# Calculating degeneracy

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### nuclear degneracy:

• atom: $2I+1$data for $I$nuclear spin number
• molecule:
• totoal degeneracy: $(2I_1+1)(2I_2+1)$
• for homonuclear molecule: $I(2I+1)$antisymmetric, $(I+1)(2I+1)$symmetric

examples:

• $N$: $I=1,\, g_N=2I+1=3$

• $N_2$: 3 antisymmetric, 6 symmetric

• $NO$: $I_1=1$, $I_2=0$, $3\times1=3$

### electronic degeneracy:

• atom: $\Sigma_J(2J+1),\, J=|L-S|...L+S$
• molecule: $^{2S+1}\Lambda_\Omega$, $\phi=1$ for $\Omega=0=\Sigma$ $\phi= 2$ for $\Omega=1,2,..=\Pi,\Delta,...$. Total degeneracy = $(2S+1)\phi=\Sigma_{\Omega}(2S+1)$

examples:

• $N: \, ^4S \quad S=3/2,\, L =0,\, J=3/2 \quad g_{elec}=4$

• $O: \, ^3P \quad S=1,\, L =1, \, J=0,1,2$correspond to $^3P_0(227cm^{-1})\; ^3P_1(158.3cm^{-1})\; ^3P_2(0cm^{-1})\quad g_{elec}=5(base)+3+1$, temperature difference$|^3P_2-^3P_1|=227.76K$, $|^3P_2-^3P_0|=326.6K$then we can get the degeneracy function $g_{elec}=5+3exp(-227.76/T)+exp(-326.6/T)$

• $O_2:\, X^3\Sigma\, S=1,\,J=2,\, g_{elec}=3$

• $N_2:\, X^1\Sigma\, S=0, \, J=1,\, g_{elec}=1$

• $NO :\,X^2\Pi\,: S=1/2\,\Lambda=1,\,\Omega=1/2,3/2$similarly, $^2\Pi_{1/2}=0K,\, ^2\Pi_{3/2}=174.237K$ $g_{elec}=2+2exp(-174.237/T)$

### rotational degeneracy $2J+1$:

• homonuclear molecule: should consider bosons or ferminus for symmetry or asymmetric
• heteronuclear molecule: just sum all
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