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One Open Shell

Given are term symbols (up to indices depending on actual case and group) and $ a$ and $ b$ coefficients. $ n$ is the number of electrons in an irrep with degeneracy $ n_{ir}$. Note that not all cases are Roothaan cases.

All single electron cases are described by:

$ a = b = 0$
$ n_{ir}$=2: e (div. groups), $ \pi$, $ \delta$ ( $ C_{\infty v}$, $ D_{\infty h}$)
$ n$ $ f$ e$ ^n$ $ \pi^n$ $ \delta^n$ $ a$ $ b$
$ ^3$A $ ^3\Sigma$ $ ^3\Sigma$ 1 2
2 $ 1/2$ $ ^1$E% latex2html id marker 39904 \setcounter{footnote}{1}\fnsymbol{footnote}% latex2html id marker 39904 ootnote}{1}<IMG WIDTH= $ ^1\Delta$ $ ^1\Gamma$ $ 1/2$ 0
$ ^1$A $ ^1\Sigma$ $ ^1\Sigma$ 0 $ -$2
3 $ 3/4$ $ ^2$E $ ^2\Pi$ $ ^2\Delta$ $ 8/9$ $ 8/9$
1 $ n_{ir}$=3: p (O(3)), t ($ T$, $ O$, $ I$)% latex2html id marker 39940 \setcounter{footnote}{2}\fnsymbol{footnote}
$ n$ $ f$ p$ ^n$ $ a$ $ b$
$ ^3$P $ 3/4$ $ 3/2$
2 $ 1/3$ $ ^1$D% latex2html id marker 39961 \setcounter{footnote}{7}\fnsymbol{footnote} $ 9/20$ $ -$$ 3/10$
$ ^1$S 0 $ -$$ 3$
$ ^4$S 1 2
3 $ 1/2$ $ ^2$D% latex2html id marker 39980 \setcounter{footnote}{7}\fnsymbol{footnote} $ 4/5$ $ 4/5$
$ ^2$P $ 2/3$ 0
$ ^3$P $ 15/16$ $ 9/8$
4 $ 2/3$ $ ^1$D% latex2html id marker 39999 \setcounter{footnote}{7}\fnsymbol{footnote} $ 69/80$ $ 27/40$
$ ^1$S $ 3/4$ 0
5 $ 5/6$ $ ^2$P $ 24/25$ $ 24/25$
only irrep g($ I$)
(mainly high spin available)
$ n$ $ f$ g$ ^n$ $ a$ $ b$
1 $ 1/8$ $ ^2$G 0 0
2 $ 1/4$ % latex2html id marker 40036 \setcounter{footnote}{8}\fnsymbol{footnote} $ 2/3$ $ 4/3$
$ ^1$A 0 $ -$$ 4$
3 $ 3/8$ $ ^4$G $ 8/9$ $ 16/9$
4 $ 1/2$ $ ^5$A $ 1$ $ 2$
5 $ 5/8$ $ ^4$G$ 24/25$$ 32/25$
6 $ 3/4$ % latex2html id marker 40074 \setcounter{footnote}{8}\fnsymbol{footnote}$ 26/27$$ 28/27$
$ ^1$A $ 8/9$ $ 4/9$
7 $ 7/8$ $ ^2$G$ 48/49$$ 48/49$
d(O3), h($ I$)
(mainly high-spin cases work)
$ n$ $ f$ d$ ^n$ $ a$ $ b$
1 $ 1/10$ $ ^2$D 0 0
2 $ 1/5$ $ ^3$F$ + ^3$P% latex2html id marker 40117 \setcounter{footnote}{8}\fnsymbol{footnote} $ 5/8$ $ 5/4$
$ ^1$S 0 $ -$$ 5$
3 $ 3/10$ $ ^4$F$ + ^4$P% latex2html id marker 40135 \setcounter{footnote}{8}\fnsymbol{footnote} $ 5/6$ $ 5/3$
4 $ 2/5$ $ ^5$D, $ ^5$H $ 15/16$ $ 15/8$
5 $ 1/2$ $ ^6$S, $ ^6$A $ 1$ $ 2$
6 $ 3/5$ $ ^5$D, $ ^5$H $ 35/36$ $ 25/18$
7 $ 7/10$ $ ^4$F$ + ^4$P% latex2html id marker 40176 \setcounter{footnote}{8}\fnsymbol{footnote} $ 95/98$ $ 55/49$
8 $ 4/5$ $ ^3$F$ + ^3$P% latex2html id marker 40187 \setcounter{footnote}{8}\fnsymbol{footnote}$ 125/128$ $ 65/64$
$ ^1$S $ 15/16$ $ 5/8$
9 $ 9/10$ $ ^2$D, $ ^2$H $ 80/81$ $ 80/81$
    % latex2html id marker 40208 \setcounter{footnote}{1}\fnsymbol{footnote}except cases (e.g. $ D_{2d}$ or $ D_{4h}$) where $ \rm e^2$ gives only one-dimensional irreps, which are not Roothaan cases.
    % latex2html id marker 40215 \setcounter{footnote}{2}\fnsymbol{footnote}only p$ ^n$ given, the state for groups $ T_d$ etc. follows from
S $ \rightarrow$ A ($ T$,$ O$,$ I$) P $ \rightarrow$ T ($ T$,$ O$,$ I$) D $ \rightarrow$ H ($ I$), E+T ($ T$,$ O$)
    % latex2html id marker 40244 \setcounter{footnote}{7}\fnsymbol{footnote}This is not a CSF in $ T$ or $ O$, ($ a,b$) describes average of states resulting from E+T
    % latex2html id marker 40251 \setcounter{footnote}{8}\fnsymbol{footnote}($ a,b$) describes weighted average of high spin states, not a CSF.


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