Molecular Orbital Theory and Bond Order

Molecular orbital basics:

• Overlap of two atomic orbitals gives one bonding MO (lower energy: $\sigma,\ \pi$) and one antibonding MO (higher energy: $\sigma^*,\ \pi^*$).
• Electrons fill MOs following the Aufbau principle, Pauli exclusion and Hund's rule — lowest energy first, one electron per degenerate orbital before pairing.
• Order for $\text{B}_2,\ \text{C}_2,\ \text{N}_2$ (and lighter): $\sigma 1s < \sigma^*1s < \sigma 2s < \sigma^*2s < \pi 2p_x = \pi 2p_y < \sigma 2p_z < \pi^*2p_x = \pi^*2p_y < \sigma^*2p_z$.
• Order swaps for $\text{O}_2,\ \text{F}_2,\ \text{Ne}_2$: here $\sigma 2p_z$ drops below the $\pi 2p$ pair, so the sequence is $\ldots \sigma^*2s < \sigma 2p_z < \pi 2p_x = \pi 2p_y < \pi^*2p_x = \pi^*2p_y < \sigma^*2p_z$.
• First count the total number of electrons (add or subtract for an ion's charge), then fill.

Bond order in molecular orbital theory:

• Bond order $= \tfrac12(N_b - N_a)$, where $N_b$ = electrons in bonding MOs and $N_a$ = electrons in antibonding MOs.
• Bond order 0 means the molecule does not exist (e.g. hypothetical $\text{He}_2$); positive bond order means a stable molecule.
• Standard values: $\text{H}_2 = 1$, $\text{Li}_2 = 1$, $\text{N}_2 = 3$, $\text{O}_2 = 2$, $\text{F}_2 = 1$, $\text{CO} = 3$ (isoelectronic with $\text{N}_2$).
• Ions give fractional bond orders: $\text{N}_2^+ = 2.5$, $\text{O}_2^+ = 2.5$, $\text{O}_2^- = 1.5$, $\text{O}_2^{2-} = 1$.

Reading properties off the MO configuration:

• Paramagnetic = one or more unpaired electrons (attracted by a magnetic field); diamagnetic = all electrons paired.
• $\text{O}_2$ has two unpaired electrons in its $\pi^*2p$ orbitals, so it is paramagnetic — a key success of MOT. $\text{N}_2$ and $\text{F}_2$ are fully paired, so diamagnetic.
• Higher bond order → shorter bond length → greater stability (more energy needed to break it): $\text{N}_2$ (BO 3) has the shortest, strongest bond; $\text{Cl}_2$ (BO 1) the longest.
• Odd-electron molecules like NO (11 valence-shell electrons) carry one unpaired electron → paramagnetic.

The bank almost always draws from the species below. Two anchors to hold it together:

• $\text{N}_2$ and CO (both 14 electrons, isoelectronic) sit at bond order 3 — the most stable.
• $\text{O}_2$ is the paramagnetic one (bond order 2, two unpaired electrons); its ions $\text{O}_2^+$ and $\text{O}_2^-$ shift the bond order up or down by $\tfrac12$.