We now use the $\mathrm{H}{2}^{+}$MOs developed in the last section to discuss many-electron homonuclear diatomic molecules. (Homonuclear means the two nuclei are the same; heteronuclear means they are different.) If we ignore the interelectronic repulsions, the zeroth-order wave function is a Slater determinant of $\mathrm{H}{2}^{+}$-like one-electron spin-orbitals. We approximate the spatial part of the $\mathrm{H}_{2}^{+}$spin-orbitals by the LCAO-MOs of the last section. Treatments that go beyond this crude first approximation will be discussed later.
The sizes and energies of the MOs vary with varying internuclear distance for each molecule and vary as we go from one molecule to another. Thus we saw how the orbital exponent $k$ in the $\mathrm{H}{2}^{+}$trial function (13.54) varied with $R$. As we go to molecules with higher nuclear charge, the parameter $k$ for the $\sigma{g} 1 s \mathrm{MO}$ will increase, giving a more compact MO. We want to consider the order of the MO energies. Because of the variation of these energies with $R$ and variations from molecule to molecule, numerous crossings occur, just
FIGURE 13.15 Formation of the $\pi{u} 2 p{x}$ and $\pi{g}^{*} 2 p{x}$ MOs. Since $\phi=0$ in the $x z$ plane, the cross sections of these MOs in the $x z$ plane are the same as for the corresponding $\pi{u} 2 p{+1}$ and $\pi{g}^{*} 2 p{-1}$ MOs. However, the $\pi 2 p_{x}$ MOs are not symmetrical about the $z$ axis. Rather, they consist of blobs of probability density above and below the nodal yz plane.
TABLE 13.1 Molecular-Orbital Nomenclature for Homonuclear Diatomic Molecules
| Separated-Atoms Description | United-Atom Description | Numbering by Symmetry |
|---|---|---|
| $\sigma_{g} 1 s$ | $1 s \sigma_{g}$ | $1 \sigma_{g}$ |
| $\sigma_{u}^{*} 1 s$ | $2 p \sigma_{u}^{*}$ | $1 \sigma_{u}$ |
| $\sigma_{g} 2 s$ | $2 s \sigma_{g}$ | $2 \sigma_{g}$ |
| $\sigma_{u}^{*} 2 s$ | $3 p \sigma_{u}^{*}$ | $2 \sigma_{u}$ |
| $\pi_{u} 2 p$ | $2 p \pi_{u}$ | $1 \pi_{u}$ |
| $\sigma_{g} 2 p$ | $3 s \sigma_{g}$ | $3 \sigma_{g}$ |
| $\pi_{g}^{*} 2 p$ | $3 d \pi_{g}^{*}$ | $1 \pi_{g}$ |
| $\sigma_{u}^{*} 2 p$ | $4 p \sigma_{u}^{*}$ | $3 \sigma_{u}$ |
as for atomic-orbital energies (Fig. 11.2). Hence we cannot give a definitive order. However, the following is the order in which the MOs fill as we go across the periodic table:
$\sigma{g} 1 s<\sigma{u}^{} 1 s<\sigma{g} 2 s<\sigma{u}^{} 2 s<\pi{u} 2 p{x}=\pi{u} 2 p{y}<\sigma{g} 2 p<\pi{g}^{} 2 p{x}=\pi{g}^{} 2 p{y}<\sigma{u}^{*} 2 p$
Each bonding orbital fills before the corresponding antibonding orbital. The $\pi{u} 2 p$ orbitals are close in energy to the $\sigma{g} 2 p$ orbital, and it was formerly believed that the $\sigma_{g} 2 p \mathrm{MO}$ filled first.
Besides the separated-atoms designation, there are other ways of referring to these MOs; see Table 13.1. The second column of this table gives the united-atom designations. The nomenclature of the third column uses $1 \sigma{g}$ for the lowest $\sigma{g} \mathrm{MO}, 2 \sigma{g}$ for the second lowest $\sigma{g} \mathrm{MO}$, and so on.
Figure 13.16 shows how these MOs correlate with the separated-atoms and unitedatom AOs. Because of the variation of MO energies from molecule to molecule, this
FIGURE 13.16 Correlation diagram for homonuclear diatomic MOs. (This diagram does not hold for $\mathrm{H}_{2}^{+}$.) The dashed vertical line corresponds to the order in which the MOs fill.
diagram is not quantitative. (The word correlation is being used here to mean a correspondence; this is a different meaning than in the term electron correlation.)
Recall (Prob. 7.29 and Fig. 6.13) that $s, d, g, \ldots$ united-atom AOs are even functions and therefore correlate with gerade $(g)$ MOs, whereas $p, f, h, \ldots$ AOs are odd functions and correlate with ungerade (u) MOs.
A useful principle in drawing orbital correlation diagrams is the noncrossing rule, which states that for MO correlation diagrams of many-electron diatomic molecules, the energies of MOs with the same symmetry cannot cross. For diatomic MOs the word symmetry refers to whether the orbital is $g$ or $u$ and whether it is $\sigma, \pi, \delta \ldots$ For example, two $\sigma_{g}$ MOs cannot cross on a correlation diagram. From the noncrossing rule, we conclude that the lowest MO of a given symmetry type must correlate with the lowest unitedatom AO of that symmetry, and similarly for higher orbitals. [A similar noncrossing rule holds for potential-energy curves $U(R)$ for different electronic states of a many-electron diatomic molecule.] The proof of the noncrossing rule is a bit subtle; see C. A. Mead, J. Chem. Phys., 70, 2276 (1979) for a thorough discussion.
Just as we discussed atoms by filling in the AOs, giving rise to atomic configurations such as $1 s^{2} 2 s^{2}$, we shall discuss homonuclear diatomic molecules by filling in the MOs, giving rise to molecular electronic configurations such as $\left(\sigma{g} 1 s\right)^{2}\left(\sigma{u}^{*} 1 s\right)^{2}$. (Recall that with a single atomic configuration there is associated a hierarchy of terms, levels, and states; the same is true for a molecular configuration; see Section 13.8.)
Figure 13.17 shows the homonuclear diatomic MOs formed from the $1 s, 2 s$, and $2 p$ AOs.
For $\mathrm{H}{2}^{+}$we have the ground-state configuration $\sigma{g} 1 s$, which gives a one-electron bond. For excited states the electron is in one of the higher MOs.
For $\mathrm{H}{2}$ we put the two electrons in the $\sigma{g} 1 s \mathrm{MO}$ with opposite spins, giving the ground-state configuration $\left(\sigma{g} 1 s\right)^{2}$. The two bonding electrons give a single bond. The ground-state dissociation energy $D{e}$ is 4.75 eV .
Now consider $\mathrm{He}{2}$. Two electrons go in the $\sigma{g} 1 s \mathrm{MO}$, thereby filling it. The other two go in the next MO, $\sigma{u}^{*} 1 s$. The ground-state configuration is $\left(\sigma{g} 1 s\right)^{2}\left(\sigma_{u}^{*} 1 s\right)^{2}$. With
FIGURE 13.17 Homonuclear diatomic MOs formed from $1 s, 2 s$, and $2 p$ AOs.
two bonding and two antibonding electrons, we expect no net bonding, in agreement with the fact that the ground electronic state of $\mathrm{He}{2}$ shows no substantial minimum in the potential-energy curve. However, if an electron is excited from the antibonding $\sigma{u}^{*} 1 s \mathrm{MO}$ to a higher MO that is bonding, the molecule will have three bonding electrons and only one antibonding electron. We therefore expect that $\mathrm{He}{2}$ has bound excited electronic states, with a significant minimum in the $U(R)$ curve of each such state. Indeed, about two dozen such bound excited states of $\mathrm{He}{2}$ have been spectroscopically observed in gas discharge tubes. Of course, such excited states decay to the ground electronic state, and then the molecule dissociates.
The repulsion of two $1 s^{2}$ helium atoms can be ascribed mainly to the Pauli repulsion between electrons with parallel spins (Section 10.3). Each helium atom has a pair of electrons with opposite spin, and each pair tends to exclude the other pair from occupying the same region of space.
Removal of an antibonding electron from $\mathrm{He}{2}$ gives the $\mathrm{He}{2}^{+}$ion, with ground-state configuration $\left(\sigma{g} 1 s\right)^{2}\left(\sigma{u}^{} 1 s\right)$ and one net bonding electron. Ground-state properties of this molecule are quite close to those for $\mathrm{H}{2}^{+}$; see Table 13.2 later in this section.
$\mathrm{Li}{2}$ has the ground-state configuration $\left(\sigma{g} 1 s\right)^{2}\left(\sigma{u}^{} 1 s\right)^{2}\left(\sigma{g} 2 s\right)^{2}$ with two net bonding electrons, leading to the description of the molecule as containing an $\mathrm{Li}-\mathrm{Li}$ single bond. Experimentally, $\mathrm{Li}{2}$ is a stable species. $\mathrm{In}{\mathrm{Li}}^{2}$ the orbital exponent of the $1 s \mathrm{AOs}$ is considerably greater than in $\mathrm{H}{2}^{+}$or $\mathrm{H}{2}$, because of the increase in the nuclear charges from 1 to 3 . This shrinks the $1 s{a}$ and $1 s{b}$ AOs in closer to the corresponding nuclei. There is thus only very slight overlap between these two AOs, and the integrals $S{a b}$ and $H{a b}$ are very small for these AOs. As a result, the energies of the $\sigma{g} 1 s$ and $\sigma{u}^{*} 1 s \mathrm{MOs}$ in $\mathrm{Li}{2}$ are nearly equal to each other and to the energy of a $1 s \mathrm{Li} \mathrm{AO}$. (For very small $R$, the $1 s{a}$ and $1 s{b}$ AOs do overlap appreciably and their energies then differ considerably.) The $\mathrm{Li}{2}$ ground-state configuration is often written as $K K\left(\sigma{g} 2 s\right)^{2}$ to indicate the negligible change in inner-shell orbital energies on molecule formation, which is in accord with the chemist's usual idea of bonding involving only the valence electrons. The orbital exponent of the $2 s \mathrm{AOs}$ in $\mathrm{Li}_{2}$ is not much greater than 1, because these electrons are screened from the nucleus by the $1 s$ electrons.
The $\mathrm{Be}{2}$ ground-state configuration $K K\left(\sigma{g} 2 s\right)^{2}\left(\sigma{u}^{*} 2 s\right)^{2}$ has no net bonding electrons.
The $\mathrm{B}{2}$ ground-state configuration $K K\left(\sigma{g} 2 s\right)^{2}\left(\sigma{u}^{*} 2 s\right)^{2}\left(\pi{u} 2 p\right)^{2}$ has two net bonding electrons, indicating a stable ground state, as is found experimentally. The bonding electrons are $\pi$ electrons, which is at variance with the notion that single bonds are always $\sigma$ bonds. We have two degenerate $\pi{u} 2 p$ MOs. Recall that when we had an atomic configuration such as $1 s^{2} 2 s^{2} 2 p^{2}$ we obtained several terms, which because of interelectronic repulsions had different energies. We saw that the term with the highest total spin was generally the lowest (Hund's rule). With the molecular configuration of $\mathrm{B}{2}$ given above, we also have a number of terms. Since the lower $(\sigma)$ MOs are all filled, their electrons must be paired and contribute nothing to the total spin. If the two $\pi{u} 2 p$ electrons are both in the same MO (for example, both in $\pi{u} 2 p{+1}$ ), their spins must be paired (antiparallel), giving a total molecular electronic spin of zero. If, however, we have one electron in the $\pi{u} 2 p{+1} \mathrm{MO}$ and the other in the $\pi{u} 2 p{-1} \mathrm{MO}$, their spins can be parallel, giving a net spin of 1 ; by Hund's rule, this term will be lowest, and the ground term of $\mathrm{B}{2}$ will have spin multiplicity $2 S+1=3$. Investigation of the electron-spin-resonance spectrum of $\mathrm{B}{2}$ trapped in solid neon at low temperature showed that the $\mathrm{B}_{2}$ ground term is a triplet with $S=1$ [L. B. Knight et al., J. Am. Chem. Soc., 109, 3521 (1987)].
The $\mathrm{C}{2}$ ground-state configuration $K K\left(\sigma{g} 2 s\right)^{2}\left(\sigma{u}^{*} 2 s\right)^{2}\left(\pi{u} 2 p\right)^{4}$ with four net bonding electrons gives a stable ground state with a double bond. As mentioned, the $\pi{u} 2 p$ and $\sigma{g} 2 p$ MOs have nearly the same energy in many molecules. The triplet term of the
$\mathrm{C}{2} K K\left(\sigma{g} 2 s\right)^{2}\left(\sigma{u}^{*} 2 s\right)^{2}\left(\pi{u} 2 p\right)^{3}\left(\sigma{g} 2 p\right)$ configuration lies only 0.09 eV above the ground $\left(\pi{u} 2 p\right)^{4}$ singlet term.
The $\mathrm{N}{2}$ ground-state configuration $K K\left(\sigma{g} 2 s\right)^{2}\left(\sigma{u}^{*} 2 s\right)^{2}\left(\pi{u} 2 p\right)^{4}\left(\sigma_{g} 2 p\right)^{2}$ with six net bonding electrons gives a triple bond, in accord with the Lewis structure : $\mathrm{N} \equiv \mathrm{N}$ :.
The $\mathrm{O}_{2}$ ground-state configuration is
\(
K K\left(\sigma{g} 2 s\right)^{2}\left(\sigma{u}^{} 2 s\right)^{2}\left(\sigma{g} 2 p\right)^{2}\left(\pi{u} 2 p\right)^{4}\left(\pi_{g}^{} 2 p\right)^{2}
\)
Spectroscopic evidence shows that in $\mathrm{O}{2}\left(\right.$ and in $\left.\mathrm{F}{2}\right)$ the $\sigma{g} 2 p \mathrm{MO}$ is lower in energy than the $\pi{u} 2 p \mathrm{MO}$. The four net bonding electrons give a double bond. The $\pi{g}^{*} 2 p{x}$ and $\pi{g}^{*} 2 p{y}$ MOs have the same energy, and by putting one electron in each with parallel spins, we get a triplet term. By Hund's rule this is the ground term. This explanation of the paramagnetism of $\mathrm{O}_{2}$ was one of the early triumphs of MO theory.
For $\mathrm{F}{2}$ the $\ldots\left(\pi{g}^{} 2 p\right)^{4}$ ground-state configuration gives a single bond.
For $\mathrm{Ne}{2}$ the $\ldots\left(\pi{g}^{} 2 p\right)^{4}\left(\sigma_{u}^{*} 2 p\right)^{2}$ configuration gives no net bonding electrons and no chemical bond.
We can go on to describe homonuclear diatomic molecules formed from atoms of the next period. Thus the lowest electron configuration of $\mathrm{Na}{2}$ is $\operatorname{KKLL}\left(\sigma{g} 3 s\right)^{2}$. However, there are some differences as compared with the corresponding molecules of the preceding period. For $\mathrm{Al}{2}$, the ground term is the triplet term of the $\ldots\left(\sigma{g} 3 p\right)\left(\pi{u} 3 p\right)$ configuration, which lies a mere 0.02 eV below the triplet term of the $\ldots\left(\pi{u} 3 p\right)^{2}$ configuration [C. W. Bauschlicher et al., J. Chem. Phys., 86, 7007 (1987)]. For $\mathrm{Si}{2}$, the ground term is the triplet term of the $\ldots\left(\sigma{g} 3 p\right)^{2}\left(\pi{u} 3 p\right)^{2}$ configuration, which lies 0.08 eV below the triplet term of the $\ldots\left(\sigma{g} 3 p\right)\left(\pi_{u} 3 p\right)^{3}$ configuration [T. N. Kitsopolous et al., J. Chem. Phys., 95, 1441 (1991)].
Table 13.2 lists $D{e}, R{e}$, and $\widetilde{\nu}{e} \equiv \nu{e} / c$ for the ground electronic states of some homonuclear diatomic molecules, where $\nu_{e}$ is the harmonic vibrational frequency (13.27). (In
TABLE 13.2 Properties of Homonuclear Diatomic Molecules in Their Ground Electronic States
| Molecule | Ground Term | Bond Order | $D_{e} / \mathrm{eV}$ | $R{e} / \AA$$\AA$ $\mathrm{H}{2}^{+}$ | ${ }^{2} \Sigma_{g}^{+}$ |
|---|---|---|---|---|---|
the research literature, $\widetilde{\nu}{e}$ is written as $\omega{e}$.) The table also lists the bond order, which is one-half the difference between the number of bonding and antibonding electrons. [For a survey of various methods to calculate bond orders, see J. J. Jules and J. R. Lombardi, THEOCHEM, 664-665, 255 (2003); see also J. F. Gonthier et al., Chem. Soc. Rev., 41, 4671 (2012).] As the bond order increases, $D{e}$ and $\nu{e}$ tend to increase and $R{e}$ decreases. (The high $\nu{e}$ of $\mathrm{H}_{2}$ is due to its small reduced mass $\mu$.) The term symbols in this table are explained in the next section.
Bonding MOs produce charge buildup between the nuclei, whereas antibonding MOs produce charge depletion between the nuclei. Hence removal of an electron from a bonding MO usually decreases $D{e}$, whereas removal of an electron from an antibonding MO increases $D{e}$. (Note that as $R$ decreases in Fig. 13.16, the energies of bonding MOs decrease, while the energies of antibonding MOs increase.) For example, the highest filled MO in $\mathrm{N}{2}$ is bonding, and Table 13.2 shows that in going from the ground state of $\mathrm{N}{2}$ to that of $\mathrm{N}{2}^{+}$the dissociation energy decreases (and the bond length increases). In contrast, the highest filled MO of $\mathrm{O}{2}$ is antibonding, and in going from $\mathrm{O}{2}$ to $\mathrm{O}{2}^{+}$the dissociation energy increases (and $R_{e}$ decreases). The designation of bonding or antibonding is not relevant to the effect of the electrons on the total energy of the molecule. Energy is always required to ionize a stable molecule, no matter which electron is removed. Hence both bonding and antibonding electrons in a stable molecule decrease the total molecular energy.
If the interaction between two ground-state He atoms were strictly repulsive (as predicted by MO theory), the atoms in He gas would not attract one another at all and the gas would never liquefy. Of course, helium gas can be liquefied. Configuration-interaction calculations and direct experimental evidence from scattering experiments show that as two He atoms approach each other there is an initial weak attraction, with the potential energy reaching a minimum at $2.97 \AA$ of 0.00095 eV below the separated-atoms energy. At distances less than $2.97 \AA$, the force becomes increasingly repulsive because of overlap of the electron probability densities. The initial attraction (called a London or dispersion force) results from instantaneous correlation between the motions of the electrons in one atom and the motions of the electrons in the second atom. Therefore, a calculation that includes electron correlation is needed to deal with dispersion attractions.
The general term for all kinds of intermolecular forces is van der Waals forces. Except for highly polar molecules, the dispersion force is the largest contributor to intermolecular attractions. The dispersion force increases as the molecular size increases, so boiling points tend to increase as the molecular weight increases.
The slight minimum in the $U(R)$ curve at relatively large intermolecular separation produced by the dispersion force can be deep enough to allow the existence at low temperatures of molecules bound by the dispersion interaction. Such species are called van der Waals molecules. For example, argon gas at 100 K has a small concentration of $\mathrm{Ar}{2}$ van der Waals molecules. $\mathrm{Ar}{2}$ has $D{e}=0.012 \mathrm{eV}, R{e}=3.77 \AA$, and has seven bound vibrational levels $(v=0, \ldots, 6)$.
For the ground electronic state of $\mathrm{He}{2}$ [corresponding to the electron configuration $\left.\left(\sigma{g} 1 s\right)^{2}\left(\sigma{u}^{*} 1 s\right)^{2}\right]$, the zero-point vibrational energy is very slightly less than the dissociation energy $D{e}$ associated with the dispersion attraction, so the $v=0, J=0$ level is the only bound level. Because of the extremely weak binding, $\mathrm{He}{2}$ exists in significant amounts only at very low temperatures. $\mathrm{He}{2}$ was detected mass spectrometrically in a beam of helium gas cooled to $10^{-3} \mathrm{~K}$ by expansion [ F . Luo et al., J. Chem. Phys., 98, 3564 (1993); 100, 4023 (1994)]. Accurate theoretical calculations on $\mathrm{He}{2}$ give $D{e}=0.000948 \mathrm{eV}, D{0}=0.00000014 \mathrm{eV}, R{e}=2.97 \AA$ and give the average internuclear distance in $\mathrm{He}_{2}$ as $\langle R\rangle \approx 47 \AA$ [M. Przybytek et al., Phys. Rev. Lett., 104, 183003 (2010); M. Jeziorska et al., J. Chem. Phys., 127, 124303 (2007)]. $\langle R\rangle$ is huge because the $v=0$ level lies so close to the dissociation limit.
Examples of diatomic van der Waals molecules and their $R{e}$ and $D{e}$ values include $\mathrm{Ne}{2}$, $3.1 \AA, 0.0036 \mathrm{eV} ; \mathrm{HeNe}, 3.2 \AA, 0.0012 \mathrm{eV} ; \mathrm{Ca}{2}, 4.28 \AA, 0.13 \mathrm{eV} ; \mathrm{Mg}{2}, 3.89 \AA, 0.053 \mathrm{eV}$. Observed polyatomic van der Waals molecules include $\left(\mathrm{O}{2}\right){2}, \mathrm{H}{2}-\mathrm{N}{2}, \mathrm{Ar}-\mathrm{HCl}$, and $\left(\mathrm{Cl}{2}\right){2}$. For van der Waals bonding, $R{e}$ is significantly greater and $D{e}$ is very substantially less than the values for chemically bound molecules. The $\mathrm{Be}{2}$ bond length of $2.45 \AA$ is much shorter than is typical for van der Waals molecules; the closeness of the $2 p$ orbitals to $2 s$ orbitals in Be allows substantial $2 s-2 p$ hybridization in $\mathrm{Be}_{2}$ and perhaps gives some amount of covalent character in addition to the dispersion attraction. For more on van der Waals molecules, see Chem. Rev., 88, 813-988 (1988); 94, 1721-2160 (1994); 100, 3861-4264 (2000).