We have concentrated mostly on the ground electronic states of diatomic molecules. In this section we consider some of the excited states of $\mathrm{H}{2}$. Figure $\mathbf{1 3 . 1 9}$ gives the potentialenergy curves for some of the $\mathrm{H}{2}$ electronic energy levels.
The lowest MO configuration is $\left(1 \sigma{g}\right)^{2}$, where the notation of the third column of Table 13.1 is used. This closed-shell configuration gives only a nondegenerate ${ }^{1} \Sigma{g}^{+}$level, designated $X^{1} \Sigma_{g}^{+}$. The LCAO-MO function is (13.93).
The next-lowest MO configuration is $\left(1 \sigma{g}\right)\left(1 \sigma{u}\right)$, which gives rise to the terms ${ }^{1} \Sigma{u}^{+}$ and ${ }^{3} \Sigma{u}^{+}$(Table 13.3). Since there is no axial electronic orbital angular momentum, each of these terms corresponds to one level. Spectroscopists have named these electronic levels $B^{1} \Sigma{u}^{+}$and $b^{3} \Sigma{u}^{+}$. By Hund's rule, the $b$ level lies below the $B$ level. The LCAO-MO functions for these levels are [see Eqs. (10.27)-(10.30)]
\(
\begin{array}{ll}
b^{3} \Sigma{u}^{+}: & 2^{-1 / 2}\left[1 \sigma{g}(1) 1 \sigma{u}(2)-1 \sigma{g}(2) 1 \sigma{u}(1)\right]\left{\begin{array}{l}
\alpha(1) a(2) \
2^{-1 / 2}[\alpha(1) \beta(2)+\alpha(2) \beta(1)] \
\beta(1) \beta(2)
\end{array}\right. \
B^{1} \Sigma{u}^{+}: \quad 2^{-1 / 2}\left[1 \sigma{g}(1) 1 \sigma{u}(2)+1 \sigma{g}(2) 1 \sigma{u}(1)\right] 2^{-1 / 2}[\alpha(1) \beta(2)-\alpha(2) \beta(1)]
\end{array}
\)
where $1 \sigma{g} \approx N\left(1 s{a}+1 s{b}\right)$ and $1 \sigma{u} \approx N^{\prime}\left(1 s{a}-1 s{b}\right)$. The $b^{3} \Sigma{u}^{+}$level is triply degenerate. The $B^{1} \Sigma{u}^{+}$level is nondegenerate. The Heitler-London wave functions for the $b$ level are given by (13.101). Both these levels have one bonding and one antibonding electron, and we would expect the potential-energy curves for both levels to be repulsive. Actually,
the $B$ level has a minimum in its $U(R)$ curve. The stability of this state should caution us against drawing too hasty conclusions from very approximate wave functions.
We expect the next-lowest configuration to be $\left(1 \sigma{g}\right)\left(2 \sigma{g}\right)$, giving rise to ${ }^{1} \Sigma{g}^{+}$and ${ }^{3} \Sigma{g}^{+}$levels. These levels of $\mathrm{H}{2}$ are designated $E^{1} \Sigma{g}^{+}$and $a^{3} \Sigma_{g}^{+}$. By Hund's rule, the triplet lies lower. The $E$ state has two substantial minima in its $U(R)$ curve, and is often called the EF state because of the two minima.
Although the $2 \sigma{u} \mathrm{MO}$ fills before the two $1 \pi{u} \mathrm{MOs}$ in going across the periodic table, the $1 \pi{u} \mathrm{MOs}$ lie below the $2 \sigma{u} \mathrm{MO}$ in $\mathrm{H}{2}$. The configuration $\left(1 \sigma{g}\right)\left(1 \pi{u}\right)$ gives rise to the terms ${ }^{1} \Pi{u}$ and ${ }^{3} \Pi{u}$, the triplet lying lower. These terms are designated $C^{1} \Pi{u}$ and $c^{3} \Pi{u}$. The $c$ term gives rise to the levels $c^{3} \Pi{2 u}, c^{3} \Pi{1 u}$, and $c^{3} \Pi{0 u}$. These levels lie so close together that they are usually not resolved in spectroscopic work. The $C$ level shows a slight hump in its potential-energy curve at large $R$. Each level is twofold degenerate, which gives a total of eight electronic states arising from the $\left(1 \sigma{g}\right)\left(1 \pi{u}\right)$ configuration.