Calculation of spin-states

In order to calculate the spin-state, the expectation value of S² is calculated.

$\begin{eqnarray*}<\Phi_k\vert S^2\vert\Phi_k> & = & S(S+1) = S_z^2 + 2 I^+I^- \\... ...ft[C_{ik}C_{jk} [\delta(\Psi_i,(I^+I^-)\Psi_j) ]\right] \right\} \end{eqnarray*}$

where C_ik is the coefficient of microstate Ψ_i in State $\Phi_k$ , 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, $O^{\alpha}_{lk}$ 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:

$\begin{displaymath}S = (1/2) [\sqrt{(1+4 S^2)} - 1 ] \end{displaymath}$

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.