Comprehensive Study Notes

We now consider the terms arising from a given diatomic molecule electron configuration.
For atoms, each set of degenerate atomic orbitals constitutes an atomic subshell. For example, the $2 p{+1}, 2 p{0}$, and $2 p{-1}$ AOs constitute the $2 p$ subshell. An atomic electronic configuration is defined by giving the number of electrons in each subshell; for example, $1 s^{2} 2 s^{2} 2 p^{4}$. For molecules, each set of degenerate molecular orbitals constitutes a molecular shell. For example, the $\pi{u} 2 p{+1}$ and $\pi{u} 2 p{-1}$ MOs constitute the $\pi{u} 2 p$ shell. Each diatomic $\sigma$ shell consists of one MO, while each $\pi, \delta, \phi, \ldots$ shell consists of two MOs; diatomic $\sigma$ shells are filled with two electrons, while non $\sigma$ shells hold up to four electrons. We define a molecular electronic configuration by giving the number of electrons in each shell, for example, $\left(\sigma{g} 1 s\right)^{2}\left(\sigma{u}^{} 1 s\right)^{2}\left(\sigma{g} 2 s\right)^{2}\left(\sigma{u}^{} 2 s\right)^{2}\left(\pi_{u} 2 p\right)^{3}$.

For $\mathrm{H}{2}^{+}$, the operator $\hat{L}{z}$ commutes with $\hat{H}$. For a many-electron diatomic molecule, one finds that the operator for the axial component of the total electronic orbital angular momentum commutes with $\hat{H}$. The component of electronic orbital angular momentum along the molecular axis has the possible values $M{L} \hbar$, where $M{L}=0, \pm 1, \pm 2, \pm \ldots$ To calculate $M_{L}$, we simply add algebraically the $m$ 's of the individual electrons. Analogous to the symbol $\lambda$ for a one-electron molecule, $\Lambda$ is defined as

\(
\begin{equation}
\Lambda \equiv\left|M_{L}\right| \tag{13.84}
\end{equation}
\)

(Some people define $\Lambda$ as equal to $M_{L}$.) The following code specifies the value of $\Lambda$ :

For $\Lambda \neq 0$, there are two possible values of $M{L}$, namely, $+\Lambda$ and $-\Lambda$. As in $\mathrm{H}{2}^{+}$, the electronic energy depends on $M{L}^{2}$, so there is a double degeneracy associated with the two values of $M{L}$. Note that lowercase letters refer to individual electrons, while capital letters refer to the whole molecule.

Just as in atoms, the individual electron spins add vectorially to give a total electronic spin $\mathbf{S}$, whose magnitude has the possible values $[S(S+1)]^{1 / 2} \hbar$, with $S=0, \frac{1}{2}, 1, \frac{3}{2}, \ldots$. The component of $\mathbf{S}$ along an axis has the possible values $M{S} \hbar$, where $M{S}=S, S-1, \ldots,-S$. As in atoms, the quantity $2 S+1$ is called the spin multiplicity and is written as a left superscript to the code letter for $\Lambda$. Diatomic electronic states that arise from the same electron configuration and that have the same value for $\Lambda$ and the same value for $S$ are said to belong to the same electronic term. We now consider how the terms belonging to a given electron configuration are derived. (We are assuming Russell-Saunders coupling, which holds for molecules composed of atoms of not-too-high atomic number.)

A filled diatomic molecule shell consists of one or two filled molecular orbitals. The Pauli principle requires that, for two electrons in the same molecular orbital, one have $m{s}=+\frac{1}{2}$ and the other have $m{s}=-\frac{1}{2}$. Hence the quantum number $M{S}$, which is the algebraic sum of the individual $m{s}$ values, must be zero for a filled-shell molecular
configuration. Therefore, we must have $S=0$ for a configuration containing only filled molecular shells. A filled $\sigma$ shell has two electrons with $m=0$, so $M{L}$ is zero. A filled $\pi$ shell has two electrons with $m=+1$ and two electrons with $m=-1$, so $M{L}$ (which is the algebraic sum of the $m$ 's) is zero. The same situation holds for filled $\delta, \phi, \ldots$ shells. Thus a closed-shell molecular configuration has both $S$ and $\Lambda$ equal to zero and gives rise to only a ${ }^{1} \Sigma$ term. An example is the ground electronic configuration of $\mathrm{H}_{2}$. (Recall that a filled-subshell atomic configuration gives only a ${ }^{1} S$ term.) In deriving molecular terms, we need consider only electrons outside filled shells.

A single $\sigma$ electron has $s=\frac{1}{2}$, so $S$ must be $\frac{1}{2}$, and we get a ${ }^{2} \Sigma$ term. An example is the ground electronic configuration of $\mathrm{H}_{2}^{+}$. A single $\pi$ electron gives a ${ }^{2} \Pi$ term, and so on.

Now consider more than one electron. Electrons that are in different molecular shells are called nonequivalent. For such electrons we do not have to worry about giving two of them the same set of quantum numbers, and the terms are easily derived. Consider two nonequivalent $\sigma$ electrons, a $\sigma \sigma$ configuration. Since both $m$ 's are zero, we have $M_{L}=0$. Each $s$ is $\frac{1}{2}$, so $S$ can be 1 or 0 . We thus have the terms ${ }^{1} \Sigma$ and ${ }^{3} \Sigma$. Similarly, a $\sigma \pi$ configuration gives ${ }^{1} \Pi$ and ${ }^{3} \Pi$ terms.

For a $\pi \delta$ configuration, we have singlet and triplet terms. The $\pi$ electron can have $m= \pm 1$, and the $\delta$ electron can have $m= \pm 2$. The possible values for $M{L}$ are thus $+3,-3,+1$, and -1 . This gives $\Lambda=3$ or 1 , and we have the terms ${ }^{1} \Pi,{ }^{3} \Pi,{ }^{1} \Phi,{ }^{3} \Phi$. (In atoms we add the vectors $\mathbf{L}{i}$ to get the total $\mathbf{L}$; hence a $p d$ atomic configuration gives $P$, $D$, and $F$ terms. In diatomic molecules, however, we add the $z$ components of the orbital angular momenta. This is an algebraic rather than a vectorial addition, so a $\pi \delta$ molecular configuration gives $\Pi$ and $\Phi$ terms and no $\Delta$ terms.)

For a $\pi \pi$ configuration of two nonequivalent electrons, each electron has $m= \pm 1$, and we have the $M{L}$ values $2,-2,0,0$. The values of $\Lambda$ are 2,0 , and 0 ; the terms are ${ }^{1} \Delta,{ }^{3} \Delta,{ }^{1} \Sigma,{ }^{3} \Sigma,{ }^{1} \Sigma$, and ${ }^{3} \Sigma$. The values +2 and -2 correspond to the two degenerate states of the same $\Delta$ term. However, $\Sigma$ terms are nondegenerate (apart from spin degeneracy), and the two values of $M{L}$ that are zero indicate two different $\Sigma$ terms (which become four $\Sigma$ terms when we consider spin).

Consider the forms of the wave functions for the $\pi \pi$ terms. We shall call the two $\pi$ subshells $\pi$ and $\pi^{\prime}$ and shall use a subscript to indicate the $m$ value. For the $\Delta$ terms, both electrons have $m=+1$ or both have $m=-1$. For $M{L}=+2$, we might write as the spatial factor in the wave function $\pi{+1}(1) \pi{+1}^{\prime}(2)$ or $\pi{+1}(2) \pi_{+1}^{\prime}(1)$. However, these functions are neither symmetric nor antisymmetric with respect to exchange of the indistinguishable electrons and are unacceptable. Instead, we must take the linear combinations (we shall not bother with normalization constants)

\(
\begin{array}{cl}
{ }^{1} \Delta: & \pi{+1}(1) \pi{+1}^{\prime}(2)+\pi{+1}(2) \pi{+1}^{\prime}(1) \
{ }^{3} \Delta: & \pi{+1}(1) \pi{+1}^{\prime}(2)-\pi{+1}(2) \pi{+1}^{\prime}(1) \tag{13.86}
\end{array}
\)

Similarly, with both electrons having $m=-1$, we have the spatial factors

\(
\begin{array}{ll}
{ }^{1} \Delta: & \pi{-1}(1) \pi{-1}^{\prime}(2)+\pi{-1}(2) \pi{-1}^{\prime}(1) \
{ }^{3} \Delta: & \pi{-1}(1) \pi{-1}^{\prime}(2)-\pi{-1}(2) \pi{-1}^{\prime}(1) \tag{13.88}
\end{array}
\)

The functions (13.85) and (13.87) are symmetric with respect to exchange. They therefore go with the antisymmetric two-electron spin factor (11.60), which has $S=0$. Thus (13.85) and (13.87) are the spatial factors in the wave functions for the two states of the doubly degenerate ${ }^{1} \Delta$ term. The antisymmetric functions (13.86) and (13.88) must go with the symmetric two-electron spin functions (11.57), (11.58), and (11.59), giving the six states of the ${ }^{3} \Delta$ term. These states all have the same energy (if we neglect spin-orbit interaction).

Now consider the wave functions of the $\Sigma$ terms. These have one electron with $m=+1$ and one electron with $m=-1$. We start with the four functions

\(
\pi{+1}(1) \pi{-1}^{\prime}(2), \quad \pi{+1}(2) \pi{-1}^{\prime}(1), \quad \pi{-1}(1) \pi{+1}^{\prime}(2), \quad \pi{-1}(2) \pi{+1}^{\prime}(1)
\)

Combining them to get symmetric and antisymmetric functions, we have

\(
\begin{array}{cl}
{ }^{1} \Sigma^{+}: & \pi{+1}(1) \pi{-1}^{\prime}(2)+\pi{+1}(2) \pi{-1}^{\prime}(1)+\pi{-1}(1) \pi{+1}^{\prime}(2)+\pi{-1}(2) \pi{+1}^{\prime}(1) \
{ }^{1} \Sigma^{-}: & \pi{+1}(1) \pi{-1}^{\prime}(2)+\pi{+1}(2) \pi{-1}^{\prime}(1)-\pi{-1}(1) \pi{+1}^{\prime}(2)-\pi{-1}(2) \pi{+1}^{\prime}(1) \
{ }^{3} \Sigma^{+}: & \pi{+1}(1) \pi{-1}^{\prime}(2)-\pi{+1}(2) \pi{-1}^{\prime}(1)+\pi{-1}(1) \pi{+1}^{\prime}(2)-\pi{-1}(2) \pi{+1}^{\prime}(1) \tag{13.89}\
{ }^{3} \Sigma^{-}: & \pi{+1}(1) \pi{-1}^{\prime}(2)-\pi{+1}(2) \pi{-1}^{\prime}(1)-\pi{-1}(1) \pi{+1}^{\prime}(2)+\pi{-1}(2) \pi{+1}^{\prime}(1)
\end{array}
\)

The first two functions in (13.89) are symmetric. They therefore go with the antisymmetric singlet spin function (11.60). Clearly, these two spatial functions have different energies. The last two functions in (13.89) are antisymmetric and hence are the spatial factors in the wave functions of the two ${ }^{3} \Sigma$ terms. The four functions in (13.89) are found to have eigenvalue +1 or -1 with respect to reflection of electronic coordinates in the $x z \sigma_{v}$ symmetry plane containing the molecular $(z)$ axis (Prob. 13.30). The superscripts + and - refer to this eigenvalue.

Examination of the $\Delta$ terms (13.85) to (13.88) shows that they are not eigenfunctions of the symmetry operator $\hat{O}{\sigma{v}}$ (Section 12.1). Since a twofold degeneracy (apart from spin degeneracy) is associated with these terms, there is no necessity that their wave functions be eigenfunctions of this operator. However, since $\hat{O}{\sigma{v}}$ commutes with the Hamiltonian, we can choose the eigenfunctions to be eigenfunctions of $\hat{O}{\sigma{v}}$. Thus we can combine the functions (13.85) and (13.87), which belong to a degenerate energy level, as follows:

\(
(13.85)+(13.87) \text { and }(13.85)-(13.87)
\)

These two linear combinations are eigenfunctions of $\hat{O}{\sigma{v}}$ with eigenvalues +1 and -1 , and we could refer to them as ${ }^{1} \Delta^{+}$and ${ }^{1} \Delta^{-}$states. Since they have the same energy, there is no point in using the + and - superscripts. Thus the + and - designations are used only for $\Sigma$ terms. However, when one considers the interaction between the molecular rotational angular momentum and the electronic orbital angular momentum, there is a very slight splitting (called $\Lambda$-type doubling) of the two states of a ${ }^{1} \Delta$ term. It turns out that the correct zeroth-order wave functions for this perturbation are the linear combinations that are eigenfunctions of $\hat{O}{\sigma{v}}$, so in this case there is a point to distinguishing between $\Delta^{+}$and $\Delta^{-}$ states. The linear combinations (13.85) $\pm$ (13.87), which are eigenfunctions of $\hat{O}{\sigma{v}}$, are not eigenfunctions of $\hat{L}{z}$ but are superpositions of $\hat{L}{z}$ eigenfunctions with eigenvalues +2 and -2 .

We can distinguish + and - terms for one-electron configurations. The wave function of a single $\sigma$ electron has no phi factor and hence must correspond to a $\Sigma^{+}$term. For a $\pi$ electron, the MOs that are eigenfunctions of $\hat{L}{z}$ are the $\pi{+1}$ and $\pi{-1}$ functions (whose probability densities are each symmetric about the $z$ axis; Fig. 13.14). The $\pi{+1}$ and $\pi{-1}$ functions are not eigenfunctions of $\hat{O}{\sigma{v}}$, but the linear combinations $\pi{+1}+\pi{-1}=\pi{x}$ and $\pi{+1}-\pi{-1}=\pi{y}$ are. The $\pi{x}$ and $\pi{y}$ MOs (whose probability densities are not symmetric about the $z$ axis; Fig. 13.15) are the correct zeroth-order functions if the perturbation of the electronic wave functions due to molecular rotation is considered. The $\pi{x}$ and $\pi{y}$ MOs have eigenvalues +1 and -1 , respectively, for reflection in the $x z$ plane, and eigenvalues -1 and +1 , respectively, for reflection in the $y z$ plane. (The operators $\hat{L}{z}$ and $\hat{O}{\sigma{v}}$ do not commute; Prob. 13.31. Hence we cannot have all the eigenfunctions of $\hat{H}$ being eigenfunctions of both these operators as well. However, since each of these operators commutes with the electronic Hamiltonian and since there is no element of choice in the wave function of a nondegenerate level, all the $\sigma$ MOs must be eigenfunctions of both $\hat{L}{z}$ and $\hat{O}{\sigma_{v}}$.)

Electrons in the same molecular shell are called equivalent. There are fewer terms for equivalent electrons than for the corresponding nonequivalent electron configuration, because of the Pauli principle. Thus, for a $\pi^{2}$ configuration of two equivalent $\pi$ electrons, four of the eight functions (13.85) to (13.89) vanish; the remaining functions give a ${ }^{1} \Delta$ term, a ${ }^{1} \Sigma^{+}$term, and a ${ }^{3} \Sigma^{-}$term. Alternatively, we can make a table similar to Table 11.1 and use it to derive the terms for equivalent electrons.

Table 13.3 lists terms arising from various electron configurations. A filled shell always gives the single term ${ }^{1} \Sigma^{+}$. A $\pi^{3}$ configuration gives the same result as a $\pi$ configuration.

For homonuclear diatomic molecules, a $g$ or $u$ right subscript is added to the term symbol to show the parity of the electronic states belonging to the term. Terms arising from an electron configuration that has an odd number of electrons in molecular orbitals of odd parity are odd $(u)$; all other terms are even $(g)$. This is the same rule as for atoms.

The term symbols given in Table 13.2 are readily derived from the MO configurations. For example, $\mathrm{O}{2}$ has a $\pi^{2}$ configuration, which gives the three terms ${ }^{1} \Sigma{g}^{+},{ }^{3} \Sigma{g}^{-}$, and ${ }^{1} \Delta{g}$. Hund's rule tells us that ${ }^{3} \Sigma{g}^{-}$is the lowest term, as listed. The $v=0$ levels of the ${ }^{1} \Delta{g}$ and ${ }^{1} \Sigma{g}^{+} \mathrm{O}{2}$ terms lie 0.98 eV and 1.6 eV , respectively, above the $v=0$ level of the ground ${ }^{3} \Sigma{g}^{-}$ term. Singlet $\mathrm{O}{2}$ is a reaction intermediate in many organic, biochemical, and inorganic reactions. [See C. S. Foote et al., eds., Active Oxygen in Chemistry, Springer, 1995; J. S. Valentine et al., eds., Active Oxygen in Biochemistry, Springer, 1995; C. Schweitzer and R. Schmidt, Chem. Rev., 103, 1685 (2003).]

Most stable diatomic molecules have a ${ }^{1} \Sigma^{+}$ground term ( ${ }^{1} \Sigma{g}^{+}$for homonuclear diatomics). Exceptions include $\mathrm{B}{2}, \mathrm{Al}{2}, \mathrm{Si}{2}, \mathrm{O}_{2}$, and NO , which has a ${ }^{2} \Pi$ ground term.

Spectroscopists prefix the ground term of a molecule by the symbol $X$. Excited terms of the same spin multiplicity as the ground term are designated as $A, B, C, \ldots$, while excited terms of different spin multiplicity from the ground term are designated as $a, b, c, \ldots$ Exceptions are $\mathrm{C}{2}$ and $\mathrm{N}{2}$, where the ground terms are ${ }^{1} \Sigma_{g}^{+}$but the letters $A, B, C, \ldots$ are used for excited triplet terms.

Just as for atoms, spin-orbit interaction can split a molecular term into closely spaced energy levels, giving a multiplet structure to the term. The projection of the total electronic

TABLE 13.3 Electronic Terms of Diatomic Molecules

$\operatorname{spin} \mathbf{S}$ on the molecular axis is $M{S} \hbar$. In molecules the quantum number $M{S}$ is called $\Sigma$ (not to be confused with the symbol meaning $\Lambda=0$ ):

\(
\Sigma=S, S-1, \ldots,-S
\)

The axial components of electronic orbital and spin angular momenta add, giving as the total axial component of electronic angular momentum $(\Lambda+\Sigma) \hbar$. (Recall that $\Lambda$ is the absolute value of $M_{L}$. We consider $\Sigma$ to be positive when it has the same direction as $\Lambda$, and negative when it has the opposite direction as $\Lambda$.) The possible values of $\Lambda+\Sigma$ are

\(
\Lambda+S, \quad \Lambda+S-1, \ldots, \quad \Lambda-S
\)

The value of $\Lambda+\Sigma$ is written as a right subscript to the term symbol to distinguish the energy levels of the term. Thus a ${ }^{3} \Delta$ term has $\Lambda=2$ and $S=1$ and gives rise to the levels ${ }^{3} \Delta{3},{ }^{3} \Delta{2}$, and ${ }^{3} \Delta{1}$. In a sense, $\Lambda+\Sigma$ is the analog in molecules of the quantum number $J$ in atoms. However, $\Lambda+\Sigma$ is the quantum number of the $z$ component of total electronic angular momentum and therefore can take on negative values. Thus a ${ }^{4} \Pi$ term has the four levels ${ }^{4} \Pi{5 / 2},{ }^{4} \Pi{3 / 2},{ }^{4} \Pi{1 / 2}$, and ${ }^{4} \Pi_{-1 / 2}$. The absolute value of $\Lambda+\Sigma$ is called $\Omega$ :

\(
\begin{equation}
\Omega \equiv|\Lambda+\Sigma| \tag{13.90}
\end{equation}
\)

The spin-orbit interaction energy in diatomic molecules can be shown to be well approximated by $A \Lambda \Sigma$, where $A$ depends on $\Lambda$ and on the internuclear distance $R$ but not on $\Sigma$. The spacing between levels of the multiplet is thus constant. When $A$ is positive, the level with the lowest value of $\Lambda+\Sigma$ lies lowest, and the multiplet is regular. When $A$ is negative, the multiplet is inverted. Note that for $\Lambda \neq 0$ the spin multiplicity $2 S+1$ always equals the number of multiplet components. This is not always true for atoms.

Each energy level of a multiplet with $\Lambda \neq 0$ is doubly degenerate, corresponding to the two values for $M_{L}$. Thus a ${ }^{3} \Delta$ term has six different wave functions [Eqs. (13.86), (13.88), (11.57) to (11.59)] and therefore six different molecular electronic states. Spin-orbit interaction splits the ${ }^{3} \Delta$ term into three levels, each doubly degenerate. The double degeneracy of the levels is removed by the $\Lambda$-type doubling mentioned previously.

For $\Sigma$ terms $(\Lambda=0)$, the spin-orbit interaction is very small (zero in the first approximation), and the quantum numbers $\Sigma$ and $\Omega$ are not defined.
$A^{1} \Sigma$ term always corresponds to a single nondegenerate energy level.