Molecular orbital ($\psi (\mathbf{r})$)

A molecular orbital is a one-electron spatial wavefunction for the combined atomic nuclei of a molecule. For a normalized molecular orbital, the Born rule gives the position probability density $|\psi (\mathbf{r}){|}^2$, with ${\int }_{{\mathbb{R}}^3}|\psi {|}^2{d}^3\mathbf{r}=1$. The molecular orbital is the amplitude, rather than its squared modulus or a classical trajectory.

In the independent-orbital approximation, a doubly occupied molecular orbital contributes $2|\psi {|}^2$ to the electron number density, whose integral is two. An empty molecular orbital contributes zero, even though its shape is still part of the model. The molecular hydrogen example therefore uses one occupied spatial molecular orbital for two electrons, distinguished by their spins. A negative real probability amplitude means a relative quantum phase of $\pi$, not positive electric charge; multiplying the whole wavefunction by $-1$ changes only its global phase.

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.

A molecular orbital whose occupation changes bonding between the selected centres relatively little. In a real complex, weak bonding or antibonding contributions may replace exact nonbonding character.

For an open-shell species, distinguish a singly occupied spatial orbital from a vacant spin orbital. Adding an electron to an oxygen molecule can fill the opposite-spin vacancy in a pi-star singly occupied molecular orbital even though the next completely empty spatial orbital is sigma-star.

A spatial molecular orbital occupied by one electron. In an open-shell molecule, its empty opposite-spin slot can accept another electron even though the spatial orbital is occupied.

A molecular orbital with little or no net bonding interaction between the atoms in the stated orbital model. A symmetry-isolated atomic orbital can be exactly nonbonding in a minimal basis: the out-of-plane nitrogen $2p_z$ orbital in planar Ammonia has no matching hydrogen symmetry orbital. A mainly localized lone pair can acquire bonding admixture when a distortion permits additional mixing.

An energy-level scheme showing the occupied molecular orbitals and their relation to atomic orbitals. Occupancies must account for degeneracy and spin.

In pyramidal Ammonia, the hydrogen symmetry orbitals span $A_1\oplus E$ of the C3v molecular point group. Nitrogen $2s$ and $2p_z$ both have $A_1$ symmetry and $(2p_x,2p_y)$ span $E$. In the planar D3h molecular point group, these become $A_1^{\prime }$, $A_2^{\prime \prime }$ and $E^{\prime }$, respectively, while the hydrogen combinations span $A_1^{\prime }\oplus E^{\prime }$. Thus the planar out-of-plane lone pair is a symmetry-isolated nonbonding orbital; pyramidalization permits mixing with the other $a_1$ functions.^[ref]

The occupied molecular orbital of greatest energy in a specified reference electron configuration. A fixed-orbital model removes an electron from this level first; relaxation and electron correlation can modify the resulting energy.

Orbital degeneracy means distinct spatial orbitals have equal energy within the specified model or symmetry. Two electrons in degenerate orbitals need not pair; exchange can favor a triplet state.

The lowest-energy unoccupied orbital of a specified reference electron configuration. In a fixed-orbital approximation, adding one electron occupies this orbital; orbital relaxation and electron correlation can alter that picture.

Occupying a bonding molecular orbital favours a chemical bond. For the two equivalent $1s$ atomic orbitals of the hydrogen atom, the sum ${\sigma }_g={\sigma }_{+}$ has constructive interference between the protons. The resulting electron density in that region helps stabilize the combined system. Its lower energy in this model follows from the Quantum Hamiltonian, not from the plus sign alone; see Field, MIT 5.61, lecture 25, pp. 1–2.

For example, $({\sigma }_g{)}^2$ in molecular hydrogen supplies $N_b=2$, hence bond order one when $N_a=0$. The sum in the molecular orbital theory toy has midpoint probability density $0.303468$, whereas the difference has zero there. The normalization factors must be included in this comparison.

Occupying an antibonding molecular orbital opposes a chemical bond. For equivalent $1s$ atomic orbitals of the hydrogen atom, the difference ${\sigma }_u^{\ast }={\sigma }_{-}$ vanishes on the plane halfway between the protons: every point there is equally far from both centres, so the amplitudes cancel. This nodal plane removes the contribution of this molecular orbital to electron density on that plane; it does not make an entire finite region empty. The $1s$-derived antibonding molecular orbital lies above its bonding molecular orbital partner in energy. See Field, MIT 5.61, lecture 25, pp. 1–2.

In the ${\mathrm{He}}_2$ minimal model, $({\sigma }_u^{\ast }{)}^2$ adds $N_a=2$ and cancels the bond order contribution of $({\sigma }_g{)}^2$. Opposite signs on the two lobes describe probability amplitudes, while both lobes have nonnegative probability density after squaring. The star is an orbital label, not an extra electron or an electric charge.