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Concept 05 / 07Chemical Bonding and Molecular Structure

Molecular Orbital Theory: Bonding and Antibonding Orbitals, Bond Order and Magnetism

Linear combination of atomic orbitals into bonding and antibonding MOs, conditions for combination, sigma and pi MOs, energy order for B2-N2 versus O2-F2 (s-p mixing), filling MO diagrams, bond order, bond length and stability, and explaining the paramagnetism of O2.

Figure 1
Molecular orbital theory (Hund, Mulliken) treats electrons in a molecule as occupying molecular orbitals (MOs) that extend over the whole molecule. MOs are formed by the linear combination of atomic orbitals (LCAO): combining nn atomic orbitals always gives nn molecular orbitals. Bonding and antibonding. Two atomic orbitals ψA\psi_{A} and ψB\psi_{B} combine in two ways. Constructive (in-phase) combination ψA+ψB\psi_{A} + \psi_{B} builds up electron density between the nuclei: a bonding MO, lower in energy than the atomic orbitals. Destructive (out-of-phase) combination ψA−ψB\psi_{A} - \psi_{B} creates a node between the nuclei: an antibonding MO (marked with an asterisk), higher in energy by slightly more than the bonding MO is lowered. Atomic orbitals combine effectively only if they have similar energies, the same symmetry about the bond axis, and appreciable overlap. Types. σ\sigma MOs are symmetric about the internuclear axis (σ1s\sigma 1s, σ∗1s\sigma^{*}1s, σ2s\sigma 2s, σ2pz\sigma 2p_{z}). π\pi MOs come from sideways overlap of pxp_{x} or pyp_{y} and occur in degenerate pairs (π2px\pi 2p_{x}, π2py\pi 2p_{y} and their antibonding partners). Energy order. For O2\mathrm{O_{2}}, F2\mathrm{F_{2}} (and Ne2\mathrm{Ne_{2}}): σ1s<σ∗1s<σ2s<σ∗2s<σ2pz<π2px=π2py<π∗2px=π∗2py<σ∗2pz\sigma 1s < \sigma^{*}1s < \sigma 2s < \sigma^{*}2s < \sigma 2p_{z} < \pi 2p_{x} = \pi 2p_{y} < \pi^{*}2p_{x} = \pi^{*}2p_{y} < \sigma^{*}2p_{z}. For B2\mathrm{B_{2}}, C2\mathrm{C_{2}}, N2\mathrm{N_{2}} (up to 1414 electrons): the π2p\pi 2p pair lies below σ2pz\sigma 2p_{z}, because in these lighter atoms the 2s2s and 2p2p energies are close and s-p mixing pushes σ2pz\sigma 2p_{z} up. Electrons fill MOs by the Aufbau principle, Pauli principle and Hund's rule, just as in atoms. Bond order =12(Nb−Na)= \frac{1}{2}(N_{b} - N_{a}), where NbN_{b} and NaN_{a} are the numbers of electrons in bonding and antibonding MOs. A positive bond order means a stable molecule; zero means it does not exist (He2\mathrm{He_{2}}, Be2\mathrm{Be_{2}}, Ne2\mathrm{Ne_{2}}). Higher bond order means shorter, stronger bonds. Unpaired electrons make a species paramagnetic. Results for second-period diatomics: H2\mathrm{H_{2}} BO 11; He2+\mathrm{He_{2}^{+}} 0.50.5 (exists); Li2\mathrm{Li_{2}} 11; B2\mathrm{B_{2}} 11 (paramagnetic, two unpaired π\pi electrons); C2\mathrm{C_{2}} 22 (diamagnetic); N2\mathrm{N_{2}} 33 (diamagnetic); O2\mathrm{O_{2}} 22 (paramagnetic); F2\mathrm{F_{2}} 11. The figure shows O2\mathrm{O_{2}}: 1616 electrons, with two in the degenerate π∗2p\pi^{*}2p orbitals occupying them singly with parallel spins (Hund's rule). This explains the paramagnetism of oxygen (liquid oxygen is attracted to a magnet), which Lewis structures with all electrons paired cannot. Removing or adding electrons: O2+\mathrm{O_{2}^{+}} BO 2.52.5, O2\mathrm{O_{2}} 22, O2−\mathrm{O_{2}^{-}} (superoxide) 1.51.5, O22−\mathrm{O_{2}^{2-}} (peroxide) 11. Removing an antibonding electron strengthens the bond; removing a bonding electron (as in N2→N2+\mathrm{N_{2} \rightarrow N_{2}^{+}}, BO 3→2.53 \rightarrow 2.5) weakens it. Key Takeaways: - nn atomic orbitals give nn MOs; bonding lower, antibonding higher. - Combination needs similar energy, same symmetry, good overlap. - Up to N2\mathrm{N_{2}}: π2p\pi 2p below σ2p\sigma 2p; for O2\mathrm{O_{2}}, F2\mathrm{F_{2}}: σ2p\sigma 2p below π2p\pi 2p. - Bond order =12(Nb−Na)= \frac{1}{2}(N_{b} - N_{a}); zero means no stable molecule. - O2\mathrm{O_{2}} paramagnetic (two unpaired π∗\pi^{*}); B2\mathrm{B_{2}} paramagnetic; N2\mathrm{N_{2}}, C2\mathrm{C_{2}}, F2\mathrm{F_{2}} diamagnetic. - O2+>O2>O2−>O22−\mathrm{O_{2}^{+}} > \mathrm{O_{2}} > \mathrm{O_{2}^{-}} > \mathrm{O_{2}^{2-}} in bond order. - Pitfall: using the O2\mathrm{O_{2}} energy order for N2\mathrm{N_{2}} or B2\mathrm{B_{2}}, which predicts the wrong magnetism for B2\mathrm{B_{2}}. Brain-Teaser 1: Compare the bond orders and magnetic behaviour of N2\mathrm{N_{2}}, N2+\mathrm{N_{2}^{+}} and N2−\mathrm{N_{2}^{-}}. Solution: N2\mathrm{N_{2}} (1414 electrons): σ1s2σ∗1s2σ2s2σ∗2s2π2p4σ2p2\sigma 1s^{2}\sigma^{*}1s^{2}\sigma 2s^{2}\sigma^{*}2s^{2}\pi 2p^{4}\sigma 2p^{2}; BO =10−42=3= \frac{10 - 4}{2} = 3, diamagnetic. N2+\mathrm{N_{2}^{+}} loses a σ2p\sigma 2p (bonding) electron: BO 2.52.5, paramagnetic. N2−\mathrm{N_{2}^{-}} gains a π∗\pi^{*} (antibonding) electron: BO 2.52.5, paramagnetic. Both ions have weaker bonds than N2\mathrm{N_{2}}, for opposite reasons. Brain-Teaser 2: Why does He2\mathrm{He_{2}} not exist while He2+\mathrm{He_{2}^{+}} does? Solution: He2\mathrm{He_{2}} would have σ1s2σ∗1s2\sigma 1s^{2}\sigma^{*}1s^{2}: BO 00, and the antibonding destabilisation slightly exceeds the bonding stabilisation, so the atoms stay apart. He2+\mathrm{He_{2}^{+}} has σ1s2σ∗1s1\sigma 1s^{2}\sigma^{*}1s^{1}: BO 0.50.5, a weak but real bond, detected spectroscopically. Brain-Teaser 3: Show that B2\mathrm{B_{2}} is paramagnetic, and explain why this proves that s-p mixing occurs. Solution: B2\mathrm{B_{2}} has 1010 electrons. With the order for light diatomics (π2p\pi 2p below σ2p\sigma 2p), the last two electrons go into the degenerate π2px\pi 2p_{x} and π2py\pi 2p_{y} singly (Hund's rule): paramagnetic, BO 11. With the O2\mathrm{O_{2}}-type order, they would pair in σ2pz\sigma 2p_{z}, making B2\mathrm{B_{2}} diamagnetic. Experiment shows B2\mathrm{B_{2}} is paramagnetic, so for boron the σ2p\sigma 2p must lie above π2p\pi 2p, which is the effect of s-p mixing.

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Molecular Orbital Theory: Bonding and Antibonding Orbitals, Bond Order and Magnetism — DPP

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