysisoftheelectrondensity, ρ ,ofamolecularsystem,but perhapsnotmanypeopleisconsciousthat ρ isnotatrulydifferentiablefield,due tothecuspsofthisfunctionatthepositionofthenuclei,whichinprincipleshould beaseriousproblemwhentryingtoanalyzeitstopology.Detailslikethis,which arecurrentlyomittedinotheranalyses,makethisworkparticularlyattractiveand
rigorous.Beautifularealsotheillustrationsusedtoexplainthelinkbetweenthe chemicalconceptofbondandthe(3, 1)criticalpointsofthedensity,andtheway inwhichthemoleculargraphs,whichareveryusefulcomponentsofthetopological analysisof ρ arebuiltup.Veryimportantly,however,theauthorsalsowarnthereader aboutthecasesinwhichthemoleculargraphdoesnotnecessarilycoincidewiththe chemicalgraph,indicatingalsothatsituationmayeasilyappearwhendealingwith molecularstructuresoutsideequilibrium.Alongthisline,itisalsoveryinteresting thedistinction,outlinedbytheauthorsfollowingthesuggestionsofProf.PaulPopelier,anauthorityinthisfield,betweenbondpath,bondinteractionlineandtheplain chemicalconceptofbond.Itisalsoacharmingnoveltytheparallelismestablished, inoneofthesectionsofthesamechapter,ofthefluxoftheelectrondensitywiththe fluxofanincompressiblefluid.Nolessinterestingisthediscussionofrelativistic effectsinthisformalism,oritsextensiontothedescriptionofthetimeevolutionof openquantumsubsystems.Allthisbackgroundwillallowinsubsequentsections todiscussconceptslikemomentumdensityanforcedensitythat,onlyinthisbook, areanalyzedindeepshowingthattheyconstitutethefundamentalideasofwhatis nowadaysknownasquantumfielddynamics.
SomethingsimilarcanbesaidasfarasthechapterdevotedtotheELF(ElectronLocalizationFunction)isconcerned,inthesensethattheauthorsprovidethe readerbeautifulargumentsjustifyingthemodel,thatarenoteasytofindinother presentationsofthetheory,playingwiththeintrinsicinteractionsbetweenfermions andthesymmetryrequirementsthewavefunctionthatrepresentthemmustfulfill, oraddressingquestionslike‘ifelectronsaredelocalized,whydoweobserveLewis pairs?’tobuildabridgebetweenthistheoreticalschemeandtheclassicalideasof LewisandLinnet,andthefundamentalroleofthespinconcept.Asamatterof fact,tobehonestImustsaythatthehistoricalintroductiontothismethodologyisa wonderfulpieceofwork.
Anotherimportantfeatureofthisbookconcerningthetopicofnon-covalentinteractionsisthesectiondedicatedtoveryrecentdevelopmentsaimingatproviding aquantificationoftheformalismsnormallyusedtoanalyzethem.Ingeneralthe differentmethodsmentionedabove,inthispreface,providedifferentwaystolookat theelectronredistributionsassociatedwiththeappearanceofnon-covalentinteractions,butstillitisachallengetoestablishaquantitativerelationshipbetweenthem andtheenergeticsoftheinteraction.Asectionofthebookaddressesthisimportant question,evenifthisquantificationremainsstilllimited.Itisalsoremarkablethe sectiondevotedtotheuseofNCIformalismstoanalyzenon-covalentinteractionsby revealingelectronlocalizationthroughtherelationshipbetweenNCIandthekinetic energydensity.Theexamplesprovidedtoillustratethispointarereallybeautiful.
Thelastsectionofthebookdedicatedtoapplicationswascarefullydesigned, tryingtopresentawidescopeoffieldsinwhichtheuseofthemethodologydescribed intheprevioussixchaptersmaybecomeanalmostmandatorytooltowellunderstand manyfeaturesofthesystemsunderscrutiny.OneimportantaddedvalueofChap. 7, thefirstchapterofthissection,wastheabilityoftheauthorstoinclude,amongthe differentexampleschosentoillustratethepotentialityofthesetheoreticaltoolsto helpintheunderstandingofthestructure,bonding,reactivityandagreatvarietyof
propertiesofmolecules,agooddealofcontroversialcaseswhereothermolecular descriptorsarenotabletoprovideaclearviewoftheaforementionedproperties.On topofthatthelastpartoftheseventhchapterisnotonlydedicatedtoexplainagreat varietyofcharacteristicsofmoleculesandclusters,butalsotopredictthebehavior ofthesesystems,mainlyinthedomainofthechemicalreactivity.Forthispurpose, again,theillustrationsusedwerecarefullyandcompetentlyselected.
Chapter 8 thatclosesthesecondpartofthisbookislikeafinalgift,becauseit addressesatopicnoteasilyfoundwellcompiledintheliterature:howtoextend thetopologicalmethodsdescribedinthepreviouschapterstothesolidstateand inparticulartoperiodiccrystals,whichrequiresacarefulanalysisoftherelationshipbetweentopologyandperiodicity.Thisopensthedoorfortheunderstandingof manyofthepropertiesofmolecularcrystalsthroughtheanalysisoftheintermolecularbehaviorinthesecrystals.Thenewviewpointsprovidedbythesetopological approachesimplytoextend,asanovelty,manyoftheideasusedinthechemical domaintotheunderstandingofpropertiesandbondinginthesolidstate,butnot restrictedonlytoperiodicdomainsbutalsoextendedtothechangesthattakeplace alongphasetransitions.
ThePartIIIofthebookwastomeawonderfulsurprise.Ingeneral,inmost oftextbooksinscience,thesectiondevotedtoexercisesisjustacompilationof wellselectednumericalproblemsorconceptualquestionsrelatedwiththetopic developedinthebook.Inthisparticularcase,Chap. 10 isanextremelyimportant partofthisbook,wherethereadercanfindadescriptionnotusuallyfoundintext books,thewayofconnectingthescientificconceptsofthedifferentformalismsto thenumericalresultsthatallowtointerpretthem.Thisrequirestobuiltupefficient algorithmswhicharesummarizedalongthischapter,payingparticularattentionto thelocationofcriticalpointsandtobasinintegration,includingimplicit,explicit, gridandadaptivealgorithms,eventhoughbasinintegrationhasnotyetreachedan optimalhandling.Thelastsectionofthisexcellentchapterprovidesanextremely usefuloverviewofmostofthecodes,freelyavailable,toanalyzethetopological featuresofavarietyofscalarfields.
ThisworkcloseswithalastchapterdevotedtoExercises;butagainthisis notaconventionalchapterinwhichdifferentquestionsornumericalproblemsare proposedtothereader,asinmosttextbooks.Heretheauthorsguidethereaderalong thedifferentstepsoftheprocess,startingwithhowtoobtainthewavefunction,and continuingwiththeroutetobefollowedtoefficientlyusethedifferentcodes,always wonderfullyillustrated,inthelaststepwithexercisesaimingatgettingthereader familiarwiththecharacteristicsofeachformalismandcode.
Myfinalwordswillbetoassurethereaderthattheexcursionalongthepagesof thisbookisgoingtobeawonderfulexperience.
June2020ManuelYáñez EmeritusProfessor,Department ofChemistry UniversidadAutónomadeMadrid
Madrid,Spain
Preface WhenIstartedmyPh.D.,Iwasconfrontedtoatopicthathadno“academic”book tostart.Iwasgivenmanyspecialized“papers”andhadtofindmywaythrough. Thisbookisintendedtoprovideathorough(evenifnotcomplete)background ontopologicalanalysisinrealspaceofelectronicstructureforstudentsatthemaster level.
Thisbookexpectstomaketheirlifeeasier.Thissaid,itisbasedontheTCCM lectureswehavebeengivingforseveralyearsnow,soitdoesnotintendtobe complete,andweapologizeforallthethingsthathavebeenleftaside!
Paris,France
Oviedo,Spain
Mayo2020
JuliaContreras-García ÁngelMartínPendás
Acknowledgements (ThisscenehappensinaParisianappartmentduringconfinement)
JuliaContreras-García: Manystudentshavedirectlyorindirectlycontributed tothisbook.Morespecifically,agoodpartoftheartworkpresentinthismonograph (inmostcases,themostvisuallyappealingone)hasbeenpossiblethankstothem. Thankyousomuchforimprovingmynatural(in)capacitytocreatenicefigures.
JulenMunarrizandRobertoABotohavehadagreatcontribution,butIalsowantto thankRubenLaplazaandFrancescaPeccatifortheircontributiontoChap. 5.
Iwanttoacknowledgethemfortheirwork,andforenlighteningmydailylife. Theyhavereallygivenmeagreatworkingenvironment.
Thisbookstartedlongtimeago,aspartofmyPh.D.thesis.Hence,Ihavetothank myfamily.Whoeverhaswrittenathesis,knowsthatyourdearestoneshavetoputup withathesis,butinthiscase,myfatherwasthemostinsistingpersononthisbook beingwritten.Bernard,forinsistingthatIshouldwritethisbook,andManuelfor crystallizingit.Ángelformakingitsoeasytoworktogether.Andagainmyfather whokeptputtingpressureonmeallthewaythrough!Andofcourse,thanksomuch toEduardforhisthoroughproof-reading!Andsincethisbookwasmainlywritten duringconfinement,IalsowanttorecalltheSopeñasforthetimespenttogetherin confinement.Butaboveall,tomyfamilywhoareaproofofunconditionalsupport, confinementbeingjustanyotherexample.
(SimultaneouslyintheNorthofSpain...) ÁngelMartínPendás: Co-writingabookseemsahardenterpriseatfirst.Inthis caseithasbeenapieceofcake,forJuliaandIhadalreadydiscussedmanytimes abouthowthemaintopicofthismonographshouldbetaughttostudents.Aprevious setofhandoutswrittenyearsagoinSpanishtohelpMasterstudentswerealsoan invaluablehelp.Ithankallthosestudentswhoconvincedmethatthosenoteswere helpful.Withoutthem,andwithoutManuel’sinsistence,thisinitiativewouldhave nevercometolight.IalsowanttothankMiguelGallegosforhishelpwithfigures.
Finally,wewanttothankBrunoLanderosandFernandoJiménezGrávalosfor makingsurethattheexerciseswheredoable!
1Introduction ..................................................1
1.1QuantumTheoryandtheChemicalBond.....................1 1.2AnalyzingScalarFields....................................3 1.3TopologicalAnalyses......................................6
2TopologicalSpaces ............................................9
2.1TopologicalSpaces........................................9
2.2DynamicalSystems(DS)...................................13
2.2.1CriticalPointsofaScalarField.......................14
2.2.2Basins,SeparatricesandInducedTopologies............19
2.3TopologicalInvariants(Advanced)...........................21
PartIDescriptors
3.3PropertiesatCriticalPoints.................................42
3.3.1TheDensityataBondCriticalPoint...................43
3.3.2Ellipticity..........................................44
3.3.3BondRadii.........................................45
3.4TheQuantumTheoryofAtomsinMolecules..................46
3.4.1SurfaceTerms......................................47
3.4.2Hermiticity........................................49
3.4.3TimeEvolutionofOpenSystems......................54
3.4.4EquationsofMotioninOpenSubsystems..............62
3.4.5One-ElectronAtomicObservables.....................66
3.4.6AtomicProperties...................................70
3.4.7TransferabilityofAtomicProperties...................75
3.5ReducedDensityMatricesandMultielectronDescriptors.......77
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