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ChemistryattheFrontierwithPhysics andComputerScience TheoryandComputation
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Chemistryatthe FrontierwithPhysics andComputerScience TheoryandComputation SergioRampino
ScuolaNormaleSuperiore
FacultyofScience
Pisa,Italy
Elsevier
Radarweg29,POBox211,1000AEAmsterdam,Netherlands
TheBoulevard,LangfordLane,Kidlington,OxfordOX51GB,UnitedKingdom 50HampshireStreet,5thFloor,Cambridge,MA02139,UnitedStates
Copyright©2022ElsevierInc.Allrightsreserved.
Nopartofthispublicationmaybereproducedortransmittedinanyformorbyanymeans, electronicormechanical,includingphotocopying,recording,oranyinformationstorageand retrievalsystem,withoutpermissioninwritingfromthepublisher.Detailsonhowtoseek permission,furtherinformationaboutthePublisher’spermissionspoliciesandourarrangements withorganizationssuchastheCopyrightClearanceCenterandtheCopyrightLicensingAgency, canbefoundatourwebsite: www.elsevier.com/permissions.
Thisbookandtheindividualcontributionscontainedinitareprotectedundercopyrightbythe Publisher(otherthanasmaybenotedherein).
Notices Knowledgeandbestpracticeinthisfieldareconstantlychanging.Asnewresearchand experiencebroadenourunderstanding,changesinresearchmethods,professionalpractices,or medicaltreatmentmaybecomenecessary.
Practitionersandresearchersmustalwaysrelyontheirownexperienceandknowledgein evaluatingandusinganyinformation,methods,compounds,orexperimentsdescribedherein.In usingsuchinformationormethodstheyshouldbemindfuloftheirownsafetyandthesafetyof others,includingpartiesforwhomtheyhaveaprofessionalresponsibility.
Tothefullestextentofthelaw,neitherthePublishernortheauthors,contributors,oreditors, assumeanyliabilityforanyinjuryand/ordamagetopersonsorpropertyasamatterofproducts liability,negligenceorotherwise,orfromanyuseoroperationofanymethods,products, instructions,orideascontainedinthematerialherein.
ISBN:978-0-323-90865-8
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ToCécile,JulieandLilas This page intentionally left blank
6.Thepotential-energysurface
7.Theoreticaltreatments
8. Fromtheorytocomputing :collinearreactivescattering withrealwavepackets
9.Fromreactiondynamicstochemicalkinetics
11.Towardscomplexity
13. Fromtheorytocomputing :theHartree–Fockmodel
13.1TheRoothaan–Hallequations
13.3Basisfunctionsandone-andtwo-electronintegrals
14.Theatomandthebond
14.1Partitioningschemes
15. Fromtheorytocomputing :analyzingthe electron-chargeredistribution
15.1Object-basedprogramming
16. Application:donationandbackdonationin coordinationchemistry
16.1Themetal–carbonylcoordinationbond
17.Relativityandchemistry 17.1Relativisticquantumchemistry
PartIV Chemistryandcomputerscience 19.Virtualreality
19.1Scientificvisualizationandvirtualreality
20.Data-drivenchemistry Biography SergioRampino SergioRampinowasborninMesagne(Apulia,Italy)in1984.Hegraduated withhonorsinChemistry(2007)andItalianLanguageandLiterature(2012)at theUniversityofPerugia,wherehealsoobtainedhisPhDinChemistry(2011). In2017hewasappointedlecturerinTheoreticalandComputationalChemistry attheScuolaNormaleSuperioreinPisa,wherehepresentlyteachesbothundergraduateandPhDstudents.Hisresearch,partlycarriedoutatseveralEuropean researchandcomputingcenters,hasfocusedonseveraltopicsofgeneral,physicalandinorganicchemistryrangingfromthequantumdynamicsofelementary reactionstorelativisticdensity-functionaltheory,theanalysisofchemicalbonding,andtheuseofvirtual-realitytechnologyforchemistry.In2016hewas awardedthe‘EoloScrocco’prizebytheDivisionofTheoreticalandComputationalChemistryoftheItalianChemicalSociety.
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Preface Ognipetraàzaparete. (‘everystonemakesthewallalittlehigher’,oldapuliansaying)
WhenIenteredtheclassroomtoattendmyfirstuniversitylesson,thiswasageneralchemistryoneandIwasaneighteen-year-oldstudentwithabackgroundin artsandhumanitiesandwithnoideaofwhattoexpectforthenextfiveyears.It tookmeseveralmonthstobecomeacquaintedwithanewparadigmofconceivingandrelatingtoreality,andacoupleofyearsofclassesandlaboratoriesof organic,physical,inorganicandbiologicalchemistrybeforemycompasscould findtheNorthinahighlyfascinating,albeitlesspopular,facetofchemistryat thecrossroadswithphysicsandcomputerscience.
Itisinthissubdisciplineofchemistry,labeledbytheadjectives‘theoretical andcomputational’,thatIhavepursuedtheacademicpathsincethestartofmy PhDprojectin2007.InaboutfifteenyearsofresearchIhavebeenluckyenough toworkonsuchdiverseandfascinatingsubjectsasthedynamicsofchemical reactions,theanalysisofchemicalbonding,theeffectsofrelativityonchemistry,andtheintegrationofimmersivevirtualrealityinchemicalapplications.I metextraordinarypeopleandhadthechanceofvisitingresearchandcomputing centersinwonderfulplacessuchasEdinburgh,Bristol,Toulouse,Paris,Zurich, BarcelonaandCracow,thefondmemoriesofwhichshallforeverbeinextricably linkedwiththetopicsofmyresearch.
InJuly2020,IreceivedaninvitationbythePublishertoauthoranewbook inthefieldoftheoreticalandphysicalchemistry,andIhadnohesitationinacceptingasIregardeditasanopportunitytosumuptheworkofalltheseyears andofferapersonalperspective,withconcreteexamplesfrommyownresearch activity,onthedisciplineonwhichIhad,overtime,specialized.Iamextremely gratefultotheacquisitionseditorAnnekaHessforthatcontact.Duringthefollowingmonths,IreceivedprecioushelpfromprojectmanagersDevlinPerson andVeronicaIIISantos,andfromthepermissionscoordinatorNarmathaMohan.Iamgratefultothemfortheirsupportduringthevariousstagesofthebook production.IwouldalsoliketothankVickyPearsonEsserforherexcellent
jobindesigningthebookcover,andtheproductionmanagerSruthiSatheesh, togetherwithanyotherpersonwhowasinvolvedintheproductionofthebook.
Theorganizationofthisbookpartlyreflectsthestructureandcontentofthe coursesthatIheldattheScuolaNormaleSuperioreinPisa,Italy,from2017to 2022,andofaseriesofoutreachlecturesthatIdeliveredforayoungeraudience withintheinitiative‘LaNormaleaScuola’organizedbytheScuolaNormale Superioreasasupporttohigh-schoolstudentsduringtheCovid-19globalhealth emergency.Iamgratefultoallthestudentswhoattendedmycoursesoverthese yearsfortheirpreciousfeedback.Inturn,thepresentationofthetheoretical aspectsofthebook,especiallythoserelatedtothefundamentalsofquantum mechanics(Chapter 2)andelectronic-structuretheory(Chapter 12 andpartof Chapter 13),drawsontheintroductorycoursesintheoreticalchemistryheldby Prof.FrancescoTarantelliattheUniversityofPerugiain2005–2007,whilethe descriptionoftheVerletalgorithminSection 7.1 drawsonusefullecturenotes byProf.HannesJónsson.
IamverygratefultomyacademicmentorsProf.AntonioLaganàand Prof.VincenzoBarone,fortheopportunityofworkingwiththemandforthe many,manystimulatingdiscussions,andtoallthepeoplewhohavecollaboratedwithmeovertheyears,manyofwhomfigureascoauthorsofthearticles citedinthisbook.Inparticular,IamgratefultoStefanoCrocchianti,Dimitrios Skouteris,andLorianoStorchiforinitialguidancein,respectively,scientific programming,quantumreactivescattering,andparallelcomputing,andtoProf. AntoninoPolimenoandProf.MircoZerbettofortheopportunityofworking oninterestingnewtopicsopeningnewperspectivesinmyresearch.Iamalso gratefultoDr.KennethLawley,Prof.KimBaldridge,Prof.StefanoEvangelisti, Prof.GabrielBalint-Kurti,Prof.ErnestoGarcía,Prof.IreneBurghardt,Dr.MariuszSterzel,andProf.CaroleMorrisonforhostingshort-termscientificvisitsto theirgroupsfinancedbytheEuropeanCOSTandHPC-Europaprogrammes. SpecialthanksarefinallyduetomyyoungercollaboratorswhoseworkIhad thechanceofsupervisingandtowhomIwishthebestfortheirfuture:Matteo DeSantis,LorenzoPaoloni,SimonePotenti,SurajitNandi,BernardoBallotta, LucaSagresti,SilviaAlessandrini,andGiovanniNottoli.
ThewritingofthisbookhasoccurredtomeatatimewhentheIrishflute enteredwildlyintomylife.IamgratefultofellowmusiciansStefanoBattaglia, YuriBernardini,DiegoCeccarelli,LorenzoDelGrande,AntonioMalacarne, NicoMarraccini,CarloRogo,andGiovanniSteaforthehundredsofjigsand reelsandforthememorable craic,andtomaestroCarloIpataforseveralfruitfulandlivelyflutelessons.Ithasalsooccurredinnotsoeasytimesofprecarity, sometimesparadoxicallymadeharderbythesametransdisciplinaryvocation thatanimatesthisbook,andthathardlyreconcileswiththehighdegreeofspecializationthatcharacterizesthecurrentacademicworld.Fortheirsupportand theirpresence,Iamgratefultomyfamily,CécilePirat,JulieandLilasRampino, tomyfamilyoforigin,GabrieleRampino,IreneandLuciaAmmaturo,Ade-
Preface xv
laide,Guido,GiulianaandDanieleRampino,toDrs.PaoloCatanzaroandAriannaLuperini,andtoAriannaFederici. TothelovingmemoryofJonMikelAzpiroz.
SergioRampino ScuolaNormaleSuperiore March2022
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Chapter1 Introductionandscope 1.1Introductionandscope Thisbookisanintroductiontotheoreticalandcomputationalchemistrythat aimsatframingchemistrybetweenitsneighboringdisciplinesphysicsandcomputerscience.Modernchemistrymaybedefinedasthebranchofsciencethat aimsatmodelingandunderstandingthepropertiesandbehaviorofmatteratthe atomicandmolecularlevel.Theoreticalandcomputationalchemistry,asopposedtoexperimentalchemistry,isasubdisciplineofchemistrypursuingthe sameobjective‘comfortably’frombehindadesk,withtheaidofpenandpaper aswellasofmoderninformationandcommunicationtechnology,ratherthanin alaboratory.Inotherwords,itusesthelawsofphysicstopredictthebehavior ofmolecularsystemsintermsoftheirconstitutingparticles,thenucleiandthe electrons.
Thiswellexplainsthefirstadjective(‘theoretical’)1 ofthedyadthatcharacterizesthisdisciplineandthatappearsasasubtitletothisbook.Thesecond adjective(‘computational’)2 owesitsraisond’êtretothefactthatthemathematicalapparatusinvolvedbythetheoryissocomplexthattheuseofacomputer ismandatoryinordertoefficientlyperformtherequirednumberofoperations andtohandletheinvolved(oftenhuge)amountofdata.
Theoryandcomputationarealsothetwoextremesofajourneywithintermediatestagesthatleadsfromphysicstochemistrythroughmathematicsand computerscience,andthatfitswellintothefollowinghierarchicalscheme:
• Theory
• Model
• Method
• Implementation
• Computation
Thesefivestages,whichreflectthedailyactivityofatheoreticalandcomputationalchemist,maybedescribedasfollows.Thephysicaltheoryrequiredfor anaccuratemodelingofatomsandmoleculesturnsouttobetoocomplexand oneroustobeusedasis,sothatapproximationsareoftenintroducedthrough
1 Theword‘theory’curiouslyderivesfromaGreekrootmeaning observation,whichisthemain activityofthe‘experiment’.Theoryisindeedconstructedbyrationalizingtheoutcomeofobservation,andthenusedtomakepredictionsonnewobservations:onemaysuccinctlysaythatitfollows observationinordertoprecedeit.
2 Theword‘computation’hasaLatinrootmeaningtocount,tocalculate.
modelsthatsimplifythephysicalandmathematicaltreatment.Modelsareput intoactionthroughmethods,whichinturnrequireaneffectiveimplementation onacomputer.Thisleadstothelaststage,computation,wherecalculationsare performedandtheiroutcomeiselaboratedandtransformedintochemicalinsight.
Thepurposeofthisbookistoillustrate–andaccompanythereader through–thisjourneybyconjugatingthebigpicturetothemostpracticalimplicationsor,inotherwords,byframingchemistryintheoverallcontextwhereit relatestophysicsandcomputersciencewhileatthesametimeofferingpractical examplesofhowtosolveachemicalproblemfromatheoreticalandcomputationalperspective,goingfromtheorytomodels,methods,implementation,and computation.Forthatpurpose,thetwocentralconceptsinchemistryofchemicalreactionandchemicalbondingwillbetakenasleadingthemestoillustrate thetreatmentofthetwobasicparticlesthatconstitutematterattheatomic andmolecularlevel:thenucleiandtheelectrons,respectively.Accordingly, afterreviewingthephysicsofmolecularsystemsandintroducingtheBorn–Oppenheimerapproximation,whichallowsfortheseparationofthenuclearand electronicmotions,theconceptsofchemicalreactionandchemicalbonding areexploredwithreferencetotheunderlyingnucleardynamicsandelectronic structure.
Constantattentionisgiventhroughoutthebooktopracticalaspectsconnectedtothenumericalresolutionoftherelevantequationsandtherelated implementationincomputerprograms,whichhaveplayedakeyroleinthe developmentofthedisciplineitself.Forachoiceofthreephysicalproblems (wavepacketdynamics,Hartree–Fockequations,electron-cloudredistribution) adetailedaccountfromtheorytoimplementationisgiventhroughtheseries ofchapters Fromtheorytocomputing,whiletheapplicationofthediscussed methodologiestorelevantchemicalproblemsfromtheauthor’sresearchareillustratedthroughthechapterseries Applications.Computer-codeexampleswill begivenintheFortran95programminglanguage,andshouldbefullyaccessible toreaderswithintermediate-levelexperienceincomputerprogramming.Onthe otherhand,therelationbetweenchemistryandcomputerscienceisnowadays rapidlyevolvingduetotheconstantprogressininformationandcommunicationtechnology,sothattheroleofcomputerscienceinchemistrygoeswell beyondthetraditionalaspectofscientificcomputing.Thefinalpartofthebook dealswiththemostrecentperspectivesduetotheconnectionofchemistrywith computerscience.
Inmoredetail,thebookisstructuredasfollows:
InPart I, Physicsandchemistry,thephysicsofmolecularsystemsisreviewedwithemphasisontheparadigmchangefromclassicaltoquantummechanics(Chapter 2).Chapter 3 explorestherelationbetweenphysicsandchemistryandprovidesanintroductiontotheconceptsofchemicalreactionsand chemicalbondingthatwillbedevelopedinParts II and III,respectively,witha focusontheperspectiveofachemistasopposedtothatofaphysicist.Chapter 4
providesabriefhistoricalaccountonthebirthandevolutionoftheoreticaland computationalchemistrywithreferencetothetopicstreatedinthisbook.
Part II, Nucleardynamicsandchemicalreactions,isdevotedtothetreatment ofthenuclearmotionandtheillustrationofthemodelingofchemical-reaction dynamics.Inparticular,Chapter 5 revisitschemicalreactionsfromthephysical perspectiveofcollisiontheory.Chapter 6 isdevotedtoacornerstoneconcept inchemistry,thatofthepotential-energysurface.Chapter 7 reviewsthemain theoreticalapproachesforthetreatmentofthenuclearmotion.Chapter 8,the firstoftheseries Fromtheorytocomputing,focusesontheimplementationof aquantummethodformodelingtheevolutionofasimplereaction.InChapter 9,therelationbetweenthemicroscopicviewofsinglecollisionalprocesses andthemacroscopiconeofchemicalkineticsisaddressed.Chapter 10 illustratesthestudyofanastrochemicalreactionthroughthepreviouslydiscussed methodologies.Finally,Chapter 11 providesanoverviewofthemainmethods tacklingnucleardynamicsinmorecomplexcontexts.
InPart III, Electronicstructureandchemicalbonding,wemoveontothe othersideoftheBorn–Oppenheimerapproximationanddealwiththetreatment oftheelectronsandtheirroleinchemicalbonding.Afterillustratingthefundamentalelectronic-structuretheory(Chapter 12),theimplementationofabasic electronic-structureprogramisdiscussedinChapter 13 (secondoftheseries Fromtheorytocomputing).Thenthechemicalconceptsofatomsandbonds arerevisitedintermsofthepreviouslyintroducedmathematicalandphysicalformalism(Chapter 14),andtheimplementationofacomputerprogram foranalyzingtheelectron-chargeredistributionoccurringuponchemicalbondingisdiscussed(Chapter 15,thirdoftheseries Fromtheorytocomputing). Chapter 16,throughastudyofchemicalbondinginadiverseclassofmetal–carbonylcomplexes,illustrateshowthechemicalconceptsof σ -donationand π -backdonationcanbemodeledthroughthepreviouslydiscussedmethodologiesrootedinthequantummechanicsoftheelectrons.Chapter 17,thelast chapterofPart III,dealswiththefrontiertopicoftheeffectsofrelativityin chemistry,withafocusonthechemistryoftheso-calledsuperheavyelements.
Part IV, Chemistryandcomputerscience,thefinalpartofthebook,explores theconnectionsbetweenchemistryandcomputerscience.Inparticular,Chapter 18 focusesonthetraditionalaspectofscientificcomputing.Afterreviewing thebasicsofcomputerprogramming,thetopicofparallelcomputing(inits high-performanceandhigh-throughputflavors)isdiscussed,andaconcreteexampleofparallelizationofarelativisticelectronic-structureprogramisgiven. Chapter 19 showshowstate-of-the-artvirtual-realitytechnologycanenhance scientificresearchandeducationinchemistrythroughthedescriptionoftwo applicationsdirectlyrelatedtothemodelingofchemicalreactionsandtheanalysisofchemicalbondingdiscussedinParts II and III ofthebook,respectively.
InChapter 20,emergingparadigmsbasedonartificialintelligenceandmachine learningforadata-drivenapproachtochemistryarediscussed.Theconcluding
chapter,Chapter 21,dealswithchallengesandimplicationsofthemostrecent perspectiveofchemistryasanopenscience.
1.2Notationandconventions Forthereader’sconvenienceasummaryofthenotationandconventionsadopted inthebookisgiveninthefollowing.Aselectionofthemostrelevantsymbols usedinthebookisgivenbelow.Symbolsaregroupedtogetheronthebasisofthe domaintowhichtheyarerelevant,andlistedinanorderthatlargelyreflectstheir appearanceinthebook.Afewsymbolsareusedwithdifferentmeaningindifferentpartsofthebooksoastoadheretoconventionalnotationsspecifictosome topics(e.g.,thescatteringmatrixinnucleardynamicsandtheoverlapmatrixin electronicstructurearebothoftennotatedwith S ).However,thisshouldhave noeffectontheoverallreadabilityasitshouldalwaysbeclearfromthecontext whattherepeatedsymbolsreferto.Insomepartsofthebook,wheredeclared, useismadeofatomicunits,wherebythemassoftheelectron me ,thecharge oftheelectron |e |,Planck’sconstant = h 2π ,andtheBohrradius a
areallsettounity(fromtheexpressionof a0 ,itfollowsthatalsotheCoulomb constant kc isequalto1inatomicunits).Lowercaseanduppercaseboldcharactersaregenerallyadoptedforvectorsandmatrices,respectively.Curlybrackets areusedtoindicatesetsofvariables.The i subscriptdenotesvariablesrelated tothe i thelectron,the a subscriptdenotesvariablesrelatedtothe a thnucleus. Whenmorethanonesubscriptisnecessary, i , j , k , l areusedforelectronsor molecularorbitals, a and b areusedfornuclei,and p , q , r , s areusedforatomic orbitals(basisfunctions).Thesymbol δij indicatestheKroneckerdelta(δij = 0 if i = j , δij = 1if i = j ).SomeintegralsareexpressedinDirac(orbra-ket) notation,whereby
Generaltheory
P(x) probabilitydistribution
p(x) probabilitydensity
ξ collectivecoordinate
p momentumvector
ˆ p momentumoperator
r positionvector
ˆ r positionoperator
Molecularquantummechanics
{r i } setofelectroniccoordinates
{r a } setofnuclearcoordinates
Ψ molecularwavefunction nuclearwavefunction
ψ electronicwavefunction
Za atomicnumberofnucleus a
ma massofnucleus a
me massofelectron
e chargeofelectron Planck’sconstant
kc Coulomb’sconstant
E totalenergy
ˆ U time-evolutionoperator
ˆ H Hamiltonianoperator
ˆ T kinetic-energyoperator
ˆ V potential-energyoperator
ˆ TN nuclear–kineticenergyoperator
ˆ He electronicHamiltonian
εn energyofelectronicstate n
Nucleardynamics
b impactparameter
σ crosssection
re diatomicequilibriumdistance
ke diatomicforceconstantatequilibriumdistance
De diatomicdissociationenergy
n bond-ordercoordinate
β bondorder–exponentialmappingparameter
rBC , rAB interatomicdistances ABCangle
R , r , θ reactantJacobicoordinates
R , r , θ productJacobicoordinates
γv vibrationalwavefunction
εv energyofvibrationalstate v
χ wavepacket
S scatteringmatrix
k(T) thermalrateconstant(functionoftemperature T )
k(E) microcanonicalrateconstant(functionofenergy E )
ρ(E) densityofstates
vR reactionrate
Electronicstructure
ϕ molecularorbital
χ atomicorbital(basisfunction)
εi energyoforbital i
ChemistryattheFrontierwithPhysicsandComputerScience
μ labelfor N -tupleofmolecularorbitals
μ Hartreeproductassociatedwith N -tuple μ
μ Slaterdeterminantassociatedwith N -tuple μ
ˆ h one-electronpartoftheelectronicHamiltonian
ˆ j (j) Coulomboperatorfororbital j
ˆ J totalCoulomboperator
ˆ k (j) exchangeoperatorfororbital j
ˆ K totalexchangeoperator
ρ electrondensity
S overlapmatrix
C molecular-orbitalcoefficientmatrix
D densitymatrix
F Fockmatrix
h one-electronpartoftheFockmatrix
J Coulombmatrix
K exchangematrix
Theoriginalfiguresofthisbookwereproducedusingthefollowingsoftware:Gnuplot,Pymol,andthePSTricksLATEXpackage.
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Chapter2 Thephysicsofmolecular systems 2.1Classicalandquantummechanics AsalreadymentionedinChapter 1,weshallbeconcernedwiththemodelingof thestructureandbehaviorofmatterattheatomicandmolecularlevel.Atthis levelofdetail,weshallconsiderthenucleiandtheelectronsasthe‘elementary’ particlesofmatter,andweshallthereforefocusonthelawsofphysicsthat governtheirbehavior.
Now,howdoweimaginenucleiandelectrons?Mostlikely,inthefirstinstance,asparticles–moreorlesspoint-like–possessingamassandacharge, andlocalizedinagivenpointofthethree-dimensionalphysicalspace.Thisis certainlyhownucleiandelectronsappearinpopulartextbookrepresentations, suchastheatommodelofRutherfordorBohr,withtheorbitingelectronsarrangedaroundanucleusandcaughtasinaphotograph,ortheLewisdiagramsof electron-pairsharinginchemicalbonding,withnonbondedelectronsdepicted asdotsneartherespectivenuclei.Boththeserepresentationsarerootedina classicalviewofphysics:withinsuchaframework,modelingandpredicting thebehaviorofamolecularsystemultimatelymeansbeingabletosimulatethe evolutionintimeofthepositionofitsconstitutingparticles.Inclassicalmechanics,thiscanbeeasilyachievedthroughNewton’slawsofmotion,which areiconicallyrepresentedbythemathematicalexpressionforNewton’ssecond law1
AccordingtoNewton’slaws,givenaparticleofmass m inagivenpointof space,theevolutionofitspositionintimecanbeeasilydeterminedoncethe forcesactingontheparticleareknown.Thinkingofnucleiandelectronsas chargedpointparticlesinteractingthroughaCoulombelectrostaticpotential andgeneralizingEq.(2.1)tothemultiparticlecase,onewouldreadilyobtainthe toolsforsimulatingtheevolutionoftheresultingmolecularsystem.However, whileclassicalmechanicsfindswidespreadusageintheoreticalandcomputationalchemistryfordealingwith‘higher-level’,coarse-grainedaspectsofchemistry(see,forinstance,Chapters 7 and 11 ofthisbook),itunfortunatelyfailsin
1 AsmentionedinChapter 1,Section 1.2,vectorswillgenerallybeindicatedwithlowercasebold charactersthroughoutthebook.Uppercaseboldcharacterswillbeusedformatrices.
