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.