Molecular Orbital Theory

Molecular orbital theory is a framework in quantum mechanics for describing the electronic structure and chemical bonding of molecules. It represents electrons using one-electron orbitals associated with the molecular system rather than exclusively with individual atoms or localized bonds. A molecular orbital describes an electron in the effective field of the nuclei and other electrons; it may extend over two or more atomic centres. The theory provides both qualitative bonding models and a foundation for quantitative electronic-structure calculations. (goldbook.iupac.org)

Origins and physical meaning

The molecular-orbital approach developed during the early application of quantum mechanics to chemistry. Robert S. Mulliken was a central contributor, using molecular spectroscopy to connect electronic states with bonding. He received the 1966 Nobel Prize in Chemistry for work on chemical bonds and molecular electronic structure through the molecular-orbital method. (nobelprize.org)

An orbital is a wave function, not a trajectory followed by an electron. Its squared magnitude gives a spatial probability density for an electron occupying that orbital. Orbital drawings commonly show surfaces of constant amplitude or regions containing a specified probability. Different colours or signs indicate the wave function’s phase, not positive and negative electric charges. (goldbook.iupac.org)

Calculations commonly employ the Born–Oppenheimer approximation, separating electronic motion from nuclear motion and initially treating nuclear positions as fixed. The electronic problem is then solved for a specified molecular geometry. (arxiv.org)

Constructing molecular orbitals

A widespread representation is the linear combination of atomic orbitals, abbreviated LCAO:

Here, $\psi_i$ is a molecular orbital, $\chi_\mu$ denotes an atom-centred basis function, and $c_{\mu i}$ specifies its contribution. The functions resemble atomic orbitals, although computational basis functions need not be exact isolated-atom orbitals. The coefficients are determined by solving an approximate electronic problem, rather than simply adding orbitals in equal proportions. (goldbook.iupac.org)

This linear-combination construction converts orbital equations into a matrix problem. In the Hartree–Fock method, the coefficients satisfy a generalized eigenvalue equation,

where $\mathbf F$ is the Fock matrix, $\mathbf S$ contains basis-function overlaps, and $\boldsymbol{\varepsilon}$ contains orbital energies. Because the effective field depends on the occupied orbitals, the equations are solved iteratively to obtain a self-consistent solution. (arxiv.org)

For qualitative orbital interactions, compatible symmetry, spatial overlap, and reasonably similar atomic orbital energies favour substantial mixing. Interactions between orbitals with widely separated energies generally produce less balanced combinations. (ocw.mit.edu)

Bonding, antibonding, and symmetry

In the simplest two-orbital model, atomic orbitals combine to produce one bonding and one antibonding molecular orbital. Constructive interference increases electron probability density between the nuclei. Destructive interference produces a node and reduces density in that region. Occupying a bonding orbital generally stabilizes bonding, whereas occupying its antibonding counterpart weakens it. Antibonding orbitals are conventionally marked with an asterisk, such as $\sigma^\ast$. (ocw.mit.edu)

Orbitals may also be approximately nonbonding when their occupation has little effect on a particular bond. In larger molecules, an orbital can have bonding character between some centres and antibonding character between others, so its classification may require specifying the interaction under discussion. (ocw.mit.edu)

For diatomic molecules, $\sigma$ orbitals have no nodal plane containing the internuclear axis, while $\pi$ orbitals have one. Both types can be bonding or antibonding. In homonuclear diatomics, the additional labels gerade and ungerade, written $g$ and $u$, indicate whether the orbital retains or reverses its sign under inversion through the molecular centre. These symmetry labels are distinct from bonding character. (ocw.mit.edu)

Electron occupation and bond order

A molecular electron configuration is constructed by occupying available orbitals. The Pauli exclusion principle permits at most two electrons in a spatial orbital, with opposite spin projections. For a ground-state qualitative diagram, lower-energy orbitals are filled first; Hund’s rule favours parallel-spin, singly occupied orbitals within a degenerate set before pairing. (ocw.mit.edu)

For simple diatomic diagrams, bond order is estimated as

where $N_{\mathrm b}$ and $N_{\mathrm a}$ count electrons in bonding and antibonding orbitals. In hydrogen, H₂, two electrons occupy the bonding $1s$-derived orbital, giving bond order one. The corresponding elementary He₂ diagram fills both bonding and antibonding orbitals, giving zero net bond order. This model addresses ordinary covalent bonding, not every possible weak interaction between helium atoms. (ocw.mit.edu)

Molecular oxygen provides a major illustration: its ground-state configuration contains two unpaired electrons in degenerate $\pi^\ast$ orbitals and has bond order two. The unpaired electrons explain its paramagnetism, which a simple closed-shell Lewis structure does not represent. (ocw.mit.edu)

Reactivity and computational limitations

Frontier molecular orbital theory emphasizes the highest occupied molecular orbital, or HOMO, and lowest unoccupied molecular orbital, or LUMO. Their energies, shapes, and symmetries help interpret electron-donor–acceptor interactions and aspects of chemical reactivity. Such descriptions are qualitative frameworks, not complete predictions of reaction rates or mechanisms. (goldbook.iupac.org)

Molecular orbitals are not necessarily uniquely delocalized objects. Appropriate transformations can produce localized orbitals, connecting the molecular-orbital description with familiar bonds and lone pairs. The contrast with valence bond theory therefore concerns how electronic states are represented and approximated, rather than two incompatible physical realities. (goldbook.iupac.org)

Hartree–Fock includes electron exchange but omits electron correlation beyond its mean-field description. Correlated methods improve the many-electron wave function using molecular orbitals as a starting representation. Accuracy consequently depends on the electronic method and basis set; a qualitative orbital diagram alone does not determine precise bond energies, excitation energies, or dissociation behaviour. (ocw.mit.edu)