Coupling of angular momenta

30 Pages·2016·0.39 MB·English

Chapter 6
Coupling of angular momenta – the vector
model
In the previouschapter, we introduced the central-field approximation as a way to
obtainseparablebasisfunctionsforthestatesofmultielectronatoms.Theseproduct
statesaresolutiontoaSchro¨dingerequationwithapurelyradialpotential,andthey
are annotated with an electron orbital nomenclature (see chapters 1.5 and 5.2.1).
Whatweneedtodonowistoapplyperturbationtheory,usingtheseelectroncon-
figurationsaszeroth-orderfunctions.
WhenweinitiallyintroducedtheCFAinchapter5,andestablishedelectroncon-
figurations,weignoredtwocontributionstothetotalHamiltonian;namelythespin-
orbitinteractionandwhat’sleftofthemutualelectronrepulsionterm,whenwehave
separatedoutitsradialpart.Theissuenowathandistoaddthesecontributionsas
perturbations.Eventually,wewillthenbeabletofurtheraddotherinteractions,for
examplewithexternalfieldsandelectromagneticmomentsfromthenucleus.
Whatwehavetodealwithpresentlyisinteractionsbetweentheorbitalandspin
angular momenta of all electrons. A model for doing this was developed by early
spectroscopists,beforequantummechanicsandbeforethesalientfeaturesofatomic
structure were known. It is a phenomenological model based on the addition of
vectors,describingthedifferentangularmomenta,withconstraintsconcerningthe
discrete nature of the vector sums. The latter rules was initially entirely based on
empiricalobservations.
Whenstudyingatomicspectra,itwasnotedthatgroupsofofenergylevels(de-
rivedfromspectrallines)appearedtogether,andthatsomespecifictypesofatoms
alwayshadsuchgroupingswithparticularmultiplicities.Forexample,alkaliatoms
had states appearing in doublets, alkaline earths were shown to have singlets and
triplets,andnitrogenhaditsenergylevelsarrangedinquartetsanddoublets.Work-
ingbackwards,itwashypothesisedthatthishastodowithinteractingangularmo-
menta, and that spin had to be included in the treaties. The model was developed
before quantum mechanics, but it turns out that when the addition of angular mo-
menta was subsequently put on a quantum mechanical footing, this vector model
stillgaveexcellentqualitativeresults,anditstillgreatlyfacilitatesaclosestudyof
basicatomicstructureandatomicspectroscopy.
73
74 6 Couplingofangularmomenta–thevectormodel
Inthefollowing,wewillintroducethevectormodel,andwewillquicklyproceed
to the application of quantum mechanical perturbation analyses of the same issue.
When doing this, we will have to isolate special cases, where some interactions
dominate over others. The extremes of these special cases are the so called LS-
coupling and jj-coupling schemes. They will be introduced in chapters 6.3 and
6.4, and then treated in detail in chapters 7 and 8. Albeit the above schemes are
limiting cases, they will help us to understand a majority of atomic structure, and
via interpolation schemes, we can get a good grasp also of various intermediated
situations.
6.1 Theconceptofthevectormodel
Thecentral-fieldapproximationisakeytothejustificationofthevectormodel.In
thepre-quantumera,andaftertheintroductionoftheatomicmodelsbyRutherford
and Bohr, the formulation was rather along the lines that only a few of the nega-
tive charges in an atom contribute to the finer features of atomic structures. With
an atomic model based on discrete orbitals, this could also be stated as that only
electronintheoutermostoccupiedorbitalsgiveangularcontributionstothestruc-
ture,whereastheinneronesgiveasphericallysymmetriccontributiontotheoverall
energyoftheatom.Thishypothesiswasforearlyspectroscopistsapostulatebased
onobservations.Today,wecangiverationalargumentsforit,andwewilldwellon
itmoreinchapter6.2.
A purely central potential gives a comparatively simple solution to the Schro¨-
dingerequation.Theobservationthatatomicspectracanstillbeverycomplicated
must then lead to the conclusion that these complications have to to with atoms
outsidethesymmetricclosedshells,i.e.thevalenceelectrons.Inner,fullyoccupied
orbitals,maycombinetoasphericalforcefield,buttheremainingvalenceelectrons
will exert torques on each other. From there, we can understand that the number
ofvalenceelectronsmusthaveanimportantbearingontheenergylevelstructure,
andthusitisunderstandablewhyatomsfoundinthesamecolumnintheperiodic
systemhavesimilarspectra.
The said angular effects must, from a Newtonian point of view, be possible to
describeasinteractionsbetweendifferentangularmomentathathavetheeffectthat
theywillallvarywithtime,whereastheirsumwillremainconstant(forthemoment
weruleoutdissipation).Wehave
dL
=τ , (6.1)
dt
andifwetaketheexampleoftheangularpartoftheelectrostaticinteractionbetween
twovalenceelectrons,itisclearthatneitherL ,norL willbeconstantintime.They
1 2
willbothprecessabouttheirvectorsum,asillustratedinfig6.1
Inquantummechanicallanguage,L2 andL2 willcommutewithL2,L andthe
1 1 z
Hamiltonian describing the interaction, but L and L will not. Likewise, for an
1z 2z
6.1 Theconceptofthevectormodel 75
Fig.6.1 Theorbitalangular L L
1 + 2
momentaoftwoelectrons,L
1
andL ,arecoupledtrough
2
electrostaticinteraction.Asa
result,theywillbothprecess,
whereastheirsumisconstant
intime.
L
1
L
2
electron spin interacting with the same electron’s orbital angular momentum, L
i
andS willprocessaroundtheirsumJ,asdescribedinchapter4.
i i
It is clear that sums of angular momenta play an important role, and thus we
havetoknowwithwhatrestrictionssuchadditionscanbemade.Tostartwith:from
classical physics it is clear that a sum of angular momenta is in itself an angular
momentum. Moreover, we know that a general quantum mechanical angular mo-
mentum J must be quantized such that the eigenvalues of J2 are j(j+1), with j
being either zero, a positive integer, or a positive half-integer, and the eigenvalues
ofJ arem ,whosepossiblevaluesliebetween−jand jandareseparatedbyinte-
z j
gernumbers(seeappendixC).
With this, we can illustrate the addition rules with a few practical examples. A
morerigorousjustificationfortheserulesisgiveninappendixI.Webeginbycon-
sideringanumberofvalenceelectronswithspinsS thatinteracttoformatotalspin
i
S=∑iSi.Weknowthatforeveryelectron,si= 21,andweassumethattheorbital
angular momenta (and/or the principal quantum numbers) are such that we do not
have tobother withthe Pauliprinciple. We writethe quantumnumbers associated
withS2andS asSandM respectively(seechapter1.5),andthepossiblevaluesof
z S
Sareillustratedinfigure6.2.
For a case with two valence electrons, the possible results for the total spin are
S=0orS=1—thetwospinsareeitheralignedoranti-parallel.Thesetwooptions
correspondtothesingletsandtripletsdescribedinchapter2.2.Withathirdelectron,
thethirdspin-vectormustbeaddedtothevectorsumofthetwofirst,eitherincreas-
ing or decreasing the total spin, and the possible values are S=1/2 and S=3/2.
Thosevaluesgivesspectrallinesappearingindoubletsandquartets,andweseethat
thereisindeedalogicallinkbetweenthetotalspinS,andthequantity2S+1,which
was assumed to be the multiplicity (see chapter 1.5). It should be noted that some
76 6 Couplingofangularmomenta–thevectormodel
Fig.6.2 Illustrationofthe
S
vectoradditionofelectron
spins.Onthex-axisisthe
number of electrons, and
onthey-axisthetotalspin
5/2
(thevectorsum).Withtwo
electrons,onlysinglets(S=
0)andtriplets(S=1)are 2
possible. Three electrons
producesdoublets(S=1/2)
and(S=3/2)quartets,andso 3/2
on.Notethatisthisdiagram,
itisassumedthatthePauli
principleisnotanimpediment 1
(someoftheotherquantum
numbersaredifferent).
1/2
N
1 2 3 4 5 e
oftheadditionpathsinfigure6.2mayinsomeinstancesbeinhibitedbythePauli
principle.
Addingorbitalangularmomentaworksinthesameway,exceptthatthedifferent
l canbeanypositiveinteger,butareneverhalf-integers.Takeanexamplewithone
i
d-electron(l =2)andonep-electron(l =1).Thepossiblevaluesofthequantum
1 2
numberforthetotalangularmomentum,L,are3,2and1.Ifthereisathirdelectron,
itsangularmomentumisaddedinthesamewaytothedifferentsumsofthefirsttwo.
Atthisstage,theorderofthesummationdoesnotmatter.
Left to do is to take into account the spin-orbit interaction. Here we are faced
withachoice.WecaneitherfirstcouplealltheindividualL andS toanumberof
i i
J =L +S,whereafteralltheindividualJ aresummedintoagrandtotalangular
i i i i
momentum,J=∑iJi.Alternatively,wefirstformL=∑iLiandS=∑iSi,andafter
thatwegettothesameJ=L+S,havingtakenadifferentroute.Thisisillustrated
infigure6.3
WhatpathwetakeinordertogettoJisactuallycrucial.Evenwhenweexclude
angular effects from inner, closed shells, we will have two angular momenta per
valenceelectrons,andtheywillallinteracteachother.Wehavealreadyresignedto
calculateenergycontributionswithperturbationtheory,usingCFAelectronconfig-
urations as zero-order states. However, if we take the Hamiltonian from (5.2) and
(5.4),andaddthespin-orbitcontributionfrom(4.16):
H=H +H +H , (6.2)
CF to SO
westillneedtotreatthelasttwotermsonebyone.Foraperturbativecalculation,
that will only work if the largest term is treated first, and if each subsequent term
inthestepwisecalculationhasanenergycontributionsignificantlysmallerthanthe
6.1 Theconceptofthevectormodel 77
~s2 ~s2
J~ S~ J~ ~j2
~s1 ~l2
L~ ~j1
~l2 ~s1
~l1 ~l1
Fig.6.3 Twodifferentwaystocouplethefourangularmomentaofatwoelectronatom(L ,S ,L
1 1 2
andS )inordertoformthetotalelectronicangularmomentum2.Intheleftpanel,thestrongest
2
interactionisthatbetweentheelectrons.TheindividualLiandSifirstcoupletoformLandS,and
thethespinorbitinteractiongivesusJ.ThisistheLS-couplingapproximation.Intherightpanel,
thespin-orbitinteractionisthemostpronounces.Inthis—the jj-couplingcase—theindividual
Jiarefirstformed,andthenthesecoupleduetotheelectron-electroninteractionandwillgiveusJ
precedingone.Thatistosaythatbothpictograminfigure6.3representsapproxi-
mationsandrepresenttwolimitingcases.
What this means is that we need to know the relative importance of H and
to
H , and the answer to that question is not universal. For light atoms, and for a
SO
considerablepartoftheperiodicsystem,theinteractionbetweenelectronsismuch
more important than the spin-orbit interaction. In that case, we first treat H , and
to
theresultarestatesreferredtoasatomicterms.Forthese,thefirststageonthepath
toJhasbeentoformLandSfromtheangularmomentaoftheindividualelectrons.
The situation is referred to as ‘LS–coupling’, and it will be treated in chapters 6.3
and7.Inthefinalstep(ignoringexternalfieldsandnucleareffects),thetermH
SO
is applied as a perturbation to the atomic terms. This reveals the ‘fine structure’,
whichwillbequantifiedinthequantumnumberJ.Therepresentationofthestates
inLS-couplingis:
|γLSJM (cid:105). (6.3)
J
The fine-structure Hamiltonian H scales as Z3 — see (4.12) — whereas H
SO to
hasalinearscalingwithZ.Asaconsequencetheapproximationthatthespin-orbit
termissmallcomparedwiththeCoulombtorqueonemaynotholdforveryheavy
atoms.
Intheotherextremelimit,wewillhavethecasewherewecansafelyinversethe
orderinwhichthetwoangularHamiltoniansareapplied.Inthesecases,theinternal
spin–orbitinteractionswampsthecouplingbetweenthevalenceelectrons.Thissitu-
ationsiswhatisillustratedintherightpartoffigure6.3.Wecallthis‘jj–coupling’,
78 6 Couplingofangularmomenta–thevectormodel
since we begin by forming the individual J, and then we add all these to form J.
i
ThequantumnumbersLandSnolongercommuteswiththetotalHamiltonian,and
therepresentationmustratherbe:
|γ j ... j JM (cid:105). (6.4)
1 N J
Wetreat jj–couplinginchapters6.4and8.
ObviouslybothLS–couplingand jj-couplingareapproximations.However,with
thesedescriptionsasabasis,weareabletodescribealsomanyintermediatecases.
Wewilllookcloseratsuchcasesinchapter9.
6.2 Closedshells
Wewilltakeacloserlookatclosedshells,andthecontributiontoatomicstructureof
electronsintheseclosedshells.Ashasbeenestablishedinchapter5,aclosedshell
isanelectronorbital,designatedbythequantumnumbersnl,whichhasexactlythe
maximumnumberofelectronsinitthatispossible.
Thefactthatorbitals,orshells,canbefilledatallisaconsequenceofthePauli
principle.Wedesignateeachelectronwithacertainsetofquantumnumbersn,l,m,
l
andm .Inastringentpointofview,thisisincorrect.Intherealmultielectronatoms,
s
allelectronsareentangled,andwhenwetalkaboutelectronconfigurations,there-
alityisthateachsetofquantumnumbersrepresentoccurrences.Assigningspecific
quantum-numberlabelstoeachelectronisstrictlyspeakingunphysical.Neverthe-
less,thinkingofelectronconfigurationisthiswaywillworkfineforfiguringoutthe
salientfeaturesoftheatomicstructure.
ThePauliprincipletellsusthatnoelectronsmaybeidentical.Thatis,theymay
not have the same set of quantum numbers — that would make the wave function
exchangesymmetric.Thismeansthatforagivensetofn,l,andm,therecanbeonly
l
two electrons — with opposite spins. For each ‘shell’, nl, we can allow 2(2l+1)
electrons.Thus,ands-shellisclosed,orfilled,whenithastwoelectrons.Ap-shell
takessixelectrons,ad-shellten,andanf-shellfourteen.
Thiswayoffillingupmeansthatinaclosedshell,foreveryelectronthathasa
positivevalueofm theremustbeanotherthatisidenticalexceptforadifferentsign
l
of m. Likewise, for every spin-up electron, there must be a spin-down one. This
l
meansthatthesumsofallm andallm forallelectronsinallclosedshellsmostbe
l s
zero. The projection quantum numbers are the only ones that can contribute to an
orientation,andthusclosedshellsmustbesphericallysymmetric.
Forthespin-orbitHamiltonian(4.16),whereallvectorsL andS areincludedin
i i
thesummation,wecanthereforeignoreallelectronsinclosedshells—theircontri-
butiontothespin-orbitcouplingswillbezero.WhenformulatingtheHamiltonian
(4.16),itwillsufficetoincludevalenceelectrons.
Itisintuitivelyreasonabletoassumethattheangularcontributionoftheelectron-
electron interaction from closed shells will also be zero, since these filled orbitals
6.3 LS-coupling 83
6.3 LS-coupling
In the LS-coupling approximation, we assume that the angular part of the interac-
tionbetweenelectronsissomuchstrongerthanthespin-orbitinteraction,intrinsic
foreachelectron,thatthelattercanbeignoredinafirststepoftheapproximation.
Oncewehavespecifiedtheelectronicconfiguration,thenextstepisthustoformthe
quantumnumbersLandS,andwewillassumethatthecorrespondingoperatorsL2
andS2 commutewiththeHamiltonian.Moregenerally,wewilltakethisasadef-
initionofLS-couplingschemes,thatisonesforwhichtheHamiltonianisdiagonal
inLandS.
Wesawinchapter6.2thatwecanignoreelectronsinclosedshells.Theprocess
ofaddingupthedifferentL andS forthevalenceelectronsisdoneascoveredin
i i
chapter6.1andasillustratedinfigurexxxxxXXX.
ThequantumnumbersLcanbezerooranypositiveinteger.Itisannotatedina
wayanalogoustothatontheorbitalangularmomentumofindividualelectrons(see
chapter 1.5), but with capital letters. A total angular momentum of L=0 will be
annotated‘S’,L=1‘P’,L=2‘D’andsoon.Thislettersymbolwillbethebasis
oftheLS-couplingterm.
Thetotalspin,S,willbehalf-integerforanoddnumberofelectrons,andinteger
orzeroforanevennumber.InanLS-couplingterm,thevalueofS isindicatedby
writingthenumericalvalueof2S+1asasuperscripttotherightofthelettersymbol
forL.2S+1iscalledthemultiplicityoftheterm,forreasonsthatwillbeclarified
inthefollowing.
6.3.1 Atomicterms
We will look more closely at the total number of quantum numbers needed to de-
scribeanatomicstate,andthemultiplicity(ordegeneracy)ofa2S+1L-term.Foran
individualelectron,weneedthefourquantumnumbersn,l,m andm todescribe
l s
astate,whichcorrespondstonunumbersofdegreesoffreedom.ForanN-electron
atom,wecanthereforeassumethatwehave4Ndegrees,andaneedfor4Nquantum
numbers.
Suppose that the number of atoms in closed shells is N and the number of va-
c
lence electrons is N , with N +N =N. Within the CFA, we describe the closed
o c o
shellswiththeircorrespondingpartoftheelectronconfiguration,whichwillgives
usN pairsofnandl.InlightofthePauliprinciple—whichistheeffectmaking
c
theseshells‘closed’inthefirstplace—thiswillactuallyspecifyallfourrelevant
quantumnumbersfortheseelectrons.Thisleavesuswithanother4N tobedeter-
o
mined.
Thevalencepartoftheconfigurationprovides2N numbers.Fortheremaining
o
2N ,thingwilldependonhowmanyvalenceelectronswehave.Ifthereisonlyone,
o
only two degrees of freedom remains, and specifying the ”total quantum numbers
basedonLandS,thatiseither|LSM M (cid:105)or|LSLM (cid:105),givesapparentlygiveustoo
L S J
84 6 Couplingofangularmomenta–thevectormodel
manyquantumnumbers.Thisisnotactuallyaproblem,sinceinthiscase,wewill
alwayshaveL=landS=s=1/2.
For two valence electrons, the sets |LSJM (cid:105) or |LSM M (cid:105) will do perfectly
J L S
to provide a full description. If we have three or more valence electrons, the LS-
coupling designation is incomplete. This latter situation diverges for cases where
all electrons are in the same orbital or not. If they are not, a ‘parent term’ can be
formedbeforethefinalatomictermisspecified.Ifthevalenceelectronsareindeed
in identical n and l, the actual number of degrees of freedoms will be limited by
i i
thePauliprinciple,andtheLS-descriptionwillbesufficientafterall.
In the LS-coupling approximation, we will thus specify an atomic term for an
N-electronatomas:
(n l ...n l )n l ...n l 2S+1L, (6.18)
1 1 Nc Nc o1 o1 N N
wherethepartoftheelectronconfigurationinsideaparenthesisrepresentstheva-
lenceelectrons.Thisismorethanoftenomittedintherepresentation.Ifparentterms
are needed, the representation will be (with the valence configuration omitted for
clarity):
n l ...n l (2Sp+1L )n 2S+1L. (6.19)
o1 o1 N−1 N−1 p N
Inacasewithamanyvalenceelectrons(forexamplead-orbitalwithfourormore
electrons),morethanoneparenttermmaybeneeded.Thetheatomictermsabove
willbeaddedspecificationfortheremainingquantumnumbers,whenmorepertur-
bationssuchasthespin-orbitcouplingareadded.
6.3.2 Twonon-equivalentelectrons
Togiveconcreteexamples,westartbyshowingthepossibleatomictermsforacase
oftwovalenceelectronsthatarenotinthesameorbitals.Thatis,wehaveeitheror
bothoftheconditionsn (cid:54)=n andl (cid:54)=l fulfilled(wenowignoretheelectronsif
1 2 1 2
theinnershellandsimplyindexthevalanceelectronsbeginningwithone).Thisis
thesimplestcase,sincethenwedonothavetobecarefulaboutobeyingthePauli
principle.Wecannevermakethesetwoelectronsequivalentmyspecifyingm ,m ,
l1 l2
m andm .Inthefollowing,wewillgivetoexamples—theconfigurations2p3p
s1 s2
and3d4p.
Forthe2p3p-configuration(oranyn pn pwithn (cid:54)=n ),wehavel =l =1and
1 2 1 2 1 2
asalwayss =s =1/2.Thevaluesform andm canbe+1,0or-1,andthespins
1 2 l1 l2
can be parallel or anti-parallel with the chosen quantization axis (‘up’ or ‘down’).
Withallpossiblepermutations,thisshouldgiveus36differentstates,wheremany
maybedegenerateinenergy.Whenl andl couples,thepossiblevaluesforLare
1 2
2, 1 or 0. The two electrons spins can be either mutually parallel or anti-parallel,
givingthepossiblevaluesof1or0forS.Thepossibletermsfortheconfiguration
2p3parethus:
1D,3D,1P,3P,1S,3S. (6.20)
6.3 LS-coupling 85
IfwechoosetospecifytherestofthetermwiththequantumnumbersJ andJ
M
(whichwillprovetobeappropriateintheabsenceofexternalfields),wecanmake
anaccountingofthenumberofstate,andshowthatitisindeed36.Weknowthefor
eachterm,thepossiblevaluesforJarebetweenL+Sand|L−S|.Thisisillustrated
intable6.1.
Table6.1 PossibleLS-couplingstatesfora2p3p-configuration
term J MJ numberof
states
1D 2 −2...2 5
3 −3...3 7
3D 2 −2...2 5
1 −1...1 3
1P 1 −1...1 3
2 −2...2 5
3P 1 −1...1 3
0 0 1
1S 0 0 1
3S 1 −1...1 3
Totalnumberofstates: 36
Forthe3d4p-configuration,Sisstill0or1,andLcanvarybetween3and1.The
possibletermsbecome:
1F,3F,1D,3D,1P,3P, (6.21)
andifwecountthenumberofpossiblestate,weinthiscaseget60.
6.3.3 Twoequivalentelectrons
Toexemplifythiscase,wetaketheconfiguration2p2(thegroundstateconfiguration
for atom number 6, C). Had it not been for the Pauli principle, the possible terms
wouldhavebeenthesameasinthecasefor2p3p,shownin(6.20).However,inthis
caseourelectronsareequivalentandwemusttakeintoaccountthatonlyexchange
anti-symmetricwavefunctionsarepossible.Wemustexcludeatomictermsthatare
onlypossibleforelectronsequivalent.Wemustalsoexcludecombinationsofm ,
l1
m ,m andm thatarethesameforequivalentelectrons.
s1 l2 s2
86 6 Couplingofangularmomenta–thevectormodel
Apracticalwaytoillustratethisistouseatablerepresentationoftheaboveindi-
vidualelectronquantumnumbers,introducedbyCondon&Shortley.Suchatable
is organised horizontally and vertically by the total angular momentum projection
numbers M =m +m and M =m +m . To illustrate this, we first show an
L l1 l2 S s1 s2
exampleforthealreadytreatedconfiguration2p3pintable6.2.Eachentryinthista-
Table6.2 ghghhghgh
M
2p3p S
1 0 -1
2 ( 1+ 1+ ) ( 1+ 1− )( 1− 1+ ) ( 1− 1− )
( 1+ 0+ ) ( 1+ 0− )( 1− 0+ ) ( 1− 0− )
1
( 0+ 1+ ) ( 0+ 1− )( 0− 1+ ) ( 0− 1− )
( 1+ −1+ ) ( 1+ −1− )( 1− −1+ ) ( 1− −1− )
ML 0 ( 0+ 0+ ) ( 0+ 0− )( 0− 0+ ) ( 0− 0− )
(−1+ 1+ ) (−1+ 1− )(−1− 1+ ) (−1− 1− )
( 0+ −1+ ) ( 0+ −1− )( 0− −1+ ) ( 0− −1− )
-1
(−1+ 0+ ) (−1+ 0− )(−1− 0+ ) (−1− 0− )
-2 (−1+ −1+ ) (−1+ −1− )(−1− −1+ ) (−1− −1− )
bleisapairofelectrons,withthenumericalvaluesshowingm andtheplus/minus
li
superscriptsindicatesm .WeseethatM canvarybetween2and-2,whichmeans
li L
thatthemaximumvalueofListo,andthatS=1isthemaximumtotalspin,since
M isbetween1and-1.Wealsoseethatwehave36statesintotal,whichisfully
S
consistentwithwhatweshowedinchapter6.3.2.
Forthecasewiththe2p3p-configuration,atablesuchastable6.2issuperfluous.
However,forthecasewithequivalentelectronsitwillhelp.Considerthesametype
oftableforthe2p2-configuration.Initially,wecancopytable6.2,butthenanumber
oftermshavetobedeleted.Wehavetoremovepairsofidenticalelectrons,suchas
(1+ 1+). Furthermore, with identical electrons we have some pairs that identical,
suchas(1+1−)and(1−1+);thesealsohavetogo.Weendupwithtable6.3.The
36stateshavebeenreducedto15.Sometermsarenolongerfeasible.Forexample
3Disnolongerpossible,sincewecannothaveM =2andM =1atthesametime.
L S
Westillneedatleastonetriplet,andwecanmakeoutfromtable6.3thatwemust
havea3P-term.IfwecontemplatethepossiblevaluesofJ andM for3P,wefind
J
thatitcorrespondsto9states.Wecanthenremove9electronpairsfromtable6.3,
withM -andM -valuesbothbetween1and-1.Forwhatremainsinthetable,we
S L
seethatwemustalsohaveaterm1D.Thiswilltakecareoffivestates,whichmeans
thattheonlythingremainingintable6.3willbealoneelectronpairforM =0and
L
M =0.Thishastocorrespondto1S.
S
WeconcludethatthepossibleLS-couplingtermsforthe2p2-configurationare: