N-electron Spin Eigenfunctions
As we know, given the number of electrons N and the spin quantum number S, there are (2S+1)N!/(N/2+S+1)!/(N/2-S)! eigenfunctions of S2 and each state has 2S+1 components. I list the spin eigenfunctions of N-electron system (N = 1 ~ 7) below. For each S, all functions with Sz = -S to +S are listed. For example, 
means that the first electron couples with the second one to obtain a singlet state (S = 0), then they couple with the third one to obtain the final doublet state (S = 1/2); 
means that the first, second and third electron is alpha, beta and alpha spin, respectively.
These graphs are generated with computers by group theories.
N = 1, S = 1/2
S = 1/2, Sz = 1/2: 
S = 1/2, Sz = -1/2: 
N = 2, S = 0
S = 0, Sz = 0: 
N = 2, S = 1
S = 1, Sz = 1: 
S = 1, Sz = 0: 
S = 1, Sz = -1: 
N = 3, S = 1/2
S = 1/2, Sz = 1/2: 
S = 1/2, Sz = -1/2: 
S = 1/2, Sz = 1/2:
S = 1/2, Sz = -1/2: 
N = 3, S = 3/2
S = 3/2, Sz = 3/2: 
S = 3/2, Sz = 1/2: 
S = 3/2, Sz = -1/2: 
S = 3/2, Sz = -3/2: 
N = 4, S = 0
S = 0, Sz = 0: 
S = 0, Sz = 0: 
N = 4, S = 1
S = 1, Sz = 1: 
S = 1, Sz = 0: 
S = 1, Sz = -1: 
S = 1, Sz = 1: 
S = 1, Sz = 0: 
S = 1, Sz = -1: 
S = 1, Sz = 1: 
S = 1, Sz = 0: 
S = 1, Sz = -1: 
N = 4, S = 2
S = 2, Sz = 2: 
S = 2, Sz = 1: 
S = 2, Sz = 0: 
S = 2, Sz = -1: 
S = 2, Sz = -2: 
N = 5, S = 1/2
S = 1/2, Sz = 1/2: 
S = 1/2, Sz = -1/2: 
S = 1/2, Sz = 1/2: 
S = 1/2, Sz = -1/2: 
S = 1/2, Sz = 1/2: 
S = 1/2, Sz = -1/2: 
S = 1/2, Sz = 1/2: 
S = 1/2, Sz = -1/2: 
S = 1/2, Sz = 1/2: 
S = 1/2, Sz = -1/2: 
N = 5, S = 3/2
S = 3/2, Sz = 3/2: 
S = 3/2, Sz = 1/2: 
S = 3/2, Sz = -1/2: 
S = 3/2, Sz = -3/2: 
S = 3/2, Sz = 3/2: 
S = 3/2, Sz = 1/2: 
S = 3/2, Sz = -1/2: 
S = 3/2, Sz = -3/2: 
S = 3/2, Sz = 3/2: 
S = 3/2, Sz = 1/2: 
S = 3/2, Sz = -1/2: 
S = 3/2, Sz = -3/2: 
S = 3/2, Sz = 3/2: 
S = 3/2, Sz = 1/2: 
S = 3/2, Sz = -1/2: 
S = 3/2, Sz = -3/2: 
N = 5, S = 5/2
S = 5/2, Sz = 5/2: 
S = 5/2, Sz = 3/2: 
S = 5/2, Sz = 1/2: 
S = 5/2, Sz = -1/2: 
S = 5/2, Sz = -3/2: 
S = 5/2, Sz = -5/2: 
N = 6, S = 0
S = 0, Sz = 0: 
S = 0, Sz = 0: 
S = 0, Sz = 0: 
S = 0, Sz = 0: 
S = 0, Sz = 0: 
N = 6, S = 1
S = 1, Sz = 1: 
S = 1, Sz = 0: 
S = 1, Sz = -1: 
S = 1, Sz = 1: 
S = 1, Sz = 0: 
S = 1, Sz = -1: 
S = 1, Sz = 1: 
S = 1, Sz = 0: 
S = 1, Sz = -1: 
S = 1, Sz = 1: 
S = 1, Sz = 0: 
S = 1, Sz = -1: 
S = 1, Sz = 1: 
S = 1, Sz = 0: 
S = 1, Sz = -1: 
S = 1, Sz = 1: 
S = 1, Sz = 0: 
S = 1, Sz = -1: 
S = 1, Sz = 1: 
S = 1, Sz = 0: 
S = 1, Sz = -1: 
S = 1, Sz = 1: 
S = 1, Sz = 0: 
S = 1, Sz = -1: 
S = 1, Sz = 1: 
S = 1, Sz = 0: 
S = 1, Sz = -1: 
N = 6, S = 2
S = 2, Sz = 2: 
S = 2, Sz = 1: 
S = 2, Sz = 0: 
S = 2, Sz = -1: 
S = 2, Sz = -2: 
S = 2, Sz = 2: 
S = 2, Sz = 1: 
S = 2, Sz = 0: 
S = 2, Sz = -1: 
S = 2, Sz = -2: 
S = 2, Sz = 2: 
S = 2, Sz = 1: 
S = 2, Sz = 0: 
S = 2, Sz = -1: 
S = 2, Sz = -2: 
S = 2, Sz = 2: 
S = 2, Sz = 1: 
S = 2, Sz = 0: 
S = 2, Sz = -1: 
S = 2, Sz = -2: 
S = 2, Sz = 2: 
S = 2, Sz = 1: 
S = 2, Sz = 0: 
S = 2, Sz = -1: 
S = 2, Sz = -2: 
N = 6, S = 3
S = 3, Sz = 3: 
S = 3, Sz = 2: 
S = 3, Sz = 1: 
S = 3, Sz = 0: 
S = 3, Sz = -1: 
S = 3, Sz = -2: 
S = 3, Sz = -3: 
N = 7, S = 1/2
S = 1/2, Sz = 1/2: 
S = 1/2, Sz = -1/2: 
S = 1/2, Sz = 1/2: 
S = 1/2, Sz = -1/2: 
S = 1/2, Sz = 1/2: 
S = 1/2, Sz = -1/2: 
S = 1/2, Sz = 1/2: 
S = 1/2, Sz = -1/2: 
S = 1/2, Sz = 1/2: 
S = 1/2, Sz = -1/2: 
S = 1/2, Sz = 1/2: 
S = 1/2, Sz = -1/2: 
S = 1/2, Sz = 1/2: 
S = 1/2, Sz = -1/2: 
S = 1/2, Sz = 1/2: 
S = 1/2, Sz = -1/2: 
S = 1/2, Sz = 1/2: 
S = 1/2, Sz = -1/2: 
S = 1/2, Sz = 1/2: 
S = 1/2, Sz = -1/2: 
S = 1/2, Sz = 1/2: 
S = 1/2, Sz = -1/2: 
S = 1/2, Sz = 1/2: 
S = 1/2, Sz = -1/2: 
S = 1/2, Sz = 1/2: 
S = 1/2, Sz = -1/2: 
S = 1/2, Sz = 1/2: 
S = 1/2, Sz = -1/2: 
N = 7, S = 3/2
S = 3/2, Sz = 3/2: 
S = 3/2, Sz = 1/2: 
S = 3/2, Sz = -1/2: 
S = 3/2, Sz = -3/2: 
S = 3/2, Sz = 3/2: 
S = 3/2, Sz = 1/2: 
S = 3/2, Sz = -1/2: 
S = 3/2, Sz = -3/2: 
S = 3/2, Sz = 3/2: 
S = 3/2, Sz = 1/2: 
S = 3/2, Sz = -1/2: 
S = 3/2, Sz = -3/2: 
S = 3/2, Sz = 3/2: 
S = 3/2, Sz = 1/2: 
S = 3/2, Sz = -1/2: 
S = 3/2, Sz = -3/2: 
S = 3/2, Sz = 3/2: 
S = 3/2, Sz = 1/2: 
S = 3/2, Sz = -1/2: 
S = 3/2, Sz = -3/2: 
S = 3/2, Sz = 3/2: 
S = 3/2, Sz = 1/2: 
S = 3/2, Sz = -1/2: 
S = 3/2, Sz = -3/2: 
S = 3/2, Sz = 3/2: 
S = 3/2, Sz = 1/2: 
S = 3/2, Sz = -1/2: 
S = 3/2, Sz = -3/2: 
S = 3/2, Sz = 3/2: 
S = 3/2, Sz = 1/2: 
S = 3/2, Sz = -1/2: 
S = 3/2, Sz = -3/2: 
S = 3/2, Sz = 3/2: 
S = 3/2, Sz = 1/2: 
S = 3/2, Sz = -1/2: 
S = 3/2, Sz = -3/2: 
S = 3/2, Sz = 3/2: 
S = 3/2, Sz = 1/2: 
S = 3/2, Sz = -1/2: 
S = 3/2, Sz = -3/2: 
S = 3/2, Sz = 3/2: 
S = 3/2, Sz = 1/2: 
S = 3/2, Sz = -1/2: 
S = 3/2, Sz = -3/2: 
S = 3/2, Sz = 3/2: 
S = 3/2, Sz = 1/2: 
S = 3/2, Sz = -1/2: 
S = 3/2, Sz = -3/2: 
S = 3/2, Sz = 3/2: 
S = 3/2, Sz = 1/2: 
S = 3/2, Sz = -1/2: 
S = 3/2, Sz = -3/2: 
S = 3/2, Sz = 3/2: 
S = 3/2, Sz = 1/2: 
S = 3/2, Sz = -1/2: 
S = 3/2, Sz = -3/2: 
N = 7, S = 5/2
S = 5/2, Sz = 5/2: 
S = 5/2, Sz = 3/2: 
S = 5/2, Sz = 1/2: 
S = 5/2, Sz = -1/2: 
S = 5/2, Sz = -3/2: 
S = 5/2, Sz = -5/2: 
S = 5/2, Sz = 5/2: 
S = 5/2, Sz = 3/2: 
S = 5/2, Sz = 1/2: 
S = 5/2, Sz = -1/2: 
S = 5/2, Sz = -3/2: 
S = 5/2, Sz = -5/2: 
S = 5/2, Sz = 5/2: 
S = 5/2, Sz = 3/2: 
S = 5/2, Sz = 1/2: 
S = 5/2, Sz = -1/2: 
S = 5/2, Sz = -3/2: 
S = 5/2, Sz = -5/2: 
S = 5/2, Sz = 5/2: 
S = 5/2, Sz = 3/2: 
S = 5/2, Sz = 1/2: 
S = 5/2, Sz = -1/2: 
S = 5/2, Sz = -3/2: 
S = 5/2, Sz = -5/2: 
S = 5/2, Sz = 5/2: 
S = 5/2, Sz = 3/2: 
S = 5/2, Sz = 1/2: 
S = 5/2, Sz = -1/2: 
S = 5/2, Sz = -3/2: 
S = 5/2, Sz = -5/2: 
S = 5/2, Sz = 5/2: 
S = 5/2, Sz = 3/2: 
S = 5/2, Sz = 1/2: 
S = 5/2, Sz = -1/2: 
S = 5/2, Sz = -3/2: 
S = 5/2, Sz = -5/2: 
N = 7, S = 7/2
S = 7/2, Sz = 7/2: 
S = 7/2, Sz = 5/2: 
S = 7/2, Sz = 3/2: 
S = 7/2, Sz = 1/2: 
S = 7/2, Sz = -1/2: 
S = 7/2, Sz = -3/2: 
S = 7/2, Sz = -5/2: 
S = 7/2, Sz = -7/2: 