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A molecular orbital near the boundary between occupied and unoccupied levels. The highest occupied molecular orbital and LUMO are central examples. Their energies, occupancies, spatial form and symmetry help predict which new chemical bonds can form when fragments approach.

The nitrogen-centered highest occupied molecular orbital of ammonia supplies a lone pair; planar borane has an empty out-of-plane $p$-derived lowest unoccupied molecular orbital. Approach along this acceptor axis gives constructive orbital overlap and donation of the nitrogen pair into the boron-centered acceptor. The resulting N-to-B coordinate covalent bond forms the ammonia borane adduct.

The occupied molecular orbital formed by this interaction is bonding, while the corresponding antibonding combination remains empty. Borane pyramidalizes as coordination increases from three to four. Ammonia can use its existing outward-pointing lone pair; its inversion to the planar transition geometry is not a prerequisite. The Lewis base and Lewis acid descriptions summarize the donor and acceptor roles, while the frontier molecular orbitals explain their directional interaction.

Work with $M=(H-\alpha I)/\beta$. Its nearest-neighbour ring entries are $1$, and its diagonal entries are $1$ on the nitrogen sites and $0$ on the carbon sites. Because $M$ commutes with each molecular symmetry operation, it only mixes symmetry-adapted linear combinations with the same irreducible representation.

In the bases $(N_{+},C_0)$ and $(N_{-},C_y)$, the two nontrivial matrix blocks areTheir characteristic polynomials are $(1-\lambda {)}^2-2$ and ${\lambda }^2-3$. The remaining single-function blocks are $M_{B_{2g}}=1$ and $M_{A_u}=-1$. Thus, ordered from lowest to highest physical energy because $\beta <0$, the molecular orbitals areAll six levels are nondegenerate in this parametrization. The right-hand ladder of the diagram in part (ii) shows these levels and occupations. The occupied $B_{2g}$ level is the highest occupied molecular orbital, and the unoccupied $B_{1u}$ level immediately above it is the lowest unoccupied molecular orbital.

The four sigma bonds are the two C–H connections, C–N and N–N, containing eight electrons. A terminal-nitrogen axial lone pair contains two more. Of the remaining six electrons, two occupy an N–N pi bond lying in the molecular plane, while four occupy a perpendicular three-centre four-electron pi bond extending over C–N–N. There are therefore 16 valence electrons in total.

For orientation, a three-site model with equal on-site energy $\alpha$ and adjacent interaction $\beta <0$ gives perpendicular pi levels at $\alpha +\sqrt{2}\beta$, $\alpha$, and $\alpha -\sqrt{2}\beta$. The lower two are doubly occupied and the upper is empty. The middle level has a terminal-end combination with little central-atom amplitude; unequal C and N orbital energies change its coefficients and make it only approximately nonbonding.

The highest occupied molecular orbital is the upper occupied perpendicular pi orbital, with substantial carbon amplitude, broadly above the strongly bonding sigma framework and the lower occupied pi orbitals. It is not the terminal axial nitrogen lone pair. Real diazomethane has C–N bonding and N–N antibonding character in this frontier orbital, rather than being an exactly nonbonding equal-site model. Photoelectron and Rydberg-state experiments support this assignment.^[ref]

An intense band near $2088\mathrm{c}{\mathrm{m}}^{-1}$ is consistent with the high-frequency coupled C–N–N stretching motion of a diazo group, with substantial N–N multiple-bond character. It lies above a typical isolated C=N double-bond stretch, and supports contributions with a strong N–N connection. It does not establish that the molecule is a literal localized C–N single bond plus an unperturbed N≡N triple bond: a normal mode moves the coupled framework, and the resonance contributors describe one electronic structure.

Protonation at the nucleophilic carbon gives methyldiazonium, $\mathrm{C}{\mathrm{H}}_3-{\mathrm{N}}^{+}\equiv \mathrm{N}$. The carbon-rich highest occupied molecular orbital and the carbon-negative resonance contributor rationalize this site. Subsequent displacement of molecular nitrogen can be strongly favourable; the initial protonation product still contains both nitrogens.

Both dimers have two electrons in a valence sigma-bonding orbital and a formal valence bond order of one. For hydrogen the minimal-basis configuration is $(1{\sigma }_g{)}^2$, with the $1s$-derived $1{\sigma }_g$ as highest occupied molecular orbital and $1{\sigma }_u^{\ast }$ as LUMO. The bonding combination has no internuclear node; the antibonding combination does.

For lithium, include the filled core pair of levels as well as the valence levels:The first two orbitals are predominantly core $1s$ combinations whose bonding and antibonding occupations cancel in a simple bond-order count. The highest occupied molecular orbital is predominantly ${\sigma }_g(2s)$, denoted $2{\sigma }_g$; the LUMO is $2{\sigma }_u$, predominantly the antibonding $2s$ combination but modified by mixing with compatible $2p$ contributions. The nearby empty $2p$-derived sigma and pi levels must also be shown. Only orbitals of matching symmetry mix, so this does not mean that every $2s$ combination mixes with every $2p$ combination.

On a common qualitative energy reference, hydrogen $1s$ is much lower than lithium's valence $2s$, while lithium's core $1s$ levels lie much deeper. The Li valence bonding–antibonding separation is smaller, consistent with its longer and weaker bond. The figure uses a compressed core region and schematic gaps, not measured orbital energies. The role of symmetry-compatible s–p orbital mixing in the first-row diatomics is illustrated in institutional molecular-orbital notes.^[ref]