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Spin operators (Read 8141 times)
Gerrit-Jan Linker
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Spin operators
09.02.11 at 21:15:33
 
Spin operators
 
When the N-electron Hamiltonian Ĥ does not contain spin-orbit coupling terms, Ĥ commutes with Ŝ2 and Ŝz. So an eigenfunction ψ of Ĥ must also be an eigenfunction of Ŝ2 and Ŝz.
Ĥψ=Eψ
Ŝ2ψ= S(S+1)ψ, with S=0,1,2,3,... for even number of electrons and S=1/2, 3/2, 5/2,... for an odd number.
Ŝzψ= Msψ, with Ms=-S,-S+1,...,S-1,S. There is a (2S+1) fold degeneracy.  
 
The eigenfunctions ψ of Ĥ can then be labeled with S and Ms: ψ(S,Ms)
It is common to write 2S+1ψ to denote the whole multiplet {ψ(S,-S), ψ(S,-S-1), ..., ψ(S,S-1), ψ(S,S)}.
 
 
The spin operator Ŝ can be written in vector form: Ŝ = ( Ŝx , Ŝy , Ŝz )
 
The total spin operator Ŝ2 is the inner product:
Ŝ2 = Ŝxx + Ŝyy + Ŝzz
 
Ŝx = Σiŝx(i)
Ŝy = Σiŝy(i)
Ŝz = Σiŝz(i)
in which ŝx(i), ŝy(i), ŝz(i) are one electron spin functions
 
We can chose one electron spin functions α(i) (i) that are eigenfunctions of ŝz(i): (we omit the index i for brevity)
ŝzα = α
ŝz = -
 
When we chose the one electron spin functions in this way, they are eigenfunctions of ŝz but not with ŝx and ŝy because ŝx, ŝy and ŝz do not commute with eachother.
 
Spin orperators working on the spin functions α and :
 
ŝxα=
ŝx
 
ŝyα=i
ŝy=-iα
 
ŝzα = α
ŝz = -
 
The spin functions α and are eigenfunctions of ŝz and also of Ŝ2:
 
Ŝ2α = α
Ŝ2 =
 
Proof:
Ŝ2α = ŝxxα + ŝyyα + ŝzzα
  = ŝx. + ŝy.i + ŝz
  = α - iiα + α =
  = α + α + α = α
 
Ŝ2 = ŝxx + ŝyy + ŝzz
  = ŝx.α - ŝy.iα - ŝz.
  = - ii + =
  = + + =
 
Source:
TC2 klapper
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« Last Edit: 18.08.12 at 15:26:54 by Gerrit-Jan Linker »  

Gerrit-Jan Linker
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Spin operators working on two particles
Reply #1 - 18.08.12 at 15:28:22
 
Spin operators working on two particles
 
Ŝ2(1,2)=Ŝx(1,2).Ŝx(1,2) + Ŝy(1,2).Ŝy(1,2) + Ŝz(1,2).Ŝz(1,2)
 
Ŝz(1,2) =ŝz(1)+ŝz(2)
 
We use the above expression for Ŝz(1,2) and the equivalent expressions for Ŝx(1,2) and Ŝy(1,2):
 
Ŝ2(1,2) = [ŝx(1)+ŝx(2)]2 + [ŝy(1)+ŝy(2)]2+[ŝz(1)+ŝz(2)]2
  = ŝx(1)2 + ŝx(2)2 + 2ŝx(1)ŝx(2)
  + ŝy(1)2 + ŝy(2)2 + 2ŝy(1)ŝy(2)
  + ŝz(1)2 + ŝz(2)2 + 2ŝz(1)ŝz(2)
  = ŝx(1)2 + ŝy(1)2 + ŝz(1)2x(2)2 + ŝy(2)2 + ŝz(2)2  
  + 2[ŝx(1)ŝx(2) + ŝy(1)ŝy(2) + ŝz(1)ŝz(2)]
,so Ŝ2(1,2) = Ŝ2(1) + Ŝ2(2) + 2[ŝx(1)ŝx(2) + ŝy(1)ŝy(2) + ŝz(1)ŝz(2)]
 
We let these formulas for Ŝz(1,2) and Ŝ2(1,2) work on two electron spin functions:
α(1)α(2), (1)(2), α(1)(2), (1)α(2), α(1)(2)-(1)α(2), α(1)(2)+(1)α(2)
 
Ŝz(1,2)α(1)α(2) = [ŝz(1)+ŝz(2)]α(1)α(2)
  = [ŝz(1)α(1)]α(2)+α(1)ŝz(2)α(2)
  = α(1)]α(2)+α(1)α(2)
  = α(1)α(2)
 
Ŝz(1,2)(1)(2) = [ŝz(1)+ŝz(2)](1)(2)
  = [ŝz(1)(1)](2)+(1)ŝz(2)(2)
  = -(1)(2)+(1)(-)(2)
  = - (1)(2)
 
Ŝz(1,2)α(1)(2) = [ŝz(1)+ŝz(2)]α(1)(2)
  = [ŝz(1)α(1)](2)+α(1)ŝz(2)(2)
  = α(1)(2)+α(1)(-)(2)
  = 0
 
Ŝz(1,2)(1)α(2)=Ŝz(1,2)α(1)(2)=0
 
Ŝ2(1,2)α(1)α(2) = {Ŝ2(1) + Ŝ2(2) + 2[ŝx(1)ŝx(2) + ŝy(1)ŝy(2) + ŝz(1)ŝz(2)]}α(1)α(2)
  = (Ŝ2(1)α(1))α(2) + α(1)(Ŝ2(2)α(2)) + 2[ŝx(1)α(1)ŝx(2)α(2) + ŝy(1)α(1)ŝy(2)α(2) + ŝz(1)α(1)ŝz(2)α(2)]
  = α(1))α(2) + α(1)α(2) + 2[β(1)β(2) + iβ(1)iβ(2) + α(1)α(2)]
  = 3α(1))α(2) + β(1)β(2) - β(1)β(2) + α(1)α(2)
  = 2α(1))α(2)
 
Ŝ2(1,2)(1)(2) = {Ŝ2(1) + Ŝ2(2) + 2[ŝx(1)ŝx(2) + ŝy(1)ŝy(2) + ŝz(1)ŝz(2)]}(1)(2)
  = (Ŝ2(1)(1))(2) + (1)(Ŝ2(2)(2)) + 2[ŝx(1)(1)ŝx(2)(2) + ŝy(1)(1)ŝy(2)(2) + ŝz(1)(1)ŝz(2)(2)]
  = (1))(2) + (1)(2) + 2[α(1)α(2) + (-i)α(1)(-i)α(2) + (-)(1)(-)(2)]
  = 3(1))(2) + α(1)α(2) - α(1)α(2) + (1)(2)
  = 2(1))(2)
 
Ŝ2(1,2)α(1)(2) = {Ŝ2(1) + Ŝ2(2) + 2[ŝx(1)ŝx(2) + ŝy(1)ŝy(2) + ŝz(1)ŝz(2)]}α(1)(2)
  = (Ŝ2(1)α(1))(2) + α(1)(Ŝ2(2)(2)) + 2[ŝx(1)α(1)ŝx(2)(2) + ŝy(1)α(1)ŝy(2)(2) + ŝz(1)α(1)ŝz(2)(2)]
  = α(1))(2) + α(1)(2) + 2[(1)α(2) + i(1)(-i)α(2) + α(1)(-)(2)]
  = 3α(1))(2) + (1)α(2) + (1)α(2) - α(1)(2)
  = α(1))(2) + (1)α(2)
 
Ŝ2(1,2)(1)α(2) = {Ŝ2(1) + Ŝ2(2) + 2[ŝx(1)ŝx(2) + ŝy(1)ŝy(2) + ŝz(1)ŝz(2)]}(1)α(2)
  = (Ŝ2(1)(1))α(2) + (1)(Ŝ2(2)α(2)) + 2[ŝx(1)(1)ŝx(2)α(2) + ŝy(1)(1)ŝy(2)α(2) + ŝz(1)(1)ŝz(2)α(2)]
  = (1))α(2) + (1)α(2) + 2[α(1)(2) + (-i)α(1)(i)(2) + (-)(1)α(2)]
  = 3(1))α(2) + α(1)(2) + α(1)(2) - (1)α(2)
  = α(1))(2) + (1)α(2)
 
Ŝ2(1,2)(α(1)(2)+(1)α(2)) = Ŝ2(1,2)α(1)(2)+Ŝ2(1,2)(1)α(2)
  = α(1))(2) + (1)α(2) + α(1))(2) + (1)α(2)
  = 2 (α(1))(2) + (1)α(2))
 
Ŝ2(1,2)(α(1)(2)-(1)α(2)) = Ŝ2(1,2)α(1)(2)-Ŝ2(1,2)(1)α(2)
  = α(1))(2) + (1)α(2) - (α(1))(2) + (1)α(2))
  = α(1))(2) + (1)α(2) - α(1))(2) - (1)α(2)
  = 0
 
To summarize, the above two electron spin functions, α(1)α(2) and (1)(2), are eigenfunctions of both spin operators. The spin functions α(1)(2) and (1)α(2) are not eigenfunctions on their own but the + and - combinations are.
 
Ŝzαα = 1αα, Ms=1
Ŝz[α-α] = 0, Ms=0
Ŝz[α+α] = 1, Ms=0
Ŝz = -1, Ms=-1
 
Ŝ2αα=2αα,             S(S+1)=2 ,so S=1
Ŝ2=2,             S(S+1)=2 ,so S=1
Ŝ2[α+α]=2[α+α], S(S+1)=2 ,so S=1
Ŝ2[α-α]=0,             S(S+1)=0 ,so S=0
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« Last Edit: 14.02.14 at 14:51:41 by Gerrit-Jan Linker »  

Gerrit-Jan Linker
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Spin operators working on a Slater determinant
Reply #2 - 25.08.12 at 17:01:16
 
Spin operators working on a Slater determinant
 
S2 |aa| = S2 (aa-aa)√ = a(1)a(2) [ S2 (α - S2 α ]√
= a(1)a(2) [ (α + α - α - α ]√ = 0
The 1 determinant, closed shell, singlet wavefunction |aa| is an eigenfunction of S2 with eivenvalue S(S+1)=0. So S=0.
 
S2 |ab| = S2 (ab-ba)√ = a(1)b(2) S2 α - b(1)a(2) S2 α √
= a(1)b(2) 2[ α + α ] - b(1)a(2) 2[α + α ] √
= 2ab + 2ab - 2ba - 2ba
= 2(ab - ba) + 2(ab -ba) √
= 2|ab| + 2|ab| √
= 2( |ab| - |ba| ) √
The 1 determinant, open shell, singlet wavefunction |ab| is not an eigenfunction of S2.
 
This means that in a1 determinant formalism an open shell singlet is not a good solution of the Schrodinger equation. The open shell singlet Slater determinant is an eigenfunction of Sz only and not of S2 or Ĥ.
 
 
The singlet determinants |(a+b)(a+b)| and |(a-b)(a-b)|, are eigenfunctions of S2 and Sz and therefore also of Ĥ. When a and b are wavefunctions on separate centers (atoms or molecules, e.g.), a 1 determinant approach forces a delocalised solution. With such a solution care should be taken whether there is an open shell siglet that is lower in energy. >
 
Note:
Short notation is used to avoid writing alpha and beta spin everywhere and using indices.
|ab| is short for:
|ab| = ( ab - ba )√ = ( a(1)α(1)b(2)(2) - a(1)(1)b(2)α(1) )√ = a(1)b(2) [ α(1)(2) - (1)α(2) ]√ = a(1)b(2) [α-α ]√
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« Last Edit: 08.11.13 at 08:22:18 by Gerrit-Jan Linker »  

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S, <S^2> and multiplicity
Reply #3 - 25.10.13 at 12:33:08
 
S, <S2> and multiplicity
 
Total spin<S2>multiplicity
SS(S+1)2S+1
001
0.50.752
123
1.53.754
265
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« Last Edit: 25.10.13 at 12:37:04 by Gerrit-Jan Linker »  

Gerrit-Jan Linker
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Total spin expectation value
Reply #4 - 08.11.13 at 07:52:17
 
Total spin expectation value
 
We calculate the expectation value of the total spin operator S2: <S2> and use a single determinantal wavefunction |ab|. We have already seen that = S2|ab|=2( |ab| - |ba|) √. The determinant |ab| is not an eigen function of S2. However |aa| is. So when b=a there is no a problem. When calculating the expectation value we see that the result can be written in terms of O=<a|b>, the overlap between a and b.
 
< |ab| |S2| |ab| > =
< ab - ba |S2| ab - ba > =
< ab |S2| ab > - < ab |S2|ba > - < ba |S2| ab > + < ba |S2|ba > =
<ab|ab><α|S2|α> - <ab|ba><α|S2|α> - <ba|ab> <α|S2|α> + <ba|ba><α|S2|α> =
<α|S2|α> - <a|b><b|a><α|S2|α> - <b|a><a|b> <α|S2|α> + <α|S2|α> =
<α|S2|α> - <a|b><b|a><α|S2|α> =            (use O=<a|b>=<b|a>)
<α|S2|α> - O2<α|S2|α>  =            (use S2α= α + α)  
<α|α + α> - O2<α|α + α> =        
<α|α><|> + <α|><|α> - O2(<α|α><| + <α|><|α>) =
1 - O2
 
If the overlap O=<a|b>=0 we get <S2>=1 and  
whe O=<a|b>=1 (when a=b) we get <S2>=0
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« Last Edit: 08.11.13 at 09:09:19 by Gerrit-Jan Linker »  

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