It is necessary to diagonalize this 2N dimensional Hamiltonian in order to find the distribution of states at thermodynamic equilibrium. The 2N operators 
, 
, and 
form a complete set of operators commuting with H and each other.
The N total spin eigenvalues 
, along with the N-1 successively larger susbsystem s values 
may be taken. Taking the total 
value m rounds out the set of 2N eigenvalues needed. The energy eigenvalue of any eigenstate depends only on its s value and is E=s(s+1)-N(1/2)(1/2+1)=s(s+1)-3N/4 for spin 1/2 particles. Since spin correlations are of interest, the basis where the 
are easily found must be used to express the energy eigenstates. Thus step one is to generate a complete list of energy eigenvectors 
with the energy eigenvalues described above. The second step is to make use of Clebsch-Gordon coefficients to transform the energy eigenbasis just generated to the 
basis. The description of how this is done is given in appendix 13.