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

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

where: $ k = \max(0,n-n_{ir})$ , $ 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

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

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


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Next: Two-component Hartree-Fock and DFT Up: Miscellaneous Previous: Totally symmetric singlets for   Contents   Index
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