Many Electron Spin Eigenfunctions - Home - Department …€¦ · · 2017-12-25Many Electron Spin...
7
J. F. Harrison 12/25/2017 1 Many Electron Spin Eigenfunctions An arbitrary Slater determinant for N electrons can be written as ˆ AΦ(1,2, ,N ) χ M (1,2, ,N ) Where Φ 1 ,2, ,N ( ) = a( 1 )b( 2 ) c( N ) is a product of N orthonormal spatial functions and χ M 1 ,2, ,N ( ) is a product of N α α spin functions and N β β spin functions and therefore has an eigenvalue of z ˆ S equal to ( ) 1 2 M N N α β = − . Note that we do not consider double occupied spatial orbital when investigating the spin characteristics of a Slater Determinant. To determine the effect of 2 ˆ S on the Slater determinant we need only investigate ˆ S 2 χ M 1 ,2, ,N ( ) . Accordingly if we want a Slater determinant to be an eigenfunction of 2 ˆ S with an eigenvalue ( ) 1 SS + we need only to write ˆ AΦ(1,2, ,N ) χ SM (1,2, ,N ) Where ˆ S 2 χ SM (1,2, ,N )= SS + 1 ( ) χ SM (1,2, ,N ) . The problem is how to find χ SM (1,2, ,N ) and there are several general approaches. Consider for example a three- electron system. We may form 3 2 8 = spin functions and we list them below labeled by the M quantum number. We assume the electron order is 1, 2, 3, so 1 2 3 ()( )( ) αβα α β α = Number Function M 1 ααα 3/2 2 ααβ 1/2 3 αβα 1/2 4 βαα 1/2 5 ββα -1/2 6 βαβ -1/2 7 αββ -1/2 8 βββ -3/2 Note that & ααα βββ must be eigenfunctions of 2 ˆ S with S=3/2 and 32 M =± . Functions 2, 3 and 4 must span the space containing one quartet (S=3/2) and two doublets (S=1/2) all with M=1/2 while functions 5, 6, and 7 lie in the 12 M = − subspace of the eigenfunctions of 2 ˆ S . So for the M=1/2 subspace we may write
Transcript of Many Electron Spin Eigenfunctions - Home - Department …€¦ · · 2017-12-25Many Electron Spin...
J. F. Harrison 12/25/2017 1
ManyElectronSpinEigenfunctionsAnarbitrarySlaterdeterminantforNelectronscanbewrittenas AΦ(1,2,,N )χM (1,2,,N ) Where Φ 1,2,,N( ) = a(1)b( 2 )c( N ) isaproductofNorthonormalspatial
functionsand χM 1,2,,N( ) isaproductofNα α spinfunctionsand Nβ β spin
functionsandthereforehasaneigenvalueof zS equalto ( )12
M N Nα β= − .Notethat
wedonotconsiderdoubleoccupiedspatialorbitalwheninvestigatingthespincharacteristicsofaSlaterDeterminant.Todeterminetheeffectof 2S ontheSlaterdeterminantweneedonlyinvestigate S
2χM 1,2,,N( ) .AccordinglyifwewantaSlaterdeterminanttobeaneigenfunctionof 2S withaneigenvalue ( )1S S + weneedonlytowrite AΦ(1,2,,N )χSM (1,2,,N ) Where S
2χSM (1,2,,N ) = S S +1( )χSM (1,2,,N ) .Theproblemishowtofind
χSM (1,2,,N ) andthereareseveralgeneralapproaches.Considerforexampleathree-electronsystem.Wemayform 32 8= spinfunctionsandwelistthembelowlabeledbytheMquantumnumber.Weassumetheelectronorderis1,2,3,so
1 2 3( ) ( ) ( )αβα α β α=
Number Function M1 ααα 3/22 ααβ 1/23 αβα 1/24 βαα 1/25 ββα -1/26 βαβ -1/27 αββ -1/28 βββ -3/2
Notethat &ααα βββmustbeeigenfunctionsof 2S withS=3/2and 3 2M = ± .Functions2,3and4mustspanthespacecontainingonequartet(S=3/2)andtwodoublets(S=1/2)allwithM=1/2whilefunctions5,6,and7lieinthe 1 2M = −subspaceoftheeigenfunctionsof 2S .SofortheM=1/2subspacewemaywrite
J. F. Harrison 12/25/2017 2
1 1 2 3
2 1 2 3
3 1 2 3
3 2 1 2
1 2 1 2
1 2 1 2
S / ,M / a a a
S / ,M / b b b
S / ,M / c c c
ϕ ααβ αβα βααϕ ααβ αβα βααϕ ααβ αβα βαα
= = = = + +
= = = = + +
= = = = + +
Orinmatrixform
1 1 2 3
2 1 2 3
3 1 2 3
a a ab b bc c c
ϕ ααβϕ αβαϕ βαα
⎛ ⎞ ⎛ ⎞⎛ ⎞⎜ ⎟ ⎜ ⎟⎜ ⎟=⎜ ⎟ ⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠⎝ ⎠
Wemayinvertthematrixandwrite
1 2 3 1
1 2 3 2
1 2 3 3
A A AB B BC C C
ααβ ϕαβα ϕβαα ϕ
⎛ ⎞ ⎛ ⎞⎛ ⎞⎜ ⎟ ⎜ ⎟⎜ ⎟=⎜ ⎟ ⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠⎝ ⎠
Where
11 2 3 1 2 3
1 2 3 1 2 3
1 2 3 1 2 3
A A A a a aB B B b b bC C C c c c
−⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟=⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
Thespinfunctionααβ isalinearcombinationofthequartet, 1ϕ andthetwodoublets, 2 3&ϕ ϕ 1 1 2 2 3 3A A Aααβ ϕ ϕ ϕ= + + Todetermine 1ϕ wewilloperateonααβwithaprojectionoperatorthatwillannihilate 2 3&ϕ ϕ ( )( ) ( ) ( )( )2
1 1 1 13 31 1 1 11 1 1 32 2 2 2 2 2S A Aααβ ϕ ϕ− + = + − + =
Andsince ( ) ( )2 2 1 4z z
ˆ ˆ ˆ ˆ ˆS S S S S S Sααβ ααβ ααβ+ − + −= − + = −
And ( ) 2ˆ ˆ ˆS S Sααβ βαβ αββ ααβ βαα αβα+ − += + = + +
J. F. Harrison 12/25/2017 3
Sowehave 2 7 4S ααβ ααβ βαα αβα= + + Andtherefore ( )2
1 13 4 3S Aααβ ααβ βαα αβα ϕ− = + + =
Afternormalizationwehave
( )113 2 1 23
S ,Mϕ ααβ βαα αβα= = = = + +
Tofindthedoubletsweannihilatethequartet. ( )( ) ( ) ( )( )( )2
2 2 3 33 3 3 31 11 1 12 2 2 2 2 2S A Aααβ ϕ ϕ− + = + − + +
Notethatsince 2 3&ϕ ϕ arebothdoubletsanylinearcombinationisadoubletso ( )( )2 3 3 1 22 2S ααβ ααβ βαα αβα− + = − + +
isadoublet.Theotherlinearlyindependentdoubletisfoundbyprojectingthequartetoutofoneoftheremainingfunctions,either or αβα βαα . ( )( )2 3 3 1 22 2S αβα αβα βαα ααβ− + = − + +
( )( )2 3 3 1 22 2S ααβ ααβ βαα αβα− + = − + +
Wethenhavethreenormalizeddoubletspinfunctions
( )
( )
( )
1 261 261 26
ααβ βαα αβα
αβα βαα ααβ
βαα αβα ααβ
− + +
− + +
− + +
Notethatthesethreedoubletscannotbelinearlyindependentbecausethereareonlytwointhespace.Wecanformtwoorthonormalfunctionsbytaking
( )1 26
ααβ βαα αβα− + +
J. F. Harrison 12/25/2017 4
asonedoubletandformingthesecondbysubtracting
( ) ( )1 12 26 6
fromαβα βαα ααβ βαα αβα ααβ− + + − + +
Wehave,afternormalizationtheseconddoublet.
( )12
αβα βαα− .
Wecanformtheremaining 1 2M = − functionsbythesameprojectionoperatortechniquebutitinstructivetousetheraisingandloweringoperators.Recallthatforanyangularmomentumeigenfunction J ,M operatingwith J± keeps J thesameandchangesM to 1M ± 1J J ,M A( J ,M ) J ,M± = ± Given 13 2 2S ,M= = wemayform 13 2 2S ,M= = − byoperatingwith S− so
( ) ( )13 2 22ˆS ,M S ααβ αβα βαα αββ βαβ ββα−= = − ≈ + + = + +
andafternormalizationwehave
( )113 2 2 3S ,M αββ βαβ ββα= = − = + +
ExerciseShowthattheeffectoftheloweringoperator S− onααα isthesamefunction
3 2 1 2S ,M= = obtainedbyusingtheprojectionoperatoronααβ .
BranchingDiagram
AveryconvenientwayofkeepingtrackofthenumberofspineigenfunctionsofagivenmultiplicityforanNelectronsystemisthebranchingdiagramshownbelow
J. F. Harrison 12/25/2017 5
Thediagramisconstructedasfollows.Atwo-electronsystemisconstructedfromaoneelectronsystembycouplingtheadditionalspin1 2 andtheoriginalspin1 2 intoeitherasinglet(down)oratriplet(up).Toformthespineigenfunctionsforathreeelectronsystemwecouplethetwoelectrontripletintoaquartet(up)andadoublet(down).Additionallythesingletmaybecoupled(up)intoadoublet.Wethenhave1quartetandtwodoublets.Toformthe4electronsystemwecouplethequartetintoaquintet(up)andintoatriplet(down).Thetwodoubletscoupleuptoformtwoadditionaltripletsanddowntoformtwosinglets.Thefourelectronsystemthencanform1quintet ( )2S = ,3triplets ( )1S = and2singlets ( )0S = foratotalof16states.Thisisofcoursetherequired 42 .Notetherapidincreaseinthenumberoflowspineiegenfunctionsasthenumberofunpairedelectronsincreases.ForN=10wehave42singlets,90triplets,75quintets,35septets,9nonetsand1bigtet.WegatherbelowafewexplicitspineigenfunctionforN=1to5andspecificallyforthecaseM S= .LowervaluesofM foraparticular S maybegeneratedusingtheloweringoperator S− .ThesearefromtheSanibelnotesofFrankEHarris.Notethatthefunctionshavebeenfactoredwherepossibletohighlightthephysicalcouplings.ForexampletheN=4singlet, ( )( ) 2/αβ βα αβ βα− − obtainsfromthesingletcouplingofelectronpairs1&2and3&4whiletheN=5doublet,( )( ) 2/αβ βα αβ βα α− − alsoobtainsfromthesingletcouplingofelectronpairs
J. F. Harrison 12/25/2017 6
1&2and3&4withthedoubletspinangularmomentumbeingcarriedbytheα spinofelectron5.
J. F. Harrison 12/25/2017 7
OrthonormalSpinFunctionsforNspinswithtotalspinSthroughN=5
N S SSχ 1 1/2 α 2 1 ( ) 2/αβ βα+ 2 0 ( ) 2/αβ βα− 3 3/2 ααα 3 1/2 ( ) 2/αβ βα α− 3 1/2 ( )2 6/βαα αβα ααβ+ − 4 2 αααα 4 1 ( ) 2/αβ βα αα− 4 1 ( )2 6/βαα αβα ααβ α+ − 4 1 ( )3 12/βααα αβαα ααβα αααβ+ + − 4 0 ( )( ) 2/αβ βα αβ βα− − 4 0 ( )2 2 12/ααββ ββαα αβαβ βαβα αββα βααβ+ − − − − 5 5/2 ααααα 5 3/2 ( ) 2/αβ βα ααα− 5 3/2 ( )2 6/βαα αβα ααβ αα+ − 5 3/2 ( )3 12/βααα αβαα ααβα αααβ α+ + − 5 3/2 ( )4 20/βαααα αβααα ααβαα αααβα ααααβ+ + + − 5 1/2 ( )( ) 2/αβ βα αβ βα α− − 5 1/2 ( )2 2 12/ααββ ββαα αβαβ βαβα αββα βααβ α+ − − − − 5 1/2 ( )( )2 12/αβ βα βαα αβα ααβ− + − 5 1/2 ( ) 6/αααββ ααβαβ ββααα αβααβ βαααβ ααββα+ + − − − 5 1/2 ( )2 2 12/ααβαβ αααββ αβαβα βααβα αββαα βαβαα− + + − −
