Spin Eigenfunctions and the Graphical Unitary Group Approach (GUGA) Part I Ron Shepard Chemical Sciences and Engineering Division Argonne Na8onal Laboratory COLUMBUS in China, Tianjin, China, Oct. 10-14, 2016
Spin Eigenfunctions and the Graphical Unitary Group Approach (GUGA)
Part I
RonShepardChemicalSciencesandEngineeringDivisionArgonneNa8onalLaboratoryCOLUMBUSinChina,Tianjin,China,Oct.10-14,2016
2
Outline
§ SpinEigenfunc8ons•GeneralBackground•ElectronicWaveFunc8ons
§ GUGA•Representa8onofCSFs•Computa8onofCouplingCoefficients
3
Spin Eigenfunctions
GUGAisBasedonGenealogicalSpinEigenfunc8ons
S =
SxSySz
⎛
⎝
⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟
Sx , Sy⎡⎣ ⎤⎦ = iSz Sy , Sz⎡⎣ ⎤⎦ = iSx Sz , Sx⎡⎣ ⎤⎦ = iSy
S2 = S ⋅ S = Sx2 + Sy
2 + Sz2
S2 , Sx⎡⎣ ⎤⎦ = Sy2 + Sz
2 , Sx⎡⎣ ⎤⎦= SySySx − SxSySy + SzSzSx − SxSzSz
= Sy SxSy − iSz( ) − SySx + iSz( ) Sy + Sz SxSz + iSy( ) − SzSx − iSy( ) Sz = 0S2 , Sx⎡⎣ ⎤⎦ = S2 , Sy⎡⎣ ⎤⎦ = S2 , Sz⎡⎣ ⎤⎦ = 0
§ Ruben Pauncz, Spin Eigenfunctions–Construction and Use, (Plenum, New York, 1979)
§ Richard N. Zare, Angular Momentum: Understanding Spatial Aspects in Chemistry and Physics, (Wiley, New York, 1988)
4
…Spin Eigenfunctions
Thereexistsabasiswithanelement:
S2 , Sz⎡⎣ ⎤⎦ = 0
S2 S,M = λS S,M ; λS ≥ 0
Sz S,M = M S,M
Sx2 + Sy
2( ) S,M = S2 − Sz2( ) S,M = λS − M
2( ) S,M
λS − M2( ) ≥ 0 ⇒∃ Mmin ,Mmax
⇒ S,M
5
…Spin Eigenfunctions
Definetheoperators:
S+ = Sx + iSy
S− = Sx − iSy
⎫⎬⎪
⎭⎪⇒
Sx = 12 S+ + S−( )
Sy = 12i S+ − S−( )
S± = S∓†
S2 , S±⎡⎣ ⎤⎦ = 0
Sz , S±⎡⎣ ⎤⎦ = ±S±
S+ , S−⎡⎣ ⎤⎦ = 2Sz
S2 = 12 S+S− + S−S+( ) + Sz2
= S∓S± + Sz Sz ±1( )
Thenitfollowsthat:
S2 S± S,M( ) = S±S2 S,M = λS S± S,M( )Sz S± S,M( ) = S±Sz ± S±( ) S,M = M ±1( ) S± S,M( )
⇒ S± S,M = C± S,M ±1
S+isaraisingoperatorS-isaloweringoperatorAlsocalledStep-up/Step-down,Ladder,andShi[Operators.
6
…Spin Eigenfunctions
0 = S−S+ S,Mmax = S2 − Sz Sz +1( )( ) S,Mmax = λS − Mmax Mmax +1( )( ) S,Mmax
0 = S+S− S,Mmin = S2 − Sz Sz −1( )( ) S,Mmin = λS − Mmin Mmin −1( )( ) S,Mmin
Mmax Mmax +1( ) = Mmin Mmin −1( )Mmax + Mmin( ) Mmax − Mmin +1( ) = 0
⇒ Mmax = −Mmin
ThepossiblevaluesofMalldifferbyintegervalues,soMmax-Mmin=2SwhereSiseitherintegerorhalf-integer.
M=S,S-1,…-S+1,-S(2S+1)possiblevaluesλS = S(S +1)
S2 S,M = S(S +1) S,M
Sz S,M = M S,M
7
…Spin Eigenfunctions
Operatefromthele[withtheadjointofbothsides:
C±2 = C±
2 S,M ±1 S,M ±1
= S,M S∓S± S,M
= S,M S2 − Sz Sz ±1( ) S,M= S S +1( ) − M M ±1( )
C± = S S +1( ) − M M ±1( )S± S,M = S S +1( ) − M M ±1( ) S,M ±1
S± S,M = C± S,M ±1
8
…Spin Eigenfunctions
SummaryofMatrixElements:
S,M S2 ′S , ′M = S S +1( )δS ′S δM ′M
S,M Sz ′S , ′M = MδS ′S δM ′M
S,M S± ′S , ′M = S S +1( ) − ′M ′M ±1( )δS ′S δM ′M ±1
S,M Sx ′S , ′M = 12 S S +1( ) − ′M ′M ±1( )δS ′S δM ′M ±1
S,M Sy ′S , ′M = ± 12i S S +1( ) − ′M ′M ±1( )δS ′S δM ′M ±1
S+ =
M \ ′M S … … −S
S!!−S
0 X … 00 0 " !! " 0 X0 … 0 0
⎛
⎝
⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟
S− =
M \ ′M S … … −S
S!!−S
0 0 … 0X 0 " !! " 0 00 … X 0
⎛
⎝
⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟
Sx =
M \ ′M S … … −S
S!!−S
0 X … 0X 0 " !! " 0 X0 … X 0
⎛
⎝
⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟
Sy =
M \ ′M S … … −S
S!!−S
0 −iX … 0iX 0 " !! " 0 −iX0 … iX 0
⎛
⎝
⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟
9
…Spin Eigenfunctions
Tobeusefulforelectronicstructuremethods,wemusthaveasecond-quan8zeddefini8onofthespinoperators:
S+ = apα† apβ
p∑
S− = apβ† apα
p∑
Sz = 12 apa
† apα − apβ† apβ
p∑
Itmaybeverifiedthattheseoperatorssa8sfythecommuta8onrela8ons
Sx , Sy⎡⎣ ⎤⎦ = iSz Sy , Sz⎡⎣ ⎤⎦ = iSx Sz , Sx⎡⎣ ⎤⎦ = iSy
thereforeeveryrela8onthatissa8sfiedfortheabstractspinoperatorsisalsosa8sfiedfortheseoperators.
10
…Spin Eigenfunctions
Aspin-eigenfunc8onbasisisusefulforthreereasons:1)Because[H,S2]=0and[H,Sz]=0,thewavefunc8onexpansioncanbelimitedtoonlythesubspaceofasingleSandM.Theexpansionlength,andnumberofvaria8onalparameters,isreducedcomparedtothefulltensor-productexpansion.2)Whenthevaria8onalspaceislimitedtoasingleSandM,thenthereisnopossibilityofbrokenspin-symmetry,spin-contamina8on,orspininstabili8es.3)ThepossibilityofallowedcrossingsbetweenstatesofdifferentSdoesnotcomplicatedthecomputa8onofpoten8alenergysurfaces.E.g.theloweststateforagivenSisalwaysthelowesteigenvalueofthecomputedHmatrix.
11
Recursion
§ re·cur·sion(ri-'kûr-zhun)n.Seerecursion.Thisdefini8ondoesnotterminate.Hereisonethatdoes:
12
§ re·cur·sion(ri-'kûr-zhun)n.Ifyous8lldon'tgetit,thenseerecursion.
Recursion…
13
§ Hofstadter'sLaw:Italwaystakeslongerthanyouexpect,evenwhenyoutakeintoaccountHofstadter'slaw.–DouglasHofstadter,Gödel,Escher,Bach:AnEternalGoldenBraid(Hofstadter'sLawdoesnotterminate)
Recursion…
14
Recursion… § tailrecursion (or tail-endrecursion) is a special case of
recursion in which the last operation of the function is a recursive call. Such recursions can be easily transformed to iterations. (from Wikipedia.org)
15
Recursion… § Factorial Function:
Recursive:
recursive function factorial(n) result(value)
if ( n .lt. 2 ) then
value = 1
else
value = n * factorial(n-1)
endif
end function factorial
16
Recursion… § Factorial Function:
Iterative:
function factorial(n) result(value)
value = 1
do i = 2, n
value = value * i
enddo
end function factorial
17
Recursion… § Factorial Function:
Iteration with Array Storage:
function factorial(n) result(value)
integer :: array(0:n)
array(0) = 1
do i = 1, n
array(i) = i * array(i-1)
enddo
value = array(n)
end function factorial
Save array(:) for future lookups.
18
Recursion and Genealogical Spin Eigenfunctions
§ InGUGA,eachCSFisagenealogicalspineigenfunc8on:
N ,S,M ;dn = Cdn ,σN −1,S ± 1
2 ,M − σ;dn−1 ⊗ 1, 12 ,σ;dnσ
± 12
∑
19
Recursion and Genealogical Spin Eigenfunctions
§ InGUGA,eachCSFisagenealogicalspineigenfunc8on:
Thetensorproductfunc8ons{|S1M1⟩}⊗ {|½,±½⟩}areeigenfunc8onsofS12,S1z,S22,S2zandhavedimension2(2S1+1)=4S1+2.Thesearecalledthe"uncoupled"basisfunc8ons.
Thecoupledfunc8ons|S'M'⟩and|S"M"⟩withS'=S1+½andS"=S1-½areeigenfunc8onsofS2,SzS12,S22andhavedimension(2S'+1)+(2S"+1)=4S1+2.
20
Recursion and Genealogical Spin Eigenfunctions
S = S1 + S2S2 = S1 + S2( ) ⋅ S1 + S2( )
= S12 + S2
2 + 2S1 ⋅ S2= S1
2 + S22 + 2S1zS2z( ) + 2S1xS2x + 2S1yS2y
= S12 + S2
2 + 2S1zS2z( ) + S1+S2− + S1−S2+
S2 S1S1; 1212 = S1 S1 +1( ) + 1
2 ⋅32 + 2S1 ⋅
12( ) S1S1; 12 12
= S1 + 12( ) S1 + 3
2( )( ) S1S1; 12 12= ′S ′S +1( ) S1S1; 12 12
ThehighestMuncoupledstatesa8sfies:
ThelowestMuncoupledstatesa8sfies:
S2 S1,−S1; 12 ,−12 = S1 S1 +1( ) + 1
2 ⋅32 − 2S1 ⋅ −
12( )( ) S1,−S1; 12 ,− 1
2
= ′S ′S +1( ) S1,−S1; 12 ,− 12
21
Recursion and Genealogical Spin Eigenfunctions
′S M , ′′S M( ) = S1M − 12 ;
12 ,
12 , S1M + 1
2 ;12 ,−
12( ) C1α C2α
C1β C2β
⎛
⎝⎜⎜
⎞
⎠⎟⎟
TheotherMstatessa8sfy:
Thecoefficientsmaybedeterminedthroughdiagonaliza8onoftheS2matrixintheuncoupledbasis.
S2C1α C2α
C1β C2β
⎛
⎝⎜⎜
⎞
⎠⎟⎟=
C1α C2α
C1β C2β
⎛
⎝⎜⎜
⎞
⎠⎟⎟
′S ′S +1( ) 00 ′′S ′′S +1( )
⎛
⎝⎜⎜
⎞
⎠⎟⎟
TheCmatrixofClebsch-Gordoncoefficientsmaybechosentobeorthonormal,CTC=1.
22
Recursion and Genealogical Spin Eigenfunctions
TheotherMstatessa8sfy:
TheeigenvectorsCare:
S2 S1M − 12 ;
12 ,
12 = S1(S1 +1) + 1
2 ⋅32 + 2(M − 1
2 )12( ) S1M − 1
2 ;12 ,
12
+ S1(S1 +1) − (M − 12 )(M + 1
2 ) S1M + 12 ;
12 ,−
12
S2 S1M + 12 ;
12 ,−
12 = S1(S1 +1) + 1
2 ⋅32 + 2(M + 1
2 )(−12 )( ) S1M + 1
2 ;12 ,−
12
+ S1(S1 +1) − (M + 12 )(M − 1
2 ) S1M − 12 ;
12 ,
12
S2 =(S1 + 1
2 )2 + M (S1 + 1
2 )2 − M 2
(S1 + 12 )2 − M 2 (S1 + 1
2 )2 − M
⎛
⎝
⎜⎜
⎞
⎠
⎟⎟
C1α C2α
C1β C2β
⎛
⎝⎜⎜
⎞
⎠⎟⎟=
S1 + 12 + M
2S1 +1−
S1 + 12 − M
2S1 +1
S1 + 12 − M
2S1 +1S1 + 1
2 + M2S1 +1
⎛
⎝
⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟
=
′S + M2 ′S
− ′′S − M +12 ′′S + 2
′S − M2 ′S
′′S + M +12 ′′S + 2
⎛
⎝
⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟
23
Recursion and Graphical Representations
§ N-electronspin-eigenfunc8onsareconstructedwithasequenceofaddi8onsteps(usingC1αandC1β)andsubtrac8onsteps(usingC2αandC2β).
§ ThisispossiblebecausetheClebsch-Gordoncoefficientsdependonlyonthe"local"S1andMvalues,notonthefullsequenceofaddi8onandsubtrac8onsteps.
§ Thissequenceofaddi8onandsubtrac8onstepsiscalledthegenealogyofthespineigenfunc8onandmaybestoredasavectordwithelementsdk={1,2}.
N ,S,M ;dn = Cdn ,σN −1,S ± 1
2 ,M − σ;dn−1 ⊗ 1, 12 ,σ;dnσ
± 12
∑
24
Recursion and Graphical Representations § BranchingDiagram:tocountthenumberofindependentspin
eigenfunc8onsforeachspa8alorbitaloccupa8on
f (N ,S) = f (N −1,S − 12 ) + f (N −1,S + 1
2 ) =2S +1N +1
N +112 N − S
⎛
⎝⎜
⎞
⎠⎟
2N = (2S +1) f (N ,S)S∑
25
Recursion and Graphical Representations § BranchingDiagram:coun8ngthenumberofindependentspin
eigenfunc8onsforeachspa8alorbitaloccupa8on
f (N ,S) = f (N −1,S − 12 ) + f (N −1,S + 1
2 ) =2S +1N +1
N +112 N − S
⎛
⎝⎜
⎞
⎠⎟
2N = (2S +1) f (N ,S)S∑
25 = 2 ⋅5 + 4 ⋅4 + 6 ⋅1 = 32
26
Recursion and Graphical Representations
DeterminantGraphsandBranching-DiagramGraphs
27
Recursion and Graphical Representations
DeterminantGraphsandBranching-DiagramGraphs
28
Recursion and Graphical Representations
DeterminantGraphsandBranching-DiagramGraphs
29
Recursion and Graphical Representations
Nonzerodeterminantcoefficientsoccuronlyinthe"allowedarea".(seePauncz,page33)
30
Recursion and Graphical Representations
TypicalTasks:§ ForagivenSlaterdeterminant|D⟩andagivenspineigenfunc8on|d⟩,
computetheoverlap⟨D|d⟩.§ Expandagiven|d⟩intermsofthefullsetofprimi8veSlater
determinants|D⟩foreitherasingleMorall(2S+1)values.§ ForagivenSlaterdeterminant|D⟩,computeallnonzero⟨D|d⟩.§ Foragivensetofsinglyoccupiedorbitals,computethefullsetof
transforma8oncoefficients⟨D|d⟩foreitherasingleMorall(2S+1)values.
This work was performed under the auspices of the Office of Basic Energy Sciences, Division of Chemical Sciences, Geosciences, and Biosciences, U.S. Department of Energy, under contract number DE-AC02-06CH11357.