In order to calculate the spin-state, the expectation value of S^{2} is
calculated.

where C_{ik} is the coefficient of microstate
Ψ_{i}
in State ,
N_{i}^{α}
is the number of alpha electrons in microstate Ψ_{i},
N_{i}^{β}
is the number of beta electrons in microstate Ψ_{i},
is the occupancy of alpha M.O. l in microstate Ψ_{k},
O_{lk}^{β}
is the occupancy of beta M.O. l in microstate Ψ_{k},
I^{+} is the spin shift up or step up operator, and I^{
} is the spin shift
down or step down operator.

The spin state is calculated from:

In practice, S is calculated to be exactly integer, or half
integer. That is, there is insignificant error due to approximations used.
This does not mean, however, that the method is accurate. The spin
calculation is completely precise, in the group theoretic sense, but the
accuracy of the calculation is limited by the Hamiltonian used, a
space-dependent function.