MathematicalPhysics inTheoretical Chemistry
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Contributors
S.M.Blinder
UniversityofMichigan,AnnArbor,MI,UnitedStates
CailaBruzzese
DepartmentofChemistry,BrockUniversity,St.Catharines,Ontario,Canada
KimberlyJordanBurch
DepartmentofMathematics,IndianaUniversityofPennsylvania,Indiana, PA,UnitedStates
AndrewL.Cooksy
DepartmentofChemistryandBiochemistry,SanDiegoStateUniversity, SanDiego,CA,UnitedStates
GuidoFano
UniversityofBologna,Bologna,Italy
JamesW.Furness
DepartmentofPhysicsandEngineeringPhysics,TulaneUniversity,NewOrleans, LA,UnitedStates
DavidZ.Goodson
DepartmentofChemistryandBiochemistry,UniversityofMassachusetts Dartmouth,NorthDartmouth,MA,UnitedStates
JustinK.Kirkland
DepartmentofChemistry,UniversityofTennessee,Knoxville,TN,UnitedStates
ErrolLewars
DepartmentofChemistry,TrentUniversity,Peterborough,ON,Canada
DevinA.Matthews
InstituteforComputationalEngineeringandSciences,TheUniversityofTexasat Austin,Austin,TX,UnitedStates
EgorOspadov
DepartmentofPhysics,BrockUniversity,St.Catharines;DepartmentofChemistry,TheUniversityofWesternOntario,London,Ontario,Canada
StuartM.Rothstein
DepartmentofPhysics;DepartmentofChemistry,BrockUniversity,St.Catharines, Ontario,Canada
JohnF.Stanton
DepartmentofChemistry,UniversityofFlorida,Gainesville,FL,UnitedStates
JianweiSun
DepartmentofPhysicsandEngineeringPhysics,TulaneUniversity,NewOrleans, LA,UnitedStates
Thedevelopmentsofphysicsinthe20thcenturymadeallofchemistryexplicable, inprinciple,byquantummechanics.AssummarizedbyDirac:“Theunderlying physicallawsnecessaryforthemathematicaltheoryofalargepartofphysicsand thewholeofchemistryarethuscompletelyknown,andthedifficultyisonlythat theexactapplicationoftheselawsleadstoequationsmuchtoocomplicatedtobe soluble”[2].Byitsverynature,quantummechanics is mathematicalphysicsand therebyweestablishtheconnectionwhichisthethemeofthisvolume.However,the loopholenotedbyDirac,theexistenceofchemicalproblemstoomathematically complextobesolvedexactly,justifiesthesurvivalofpartsofchemistryasan empiricalscience.Inthiscategoryaresemiempiricalconceptsofchemicalbonding andreactivity.Thishasalsoledtocomputationalmodelspromotingrationaldrug design.Thesehavealsostimulatedapplicationsofotherbranchesofmathematics, forexample,informationtheoryandgraphtheoryappliedtothedefinitionofvarious chemicalindices.
Theprimaryobjectiveoftheoreticalchemistryistoprovideacoherentaccount forthestructureandpropertiesofatomicandmolecularsystems.Techniques adaptedfrommathematicsandtheoreticalphysicsareappliedinattemptstoexplain andcorrelatethestructuresanddynamicsofchemicalsystems.Inviewofthe immensecomplexityofchemicalsystems,theoreticalchemistry,incontrastto theoreticalphysics,generallyusesmoreapproximatemathematicaltechniques,often supplementedbyempiricalorsemiempiricalmethods.
Thisvolumebeginswithanintroductiontothequantumtheoryforatomsand smallmolecules,expandingupontheoriginalapplicationsofmathematicalphysics inchemistry.Thisfieldisnowlargelysubsumedwithinasubdisciplineknownas computationalchemistry. Chapter1 beginswithanintroductiontotheHartree-Fock method,whichistheconceptualfoundationforcomputationalchemistry. Chapter2 discussesthebasisfunctionsemployedinthesecomputations,nowlargelydominated byGaussianfunctions. Chapter3 describessomepost-Hartree-Fockmethods,which seektoattain“chemicalaccuracy”inatomicandmolecularcomputations,inparticular,configurationinteraction,many-bodyperturbationtheory,andcoupled-cluster theory. Chapter10 discussesdiagrammatictechniquesborrowedfromtheoretical physics,whichcanenhancetheefficiencyofcomputations. Chapter7 isanaccount ofthedevelopmentofpersonalcomputersandtheirapplicationstocomputational chemistry.
Forlargermoleculesandcondensedmatter,alternativeapproaches,including densityfunctionaltheory(Chapter4)andquantumMonte-Carlo(Chapter6),are becomingpopularcomputationalmethods.Someadditionaltopicscoveredinthis volumearevibrationalpartitionfunctions(Chapter5),singularityanalysisofperturbationtheories(Chapter9),andchemicalapplicationsofgraphtheory(Chapter8).
Finally, Chapter11 introducestheprinciplesofthequantumcomputer,whichhas thespeculativepossibilityofexponentialenhancementofcomputationalpowerfor theoreticalchemistry,aswellasmanyotherapplications.
REFERENCES
[1] NyeMJ.Fromchemicalphilosophytotheoreticalchemistry.Berkeley:Universityof CaliforniaPress;1993.
[2] DiracPAM.Quantummechanicsofmany-electronsystems.ProcRSocA(Lond) 1929;123:714–33.
V (r ) =− Ze2 r .(1)
(WeuseGaussianunitstoavoidtheunnecessaryfactors4π 0 ,and,inanyevent,we willsoonbeswitchingtoatomicunits.)Toreview,theSchrödingerequationfora hydrogen-likeatomcanbewritten
),(2)
wheretheenergyforprincipalquantumnumber n isgivenby n =−Z 2 e2 /2a0 n2 , with a0 equaltotheBohrradius h2 /me2 .Theone-electronfunctions ψ(r),whenused inthecontextofamultielectronsystem,arecalled orbitals [5],anadjective,usedas anoun,todenotethequantum-mechanicalanalogofclassical orbits.Foranelectron atpoint r interactingwiththechargedistributionofasecondelectroninanatomic orbital ψ(r ),thepotentialenergyisgivenby
Thus,thetotalpotentialenergyforelectron1isgivenby
wherethenotation V1 [ψ2 ] indicatesthat V1 isa functional of ψ2 ,emphasizingthe dependanceonthechargedistributionofelectron2.InHartree’smethod,electron1 obeystheeffectiveone-particleSchrödingerequation
where 1 istheorbitalenergyofelectron1,negativeforboundstates.Analogously, interchangingthelabels1and2,theorbitalfunctionforelectron2isthesolutionof
Thecoupledintegro-differentialequations(5),(6),knownasthe Hartreeequations, canberepresentedinsymbolicformby
Thesearecoupledinthesensethatthesolutiontothefirstequationentersthesecond equation(viatheeffectiveHamiltonianoperator H eff 2 containing V2 [ψ1 ]),andvice versa.Asolutiontotheseequationscanbefound,inprinciple,byasuccessive approximationprocedure.Aninitial“guess”ofthefunctions ψ1 and ψ2 isusedto
computethepotentialenergies V1 [ψ2 ] and V2 [ψ1 ].EachHartreeequationscanthen besolvedtogive“first-improved”orbitalfunctions ψ (1) 1 and ψ (1) 2 .These,inturn, areusedtorecompute V (1) 1 and V (1) 2 ,andthenewHartreeequationsaresolvedto givesecond-improvedorbitalfunctions.Theiterativeprocedureiscontinueduntil theinputandoutputfunctionsagreetowithinsomedesiredaccuracy.Theorbital functionsandpotentialfieldsarethensaidtobe self-consistent.Theusualquantummechanicalrestrictionsonaboundstatewavefunction—thatitbeeverywheresinglevalued,finite,andcontinuous—applyateachstageofthecomputation.EachHartree equationisthusaneigenvalueproblem,solubleonlyforcertaindiscretevaluesof i (ingeneral,differentineachstage).Fortheheliumatomtheorbitalfunctions ψ1 and ψ2 turnouttobeidentical.ThisdoesnotviolatethePauliprinciplesincethetwo orbitalscanhaveoppositespins.NotethattheHartreemethoddoesnotitselftake spinintoaccount.
ExtensionoftheHartreemethodtoan N -electronatomisstraightforward.Each electronnowmovesinthepotentialfieldofthenucleusplustheoverlappingcharge cloudsof N 1otherelectrons.Now N coupledintegro-differentialequationsareto besolved:
Eachsetoforbitalfunctions ψ1 ...ψN canbeidentifiedwithanelectronicconfiguration,forexample,1s2 2s2 2p6 3s fortheNaatom.Itislefttothegoodsenseoftheuser nottoallowmorethantwooftheorbitals ψ1 ...ψN tobethesame.1 Thedifferent orbitalpairsshouldalsobeconstructedtobemutuallyorthogonal.Theeigenvalues i shouldbenegativeforboundorbitals.Theirmagnitudesareapproximationstothe ionizationenergiesofthecorrespondingelectrons.
Atthispoint,itisconvenienttointroduceatomicunits,whichsimplifiesallofthe previousformulasbyremovingtherepetitivephysicalconstants.Weset
1 Ignoringthisrestrictionhasbeendubbed“inconsistentfieldtheory.”
Thevariationof L [ψ , ψ ∗ ] intermsofvariationsinallthe ψi and ψ ∗ i isgivenby
Sincetheminimumin L isunconditional,thisresultmustholdforarbitrary variationsofallthe δψi and δψ ∗ i .Thisispossibleonlyifeachofthecoefficients ofthesevariationsvanish,thatis,
Letusfocusononeparticularterminthevariation δ L ,namelythetermlinearin δψ ∗ k forsome i = k .Fromthecondition ∂ L ∂ψ ∗ k = 0appliedtoEq.(23),weareledto theHartreeequations2
inagreementwithEqs.(8)–(10).Wehaveusedthefactsthatthefirstsummation i reducestoasingletermwith i = k andthevanishingoftheintegral d 3 r for arbitraryvaluesof δψ ∗ k impliesthattheremainingintegrandisidenticallyequalto0.
2 DETERMINANTALWAVEFUNCTIONS
Theelectronineachorbital ψi (r) isaspin 1 2 particleandthushastwopossiblespin orientationsw.r.t.anarbitraryspatialdirection, ms =+ 1 2 or ms =− 1 2 .Thespin functionisdesignated σ ,whichcancorrespondtooneofthetwopossiblespinstates σ = α or σ = β .Wedefineacompositefunction,knownasa spin-orbital
(27)
denotingby x thefour-dimensionalmanifoldofspaceandspincoordinates.For example,ahydrogen-likespin-orbitalislabeledbyfourquantumnumbers,so a = {n, l, m, ms }.Wewillabbreviatecombinedintegrationoverspacecoordinatesand summationoverspincoordinatesby spin
d 3 r = dx (28)
2 TheHartreeequationsmightappeartodaytohaveonlyhistoricalsignificance,buttheirgeneralization leadstotheKohn-Shamequationsofmoderndensity-functionaltheory.
AHartreeproductofspin-orbitalsnowtakestheform
Ψ(1 ... N ) = φa (1)φb (2)...φn (N ). (29)
Forfurtherbrevity,wehavereplacedthevariables xi simplybytheirlabels i Tobephysicallyvalid,asimpleHartreeproductmustbegeneralizedtoconform totwoquantum-mechanicalrequirements.FirstisthePauliexclusionprinciple, whichstatesthatnotwospin-orbitalsinanatomcanbethesame.Thisallows anorbitaltooccurtwice,butonlywithoppositespins.Second,themetaphysical perspectiveofthequantumtheoryimpliesthatindividualinteractingelectronsmust beregardedasindistinguishableparticles.Onecannotuniquelylabelaspecific particlewithanordinalnumber;theindicesgivenmustbeinterchangeable.Thus eachofthe N electronsmustbeequallyassociatedwitheachofthe N spin-orbitals. Sincewehavenowundonetheuniqueconnectionbetweenelectronnumberand spin-orbitallabel,wewillhenceforthdesignatethespin-orbitallabelsaslowercaseletters a, b, ... , n whileretainingthelabels1,2, ... , N forelectronnumbers. Thesimplestexampleisagainthe1s2 groundstateofheliumatom.Letthetwo occupiedspin-orbitalsbe φa (1) =
2).Tofulfillthe necessaryquantumrequirements,wecanconstructthe(approximate)groundstate wavefunctionintheform
Inclusionofthetermwithinterchangedparticlelabels, φa (2)φb (1),fulfillstheindistinguishabilityrequirement.Thefactor 1 √2 preservesnormalizationforthelinear combination(assumingthat φa and φb areindividuallyorthonormalized).The exclusionprincipleisalsosatisfied,sincethefunctionwouldvanishidenticallyif spin-orbitals a and b werethesame.AgeneralconsequenceofthePauliprincipleis the antisymmetryprinciple foridenticalfermions,whereby Ψ(2,1)
Thefunction(30)hastheformofa2 × 2determinant
Thegeneralizationforafunctionof N spin-orbitals,whichisconsistentwiththe Pauliandindistinguishabilityprinciples,isan N × N Slaterdeterminant3
3 ThedeterminantalformwasfirstproposedbyHeisenberg[8,9]andDirac[10].Slaterfirstuseditin theapplicationtoamany-electronsystem[11].
Thecorecontributionstotheenergyinvolvestermsintheone-electronsumin Eq.(39).Definingthecoreoperator
theexpressionforthecoreintegral Hr reducesto
InanalogywithEq.(43)forcaseofthenormalizationbra-ket,alltheotherfactors φb |φs , s = r areequalto1.ThisisanalogoustoEq.(19),thedefinitionofthe coreintegralintheHartreemethod,exceptthatnowspin-orbitals,ratherthansimple orbitalsarenowused.Actually,thescalarproductsofthespinfunctions σr give factorsof1,sothatonlythespace-dependentorbitalfunctionsareinvolvedinthe computation,justasintheHartreecase.
Weconsidernexttheinterelectronicrepulsions r 1 ij .Followingananalogous calculation,allcontributionsexceptthosecontainingparticlenumbers i or j give factorsof1.Whatremainsis
Theminussignreflectsthefactthatinterchangingtwoparticlelabels i, j multiplies thewavefunctionby 1.ThefirsttermearliercorrespondstoaCoulombintegral (20);againthesearelabeledbyspin-orbitals,butthecomputationinvolvesonly space-dependentorbitalfunctions:
ThesecondterminEq.(46)givesrisetoan exchangeintegral:
Thisrepresentsapurelyquantum-mechanicaleffect,havingnoclassicalanalog,and arisingfromtheantisymmetryprinciple.Intermsoftheorbitals ψ(r),aftercarrying outtheformalintegrationsoverthespin,wecanwrite
Unlike Jij , Kij involvestheelectronspin.Becauseofthescalarproductofthespins associatedwith φi and φj ,theexchangeintegralvanishesif σi = σj ,inotherwords, ifspin-orbitals i and j haveoppositespins, α , β or β , α Theexpressionfortheapproximatetotalenergycannowbegivenbythe summation
Notethat Kii = Jii ,whichwouldcancelanypresumedelectrostaticself-energyof aspin-orbital.Theeffectiveone-electronequationsfortheHFspin-orbitalscanbe derivedbyaprocedureanalogoustothatofEqs.(22)–(26).Anewfeatureisthatthe Lagrangemultipliersmustnowtakeaccountof N 2 orthonormalizationconditions
φi |φj = δij ,leadingto N 2 multipliers λij .Accordingly
TheLagrangemultipliers λij canberepresentedbyaHermitianmatrix.Itshould thereforebepossibletoperformaunitarytransformationtodiagonalizethe λ-matrix. Fortunately,wedonothavetodothistransformationexplicitly;wecanjustassume thatthesetofspin-orbitals φi aretheresultsafterthisunitarytransformationhasbeen carriedout.Thenewdiagonalmatrixelementscanbedesignated i = λii .Again,we seethatthe i willcorrespondtotheone-electronenergiesinthesolutionsoftheHF equations.AsageneralizationofEq.(26),thecontributiontothevariation δ L linear in δφ ∗ i isgivenby
TheeffectiveHFHamiltonian HHF isknownasthe Fockoperator,designated F . Finally,theHFequationscanbewritten
IncontrasttotheHartreeequations(26), F φi (x) alsoproducestermslinearinthe otherspin-orbitals φj , j = i.JustasintheHartreecase,thecoupledsetofHFintegrodifferentialequationscan,inprinciple,besolvednumerically,usingtheanalogous self-consistencyapproach,withiterativelyimprovedsetsofspin-orbitals.
Thesignificanceoftheone-electroneigenvalues i canbefoundbypremultiplyingtheHFequation(53)by φ ∗ i (x) andintegratingover x.Usingthedefinitionsof Hi , Jij ,and Kij ,wefind
E = i Hi +
Kij . (51)
Considernowthedifferenceinenergiesofthe N -electronsystemandthe (N 1)electronsystemwiththespin-orbital φk removed
Therefore,themagnitudesoftheeigenvalues k areapproximationsfortheionization energiesofthecorrespondingspin-orbitals φk .Sincethe k arenegative,IPk =| k |. Thisresultisknownas Koopmans’theorem.Itisnotexactsinceitassumes“frozen” spin-orbitals,whenthe N -electronsystembecomesan (N 1)-electronpositiveion. Inactualfact,theseparatelyoptimizedorbitalsforanatomormoleculeandits positiveionwillbedifferent.
ItcanbeshownthatthemagnitudesoftheCoulombandexchangeintegrals satisfytheinequalities
Ingeneral, Kij isanorderofmagnitudesmallerthanthecorrespondingCoulomb integral Jij .HFexpressionsforthetotalenergycanreadilyexplainwhythetriplet stateof,forexample,the1s2s 3 S configurationofheliumatomislowerinenergy thanthesingletofthesameconfiguration1s2s 1 S.Denotingthetwo-determinant functionsinEq.(38)as Ψ(1,3 S) forthe(+)and( )signs,respectively,wecompute theexpectationvalueofthetwo-electronHamiltonianforhelium(with Z = 2).After somealgebra,thefollowingresultisfound:
Therefore,since K > 0,thetriplet,withthe( )sign,hasthelowerenergy.One caution,however,isagainthefactthatthesingletandtripletstateshavedifferent optimizedorbitals,sothatthevaluesof K1s,2s (aswellas J1s,2s , H1s ,and H2s )arenot equal.Butevenwithseparatelyoptimizedorbitals,theconclusionremainsvalid.
OnecanalsogiveasimpleexplanationofHund’sfirstrulebasedonexchange integrals.Foragivenelectronconfiguration,thetermwithmaximummultiplicity hasthelowestenergy.Themultiplicity2S + 1ismaximizedwhenthenumberof parallelspinsisaslargeaspossible,whileconformingtothePauliprinciple.But moreparallelspinsgivemorecontributionsoftheform Kij ,thuslowerenergy.
Thisiscalledthe occupation-numberrepresentation,the n-representation,or Fock space.Inthelanguageofsecondquantization,onedoesnotask“whichparticleis inwhichstate,”butrather,“howmanyparticlesarethereineachstate.”The vacuum state,inwhichall ni = 0,willbeabbreviatedby
(Anothercommonnotationis |vac .)
Therewillexistcreationandannihilationoperators a† 1 , a† 2 ... , a1 , a2 ....Assumingthatthebosonsdonotinteract, a and a† operatorsfordifferentvarietieswill commute.Thefollowinggeneralizedcommutationrelationsaresatisfied:
Forbosons,theoccupationnumbers ni arenotrestrictedandthewavefunctionofa compositestateissymmetricw.r.t.anypermutationofindices.Thingsare,ofcourse, quitedifferentforthecaseofelectrons,orotherfermions.Theexclusionprinciple limitstheoccupationnumbersforfermions, ni toeither0or1.Also,aswehaveseen, thewavefunctionofthesystemisantisymmetricforanyoddpermutationofparticle indices.
Thebehavioroffermionscanbeelegantlyaccountedforbyreplacingthe bosoncommutationrelations(75)bycorresponding anticommutationrelations.The anticommutatoroftwooperatorsisdefinedby
{A, B}≡ AB + BA,(76)
andthebasicanticommutationrelationsforfermioncreationandannihilation operatorsaregivenby
Theserelationsareintuitivelyreasonable,sincetherelation
isan alternativeexpressionoftheantisymmetryprinciple(34),while
0accords withtheexclusionprinciple.
Thestate(73)canbeconstructedbysuccessiveoperationsofcreationoperators onthe N -particlevacuumstate
Letusnextconsidertherepresentationofmatrixelementsinsecond-quantized notation.Wewishtoreplacetheexpectationvalueofanoperator A forthestate |Ψ withoneevaluatedforthevacuumstate |O :
CHAPTER XXIX.
"SHOULD OLD ACQUAINTANCE BE FORGOT?"
If Dean was surprised to see his old enemy in such an out of the way place, Kirby was no less surprised to see his former traveling companion. There was this difference: the encounter brought him pleasure, while to Dean it carried dismay. Neither could understand where on earth the other had sprung from.
"Oho!" laughed Kirby, "so we meet again."
Dan looked surprised, thinking the words were addressed to him, but following the direction of Kirby's eyes, he saw that he was mistaken.
"Do you know this boy?" he asked.
"Do I know him? Why, we started from the East together."
"How is that?"
"It was at the request of a friend of ours."
"The captain?"
"Yes."
"And why did you separate?"
"Well, I mustn't tell tales out of school. I am very glad to meet you again, youngster. Is the pleasure mutual?"
"No, it isn't," said Dean, bluntly.
"So I should judge, after the trick you played upon me at our last meeting."
"What do you refer to?"
"You know well enough. You cautioned Dr. Thorp against me. Don't deny it, for I know it is true."
"I don't deny it. What happened that night showed that I had good reason."
"Be that as it may," said Kirby with an ugly scowl, "you did a bad thing for yourself. You probably thought you would never meet me again."
Dean was silent, but Dan, whose curiosity was aroused, interposed with an inquiry.
"What are you two talkin' about," he said. "Is this boy a friend or an enemy?"
"He is an enemy of our association," replied Kirby. "I am glad to have him in my power."
"So there is an association?" thought Dean. "These two men belong to it, and Squire Bates is the captain. I shall soon know all about it."
But in the meanwhile the evident hostility of Kirby, reflected in the face of his new acquaintance Dan, was ominous of danger. Dean felt that he would gladly pass the night out in the woods exposed to the night air if he could only get away. But he saw clearly that escape was not at present practicable.
"Have you seen the old woman?" asked Dan, meaning his mother.
"Yes, she told me that she had taken in a kid for the night, but I had no idea it was any one I knew. The old lady wears well, Dan."
"Yes, she's tough," said the affectionate son carelessly. "I'll go in and see whether she's got supper ready."
He entered the house, leaving Dean and his old employer together.
"Come here, boy, and sit down," said Kirby smiling, and eying Dean very much as a cat eyes the mouse whom she proposes soon to devour. "You must be tired."
"Thank you," said Dean calmly, as he went forward and seated himself on the settee beside Peter Kirby.
"What brought you so far West as Colorado?" proceeded Kirby, giving vent to his curiosity.
"I kept coming West. Besides I heard there were mines in Colorado, and I thought I might find profitable work."
"So you gave up playing on that harmonica of yours?" "Yes."
"Couldn't you make it pay?"
"I needed a partner like the one I started with—Mr. Montgomery. I couldn't give an entertainment alone."
"Then you haven't been making any money lately?" "No."
"Where did you get that watch?"
"From Dr. Thorp."
"When did he give it to you?"
"Just before I left town."
"It was a present to you for informing on me, I suppose?" said Kirby, his face again assuming an ugly frown.
"I believe it was for saving him from being robbed."
"Then he had considerable money and bonds in the house?"
"Yes."
"Were they in the cabinet?"
"He removed them."
