*N*-electron Spin Eigenfunctions

As we know, given the number of electrons *N* and the spin quantum number *S*, there are (2*S*+1)*N*!/(*N*/2+S+1)!/(*N*/2-S)! eigenfunctions of *S ^{2}* and each state has 2

These graphs are generated with computers by group theories.

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*N*= 1,*S*= 1/2*N*= 2,*S*= 0*N*= 2,*S*= 1*N*= 3,*S*= 1/2*N*= 3,*S*= 3/2*N*= 4,*S*= 0*N*= 4,*S*= 1*N*= 4,*S*= 2*N*= 5,*S*= 1/2*N*= 5,*S*= 3/2*N*= 5,*S*= 5/2*N*= 6,*S*= 0*N*= 6,*S*= 1*N*= 6,*S*= 2*N*= 6,*S*= 3*N*= 7,*S*= 1/2*N*= 7,*S*= 3/2*N*= 7,*S*= 5/2*N*= 7,*S*= 7/2

*N* = 1, *S* = 1/2

*S* = 1/2, *S _{z}* = 1/2:

*S* = 1/2, *S _{z}* = -1/2:

*N* = 2, *S* = 0

*S* = 0, *S _{z}* = 0:

*N* = 2, *S* = 1

*S* = 1, *S _{z}* = 1:

*S* = 1, *S _{z}* = 0:

*S* = 1, *S _{z}* = -1:

*N* = 3, *S* = 1/2

*S* = 1/2, *S _{z}* = 1/2:

*S* = 1/2, *S _{z}* = -1/2:

*S* = 1/2, *S _{z}* = 1/2:

*S* = 1/2, *S _{z}* = -1/2:

*N* = 3, *S* = 3/2

*S* = 3/2, *S _{z}* = 3/2:

*S* = 3/2, *S _{z}* = 1/2:

*S* = 3/2, *S _{z}* = -1/2:

*S* = 3/2, *S _{z}* = -3/2:

*N* = 4, *S* = 0

*S* = 0, *S _{z}* = 0:

*S* = 0, *S _{z}* = 0:

*N* = 4, *S* = 1

*S* = 1, *S _{z}* = 1:

*S* = 1, *S _{z}* = 0:

*S* = 1, *S _{z}* = -1:

*S* = 1, *S _{z}* = 1:

*S* = 1, *S _{z}* = 0:

*S* = 1, *S _{z}* = -1:

*S* = 1, *S _{z}* = 1:

*S* = 1, *S _{z}* = 0:

*S* = 1, *S _{z}* = -1:

*N* = 4, *S* = 2

*S* = 2, *S _{z}* = 2:

*S* = 2, *S _{z}* = 1:

*S* = 2, *S _{z}* = 0:

*S* = 2, *S _{z}* = -1:

*S* = 2, *S _{z}* = -2:

*N* = 5, *S* = 1/2

*S* = 1/2, *S _{z}* = 1/2:

*S* = 1/2, *S _{z}* = -1/2:

*S* = 1/2, *S _{z}* = 1/2:

*S* = 1/2, *S _{z}* = -1/2:

*S* = 1/2, *S _{z}* = 1/2:

*S* = 1/2, *S _{z}* = -1/2:

*S* = 1/2, *S _{z}* = 1/2:

*S* = 1/2, *S _{z}* = -1/2:

*S* = 1/2, *S _{z}* = 1/2:

*S* = 1/2, *S _{z}* = -1/2:

*N* = 5, *S* = 3/2

*S* = 3/2, *S _{z}* = 3/2:

*S* = 3/2, *S _{z}* = 1/2:

*S* = 3/2, *S _{z}* = -1/2:

*S* = 3/2, *S _{z}* = -3/2:

*S* = 3/2, *S _{z}* = 3/2:

*S* = 3/2, *S _{z}* = 1/2:

*S* = 3/2, *S _{z}* = -1/2:

*S* = 3/2, *S _{z}* = -3/2:

*S* = 3/2, *S _{z}* = 3/2:

*S* = 3/2, *S _{z}* = 1/2:

*S* = 3/2, *S _{z}* = -1/2:

*S* = 3/2, *S _{z}* = -3/2:

*S* = 3/2, *S _{z}* = 3/2:

*S* = 3/2, *S _{z}* = 1/2:

*S* = 3/2, *S _{z}* = -1/2:

*S* = 3/2, *S _{z}* = -3/2:

*N* = 5, *S* = 5/2

*S* = 5/2, *S _{z}* = 5/2:

*S* = 5/2, *S _{z}* = 3/2:

*S* = 5/2, *S _{z}* = 1/2:

*S* = 5/2, *S _{z}* = -1/2:

*S* = 5/2, *S _{z}* = -3/2:

*S* = 5/2, *S _{z}* = -5/2:

*N* = 6, *S* = 0

*S* = 0, *S _{z}* = 0:

*S* = 0, *S _{z}* = 0:

*S* = 0, *S _{z}* = 0:

*S* = 0, *S _{z}* = 0:

*S* = 0, *S _{z}* = 0:

*N* = 6, *S* = 1

*S* = 1, *S _{z}* = 1:

*S* = 1, *S _{z}* = 0:

*S* = 1, *S _{z}* = -1:

*S* = 1, *S _{z}* = 1:

*S* = 1, *S _{z}* = 0:

*S* = 1, *S _{z}* = -1:

*S* = 1, *S _{z}* = 1:

*S* = 1, *S _{z}* = 0:

*S* = 1, *S _{z}* = -1:

*S* = 1, *S _{z}* = 1:

*S* = 1, *S _{z}* = 0:

*S* = 1, *S _{z}* = -1:

*S* = 1, *S _{z}* = 1:

*S* = 1, *S _{z}* = 0:

*S* = 1, *S _{z}* = -1:

*S* = 1, *S _{z}* = 1:

*S* = 1, *S _{z}* = 0:

*S* = 1, *S _{z}* = -1:

*S* = 1, *S _{z}* = 1:

*S* = 1, *S _{z}* = 0:

*S* = 1, *S _{z}* = -1:

*S* = 1, *S _{z}* = 1:

*S* = 1, *S _{z}* = 0:

*S* = 1, *S _{z}* = -1:

*S* = 1, *S _{z}* = 1:

*S* = 1, *S _{z}* = 0:

*S* = 1, *S _{z}* = -1:

*N* = 6, *S* = 2

*S* = 2, *S _{z}* = 2:

*S* = 2, *S _{z}* = 1:

*S* = 2, *S _{z}* = 0:

*S* = 2, *S _{z}* = -1:

*S* = 2, *S _{z}* = -2:

*S* = 2, *S _{z}* = 2:

*S* = 2, *S _{z}* = 1:

*S* = 2, *S _{z}* = 0:

*S* = 2, *S _{z}* = -1:

*S* = 2, *S _{z}* = -2:

*S* = 2, *S _{z}* = 2:

*S* = 2, *S _{z}* = 1:

*S* = 2, *S _{z}* = 0:

*S* = 2, *S _{z}* = -1:

*S* = 2, *S _{z}* = -2:

*S* = 2, *S _{z}* = 2:

*S* = 2, *S _{z}* = 1:

*S* = 2, *S _{z}* = 0:

*S* = 2, *S _{z}* = -1:

*S* = 2, *S _{z}* = -2:

*S* = 2, *S _{z}* = 2:

*S* = 2, *S _{z}* = 1:

*S* = 2, *S _{z}* = 0:

*S* = 2, *S _{z}* = -1:

*S* = 2, *S _{z}* = -2:

*N* = 6, *S* = 3

*S* = 3, *S _{z}* = 3:

*S* = 3, *S _{z}* = 2:

*S* = 3, *S _{z}* = 1:

*S* = 3, *S _{z}* = 0:

*S* = 3, *S _{z}* = -1:

*S* = 3, *S _{z}* = -2:

*S* = 3, *S _{z}* = -3:

*N* = 7, *S* = 1/2

*S* = 1/2, *S _{z}* = 1/2:

*S* = 1/2, *S _{z}* = -1/2:

*S* = 1/2, *S _{z}* = 1/2:

*S* = 1/2, *S _{z}* = -1/2:

*S* = 1/2, *S _{z}* = 1/2:

*S* = 1/2, *S _{z}* = -1/2:

*S* = 1/2, *S _{z}* = 1/2:

*S* = 1/2, *S _{z}* = -1/2:

*S* = 1/2, *S _{z}* = 1/2:

*S* = 1/2, *S _{z}* = -1/2:

*S* = 1/2, *S _{z}* = 1/2:

*S* = 1/2, *S _{z}* = -1/2:

*S* = 1/2, *S _{z}* = 1/2:

*S* = 1/2, *S _{z}* = -1/2:

*S* = 1/2, *S _{z}* = 1/2:

*S* = 1/2, *S _{z}* = -1/2:

*S* = 1/2, *S _{z}* = 1/2:

*S* = 1/2, *S _{z}* = -1/2:

*S* = 1/2, *S _{z}* = 1/2:

*S* = 1/2, *S _{z}* = -1/2:

*S* = 1/2, *S _{z}* = 1/2:

*S* = 1/2, *S _{z}* = -1/2:

*S* = 1/2, *S _{z}* = 1/2:

*S* = 1/2, *S _{z}* = -1/2:

*S* = 1/2, *S _{z}* = 1/2:

*S* = 1/2, *S _{z}* = -1/2:

*S* = 1/2, *S _{z}* = 1/2:

*S* = 1/2, *S _{z}* = -1/2:

*N* = 7, *S* = 3/2

*S* = 3/2, *S _{z}* = 3/2:

*S* = 3/2, *S _{z}* = 1/2:

*S* = 3/2, *S _{z}* = -1/2:

*S* = 3/2, *S _{z}* = -3/2:

*S* = 3/2, *S _{z}* = 3/2:

*S* = 3/2, *S _{z}* = 1/2:

*S* = 3/2, *S _{z}* = -1/2:

*S* = 3/2, *S _{z}* = -3/2:

*S* = 3/2, *S _{z}* = 3/2:

*S* = 3/2, *S _{z}* = 1/2:

*S* = 3/2, *S _{z}* = -1/2:

*S* = 3/2, *S _{z}* = -3/2:

*S* = 3/2, *S _{z}* = 3/2:

*S* = 3/2, *S _{z}* = 1/2:

*S* = 3/2, *S _{z}* = -1/2:

*S* = 3/2, *S _{z}* = -3/2:

*S* = 3/2, *S _{z}* = 3/2:

*S* = 3/2, *S _{z}* = 1/2:

*S* = 3/2, *S _{z}* = -1/2:

*S* = 3/2, *S _{z}* = -3/2:

*S* = 3/2, *S _{z}* = 3/2:

*S* = 3/2, *S _{z}* = 1/2:

*S* = 3/2, *S _{z}* = -1/2:

*S* = 3/2, *S _{z}* = -3/2:

*S* = 3/2, *S _{z}* = 3/2:

*S* = 3/2, *S _{z}* = 1/2:

*S* = 3/2, *S _{z}* = -1/2:

*S* = 3/2, *S _{z}* = -3/2:

*S* = 3/2, *S _{z}* = 3/2:

*S* = 3/2, *S _{z}* = 1/2:

*S* = 3/2, *S _{z}* = -1/2:

*S* = 3/2, *S _{z}* = -3/2:

*S* = 3/2, *S _{z}* = 3/2:

*S* = 3/2, *S _{z}* = 1/2:

*S* = 3/2, *S _{z}* = -1/2:

*S* = 3/2, *S _{z}* = -3/2:

*S* = 3/2, *S _{z}* = 3/2:

*S* = 3/2, *S _{z}* = 1/2:

*S* = 3/2, *S _{z}* = -1/2:

*S* = 3/2, *S _{z}* = -3/2:

*S* = 3/2, *S _{z}* = 3/2:

*S* = 3/2, *S _{z}* = 1/2:

*S* = 3/2, *S _{z}* = -1/2:

*S* = 3/2, *S _{z}* = -3/2:

*S* = 3/2, *S _{z}* = 3/2:

*S* = 3/2, *S _{z}* = 1/2:

*S* = 3/2, *S _{z}* = -1/2:

*S* = 3/2, *S _{z}* = -3/2:

*S* = 3/2, *S _{z}* = 3/2:

*S* = 3/2, *S _{z}* = 1/2:

*S* = 3/2, *S _{z}* = -1/2:

*S* = 3/2, *S _{z}* = -3/2:

*S* = 3/2, *S _{z}* = 3/2:

*S* = 3/2, *S _{z}* = 1/2:

*S* = 3/2, *S _{z}* = -1/2:

*S* = 3/2, *S _{z}* = -3/2:

*N* = 7, *S* = 5/2

*S* = 5/2, *S _{z}* = 5/2:

*S* = 5/2, *S _{z}* = 3/2:

*S* = 5/2, *S _{z}* = 1/2:

*S* = 5/2, *S _{z}* = -1/2:

*S* = 5/2, *S _{z}* = -3/2:

*S* = 5/2, *S _{z}* = -5/2:

*S* = 5/2, *S _{z}* = 5/2:

*S* = 5/2, *S _{z}* = 3/2:

*S* = 5/2, *S _{z}* = 1/2:

*S* = 5/2, *S _{z}* = -1/2:

*S* = 5/2, *S _{z}* = -3/2:

*S* = 5/2, *S _{z}* = -5/2:

*S* = 5/2, *S _{z}* = 5/2:

*S* = 5/2, *S _{z}* = 3/2:

*S* = 5/2, *S _{z}* = 1/2:

*S* = 5/2, *S _{z}* = -1/2:

*S* = 5/2, *S _{z}* = -3/2:

*S* = 5/2, *S _{z}* = -5/2:

*S* = 5/2, *S _{z}* = 5/2:

*S* = 5/2, *S _{z}* = 3/2:

*S* = 5/2, *S _{z}* = 1/2:

*S* = 5/2, *S _{z}* = -1/2:

*S* = 5/2, *S _{z}* = -3/2:

*S* = 5/2, *S _{z}* = -5/2:

*S* = 5/2, *S _{z}* = 5/2:

*S* = 5/2, *S _{z}* = 3/2:

*S* = 5/2, *S _{z}* = 1/2:

*S* = 5/2, *S _{z}* = -1/2:

*S* = 5/2, *S _{z}* = -3/2:

*S* = 5/2, *S _{z}* = -5/2:

*S* = 5/2, *S _{z}* = 5/2:

*S* = 5/2, *S _{z}* = 3/2:

*S* = 5/2, *S _{z}* = 1/2:

*S* = 5/2, *S _{z}* = -1/2:

*S* = 5/2, *S _{z}* = -3/2:

*S* = 5/2, *S _{z}* = -5/2:

*N* = 7, *S* = 7/2

*S* = 7/2, *S _{z}* = 7/2:

*S* = 7/2, *S _{z}* = 5/2:

*S* = 7/2, *S _{z}* = 3/2:

*S* = 7/2, *S _{z}* = 1/2:

*S* = 7/2, *S _{z}* = -1/2:

*S* = 7/2, *S _{z}* = -3/2:

*S* = 7/2, *S _{z}* = -5/2:

*S* = 7/2, *S _{z}* = -7/2: