Average of high-spin states: n electrons in MO with
degenerate n_ir.

$${\frac{{n_{ir}\bigl(4k(k+l-1)+l(l-1)\bigr)}}{{(n_{ir}-1)n^2}}}$$ a

$${\frac{{2 n_{ir}\bigl(2k(k+l-1)+l(l-1)\bigr)}}{{(n_{ir}-1)n^2}}}$$ b

where: k = max(0, n - n_ir) , l = n - 2k = 2S    (spin)

This covers most of the cases given above. A CSF results only if n = {1,(n_ir - 1), n_ir, (n_ir + 1), (2n_ir -1)} since there is a single high-spin CSF in these cases.

The last equations for a and b can be rewritten in many ways, the probably most concise form is

$${\frac{{n(n-2)+2S}}{{(n-2f)n}}}$$ a

$${\frac{{n(n-2)+(2S)^2}}{{(n-2f)n}}}$$ b

This applies to shells with one electron, one hole, the high-spin couplings of half-filled shells and those with one electron more ore less. For d², d³, d⁷, and d⁸ it represents the (weighted) average of high-spin cases: ³F + ³P for d²,d⁸, ⁴F + ⁴P for d³, d⁷.