Average of high-spin states: $n$ mathend000# electrons in MO with
degenerate $n_{ir}$ mathend000#.

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

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

where: $k = \max(0,n-n_{ir})$ mathend000# , $l = n-2k = 2S$   (spin) mathend000#

This covers most of the cases given above. A CSF results only if $n = \{1, (n_{ir}-1),$ mathend000# $n_{ir},$ mathend000# $(n_{ir}+1),$ mathend000# $(2n_{ir}-1)\}$ mathend000# since there is a single high-spin CSF in these cases.

The last equations for $a$ mathend000# and $b$ mathend000# can be rewritten in many ways, the probably most concise form is

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

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

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 $d^2$ mathend000#, $d^3$ mathend000#, $d^7$ mathend000#, and $d^8$ mathend000# it represents the (weighted) average of high-spin cases: $^3$ mathend000#F + $^3$ mathend000#P for $d^2$ mathend000#,$d^8$ mathend000#, $^4$ mathend000#F + $^4$ mathend000#P for $d^3$ mathend000#, $d^7$ mathend000#.