Molecular Orbital Theory (MOT) AOs and MOs

1. Jens Martensson 1
Molecular Orbital
Theory (MOT)

2. MolecularOrbitalTheory
Jens Martensson 2
• TheMolecularOrbitalTheory(oftenabbreviatedtoMOT)isa theoryonchemicalbondingdevelopedatthe beginningofthetwentieth
centurybyF
.HundandR.S.Mullikentodescribethestructureandpropertiesof differentmolecules.
• Thevalence-bondtheoryfailedtoadequatelyexplainhowcertainmoleculescontaintwoormoreequivalent bondswhosebondorders
lie betweenthatof a singlebondandthatof a doublebond,suchasthebondsin resonance-stabilizedmolecules.Thisiswherethe
molecularorbitaltheoryprovedtobemorepowerfulthan thevalence-bondtheory(sincetheorbitalsdescribedbytheMOTreflectthe
geometriesofthemoleculesto whichitisapplied).

3. MolecularOrbitalTheory
Jens Martensson 3
Thekeyfeaturesofthemolecularorbitaltheoryarelistedbelow.
• Thetotalnumberofmolecularorbitalsformedwillalwaysbeequaltothetotalnumberofatomicorbitals offeredbythebonding
species.
• Thereexistdifferenttypesofmolecularorbitalsviz;bondingmolecularorbitals,anti-bondingmolecular orbitals,andnon-bonding
molecular orbitals. Of these,anti-bonding molecular orbitals will always have higher energy thantheparent orbitals whereas
bondingmolecularorbitalswillalwayshavelowerenergy thantheparentorbitals.
• Theelectronsarefilled intomolecularorbitalsintheincreasingorderof orbitalenergy(fromtheorbital withthelowestenergyto
theorbitalwiththehighestenergy).
• Themosteffectivecombinationsofatomicorbitals(fortheformationofmolecularorbitals)occurwhenthe combiningatomicorbitals
havesimilarenergies.

4. MolecularOrbitalTheory
Jens Martensson 4
• Insimpleterms,themolecularorbitaltheorystatesthateachatomtendstocombinetogetherandformmolecularorbitals.Asaresultof
sucharrangement,electronsarefoundinvariousatomicorbitals andtheyareusuallyassociatedwithdifferentnuclei.Inshort,an
electroninamoleculecanbepresent anywhereinthemolecule.
• Oneofthemainimpactsofthemolecularorbitaltheoryafteritsformulationisthatitpaveda newway tounderstandtheprocessof
bonding.Withthistheory
,themolecularorbitalsarebasicallyconsideredas LinearCombinationsofAtomicOrbitals(LCAO).The
approximationsare furtherdoneusingthe Hartree–Fock(HF)or thedensityfunctionaltheory(DFT)modelstotheSchrödinger
equation.

5. MolecularOrbitalTheory
• Molecular orbital theory approximation of the molecular orbitals as linear combinations ofatomic
orbitalscanbeillustratedas follows.
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6. LinearCombinationofAtomicOrbitals (LCAO)
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• Molecular orbitals can generally be expressed through a linear combination of ato
m
ic orbitals
(abbreviatedtoLCAO).TheseLCAOsareusefulintheestimationoftheformationoftheseorbitalsinthe
bondingbetweentheatomsthatmakeupamolecule.
• TheSchrodingerequationusedtodescribetheelectronbehaviorformolecularorbitalscanbewrittenin a methodsimilartothatfor
atomicorbitals.
• It is an approximate method for representing molecular orbitals. It’s more of a superimposition method where constructive
interferenceof twoatomicwavefunction producesa bondingmolecularorbital whereasdestructiveinterferenceproducesnon-
bondingmolecular orbital.

7. ConditionsforLinearCombinationofAtomicOrbitals
Jens Martensson 7
Theconditionsthatarerequiredforthelinearcombinationof atomicorbitalsareasfollows:
1
. S
a
m
eEnergyofCombiningOrbitals
Theatomicorbitalscombiningtoformmolecularorbitalsshouldhavecomparableenergy
.Thismeansthat 2porbitalofanatomcan
combinewith another 2p orbital of another atombut 1sand 2p cannot combinetogether as theyhaveappreciable energy
difference.
2
. S
a
m
eSy
mmetryaboutMolecularAxis
Thecombiningatomsshouldhavethesamesymmetryaroundthemolecularaxisfor propercombination, otherwise,theelectron
densitywillbesparse.Fore.g.allthesub-orbitalsof2phavethesameenergy butstill,2pzorbitalofanatomcanonlycombine
witha 2pz orbital of anotheratombutcannotcombine with2px and 2py orbital astheyhavea different axis of symmetry
.In
general,thez-axisisconsidered asthemolecularaxisof symmetry
.

8. ConditionsforLinearCombinationofAtomicOrbitals
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3.ProperOverlapbetweenAtomicOrbitals
Thetwoatomicorbitalswillcombinetoformmolecularorbitalif theoverlapisproper.Greatertheextentofoverlapoforbitals,
greaterwillbethenucleardensitybetweenthenucleiofthetwoatoms.
• Theconditioncanbeunderstoodbytwosimplerequirements.Forthe formationof propermolecular orbital,properenergyand
orientationarerequired.Forproperenergy
,thetwoatomicorbitalsshould havethesameenergyandfortheproperorientation,the
atomicorbitalsshouldhaveproperoverlap andthesamemolecularaxisofsymmetry
.

9. MolecularOrbitals
Jens Martensson 9
• Thespaceina moleculeinwhichtheprobabilityoffindinganelectronismaximumcanbecalculated usingthe molecularorbital
function.Molecularorbitalsarebasicallymathematicalfunctionsthat describethewavenatureofelectronsina givenmolecule.
• Theseorbitalscanbeconstructedviathe combination ofhybridizedorbitalsoratomicorbitalsfrom eachatombelongingtothe
specific molecule. Molecular orbitals provide a great model via the molecular orbital theory to demonstrate thebonding of
molecules.

10. TypesofMolecularOrbitals
Jens Martensson 10
Accordingtothemolecularorbitaltheory
,thereexistthreeprimarytypesof molecularorbitalsthat areformedfromthe
linearcombinationofatomicorbitals.Theseorbitalsaredetailedbelow.
• BondingMolecularOrbital
Bondingmolecularorbitalisformedbythe additionof overlappingof atomicorbitals.Thewave functionof thebondingMO
maybewrittenas:
Ψ(MO)=ΨA+ΨB
AtomicorbitalshavingsamesignformstheBondingmolecularorbitalwhenthelobesof themcombines. Theelectrondensityis
greaterintheregionbetweenthetwonucleiofbondedatoms.Theforcesinthis orbitaltendtobringthe twonuclei ofthe atoms
together.Therefore,electronsinthe bondingMO contributetoattractionbetweenthetwoatoms.Ithaslowerenergythanthe
isolatedatomicorbitals.

11. TypesofMolecularOrbitals
Jens Martensson 11
• Non-BondingMolecularOrbitals
Inthecaseof non-bondingmolecularorbitals, duetoa completelack of symmetryinthecompatibility of twobondingatomic
orbitals,themolecularorbitalsformedhavenopositiveornegativeinteractions witheachother.Thesetypesof orbitalsdonot
affectthebondbetweenthetwoatoms.
• AntiBondingMolecularOrbitals
Anti-bonding molecularorbital isformedby thesubtractionof overlapping of atomicorbitals. Thewave functionfor theanti-
bondingMO maybewrittenas:
Ψ(MO)=ΨA–ΨB
Atomicorbitalshavingoppositesignformsthe Bondingmolecularorbitalwhenthe lobesofthemcombines.Theelectrondensityis
lesserintheregionbetweenthetwonucleioftheatoms.Theforcesin thisorbitalpushthe nucleiapart.Therefore,theelectronsin
antibondingMO contributetorepulsion betweentheatoms.Ithashigherenergythantheisolatedatomicorbitals.

12. FormationofMolecularOrbitals
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Anatomicorbitalisanelectronwave;thewavesofthetwoatomicorbitalsmaybeinphaseoroutof phase.SupposeΨAandΨB
representtheamplitudeoftheelectronwaveoftheatomicorbitalsofthetwoatomsAandB.
• Case1:WhenthetwowavesareinphasesothattheyaddupandamplitudeofthewaveisΦ=ΨA+
ΨB

13. FormationofMolecularOrbitals
• Case2:whenthetwowavesareoutofphase,thewavesaresubtractedfromeachothersothatthe amplitudeofthenewwaveisΦ
´=ΨA–ΨB
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14. CharacteristicsofBondingMolecularOrbitals
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• Theprobabilityof finding the electron in the internuclear region of the bonding m
o
lecular orbital is
greaterthanthatofcombiningatomicorbitals.
• Theelectronspresentinthebondingmolecularorbitalresultintheattractionbetweenthetwoatoms.
• Thebondingmolecularorbitalhaslowerenergyasa resultofattractionandhencehasgreaterstability thanthatofthecombining
atomicorbitals.
• Theyareformedbytheadditiveeffectoftheatomicorbitalssothattheamplitudeofthenewwaveis givenbyΦ=ΨA+ΨB
• Theyarerepresentedbyσ,π,andδ.

15. CharacteristicsofAnti-bondingMolecularOrbitals
• Theprobabilityoffindingtheelectronintheinternuclearregiondecreasesintheanti-bondingmolecular orbitals.
• Theelectronspresentintheanti-bondingmolecularorbitalresultintherepulsionbetweenthetwo atoms.
• The anti-bonding m
o
lecular orbitals have higher energy because of therepulsive forcesand lower
stability.
• Theyareformedbythesubtractiveeffectoftheatomicorbitals.Theamplitudeofthenewwaveisgiven byΦ´=ΨA– ΨB
• Theyarerepresentedbyσ∗
,π∗
,δ∗
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16. W
h
yareAntibondingOrbitalsHigherinEnergy?
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• Theenergylevelsofbondingmolecularorbitalsarealwayslowerthanthoseofanti-bondingmolecular orbitals.Thisisbecausethe
electronsintheorbitalareattractedbythenuclei inthe caseofbonding MolecularOrbitalswhereasthenucleirepeleachotherin
thecaseoftheanti-bondingMolecularOrbitals.
• Theloweringoftheenergyofbondingmolecularorbitalthanthecombiningatomicorbitalis calledstabilizationenergyandsimilarly
increaseinenergyoftheanti-bondingmolecularorbitalsis calleddestabilizationenergy.

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18. FeaturesofMolecularOrbitalTheory
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• Theatomicorbitalsoverlaptoformneworbitalscalledmolecularorbitals.Whentwoatomicorbitals overlaptheylosetheiridentity
andformneworbitalscalled molecularorbitals.
• TheelectronsinthemoleculesarefilledinthenewenergystatescalledtheMolecularorbitalssimilarto theelectronsinanatom
beingfilled inanenergystatecalledatomicorbitals.
• Theprobabilityoffindingtheelectronicdistributioninamoleculearounditsgroupofnucleiisgivenby themolecularorbital.
• Thetwocombiningatomicorbitalsshouldpossessenergiesofcomparablevalueandsimilarorientation. Forexample,1scan
combinewith1sandnotwith2s.
• Thenumberofmolecularorbitalsformedisequaltothenumberofatomicorbitalscombining.
• Theshapeofmolecularorbitalsformeddependsupontheshapeofthecombiningatomicorbitals.

19. Accordingtot
heMolecularOrbitalTheory,t
hefillingoforbitalstakesplaceaccordingtot
h
efollowingrules:
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• Aufbau’
sprinciple:Molecularorbitalsarefilledintheincreasingorderofenergylevels.
• Pauli’
sexclusionprinciple:Inanatomoramolecule,notwoelectronscanhavethesamesetoffour quantumnumbers.
• H
und’
s rule of m
aximum m
ultip
licity
: Pairing of electrons doesn’t take place until all the ato
m
ic or
molecularorbitalsaresinglyoccupied.

20. BondOrder
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• T
wentyyearsbeforethedevelopmentofquantummechanics,AmericanscientistG.N.Lewisrealized thatmolecules m
u
s
tcombine
andorganizethemselves accordingtoa certainprinciple.His theory centeredaroundcovalentbonds,which,unlike ionicbonds,
didn’tdemandanexchangeofvalence electrons,butinsteadrequiredatoms tosharesomeofthem.Theatoms shareasmany
electronsas requiredtocompletelyfill theirvalenceshell.
• Bondorderisa measurementof thenumberof electronsinvolvedinbondsbetweentwoatoms ina molecule.Itisusedasan
indicatorofthestabilityofa chemicalbond. Usually,the higherthebond order,thestrongerthechemicalbond.Mostofthetime,
bondorderisequaltothenumberofbondsbetweentwoatoms.

21. BondorderaccordingtoValenceBondtheory
• Lewisdrewsmalldiagramstoillustratethiscamaraderie,whicharenowcalledLewisstructures.ALewis structuredescribesthe
structureof a moleculeby connectingtheatomswithlines.Thelinesrepresentthe numberof electronsthathavebeenshared
betweentwoormoreatoms.Thus,whentwoatomssharetwo electrons,wedepictitbyconnectingthemwithtwolines.Thenumberof
lines,ormoreprecisely,thenumberofchemicalbondsthatcompriseamolecule,iscalleditsbondorder.
• For instance, the bond order of carbondioxide and me
t
ha
neis 4, which can easily be discerned by examining their Lewis
structures.Noticehowthe magnitudeof electronsshared betweeneachpair adequatelyfills thevalenceshellof bothatoms.
Hydrogenonlyrequiresa singleelectron,asitsshellis filledwhenitcontains2electrons,not8.
Lewis structuresof carbon dioxide and methane.
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22. • TherealsoexistmoleculesthatcanbedescribedbymorethanoneLewisstructure,suchassulfurdioxide. The bondorderof sucha
moleculeistheaverageofthebondordersofallthepossiblestructuresthatdescribeit. Thebondorderofsulfurdioxideistherefore1.5,
not3.
Sulfur dioxide exhibits two Lewis
structures
• However,computingbondorderbysimplyreferringtothenumberoflinesintheLewisstructureofa molecule isonlyacceptableunderthe
V
alenceBond(VB)theory
.WhenitcomestotheMolecularOrbital(MO)theory
, thealternativetheory thatdescribes molecularbonding,
thebondordermightbethesame,buttheimplicationsaredrasticallydifferent.
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23. ValenceBondtheoryvsMolecularOrbitaltheory
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TheValenceBondtheoryprovidesa cruderepresentationof molecularstructures,whereastheMolecular Orbital theorygivesa
moreaccurateandrealisticrepresentationofhowmoleculesareformed.Inthe former,onecanobservehowtheelectronsdepicted
bydotsabovethe atoms arelocalized, meaning thattheirlocationisdefinite.Ontheotherhand,MO theoryisbasedonthe
quantummechanicaltheoryof atoms.Ittakesintoaccount theprobabilisticornon-localized natureof electronsandthedistinct
energy levelsinvolved.
Thereisnowinneramongthetwo:VBtheoryiseasiertounderstand,butdoesn’texplaintheminute,yet highlycrucialdetailsthatMO
theoryexplainswiththehelpofhighlyesotericandsophisticatedconcepts. However,thiscomprehensionofgeometryandthedetail
ofamolecularstructurecomeattheexpenseof theeasewithwhichalaymancanvisualize it.

24. Thisiswhytheimplicationsof MO theorycanoftenbedrasticallydifferent.Forinstance,thebondorderof anoxygenmoleculeis
foundtobe2inboththeories,butVB theorydoesn’t explaintheparamagnetic propertiesthatoxygenexhibits.
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Experimentshavedemonstratedhowliquidoxygenisweaklyaffectedbya magneticfield.Anatomexhibits anytypeofmagnetism
whenitcontainsunpairedelectrons.If yourefertotheLewisstructureofoxygen, youwillfindthatall theelectronsarepaired,which
wouldrendertheatomdiamagneticorunaffectedbya magneticfield.However,MO theoryrevealsoxygen’sstructureinitstruedetail.
TheMO theoryrightly predictsthepresenceofunpairedelectronsinoxygen’sorbitals,andthereforeitsparamagnetism.

25. HowtofindbondorderusingMolecularOrbitaltheory
• However,before determiningthenumberof electronsincertain orbitals, onemu
s
tfill theseorbitals with electronsfirst. T
ofill the
orbitals,onemu
stknowthe rulesaccordingtowhich orbitalsareoccupied. Withoutunderstandingthisrule,computingamolecule’s
bondorderwouldbeimpossible.Thisexpression isusedtocomputethebondorderofamolecule:
• Thosewhoaren’tawarehavenooptionbuttolearnthem.If ithelps,onecansimplylearntherulesfor fillingatomicorbitals.Therules
tofill molecularorbitals are thesame,exceptthateach“bonding”orbital isfollowedbyan“anti-bonding”orbital. While atomic
orbitalsarefilled as1s2s2p… molecularorbitals arefilled as1s1s*2s2s*2p…. Theasteriskedorbitalsrepresentanti-
bondingorbitals.
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26. Example:
• Intotal,asinglemoleculeofoxygenconsistsof12 valenceelectrons.
Now,according to therules, the electrons m
u
stbe arranged in this
manner:
• We see that there are 8 electrons distributed in bonding orbitals,
while 4 are in anti-bonding orbitals. Substitute the numbers in the
expressionandyouwill findthatthebondorderofoxygenis2.
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27. • ThetwoorbitalsarelikeJekyllandHyde– theyinhabitthesamegeometry
,buttheanti-bondingorbital existsatanotoriouslyhigh
energy level, whichdenies thecombination of electrons any stability, whereas the bonding orbital exists at an energy where
electronscancomfortablybind,allowingthe resulting moleculetoattainstability.Infact,theenergyofa bondingorbitalislower
thantheenergiesof the individual levels thatthe electronsinhabit ina single atom.Thismeansthatthe electronswouldrather
combineandforma moleculethanexistinunpaired.
• The bonding orbital of a hydrogen molecule exists at a lower
energy than the individual levels of single hydrogen atoms. This
meansthatthe electronswould rathercombineandforma molecule
thanexistinan unpairedstate.
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28. • T
osummarize,if you’renotfamiliarwiththeMO theory
,youcancomputethebondorderofa moleculebydrawingits
Lewisstructureandthencheckingthetotalnumberofelectronsthat havebeensharedHowever,if you’refamiliarwiththe
MO theory
, you can compute the bond order by first filling the bonding and anti-bonding molecular orbitals with
valenceelectrons accordingtotherules,andthenrefertotheexpression.
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29. SignificanceofBondOrder
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• Thebondorderofa moleculegivesusa measureorindexofthestrengthofthebondsthatbindit.The bondsbindtheatomslikea
rubberbandbindingyourtwohands.Adouble-bondwouldmeanthatthe rubberbandisnowfoldedinhalfandtiedaroundyour
hands.Duetothestrengthofthisnewdouble- bond,yourhands havemoved closertoeachother. Themoleculeisnowmorestable.
Similarly,three bondsorthreefoldswouldmoveyourhandsevencloser.
• Furthermore,thestrongerthebond,themoreenergyisrequiredtobreakit— thehandsarem
u
c
heasier tofreewhenboundbya
single-fold rubber band than a double- or triple-fold band. Theenergy required to break the bond is referred to as the
molecules’bondenergy
.

30. HomonuclearDiatomicMolecules
• Bondorderis1.
• Diamagnetic
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36. HeteronuclearDiatomicMolecules
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38. combination of the two wave functions
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unpairedelectrons.
VBT
• Atomicorbitalsaremono-centric.
• Ato
m
s, which are involved in the bond formation,
maintaintheirindividualnature.
• VBThassimplicityandconvenience.
• R
esu
lting Molecular Orbital is obtained by the
of two
• InVBT
,someofthevalenceelectronsareindicated asunshared
anduninvolvedintheformationofthemolecule.
• VBTfailedtoexplainthe paramagneticbehavior of oxygen.
MOT
• Molecularorbitalsare poly-centric.
• Atomicorbitals,which formmolecularorbitals,do notretaintheir
individualnature.
• MOTissomewhatcomplex,especiallymolecules containmore
thantwoatoms.
• TheformationoftheMolecularOrbitalsisbasedon theLCAO
approximation method. Atomic Orbitals corresponding of the
Valency Shell of two only takes part in the formation of
MolecularOrbitals.
• MOTmethodshowsthatallthe electronsofthe valenceshellsof
theatomsofa moleculetakepart inthebonding.
• MOT successfully explains the paramagnetic behavior of
oxygen.

39. Jens Martensson 39