252 CHAPTER15. IDENTICALPARTICLES
2.ForagiventotalspinS,thelowestenergystateisthestatewiththelargestL.- ForagiventotalSandL, ifthevalencelevel isnotmorethanhalf-filled,the
lowestenergy statehastheminimumJ =|L−S|;iftheshellismorethan
half-filled,thelowestenergystatehasJ=L+S.
AsanexampleoftheserulesweconsiderCarbon,whichhasanatomicconfigura-
tion 1 s^22 s^22 p^2 .Thefilledshellshavezerospinandzeroorbitalangularmomentum
(because thereisnostateonecanmakewithmtotal += 0 , Sztotal +=0),so onlythe
twovalenceelectronsarerelevantfordeterminingtheoverallspin,orbital,andtotal
angularmomentum.Thetwospin^12 electronsinthe 2 pstatescancombineintoS= 0
orS=1.ByHund’sFirstRule,theS= 1 valueischosen.Thismeans(forSz=+1)
bothelectronsarespinup. They arealsobothinanl= 1 orbital,soadditionof
angularmomentumcouldleadtoanyofthevaluesL= 2 , 1 ,0. Butbecauseof the
exclusionprinciple,thetwospin-upelectronscan’tpossiblyhavethesamemvalues.
ThismeansthatthehighestandlowestweightstatesforaddingL 1 = 1 andL 2 = 1
areexcluded,sincewecan’thavem 1 =m 2 =±1.SothisrulesoutL=2.Thehigh-
estLleft,byHund’ssecondrule,isL=1. ByHund’sthirdrule,sincethevalence
shellislessthanhalffilled,J=|L−S|=0.Sothetermnotation(or”spectroscopic
description”)forthegroundstateofCarbonis^3 P 0.
ThereismuchmorethatcanbededucedaboutthePeriodicTable,atomicphysics,
andchemicalreactionsfromquantum-mechanicalprinciples.Thisisthesubjectmat-
terofacourseonTheoreticalChemistry. Buttherearetwopointsthat,onceagain,
deservespecialemphasis. First,theexistenceofchemicalreactionsisaconsequence
oftheexclusionprinciple. Wereitnotfortheexclusionprinciple,allatomicelectrons
wouldsitinthe 1 sstate,becomingmoreandmoretightlyboundasZincreases. The
secondpointisthattheobservedrepetitionofchemicalpropertiesisconnectedtothe
2(2l+1)occupancyofthes,p,d,forbitals,andthisinturnisconnectedtothepe-
culiaralgebraoftheangularmomentumoperators. Inshort,thechemicalproperties
oftheelementsaredueessentiallytotheexclusionprinciple,andtothefactthattwo
rotations,doneindifferentorders,endupindifferentplaces. Lifeisabyproductof
chemistry;it’sacuriousthoughtthatweoweourexistencetotheabsoluteidentity
ofelectrons,andtothenon-commutativepropertyofrotations.